Ullmann’s Chemical Engineering and Plant Design [1,2] 9783527311118, 3527311114

Table of contents :
MATHEMATICS AND PHYSICS IN CHEMICAL ENGINEERING.Mathematics in Chemical Engineering.Mathematical Modeling.Transport Phenomena.Fluid Mechanics.Design of Experiments.Computational Fluid Dynamics.Model Reactors and Their Design Equations.FUNDAMENTALS OF CHEMICAL ENGINEERING.Estimation of Physical Properties.Construction Materials in Chemical Industry.Corrosion.Abrasion and Erosion.Mechanical Properties and Testing of Metallic Materials.Nondestructive Testing.On-Line Monitoring of Chemical Reactions.PLANT AND PROCESS DESIGN.Process Development.Separation Processes.Solid-Liquid Separation.Solid-Solid Separation.Mixing.Solids Technology.Reactor Types and Their Industrial Applications.Chemical Plant Design and Construction.Pinch Technology.Pilot Plants.Scale-Up in Chemical Engineering .Biochemical Engineering.Energy Management in Chemical Industry.Plant and Process Safety.Environmental Management in the Chemical Industry.

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ULLMANN’S Chemical Engineering and Plant Design

Volume 1

Mathematics and Physics in Chemical Engineering Fundamentals of Chemical Engineering

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: Applied for. British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by The Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de

ISBN 978-3-527-31111-8

c 2005 WILEY-VCH Verlag GmbH & Co. KGaA,
Weinheim Printed on acid-free paper. All rights reserved (including those of translation in other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition: Steingraeber Satztechnik GmbH, Dossenheim Printing: Strauss GmbH, Mörlenbach Bookbinding: Litges & Dopf Buchbinderei GmbH, Heppenheim Cover Design: Gunther Schulz, Fußg¨onheim Printed in the Federal Republic of Germany.

V

Preface Since the unabridged 40-volume Ullmann’s Encyclopedia is inaccessible to many readers – particularly individuals, smaller companies or institutes – all the information on chemical engineering and plant design has been condensed into this convenient two-volume set. Based on the very latest print edition of Ullmann’s, this ready reference is the one-stop resource for the plant design engineering community. Starting with the quantitative treatment and fundamentals of chemical engineering, it combines all aspects of process development and reactor technology, as well as detailing their practical applications in sections devoted to plant design, scale-up and plant safety. Each of the detailed and carefully edited articles is written by relevant experts from industry or academia. A keyword and an author index complete the contents of this handbook. Throughout, readers benefit from the rigorous and cross-indexed nature of the parent reference, and will find both broad introductory information as well as in-depth details of significance to industrial and academic environments. The Publisher

Contents VII

Contents Volume 1 Symbols and Units . . . . . . . . . . . . . . . . . . . . IX Conversion Factors . . . . . . . . . . . . . . . . . . . XI Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . XII Country Codes . . . . . . . . . . . . . . . . . . . . . . . XVII Periodic Table of Elements . . . . . . . . . . . . XVIII

Mathematics and Physics in Chemical Engineering Mathematics in Chemical Engineering . . Mathematical Modeling . . . . . . . . . . . . . . . Transport Phenomena . . . . . . . . . . . . . . . . . Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . Design of Experiments . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . Model Reactors and Their Design Equations . . . . . . . . . . . . . . . . . . . . . . . .

3 165 271 371 423 463 487

Fundamentals of Chemical Engineering Estimation of Physical Properties . . . . . . . Construction Materials in Chemical Industry . . . . . . . . . . . . . . . . . . . . . . . . . . Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abrasion and Erosion . . . . . . . . . . . . . . . . . Mechanical Properties and Testing of Metallic Materials . . . . . . . . . . . . . . . .

537

Nondestructive Testing . . . . . . . . . . . . . . . . 785 On-Line Monitoring of Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . 821

Volume 2 Plant and Process Design Process Development . . . . . . . . . . . . . . . . . Separation Processes, Introduction. . . . . . Solid – Liquid Separation, Introduction . . Solid – Solid Separation, Introduction . . . Mixing, Introduction . . . . . . . . . . . . . . . . . . Solids Technology, Introduction . . . . . . . . Reactor Types and Their Industrial Applications. . . . . . . . . . . . . . . . . . . . . . Chemical Plant Design and Construction Pinch Technology . . . . . . . . . . . . . . . . . . . . . Pilot Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up in Chemical Engineering . . . . . Biochemical Engineering . . . . . . . . . . . . . . Energy Management in Chemical Industry . . . . . . . . . . . . . . . . . . . . . . . . . . Plant and Process Safety . . . . . . . . . . . . . . . Environmental Management in the Chemical Industry . . . . . . . . . . . . . . . .

873 915 923 931 939 941 953 987 1075 1083 1093 1117 1187 1205 1331

605 653 735

Author Index . . . . . . . . . . . . . . . . . . . . . . . 1359

761

Subject Index . . . . . . . . . . . . . . . . . . . . . . . 1363

Mathematics and Physics in Chemical Engineering

Mathematics in Chemical Engineering

3

Mathematics in Chemical Engineering Bruce A. Finlayson, Department of Chemical Engineering, University of Washington, Seattle, Washington 98195, United States (Chaps. 1 – 9, 11, 12) Lorenz T. Biegler, Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States (Chap. 10) Ignacio E. Grossmann, Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States (Chap. 10) Arthur W. Westerberg, Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States (Chap. 10)

1. 1.1. 1.2. 1.3. 1.4. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 3. 3.1. 3.2. 3.3. 3.4. 3.5. 4. 4.1. 4.2. 4.3. 5. 6. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9.

Solution of Equations . . . . . . . . Linear Algebraic Equations . . . . Nonlinear Algebraic Equations . . Linear Difference Equations . . . . Eigenvalues . . . . . . . . . . . . . . . Approximation and Integration . . Introduction . . . . . . . . . . . . . . Global Polynomial Approximation Piecewise Approximation . . . . . . Quadrature . . . . . . . . . . . . . . . Linear Least Squares . . . . . . . . . Nonlinear Least Squares . . . . . . Fourier Transforms of Discrete Data . . . . . . . . . . . . Two-Dimensional Interpolation and Quadrature . . . . . . . . . . . . Complex Variables . . . . . . . . . . Introduction to the Complex Plane Elementary Functions . . . . . . . . Analytic Functions of a Complex Variable . . . . . . . . . . . . . . . . . Integration in the Complex Plane Other Results . . . . . . . . . . . . . . Integral Transforms . . . . . . . . . Fourier Transforms . . . . . . . . . . Laplace Transforms . . . . . . . . . Solution of Partial Differential Equations by Using Transforms . . Vector Analysis . . . . . . . . . . . . Ordinary Differential Equations as Initial Value Problems . . . . . . . . Solution by Quadrature . . . . . . . Explicit Methods . . . . . . . . . . . Implicit Methods . . . . . . . . . . . Stiffness . . . . . . . . . . . . . . . . . Differential – Algebraic Systems . Computer Software . . . . . . . . . Stability, Bifurcations, Limit Cycles . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . Eigenvalues and Roots by Initial Value Techniques . . . . . . . . . . .

6 6 12 14 15 16 16 16 18 21 24 26

7. 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10.

27 29 30 30 31 33 34 37 37 37 42

8. 8.1. 8.2. 8.3. 8.4. 8.5. 9. 9.1. 9.2. 9.3.

48 51 9.4. 61 62 63 67 68 69 71 72 74 75

9.5. 9.6. 10. 10.1. 10.2. 10.3. 10.4. 10.5. 10.5.1.

Ordinary Differential Equations as Boundary Value Problems . . . . . Solution by Quadrature . . . . . . . Shooting Methods . . . . . . . . . . . Finite Difference Method . . . . . . Orthogonal Collocation . . . . . . . Orthogonal Collocation on Finite Elements . . . . . . . . . . . . . . . . . Galerkin Finite Element Method . Cubic B-Splines . . . . . . . . . . . . Adaptive Mesh Strategies . . . . . . Comparison . . . . . . . . . . . . . . . Singular Problems and Infinite Domains . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . Classification of Equations . . . . . Hyperbolic Equations . . . . . . . . Parabolic Equations in One Dimension . . . . . . . . . . . Elliptic Equations . . . . . . . . . . . Parabolic Equations in Two or Three Dimensions . . . . . . . . . Integral Equations . . . . . . . . . . Classification . . . . . . . . . . . . . . Numerical Methods for Volterra Equations of the Second Kind . . . Numerical Methods for Fredholm, Urysohn, and Hammerstein Equations of the Second Kind . . . . . . Numerical Methods for Eigenvalue Problems . . . . . . . . . . . . . . . . . Green’s Functions . . . . . . . . . . . Boundary Integral Equations and Boundary Element Method . . . . . Optimization . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . Conditions for Optimality . . . . . Strategies of Optimization . . . . . Successive Quadratic Programming (SQP) . . . . . . . . . Linear Programming . . . . . . . . . Basic Properties . . . . . . . . . . . . .

76 76 77 79 82 87 88 90 91 92 93 94 94 96 98 104 108 109 109 111

113 115 115 117 118 118 119 123 127 130 131

4

Mathematics in Chemical Engineering

10.5.2. 10.6. 10.7. 11. 11.1. 11.2. 11.3.

Simplex Algorithm . . . . . . . . . . . Mixed-Integer Programming . . . Solution of Dynamic Optimization Problems . . . . . . . . . . . . . . . . . Probability and Statistics . . . . . . Concepts . . . . . . . . . . . . . . . . . Sampling and Statistical Decisions Error Analysis in Experiments . .

132 133

12. 136 142 143 146 150

Symbols a A b

B

Bi c Co d D Da f

F g

11.4.

Variables scalar constant in quadratic approximation for F, the objective function m×n matrix of constant coefficients for equality constraints in linear programming model vector of constants premultiplying the r independent variables u in the quadratic approximation for F, the objective function. Also vector of m right-hand-sides for equality constraints in linear programming problem approximation of the n×n Hessian matrix for the Lagrange function for the successive quadratic programming algorithm. Also m×m non-singular matrix corresponding to the basis variables in a linear programming problem. Also coefficient matrix for binary variables y in the set of equality constraints for a mixed integer programming problem Biot number vector of n constant cost coefficients for all variables z in linear programming problem Courant number search direction in the space of all n variables z (both dependent and independent variables) diffusion constant Damk¨ohler number nonlinear scalar contribution to objective function which is a function only of the continuous variables, y, for a mixed integer programming problem scalar objective function for an optimization problem set of p inequality constraints for an optimization problem

12.1. 12.2. 12.3. 13.

h H I k L m n N p

Pe Q r Re R

Ren s

Sh u

Factorial Design of Experiments and Analysis of Variance . . . . . . Multivariable Calculus Applied to Thermodynamics . . . . . . . . . . . State Functions . . . . . . . . . . . . . Applications to Thermodynamics Partial Derivatives of All Thermodynamic Functions . . . . . . . . . . References . . . . . . . . . . . . . . . .

150 153 153 154 155 156

set of m equality constraints for an optimization problem the inverse of matrix Q. Also the scalar Hamiltonian function for a dynamic optimization problem identity matrix iteration matrix scalar Lagrange function number of equality constraints for an optimization problem number of total variables in an optimization problem (n-m)×m matrix corresponding to the non-basis variables in a linear programming problem number of inequality constraints for an optimization problem. Also vector of parameters (do not vary with time) for dynamic optimization problem Peclet number an r×r matrix of constants used in defining the quadratic term for a quadratic approximation for F, the objective function number of independent variables for an optimization problem (r = n-m) the space of real numbers matrix defined in the quadratic programming algorithm. Also the vector of residual equations in the finite element approach to solving dynamic optimization problems real number vector space of dimension n change in the independent variables u (a vector with r elements in it). Also sensitivity of the functions f with respect to the parameter p for a dynamic optimization problem Sherwood number vector of r independent variables for an optimization problem. For a time-varying

Mathematics in Chemical Engineering

W x y

Y

z z¯ Z

problem, the independent time-varying control variables r×r weighting matrix used in computing Frobenius norm vector of m dependent variables for an optimization problem (the state variables for a dynamic optimization problem) variables used in linear programming problem. Also binary variables in the formulation of a mixed integer programming problem n×m matrix whose columns are vectors that span the range space of the linearized constraints (see section on successive quadratic programming algorithm) vector of all n variables – both the m dependent variables x and the r independent variables u – for an optimization problem interior point for linear programming problem n×(n-m) matrix whose columns are vectors that span the null space of the linearized constraints (see section on successive quadratic programming algorithm)

µ ν

τ Y

| :

∈ →

γ δ ∆ ε φ

κ λ

subject to mapping. For example, h : Rn → Rm , states that functions h map real numbers into m real numbers. There are m functions h written in terms of n variables member of maps into Subscripts

Greek symbols scalar for setting the step size in a line search algorithm for the successive quadratic programming algorithm. Also a vector of coefficients to form a convex combination of extreme points for a linear programming problem. Also scalar variable used as the objective for the mixed integer programming problem in equation 76. Also length of an element in the finite element approach to solving dynamic optimization problems change in gradient of the objective function in moving from point k to point k + 1. γ is a vector with r elements in it Kronecker delta sampling rate small scalar vector of polynomial approximation functions for the state variables for finite element method for dynamic optimization problem, also Thiele modulus condition number vector of m Lagrange multipliers for an optimization problem. Also for a dynamic

optimization problem, the Lagrange multiplier for the state variable constraints. vector of p Kuhn – Tucker multipliers vector of Lagrange multipliers (vary with time) for the algebraic equality constraints for a dynamic optimization problem rescaled time so it lies in range [0,1] vector of polynomial approximation function for the control variables for finite element method for dynamic optimization problem Special symbols

A α

5

B f i N NE OA o

gA are the active constraints (i.e., precisely equal to zero) among the inequality constraints basis variables in a linear programming problem value at the final time for a dynamic optimization problem i-th element of a vector of variables non-basis variables in a linear programming problem number of elements in a finite element model outer approximation value at the initial time for a dynamic optimization problem Superscript

k L T U ˆ

iteration index lower bound transpose of a vector or matrix upper bound base point for a Taylor series expansion

6

Mathematics in Chemical Engineering

1. Solution of Equations

1.1. Linear Algebraic Equations

Mathematical models of chemical engineering systems can take many forms: they can be sets of algebraic equations, differential equations, or integral equations. Mass and energy balances of chemical processes typically lead to large sets of algebraic equations:

Consider the n×n linear system a11 x1 + a12 x2 + . . . + a1n xn = f1 a21 x1 + a22 x2 + . . . + a2n xn = f2 ...

a11 x1 + a12 x2 = b1 an1 x1 + an2 x2 + . . . + ann xn = fn a21 x1 + a22 x2 = b2

Mass balances of stirred tank reactors may lead to ordinary differential equations: dy = f [ y (t)] dt

Radiative heat transfer may lead to integral equations:

n 

aij xj = fj or A x = f

j=1


1 y (x) = g (x) + λ

In this equation a11 , . . . , ann are known parameters, f 1 , . . . , fn are known, and the unknowns are x 1 , . . . , xn . The values of all unknowns that satisfy every equation must be found. This set of equations can be represented as follows:

K (x, s) f (s) ds 0

Even when the model is a differential equation or integral equation, the most basic step in the algorithm is the solution of sets of algebraic equations. The solution of sets of algebraic equations is the focus of Chapter 1. A single linear equation is easy to solve for either x or y: y = ax + b

The most efficient method for solving a set of linear algebraic equations is to perform a lower – upper (LU) decomposition of the corresponding matrix A. This decomposition is essentially a Gaussian elimination, arranged for maximum efficiency. The LU decomposition is done by calculating in turn for i = 1, n do for j = 1, n do

If the equation is nonlinear, (1)

f (x) = 0

it may be more difficult to find the x satisfying this equation. These problems are compounded when there are more unknowns, leading to simultaneous equations. If the unknowns appear in a linear fashion, then an important consideration is the structure of the matrix representing the equations; special methods are presented here for special structures. They are useful because they increase the speed of solution. If the unknowns appear in a nonlinear fashion, the problem is much more difficult. Iterative techniques must be used (i.e., make a guess of the solution and try to improve the guess). An important question then is whether such an iterative scheme converges. Other important types of equations are linear difference equations and eigenvalue problems, which are also discussed.

aij

= aij

enddo enddo for k = 2, n do for i = k + 1, n do for j = k + 1, n do (k)

(k− 1)

aij = aij enddo enddo enddo

−

(k−1) ai, k−1 (k−1) ak−1,k−1

(k− 1)

ak−1,j

Mathematics in Chemical Engineering Then two matrices are formed. The upper matrix U = A(n) is defined as  (n)

U ij = Aij =

0

for i = n − 1, 1 do 

i >j (i)

f

i ≤ j, 1 ≤ j ≤ n

a ij

(i) i

n

−

j=i+1

xi =

Lij

(i) aij xj

(i)

aii

The lower matrix L is defined as   0   1 = (j) aij     (j)

7

enddo ij

ajj

The U is upper triangular; it has zero elements below the main diagonal and possibly nonzero values along the main diagonal and above it (see Fig. 1). The L is lower triangular. It has the value 1 in each element of the main diagonal, nonzero values below the diagonal, and zero values above the diagonal (see Fig. 1). Thus, A = LU

The original problem can be solved in two steps:

When f is changed, the last steps can be done without recomputing the LU decomposition. Thus, multiple right-hand sides can be computed efficiently. The number of multiplications and divisions necessary to solve for m right-hand sides is: Operation count =

1 3 1 n − n + m n2 3 3

The actual algorithm used will be different for parallel computers. The determinant is given by DetA =

n 

(i)

aii

i=1

Ly = f , U x = y solves Ax = LU x = f

Each of these steps is straightforward because the matrices are upper triangular or lower triangular. The solution is performed using the equations for i = 1, n do (1)

fi

= fi

This should be calculated as the LU decomposition is performed. If the value of the determinant is a very large or very small number, it can be divided or multiplied by 10 to retain accuracy in the computer; the scale factor is then accumulated separately. The condition number κ can be defined in terms of the singular value decomposition as the ratio of the largest wi to the smallest wi (see below). It can also be expressed in terms of the norm of the matrix:

enddo

κ (A) = A A−1 

for k = 2, n do

where the norm is defined as

for i = k + 1, n do

 A ≡ supx=0 (k−1)

(k)

(k−1)

f i =f i

−

ai,k−1 (k−1)

ak−1,k−1

(k−1)

f k−1

enddo enddo (n)

n   A x |ajk | = maxk x j=1

(n)

xn = f n /ann

(1)

If this number is infinite, the set of equations is singular. If the number is too large, the matrix is said to be ill-conditioned. The definition of “large” refers to the accuracy of the computer being used; the criterion is the inverse of the floating point precision. If the machine’s floating point precision is 10−6 then 106 is large; if double precision is used then the precision is 10−12 and 1012 is large. Calculation of the condition number can be lengthy so another criterion is

8

Mathematics in Chemical Engineering

also useful. Compute the ratio of the largest to the smallest pivot and make judgments on the ill-conditioning based on that. (k)

Ratio =

maxk |akk | (k) mink |akk |

Another empirical test is the quantity V ; when V is small the matrix is ill-conditioned. V =

1/2 |detA| , αi = a2i1 + a2i2 +. . . + a2in α1 α2 . . . α n

element can be obtained from only the diagonal entries (partial pivoting) or from all the remaining entries. If the matrix is nonsingular, Gaussian elimination (or LU decomposition) could fail if a zero value were to occur along the diagonal and were to be a pivot. With full pivoting, however, the Gaussian elimination (or LU decomposition) cannot fail because the matrix is nonsingular. A matrix is symmetric if aij = aji

and it is positive definite if xT A x =

n n  

aij xi xj ≥ 0

i=1 j=1

for all x and the equality holds only if x = 0 (where xT is the transpose of the matrix, see page 52) The Cholesky decomposition can be used for real, symmetric, positive definite matrices. This algorithm saves on storage (divide by about 2) and reduces the number of multiplications (divide by 2), but adds n square roots. The inverse of a matrix can also be used to solve sets of linear equations. The inverse is a matrix such that when A is multiplied by its inverse the result is the identity matrix, a matrix with 1.0 along the main diagonal and zero elsewhere. A A−1 = I

Then the linear equations are solved by x = A−1 f

Figure 1. Structure of L and U matrices

When a matrix is ill-conditioned the LU decomposition must be performed by using pivoting (or the singular value decomposition described below). With pivoting, the order of the elimination is rearranged. At each stage, one looks for the largest element (in magnitude); the next stages if the elimination are on the row and column containing that largest element. The largest

Generally, the inverse is not used in this way because it requires three times more operations than solving with an LU decomposition. However, if an inverse is desired, it is calculated most efficiently by using the LU decomposition and then solving A x(i) =b(i) (i) bj

 0 = 1

j=i j=i

Then set   A−1 = x(1) |x(2) |x(3) |. . .|x(n)

Mathematics in Chemical Engineering Generally software packages are available to solve this problem, and the user must submit the elements of the matrix in proper fashion. Remembering that the information passed from calling program to subroutine is a linear array of numbers, and that actually only the location of the first number is passed to the computer subroutine, is useful for interpretating user instructions. If the matrix is an n×n matrix stored in an m×m array, then what is passed is the location of a11 and the numbers are expected to be in the arrangement >

9

The number of multiplications and divisions for a problem with n unknowns and m right-hand sides is Operation count = 2 (n − 1) + m (3 n − 2)

If |bi | >|ai | +|ci |

no pivoting is necessary. For solving two-point boundary value problems and partial differential equations this is often the case.

a11 , a21 ,. . . , an1 -, -, -, a12 , a22 ,. . . , an2 -, -, -, . . . Entry 1, 2, . . . , n, -, -, m, m+1, m+2,. . . , m+n, -, -, 2n, . . .

Solutions of Special Matrices. Special matrices can be handled even more efficiently. A tridiagonal matrix is one with nonzero entries along the main diagonal, and one diagonal above and below the main one (see Fig. 2). The corresponding set of equations can then be written as ai xi−1 + bi xi + ci xi+1 = di

The LU decomposition algorithm for solving this set is b1 = b1 for k = 2, n do ak =

ak ak , bk = bk −  ck−1 bk−1 bk− 1

enddo d1 = d1 for k = 2, n do dk = dk − ak dk−1 enddo xn = dn /bn for k = n − 1, 1 do xk = enddo

dk − ck xk+1 bk

Figure 2. Structure of tridiagonal matrices

A block tridiagonal matrix (see Fig. 3) frequently arises in solving multiple differential equations, either two-point boundary value problems or partial differential equations. When solving several simultaneous equations the unknowns must be arranged in a particular way to achieve the block tridiagonal form of the matrix and the resulting efficiencies. Suppose we are solving for two unknowns, for example, temperature T and concentration c, and a value for each exists at each node i = 1 to n, the unknowns are {c1 , c2 , . . . , cn , T 1 , T 2 , . . . , Tn }. If the unknowns are ordered in this way, however, the resulting linear equations to be solved, when solving two-point boundary value problems or partial differential equations, will be dense, that is, the nonzero entries of the matrix A will have no special pattern. However, if we arrange the unknowns in the pattern {c1 , T 1 , c2 , T 2 , . . . , cn , Tn }, then special patterns emerge. If the finite difference method is used, the block tridiagonal matrix arises. For

10

Mathematics in Chemical Engineering

other methods, other special patterns arise. Solution algorithms are most efficient if these patterns are taken into account in the LU decomposition. Clearly, a trade-off occurs between the programming time (needed to exploit any special structure) and the value received from a more efficient solution. If the problem is being solved only a few times or the time is extremely small, the special structure need not be exploited and usual software packages are suitable. If the problem is being solved thousands of times or the computer time is large (when n is large, e.g., 1000), the special structure must be exploited. Suppose the block tridiagonal matrix is composed of blocks that are nb ×nb , with n columns of blocks (along the diagonal), then the number of multiplications and divisions is (for large n and nb ) [3, p. 61] 5 Operation count = n n3b 3

and Ax = f

is solved by using the same LU decomposition of A, Ay = B

These problems are sparse and the special methods are applied to them. Next, the dense matrix problem (D − C y) v = −C x + g

is solved and u = x − yv

is evaluated, the solution is then (uT , v T ).

If n = 100 and nb = 3, the block tridiagonal technique requires only 4500 operations, whereas the dense matrix technique requires 9×106 operations. Thus, the block tridiagonal technique is 2000 times faster.

Figure 4. Structure of arrow matrix

Figure 3. Structure of block tridiagonal matrix

Another matrix with special structure is the arrow matrix (see Fig. 4). This matrix has a large, banded portion, with a much smaller number of columns at one side and rows at the bottom. This matrix arises when solving sets of fluid flow problems with free surfaces, when using continuation methods or parameter estimation methods. The matrix problem is subdivided as 

A C

B D

   f u = v g

Sparse matrices are ones in which the majority of elements are zero. If the zero entries occur in special patterns, efficient techniques can be used to exploit the structure, as was done above for tridiagonal matrices, block tridiagonal matrices, arrow matrices, etc. These structures typically arise from numerical analysis applied to solve differential equations. Other problems, such as modeling chemical processes, lead to sparse matrices but without such a neatly defined structure – just a lot of zeros in the matrix. For matrices such as these, special techniques must be employed: efficient codes are available [4]. These codes usually employ a symbolic factorization, which must be repeated only once for each structure of the matrix. Then an LU factorization is performed, followed by a solution step

Mathematics in Chemical Engineering using the triangular matrices. The symbolic factorization step has significant overhead, but this is rendered small and insignificant if matrices with exactly the same structure are to be used over and over [5–9]. The efficiency of a technique for solving sets of linear equations obviously depends greatly on the arrangement of the equations and unknowns because an efficient arrangement can reduce the bandwidth, for example. Techniques for renumbering the equations and unknowns arising from elliptic partial differential equations are available for finite difference methods [10] and for finite element methods [11]. Solutions with Iterative Methods. Sets of linear equations can also be solved by using iterative methods; these methods have a rich historical background. Some of them are discussed in Chapter 8 and include Jacobi, Gauss – Seidel, and overrelaxation methods. As the speed of computers increases, direct methods become preferable for the general case, and those are the methods presented here. One iterative method deserves special mention, however, because of its importance in solving partial differential equations. The conjugate gradient method is an iterative method that can solve a set of n linear equations in n iterations. The method primarily requires multiplying a matrix by a vector, which can be done very efficiently on parallel computers: for sparse matrices this is a viable method. The original method was devised by Hestenes and Stiefel [12]; however, more recent implementations use a preconditioned conjugate gradient method because it converges faster, provided a good “preconditioner” can be found. An efficient implementation of such a method is described by Eisenstat [13]. The system of n linear equations A x =f

where A is symmetric and positive definite, is to be solved. A preconditioning matrix M is defined in such a way that the problem M t =r

is easy to solve exactly (M might be diagonal, for example). Then the preconditioned conjugate gradient method is

11

Guess x0 Calculate r 0 = f − Ax0 Solve M t0 = r 0 , and set p0 = t0 for k = 1, n (or until convergence) ak =

rT k tk pT k A pk

xk+1 = xk + ak pk r k+1 =r k − ak A pk Solve M tk+ 1 = r k+1 bk =

rT k+1 tk+1 rT k tk

pk+1 = tk+1 + bk pk test for convergence enddo

Additional information is available in [14] and [15]. The singular value decomposition is useful when the matrix is singular or nearly singular, or when the system of equations is overdetermined. An m×n matrix A can be represented by 



w1

  A =U W V T , W =  

   

w2 ... wn

where the matrices U and V are orthogonal in the following sense: m 

Uik Uij = δkj , 1 ≤ k ≤ n

i=1 n 

Vik Vij = δkj , 1 ≤ j ≤ n

i=1

In addition V V T =I

This decomposition can always be performed, even for singular matrices. The condition number is the ratio of the largest wj to the smallest wj . The inverse of A is

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Mathematics in Chemical Engineering

A−1 =V W −1 U T

The rank r of a matrix is a value such that all r + 1×r + 1 determinants are zero. If an n×n matrix A is singular, the rank of the matrix is r < n. The columns of U whose same-numbered elements wj are nonzero are an orthonormal set of basis vectors that span the range. The columns of V whose same-numbered wj are zero provide an orthonormal basis for the nullspace. The solution to the problem A x = f is

    xk+1 = xk + β f xk ≡g xk

x =V W −1 U T f

In this Equation if wj = 0, 1/wj is replaced with zero [16]. This is also the least-squares solution to a set of overdetermined equations (i.e., A is an m×n matrix, x is an n×1 vector, and f is an m×1 vector). In dimensional analysis if the dimensions of each physical variable Pj (there are n of them) are expressed in terms of fundamental measurement units mj (such as time, length, mass; there are m of them): α

α

α

[Pj ] = m1 1j m2 2j . . . mmmj

then a matrix can be formed from the αij . If the rank of this matrix is r, n − r independent dimensionless groups govern that phenomenon. In chemical reaction engineering the chemical reaction stoichiometry can be written as n 

supply various x until f (x) = 0. Another alternative is to program a microcomputer to make a table of x, f (x) and look for the root in the table, refining the table where needed. These methods can be employed when the problem is not too hard and it need be solved only once or twice. Even experienced programmers may do this to save time in some problems. Iterative methods are also employed to solve the equation, and the k-th iteration is denoted as xk . The successive substitution method is written as

αij Ci = 0, j = 1, 2, . . . , m

If the constant β is chosen correctly these iterations will converge to a solution. The conditions are that [17]  dg    (x)  ≤ µ for |x − α| < h  dx

where µ and h are constants, 0 < µ < 1, and α is the unknown solution. In other words, if the derivative can be bounded by 1.0 for all x then the method converges. The convergence is linear and may be slow, requiring many iterations, but the method is easy to program. This method is difficult to apply to systems of equations, but it is applied in the solution of ordinary differential equations. In that case, the constant β is proportional to ∆t, and ∆t is reduced until the method converges. The Newton – Raphson method is based on a Taylor series of the equation about the k-th iterate:

i=1

where there are n species and m reactions. Then if a matrix is formed from the coefficients αij , which is an n×m matrix, and the rank of the matrix is r, there are r independent chemical reactions. The other n − r reactions can be deduced from those r reactions.

1.2. Nonlinear Algebraic Equations In considering a single nonlinear equation in one unknown,

 f

   df     xk+1 = f xk +  k xk+1 − xk dx x +

2 d2 f  1  k+1 − xk + . . . x  k 2 x dx 2

The second and higher-order terms are neglected and f (xk+1 ) = 0. Rearrangement gives x

k+1

f xk = x − df /dx xk k

This method converges if the following inequalities are satisfied [3, p. 115].

f (x) = 0

df   f x0        0 > 0 , |x1 − x0 | =  ≤ b, dx x df /dx (x0 ) x0

one quick method for finding a root is to program a calculator to evaluate f for a given x and then

 d2 f    and  2  ≤ c dx

Mathematics in Chemical Engineering where b and c are bounded constants. In practice the method may not converge unless the initial guess is good, or it may converge for some parameters and not others. Unfortunately, when the method is nonconvergent the results look as though a mistake occurred in the computer programming; distinguishing between these situations is difficult, so careful programming and testing are required. If the method converges the difference between successive iterates is something like 0.1, 0.01, 0.0001, 10−8 . The error (when it is known) goes the same way; the method is said to be quadratically convergent when it converges. The method is not robust because it can fail for poor initial guesses. If the derivative is difficult to calculate a numerical approximation may be used.



f xk + ε − f x d f   k= dx x ε

k

In the secant method the same formula is used as for the Newton – Raphson method, except that the derivative is approximated by using the values from the last two iterates:

f xk − f xk−1 d f   = dx xk xk − xk−1

This is equivalent to drawing a straight line through the last two iterate values on a plot of f (x) versus x. The Newton – Raphson method is equivalent to drawing a straight line tangent to the curve at the last x. In the method of false position (or regula falsi), the secant method is used to obtain xk+1 , but the previous value is taken as either xk−1 or xk . The choice is made so that the function evaluated for that choice has the opposite sign to f (xk+1 ). This method is slower than the secant method, but it is more robust and keeps the root between two points at all times. In all these methods, appropriate strategies are required for bounds on the function or when df /dx = 0. Brent’s method combines bracketing, bisection, and an inverse quadratic interpolation to provide a method that is fast and guaranteed to converge, if the root can be bracketed initially [16, p. 251]. In the method of bisection, if a root lies between x 1 and x 2 because f (x 1 ) < 0 and f (x 2 ) > 0, then the function is evaluated at the center, x c = 0.5 (x 1 + x 2 ). If f (x c ) > 0, the root lies between x 1 and x c . If f (x c ) < 0, the root lies between x c and x 2 . The process is then repeated.

13

If f (x c ) = 0, the root is x c . If f (x 1 ) > 0 and f (x 2 ) > 0, more than one root may exist between x 1 and x 2 (or no roots). For systems of equations the Newton – Raphson method is widely used, especially for equations arising from the solution of differential equations. fi ({xj }) = 0, 1 ≤i, j ≤n, where {xj } = (x1 , x2 , . . . , xn ) = x

Then, an expansion in several variables occurs: n      ∂ fi  fi xk+1 = fi xk +  ∂ xj xk j=1



 xk+1 − xkj + . . . j

The Jacobian matrix is defined as J kij =

∂ fi   ∂xj xk

and the Newton – Raphson method is n 

    J kij xk+1 − xk = − fi xk

j=1

For convergence, the norm of the inverse of the Jacobian must be bounded, the norm of the function evaluated at the initial guess must be bounded, and the second derivative must be bounded [3, p. 115], [17, p. 12]. A review of the usefulness of solution methods for nonlinear equations is available [18]. This review concludes that the Newton – Raphson method may not be the most efficient. Broyden’s method approximates the inverse to the Jacobian and is a good all-purpose method, but a good initial approximation to the Jacobian matrix is required. Furthermore, the rate of convergence deteriorates for large problems, for which the Newton – Raphson method is better. Brown’s method [18] is very attractive, whereas Brent’s is not worth the extra storage and computation. Homotopy methods can be used to ensure finding the solution when the problem is especially complicated. Suppose an attempt is made to solve f (x) = 0, and it fails; however, g (x) = 0 can be solved easily, where g (x) is some function, perhaps a simplification of f (x). Then, the

14

Mathematics in Chemical Engineering

two functions can be embedded in a homotopy by taking

curve. Then the homotopy equation is written along with the arc-length equation.

h (x, t) =t f (x) + (1 −t) g (x)

∂h dx ∂h dt + = 0 ∂x ds ∂t ds

In this equation, h can be a n×n matrix for problems involving n variables; then x is a vector of length n. Then h (x, t) = 0 can be solved for t = 0 and t gradually changes until at t = 1, h (x, 1) = f (x). If the Jacobian of h with respect to x is nonsingular on the homotopy path (as t varies), the method is guaranteed to work. In classical methods, the interval from t = 0 to t = 1 is broken up into N subdivisions. Set ∆t = 1/N and solve for t = 0, which is easy by the choice of g (x). Then set t = ∆t and use the Newton – Raphson method to solve for x. Since the initial guess is presumably pretty good, this has a high chance of being successful. That solution is then used as the initial guess for t = 2 ∆t and the process is repeated by moving stepwise to t = 1. If the Newton – Raphson method does not converge, then ∆t must be reduced and a new solution attempted. Another way of using homotopy is to create an ordinary differential equation by differentiating the homotopy equation along the path (where h = 0). dh [x (t) , t] ∂h dx ∂h = + = 0 dt ∂x dt ∂t

This can be expressed as an ordinary differential equation for x (t): ∂h dx ∂h = − ∂x dt ∂t

If Euler’s method is used to solve this equation, a value x 0 is used, and dx/dt from the above equation is solved for. Then x1,0 = x0 +∆t

dx dt

dxT dx + ds ds



dt ds

2 = 1

The initial conditions are x (0) =x0 t (0) = 0

The advantage of this approach is that it works even when the Jacobian of h becomes singular because the full matrix is rarely singular. Illustrations applied to chemical engineering are available [19].

1.3. Linear Difference Equations Difference equations arise in chemical engineering from staged operations, such as distillation or extraction, as well as from differential equations modeling adsorption and chemical reactors. The value of a variable in the n-th stage is noted by a subscript n. For example, if yn,i denotes the mole fraction of the i-th species in the vapor phase on the n-th stage of a distillation column, xn,i is the corresponding liquid mole fraction, R the reflux ratio (ratio of liquid returned to the column to product removed from the condenser), and Kn,i the equilibrium constant, then the mass balances about the top of the column give yn+1,i =

R 1 xn,i + x0,i R+1 R+1

and the equilibrium equation gives yn,i =Kn,i xn,i

is used as the initial guess and the homotopy equation is solved for x1 .

If these are combined,

   ∂h  1,k+1 − x1,k = − h x1,k , t x ∂x

Kn+1,i xn+1,i =

Then t is increased by ∆t and the process is repeated. In arc-length parameterization, both x and t are considered parameterized by a parameter s, which is thought of as the arc length along a

is obtained, which is a linear difference equation. This particular problem is quite complicated, and the interested reader is referred to [20, Chap. 6]. However, the form of the difference equation is clear. Several examples are given

R 1 xn,i + x0,i R+1 R+1

Mathematics in Chemical Engineering here for solving difference equations. More complete information is available in [21]. An equation in the form

xn+1 −fn xn = 0 x0 = c

xn+1 −xn = fn

the general solution is

can be solved by

xn = c

xn =

n 

n 

fi

i=1

fi

This can then be used in the method of variation of parameters to solve the equation

i=1

Usually, difference equations are solved analytically only for linear problems. When the coefficients are constant and the equation is linear and homogeneous, a trial solution of the form xn = ϕ n

c xn−1 + b xn + a xn+1 = 0

coupled with the trial solution would lead to the equation a ϕ2 + b ϕ + c = 0

This gives −b ±

xn+1 −fn xn = gn

1.4. Eigenvalues

is attempted; ϕ is raised to the power n. For example, the difference equation

ϕ1,2 =

15

√

b2 − 4 a c 2a

and the solution to the difference equation is n xn = A ϕ n 1 + B ϕ2

where A and B are constants that must be specified by boundary conditions of some kind. When the equation is nonhomogeneous, the solution is represented by the sum of a particular solution and a general solution to the homogeneous equation. xn = xn,P + xn,H

The general solution is the one found for the homogeneous equation, and the particular solution is any solution to the nonhomogeneous difference equation. This can be found by methods analogous to those used to solve differential equations: the method of undetermined coefficients and the method of variation of parameters. The last method applies to equations with variable coefficients, too. For a problem such as

The n×n matrix A has n eigenvalues λi , i = 1, . . . , n, which satisfy det (A −λi I ) = 0

If this equation is expanded, it can be represented as Pn (λ) = (−λ)n + a1 (−λ)n−1 + a2 (−λ)n−2 + . . . + an−1 (−λ) + an = 0

If the matrix A has real entries then ai are real numbers, and the eigenvalues either are real numbers or occur in pairs as complex numbers with their complex conjugates (for definition of complex numbers, see Chap. 3). The Hamilton – Cayley theorem [20, p. 127] states that the matrix A satisfies its own characteristic equation. Pn (A) = (−A)n + a1 (−A)n−1 + a2 (−A)n−2 + . . . + an−1 (−A) + an I = 0

A laborious way to find the eigenvalues of a matrix is to solve the n-th order polynomial for the λi – far too time consuming. Instead the matrix is transformed into another form whose eigenvalues are easier to find. In the Givens method and the Housholder method the matrix is transformed into the tridiagonal form; then, in a fixed number of calculations the eigenvalues can be found [16]. The Givens method requires 4 n3 /3 operations to transform a real symmetric matrix to tridiagonal form, whereas the Householder

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Mathematics in Chemical Engineering

method requires half that number [3]. Once the tridiagonal form is found, a Sturm sequence is applied to determine the eigenvalues. These methods are especially useful when only a few eigenvalues of the matrix are desired. If all the eigenvalues are needed, the Q R algorithm is preferred. In the Q R algorithm [22] a modified Householder transformation is applied to A, transforming it to the form

methods for parameter estimation for both linear and nonlinear models are given in Sections 2.5 and 2.6. Fourier transforms to represent discrete data are described in Section 2.7. The chapter closes with extensions to two-dimensional representations.

A = QR

A global polynomial Pm (x) is defined over the entire region of space

where Q is orthogonal and R is upper triangular. If A is banded, then Q and R are banded. The eigenvalues of a certain tridiagonal matrix can be found analytically. If A is a tridiagonal matrix with aii = p, ai,i+1 = q, ai+1,i = r, q r > 0

then the eigenvalues of A are [23] λi = p + 2 (q r)1/2 cos

iπ i = 1, 2, . . . , n n+1

2.2. Global Polynomial Approximation

Pm (x) =

m 

c j xj

j=0

This polynomial is of degree m (highest power is xm ) and order m + 1 (m + 1 parameters {cj }). If a set of m + 1 points is given, y1 =f (x1 ) , y2 =f (x2 ) , . . . , ym+1 =f (xm+1 )

then Lagrange’s formula yields a polynomial of degree m that goes through the m + 1 points:

This result is useful when finite difference methods are applied to the diffusion equation.

Pm (x) =

2. Approximation and Integration

(x − x1 ) (x − x3 ) . . . (x − xm+1 ) y2 + . . . + (x2 − x1 ) (x2 − x3 ) . . . (x2 − xm+1 )

2.1. Introduction Two types of problems arise frequently: 1) A function is known exactly at a set of points and an interpolating function is desired. The interpolant may be exact at the set of points, or it may be a “best fit” in some sense. Alternatively it may be desired to represent a function in some other way. 2) Experimental data must be fit with a mathematical model. The data have experimental error, so some uncertainty exists. The parameters in the model as well as the uncertainty in the determination of those parameters is desired. These problems are addressed in this chapter. Section 2.2 gives the properties of polynomials defined over the whole domain and Section 2.3 of polynomials defined on segments of the domain. In Section 2.4, quadrature methods are given for evaluating an integral. Least-squares

(x − x2 ) (x − x3 ) . . . (x − xm+1 ) y1 + (x1 − x2 ) (x1 − x3 ) . . . (x1 − xm+1 )

(x − x1 ) (x − x2 ) . . . (x − xm ) ym+1 (xm+1 − x1 ) (xm+1 − x2 ) .. . (xm+1 − xm )

Note that each coefficient of yj is a polynomial of degree m that vanishes at the points {xj } (except for one value of j ) and takes the value of 1.0 at that point, i.e., Pm (xj ) =yj

j = 1, 2, . . . , m + 1

If the function f (x) is known, the error in the approximation is [24] |error (x) | ≤

|xm+1 − x1 |m+1 (m + 1)!

maxx1 ≤x≤xm+1 | f (m+1) (x) |

The evaluation of Pm (x) at a point other than the defining points can be made with Neville’s algorithm [25]. Let P1 be the value at x of the unique function passing through the point (x 1 , y1 ); i.e., P1 = y1 . Let P12 be the value at x of the unique polynomial passing through the points x 1 and

Mathematics in Chemical Engineering x 2 . Likewise, Pijk...r is the unique polynomial passing through the points xi , xj , xk , . . . , xr . The following scheme is used:

17


b W (x) [ f (x) − p (x)]2 dx

I = a

when b cj =

These entries are defined by using

W (x) f (x) Pj (x) dx

a

Wj
b W (x) Pj2 (x) dx

Wj =

Pi(i+1)...(i+m) =

,

a

(x − xi+m ) Pi(i+1)...(i+m−1) + (xi − x) P(i+1)(i+2)...(i+m) xi − xi+m

Consider P1234 : the terms on the right-hand side of the equation involve P123 and P234 . The “parents,” P123 and P234 , already agree at points 2 and 3. Here i = 1, m = 3; thus, the parents agree at xi+1 , . . . , xi+m−1 already. The formula makes Pi(i+1)...(i+m) agree with the function at the additional points xi+m and xi . Thus, Pi(i+1)...(i+m) agrees with the function at all the points {xi , xi+1 , . . . , xi+m }. Orthogonal Polynomials. Another form of the polynomials is obtained by defining them so that they are orthogonal. It is required that Pm (x) be orthogonal to Pk (x) for all k = 0, . . . , m − 1.
b W (x) Pk (x) Pm (x) dx = 0 a

Note that each cj is independent of m, the number of terms retained in the series. The minimum value of I is
b a

k = 0, 1, 2, . . . , m − 1


b 2 W (x) Pm (x) dx = 1 a

The polynomial Pm (x) has m roots in the closed interval a to b. The polynomial

W

minimizes

Wj c2j

j=0

x2 Pk x2 Pm x2 xa−1 dx = 0



0

k = 0, 1, . . . , m − 1

where a = 1 is for planar, a = 2 for cylindrical, and a = 3 for spherical geometry. These functions are useful if the solution can be proved to be an even function of x. Rational Polynomials. Rational polynomials are ratios of polynomials. A rational polynomial Ri(i+1)...(i+m) passing through m + 1 points yi = f (xi ) , i = 1, . . . , m + 1

p (x) = c0 P0 (x) + c1 P1 (x) + . . . cm Pm (x)

m 

Such functions are useful for continuous data, i.e., when f (x) is known for all x. Typical orthogonal polynomials are given in Table 1. Chebyshev polynomials are used in spectral methods (see Chap. 6). The last two rows of Table 1 are widely used in the orthogonal collocation method in chemical engineering. The last entry (the shifted Legendre polynomial as a function of x 2 ) is defined by
1

The orthogonality includes a nonnegative weight function, W (x) ≥ 0 for all a ≤ x ≤ b. This procedure specifies the set of polynomials to within multiplicative constants, which can be set either by requiring the leading coefficient to be one or by requiring the norm to be one.

W (x) f 2 (x) dx −

Imin =

is

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Mathematics in Chemical Engineering

Table 1. Orthogonal polynomials [24] a

b

W (x)

Name

Recursion relation

−1

1

1

Legendre

(i+1) Pi +1 =(2 i+1)xPi − i Pi −1

−1

1

√

Chebyshev

Ti +1 =2xTi −Ti −1

0

1

x q −1 (1−x)p−q

−∞ 0 0 0

∞ ∞ 1 1

e−x x c e−x 1 1

1 1−x2

Ri(i+1)...(i+m) =

2

Jacobi ( p, q) Hermite Laguerre (c) shifted Legendre shifted Legendre, function of x 2

Pµ (x) p0 + p1 x+ . . . +pµ xµ , = Qν (x) q0 + q1 x + ... + qν xν

An alternative condition is to make the rational polynomial agree with the first m + 1 terms in the power series, giving a Pad´e approximation, i.e., dxk

=

dk f (x) dxk

 1−

R(i+1) ...(i+m) − R(i+ 1)...(i+m−1)

xα − x 0 ∆x

and the finite interpolation formula through the points y0 , y1 , . . . , yn is written as follows:



R(i+1)...(i+m) Ri(i+ 1)...(i+m−1)

∆2 yn = ∆yn+1 − ∆yn = yn+2 − 2yn+1 + yn

α =

Den

Den =

∆yn = yn+1 −yn

Then, a new variable is defined

R(i+1)...(i+m) − Ri(i+ 1)...(i+m−1)

x − xi x − xi+m

yn = y (xn )

forward differences are defined by

Ri(i+ 1)...(i+m) = R(i+1) ...(i+m)



xn+1 −xn = ∆x

k = 0, . . . , m

The Bulirsch – Stoer recursion algorithm can be used to evaluate the polynomial:

+

2.3. Piecewise Approximation Piecewise approximations can be developed from difference formulas [27]. Consider a case in which the data points are equally spaced

m + 1 = µ+ν + 1

dk Ri(i+1)...(i+m)

Hi +1 =2xHi −2 i Hi −1 (i+1) L i +1 c =(−x+2 i+c+1) L ci −(i+c) L i −1 c



yα = y0 + α∆y0 + −1

Rational polynomials are useful for approximating functions with poles and singularities, which occur in Laplace transforms (see Section 4.2). Fourier series are discussed in Section 4.1. Representation by sums of exponentials is also possible [26]. In summary, for discrete data, Legendre polynomials and rational polynomials are used. For continuous data a variety of orthogonal polynomials and rational polynomials are used. When the number of conditions (discrete data points) exceeds the number of parameters, then see Section 2.5.

α (α − 1) 2 ∆ y0 + . . . + 2!

α (α − 1) . . . (α − n + 1) n ∆ y0 n!

(1)

Keeping only the first two terms gives a straight line through (x 0 , y0 ) − (x 1 , y1 ); keeping the first three terms gives a quadratic function of position going through those points plus (x 2 , y2 ). The value α = 0 gives x = x 0 ; α = 1 gives x = x 1 , etc. Backward differences are defined by ∇yn =yn − yn−1 ∇2 yn = ∇yn − ∇yn−1 = yn − 2yn−1 + yn−2

Mathematics in Chemical Engineering The interpolation polynomial of order n through the points y0 , y−1 , y−2 , . . . is yα = y0 + α∇y0 +

α (α + 1) 2 ∇ y0 + . . . + 2!

19

NI=1 = 1 − u , NI=2 = u

and the expansion as is rewritten as 2 

α (α + 1) . . . (α + n − 1) n ∇ y0 n!

y (x) =

The value α = 0 gives x = x 0 ; α = − 1 gives x = x −1 . Alternatively, the interpolation polynomial of order n through the points y1 , y0 , y−1 , . . . is

x in e-th element and ci = ceI within the element e. Thus, given a set of points (xi , yi ), a finite element approximation can be made to go through them.

yα = y1 + (α − 1) ∇y1 +

ceI NI (u)

(2)

I=1

α (α − 1) 2 ∇ y1 + . . . + 2!

(α − 1) α (α + 1) . . . (α + n − 2) n ∇ y1 n!

Now α = 1 gives x = x 1 ; α = 0 gives x = x 0 . The finite element method can be used for piecewise approximations [28]. In the finite element method the domain a ≤ x ≤ b is divided into elements as shown in Figure 5. Each function Ni (x) is zero at all nodes except xi ; Ni (xi ) = 1. Thus, the approximation is y (x) =

NT 

ci Ni (x) =

i=1

NT 

y (xi ) Ni (x)

i=1

where ci = y (xi ). For convenience, the trial functions are defined within an element by using new coordinates: u =

x − xi ∆xi

The ∆xi need not be the same from element to element. The trial functions are defined as Ni (x) (Fig. 5 A) in the global coordinate system and NI (u) (Fig. 5 B) in the local coordinate system (which also requires specification of the element). For x i < x < xi+1 y (x) =

NT 

ci Ni (x) = ci Ni (x) + ci+1 Ni+1 (x)

Figure 5. Galerkin finite element method – linear functions A) Global numbering system; B) Local numbering system

Quadratic approximations can also be used within the element (see Fig. 6). Now the trial functions are   1 NI=1 = 2 (u − 1) u − 2 NI=2 = 4 u (1 − u)  NI=3 = 2 u

u−

1 2



The approximation going through an odd number of points (xi , yi ) is then

i=1 3 

because all the other trial functions are zero there. Thus

y (x) =

y (x) = ci NI=1 (u) + ci+1 NI=2 (u) ,

with ceI = y (xi ) , i = (e − 1) 2 + I

xi < x < xi+1 , 0 < u < 1

Then

(3)

ceI NI (u) x in e−th element

I=1

in the e−th element

20

Mathematics in Chemical Engineering Splines. Splines are functions that match given values at the points x 1 , . . . , xN T , shown in Figure 7, and have continuous derivatives up to some order at the knots, or the points x 2 , . . . , xN T −1 . Cubic splines are most common. In this case the function is represented by a cubic polynomial within each interval and has continuous first and second derivatives at the knots.

Figure 6. Finite elements approximation – quadratic elements A) Global numbering system; B) Local numbering system

Hermite cubic polynomials H which have continuous first derivatives can be used for the NI [28]. Now, write y (x) =

4 

ceI HI (u) x in e−th element

I=1

where H1 = (1 −u)2 (1 + 2u)

Figure 7. Finite elements for cubic splines A) Notation for spline knots. B) Notation for one element

Consider the points shown in Figure 7 A. The notation for each interval is shown in Figure 7 B. Within each interval the function is represented as a cubic polynomial. Ci (x) = a0i + a1i x + a2i x2 + a3i x3

H2 = u (1 − u)2 ∆xe

The interpolating function takes on specified values at the knots.

H3 = u2 (3 − 2u)

Ci−1 (xi ) = Ci (xi ) =f (xi )

H4 = u2 (u − 2) ∆xe

Given the set of values {xi , f (xi )}, the objective is to pass a smooth curve through those points, and the curve should have continuous first and second derivatives at the knots.

Identify the points within the e-th element as xi : u = 0; xi+1 : u = 1. Then ce1 = y (xi ) , ce2

ce3 = y (xi+1 )

dy dy = (xi ) ∆xe , ce4 = (xi+1 ) ∆xe dx dx

Thus, at the points x1 , x 2 , . . . , x N T

both the function and first derivative are necessary: y (x1 ) ,

y (x2 ) , . . . , y (xN T )

dy dy dy (x1 ) , (x2 ) , . . . , (xN T ) dx dx dx

 Ci−1 (xi ) = Ci (xi )  (xi ) = Ci (xi ) Ci−1

The formulas for the cubic spline are derived as follows for one region. Since the function is a cubic function the third derivative is constant and the second derivative is linear in x. This is written as   x − xi Ci (x) = Ci (xi ) + Ci (xi+1 ) − Ci (xi ) ∆xi

Mathematics in Chemical Engineering and integrated once to give Ci

(x) =

Ci

(xi ) +



Ci

(xi ) (x − xi ) +  (x − xi )2 2∆xi

Ci (xi+1 ) − Ci (xi )

and once more to give Ci (x) = Ci (xi ) + Ci (xi ) (x − xi ) + Ci (xi )   (x − xi )3 (x − xi )2 + Ci (xi+1 ) − Ci (xi ) 2∆xi 6∆xi

Now is defined so that Ci (x) = yi + yi (x − xi ) +

1 y  (x − xi )2 2∆xi i

A number of algebraic steps make the interpolation easy. These formulas are written for the i-th element as well as the i − 1-th element. Then the continuity conditions are applied for the first and second derivatives, and the values y
i and y
i −1 are eliminated [29, p. 163]. The result is   ∆xi−1 + yi 2 (∆xi−1 +∆xi ) + yi+1 ∆xi = yi−1



yi+1 − yi yi − yi−1 − ∆xi−1 ∆xi



This is a tridiagonal system for the set of {y

i } in terms of the set of {yi }. Since the continuity conditions apply only for i = 2, . . . , NT − 1, only NT − 2 conditions exist for the NT values of y

i . Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y

1 , y

N T . If these values are zero, the natural cubic splines are obtained; they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero, in the natural cubic spline, an NT − 2 system of tridiagonal equations exists, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by yi =

y1 =y (x1 )

and then at the NCOL interior points to each element yIe = yi = y (xi ) , i = (N COL + 1) e +I

The actual points xi are taken as the roots of the orthogonal polynomial.

1  y − yi (x − xi )3 6∆xi i+1

−6

Orthogonal Collocation on Finite Elements. In the method of orthogonal collocation on finite elements the solution is expanded in a polynomial of order NP = NCOL + 2 within each element [28]. The choice NCOL = 1 corresponds to using quadratic polynomials, whereas NCOL = 2 gives cubic polynomials. The notation is shown in Figure 8. Set the function to a known value at the two endpoints

yN T = y (xN T )

yi = Ci (xi ) , yi = Ci (xi ) , yi = Ci (xi )

+

21

∆xi yi+1 − yi  ∆xi − yi − yi+1 ∆xi 3 6

The function itself is then known within each element.

PN COL (u) = 0 gives u1 , u2 , . . . , uN COL

and then xi = x(e) + ∆xe uI ≡ xeI

The first derivatives must be continuous at the element boundaries: dy  dy  =   dx x=x(2)− dx x=x(2)+

Within each element the interpolation is a polynomial of degree NCOL + 1. Overall the function is continuous with continuous first derivatives. With the choice NCOL = 2, the same approximation is achieved as with Hermite cubic polynomials.

2.4. Quadrature To calculate the value of an integral, the function can be approximated by using each of the methods described in Section 2.3. Using the first three terms in Equation (1) gives x
0 +h


1 y (x) dx =

x0

=

yα h dα 0

1 3  h ( y0 + y 1 ) − h y0 (ξ) , x0 ≤ ξ ≤ x0 + h 2 12

22

Mathematics in Chemical Engineering

Figure 8. Notation for orthogonal collocation on finite elements • Residual condition;
Boundary conditions; | Element boundary, continuity NE = total no. of elements. NT = (NCOL + 1) NE + 1

This corresponds to passing a straight line through the points (x 0 , y0 ), (x 1 , y1 ) and integrating under the interpolant. For equally spaced points at a = x 0 , a + ∆x = x 1 , a + 2 ∆x = x 2 , . . . , a + N ∆x = xN , a + (N + 1) ∆x = b = xn+1 , the trapezoid rule is obtained. Trapezoid Rule.
b a

h y (x) dx = ( y0 + 2 y1 + 2 y2 + . . . + 2 yN 2

+ yN +1 ) + O h3

The first five terms in Equation (1) are retained and integrated over two intervals. x0
+2h


2 y (x) dx =

x0

0

−

h yα h dα = ( y0 + 4 y1 + y2 ) 3

h5 (IV) (ξ) , x0 ≤ ξ ≤ x0 + 2 h y 90 0

This corresponds to passing a quadratic function through three points and integrating. For an even number of intervals and an odd number of points, 2 N + 1, with a = x 0 , a + ∆x = x 1 , a + 2 ∆x = x 2 , . . . , a + 2 N ∆x = b, Simpson’s rule is obtained. Simpson’s Rule.
b y (x) dx = a

h ( y0 + 4 y1 + 2 y2 + 4 y3 + 2 y4 3

+ . . . + 2 y2N −1 + 4 y2N + y2N +1 ) + O h5

Within each pair of intervals the interpolant is continuous with continuous derivatives, but only the function is continuous from one pair to another. If the finite element representation is used (Eq. 2), the integral is x
i+1

y (x) dx =

ceI NI (u) (xi+1 − xi ) du

0 I=1

xi

= ∆xi

2 




1 ceI

I=1

=


1  2

NI (u) du = ∆xi 0

ce1

1 1 + ce2 2 2



∆xi ( yi + yi+1 ) 2

Since ce1 = yi and ce2 = yi+1 , the result is the same as the trapezoid rule. These formulas can be added together to give linear elements:
b y (x) dx = a

 ∆xe ( y1e + y2e ) 2 e

If the quadratic expansion is used (Eq. 3), the endpoints of the element are xi and xi+2 , and xi+1 is the midpoint, here assumed to be equally spaced between the ends of the element:

Next Page Mathematics in Chemical Engineering x
i+2

y (x) dx =


1  3

The quadrature is exact when y is a polynomial of degree 2 m − 1 in x. The m weights and m Gauss points result in 2 m parameters, chosen to exactly represent a polynomial of degree 2 m − 1, which has 2 m parameters. The Gauss points and weights are given in Table 2. The weights can be defined with W (x) in the integrand as well.

ceI NI (u) (xi+2 − xi ) du

0 I=1

xi

= ∆xi

3 


1 ceI

I=1

 = ∆xe

ce1

NI (u) du 0

1 2 1 + ce2 + ce3 6 3 6



Table 2. Gaussian quadrature points and weights ∗

For many elements, with different ∆xe , quadratic elements :
b y (x) = a

 ∆xe ( y1e + 4 y2e + y3e ) 6 e

y (x) dx =



 ∆xe

e

a

x
i+1

xi

1 Ci (x) dx = ∆xi ( yi + yi+1 ) 2

−



1  ∆x3i yi + yi+1 24

For the entire interval the quadrature formula is x
N T

y (x) dx = x1

−

1 24

N T −1 1  ∆ xi ( yi + yi+1 ) 2 i=1

N T− 1



 ∆x3i yi + yi+1

i=1

with y

1 = 0, y

N T = 0 for natural cubic splines. When orthogonal polynomials are used, as in Equation (1), the m roots to Pm (x) = 0 are chosen as quadrature points and called points {xj }. Then the quadrature is Gaussian :

xi

Wi

1 2

0.5000000000 0.2113248654 0.7886751346 0.1127016654 0.5000000000 0.8872983346 0.0694318442 0.3300094783 0.6699905218 0.9305681558 0.0469100771 0.2307653450 0.5000000000 0.7692346551 0.9530899230

0.6666666667 0.5000000000 0.5000000000 0.2777777778 0.4444444445 0.2777777778 0.1739274226 0.3260725774 0.3260725774 0.1739274226 0.1184634425 0.2393143353 0.2844444444 0.2393143353 0.1184634425

4

5



1 e e ∆xe e (c + c ) + (c2 − ce4 ) 2 1 3 12

For cubic splines the quadrature rule within one element is

N

3

If the element sizes are all the same this gives Simpson’s rule. For Hermite cubic functions the quadrature rule is
b

∗

For a given N the quadrature points x 2 , x 3 , . . . , xN P − 1 are given above. x 1 = 0, xN P = 1. For N = 1, W 1 = W 3 = 1/6 and for N ≥ 2, W 1 = WN P = 0.

For orthogonal collocation on finite elements the quadrature formula is
1 y (x) dx =

y (x) dx = 0

m  j=1

Wj y (xj )



∆xe

e

0

NP 

Wj y (xeJ )

j=1

Each special polynomial has its own quadrature formula. For example, Gauss – Legendre polynomials give the quadrature formula
∞

e−x y (x) dx =

n 

Wi y (xi )

i=1

0

(points and weights are available in mathematical tables) [24]. For Gauss – Hermite polynomials the quadrature formula is
∞ −∞


1

23

2

e−x y (x) dx =

n 

Wi y (xi )

i=1

(points and weights are available in mathematical tables) [24].

Previous Page 24

Mathematics in Chemical Engineering

Romberg’s method uses extrapolation techniques to improve the answer [25]. If I 1 is the value of the integral obtained by using interval size h = ∆x, I 2 the value of I obtained by using interval size h/2, and I 0 the true value of I, then the error in a method is approximately hm , or I1 ≈ I0 +c h

m

P =

 m h I2 ≈ I0 + c 2

2m I2 − I1 2m − 1

This process can also be used to obtain I 1 , I 2 , . . . , by halving h each time, calculating new estimates from each pair, and calling them J 1 , J 2 , . . . (i.e., in the formula above, I 0 is replaced with J 1 ). The formulas are reapplied for each pair of J ’s to obtain K 1 , K 2 , . . . . The process continues until the required tolerance is obtained. I1

I2 J1

I3 J2 K1

I4 J3 K2 L1

Romberg’s method is most useful for a loworder method (small m) because significant improvement is then possible. When the integrand has singularities, a variety of techniques can be tried. The integral may be divided into one part that can be integrated analytically near the singularity and another part that is integrated numerically. Sometimes a change of argument allows analytical integration. Series expansion might be helpful, too. When the domain is infinite, Gauss – Legendre or Gauss – Hermite quadrature can be used. Also a transformation can be made [25]. For example, let u = 1/x and then
b

1/a


f (x) dx = a

N 



  1 1 f du u2 u



1 exp − 2

i=1

Replacing the ≈ by an equality (an approximation) and solving for c and I 0 give I0 =

squares [25]. If each measurement point yi has a measurement error ∆yi that is independently random and distributed with a normal distribution about the true model y (x) with standard deviation σi , then the probability of a data set is 

yi − y (xi ) σi

2

(For definition of probability, normal distribution, and standard deviation, see Chap. 11.) Here yi is the measured value, σi is the standard deviation of the i-th measurement, and ∆y is needed in order that a measured value ± ∆y has a certain probability. Given a set of parameters (maximizing this function), the probability that this data set ( plus or minus ∆y) could have occurred is P. This probability is maximized (giving the maximum likelihood) if the negative of the logarithm is minimized. − log P =

 N   yi − y (xi ) 2 √ − N log ∆y 2 σi i=1

Because N, σi , and ∆y are constants, this is the same as minimizing χ2 : χ2 =

 N   yi − y (xi ; a1 , . . . , aM ) 2 σi i=1

 P = 1−Q

ν 1 2 , χ 2 2 0



where Q (a, x) is the incomplete gamma function 1 Γ (a)


x

e−t ta−1 dt (a > 0)

0

a, b > 0

and Γ (a) is the gamma function
∞ Γ (a) =

2.5. Linear Least Squares The following description of maximum likelihood applies to both linear and nonlinear least

(4)

with respect to the parameters {aj }. Note that the standard deviations {σi } of the measurements are expected to be known. The goodness of fit is related to the number of degrees of freedom, ν = N − M. The probability P that χ2 would exceed a particular value (χ0 )2 is

Q (a, x) =

1/b

∆y

0

ta−1 e−t dt

Mathematics in Chemical Engineering Both functions are tabulated in mathematical handbooks. The function P gives the goodness of fit. After Equation (4) is minimized, let χ20 be the value of χ2 at the minimum. Then Q > 0.1 represents a believable fit; Q > 0.001 might be an acceptable fit; smaller values of Q indicate that the model may be in error (or the σi are really larger). A “typical” value of χ2 for a moderately good fit is χ2 ∼ ν. Asymptotically for large ν, the statistic χ2 becomes normally distributed √ with a mean ν and a standard deviation 2ν. If values σi are not known in advance, assume σi = σ (then its value does not affect the minimization of χ2 ). Find the parameters by minimizing χ2 and compute N  [ yi − y (xi )]2 N i=1

σ2 =

This gives some information about the errors (i.e., the variance and standard deviation of each data point), although the goodness of fit P cannot be calculated. The minimization of χ2 requires  N   yi − y (xi ) ∂y (xi ; a1 , . . . , aM ) = 0, σi2 ∂ak i=1 k = 1, . . . , M

(5)

 N   yi − a − b xi 2 σi i=1

Define S =

i=1

ti =

N 

1 xi , Sx = , Sy = 2 σi2 σ i=1 i

N  i=1

yi σi2

N N   x2i xi y i , S = , xy σ2 σi2 i=1 i i=1

1 σi



Sx S

xi −

 , Stt =

N 

t2i

i=1

Then N 1  ti yi Sy − Sx b b = , a = , Stt i=1 σi S

σa2 =

1 S

N

 χ2 = 1 − r2 ( yi − y¯)2 i=1

Here N

r = !

i=1 N

i=1

(xi − x ¯)( yi − y ¯) σi2

!

¯ )2 (xi − x σi2

N

i=1

¯)2 (yi − y σi2

Values of r near 1 indicate a positive correlation, r near − 1 means a negative correlation, and r near zero means no correlation. A general linear model is one expressed as y (x) =

M 

ak Xk (x)

k=1

where the parameters are {ak }, and the expression is linear with respect to them, and Xk (x) can be any (nonlinear) functions of x, not depending on the parameters {ak }. Equation (5) is then

k = 1, . . . , M

N 

Sxx =

Thus, the values of a and b with maximum likelihood are obtained: the variances of a and b. By using the value of χ2 for this a and b, the goodness of fit P can also be calculated. In addition, the linear correlation coefficient r is related by

  N M   1  yi − aj Xj (xi ) Xk (xi ) = 0 , σ2 i=1 i j=1

When the model is a straight line χ2 (a , b) =

25

 1+

Sx2 S Stt

Cov (a, b) = −

 , σb2 =

1 Stt

Sx Cov (a, b) , rab = S Stt σ a σb

which is rewritten as

N M N   1  yi Xj (xi ) Xk (xi ) aj = Xk (xi ) 2 2 σ σ j=1 i=1 i i=1 i

or as M 

αkj aj = βk

(6)

j=1

Solving this set of equations gives the parameters {aj }, which maximize the likelihood. The variance of aj is σ 2 (aj ) = Cjj

26

Mathematics in Chemical Engineering

−1 where Cjk = αjk , i.e., C is the inverse of α. The covariance of aj and ak is given by Cjk . If roundoff errors affect the result, try to make the functions orthogonal. For example, using

be added to the criterion and the equation minimized:
b  I = α1 χ2 + α2 a

k−1

Xk (x) = x

will cause round-off errors for a smaller M than Xk (x) = Pk−1 (x)

M 

NP 

ak Nk (x) =

aej NJ (u)

e J=1

k=1

and then minimize  N    χ =   2

i=1

yi −

P N

2 aeJ NJ (uI )    

e J=1 σi2

Equation (6) is still solved, but with Xj (xi ) given by Nj (xi ), as shown in Figure 9 for NP = 2.

N  N 

2 dx =


b aj ai

j=1 i=1

(7)

where Pk−1 are orthogonal polynomials. If necessary, a singular value decomposition can be used rather than solving Equation (6) directly. Various global and piecewise polynomials can be used to fit the data. Some of the approximations only make sense for N = M, leading to a perfect fit of the data but perhaps a highly oscillatory model. These include the Lagrangian polynomial, and the forward and backward differences. However, the other approximations can be used with M < N. Consider orthogonal polynomials as expressed in Equation (7). Simply evaluate the polynomials at the points {xi }, and solve Equation (6) for the coefficients. When piecewise polynomials are used, write y (x) =

α1 χ2 + α2

dy dx

a

dNi dNj dx dx dx

Generally, small values of α2 are used. The format for the last term, in terms of finite elements (see Section 7.6), is
b a

 1 dNi dNj dx = dx dx ∆xe e


1 0

dNI dNJ du du du

The effect of this smoothing on the statistics for {ak } is unknown.

2.6. Nonlinear Least Squares The Levenberg – Marquardt method is used when the parameters of the model appear nonlinearly. Define χ2 (a) =

 N   yi − y (xi ; a) 2 σi i=1

and, near the optimum, represent χ2 by χ2 (a) = χ20 − dT ·a +

1 T a ·D · a 2

where d is an M×1 vector and D is an M×M matrix. Then calculate iteratively     D · ak+1 −ak = − ∇χ2 ak

(8)

The notation akl means the l-th component of a evaluated on the k-th iteration. If ak is a poor approximation to the optimum, steepest descent might be used instead   ak+1 −ak = −constant × ∇χ2 ak

Figure 9. Fitting discrete data using linear elements

Sometimes, particularly in two-dimensional problems, elements may have no, or too few, data points in them. Then the matrix αkj would be singular. In that case a smoothing term can

(9)

and the constant chosen somehow to decrease χ2 as much as possible. The gradient of χ2 is  yi − y (xi ; a) ∂y (xi ; a) ∂χ2 = −2 σi2 ∂ak ∂ak i=1 N

k = 1, 2, . . . , M

Mathematics in Chemical Engineering The second derivative (in D) is ( N  ∂ 2 χ2 1 ∂y (xi ; a) ∂y (xi ; a) = 2 2 ∂ak ∂al σ ∂ak ∂al i=1 i − [ yi − y (xi ; a)]

∂ 2 y (xi ; a) ∂ak ∂al

)

Equations (8) and (9) are included in M 

  αkl ak+1 − akl = βk l

(10)

l=1

where akl =

N  1 ∂y (xi ; a) ∂y (xi ; a) 2 σ ∂ak ∂al i=1 i

akk =

  N  1 ∂y (xi ; a) 2 (1 + λ) σ2 ∂ak i=1 i

βk =

k= l

27

For normally distributed errors the parameter region in which χ2 = constant can give boundaries of the confidence limits. The value of a obtained in the Marquardt method gives the minimum χ2min . Setting χ2 = χ2min + ∆χ2 for some ∆χ21 and looking at contours in parameter space where χ21 = constant give confidence boundaries at the probability associated with χ21 . For example, in a chemical reactor with radial dispersion, the heat-transfer coefficient and radial effective heat conductivity are closely connected: decreasing one and increasing the other can still give a good fit. Thus, the confidence boundaries may look something like Figure 10. The ellipse defined by ∆χ2 = 2.3 contains 68.3 % of the normally distributed data. The curve defined by ∆χ2 = 6.17 contains 95.4 % of the data.

N  yi − y (xi ; a) ∂y (xi ; a) σi2 ∂ak i=1

The second term in the second derivative is dropped because it is usually small [remember that yi will be close to y (xi , a)]. The Levenberg – Marquardt method then iterates as follows:

Figure 10. Parameter estimation for heat transfer

2

1) Choose a and calculate χ (a). 2) Choose λ, say λ = 0.001. 3) Solve Equation (10) for ak +1 and evaluate χ2 (ak +1 ). 4) If χ2 (ak +1 ) ≥ χ2 (ak ), increase λ by a factor of 10, for example, and go back to step 3. This makes the step more like a steepest descent. 5) If χ2 (ak +1 ) < χ2 (ak ), then update a (i.e., use a = ak +1 ), decrease λ by a factor of 10, and go back to step 3. 6) Stop the iteration when the decrease in χ2 from one step to another is not statistically meaningful (i.e., less than 0.1 or 0.01 or 0.001). 7) Set λ = 0 and compute the estimated covariance matrix.

2.7. Fourier Transforms of Discrete Data [25] Suppose a signal y (t) is sampled at equal intervals yn = y (n∆) , n = . . . , − 2, − 1, 0, 1, 2, . . . ∆ = sampling rate (e.g., number of samples per second)

The Fourier transform and inverse transform are
∞ y (t) eiωt dt

Y (ω) = −∞

C =α−1

This gives the standard errors in the fitted parameters a.

y (t) =

1 2π


∞ −∞

Y (ω) e−iωt dω

28

Mathematics in Chemical Engineering

(For definition of i, see Chap. 3.) The Nyquist critical frequency or critical angular frequency is 1 π , ωc = 2∆ ∆

fc =

If a function y (t) is bandwidth limited to frequencies smaller than f c , i.e., Y (ω) = 0 for ω > ωc

then the function is completely determined by its samples yn . Thus, the entire information content of a signal can be recorded by sampling at a rate ∆−1 = 2 f c . If the function is not bandwidth limited, then aliasing occurs. Once a sample rate ∆ is chosen, information corresponding to frequencies greater than f c is simply aliased into that range. The way to detect this in a Fourier transform is to see if the transform approaches zero at ± f c ; if not, aliasing has occurred and a higher sampling rate is needed. Next, for N samples, where N is even yk = y (tk ) , tk = k∆, k = 0, 1, 2, . . . , N − 1

and the sampling rate is ∆; with only N values {yk } the complete Fourier transform Y (ω) cannot be determined. Calculate the value Y (ωn ) at the discrete points ωn =

Yn =

2πn N N , n = − , . . . , 0, . . . , N∆ 2 2

N −1 

yk e2πikn/N

k=0

Y (ωn ) = ∆Yn

The discrete inverse Fourier transform is N −1 1  yk = Yn e−2πikn/N N k=0

The fast Fourier transform (FFT ) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. Yk = Yke +W k Y

o k

Here Yk is the k-th component of the Fourier transform of y, and Y ek is the k-th component of

the Fourier transform of the even components of {yj } and is of length N/2; similarly Y ok is the k-th component of the Fourier transform of the odd components of {yj } and is of length N/2; and W is a constant, taken to the k-th power. W = e2πi/N

Because Yk has N components, whereas Y ek and Y ok have N/2 components, Y ek and Y ok are repeated once to give N components in the calculation of Yk . This decomposition can be used recursively. Thus, Y ek is split into even and odd terms of length N/4. Yke =Ykee +W k Ykeo Yko =Ykoe +W k Ykoo

This process is continued until only one component remains. For this reason the number N is taken as a power of 2. The vector {yj } is filled with zeroes, if need be, to make N = 2p for some p. For the computer program, see [25, p. 381]. The standard Fourier transform takes N 2 operations to calculate, whereas the fast Fourier transform takes only N log2 N. For large N the difference is significant; at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100. The discrete Fourier transform can also be used for differentiating a function; this is used in the spectral method for solving differential equations. Consider a grid of equidistant points: xn = n∆x , n = 0, 1, 2, . . . , 2 N − 1 , ∆x =

L 2N

the solution is known at each of these grid points {Y (xn )}. First, the Fourier transform is taken: yk =

2N −1 1  Y (xn ) e−2ikπxn /L 2 N n=0

The inverse transformation is Y (x) =

1 L

N 

yk e2ikπx/L

k=−N

which is differentiated to obtain dY 1 = dx L

N  k=−N

yk

2 πi k 2ikπx/L e L

Thus, at the grid points

Mathematics in Chemical Engineering dY  1  = dx n L

N 

yk

k=−N

2 πi k 2ikπxn /L e L

The process works as follows. From the solution at all grid points the Fourier transform is obtained by using FFT {yk }. This is multiplied by 2 π i k/L to obtain the Fourier transform of the derivative: yk = yk

2 πi k L

The inverse Fourier transform is then taken by using FFT, to give the value of the derivative at each of the grid points: dY  1  = dx n L

N 

gest computing NY different cubic splines of size NX along lines of constant y, for example, and storing the derivative information. To obtain the value of f at some point x, y, evaluate each of these splines for that x. Then do one spline of size NY in the y direction, doing both the determination and the evaluation. If the points are distributed randomly within the domain, the finite element method can be used, which is especially useful if the domain is irregular. The expansion is z (x, y) =

 N  zi  i=1

2.8. Two-Dimensional Interpolation and Quadrature If the domain is square, a ≤ x ≤ b, c ≤ y ≤ d, then the approximation can be made by using tensor products of orthogonal polynomials. cij Pi (x) Pj ( y)

i=0 j=0

The coefficients are chosen to minimize I.

N  1 2 σ i=1 i

 zi −

M 

z˜l Nl (xi yi ) Nk (xi ; yi ) = 0

l=1

k = 1, . . . , M

With two-dimensional domains, ensuring enough data points with each element is difficult. Thus, minimize
∇z ·∇z dx dy

The minimum is achieved for

 z (x, y) −

m2 m1  

2 cij Pi (x) Pj ( y) dxdy

i=0 j=0

ckl =

M 

 α1

l=1

N  1 1 Nl (xi , yi ) Nk (xi , yi ) 2 σ2 σ k i=1 i





b
d W (x) W ( y) z (x, y) Pk (x) Pl (x) dxdy /

∇Nl ·∇Nk dx dy

+ α2

  

z˜l

Ω

c


b


d W (x)

a

l=1

2 z˜l Nl (xi ; yi )     σi2

Ω

c

a

M

and this is minimized with respect to zk .

W (x) W ( y) a

  

−

I = αl χ2 + α2


b
d I =

z˜l Nl (x, y)

Let the data points be zi = z (xi , yi ), i = 1, . . . , N. The maximum likelihood of the parameter fit is

yk e2ikπxn /L

χ2 =

z (x, y) =

M  l=1

k=−N

m2 m1  

29

Pk2

(x) dx

W ( y)

Pl2

= α1 ( y) dy

c

Bicubic splines can be used to interpolate a set of values on a regular array, f (xi , yj ). Suppose NX points occur in the x direction and NY points occur in the y direction. Press et al. [25] sug-

N  z˜i Nk (xi , yi ) σ2 σk2 i=1 i

The finite element integrals are calculated by [28]
∇Nl ·∇Nk dx dy = Ω


e

Ωe

∇NL ·∇NK dx dy

30

Mathematics in Chemical Engineering

If linear elements on triangles are used and the nodes are (xJ , yJ ), (xK , yK ), and (xL , yL ), the integral is


1 (bL bK + cL cK ) 4∆

∇NL ·∇NK dx dy = Ωe

where

3.1. Introduction to the Complex Plane A complex number is an ordered pair of real numbers, x and y, that is written as z = x +i y



1 xJ  ∆ = 2 det 1 xK 1 xL



yJ  yK  , yL

The variable i is the imaginary unit which has the property

bK = yL − yJ , and cK = xJ − xL

plus permutations on J, K, L that are arranged in a counterclockwise orientation. Quadrature follows directly from the approximation. If orthogonal polynomials are used, then
b
d z (x, y) dxdy = a

3. Complex Variables

m1−1  m2−1  j=1

c

Wxj Wjk z (xj , yk )

k=1

i2 = −1

The real and imaginary parts of a complex number are often referred to: Re (z) =x, Im (z) = y

A complex number can also be represented graphically in the complex plane, where the real part of the complex number is the abscissa and the imaginary part of the complex number is the ordinate (see Fig. 11).

and the limits of integration must be transformed to coincide with those of the defining polynomial. If finite elements are used, then
z (x, y) dxdy =

M 


z˜l

l=1

Ω

=

NP  e L=1

Nl (x, y) dxdy Ω


e z˜L

NL (x, y) dxdy Ωe

For the same linear elements on triangles
NL (x, y) dxdy =

∆ 3

Ωe

Multidimensional integrals can also be broken down into one-dimensional integrals. For example,
b f
2 (x)


b z (x, y) dxdy =

a f1 (x)

G (x) dx; a

f
2 (x)

G (x) =

Figure 11. The complex plane

Another representation of a complex number is the polar form, where r is the magnitude and θ is the argument. r = |x +i y| =

-

x2 + y 2 , θ = arg (x +i y)

Write z (x, y) dx

f1 (x)

z = x +i y = r (cos θ + i sin θ)

Mathematics in Chemical Engineering so that x = rcos θ, y = r sin θ

and θ = arctan

y x

Since the arctangent repeats itself in multiples of π rather than 2 π, the argument must be defined carefully. For example, the θ given above could also be the argument of − (x + i y). The function z = cos θ + i sin θ obeys | z | = | cos θ + i sin θ | = 1. The rules of equality, addition, and multiplication are z1 =x1 + i y1 , z2 =x2 + i y2

31

The complex conjugate is z∗ = x − i y when z = x + i y and | z∗ | = | z |, arg z∗ = − arg z For complex conjugates then z ∗ z = |z|2

The reciprocal is 1 1 z∗ = (cos θ − i sin θ) , arg = z r |z|2

  1 = −argz z

Then z1 x1 +i y1 x2 − i y2 = = (x1 +i y1 ) 2 z2 x2 +i y2 x2 + y22 x1 x2 + y1 y2 x2 y1 − x1 y2 +i x22 + y22 x22 + y22

Equality :

=

z1 = z2 if and only if x1 = x2 and y1 = y2

and

Addition :

z1 r1 = [cos (θ1 − θ2 ) +i sin (θ1 − θ2 )] z2 r2

z1 + z2 = (x1 + x2 ) + i (y1 + y2 ) Multiplication : z1 z2 = (x1 x2 − y1 y2 ) + i (x1 y2 + x2 y1 )

The last rule can be remembered by using the standard rules for multiplication, keeping the imaginary parts separate, and using i2 = − 1. In the complex plane, addition is illustrated in Figure 12. In polar form, multiplication is

3.2. Elementary Functions Properties of elementary functions of complex variables are discussed here [31]. When the polar form is used, the argument must be specified because the same physical angle can be achieved with arguments differing by 2 π. A complex number taken to a real power obeys

z1 z2 =r1 r2 [cos (θ1 +θ2 ) + i sin (θ1 +θ2 )]

The magnitude of z1 + z2 is bounded by

u = z n , |z n | =|z|n , arg (z n ) =n argz (mod 2π)

|z1 ±z2 | ≤ |z1 | + |z2 | and |z1 | − |z2 | ≤ |z1 ± z2 |

u = z n = r n (cos nθ + i sin nθ)

as can be seen in Figure 12. The magnitude and arguments in multiplication obey

Roots of a complex number are complicated by careful accounting of the argument

|z1 z2 | =|z1 | |z2 |, arg (z1 z2 ) = argz1 + argz2

z = w1/n with w = R (cosΘ+ i sinΘ) , 0 ≤Θ ≤ 2π

then ( zk = R1/n  +i sin

Θ 2π + (k − 1) n n

Θ 2π + (k − 1) n n

such that Figure 12. Addition in the complex plane

 cos

)



32

Mathematics in Chemical Engineering

(zk )n = w for every k

cos z =

eiz + e−iz eiz − e−iz , sin z = , 2 2

tan z =

sin z cos z

z = r (cos θ + i sin θ) n

r = R, nθ = Θ (mod 2 π)

and satisfy The exponential function is ez = ex (cos y +i sin y)

sin (−z) = − sin z, cos (−z) = cos z sin (iz) = i sinh z,

Thus,

cos (iz) = −cosh z

can be written

All trigonometric identities for real, circular functions with real arguments can be extended without change to complex functions of complex arguments. For example,

z = reiθ

sin2 z + cos2 z = 1,

z = r (cos θ +i sin θ)

and

sin (z1 +z2 ) = sin z1 cos z2 + cos z1 sin z2 x

|e | = e , arg e = y (mod 2π) z

z

The exponential obeys ez =0 for every finite z

and is periodic with period 2 π: z+2πi

e

=e

z

Trigonometric functions can be defined by using eiy = cos y + i sin y, and e−iy = cos y − i sin y

The same is true of hyperbolic functions. The absolute boundaries of sin z and cos z are not bounded for all z. Trigonometric identities can be defined by using eiθ = cos θ + i sin θ

For example, ei(α+β) = cos (α + β) + i sin (α + β) = eiα eiβ = (cos α + i sin α)

Thus, (cos β + i sin β) eiy + e−iy cos y = = cosh i y 2 sin y =

eiy − e−iy = − i sinh i y 2i

The second equation follows from the definitions cosh z ≡

ez + e−z ez − e−z , sinh z ≡ 2 2

The remaining hyperbolic functions are tanh z ≡

sinh z 1 , coth z ≡ cosh z tanh z

sech z ≡

1 1 , csch z ≡ cosh z sinh z

The circular functions with complex arguments are defined

= cos α cos β − sin α sin β + i (cos α sin β + cos β sin α)

Equating real and imaginary parts gives cos (α + β) = cos α cos β − sin α sin β sin (α + β) = cos α sin β + cos β sin α

The logarithm is defined as ln z = ln |z| + i arg z

and the various determinations differ by multiples of 2 π i. Then, eln z =z ln (ez ) − z ≡ 0 (mod 2 πi)

Mathematics in Chemical Engineering Also, ln (z1 z2 ) − lnz1 − lnz2 ≡ 0 (mod 2πi)

is always true, but ln (z1 z2 ) = ln z1 + ln z2

holds only for some determinations of the logarithms. The principal determination of the argument can be defined as − π < arg ≤ π.

3.3. Analytic Functions of a Complex Variable Let f (z) be a single-valued continuous function of z in a domain D. The function f (z) is differentiable at the point z0 in D if f (z0 + h) − f (z0 ) lim h→0 h

exists as a finite (complex) number and is independent of the direction in which h tends to zero. The limit is called the derivative, f
(z0 ). The derivative now can be calculated with h approaching zero in the complex plane, i.e., anywhere in a circular region about z0 . The function f (z) is differentiable in D if it is differentiable at all points of D; then f (z) is said to be an analytic function of z in D. Also, f (z) is analytic at z0 if it is analytic in some ε neighborhood of z0 . The word analytic is sometimes replaced by holomorphic or regular [31]. The Cauchy – Riemann equations can be used to decide if a function is analytic. Set f (z) =f (x +i y) =u (x, y) + i v (x, y)

Theorem [31]. Suppose that f (z) is defined and continuous in some neighborhood of z = z0 . A necessary condition for the existence of f
(z0 ) is that u (x, y) and v (x, y) have first-order partials and that the Cauchy – Riemann conditions (see below) hold. ∂u ∂v ∂u ∂v = and =− at z0 ∂x ∂y ∂y ∂x

33

Theorem [32]. The function f (z) is analytic in a domain D if and only if u and v are continuously differentiable and satisfy the Cauchy – Riemann conditions there. If f 1 (z) and f 2 (z) are analytic in domain D, then α1 f 1 (z) + α2 f 2 (z) is analytic in D for any (complex) constants α1 , α2 . f1 (z) + f2 (z) is analytic in D f1 (z) /f2 (z) is analytic in D except where f2 (z) = 0

An analytic function of an analytic function is analytic. If f (z) is analytic, f
(z) = 0 in D, f (z1 ) = f (z2 ) for z1 = z2 , then the inverse function g (w) is also analytic and g  (w) =

1 where w = f (z) , f  (z)

g (w) = g [ f (z)] = z

Analyticity implies continuity but the converse is not true: z∗ = x − i y is continuous but, because the Cauchy – Riemann conditions are not satisfied, it is not analytic. An entire function is one that is analytic for all finite values of z. Every polynomial is an entire function. Because the polynomials are analytic, a ratio of polynomials is analytic except when the denominator vanishes. The function f (z) = | z2 | is continuous for all z but satisfies the Cauchy – Riemann conditions only at z = 0. Hence, f
(z) exists only at the origin, and | z |2 is nowhere analytic. The function f (z) = 1/z is analytic except at z = 0. Its derivative is − 1/z2 , where z = 0. If ln z = ln | z | + i arg z in the cut domain − π < arg z ≤ π, then f (z) = 1/ln z is analytic in the same cut domain, except at z = 1, where log z = 0. Because ez is analytic and ± i z are analytic, e±iz is analytic and linear combinations are analytic. Thus, the sine and cosine and hyperbolic sine and cosine are analytic. The other functions are analytic except when the denominator vanishes. The derivatives of the elementary functions are d z e = ez , dz

d n z = n z n−1 dz

d 1 (ln z) = , dz z

d sin z = cos z , dz

d cos z = − sin z dz

34

Mathematics in Chemical Engineering

In addition, d dg df ( f g) = f +g dz dz dz d f dg d f [g (z)] = dz dg dz d dw sin w = cos w , dz dz

d dw cos w = − sin w dz dz

Define z a = ea ln z for complex constant a. If the determination is −π < arg z ≤ π, then z a is analytic on the complex plane with a cut on the negative real axis. If a is an integer n, then e2πin = 1 and zn has the same limits approaching the cut from either side. The function can be made continuous across the cut and the function is analytic there, too. If a = 1/n where n is an integer, then

and only if, it is the real part of an analytic function. The imaginary part is also harmonic. Given any harmonic function u, a conjugate harmonic function v can be constructed such that f = u + i v is locally analytic [31, p. 85]. Maximum Principle. If f (z) is analytic in a domain D and continuous in the set consisting of D and its boundary C, and if | f (z) | ≤ M on C, then | f (z) | < M in D unless f (z) is a constant [32].

3.4. Integration in the Complex Plane Let C be a rectifiable curve in the complex plane C:z = z (t) , 0 ≤t ≤ 1

z 1/n = e(lnz)/n =|z|l/n ei(argz)/n

So w = z1/n has n values, depending on the choice of argument. Laplace Equation. If f (z) is analytic, where

where z (t) is a continuous function of bounded variation; C is oriented such that z1 = z (t 1 ) precedes the point z2 = z (t 2 ) on C if and only if t 1 < t 2 . Define



1

f (z) =u (x, y) + i v (x, y)

the Cauchy – Riemann equations are satisfied. Differentiating the Cauchy – Riemann equations gives the Laplace equation: ∂2u ∂2v ∂2u ∂2v = = − = or 2 ∂x ∂x ∂y ∂y ∂x ∂y 2

f (z) dz =

f [z (t)] dz (t) 0

C

The integral is linear with respect to the integrand:
[α1 f1 (z) + α2 f2 (z)] dz C

∂2u ∂2u + = 0 ∂x2 ∂y 2


= α1 C

Similarly, ∂2v ∂2v + = 0 2 ∂x ∂y 2

Thus, general solutions to the Laplace equation can be obtained from analytic functions [31, p. 83], [32, p. 223]. For example, ln

1 |z −z0 |

is analytic so that a solution to the Laplace equation is /−1/2 . ln (x − a)2 + (y − b)2

A solution to the Laplace equation is called a harmonic function. A function is harmonic if,


f1 (z) dz + α2

f2 (z) dz C

The integral is additive with respect to the path. Let curve C 2 begin where curve C 1 ends and C 1 + C 2 be the path of C 1 followed by C 2 . Then,



f (z) dz =

C1 +C2


f (z) dz +

C1

f (z) dz

C2

Reversing the orientation of the path replaces the integral by its negative:



f (z) dz = −

−C

f (z) dz C

If the path of integration consists of a finite number of arcs along which z (t) has a continuous derivative, then

Mathematics in Chemical Engineering



1 f (z) dz =

Power Series. If f (z) is analytic interior to a circle | z − z0 | < r 0 , then at each point inside the circle the series

f [z (t)] z  (t) dt

0

C

Also if s (t) is the arc length on C and l (C ) is the length of C  
  f (z) dz  ≤ max | f (z) | l (C) 

f (z) = f (z0 ) +

C

and
1
 
  f (z) dz  ≤ | f (z) | |dz| = | f [z (t)] |ds (t) 

ez = 1 +

0

C

Cauchy’s Theorem [31]. Suppose f (z) is an analytic function in a domain D and C is a simple, closed, rectifiable curve in D such that f (z) is analytic inside and on C. Then 0 (11)

C

∞  zn n! n=1

represents the function for all z. Another result of Cauchy’s integral formula is that if f (z) is analytic in an annulus R, r 1 < | z − z0 | < r 2 , it is represented in R by the Laurent series f (z) =

f (z) dz = 0

∞  f (n) (z0 ) (z − z0 )n n! n=1

converges to f (z). This result follows from Cauchy’s integral. As an example, ez is an entire function (analytic everywhere) so that the MacLaurin series

z∈C

C

35

∞ 

An (z − z0 )n , r1 < |z − z0 | ≤ r2

n=−∞

where

If D is simply connected, then Equation (11) holds for every simple, closed, rectifiable curve C in D. If D is simply connected and if a and b are any two points in D, then
b a

is independent of the rectifiable path joining a and b in D. Cauchy’s Integral. If C is a closed contour such that f (z) is analytic inside and on C, z0 is a point inside C, and z traverses C in the counterclockwise direction, 1 2 πi

0 C

1 f (z0 ) = 2 πi 

1 2 πi




f (z) (z − z0 )n+1

C

dz ,

n = 0, ± 1, ± 2, . . . ,

and C is a closed curve counterclockwise in R.

f (z) dz

f (z0 ) =

An =

f (z) dz z − z0

f (z) = . . . +

A−2 (z − z0 )2

+

A−1 + A0 z − z0

+A1 (z − z0 ) + . . . 0 < |z − z0 | ≤ r0

(12)

In particular,

0 C

Singular Points and Residues [33, p. 159]. If a function in analytic in every neighborhood of z0 , but not at z0 itself, then z0 is called an isolated singular point of the function. About an isolated singular point, the function can be represented by a Laurent series.

f (z) (z − z0 )2

dz

A−1 =

1 2 πi

0 f (z) dz C

Under further restriction on the domain [31, p. 178], f (m) (z0 ) =

m! 2 πi

0 C

f (z) (z − z0 )m+1

dz

where the curve C is a closed, counterclockwise curve containing z0 and is within the neighborhood where f (z) is analytic. The complex number A−1 is the residue of f (z) at the isolated singular point z0 ; 2 π i A−1 is the value of the

36

Mathematics in Chemical Engineering

integral in the positive direction around a path containing no other singular points. If f (z) is defined and analytic in the exterior | z − z0 | > R of a circle, and if  v (ζ) = f

z0 +

1 ζ

 obtained by ζ =

1 z − z0

has a removable singularity at ζ = 0, f (z) is analytic at infinity. It can then be represented by a Laurent series with nonpositive powers of z − z0 . If C is a closed curve within which and on which f (z) is analytic except for a finite number of singular points z1 , z2 , . . . , zn interior to the region bounded by C, then the residue theorem states 0 f (z) dz = 2 πi (1 + 2 + . . . n ) C

where n denotes the residue of f (z) at zn . The series of negative powers in Equation (12) is called the principal part of f (z). If the principal part has an infinite number of nonvanishing terms, the point z0 is an essential singularity. If A−m = 0, A−n = 0 for all m < n, then z0 is called a pole of order m. It is a simple pole if m = 1. In such a case, f (z) =

∞  A−1 + An (z − z0 )n z − z0 n=0

If a function is not analytic at z0 but can be made so by assigning a suitable value, then z0 is a removable singular point. When f (z) has a pole of order m at z0 , ϕ (z) = (z − z0 )m f (z) , 0 0). The ray θ = π is a branch cut. Analytic Continuation [33, p. 165]. If f 1 (z) is analytic in a domain D1 and domain D contains D1 , then an analytic function f (z) may exist that equals f 1 (z) in D1 . This function is the analytic continuation of f 1 (z) onto D, and it is unique. For example, f1 (z) =

∞ 

z n , |z| < 1

n=0

is analytic in the domain D1 : | z | < 1. The series diverges for other z. Yet the function is the MacLaurin series in the domain f1 (z) =

1 , 1−z

|z| < 1

Thus, f1 (z) =

1 1−z

is the analytic continuation onto the entire z plane except for z = 1. An extension of the Cauchy integral formula is useful with Laplace transforms. Let the curve C be a straight line parallel to the imaginary axis and z0 be any point to the right of that

Mathematics in Chemical Engineering

37

(see Fig. 13). A function f (z) is of order zk as | z | → ∞ if positive numbers M and r 0 exist such that

Conformal Mapping. Let u (x, y) be a harmonic function. Introduce the coordinate transformation

|z −k f (z) | < M when |z| > r0 , i.e.,

x = x ˆ (ξ, η) , y = yˆ (ξ, η)

|f (z) | < M |z|k for

It is desired that

|z| sufficiently large

U (ξ, η) =u [ˆ x (ξ, η) , yˆ (ξ, η)]

be a harmonic function of ξ and η. Theorem [32, p. 237]. The transformation z = f (ζ)

(13)

takes all harmonic functions of x and y into harmonic functions of ξ and η if and only if either f (ζ) or f ∗ (ζ) is an analytic function of ζ = ξ + i η. Equation (13) is a restriction on the transformation which ensures that if

Figure 13. Integration in the complex plane

Theorem [33, p. 167]. Let f (z) be analytic when R (z) ≥ γ and O (z−k ) as | z | → ∞ in that half-plane, where γ and k are real constants and k > 0. Then for any z0 such that R (z0 ) > γ 1 f (z0 ) = − lim 2 πi β→∞

γ+iβ


f (z) dz , z − z0

∂2u ∂ 2u ∂2 U ∂2U + 2 = 0 then + 2 = 0 2 ∂x ∂y ∂ζ 2 ∂η

Such a mapping with f (ζ) analytic and f
(ζ) = 0 is a conformal mapping. If Laplace’s equation is to be solved in the region exterior to a closed curve, then the point at infinity is in the domain D. For flow in a long channel (governed by the Laplace equation) the inlet and outlet are at infinity. In both cases the transformation ζ =

az + b z − z0

i.e., integration takes place along the line x = γ.

takes z0 into infinity and hence maps D into a bounded domain D∗ .

3.5. Other Results

4. Integral Transforms

Theorem [31, p. 84]. Let P (z) be a polynomial of degree n having the zeroes z1 , z2 , . . . , zn and let π be the least convex polygon containing the zeroes. Then P
(z) cannot vanish anywhere in the exterior of π. If a polynomial has real coefficients, the roots are either real or form pairs of complex conjugate numbers. The radius of convergence R of the Taylor series of f (z) about z0 is equal to the distance from z0 to the nearest singularity of f (z).

4.1. Fourier Transforms

γ−iβ

Fourier Series [34]. Let f (x) be a function that is periodic on − π < x < π. It can be expanded in a Fourier series f (x) =

where

∞  a0 (an cos n x + bn sin n x) + 2 n=1

38

a0 =

Mathematics in Chemical Engineering 1 π

1 bn = π


π f (x) dx , an = −π

1 π


π f (x) cos n x dx , −π

T (x, 0) = f (x)


π f (x) sin n x dx

T (−π, t) = T (π, t)

−π

The values {an } and {bn } are called the finite cosine and sine transform of f , respectively. Because cos n x =

 1  inx + e−inx e 2

 1  inx − e−inx and sin n x = e 2i

the Fourier series can be written as ∞ 

f (x) =

∂T ∂2T = ∂t ∂x2

Let T =

∞ 

Then, ∞ ∞ 

dcn −inx  = cn (t) −n2 e−inx e dt −∞ −∞

Thus, cn (t) satisfies 2 dcn = − n2 cn , or cn = cn (0) e−n t dt

cn e−inx

Let cn (0) be the Fourier coefficients of the initial conditions:

n=−∞

where  cn =

1 2 1 2

for n ≥ 0 for n < 0

(an + i bn ) (a−n − i b−n )

f (x) =

1 2π

−π

Fourier Transform [34]. When the function f (x) is defined on the entire real line, the Fourier transform is defined as
∞

If f is continuous and piecewise continuously differentiable ∞ 

F [ f ] ≡ fˆ (ω) =

f (x) eiωx dx

−∞

This integral converges if (−i n) cn e−inx

−∞


∞ | f (x) |dx

If f is twice continuously differentiable f (x) =

2

cn (0) e−n t e−inx

−∞

c−n = c∗n .



∞ 

f (x) einx dx

If f is real

f  (x) =

cn (0) e−inx

−∞

T =


π

∞ 

The formal solution to the problem is

and cn =

cn (t) e−inx

−∞

∞ 

−n

2

−∞

does. The inverse transformation is cn e

−inx

−∞

Inversion. The Fourier series can be used to solve linear partial differential equations with constant coefficients. For example, in the problem

f (x) =

1 2π


∞

fˆ (ω) e−iωx dω

−∞

If f (x) is continuous and piecewise continuously differentiable,
∞ f (x) eiωx dx −∞

Mathematics in Chemical Engineering converges for each ω, and lim

x→±∞


∞



n→∞ −∞

f (x) = 0

The sequence also satisfies the Cauchy criterion

df dx




∞

= − i ωF [ f ]

| fn − fm |2 dx = 0

lim n→∞ m→∞

If f is real F [ f (− ω)] = F [ f (ω)∗ ]. The real part is an even function of ω and the imaginary part is an odd function of ω. A function f (x) is absolutely integrable if the improper integral
∞ | f (x) |dx −∞

has a finite value. Then the improper integral
∞ f (x) dx −∞

converges. The function is square integrable if
∞

−∞

Theorem [34, p. 307]. If a sequence of square integrable functions fn (x) converges to a function f (x) uniformly on every finite interval a ≤ x ≤ b, and if it satisfies Cauchy’s criterion, then f (x) is square integrable and fn (x) converges to f (x) in the mean. Theorem (Riesz – Fischer) [34, p. 308]. To every sequence of square integrable functions fn (x) that satisfy Cauchy’s criterion, there corresponds a square integrable function f (x) such that fn (x) converges to f (x) in the mean. Thus, the limit in the mean of a sequence of functions is defined to within a null function. Square integrable functions satisfy the Parseval equation.
∞

| f (x) |2 dx


∞ | fˆ (ω) |2 dω = 2 π

−∞

−∞

has a finite value. If f (x) and g (x) are square integrable, the product f (x) g (x) is absolutely integrable and satisfies the Schwarz inequality: 
∞ 2   f (x) g (x) dx 

−∞

This is also the total power in a signal, which can be computed in either the time or the frequency domain. Also
∞ fˆ (ω) gˆ∗ (ω) dω = 2 π −∞


∞


∞ | f (x) |2 dx

≤ −∞

|g (x) |2 dx −∞

The triangle inequality is also satisfied:  ∞ 


1/2 

 ∞ 


−∞

Fourier transforms can be used to solve differential equations too. Then it is necessary to find the inverse transformation. If f (x) is square integrable, the Fourier transform of its Fourier transform is 2 π f (− x), or f (x) = F



f (x) g ∗ (x) dx −∞

.

| f + g|2 dx

−∞

| f (x) |2 dx


∞

−∞

≤

| f (x) − fn (x) |2 dx = 0

lim

then F

39


∞ / 1 fˆ (ω) = fˆ (ω) e−iωx dω 2π −∞

| f |2 dx

1/2  ∞ 
 



−∞

|g|2 dx

1/2  

A sequence of square integrable functions fn (x) converges in the mean to a square integrable function f (x) if

=

1 2π


∞
∞

f (x) eiωx dxe−iωx dω

−∞ −∞

f (x) =

1 1 F [F f (−x)] or f (−x) = F [F [ f ]] 2π 2π

40

Mathematics in Chemical Engineering

Properties of Fourier Transforms [34, p. 324], [35].  F

df dx

 = − i ωF [ f ] = i ω fˆ

F [i x f (x)] =

d ˆ d F [f ]= f dω dω

1 iωb/a ˆ  ω  e F [ f (a x − b)] = f |a| a . / F eicx f (x) = fˆ (ω + c) F [cos ω0 xf (x)] =

/ 1 .ˆ f (ω + ω0 ) + fˆ (ω − ω0 ) 2

/ 1 .ˆ f (ω + ω0 ) − fˆ (ω − ω0 ) F [sin ω0 xf (x)] = 2i . / F e−iω0 x f (x) = fˆ (ω − ω0 )

If f (x) is real, then f (− ω) = fˆ ∗ (ω). If f (x) is imaginary, then fˆ (− ω) = − fˆ ∗ (ω). If f (x) is even, then fˆ (ω) is even. If f (x) is odd, then fˆ (ω) is odd. If f (x) is real and even, then fˆ (ω) is real and even. If f (x) is real and odd, then fˆ (ω) is imaginary and odd. If f (x) is imaginary and even, then fˆ (ω) is imaginary and even. If f (x) is imaginary and odd, then fˆ (ω) is real and odd.

∂T ∂2T = ∂t ∂x2 − ∞ < x x0 dsn

s2 y (s) − s Y (0) − Y  (0) − 2 [s y (s) − Y (0)]

and f ∗ (s) = f (s∗ )

+ y (s) =

The functions | f (s) | and | x f (s) | are bounded in the half-plane x ≥ x 1 > x 0 and f (s) → 0 as | y | → ∞ for each fixed x. Thus,

1 s−2

and combining terms

s2 − 2 s + 1 y (s) =

y (s) =

1 (s − 2) (s − 1)2

lead to Y (t) = e2t − (1 + t) et

1 s−2

| f (x + i y) | < M ,

|x f (x + i y) | < M ,

x ≥ x 1 > x0 lim

y→±∞

f (x + i y) = 0, x > x0

If F (t) is continuous, F
(t) is piecewise continuous, and both functions are O [exp (x 0 t)], then | f (s) | is O (1/s) in each half-plane x ≥ x 1 > x 0 .

Next Page Mathematics in Chemical Engineering

|s f (s) | 0, is f (s) =

1 lim 2 πi β→∞

γ+iβ


γ−iβ

f (z) dz , Re (s) > γ s−z

Applying the inverse Laplace transformation on either side of this equation gives 1 F (t) = lim 2 πi β→∞

γ+iβ


ezt f (z) dz

ε→∞

1 [F (t0 + ε) + F (t0 − ε)] 2

When t = 0 the inversion integral represents 0.5 F (O +) and when t < 0, it has the value zero. If f (s) is a function of the complex variable s that is analytic and of order O (s−k−m ) on R (s) ≥ x 0 , where k > 1 and m is a positive integer, then the inversion integral converges to F (t) and dn F 1 = lim dtn 2 πi β→∞

∞ 

n (t)

n=1

When sn is a simple pole

If F (t) is of order O [exp (x 0 t)] and F (t) and F
(t) are piecewise continuous, the inversion integral exists. At any point t 0 , where F (t) is discontinuous, the inversion integral represents the mean value F (t0 ) = lim

Series of Residues [36]. Let f (s) be an analytic function except for a set of isolated singular points. An isolated singular point is one for which f (z) is analytic for 0 < | z − z0 | <  but z0 is a singularity of f (z). An isolated singular point is either a pole, a removable singularity, or an essential singularity. If f (z) is not defined in the neighborhood of z0 but can be made analytic at z0 simply by defining it at some additional points, then z0 is a removable singularity. The function f (z) has a pole of order k ≥ 1 at z0 if (z − z0 )k f (z) has a removable singularity at z0 whereas (z − z0 )k−1 f (z) has an unremovable isolated singularity at z0 . Any isolated singularity that is not a pole or a removable singularity is an essential singularity. Let the function f (z) be analytic except for the isolated singular point s1 , s2 , . . . , sn . Let n (t) be the residue of ezt f (z) at z = sn (for definition of residue, see Section 3.4). Then F (t) =

γ−iβ

47

z→sn

= esn t lim (z − sn ) f (z) z→ sn

When f (z) =

p (z) q (z)

where p (z) and q (z) are analytic at z = sn , p (sn ) = 0, then n (t) =

p (sn ) sn t e q  (sn )

If sn is a removable pole of f (s), of order m, then

γ+iβ


e

n (t) = lim (z − sn ) ezt f (z)

zt n

z f (z) dz ,

ϕn (z) = (z − sn )m f (z)

γ−iβ

n = 1, 2, . . . , m

Also F (t) and its n derivatives are continuous functions of t of order O [exp (x 0 t)] and they vanish at t = 0.

is analytic at sn and the residue is n (t) =

Φn (sn ) (m − 1)!

where Φn (z) = F (0) =F  (0) = . . . F (m) (0) = 0

 ∂ m−1  ϕn (z) ezt ∂z m−1

An important inversion integral is when

Previous Page 48

Mathematics in Chemical Engineering

f (s) =

Applied to the differential equation

  1 exp −s1/2 s

 F

The inverse transform is  F (t) = 1 − erf

2

1 √

 t

 = erfc

2

1 √

 t

∂2T ∂x2

 = − ω 2 αF [T ]

∂ Tˆ + ω 2 α Tˆ = 0, Tˆ (ω, 0) = fˆ (ω) ∂t

where erf is the error function and erfc the complementary error function.

By solving

4.3. Solution of Partial Differential Equations by Using Transforms

the inverse transformation gives [34, p. 328], [40, p. 58]

A common problem facing chemical engineers is to solve the heat conduction equation or diffusion equation ∂T ∂2T  Cp = k ∂t ∂x2

∂c ∂2c or = D ∂t ∂x2

The equations can be solved on an infinite domain − ∞ < x < ∞, a semi-infinite domain 0 ≤ x < ∞, or a finite domain 0 ≤ x ≤ L. At a boundary, the conditions can be a fixed temperature T (0, t) = T 0 (boundary condition of the first kind, or Dirichlet condition), or a fixed flux −k ∂T ∂x (0, t) = q0 (boundary condition of the second kind, or Neumann condition), or a combination −k ∂T ∂x (0, t) = h [T (0, t) − T0 ] (boundary condition of the third kind, or Robin condition). The functions T 0 and q0 can be functions of time. All properties are constant (, Cp , k, D, h), so that the problem remains linear. Solutions are presented on all domains with various boundary conditions for the heat conduction problem. ∂T k ∂2T , α= = α ∂t ∂x2  Cp

2 Tˆ (ω, t) = fˆ (ω) e−ω αt

T (x, t) =

1 lim 2 π L→∞


L

e−iωx fˆ (ω) e−ω

2

αt

dω

−L

Another solution is via Laplace transforms; take the Laplace transform of the original differential equation. s t (s, x) − f (x) = α

∂2t ∂x2

This equation can be solved with Fourier transforms [34, p. 355] t (s, x) =

1 √ 2 sα

)
∞ ( 1 s e − |x − y| f ( y) dy α

−∞

The inverse transformation is [34, p. 357], [40, p. 53] 1 T (x, t) = √ 2 π αt


∞

e−(x−y)

2

/4αt

f ( y) dy

−∞

Problem 2. Semi-infinite domain, boundary condition of the first kind, on 0 ≤ x ≤ ∞ T (x, 0) =T0 = constant

Problem 1. Infinite domain, on − ∞ < x < ∞.

T (0, t) =T1 = constant

T (x, 0) =f (x) , initial conditions

The solution is

T (x, t) bounded

.  √ / T (x, t) =T0 + [T1 −T0 ] 1 − erf x/ 4α t

Solution is via Fourier transforms
∞ Tˆ (ω, t) =

T (x, t) eiωx dx −∞

 √  or T (x, t) =T0 + (T1 −T0 ) erfc x/ 4α t

Mathematics in Chemical Engineering Problem 3. Semi-infinite domain, boundary condition of the first kind, on 0 ≤ x < ∞ T (x, 0) =f (x)

T2 (x, t) =

2α π


∞


t ω sin ωx

0

e−ω

2

α(t−τ )

49

g (τ ) dτ dω

0

T (0, t) =g (t)

The solution for T 1 can also be obtained by Laplace transforms.

The solution is written as

t1 =L [T1 ]

Applying this to the differential equation

T (x, t) =T1 (x, t) +T2 (x, t)

∂ 2 t1 , t1 (0, s) = 0 ∂x2

where

s t1 − f (x) = α

T1 (x, 0) =f (x) , T2 (x, 0) = 0

and solving gives

T1 (0, t) = 0, T2 (0, t) =g (t)

1 t1 = √ sα

Then T 1 is solved by taking the sine transform


x

e−

√

s/α(x −x)

f x dx

0

and the inverse transformation is [34, p. 437], [40, p. 59]

U1 =Fsω [T1 ]

T1 (x, t) = √

∂U1 = − ω 2 α U1 ∂t

−e−(x+ξ)

2


∞.

1 4 π αt

/4αt

e−(x−ξ)

2

/4αt

0

/ f (ξ) dξ

U1 (ω, 0) =Fsω [ f ]

Thus, U1 (ω, t) = Fsω [ f ] e−ω

2

Problem 4. Semi-infinite domain, boundary conditions of the second kind, on 0 ≤ x < ∞.

αt

T (x, 0) = 0

and [34, p. 322] 2 T1 (x, t) = π


∞

−k Fsω [ f ] e−ω

2

αt

sin ωx dω

0

Solve for T 2 by taking the sine transform U2 =Fsω [T2 ]

Take the Laplace transform t (x, s) =L [T (x, t)] st = α −k

∂U2 = − ω 2 α U2 + α g (t) ω ∂t U2 (ω, 0) = 0

U2 (ω, t) =

∂2t ∂x2

∂t q0 = ∂x s

The solution is t (x, s) =

Thus,
t

∂T (0, t) = q0 = constant ∂x

e−ω

0

and [34, p. 435]

2

√ q0 α −x√s/α e ks3/2

The inverse transformation is [36, p. 131], [40, p. 75] α(t−τ )

αω g (τ ) dτ q0 T (x, t) = k

 1 2

α t −x2 /4αt − x erfc e π

 √

x 4 αt



50

Mathematics in Chemical Engineering

Problem 5. Semi-infinite domain, boundary conditions of the third kind, on 0 ≤ x < ∞

s t (x, s) − T0 = α

T (x, 0) =f (x) k

The solution is

Take the Laplace transform

k

2 2 s s T0 sinh α x T0 sinh α (L − x) 2 2 t (x, s) = − − s s sinh s L s sinh α L α

∂2t ∂x2

∂t (0, s) = h t (0, s) ∂x

+

The solution is f (ξ) g (x, ξ, s) dξ

T (x, t) =

0

where [36, p. 227] 2

T0 s

The inverse transformation is [36, p. 220], [40, p. 96]


∞ t (x, s) =

  s/α g (x, ξ, s) = exp − |x − ξ| s/α

 2 −n2 π2 αt/L2 2 nπ x sin T0 e π n L n=1,3,5,...

or (depending on the inversion technique) [34, pp. 362, 438]

√ √ / . s − h α/k +√ exp − (x + ξ) s/α √ s + h α/k

T (x, t) = √

One form of the inverse transformation is [36, p. 228]

e−[(x−ξ)+2nL]

T (x, t) =

2 π


∞

e−β

∂2t ∂x2

t (0, s) =t (L, s) = 0

∂T (0, t) = h T (0, t) ∂x

s t − f (x) = α

Take the Laplace transform

2


L  ∞

T0 4 π αt 2

[

0 n=−∞

/4αt

− e−[(x+ξ)+2nL]

2

/4αt

/ dξ


∞ αt

cos [β x − µ (β)]

0

f (ξ) 0

cos [β ξ − µ (β)] dξdβ µ (β) = arg (β + i h /k)

Problem 7. Finite domain, boundary condition of the first kind T (x, 0) = 0 T (0, t) = 0

Another form of the inverse transformation when f = T 0 = constant is [36, p. 231], [40, p. 71]    2 2 x +ehx/k eh αt/k T (x, t) = T0 erf √ 4 αt erfc

 √ 

h αt x +√ k 4 αt

T (L, 0) = T0 = constant

Take the Laplace transform s t (x, s) = α

∂2t ∂x2

t (0, s) = 0, t (L , s) =

T0 s

The solution is Problem 6. Finite domain, boundary condition of the first kind

x T0 sinh L s/α t (x, s) = s sinh s/α

T (x, 0) =T0 = constant T (0, t) = T (L, t) = 0

and the inverse transformation is [36, p. 201], [40, p. 313]

Mathematics in Chemical Engineering T (x, t) =  T0

∞ nπ x x 2  (−1)n −n2 π 2 αt/L2 sin + e L π n=1 n L



An alternate transformation is [36, p. 139], [40, p. 310] T (x, t) = T0

∞  

 erf

n=0

 − erf

(2 n + 1) L + x √ 4 αt

(2 n + 1) L − x √ 4 αt





Problem 8. Finite domain, boundary condition of the second kind T (x, 0) =T0 ∂T (0, t) = 0, T (L , t) = 0 ∂x

Take the Laplace transform s t (x, s) − T0 = α

∂2t ∂x2

The solution is t (x, s) =

T0 s

m (n u) = (m n) u



cosh x s/α 1− cosh L s/α

(m + n) u = m u + n u m (u + v) = m u + m v

Its inverse is [36, p. 138] T (x, t) = T0 − T0

∞ 

and two directions associated with it, as defined precisely below. The most common examples are the stress dyadic (or tensor) and the velocity gradient (in fluid flow). Vectors are printed in boldface type and identified in this chapter by lower-case Latin letters. Second-order dyadics are printed in boldface type and are identified in this chapter by capital or Greek letters. Higher order dyadics are not discussed here. Dyadics can be formed from the components of tensors including their directions, and some of the identities for dyadics are more easily proved by using tensor analysis, which is not presented here (see also, → Transport Phenomena, Chap. 1.1.4.). Vectors are also first-order dyadics. Vectors. Two vectors u and v are equal if they have the same magnitude and direction. If they have the same magnitude but the opposite direction, then u = − v. The sum of two vectors is identified geometrically by placing the vector v at the end of the vector u, as shown in Figure 16. The product of a scalar m and a vector u is a vector m u in the same direction as u but with a magnitude that equals the magnitude of u times the scalar m. Vectors obey commutative and associative laws. u+v=v+u u + (v + w) = (u + v) + w mu=um

∂t (0, s) = 0, t (L , s) = 0 ∂x

51

Commutative law for addition Associative law for addition Commutative law for scalar multiplication Associative law for scalar multiplication Distributive law Distributive law

The same laws are obeyed by dyadics, as well.

(−1)n

n= 0



 erfc

(2n + 1) L − x √ 4αt



 + erfc

(2n + 1) L + x √ 4αt



Figure 16. Addition of vectors

5. Vector Analysis Notation. A scalar is a quantity having magnitude but no direction (e.g., mass, length, time, temperature, and concentration). A vector is a quantity having both magnitude and direction (e.g., displacement, velocity, acceleration, force). A second-order dyadic has magnitude

A unit vector is a vector with magnitude 1.0 and some direction. If a vector has some magnitude (i.e., not zero magnitude), a unit vector eu can be formed by eu =

u |u|

The original vector can be represented by the product of the magnitude and the unit vector.

52

Mathematics in Chemical Engineering Dyadics. The dyadic A is written in component form in cartesian coordinates as

u = |u|eu

In a cartesian coordinate system the three principle, orthogonal directions are customarily represented by unit vectors, such as {ex , ey , ez } or {i, j, k}. Here, the first notation is used (see Fig. 17). The coordinate system is right handed; that is, if a right-threaded screw rotated from the x to the y direction, it would advance in the z direction. A vector can be represented in terms of its components in these directions, as illustrated in Figure 18. The vector is then written as u =ux ex + uy ey + uz ez

The magnitude is u = |u| =

2

u2x +u2y + u2z

The position vector is r =xex + yey + zez

with magnitude r = |r| =

-

x2 +y 2 +z 2

A =Axx ex ex + Axy ex ey + Axz ex ez + Ayx ey ex + Ayy ey ey + Ayz ey ez + Azx ez ex + Azy ez ey + Azz ez ez

Quantities such as ex ey are called unit dyadics. They are second-order dyadics and have two directions associated with them, ex and ey ; the order of the pair is important. The components Axx , . . . , Azz are the components of the tensor Aij which here is a 3×3 matrix of numbers that are transformed in a certain way when variables undergo a linear transformation. The y x momentum flux can be defined as the flux of x momentum across an area with unit normal in the y direction (→ Fluid Mechanics, Chap. 2.2.; introduction to → Transport Phenomena, Chap. 2.2.). Since two directions are involved, a secondorder dyadic (or tensor) is needed to represent it, and because the y momentum across an area with unit normal in the x direction may not be the same thing, the order of the indices must be kept straight. The dyadic A is said to be symmetric if Aij = Aji

Here, the indices i and j can take the values x, y, or z; sometimes (x, 1), ( y, 2), (x, 3) are identified and the indices take the values 1, 2, or 3. The dyadic A is said to be antisymmetric if Aij = −Aji

The transpose of A is Figure 17. Cartesian coordinate system

AT ij = Aji

Any dyadic can be represented as the sum of a symmetric portion and an antisymmetric portion. Aij = B ij + C ij , B ij ≡ C ij ≡

1 (Aij + Aji ) , 2

1 (Aij − Aji ) 2

An ordered pair of vectors is a second-order dyadic. Figure 18. Vector components uv =

 i

e i e j ui v j

j

The transpose of this is

Mathematics in Chemical Engineering

(u v)T = v u

A:B =

 i

but u v = v u

The Kronecker delta is defined as ( δij =

1 if i = j 0 if i = j

)

δ =

i

Aij Bji

j

Because the dyadics may not be symmetric, the order of indices and which indices are summed are important. The order is made clearer when the dyadics are made from vectors. (u v) · (w x) =u (v· w) x =u x (v· w)

and the unit dyadic is defined as 

(u v) : (w x) = (u· x) (v· w)

The dot product of a dyadic and a vector is ei ej δij

j

A·u =







ei 

i

Operations. The dot or scalar product of two vectors is defined as u· v =|u| |v| cos θ, 0 ≤θ ≤π

where θ is the angle between u and v. The scalar product of two vectors is a scalar, not a vector. It is the magnitude of u multiplied by the projection of v on u, or vice versa. The scalar product of u with itself is just the square of the magnitude of u.

 Aij uj 

j

The cross or vector product is defined by c =u×v =a |u||v| sin θ, 0 ≤θ ≤π

where a is a unit vector in the direction of u×v. The direction of c is perpendicular to the plane of u and v such that u, v, and c form a righthanded system. If u = v, or u is parallel to v, then θ = 0 and u×v = 0. The following laws are valid for cross products. u×v =−v×u

u· u = |u2 | =u2

u×(v×w) = (u×v)×w

The following laws are valid for scalar products

u×(v + w) = u×v + u×w

u·v=v·u

Commutative law for scalar products Distributive law for scalar products

u · (v + w) = u · v + u · w ex · ex = ey · ey = ez · ez = 1 ex · ey = ex · ez = ey · ez = 0

If the two vectors u and v are written in component notation, the scalar product is

If u · v = 0 and u and v are not null vectors, then u and v are perpendicular to each other and θ = π/2. The single dot product of two dyadics is  i

j

 ei ej



Commutative law fails for vector product Associative law fails for vector product Distributive law for vector product

ex ×ex = ey ×ey = ez ×ez = 0 ex ×ey = ez , ey ×ez = ex , ez ×ex = ey

 ex  u×v= det ux vx

ey uy vy

 ez  uz  vz

= ex (uy vz − vy uz ) + ey (uz vz − ux vz ) + ez (ux vy − uy vx )

u· v =ux vx + uy vy + uz vz

A ·B =

53



This can also be written as u×v =

i

The double dot product of two dyadics is

εkij ui vj ek

j

where

Aik Bkj

k



εijk

  1 = −1  0

if i, j, k is an even permutation of 123 if i, j, k is an odd permutation of 123 if any two of i, j, k are equal

54

Mathematics in Chemical Engineering

Thus ε123 = 1, ε132 =- 1, ε312 = 1,ε112 = 0, for example. The magnitude of u×v is the same as the area of a parallelogram with sides u and v. If u×v = 0 and u and v are not null vectors, then u and v are parallel. Certain triple products are useful. (u· v) w =u (v· w)

u× (v× w) = (u·w) v − (u·v) w (u× v) × w = (u·w) v − (v·w) u

The cross product of a dyadic and a vector is defined as  i

 ei ej

j

 k

 εklj Aik ul

l

The magnitude of a dyadic is 1 |A| =A =

3 4  1 41 T A2ij (A :A ) = 5 2 2 i j

There are three invariants of a dyadic. They are called invariants because they take the same value in any coordinate system and are thus an intrinsic property of the dyadic. They are the trace of A, A2 , A3 [41]. I = trA =



 i

III = trA3 =

Aij Aji

j

 i

j

An important theorem of Hamilton and Cayley [43] is that a second-order dyadic satisfies its own characteristic equation. (14)

Thus A3 can be expressed in terms of δ, A, and A2 . Similarly, higher powers of A can be expressed in terms of δ, A, and A2 . Decomposition of a dyadic into a symmetric and an antisymmetric part was shown above. The antisymmetric part has zero trace. The symmetric part can be decomposed into a part with a trace (the isotropic part) and a part with zero trace (the deviatoric part). A = 1/3 δδ : A Isotropic

+ 1/2 [A + AT - 2/3 δδ : A] + 1/2 [A - AT ] Deviatoric Antisymmetric

Differentiation. The derivative of a vector is defined in the same way as the derivative of a scalar. Suppose the vector u depends on t. Then du u (t +∆t) − u (t) = lim dt ∆t→0 ∆t

If the vector is the position vector r (t), then the difference expression is a vector in the direction of ∆ r (see Fig. 19). The derivative

Aii

i

II = trA2 =

λ3 −I1 λ2 +I2 λ −I3 = 0

A3 −I1 A2 +I2 A −I3 δ = 0

u· (v× w) = v· (w× u) = w· (u× v)

A×u =

can be formed where λ is an eigenvalue. This expression is

Aij Ajk Aki

k

The invariants can also be expressed as

dr ∆r r (t +∆t) − r (t) = lim = lim ∆t→0 dt ∆ t→0 ∆t ∆t

is the velocity. The derivative operation obeys the following laws. d du dv (u + v) = + dt dt dt

I1 = I

1 2 I − II I2 = 2

1 3 I − 3 I · II + 2 III = detA I3 = 6

Invariants of two dyadics are available [42]. Because a second-order dyadic has nine components, the characteristic equation det (λ δ − A) = 0

d du dv (u· v) = ·v+u· dt dt dt du dv d (u×v) = ×v+u× dt dt dt d dϕ du (ϕu) = u+ ϕ dt dt dt

If the vector u depends on more than one variable, such as x, y and z, partial derivatives are defined in the usual way. For example,

Mathematics in Chemical Engineering

55

then

if u (x, y, z) , then

ds2 = dr · dr = dx2 + dy 2 + dz 2

u (x +∆x, y, z) − u (x, y, z) ∂u = lim ∆t→0 ∂x ∆x

Rules for differentiation of scalar and vector products are

The derivative dr/dt is tangent to the curve in the direction of motion dr

u =  dt   dr   dt 

∂ ∂u ∂v (u·v) = ·v+u· ∂x ∂x ∂x ∂ ∂u ∂v (u×v) = xv +ux ∂x ∂x ∂x

Also,

Differentials of vectors are

u=

dr ds

du= dux ex +duy ey +duz ez

Differential Operators (see also, → Transport Phenomena, Chap. 1.1.4.). The vector differential operator (del operator) ∇ is defined in cartesian coordinates by

d (u ·v) = du ·v+u ·dv d (u×v) = du×v+u×dv du =

∂u ∂u ∂u dx + dy + dz ∂x ∂y ∂z

∇ =ex

∂ ∂ ∂ + ey + ez ∂x ∂y ∂z

The gradient of a scalar function is defined ∇ϕ = ex

Figure 19. Vector differentiation

If a curve is given by r (t), the length of the curve is [44, p. 373]
b 1 L = a

dr dr · dt dt dt

The arc-length function can also be defined:
t

1

s (t) = a

∂ϕ ∂ϕ ∂ϕ + ey + ez ∂x ∂y ∂z

and is a vector. If ϕ is height or elevation, the gradient is a vector pointing in the uphill direction. The steeper the hill, the larger is the magnitude of the gradient. The divergence of a vector is defined by ∇ ·u =

∂ux ∂uy ∂uz + + ∂x ∂y ∂z

and is a scalar. For a volume element ∆V , the net outflow of a vector u over the surface of the element is
u ·ndS

dr dr ∗ · dt dt∗ dt∗

∆S

This is related to the divergence by [45, p. 411]

This gives 

ds dt

2

dr dr · = = dt dt



dx dt

Because dr = dxex +dyey + dzez

2

 +

dy dt

2

 +

dz dt

2

∇ ·u = lim

∆V →0

1 ∆V


u ·ndS ∆S

Thus, the divergence is the net outflow per unit volume. The curl of a vector is defined by

56

Mathematics in Chemical Engineering is called the Laplacian operator ∇×(∇ϕ) = 0. The curl of the gradient of ϕ is zero.

  ∂ ∂ ∂ × ∇×u = ex + ey + ez ∂x ∂y ∂z

∇ · (∇×u) = 0 (ex ux + ey uy + ez uz )  = ex  + ez

∂uz ∂uy − ∂y ∂z ∂uy ∂ux − ∂x ∂y



 +ey

∂ux ∂uz − ∂z ∂x



∇ · (∇v)T = ∇ (∇ ·v)



∇ · (τ ·v) = v · (∇ ·τ ) +τ : ∇v

and is a vector. It is related to the integral





u ·ds = C

us ds C

which is called the circulation of u around path C. This integral depends on the vector and the contour C, in general. If the circulation does not depend on the contour C, the vector is said to be irrotational; if it does, it is rotational. The relationship with the curl is [45, p. 419] n · (∇×u) = lim

∆S→0

The divergence of the curl of u is zero. Formulas useful in fluid mechanics are

1 ∆S




u ·ds C

Thus, the normal component of the curl equals the net circulation per unit area enclosed by the contour C. The gradient, divergence, and curl obey a distributive law but not a commutative or associative law. ∇ (ϕ + ψ) = ∇ϕ +∇ψ

1 v ·∇v= ∇ (v ·v) − v× (∇×v) 2

If a coordinate system is transformed by a rotation and translation, the coordinates in the new system (denoted by primes) are given by    x l11    y  = l21 z l31

Any function that has the same value in all coordinate systems is an invariant. The gradient of an invariant scalar field is invariant; the same is true for the divergence and curl of invariant vectors fields. The gradient of a vector field is required in fluid mechanics because the velocity gradient is used. It is defined as ∇v=

 i

∇ ·ϕ = ϕ∇

(∇v)T =

 i

∂vj and ∂xi

e i ej

j

∂vi ∂xj

The divergence of dyadics is defined

∇ ·u = u ·∇

Useful formulas are [46]

∇ ·τ =

 i

  ∂τji  and ei  ∂xj j

∇ · (ϕu) = ∇ϕ ·u + ϕ∇ ·u ∇ × (ϕu) = ∇ϕ×u + ϕ ∇ × u ∇ · (u×v) = v · (∇×u) − u · (∇×v) ∇ × (u×v) = v ·∇u − v (∇ ·u) − u · ∇v+u (∇ ·v) ∇ × (∇ × u) = ∇ (∇ ·u) − ∇2 u ∇ · (∇ϕ) = ∇2 ϕ =

ei e j

j

∇ · (u + v) = ∇ ·u + ∇ ·v ∇× (u + v) = ∇×u+∇×v

    l13 x a1     l23  y  + a2  z l33 a3

l12 l22 l32

∂2ϕ ∂2ϕ ∂2ϕ + + , where ∇2 2 2 ∂x ∂y ∂z 2

∇ · (ϕu v) =



 i

  ∂ ei  (ϕ uj vi ) ∂xj j 

where τ is any second-order dyadic. Useful relations involving dyadics are (ϕ δ : ∇v) = ϕ (∇ ·v) ∇ · (ϕ δ) = ∇ϕ

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Vector Integration [45, pp. 206 – 212]. If u is a vector, then its integral is also a vector.

∇ · (ϕτ ) = ∇ϕ · τ + ϕ∇ ·τ




n t:τ = t ·τ ·n = τ : n t




u (t) dt = ex

A surface can be represented in the form

The normal to the surface is given by

provided the gradient is not zero. Operations can be performed entirely within the surface. Define [47]

v II ≡ δ II ·v,

∂ ≡ n ·∇ ∂n

vn ≡ n ·v

Then a vector and del operator can be decomposed into v =v II +nvn , ∇ = ∇II + n

∂ ∂n

The velocity gradient can be decomposed into ∇v= ∇II v II + (∇II n) vn + n∇II vn + n

+n n

∂v II ∂n

∂vn ∂n

The surface gradient of the normal is the negative of the curvature dyadic of the surface. ∇II n = −B

The surface divergence is then ∇II ·v = δ II : ∇v= ∇II ·v II − 2 H vn

where H is the mean curvature. H =

uy (t) dt

uz (t) dt

If the vector u is the derivative of another vector, then

∇f |∇ f |

δ II ≡ δ − n n , ∇II ≡ δ II ·∇,




ux (t) dt +ey


+ ez

f (x, y, z) = c = constant

n=

57

1 δ II : B 2

u=

dv , dt







u (t) dt =

dv dt = v+constant dt

If r (t) is a position vector that defines a curve C, the line integral is defined by





u · dr = C

(ux dx + uy dy + uz dz) C

Theorems about this line integral can be written in various forms. Theorem [44, p. 460]. If the functions appearing in the line integral are continuous in a domain D, then the line integral is independent of the path C if and only if the line integral is zero on every simple closed path in D. Theorem [44, p. 461]. If u = ∇ϕ where ϕ is single-valued and has continuous derivatives in D, then the line integral is independent of the path C and the line integral is zero for any closed curve in D. Theorem [44, p. 460]. If f, g, and h are continuous functions of x, y, and z, and have continuous first derivatives in a simply connected domain D, then the line integral


( f dx + gdy + hdz) C

is independent of the path if and only if ∂h ∂g = , ∂y ∂z

∂f ∂h = , ∂z ∂x

∂g ∂f = ∂x ∂y

The surface curl can be a scalar

or if f, g, and h are regarded as the x, y, and z components of a vector v:

∇II × v= − εII : ∇v

∇ × v= 0

= − εII : ∇II v II = − n· (∇ × v) , εII = n·ε

or a vector ∇II × v ≡ ∇II × v II = n∇II × v=n n ·∇ × v

Consequently, the line integral is independent of the path (and the value is zero for a closed contour) if the three components in it are regarded as the three components of a vector and the vector is derivable from a potential (or zero curl). The conditions for a vector to be derivable from a potential are just those in the third theorem. In two dimensions this reduces to the more usual theorem.

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Mathematics in Chemical Engineering

Theorem [45, p. 207]. If M and N are continuous functions of x and y that have continuous first partial derivatives in a simply connected domain D, then the necessary and sufficient condition for the line integral
(M dx + N dy) C

to be zero around every closed curve C in D is ∂M ∂N = ∂y ∂x

If a vector is integrated over a surface with incremental area d S and normal to the surface n, then the surface integral can be written as





u · dS =

S

u ·ndS

volume, divided by the volume. If the vector represents the flow of energy and the divergence is positive at a point P, then either a source of energy is present at P or energy is leaving the region around P so that its temperature is decreasing. If the vector represents the flow of mass and the divergence is positive at a point P, then either a source of mass exists at P or the density is decreasing at the point P. For an incompressible fluid the divergence is zero and the rate at which fluid is introduced into a volume must equal the rate at which it is removed. Various theorems follow from the divergence theorem. Theorem. If ϕ is a solution to Laplace’s equation ∇2 ϕ = 0

S

If u is the velocity then this integral represents the flow rate past the surface S. Divergence Theorem [45], [46]. If V is a volume bounded by a closed surface S and u is a vector function of position with continuous derivatives, then



∇ ·udV =

V


n ·udS =

S


u ·ndS =

S

u · dS S

where n is the normal pointing outward to S. The normal can be written as

in a domain D, and the second partial derivatives of ϕ are continuous in D, then the integral of the normal derivative of ϕ over any piecewise smooth closed orientable surface S in D is zero. Suppose u = ϕ ∇ψ satisfies the conditions of the divergence theorem: then Green’s theorem results from use of the divergence theorem [44, p. 451].
V




∂ux ∂uy ∂uz + + ∂x ∂y ∂z

 dxdydz =

V


[ux cos (x, n) + uy cos ( y, n) + uz cos (z, n)] dS S

∇ ·u = lim

∆V →0

1 ∆V

ϕ

∂ψ dS ∂n

ϕ

∂ψ ∂ϕ −ψ ∂n ∂n

S






ϕ∇2 ψ − ψ∇2 ϕ dV =

V




 dS

S

Also if ϕ satisfies the conditions of the theorem and is zero on S then ϕ is zero throughout D. If two functions ϕ and ψ both satisfy the Laplace equation in domain D, and both take the same values on the bounding curve C, then ϕ = ψ; i.e., the solution to the Laplace equation is unique. The divergence theorem for dyadics is



∇ · τ dV =

If the divergence theorem is written for an incremental volume




and

n =ex cos (x, n) +ey cos (y, n) +ez cos (z, n)

where, for example, cos (x, n) is the cosine of the angle between the normal n and the x axis. Then the divergence theorem in component form is



ϕ∇2 ψ +∇ϕ · ∇ψ dV =

V

n · τ dS S


un dS ∆S

the divergence of a vector can be called the integral of that quantity over the area of a closed

Stokes Theorem [45], [46, p. 106]. Stokes theorem says that if S is a surface bounded by a closed, nonintersecting curve C, and if u has continuous derivatives then

Mathematics in Chemical Engineering 0



u ·dr =

C



(∇ × u) ·n dS =

S

(∇ × u) ·dS S

The integral around the curve is followed in the counterclockwise direction. In component notation, this is 0 [ux cos (x, s) + uy cos ( y, s) + uz cos (z, s)] ds = C





∂uz ∂uy − ∂y ∂z



 cos (x, n) +

∂ux ∂uz − ∂z ∂x



S

 cos ( y, n) +

∂uy ∂ux − ∂x ∂y



 cos (z, n) dS

Applied in two dimensions, this results in Green’s theorem in the plane:



0 (M dx + N dy) = C

∂N ∂M − ∂x ∂y

S

The unit vectors are related by er = cosθex + sinθey eθ = − sinθex + cosθey ez = ez

0 n · (∇ × τ ) dS =

 r = x2 + y 2 y θ = arctan x z=z

x = rcosθ y = rsinθ z=z

dxdy

The formula for dyadics is



Curvilinear Coordinates. Many of the relations given above are proved most easily by using tensor analysis rather than dyadics. Once proven, however, the relations are perfectly general in any coordinate system. Displayed here are the specific results for cylindrical and spherical geometries. Results are available for a few other geometries: parabolic cylindrical, paraboloidal, elliptic cylindrical, prolate spheroidal, oblate spheroidal, ellipsoidal, and bipolar coordinates [41], [48]. For cylindrical coordinates, the geometry is shown in Figure 20. The coordinates are related to cartesian coordinates by



S

59

ex = cosθer − sinθeθ ey = sinθer + cosθeθ ez = ez

τ T ·dr

Derivatives of the unit vectors are

C

deθ = − er dθ , der =eθ dθ, dez = 0

Representation. Two theorems give information about how to represent vectors that obey certain properties. Theorem [45, p. 422]. The necessary and sufficient condition that the curl of a vector vanish identically is that the vector be the gradient of some function. Theorem [45, p. 423]. The necessary and sufficient condition that the divergence of a vector vanish identically is that the vector is the curl of some other vector. Leibniz Formula. In fluid mechanics and transport phenomena, an important result is the derivative of an integral whose limits of integration are moving. Suppose the region V (t) is moving with velocity v s . Then Leibniz’s rule holds: d dt















ϕdV = V (t)



ϕv s ·ndS

+ S

V (t)

∂ϕ dV ∂t

Differential operators are given by [41] ∇ =er

∂ ∂ eθ ∂ + + ez , ∂r r ∂θ ∂z

∂ϕ ∂ϕ eθ ∂ϕ + + ez ∂r r ∂θ ∂z   2ϕ ∂ ∂ϕ 1 1 ∂2ϕ ∂ r + 2 + ∇2 ϕ = r ∂r ∂r r ∂θ2 ∂z 2

∇ϕ = er

=

∂ 2 ϕ 1 ∂ϕ ∂2ϕ 1 ∂2ϕ + + + 2 ∂r2 r ∂r r ∂θ2 ∂z 2

∇ ·v=

1 ∂ 1 ∂vθ ∂vz (r vr ) + + r ∂r r ∂θ ∂z 

∇ × v = er  +ez

1 ∂vz ∂vθ − r ∂θ ∂z

1 ∂ 1 ∂vr (r vθ ) − r ∂r r ∂θ







+eθ

∂vr ∂vz − ∂z ∂r



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Mathematics in Chemical Engineering 

∇ · τ = er  +eθ  +ez

1 ∂ 1 ∂τθz ∂τzz (r τrz ) + + r ∂r r ∂θ ∂z

 +eθ er

+ e θ ez

+

1 ∂vr vθ − r ∂θ r



 +eθ eθ



1 ∂vθ vr + r ∂θ r

∂ ∂r



 +

+eθ

∂ ∂r



1 ∂ r ∂r

∂ 1 ∂ 1 ∂ + eθ + eφ , ∂r r ∂θ r sin θ ∂ϕ

∇ψ = er

∂ψ 1 ∂ψ 1 ∂ψ + eθ + eφ ∂r r ∂θ r sin θ ∂ϕ

 ∇2 ψ =

+

1 ∂ (r vθ ) r ∂r

 ∂ 2 vθ 2 ∂vr 1 ∂ 2 vθ + + 2 + 2 2 2 r ∂θ ∂z r ∂θ

+ez

∇ = er

1 ∂ (r vr ) r ∂r





∂er ∂eθ =eφ sin θ, = eφ cos θ, ∂ϕ ∂ϕ

Others 0 Differential operators are given by [41]

 1 ∂ 2 vr ∂ 2 vr 2 ∂vθ + − + r2 ∂θ2 ∂z 2 r 2 ∂θ 

+

∂eφ = − er sin θ − eθ cos θ ∂ϕ

1 ∂vz ∂vr ∂vθ ∂vz + e z er + ez eθ + ez ez r ∂θ ∂z ∂z ∂z 

∂er ∂eθ = eθ , = − er ∂θ ∂θ



∂vr ∂vθ ∂vz + e r eθ + e r ez + ∂r ∂r ∂r

∇2 v=er

+

Derivatives of the unit vectors are



1 ∂τθθ 1 ∂ 2 ∂τzθ τθr − τrθ r τrθ + + + r2 ∂r r ∂θ ∂z r

∇v=er er

+

1 ∂ 1 ∂τθr ∂τzr τθθ (r τrr ) + + − r ∂r r ∂θ ∂z r

   ∂ 2 vz ∂vz 1 ∂ 2 vz + r + 2 ∂r r ∂θ2 ∂z 2

For spherical coordinates, the geometry is shown in Figure 21. The coordinates are related to cartesian coordinates by x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ

 x2 + y2 + z2   θ = arctan x2 + y 2 /z y ϕ = arctan x

1 1 ∂ 2 ∂ r vr + (vθ sin θ) + r 2 ∂r r sin θ ∂θ

1 ∂vφ r sin θ ∂ϕ 

∇ × v=er 

r =

The unit vectors are related by

+eθ  +eφ

eφ = − sin ϕex + cos ϕey

1 ∂ 1 ∂vr (r vθ ) − r ∂r r ∂θ

ex = sin θ cos ϕer + cos θ cos ϕeθ − sin ϕeφ ey = sin θ sin ϕer + cos θ sin ϕeθ + cos ϕeφ ez = cos θer − sin θeθ

 ∇ · τ = er

+



1 1 ∂vθ ∂ vφ sin θ − + r sin θ ∂θ r sin θ ∂ϕ



1 ∂vr 1 ∂ − r vφ + r sin θ ∂ϕ r ∂r

er = sin θcos ϕex + sin θ sin ϕey +cos θez eθ = cos θcos ϕex +cos θ sin ϕey − sin θez

    ∂ψ 1 ∂ ∂ψ r2 + 2 sin θ + ∂r r sin θ ∂θ ∂θ

∂2ψ 1 r 2 sin2 θ ∂ϕ2

∇ ·v=

+

1 ∂ r2 ∂r

1 ∂ 2 1 ∂ r τrr + (τθr sin θ) r 2 ∂r r sin θ ∂θ

τθθ + τφφ 1 ∂τφr − r sin θ ∂ϕ r 

+eθ



 +

1 ∂ 3 1 ∂ r τrθ + (τθθ sin θ) r3 ∂r r sin θ ∂θ

Mathematics in Chemical Engineering

+

 τθr − τrθ − τφφ cotθ 1 ∂τφθ + + r sin θ ∂ϕ r 

+eφ

+

61

1 ∂ 3 1 ∂ r τrφ + τθφ sin θ r3 ∂r r sin θ ∂θ

τφr − τrφ + τφθ cotθ 1 ∂τφφ + r sin θ ∂ϕ r

∇v=er er 

∂vφ ∂vr ∂vθ + e r eθ + e r eφ + ∂r ∂r ∂r

+eθ er

1 ∂vr vθ − r ∂θ r

+eθ eφ

1 ∂vφ + eφ er r ∂θ 



 +eθ eθ 

 +eφ eφ

∂ ∂r



 +

 +

 +

1 ∂vφ vr vθ + + cotθ r sin θ ∂ϕ r r

 ∇2 v=er

1 ∂vθ vr + r ∂θ r

vφ 1 ∂vr − r sin θ ∂ϕ r

vφ 1 ∂vθ − cotθ r sin θ ∂ϕ r

+eφ eθ





Figure 20. Cylindrical coordinate system

 1 1 ∂ 2 ∂ + 2 v r r r 2 ∂r r sin θ ∂θ

  ∂ 2 vr ∂vr 1 sin θ + 2 2 ∂θ r sin θ ∂ϕ2 −

2 r2 sin θ 

+eθ 

+

 r2

∂vθ ∂r

 +

2 ∂vr 2 cotθ ∂vφ − 2 r2 ∂θ r sin θ ∂ϕ 

+

1 ∂ r2 ∂r



1 ∂ r2 ∂θ

 ∂ 2 vθ 1 ∂ 1 (vθ sin θ) + 2 2 sin θ ∂θ r sin θ ∂ϕ2

+eφ 

∂vφ 2 ∂ (vθ sin θ) − 2 ∂θ r sin θ ∂ϕ

1 ∂ r2 ∂r



Figure 21. Spherical coordinate system

  ∂vφ 1 ∂ r2 + 2 ∂r r ∂θ



∂ 2 vφ 1 1 ∂ vφ sin θ + 2 2 sin θ ∂θ r sin θ ∂ϕ2 2 r2 sin θ

∂vr 2 cotθ ∂vθ + 2 ∂ϕ r sin θ ∂ϕ



6. Ordinary Differential Equations as Initial Value Problems A differential equation for a function that depends on only one variable (often time) is called an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are required to specify which of those

62

Mathematics in Chemical Engineering

are desired. If all conditions are at one point, the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, the ordinary differential equations become two-point boundary value problems, which are treated in Chapter 7. Initial value problems as ordinary differential equations arise in control of lumped-parameter models, transient models of stirred tank reactors, and generally in models where no spatial gradients occur in the unknowns.

1 1 = kt + c c0

For other ordinary differential equations an integrating factor is useful. Consider the problem governing a stirred tank with entering fluid having concentration cin and flow rate F, as shown in Figure 22. The flow rate out is also F and the volume of the tank is V . If the tank is completely mixed, the concentration in the tank is c and the concentration of the fluid leaving the tank is also c. The differential equation is then V

6.1. Solution by Quadrature When only one equation exists, even if it is nonlinear, solving it by quadrature may be possible. For dy = f ( y) dt y (0) = y0

dc = F (cin − c) , c (0) = c0 dt

Upon rearrangement, dc F F c = cin + V V dt

is obtained. An integrating factor is used to solve this equation. The integrating factor is a function that can be used to turn the left-hand side into an exact differential and can be found by using Fr´echet differentials [49]. In this case, 

the problem can be separated exp dy = dt f ( y)

y0

dy  = f ( y )




t dt = t

       d Ft Ft F exp c = exp cin dt V V V

This can be integrated once to give

0



Ft V



If the quadrature can be performed analytically then the exact solution has been found. For example, consider the kinetics problem with a second-order reaction.

exp

dc = − k c2 , c (0) = c0 dt

c (t) = exp

To find the function of the concentration versus time, the variables can be separated and integrated. dc = − kdt , c2 −

     dc F d Ft + c = exp c dt V dt V

Thus, the differential equation can be written as

and integrated:
y

Ft V

1 = − kt+D c

Application of the initial conditions gives the solution:

c = c (0) +

F V




t exp 0

F t V



cin t dt

or

+

F V


t exp 0

  Ft c0 − V   F (t − t ) − cin t dt V

If the integral on the right-hand side can be calculated, the solution can be obtained analytically. If not, the numerical methods described in the next sections can be used. Laplace transforms can also be attempted. However, an analytic solution is so useful that quadrature and an integrating factor should be tried before resorting to numerical solution.

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63

The initial conditions would have to be specified for variables y1 (0), . . . , yn (0), or equivalently y (0), . . . , y(n−1) (0). The set of equations is then written as dy =f ( y, t) dt

All the methods in this chapter are described for a single equation; the methods apply to the multiple equations as well. Taking the single equation in the form Figure 22. Stirred tank

dy = f ( y) dt

multiplying by dt, and integrating once yields

6.2. Explicit Methods

t
n+1

Consider the ordinary differential equation tn

dy = f ( y) dt

f



 y t dt

tn

This is

Multiple equations that are still initial value problems can be handled by using the same techniques discussed here. A higher order differential equation   y (n) +F y (n−1) , y (n−2) , . . . , y  , y = 0

with initial conditions   Gi y (n−1) (0) , y (n−2) (0) , . . . , y  (0) , y (0) = 0

t
n+1

y

n+1

n

= y + tn

dy  dt dt

The last substitution gives a basis for the various methods. Different interpolation schemes for y (t) provide different integration schemes; using low-order interpolation gives low-order integration schemes [50], [51]. Euler’s method is first order y n+1 = y n + ∆tf (y n )

i = 1, . . . , n

can be converted into a set of first-order equations. By using yi ≡ y (i−1) =

t
n+1

dy  dt = dt

d(i−1) y d (i−2) dyi−1 y = = dt dt dt(i−1)

the higher order equation can be written as a set of first-order equations:

Adams – Bashforth Methods. The order Adams – Bashforth method is y n+1 = y n +

y n+1 = y n +

dy2 = y3 dt

+ 37 f

... dyn = − F ( yn−1 , yn−2 , . . . , y2 , y1 ) dt

 ∆t  3 f ( y n ) − f y n−1 2

The fourth-order Adams – Bashforth method is

dy1 = y2 dt

dy3 = y4 dt

second-



y n−2

∆t  55 f ( y n ) − 59 f y n−1 24

− 9f



y n−3



Notice that the higher order explicit methods require knowing the solution (or the right-hand side) evaluated at times in the past. Because these were calculated to get to the current time, this presents no problem except for starting the evaluation. Then, Euler’s method may have to be used with a very small step size for several steps

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Mathematics in Chemical Engineering

Table 6. Properties of integration methods for ordinary differential equations Error term

Order

Function evaluations per step

Stability limit, λ ∆t≤

Euler

h2 2

y 

1

1

2.0

Second-order Adams – Bashforth

3  5 12 h y

2

1

4

1

0.3

2 4 5

2 4 6

2.0 2.8 3.0

2 2

2 2

2.0 1.3

1 2 4

many, iterative many, iterative many, iterative

∞∗ 2∗ 3∗

Method Explicit methods

5 (5)

251 720 h

Fourth-order Adams – Bashforth Second-order Runge – Kutta (midpoint) Runge – Kutta – Gill Runge – Kutta – Feldberg

y

n +1

y

−z

n +1

Predictor – corrector methods Second-order Runge – Kutta Adams, fourth-order Implicit methods, stability limit ∞ Backward Euler Trapezoid rule Fourth-order Adams – Moulton ∗

1 − 12 h3 y 

Oscillation limit, λ ∆t ≤.

to generate starting values at a succession of time points. The error terms, order of the method, function evaluations per step, and stability limitations are listed in Table 6. The advantage of the fourth-order Adams – Bashforth method is that it uses only one function evaluation per step and yet achieves high-order accuracy. The disadvantage is the necessity of using another method to start. Runge – Kutta Methods. Runge – Kutta methods are explicit methods that use several function evaluations for each time step. The general form of the methods is y n+1 = y n +

v 

The midpoint scheme is another second-order Runge – Kutta method:  y n+1 = y n +∆t f

with i−1 

 aij kj 

∆t n ∆t n ,y + f 2 2



k1 = ∆t f (tn , y n )  k2 = ∆t f



tn +

∆t n k1 ,y + 2 2

tn +

∆t n , y + a k 1 + b k2 2



k4 = ∆t f (tn + ∆t, y n + c k2 + d k3 ) y n+1 = y n +

j=1

Runge – Kutta methods traditionally have been writen for f (t, y) and that is done here, too. If these equations are expanded and compared with a Taylor series, restrictions can be placed on the parameters of the method to make it first order, second order, etc. Even so, additional parameters can be chosen. A second-order Runge – Kutta method is

tn +

A popular fourth-order method is the Runge – Kutta – Gill method with the formulas

k3 = ∆t f

i=1

ki = ∆t f tn + ci ∆t , y n +

∆t n [ f + f (tn +∆t , y n +∆t f n )] 2



w i ki



y n+1 = y n +

√ a =

2−1 , 2

1 1 (k1 + k4 ) + (b k2 + d k3 ) 6 3 b =

√ 2− 2 , 2

√

√ 2 2 c = − , d=1 + 2 2

Another fourth-order Runge – Kutta method is given by the Runge – Kutta – Feldberg formulas [52]; although the method is fourth-order, it

Mathematics in Chemical Engineering achieves fifth-order accuracy. The popular integration package RKF 45 is based on this method. k1 = ∆t f (tn , y n )  tn +

k2 = ∆t f 

tn +

k3 = ∆t f 

tn +

k4 = ∆t f

∆t n k1 ,y + 4 4

3 3 9 ∆t , y n + k1 + k2 8 32 32

439 k1 216

tn +∆t , y n +

− 8 k2 +

3680 845 k3 − k4 513 4104

−



z n+1 = y n +

+

and determine the formula for rmn :

rmn (z) =

pn (z) ≈ e−z qm (z)

and is an approximation to exp (− z), called a Pad´e approximation. The stability limits are the largest positive z for which

∆t n 8 ,y − k1 + 2 k2 2 27

3544 1859 11 k3 + k4 − k5 2565 4104 40

y n+1 = y n +

dy = − λ y , y (0) = 1 dt

The rational polynomial is defined as

k5 = ∆t f

tn +

For other problems the most important criterion for choosing a method is probably the time the user spends setting up the problem. The stability of an integration method is best estimated by determining the rational polynomial corresponding to the method. Apply this method to the equation

y k+1 =rmn (λ ∆t) y k









12 1932 ∆t , y n + k1 13 2197

7200 7296 − k2 + k3 2197 2197

k6 = ∆t f



65



25 1408 2197 1 k1 + k3 + k4 − k 5 216 2565 4104 5

16 6656 k1 + k3 135 12 825

28 561 9 2 k4 − k5 + k6 56 430 50 55

The value of yn+1 − zn+1 is an estimate of the error in yn+1 and can be used in step-size control schemes. Generally, a high-order method should be used to achieve high accuracy. The Runge – Kutta – Gill method is popular because it is high order and does not require a starting method (as does the fourth-order Adams – Bashforth method). However, it requires four function evaluations per time step, or four times as many as the Adams – Bashforth method. For problems in which the function evaluations are a significant portion of the calculation time this might be important. Given the speed of computers and the widespread availability of microcomputers, the efficiency of a method is most important only for very large problems that are going to be solved many times.

|rmn (z) | ≤ 1

The method is A acceptable if the inequality holds for Re z > 0. It is A (0) acceptable if the inequality holds for z real, z > 0 [60]. The method will not induce oscillations about the true solution provided rmn (z) > 0

A method is L acceptable if it is A acceptable and lim rmn (z) = 0

z→∞

For example, Euler’s method gives y n+1 =y n −λ ∆t y n or y n+1 = (1 −λ ∆t) y n or rmn = 1 − λ∆t

The stability limit is then λ ∆t ≤ 2

The Euler method will not oscillate provided λ ∆t ≤ 1

The stability limits listed in Table 6 are obtained in this fashion. The limit for the Euler method is 2.0; for the Runge – Kutta – Gill method it is

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Mathematics in Chemical Engineering

2.785; for the Runge – Kutta – Feldberg method it is 3.020. The rational polynomials for the various explicit methods are illustrated in Figure 23. As can be seen, the methods approximate the exact solution well as λ ∆t approaches zero, and the higher order methods give a better approximation at high values of λ ∆t. LIVE GRAPH

Click here to view

the two-step calculation y2 . Then an improved solution at the new time is given by y =

2p y 2 − y 1 2p − 1

This gives a good estimate provided ∆t is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg’s method was used for quadrature (see Section 2.4). The accuracy of a numerical calculation depends on the step size used, and this is chosen automatically by efficient codes. For example, in the Euler method the local truncation error LTE is LTE =

∆t2  y 2 n

Yet the second derivative can be evaluated by using the difference formulas as



   = ∆t yn = yn = ∇ ∆t yn − yn−1 Figure 23. Rational approximations for explicit methods a) Euler; b) Runge – Kutta – 2; c) Runge – Kutta – Gill; d) Exact curve; e) Runge – Kutta – Feldberg

In solving sets of equations dy = A y + f , y (0) = y0 dt

all the eigenvalues of the matrix A must be examined. Finlayson [50] and Amundson [53, p. 197 – 199] both show how to transform these equations into an orthogonal form so that each equation becomes one equation in one unknown, for which single equation analysis applies. For linear problems the eigenvalues do not change, so the stability and oscillation limits must be satisfied for every eigenvalue of the matrix A. When solving nonlinear problems the equations are linearized about the solution at the local time, and the analysis applies for small changes in time, after which a new analysis about the new solution must be made. Thus, for nonlinear problems the eigenvalues keep changing. Richardson extrapolation can be used to improve the accuracy of a method. Step forward one step ∆t with a p-th order method. Then redo the problem, this time stepping forward from the same initial point but in two steps of length ∆t/2, thus ending at the same point. Call the solution of the one-step calculation y1 and the solution of

∆t ( fn − fn−1 )

Thus, by monitoring the difference between the right-hand side from one time step to another, an estimate of the truncation error is obtained. This error can be reduced by reducing ∆t. If the user specifies a criterion for the largest local error estimate, then ∆t is reduced to meet that criterion. Also, ∆t is increased to as large a value as possible, because this shortens computation time. If the local truncation error has been achieved (and estimated) by using a step size ∆t 1 LTE = c ∆tp1

and the desired error is ε, to be achieved using a step size ∆t 2 ε = c ∆tp2

then the next step size ∆t 2 is taken from LTE = ε



∆t1 ∆t2

p

Generally, things should not be changed too often or too drastically. Thus one may choose not to increase ∆t by more than a factor (such as 2) or to increase ∆t more than once every so many steps (such as 5) [54]. In the most sophisticated codes the alternative exists to change the order

Mathematics in Chemical Engineering of the method as well. In this case, the truncation error of the orders one higher and one lower than the current one are estimated, and a choice is made depending on the expected step size and work.

6.3. Implicit Methods By using different interpolation formulas, involving yn+1 , implicit integration methods can be derived. Implicit methods result in a nonlinear equation to be solved for yn+1 so that iterative methods must be used. The backward Euler method is a first-order method:

y n+1 =y n + ∆t f y n+1

Here, the superscript k refers to an iteration counter. The successive substitution method is guaranteed to converge, provided the first derivative of the function is bounded and a small enough time step is chosen. Thus, if it has not converged within a few iterations, ∆t can be reduced and the iterations begun again. The Newton – Raphson method (see Section 1.2) would solve the problem as y n+1, k+1 = ∆t β0 f 

I − ∆t β0

When the trapezoid rule is used with the finite difference method for solving partial differential equations it is called the Crank – Nicolson method. Adams methods exist as well, and the fourth-order Adams – Moulton method is y n+1 = y n + −5 f



∆t  9 f y n+1 + 19 f ( y n ) 24



 y n−1 + f y n−2

The properties of these methods are given in Table 6. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. An application to dynamic distillation problems is given in [55]. All these methods can be written in the form y n+1 =

k 

αi y n+1−i +∆t

i=1

k 

βi f



y

n+1−i

i=0

or y n+1 = ∆t β0 f



y n+1 + wn



y n+1, k+1 = ∆t β0 f y n+1, k



 ∆t β0 f

+wn

∂ f   ∂y yn+1,k



y n+1,k+1 − y n+1, k

 =

 ∂ f  y n+1,k + wn −∆t β0 y n+1,k  ∂y yn+1,k

For multiple equations, I is the Dirac Kronecker function δij , and ∂f /∂y is the Jacobian ∂fi /∂yj . As ∆t becomes smaller the conditions for convergence are more likely to be satisfied, but if the solution does not converge, ∆t can be reduced and the process repeated. In many computer codes, iteration is allowed to proceed only a fixed number of times (e.g., three) before ∆t is reduced. Because a good history of the function is available from previous time steps, a good initial guess is usually possible. The best software packages for stiff equations (see Section 6.4) use Gear’s backward difference formulas. The formulas of various orders are [56], [57, p. 263] 1 : y n+1 = y n +∆t f



y n+1



4 1 2 2 : y n+1 = y n − y n−1 + ∆t f y n+1 3 3 3 3 : y n+1 =

+

where wn represents known information. This equation (or set of equations for more than one differential equation) can be solved by using successive substitution:

 ∂ f  y n+1,k +∆t β0  ∂y yn+1,k

or 

 ∆t  f ( y n ) + f y n+1 2



 y n+1,k+1 − y n+1,k + wn

The trapezoid rule (see Section 2.4) is a secondorder method: y n+1 = y n +

67

6 ∆t f y n+1 11

4 : y n+1 =

+

18 n 9 n−1 2 n−2 + y − y y 11 11 11

48 n 36 n−1 16 n−2 3 n−3 + − y − y y y 25 25 25 25

12 ∆t f y n+1 25

68

Mathematics in Chemical Engineering

5 : y n+1 =

300 n 300 n− 1 200 n−2 + y − y y 137 137 137

−

75 n−3 12 n− 4 y y + 137 137

+

60 ∆t f y n+1 137

The stability properties of these methods are determined in the same way as explicit methods. They are always expected to be stable, no matter what the value of ∆t is, and this is confirmed in Figure 24. LIVE GRAPH

Click here to view

y n+1 = y n +

 ∆t  9 f y¯ n+1 + 19 f ( y n ) + . . . 24

The last step can be applied over and over, if desired. The stability properties of these methods are listed in Table 6. Step-size control can also be employed for these methods.

6.4. Stiffness Why is it desirable to use implicit methods that lead to sets of algebraic equations that must be solved iteratively whereas explicit methods lead to a direct calculation? The reason lies in the stability limits; to understand their impact, the concept of stiffness is necessary. When modeling a physical situation, the time constants governing different phenomena should be examined. Consider flow through a packed bed, as illustrated in Figure 25.

Figure 25. Flow through packed bed Figure 24. Rational approximations for implicit methods a) Backward Euler; b) Exact curve; c) Trapezoid; d) Euler

The superficial velocity u is given by u =

Predictor – corrector methods can be employed in which an explicit method is used to predict the value of yn+1 . This value is then used in an implicit method to evaluate f ( yn+1 ). The first-order method is merely the Runge – Kutta method described above.

Q Aϕ

where Q is the volumetric flow rate, A is the cross-sectional area, and ϕ is the void fraction. A time constant for flow through the device is then tflow =

n+ 1

= y +∆t f ( y )

n+1

 ∆t  n+1 f y¯ + f ( yn ) = y + 2

y¯ y

n

n

n

Note we do not iterate on yn+1 , as we would in implicit methods. The fourth-order Adams predictor – corrector method uses the Adams – Bashforth method to provide a predicted value: y¯n+ 1 = y n +

∆t [55 f ( y n ) + . . . ] 24

and the Adams – Moulton method to correct it:

ϕAL L = u Q

where L is the length of the packed bed. If a chemical reaction occurs, with a reaction rate given by Moles = − kc Volume time

where k is the rate constant (time−1 ) and c is the concentration (moles/volume), the characteristic time for the reaction is trxn =

1 k

If diffusion occurs inside the catalyst, the time constant is

Mathematics in Chemical Engineering tinternal diffusion =

ε R2 De

where ε is the porosity of the catalyst, R is the catalyst radius, and De is the effective diffusion coefficient inside the catalyst. The time constant for heat transfer is tinternal heat transfer =

s Cs R2 R2 = α ke

where s is the catalyst density, C s is the catalyst heat capacity per unit mass, k e is the effective thermal conductivity of the catalyst, and α is the thermal diffusivity. The time constants for diffusion of mass and heat through a boundary layer surrounding the catalyst are texternal diffusion =

R kg

texternal heat transfer =

s Cs R hp

where k g and hp are the mass-transfer and heattransfer coefficients, respectively. The importance of examining these time constants comes from realization that their orders of magnitude differ greatly. For example, in the model of an automobile catalytic converter [58] the time constant for internal diffusion was 0.3 s, for internal heat transfer 21 s, and for flow through the device 0.003 s. Flow through the device is so fast that it might as well be instantaneous. Thus, the time derivatives could be dropped from the mass balance equations for the flow, leading to a set of differential-algebraic equations (see below). If the original equations had to be solved, the eigenvalues would be roughly proportional to the inverse of the time constants. The time interval over which to integrate would be a small number (e.g., five) multiplied by the longest time constant. Yet the explicit stability limitation applies to all the eigenvalues, and the largest eigenvalue would determine the largest permissible time step. If that eigenvalue was very large, very small time steps would have to be used, but a long integration would be required to reach steady state. Such problems are termed stiff, and implicit methods are very useful for them. In that case the stable time constant is not of any interest, because any time step is stable. What is of interest is the largest step for which a solution

69

can be found. If a time step larger than the smallest time constant is used, then any phenomena represented by that smallest time constant will be overlooked – at least transients in it will be smeared over. However, the method will still be stable. Thus, if the very rapid transients of part of the model are not of interest, they can be ignored and an implicit method used. The idea of stiffness is best explained by considering a system of linear equations: dy =A y dt

Let λi be the eigenvalues of the matrix A. This system can be converted into a system of n equations, each of them having only one unknown; the eigenvalues of the new system are the same as the eigenvalues of the original system [50, pp. 39 – 42], [53, pp. 197 – 199], [59]. Then the stiffness ratio SR is defined as [60, p. 32] SR =

maxi |Re (λi ) | mini |Re (λi ) |

SR = 20 is not stiff, SR = 103 is stiff, and SR = 106 is very stiff. If the problem is nonlinear, the solution is expanded about the current state: n  dyi ∂fi [ yj − yj (tn )] = fi [y (tn )] + dt ∂y j j=1

The question of stiffness then depends on the eigenvalue of the Jacobian at the current time. Consequently, for nonlinear problems the problem can be stiff during one time period and not stiff during another. Packages have been developed for problems such as these. Although the chemical engineer may not actually calculate the eigenvalues, knowing that they determine the stability and accuracy of the numerical scheme, as well as the step size employed, is useful.

6.5. Differential – Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are

70 M

Mathematics in Chemical Engineering

dxn = V dt

n+1 n+1

y

− Ln xn − V

n n

y

+ Ln−1 xn−1 xn−1 N 



− xn = E n xn−1 − x∗,n

xi = 1

F

  dy = 0 t, y, dt

or the variables and equations may be separated according to whether they come primarily from differential or algebraic equations: dy = f (t, y, x) , g (t, y, x) = 0 dt

i=1

where x and y are the mole fractions in the liquid and vapor, respectively; L and V are liquid and vapor flow rates, respectively; M is the holdup; and the superscript n is the stage number. The efficiency is E, and the concentration in equilibrium with the vapor is x ∗ . The first equation is an ordinary differential equation for the mass of one component on the stage, whereas the third equation represents a constraint that the mass fractions add to one. As a second example, the following kinetics problem can be considered: dc1 = f (c1 , c2 ) dt dc2 = k1 c1 − k2 c22 dt

The first equation could be the equation for a stirred tank reactor, for example. Suppose both k 1 and k 2 are large. The problem is then stiff, but the second equation could be taken at equilibrium. If c1  2 c2

The equilibrium condition is then c22 k1 = ≡ K c1 k2

Under these conditions the problem becomes dc1 = f (c1 , c2 ) dt 0 = k1 c1 − k2 c22

Thus, a differential-algebraic system of equations is obtained. In this case, the second equation can be solved and substituted into the first to obtain differential equations, but in the general case that is not possible. Differential-algebraic equations can be written in the general notation

Another form is not strictly a differentialalgebraic set of equations, but the same principles apply; this form arises frequently when the Galerkin finite element is applied: A

dy = f ( y) dt

Suppose the general problem is to be solved by using the backward Euler method. Then, the nonlinear differential equation is replaced by the nonlinear algebraic equation for one step: F

  y n+1 − y n = 0 t, y n+1 , ∆t

This equation must be solved for yn+1 . The Newton – Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, higher order backwarddifference Gear methods are used in the computer program DASSL [61]. Differential-algebraic systems are more complicated than differential systems because the solution may not always be defined. Pontelides et al. [62] introduced the term “index” to identify possible problems. The index is defined as the minimum number of times the equations must be differentiated with respect to time to convert the system to a set ofordinary differential equations. These higher derivatives may not exist, and the process places limits on which variables can be given initial values. Sometimes the initial values must be constrained by the algebraic equations [62]. For a differential-algebraic system modeling a distillation tower, the index depends on the specification of pressure for the column [62]. Several chemical engineering examples of differential-algebraic systems and a solution for one involving two-phase flow are given in [63].

Next Page Mathematics in Chemical Engineering

6.6. Computer Software Efficient software packages are widely available for solving ordinary differential equations as initial value problems. Three of them – RKF 45, LSODE, and EPISODE – are discussed here. In each of the packages the user specifies the differential equation to be solved and a desired error criterion. The package then integrates in time and adjusts the step size to achieve the error criterion, within the limitations imposed by stability. A popular explicit Runge – Kutta package is RKF 45, developed by Forsythe et al. [52, Chap. 6]. The method is based on the Runge – Kutta – Feldberg formulas (see Section 6.2). An estimate of the truncation error at each step is available. Then the step size can be reduced until this estimate is below the userspecified tolerance. The method is thus automatic, and the user is assured of the results. Note, however, that the tolerance is set on the local truncation error, namely, from one step to another, whereas the user is generally interested in the global trunction error, i.e., the error after several steps. The global error is generally made smaller by making the tolerance smaller, but the absolute accuracy is not the same as the tolerance. If the problem is stiff, then very small step sizes are used and the computation becomes very lengthy. The RKF 45 code discovers this and returns control to the user with a message indicating the problem is too hard to solve with RKF 45. A popular implicit package is LSODE, a version of Gear’s method [56] written by Alan Hindmarsh at Lawrence Livermore Laboratory [64]. Earlier versions of this were GEAR and GEARB [65]. In this package, the user specifies the differential equation to be solved and the tolerance desired. Now the method is implicit and, therefore, stable for any step size. The accuracy may not be acceptable, however, and sets of nonlinear equations must be solved. Thus, in practice, the step size is limited but not nearly so much as in the Runge – Kutta methods. In these packages both the step size and the order of the method are adjusted by the package itself. Suppose a k-th order method is being used. The truncation error is determined by the (k + 1)-th order derivative. This is estimated by using difference formulas and the values of

71

the right-hand sides at previous times. An estimate is also made for the k-th and (k + 2)-th derivative. Then, the errors in a (k − 1)-th order method, a k-th order method, and a (k + 1)-th order method can be estimated. Furthermore, the step size required to satisfy the tolerance with each of these methods can be determined. Then the method and step size for the next step that achieves the biggest step can be chosen, with appropriate adjustments due to the different work required for each order. The package generally starts with a very small step size and a first-order method – the backward Euler method. Then it integrates along, adjusting the order up (and later down) depending on the error estimates. The user is thus assured that the local truncation error meets the tolerance. A further difficulty arises because the set of nonlinear equations must be solved. Usually a good guess of the solution is available, because the solution is evolving in time and past history can be extrapolated. Thus, the Newton – Raphson method will usually converge. The package protects itself, though, by only doing a few (i.e., three) iterations. If convergence is not reached within these iterations, the step size is reduced and the calculation is redone for that time step. The convergence theorem for the Newton – Raphson method (Chap. 1) indicates that the method will converge if the step size is small enough. Thus, the method is guaranteed to work. Further economies are possible. The Jacobian needed in the Newton – Raphson method can be fixed over several time steps. Then if the iteration does not converge, the Jacobian can be reevaluated at the current time step. If the iteration still does not converge, then the step size is reduced and a new Jacobian is evaluated. The successive substitution method can also be used – which is even faster, except that it may not converge. However, it too will converge if the time step is small enough. Comparisons of the methods and additional details are provided for chemical engineering problems by Finlayson [50, pp. 54 – 56] and Carnahan and Wilkes [57]. Generally the Runge – Kutta methods give extremely good accuracy, especially when the step size is kept small for the stability reason. When the computation time is comparable for LSODE and RKF 45, the RKF 45 package generally gives much more accurate results. The RKF 45 package is unsuitable, however, for many chemi-

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Mathematics in Chemical Engineering

cal reactor problems because these problems are so stiff. Other comparisons are available by Hull et al. [66], Shampine et al. [67], Byrne et al. [68], and Junce et al. [69]. Other packages are available through IMSL etc; see [54, pp. 291 – 292] and [70, pp. 439, 451]. Generally, standard packages must have a highorder explicit method (usually a version of Runge – Kutta) and a multistep, implicit method (usually a version of GEAR, EPISODE, or LSODE). The package DASSL [61] uses similar principles to solve the differential-algebraic systems. The packages LSODA and LSODAR are sister programs to LSODE. They have the feature of switching between stiff and nonstiff methods automatically; LSODAR also has a stopping feature when some condition is satisfied, which can be used for finding roots. [The software described here is available by electronic mail over BITNET. Sending the message: “mail [email protected]”, “send index” will retrieve an index and descriptions of how to obtain the software. The packages DASSL and ODEPACK (containing LSODE . . . LSODAR) can be obtained for a nominal fee on a computer tape from Professor W. E. Schiesser, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015.]

6.7. Stability, Bifurcations, Limit Cycles

The variable u can be a vector, which makes F a vector, too. Here, F represents a set of equations that can be solved for the steady state: F (u, λ) = 0

If the Newton – Raphson method is applied, F su δ us = − F (us , λ) us+1 = us + δu2

is obtained, where Fus =

∂F (us ) ∂u

is the Jacobian. Look at some property of the solution, perhaps the value at a certain point or the maximum value or an integral of the solution. This property is plotted versus the parameter λ; typical plots are shown in Figure 26. At the point shown in Figure 26 A, the determinant of the Jacobian is zero: det Fu = 0

For the limit point, ∂F = 0 ∂λ

whereas for the bifurcation-limit point ∂F = 0 ∂λ

In this section, bifurcation theory is discussed in a general way. Some aspects of this subject involve the solution of nonlinear equations; other aspects involve the integration of ordinary differential equations; applications include chaos and fractals as well as unusual operation of some chemical engineering equipment. An excellent introduction to the subject and details needed to apply the methods are given in [71]. For more details of the algorithms described below and a concise survey with some chemical engineering examples, see [72] and [73]. Bifurcation results are closely connected with stability of the steady states, which is essentially a transient phenomenon. Consider the problem

The stability of the steady solutions is also of interest. Suppose a steady solution uss ; the function u is written as the sum of the known steady state and a perturbation u
:

∂u = F (u, λ) ∂t

∂u = Fuss u ∂t

u = uss +u

This expression is substituted into the original equation and linearized about the steady-state value:

∂uss ∂u + = F uss +u , λ ∂t ∂t ≈ F (uss , λ) +

The result is

∂F   u + . . . ∂u uss

Mathematics in Chemical Engineering

73

Figure 26. Limit points and bifurcation – limit points A) Limit point (or turning point); B) Bifurcation-limit point (or singular turning point or bifurcation point)

A solution of the form u (x, t) = eσt X (x)

gives σt σeσt X = F ss ue X

The exponential term can be factored out and (F ss u − σ δ) X = 0

A solution exists for X if and only if det |F ss u − σ δ| = 0

The σ are the eigenvalues of the Jacobian. Now clearly if Re (σ) > 0 then u
grows with time, and the steady solution uss is said to be unstable to small disturbances. If Im (σ) = 0 it is called stationary instability, and the disturbance would grow monotonically, as indicated in Figure 27 A. If Im (σ) = 0 then the disturbance grows in an oscillatory fashion, as shown in Figure 27 B, and is called oscillatory instability. The case in which Re (σ) = 0 is the dividing point between stability and instability. If Re (σ) = 0 and Im (σ) = 0 – the point governing the onset of stationary instability – then σ = 0. However, this means that σ = 0 is an eigenvalue of the Jacobian, and the determinant of the Jacobian is zero. Thus, the points at which the determinant of the Jacobian is zero (for limit points and bifurcationlimit points) are the points governing the onset of stationary instability. When Re (σ) = 0 but Im (σ) = 0, which is the onset of oscillatory instability, an even number of eigenvalues pass from the left-hand complex plane to the righthand complex plane. The eigenvalues are complex conjugates of each other (a result of the

original equations being real, with no complex numbers), and this is called a Hopf bifurcation. Numerical methods to study Hopf bifurcation are very computationally intensive and are not discussed here [71]. To return to the problem of solving for the steady-state solution: near the limit point or bifurcation-limit point two solutions exist that are very close to each other. In solving sets of equations with thousands of unknowns, the difficulties in convergence are obvious. For some dependent variables the approximation may be converging to one solution, whereas for another set of dependent variables it may be converging to the other solution; or the two solutions may all be mixed up. Thus, solution is difficult near a bifurcation point, and special methods are required. These methods are discussed in [72]. The first approach is to use natural continuation (also known as Euler – Newton continuation). Suppose a solution exists for some parameter λ. Call the value of the parameter λ0 and the corresponding solution u0 . Then F (u0 , λ0 ) = 0

Also, compute uλ as the solution to F ss u uλ = − F λ

at this point [λ0 , u0 ]. Then predict the starting guess for another λ using u0 =u0 +uλ (λ − λ0 )

and apply Newton – Raphson with this initial guess and the new value of λ. This will be a

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Mathematics in Chemical Engineering

Figure 27. Stationary and oscillatory instability A) Stationary instability; B) Oscillatory instability

much better guess of the new solution than just u0 by itself. Even this method has difficulties, however. Near a limit point the determinant of the Jacobian may be zero and the Newton method may fail. Perhaps no solutions exist at all for the chosen parameter λ near a limit point. Also, the ability to switch from one solution path to another at a bifurcation-limit point is necessary. Thus, other methods are needed as well: arclength continuation and pseudo-arc-length continuation [72]. The latter method can use the arrow matrix LU decomposition described in Chapter 1.

6.8. Sensitivity Analysis Often, when solving differential equations, the solution as well as the sensitivity of the solution to the value of a parameter must be known. Such information is useful in doing parameter estimation (to find the best set of parameters for a model) and in deciding whether a parameter needs to be measured accurately. The differential equation for y (t, α) where α is a parameter, is dy = f ( y, α) , y (0) = y0 dt

If this equation is differentiated with respect to α, then because y is a function of t and α ∂ ∂α



dy dt

 =

∂ f ∂y ∂f + ∂y ∂α ∂α

Exchanging the order of differentiation in the first term leads to the ordinary differential equation d dt



∂y ∂α

 =

∂ f ∂y ∂f + ∂y ∂α ∂α

The initial conditions on ∂y/∂α are obtained by differentiating the initial conditions ∂ ∂y [ y (0, α) = y0 ] , or (0) = 0 ∂α ∂α

Next, let y1 = y, y2 =

∂y ∂α

and solve the set of ordinary differential equations dy1 = f ( y1 , α) dt

y1 (0) = y0

dy2 ∂f ∂f = ( y1 , α) y2 + dt ∂y ∂α

y2 (0) = 0

Thus, the solution y (t, α) and the derivative with respect to α are obtained. To project the impact of α, the solution for α = α1 can be used: y (t, α) = y1 (t, α1 ) +

∂y (t, α1 ) (α − α1 ) + . . . ∂α

= y1 (t, α1 ) + y2 (t, α1 ) (α − α1 ) + . . .

This is a convenient way to determine the sensitivity of the solution to parameters in the problem.

Mathematics in Chemical Engineering

6.9. Eigenvalues and Roots by Initial Value Techniques

The solution is a zero-order Bessel function.

Initial value methods can also be used to find roots of equations. Doing this requires a code that stops when the dependent variable (e.g., y) takes a certain value, as opposed to stopping when the independent variable takes a certain value (e.g., t). The RKF 45 code has been modified in this way [74], [75], as has the LSODAR code. To find the values of t that satisfy

Now a new variable is defined

75

R (r) = J0 (αk r) because J0 (αk ) = 0

z = αr

and the equation rearranged for R: d2 R 1 dR + + R = 0, dz 2 z dz

dR (0) = 0, R (0) = 1 dz

(arbitrary, take = 1) f (t) = 0

set

This equation can be solved as an initial value problem. The solution is continued until

y = f (t)

R=0

and differentiate to get

Suppose this occurs for zk . To have

dy df = dt dt

R=0

Now integrate the problem

when

dy df = ≡ g (t) , y (0) = y0 (arbitrary) dt dt

r=1

until y = 0. The t for which this occurs gives the root to the equation. Continued integration will give multiple roots. This technique can also be used to solve certain eigenvalue problems. If separation of variables is applied to unsteady heat conduction   ∂T r ∂r

∂T 1 ∂ = ∂t r ∂r

T (1, t) = 0,

∂T (0, t) = 0 ∂r

the following is obtained:  r

dR dr

 = constant = − α2

Then the function R (r) must satisfy 1 d r dr



dR r dr

αk = z k

The function R (αk r) = R (z)

then satisfies the correct differential equation and the appropriate boundary conditions. This gives a method of finding the Bessel functions of order zero. When the Graetz problem for heat transfer to a fluid flowing in a tube is considered,

∂T 1 ∂ 1 − r2 = ∂t r ∂r

T (r, t) = T (t) R (r) 1 dT 1 d = T dt r R dr

choose

 + α2 R (r) = 0

T (1, t) = 0,

  ∂T r ∂r

∂T (0, t) = 0 ∂r

the corresponding eigenvalue problem is

d2 R 1 dR + + α2 1 − r 2 R (r) = 0, dr 2 r dr

or d2 R 1 dR + + α2 R (r) = 0, R (1) = 0, dr 2 r dr dR (0) = 0 dr

R (1) = 0,

dR (0) = 0 dr

Now the initial value methods apparently cannot be used because the transformation is not possible.

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7. Ordinary Differential Equations as Boundary Value Problems

∆p rdr L

d (r τ ) = −

Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial locations: at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations the problems are not initial value in character. To begin at one position and integrate directly is impossible because at least one of the conditions is specified somewhere else and not enough conditions are available to begin the calculation. Thus, methods have been developed especially for boundary value problems.

and then integrated to give

7.1. Solution by Quadrature

−η v = −

When only one equation exists, even if it is nonlinear, it may possibly be solved by quadrature. For

Now the two unknowns must be specified from the boundary conditions. This problem is a twopoint boundary value problem because one of the conditions is usually specified at r = 0 and the other at r = R, the tube radius. However, the technique of separating variables and integrating works quite well.

dy = f ( y) dt y (0) = y0

rτ = −

∆p r2 + c1 L 2

Proceeding further requires choosing a constitutive relation relating the shear stress and the velocity gradient as well as a condition specifying the constant. For a Newtonian fluid τ = −η

dv dr

where v is the velocity and η the viscosity. Then the variables can be separated again and the result integrated to give ∆p r 2 + c1 lnr + c2 L 4

the problem can be separated dy = dt f (y)

and integrated
y y0

dy  = f ( y )


t dt = t 0

If the quadrature can be performed analytically, the exact solution has been found. As an example, consider the flow of a nonNewtonian fluid in a pipe, as illustrated in Figure 28. The governing differential equation is [76] 1 d ∆p (r τ ) = − r dr L

where r is the radial position from the center of the pipe, τ is the shear stress, ∆ p is the pressure drop along the pipe, and L is the length over which the pressure drop occurs. The variables are separated once

Figure 28. Flow in pipe

When the fluid is non-Newtonian, it may not be possible to do the second step analytically. For example, for the Bird – Carreau fluid [77, p. 171], stress and velocity are related by τ = 

 1+λ

η0 2 (1−n)/2

dv dr

where η 0 is the viscosity at v = 0 and λ the time constant. Putting this value into the equation for stress as a function of r gives 

 1+λ

η0 ∆p r c1 = − + 2 (1−n)/2 L 2 r dv dr

Mathematics in Chemical Engineering This equation cannot be solved analytically for dv/dr, except for special values of n. For problems such as this, numerical methods must be used.

7.2. Shooting Methods A shooting method is one that utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Suppose the nonlinear boundary value problem d2 y = f dx2

 x, y,

dy dx



b0 y (1) + b1

which should preferably be zero. Note that the solution depends on s. χ (s) = 0

Iteration on s is required to find the solution to the problem. The condition at x = 0 is satisfied for any s; the differential equation is satisfied by the integration routine; and all that must be ensured is that the last boundary condition is satisfied. A successive substitution method would use   sk+1 = sk − m χ sk

Keller [78] showed that if

with the boundary conditions a0 y (0) − a1

77

dy (0) = α, ai ≥ 0 dx

dy (1) = β, bi ≥ 0 dx

∂f ≤ N ∂y

for some N and 0 < m < 2 Γ , where Γ increases as N increases, then the iteration scheme converges as k approaches infinity. Newton’s method of iteration would use χ sk dχ k s ds

Convert this second-order equation into two first-order equations along with the boundary conditions written to include a parameter s.

sk+1 = sk −

du = v dx

The function dχ/ds is determined by integrating two more equations obtained by differentiating the original equations with respect to s. If

dv = f (x, u, v) dx

ζ =

∂u (x, s) ∂v (x, s) , η = ∂s ∂s

u (0) = a1 s − c1 α

the additional differential equations are

v (0) = a0 s − c0 α

dζ dη ∂ f ∂f = η, = ζ+ η dx dx ∂u ∂v

The parameters c0 and c1 are specified by the analyst such that a1 c0 −a0 c1 = 1

ζ (0) = a1 , η (0) =a0

This ensures that the first boundary condition is satisfied for any value of parameter s. If the proper value for s is known, u (0) and u
(0) can be evaluated and the equation integrated as an initial value problem. The parameter s should be chosen so that the last boundary condition is satisfied. Define the function χ (s) = b0 u (1, s) + b1

These equations are integrated along with the initial conditions

du (1, s) − β dx

= b0 u (1, s) + b1 v (1, s) − β

Then the derivative is dχ = b0 ζ (1, s) + b1 η (1, s) ds

The process can be used when multiple conditions must be satisfied, such as might arise when more than one dependent variable exists. Suppose two equations must be satisfied at one boundary χ1 (s1 , s2 ) = 0, χ2 (s1 , s2 ) = 0

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Mathematics in Chemical Engineering

Then two variables s1 and s2 are introduced, and the Newton – Raphson method is applied in the form  ∂χ

1

∂s1 ∂χ2 ∂s1

∂χ1 ∂s2 ∂χ2 ∂s2





sk+1 χ1 sk1 , sk2 − sk1 1

= − χ2 sk1 , sk2 sk+1 − sk2 2

The problem for reaction and diffusion in a catalyst pellet is 1 ra−1

d dr

 r a−1

dc dr

 = ϕ2 R (c)

dc (0) = 0, c (1) = 1 dr

(15)

where ϕ is the Thiele modulus. Weisz and Hicks [79] showed how to transform this problem into one that can be solved by using initial value methods. The problem can be written as 1 z a−1

d dz

  dc z a−1 = d2 R (c) dz

dc (0) = 0, c (b z = 1) = 1 dz

by using r = b z and d = b ϕ. For any d, choose an arbitrary c (0) = c0 and integrate until the concentration reaches 1.0. Suppose that occurs at z = z1 . Then, let b =

1 d , ϕ = = d z1 z1 b

and the solution for the problem is obtained without iteration. We do not know in advance what ϕ the problem has been solved for, so this technique is especially useful to solve the problem for a range of ϕ. Employing this method requires a code that stops when the dependent variable (here c) equals some value, rather than one that stops when the independent variable (here z) equals some value. The model for a chemical reactor with axial diffusion is 1 dc2 dc − = Da R (c) , P e dz 2 dz dc 1 dc (0) + c (0) = cin , (1) = 0 − P e dz dz

where Pe is the P´eclet number and Da the Damk¨ohler number. The boundary conditions are due to Danckwerts [80] and to Wehner and Wilhelm [81].

This problem can be treated by using initial value methods also, but the method is highly sensitive to the choice of the parameter s, as outlined above. Starting at z = 0 and making small changes in s will cause large changes in the solution at the exit, and the boundary condition at the exit may be impossible to satisfy. By starting at z = 1, however, and integrating backwards, the process works and an iterative scheme converges in many cases [82]. However, if the problem is extremely nonlinear the iterations may not converge. In such cases, the methods for boundary value problems described below must be used. Computer software exists for solving twopoint boundary value problems: for example, the IMSL program DTPTB used DVERK, which employs Runge – Kutta integration to integrate the ordinary differential equations [83, p. 301]. Initial value methods can also be used when the sensitivity of the solution to parameters must be determined. The sensitivity might be desired for studying the stability of the equations, especially when the solution bifurcates into more than one solution, or it might be required for parameter estimation. Knowing the sensitivity gives clues as to what experimental measurements are most important for verifying a mathematical model. As an example, suppose the sensitivity of the solution to Equation (15) with respect to the Thiele modulus squared ϕ2 must be known. The variables are defined u (r) = c (r) , v (r) =

du dr

and the differential equation is written as du = v dr a−1 dv = ϕ2 R (u) − v dr r

Next, the equation is differentiated with respect to ϕ2 . Then the following variables and equations are added: ζ =

∂u ∂v , η = ∂ϕ2 ∂ϕ2

dζ = η dr dR a−1 dη = R (u) + ϕ2 ζ − η dr du r

Mathematics in Chemical Engineering These can be solved along with the original problem (on each iteration) or only after the proper s is found from the iterations. The result of this calculation is that we have c (x, ϕ2 ) for some ϕ2 and we also know ∂c/∂ϕ2 at the same value of ϕ2 . Then,

c r,

ϕ22



≈ c r,

ϕ21



+ ζ r,

ϕ21



ϕ22 −ϕ21

7.3. Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 29 shows a uniform mesh of n points (nonuniform meshes are described below). The unknown, here c (x), at a grid point xi is assigned the symbol ci = c (xi ). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point.

dc  d2 c  ∆x2 −...   ∆x + dx i dx2 i 2

Figure 29. Finite difference mesh; ∆x uniform

d2 c  ci+1 − 2 ci + ci−1 d4 c  2 ∆x2 − + . .(20) .  =  2 2 ∆x dx4 i 4! dx i

The truncation error is proportional to ∆x 2 . To see how to solve a differential equation, consider the equation for convection, diffusion, and reaction in a tubular reactor: 1 d2 c dc − = Da R (c) P e dx2 dx

To evaluate the differential equation at the i-th grid point, the finite difference representations of the first and second derivatives can be used to give

dc  d2 c  ∆x2 + ... = ci +  ∆x + 2  dx i dx i 2

ci−1 = ci −

last equation is chosen to ensure the best accuracy.

The finite difference representation of the second derivative can be obtained by adding the two expressions in Equation (16). Rearrangement and division by ∆x 2 give

can be used.

ci+1

79

(16)

These formulas can be rearranged and divided by ∆x to give

1 ci+1 − 2 ci + ci−1 ci+1 − ci−1 = Da R (ci ) − Pe ∆x2 2∆x (21)

which are representations of the first derivative. Alternatively the two equations can be subtracted from each other, rearranged and divided by ∆x to give

This equation is written for i = 2 to n − 1 (i.e., the internal points). The equations would then be coupled but would involve the values of c1 and cn , as well. These are determined from the boundary conditions. If the boundary condition involves a derivative, the finite difference representation of it must be carefully selected; here, three possibilities can be written. Consider a derivative needed at the point i = 1. First, Equation (17) could be used to write

dc  ci+1 − ci−1 d3 c  ∆x2 −   = dx i 2 ∆x dx3 i 3!

dc  c2 − c1  = dx 1 ∆x

dc  d2 c  ∆x ci+1 − ci − + ...   = dx i ∆x dx2 i 2

(17)

dc  d2 c  ∆x ci − ci−1 + +...   = dx i ∆x dx2 i 2

(18)

(19)

If the terms multiplied by ∆x or ∆x 2 are neglected, three representations of the first derivative are possible. In comparison with the Taylor series, the truncation error in the first two expressions is proportional to ∆x, and the methods are said to be first order. The truncation error in the last expression is proportional to ∆x 2 , and the method is said to be second order. Usually, the

(22)

Then a second-order expression is obtained that is one-sided. The Taylor series for the point ci+2 is written: ci+2 = ci +

+

dc  d2 c  4∆x2   2∆x + dx i dx2 i 2!

d3 c  8∆x3 + ...  dx3 i 3!

80

Mathematics in Chemical Engineering 1 c2 − c 1 + c1 = cin P e ∆x

Four times Equation (16) minus this equation, with rearrangement, gives

−

dc  −3 ci + 4 ci+1 − ci+2 + O ∆x2  = dx i 2∆x

cn − cn−1 =0 ∆x

plus Equation (21) at points i = 2 through n − 1; second order in ∆x, by using a three-point onesided derivative

Thus, for the first derivative at point i = 1 dc  −3 ci + 4 c2 − c3  = dx i 2∆x

(23)

This one-sided difference expression uses only the points already introduced into the domain. The third alternative is to add a false point, outside the domain, as c0 = c (x = − ∆x). Then the centered first derivative, Equation (19), can be used: dc  c2 − c0  = dx 1 2∆x

Because this equation introduces a new variable, another equation is required. This is obtained by also writing the differential equation (Eq. 21), for i = 1. The same approach can be taken at the other end. As a boundary condition, any of three choices can be used: dc  cn − cn−1  = dx n ∆x dc  cn−2 − 4 cn−1 + 3 cn  = dx n 2∆x dc  cn+1 − cn−1  = dx n 2∆x

The last two are of order ∆x 2 and the last one would require writing the differential equation (Eq. 21) for i = n, too. Generally, the first-order expression for the boundary condition is not used because the error in the solution would decrease only as ∆x, and the higher truncation error of the differential equation (∆x 2 ) would be lost. For this problem the boundary conditions are −

1 dc (0) + c (0) = cin P e dx

dc (1) = 0 dx

Thus, the three formulations would give first order in ∆x

−

1 −3 c1 + 4 c2 − c3 + c1 = cin Pe 2 ∆x

cn−2 − 4 cn−1 + 3 cn = 0 2 ∆x

plus Equation (21) at points i = 2 through n − 1; second order in ∆x, by using a false boundary point −

1 c2 − c 0 + c1 = cin P e 2 ∆x

cn+1 − cn−1 = 0 2 ∆x

plus Equation (21) at points i = 1 through n. The sets of equations can be solved by using the Newton – Raphson method, as outlined in Section 1.2. The form of the equations is important, however, because the equations give a tridiagonal structure. For the first-order method the equations have the form shown in Figure 30 A and the standard routines for solving tridiagonal equations suffice. For the second-order method with a one-sided three-point derivative the form of the equations is shown in Figure 30 B. Now the tridiagonal structure is broken. However, if the second equation is multiplied by a constant and added to the first one, a zero can be obtained in the third column of the first row. Then the standard routines for tridiagonal matrices can be called. Similar manipulations are necessary in the last row of the matrix to obtain the tridiagonal form of the equations. For the second-order method with a false boundary point, the form of the equations is shown in Figure 30 C. Now, the two additional equations are almost tridiagonal. With the manipulations described above, they can be put into a tridiagonal form and the standard routines can be called. Frequently, the transport coefficients (e.g., diffusion coefficient D or thermal conductivity) depend on the dependent variable (concentration or temperature, respectively). Then the differential equation might look like d dx

  dc D (c) =0 dx

Mathematics in Chemical Engineering

81

Figure 30. Equation structure for different boundary conditions A) First-order method; B) Second-order method with one-sided three-point derivative; C) Second-order method using false boundary point

This could be written as −

dJ = 0 dx

(24)

in terms of the mass flux J, where the mass flux is given by J = − D (c)

dc dx

Because the coefficient depends on c the equations are more complicated. A finite difference method can be written in terms of the fluxes at the midpoints, i + 1/2. Thus, −

Ji+1/2 − Ji−1/2 ∆x

The advantage of this approach is that it is easier to program than a full Newton – Raphson method. If the transport coefficients do not vary radically, the method converges. If the method does not converge, use of the full Newton – Raphson method may be necessary. Three ways are commonly used to evaluate the transport coefficient at the midpoint. The first one employs the transport coefficient evaluated at the average value of the solutions on either side:

D ci+1/2 ≈ D





1 (ci+1 + ci ) 2

=0

Then the constitutive equation for the mass flux can be written as

The truncation error of this approach is ∆x 2 [84, Chap. 14]. The second approach uses the average of the transport coefficients on either side:

ci+1 − ci Ji+1/2 = − D ci+1/2 ∆x

1 D ci+1/2 ≈ [D (ci+1 ) + D (ci )] 2

If these are combined,

D ci+1/2 (ci+1

− ci ) − D ci−1/2 (ci − ci−1 ) ∆x2

The truncation error of this approach is also ∆x 2 [84, Chap. 14], [85, p. 215]. The third approach = 0 employs an “upstream” transport coefficient.

This represents a set of nonlinear algebraic equations that can be solved with the Newton – Raphson method. However, in this case a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve

D ci+1/2 ≈ D (ci+1 ) , when D (ci+1 ) >D (ci )

D ci+1/2 ≈ D (ci ) , when D (ci+1 ) < D (ci )

This approach is used when the transport coefficients vary over several orders of magnitude and the “upstream” direction is defined as the

      k+1 ck+1 − D cki−1/2 ck+1 D cki+1/2 − ck+1 i+1 − ci i i−1 ∆x2

(25)

= 0

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Mathematics in Chemical Engineering

one in which the transport coefficient is larger. The truncation error of this approach is only ∆x [84, Chap. 14] , [85, p. 253], but this approach is useful if the numerical solutions show unrealistic oscillations [84], [85]. If the grid spacing is not uniform the formulas must be revised. The notation is shown in Figure 31. The finite difference form of Equation (24) is then −

Ji+1/2 − Ji−1/2 1 2

(∆xi +∆ xi+1 )

= 0

y (x) =

N +2 

ai yi (x)

Suppose the differential equation is N [ y] = 0

Then the expansion is put into the differential equation to form the residual: Residual = N

Ji−1/2

ci+1 − ci , ∆xi+1

ci − ci−1 = − Di−1/2 ∆xi

Combined with Equation (25), this gives the following representation for Equation (24) 1 ∆xi+1 +∆xi −



1 (Di+1 + Di ) (ci+1 − ci ) ∆xi+1

 1 (Di + Di−1 ) (ci − ci−1 ) = 0 ∆xi

7.4. Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method [85], the dependent variable is expanded in a series.

ai yi (x)

In the collocation method, the residual is set to zero at a set of points called collocation points: N

N +2 

ai yi (xj ) = 0, j = 2, . . . , N + 1

i=1

This provides N equations; two more equations come from the boundary conditions, giving N + 2 equations for N + 2 unknowns. This procedure is especially useful when the expansion is in a series of orthogonal polynomials, and when the collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [87], [88]. A major improvement was the proposal by Villadsen and Stewart [16] that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. Thus, Equation (26) would be evaluated at the collocation points

Figure 31. Finite difference grid with variable spacing

Rigorous error bounds for linear ordinary differential equations solved with the finite difference method are dicussed by Isaacson and Keller [86, p. 431]. Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method, with a variable step size [83, p. 301]; FDRXN is given for reaction problems in [85, p. 335].

N +2  i=1

and the constitutive relations are Ji+1/2 = − Di+1/2

(26)

i=1

y (xj ) =

N +2 

ai yi (xj ) , j = 1, . . . , N + 2

i=1

and solved for the coefficients in terms of the solution at the collocation points: ai =

N +2 

[ yi (xj )]−1 y (xj ) , i = 1, . . . , N + 2

j=1

Furthermore, if Equation (26) is differentiated once and evaluated at all collocation points, the first derivative can be written in terms of the values at the collocation points: N +2  dy dyi ai (xj ) = (xj ) , j = 1, . . . , N + 2 dx dx i=1

This can be expressed as

Mathematics in Chemical Engineering

and the set of algebraic equations solved, perhaps with the Newton – Raphson method. If orthogonal polynomials are used and the collocation points are the roots to one of the orthogonal polynomials, the orthogonal collocation method results. In the orthogonal collocation method the solution is expanded in a series involving orthogonal polynomials, where the polynomials Pi−1 (x) are defined in Section 2.2.

N +2  dy dyi [ yi (xk )]−1 y (xk ) (xj ) = (xj ) , dx dx i,k=1

j = 1, . . . , N + 2

or shortened to N +2  dy Ajk y (xk ) , (xj ) = dx k=1

Ajk =

N +2 

[ yi (xk )]−1

i=1

83

dyi (xj ) dx

y = a + b x + x (1 − x)

N 

ai Pi−1 (x)

i=1

Similar steps can be applied to the second derivative to obtain

=

N +2 

bi Pi−1 (x)

(30)

i=1

which is also

N +2  d2 y (x ) = Bjk y (xk ) , j dx2 k=1

Bjk =

N +2 

y =

N +2 

di xi−1

i=1

[ yi (xk )]−1

i=1

d 2 yi (xj ) dx2

This method is next applied to the differential equation for reaction in a tubular reactor, after the equation has been made nondimensional so that the dimensionless length is 1.0. 1 d2 c dc − = Da R (c) , P e dx2 dx

The collocation points are shown in Figure 32. There are N interior points plus one at each end, and the domain is always transformed to lie on 0 to 1. To define the matrices Aij and Bij this expression is evaluated at the collocation points; it is also differentiated and the result is evaluated at the collocation points. y (xj ) =

N +2 

di xi−1 j

i=1

−

dc dc (0) = P e [c (0) − cin ] , (1) = 0 dx dx

(27)

The differential equation at the collocation points is 1 Pe

N +2 

Bjk c (xk ) −

N +2 

k=1

Ajk c (xk ) = Da R (cj )

k=1

(28)

and the two boundary conditions are −

N +2 

N +2  dy di (i − 1) xi−2 (xj ) = j dx i=1 N +2  d2 y (x ) = di (i − 1) (i − 2) xi−3 j j dx2 i=1

These formulas are put in matrix notation, where Q, C, and D are N + 2 by N + 2 matrices. y = Q d,

dy d2 y = Dd = C d, dx dx2

, Cji = (i − 1) xi−2 , Qji = xi−1 j j

Alk c (xk ) = P e (c1 − cin ) ,

k=1 N +2 

Dji = (i − 1) (i − 2) xi−3 j AN +2,k c (xk ) = 0

(29)

k=1

Note that 1 is the first collocation point (x = 0) and N + 2 is the last one (x = 1). To apply the method, the matrices Aij and Bij must be found

In solving the first equation for d, the first and second derivatives can be written as d = Q−1 y,

dy =C Q−1 y = A y, dx

d2 y = D Q−1 y = B y dx2

(31)

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Thus the derivative at any collocation point can be determined in terms of the solution at the collocation points. The same property is enjoyed by the finite difference method (and the finite element method described below), and this property accounts for some of the popularity of the orthogonal collocation method. In applying the method to Equation (27), the same result is obtained; Equations (28) and (29), with the matrices defined in Equation (31). To find the solution at a point that is not a collocation point, Equation (30) is used; once the solution is known at all collocation points, d can be found; and once d is known, the solution for any x can be found.


1

W x2 Pk x2 Pm x2 xa−1 dx = 0,

0

k ≤ m−1

(33)

where the power on xa−1 defines the geometry as planar or cartesian (a = 1), cylindrical (a = 2), and spherical (a = 3). An analogous development is used to obtain the (N + 1)×(N + 1) matrices y (xj ) =

N +1 

di xj2i− 2

i=1 N +1  dy 3 di (2 i − 2) x2i− (xj ) = j dx i=1

∇2 y (xi ) =

To use the orthogonal collocation method, the matrices are required. They can be calculated as shown above for small N (N < 8) and by using more rigorous techniques, for higher N (see Chap. 2). However, having the matrices listed explicitly for N = 1 and 2 is useful; this is shown in Table 7. For some reaction diffusion problems, the solution can be an even function of x. For example, for the problem dc (0) = 0, c (1) = 1 dx

(32)

The solution can be proved to involve only even powers of x. In such cases, an orthogonal collocation method, which takes this feature into account, is convenient. This can easily be done by using expansions that only involve even powers of x. Thus, the expansion N

 y x2 = y (1) + 1 − x2 ai Pi−1 x2 i=1

y = Q d,

dy = C d, ∇2 y = D d dx

, Cji = (2 i − 2) x2i−3 , Qji = x2i−2 j j

Dji = ∇2 x2i−2 |xj d = Q−1 y,

dy = C Q−1 y = A y, dx

∇2 y=D Q−1 y = B y

In addition, the quadrature formula is W Q = f , W = f Q−1

where
1 x2i−2 xa−1 dx =

Wj x2i−2 j

j=1

0

=

N +1 

1 ≡ fi 2i − 2 + a

As an example, for the problem

is equivalent to

1

+1 +1 N N y x2 = bi Pi−1 x2 = di x2i−2 i=1



di ∇2 x2i−2 |xj

i=1

Figure 32. Orthogonal collocation points

d2 c = k c, dx2

N +1 

i=1

The polynomials are defined to be orthogonal with the weighting function W (x 2 ).

xa− 1

d dx

  dc xa−1 = ϕ2 R (c) dx

dc (0) = 0 , c (1) = 1 dx

orthogonal collocation is applied at the interior points

Mathematics in Chemical Engineering

85

Table 7. Matrices for orthogonal collocation

N +1 

Bji ci = ϕ2 R (cj ) , j = 1, . . . , N

the center (a frequent situation because the center concentration can be the most extreme one), it is given by

i=1

and the boundary condition solved for is cN +1 = 1

The boundary condition at x = 0 is satisfied automatically by the trial function. After the solution has been obtained, the effectiveness facor η is obtained by calculating 1 η ≡

0 1

N +1

R [c (x)] xa−1 dx = R [c (1)] xa− 1 dx

0

i=1 N +1

Wj R (cj ) Wj R (1)

i=1

Note that the effectiveness factor is the average reaction rate divided by the reaction rate evaluated at the external conditions. Error bounds have been given for linear problems [90, p. 356]. For planar geometry the error is Error in η =

ϕ2(2N +1) (2 N + 1)! (2 N + 2) !

This method is very accurate for small N (and small ϕ2 ); note that for finite difference methods the error goes as 1/N 2 , which does not decrease as rapidly with N. If the solution is desired at

c (0) = d1 =

N +1 



Q− 1

 1i

yi

i=1

Table 8. Collocation points for orthogonal collocation with symmetric polynomials and W =1 Geometry N

Planar

Cylindrical

Spherical

1 2

0.5773502692 0.3399810436 0.8611363116 0.2386191861 0.6612093865 0.9324695142 0.1834346425 0.5255324099 0.7966664774 0.9602898565 0.1488743390 0.4333953941 0.6794095683 0.8650633667 0.9739065285

0.7071067812 0.4597008434 0.8880738340 0.3357106870 0.7071067812 0.9419651451 0.2634992300 0.5744645143 0.8185294874 0.9646596062 0.2165873427 0.4803804169 0.7071067812 0.8770602346 0.9762632447

0.7745966692 0.5384693101 0.9061793459 0.4058451514 0.7415311856 0.9491079123 0.3242534234 0.6133714327 0.8360311073 0.9681602395 0.2695431560 0.5190961292 0.7301520056 0.8870625998 0.9782286581

3

4

5

The collocation points are listed in Table 8. For small N the results are usually more accurate when the weighting function in Equation (33) is 1 − x 2 . The matrices for N = 1 and N = 2 are given in Table 9 for the three geometries. Computer programs to generate matrices and a

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Table 9. Matrices for orthogonal collocation with symmetric polynomials and W =1−x 2

Mathematics in Chemical Engineering program to solve reaction diffusion problems, OCRXN, are available [85, p. 325, p. 331]. Orthogonal collocation can be applied to distillation problems. Stewart et al. [91], [92] developed a method using Hahn polynomials that retains the discrete nature of a plate-to-plate distillation column. Other work treats problems with multiple liquid phases [93].

7.5. Orthogonal Collocation on Finite Elements In the method of orthogonal collocation on finite elements, the domain is first divided into elements, and then within each element orthogonal collocation is applied. Figure 33 shows the domain being divided into NE elements, with NCOL interior collocation points within each element, and NP = NCOL + 2 total points per element, giving NT = NE ∗ (NCOL + 1) + 1 total number of points. Within each element a local coordinate is defined u =

x − x(k) ∆xk

, ∆xk = x(k+1) − x(k)

The reaction – diffusion equation is written as 1 xa− 1

d dx

 a−1

x

dc dx



d2 c a − 1 dc = 2 + dx x dx

= ϕ2 R (c)

and transformed to give 1 d2 c 1 dc a−1 = ϕ2 R (c) + ∆x2k du2 x(k) + u∆xk ∆xk du

The boundary conditions are typically dc (0) = 0 , dx

dc − (1) = Bim [c (1) − cB ] dx

where Bim is the Biot number for mass transfer. These become 1 dc (u = 0) = 0, ∆x1 du

in the first element; −

dc 1 (u = 1) = Bim [c (u = 1) − cB ] , ∆xN E du

in the last element. The orthogonal collocation method is applied at each interior collocation point.

87

NP NP 1  1  a−1 BIJ cJ + AIJ cJ = 2 ∆xk J=1 x(k) + uI ∆xk ∆xk J=1

= ϕ2 R (cJ ) , I = 2, . . . , N P − 1

The local points i = 2, . . . , NP − 1 represent the interior collocation points. Continuity of the function and the first derivative between elements is achieved by taking NP   1  AN P,J cJ  element k−1 ∆xk−1 J=1

=

NP  1   A1,J cJ  element k ∆xk J= 1

at the points between elements. Naturally, the computer code has only one symbol for the solution at a point shared between elements, but the derivative condition must be imposed. Finally, the boundary conditions at x = 0 and x = 1 are applied: NP 1  A1,J cJ = 0, ∆xk J= 1

in the first element; −

NP  1 AN P,J cJ = Bim [cN P − cB ] , ∆xN E J= 1

in the last element. These equations can be assembled into an overall matrix problem AAc = f

The form of these equations is special and is discussed by Finlayson [85, p. 116], who also gives the computer code to solve linear equations arising in such problems. Reaction – diffusion problems are solved by the program OCFERXN [85, p. 337]. See also the program COLSYS described below. Another way to approach the same problem is to use the Hermite polynomials (see Section 2.3). Then the continuity of the first derivative between elements is already part of the basis set, leading to a smaller set of equations. If cubic Hermite polynomials are used, the solution is

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Figure 33. Grid for orthogonal collocation on finite elements

identical to that obtained above with the orthogonal collocation method when using cubic polynomials (NCOL = 2). This is not usually done in chemical engineering; the interested reader is referred to [85, p. 121]. The error bounds of DeBoor [94] give the following results for second-order problems solved with cubic trial functions on finite elements with continuous first derivatives. The error at all positions is bounded by  i    d    dxi (y − yexact ) 

∞

≤ constant |∆x|2

The series (the trial solution) is inserted into the differential equation to obtain the residual: Residual =

NT 

ai

i=1

−ϕ R 2

1 xa−1

NT 

d dx

  dbi xa−1 dx

ai bi (x)

i= 1

The residual is then made orthogonal to the set of basis functions.
1 bj (x)

The error at the collocation points is more accurate, giving what is known as superconvergence.

0

  di   ≤ constant |∆x|4  i (y − yexact )  collocation points dx

−ϕ R

N T 

ai

i=1

2

N T 

1 xa−1

d dx

  dbi xa−1 dx

ai bi (x)

xa−1 dx = 0

i=1

j = 1, . . . , N T

7.6. Galerkin Finite Element Method In the finite element method the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced: the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced. To solve the problem 1 xa− 1

d dx

 xa−1

dc (0) = 0 , dx

−

dc dx

 = ϕ2 R (c)

dc (1) = Bim [c (1) − cB ] dx

the unknown solution is expanded in a series of known functions {bi (x)}, with unknown coefficients {ai }. c (x) =

NT  i=1

ai bi (x)

(34)

This process makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete, and an infinite number of them are used, then the residual is zero. Once the residual is zero the problem is solved. It is necessary also to allow for the boundary conditions. This is done by integrating the first term of Equation (34) by parts and then inserting the boundary conditions:
1 bj (x) 0


1 = 0

1 xa−1

d dx

  dbi xa−1 xa−1 dx dx

 
1 dbj dbi a−1 d a−1 dbi dx bj (x) x dx − x dx dx dx dx 0

Mathematics in Chemical Engineering  
1 dbj dbi a−1 dbi 1 = bj (x) xa−1 − dx x dx 0 dx dx

dNj 1 dNJ = , dx = ∆xe du dx ∆xe du

0


1 = − 0

dbj dbi a−1 dx − Bim bj (1) [bi (1) − cB ] x dx dx (35)

in the e-th element. Then the Galerkin method is 

−

e

Combining this with Equation (34) gives NT
 dbj dbi a−1 − dx ai x dx dx i=1

NP
1  dNJ dNI (xe + u∆xe )a−1 du ceI ∆xe I=1 du du 1

0

1

− Bim



NJ (1)

N P 

e

− Bim bj (1)

=ϕ

ai bi (1) − cB


1 bj (x) R

N T 



j = 1, . . . , N T


1 ∆xe

NJ (u) R

N P 

(36)

c (1) =cB

(u)

I=1

0

(37)

e BJI = −

1 ∆xe


1 0

dNJ dNI (xe + u∆xe )a−1 du, du du


1 FJe

2

= ϕ ∆xe

NJ (u) R

N P 

ceI NI

(u)

I=1

0

(xe + u∆xe )a−1 du

whereas the boundary element integrals are

then the boundary condition is used (instead of Eq. 36) for j = NT ,

e B BJI = − Bim NJ (1) NI (1) ,

NT 

F FJe = −Bim NJ (1) c1

ai bi (1) = cB

i=1

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. Both linear and quadratic finite element approximations are described in Chapter 2. The trial functions bi (x) are then generally written as Ni (x). NT 

ceI NI

The element integrals are defined as

This equation defines the Galerkin method, and a solution that satisfies this equation (for all j = 1, . . . , ∞) is called a weak solution. For an approximate solution the equation is written once for each member of the trial function, j = 1, . . . , NT . If the boundary condition is

c (x) =

(1) − c1

(xe + u∆xe )a−1 du

ai bi (x) xa−1 dx

i=1

0

 e

i=1

= ϕ2

2

ceI NI

I=1

0

N T 

89

ci Ni (x)

i=1

Each Ni (x) takes the value 1 at the point xi and zero at all other grid points (Chap. 2). Thus ci are the nodal values, c (xi ) = ci . The first derivative must be transformed to the local coordinate system, u = 0 to 1 when x goes from xi to xi + ∆x.

Then the entire method can be written in the compact notation 

e BJI ceI +

e

 e

e B BJI ceI =

 e

FJe +



F FJe

e

The matrices for various terms are given in Table 10. This equation can also be written in the form AA c = f

where the matrix AA is sparse. If linear elements are used the matrix is tridiagonal. If quadratic elements are used the matrix is pentadiagonal. Naturally the linear algebra is most efficiently carried out if the sparse structure is taken into

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Table 10. Element matrices for Galerkin method

account. Once the solution is found the solution at any point can be recovered from ce (u) = ceI=1 (1 − u) + ceI=2 u

for linear elements c

e

(u) = ceI=1 2



1 (u − 1) u − 2

+ ceI=2 4 u (1 − u) + ceI=2 2 u

y =





NJ (u) R



1 u− 2

ceI NI (u) (xe + u∆xe )a−1 du

I=1

0

=

N P 

NG 

Wk NJ (uk ) R

k=1

 NP 

N T+ 1

a i Si

i=1

for quadratic elements Because the integrals in Equation (36) may be complicated, they are usually formed by using Gaussian quadrature. If NG Gauss points are used, a typical term would be
1

but the treatment for differential equations follows Sincovec [95]. The trial function is taken as

The values of the function and the first and second derivatives at the knots are listed in Table 11. The knots are the points between elements. If a differential equation d2 y = f dx2

(uk )

For an application of the finite element method in fluid mechanics, see → Fluid Mechanics, Chap. 5.2.

=

Cubic B-splines have cubic approximations within each element, but first and second derivatives continuous between elements. The functions are the same ones discussed in Chapter 2,



3 (ai−1 − 2 ai + ai+1 ) 2 h2

The differential equation is then satisfied at each knot. ai−1 − 2 ai + ai+1 =  f

7.7. Cubic B-Splines

dy dx

d2 y  3 3 3 ai−1 − 2 ai + ai+1  = dx2 i 2 h2 h 2 h2

I= 1

(xe + uk ∆xe )a−1

x, y,

must be solved, the derivatives are approximated by

ceI NI



2 h2 3 

xi ,

1 1 3 ai−1 + ai + ai+1 , (ai+1 − ai−1 ) 4 4 4h

i = 1, . . . , N T

Mathematics in Chemical Engineering The boundary conditions must also be applied. If the boundary condition at the left is

91

Ej < 0.1 Ea remove node j 0.1 Ea ≤ Ej ≤ 10 Ea keep node j

y (x1 ) =yleft 10 Ea < Ej ≤ 100 Ea add 1 node

it is satisfied by y (x1 ) =

N T +1

100 Ea < Ej ≤ 1000 Ea add 2 nodes ai S (x1 ) = a0 S0 (x1 ) + a1 S1 (x1 )

i=0

+ a2 S2 (x1 ) = a0

1 1 + a1 + a2 = yleft 4 4

Ea =

Table 11. Function and derivative values at knots of cubic B-spline x i−1

xi

Si

0

1 4

1

Si

0

3 4h

0

S i

0

3 2 h2

−

xi +1

3 h2

−

n−1 

add 3 nodes

Ej / (n − 2)

j=2

Similar considerations apply at the right-hand side. To preserve the tridiagonal nature of the equations, the manipulations discussed in Section 7.3 must be used.

x i−2

1000 Ea < Ej

xi +2

1 4

0

3 4h

0

3 2 h2

0

7.8. Adaptive Mesh Strategies In many two-point boundary value problems, the difficulty in the problem is the formation of a boundary layer region, or a region in which the solution changes very dramatically. In such cases small mesh spacing should be used there, either with the finite difference method or the finite element method. If the region is known a priori, small mesh spacings can be assumed at the boundary layer. If the region is not known though, other techniques must be used. These techniques are known as adaptive mesh techniques. A simple technique that has proven useful [84, Chap. 7] is presented first, and more complicated (and more robust) techniques that have been implemented are then described. The adaptive mesh technique requires some criteria for deciding whether to add or remove points. The policy is taken as

Finlayson [84, Chap. 7] tried several criteria: the residual, the first derivative, the second derivative, etc. The first and second derivative worked best and are the easiest so they are described. If a finite difference method or a linear finite element method is used, the truncation error in the method is [96]:  2  d c Errorj = C∆x2j  2 dx

   

j

Thus, the pointwise value of the second derivative is used as the criterion. Because the grid points or mesh points are spaced at irregular intervals, an expression must be used for the second derivative that accounts for the irregularity. For the notation shown in Figure 30, the criterion is then Ej = ∆x2j

d2 c = ∆x2j dx2

cj+1 − cj c −c − xj − xj−1 xj+1 − xj j j−1 1 (x − x ) j+1 j−1 2

and this should be made uniform throughout the domain. The first derivative for the irregular mesh is cj+1 − cj dc   = dx j xj+1 − xj

However, the mean square derivative can also be used.   Ej = 


xj 

xj−1

 = ∆xj−1



dc dx

2

1/2  dx

cj − cj−1 ∆xj−1

2 1/2

|cj − cj−1 | = ∆xj−1

These adaptive mesh methods work as follows. An initial mesh is assumed. This might be a uniform mesh with only a few points, or it could

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have some features of the solution built into it. Then the problem is solved on this mesh. One of the criteria is then applied to decide if more points should be added or if points should be removed. Once an entire new mesh is found, the points can be smoothed somewhat by using [97] xk =

1 (xk + xk+1 ) 2

This ensures that the mesh does not change too drastically. Then the old solution is interpolated onto the new mesh and the problem is resolved. The process is continued until the solution is good enough, which might be defined as making the second (or first) derivative smaller than some fixed number over the entire domain. The adaptive mesh strategy was employed by Ascher et al. [98] and by Russell and Christiansen [99]. For a second-order differential equation and cubic trial functions on finite elements, the error in the i-th element is given by

Because cubic elements do not have a nonzero fourth derivative, the third derivative in adjacent elements is used [85, p. 166]: d3 ci+1 1 d 3 ci 1 , ai+1 = 3 3 3 ∆xi du ∆xi+1 du3

 u(4) i ≈

1 2

 1 2

ai − ai−1 + (xi+1 − xi−1 )

y (∆x) =yexact +c2 ∆x2

for small enough (and uniform) ∆x. A computer code should be run for varying ∆x to confirm this. For quadratic elements, the error is y (∆x) =yexact +c3 ∆x3

If orthogonal collocation on finite elements is used with cubic polynomials, then y (∆x) =yexact +c4 ∆x4

However, the global methods, not using finite elements, converge even faster [100], for example,  y (N ) = yexact +cN

 Error i = C∆x4i  u(4) i

ai =

power of ∆x, and the power is higher for the higher order methods, which suggests that the error is less. For example, with linear elements the error is

ai+1 − ai 1 (xi+2 − xi ] 2



Element sizes are then chosen so that the following error bounds are satisfied C ∆x4i  u(4) i ≤ε for all i

These features are built into the code COLSYS. The error expected from a method one order higher and one order lower can also be defined. Then a decision about whether to increase or decrease the order of the method can be made by taking into account the relative work of the different orders. This provides a method of adjusting both the mesh spacing (∆x, sometimes called h) and the degree of polynomial ( p). Such methods are called h – p methods.

7.9. Comparison What method should be used for any given problem? Obviously the error decreases with some

1 N COL

N COL

Yet the workload of the methods is also different. These considerations are discussed in [85]. Here, only sweeping generalizations are given. If the problem has a relatively smooth solution, then the orthogonal collocation method is preferred. It gives a very accurate solution, and N can be quite small so the work is small. If the problem has a steep front in it, the finite difference method or finite element method is indicated, and adaptive mesh techniques should probably be employed. Consider the reaction – diffusion problem: as the Thiele modulus ϕ increases from a small value with no diffusion limitations to a large value with significant diffusion limitations, the solution changes as shown in Figure 34. The orthogonal collocation method is initially the method of choice. For intermediate values of ϕ, N = 3 – 6 must be used, but orthogonal collocation still works well (for η down to approximately 0.01). For large ϕ, use of the finite difference method, the finite element method, or an asymptotic expansion for large ϕ is better. The decision depends entirely on the type of solution that is obtained. For steep fronts the finite difference method and finite element method with adaptive mesh are indicated.

Mathematics in Chemical Engineering

2

93

d3 f d2 f +f = 0 3 dη dη 2

f =

df = 0 at η = 0 dη

df = 1 at η → ∞ dη

Because one boundary is at infinity using a mesh with a constant size is difficult! One approach is to transform the domain. For example, let z = e−η

Then η = 0 becomes z = 1 and η = ∞ becomes z = 0. The derivatives are Figure 34. Concentration solution for different values of Thiele modulus

7.10. Singular Problems and Infinite Domains If the solution being sought has a singularity, a good numerical solution may be hard to find. Sometimes even the location of the singularity may not be known [101, pp. 230 – 238]. One method of solving such problems is to refine the mesh near the singularity, by relying on the better approximation due to a smaller ∆x. Another approach is to incorporate the singular trial function into the approximation. Thus, if the solution approaches f (x) as x goes to zero, and f (x) becomes infinite, an approximation may be taken as y (x) = f (x) +

N 

ai yi (x)

i=1

This function is substituted into the differential equation, which is solved for ai . Essentially, a new differential equation is being solved for a new variable: u (x) ≡y (x) −f (x)

The differential equation is more complicated but has a better solution near the singularity (see [102, pp. 189 – 192], [103, p. 611]). Sometimes the domain is infinite. Boundary layer flow past a flat plate is governed by the Blasius equation for stream function [104, p. 117].

d2 z = e−η = z dη 2

dz = − e−η = − z, dη df d f dz df = = −z dη dz dη dz d2 f d2 f = 2 dη dz 2



dz dη

2 +

d2 f d f d2 z df = z 2 2 +z 2 dz dη dz dz

The Blasius equation becomes   d3 f d2 f df 2 −z 3 3 − 3z 2 2 − z dz dz dz  +f

z2

d2 f df +z dz 2 dz

 = 0

The differential equation now has variable coefficients, but these are no more difficult to handle than the original nonlinearities. Another approach is to use a variable mesh, perhaps with the same transformation. For example, use z = e−η and a constant mesh size in z. Then with 101 points distributed uniformly from z = 0 to z = 1, the following are the nodal points: z = 0., 0.01, 0.02, . . . , 0.99, 1.0 η = ∞, 4.605, 3.912, . . . , 0.010, 0 ∆η = ∞, 0.693, . . . , 0.01

Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution [105]. A location that is far enough away must be found by trial and error.

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Mathematics in Chemical Engineering

8. Partial Differential Equations Partial differential equations are differential equations in which the dependent variable is a function of two or more independent variables. These can be time and one space dimension, or time and two or more space dimensions, or two or more space dimensions alone. Problems involving time are generally either hyperbolic or parabolic, whereas those involving spatial dimensions only are often elliptic. Because the methods applied to each type of equation are very different, the equation must first be classified as to its type. Then the special methods applicable to each type of equation are described. For a discussion of all methods, see [106]; for a discussion oriented more toward chemical engineering applications, see [107]. Many chemical engineering examples that use some of the methods described below are given in [108].

α = (α0 , α1 , . . . , αn ) , |α| =

n 

αi

i=0

∂α =

∂ |α| αn 1 ∂tα0 ∂xα 1 . . . ∂xn

the characteristic equation for P is 

aα σ α = 0, σ = (σ0 , σ1 , . . . , σn )

|α|=m αn σ α = σ0α0 σ1α1 . . . σn

(38)

where σ represents coordinates. Thus only the highest derivatives are used to determine the type. The surface is defined by this equation plus a normalization condition: n 

σk2 = 1

k=0

8.1. Classification of Equations A set of differential equations may be hyperbolic, elliptic, or parabolic, or it may be of mixed type. The type may change for different parameters or in different regions of the flow. This can happen in the case of nonlinear problems; an example is a compressible flow problem with both subsonic and supersonic regions. Characteristic curves are curves along which a discontinuity can propagate. For a given set of equations, it is necessary to determine if characteristics exist or not, because that determines whether the equations are hyperbolic, elliptic, or parabolic.

The shape of the surface defined by Equation (38) is also related to the type: elliptic equations give rise to ellipses; parabolic equations give rise to parabolas; and hyperbolic equations give rise to hyperbolas. σ2 σ12 + 22 = 1 , 2 a b σ0 = a σ12 , σ02 − a σ12 = 0,

Ellipse Parabola Hyperbola

Linear Problems. For linear problems, the theory summarized by Joseph et al. [109] can be used.

If Equation (38) has no nontrivial real zeroes then the equation is called elliptic. If all the roots are real and distinct (excluding zero) then the operator is hyperbolic. This formalism is applied to three basic types of equations. First consider the equation arising from steady diffusion in two dimensions:

∂ ∂ ∂ , ..., , ∂t ∂xi ∂xn

∂2c ∂2c + = 0 2 ∂x ∂y 2

is replaced with the Fourier variables

−ξ12 −ξ22 = − ξ21 +ξ22 = 0

i ξ0 , i ξ1 , . . . , i ξn

If the m-th order differential equation is P =

 |α|=m

where

aα ∂ α +

This gives

 |α|≤m

bα ∂ α

Thus, σ12 +σ22 = 1 (normalization) σ12 +σ22 = 0 (equation)

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95

These cannot both be satisfied so the problem is elliptic. When the equation is

The roots are real and the equation is hyperbolic. When β = 0

∂2u ∂2u − =0 ∂t2 ∂x2

ξ12 = 0

and the equation is parabolic. First-order quasi-linear problems are written in the form

then −ξ02 +ξ12 = 0

n 

Now real ξ 0 can be solved and the equation is hyperbolic σ02 +σ12 = 1 (normalization) −σ02 +σ12 = 0 (equation)

When the equation is ∂c =D ∂t



∂2 c ∂2c + ∂x2 ∂y 2



then

A1

l=0

∂u = f , x = (t , x1 , . . . , xn ) ∂x1

u = (u1 , u2 , . . . , uk )

The matrix entries A1 is a k×k matrix whose entries depend on u but not on derivatives of u. Equation (39) is hyperbolic if A = Aµ

is nonsingular and for any choice of real λ1 , l = 0, . . . , n, l = µ the roots αk of 

σ02 +σ12 +σ22 =

1 (normalization)

 det  αA −

σ12 +σ22 = 0 (equation)

thus we get σ02 = 1 (for normalization)

and the characteristic surfaces are hyperplanes with t = constant. This is a parabolic case. Consider next the telegrapher’s equation: ∂T ∂2T ∂2T = +β 2 ∂t ∂t ∂x2

Replacing the derivatives with the Fourier variables gives

(39)

 n  l=0 l=µ

 λ1 A1  = 0

are real. If the roots are complex the equation is elliptic; if some roots are real and some are complex the equation is of mixed type. Apply these ideas to the convection equation ∂u ∂u + F (u) = 0 ∂t ∂x

Thus, det (αA0 − λ1 A1 ) = 0 or det (αA1 − λ0 A0 ) = 0

In this case, n = 1, A0 = 1, A1 =F (u)

i ξ0 −β ξ02 +ξ12 = 0

Using the first of the above equations gives

The equation is thus second order and the type is determined by −β σ02 +σ12 = 0

The normalization condition σ02 +σ12 = 1

1 − (1 +

Thus, the roots are real and the equation is hyperbolic. The final example is the heat conduction problem written as  Cp

is required. Combining these gives β) σ02 =

det (α − λ1 F (u)) = 0, or α = λ1 F (u)

0

∂T ∂q ∂T = − , q = −k ∂t ∂x ∂x

In this formulation the constitutive equation for heat flux is separated out; the resulting set of equations is first order and written as

Previous Page 96

Mathematics in Chemical Engineering

 Cp

∂c ∂c ∂2c +u = D ∂t ∂x ∂x2

∂T ∂q + = 0 ∂t ∂x k

and Burgers viscosity equation

∂T = −q ∂x

∂u ∂u ∂2u +u = ν ∂t ∂x ∂x2

In matrix notation this is 

 Cp 0



0 0

∂T ∂t ∂q ∂t

+

 0 k





∂T 1 0 ∂x = ∂q 0 −q ∂x

∂u ∂u + A1 =f ∂x0 ∂x1

ϕ

In this case A0 is singular whereas A1 is nonsingular. Thus, det (αA1 −λ0 A0 ) = 0

is considered for any real λ0 . This gives − C λ  p 0  kα

α = 0 0

or α2 k = 0

Thus the α is real, but zero, and the equation is parabolic.

8.2. Hyperbolic Equations The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. A prototype equation is

∂c d f ∂c ∂c + ϕu + (1 − ϕ) = 0 ∂t ∂x dc ∂t

(43)

where ϕ is the void fraction and f (c) gives the equilibrium relation between the concentrations in the fluid and in the solid phase. In these examples, if the diffusion coefficient D or the kinematic viscosity ν is zero, the equations are hyperbolic. If D and ν are small, the phenomenon may be essentially hyperbolic even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. Equations for several methods are given here, as taken from [111]. If the convective term is treated with a centered difference expression the solution exhibits oscillations from node to node, and these vanish only if a very fine grid is used. The simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give the following for Equations (40): dci F (ci ) − F (ci−1 ) ci+1 − 2 ci + ci−1 + = D dt ∆x ∆x2

for Equation (42):

∂c ∂F (c) + = 0 ∂t ∂x

Depending on the interpretation of c and F (c), this can represent accumulation of mass and convection. With F (c) = u c, where u is the velocity, the equation represents a mass balance on concentration. If diffusive phenomenon are important, the equation is changed to ∂c ∂F (c) ∂2c + = D ∂t ∂x ∂x2

(42)

where u is the velocity and ν is the kinematic viscosity. This is a prototype equation for the Navier – Stokes equations (→ Fluid Mechanics, Chap. 3.2.). For adsorption phenomena [110, p. 202],

This compares with A0

(41)

(40)

where D is a diffusion coefficient. Special cases are the convective diffusive equation

dui ui − ui−1 ui+1 − 2 ui + ui−1 + ui = ν dt ∆x ∆x2

and for Equation (43): ϕ

ci − ci−1 d f  dci dci + ϕ ui + (1 − ϕ) = 0  dt ∆x dc i dt

If the flow were from right to left, then the formula would be dci F (ci+1 ) − F (ci ) ci+1 − 2 ci + ci−1 + =D dt ∆x ∆x2

If the flow could be in either direction, a local determination must be made at each node i and

Mathematics in Chemical Engineering the appropriate formula used. The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by taking the finite difference form of the convective diffusion equation

  1  n+1 2  n+1 ci+1 − cn ci − cn i+1 + i 6 3  1  n+1 c − cn i−1 6 i−1

+

= −

dci ci − ci−1 ci+1 − 2 ci + ci−1 +u =D dt ∆x ∆x2

and rearranging dci ci+1 − ci−1 +u dt 2∆x  =

D+

u∆x 2



ci+1 − 2 ci + ci−1 ∆x2

Thus the diffusion coefficient has been changed from D to D +

u∆x 2

Another method often used for hyperbolic equations is the MacCormack method. This method has two steps; it is written here for Equation (41). c∗n+1 i

=

cn i

u∆t n − − cn c i ∆x i+1

∆t D n n c − 2 cn + i + ci−1 ∆x2 i+1 = cn+1 i +

  u∆t  ∗n+1 1 n − c + c∗n+1 c − c∗n+1 i i−1 2 i 2∆x i

 ∆t D  ∗n+1 c − 2 c∗n+1 + c∗n+1 i i−1 2∆x2 i+1

The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method; then they are combined to add just enough diffusion to eliminate the oscillations without smoothing the solution too much. The algorithm is complicated and lengthy but well worth the effort [111–113]. If finite element methods are used, an explicit Taylor – Galerkin method is appropriate. For the convective diffusion equation the method is

97

+

u∆t n c − cn i−1 + 2∆x i+1

u2 ∆t2 2∆x2







n n cn i+1 − 2 ci + ci−1

∆t D ∆x2

Leaving out the u2 ∆t 2 terms gives the Galerkin method. Replacing the left-hand side with cn+1 − cn i i

gives the Taylor finite difference method, and dropping the u2 ∆t 2 terms in that gives the centered finite difference method. This method might require a small time step if reaction phenomena are important. Then the implicit Galerkin method (without the Taylor terms) is appropriate  2  1  n+1 ci+1 − cn cin+1 − cn i+1 + i 6 3 +

 1  n+1 ci−1 − cn i−1 6

=−

  u∆t n [1 − θ] cn i+1 − ci−1 2∆x

−

/ u∆t . n+1 θ ci+1 − cn+1 i−1 2∆x

+

  ∆t D n n [1 − θ] cn i+1 − 2 ci + ci−1 ∆x2

+

/ ∆t D . n+1 n+1 n+1 θ c − 2 c + c i+1 i i−1 ∆x2

The Taylor terms are not required because the implicit time step provides the same effect as diffusion. For the nonlinear Equation (40) the Taylor – Galerkin method is

98

Mathematics in Chemical Engineering

n+1 n+1 n n − cn 1 ci+1 − ci+1 2 cn+1 1 ci−1 − ci−1 i i + + 6 ∆t 3 ∆t 6 ∆t

=

 ∆t 1  n n + + Fi−1 −Fi+1 2∆x 4∆x2



dF dc

 −

+

2

dF dc

 +

i+ 1

2

 +

i

dF dc

dF dc

2

 n  ci+1 − cn i

i

2



 n  ci − cn i−1

i−1

D n n c − 2 cn i + ci− 1 ∆x2 i+1

A stability diagram for the explicit methods applied to the convective diffusion equation is shown in Figure 35. Notice that all the methods require Co =

u∆t ≤ 1 ∆x

where Co is the Courant number. How much Co should be less than one depends on the method and on r = D ∆t/∆x 2 , as given in Figure 35. The MacCormack method with flux correction requires a smaller time step than the MacCormack method alone (curve a), and the implicit Galerkin method (curve e) is stable for all values of Co and r shown in Figure 35 (as well as even larger values). LIVE GRAPH

Click here to view

Each of these methods tries to avoid oscillations that would disappear if the mesh were fine enough. For the steady convective diffusion equation these oscillations do not occur provided u∆x ≤ 1 2D

(44)

For large u, ∆x must be small to meet this condition. An alternative is to use a small ∆x in regions where the solution changes drastically. Because these regions change in time, the elements or grid points must move. The criteria to move the grid points can be quite complicated, and typical methods are reviewed in [111]. The criteria include moving the mesh in a known way (when the movement is known a priori), moving the mesh to keep some property (e.g., first- or second-derivative measures) uniform over the domain, using a Galerkin or weighted residual criterion to move the mesh, and Euler – Lagrange methods which move part of the solution exactly by convection and then add on some diffusion after that. The final illustration is for adsorption in a packed bed, or chromatography. Equation (43) can be solved when the adsorption phenomenon is governed by a Langmuir isotherm. f (c) =

αc 1+K c

Similar numerical considerations apply and similar methods are available [111].

8.3. Parabolic Equations in One Dimension

Figure 35. Stability diagram for convective diffusion equation (stable below curve) a) MacCormack; b) Centered finite difference; c) Taylor finite difference; d) Upstream; e) Galerkin; f) Taylor – Galerkin

In this section several methods are applied to parabolic equations in one dimension: separation of variables, combination of variables, finite difference method, finite element method, and the orthogonal collocation method. Separation of variables is successful for linear problems, whereas the other methods work for linear or nonlinear problems. The finite difference, the finite element, and the orthogonal collocation methods are numerical, whereas the separation or combination of variables can lead to analytical solutions.

Mathematics in Chemical Engineering Analytical Solutions. Consider the diffusion equation

99

c (x, t) = f (x) + u (x, t) u (0, t) = 0

∂c ∂2c =D ∂t ∂x2

u (1, t) = 0

with boundary and initial conditions c (x, 0) = 0 c (0, t) = 1,

c (1, t) = 0

A solution of the form c (x, t) =T (t) X (x)

is attempted and substituted into the equation, with the terms separated to give 1 dT 1 d2 X = D T dt X dx2

One side of this equation is a function of x alone, whereas the other side is a function of t alone. Thus, both sides must be a constant. Otherwise, if x is changed one side changes, but the other cannot because it depends on t. Call the constant − λ and write the separate equations dT = − λ D T, dt

d2 X = − λX dx2

The first equation is solved easily T (t) =T (0) e−λDt

Thus, f (0) = 1 and f (1) = 0 are necessary. Now the combined function satisfies the boundary conditions. In this case the function f (x) can be taken as f (x) = 1 −x

The equation for u is found by substituting for c in the original equation and noting that the f (x) drops out for this case; it need not disappear in the general case: ∂u ∂2u =D ∂t ∂x2

The boundary conditions for u are u (0, t) = 0 u (1, t) = 0

The initial conditions for u are found from the initial condition u (x, 0) = c (x, 0) −f (x) = x − 1

Separation of variables is now applied to this equation by writing u (x, t) =T (t) X (x)

and the second equation is written in the form d2 X + λX = 0 dx2

Next consider the boundary conditions. If they are written as

The same equation for T (t) and X (x) is obtained, but with X (0) = X (1) = 0. d2 X + λX = 0 dx2 X (0) = X (1) = 0

c (1, t) = 1 =T (t) X (1) c (0, t) = 0 =T (t) X (0)

the boundary conditions are difficult to satisfy because they are not homogeneous, with a zero right-hand side. Thus, the problem must be transformed to make the boundary conditions homogeneous. The solution is written as the sum of two functions, one of which satisfies the nonhomogeneous boundary conditions, whereas the other satisfies the homogeneous boundary conditions.

Next X (x) is solved for. The equation is an eigenvalue problem. The general solution is obmx 2 tained by using √ e and finding that m + λ = 0; thus m= ±i λ. The exponential term √

e±i

λx

is written in terms of sines and cosines, so that the general solution is √ √ X = B cos λ x +E sin λ x

100

Mathematics in Chemical Engineering

The boundary conditions are √

The Galerkin criterion for finding An is the same as the least-squares criterion [114, p. 183]. The solution is then

√

X (1) = B cos λ + E sin λ= 0 X (0) = B = 0

If B = 0, then E = 0 is required to have any solution at all. Thus, λ must satisfy √

sin λ = 0

This is true for certain values of λ, called eigenvalues or characteristic values. Here, they are λn = n2 π 2

Each eigenvalue has a corresponding eigenfunction Xn (x) =E sin n π x

The composite solution is then Xn (x) Tn (t) = E A sin n π x e−λn Dt

This function satisfies the boundary conditions and differential equation but not the initial condition. To make the function satisfy the initial condition, several of these solutions are added up, each with a different eigenfunction, and E A is replaced by An . u (x, t) =

∞ 

2

An sin n π x e−n

The residual R (x) is defined as the error in the initial condition: R (x) = x − 1 −

∞ 

An sin n π x

n=1

Next, the Galerkin method is applied, and the residual is made orthogonal to a complete set of functions, which are the eigenfunctions.
1 (x − 1) sin m π xdx

n=1


1 sin m π x sin n π xdx =

An 0

n=1

This is an “exact” solution to the linear problem. It can be evaluated to any desired accuracy by taking more and more terms, but if a finite number of terms are used, some error always occurs. For large times a single term is adequate, whereas for small times many terms are needed. For small times the Laplace transform method is also useful, because it leads to solutions that converge with fewer terms. For small times, the method of combination of variables may be used as well. For nonlinear problems, the method of separation of variables fails and one of the other methods must be used. The method of combination of variables is useful, particularly when the problem is posed in a semi-infinite domain. Here, only one example is provided; more detail is given in [114– 116] The method is applied here to the nonlinear problem     ∂c ∂ ∂c ∂ 2 c d D (c) ∂c 2 + = D (c) = D (c) ∂t ∂x ∂x ∂x2 dc ∂x

c (x, 0) = 0

The transformation combines two variables into one c (x, t) = f (η) where η = √

Am 2

x 4 D0 t

The use of the 4 and D0 makes the analysis below simpler. The equation for c (x, t) is transformed into an equation for f (η) ∂c d f ∂η = , ∂t dη ∂t ∂2c d2 f = 2 ∂x dη 2

0 ∞ 

π 2 Dt

c (0, t) = 1, c (∞, t) = 0 An sin n π x = x − 1

n=1

=

2

An sin n π x e−n

with boundary and initial conditions

The constants An are chosen by making u (x, t) satisfy the initial condition. ∞ 

∞ 

π 2 Dt

n=1

u (x, 0) =

c (x, t) = 1 − x +

∂c d f ∂η = ∂x dη ∂x 

∂η ∂x

2

∂η x/2 , = −∂t 4 D0 t3

+

d f ∂2η dη ∂x2

∂η 1 = √ , ∂x 4 D0 t

∂2η = 0 ∂x2

Mathematics in Chemical Engineering

the theory presented in Chapter 6. The equations are written as

The result is 

d df K (c) dη dη

 + 2η

101

df = 0 dη

n+1 dci D  ci+1 − 2 ci + ci−1 = Bij cj = D 2 2 dt ∆x ∆x j=1

K (c) = D (c) /D0

The boundary conditions must also combine. In this case the variable η is infinite when either x is infinite or t is zero. Note that the boundary conditions on c (x, t) are both zero at those points. Thus, the boundary conditions can be combined to give

where the matrix B is tridiagonal. The stability of the integration of these equations is governed by the largest eigenvalue of B. If Euler’s method is used for integration,

f (∞) = 0

The largest eigenvalue of B is bounded by the Gerschgorin theorem [119, p. 135].

The other boundary condition is for x = 0 or η = 0,

∆t

D 2 ≤ ∆x2 |λ|max

|λ|max ≤ max2 0)

In solving a differential equation for f (x, t), for example, the function f is expanded in Chebyshev polynomials ∞ 

∂c = C Q−1 c ≡ A c ∂x

f =

Now both c and ∂c/∂x are functions of time, but the matrix A is constant in time. For the time derivatives,

Various derivatives are also expanded in Chebyshev polynomials. For differential operator L, formally

dc (xj , t) dcj ∂c  =  = ∂t xj dt dt

Lf =

an Tn , |x| ≤ 1

n=0

∞ 

bn Tn , |x| ≤ 1

n=0

Thus, for diffusion problems dcj = dt

N +2 

Bji ci , j = 2, . . . , N + 1

For specific cases [121, p. 160] the relations between the coefficients of the function (an ) and the derivatives (bn ) are given by

i=1

This can be integrated by using the standard methods for ordinary differential equations as initial value problems. Stability limits for explicit methods are available [114, p. 204]. The method of orthogonal collocation on finite elements can also be used, and details are provided elsewhere [114, pp. 228 – 230]. An application to chemical reactors, where the radial direction is handled by using the method of orthogonal collocation on finite elements and the axial direction is handled by using the method of lines (PDECOL), is given by Pirkle et al. [120]. Spectral methods employ Chebyshev polynomials and the fast Fourier transform, and are

Lf =

df , cn bn = 2 dx

Lf =

d2 f , cn bn = dx2

∞ 

p ap

p=n+1 p+nodd

∞ 

p p2 − n2 ap

p=n+2 p+neven

Derivatives can be evaluated even more efficiently by using a recursion relation. When Sn =

∞ 

p ap

p=n+1 p+nodd

the following can be used [121, p. 117]:

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Mathematics in Chemical Engineering

Sn = Sn+2 + (n + 1) an+1 , 0 ≤ n ≤ N − 1 SN = 0, SN +1 = 0

This recurrence relation is assured by 1 dTn+ 1 1 dTn−1 − , n>1 n + 1 dx n − 1 dx

2 Tn =

In the Chebyshev collocation method, N + 1 collocation points are used xj = cos

πj , j = 0, 1, . . . , N N

As an example, consider the equation ∂u ∂u + f (u) = 0 ∂t ∂x

An explicit method in time can be used un+1 − un ∂u n + f (un )  = 0 ∆t ∂x

|λ|max =

and evaluated at each collocation point un+1 − un j j ∆t

∂u n + f un  = 0 j ∂x j

N 

ap (t) cos

p=0

π pj n , un j = uj (t ) N

(46)

Assume that the values unj exist at some time. Then invert Equation (46) using the fast Fourier transform to obtain {ap } for p = 0, 1, . . . , N; then calculate Sp Sp = Sp+2 + (p + 1) ap+1 , 0 ≤ p ≤ N − 1 SN = 0, SN +1 = 0

and finally (1)

ap

=

LB ∆x2

(47)

where the values of LB are as follows:

The trial function is taken as uj (t) =

collocation point the solution can be advanced forward to the n + 1-th time level. The advantage of the spectral method is that it is very fast and can be adapted quite well to parallel computers. It is, however, restricted in the geometries that can be handled. Software packages exist that use various discretizations in the spatial direction and an integration routine in the time variable: PDECOL uses B-splines for the spatial direction and various GEAR methods in time [122, p. 346]; PDEPACK and DSS [122, p. 351] use finite differences in the spatial direction and GEARB in time [123, p. 163]; and REACOL [114, p. 191] use orthogonal collocation in the radial direction and LSODE in the axial direction, whereas REACFD uses finite difference in the radial direction (both codes are restricted to modeling chemical reactors). The maximum eigenvalue for all the methods is given by

2 Sp cp

Thus, the first derivative is given by N  ∂u  π pj (1) ap (t) cos  = ∂x j N p=0

This is evaluated at the set of collocation points by using the fast Fourier transform again. Once the function and the derivative are known at each

Finite difference Galerkin, linear elements, lumped Galerkin, linear elements Galerkin, quadratic elements Orthogonal collocation on finite elements, cubic

4 4 12 60 36

8.4. Elliptic Equations Elliptic equations can be solved with both finite difference and finite element methods. Onedimensional elliptic problems are two-point boundary value problems and are covered in Chapter 7. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and with direct methods when the finite element method is used. Thus, two aspects must be considered: how the equations are discretized to form sets of algebraic equations and how the algebraic equations are then solved. The prototype elliptic problem is steady-state heat conduction or diffusion,  k

∂2T ∂2T + 2 ∂x ∂y 2

 = Q

Mathematics in Chemical Engineering possibly with a heat generation term per unit volume, Q. The boundary conditions can be Dirichlet or 1st kind: Neumann or 2nd kind:

k ∂T ∂n = q2 on boundary S 2

Robin, mixed, or 3rd kind:

−k ∂T ∂n = h (T − T3 ) on boundary S 3

from low to high j, the Gauss – Seidel method can be used: 

T = T 1 on boundary S 1

2

Ti,j+1 − 2 Ti,j + Ti,j−1 = Qi,j ∆y 2

(48)

Ti,j = T1 for i, j on boundary S1 ∂T   = q2 for i, j on boundary S2 ∂n i,j

∂T  −k  = h (Ti,j − T3 ) for i, j on boundary S3 ∂n i,j

If the boundary is parallel to a coordinate axis the boundary slope is evaluated as in Chapter 7, by using either a one-sided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line, more complicated expressions are needed and the finite element method may be the better method. Equation (48) is rewritten in the form  1+

∆x2 ∆y 2

 Ti,j = Ti+1,j + Ti− 1,j +

(Ti,j+1 + Ti,j−1 ) − ∆x2

2

1+

T

 ∆x2 ∆y 2

s i,j+1

∆x2 ∆y 2

Qi,j k

+T

1+

∆x2 ∆y 2



T ∗i,j = T si+1,j + T s+1 i−1,j +



− ∆x



∆x2 ∆y 2

Qi,j k

s

− ∆x2

If β = 1, this is the Gauss – Seidel method. If β > 1, it is overrelaxation; if β < 1, it is underrelaxation. The value of β may be chosen empirically, 0 < β < 2, but it can be selected theoretically for simple problems like this [124, p. 100], [114, p. 282]. In particular, the optimal value of the iteration parameter is given by − ln (βopt − 1) ≈R

and the error (in solving the algebraic equation) is decreased by the factor (1 − R)N for every N iterations. For the heat conduction problem and Dirichlet boundary conditions, R =

π2 2 n2

(when there are n points in both x and y directions). For Neumann boundary conditions, the value is R =

1 π2 2 n2 1 +max [∆x2 /∆y 2 , ∆y 2 /∆x2 ]

These iterative methods are point iterative methods. Line iterative methods [124, p. 113] use 

s s T s+1 i,j = T i+1,j + T i−1,j +

s i,j−1

 Qi,j ∆x2  s T i,j+1 + T s+1 − ∆x2 i,j−1 2 ∆y k

 2

The Jacobi method is 

s+1 s T s+1 i,j = T i+1,j + T i−1,j

∗ s T s+1 i,j = T i,j + β T i,j − T i,j

Ti+1,j − 2 Ti,j + Ti−1,j ∆x2

2



T si,j+1 + T s+1 i,j−1

The finite difference formulation is then

k

∆x2 ∆y 2

This method converges twice as fast as the Jacobi method. The relaxation method uses

Ti,j = T (i ∆x, j ∆y)

+

1+

+

Illustrations are given for constant physical properties k, h, while T 1 , q2 , T 3 are known functions on the boundary and Q is a known function of position. The finite difference formulation is given by using the following nomenclature

105

∆x2

2

1+

∆x2 ∆y 2



T ∗i,j = T ∗i+1,j + T ∗i−1,j

∆y 2

2 Qi,j

k

If the points are located in a regular order, and the calculations proceed from low to high i, then

Qi,j ∆x2 s T i,j+1 + T si,j−1 − ∆x2 k ∆y 2

s ∗ s T s+1 i,j = T i,j + β T i,j − T i,j +

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Mathematics in Chemical Engineering

The last equation is applied after the solution for the entire line (current value of i). Now the value of R is R =

2π n

1+


FIe =

J

AeIJ T eJ =

h3 NI NJ dC


NI QdA +


NI q2 dC −

C2

NI h3 T3 dC C3

Also, a necessary condition is that

for Neumann boundary conditions. Line Jacobi converges twice as fast as point Jacobi (when ∆x = ∆y), and line successive overrelaxation converges twice as fast as point successive overrelaxation [124, pp. 110, 114]. The alternating direction method can be employed for elliptic problems by using sequences of iteration parameters [114, pp. 283 – 286], [124, pp. 120 – 127]. The method is well suited to transient problems. These are the classical iterative techniques. Recently, preconditioned conjugate gradient methods have been developed (see Chap. 1). In these methods a series of matrix multiplications are done iteration by iteration; and the steps lend themselves to the efficiency available in parallel computers. In the multigrid method the problem is solved on several grids, each more refined than the previous one. In iterating between the solutions on different grids, one converges to the solution of the algebraic equations. A chemical engineering application is given in [125]. Another way to solve the elliptic equations is to convert them to parabolic equations and integrate them to “steady state”. One method of doing this, operator splitting, is illustrated for the Navier – Stokes equations in Section 8.5. The Galerkin finite element method (FEM) is useful for solving elliptic problems and is particularly effective when the domain or geometry is irregular. As an example, cover the domain with triangles and define a trial function on each triangle. The trial function takes the value 1.0 at one corner and 0.0 at the other corners, and is linear in between (see Fig. 36). These trial functions on each triangle are pieced together to give a trial function on the whole domain. General treatments of the finite element method are available [126–128]. For the heat conduction problem the method gives [114] e


k∇NI ·∇NJ dA − C3

∆y ∆x2

2π n




AeIJ = −

2 1/2



for Dirichlet boundary conditions; R =

where

 e

J

FIe

(49)

Ti =T1 on C1

In these equations I and J refer to the nodes of the triangle forming element e and the summation is made over all elements. These equations represent a large set of linear equations. They are solved simultaneously, with only one iteration, by using either the frontal solution method [129] or various sparse matrix techniques (Chap. 1).

Figure 36. Finite elements trial function: linear polynomials on triangles

If the problem is nonlinear, e.g., with k a function of temperature,     ∂ ∂T ∂ ∂T k (T ) + k (T ) = Q (T ) ∂x ∂x ∂y ∂y

then the equations must be solved iteratively. Let T sI be the known temperature at node I. Then Equation (49) can be used with
AeIJ (T e ) = −

k (T e ) ∇NI ·∇NJ dA


−

h3 NI NJ dC C3

Mathematics in Chemical Engineering
FIe (T e ) =


NI Q (T e ) dA +

NI q2 dC

C2

aI + aJ + aK = 1 bI + bJ + bK = 0


−

NI h3 T3 dC

cI + cJ + cK = 0

C3

AeIJ = −

or  e

AeIJ (T e,s ) T e,s+1 = J

 e

J

FJe (T e,s )

J

This is a successive substitution method. The Newton – Raphson method can also be used in the form  e

=

AeIJ (T e,s ) T e,s+1 J

J

+

    dAe IJ (T e,s ) T e,s+1 − T e,s K K dTK e K



 FIe (T e,s ) +

e

 dF e I dT K K

/  (T e,s ) T e,s+1 − T e,s K K

The integrals in these formulas can be calculated analytically when the physical properties are constant. The results are given for a triangle with nodes I, J, and K in counterclockwise order. Within an element, T = NI (x, y) TI + NJ (x, y) TJ + NK (x, y) TK NI =

aI + bI x + cI y 2∆

a I = x J yK − x K yJ bI = yI − yK cI = x K − x J plus permutation on I, K, J 

1 xI  2 ∆ = det 1 xJ 1 xK

107

 yI  yJ  = 2 (area of triangle) yK

e = FIJ

x ¯=

k (bI bJ + cI cJ ) 4∆

Q∆ Q ¯ + cI y¯) = (aI + bI x 2 3

yI + y J + y K xI + x J + x K , y¯ = , 3 3

¯ + cI y¯ = aI + bI x

2 ∆ 3

One nice feature of the finite element method is the use of natural boundary conditions. In this problem the natural boundary conditions are the Neumann or Robin conditions. When using Equation (49), the problem can be solved on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in [111] and [130]. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. Use of the collocation method on finite elements is discussed in [106, pp. 330 – 339]. The methods can also be combined by using one method for one direction and another method for the other direction. Klein [131, p. 603] treated two-dimensional reactors using orthogonal collocation in the radial direction and a shooting method in the axial direction. Pirkle et al. [132] were able to derive an equivalent one-dimensional model of a chemical reactor to avoid solving the two-dimensional problem. Degreve et al. [133] used an adaptive mesh calculation for combustion problems (see also the examples in [111]). A general-purpose package for general twodimensional domains and rectangular threedimensional domains is ELLPACK [116, p. 348]. This package allows choice of a variety of methods: finite difference, Hermite collocation, spline Galerkin, collocation, and others. Comparisons of the various methods are available [134]. The program FISHPAK solves

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the Helmholtz equation in multiple dimensions when the domain is separable (because fast methods like FFT are used) [122, p. 346].

8.5. Parabolic Equations in Two or Three Dimensions Computations become much more lengthy with two or more spatial dimensions, for example, the unsteady heat conduction equation ∂T  Cp = k ∂t



∂2T ∂2T + ∂x2 ∂y 2



∂2c ∂2c + ∂x2 ∂y 2

−Q



=

T n+1 − Tn i,j i,j ∆t

k n n T i+1,j − 2 T n i,j + T i−1,j ∆x2

+

k n n T i,j+1 − 2 T n i,j + T i,j−1 − Q ∆y 2

When Q = 0 and ∆x = ∆y, the time step is limited by ∆t ≤

∆x2  Cp ∆x2 or 4k 4D

These time steps are smaller than for onedimensional problems. For three dimensions, the limit is ∆t ≤

− Tn i,j

∆t/2

=

 k  n+1/2 n+1/2 n+1/2 T i+1,j − 2 T i,j + T i−1,j 2 ∆x +

k n n T i,j+1 − 2 T n i,j + T i,j−1 − Q ∆y 2

and then another ∆t/2 using an implicit method  Cp

∆x2 6D

To avoid such small time steps, which become smaller as ∆x decreases, an implicit method could be used. This leads to large sparse matrices, rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful [124, pp. 57 – 63]. This reduces a problem on an n×n grid to a series of 2 n one-dimensional problems on an n-grid. Here, step forward ∆t/2 by using the Crank – Nicolson method

T n+1 − T i,j i,j ∆t/2

=

 k  n+1/2 n+1/2 n+1/2 T i+1,j − 2 T i,j + T i−1,j 2 ∆x

− R (c)

In the finite difference method an explicit technique would evaluate the right-hand side at the n-th time level:  Cp

T i,j

n+1/2



or the unsteady diffusion equation ∂c =D ∂t

n+1/2

 Cp

+

 k  n+1 n+1 T i,j+1 − 2 T n+1 −Q i,j + T i,j−1 ∆y 2

This method is second order in ∆t errors [O (∆t 2 )] and reduces the approximate computational burden from n4 to 6 n2 . The method of operator splitting is also useful when different terms in the equation are best evaluated by using different methods or as a technique for reducing a larger problem to a series of smaller problems. Here the method is illustrated by using the Navier – Stokes equations. In vector notation the equations are 

∂u + u ·∇u =  f − ∇p + µ∇2 u ∂t

The equation is solved in the following steps 

u∗ − un = − un ·∇un + f + µ∇2 un ∆t

∇2 pn+1 =



1 ∇ ·u∗ ∆t

un+1 − u∗ = − ∇p ∆t

This can be done by using the finite difference [135, p. 162] or the finite element method [136]. In the finite element method, the need to solve large sets of equations simultaneously must be eliminated. To do this, the matrices multiplying time derivatives are approximated by moving all terms to the diagonal (called lumping). The effects of this have been examined in detail in [137].

Mathematics in Chemical Engineering

109

9. Integral Equations [138–141]
t

If the dependent variable appears under an integral sign an equation is called an integral equation; if derivatives of the dependent variable appear elsewhere in the equation it is called an integrodifferential equation. This chapter describes the various classes of equations, gives information concerning Green’s functions, and presents numerical methods for solving integral equations.

1 T (x, t) = √ π

9.1. Classification

The solution to this problem is

Volterra integral equations have an integral with a variable limit, whereas Fredholm integral equations have a fixed limit. Volterra equations are usually associated with initial value or evolutionary problems, whereas Fredholm equations are analogous to boundary value problems. The terms in the integral can be unbounded, but still yield bounded integrals, and these equations are said to be weakly singular. A Volterra equation of the second kind is

1 T (x, t) = √ π


t y (t) = g (t) + λ

K (t, s) y (s) ds

(50)

a

whereas a Volterra equation of the first kind is
t y (t) = λ

K (t, s) y (s) ds a

Equations of the first kind are very sensitive to solution errors so that they present severe numerical problems. An example of a problem giving rise to a Volterra equation of the second kind is the following heat conduction problem:  Cp

∂T ∂2T , 0 ≤ x < ∞, t > 0 = k ∂t ∂x2

T (x, 0) = 0,

∂T (0, t) = − g (t) , ∂x

lim T (x, t) = 0,

x→∞

lim

x→∞

∂T = 0 ∂x

If this is solved by using Fourier transforms the solution is

g (s) √ 0

2 1 e−x /4(t−s) ds t−s

Suppose the problem is generalized so that the boundary condition is one involving the solution T , which might occur with a radiation boundary condition or heat-transfer coefficient. Then the boundary condition is written as ∂T = − G (T, t) , x = 0, t > 0 ∂x

2

e−x

/4(t−s)


t G (T (0, s) , s) √ 0

1 t−s

ds

If T (t) is used to represent T (0, t), then 1 T (t) = √ π


t G (T (s) , s) √ 0

1 ds t−s

Thus the behavior of the solution at the boundary is governed by an integral equation. Nagel and Kluge [142] use a similar approach to solve for adsorption in a porous catalyst. The existence and uniqueness of the solution can be proved. One example is the following: Theorem [140, p. 30]. If K (t, s) is continuous in 0 ≤ s ≤ t ≤ α and g (t) is continuous in 0 ≤ t ≤ α, then the integral equation (Eq. 50) possesses a unique continuous solution for 0 ≤ t ≤ α. More general theorems are also available [140, p. 32]. The solution to the integral equation can be obtained by using a Picard or successive substitution method (this is how the existence theorem is proved).
t yn (t) = g (t) + λ

K (t, s) yn−1 (s) ds a

Sometimes the kernel is of the form K (t, s) =K (t − s)

Equations of this form are called convolution equations and can be solved by taking the Laplace transform. For the integral equation

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t


t K (t − τ ) Y (τ ) dτ

Y (t) = G (t) + λ

y (t) = y (0) +

which is a nonlinear Volterra equation. The general nonlinear Volterra equation is


t

∗

K (t) Y (t) ≡

F (s, y (s)) ds 0

0

K (t − τ ) Y (τ ) dτ 0


t y (t) = g (t) +

the Laplace transform is

K (t, s, y (s)) ds

(53)

0

y (s) = g (s) + k (s) y (s)

A successive substitution method for its solution is

k (s) y (s) =L [K (t) ∗ Y (t)]


t

Solving this for y (s) gives

yn+1 (t) = g (t) +

K [t, s, yn (s)] ds 0

g (s) y (s) = 1 − k (s)

If the inverse transform can be found, the integral equation is solved. A Fredholm equation of the second kind is

|K (t, s, y) − K (t, s, z) | ≤ L |y − z|


b y (x) = g (x) + λ

Theorem [140, p. 55]. If g (t) is continuous, the kernel K (t, s, y) is continuous in all variables and satisfies a Lipschitz condition

K (x, s) y (s) ds

(51)

a

whereas a Fredholm equation of the first kind is

then the nonlinear Volterra equation has a unique continuous solution. Nonlinear Fredholm equations have special names. The equation


b


1

K (x, s) y (s) ds = g (x)

f (x) =

a

K [x, y, f ( y)] dy 0

The limits of integration are fixed, and these problems are analogous to boundary value problems. An eigenvalue problem is a homogeneous equation of the second kind.


1 f (x) =


b y (x) = λ

is called the Urysohn equation [139, p. 208]. The special equation K [x, y] F [y, f ( y)] dy 0

K (x, s) y (s) ds

(52)

a

Solutions to this problem occur only for specific values of λ, the eigenvalues. Usually the Fredholm equation of the second or first kind is solved for values of λ different from these, which are called regular values. Nonlinear Volterra equations arise naturally from initial value problems. For the initial value problem

is called the Hammerstein equation [139, p. 209]. Iterative methods can be used to solve these equations, and these methods are closely tied to fixed point problems. A fixed point problem is x = F (x)

and a successive substitution method is xn+1 =F (xn )

dy = F (t, y (t)) dt

both sides can be integrated from 0 to t to obtain

Local convergence theorems prove the process convergent if the solution is close enough to the answer, whereas global convergence theorems

Mathematics in Chemical Engineering are valid for any initial guess [139, p. 229 – 231]. The successive substitution method for nonlinear Fredholm equations is
1 yn+1 (x) =

K [x, s, yn (s)] ds 0

Typical conditions for convergence include that the function satisfies a Lipschitz condition.

9.2. Numerical Methods for Volterra Equations of the Second Kind

dy = − λ y, y (0) = 1 dt

the stability results are identical to those for initial value methods. In particular, using the trapezoid rule for integral equations is identical to using this rule for initial value problems. The method is A-stable. The next highest quadrature method is Simpson’s rule. When expressed for three points, each ∆t apart, this is t
i+2

y (s) ds = ti

Volterra equations of the second kind are analogous to initial value problems. An initial value problem can be written as a Volterra equation of the second kind, although not all Volterra equations can be written as initial value problems [140, p. 7]. Here the general nonlinear Volterra equation of the second kind is treated (Eq. 53). The simpliest numerical method involves replacing the integral by a quadrature using the trapezoid rule. ( yn ≡ y (tn ) = g (tn ) +∆t

+

n−1 

K (tn , ti , yi ) +

i=1

1 K (tn , t0 , y0 ) 2

1 K (tn , tn , yn ) 2

This equation is a nonlinear algebraic equation for yn . Since y0 is known it can be applied to solve for y1 , y2 , . . . in succession. For a single integral equation, at each step one must solve a single nonlinear algebraic equation for yn . Typically, the error in the solution to the integral equation is proportional to ∆t µ , and the power µ is the same as the power in the quadrature error [140, p. 97]. The stability of the method [140, p. 111] can be examined by considering the equation
t y (t) = 1 − λ

y (s) ds 0

whose solution is y (t) = e−λt

Since the integral equation can be differentiated to obtain the initial value problem

111

∆t [ y (ti ) + 4 y (ti+1 ) + y (ti+2 )] 3

ti+1 = ti +∆t, ti+2 = ti + 2∆t

When applied to an integral equation in the first two intervals, yi+2 = g (ti+2 ) +

∆t {K (ti+2 , ti , yi ) 3

+ 4 K (ti+2 , ti+1 , yi+1 ) + K (ti+2 , ti+2 , yi+2 )}

This is the equation for yi+2 , but now there is no equation for yi+1 . This is obtained by subdividing the region from ti to ti+1 with another point at ti+1/2 . Then an integral equation can be written as yi+1 = g (ti+1 ) +

∆t {K (ti+1 , ti , yi ) 3

+ 4 K ti+1 , ti+1/2 , yi+1/2 + K (ti+1 , ti+1 , yi+1 )}

To obtain yi+1/2 a quadratic interpolation is used through the original three points: yi+1/2 =

3 3 1 yi + yi+1 − yi+2 8 4 8

Now two equations must be solved for the two points yi+1 and yi+2 . In this block-by-block method [140, p. 114], the solution proceeds to solve for two unknowns at a time, block by block. Explicit Runge – Kutta methods have an analogous formula for integral equations. When solving nonstiff differential equations, Runge – Kutta methods are fast and efficient until the time step is very small to meet the stability

112

Mathematics in Chemical Engineering

requirements. Then implicit methods are used, even though the set of simultaneous algebraic equations must be solved. This time-consuming step can be justified only for stiff problems. When solving integral equations, however, the time-consuming part of the calculation is the repeated approximation of the integrals; thus the effort needed to solve the algebraic equations is not as large a fraction of the total effort for integral methods. Thus, implicit methods, like the trapezoid rule given above, tend to be preferred for integral equations [140, p. 124]. Predictor-corrector methods can be used, however. Using fourth-order Adams – Moulton and Adams – Bashforth methods gives y¯n+1

∆t = gn+1 + [55 K (tn+1 , tn , yn ) 24

functional form of p is included in the quadrature rule. For example, the integral
b I =

p (s) ψ (s) ds a

where p is singular at some points. Each quadrature method corresponds to some method of interpolating the solution. That interpolation is written as ψ (t) =



ai ψi (t)

Integrals of the form
b p (s) ψi (s) ds a

− 59 K (tn+1 , tn−1 , yn−1 )

− 5 K (tn+1 , tn−1 , yn−1 )

must be integrated exactly. Then methods can be constructed to handle weakly singular problems. The product trapezoid rule and product Simpson’s rule are contained in the following algorithm. Subdivide the interval into n equal subintervals of length ∆t, having points at t 0 , t 1 , . . . , tn . Within each subinterval, introduce a further subdivision with points ti ≤ ti0 < ti1 < . . . tim ≤ ti+1 . Approximate the function with a polynomial of degree m in each subinterval using a Lagrangian interpolation (see Section 2.2).

+ K (tn+1 , tn−2 , yn−2 )]

ψ (t) =

+ 37 K (tn+1 , tn−2 , yn−2 ) − 9 K (tn+1 , tn−3 , yn−3 )]

and yn+1 = gn+1 +

∆t [9 K (tn+1 , tn+1 , y¯n+1 ) 24

+ 19 K (tn+1 , tn , yn )

m 

lij (t) ψ (tij ) , ti0 ≤ t ≤ tim

i=0

lij =

(t − ti0 ) . . . (t − ti,j−1 ) (t − ti,j+1 ) . . . (t − tim ) (tij − ti0 ) . . . (tij − ti,j−1 ) (tij − ti,j+1 ) . . . (tij − tim )

The next situation that must be addressed is how to solve problems when the kernel is infinite at certain points, i.e., when the problem has a weak singularity. Clearly the formulas given above would be unsuitable when the kernel is infinite at the quadrature point. To handle problems like this, product equations are employed. The problem is written as

I =

n−1 m 

where t
i+2

where the kernel K is Lipschitz continuous and the singular behavior is contained in p. Then, the

p (s) lij (s) ds ti

p (t, s) K [t, s, y (s)] ds 0

wij ψ (tij )

i=0 j=0

wij =


t y (t) = g (t) +

Inserting this approximation into the integral gives

If the following integrals can be calculated
sµ p (s) ds, µ = 0, 1, . . . , m

Mathematics in Chemical Engineering then the weights can be found explicitly. An example is the function as piecewise constant [140, p. 132]. Then the integrals are

113

where αn,i+1 =

t
i+1

1 ∆t

t
i+1

(ti+1 − s) p (tn , s) ds, ti

K [tn , s, y (s)] p (s) ds = ti

βn,i+1 = t
i+1

K [tn , ti , y (ti )]

p (s) ds ti

In using this expression the numerical form of the integral equation is y n = gn +

n−1 

wni K [tn , ti , yi ] , n = 1, 2, . . .

where the weights are t
i+1

p (s) ds ti

When the function p (s, t) = (s − t)−1/2 , the weights are wni = 2

/ .√ tn − ti − tn − ti+1

By starting with y0 = g (0) successive values of y1 , y2 , etc. can be calculated. This corresponds to the Euler method of integration and is only accurate to O (∆t). To improve the accuracy, the Richardson extrapolation techniques described in Section 6.2 can be used. Next, consider the product trapezoid rule [140, p. 135]. The kernel is approximated by piecewise linear functions K [t, s, y (s)] =

+

t
i+1

(s − ti ) p (tn , s) ds ti

The numerical version of the integral equation is yn = gn + αn K [tn , t0 , y (t0 )]

+

N −1 

(αn,i+1 + βni ) K [tn , ti , y (ti )]

i=1

i=0

wni =

1 ∆t

s − ti K [t, ti+1 , y (ti+1 )] ∆t

ti+1 − s K [t, ti , y (ti )] , ti ≤ s ≤ ti+1 ∆t

+ βnn K [tn , tn , y (tn )]

Higher order methods can be solved block by block in a similar fashion. Volterra equations of the first kind are not well posed, and small errors in the solution can have disastrous consequences [140, p. 71]. Only lowest order methods have been recommended for problems of the first kind [140, p. 151].

9.3. Numerical Methods for Fredholm, Urysohn, and Hammerstein Equations of the Second Kind Whereas Volterra equations could be solved from one position to the next, like initial value differential equations, Fredholm equations must be solved over the entire domain, like boundary value differential equations. Thus, large sets of equations will be solved and the notation is designed to emphasize that. The methods are also based on quadrature formulas. For the integral
b

Then the integration formula is I (ϕ) =


tn

ϕ ( y) dy a

p (tn , s) K [tn , s, y (s)] ds = αn1 K [tn , t0 , y (t0 )] 0

+

N −1 

(αn,i+1 + βni ) K [tn , ti , y (ti )]

a quadrature formula is written: I (ϕ) =

n 

wi ϕ ( y i )

i=0

i=1

+ βnn K [tn , tn , y (tn )]

Then the integral Fredholm equation can be rewritten as

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Mathematics in Chemical Engineering

f (x) − λ

n 


b

wi K (x, yi ) f ( yi ) = g (x) ,


b

i=0 a

a ≤ x ≤ b

(54)

If this equation is evaluated at the points x = yj , f ( yj ) − λ

n 

w i K ( yj , y i ) f ( y i ) = g ( y i )

is obtained, which is a set of linear equations to be solved for { f ( yj )}. The solution at any point is then given by Equation (54). Because the quadrature method is often defined on subintervals, the equation is rewritten in terms of information obtained on the subintervals. The finite element notation is a natural notation. On a subinterval with points yI =1 , yI =2 , . . . the quadrature formula is x
i+2


1 ϕ ( y) dy = ∆x

xi

ϕ (u) du = 0


b


b K (x, y) f (x) dy =

+

K (x, y) [f (y)

NP 

wI ϕ ( yI )

I=1

 . 0 . 0    . .  . wn

0 w2 . 0

a

−f (x)] dy + f (x) H (x)

where
b H (x) =

K (x, y) f (x) dy a

is a known function. The integral equation is then replaced by f (x) = g (x) +

n 

wi K (x, yi ) [ f ( yi )

i=0

An element diagonal matrix is then constructed: w1 0  D= . 0

a

a

i=0



K (x, y) [f (y) − f (x)] dy

K (x, y) f (y) dy=

− f (x)] + f (x) H (x)

In matrix notation this is (I − λM ) f = g

and the matrix M is the same as K D except for the diagonal elements, which are 

and the element kernel

Mii = Hi −

e KiJ = K (xi , xe +∆ xe uJ )

The product quadrature can be used to handle weak singularities, as with Volterra equations (see [138, p. 540] for an example). Collocation methods can be applied as well [138, p. 396]. To solve integral Equation (51) expand f in the function

The numerical version of the integral equation is then fi − λ

NP 

e e KiJ DJJ f eJ = gi

e J=1

In matrix notation this can be written as

f=

n 

Kik wk

ai ϕi (x)

i=0

(I −λK D) f =g

The structure of this matrix is typically dense, because the kernel is nonzero even when the points x and y are in different elements. Thus, the economies obtained by using sparse matrix routines to solve differential equations do not always hold for integral equations. A common type of integral equation has a singular kernel along x = y. This can be transformed to a less severe singularity by writing

Substitute f into the equation to form the residual n 

ai ϕi (x) − λ

i=0

n  i=0


b K (x, y) ϕi ( y) dy

ai a

= g (x)

Evaluate the residual at the collocation points

Mathematics in Chemical Engineering

n 

ai ϕi (xj ) − λ

i=0

n  i=0

n 


b K (xj , y) ϕi ( y) dy

ai

115

w i K ( yi , y i ) f ( y i ) = λ f ( y j ) ,

i=1

a

i = 0, 1, . . . , n = g (xj )

leads to the matrix eigenvalue problem

The expansion can be in piecewise polynomials, leading to a collocation finite element method, or global polynomials, leading to a global approximation. If orthogonal polynomials are used then the quadratures can make use of the accurate Gaussian quadrature points to calculate the integrals. Galerkin methods are also possible [138, p. 406]. Mills et al. [143] consider reaction – diffusion problems and say the choice of technique cannot be made in general because it is highly dependent on the kernel. When the integral equation is nonlinear, iterative methods must be used to solve it. Convergence proofs are available, based on Banach’s contractive mapping principle. Consider the Urysohn equation, with g (x) = 0 without loss of generality:

9.5. Green’s Functions Integral equations can arise from the formulation of a problem by using Green’s function. For example, the equation governing heat conduction with a variable heat generation rate is represented in differential forms as d2 T Q (x) = , T (0) = T (1) = 0 dx2 k

In integral form the same problem is [138, pp. 57 – 60] 1 k

T (x) =


b f (x) =

K D f =λf

F [x, y, f ( y)] dy a


1 G (x, y) Q ( y) dy 0

 −x (1 − y) −y (1 − x)

x ≤ y y ≤ x

The kernel satisfies the Lipschitz condition

G (x, y) =

maxa≤x,y≤b |F [x, y, f ( y)] − F [x, z, f (z)] |

Green’s functions for typical operators are given below. For the Poisson equation with solution decaying to zero at infinity

≤ K | y − z|

Theorem [139, p. 214]. If the constant K is < 1 and certain other conditions hold, the successive substitution method

the formulation as an integral equation is


ψ (r) =


b fn+1 (x) =

∇2 ψ = − 4π 

F [x, y, fn ( y)] dy, n = 0, 1, . . . a

converges to the solution of the integral equations.

 (r 0 ) G (r, r 0 ) dV0 V

where Green’s function is [144, p. 891] G (r, r 0 ) =

1 in three dimensions r

= − 2 ln r in two dimensions

9.4. Numerical Methods for Eigenvalue Problems Eigenvalue problems are treated similarly to Fredholm equations, except that the final equation is a matrix eigenvalue problem instead of a set of simultaneous equations. For example,

2 where r =

(x − x0 )2 + ( y − y0 )2 + (z − z0 )2

in three dimensions and r =

2 (x − x0 )2 + ( y − y0 )2

in two dimensions

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For the problem

Green’s function for the differential operators are [146]

∂u = D ∇2 u, u = 0 on S, ∂t

a = 1

with a point source at x0 , y0 , z0

Green’s function is [145, p. 355] u =

e

1



2

−[(x−x0 ) +(y−y0 ) +(z−z0 )

2

]/4D(t−τ )

When the problem is

G (x, y, Sh) =

 G (x, y, Sh) =

c = f (x, y, z) in S at t = 0 c = ϕ (x, y, z) on S, t > 0

the solution can be represented as [145, p. 353]


c =

K (x, y) =
t

ϕ (x, y, z, τ ) 0

∂u dSdt ∂n

When the problem is two dimensional, 2 2 1 u = e−[(x−x0 ) +(y−y0 ) ]/4D(t−τ ) 4 π D (t − τ )

(u)τ =0 f (x, y) dxdy
t
ϕ (x, y, τ ) 0

∂u dCdt ∂n

For the following differential equation and boundary conditions d dx

2 Sh 2 Sh

+ +

y ≤ x x 0, because then the initial basis to be selected can be the slack variables for which B = I. Otherwise, one has to resort to the use of artificial variables and a twophase procedure [181]. Second, the optimality condition is verified by checking if

such that − z N ≤ 0

From the Kuhn – Tucker conditions presented earlier, −1 T T −cT N + cT BB N − µ = 0

−z N ≤ 0, µ ≥ 0 , µT z N = 0

Thus, because the multipliers µ ≥ 0, Equation (71) holds. 10.5.2. Simplex Algorithm The simplex algorithm [182] exploits the basic properties described in the previous section by searching only among the vertex points in the feasible space. It verifies whether Equation (73) holds to identify the optimum solution. Rather than enumerating all possible vertices, the algorithm proceeds by moving sequentially from one vertex to another, ensuring that the objective function value decreases. To explain the essence of the simplex algorithm, rewrite the objective function of the equation F − cT z = 0

Then, for a given selection of the basis, the problem can be presented through the following tableau [see the above equations and Equation (72)]:

T

−1 cT ni − cT BB Ni ≤ 0

for each nonbasic variable zNi . If this condition is not satisfied, move to a new vertex which requires the selection of a new basis. This is accomplished as follows: 1) Introduce in the basis the nonbasic variable zNj with largest positive value for the above optimality condition. This ensures a move to a neighboring vertex that will lead to the largest decrease in the objective per unit of distance moved. 2) Remove from the basis the basic variable zBr that will become zero as a step is taken along the j-th component of the reduced gradient. The r-th basic variable can be determined by computing the smallest step size a from among all the basic variables zBi :

Mathematics in Chemical Engineering

a = min {bi /yij } = br /yrj ≥ 0

where yij is the i-th in row in the vector B−1 nj which corresponds to the j-th column of the nonbasic variables. 3) Pivot on the r-th row and the j-th column of the tableau to obtain the identity matrix for the basic variables. To illustrate the application of the simplex algorithm, construct the tableau for the example problem. From Equation (69), F

z1

z2

z3

z4

Right-hand side

1

1

3

0

0

0

0

1

1

1

0

4

0

1

2

0

1

6

If the initial basis z3 and z4 is chosen, the point is feasible because z1 = z2 = 0 (nonbasic) and z3 = 4, z4 = 6 (vertex A in Fig. 41). Furthermore, because the matrix in the basis corresponds to the identity matrix, the tableau has the form of (T 2). The entries in the first row under z1 and z2 correspond to the negative of the reduced gradient. Select z2 as the variable to enter the basis because it has the largest negative reduced gradient, − 3. Now compute the step size a: a = min {4/1, 6/2} = 6/2 = 3

Because this corresponds to the right-hand side element that is the value of z4 , select z4 as the variable to leave the basis. The second constraint will become active as its slack variable z4 is set to zero when it becomes nonbasic. Notice from Figure 41 that a = 3 along the direction (0, 1) corresponds to the boundary of the second constraint; a = 4 would lead to the boundary of the first constraint, which would then violate the second constraint. By selecting z2 as the variable to enter the basis and z4 as the one to leave it, one pivots on the entry with the value of “2” in the column beneath z2 in the bottom row above. This then leads to the following new tableau:

133

F

z1

z2

z3

z4

−9

−1/2

0

0

−3/2

Right-hand side

0

1/2

0

1

−1/2

1

0

1/2

1

0

1/2

3

Then, because the elements of the negative of the reduced gradient (− 1/2, − 3/2) (variables z1 and z4 are the nonbasic variables) are negative, stop. Thus, the optimal solution is given at z1 = 0, z2 = 3, and z3 = 1 (slack for first constraint), z4 = 0 (second constraint is active) with F = − 9. Finally, note that most commercial codes for LP (MPSX, SCICONIC, MINOS) are based on extensions of the simplex algorithm and can typically handle problems with up to 15 000 constraints. The new interior point methods (see [159] for a review) seem to offer the potential of solving larger problems more efficiently.

10.6. Mixed-Integer Programming Up to this point, this chapter has assumed that only continuous variables are involved in the optimization problems. A number of important applications, however, require that all or a subset of the variables be constrained to take only integer or discrete values. Simple examples occur in modeling the number of batches to be produced or discrete sizes to be selected for a piece of equipment. Another example is modeling the selection of process units in a flow sheet (i.e., yes or no decisions). When an optimization problem involves both discrete and continuous variables, it is called a mixed-integer programming problem. This section considers the case in which the objective function and constraints are linear (MILP) first and then the case in which nonlinearities are involved (MINLP). Also, for convenience, assume that all the discrete variables are of the 0 – 1 type. Mixed-Integer Linear Programming (MILP). Assume a linear programming problem in which a subset of the variables y is restricted to take only 0 or 1 values. This then gives rise to the MILP problem:

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min F = cT z + bT y such that Az + By ≤ d z≥0, y ∈ {0, 1}

(74)

A first approach to solve this MILP problem is to solve a linear programming problem for every combination of 0 – 1 variables and pick as the solution the 0 – 1 combination with the lowest value for the objective function. The major drawback with such an approach is that the number of 0 – 1 combinations can be very large. For example, an MILP problem with ten 0 – 1 variables would require the solution of 210 = 1024 linear programs, whereas a problem with fifty 0 – 1 variables would require the solution of 250 = 1.13×1015 programs. This approach is, in general, computationally infeasible. A second alternative is to relax the 0 – 1 requirements and treat the variables y as continuous with bounds 0 ≤ y ≤ 1. The problem with such an approach, however, is that except for few special cases (e.g., assignment problems), there is no guarantee that the variables y will take integer values at the relaxed LP solution. As an example, consider the pure integer program min F = − 1.2y1 −y2 such that 1.2y1 + 0.5y2 ≤ 1 y1 , y2 = 0.1

By relaxing y1 and y2 to be continuous the solution yields y1 = 0.715, y2 = 0.285, F = 1.148. It might then be tempting to simply round the variables to the nearest integer value, namely y1 = 1, y2 = 0. This point is an infeasible solution because it violates the first constraint. In fact, the optimal solution is y1 = 0, y2 = 1, F = − 1. Thus, solving the MILP problem by relaxation of the y variables will in general not lead to the correct solution. Note, however, that the relaxed LP has the property that its optimal objective value provides a lower bound to the integer solution. To obtain the solution to Equation (73), the most common approach is the branch and bound method [183], where the basic objective is to perform an enumeration without having to examine all the 0 – 1 combinations. The basic idea is to represent all the 0 – 1 combinations through a

binary tree such as the example shown in Figure 42. At each node of the tree the solution of the linear program is considered subject to integer constraints for the subset of the y variables that are fixed in previous branches. For example, in node A the root of the tree involves the solution of the relaxed LP, whereas node B involves the solution of the LP with fixed y1 = 0, y2 = 1 and with 0 ≤ y3 ≤ 1.

Figure 42. Binary tree for three 0 – 1 variables

To avoid enumeration of all the nodes in the binary tree, the following basic properties can be exploited. Let k denote a descendent node of another node l in the tree (e.g., k = B, l = A) and let (P k ) and (P l ) denote the corresponding LP subproblems. Then the following properties can be easily established: 1) If (P l ) is infeasible then (P k ) is also infeasible. 2) If (P k ) is feasible than (P l ) is also feasible, and (F l )∗ ≤ (F k )∗ . That is, the optimal value of the objective of subproblem (P l ) corresponds to a lower bound of the optimal value of the objective for subproblem (P k ). 3) If the optimal solution of subproblem (P k ) is such that all y = 0 or 1, then (F k )∗ ≥ F ∗ . That is, the optimal objective of subproblem (P k ) corresponds to an upper bound of F ∗ , the optimal MILP solution. These properties can be used to fathom nodes in the tree within an enumeration procedure. The question of how actually to enumerate the tree involves the use of branching nodes. It is not necessary to follow the order of the index of the variables y for branching as might be implied in Figure 42. A simple alternative is to branch instead on the 0 – 1 variable that is closest to 0.5. Alternatively, a priority ordering for the 0 – 1 variables can be specified, or else a more sophisticated scheme can be used, such as the penalties described in [184]. After solving the

Mathematics in Chemical Engineering LP at a node in the tree, decide which node to examine next. Here the two primary alternatives are to use a depth-first (last in – first out) or a breadth-first (best second rule) enumeration. In the former case, one of the branches of the most recent node is expanded first: if all of them have been examined, backtrack to another node. In the latter case, the two branches of the node with the lowest bound are expanded successively; in this case, no backtracking is required. Although the depth-first enumeration requires less storage, the breadth-first enumeration generally requires examination of fewer nodes. In summary, the branch and bound method consists of first solving the relaxed LP problem. If all the integer variables y take integer values, stop. Otherwise, proceed to enumerate the nodes in the tree according to some prespecified branching rules. At each node the corresponding LP subproblem is solved. (Typically the dual problem is updated because it requires less work. The dual problem for an LP is not discussed in this chapter.) By making use of the properties stated above, either the node is fathomed (i.e., terminate looking at it and all nodes emanating from it) if the LP for it is infeasible or if its lower bound is greater or equal to its upper bound) or it is kept open for further examination. As an example, consider the following MILP problem involving one continuous variable and three 0 – 1 variables: min F = z +y1 + 3y2 + 2y3 such that − z + 3y1 + 2y2 + y3 ≤ 0 − 5y1 − 8y1 − 3y3 ≤ −9 z ≥ 0, y1 , y2 , y3 = 0.1

The branch and bound enumeration using a breadth-first enumeration is shown in Figure 43, where the number in the circles represents the order in which 9 out of the 15 nodes in the tree are examined to find the optimum. Note that the relaxed solution (node 1) has a lower bound of F = 5.8 and that the optimum is found in node 9 where F = 8, y1 = 0, y2 = y3 = 1, and z = 3.

135

Figure 43. Branch and bound search

Mixed-Integer Nonlinear Programming (MINLP). Consider now the case of the nonlinear optimization problem involving both continuous and 0 – 1 variables (MINLP, Eq. 75) min F = F (z, y) such that h (z, y) = 0 g (z, y) ≤ 0 z∈Ren y ∈ {0, 1}m

(75)

This mixed-integer nonlinear program can in principle also be solved with the branch and bound method presented in the previous section. The major difference here would be that the examination of each node would require the solution of a nonlinear program rather than the solution of an LP. Provided the solution of each NLP subproblem is unique, similar properties as in the case of the MILP would hold with which the rigorous global solution of the MINLP can be guaranteed. An important drawback of the branch and bound method for MINLP is that solution of the NLP subproblems is much more expensive, and they cannot be updated readily as in the case of the MILP. Therefore, to reduce the computational expense involved in solving many NLP subproblems, two other methods can be used: generalized Benders decomposition [185] and outer-approximation [186]. In both these methods the MINLP (Eq. 75) is assumed to be linear in the 0 – 1 variables and

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Mathematics in Chemical Engineering

nonlinear in the continuous variables z; that is, MINLP (Eq. 76) min F = f (z) +cT y such that h (z) = 0 g (z) + B y ≤ 0 z ∈ Ren , y ∈ {0, 1}m

(76)

The basic idea in both methods is to solve an alternating sequence of NLP subproblems and MILP master problems. The NLP subproblems are solved by optimizing the continuous variables z for a given fixed value of y, and their solution yields an upper bound to the optimal solution of Equation (76). The MILP master problems consist of a linear approximation that is refined as iterations proceed, and they have the objective of predicting new values of the binary variables y as well as a lower bound on the optimal solution. The alternate sequence of NLP subproblems and MILP master problems is continued up to the point where the predicted lower bound of the MILP master is greater than or equal to the best upper bound obtained from the NLP subproblems. The MILP master problem in generalized Benders decomposition is given at any iteration K by Equation (77): K FGB = min α

such that α≥ f

 /  T .   g z k +B y z k +cT y + µk



k = 1, 2, . . . K α ∈ Re1 , y ∈ {0, 1}m

(77)

where α is the largest Lagrangian approximation obtained from the solution of the K NLP subproblems; zk and µk correspond to the optimal solution and multiplier of the k-th NLP subproblem; F K GB corresponds to the predicted lower bound at iteration K. In the case of the outer-approximation method, the MILP master problem is given by Equation (78)

where α is the largest linear approximation of the objective function subject to linear approximations of the feasible region obtained from the solution of the K NLP subproblems; T k is a diagonal matrix whose entries t kii = sign (λki ), where λki is the Lagrange multiplier of equation hi at iteration k and is used to relax the equations in the form of inequalities [187]. Note that in both master problems the preK dicted linear bounds F K GB , and F OA increase monotonically as iterations K proceed, because the linear approximations are refined by accumulating the Lagrangian (in Eq. 77) or linearizations (in Eq. 78) of previous iterations. Also, in both cases, rigorous lower bounds can only be ensured when certain convexity conditions hold [185], [186]. In comparing the two methods, the lower bounds predicted by the outer-approximation method are always greater or equal to the lower bounds predicted by generalized Benders decomposition. Hence, the outer-approximation method requires the solution of fewer NLP subproblems and MILP master problems. On the other hand, the MILP master in the outerapproximation method is more expensive to solve, so the generalized Benders method may require less time if the NLP subproblems are inexpensive to solve. For a more extensive discussion and computational experience in chemical engineering applications, see [187–189].

10.7. Solution of Dynamic Optimization Problems This section deals with the solution of optimization problems that include differential equations as well as algebraic constraints. In addition to parametric decision variables, they can include control profiles that are functions of time. Differential – algebraic models appear in all aspects of process engineering. Optimiza-

Mathematics in Chemical Engineering tion problems that include these models are extremely common, covering problems as fundamental as the design of a single catalyst pellet to the optimal design and operation of an entire chemical plant. However, solving even the simplest and smallest of these optimization problems is typically difficult and timeconsuming. Consequently, the development of efficient methods for these problems represents an interesting research area with a wealth of important and challenging applications. The general optimization problem under consideration can be represented by Equation (79): Min

p,x(t),u(t)

F





p, x tf

137

Optimality conditions for the above problem formulation can be derived in a manner similar to nonlinear programming problems. Here, a Lagrange function is formed that consists of the objective function and the weighted sum of the constraints evaluated at each point in time. This weighted sum is expressed more concisely as an integral and the resulting function can be written as

L (x, u, p, µ, ν, λ) = F x tf
tf 6

λT ( f (x, u) − x) ˙

+ 0

7 +µT g (x, u) + ν T h (x, u) dt

such that h (p, u (t) , x (t)) = 0

hf p, x tf = 0





+ νT +µT f g f x tf f hf x tf

g (p, u (t) , x (t)) ≤ 0

To express this function entirely in terms of parameters and state and control variables, the following transformation is applied:



g f p, x tf ≤ 0


tf x˙ (t) = f (p, u (t) , x (t))

T λT x˙ dt = λ tf x tf − λ (0)T x (0)

0

x (0) = x0
tf

pL ≤ p ≤ pU

− 0

uL ≤ u (t) ≤ uU xL ≤ x (t) ≤ xU

˙ T x dt λ

and the Lagrange function becomes

(79)

where F is the objective function; g, g f represent the design inequality constraint vector; h, hf the design equality constraint vector; p is a parameter, decision variable vector; x (t) the state profile vector; u (t) the control profile vector; pL , pU are the parameter variable bounds; xL , xU the state profile bounds; and uL , uU the control profile bounds. Note that algebraic constraints have been classified into conditions at final time tf as well as constraints that must be enforced over the time domain. Also, although the model is given as an initial value problem for convenience, general ordinary differential equations can be handled in a straightforward manner. Finally, note the distinction between continuous variables p ( parameters that do not vary with time) and u (t) (time-varying control profiles) as decision variables.

.

T − λ tf L (x, u, p, µ, ν, λ) = F x tf x tf



/ + νT + µT f g f x tf f hf x tf

T


tf 6

λT f (x, u)

+λ (0) x (0) + 0

7 +λT x + µT g (x) + ν T h (x) dt

Note that multipliers λ, µ, and ν have been introduced, which correspond to f (x, u, p) − x˙ , g (x, u, p), and h (x, u, p), and that all are functions of t. Because the multipliers λ perform a special function for the sensitivity of the differential equations, they are denoted as adjoint variables. On the other hand, multipliers µf and ν f , which correspond to the final time conditions, do not vary with time and function in the same manner as multipliers in nonlinear programming.

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Mathematics in Chemical Engineering

Stationary conditions of this Lagrange function can be given as follows with respect to x (t), u (t), and p. In addition, complementarity and nonnegativity conditions relating to the inequality constraints and feasibility conditions make up the balance of the optimality conditions given below:

between 0 and 1. Note that a new parameter pf , whose value is determined as part of the optimization, is introduced to represent final time. By writing the problem with τ substituted for t, differential equations are merely rewritten as

˙ + ∇x f λ + ∇x g µ (t) + ∇x h ν (t) = 0 x (t) : λ

and final conditions are evaluated at τ = 1. Finally, a large body of literature appears for this problem in which the algebraic constraints H and the profile bounds are deleted. For this case, the Hamiltonian is defined as λT f and the resulting simplified optimality conditions are denoted as the Euler – Lagrange equations:

x tf : ∇xf F − λf +∇xf g f µf + ∇xf hf ν f = 0 u (t) : ∇u f λ + ∇u g µ (t) + ∇u h ν (t) = 0 /
f T µ +∇ h ν [∇p f λ ∇p F + ∇p g T p f + f f

. p :

t

0

+∇p g µ + ∇p h ν] dt = 0 µT f g f = 0 µ, µf ≥ 0 g, g f ≤ 0 µT g= 0 h, hf = 0 x˙ = f (x, u) , x (0) = x0

The derivation of these conditions is very similar to the Kuhn – Tucker conditions for the nonlinear programming problem. The additional differential equations encountered for λ are known as adjoint equations and, in this case, have final conditions associated with them. In the same manner as for nonlinear programs, second-order conditions can also be derived, based on constrained projections of the second variations of the Lagrange functions. A detailed derivation of these conditions is given in [190]. Moreover, several straightforward modifications can be made to extend this problem. For example, if some or all initial conditions are not specified, they can be represented by parameters p0 that are determined as part of the optimization. In this case, an additional term νT 0 (x (0) − p0 ) is appended to the Lagrange function, and the corresponding stationary condition with respect to p0 becomes

dx = pf f (x, u) ; x (0) = x0 dτ

∂H ∂f = λ= 0 ∂u ∂u ˙ = − ∂H = ∇x f λ λ f ∂x λ tf = − ∇xf F

To illustrate these optimality conditions, the simple example given below is considered: Min x2 tf such that x˙ 1 = − 2x1 + u2 − u x˙ 2 = x1 x1 (0) = 1 x2 (0) = 1 tf = 1

Here, the optimal control profile over a 1-h period must be determined for this dynamic system. This problem has no algebraic side conditions and no parameters. The Lagrange function for this system is given by L = x2 tf +


tf

8 2

λ1 u − u − 2x1 − x˙ 1

0

x0 : ν0 − λ0 = 0

+λ2 (x1 − x˙ 2 )} dt =

p0 : ν0 = 0

x2 tf + λ1 tf x1 tf + λ2 tf x2 tf

λ0 = 0, an additional boundary condition for

− λ1 (0) x1 (0) − λ2 (0) x2 (0)
tf

the adjoint equations

In addition, if final time tf is not specified for this problem, t can be normalized by τ = t/pf

+ 0

8 2

λ1 u − u − 2x1

Mathematics in Chemical Engineering 7 ˙ 1 x1 + λ2 x1 + λ ˙ 2 x2 dt +λ

and the stationary conditions with respect to x (t), x (1.0), and u (t) are given by: x1 :

λ˙ 1 = 2λ1 − λ2 λ1 (1) = 0

x2 :

λ˙ 2 = 0

u:

λ2 (1) = 1

λ (2u − 1) = 0

The adjoint variables λ1 and λ2 can be determined directly from the adjoint equations and the associated final conditions. These are λ1 (t) = {1 − exp (2t − 2)} /2 λ2 (t) = 1

Because λ1 is zero only at t = 1, the stationary condition for u (t) is satisfied only when u = 0.5. Consequently, it can be verified that u (t) = 0.5 minimizes x2 (1). Solution of the differential – algebraic optimization problem becomes especially difficult in the presence of state variable inequality and equality constraints. Because many process problems are constrained, optimization problems of this type are not considered frequently. Methods for tackling these problems can be divided into three basic types: 1) iterative methods based on variational conditions; 2) feasible path nonlinear programming methods; and 3) simultaneous nonlinear programming methods. The analytical approach to solving small design problems, such as the above example, naturally leads to iterative algorithms based on variational conditions. Based on the optimality conditions derived above, control vector iteration algorithms were proposed that involve the solution of model equations forward in time, adjoint equations backward in time, and intermittent updating of the control profile to minimize the Hamiltonian function, λT f (x, u) (or which make dH/du = 0). Methods that are analogous to those for solving unconstrained optimization problems can be applied here. For example, using the gradient of the Hamiltonian with respect

139

to u (t), Bryson and Denham [191] proposed a steepest descent method. Lasdon, Mitter, and Warren [192] and Lasdon [193] accelerated this approach by proposing conjugate gradient and variable metric methods, respectively, for these types of problems. Finally, a Newton-type extension to optimal control problems is presented in [194]. These methods work best on problems with initial value ordinary differential equation models without state variable and final time constraints. For these simple problems, solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations [195], [196]. Moreover, any additional constraints in this problem require the search for appropriate multiplier values, µ and ν, which often requires an outer loop in the solution algorithm and can easily lead to a prohibitive number of model evaluations, even for small systems. Consequently, the control vector iteration methods are limited to the simplest optimal control problems. Instead, the optimal control problem can be approached as a nonlinear programming problem which results after the control variable is discretized. Here, the ordinary differential equation model is solved repeatedly in an inner loop while parameters representing u (t) are updated on the outside. Initially, the updating was performed by direct search or “hill climbing” algorithms and could become costly for large problems. Use of more sophisticated methods, on the other hand, must lead to a consideration of how gradients can be calculated efficiently from the differential equation model. Using this nonlinear programming approach (also termed the feasible path approach), denote up as the vector of parameters representing u (t). For example, if u (t) is assumed piecewise constant over a variable distance, include ui and ti in up . The original optimization problem then becomes the following nonlinear programming problem: Min F (up , p) up ,p

such that h (up , p) = 0 g (up , p) ≤ 0

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Mathematics in Chemical Engineering

pL ≤ p ≤ p U uL p ≤ up ≤ up

and the ordinary differential equation model: x˙ (t) = f (p, up , x (t)) x (0) = x0

These equations are very similar to the optimality conditions derived above. Gradients with respect to up can also be calculated through sensitivity equations derived from the ordinary differential equation model. Here, for continuous state variable profiles, the sensitivity equations with respect to parameters p, for example, are



xK+1 ti−1,f = xio i = 2, N E

∂ ∂ {x˙ = f (x, u)} → ∂p ∂p



xK+1 tN E,f = xf

+



xL ≤ xil ≤ xU

∂f ∂x

 =

∂f ∂p



s˙ = ∂f /∂p + s (∂f /∂x)

pL ≤ p ≤ p U

is solved. Note that the independent variable, time, disappears from this problem, and constraints imposed at final time appear naturally in the above problem; other constraints that must be enforced over time have to be treated in a more complex manner. For example, they can be converted to final time constraints by integrating the square of the constraint violations and forcing these to be less than a tolerance at the final time. To solve the nonlinear program, gradients can be calculated in a number of ways with respect to up from the ordinary differential equation model. The easiest, but least efficient and accurate, way is simply to resolve the model for each perturbation of the parameters. Sensitivity information can also be obtained by solving the adjoint equations ˙ i =∇x f (x, u) λi , i = 0, m λ λ0 =∇xf F ; λi = ∇xf g fi or ∇xf hfi

and evaluating parameteric sensitivities by using the following relation:
tf .

/ ∇p f T λ0 dt

0

∂g fi ∂hfi or = ∂p ∂p



dx dt

which, upon changing orders of differentiation and defining s = ∂x/∂p, leads to

uL ≤ uil ≤ uU

∂F = ∂p

∂x ∂p




tf ∇p f T λi dt 0

Either approach results in gradient calculations with costs proportional to problem size; effort for evaluating gradients with adjoint approaches is proportional to the number of (objective and constraint) functions evaluated at final time, whereas effort for sensitivity equations is proportional to the number of parameters up and p. Consequently, the choice of gradient calculation approach depends on the structure of the particular feasible path problem. Nevertheless, both Carcotsios and Stewart [197] and Sargent and Sullivan [198] demonstrated that gradient calculation approaches can be accelerated considerably by tailoring the ordinary differential equation solver to include sensitivity or adjoint equations. The feasible path approach has been very successful in solving large process problems. However, this approach still requires repeated solution of the process model (and sensitivities). For large processes or for processes that require the solution of rigorous underlying procedures, this approach can become expensive. Moreover, for stiff or otherwise difficult systems, this approach is only as reliable as the ordinary differential equation solver. The feasible path approach also offers only indirect ways of handling time-dependent constraints. Finally, the optimal solution with this approach is only as good as its control-variable parameterization, which often can only be improved by a priori information about the specific problem. Consequently, a simultaneous nonlinear programming

Next Page Mathematics in Chemical Engineering approach is also considered as an alternative solution method. Instead of parameterizing the control profile and solving the system as a nonlinear program, the simultaneous approach begins with a parameterization of both the control and the state variable profiles, and solves a mathematical programming problem consisting of algebraic equations. However, early application of the simultaneous approach suffered from two drawbacks. First, simultaneous approaches lead to much larger nonlinear programs than feasible path approaches. Consequently, nonlinear programming methods must be very efficient to compete with smaller feasible path formulations. Here, a trade-off occurs between the expense of repeated ordinary differential equation model solution and solution of a larger nonlinear program. Second, care must be taken in the formulation to yield an accurate algebraic representation of the differential equations. Recently, the SQP algorithm, described in a previous section, and orthogonal colloction have been applied successfully to a number of optimal control problems; this approach shows considerably better performance than control vector iteration methods. Moreover, through appropriate discretization of the differential equations, this simultaneous approach can handle constraints on state and control profiles directly. Moreover, by using the range and null space decomposition technique for SQP, very efficient solutions can be achieved with the simultaneous approach. Because their approximation and stability properties are well studied, various ordinary differential equation solution methods can be applied directly to discretize the differential equations to algebraic constraints. Here, implicit Runge – Kutta methods are considered, which coincidentally also include the method of orthogonal collocation. By defining state and control profiles at τi ε [0,1] [e.g., the shifted roots of orthogonal (Legendre) polynomials], these profiles are parameterized as Lagrange-form polynomials over τ ε [0,1]: xK+1 (τ ) =

K 

xl ϕl (τ )

uK (τ ) =

K 

141

ul ψl (τ )

l=1

ϕl (τ ) =

K K   (τ − τk ) (τ − τk ) , ψl (τ ) = (τ − τ ) (τ l k l − τk ) k=0,l k=1,l

where x K +1 > (τ ) and uK are (K + 1)-th order (degree < K + 1) and K-th order polynomials, respectively. (Here the notation k = 0, l refers to the index k starting at zero but not equal to l.) The state variables are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of x (t) and u (t) this approach is extended to piecewise polynomials and orthogonal collocation on NE finite elements (of length ∆αi ) is applied. The differential equations are now represented by the following nonlinear algebraic equations: R (til , ∆αi ) =

K 

xij ϕj (τl ) − ∆αi f (p, uil ) = 0

j=0

l = 1, . . . K, i = 1 , . . . N E

with

xi,0 = x0 and xK+1 ti−1,f = xi0 i = 2, N E

where til represent shifted roots of Legendre polynomials over the i-th element length and ti−1, f is the time at the end of the (i − 1)-th element. Note that state profiles satisfy continuity conditions across finite elements ∆αi , whereas this property is not enforced for control profiles. Substituting this representation of the ordinary differential equation model into the dynamic optimization problem with ∆αi fixed, leads to the following nonlinear programming problem: Min

p,xil ,uil

F p, xf

such that h (p, uil , xil ) = 0 g (p, uil , xil ) ≤ 0 R (til , ∆αi ) = 0

l=0

i = 1, . . . N E, l = 1 , . . . K xi0 = x0

Previous Page 142

Mathematics in Chemical Engineering

By taking advantage of the orthogonal properties of the polynomial representation, the Kuhn – Tucker conditions of this nonlinear program can be shown to be parameterizations of the optimality conditions of the dynamic problem. Therefore, the only requirement is that ∆α be chosen appropriately (i.e., be sufficiently small for accurate approximation of the differential equations) and that the breakpoints for the optimal control profile can be located. Choosing the element lengths to render an accurate discretization of the differential equations can also be performed automatically by the nonlinear programming problem. For example, the element lengths ∆αi could be made decision variables, and a measure of the approximation error could be included as an inequality constraint. Here, a suitable approximation measure is the residual of the differential equation evaluated at a noncollocation point t¯ within each element, i.e., | R (t¯, ∆αi ) | ≤ ε for some ε tolerance. For dynamic optimization problems without control profiles and with a sufficiently large number of elements, this simultaneous nonlinear programming approach can be shown to yield accurate solutions to difficult dynamic optimization problems. Therefore, the choice of approach for parameter optimization problems depends on the difficulty of solving the ordinary differential equation model. If the model and its sensitivity information can be determined quickly, then a feasible path approach is probably more efficient than a simultaneous strategy. On the other hand, if the model is expensive and difficult to solve at intermediate points, and if state variable constraints must be enforced over time, then a simultaneous approach with an efficient nonlinear programming strategy should be considered. Here, the advantage of this approach is that the ordinary differential equation model is solved only once and state variable constraints can be enforced directly. For problems with control profiles, on the other hand, one must be especially careful about the stability properties of the ordinary differential equation discretization. Without control profiles, stability properties of the ordinary differential equation discretization are determined by properties of the corresponding ordinary differential equation solver (e.g., implicit Runge – Kutta methods). However, for dy-

namic optimization problems with control profiles, the optimality conditions form a set of differential – algebraic equations that may lead to different approximation and stability properties for the same discretization method. Difficulties in solving differential – algebraic equation systems normally occur when the discretized system of algebraic equations becomes singular. This can occur, for example, if timedependent equalities (h) and active inequalities (g) are not functions of u (t). Although solutions to the differential – algebraic equation system (and the optimization problem) do exist, instabilities and loss of accuracy may result unless the discretization is of reasonably high order and has very strong stability properties. Brenan and Petzold [199] and Logsdon and Biegler [200] mention that collocation formulas can deal with these difficult systems, but the orthogonal colloction method may have to be modified before it can be applied. Consequently, to deal with dynamic optimization problems with control profiles, it is necessary to determine the difficulty of the system (a straightforward analysis can be found in [199] and [201]) to see if an appropriately accurate and stable discretization (e.g., orthogonal collocation, implicit Runge – Kutta, etc.) is available. If one cannot be found, then the only recourse may be to parameterize the control profile and settle for a suboptimal solution. The advantage of this, however, is that the methods described above for parameter problems can be applied directly without any difficulties due to discretization of differential – algebraic equation systems.

11. Probability and Statistics [202–205] The models treated thus far have been deterministic, that is, if the parameters are known the outcome is determined. In many situations, all the factors cannot be controlled and the outcome may vary randomly about some average value. Then a range of outcomes has a certain probability of occurring, and statistical methods must be used. This is especially true in quality control of production processes and experimental measurements. This chapter presents standard statistical concepts, sampling theory and statistical

Mathematics in Chemical Engineering decisions, and factorial design of experiments or analysis of variances. Multivariant linear and nonlinear regression is treated in Chapter 2.

is used in calculating the sample variance n 2

s =

Suppose N values of a variable y, called y1 , y2 , . . . , yN , might represent N measurements of the same quantity. The arithmetic mean E ( y) is

E ( y) =

yi

N

The median is the middle value (or average of the two middle values) when the set of numbers is arranged in increasing (or decreasing) order. The geometric mean y¯ G is y¯G = (y1 y2 . . . yN )1/N

The root-mean-square or quadratic mean is Root − mean − square =

2 E

( y2 )

3 4N 4 =5 yi2 /N i=1

The range of a set of numbers is the difference between the largest and the smallest members in the set. The mean deviation is the mean of the deviation from the mean. N

Mean − deviation =

var ( y) = σ =

N

( yi − E ( y))2

i=1

N

N

If the set of numbers {yi } is a small sample from a larger set, then the sample average

n

yi

( yi − y¯) = 0

Thus, knowing n − 1 values of yi − y¯ and the fact that there are n values automatically gives the nth value. Thus, only n − 1 degrees of freedom ν exist. This occurs because the unknown mean E ( y) is replaced by the sample mean y derived from the data. If data are taken consecutively, running totals can be kept to permit calculation of the mean and variance without retaining all the data: n 

( yi − y¯)2 =

i=1

n 

y12 − 2 y¯

i=1 n 

n 

yi + (¯ y )2

i=1

yi /n

i=1

Thus, n  i=1

3 4 N 4 4 ( y − E ( y))2 5 i=1 i

i=1

n−1

i=1

|yi − E ( y) |

i=1

and the standard deviation σ is the square root of the variance.

y¯ =

n 

n, N

n

s =

3 4 4 n ( y − y¯)2 4 5 i=1 i

y¯ =

The variance is

σ =

n−1

The value n − 1 is used in the denominator because the deviations from the sample average must total zero:

i=1

2

( yi − y¯)2

i=1

and the sample standard deviation

11.1. Concepts

N

143

y12 , and

n 

yi

i=1

are retained, and the mean and variance are computed when needed. Repeated observations that differ because of experimental error often vary about some central value in a roughly symmetrical distribution in which small deviations occur more frequently than large deviations. In plotting the number of times a discrete event occurs, a typical curve is obtained, which is shown in Figure 44. Then the probability p of an event (score) occurring can be thought of as the ratio of the number of times it was observed divided by the total number of events. A continuous representation of this probability density function is given by the normal distribution

144

p ( y) =

Mathematics in Chemical Engineering

2 2 1 √ e−[y−E(y)] /2σ . σ 2π

(80)

This is called a normal probability distribution function. It is important because many results are insensitive to deviations from a normal distribution. Also, the central limit theorem says that if an overall error is a linear combination of component errors, then the distribution of errors tends to be normal as the number of components increases, almost regardless of the distribution of the component errors (i.e., they need not be normally distributed). Naturally, several sources of error must be present and one error cannot predominate (unless it is normally distributed). The normal distribution function is calculated easily; of more value are integrals of the function, which are given in Table 12; the region of interest is illustrated in Figure 45. Table 12. Area under normal curve ∗ F (z) = √12 π 


z

e−z

0

2 /2

dz

z

F (z)

z

F (z)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

0.0000 0.0398 0.0793 0.1179 0.1554 0.1915 0.2257 0.2580 0.2881 0.3159 0.3413 0.3643 0.3849 0.4032 0.4192

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.7 3.0 4.0 5.0

0.4332 0.4452 0.4554 0.4641 0.4713 0.4772 0.4821 0.4861 0.4893 0.4918 0.4938 0.4965 0.4987 0.499968 0.4999997

∗ Table gives the probability F that a random variable will fall in the shaded region of Figure 45. For a more complete table (in slightly different form), see [207, Table 26.1].

Figure 45. Area under normal curve

For a small sample, the variance can only be estimated with the sample variance s2 . Thus, the normal distribution cannot be used because σ is not known. In such cases Student’s t-distribution, shown in Figure 46 [202, p. 70], is used: y0

p ( y) =  1+

t2 n−1

n/2 , t =

y¯ − E ( y) √ s/ n

and y0 is chosen such that the area under the curve is one. The number ν = n − 1 is the degrees of freedom, and as ν increases, Student’s t-distribution approaches the normal distribution. The normal distribution is adequate (rather than the t-distribution) when ν > 15, except for the tails of the curve which require larger ν. Integrals of the t-distribution are given in Table 13, the region of interest is shown in Figure 47.

Table 13. Percentage points of area under Students t-distribution ∗ ν

α=0.10

α=0.05

α=0.01

α=0.001

1 2 3 4 5 6 7 8 9 10 15 20 25 30 ∞

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.753 1.725 1.708 1.697 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.131 2.086 2.060 2.042 1.960

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 2.947 2.845 2.787 2.750 2.576

636.619 31.598 12.941 8.610 6.859 5.959 5.408 5.041 4.781 4.587 4.073 3.850 3.725 3.646 3.291

∗

Table gives t values such that a random variable will fall in the shaded region of Figure 47 with probability α. For a one-sided test the confidence limits are obtained for α/2. For a more complet table (in slightly different form), see [207, Table 26.10].

Figure 44. Frequency of occurrence of different scores

Mathematics in Chemical Engineering

145

Figure 46. Student’s t-distribution. For explanation of ν see text

where dF =p dx

is the probability of x being between x and x + dx. The probability density function satisfies p (x) ≥ 0
∞ p (x) dx = 1 Figure 47. Percentage points of area under Student’s t-distribution

Other probability distribution functions are useful. Any distribution function must satisfy the following conditions:

−∞

The Bernoulli distribution applies when the outcome can take only two values, such as heads or tails, or 0 or 1. The probability distribution function is

0 ≤ F (x) ≤ 1

p (x = k) =pk (1 −p)1−k , k = 0 or 1

F (−∞) = 0, F (+∞) = 1

and the mean of a function g (x) depending on x is

F (x) ≤ F (y) when x ≤ y

The probability density function is p (x) =

dF (x) dx

E [g (x)] =g (1) p +g (0) (1 −p)

The binomial distribution function applies when there are n trials of a Bernoulli event; it gives the

146

Mathematics in Chemical Engineering

probability of k occurrences of the event, which occurs with probability p on each trial

  0 F (x) =

p (x = k) = n! pk (1 − p)n−k k! (n − k)!

1

The mean and variance are E (x) =

The mean and variance are

a+b 2

var (x) =

E (x) =n p var (x) = np (1 − p)

The hypergeometric distribution function applies when there are N objects, of which M are of one kind and N − M are of another kind. Then the objects are drawn one by one, without replacing the last draw. If the last draw had been replaced the distribution would be the binomial distribution. If x is the number of objects of type M drawn in a sample of size n, then the probability of x = k is

x> 3 βr = 1+

νA mBL DB νB mAPh DA

The accelerating effect of the reaction is controlled by the limited availability of B. LIVE GRAPH

Click here to view

239

other is not rate determining. Many heterogeneous systems also come within the term “single phase”, for example heterogeneous catalytic reactions, which are dealt with as quasi-single phase. The mathematical treatment of singlephase reactors differs from the problems discussed in the earlier chapters due to the boundary conditions and the models for the processes occurring at and in the phase boundary layers. Definition of the state of the reactive mixture in chemically reactive flows with the fields u (x, t), p (x, t), mi (x, t), i = 1, . . ., N and h (x, t) requires the solution of the conservation equations for the total mass, the components of the momentum, the mass of the individual chemical species, and the enthalpy. For closure of the system of equations other fundamental equations may have to be incorporated. With the simplifications introduced in the earlier sections for the molecular transport processes this system of equations can be given in the form ∂ ∂ (uk ) = 0 + ∂t ∂xk

(131)

for the total mass, ∂ui ∂ui ∂τik ∂p + uk = − + gi ∂t ∂xk ∂xk ∂xi i = 1, 2, 3 

Figure 39. Acceleration factor β r as a function of the Hatta number Ha for various stoichiometric characteristic values [110] ν = stoichiometric coefficient; D = diffusion coefficient The subscripts A and B denote components, L and Ph denote liquid and phase boundary, respectively.

for the components of the momentum,

Introduction of the acceleration factor β r means that the effect of the chemical reactions is incorporated into the mass transfer; the chemical reactions are no longer coupled with the mass transfer. Further aspects of mass transfer with chemical reactions are discussed in Section 5.3.3.1.



5.3.3. Chemical Reactions in the Homogeneous Phase Many chemical reactors are single phase. The term single phase should be understood here to mean that mass transfer from one phase to the



∂h ∂h ∂ ∂T = λ + Sh + uk ∂t ∂xk ∂xk ∂xk

(204)

(240b)

for the enthalpy, and   ∂mi ∂mi ∂mi ∂ Di + Si = + uk ∂t ∂xk ∂xk ∂xk i = 1, ...,N − 1

(265b)

for the chemical species. Since the system of Equations (131), (204), (240 b), and (265 b) is coupled, simultaneous solution is necessary. This is usually done numerically. Before discussing a numerical procedure, some fundamental characteristics of the modeling of reactors will be shown by a simplified version of the system of Equations (131), (204), (240 b), and (265 b). The first simplification is the assumption of constant density, isothermal conditions, chemical reactions that do not involve volume changes, and frictionless flow.

240

Mathematical Modeling

5.3.3.1. Isothermal Reactors with Frictionless Flow, Constant Density, and Reactions Without Volume Changes

Here however the numerical procedure analogous to Section 5.1.2.2 is discussed in order to compare various concepts of mathematical modeling.

Under the above conditions it is easy to show that the equation system 

  ∂mi ∂mi ∂ ∂mi + u = Di + Si ∂t ∂x ∂x ∂x i = 1, ...,N − 1

(305)

is adequate for the definition of the problem provided that pressure effects and physical forces are neglected and the assumption is made that v = w = 0 (one-dimensional formulation). If only two chemical species A and B are considered which react in a first-order reaction A + B → products, Equation (305) for the steady-state case and with S A = − k mA is u

d2 m dm − D +km=0 dx dx2

(306)

In Equation (306) m = mA , and thus also S A = S = − k m. Analytical solutions can be found for this equation for various boundary conditions. If the equation is, for example, normalized after dividing by the density  in the form η = x/L, then d2 m dm −P e − DaI P em= 0 dη 2 dη

Here Pe = u L/D, and DaI is the Damk¨ohler number of the first kind, DaI = k τ , where τ = L/u is the mean hydrodynamic residence time. For DaI < Pe, the characteristic equation of the above differential equation has two real roots ! 1 λ1,2 = P e ± 2

P e2 +DaI P e 4

The general solution is thus m = C 1 e λ1 x + C 2 e λ2 x

where C 1 and C 2 are found from the boundary conditions m = m0 for x = 0 and dm/dx = 0 for x = L as e λ2 L

λ2 and λ1 eλ1 L − λ2 eλ2 L   λ2 eλ2 L C2 = m0 − C1 = m0 1 + λ1 eλ1 L − λ2 eλ2 L C1 = − m 0

Figure 40. “Up-wind” difference scheme for a second-order differential equation For explanation of symbols see Figure 26.

To convert the differential operators into differences, Equation (306) is subjected to the same manipulations as Equation (226) and brought into discrete form with an “up-wind” difference scheme. For a one-dimensional problem a line along the x-direction of the grid in Figure 26 is adequate (Fig. 40). In Figure 40 the profile for m is also shown. When the operations carried out in Section 5.1.2.2 are applied to Equation (306) with the grid given in Figure 40, then the system of equations −al−1 ml−1 +bl ml − al+1 ml+1 −S∆Vl = 0

(307)

is obtained. The coefficients of this tridiagonal system of equations, with the notation from Figure 40 are given by  1 Fw xl − xl−1   1 al+1 = +D Fe xl+1 − xl  al−1 = + (u)w +D

bl =al−1 +al+1

(308a) (308b) (308c)

Under the conditions given above u > 0. Assuming that (u F ) = ∆Vl /τl where τl is the hydrodynamic residence time over a distance l, and Pel is a local P´eclet number

Mathematical Modeling Pel = u (xl − xl−1 )/D, the coefficients can be converted into   ∆Vl + ∆Vl τl τl P el  al+1 = ∆Vl τl P el+1    bl = ∆Vl + ∆Vl + ∆Vl τl τl P el τl P el+1

al−1 =

(308d) (308e) (308f)

From Equations (307) and (308 d – f ) the following tridiagonal equation system      1 1  1 1+ ml−1 − 1+ ml + τl P el τl P el P el+1 1  + ml+1 + S (ml ) = 0 (309) τl P el+1

is obtained. If the case of convectively dominated flows is considered where Pe → ∞, Equation (309) becomes  (ml−1 − ml ) +S (ml ) = 0 τl

(310)

This is the mass balance for a cascade of ideally mixed reactors. This analysis again shows that reactors with very large Bodenstein or P´eclet numbers (ideally unmixed reactors) can be represented by a cascade of ideally mixed reactors (see Fig. 41 and Section 4.3.1.1).

241

Numerically this physical model finds its equivalent in the discrete form of differential Equation (306) using an “up-wind” difference scheme for the convectively dominated case. Equation (310) can easily be solved either by elimination or graphically by plotting both terms of the equation against ml . The graphical solution is shown in Figure 41 for a nonequidistant grid, i.e., for ideally mixed reactors with different hydrodynamic residence times. It is important that there is no feedback from elements further downstream. If the flow becomes predominantly diffusive, then Pe is very small and Equation (309) is written as    1 1  1 ml ml−1 − + τl P el τl P el P el+1 1  + ml+1 +S (ml ) = 0 τl P el+1

(311)

Equation (311) has the same structure as Equation (310) but the feed for the ideally mixed reactor with the number l consists of a flow with the composition ml−1 and a flow with the composition ml+1 (Fig. 42). There is therefore a feedback from the elements further downstream, and the volumetric mass flows are increased by the factor 1/Pel etc., as compared with the convective mass flow  m l /τl . Algorithms for the solution of the tridiagonal Equation (311) on the basis of LU decomposition of the system matrix can be found in [29–32].

Figure 42. Cascade of ideally mixed reactors with feedback, equivalent to a tubular reactor with strong back-mixing Pe = P´eclet number. For explanation of other symbols see Figure 41. Figure 41. Cascade of ideally mixed reactors equivalent to an ideal tubular reactor τ = residence time;  = density; l = number of reactor.

The general case where 1/Pel = O (1) has already been formulated with Equation (309).

242

Mathematical Modeling

Feedback dependent on the local P´eclet number exists from the element l + 1 and the volumetric mass flows are increased as against the convective flows by the factor [1 + (1/Pel )] etc. (Fig. 43). For solution, the modeling parameter Pe is required, which contains the molecular diffusion coefficients or experimentally determined effective mixing coefficients (see page 207).

With the definitions given in Figure 44, the mass fraction me at the outlet of the reactor can be obtained from Equation (310) me =m0



1 (1+kτl )n

 (312)

For the sake of simplicity the ideal plug flow reactor in n equidistant sections so that τl = τ = τ tot /n. Thus from Equation (312)

 me =m0

 1

 1+kτtot +

n−1 n2!

(kτtot )2 +

(n−1)(n−2) n2 3!

(kτtot )3 +...



(313)

For n  1 and k τ tot  1  me ≈m0

 1

 1+kτtot +

In the foregoing analysis real reactors were described by numerical solution of differential Equation (306). This is equivalent to a representation by elements of ideally mixed reactors. Alternative structures of matrices of elements of ideal reactors can be derived by physical analysis of complex reactors [4], [11], [12], [24], [111], [112]. An example which is of some importance in chemical engineering will be described here. Example of Combinations of Elements of Ideal Reactors: the Loop Reactor. The layout of an ideal plug flow reactor with recirculation (loop reactor) is shown in Figure 44, part of the product flow is added to the feed. For analysis the same assumptions apply as in Section 5.3.3.1.

1 2!

1 (kτtot )2 + 3! (kτtot )3 +...

  =m0 e−kτtot

(314)

where m
0 and τ tot are unknown and depend on the magnitude of the recirculated volume qr .

Figure 44. Schematic representation of a loop reactor q = flow rate; m = mass fraction The subscripts 0, r, and e denote inlet, loop (recirculated), and final, respectively; the superscript ( ) denotes conditions after mixing of inlet and backflow.

The term m
0 results from the combined mass fractions of the flows entering the plug flow reactor and τ tot from the volumes of the flow reactor and the total volumetric flow. If ϕ = qr /q0 (the recirculation ratio), then m0 =m0

1 ϕ + me 1+ϕ 1+ϕ

(315a)

and τtot = τ0

Figure 43. Cascade of ideally mixed reactors with feedback, equivalent to a tubular reactor with back-mixing, general case Pe = P´eclet number. For explanation of symbols see Figure 41.

1 1+ϕ

(315b)

where τ 0 is the hydrodynamic residence time without recirculation. Substitution into Equation (314) gives me =m0

e−kτ0 /(1+ϕ)

1+ϕ 1 + e−kτ0 /(1+ϕ)

(316)

Mathematical Modeling For ϕ → 0 the loop reactor takes on the character of the ideal plug flow reactor and me = m0 e−kτ0 . For k τ 0 = DaI = 0.1, me /m0 ≈ 90 %. For ϕ → ∞, e−kτ 0 /ϕ = 1 − (k τ 0 /ϕ) and me = m0 /(1 + k τ 0 ). The loop reactor is similar to an ideally stirred reactor with correspondingly lower conversions. The adjustable recirculation ratio of the loop reactor is fully exploited in important engineering applications. When measuring kinetic data in laboratory reactors the lowest possible conversion is required in the reactor so that the measured reaction rates can be assigned to specific temperatures and concentrations. Thus DaI  1. On the other hand the concentration differences measured for evaluating the reaction rates should be as large as possible to minimize statistical errors. As may be calculated from Equations (314) and (316), for DaI = 0.1 and ϕ = 10 the conversion in the plug flow reactor is 1 − (me /m
0 ) ≈ 1 %. The measured difference in the mass fractions is, however, me /m0 ≈ 90 %. For reactions with positive reaction order relative to the reactants, the reaction rate falls off as the concentration decreases. Recirculation of the product lowers the concentration in the feed due to dilution. There is however a class of reactions in which the reaction rate first increases with decreasing concentration and then decreases after passing through a maximum (e.g., in heterogeneous catalysis, enzymatic catalysis, and autocatalysis). Recirculation has a positive effect on conversion in these cases. 5.3.3.1.1. Stability of Isothermal Reactors Steady-State Cases. The importance of the stability of chemical reactors for safety and economic reasons is obvious. The description of stability analysis given here refers to the description of processes in physical space, location x = (x 1 , x 2 , x 3 ) and time t. Stability analysis and theories often refer to the description of chemical processes in the form of transfer functions in the space of Laplace-transformed variables or to the frequency behavior. Detailed discussion of these relationships can be found in [3], [9], [113–116]. To illustrate stability analysis in isothermal, steady-state reactors one element of the ideal plug flow reactor will be used (Fig. 40).

243

This can be regarded as an ideally mixed reactor where the essentials of stability analysis can be shown. Three types of chemical reactions will be considered: 1)

2)

3)

Normal reactions with declining reaction rate A + B → products where dmA /dt =− k 2 mA mB and mB = mB0 − (mA0 − mA ) Autocatalysis A+2B→3B where dmA /dt = − k 3 mA m2B and mB = mB0 + (mA0 − mA ) Self poisoning A→B where dmA /dt =− k 1 mA /(1 + K mA )2 (Section 3.2.1.1, Eq. 53).

Along with the assumptions from Section 5.3.3.1, equal molar masses of A and B are also assumed. Using the abbreviations α = mA /mA0 and β 0 = mB0 /mA0 and Equation (310), the following equations are obtained for the three cases 1−α =α2 + (β0 −1) α τ k2 mA0 1−α =α [(β0 +1) −α]2 τ k3 m2A0 

1−α 1

=α τ k1 1+αmA0 K 2+αmA0 K

(317a) (317b) (317c)

where 1 − α is the conversion. The term τ k 2 mA0 , τ k 3 m2A0 , or τ k 1 can be regarded as the ratio of the hydrodynamic residence time to a characteristic chemical time and is denoted as the Damk¨ohler number of the first kind DaI . Equations (317) can conveniently be solved graphically by plotting both sides against 1 − α. Figures 45, 46, 47 show such plots for the three cases above. For the normal case (Fig. 45 A), decreasing reaction rate with decreasing reactant concentration, an unambiguous steady-state solution exists for each Damk¨ohler number. With increasing Damk¨ohler number (i.e., increasing hydrodynamic residence time), the steady-state solutions give higher conversions. All steadystate solutions, which are plotted against the Damk¨ohler number in Figure 45 B, are stable. If a small disturbance occurs to the left or right on the conversion axis in the steady-state solution, opposing changes are produced in the right-hand side (RHS) or left-hand side (LHS) respectively of Equation (317 a).

244

Mathematical Modeling for stability (Eq. 318). For Equation (317 b) d d(1−α) [RHS] = [β0 + (1−α)] [2 − 3 (1−α) −β0 ], so that the stability criterion is not met for small values of (1 − α).

Figure 45. Mono-steady-state conversion points for “normal” chemical reactions (A) and bifurcation diagram (B) for isothermal conditions in a well-stirred reactor LHS = left-hand side (Eq. 317 a); RHS = right-hand side (Eq. 317 a); 1 − α = conversion; DaI = Damk¨ohler number of first kind; β 0 = mB0 /mA0

The “convective term” on the LHS decreases (increases), whilst the “reaction term” on the RHS increases (decreases). A lesser (greater) availability of A due to convection therefore counterbalances a greater (lesser) consumption of A due to reaction, so that the conversion is shifted to the right or to the left. The general condition for stability can be given in the terminology of Equation (317) as d d [LHS] > [RHS] d (1−α) d (1−α) at LHS = RHS d d(1−α)

(318a)

For Equation (317 a), [LHS] = 1/DaI d and d(1−α) [RHS] =2 (1−α) −β − 1; this satisfies the condition for stability for all possible values of 1 − α based on the inlet condition β 0 . A completely different picture results for autocatalysis (Fig. 46 A) or self poisoning (Fig. 47 A). In these cases multiple steadystate solutions occur for a particular range of Damk¨ohler numbers. Some of the solutions do not comply with the conditions

Figure 46. Multiple steady-state conversion points for autocatalytic chemical reactions (A) and bifurcation diagram (B) for isothermal conditions in a well-stirred reactor LHS = left-hand side (Eq. 317 b); RHS = right-hand side (Eq. 317 b) For explanation of other symbols see Figure 45.

In this range a small disturbance of (1 − α) to the right leads to a larger increase in the “reaction term” than in the availability due to convection; conversion therefore increases. The reaction system thus always heads for the nearest adjacent stable operating point. In the bifurcation diagrams shown in Figures 46 B and 47 B, these unstable steady-state solutions lie on an Sshaped branch. With a continuous increase in the Damk¨ohler number (residence time) however the system cannot pass through this S-shaped branch, but jumps from a condition with low conversion to one with high conversion. The bifurcation diagrams show that in both cases with

Mathematical Modeling multiple steady-state solutions the Damk¨ohler number for “igniting” the reaction is higher than that for “extinguishing” it. Thus hysteresis occurs. The jumping points are given by the condition for the coalescence of an unstable with a stable steady-state solution: d d [LHS] = [RHS] d (1−α) d (1−α) at LHS = RHS

(318b)

In the case of autocatalysis this condition occurs for 8 = 1 + 20β0 −8β02 ± (1 − 4β0 ) (1 − 8β0 ) DaI

245

concentration of the catalyst β 0 , the S-shaped curve unfolds. Increasing the concentration of the catalyst at the reactor inlet results in stable reactor behavior, shown by the disappearance of multiple steady-state solutions. An analogous picture results for self poisoning. The appropriate conditions can easily be calculated from Equation (317 c) and the stability condition. Here the parameter controlling the nonambiguity of the solutions is K mA0 which can be regarded as a measure of the poisoning. A decrease in this quantity (lower pressure or lower inlet concentration of A in heterogeneous catalytic reactions) leads to a range of conversions that exhibits unambiguous, stable, steadystate solutions (compare Figs. 45 B, 46 B, 47 B). Nonsteady-State Cases. The stability analysis of steady-state, isothermal, ideally mixed reactors has shown that the phenomenon of multiple steady states with hysteresis always occurs when there is feedback. A feedback system will now be considered for the stability analysis of nonsteady-state reactions. The chemical reaction is formulated as

where the step A → B is a noncatalytic firstorder reaction and a cubic (third-order) reaction catalyzed with B. The feedback mechanism in this reaction is clear: the more B is formed, the faster A is consumed. Since the supply of A from P remains limited, at high concentrations of catalyst B the reaction A → B breaks down until adequate A is again supplied. Analysis will be performed on a batch reactor so that the system of Equations (305) can be written in the form

Figure 47. Multiple steady-state conversion points for chemical reactions with self poisoning (A) and bifurcation diagram (B) for isothermal conditions in a well-stirred reactor K = adsorption coefficient; mA0 = inlet mass fraction of component A; LHS = left-hand side (Eq. 317 c), RHS = right-hand side (Eq. 317 c) For explanation of other symbols see Figure 45.

The positions of the two jumping points move closer together and, with an increase in the inlet

dmP = −k0 mP dt

(319a)

dmA = k0 mP −k1 mA m2B −ku mA dt

(319b)

dmB = k1 mA m2B +ku mA −k2 mB dt

(319c)

A condition for mC results from the mass balance of the system. The quantity (k 2 /k 1 )1/2 is

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Mathematical Modeling

used to normalize the mass fraction, and the normalization of the time is achieved through (1/k 2 ). The system of equations can then be written as dπ = −επ dτ

(320a)

For the stability analysis α and β are regarded as being subject to small disturbances ∆α and ∆β, respectively. The time evolution of these disturbances are then considered. The perturbed quantities thus become 

1/2

where π = mP (k 2 /k 1 ) τ = t/k 2

, ε = (k 0 /k 2 ), and

α = α0 +∆α, f (α,β) =f (α,β)0 +  +

dα = επ − αβ 2 − κα dτ

(320b) 1/2

1/2

where α = mA (k 2 /k 1 ) , β = mB (k 2 k 1 ) κ = (k u /k 2 ), and finally dβ = αβ 2 + κα − β dτ

, and

π = π0 e−ετ

is used in the system of Equations (320) which thus become a system of coupled first-order differential equations dα = µ − αβ 2 −κα=f (α, β) dτ

(322a)

dβ = αβ 2 +κα − β = g (α, β) dτ

(322b)

where the abbreviation µ denotes µ = ε π 0 e−ετ = µ0 e−ετ . For k 0  k 1 , k u , k 2 the principle of quasi-steady state is applicable for the mass fractions α and β. Thus Equations (322 a, b) are solved without integration; approximate solutions for α and β can be obtained from the conditions dα dβ = =0 dτ dτ

∆α 0

∆β+...

(325a)

0

 β = β0 +∆β,g (α,β) =g (α,β)0 + +

(320c)

(321)

∂f (α,β) ∂β



and 

The initial conditions are π = π 0 , α = β = 0 for τ = 0. Thus Equation (320 a) can easily be solved. The result



∂f (α,β) ∂α

∂g (α, β) ∂β

 ∆β+...

∂g (α,β) ∂α

 ∆α 0

(325b)

0

The functions f (α, β) and g (α, β) in Equations (325) are developed in Taylor series at α0 and β 0 that are truncated after the first term. This is a reasonable approximation for small disturbances. If Equation (325) is substituted into Equation (322), the time evolution of the disturbances can be expressed as   ∂α0 ∂∆α ∂f (α, β) + = f (α, β)0 + ∆α ∂τ ∂τ ∂α 0   ∂f (α, β) + ∆β (326a) ∂β 0

and   ∂β0 ∂g (α, β) ∂∆β ∆α + = g (α, β)0 + ∂τ ∂τ ∂α 0   ∂g (α, β) ∆β (326b) + ∂β 0

If the quasi-steady-state solutions are considered, then Equation (326) becomes the system of coupled ordinary differential equations 

d∆α dτ d∆β dτ



 − J qs

∆α ∆β

 =0

(327a)

where Jqs is the Jacobi matrix of Equation (323)

From

 ∂f (α,β)

µ − αβ 2 −κα= 0

(323a)

and αβ +κα − β= 0 2

(323b)

the quasi-steady-state mass fractions αqs , β qs are given by

2

αqs =µ/ µ +κ and βqs =µ

(324a,b)

J qs =

∂α ∂g(α,β) ∂α

∂f (α,β) ∂β ∂g(α,β) ∂β

 (327b) qs

Equation (327) can easily be solved analytically after being transformed into a second-order differential equation (see Section 5.3.1.3). The general solution is ∆α (τ ) =C1 eλ1 τ + C2 eλ2 τ

(328a)

Mathematical Modeling ∆β (τ ) = C3 eλ1 τ +C4 eλ2 τ

(328b)

where the eigenvalues λ1 and λ2 are determined from the characteristic equation λ2 −Tr (J qs ) λ+Det (J qs ) = 0

(329)

This expression has already been used in Sections 5.3.1.3, 5.3.2, and 5.3.3.1. The trace of the Jacobi matrix is the sum of the diagonal elements  Tr (J qs ) =

∂f (α, β) ∂α



 + qs

∂g (α, β) ∂β



   ∂g (α, β) ∂f (α, β) ∂α ∂β qs qs     ∂f (α, β) ∂g (α, β) − ∂α ∂β qs qs 

Det (J qs ) =

so that the solution of the characteristic equation is 6 1. Tr (J qs ) ± Tr (J qs )2 2

−4Det (J qs )}1/2

The different domains of the solutions of Equations (327) are shown in Figure 48 in a stability diagram. To demonstrate some of the possible cases the simplifying assumption is made that the firstorder noncatalytic reaction (A → B) proceeds very slowly, i.e., κ  1. Equations (323) and (324) can then be further simplified

Tr (J qs ) = 1−µ2

(331a)

Det (J qs ) =µ2

(331b)

Tr (J qs )2 −4Det (J qs ) =µ4 −6µ2 +1

(331c)

qs

and its determinant is

λ1,2 =

247

/ (330)

Figure 48. Stability diagram and nature of the steady-state solutions for the example in Section 5.3.3.1 λ1, 2 = eigenvalues of Equations (328).

The time evolution of the disturbances ∆α and ∆β is affected by the eigenvalues of λ1, 2 , which depend on the values of Tr (Jqs ) and Det (Jqs ).

As µ is a function of time, it is to be expected that during the course of the reaction with the depletion of the reservoir P, the discriminants, Equation (331 c), as well as Tr (Jqs ) and Det (Jqs ) will change their values and signs, so that λ1, 2 take all possible combinations √ 1) Initial condition, µ>1+ 2 Initially only P is present so µ is very large. Then Tr (Jqs ) is negative, Det (Jqs ) is positive, and the discriminant is also positive. Thus λ1, 2 become both real and negative. Small disturbances ∆α and ∆β decay monotonically, and the quasi-steady-state mass fractions represent a stable node of the system. The terms node (and focus) are derived from the phase diagrams of ∆α and ∆β, see [114]. In Figure 49 this range is shown in the plot of αqs and β qs against µ. √ 2) Progressive reaction, 1 1 are consistent with the definition of the internal effectiveness factor. Due to the exothermic reaction the temperature inside the catalyst can be higher than temperature on the outer surface. By introducing an effective reaction rate r eff = η eff r G , the rate of heterogeneous catalytic reaction is related to the conditions in the gas

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Mathematical Modeling

phase and is thus no longer coupled with the internal transport in the catalyst. This permits the description of reactors with heterogeneous catalysis as quasi-single-phase systems. The effectiveness factors η int or η ext can be calculated from the variables of state at the surface of the catalyst or in the gas phase, respectively. For the isothermal case and spherical geometry  ηeff =

1 ηint

+

Φ2K 3Bi

−1

where the Biot number Bi = β R Cat /Deff . Nonisothermal conditions and other geometries are discussed in [24], [123], [124]. LIVE GRAPH

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pression is of the first order with respect to the mass fraction of A. Inhibition by mass transport shifts the order of reaction in the direction of first order. Selectivity in parallel or secondary reactions will be similarly affected. The temperature dependence of the reaction rate is also influenced by inhibition by mass transport. If the expression r eff = − η eff k m G = − k∗ m G is used for a firstorder reaction, and the temperature dependence ∗ is given as usual by k∗ = k ∗0 e−E a /RT , then lnk∗ = lnk0∗ −

Ea∗ RT

(367)

For η eff = 1 there is chemical control, k ∗0 = k 0 and E ∗a = E a . If diffusion in the catalyst pores is rate controlling, η ≈ 3/ΦK . Then ln k ∗ = ln k 0 + ln 3  − ln ΦK − E a /R T . Substitution of ΦK =RCat k/Deff gives 1 1 1 Ea lnk∗ = lnk0 +ln3+ lnDeff − 2 2 2 RT

Figure 53. Effectiveness factor of a catalyst as a function of the Thiele modulus ΦK and the Prater number β H for spherical catalyst geometry (Arrhenius number γ = 20) [124] γ = Arrhenius number; η int = internal effectiveness factor.

The discussion in this section was based on first-order reactions with no changes in the mole number. Very few reactions in heterogeneous catalysis can be described with these expressions, higher order reactions are much more frequent. Furthermore the catalysts may be “poisoned” by adsorption of the reactants or products (see page 249) yielding nonintegral orders of reaction. Inhibition by mass transport influences the order of the catalytic reaction. In the extreme case r eff = β s (m G − m S ). This rate ex-

√ thus k0∗ = 3 k0 Deff and Ea∗ = 12 Ea . If the temperature dependence of the effective diffusion coefficients is neglected, the activation energy for pore diffusion-controlled reaction is about half that for chemical control. If control is exclusively due to mass transfer, then r eff ≈ − β s m G and k ∗ = β s ∼ Deff . Taking into account the temperature dependence of Deff , then lnk ∗ = lnk0∗ + 32 lnT which leads to an activation energy E ∗a of ca. 5 kJ/mol for this case. Thus the experimental determination of the effective formal activation energy for a heterogeneous catalytic reaction clarifies the extent of control by different transport processes. For further discussion of these relationships see [24], [123], [124]. In addition to the formulation of conservation equations for reactor models for heterogeneous catalytic reactions, combinations of simple reactor elements are frequently used for modeling these systems. For further information, see [12], [125]. Other complicated reactors (e.g., fluidized bed reactors) for heterogeneous catalytic reactions can be similarly dealt with [24], [123]. 5.3.3.2.2. Stability Analysis of Nonisothermal Reactors Steady-State Cases. The stability analysis of isothermal reactors discussed in Section

Mathematical Modeling 5.3.3.1 demonstrated the close connection of the phenomenon of multiple steady states in steadystate systems and the instability of nonsteadystate systems to a feedback mechanism. For the isothermal cases a chemical feedback mechanism was discussed in the form of autocatalysis and self-poisoning. Another feedback mechanism is the heat release from a chemical reaction. This will be discussed for a simple first-order reaction, A → B + heat, the temperature dependence of the rate coefficient is of the Arrhenius form, k = k0 e−Ea /RT . With increasing conversion of A heat is released. The released heat increases the temperature of the reaction mixture and the reaction rate. This in turn increases the rate of heat release. The reaction rate is an exponential function of the temperature. Therefore the feedback mechanism is nonlinear in temperature. Stability analysis for nonisothermal steadystate reactors will again be carried out for a wellstirred reactor (one element of a reactor cascade of ideally mixed reactors which is used to represent a flow reactor with convectively dominated flow). In analogy with Equation (310), the mass balance for A gives  (m0 −m) −mk0 e−Ea /RT =0 τ

(368)

From the enthalpy balance and transformation of Equation (240 b) into an equation for temperature, the following equation is obtained: cp  (T0 −T ) + (−∆hr ) mk0 e−Ea /RT τ − αs (T − TA ) =0

τN =

257

cp  cp k0 = αstchem αs

can be defined from the heat transfer. A reciprocal Arrhenius number γ r = RT A /E a is also introduced. With these abbreviations Equations (368) and (369) become   1−α θ =0 (370a) −αexp DaI 1+γr θ     θC 1 θ 1 θ+ − θad αexp + =0 1+γr θ DaI τN τN (370b)

Equations (370 a, b) contain the variables α and θ as a function of six parameters. To reduce the number of parameters, the case of very high activation energy, γ r  1, T A = T 0 , and θC = 0 is considered. With these assumptions the following simplified equations are obtained 1−α −αeθ = 0 DaI   1 1 θ=0 + θad αeθ − DaI τN

(371a) (371b)

In order to remain as close as possible to the notation used in Section 5.3.3.2, θ in Equation (371 a) is replaced by Equation (371 b) so that   1−α (1−α) θ ad  =αexp  DaI 1+ DaI

(372a)

τN

The temperature increase can easily be attributed to α from Equations (371 a, b): 

(369)

where the specific surface area s is the effective surface for heat transfer to the surroundings divided by the total volume of the ideally mixed reactor. The source term in Equation (240 b) is thus represented by a heat-transfer term. As in Section 5.3.3.1, Equations (368) and (369) are transformed into normalized variables: α = mA /mA0 , θ = (T − T 0 ) E a /RT 20 , θC = (T A − T 0 ) E a /RT 20 , and DaI = τ k 0 . The Damk¨ohler number (the ratio of the mean residence time to a characteristic chemical time, the time required until m/m0 = e−1 ) is defined with the inlet conditions. If the adiabatic temperature difference ∆T ad = (− ∆hr ) m0 /cp is also defined with E a /RT 20 , then θad = (− ∆hr ) m0 E a /cp RT 20 . Finally a “cooling time”

θ=

 (1−α) θ ad   I 1+ Da τ

(372b)

N

Equation (372 a) can be solved graphically by plotting both sides against the progress of the reaction 1 − α (conversion). In order to further reduce the number of parameters, ideal adiabatic conditions are assumed, i.e., τ N → ∞. Thus the left-hand side (LHS) of Equation (372 a) is only dependent on DaI , whilst the right-hand side (RHS) is only dependent on the adiabatic temperature increase. The solutions for the case discussed here (Fig. 54) are qualitatively similar to the solutions for the isothermal case with self poisoning (see page 243). Here too for a given residence time (Damk¨ohler number) multiple steady states occur for particular areas of

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Mathematical Modeling

the adiabatic temperature increase θad . Using the same argumentation as in Section 5.3.3.2, the stability condition Equation (318 a) d d (LHS) > (RHS) d (1−α) d (1−α) at LHS = RHS

(318a)

F (x, y;p, q, r, s, ...) = 0

where x is the variable for describing the steady state, y the bifurcation parameter which controls the multiple steady state, and p, q, r, s, . . . are additional parameters which produce folding or unfolding of the surface of x in space y and p, q, r, s, . . . LIVE GRAPH

is not complied with for the middle one of the multiple solutions. In this state a small perturbation of the conversion causes an increase or decrease in the “convective term” (LHS). The “reaction term” (RHS) however reacts in the same way so that a greater (smaller) availability of A due to convection is followed by a greater (smaller) consumption due to reaction. The system therefore tends towards one of the external stable steady-state solutions. As can be seen from Figure 54 A, the shape of the curve depends on the adiabatic temperature increase θad . Figure 54 B shows that as θad becomes smaller the S-shaped profile vanishes and the surface of solutions unfolds. Figure 54 B also demonstrates that the unstable branch of the solutions is not passed. The system jumps from low conversion to high conversion with increasing residence time DaI . On igniting and extinguishing the reaction, hysteresis occurs and the jumping points can be calculated from Equation (318 b) d d (LHS) = (RHS) d (1−α) d (1−α) at LHS = RHS



(373)

Since θad is always positive for exothermic reactions, then for the point at which the discriminant in Equation (373) disappears, θad = 4. If θad > 4, multiple steady states are observed. If the simplification γ r → 0 is discarded, the following equation is obtained for the steadystate solutions   1−α (1−α) θad =αexp DaI 1+γr θad (1−α)

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(318b)

For the example discussed here the jumping points are given by ! 

4 1 1± 1− (1−α) = 2 θad

(374b)

(374a)

The right-hand side of Equation (374 a) now depends on two parameters. In general it can be written in the form

Figure 54. Multiple steady-state conversion points for “normal” exothermic chemical reactions (A) and bifurcation diagram (B) for nonisothermal conditions in a well-stirred reactor θad = adiabatic temperature increase; for explanation of other symbols see Figure 45.

One or more steady-state solutions exist for the condition given by Equation (374 b). The steady-state solution is stable for F (x, y; p, q, r, s, ...) = 0 and Fx >0

(374c)

For the unfolding of the hysteresis loop the following applies F (x, y;p, q, r, s, ...) = 0, Fx >0, and Fxx = 0

(374d)

and “islands” finally occur in the bifurcation diagram which grow into “mushrooms” when

Mathematical Modeling mP =m0P e−k0 t or π = π0 e−ετ

F (x, y;p, q, r, s, ...) = 0, Fx = 0, Fy = 0, Fxy = 0, Fxx = 0, and Fyy = 0

(374e)

Depending on the process under consideration these conditions apply for various parameter combinations. For Equation (374 a) the condition for unfolding occurs when θad (4 γ r − 1) + 4 = 0. In the system of Equations (374), Fx , Fy , etc. denote the derivatives according to the variable in the index. Nonsteady-State Cases. As in the stability analysis of isothermal reactors, the stability of nonsteady-state nonisothermal reactors is investigated for a consecutive reaction P → A → B where A is produced from a large reservoir of P. The second reaction A → B is exothermic and its rate coefficient k1 is temperature dependent. For the reaction sequence P → A → B where k1 = k01 e−E a /RT , the mass and enthalpy balances for a batchwise operating ideally mixed reactor are given by dmP = −k0 mP dt dmA = k0 mP −k1 mA dt dT cp = (−∆hr ) k1 mA −αs (T − TA ) dt

(375) (376a) (376b)

(see page 245). The feedback in this system results from the temperature dependence of the second reaction rate which increases the reaction rate with increasing conversion. As previously, normalized variables are used: α = mA /mref , π = mP /mref , θ = (T − T A ) E a /RT 2A and γ r = RT A /E a . For the temperature-dependent reaction rate coefficient it thus follows that k1 =k01 e

−Ea /RTA

exp [θ/ (1+γr θ)]

The cooling time is not related to a characteristic chemical time τ N =  cp /α s, so that a normalized time can be given as τ = t/τ N . Thus the rate coefficients k 0 and k 1 are normalized so that ε = k 0 τ N and κ = τ N k 01 e−E a /RT A . Finally mref is established from mref =

2 αsRTA

Ea (−∆hr ) k01 e−Ea /RTA

There is no feedback to reservoir P thus Equation (375) can be integrated separately:

259 (377)

Using this Equations (376 a) and (377 b) give   dα θ = επ0 e−ετ − καexp dτ (1+γr θ)   θ =µ − καexp (1+γr θ)   dθ θ =αexp −θ dτ (1+γr θ)

(378a)

(378b)

where µ = ε π 0 e−ετ = µ0 e−ετ . For the sake of simplicity the case of high activation energy (i.e., γ r  1) will be investigated. Equations (378 a, b) are thus again simplified into dα = µ − καeθ dτ

(379a)

dθ = αeθ − θ dτ

(379b)

For very small values of ε (i.e., for a very much quicker time scale for the changes in α and θ than for the consumption of P) the principle of quasi-steady-state can again be used, and hence Equations (379 a, b) do not have to be integrated. The quasi-steady-state solutions for α and θ result from dα/ dτ = 0 and dθ/ dτ = 0, hence µ − καeθ = 0

(380a)

αeθ − θ= 0

(380b)

giving µ κ µ αqs = e−µ/κ κ

θqs =

(381a) (381b)

The approximate solutions αqs and θqs are a function of the ratio µ/κ, and also of time since µ is a function of time. With these operations and simplifications the quasi-steady-state solutions αqs and θqs in Equation (381) have the same form as for the isothermal case (Eq. 324). The stability analysis which now follows is carried out in the same way as for the isothermal case (Section 5.3.3.1). The development of Equation (380 a) into a Taylor series around the quasi-steady-state solutions provides a system of coupled differential

260

Mathematical Modeling

equations for the disturbances ∆α and ∆θ, the general solution of which is given by ∆α (τ ) =C1 eλ1 τ + C2 eλ2 τ

(382a)

∆θ (τ ) =C3 eλ1 τ + C4 eλ2 τ

(382b)

The eigenvalues λ1 , λ2 once again determine the characteristics of the time evolution of the disturbances. They are given by the solutions to the characteristic Equation (329) λ2 −Tr (J qs ) λ+Det (J qs ) = 0

(329)

i.e., λ1,2 =

6 1. Tr (J qs ) ± Tr (J qs )2 2

−4Det (J qs )}1/2

/ (330)

In this case the Jacobi matrix takes the form  J qs =

∂f (α,θ) ∂α ∂g(α,θ) ∂α

∂f (α,θ) ∂θ ∂g(α,θ) ∂θ

 (383)

µ κ

2 θqs ±1 e−θqs

(386a)

which satisfy the above condition for the discriminant to be zero. The appropriate values for µ are µ1,2 =κ1,2 θqs

(386b)

The stability diagram is best displayed in the form given in Figure 55 with the two parameters µ and κ. Two closed loops are obtained, the outer of which corresponds to the larger root from Equation (386 a) and the inner to the smaller. The area outside the outer curve represents stable nodes, stable or unstable foci lie between the two curves, and the area within the inner curve gives unstable nodes. The second case is the loss of stability of the system given by the condition Tr (Jqs ) = 0. This is also a condition for Hopf bifurcation. If µ in Equation (384 a) is replaced by Equation (381 a), this condition is given as

qs

The three quantities which decide the nature of the solutions are the discriminant of the characteristic equation Tr (Jqs )2 − 4 Det (Jqs ), the trace of the Jacobi matrix Tr (J qs ) =

κ1,2 =

−1−κeµ/κ

(384a)

and also their determinant Det (J qs ) =κeµ/κ

(384b)

As µ/κ is always positive, not all the combinations of λ1 , λ2 given in the stability diagram (Fig. 48) will be passed through with increasing conversion of P. Two states will now be discussed that are of practical importance. The first case is the transition of the solutions from a node of the system into a focus. The condition for this is that λ1 , λ2 become conjugate complex, the discriminant thus passes through zero. If µ in Equations (384 a, b) is replaced by Equation (381 a), the result is the following equation for the discriminant κ2 e2θqs − 2κ (1+θqs ) eθqs + (θqs − 1)2 = 0

(385)

For each positive value of θqs , there are thus two values of κ

κ= (θqs −1) e−θqs and µ=κθqs

(387a,b)

The pertinent curve in the stability diagram is a closed loop which lies between the two loops for the transition from nodes to foci. The region within this curve denotes unstable foci, see the enlarged section of Figure 55 in the area of the origin. The maximum of this curve is at e−2 . Stability analysis for nonisothermal nonsteady-state batchwise operating models exhibits the following important results: for each combination of the experimental conditions µ and κ there is an unambiguous solution for the quasi-steady-state concentration of the intermediate αqs and the normalized temperature increase θqs . If the normalized reaction rate coefficient is κ > e−2 , then the quasi-steady-state solutions are always stable. For κ ≈ 1.78 transition from stable nodes to stable foci occurs. For κ < e−2 the quasi-steady-state solutions are unstable. At the location of the Hopf bifurcations two solutions occur (Fig. 55). Between the two bifurcation points for αqs and θqs as functions of the conversion (i.e., as functions of µ), oscillations of varying frequency and amplitude occur.

Mathematical Modeling

261

is given in Section 5.1.2.1. The averaged equations then take the form

LIVE GRAPH

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Figure 55. Stability diagram for exothermic consecutive reactions for quasi-steady-state nonisothermal conditions [113] For explanation of symbols see Figure 49 and text.

For engineering purposes it is important that, for the simplified case discussed here, stability can be controlled through the parameter κ. For κ > e−2 the system remains stable. If the properties of the chemical reaction are given by k 1 = k 01 e−E a /RT , stability can only be achieved by altering the cooling time by means of increased heat transfer to the surroundings. Detailed discussions on the stability of nonisothermal reactors can be found in [113], [114].

∂ 9 :9 : ∂ 9 :9 :9 :  Uk Ui  Ui + ∂t ∂xk 9 : : 9 :9 : ∂ p ∂ 9 τtik − +  gi , = ∂xk ∂xi i = 1, 2, 3

(388)

which is a transport equation for the first moment of the velocities. The accompanying problem of describing the turbulent viscosity was solved using the eddy viscosity hypothesis, µt = Cµ  (k2 /ε) . As well as the solution of Equation (388), the model requires the solution of the equations for the turbulence energy, k (Eq. 222), and its dissipation rate, ε (Eq. 225), thereby defining a turbulent time scale according to τ = k/ε . A numerical solution that takes into account the special nature of the averaged conservation equations of momentum is demonstrated in Section 5.1.2.2. For nonisothermal reacting systems this concept has to be expanded to the equations for the conservation of enthalpy (Eq. 134 or 240 a) and for the conservation of mass of the chemical species (Eq. 133 or 265 a). Averaging Equations (240 a) and (265 a) using the same assumptions as in Section 5.1.2.1 results in the equations ∂ 9 :9 : ∂ 9 :9 :9 :  h +  Uk h ∂t ∂xk  : 9 h : 9 :9 f f : 9 ∂ = − J +  h Uk + Sh ∂xk

(389)

and 5.3.3.2.3. Use on Statistical Processes The description of statistical processes by averaging the conservation equations of momentum is discussed in Section 5.1.2 for isothermal inert systems. The averaged equations however contain terms with higher moments which must be modeled; in Equation (216) these are the Reynolds stresses  U fi U fj . A model for the Reynolds stresses which shows them in analogy to the laminar viscous stresses as an isotropic tensor of turbulent viscous stresses τtij =µt

 9 : ∂ Ui ∂xj

+

9 : ∂ Uj ∂xi

−

9 : 2 ∂ Uk ∂ij µt 3 ∂xk

∂ 9 :9 : ∂ 9 :9 :9 :  mi +  Uk mi ∂t ∂xk  : 9 m : 9 :9 f f : 9 ∂ =− J i +  mi Uk + Si ∂xk i=1, ...,N − 1

(390)

The problem now lies in modeling the Reynolds fluxes  hf U fj  and  mfi U fj , and also the expected values of the source term  Si  and  Sh . Modeling the Reynolds Fluxes. The problem of modeling the Reynolds fluxes  hf U f j  and  mf i U f j  can be solved analogously to the modeling of the Reynolds stresses. As with the transport equations for

262

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the Reynolds stresses (see Section 5.1.2.1, Eq. 221), the analogous transport equation for the Reynolds fluxes is not closed however. The same reasons as were discussed in Section 5.1.2.1 for the Reynolds stresses, lie behind the fact that transport equations for Reynolds fluxes are usually not solved in engineering but are modeled directly. The direct modeling of Reynolds fluxes is based on the Boussinesq hypothesis and assumes that there is a similarity between the turbulent transport of momentum, and enthalpy, and mass. This principle has already been used in Sections 5.2.2.1 and 5.3.1.1 for deterministic systems. The turbulent flux of a scalar quantity is thus related to the gradients of the expected values of the scalar. The proportionality number is a turbulent transport coefficient which is obtained from the turbulent viscosity: 9 : : 9 :9 µt ∂ Φ −  Φf Ukf = σϕ ∂xk

(391)

This expression has already been used in the equation for turbulence energy (Eq. 223); σϕ is a turbulent Prandtl or Schmidt number, which relates the turbulent transfer of the scalar in question and the turbulent transfer of momentum. With the help of the eddy viscosity hypothesis, σϕ can be interpreted as the ratio of the mixing length of the velocity fluctuations to the mixing length of the fluctuations in the scalar. With this hypothesis and neglecting the proportion of laminar flux to turbulent flux, Equations (389) and (390) become ∂ 9 :9 : ∂ 9 :9 :9 :  h +  Uk h ∂t ∂xk 9 : : 9 ∂ µt ∂ h + Sh = ∂xk σh ∂xk

(392)

and ∂ 9 :9 : ∂ 9 :9 :9 :  Uk mi  mi + ∂t ∂xk 9 : : 9 ∂ µt ∂ mi = + Si ∂xk σm ∂xk i = 1, ...,N − 1

(393)

The remaining problem is now the modeling of the terms  Sh  and  Si . If the change in enthalpy due to mechanical work and radiation are neglected in Equation (392), then the term  Sh  vanishes. A modeling expression then only needs to be found for  Si .

Modeling of the Mean Reaction Rates. The discussion of models for the mean reaction rates in turbulent (statistical) flows is of similar importance in engineering to the modeling of Reynolds stresses and Reynolds fluxes. Reviews are given in [126–128]. The reaction rates are generally a function of mass fractions and temperature (Eq. 137 a). The expected values  Si  are obtained from Equation (109) since the averaging rule (Eq. 107) is applicable to functions. If changes in density are neglected, then Equation (137 a) gives 9

: Si =


T0
1

Tu 0


1 P (ml , ...,mo , T )

... 0

·Si (ml , ...,mo ,T ) dml ...dmo dT

(394)

In Equation (394) P (ml , . . ., mo , T ) is a joint probability density function of temperature and the mass fractions of the chemical species that are contained in the expression for the reaction rates. Si (ml , . . ., mo , T ) are usually known functions and hence to determine Si , the joint probability density functions have to be established. The example shown in Figure 14 demonstrates that a priori determinations for this are difficult; there are intermittencies in the region at the jet boundary so that P (ml , . . ., mo , T ) becomes “bimodal”. The bimodal structure is not dominant further downstream and within the turbulent jet. An acceptable assumption for the form of P (ml , . . ., mo , T ) is a multidimensional normal distribution, which must be suitably clipped due to the restricted domain of definition of temperature and mass fractions. Although this form does not cover the intermittencies at the boundary of the turbulent jet, it is a good approximation for the inside of the turbulent jet and further downstream. For many applications the deficiencies in representation by a multidimensional normal distribution is of no consequence because the mean reaction rates in the areas with strong intermittencies are mostly small compared with those inside the turbulent jet. If the approximation of P (ml , . . ., mo , T ) by means of a multidimensional normal distribution is adequate, the form of P (ml , . . ., mo , T ) must be established quantitatively. As in the one-dimensional case, this is achieved through the expected values

Mathematical Modeling and the second moments. Consequently in order to determine P (ml , . . ., mo , T ) calculation of ml , . . . , mo , T , mfi mfj , i, j = l, o, and mfi T f , i = l, o is necessary. Equation (394) can be presented in the form 9

: 9 : 9 : 9 : Si =Si ml , ..., mo , T  9 :9 : ·F mfi mfj , mfi T f ,i, j=l,o

(395)

in which F (mfi mfi , mfi T f , i, j = l, o) are correction functions for reproducing the effect of turbulent fluctuations in temperature and mass fractions in the averaging procedure of weighting with the joint probability density functions. These correction functions can be given as polynomials of the second moments mfi mfj , mfi T f  (or more precisely of the turbulence intensities and correlation coefficients) and of the activation energies for the reactions in question [129], [130]. The coefficients of these polynomials can be obtained by applying the averaging procedure (Eq. 394) for a number of predefined values of mfi mfj  mfi T f  and E a ; Si  is then represented as a function of these quantities by an empirical polynomial expression [129]. Thus integration of Equation (394) is unnecessary for the numerical solution of the resulting system of equations. Calculation of the expected values Si  of the reaction rates from Equation (395) assumes knowledge of the second moments mfi mfj , i, j = 1, . . . , N and mfi T f , i = 1, . . . , N. In this case too closure of the equation system at the level of the expected values necessitates calculation of the second moments. The relevant transport equations can be derived by similar considerations and modeling assumptions used for deriving the equation for other second moments, see Section 5.1.2.1 [52], [126–128], [130]. The numerical work involved in calculating a statistical reacting nonisothermal flow is considerable, despite the relatively simple structure of the modeling components. In addition to the solution of the balances of momentum, direct modeling of Reynolds stresses requires the solution of two further equations for turbulence energy and the turbulent time scale. The Reynolds fluxes are modeled on the same basis as the Reynolds stresses, so that apart from the enthalpy balance (Eq. 392) and the transport equations for the mass of the chemical species

263

(Eq. 393), no other equations originate from this part of the model. Modeling of the mean chemical reaction rates through the simple expression (Eq. 394) with presumed shape joint probability density functions requires the solution of equations for the variances and covariances mfi mfj , i, j = 1, . . . , N, and mfi T f , i = 1, . . . , N. In addition the density has to be calculated from the averaged form of the thermal equation of state 9 : N 9 : p  mi 9 :9 : 9 :  = R i=1 T mi + mfi T f

(396)

which also contains the second moments mfi T f , i = 1, . . . , N. Equation (396) can easily be derived from the ideal gas equation. The temperature must be determined from the expected value of the enthalpy of the mixture through the definition of the enthalpy: N 9 :  9 : h = mi hi = i=1 N 



9

mi




h0i +

cpi dT

:

(397)

i=1

Equation (397) is not solved here for the temperature, for further details see [129–131]. Example: Combustion of Hydrogen in a Turbulent Diffusion Flame. The simulation of a turbulent nonisothermal reacting flow by the system of equations outlined above will be demonstrated briefly on an example. A circular jet of hydrogen with a Reynolds number of 12 000 relative to the nozzle outlet is injected into virtually stationary air and is burned in a horizontal combustion chamber. In the experiments described in [132–134], velocities, temperatures, and concentrations of stable chemical species and of OH radicals are measured downstream of the burner nozzle. The system is axially symmetrical. The numerical solution contains the solutions of Equations (223), (225), (388), (392), (393) and also of the equations for the variances and covariances mfi mfj , i, j = 1, . . . , N and mf i T f , i = 1, . . . , N in axially symmetrical formulation. The numerical procedure is the same as that outlined for the isothermal case in Section 5.1.2.3. The chemical reactions are described by 44 elementary reactions between

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eleven chemical species: H2 , O2 , H2 O, N2 , H, O, OH, HO2 , H2 O2 and also N and NO [135]. The reaction rates of each of these components consists of the summation of reaction rates of all the elementary reactions j in which the compoR
nent in question i occurs, i.e., ri = rij . The

lates them to the gradients of the mean velocities, large sensitivities inevitably result in response to changes in the apparent turbulent viscosity in areas with high velocity gradients.

j=1

rate coefficients of the reactions are expressed in the form kj = k0j T αj e−E a /RT and are thus not freely selectable modeling parameters. For details of the reaction rate coefficients of the H2 –O2 system see [130], [135]. The modeling parameters used are the coefficients Cµ = 0.09, C ε1 = 1.4, C ε2 = 0.925, σ k eff = 1.0, σ ε eff = 1.3; the turbulent Prandtl numbers in Equations (392) and (393); and also the equations for the second moments which are defined as σϕ = 0.7 [91]. Figure 56 shows the result of the simulations from the model compared with the measured results given in [132–134]. The correspondence between the measured results and the model is satisfactory. In particular the penetration of oxygen to the axis of the turbulent jet close to the nozzle is well predicted. Along with the calculation of the OH radical concentrations, this is a particular potential of the model that is based on an elementary reaction scheme for describing the chemical reactions for the H2 – O2 – N2 system. Another aspect of the model presented here can be shown with the results of sensitivity analysis. The first-order sensitivity coefficients are calculated as described in Section 5.3.3.1.2. Due to the elliptical form of the conservation equations and the difference form which follows from this (see Section 5.1.2.2, Equation 232), the resulting system of equations has a block pentadiagonal structure and is solved by an ADI procedure. Figure 57 A shows the results in the form of the gradients of the axial velocity U  with respect to the model parameter Cµ , which controls the turbulent viscosity. The sensitivity coefficients are given in relative form (∂U/U)/ (∂Cµ /Cµ ) as in Equations (29) and (30). The sensitivity with respect to a change in the parameter Cµ is particularly large in areas with high velocity gradients. The sensitivity analysis thus reflects the physical background and the limitations of the model used: since the turbulent stresses are described by an approach which re-

Figure 57. Relative sensitivity coefficient for a H2 –O2 diffusion flame A) Coefficients for the axial velocity U  with respect to the modeling parameter Cµ ; B) Coefficients for the mole fraction of oxygen X O2  with respect to the rate coefficient k 01 x = axial distance; r = radial distance.

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Mathematical Modeling

265 LIVE GRAPH

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Figure 56. Measured (upper section) and calculated (lower section) profiles for temperature, mole fraction of oxygen and water, and mass fraction of OH radicals for a turbulent hydrogen – air diffusion flame [130], [131] x = axial distance; y = radial distance; D = nozzle diameter

Figure 57 B shows the gradients of the molar fractions for oxygen with respect to the rate coefficient for the reaction O2 + H → OH + O, this is one of the most important oxygenconsuming reactions. Here to, the sensitivity coefficients are given in relative form (∂X O2 /X O2 )/(∂k 01 / k 01 ). Sensitivity is only obvious in the main reaction zones of the turbulent hydrogen flame. Thus, the molar fraction of oxygen only reacts sensitively to the change in these rate coefficients in the main reaction zone of the flame.

The result of averaging the reaction rates with Equations (394) and (395) can be interpreted as the change in the rate coefficients due to the correction functions. The unsatisfactory representation of the joint probability density function P (ml , . . . , mo , T ) by a multidimensional normal distribution used to calculate these correction functions is therefore no longer of importance at the edge of the jet for the prediction of the oxygen molar fraction. This is due to the low reaction rates for the consumption of oxygen in this region.

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Mathematical Modeling

For further discussion of sensitivity analysis, especially for discussion of the possibility of reduction of the chemical models, see [131].

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Mathematical Modeling 122. E. Kreyszig: Advanced Engineering Mathematics, J. Wiley and Sons, New York 1988. 123. G. F. Fromment, K. B. Bishoff: Chemical Reactor Analysis and Design, J. Wiley and Sons, New York 1990. 124. P. B. Weisz, J. S. Hicks, Chem. Eng. Sci. 17 (1962) 265. 125. J. Ganoulis, F. Durst: “Finite Elements in Water Resources,” Proceedings VI Int. Conf., Lisbon 1986, p. 655. 126. S. N. B. Murthy (ed.): Turbulent Mixing in Nonreactive and Reactive Flows, Plenum Press, New York – London 1975. 127. R. W. Bilger in P. A. Libby, F. A. Williams (eds.): Turbulent Reacting Flows, Springer Verlag, Berlin – Heidelberg 1980, p. 65. 128. R. Borghi, Prog. Energy Combust. Sci. 14 (1988) 245.

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129. H. Bockhorn in C.-M. Brauner, C. Schmidt-Lain´e (eds.): Mathematical Modeling in Combustion and Related Topics, Martinus Nijhoff Publishers, Dordrecht 1988, p. 411. 130. H. Bockhorn: Twenty-Second Symposium (International ) on Combustion, The Combustion Institute, Pittsburgh 1988, p. 665. 131. H. Bockhorn: Twenty-Third Symposium (International ) on Combustion, The Combustion Institute, Pittsburgh. 132. M. C. Drake et al.: Twentieth Symposium (International ) on Combustion, The Combustion Institute, Pittsburgh 1984, p. 327. 133. M. C. Drake, R. W. Pitz, M. Lapp: AIAA Pap. 84-0544 (1984). 134. G. M. Faeth, G. S. Samuelson, Prog. Energy Combust. Sci. 12 (1986) 305. 135. J. Warnatz in W. C. Gardiner, Jr. (ed.): Combustion Chemistry, Springer Verlag, New York 1984, p. 196.

Transport Phenomena

271

Transport Phenomena See also → Mathematics in Chemical Engineering and Fluid Mechanics Raymond W. Flumerfelt, Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843, United States Charles J. Glover, Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843, United States 1. 1.1. 1.1.1. 1.1.2. 1.1.3. 1.1.4. 1.1.5. 1.1.6. 1.1.7. 1.1.8. 1.2. 1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.2.5. 1.2.6. 1.2.7. 1.2.8. 1.2.9. 1.2.10. 1.2.11. 1.2.12. 1.2.13. 1.2.14. 1.2.15. 1.3. 1.4. 1.5. 1.6. 2. 2.1. 2.1.1. 2.1.2.

Foundations . . . . . . . . . . . . . . . Mathematical Preliminaries . . . . Coordinate Systems . . . . . . . . . . Vector and Tensor Operations . . . . The Jacobian . . . . . . . . . . . . . . . Calculus of Vectors and Tensors . . . Divergence Theorem . . . . . . . . . . Kinematic Relations . . . . . . . . . . Partial and Total Derivatives . . . . . Relation Between Different Time Derivatives . . . . . . . . . . . . . . . . Basic Equations for Compositionally Homogeneous Systems . . . . . The Reynolds Transport Theorem . . Conservation of Total Mass . . . . . . Conservation of Linear Momentum . Condition of Local Stress Equilibrium . . . . . . . . . . . . . . . . . . . . Stress Tensor . . . . . . . . . . . . . . . Equation of Motion . . . . . . . . . . . Conservation of Angular Momentum Mechanical Energy Accounting Equation . . . . . . . . . . . . . . . . . Conservation of Total Energy . . . . Thermal Energy Accounting Equation . . . . . . . . . . . . . . . . . . . . . Forms of the Governing Equations . Entropy Inequality . . . . . . . . . . . Linear Transport Fluxes and Relations . . . . . . . . . . . . . . . . . . . . Transport Properties from Molecular Theories . . . . . . . . . . . . . . . . . . Non-Newtonian Fluids . . . . . . . . Summary of Basic Equations . . . Boundary Conditions . . . . . . . . Solution Philosophy . . . . . . . . . . Dimensionless Equations of Change . . . . . . . . . . . . . . . . Transport in Compositionally Homogeneous Systems . . . . . . . . Heat Conduction in Solids . . . . . Heat Conduction Equations . . . . . Initial and Boundary Conditions . . .

278 279 279 279 280 281 282 282 283 283 283 283 284 284 285 285 285 285 286 286 286 287 287 287 288 289 290 291 292 292 294 294 294 295

2.1.3. 2.1.4. 2.1.5. 2.1.6. 2.1.7. 2.1.8. 2.2. 2.2.1. 2.2.2. 2.2.3. 2.2.4. 2.2.5. 2.3. 2.3.1. 2.3.2. 2.3.3. 2.4. 2.4.1. 2.4.2. 2.4.3. 2.5. 2.5.1. 2.5.2. 2.5.3. 2.5.4.

Steady, One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . Convective Boundary Conditions . . Composite Systems . . . . . . . . . . Steady Conduction with Heat Generation and in Multidimensions . . . . Transient Heat Conduction – Lumped Capacity Systems . . . . . . . . . . . . Transient Heat Conduction – More General Solutions . . . . . . . . . . . . Steady, One-Dimensional Flows . . Generalized Couette Flow . . . . . . One-Dimensional Poiseuille Flows . Flow in Channels of Arbitrary Cross Section . . . . . . . . . . . . . . . . . . Poiseuille Flow of Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . Two-Phase Concentric Flow in a Tube – Segregated Flow . . . . . Multidimensional Momentum Transfer . . . . . . . . . . . . . . . . . Two-Dimensional Flows – Stream Function Equations . . . . . Creeping Flow Around a Sphere and Other Bodies of Revolution . . . . . . Flow in Channels with Varying Cross Sections – Lubrication Analysis . . . Coupled Momentum and Energy Transfer . . . . . . . . . . . . . . . . . Heat Transfer in Laminar Tube Flow Momentum and Heat Transfer in Laminar Boundary Layers . . . . . . Free Convection on a Vertical Plate . Turbulent Momentum and Energy Transfer . . . . . . . . . . . . . . . . . Physical Characteristics . . . . . . . . Time-Smoothed Momentum and Energy Balances for Turbulent Flow . . Mixing Length Theories . . . . . . . . Turbulent Heat Transfer in Tubes – Reynolds, Prandtl, and von Karman Analogies . . . . . . . .

296 297 298 298 299 299 302 303 304 305 306 308 309 309 310 313 316 317 319 323 324 324 327 328

329

272 2.5.5. 2.6. 3. 3.1. 3.1.1. 3.1.2. 3.1.3. 3.1.4. 3.1.5. 3.1.6. 3.1.7. 3.1.8. 3.1.9. 3.2. 3.2.1. 3.2.2. 3.3. 3.3.1.

Transport Phenomena Turbulent Momentum and Heat Transferon a Flat Plate . . . . . . . . . Summary of Heat-Transfer Relations . . . . . . . . . . . . . . . . . Transport in Multicomponent Systems . . . . . . . . . . . . . . . . . . Diffusive Mass Transfer in Binary Systems . . . . . . . . . . . . . . . . . . Species Mass Balances . . . . . . . . Species Diffusion Fluxes . . . . . . . Fick’s First Law of Diffusion . . . . Special Cases of Diffusion . . . . . . Diffusivities of Gases and Liquids . Theoretical Foundations of SteadyState Measurement of Diffusion . . . Diffusion with Homogeneous Chemical Reaction . . . . . . . . . . . . . . . Diffusion with Heterogeneous Reaction . . . . . . . . . . . . . . . . . . . . . Perspective . . . . . . . . . . . . . . . . Convective Mass Transport . . . . Gas Absorption in a Falling Film with Reaction . . . . . . . . . . . . . . . . . Mass Transfer in Laminar and Turbulent Boundary Layers . . . . . . . . Mass Transfer Across Interfaces . Mass-Transfer Coefficients . . . . . .

3.3.2. 331 3.3.3. 332 3.3.4. 332 332 332 336 336 337 338 339 342 343 345 345

Aw Aξ Aˆ ATSB b b∗ b 0 , bL B Bi

4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.9.1.

345 348 349 350

Symbols a, b a a, b, c A A A, B Ac An

4. 4.1. 4.2.

major and minor axes of spheroids acceleration arbitrary vectors area area vector (magnitude and direction) arbitrary second-order tensors cross-sectional area numerical parameters defined by Equation (2.174) wall surface area area perpendicular to ξ-direction Helmholtz free energy per unit mass A through stagnant B diffusion situation flow geometry height dimensionless flow channel height, defined below Equation (2.174) flow channel vertical dimensions at x = 0 and x = L, Figure 13 applied load, Figure 13 Biot number; Equation (2.19) and definition below Equation (2.27)

4.9.2. 4.9.3. 5.

Br c

Functional Forms of Mass-Transfer Coefficient Relations . . . . . . . . . . Relations for Low Mass-Transfer Rates . . . . . . . . . . . . . . . . . . . . Relations for High Mass-Transfer Rates . . . . . . . . . . . . . . . . . . . . Macroscopic Systems . . . . . . . . . Conservation of Total Mass . . . . Conservation of Linear Momentum . . . . . . . . . . . . . . . Conservation of Angular Momentum . . . . . . . . . . . . . . . An Accounting of Mechanical Energy . . . . . . . . . . . . . . . . . . Conservation of Total Energy . . . An Accounting of Thermal Energy The Second Law of Thermodynamics . . . . . . . . . Tabular Summary of the Macroscopic Equations . . . . . . . Some Examples . . . . . . . . . . . . Pressure Drops and Temperature Changes for Closed Channel Flow . Compressible Flow in a Tube . . . . Heterogeneous Reaction in a Fluidized-Bed Reactor . . . . . . . . . References . . . . . . . . . . . . . . . .

352 353 354 356 356 357 358 358 359 360 360 362 362 362 363 364 366

Brinkman number, Equation (1.118a) molar concentration, moles per volume cp constant pressure heat capacity per unit mass cv constant volume heat capacity per unit mass cA , cB , ci moles of A, B, i in a mixture per unit volume of mixture c∗A dimensionless molar concentration of species A c¯ A time-averaged value of cA cA0 molar concentration at interface cA∞ molar concentration of A in free stream, or in bulk fluid in contact with surface C constant CD drag coefficient defined by Equation (2.155) C Dx local drag coefficient defined in Equation (2.235) C DL overall drag coefficient defined by Equation (2.236)

Transport Phenomena Ck coefficients in Equation (2.49) C 1 , C 2 , Cˆ 1 , Cˆ 2 integrating constants d diameter d/dt time derivative seen by an observer moving at arbitrary velocity; the total time derivative of a function D/Dt time derivative seen by observer maintaining constant material coordinates (following the motion of an element, Eq. 1.50) DAB diffusivity of species A in B D◦AB diffusivity of infinitely dilute A in solvent B Daδ Damk¨ohler number based on film thickness, Equation (3.100) DaL Damk¨ohler number based on film length e height of wall roughness (protuberances) eijk permutation symbol, Equation (1.12) E2 differential operator, Equation (2.137) E (k) complete elliptic integral with argument k ECD equimolar counter diffusion situation f in Section 2.4.1, function of r defined by Equation (2.197); in Section 2.3.2, function of r defined by Equations (2.133) and (2.134); in Section 2.3.3, a function defined below Equation (2.162); friction factor defined by Equation (2.79) F a function of time (1.56) F force, a vector FB buoyancy force FD drag force external force Fext F in , F out volumetric flow rate entering and leaving a system Fx , Fy , Fz force components F∗x , F∗y , F∗z dimensionless force components F in Chapter 1, an arbitrary scalar, vector, or tensor function of time and position; in Section 2.5.2, any linear time or spatial derivative operator Fr Froude number, Equation (1.114) Fˆ conversion of mechanical energy to thermal energy by viscous dissipation, per unit mass (fluid friction losses) g body force per unit mass or gravitational acceleration

273

g i , g j , g k i, j, k base vectors in a curvilinear coordinate system gx , gy , gz components of gravitational force per unit mass, in x, y, z directions G scalar function in Equation (2.125) Grx local Grashoff number, Equation (2.273) GrL Grashoff number associated with surface of length L Gˆ Gibbs free energy per unit mass h heat-transfer coefficient, Equation (2.6) hk heat-transfer coefficient associated with surface at position xk hlm log mean heat-transfer coefficient, Equation (2.206) hx,lam local heat-transfer coefficient, laminar boundary layer hx, turb local heat-transfer coefficient, turbulent boundary layer hL overall heat-transfer coefficient over surface of length L, Equation (2.254) h¯ average heat-transfer coefficient over surface A Hˆ enthalpy per unit mass ii , ij , ik base vectors in a rectangular cartesian coordinate system ix , iy , iz unit base vectors in x, y, z directions ir , iθ , iz unit base vectors in cylindrical coordinates ir , iθ , iφ unit base vectors in spherical coordinates I identity tensor I∆ , II∆ , III∆ first, second, and third scalar invariants of second-order tensor ∆, defined by Equations (1.107) – (1.109) Iu intensity of turbulence, Equation (2.280) Iφ a measure of the fluctuating part of φ in a turbulent flow, Equation (2.279) jA mass diffusion flux of species A relative to u, Equation (3.16) j˜A mass diffusion flux of species A relative to u, ˜ Equation (3.19) J Jacobian relating differential volume elements in two coordinate systems and defined by Equations (1.23) and (1.24) Jn ( ) Bessel functions of the first kind with argument ( ) and n = 0, 1, . . . JA molar diffusion flux of species A, relative to u, Equation (3.18)

274 J˜ A k

k
k
y kc k cL kG kx ky k1 K K∗ Kc KG Kx Ky ˆ KE l L L m, n n

Transport Phenomena molar diffusion flux of species A, relative to u, ˜ Equation (3.14) arbitrary constant, Equation (1.4); in Section 2.2.3, defined below Equation (2.94); in Section 2.5.3, proportionality constant in Equation (2.299); reaction rate constant, Equation (4.29) reaction rate constant, Equation (3.98) drift factor-corrected gas-phase masstransfer coefficient single-phase mass-transfer coefficient based on molar concentrations average mass-transfer coefficient defined by Equation (3.112) gas-phase mass-transfer coefficient based on gas-phase partial pressures, Equation (3.132) liquid-phase mass-transfer coefficient based on liquid-phase mole fractions gas-phase mass-transfer coefficient based on gas-phase mole fractions first-order reaction rate constant pressure gradient defined by Equation (2.68); overall reaction rate constant, Equation (4.31) dimensionless pressure gradient, defined by Equation (2.73) overall two-phase mass-transfer coefficient based on liquid-phase molar concentrations overall two-phase mass-transfer coefficient based on gas-phase partial pressures, Equation (3.134) overall two-phase mass-transfer coefficient based on liquid-phase mole fractions overall two-phase mass-transfer coefficient based on gas-phase mole fractions kinetic energy per unit mass characteristic length, Equation (2.82); in Section 2.5.3, the Prandtl mixing length angular momentum characteristic length for a process or flow, Chapter 1; length of body, surface, or flow channel material parameters for power-law non-Newtonian fluid, Equation (2.97) coordinate in direction normal to surface, Equation (2.6)

n

outwardly directed unit normal on surface enclosing volume V NA molar flux (vector) of species A N Ay mole flux of A in y-direction N¯ Ay time-averaged value of N Ay NuAB mass-transfer Nusselt number, NuAB = kx d/c DAB Nud Nusselt number based on diameter, Equation (2.202) Nud, lm log mean Nusselt number, Equations (2.205) and (2.206) NuL overall Nusselt number associated with surface of length L, Equation (2.255) Nux local heat-transfer Nusselt number, Equation (2.251) O() order of magnitude of ( ) p linear momentum p pressure, the isotropic stress p0 reference pressure pA , pB partial pressure of species A, B pH hydrostatic pressure, Equation (2.63) p¯ time-averaged pressure P dynamic pressure, Equation (2.62) P0 reference dynamic pressure ∆P pressure drop across flow channel in direction of flow Pe P´eclet number, Equation (1.117) PeL mass-transfer P´eclet number based on film length Peδ mass-transfer P´eclet number based on film thickness, Equation (3.83) Pr Prandtl number ( ≡ ν/α) PˆE potential energy per unit mass q a dimensionless parameter defined by Equation (3.103) q heat flux vector (heat flow rate per unit area) q¯ time-averaged heat flux qn heat flux component normal to surface, Equation (2.6) qr , qθ , qz heat flux components in cylindrical coordinates qw wall heat flux qx , qy , qz heat flux components in x-, y-, and z-directions (t) qy turbulent heat flux component in ydirection, Equation (2.301) qξ heat flux in ξ-direction Q energy transferred as heat, Chapter 4; volumetric flow rate, Chapter 2

Transport Phenomena Q, Qˆ

volumetric flow rates in inner and outer regions, respectively, Section 2.2.5 Q∗ defined by Equation (2.110) Qˆ heat transfer per unit mass Q˙ heat transfer rate r molecular separation, Lennard – Jones function r position vector r∗ dimensionless radial position r A , r B , ri mass rate of production of species A, B, i per unit volume due to reaction (r A > 0 for generation of A; r A < 0 for consumption of A) rc radial position of interface between inner and outer phases r ∗c dimensionless radial position, defined below Equation (2.107) rH hydraulic radius defined below Equation (2.96) rY radial position of yield surface r0, r1 radii associated with cylinder or spherical surfaces R ideal gas constant; in Section 2.3.3, the flow channel radius RA , RB , Ri molar rate of production of species A, B, i per unit volume of mixture due to reaction (RA > 0 for production of A; RA < 0 for consumption of A) Rcond conductive resistance, Equation (2.14) Rconv,i convective heat-transfer resistance, Equation (2.24) Re Reynolds number, Equations (1.113) and (2.81) Recr critical Reynolds number for laminar to turbulent transition Red Reynolds number based on diameter d, defined below Equation (2.127) ReL Reynolds number based on length L Rex local Reynolds number, Equation (2.231) Reδ Reynolds number based on length scale δ r, θ, z coordinates in cylindrical coordinate system r, θ, φ coordinates in spherical coordinate system s defined below Equation (2.98) s displacement vector S control volume surface Sc Schmidt number ( ≡ v/DAB )

275

Sherwood number ( ≡ kx d/cDAB ) Sherwood number based on length L, Equation (3.95) Shx local Sherwood number defined below Equation (3.110) St Stanton number, Equation (2.311) Sˆ entropy per unit mass S˙ gen entropy generation rate S˙ in rate of entropy addition t time t exp diffusion penetration theory exposure time, Equations (3.136) and (3.137) t(n) traction vector acting on surface A, Equation (1.67) t0 in Section 2.5.1, time over which quantities are averaged; elsewhere, a reference time t∗ dimensionless time, defined below Equation (2.27) tr trace operation of a second-order tensor, Equation (1.8) T absolute temperature T total stress tensor, a symmetric second-order tensor T∗ dimensionless temperature, Equation (2.34) Tb bulk average temperature, Equation (2.195) Tf film temperature [≡ (T w + T ∞ )/2] in Section 2.5.5 and Table 7; bulk fluid temperature Tfk temperature of bulk fluid next to surface at xk Ti initial temperature Tk temperature at position xk T0 characteristic reference temperature for a process Tw wall temperature T w0 , T wL wall temperatures at positions z = 0 and z = L Tx , Ty , Tz , Tr solutions defined below Equations(2.54) – (2.56) Tξ time-averaged temperature at y = ξ T∞ temperature in free stream T 0 , T 1 , T 2 temperatures at positions x 0 , x 1 , and x2 ∆T ratio of momentum and thermal boundary layer thicknesses ( ≡ δ m /δ T ) ∆T lm log mean temperature difference, Equation (2.207) Sh ShL

276 T¯

Transport Phenomena

average spatial temperature in a region, Section 2.1.7; time-averaged temperature, Section 2.5.2 u, u velocity vector (momentum per unit mass) of magnitude u; in a mixture: a mass average velocity, Equation (3.8) u
fluctuating part of velocity in turbulent field u
x , u
y , u
z fluctuating velocity components uavg average velocity uc characteristic velocity, Equation (2.268) ui velocity of species i in a mixture uinterface fluid velocity at a gas–liquid interface umax maximum velocity ux , uy , uz components of velocity in rectangular cartesian system u0 characteristic velocity, Equation (2.159) u∞ free stream velocity; see Figures 10, 11, 14, 15, 16 u∗ friction velocity, Equation (2.286) u∗ dimensionless velocity u+ dimensionless velocity, Equation (2.284) u˜ mixture molar average velocity, Equation (3.12) u˜ time-averaged velocity in turbulent field defined by Equation (2.277) u¯ x , u¯ y , u¯ z time-averaged velocity components uz , uˆ z velocity components in inner and outer regions, respectively, Section 2.2.5 u¯ ξ time-averaged velocity at y = ξ U characteristic velocity for a process or flow; velocity of moving surface, Figure 5; velocity of lower plate, Figure 13 Uˆ internal energy per unit mass V volume VA molar volume of species A, volume per mole V0 material volume in the reference coordinates Vˆ volume per unit mass ( ≡ 1/) w arbitrary velocity of an observer, Equation (1.55) w flow geometry width wA mass fraction of species A in a mixture: wA = A / W film width (in x direction), Figure 25; work, Equation (4.8)

WA ˙ W ˆ es W

rate of mass transfer, moles/time work rate (power) work done on the fluid at the entrance and exit points by extra stress, per unit mass ˙ np W nonpressure work rate, Section 4.5 ˆs W shaft work per unit mass x, xi , xj , xk i, j, k components of a position vector x in a curvilinear coordinate system xA mole fraction of species A in a mixture: x A = cA /c x Ae mole fraction of species A in the liquid phase required for equilibrium with a given bulk gas-phase mole fraction, Figure 29 x Ai , x Ab mole fraction of species A at the interface in the liquid phase and in the bulk liquid phase, Figure 29 x cr position on surface where boundary layer becomes turbulent, Figure 14 x∗ dimensionless coordinate position x 0 , y0 principal axes of elliptic cross section; defined above Equation (2.92) x 0 , y0 , z0 linear half-thickness associated with rectangular parallelepiped or rectangular bar, Section 2.1.8; reference position, Section 2.2 ∆xk thickness of kth slab (xk − xk−1 ) Xi , Xj , Xk material coordinates (yi , yj , yk at a reference time) of an element of mass yA mole fraction of species A in a gas mixture yAe mole fraction of species A in the gas phase required for equilibrium with a given bulk liquid-phase mole fraction, Figure 29 yAi , yAb mole fraction of species A at the interface in the gas phase and in the bulk gas phase, Figure 29 yi , yj , yk i, j, k components of a vector y or position vector in a rectangular cartesian coordinate system y
y coordinate measured from channel centerline, defined above Equation (2.76) y∗ dimensionless y coordinate y+ dimensionless distance from wall, Equation (2.285) ze thermal entry length defined above Equation (2.204)

Transport Phenomena z1 , z2 z∗

elevation at points 1 and 2 in a flow conduit dimensionless z coordinate

Greek Symbols α αKE

thermal diffusivity (≡ λ/ cp ) factor representing deviation from a flat velocity profile, Equation (4.6) αmom factor representing deviation from a flat velocity profile, Equation (4.4) β dimensionless quantity defined by Equation (2.43); isobaric coefficient of thermal expansion, Equation (2.185) βn eigenvalues given by Equations (2.48), (2.51), or (2.53) with n = 1, 2,... β∞ coefficient of thermal expansion at T∞ γ defined below Equation (3.69) γ˙ shear rate, Equation (1.110c) ∆ rate of deformation tensor, Equation (1.105) δ film thickness δ, δij Kronecker delta defined by Equations (1.10) and (1.11) δc boundary layer thickness for mass transfer analogous to Equations (2.211) and (2.237) δm momentum boundary layer thickness, Equation (2.211) δT thermal boundary layer thickness, Equation (2.237) ε fractional penetration of a gas into a film, used for nondimensionalizing the penetration, Equation (3.81a); smallness parameter, Equation (2.158); in Section 2.5.3, the eddy diffusivity; in Section 3.2.1, a smallness parameter defined by Equation (3.84); Lennard–Jones potential function energy parameter, Chapter 1 εA , εB , εAB Lennard–Jones potential function energy parameters for species A and B and for the mixture, εAB = (εA εB )1/2 εc , εm , εT eddy mass, momentum, and thermal diffusivities, Equations (3.113), (2.296), and (2.297) εM eddy diffusivity associated with transfer of quantity M by turbulent fluctuations

εs ξ η

277

surface shear viscosity dependent variable, Equation (2.35) viscosity; catalyst effectiveness factor, Equation (4.28) η, ηˆ in Section 2.2.5, viscosities of core fluid and outer fluid, respectively; in Section 2.3.1, viscosities of continuous and drop phases, respectively ηB viscosity of solvent B, Equation (3.29) ηp plastic viscosity in Bingham plastic model η ∞ , η w viscosities evaluated at bulk and wall temperatures, respectively θ angle subtended by two vectors, Equation (1.5) κ heat capacity ratio, cp /cv ; equal to bL /b0 in Section 2.3.3 Λ interference parameter associated with hindered settling (see Eq. 2.157) λ thermal conductivity ¯ λ average thermal conductivity similar to Equation (2.10) with T replacing T1 ¯1 average thermal conductivity, Equaλ tion (2.10) ¯k λ average thermal conductivity, Equation (2.10) with Tk replacing T 1 ν kinematic viscosity (≡ µ/) ξ coordinate direction; dummy integration variable; y position associated with edge of sublayer in turbulent flow  density, mass per unit volume A , B , i mass per unit volume of A, B, i in a mixture s density of sphere density at T ∞ ∞ σ hard-sphere diameter, Equation (1.99); Lennard–Jones potential function separation parameter, Equation (1.100) σ A , σ B , σ AB Lennard–Jones hard-sphere diameters for species A and B and for the mixture, σ AB = (σ A + σ B )/2 τ extra stress second-order tensor, Equation (1.84) τij stress components τrz , τˆrz stress components in inner and outer regions, respectively, Section 2.2.5 τw magnitude of wall shear stress τY yield stress in Bingham plastic model

278

Transport Phenomena

(t)

τ yx

turbulent stress component, Equation (2.300) τ0 shear stress at y = 0 Φ arbitrary flux vector or tensor, Equation (1.39); in Equations (2.289) and (2.290), represents a scalar, vector, or tensor function; rate of heat production and rate of radiation energy absorption, per unit volume Φv rate of conversion of mechanical energy to thermal energy by viscous dissipation, per unit mass (Eq. 1.97) for a constant-density Newtonian material Φ0 , Φ1 , Φ2 coefficients for the general viscous fluid, Equation (1.106) φ function described by Equations (2.90) and (2.91); association factor, Equation (3.29); fractional flow, Equation (2.109) φ ( y ∗, z∗) in Section 3.2.1, the cA solution from Equation (3.91) χ ratio of concentric cylinder inner and outer radii ψ stream function defined by Equation (2.125); in Section 2.4.1, function defined by Equation (2.210); in Equations (2.289) and (2.290), represents a scalar, vector, or tensor function ΩD collision integral correction factor for intermolecular interactions, Equation (3.28) Ωv collision integral correction factor for intermolecular interactions, Equations (1.101) and (1.102) ∂/∂t time derivative seen by an observer maintaining constant spatial coordinates (position) Others ∇ Del operator for spatial differentiation, defined by Equation (1.29) for rectangular cartesian coordinate systems ∇2 Laplacian operator, ∇2 = ∇ · ∇, Equation (1.38) ∼ order of magnitude estimate

1. Foundations The objective of this article is to provide the foundations for understanding and analyzing

phenomena that occur in continua as a result of driving forces in momentum, mass, and heat transfer. The descriptions can be concerned with continua on either differential or macroscopic scales; the results can be used for process analysis or for design to improve the efficiency of existing processes and to develop new processes. The analysis of transport processes relies on a small set of fundamental laws. These laws, together with specific information and data about the properties and behavior of the process materials and about the constitutive relations between driving forces and their resulting fluxes, are used to provide a mathematical description of the behavior of the process or phenomenon. This mathematical description, together with process constraints such as geometry and boundary conditions, defines the mathematical problem. Its solution, then, is approached by using a variety of mathematical and engineering strategies for solutions to algebraic, ordinary, and partial differential equations. The governing laws that establish the behavior of continua are the conservation laws for mass, energy, and linear and angular momentum (and their subset accounting equations for species mass, mechanical energy, and thermal energy) plus the second law of thermodynamics. These laws require a knowledge of the properties and behavior of materials in the form of (1) volumetric (pressure – volume – temperature) property relations and data; (2) thermodynamic property relations and data (such as internal energy, enthalpy, and entropy); and (3) transport flux relations and data (for heat, mass, and momentum fluxes in terms of their driving forces: temperature gradient, mole fraction gradient, and velocity gradient). The foundations establish conventions and identities that are used throughout this article: (1) kinematic relationships used in describing the motion of continua, (2) development of the conservation laws and their corollary equations related to mechanical and thermal energy balances, (3) linear transport relations for heat and momentum fluxes, and (4) discussion of how this material is combined in a methodology establishing quantitative models for describing the behavior of continua. Additionally, the molecular foundations for transport properties of materials are discussed. Theories for these properties, used to quantify the relations between the

Transport Phenomena transport fluxes and their driving forces, work reasonably well for gases to calculate properties such as thermal conductivity, viscosity, and diffusivity. However, for more dense phases in which intermolecular interactions play a significant role the theories serve primarily as a guiding framework for empirical relations. Finally, a brief discussion of non-Newtonian fluids and their constitutive relations is presented. After development of the basic equations, boundary conditions and a solution philosophy for attacking problems are discussed briefly. Also, the equations of change in dimensionless form are presented as the basis for dimensionless group correlations and analogies between different modes of transport. Chapter 2 is limited to compositionally homogeneous systems, whereas transport in multicomponent systems is discussed in Chapter 3.

279

held constant). A coordinate system is referred to as curvilinear if it is not rectangular. 1.1.2. Vector and Tensor Operations Useful references for vector and tensor notation, and their operation are given in [1] and [2]. The relations presented below summarize operations for rectangular cartesian systems for the purpose of setting notation and conventions. Accordingly, any vector a may be written as a linear combination of its components and base vectors in the following way: 3



a = a i ii ≡

a i ii

(1.1)

i=1

A second-order tensor A is written in terms of its base vectors and components in the following way:

1.1. Mathematical Preliminaries 1.1.1. Coordinate Systems The study of transport processes is concerned with scalars, which are represented by a single number (their magnitude), and with vectors and tensors, which have directions in space as well as magnitude associated with them. To describe vectors and tensors a coordinate system is defined for representing positions in three-dimensional space (direction and magnitude). A variety of systems are possible and can be used. Each coordinate system has defined base vectors, and all other vectors can be described as linear combinations of these base vectors. Any three noncoplanar vectors can be used as base vectors. However, orthogonal vectors (mutually perpendicular) that form a righthanded system are normally used. Base vectors are cartesian if they are of unit length. If they are both orthogonal and cartesian, then they are orthonormal. Base vectors are rectangular, cartesian if they are orthonormal and everywhere the same (in both magnitude and direction). In general, the base vectors for a coordinate system are not independent of position. For example, the base vectors for a cylindrical or spherical system vary with position (in a cylindrical system, the radius vector changes direction as the angular coordinate varies and the z coordinate is

A = Aij ii ij

(1.2)

Here summation occurs over both indices i and j. The addition of two vectors is a vector that is obtained by adding corresponding components: 3

a+b=



(ai + bi ) ii

(1.3)

i=1

Multiplication of a vector or tensor by a scalar k is performed by multiplying each component by that scalar: ka = k (ai ii ) = (k ai ) ii

(1.4)

The dot product of two vectors (which is a scalar) by definition (→ Mathematics in Chemical Engineering) is the product of their magnitudes and the cosine of their subtended angle, which implies a sum of the products of the corresponding components (for an orthonormal system): a ·b = |a| |b|cos θ = ai bi

(1.5)

The dot product of a vector and tensor is calculated by a similar sum over adjacent indices according to a ·B = ai Bij ij B ·a = Bij aj ii

(1.6)

280

Transport Phenomena

The components thus obtained are analogous to the elements of a vector obtained from the matrix multiplication of a vector and a square matrix. A different result is obtained for pre-versus postmultiplication. The dot product of two secondorder tensors is a second-order tensor given by A ·B = Aij Bjk ii ik

(1.7)

The trace of a second-order tensor is a scalar given by the sum of its diagonal terms tr (A) = Aii

(1.8)

and the trace of the dot product of two secondorder tensors is a double sum performed in the following way: tr (A ·B) = Aij Bji

(1.9)

The identity second-order tensor is defined such that δ = δij ii ij

δij

for i = j fori = j

(1.11)

This identity tensor, either pre- or post-dotted with a vector, returns the original vector. Likewise, when it is dotted with a second-order tensor, the original second-order tensor is obtained. The permutation symbol eijk is defined such that eijk

eijk emjk = 2 δ im

(1.16)

The dyadic or tensor product of two vectors is a second-order tensor whose dot product with a vector is given by (a b) ·c = a (b ·c)

(1.17)

In terms of index notation the tensor product a b is written a b = ai bj ii j j

(1.18)

The transpose AT of a second-order tensor is defined so that the i j component of A is the j i component of the second-order tensor AT . Accordingly, AT = Aij ij ii

(1.19)

(1.10)

where δij is the Kronecker delta defined by  1 = 0

and

  for an even permutation of 123 1 = 0 when any two indices are repeated (1.12)  −1 for an odd permutation of 123

The cross product of two vectors is then defined by a × b = ai bj eijk ik

(1.13)

and the cross product of a vector and a secondorder tensor is similarly given by a × B = ai Bjm eijk ik im

(1.14)

Two convenient identities between the permutation symbol and the Kronecker delta are eijk emnk = (δim δjn − δin δjm )

(1.15)

1.1.3. The Jacobian The Jacobian is the ratio of differential volume elements represented in two coordinate systems. It has special value when one of the coordinate systems involves material coordinates and the other involves the spatial coordinate system discussed below in kinematics. In rectangular cartesian coordinates, the differential volume element can be written as dV = dy1 dy2 dy3

(1.20)

which can be expressed equivalently in terms of the triple scalar product: dV = dy1 i1 · (dy2 i2 ×dy3 i3 )

A position vector in space can be represented in terms of a rectangular coordinate system or in terms of an alternate system according to r = y i ii = x i g i

(1.21)

where xi are the components of the position vector in the alternate system with base vectors g i . Each of the cartesian coordinates can be expressed as a function of the other system coordinates, yi = yi (x 1 , x 2 , x 3 ). Then, the differential volume element can be expressed in terms of the

Transport Phenomena triple scalar product of three vectors written in terms of the derivatives with respect to each of the new coordinate system’s three coordinates: dV =

∂y dx1 · ∂x1



∂y ∂y dx2 × dx3 ∂x2 ∂x3



Derivatives with Respect to Position. The del operator ∇ is defined to differentiate quantities that are functions of position. In rectangular cartesian coordinate systems the del operator is

(1.22) ∇ ≡ii

or dy1 dy2 dy3 = Jdx1 dx2 dx3

(1.23)

and the Jacobian is the determinant of the partial derivatives relating changes in coordinates between the two systems:  ∂y1   ∂x1   ∂y1 J ≡  ∂x  2  ∂y  1 ∂x3

∂y2 ∂x1 ∂y2 ∂x2 ∂y2 ∂x3

∂y3  ∂x1 

 ∂y3  ∂x2  ∂y3 

281

∂ ∂yi

(1.29)

The del operator can operate in dyadic, dot, or cross product forms on scalars, vectors, or tensors. As a dyadic operation, grad (ϕ) is ∇ϕ =

∂ϕ ii ∂yi

(1.30)

and grad (a) (1.24)

∇a =

∂aj ii ij ∂yi

(1.31)

∂x3

The Jacobian only exists if there is a unique transformation between the two coordinate systems. 1.1.4. Calculus of Vectors and Tensors In transport phenomena generally, two kinds of vector and tensor derivatives exist with respect to time and with respect to position. Derivatives with Respect to Time. The derivative of a vector that is a function of time is a vector; a derivative of a second-order tensor with respect to time is a second-order tensor. Derivatives are obtained straightforwardly by differentiating each of their components with respect to time: da dai = ii dt dt

(1.25)

dAij dA = ii ij dt dt

(1.26)

Differentiating products of vectors is similar to differentiating products of scalars: d da db (a ·b) = ·b + a · dt dt dt

(1.27)

d da db (a×b) = ×b+ a× dt dt dt

(1.28)

For grad (a), note the order of the index corresponding to the del operator with respect to that corresponding to the vector. The opposite order is sometimes used in the literature, and care must be taken to follow a consistent convention throughout a given work. The dot product of the del operator with a vector a or tensor A is given by ∇ ·a =

∂ai ∂yi

(1.32)

∇ ·A =

∂Aij ij ∂xi

(1.33)

and is called the divergence of the vector or tensor. Again, note the order of the indices corresponding to the del operator and the secondorder tensor. The divergence of a vector is a scalar, so the order is immaterial. The del operator can also be crossed with a vector or tensor, and these operations are given by ∇×a =

∂aj eijk ik ∂yi

(1.34)

∇×A =

∂ Ajm eijk ik im ∂yi

(1.35)

and called the curl of the vector or tensor. Here, only one definition with respect to the order of the indices seems to exist in the literature. Because of its dual vector – operator role, the del operator may be dotted with one entity and operate on (differentiate) another. For example, in the thermal energy equation in Section 1.2 (Eq. 1.96), the term

282

Transport Phenomena

u ·∇F

(1.36)

appears, in which the del operator is pre-dotted with the velocity vector u but operates (by differentiation) on the temperature (see also Eq. 1.53): u ·∇T = ui

∂T ∂yi

(1.37)

The Laplacian is another operator, a scalar operator, defined by ∇ · ∇. For rectangular cartesian systems it is given by ∇2 ϕ = ∇ · ∇ϕ =

∂2ϕ ∂yi ∂yi

(1.38)

1.1.6. Kinematic Relations A fundamental concept of kinematics, the description of motion without reference to its cause, is that of material coordinates. Given a fluid or material undergoing motion (translation, rotation, deformation), each particle of that material has a current position y y = y i ii

Also, each element of the material can be identified with the position it had at some reference time (t = 0, for example): X = Xi ii

1.1.5. Divergence Theorem The divergence theorem is extremely important in deriving and understanding the basic transport relations. This theorem equates an integral over the entire volume of a region of space enclosed by a closed surface to an integral over that surface:



∇ ·ΦdV =

V

n ·Φ dA

(1.39)

A

In this equation, Φ can be a vector or a secondorder tensor. If it is an isotropic second-order tensor ( p I, for example), then a form involving a scalar function is obtained:



∇pdV =

V

pn dA

(1.40)

A

In transport phenomena, Φ represents the flux of an extensive property and the surface integral represents the rate at which the property (as written here using the outward normal vector n) is leaving the volume element across the surface. With the del operator defined as above, the surface integral must be written with the surface outward normal vector pre-dotting the quantity Φ.

(1.41)

(1.42)

The spatial coordinates yi are used to identify a position in space, and the material coordinates Xi are used to identify elements of the material. Furthermore, the position in space can be defined in terms of the element of the material that resides there (identified by its position at time t = 0, its reference position) at time t: y = y (X, t)

(1.43)

Inversely, the material coordinates X can be defined in terms of the spatial coordinates and time t; each element of mass can be defined in terms of its location at time t X = X (y, t)

(1.44)

This inversion depends on 0 < J < ∞ where J is the Jacobian. Every quantity expressed in terms of spatial coordinates and time can just as well be expressed in terms of material coordinates and time. For example, for a flowing fluid the density, velocity, etc., vary with position  =  (y, t)

(1.45)

u = u (y, t)

(1.46)

or, equivalently, vary from material element to material element:  =  (X, t)

(1.47)

u = u (X, t)

(1.48)

Transport Phenomena 1.1.7. Partial and Total Derivatives The partial derivative with respect to time [of a function F (x, t)], if position is held constant, is written as ∂F(x, t) ≡ ∂t



∂F ∂t

 (1.49) x

and represents changes in the variable of interest with time at a fixed point in space. The partial derivative with respect to time [of a function F (X, t)], if the material coordinates are kept constant, is called the material derivative or substantial derivative, and represents changes observed while following the motion of the material: 

D F(X, t) ≡ Dt

∂F ∂t



∂x ∂t

 = X

(1.50) X

Dx Dt

(1.51)

The total derivative represents changes that would be observed while moving around at an arbitrary velocity w; it is represented by dF/dt. This velocity w then represents the total derivative of the position vector: w = dx/dt

1.1.8. Relation Between Different Time Derivatives A property F written in terms of x and t has the total differential  dF =

∂F ∂t

 x

∂F ∂t

 =

 dt + dxi

X

DF = Dt

∂F ∂t

 + x

dx ·∇F dt

(1.54)

or dF = dt



∂F ∂t

 x

+ w ·∇F

(1.55)

A physical interpretation of the three derivatives follows [3]. An observer interested in determining the changes in the concentration of fish in a river cf can do so while staying at a fixed position in the river ∂cf /∂t, while drifting in a canoe at the velocity of the river D cf /D t, or while moving around in a motorboat with velocity w (dcf /dt).

∂F ∂xi



∂F ∂t

 (1.52) t

 x

1.2. Basic Equations for Compositionally Homogeneous Systems [3–7] 1.2.1. The Reynolds Transport Theorem In deriving the conservation equations of mass, energy, and momentum for a continuum, a volume (system) that always encloses the same elements of mass should be defined. This volume must then move and deform with the fluid, which is convenient because no convected input or output term need be considered. This concept, and the substantial derivative and Jacobian from kinematics, are very useful in deriving the desired conservation equations. If F (x, t) is a function of position and time (scalar, vector, or tensor), representing the amount of the desired property per unit volume, and V (t) is the closed volume moving with the fluid, then
F (t) ≡

F(x, t) dV

(1.56)

V

from which the relation between the substantial and partial derivatives and u may be determined: 





Note that if F is the spatial coordinate of an element of fluid, then D F/D t = D x/D t is the change in position with time of a material element, i.e., the fluid velocity: u ≡

dF = dt

283

+ u ·∇F

(1.53)

In an analogous way, the total derivative is related to the partial and substantial derivatives and to w by

represents the total amount of the desired property contained within the system V (t). For the conservation equation, then, the accumulation rate of the property within the system is required D F (t)/D t. Here the substantial derivative must be used because it is the derivative “following the motion.” Then

284

Transport Phenomena

DF D = Dt Dt


F(x, t) dV

(1.57)

or leaves the system, so the mass conservation equation is

V

However, the derivative cannot be taken across the integral in this form because the volume is a function of time. This difficulty is circumvented by transforming the system to the reference coordinates, differentiating, and then transforming back. In the reference coordinates, the volume element is a constant. Now, dV = JdV0

(1.58)

DF D = Dt Dt =




dV = 0

(1.62)

V

By the Reynolds transport theorem then,


 D + ∇ ·u dV = 0 Dt

V0

DF DJ J +F Dt Dt

The volume chosen as the system is completely arbitrary, so this result implies

(1.59)

DJ = J∇·u so Dt


(1.64)

∂ = − ∇ ·u ∂t

 dV0





 G dV = V

J dV0

(1.65)

This equation (in either form) is called the continuity equation (see also, → Fluid Mechanics, Chap. 2.1.). Note that if F in Equation (1.61) is proportional to  (i.e., F =  G), then by continuity D Dt



DF +F∇ ·u Dt

(1.63)

V

or, in terms of the partial derivative,

F[x (X, t) , t] J dV0

V0

DF = Dt




D = − ∇ ·u Dt

So




D Dt



DG dV Dt

(1.66)

V

(1.60)

V0

Finally, converting back to spatial coordinates gives DF = Dt




 DF +F∇ ·u dV Dt

(1.61)

V

which is known as the Reynolds transport theorem (RTT). 1.2.2. Conservation of Total Mass In the absence of nuclear conversions, total mass is conserved; no generation or consumption occurs. (The term balance is commonly used to apply to both conservation and accounting equations. In this section, and in Chapter 4 on macroscopic systems, the term “accounting” is used for properties that may be generated or consumed and the term “conservation” for properties that may not be generated or consumed.) Now, for the volume that moves and deforms with the flowing fluid, as described above, no mass enters

1.2.3. Conservation of Linear Momentum An equation for the conservation of momentum or energy for a continuum must be able to express the forces that are exerted on an element in the continuum. These forces are of two kinds: 1) Body forces act upon the entire volume (reach inside the volume and act upon each element) and may be gravitational, electromagnetic, etc. 2) Contact (or internal) forces act at the surface of the element of the continuum. Cauchy’s stress principle states that if t(n ) is the force per unit area (stress vector or traction) at a point on the surface of a system volume (surface orientation indicated by n), then t(n ) is a function of x, t, and the orientation of the surface. The stress vector at a point x at a particular time is not sufficient; the surface upon which the stress vector acts must also be defined. Note that t(n ) is a vector force per area (the stress vector or traction vector) and that t(n ) dA

Transport Phenomena is the force acting from outside the surface. For a volume (with a surface) the total force acting on this volume due to these contact forces is the integral over the entire surface A:
t(n) dA

(1.67)

285

1.2.6. Equation of Motion Now, D Dt





 u dV = V


n ·T dA

 g dV + V

(1.71)

A

A

Body forces can be summed by a volume integral (g is force per unit mass)
 g dV

or, by the divergence theorem and Reynolds transport theorem,


(1.68)

Du dV = Dt







V

V

and the conservation of linear momentum (for a system that moves with the material) can be expressed as

so (again, V is arbitrary)

D Dt





 u dV = V


 g dV +

V

t(n) dA

(1.69)

V

(1.72)

V

Du =  g +∇ ·T Dt

(1.73)

or

A

a = g+∇ ·T

This is known as Euler’s first law. 1.2.4. Condition of Local Stress Equilibrium As a direct consequence of the conservation of linear momentum, the stress vector (traction) acting at a point on a surface is matched by an opposite and equal reaction force on the surroundings. That is, if momentum is conserved, the transfer of momentum between the system and its surroundings due to forces is mutual (condition of local stress equilibrium). 1.2.5. Stress Tensor The condition of local stress equilibrium leads to an important decoupling of stress from the surface orientation. Consideration of the stresses acting on the surfaces of a tetrahedron shows that t(n) =n ·T



∇ ·T dV

 g dV +

(1.70)

where T is the second-order stress tensor (→ Fluid Mechanics, Chap. 3.1.), which may be a function of position and time but not of surface orientation. The state of stress in a continuum at a point is the totality of all possible pairs of traction vectors t(n ) and surface orientations n (i.e., an infinite number). However, the nine-component stress tensor can be used to obtain t(n ) if n is given.

(1.74)

where a = D u/D t, the acceleration. This is known as Cauchy’s first law, the conservation of linear momentum, or simply the equation of motion for a continuum. For further details, see → Fluid Mechanics, Chap. 2.2. 1.2.7. Conservation of Angular Momentum If angular momentum is conserved locally in a continuum (the usual situation),

D  (u × x) dV =  (g × x) dV + Dt V V


t(n) × x dA

(1.75)

A

which is Euler’s second law. By writing t(n ) in terms of T (Eq. 1.70) and using the divergence theorem, Reynold’s transport theorem, and vector – tensor identities, this statement of the conservation of angular momentum reduces to the result that the stress tensor is symmetric. That is, a necessary and sufficient condition for satisfying the conservation of angular momentum is: T = TT

(1.76)

286

Transport Phenomena

However, this is restricted to fluids that have no couple stresses (i.e., torques on the fluid are the result only of forces) [5], [6]. Also, if T is a symmetric second-order tensor (nondiagonal, in general), a rotated coordinate system exists in which T is diagonal. In this system, the coordinate axes are called the principal axes of stress and the three normal stresses (eigenvalues) are called the principal stresses. In mechanics, analysis of the principal axes of stress provides the basis for Mohr’s stress circle technique [6], [7].

1.2.9. Conservation of Total Energy If Uˆ is the internal energy per unit mass, then a total energy balance is  

D 1 2 ˆ dV =  g ·u dV + u +U Dt 2 V V


t(n) ·u dA − n ·q dA + Φ dV A

A

where q is the heat flux at the system boundary. The left side of this equation represents changes in kinetic and internal energy, and the terms on the right side have the following meaning:


1.2.8. Mechanical Energy Accounting Equation

(1.81)

V

g ·udV

V


t(n) ·udA
− n ·qdA
A ΦdV

rate at which work is done by body forces, rate at which work is done due to traction,

A

Now, because of the conservation of linear momentum (Eq. 1.74), (a − g − ∇ ·T ) ·u = 0

(1.77)

or   D 1 u ·u − g ·u − (∇ ·T ) ·u = 0  Dt 2  D (u·u) Du Du = ·u + u · = Dt Dt Dt  a ·u+u ·a= 2a ·u

(1.78)

Now, as an identity, for T symmetric

rate of energy transfer due to q, and rate of radiant energy absorption.

V

This last term may also be used to represent heat generation (conversion of energy from other forms to thermal energy) by mechanisms such as electrical heating and chemical or nuclear reactions instead of accounting for these conversions through changes in, e.g., changes in internal energy from one form to another within the material volume. From Equation (1.81) the continuum statement of total energy conservation is   D 1 2 ˆ u +U Dt 2 =  g ·u+∇ · (T ·u) − ∇ ·q + Φ

 ∇ · (T ·u) = (∇ ·T ) ·u+ tr (T ·∇u)

(1.79)

(1.82)

so that 

D Dt





1 u ·u 2

= g ·u+∇ · (T ·u) − tr (T ·∇u)(1.80)

This scalar equation is a rate accounting equation of mechanical energy in the form of kinetic energy (the left-hand side), work done by body forces ( g · u), work done by traction forces [∇ · (T · u)], and the conversion of mechanical energy to thermal energy due to traction through both fluid compression and viscous dissipation [tr (T · ∇ u)], all per unit volume. Obviously, this result is not independent of the equation of motion and, in fact, as a scalar equation must contain less information than the vector equation of motion.

1.2.10. Thermal Energy Accounting Equation Subtracting the mechanical energy accounting equation from the total energy conservation equation gives an accounting of thermal energy: 

ˆ DU = − ∇ ·q+tr (T ·∇u) +Φ Dt

(1.83)

In this result, tr (T · ∇u) represents the conversion of mechanical energy to thermal energy. Equation (1.83) is not an independent equation but has the advantage of not involving mechanical energy terms. It can be used in place of the total energy equation.

Transport Phenomena 1.2.11. Forms of the Governing Equations Many possible forms of the continuity, motion, and energy equations exist, depending, for example, on whether partial or substantial derivatives are used and whether heat capacities (cp or cv ) are used instead of internal energy. A convenient table for compositionally homogeneous continua summarizing these forms appears in [3]. Further summary of the governing equations is given in Section 1.4. In these forms, the stress tensor is commonly separated into isotropic stress p I and extra stress τ (→ Fluid Mechanics, Chap. 3.1.). Following the convention of Bird, T = − τ − pI

(1.84)

where p ≡ − tr (T )/3. As a result ∇ · T becomes − ∇ · τ − ∇p and tr (T · ∇u) becomes − tr (τ · ∇u) − p∇ · u. 1.2.12. Entropy Inequality If T = − τ − pI for a compressible fluid, then the thermal energy equation is 

ˆ DU = − ∇ ·q − p∇ ·u − tr (τ ·∇u) + Φ Dt

(1.85)

ˆ Vˆ ), where Sˆ is the entropy per Now if Uˆ = Uˆ (S, ˆ unit mass and V the volume per unit mass, then ˆ = T dSˆ − pdVˆ dU

(1.86)

or by the equation of continuity 

ˆ DSˆ DVˆ DSˆ DU = T − p = T − p∇ ·u (1.87) Dt Dt Dt Dt

This gives for the thermal energy equation T

D Sˆ = − ∇ ·q − tr (τ ·∇u) + Φ Dt

Now, from experience, q is known to be in the opposite direction of ∇T , so − q · ∇T ≥ 0. Furthermore, and also from experience, − tr (τ · ∇u) ≥ 0. Consequently, an inequality results that must hold for all continuum processes:



q DSˆ Φ +∇ · − ≥0 Dt T T

Alternatively, − q · ∇T ≥ 0 and − tr (τ · ∇v) ≥ 0 may be viewed as constraints on allowable constitutive equations for q and T. 1.2.13. Linear Transport Fluxes and Relations The above transport equations describe the conservation of mass, linear momentum, and total energy, and include terms representing heat and momentum fluxes. The flux relations that are normally adopted follow the observation that the flux is proportional to a gradient driving force; i.e., the heat flux is proportional to the temperature gradient and the momentum flux is proportional to the velocity gradient (actually, the symmetric part of the velocity gradient tensor). Fourier‘s Law. For heat conduction, q = − λ∇T

(1.91)

where λ is the thermal conductivity and, in general, may be a function of location in the material and temperature. (To be more general, the heat flux can be written in terms of an anisotropic thermal conductivity tensor to allow for directionality in the thermal conductivity.) In rectangular cartesian coordinates the components of the heat flux are  qx = − λ

q DSˆ ∇T  = − ∇· − q· 2 Dt T T 1 Φ tr (τ ·∇u) + T T

(1.90)

(1.88)

or

−

287

 qy = − λ (1.89)

∂T ∂y

 qz = − λ

∂T ∂x

 y,z



∂T ∂z

x,z

 (1.92) x,y

288

Transport Phenomena

Momentum Flux Relations. For viscous materials the most common momentum flux relation is Newton’s law of viscosity, which for an incompressible material is written in terms of the rate of deformation tensor ∆ as τ = − η∆

(1.93)

where η is the viscosity. For Newtonian materials the viscosity is constant at constant temperature (i.e., it is independent of shear rate), resulting in this linear flux relation. For onedimensional shear flow with only one velocity gradient component, this result becomes τyx = − η

dux dy

(1.94)

These linear relations can be used together with the conservation equations to model the behavior of continua. For a Newtonian fluid of constant viscosity and density (or for isochoric flow for which ∇ · u = 0), the equation of motion becomes the Navier – Stokes equation (for further details, see → Fluid Mechanics, Chap. 3.2., → Fluid Mechanics, Chap. 3.3.) 





∂u + u ·∇u = g − ∇p + η∇2 u ∂t



∂T + u ·∇T ∂t

 = λ∇2 T + Φv

(1.96)

where the distinction between constant pressure and constant volume heat capacities vanishes for true incompressibility. In this result, Φv accounts for the conversion of mechanical energy to thermal energy through viscous dissipation and is given by 6

.

/7

Φv = η tr [∇u ·∇u] +tr (∇u)T · ∇u

(1.97)

Further simplifications are often appropriate, such as for a solid material whose velocity terms in the thermal energy equation become insignificant, giving   cp

∂T ∂t

 = λ∇2 T

1.2.14. Transport Properties from Molecular Theories The calculation of transport properties by using molecular transport theories with parameters estimated from experimental data is quite reliable for dilute (low-density) gases. This is a direct result of the fact that at low densities, intermolecular collisions and forces are minimized. Details of the theories are available in the original references or in summaries such as [8]. In this section, theories for viscosity and thermal conductivity are considered. Section 3.1.5 discusses theories for diffusivities. Viscosity. Based on two-body, hard-sphere collisions the viscosity of a gas (in micropascal seconds, i.e., 10−6 Pa · s) theoretically is given by η =

5 16

(π Mr R T )1/2

(1.95)

Also, for a Newtonian fluid of constant density, thermal conductivity, and viscosity, the thermal energy equation becomes  cp

non-Newtonian stress relations are given in the following paragraphs.

(1.98)

Understanding material behavior with respect to thermal conductivity and viscosity is an important factor in understanding transport phenomena. Theoretical foundations for calculating these properties for low-density gases and

π σ2

√ = 26.69

Mr T σ2

(1.99)

where T is in kelvin and the hard-sphere diameter σ is in a˚ ngstr¨oms (i.e., 10−10 m); M r is the molecular mass of the gas. To adjust for non-hard-sphere interactions, Chapman and Enskog introduced the use of spherically symmetric molecular potential functions [9], [10]. The most commonly used function is the Lennard – Jones 12 – 6 potential ψ (r) = 4 ε

  σ 12 r

−

 σ 6  r

(1.100)

from which can be calculated the interaction forces between molecules as a result of their separation r and model parameters σ and ε. These forces are then used to adjust the viscosity calculations through the dimensionless collision integral Ωv , which is a function of k T/ε where k is the Boltzmann constant. Tabulations of this quantity are readily found in the literature, and a convenient and accurate computational form for Ωv is given by Neufeld et al. [8], [11]. The value of Ωv is unity for the ideal hard sphere; thus, deviations are a measure of the departure from this ideal model. The Chapman – Enskog relation for viscosity, then, is

Transport Phenomena η =

5 16

(π Mr R T )1/2 (π σ 2 ) Ωv

√ = 26.69

Mr T σ 2 Ωv

(1.101)

where again η is in 1 micropascal seconds, T in kelvin, and σ in a˚ ngstr¨oms. Other potential functions have been used. The Stockmayer potential adjusts for polarity and should be used instead of the Lennard – Jones potential for polar molecules. Calculation of the Stockmayer potential by using the analytical correction to the Lennard – Jones potential provided by Brokaw is recommended [8], [12]. These potential energy functions rely on molecular parameters that must be determined from data. In practice, tabulations of these parameters, which are available for individual materials, are derived from experimental data for viscosity by using the appropriate potential function with the Chapman – Enskog theory. Using the parameters, then, along with the potential function (collision integral) to calculate viscosity, amounts to adjusting the measured values to the desired temperature by using the squareroot temperature dependence of the molecular theory. As long as the form of the theory is correct, the calculations should be very good. In fact, when tabulated potential function parameters are used for a dilute gas, calculated viscosity values are correct to within ca. 1 %. Methods also exist for estimating the parameters; these too give quite reasonable viscosity values, with errors from 1 to 3 % [8]. Thermal Conductivity. The molecular theory for thermal conductivity is complicated by the fact that internal energy is stored in molecules in internal vibrational and rotational modes as well as in the translational mode, unlike viscosity (momentum) which is manifest only in translation. Consequently, although the basic molecular theory parallels that for viscosity, an additional term for these internal energy modes is necessary for polyatomic molecules. For monatomic gases (which do not have the internal modes) the Chapman – Enskog thermal conductivity relation is λ =

25 32

(π Mr R T )1/2 cv Mr (π σ 2 ) Ωv

= 0.0833

T /Mr (1.102) σ 2 Ωv

where T is expressed in K, σ in ˚A, and λ in J m−1 s−1 K−1 . In this relation a value of cv = 3 k/2 (monatomic ideal gas) was used. By

289

using the viscosity relation (Eq. 1.101), the thermal conductivity can be expressed in terms of the viscosity and constant volume heat capacity as λ Mr = 2.5 η cv

(1.103)

This result works quite well for monatomic gases. For polyatomic gases it must be adjusted to account for the additional degrees of freedom. The Eucken equation λ Mr = cv + 1.872 × 104 η

(1.104)

is a simple, yet reasonably accurate method for this correction. Other methods are available and are discussed and evaluated by Reid et al. [8]. 1.2.15. Non-Newtonian Fluids (see also → Fluid Mechanics, Chap. 4.) In the momentum and energy balances, q and τ represent heat and momentum fluxes. Before problems can be solved, constitutive relations must be selected for these fluxes in terms of appropriate driving forces (e.g., ∇T or ∇u). The specific relation for τ that is appropriate for a particular material is the subject of rheology. Specific rules dictate the form of constitutive equations or set constraints on them. Useful references on non-Newtonian rheology are [13– 15]. The rheological constitutive equation that is applicable in a given circumstance can be viewed as depending on either the type of material or on the specific flow situation. That is, the material may behave in a certain way because it behaves according to that model in all circumstances (i.e., it is that kind of material) or because all materials behave in that way for a particular flow situation. From a materials viewpoint, behavior is normally classified as viscous, elastic, or viscoelastic. Complicating this classification by material type is the presence of history or memory effects (→ Fluid Mechanics, Chap. 4.1.4.). One very common type of constant stretch history flow is viscometric flow. The importance of viscometric flows with respect to the characterization of materials is that all materials,

290

Transport Phenomena

whether they are viscous, elastic, or viscoelastic, can be characterized by the same set of viscometric functions. The functions are referred to as the shear stress (or, equivalently, the viscosity) and the first and second normal stress differences (for further details, see → Fluid Mechanics, Chap. 4.3.1.). Other kinds of flows are elongational, slow, and small deformation oscillatory flows. The last are periodic and yield a set of dynamic viscoelastic functions such as the dynamic elasticity and the dynamic viscosity (also referred to as the storage and loss moduli, respectively). A Newtonian fluid has a constant viscosity independent of shear rate (at constant temperature). Other materials may show a decrease (pseudoplastic) or an increase (dilatant) in viscosity with increasing shear rate or exhibit a yield stress (e.g., the Bingham plastic). Complicating this picture is the fact that time-dependent behavior may also be observed and such fluids are referred to as thixotropic (decrease in viscosity with time at fixed shear stress) or rheopectic (increase). In addition to shear stress, normal stresses may be generated by steady shear flow. These are stresses that are normal to the flow direction and are additional to the isotropic stress (pressure). The generation of normal stresses can have some rather dramatic effects on the flow of fluids and may have to be considered in process design. Experimentally, the only materials that have been observed to exhibit normal stresses are viscoelastic, such as polymer melts and polymer solutions. More quantitatively, a viscous fluid is one whose stress tensor is a function only of the rate of deformation tensor ∆ (a function of strain rate and not of strain), where ∆ ≡ ∇u + (∇u)T

(1.105)

A completely general form for a purely viscous fluid is T = Φ0 I + Φ1 ∆ + Φ2 ∆2

/ 1. (tr (∆))2 − tr ∆2 2

III∆ = det∆

(1.108) (1.109)

Note that I∆ = tr (∆) = 2 ∇ · u, which is zero for an incompressible fluid or for isochoric flow. For incompressible flow then, the general viscous equation reduces to T = − pI + Φ1 ∆ + Φ2 ∆2

(1.110)

which is called the Reiner – Rivlin equation. In this case, Φ1 and Φ2 are functions only of II∆ and III∆ because I∆ = 0. Apparently, actual fluids of this type have not been documented (i.e., fluids with a nonzero second-order term and which are purely viscous have not been observed). Nevertheless, it may be a useful model because the second-order term provides a mechanism for generating normal stress differences in steady shear flow. However, because normal stress differences are not generally observed with viscous fluids, the second-order term is usually dropped (Φ2 = 0), and the model then reduces to the form generally used for incompressible Newtonian and non-Newtonian (non-constantviscosity) fluids: T = − pI − η∆

(1.110a)

or, in terms of the extra stress τ , τ = − η∆

(1.110b)

where η = η (II∆ , III∆ ) is the non-Newtonian viscosity. For shear flows, [u1 = u1 (x 2 )], III∆ is zero so that η is only a function of the second invariant which can be related to the shear rate γ˙ by  γ˙ ≡ |du1 /dx2 | =

1/2

1 |II∆ | 2

(1.110c)

Equation (1.110 b) is commonly used with η = η (γ) ˙ for other kinds of flow (nonshear), even though to do so is not well-founded theoretically.

(1.106)

where the coefficients Φ0 , Φ1 , and Φ2 are functions of the three scalar invariants of ∆ (I∆ , II∆ , and III∆ ) and are defined in the following way: I∆ = tr (∆)

II∆ =

(1.107)

1.3. Summary of Basic Equations A handful of conservation laws govern physical behavior and, as such, provide the basics of transport phenomena: total mass, linear momentum, angular momentum, and total energy. In

Transport Phenomena

291

Table 1. Summary of the basic transport relations Equation name

Conservation law

Equation

Equation number

Continuity

total mass

D Dt = ∂ ∂t =

− ∇ ·u

(1.64)

−∇ · (u)

(1.65)

total mass Motion

linear momentum linear momentum linear momentum angular momentum total energy

Thermal energy accounting

Entropy inequality

a= g+∇ ·T   ∂u ∂t + u ·∇u = g+∇ ·T  ∂u  ∂t + u ·∇u =  g − ∇ ·τ − ∇p

(1.74)

T=T T   D 1 2 ˆ  Dt 2 u + U = g ·u+∇ · (T ·u) − ∇ ·q + Φ

(1.82)

u ˆ D Dt = − ∇ ·q+tr (T · ∇u) + Φ

(1.83)



(1.76)



∂p  cv DT (∇ ·u) − tr (τ ·∇u) + Φ Dt = − ∇ ·q + T ∂T   ˆ Dp ∂lnV DT  cp Dt = − ∇ ·q + ∂lnT Dt − tr (τ ·∇u) + Φ p q ˆ Φ DS  Dt +∇ · T − T ≥0 (1.90)

addition, subsets of energy give useful accounting relations, and the entropy inequality (second law of thermodynamics) provides constraints on the conservation laws. These relations are summarized in Table 1 along with some forms not presented in the text.

1.4. Boundary Conditions The transport conservation relations (mass, total energy – or an accounting of thermal energy – and linear and angular momentum), along with the second law of thermodynamics and relations that describe material behavior, are not completely adequate for describing transport processes. With the addition of appropriate initial and boundary conditions the transport partial differential equations can be solved to obtain future behavior of the process over the required region of space. Specific examples appear in later sections. The principal types of conditions that are encountered and applied are outlined briefly in this section. The conditions apply to the dependent variables of the transport equations (such as temperature or velocity), their derivatives, or other aspects of their behavior. The following are the types of conditions encountered:

1) A dependent variable is known throughout the region of space of interest at time zero; values at subsequent times result from the behavior of the system as destined through transport principles. For example, the temperature of a fluid over a region of space may be initially set to a prescribed value or function of position. Likewise, the velocity of the fluid may be known initially over the region of interest (e.g., if the fluid initially is stagnant, then u = 0 at t = 0). 2) The dependent variable at a boundary of the region of space of interest is known; it may be fixed at a constant value over time or it may be a known variable as a function of time. For example, in fluid mechanics a fluid in contact with a solid usually meets the noslip condition: the velocity of the fluid at this solid boundary is equal to the velocity of the solid surface. Similarly, the temperature at a boundary could be a known value or function of time. 3) The transport flux at a boundary of the region of interest is known as a function of time. This is not a condition on a dependent variable directly; however, it is an indirect condition. For example, in Fourier’s law the heat flux is proportional to the temperature gradient; thus, specifying the heat flux at the boundary places a condition on the behav-

292

Transport Phenomena

ior of the temperature at the boundary. If the component of the heat flux normal to the surface is known, a condition on n · q is established. For momentum transport, a known traction (stress vector) at the boundary is stated in terms of n · τ . As an example, at gas – liquid interfaces the stresses supported by the gas phase are very low (low viscosity), the momentum flux (traction) due to the extra stress at the interface is therefore zero: n · τ = 0. 4) The continuity of properties across the surface can also serve as a boundary condition. For example, at a surface (boundary) velocity, temperature, traction, and heat flux are normally continuous. The continuity of heat transfer across the interface holds as long as no thermal energy accumulates at the surface of interest. This, however, does not imply that temperature gradients are continuous; thermal conductivities may change at a boundary. Likewise, the continuity of traction is necessitated by the condition of local stress equilibrium (i.e., by the conservation of linear momentum). 5) The stress in fluids is finite, that is, fluids cannot sustain stresses without deforming so as to reduce those stresses. This condition is applicable, for example, in pipe flow problems where, if such a condition is not imposed a priori, solutions to the equation of motion could yield an infinite stress at the center of the pipe.

1.5. Solution Philosophy We now have a total of five equations and one inequality. The equations are (1) continuity (mass conservation), (2) linear momentum conservation (three equations, one for each spatial direction), and (3) total energy conservation. Additionally, the conservation of angular momentum constrains T to be symmetric, and the entropy inequality places further constraints on q and T. Also, linear momentum conservation can be used to obtain an accounting of mechanical energy (not an independent equation), which can be employed with total energy conservation to give an accounting of thermal energy that can be used in place of the total energy conservation

law. The above equations have six independent variables: , ux , uy , uz (or x, y, and z), p, and T . [The stress T and heat flux q are expressed in terms of position derivatives and temperature gradients, respectively. Furthermore, the therˆ H, ˆ and S) ˆ are exmodynamic properties (U, pressed as functions of T and pressure (or ) through heat capacity and p –  – T data or equation of state.] The five equations plus one volumetric equation of state [p = p (, T )] along with appropriate boundary or initial conditions dictate the subsequent behavior of the system. If temperature is constant, the number of variables is reduced by one, as is the number of equations (energy conservation reduces to the mechanical energy accounting equation, which is not an independent equation). If density is constant, the number of variables and equations (volumetric equation of state) both decrease by one.

1.6. Dimensionless Equations of Change (→ Scale-Up in Chemical Engineering) As discussed above, for a given flow situation involving momentum and heat transfer, a scalar equation of continuity, a vector equation of motion, and a scalar equation of energy (either total energy or a thermal energy accounting equation) are available. These laws govern the behavior of the process at hand. In these equations, each term has a specific physical meaning or origin important to the current situation. Consequently, each term carries with it dimensional characteristics that result from those physical meanings. Frequently, from both computational and conceptual perspectives, equations are usefully recast into a dimensionless form by means of dimensions or parameters that are characteristic of the process. These dimensionless forms have several advantages. First, grouping the characteristic parameters or dimensions into a reduced number of parameters allows parametric calculations for a reduced number of cases. Second, the dimensionless equations allow comparison of processes of different size, which otherwise are the same; the dimensionless equations clarify the adjustments that must be made between the two processes to have them behave the same dynamically. In this way, a small-scale process can

Transport Phenomena give an accurate prediction of a large-scale process for design purposes. Third, correlations of experimental data can be made very efficiently by using dimensionless groups and correlations. Correlations developed for one transport phenomenon can even be used for another one when the situations (process, apparatus, and boundary conditions) are analogous (e.g., heat-transfer results can be used to predict mass-transfer behavior). This is the very important Reynolds analogy, which rests upon the similarity of conservation laws (momentum, mass, and energy) and flux relations (Fourier’s law, Newton’s law of viscosity, and Fick’s first law of diffusion) for certain situations. Finally, dimensionless forms of the transport laws have considerable value in understanding the relative importance of the various contributions to process behavior (e.g., convective versus conductive heat transfer). This order-of-magnitude scaling is discussed further in Chapter 2. The objective of dimensional analysis using the governing equations, then, is to convert the equation to a dimensionless form, thereby extracting dimensionless groups. First, the variables of the equations are expressed in terms of dimensionless quantities (herein denoted by ∗) by dividing each of them by a characteristic parameter of the same dimensions. For example, a temperature-dependent variable would be divided by a reference characteristic temperature or, as is more frequently used, a temperature difference (T 1 − T 0 ); velocity, by a velocity U that is characteristic of the process; variables involving distance, such as a derivative with respect to a distance coordinate, by a characteristic length L; time, either by a characteristic process time (e.g., t c , a relaxation time) or by the ratio of the characteristic distance to the characteristic velocity L/U; mass, by a characteristic mass or, if the material density is constant, by the density (M/L 3 ); and pressure, by  U 2 . Then, when all the variables have been nondimensionalized by appropriate constants, the resulting coefficients of the various terms can be grouped into a smaller number of coefficients. When this is carried out, some terms in a given equation will consist solely of the nondimensional variables of the problem, and other terms will consist of quantities involving nondimensional variables multiplied by nondimensional groups of the constants and characteristic parameters of the equation.

293

The selection of specific constants for the nondimensionalizing process is not necessarily unique. In comparing two processes of geometrical similarity and comparable boundary conditions, however, the same physical dimension must be selected for the two processes (e.g., the diameter of a process tank for both situations). Also, for order-of-magnitude scaling, the characteristic dimensions chosen must represent the physical phenomena in a quantitatively meaningful way. Nondimensionalizing the equation of motion for an incompressible, Newtonian material provides a basis for comparing the relative importance of viscous forces, inertial forces, body forces, etc. The Navier – Stokes equation (Eq. 1.95),  

 ∂u + u ·∇u = g − ∇p + η∇2 u ∂t

(1.111)

can be nondimensionalized by the three characteristic dimensions L, U, and  (time is nondimensionalized by L/U) to give ∂u∗ + u∗ ·∇u∗ ∂t∗  ∗    g Lg η − ∇∗ p∗ + ∇∗2 u∗ = 2 U g LU (1.112)

where in Equation (1.111) the  u · ∇ u term represents inertial forces and the viscosity term represents viscous forces. Body forces such as gravity are accounted for by  g. In obtaining the dimensionless form (Eq. 1.112) the coefficient of the inertial term has been set to unity by division so that the remaining coefficients represent a ratio to the inertial forces. For example, the coefficient of the viscous term in Equation (1.112) is a dimensionless group representing the ratio of the viscous force to the inertial force characteristic quantities. The inverse of this group is the Reynolds number Re =

LU η

(1.113)

The coefficient of the body force term represents the ratio of the body forces to the inertial forces and is the inverse of the Froude number: Fr =

U2 gL

(1.114)

294

Transport Phenomena

These dimensionless groups represent the relative importance of these terms in the equation of motion. A large Reynolds number signifies large inertial forces compared to viscous forces, and a large Froude number signifies a greater importance of inertial forces relative to the body forces (e.g., forced convection versus natural convection). The equation of energy can be nondimensionalized in a similar way to obtain analogous dimensionless groups. In dimensional form the equation of thermal energy for an incompressible Newtonian fluid with Fourier’s law of heat conduction is   cp

∂T + u ·∇T ∂t

 = λ∇2 T + Φv

(1.115)

where Φv is the viscous dissipation defined by Equation (1.97). This energy equation can be nondimensionalized to give ∂T ∗ 1 ∗2 ∗ Br ∗ + u∗ ·∇∗ T ∗ = ∇ T + Φ ∂t∗ Pe Pe v

(1.116)

In Equation (1.115) the u · ∇T term is the transfer of heat as a result of fluid convection, λ ∇2 T represents heat transfer by conduction, and Φv is the irreversible conversion of mechanical energy to thermal energy by viscous dissipation. When these terms are nondimensionalized and the coefficient of the convection heat-transfer term is made equal to unity, the inverse of the coefficient of the conduction term represents the relative importance of convention to conduction and is termed the P´eclet number: Pe =

L U  cp λ

(1.117)

It may also be viewed as the product of the Reynolds number and another dimensionless group termed the Prandtl number, given by Pr =

cp η λ

coefficient of the viscous dissipation term represents the relative importance of viscous dissipation to convection and is normally expressed as the ratio of the Brinkman to the P´eclet number, where the Brinkman number is Br =

η U2 λ T0

(1.118a)

and T 0 is a reference temperature or temperature difference. Use of dimensionless forms of the governing equations provides an excellent basis for comparing the same process at different scales. If two processes are the same, that is, if they accomplish the same task in geometrically similar apparatuses and with comparable boundary conditions, then they are mathematically identical with respect to these governing equations. For further details, see → Scale-Up in Chemical Engineering.

2. Transport in Compositionally Homogeneous Systems 2.1. Heat Conduction in Solids Heat transfer by conduction arises from spatial variations of temperature in a body. If the body is a stationary, opaque solid, conduction is the only mechanism for heat transfer. In this section the basic equations for heat conduction are summarized and specific results are presented that illustrate the effects of geometry, thermal properties, initial state, and boundary conditions. Good general references for this material are [16, Chaps. 2 – 4], [17, Chaps. 9 – 11], [18, Chaps. 1 – 5], and [19]. 2.1.1. Heat Conduction Equations

(1.118)

Evidently, the Prandtl number may be thought of as a conversion factor relating momentum transport (the ratio of transport by convection to that by molecular processes, i.e., stresses) to heat transport (the ratio of transport by convection to that by molecular processes, i.e., conduction) and expressed alternatively as the ratio of momentum diffusivity (kinematic viscosity) ν (= µ/) to thermal diffusivity, α (= λ/ cp ). The

The general energy balance was given in Equation (1.83). For heat conduction in solids it is simplified in several ways: In the case of stationary materials where u = 0, only the accumulation, conduction, and generation terms are retained. Also, the continuity equation (Eq. 1.65) implies that the density  is time-invariant. Further, local equilibrium is assumed, and the specific internal energy Uˆ is represented in terms of the density and temperature. Then,

Next Page Transport Phenomena ∂ ˆ ∂T U (, T ) = cv ∂t ∂t

(2.1)

For solids, the specific heat capacity at constant ˆ volume cv ≡ (∂ U/∂T ) is approximately equal to the specific heat capacity at constant presˆ sure cp = (∂ H/∂T )p . The latter quantity is usually measured and found in data tables. With these simplifications, the energy balance (Eq. 1.83) reduces to  cp

∂T = − ∇ ·q + Φ ∂t

(2.2)

This equation is the differential form of the heat conduction equation. The term on the left is the accumulation term, and the terms on the right are the conductive flux and thermal energy generation and radiation terms, respectively. To use the above equations to determine temperature fields and heat-transfer rates, the heat flux vector q must be related to the temperature gradient through Fourier’s law, i.e., q = − λ∇T

(2.3)

which is written here for isotropic materials. In general, the thermal conductivity λ depends on temperature, pressure, and composition. For homogeneous stationary solids, only the temperature dependence is important. Substituting Fourier’s law into the heat conduction equation gives  cp

∂T = ∇ · (λ∇T ) + Φ ∂t

(2.4)

which, with the specifications of λ (T ), cp (T ), and Φ, as well as the necessary boundary and initial conditions, can be used to determine the temperature field. Once the temperature field is known, the directional heat-transfer rates can be determined from Fourier’s law. When a temperature-dependent thermal conductivity is used, the conduction equation is nonlinear and advanced analytical or numerical methods are generally required for solution [19, pp. 10 – 12]. For many materials, the dependence of λ on T is weak and λ can be taken as constant. Under these conditions, the heat conduction equation reduces to the nonhomogeneous form of the classical heat conduction (or diffusion) equation: ∂T Φ = α∇2 T + ∂t  cp

(2.5)

295

The quantity α ≡ λ/ cp is the thermal diffusivity, which plays the same role in heat transfer that mass diffusivity plays in mass transfer; it also has the same units (m2 /s). Typical values of the thermal properties of various solid materials are given in Table 2. Properties can vary widely depending on the material, composition, and structural form. Table 2. Thermal properties of various solid materials∗ at θ = 20 ◦ C Material

λ, cp , , W m−1 K−1 J kg−1 K−1 kg/m3

α×106 , m2 /s

Aluminum Copper Cast iron (4 % C) Stainless steel (304) Lead Silver Common red brick Concrete Glass Plaster Wood, pine Asbestos Glass wool Corkboard

237 398 52

905 384 420

2 707 8 954 7 272

96.7 115.7 17.0

13.8

400

8 000

4.0

35 427 0.69

130 236 840

11 373 10 524 1 600

23.4 171.9 0.51

800 800 800 2 500

2 100 2 690 1 400 600 580 24 160

0.6 0.35 0.43 0.07

1 0.76 0.48 0.1 0.2 0.038 0.043

∗ Adapted from [16, p. 543] and [21, p. 592]. The values for construction materials can vary significantly depending on the specific composition and packing of the material.

2.1.2. Initial and Boundary Conditions The initial condition required in the solution of the conduction equation involves the specification of temperature as a function of position at some initial time, say t = 0. The boundary conditions can take several forms: 1) The temperature on the bounding surface of the solid can be specified as a function of position and time. Although this is the most convenient boundary condition from a solution standpoint, surface temperatures are very difficult to measure. More commonly, the temperature of the fluid in contact with the surface is known (see condition 3 below). 2) The heat flux normal to the bounding surface is specified as a function of position and time. A special case arises with a perfectly

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insulated surface where the normal heat flux component is zero. 3) For a fluid – solid boundary, the normal component of the conductive heat flux on the solid side is equated to the “convective” flux away from the surface on the fluid side. This is expressed as On A :

qn = − λ

∂T = h (T − Tf ) ∂n

(2.6)

where the conductive flux has been represented by Fourier’s law and the “convective” flux by Newton’s law of cooling. The latter relation represents the “convective” flux in terms of the temperature difference between the surface and the fluid (T f ) and the quantity h, which is called the heat-transfer coefficient. Actually, the flux on the fluid side arises from complex, coupled, conduction, and convection mechanisms that depend on the flow field in the neighborhood of the surface. Such effects are not represented explicitly in this relation but are embedded in the heat-transfer coefficient h, which is an effective thermal conductance that depends on fluid properties, geometry, and fluid motion. Large values of h are associated with efficient fluid mixing and efficient convective – conductive heat transfer in the neighborhood of the surface (e.g., turbulent transport in high-conductivity liquids). Small values of h are associated with nearly stagnant fluids of low conductivity. Typical values of h for various physical situations are given in Table 3. Table 3. Order of magnitude of heat-transfer coefficients Physical situation

h, W m−2 K−1

Free convection, air Forced convection, air Forced convection, oil Forced convection, water Boiling water Condensing steam

4 – 50 10 – 300 60 – 1800 250 – 15 000 300 – 100 000 4000 – 160 000

2.1.3. Steady, One-Dimensional Conduction In many problems, the temperature field is timeinvariant and depends on only one spatial variable. Consider first the plane slab case illustrated

Figure 1. Steady one-dimensional heat conduction in various geometries A) Conduction in a plane wall; B) Radial conduction in a cylindrical shell; C) Radial conduction in a spherical shell Conductive resistance Rcond obtained from Equation (2.14); heat-transfer rate Q˙ from Equation (2.13).

Transport Phenomena in Figure 1 A where no heat is generated, conduction is in the x direction, and the temperature dependence is T (x). In this case, Equation (2.2) reduces to dqx = 0 dx

(2.7)

or qx =

Q˙ = constant A

(2.8)

where Q˙ is the heat-transfer rate in joules per second across the slab face of area A. If the temperatures T 0 and T 1 are specified at x 0 and x 1 , and Fourier’s law is used for the heat flux, the heat-transfer rate Q˙ can be obtained by direct integration: λ1 A (T0 − T1 ) Q˙ = ∆x1

(2.9)

¯ 1 is the average therwhere ∆x 1 ≡ x 1 − x 0 and λ mal conductivity defined by 1 λ1 ≡ (T1 − T0 )

λdT

Q˙ λA

λ1

λ∆x1

(2.11)

(x − x0 )

dξ Aξ

(2.14)

Here Aξ is the cross-sectional area perpendicular to the ξ direction. For radial conduction in cylindrical shells (see Fig. 1 B), ξ corresponds to r and Aξ is 2 π rL; for radial conduction in spherical shells (see Fig. 1 C), ξ corresponds to r and Aξ is 4 π r 2 . The temperature profile is obtained in the same way as for the plane slab: T = T0 −


ξ

Q˙ λ (T )

ξ0

dξ Aξ

(2.15)

If the dependence of thermal conductivity on temperature is sufficiently weak, where λ (T ) ≈ λ (T 1 ), using Equations 2.13 and 2.14 gives (T0 − T1 ) ξ 1

dξ Aξ


ξ

dξ Aξ

ξ0

(2.16)

For the cylindrical and spherical geometries illustrated in Figure 1, the conductive resistances and the heat-transfer rates are given by: (Cylindrical) Rcond =

ln (r1 , r0 ) 2π L λ1

2π λ1 (T0 −T1 ) Q˙ = ln (r1 /r0 ) (Spherical)

Rcond =

(2.16a) (2.16b)

(r1 −r0 ) 4π r0 r1 λ1

4π r0 r1 λ1 (T0 −T1 ) Q˙ = (r1 −r0 )

(2.16c) (2.16d)

(2.12)

If the dependence of λ on T is sufficiently weak ¯≈λ ¯ 1 , a linear temperature profile results. and λ If λ is a stronger function of temperature, nonlinear profiles develop. These results can be extended to more general geometries with a few modifications. If ξ is the direction of the nonzero heat flux component, the heat-transfer rate can be expressed as (T0 − T1 ) Q˙ = qξ Aξ = Rcond

ξ0

ξ0

(x − x0 )

λ1 (T0 − T1 )


ξ1

1

T0

¯ = λ (T ) is the average conductivity in where λ the range from T 0 to T and is defined by an equation similar to Equation (2.10) except that T 1 is replaced everywhere by T . If Equation (2.9) is substituted into Equation (2.11), then T = T0 −

Rcond =

(2.10)

The temperature field is obtained similarly except that the range of integration is from x 0 to any position x: T = T0 −

with the conductive resistance Rcond given by

T = T0 −


T1

297

(2.13)

2.1.4. Convective Boundary Conditions As noted previously, the temperature of the fluid in contact with a surface is more commonly known than the surface temperature. Consider again the plane slab case in Figure 1 A, but now assume that the left face is in contact with a fluid of temperature T f0 , then from Newton’s law of cooling,

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Transport Phenomena

Tf0 − T0 =

qx Q˙ = h0 h0 A

(2.17)

Eliminating T 0 between Equations (2.9) and (2.17) gives (Tf0 − T1 )  Q˙ =  1 + ∆x1 h A

(2.18)

λ1 A

0

where ∆x 1 = x 1 − x 0 . This equation gives the total heat-transfer rate in terms of the temperature driving force T f0 − T 1 and the series of convective and conductive resistances. The ratio of the ¯ 1 A) and the convective resisconductive (∆x 1 /λ tances (1/h0 A) is the dimensionless Biot number: Bi ≡

h0 ∆x1

(2.19)

λ1

When the Biot number is large, conduction is the controlling heat-transfer mechanism and T 0 ≈ T f0 . When the Biot number is small, convection is controlling and T 0 ≈ T 1 . 2.1.5. Composite Systems The results of Sections 2.1.1, 2.1.2, 2.1.3, and 2.1.4 can be extended to composite or multilayered systems by simply applying equations similar to Equation (2.9) across each solid layer and Newton’s law of cooling at each fluid – solid boundary. For the two-layer slab shown in Figure 2 λ1 (T0 − T1 ) λ2 (T1 − T2 ) = Q˙ = ∆x1 ∆x2 = α0 (Tf0 − T0 ) = α2 (T2 − Tf2 )

1 α0 A

+

(Tf0 ∆x1 λA

− Tf2 ) +

∆x2 λ2 A

+

1



(2.21)

α2 A

or (Tf0 − Tf2 ) Q˙ = Rconv + Rcond

(2.22)

Generalizing to N layers gives Q˙ =  1 α0 A

(Tf0 − TfN )  N  ∆xi + + i=1

λi A

1 αN A

Similar expressions can be written for composite systems in other geometries (cylinders, spheres, etc.). In these cases, the heat-transfer rate is given by Equation (2.22), with the Rcond,i contributions for each layer obtained from Equation (2.14) and the Rconv, i contributions from Rconv,i =

1 hi Ai

(2.24)

where hi is the heat-transfer coefficient at the “ith” fluid – solid surface and Ai the corresponding surface area.

(2.20)

where ∆xk = xk − xk−1 . Using these equations to eliminate T 0 , T 1 , and T 2 yields Q˙ = 

Figure 2. Conduction through a composite two-layer slab Left face is exposed to a fluid at temperature T f0 ; right face is exposed to a fluid of temperature T f2 . The overall driving force for the heat transfer and the conductive resistances are shown in the equivalent electronic circuit.

(2.23)

2.1.6. Steady Conduction with Heat Generation and in Multidimensions Heat is commonly generated in solids by electrical heating as well as by chemical and nuclear reactions. If heat generation is uniform and conduction is one-dimensional, the temperature field can be determined by direct integration. For more complex problems involving multidimensional conduction and nonuniform generation, advanced mathematical methods are required. These are outside the scope of this article but are discussed in [19].

Transport Phenomena

299

2.1.7. Transient Heat Conduction – Lumped Capacity Systems

2.1.8. Transient Heat Conduction – More General Solutions

When a solid body with an initial temperature T i is exposed to a fluid of temperature T f , the temperature field in the body changes with time to decrease the magnitude of the temperature difference. Under the conditions of small Biot numbers, the determination of the resulting transient temperature change can be analyzed by using a lumped parameter approach. In particular, if Newton’s law of cooling describes the conditions at the fluid – solid boundary, an order-ofmagnitude estimate (symbolized by “∼”) gives

At higher Biot numbers, the spatial variations of temperature cannot be neglected and the problem is mathematically more complex than that given in the previous section. In this section, solutions for the semi-infinite body, the plane slab, the infinite cylinder, and the sphere are summarized. Mathematical details are presented in [19, pp. 50 – 73 (semi-infinite solid), pp. 119 – 127 (plane slab), pp. 201 – 203 (infinite cylinder), and pp. 237 – 238 (sphere)]. Consider first the transient heating (or cooling) of a body bounded by the plane x = 0 and extending to infinity in the positive x direction. The body is initially at temperature T i and its face at x = 0 is held at temperature T f . If constant properties are assumed, the governing equations for T (x, t) are

 ∆T ∼

hl λ

 (T0 − Tf )

(2.25)

where l is the characteristic length scale of the body and ∆T is the approximate temperature change occurring over this length scale. Clearly, when Bi ≡ (h l/λ)  1, the spatial temperature change ∆T in the body is much smaller than the temperature changes T 0 − T f across the fluid layer adjacent to the bounding face. Under such conditions, the surface temperature of the body (T 0 ) can be assumed to be approximately equal to the average body temperature (T¯ ). The energy balance then becomes  cp V

dT¯ ¯ A T¯ − Tf = −h dt

(2.26)

where h¯ and T¯ represent average values over the surface and volume V of the body, respectively. If the material properties are taken to be constant, this equation can be integrated to give T¯ = Tf + (Ti − Tf ) exp {−Bi t∗ }

(2.27)

where, in this case, the characteristic length is l ≡ V/A, the Biot number is Bi ≡ l h/λ, and the dimensionless time is t∗ ≡ α t/l 2 . The corresponding rate of heat transfer to the fluid is given by ¯ A (Ti − Tf ) exp {−Bi t∗ } Q˙ = h

(2.28)

The significance of these results for Bi  1 is that the rate of heat transfer given by Equation (2.28) represents an upper bound; i.e., the cooling or heating rates found at higher Biot numbers will be slower and the characteristic time for temperature accommodation of the body with the fluid will be longer.

∂T ∂2T = α ∂x2 ∂t

(2.29)

T (x, 0) = Ti , x ≥ 0

(2.30)

T (0, t) = Tf , t > 0

(2.31)

T (∞, t) = Ti ,

t≥0

(2.32)

These equations suggest a solution of the form T − Tf = f (x, t, α, Ti − Tf )

(2.33)

If dimensional analysis is used, the dimensionless temperature can be expressed in terms of a single independent variable, i.e., T∗ ≡

T − Tf = T ∗ (ζ) Ti − Tf

(2.34)

where ζ = √

x 4αt

(2.35)

When Equation (2.34) is substituted into Equations (2.29) – (2.32), the problem is reduced to the solution of an ordinary differential equation d2 T ∗ dT ∗ + 2ζ = 0 dζ 2 dζ

subject to the boundary conditions

(2.36)

300

Transport Phenomena

T ∗ (0) = 0

(2.37)

T ∗ (∞) = 1

(2.38)

Integrating Equation (2.36) first for dT ∗/dζ and then again for T ∗ (ζ) and using the boundary conditions to determine the two integrating constants give 2 T∗ = √ π


ζ

√

  1 T ∗ = erf (ζ) + exp ζ β + β 2 · 4    1 erfc ζ + β 2

2

e−ξ dξ ≡ erf (ζ)

(2.39)

0

4α t = 3.6

√

αt

(2.40)

This is illustrated in Figure 3 where the time evolution of the temperature profile is plotted for the transient heating of a semi-infinite body.

√ αth λ

2

(2.43)

and erfc ( y) = 1 − erf ( y) is the complementary error function. Although these results have been developed in the context of a semi-infinite body, they are also valid for a body of finite thickness (e.g., a slab of thickness 2 l ) for all times when δ T ≤ l. For times longer than this, the boundary layer exceeds the half thickness and a semi-infinite solution is no longer valid. Then solutions for finite bodies must be used. Solutions of the latter type are now presented for the plane slab, the infinite cylinder, and the sphere. Consider one-dimensional heat conduction in a plane slab bounded by the surfaces at x = ± l. For constant thermal properties the governing energy equation is Equation (2.29); the initial and boundary conditions take the form T (x, 0) = Ti ,

−l ≤ x ≤ l

∂T (0, t) = 0, all t ∂x −λ

Figure 3. Time evolution of the temperature profile during heating of a semi-infinite slab (x > 0) The x position where T = T f − 0.99 (T f − T i ) defines the to Equation thermal boundary layer δT which according √ (2.40) increases in proportion to α t α t.

If the temperature boundary condition at x = 0 (Eq. 2.31) is replaced by a convective flux condition, i.e., λ

∂T (0, t) = h [T (0, t) − Tf ] ∂x

(2.41)

(2.42)

where β ≡

The middle term is called the error function [erf (ζ)]. Tabulated values can be found in various sources [19, pp. 485 – 487]. Of particular interest here is the fact that the error function (as well as T ∗) takes on a value of approximately 0.99 when its argument ζ takes on a value of 1.8. Physically, this implies that the temperature change (T − T f ) reaches 99 % of the maximum change (T i − T f ) within a thermal boundary layer defined by δT = 1.8

the solution takes the form

∂T (l, t) = h [T (l, t) − Tf ] , t > 0 ∂x

(2.44) (2.45) (2.46)

The second equation arises from the fact that the midplane (x = 0) is a plane of symmetry and the flux across it is zero. The solution to this problem in terms of the dimensionless position x∗ ≡ x/l and the dimensionless time t∗ ≡ α t/l 2 is obtained as a Fourier series: T∗ ≡

T − Tf Ti − Tf

∞

= 2

 sin βn cos (βn x∗ ) 8 2 ∗; exp −βn t βn + sin βn cos βn n=1

(2.47)

where the eigenvalues βn are obtained from βn tanβn = Bi

(2.48)

Transport Phenomena For short times the convergence of Equation (2.47) is slow and many terms must be taken before an accurate solution can be obtained. However, for t∗ > 0.25, only the first term of this equation is needed to obtain results with errors less than 1 %. Even in these cases the solution is not convenient because β 1 must be determined from an implicit relation (Eq. 2.48). Fortunately, the roots of the latter equation can be accurately fitted to an expression of the form

301

As before, for t∗ > 0.25, only the first term of Equation (2.50) is important. Also, the first root of Equation (2.51) can be obtained from Equation (2.49) with the coefficients Ck given in the second column of Table 4. For a sphere, the solution is T∗ = ∞

8 2 ∗; 2 Bi  sin βn sin (βn r ∗ ) t (2.52) exp −βn r ∗ n=1 βn [βn − sin βn cos βn ]

5



β1 =

Ck (logBi)k−1 for 0.02 ≤ Bi ≤ 8 (2.49)

with the eigenvalues given by

k=1

For the plane slab case, the coefficients Ck are given in the first column of Table 4. When this equation is used with the first term of Equation (2.47), temperatures can be predicted to within 5 % and, in most cases, depending on the Biot number, to within 1 %. When Bi > 8, tabulated values of βn can be used [19, p. 491]. Such approaches are preferred to the time – temperature charts used by many investigators [22–25] because these charts are often difficult to read and to interpolate accurately. Table 4. Coefficients for Equation (2.49) Coefficient

Plane slab

Cylinder

Sphere

C1 C2 C3 C4 C5

0.859777 0.702996 0.050636 −0.144345 −0.044631

1.256701 1.112492 0.148517 −0.260841 −0.111339

1.573305 1.450377 0.251563 −0.306763 −0.139485

Similar types of solutions are available for the infinite cylinder (radius r 0 ) and the sphere (also radius r 0 ). Here again, the body is assumed to be at an initial temperature T i and at t = 0 is immersed in a fluid at temperature T f . For the cylinder, the solution in terms of the first- and second-order Bessel functions of the first kind is T∗ = ∞

2

 n=1

8 2 ∗; J1 (βn ) J0 (βn r∗ )   exp −βn t βn J02 (βn ) + J12 (βn )

(2.50)

where T ∗ is defined as in Equation (2.47), r∗ ≡ r/r 0 and t∗ ≡ α t/r 20 . The eigenvalues βn are obtained from J1 (βn ) βn = Bi J0 (βn )

(2.51)

βn cotβn = 1 − Bi

(2.53)

The one-term solution is again valid for t∗ > 0.25, and the first eigenvalue can be obtained from Equation (2.49) by using the coefficients in the third column of Table 4. The solutions of these one-dimensional unsteady-state problems can be used in simple ways to obtain solutions to multidimensional, unsteady-state problems. These are now summarized for several cases. Consider first the case of an infinite rectangular bar bounded by the surfaces at x = ± x 0 and y = ± y0 . The solution at any position (x, y) at time t is given by T (x, y, t) = Tx (x, t) Ty (y, t)

(2.54)

Here Tx (x, t) is obtained from Equation (2.47) with x 0 replacing l as the length scale. Similarly, Ty ( y, t) is obtained from Equation (2.47) with y replacing x in that equation and y0 replacing l. For a rectangular parallelepiped with bounding surfaces at x = ± x 0 , y = ± y0 , and z = ± z0 , the solution is given by T (x, y, z, t) = Tx (x, t) Ty ( y, t) Tz (z, t)

(2.55)

The Tx and Ty solutions are obtained in the manner just described for the infinite rectangular bar. The Tz solution is obtained from Equation (2.47) with z replacing x in that equation and z0 replacing l. Finally, for a finite cylinder bounded by the surfaces r = r 0 and z = ± l, the solution is T (r, z, t) = Tr (r, t) Tz (z, t)

(2.56)

Here Tr is obtained from Equation (2.50) and Tz from Equation (2.47) (z replaces x).

302

Transport Phenomena

Examples of the use of Equations (2.54) – (2.56) are given in many texts [26, pp. 307 – 310], [27, pp. 174 – 177]. The approach suggested here is different in that the onedimensional solutions are obtained from oneterm analytical solutions instead of time – temperature charts.

2.2. Steady, One-Dimensional Flows (→ Fluid Mechanics, Chap. 3.3.1.) As in the sections above where heat transfer arose from the existence of temperature gradients, momentum transfer arises from fluiddeforming velocity gradients. In particular, for the one-dimensional unidirectional flow u ≡ (ux , uy , uz ) = [0, 0, uz ( y)]

(2.57)

the resulting momentum transfer involves zmomentum being transferred in the y-direction. This is illustrated in Figure 4 where faster moving fluid layers transfer momentum to adjacent slower moving layers. The nonzero stress components associated with this velocity field can be interpreted as momentum fluxes [17, Chap. 2], and if the fluid is Newtonian, these components can be expressed as τyz = τzy = − η

duz dy

(2.58)

which is analogous to Fourier’s law for onedimensional heat conduction in the y-direction.

flows commonly arise with internal flows in straight channels where the cross-sectional area is invariant in the flow direction. Under such conditions, the velocity uz does not vary in the z-direction and the most general form of the velocity field is u = [0, 0, uz (x, y)]. The latter two-dimensional problem arises with systems of arbitrary cross section. For simpler systems such as flow between parallel plates or flow in cylindrical tubes, the problem becomes onedimensional and uz is dependent on only one spatial variable. The driving force for unidirectional flows can be pressure gradients, gravity, or moving boundaries. The pressure gradient and gravity effects appear directly in the equation of motion and can be combined into an overall driving force term ∇P: ∇P = ∇p − g

(2.59)

To achieve a unidirectional flow without moving boundaries, ∇P must be nonzero and the flow direction must be coincident with the directions of ∇P. For flow in the z-direction where P (z), the only nonzero component of Equation (2.59) is dp dP = −  gz dz dz

(2.60)

where gz is the component of gravity in the zdirection. Integration of this equation along the flow channel from z = z0 to any point z gives P − P0 = p − p0 −  gz (z − z0 )

(2.61)

That is, the driving force for flow arises from variations of pressure and potential energy along the flow direction. The quantity P is sometimes referred to as the dynamic pressure because it can be written as the difference between the pressure existing during flow and that arising under no flow or hydrostatic conditions: P = p − pH

(2.62)

Figure 4. Flow resulting from the movement of a surface In this case, z-momentum is being transferred from layer to layer in the y-direction. The stress component τyz can be interpreted as a momentum flux (z momentum transferred in the y direction).

Here, pH is the hydrostatic pressure which is obtained from

The flow above is a special case of the general class of unidirectional flows. Unidirectional

In Equation (2.61) the hydrostatic pressure part is p0 +  gz (z − z0 ).

∇pH −  g=0

(2.63)

Transport Phenomena

303

Another characteristic of steady unidirectional flows is that these are noninertial flows. Specifically, a fluid element in this flow does not accelerate along its path, and the nonlinear inertial term u · ∇u in the equation of motion is zero. The z-component of the equation of motion then takes the form (∇ ·τ ) ·iz +

dP = 0 dz

(2.64)

where iz is the unit base vector in the z direction. If the fluid is an incompressible Newtonian fluid, this equation can be expressed as η ∇ 2 uz −

dP = 0 dz

(2.65)

With respect to Newton’s laws of motion, this equation can be interpreted as a balance of forces per unit volume on any fluid element in the flow; specifically, the driving force for the flow is (− dP/dz) and the viscous resisting force is η ∇ 2 uz . The only material parameter that arises in the treatment of steady, unidirectional flows is the viscosity η. In more general flows, the density  also arises. In Table 5, typical values of density and viscosity for a number of fluids are presented. Values for the heat capacity cp , the thermal conductivity λ, and the Prandtl number Pr ≡ ν/α are also provided. All of these quantities arise later in the treatment of convective heat transfer. 2.2.1. Generalized Couette Flow A flow pattern arising in many different applications (polymer extrusion, screw pumping and expression, journal-bearing lubrication, etc.) is the unidirectional plane parallel flow illustrated in Figure 5. The flow is driven by both the movement of the upper plate at velocity U and the application of a pressure gradient (P0 − PL )/L in the direction of flow. The flow geometry is assumed to be of thickness b in the y-direction, width w in the x-direction, and length L in the z-direction. The thickness-to-width ratio (b/w) and the thickness-to-length ratio (b/L) are assumed to be sufficiently small that edge effects and entrance or exit effects can be neglected, and the velocity field can be represented by u = (ux , uy , uz ) = [0, 0, uz (y)].

Figure 5. Generalized Couette flow between parallel surfaces The upper surface moves with velocity U; the lower surface is stationary. The nature of the flow depends on the magnitude and sign of the applied pressure gradient dP/dz. In the case above, DP/Dz > 0.

The nonzero stress components for this flow are τyz and τzy , and Equation (2.64) reduces to dτyz dP + = 0 dy dz

(2.66)

Since the first term on the left-hand side is at most a function of y and the second term at most a function of z, this equation is satisfied only if both terms are equal to a constant. Integration then gives τyz = K y + τ0 K ≡

(P0 − PL ) L

(2.67) (2.68)

where τ 0 is the shear stress at y = 0. At this point, the quantity τ 0 is unknown. It is determined by using the velocity boundary conditions. If the fluid is an incompressible Newtonian fluid, Equation (2.58) can be combined with Equation (2.67) and integrated to give the velocity profile uz =

y b2 ∆P  y   y 1− +U 2η L b b b

(2.69)

where the quantity τ 0 and the additional constant that arises from the integration are evaluated by using the boundary conditions uz (0) = 0

(2.70)

uz (b) = U

(2.71)

In dimensionless form, the velocity is y uz y y = K∗ 1− + U b b b

(2.72)

304

Transport Phenomena

Table 5. Thermophysical properties of various gases and liquids∗ Fluid Gases (101.3 kPa) Air Carbon dioxide Helium Hydrogen Nitrogen Oxygen Liquids (saturated) Dichlorodifluoromethane Glycerol Mercury Light machine oil Water

θ, ◦ C

, kg/m3

η, Pa s

λ, W m−1 K−1

cp , J kg−1 K−1

Pr

27 27 −18 27 27 27

1.183 1 .7973 0.1906 0.08185 1.1233 1.3007

1.853×10−5 1.4958×10−5 1.817×10−5 0.8963×10−5 1.784×10−5 2.063×10−5

0.02614 0.016572 0.1357 0.182 0.0259 0.02676

1.003×103 0.871×103 5.200×103 14.314×103 1.0408×103 0.09203×103

0.711 0.770 0.70 0.706 0.715 0.709

1305 1261 13 611 907 996.6

0.2545×10−3 1412×10−3 1.633×10−3 145.1×10−3 0.8232×10−3

0.0690 0.285 8.34

980 2350 139.1

3.62 11 630 0.027

0.6084

4177

5.65

27 20 27 16 27

∗ Adapted from [21, pp. 595 – 601].

where K∗ =

b2 ∆P 2η U L

(2.73)

When K ∗ = − 3, the volumetric flow rate associated with the moving boundary is balanced by the flow rate arising from the adverse pressure gradient and Q = 0.

is the dimensionless pressure gradient. Depending on the magnitude and sign of K ∗, the velocity profile can take various forms (see Fig. 6). When K ∗ = 0, the profile is linear and corresponds to simple Couette flow: uz = U

y

(2.74)

b

When K ∗ > 0, the pressure gradient and the moving upper boundary both contribute to flow in the positive z direction, and if K ∗ is sufficiently large a maximum occurs in the velocity profile (see the K ∗ ≥ 2 cases in Fig. 6). When K ∗ < 0, the pressure gradient causes a reverse flow. If the magnitude of this adverse gradient is sufficiently large, the fluid moves in the positive z-direction in the upper part of the flow channel (due to motion of the upper boundary) and in the negative z-direction in the lower part of the channel (see the K ∗ = − 4 case in Fig. 6). The volumetric flow rate Q and the average velocity uavg associated with this flow are given by
b

Q = uavg b w =

uz wdy = 0

1 b3 w∆P + bwU 12 η L 2 (2.75)

The first term on the right represents the flow contribution arising from the pressure gradient and the second that from the moving boundary.

Figure 6. Dimensionless velocity profiles for generalized Couette flow

If the fluid is non-Newtonian, significantly more complex flow equations are obtained for generalized Couette flow. For the case of powerlaw, viscous non-Newtonian behavior, see [28], [29] (see also → Fluid Mechanics, Chap. 4.1.2.). 2.2.2. One-Dimensional Poiseuille Flows (→ Fluid Mechanics) When U = 0, the flow illustrated in Figure 5 becomes a pressure-driven Poiseuille flow. Under such conditions, the velocity profile becomes parabolic with a plane of symmetry at y = b/2. Specifically, in terms of a y coordi-

Transport Phenomena nate measured from the channel centerline, i.e., y
= y − b/2, Equation (2.69) takes the form  uz =

b2 ∆P 8η L

   2

y 1−4 b

(2.76)

and the volumetric flow rate reduces to a Hagen – Poiseuille relation where Q is proportional to (P0 − PL )/L, i.e., Q = uavg b w =

b3 w∆P 12 η L

(2.77)

In the case of one-dimensional Poiseuille flows in channels with circular and concentric annulus cross-sectional geometries, analogous results can be obtained . In all cases the volumetric flow rate is proportional to the pressure drop (for further details, see → Fluid Mechanics, Chap. 3.3.1.). An alternate representation of flow rate – pressure drop results for unidirectional flows is in terms of the friction factor relation. In particular, the wall shear stress τ w ≡ − η | duz /dy
|w is assumed to be proportional to the kinetic energy per unit volume expressed in terms of the average velocity uavg :  τw = f

1  u2avg 2

 (2.78)

Here, the constant of proportionality is the friction factor f . Since the pressure force must be balanced by the drag force on the walls  ∆P Ac = τw Aw = f Aw

1  u2avg 2

 (2.79)

Here Ac is the cross-sectional area and Aw is the wall surface area. The friction factor can then be expressed as f Re = C ≡

l2

(∆P/L ) 2 η uavg

 uavg l η

(2.81)

and l =

f Re = C = 24

where Re ≡ 2  uavg b/η. Another flow that arises in many industrial applications is Poiseuille flow in a tube. If the tube has radius r 1 and length L, the stress and velocity fields for incompressible Newtonian fluids are  ∆P r L    2

r 2 ∆P r uz = 1 1− 4η L r1 τrz = τzr =

1 2



(2.84)

(2.85)

where r is the radial coordinate. The volumetric flow rate is then
r1 Q = uavg π r12 = 2 π

uz rdr = 0

π r14 8η



∆P L

 (2.86)

If the latter result is used in Equation (2.80), then f Re = 16

(2.87)

where the characteristic length l is 2 r 1 , and the Reynolds number is defined by Re ≡ 2  uavg r 1 /η. Similar results can be obtained for axial flow in the annular region between concentric cylinders of radii r 0 and r 1 where r 1 > r 0 . In this case, the characteristic length is l = 2 (r 1 − r 0 ), and the value of C in Equation (2.80) varies with the ratio χ ≡ r 0 /r 1 . In the limit of small values of r 0 /r 1 , C takes on the value of 16; it approaches an upper limit of 24 as r 0 /r 1 approaches 1. 2.2.3. Flow in Channels of Arbitrary Cross Section For flows in channels with more complex cross sections (e.g., flow in rectangular, elliptic, or eccentric annular channels) or flow in channels with irregular cross sections, the velocity uz is a function of two spatial variables, i.e., uz = uz (x, y)

4 Ac L Aw

(2.83)

(2.80)

where the Reynolds number Re and the characteristic length l are defined by Re =

305

(2.82)

For plane Poiseuille flow where l = 2 b, Equation (2.77) can be used in Equation (2.80) to obtain

(2.88)

For incompressible Newtonian fluids, Equation (2.65) is applicable, and the solution to this equation is given by

306 uz =

Transport Phenomena

1 2 x + y2 4η



dP dz

 + ϕ (x, y)

(2.89)

where the function ϕ (x, y) is a solution of Laplace’s equation ∂2ϕ ∂2ϕ + = 0 2 ∂x ∂y 2

(2.90)

If a no-slip velocity condition is assumed (i.e., uz ,S = 0), then   

dP 1 2 ϕ = ϕS = − x + y2 4η dz S

(2.91)

on the bounding surfaces. For a channel with an elliptic cross section defined by (x/x 0 )2 + (y/y0 )2 = 1, the velocity is given by [30, p. 38] uz =

x2 y 2 ∆P 02 0 2 2 η x0 + y 0 L





1−

x x0

2

 −

y y0

2

a hydraulic radius approach where an effective radius is defined for the channel, i.e., r H ≡ l/2 = 2 Ac L/Aw , and C in Equation (2.80) is taken to be 16, the value corresponding to Poiseuille flow in a tube. However, this approach is approximate and can lead to sizable errors if the actual C for the geometry differs significantly from 16. For flow in channels with elliptic cross sections, the maximum error would be ca. 19 %; for square channels, the error would be 12 %. All of these results are for laminar flow. When the Reynolds number reaches sufficiently high values, the flow becomes unstable and turbulent flow arises. For flow in circular tubes, turbulent flow is generally observed when Re ≡ 2  uavg r 1 /η > 2100. For other geometries, the transition values are different and must be determined in each individual case.

(2.92)

2.2.4. Poiseuille Flow of Non-Newtonian Fluids (→ Fluid Mechanics, Chap. 4.2.)

and the volumetric flow rate by Q = uavg π x0 y0 =

π x30 y03 ∆P

4 η x20 + y02 L

(2.93)

In terms of the friction factor, this result can be written f Re = (



2 π2 [E (k)]2

function of

y0 x0

1+ )



y0 x0

2

= (2.94)

where E (k) is the complete elliptic integral with argument k = [1 − (y0 /x 0 )2 ] [31, p. 509]. When y0 /x 0 → 0, f Re → 19.74, and when y0 /x 0 = 1, f Re = 16 (the cylindrical tube result). For flow in channels with square and equilateral-triangle cross sections with sides a, the flow rate – pressure drop relations are given by Square : f Re = 14.2

(2.95)

Triangular : f Re = 13.3

(2.96)

Results for other cross sections (rectangular and eccentric annuli) are given in [30, pp. 33 – 39]. For even more complex and less regular cross-sectional geometries, estimates (sometimes crude) can be obtained by using

The flow characteristics of non-Newtonian fluids in tubes and planar channels can be quite different from those observed with Newtonian fluids. This is illustrated by considering the steady flow of a power-law, non-Newtonian fluid in a tube of radius r 0 . As in the Newtonian case, the only nonzero stress components are τrz = τzr , and these are described by Equation (2.84). Because duz /dr < 0 for this flow, the shear stress can be written as  τrz = m

−

duz dr

n (2.97)

Combining this equation with Equation (2.84), integrating, and using the no-slip boundary condition uz = 0 at the tube wall give uz =

 τ s w

m

  s+1

r1 r 1− s+1 r1

(2.98)

where s ≡ 1/n, and τ w is the wall shear stress given by τw ≡ |τrz |r=r1 =

1 2



∆P L

 r1

(2.99)

Because the maximum velocity occurs at r = 0, Equation (2.98) can also be written as

Transport Phenomena 



uz = umax 1 −

r r1

s+1

(2.100)

with umax given by umax =

 τ s w

m

r1 (s + 1)

For s = 1 (or n = 1), the behavior is Newtonian and the profile is parabolic. For pseudoplastic fluids where s > 1 (or n < 1), the profile becomes more blunted and pluglike as s takes on larger and larger values (see Fig. 7). For dilatant fluids where s < 1 (or n > 1), the velocity profile becomes more and more pointed as s decreases.

Figure 7. Velocity profiles for the flow of a power-law fluid in a tube The parameter s is the reciprocal of the power-law slope parameter n. When s = 1, Newtonian behavior is observed. When s > 0, the behavior is dilatant.

Comparison of the average velocity with the maximum velocity gives uavg s+1 = umax s+3

the yield surface, i.e., the position r = r Y where | τrz | = τ Y . From Equation (2.84) rY =

(2.101)

(2.102)

For Newtonian fluids (s = 1), this reduces to uavg /umax = 1/2. For pseudoplastic fluids (s > 1), this ratio increases with s and approaches the limit of 1 ( plug flow) as s gets very large (n → 0). For dilatant fluids, uavg /umax ranges from 1/2 to 1/3 as s decreases (or n increases). These results are illustrated in Figure 7. The flow rate – pressure drop expression for this flow is obtained directly from Q = π r 21 uavg . For values of s different from 1, this relation is nonlinear, with d (∆ P/L)/dQ increasing with Q for dilatant fluids and d (∆ P/L)/dQ decreasing with Q for pseudoplastic fluids. In the case of a Bingham plastic with a yield stress τ Y and a plastic viscosity η p , the nature of the flow depends on the radial position of

307

τY r1 2 τY = τw (∆P/L )

(2.103)

At radial positions outside the yield surface (r > r Y ), the magnitude of the shear stress exceeds the yield stress and a shear flow [uz = uz (r)] results. At radial positions inside the yield surface (r < r Y ), the magnitude of the shear stress is less than τ Y and a uniform plug flow occurs with velocity uz (r Y ). As (∆ P/L) decreases, the radial position of the yield surface moves closer and closer to the tube wall. At a value of (∆ P/L) = 2 τ Y /r 1 , the position of the yield surface coincides with the wall and flow ceases. Hence, application of a finite pressure gradient does not ensure flow in the case of materials with plastic behavior. Flow rate results for Newtonian fluids, power-law fluids, and Bingham plastics may be summarized as follows: Flow geometry: Circular tube radius r 1 , length L Newtonian fluid (viscosity η) Q =

π r14 ∆P 8η L

Power-law fluid (parameters m, n) Q =

π r13 (s + 3)



∆P r1 2L m

s , s ≡

1 n

Bingham plastic (parameters τ Y , η p ) π r13 τw Q = 4 ηp



4 1− 3



τy τw



1 + 3



τy τw

4

,

when τw >τy

where ∆P r1 2L Q = 0, when τw ≤τy τw ≡

Expressions for other types of non-Newtonian fluids can be found in various sources. These developments are described and the results summarized for various non-Newtonian fluid models [32, Chap. 5], [33–35]. Further details are given in → Fluid Mechanics, Chap. 4.2.1.

308

Transport Phenomena

2.2.5. Two-Phase Concentric Flow in a Tube – Segregated Flow An interesting example of viscous, onedimensional laminar flow in a tube takes place when two immiscible fluids are injected into a tube of radius r 1 such that one fluid occupies the core region r < r c and the other fluid the outer region r > r c (see Fig. 8). The core fluid has viscosity η and the outer fluid viscosity ηˆ.

The integrating constants can be evaluated by using the conditions (1) that the stress τrz is finite at r = 0, (2) that the velocity uz is zero at the tube wall, and (3) that the stress and velocity are continuous at the boundary r = r c between the phases [τrz (r c ) = τˆrz (r c ) and uz (r c ) = uˆ z (r c )]. Integrating the respective velocity profiles over the inner and outer fluid regions according to Equation (2.86) gives the corresponding volumetric flow rates Q =

π 8 ηˆ



∆P L



 r14

ηˆ ∗4 r + 2 1 − rc∗2 rc∗2 η c

 (2.106)

ˆ = π Q 8 ηˆ



∆P L





2 r14 1 − rc∗2

(2.107)

Here, r c ∗ ≡ r c /r 1 is the dimensionless radial position of the boundary between the inner and outer fluid regions. Eliminating ∆ P/L between these equations gives an implicit expression for r c ∗:  

2

ηˆ ∗2 (1 − ϕ) 1 − rc∗2 − ϕrc∗2 rc + 2 1 − rc∗2 = 0 η (2.108) Figure 8. Segregated two-phase flow in a tube

where

Such problems are often encountered in pipelines where the flow of a viscous fluid is facilitated by the injection of a second lowviscosity immiscible phase. The interest here is in the flow rate enhancement that results from the injection of this second fluid. The analysis of flow in the separate fluid regions is similar to that for Poiseuille flow in cylindrical tubes and annuli. In particular, for Newtonian fluids, Equations (2.64) and (2.65) can be integrated to give the stress and velocity fields for the inner fluid region as τrz

1 = 2

uz = −



∆P L

1 4η





C1 r+ r

∆P L

 r2 −

(2.104)

C1 lnr + C2 η

(2.105)

where C 1 and C 2 are the integrating constants. Identical expressions are obtained for the outer fluid region except that τrz , uz , η, C 1 , and C 2 are replaced by τˆrz , uˆ z , ηˆ, Cˆ 1 , and Cˆ 2 . It writing these expressions, the axial pressure distributions in the two phases are assumed to be equal.

ϕ =

ˆ Q ˆ Q+Q

(2.109)

is the fractional flow of the outer fluid. Specification of ϕ and the viscosity ratio ηˆ/η then allows the determination of r c ∗ by using simple numerical root finding methods. The flow rate enhancement for the inner fluid is Q∗ ≡

Q = rc∗4 + 2 (Q)r∗ =1 c

 

η 1 − rc∗2 rc∗2 ηˆ (2.110)

Here (Q)r ∗c =1 is the volumetric flow rate when the inner fluid occupies the entire flow region (no outer fluid present). For a specified viscosity ratio ηˆ/η, Equations (2.108) – (2.110) can be used to determine the flow rate enhancement in terms of the fractional flow of the outer fluid and the viscosity ratio. The numerical results for several values of viscosity ratio are given in Figure 9. In particular, for the case in which the outer fluid is 100 times less viscous than the inner fluid (ˆ η /η = 0.01) and the

Transport Phenomena fractional flow is 10 %, the flow rate of the more viscous inner fluid is enhanced by a factor of 30. For higher fractional flows and lower viscosity ratios, even greater enhancement factors are obtained. Clearly, when such segregated flows are possible, significant increases in throughput can be achieved per unit of applied pressure drop. In practice, the application of such approaches is limited only by the ability to create and maintain a stable outer concentric region of the lower viscosity fluid.

309

vides the velocity field. Physically, lines of constant ψ represent streamlines of the flow. To illustrate this more explicitly, consider the flow ux = ux (x, y, t) , uy = uy (x, y, t) , uz = 0

(2.111)

where the governing equations of continuity and motion are ∂ux ∂uy + = 0 ∂x ∂y

(2.112)

and  ∂ux ∂ux ∂ux + ux + uy ∂t ∂x ∂y   2 ∂ 2 ux ∂ ux ∂P + +η =− 2 2 ∂x ∂x ∂y 



(2.113)



 ∂uy ∂uy ∂uy + ux + uy ∂t ∂x ∂y   2 2 ∂P ∂ uy ∂ uy =− + +η ∂y ∂x2 ∂y 2 

Figure 9. Flow rate enhancement associated with the injection of an immiscible low-viscosity fluid into a transport line

If the stream function ψ is defined in terms of the velocity components by ux = −

2.3. Multidimensional Momentum Transfer 2.3.1. Two-Dimensional Flows – Stream Function Equations The flow examples considered to this point have involved only one nonvanishing velocity component. More complex flows are now considered, particularly two-dimensional incompressible flows of Newtonian fluids having two nonvanishing velocity components that depend on two spatial variables and time. The governing equations are the continuity equation and the two nonvanishing components of the equation of motion. Rather than solving these equations directly, the equations are more conveniently transformed into a single equation in terms of a stream function ψ (→ Fluid Mechanics, Chap. 2.1.). Because the stream function is defined in terms of the nonvanishing velocity components, solution of this equation then pro-

(2.114)

uy =

∂ψ ∂y

(2.115)

∂ψ ∂x

(2.116)

the continuity equation (Eq. 2.112) is satisfied identically. To obtain the desired stream function equation, the motion equations are combined in the following way. First operate on Equation (2.113) by ∂/∂y and Equation (2.114) by ∂/∂x; then subtract the resulting equations to eliminate the pressure terms; finally, replace ux and uy with Equations (2.115) and (2.116). After simplification, the resulting equation is

∂ ∇2 ψ ∂t

+

∂ ψ, ∇2 ψ ∂ (x, y)

= ν∇4 ψ

(2.117)

where ν ≡ η/ and ∇2 ≡

∂2 ∂2 + , ∇ 4 ≡ ∇ 2 ∇2 2 ∂x ∂y 2

(2.118)

Also,  ∂f ∂ ( f , g)  ∂x ≡  ∂g ∂ (x, y)  ∂x



∂f  ∂y  ∂g   ∂y

(2.119)

310

Transport Phenomena

where a determinant operation is implied by the straight brackets. Solution of Equation (2.117) gives the stream function ψ (x, y, t). The velocity components are then obtained from Equations (2.115) and (2.116), and the pressure P is obtained from these velocity components and Equations (2.113) and (2.114). A similar approach can be used in other two-dimensional flows, but the velocity – stream function relations and the stream function equation may take different forms depending on flow characteristics. For example, for axisymmetric flow around a sphere (see Fig. 10) where ur = ur (r, θ, t) , uθ = uθ (r, θ, t) , uϕ = 0 (2.120)

the velocity – stream function relations are 1 ∂ψ r2 sin θ ∂θ 1 ∂ψ uθ = − r sin θ ∂r ur =

(2.121) (2.122)

and the stream function equation is

∂ ψ, E 2 ψ 1 2 E2 ψ − 2 + 2 2 ∂t r sin θ ∂ (r, θ) r sin θ  

1 ∂ψ ∂ψ 2 cos θ − sin θ = ν E E 2 ψ · ∂r r ∂θ



∂ E2ψ

(2.123)

∂2 sin θ ∂ + 2 ∂r2 r ∂θ



1 ∂ sin θ ∂θ

u = ∇ψ × ∇G

(2.125)

where G is a scalar function. This form satisfies the continuity equation (∇ · u = 0) for incompressible flow [20, pp. 227 – 228]. For axisymmetric flow, the scalar function G is simply the coordinate ξ corresponding to the azimuthal angle about the axis of symmetry. For axisymmetric flow around a sphere, G = ϕ and the resulting components of Equation (2.125) are the same as those of Equations (2.121) and (2.122). For two-dimensional axisymmetric flow in a cylindrical geometry with uθ = 0, G = θ is used, and the velocity – stream function relations take the forms ur = −

1 ∂ψ , r ∂z

uz =

1 ∂ψ r ∂r

 (2.124)

(2.126)

If Equation (2.123) is put in dimensionless form by scaling t, r, and ψ using d/u∞ , d, and u∞ d 2 , respectively, where d is the sphere diameter and u∞ the free stream velocity, then Red {left − hand side terms} = E ∗4 ψ ∗

where E2 is an operator defined by E2 ≡

Similar relations for other flows are given in [17, p. 131], [36, pp. 114 – 115]. Specification of the velocity in terms of the stream function ψ is given by

(2.127)

where Red ≡  u∞ d/η is the Reynolds number. Since all terms except Red are scaled to order 1 by the selections of d and u∞ , when Red  1 the left-hand side can be set to zero and a creeping flow solution can be obtained. The latter is generally associated with small spheres, viscous fluids, and low velocities. When Red 1, the right-hand side can be set to zero and a nonviscous flow solution can be obtained. 2.3.2. Creeping Flow Around a Sphere and Other Bodies of Revolution

Figure 10. Axisymmetric flow past a sphere

Consider the steady, axisymmetrical flow of an incompressible Newtonian fluid past a sphere as illustrated in Figure 10. If the coordinate system is attached to the sphere, this representation is valid either for flow past a stationary sphere as illustrated or for the steady fall of a sphere (at terminal velocity u∞ ) in a fluid at rest. The velocity and pressure fields under creeping flow

Transport Phenomena conditions (Red  1) are of particular interest, along with the drag force associated with viscous and pressure effects. Under creeping flow conditions, Equation (2.123) reduces to

P =

dP = ∞

0


θ 

∂P ∂θ

∂P ∂r

 dr + θ=π/2

 dθ

(2.137)

r

π/2

E4 ψ = 0


r 

P
(r,θ)

(2.128)

with boundary conditions at r = r1 ,

ur =

1 ∂ψ =0 r 2 sin θ ∂θ

(2.129)

If (∂P/∂r) and (∂P/∂θ) are obtained from the r and θ components of the equation of motion (creeping flow conditions) by using Equations (2.135) and (2.136), then 3 η u∞  r1 2 cos θ 2 r1 r

P = − 1 ∂ψ = 0 uθ = − rsin θ ∂r

at r = r1 ,

(2.130)

as r → ∞,

uz =

uθ ur = − → u∞ (2.131) cos θ sin θ

By using Equations (2.121) and (2.122), the latter condition can be written as 1 ψ → u∞ r 2 sin2 θ 2

(2.132)

Because Equation (2.132) is valid at all θ, the θ dependence at smaller r values might be expected to be the same. This suggests a solution to Equation (2.128) of the form ψ = f (r) sin2 θ

(2.133)

Substituting this expression into Equation (2.128) and using Equations (2.129), (2.130), and (2.132) give f = −



1 u∞ 4

r13 + 3 r1 r − 2 r 2 r

 ur = − u ∞

 3  r1  1  r1 3 + 1− cos θ (2.135) 2 r 2 r

 uθ = − u ∞

p = p0 −  g z −

 3  r1  1  r 1 3 1− sin θ (2.136) − 4 r 4 r

The dynamic pressure P is obtained from a line integration between the position r → ∞ and θ = π/2 (where P = 0) and any position r, θ [where P = P (r, θ)], i.e.,

3 η u∞  r1 2 cos θ 2 r1 r

(2.140)

The drag force of the fluid on the sphere is then obtained from
n · (−pI − τ ) dS

F =

(2.141)

S

where n = ir is the outwardly directed unit normal to the sphere surface. Expanding Equation (2.141) and finding the component in the zdirection give
2π
π [( p + τrr ) cos θ − 0

(2.134)

Then, from Equations (2.133), (2.121), and (2.122),

(2.139)

where p0 is the pressure at r → ∞ and θ = π/2. Combining the dynamic and hydrostatic results (Eq. 2.62) gives the pressure field

Fz = −



(2.138)

Also, the hydrostatic pressure is pH = p0 −  g z

r → ∞,

311

0

τrθ sin θ]r1 r12 sin2 θ dθdϕ

(2.142)

Then, by using Equations (2.135) and (2.136), the stress components and the pressure are evaluated at r 1 with the results   τrr 

r1

  τrθ 

r1

∂ur  = − 2η  = 0 ∂r r1   ∂  ur  1 ∂ur = −η r + ∂r r r ∂θ r1

3 η u∞ = sin θ r1

(2.143)

(2.144)

and  3 η u∞  p  = p0 −  g r1 cos θ − cos θ r1 2r1

(2.145)

312

Transport Phenomena

Thus, Fz =

r 31 

4/3 π g+ 2 π η r 1 u∞ + 4 π η r 1 u∞ Buoyancy Form drag Viscous drag from pH from P from τrθ

(2.146)

If surface-active agents are present that give rise to a surface shear viscosity εs , the drag force for a fluid sphere is [30, pp. 129]:  F D = 6 π η r 1 u∞

The first term arises from hydrostatic pressure effects around the sphere (pressure is higher as θ → π causing a net upward force); the second term, from dynamic pressure effects (again from higher pressure on the sphere bottom); and the third, from viscous effects associated with τrθ on the sphere surface. If this result is represented in terms of a buoyancy force F B (static effect) and a drag force F D (dynamic effect), i.e., Fz = F B + F D

(εs /r1 ) + 3 ηˆ + 2 η (εs /r1 ) + 3 ηˆ + 3 η

 (2.152)

The important point here is that if r 1 is sufficiently small, surface viscosity effects dominate to give rigid sphere results, even when ηˆ is small (gas bubble case). Small bubbles often behave as rigid spheres when contaminant amounts of surface-active agents are present.

(2.147)

then F B is just the first term of Equation (2.146) and F D the sum of the other two terms, i.e., F D = 6 π η r1 u ∞

(2.148)

This equation is known as Stokes law and indicates that the drag force on a solid sphere increases in direct proportion to viscosity, radius, and terminal (or free stream) velocity. If a sphere of density s settles under its own weight in a fluid, the terminal velocity can be determined by equating the sum of F B and F D to the gravitational force on the sphere (4/3 π r 31 s g). The result is u∞ =

2 (s − ) r12 g 9η

(2.149)

This result indicates that for particle settling in viscous fluids, the terminal velocity is most sensitive to variations in particle size. Finally, the drag force results for several variations of this problem are noted. In particular, for creeping flow past a fluid sphere (or droplet) of viscosity ηˆ [30, pp. 127 – 129]

Figure 11. Drag force for creeping flow around oblate and prolate spheroids A) Oblate spheroid, drag force F D = 6 π ηa u∞ K; B) Prolate spheroid, drag force F D = 6 π η b u∞ K b/a 0.0 0.1 0.5 1.0

K (oblate) 0.8488 0.8525 0.9053 1.0000

K (prolate) ∞ 2.6471 1.2039 1.0000

For a very viscous droplet or solid sphere where ηˆ  η, this result reduces to Stokes law. For gas bubbles where ηˆ  η

For nonspherical solid bodies, somewhat different drag force results are obtained. For oblate and prolate spheroids, the results are summarized in Figure 11. Note that the drag force for the prolate spheroid is particularly sensitive to the ratio of the minor and major axes. In the limit as b/a goes to zero, the oblate spheroid reduces to a circular flat disk with the drag force given by

F D = 4 π η r1 u ∞

FD = 5.093 π η a u∞

 F D = 6 π η r1 u ∞

3 ηˆ + 2 η 3 ηˆ + 3 η

 (2.150)

(2.151)

In this case, the drag is two-thirds that of a rigid sphere.

(2.153)

where a is the radius of the flat disk. In the limit as b/a goes to zero for the prolate spheroid, the

Transport Phenomena results for an elongated rod of radius b and length 2 a are FD =

4 π η a u∞ ln (a/b) + 0.1932

(2.154)

Additional drag force results for creeping flow around bodies are given in [30]. An alternate way of representing drag force results around bodies is in terms of a drag coefficient C D (→ Fluid Mechanics, Chap. 3.3.3.): FD = CD (Characteristic area of body) · (Characteristic kinetic energy of fluid per unit volume)

(2.155)

For flow past a sphere FD = C D CD =

π

24 Red

4

d2

1 2

2  u∞

(2.156) (2.156a)

For the settling of multiple bodies, interaction effects can be quite important, even over fairly large separation distances. In the case of two identical spheres of diameter d settling either parallel or perpendicular to their line of centers, the drag coefficient on each sphere is CD =

24 Λ Red

2.3.3. Flow in Channels with Varying Cross Sections – Lubrication Analysis In many problems the flow geometry is sufficiently complex that exact solutions of the motion equations are not possible, and approximate or numerical methods are required to obtain the solutions desired. In this section an approximate analysis is described, called lubrication theory. (The name lubrication theory arose from early applications of this approximate analysis to thinfilm lubrication problems). This method is applicable when the change in flow channel geometry in the direction of flow is weak. Consider steady two-dimensional flow u = [ux (x, y), uy (x, y), 0] in a channel with a slowly changing cross-sectional area as shown in Figure 12. The flow channel gap b varies with x, but the magnitude of this change at any x is small, i.e., ε≡

db 1 dx

(2.158)

The equation of continuity and the x and y components of the equation of motion are the same as Equations (2.112) – (2.114), but without the time derivative terms. To scale these equations and determine the most dominant terms, the velocity component ux must first be scaled to the characteristic average velocity u0 in the channel:

(2.157)

where Λ is an interference parameter which is a function of the ratio l/d of the distance between the particles and the particle diameter. When l/d → ∞, Λ → 1. For two spheres settling side by side, Λ is found to be < 1 for finite values of l/d. As a result, two interfering spheres fall faster than isolated spheres. Even though two spheres may fall faster than an isolated sphere, a suspension of many spheres usually experiences a higher drag than the isolated sphere case. As a result, when a suspension settles in a beaker, a sharp line is observed separating the clear fluid from the particle-laden fluid. This hindered settling phenomenon arises because particles in the low-concentration regions near the top of the settling suspension tend to overtake the more slowly settling particles in the underlying high-concentration regions.

313

ux ∼ uavg (x) =

Q Q ∼ ≡ u0 b (x) w b0 w

(2.159)

Here Q is the volumetric flow rate, w the channel width, b0 the channel gap at x = 0, and the symbol ∼ indicates an order-of-magnitude estimate. From this result, ∂ux Q db u0 ∼ 2 ∼ ε ∂x b w dx b0

(2.160)

and from the equation of continuity, ∂uy ∂ux u0 = − ∼ ε ∂y ∂x b0

(2.161)

or uy ∼ u0 ε,

∂f ε f, ∼ ∂x b0

∂f 1 f ∼ ∂y b0

(2.162)

where f may be ux or uy , or higher derivatives of these variables. Based upon these scalings, the magnitudes of the various terms in the x component of the equation of motion can now be compared:

314

Transport Phenomena The solution of Equation (2.166) with the boundary conditions at y = 0,

ux = U

at y = h (x) ,

The inertial term will be negligible compared to the y derivative viscous term if

is

 u20

ux = U

η u0 ε 2 b0 b0  u0 b0 ε 1 η

Similarly, the x-derivative viscous term will be negligible compared to the y-derivative viscous term if η u0 2 η u 0 ε  2 b0 b20

(2.165)

or ε2  1

(2.165a)

(2.166)

In a similar way, the y-component of the equation of motion can be simplified to ∂P ∂ 2 uy 0 = − +η ∂y ∂y 2

(2.167)

From these equations, ∂P ∂P η u0 ∼ 2 ε ∼ ε ∂y b0 ∂x

(2.168)

which indicates that when ε  1, pressure changes in the y-direction are negligible compared to those in the x-direction. Hence, changes of P in the y-direction can be neglected and the assumption made that P = P (x).

Figure 12. Flow in a two-dimensional channel with varying cross section Flow can arise from an applied pressure gradient or from moving wall(s).

(2.170)

with the volumetric flow rate being
b Q =

ux wdy = 0

1 b3 w U bw − 2 12 η



dP dx

 (2.172)

If this equation is solved for (dP/dx) and then integrated, the pressure drop between x = 0 and x = L is ∆P ≡

Equation (2.163) then reduces to ∂ 2 ux ∂P +η 0 = − ∂x ∂y 2

ux = 0

    y 2  y b2 dP y 1− − − b 2 η dx b b (2.171)

(2.164) (2.164a)

(2.169)

12 A3 η Q L 6 A2 η U L − w h30 h20

(2.173)

Here A2 and A3 are numerical parameters that depend on the geometry of the flow channel [h (x)]; they are given by
1 An = 0

dx∗ , h∗n

n = 2, 3

(2.174)

where x∗ ≡ x/L and b∗ ≡ b/b0 . Equation (2.173) is the pressure drop – flow rate relation for flow in a two-dimensional channel with a weakly varying cross section. Similar relations can be obtained for axisymmetric flows where the cross section varies along the axis of flow [39]. This includes flows in sinusoidal tubes, and converging or diverging ducts. Lubrication theory is applicable in such problems as long as ε ≡ dR/dz is sufficiently small, where R (Z) is the local channel radius. Good engineering approximations are often possible even for values of ε as high as 0.2. To conclude this section an application of lubrication theory is presented which involves the analysis of slipper block performance. Consider the slipper block shown in Figure 13, which could represent a wiping block or ring on a moving piston. In some applications, the surface moves and the block is stationary (see Fig. 13);

Transport Phenomena in others, the block moves and the surface is stationary. The thickness of the film gap between the block and the moving surface, as well as the drag force on the block, depends on the applied load B, the speed U, the viscosity of the film fluid η, the width w, the length L, and the variation of the gap thickness with x. Lubrication theory is used to develop the appropriate relations between these variables.

315

If the gap geometry is described by the linear relation b = b0 + (bL − b0 )

x L

(2.178)

the dimensionless pressure distribution can be obtained from Equations (2.175) and (2.177) with the result P∗ ≡

(P − P0 ) L (b∗ − 1) (b∗ − κ) = 6η U (b0 /L )2 (κ2 − 1)

(2.179)

where κ ≡ bL /b0 . As depicted in Figure 13, the pressure is positive in the gap and exhibits a maximum toward the narrow gap end. The magnitude of the pressure increases with U and η, and decreases with (b0 /L). The pressure also causes a vertical force Fy on the block, which is obtained by integrating P over the block length. The result is 1 Fy = 6η U w (b0 /L )2 (κ − 1)2   2 (κ − 1) − ln κ κ+1 Fy∗ ≡

LIVE GRAPH

Click here to view

(2.180)

For a given b0 /L, the maximum vertical force is obtained for a block geometry where κ = 2.19. Also, because the vertical pressure force must be equal to the load B, the resulting thickness b∗0 is Figure 13. Lubrication flow between block and moving surface A) Geometry; B) Pressure distribution

Begin by using Equation (2.172): dP 12 η = 3 dx b



Q Ub − 2 w

 (2.175)

In this problem, Q is not known and must be determined from known pressure conditions. In particular, at x = 0 and x = L the pressure is assumed to be zero. Integration of Equation (2.175) between these points then yields
L dP = 0 = 0

12 η L h20



1 A3 Q A2 U − 2 h0 w

 (2.176)

where the An are defined by Equation (2.174). The volumetric flow rate is then A2 w b0 U Q = 2 A3

(2.177)

b∗0 ≡

b0 = L

(



6η U w B (κ − 1)

2

)1/2 2 (κ − 1) − ln κ κ+1 (2.181)

Here, for a given κ, the gap thickness increases with increasing U and η, and decreases with increasing load B, and each case has a square-root dependence. The drag force Fx on the moving plate is obtained by integrating τyx over this surface: 1 Fx = 6η wU (b0 /L ) (κ − 1)   (κ − 1) 2 − lnκ κ+1 3 Fx∗ ≡

(2.182)

Finally, if Equation (2.181) is used to eliminate b0 /L,  Fx =

2 ηU wB 3

    1/2  3 κ − 1 − 2 lnκ  κ+1    2 κ − 1 − lnκ  κ+1 (2.183)

316

Transport Phenomena

which shows that the drag force is proportional to the square root of the applied load. This halfpower dependence when a lubricating film is present is in contrast to the first-power dependence generally assumed between sliding solid surfaces with no lubricant. Also, the effective coefficient of friction for lubricated surfaces is orders of magnitude smaller than that associated with direct – contact solid surfaces.

Consider now the case of steady flow of a Newtonian fluid in a heated tube where the temperature of the fluid varies in the radial and axial directions because of heat exchange with the walls, viscous heat generation, and reversible work effects. If free convection, generation, and radiation effects are neglected, and the velocity field is assumed to be u = [0, 0, uz (r)], Equation (2.184) takes the form  cp uz

2.4. Coupled Momentum and Energy Transfer This discussion has been concerned with examples that involve only heat or momentum transfer, but not both. In this section, examples are considered in which heat and momentum are transferred simultaneously and, specifically, heat is being transferred in a flowing fluid. When such heat transfer results directly from fluid movement, the transfer is termed heat convection. If the flow arises from applied pressure gradients or moving boundaries, the heat transfer is called forced convection. When the flow arises from buoyancy effects associated with density variations in nonisothermal fluids, it is called free or natural convection. In analyzing heat transfer with flow, the energy balance is expressed in the form [16, pp. 215 – 216]:  ∂T + u ·∇T = ∇ · (λ∇T ) ∂t   ∂p + u ·∇p + tr (τ ·∇u) + Φ + Tβ ∂t 

 cp

(2.184)

which is equivalent to Equation (1.85) with Fourier’s law (Eq. 1.91) used for the conductive heat flux q; β is the coefficient of thermal expansion defined by β ≡ −

1 



∂ ∂T

 (2.185) p

In Equation (2.184) the second term on the left, which involves velocity, accounts for convective heat transfer, and the terms on the right are the conductive, reversible work (fluid expansion – contraction), viscous dissipation, and thermal generation terms, respectively.

 2η

∂T dp = ∇ · (λ∇T ) + T β uz + ∂z dz

duz dr

2 (2.186)

The contribution of viscous dissipation to the temperature gradient in the z-direction can be estimated from 

 ∂T 2 η (duz /dr)2 = ∼ ∂z viscous dissipation  cp uz   ν2 (2.187) O Red cp d3

where Red ≡ (uavg d/ν) is the Reynolds number; ν ≡ η/, the momentum diffusivity, d the tube diameter; and uavg the average velocity in the tube. For air at room temperature (ν ≈ 2× 10−5 m2 /s, cp ≈ 103 J kg−1 K−1 ) flowing in a 0.025-m tube at a Reynolds number Red = 103 , the temperature gradient due to viscous effects is on the order of 3×10−5 K/m. For water under the same conditions (ν ≈ 8×10−7 m2 /s, cp ≈ 4 ×103 J kg−1 K−1 ), the gradient is on the order of 1×10−8 K/m; for an oil with properties ν ≈ 9×10−4 m2 /s, cp ∼ 2×103 J kg−1 K−1 , the gradient is ca. 3×10−2 K/m. In most heatexchange applications, such effects are negligible compared to the conduction contributions (first term on right-hand side, Eq. 2.186) which can be on the order of several kelvin per meter. Therefore, viscous dissipation effects are generally neglected in most heat-exchange applications. However, viscous dissipation can be important in lubrication problems where viscous oils are being sheared at high rates between moving surfaces. If a similar order-of-magnitude analysis is carried out for the contribution of the reversible work term to the axial temperature gradient:

Transport Phenomena  ∂T T β (dp/dz) = ∼ ∂z rev. work  cp   T β ν2 O 32 Red cp d3



2.4.1. Heat Transfer in Laminar Tube Flow (2.188)

Here, the pressure gradient is related to velocity through the Hagen – Poiseuille equation (Eq. 2.86). Comparing Equations (2.187) and (2.188) gives 

∂T ∂z



 ∼ ± 32 T β rev. work

∂T ∂z

uz viscous dissipation

where the sign depends on whether dp/dz in Equation (2.188) is positive or negative. When dp/dz < 0, the reversible work results in a decrease in temperature in the direction of flow; in contrast, the viscous dissipation causes an increase in temperature. For air at room temperature where β = − 1/T (ideal gas), the temperature gradient contribution associated with reversible work effects is on the order of 30 times that associated with viscous dissipation effects. For liquids where β ∼ 10−3 K−1 , the reversible work contribution to the temperature gradient is on the order of 10 times that arising from viscous dissipation. Still, in most applications, these effects are small compared to conduction or convection effects and can be neglected. The other effect that is not commonly considered in convection analyses is the temperature dependence of thermal conductivity. For most fluids, the magnitude of (1/λ) (∂λ/∂T ) is on the order of 10−3 K−1 and 1 ∇ · (λ∇T ) = λ ∇ T + λ 2

∼ λ∇2 T



∂λ ∂T



 |∇T |

2

(2.190)

Hence, if viscous dissipation and reversible work effects are neglected, the heat generation term is zero, and the conductivity is assumed to be temperature-independent, the energy balance (Eq. 2.184) reduces to ∂T +u ·∇T = a∇2 T ∂t

Consider the heat exchange that occurs when a fluid is passed through a heated or cooled tube under steady laminar flow conditions. If a Newtonian fluid, constant physical properties, negligible axial conduction, and fully developed velocity and temperature profiles are assumed, the energy balance can be written as



(2.189)



317

(2.191)

where α ≡ λ/ cp is the thermal diffusivity. This is the equation used in the forced convection examples considered here.

  ∂T r ∂r

∂T 1∂ = α ∂z r ∂r

(2.192)

where   2

1 r uz = uavg 1 − 2 r1

(2.193)

with r 1 being the tube radius and uavg the average velocity. Depending on the boundary conditions, various solutions are possible. When the heat flux at the wall qw is constant, a particularly simple solution is possible. In particular, if Equation (2.192) is multiplied by r and integrated with respect to r, then  dTb = dz

2α

∂T ∂r

 r1

r1 uavg

=

2 α qw r1 uavg λ

(2.194)

where T b is the bulk average temperature defined by Tb =

1 π r12 uavg


r1 uz T 2 π rdr

(2.195)

0

The bulk average temperature is simply the “cup-mixing temperature” that would be measured if the tube were chopped off at z and the fluid issuing forth were collected and thoroughly mixed in a container. Equation (2.194) implies that T b varies linearly with z. The achievement of a fully developed temperature profile requires that ∂ ∂z



Tw − T Tw − Tb

 =0

(2.196)

or Tw − T = f (r) Tw − Tb

From Newton’s law of cooling,

(2.197)

Next Page 318

Transport Phenomena  λ

h = −

∂T ∂r

 r1

(Tw − Tb )

 = λ

∂ ∂r



Tw − T Tw − Tb



= λ f (r1 ) = const.

r1

(2.198)

or Tw − Tb =

qw = const. h

(2.199)

By using this equation with Equations (2.194) and (2.197), dTw dTb ∂T dT 2 qw α = = = = dz dz ∂z dz r1 uavg λ

(2.200)

Substituting this result into Equation (2.192) and integrating over r give Tw − Tb =

11 qw d 11 qw r1 = 24 λ 48 λ

(2.201)

hd = 4.364 (qw = const., large z) λ

N ud,lm ≡

hlm d = λ

1/3

The latter result indicates that when the fluid is heated (or cooled) under constant wall heat flux conditions, the temperature difference between the wall and the bulk fluid is constant. If Equations (2.201) and (2.199) are combined, N ud ≡

long and fully developed thermal profiles are seldom achieved. In the latter cases, more complete solutions must be used to describe heat transfer in the developing regions, and these are considerably more complicated than the fully developed solutions presented above [38–42]. The classical Graetz solution [38], which involves the solution of Equation (2.192) for parabolic and flat velocity profiles and constant wall temperatures are the best known among these solutions. A useful empirical expression is given by Sieder and Tate [43]; it is an empirical modification of the Graetz solution, ( parabolic velocity) i.e.,

(2.202)

where Nud is the dimensionless Nusselt number based upon the tube diameter d. This result provides a simple expression for estimating h under constant wall heat flux conditions and in zones (large z) where the profiles are fully developed. For the case of constant wall temperature, the solution is somewhat more involved [38]; however, Nud is still constant but equal to a different value

1.86 Red

P r1/3 (L /d)−1/3 (ηbm /ηw )0.14

Here, the physical properties of the fluid are evaluated at the mean bulk fluid temperature, T bm ≡ (T b0 + T bL )/2, where T b0 and T bL are the entering and exit bulk temperatures for a tube section of length L. The viscosity ratio (η bm /η w ) is the ratio of the viscosities evaluated at the mean bulk temperature and at the mean wall temperature; the quantity hlm is the log-mean film heat-transfer coefficient defined by L hlm

Q˙ = = − λ L d∆Tlm

0

L∆Tlm

(2.206)



 − T ) − (T − T ) (T w0 wL bL b0    ∆Tlm ≡  − TbL ln TTwL − T

(2.207)

b0

(2.203)

Use of this result is limited to z values greater than the thermal entry length ze which is the position from the tube entrance where Equation (2.196) is approximately satisfied, say ∂ (T w − T )/∂z ∼ 0.01. For laminar flow, this entry length can be estimated from ze ≈ 0.05 Red P r d

(qr )r1 dz

where

w0

N ud = 3.658 (Tw = const., large z)

(2.205)

(2.204)

In particular, for water at room temperature (Pr ≡ ν/α ∼ 10) flowing at a Reynolds number of 103 , the entry length would be 500 diameters; for oils with Pr ∼ 104 , the entry lengths are very

is the log-mean temperature difference. The heat-transfer rate Q˙ across the wall can be expressed in terms of the difference between the convected energy input and output to the heated (or cooled) section by   π d2 Q˙ = uavg ( cp T )L − ( cp T )0 4

(2.208)

For large values of L/d, Equation (2.205) predicts that the log-mean Nusselt number goes to zero, which is not consistent with results for the fully developed regions. Whitaker [16,

Previous Page Transport Phenomena pp. 324 – 325] suggests the use of the following expression, which is based on the work of Hausen [42]: N ud,lm = 3.66 +

0.0745 ψ 3 1 + 0.04 ψ 2

(2.209)

where ψ ≡ (Re P r)1/3 (L /d)−1/3 (ηbm /ηw )0.14

(2.210)

This result is accurate to ca. ± 10 % for situations in which the wall temperature is constant or nearly constant. In the case of constant wall heat flux, variations of T w can be large; however, the local Nusselt numbers are only ca. 20 % higher than those for constant-temperature cases. As a result, Equation (2.209) is used even for the case of constant heat flux and surface temperature variations are ignored. When the entering fluid temperature and the wall temperature are known, the exit temperature TL and the overall heat transfer rate Q˙ can be calculated by using Equations (2.206) – (2.210). Since the evaluation of the properties requires a knowledge of TL , an iterative procedure is required in which TL is assumed and hlm is calculated from Equation (2.209). Then this value of the heat-transfer coefficient is used with Equa˙ tions (2.207) and (2.208) to determine TL and Q. The new value of TL is then employed to update the value of hlm , and the procedure is repeated until convergence is obtained.

319

fluid-deforming velocity gradients) are negligible. When applicable, such an approach simplifies the solution considerably, and approximate analytical solutions are possible in many cases. Momentum Transfer in a Laminar Boundary Layer. To illustrate this approach, consider first the momentum transfer associated with flow past the surface shown in Figure 14. In this case, the flow is two dimensional where u = [ux (x, y), uy (x, y), 0], with the x-coordinate measured along the surface and the y-coordinate measured normal to the surface. As the fluid moves over the surface with velocity u, a boundary layer is formed that encompasses most of the velocity changes resulting from interactions of the fluid with the surface. Although many different ways of defining the boundary layer thickness exist [44], in this article thickness is defined as the y-position at which the velocity is 99 % of the mainstream velocity u∞ , i.e., δm (x) ≡ {The y−position at each x where ux = 0.99 u∞ }

(2.211)

The subscript m indicates the boundary layer thickness for momentum transfer; boundary layer thicknesses for heat- and mass-transfer processes are defined later and denoted by δT and δc , respectively.

2.4.2. Momentum and Heat Transfer in Laminar Boundary Layers A characteristic of real fluids moving around solid bodies is the adherence of the fluid to the solid surface. This no-slip condition causes fluid-deforming velocity gradients that tend to be large in the neighborhood of the body and to diminish far away from it. If the Reynolds number based on the characteristic length of the body is sufficiently high, the resulting velocity gradient is restricted largely to a thin region in the neighborhood of the boundary. Under such conditions, a boundary layer approach is possible where the flow field is divided into two regions: (1) a thin boundary layer in which both viscous and inertial effects are important, and (2) an external region in which viscous effects (and

Figure 14. Momentum boundary layer on a surface a) Fluid; b) Solid body; c) Laminar boundary layer; d) Turbulent boundary layer; e) Laminar sublayer

As illustrated in Figure 14, the boundary layer thickness increases with distance x along the surface. For a certain distance x < x cr , flow in the boundary layer is laminar. At the position x cr , which can be estimated from Recr ≡

u∞ xcr ≈ 2 × 105 ν

(2.212)

inertial effects become sufficiently large compared to viscous effects to destabilize the flow and except for a very thin laminar sublayer next

320

Transport Phenomena

to the surface of the body, the flow exhibits sporadic vortex-like instabilities that eventually grow into fully developed turbulence. Physically, this is observed in flow visualization when the boundary layer thickness increases rather abruptly as shown in Figure 14 (in graphically depicting a boundary layer as in Fig. 14, the thickness is exaggerated and not to scale). Generally, x cr may be on the order of meters and δ m on the order of millimeters. The velocity field in the laminar portion of the boundary layer, and the associated drag force on the surface are now determined. If the radius of curvature of the surface is sufficiently large compared to the boundary layer thickness, the flow can be represented as that along a flat plate as shown in Figure 15. If the scalings ux ∼ u ∞

(2.213)

∂ux u∞ ∼ ∂x L

(2.214)

are used, the equation of continuity ∂ux ∂uy + = 0 ∂x ∂y

(2.215)

uy ∼ u ∞

δm L

(2.217)

∂ux ∂ux 1 dp ∂ 2 ux + gx (2.218) + uy = − +ν ∂x ∂y  dx ∂y 2

The y-component simply reduces to ∂p/∂y = 0, which indicates that the pressure is independent of y and hence is simply the pressure p (x) outside the boundary layer. If the external flow is assumed to be nonviscous the pressure field can be determined from: 1 p = −  u2∞ −  gx x + constant 2

∂ux ∂ux du∞ ∂ 2 ux + uy = u∞ +ν ∂x ∂y dx ∂y 2

uy

∂ux ∂uy ∂ = (uy ux ) − ux ∂y ∂y ∂y

(2.221)

If (∂uy /∂y) is replaced with − (∂ux /∂x) by using the equation of continuity, Equation (2.220) can then be written as

When this equation is integrated over y from y = 0 to y = ∞, and the conditions ux = 0,

(2.220)

uy = 0 at y = 0

ux = u ∞ ,

∂ux = 0 at y ≥ δm ∂y

(2.223) (2.224)

along with (from Eq. 2.215)
δm uy (x, ∞) = − 0

∂ux dy ∂x

(2.225)

are used, the result is d dx


δm ux (u∞ − ux ) dy + 0

(2.219)

using this equation in Equation (2.218) gives ux

In particular, note that

(2.216)

By using these scalings, the x- and y-components of the equation of motion can be simplified. In particular, if δ m /L  1, the x component takes the form [44, p. 131] ux

Figure 15. Laminar momentum and thermal boundary layers on a flat plate

∂ 2 ∂ ∂ 2 ux 1 d 2 u + u +ν (uy ux ) = ∂x x ∂y 2 dx ∞ ∂y 2 (2.222)

yields ∂uy u∞ ∼ ∂y L

In principle, if u∞ is known (obtained from the solution of the nonviscous equations for the flow geometry [45]), Equations (2.215) and (2.220) can be solved for the velocity components ux and uy . Although these equations can be solved numerically [44], [46], here the focus is on the use of approximate integral methods.

du∞ dx


δm (u∞ − ux ) dy = 0

τw 

(2.226)

where τ w is the wall shear stress given by

Transport Phenomena  τw = η

∂ux ∂y

 (2.227) y=0

Equation (2.226) is the von Karman momentum integral equation. For a given external flow u∞ (x), its solution requires a knowledge of the velocity ux as a function of y. Even with crude approximations for ux (y), reasonably accurate estimates of the drag force can be obtained. As an example, consider the case of a flat plate in a free stream flow where u∞ = constant. If a cubic relation is assumed for ux (y), i.e., ux = a + b y + c y 2 + d y 3

(2.228)

and the boundary conditions of Equations (2.223) and (2.224) are used, along with (∂ 2 ux /∂y2 ) = 0 at y = δ m to evaluate the constants, then  ux = u ∞

1 3 y − 2 δm 2



y δm

3

(2.229)

If this assumed profile is substituted in the momentum integral balance (Eq. 2.226) and the necessary integrations are performed, the following differential equation for boundary layer thickness is obtained: dδm 140 ν 1 = dx 13 u∞ δm

(2.230)

Integration with the condition δ m = 0 at y = 0 gives the boundary layer thickness δm 4.64 = 1/2 x Rex

(2.231)

where Rex ≡ u∞ x/ν is the local Reynolds number. The corresponding drag coefficient is given by τw 2 = 2 1 2  u  u ∞ ∞ 2

CDx ≡

3 η u∞ 0.646 = 1/2 2 δm Rex

(2.232)

where the subscript x on C D indicates the coefficient is a local value that varies with x. The drag force per unit width is then 

L

FD = w




τw dx = CDL L 0

1 2 u 2 ∞

 (2.233)

where the overall film coefficient C DL is given by

CDL ≡

1 L


L CDx dx = 0

1.292 1/2

321

(2.234)

ReL

and ReL ≡ u∞ L/ν. The exact solution for a laminar boundary layer on a flat plate with a uniform free stream velocity was obtained by Blasius [44]. The corresponding local and overall drag coefficient results are CDx = CDL =

0.664 1/2

Rex 1.328 1/2

(2.235) (2.236)

ReL

The approximate results of Equations (2.232) and (2.234) compare favorably with the exact results. This demonstrates the utility of the integral method. Even more accurate results could have been obtained if higher order approximations (beyond the cubic) were used for the velocity profile. In general, such integral approaches are valuable tools in engineering analysis and provide excellent approximations with comparable ease relative to the numerical methods required to obtain exact results. Heat Transfer in a Laminar Boundary Layer. Just as the momentum boundary layer thickness was defined as the region in which most of the fluid-deforming velocity gradients occur, a thermal boundary layer thickness δT can be defined as that region next to a heated (or cooled) surface in which most of the temperature gradients occur. Specifically, for heat transfer from a flat plate at temperature T w to a flowing fluid with free stream temperature T ∞ , the thermal boundary layer is defined as δt ≡ {The y−position where T − Tw = 0.99 (T∞ − Tw )}

(2.237)

In general, as shown in Figure 15, the thermal boundary layer thickness is not equal to the momentum boundary layer thickness. However, the relation between the thermal and momentum boundary layers in laminar flow can be approximated by [47] δm ≡ ∆T = const. δT

(2.238)

322

Transport Phenomena

Equation (2.238) implies that the dependence of δT on x is the same as that of δ m on x. For heat transfer in a laminar boundary layer on a flat plate where δT /L  1, the energy balance (Eq. 2.191) can be simplified to

where ∆T ≡ δ m /δT has been assumed to be constant. By integrating this equation with the condition δT = 0 at y = 0, the thermal boundary layer thickness is found to be 3/2

∂T ∂T ∂2T ux + uy = α ∂x ∂y ∂y 2

(2.239)

By integrating over y from 0 to ∞ and using the equation of continuity to replace uy (the steps involved are analogous to Eqs. 2.221 – 2.226), the energy balance can be put into the integral form d dx


δT 0

qw ux (T − T∞ ) dy =  cp

(2.240)

∂T   ∂y y=0

(2.241)

Equation (2.240) is analogous to the momentum integral equation (Eq. 2.226). To obtain a solution to this integral energy balance, a temperature profile must be assumed that satisfies the conditions at y = 0,

T = Tw

(2.242)

at y = δT , T = T∞ at y = δT ,

(2.243)

∂nT = 0 for n = 1, 2, . . . ∂y n

(2.244)

If a cubic power relation is assumed for the temperature profile T = a + b y + c y2 + d y3

(2.245)

the boundary conditions of Equations (2.242) – (2.244) can be used to determine the coefficients, with the result  T = Tw + (T∞ − Tw )

3 2



y δT

 −

1 2



y δT

3

(2.246)

When this profile is substituted into the integral form of the energy balance, Equation (2.229) is used for ux , and Equation (2.238) is used for δ m , then δT

dδT 10 α = dx u∞



14 ∆3T 14 ∆2T − 1



14 14 ∆2T − 1

 (2.247)

1/2 (2.248)

where Pr = ν/α is the Prandtl number, the ratio of the momentum and thermal diffusivities. Then, from Equations (2.231), (2.238), and (2.248), ∆T ≡

δm δT



14 13

 1−

1 14 ∆2T

 1/3

≈ P r1/3

where qw is the wall heat flux given by qw = − λ

4.47 ∆T δT = 1/2 x Rex P r1/2

P r1/3

(2.249)

and δT −1/2 = 4.47 Rex P r1/3 x

(2.250)

where it has been assumed 14 ∆2T  1

Based upon these results, the momentum boundary layer thickness increases relative to the thermal boundary layer as the momentum diffusivity ν increases relative to the thermal diffusivity a (or Pr increases). For gases, the Prandtl number is on the order of 1 and the boundary layer thickness δ m and δT have similar magnitudes. For liquids, the Prandtl number can be much higher than 1 and δ m can be larger than δT . For liquid metals, Pr  1 and the momentum boundary layer thickness is much smaller than the thermal boundary layer thickness. Thus, from Equations (2.241), (2.246), and Newton’s law of cooling, N ux ≡

hx χ 1/2 = 0.36 Rex P r1/3 λ

(2.251)

where Nux is the dimensionless local Nusselt number. This result is ca. 8 % higher than the exact result from numerical solutions, which is given by 1/2

N ux = 0.332 Rex

P r1/3

(2.252)

Heat transfer from the plate can then be determined from

Transport Phenomena
L Q˙ =

h (Tw − T∞ ) wdx = hL w L (Tw − T∞ ) 0

(2.253)

where 1 hL ≡ L


L hdx

(2.254)

0

N uL ≡

hL L 1/2 = 0.664 ReL P r1/3 λ

1/2

P r1/3

(2.255)

(2.256)

In this case, wall temperature varies with position x on the plate according to Tw

qw = T∞ + hx

dp = − ∞ g dx

(2.257)

Although the above relations have been derived for flat surfaces, they are also useful as approximations for curved surfaces if the characteristic radius of curvature for the surface is much larger than the boundary layer thickness. 2.4.3. Free Convection on a Vertical Plate When a stagnant fluid is in contact with a heated surface, the fluid layers near the surface are at higher temperatures than the fluid layers further removed from the surface. Because of their higher temperatures, the layers near the surface have lower densities and will rise relative to the stagnant bulk fluid. Such free convection processes can be analyzed in much the same way as forced convection on a flat plate was analyzed by using boundary layer methods. The velocity and temperature field inside the momentum and thermal boundary layers are illustrated in Figure 16. The boundary layer equations, Equations (2.218) and (2.239), describe

(2.258)

where ∞ is the density of the fluid at T ∞ . Inside the thermal boundary layer, the density can be related to the temperature by  = ∞ +

where ReL ≡ (u∞ L/ν). Hence, if the properties are determined from the average film temperature, 1/2 (T w + T ∞ ), the total heat-transfer rate from the plate can be determined from Equation (2.253), with hL estimated from Equation (2.255). For the case of constant heat flux at the plate surface, the local Nusselt number is given by N ux = 0.453 Rex

the behavior of these variables. Outside the boundary layer, the pressure gradient is simply the hydrostatic gradient



Using Equation (2.252) yields

323

∂ ∂T

 (T − T∞ ) T∞

= ∞ [1 − β∞ (T − T∞ )]

(2.259)

Substituting Equations (2.258) and (2.259) into Equation (2.218) gives the momentum equation for free convection, i.e., ux

∂ux ∂ux ∂ 2 ux + g β∞ (T − T∞ ) + uy = ν ∂x ∂y ∂y 2 (2.260)

In integral form, this equation becomes d dx


δm u2x dy = − ν 0

∂ux  +  ∂y y=0


δT (T − T∞ ) dy

g β∞

(2.261)

0

The integral form of the energy balance is the same as Equation (2.240). In selecting appropriate velocity and temperature expressions to use in these equations, the following boundary conditions are applicable: at y = 0, ux = 0

(2.262)

T = Tw

(2.263)

at y = δm , ux = 0

(2.264)

∂ n ux = 0, n = 1, 2, . . . ∂y n

(2.265)

at y = δT , T = T∞

(2.266)

∂nT = 0, n = 1, 2, . . . ∂y n

(2.267)

In particular, if a cubic expression is used for ux ( y) and a quadratic expression for T ( y), the resulting profiles that are consistent with the boundary conditions are

324

Transport Phenomena N uL ≡

hL L λ



= 0.678 (P r GrL )1/4

Pr 0.952 + P r

1/4 (2.272)

Here Grx is the local Grashof number given by Grx =

g β∞ (Tw − T∞ ) x3 ν

(2.273)

and Pr Grx is the Rayleigh number given by Rax ≡ P r Grx =

g β (Tw − T∞ ) x3 να

(2.273a)

The assumption of δ m = δT would seem to limit the results to Prandtl numbers on the order of 1; however, because the effect of the velocity profile on the temperature profile is small as Pr increases, the result is also useful at higher Prandtl numbers. Empirically, Churchill and Chu [47] have found that free convection data on vertical surfaces correlate quite accurately with N uL = 0.68 + 0.67 (P r GrL )1/4 

−4/9   0.492 9/16 · 1+ Pr Figure 16. Free convection from a vertical heated plate, development of momentum, and thermal boundary layers ux = uc (x)

 y 2 δm   y 2 − T∞ ) 1 − δT

y δm

T = T∞ + (Tw



1−

(2.268) (2.269)

where uc is a characteristic velocity that varies with x. If uc = A xm and δ m = δT = B x n , and these relations are substituted into Equations (2.261) and (2.240), the boundary layer thickness δ m and the local and overall heat-transfer coefficients can be determined [21, pp. 371 – 376]. The results are δ = 3.936 x

N ux ≡



0.952 + P r P r2

hx x λ

= 0.508 (P r Grx )1/4

and



1/4

1 1/4

(2.270)

Grx

Pr 0.952 + P r

1/4 (2.271)

(2.274)

In using this correlation, the thermal properties should be evaluated at the mean boundary layer temperature, except for β ∞ which should be evaluated at T ∞ if the fluid is a gas. Equation (2.272) is quite accurate at Pr ≈ 1 or greater; however, it is not valid at low Prandtl numbers (liquid metals), and Equation (2.274) must be used instead.

2.5. Turbulent Momentum and Energy Transfer 2.5.1. Physical Characteristics Much flow of industrial importance is turbulent flow. Momentum and energy transport in turbulent flow are characterized by time-varying velocity and temperature fields. In particular, for turbulent flow in a tube under constant flow rate conditions, the axial velocity component at a specific position in the flow appears as in Figure 17 A with the velocity fluctuating around a time-averaged value. The same fluctuating behavior is also observed for the other velocity

Transport Phenomena components. Also, if heat is convected between the wall and the fluid, the temperature at each spatial position in the flow also fluctuates.

325

average is taken is long compared to the reciprocal frequency of the fluctuations, but short compared to the time of macroscopic changes in pressure drop, wall temperature, and other externally controlled variables. Although turbulence is inherently unsteady, steady conditions in turbulent flow can be considered if ∂ϕ ¯ = 0 ∂t

(2.278)

In Figure 17 A, steady turbulent flow is illustrated; in Figure 17 B, unsteady turbulent flow. The latter might arise in a flow field if the applied pressure drop decreases with time. To obtain a measure of the relative magnitude of the fluctuating part of a turbulent flow variable, the quantity Iϕ ≡

2 (ϕ )2

(2.279)

ϕ ¯

can be defined. In particular, for a shear flow [¯ux = u¯ x (y)] the intensity of turbulence is defined by Iu ≡

2 2 (ux )2 + uy + (uz )2

(2.280)

u ¯

Figure 17. Illustration of velocity fluctuations in turbulent flow The velocity consists of a time-average part (¯uz) and a fluctuating part (u z). A) Steady turbulent flow, u¯ z constant with time; B) Unsteady turbulent flow, u¯ z varying with time

LIVE GRAPH

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In describing this behavior, the velocity and temperature variables are conveniently separated into time-averaged values plus fluctuating parts, i.e., u=u ¯ + u

(2.275)

and T = T¯ + T 

(2.276)

where u¯ and T¯ are the time-averaged values and u
and T
are the fluctuating parts. The timeaveraged value at time t of a variable ϕ is defined by 1 ϕ ¯ ≡ t0

t+t
0

ϕdt

(2.277)

Figure 18. Distribution of intensities of the different velocity components in a turbulent boundary layer a

 2   2 ux (ux )2 + uy +(uz )2 , = u∞     

b=

t

where ϕ can correspond to u, T, p, or any other dependent variable. The time t 0 over which the

c=

uz

 uy

2

2 

(ux )2 +

uy

2

+(uz )

2

u∞  2 2 uy +(uz )

(ux )2 +

u∞

,

326

Transport Phenomena

In Figure 18, the intensities of individual velocity components are shown for a turbulent boundary layer on a flat plate as observed by Klebanoff [16, p. 275], [49]. As can be seen, the largest component intensity is that associated with the contribution in the direction of flow (ux ), with a maximum value of 0.11 which oc(t) curs very close to the wall ( y/δ m ≈ 0.007). The smallest component intensity is that associated with uy , the velocity component normal to the wall, where a value of 0.04 is observed (t) at y/δ m = 0.15. Although the fluctuating intensities may appear small, these quantities are important in the heat- and mass-transfer rates observed in turbulent flow. In Figure 19, the normalized velocity profile u¯ x /u∞ for a turbulent boundary layer on a flat plate (b) is compared with that for a laminar boundary layer (a), both corresponding to a local Reynolds number Rex of 106 . The turbulent boundary layer thickness is nearly five times that of the laminar boundary layer, and in turbulent flow the velocity profile is more uniform with a higher ratio of average velocity to maximum velocity. Also, the wall velocity gradient and the corresponding wall friction drag are higher in the turbulent case. The latter might suggest that a laminar boundary layer would be more desirable in minimizing frictional drag on surfaces; however, in most cases a turbulent boundary layer is preferred because it is more effective in resisting boundary layer separation and increased form drag. For turbulent flow in tubes and along flat surfaces, various flow regions can be identified. At distances sufficiently close to the wall, the turbulent fluctuations diminish and a laminar sublayer is observed. In this region, momentum or energy transfer occurs largely from conventional molecular and laminar convection mechanisms. At distances sufficiently remote from the wall, a fully developed turbulent core is present where the momentum- and energy-transfer mechanisms are largely a result of convective mixing of the turbulent eddies. In the buffer zone between the laminar sublayer and the turbulent core, the flow is neither laminar nor fully developed turbulent flow. In this region, both types of molecular and turbulent transport mechanisms are present.

LIVE GRAPH

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Figure 19. Dimensionless laminar and turbulent velocity profiles for flow on a flat plate at a local Reynolds number Rex = 106 (l) (t) Here, δ m and δ m denote the momentum boundary layer thickness for laminar (a) and turbulent flow, (b) respectively; (t) (l) δ m /δ m ≈ 4.8.

The velocity distribution across these three regions can be described by the semiempirical relations Laminar sublayer: u+ = y + for y + < 5

(2.281)

Buffer zone: u+ = − 3.05 + 5 ln y + for 5 ≤ y + < 30

(2.282)

Turbulent core: u+ = 5.5 + 2.5ln y + for 30 ≤ y + +

(2.283)

+

Here, u and y are defined by u+ ≡ ux /u∗

(2.284)

y + ≡ y u∗ /ν,

(2.285)

and u∗ is the friction velocity defined by u∗ ≡

τw /

(2.286)

where τ w is the wall shear stress. The velocity distribution represented by Equations (2.281) – (2.283) is commonly referred to as the universal velocity distribution. Another form of the velocity distribution that is commonly used to describe turbulent flow in tubes is u ¯x = umax



y r1

1/n (2.287)

Transport Phenomena where y is the distance measured from the wall and r 1 is the tube radius. Nikuradse [50] obtained excellent agreement with experimental observations when n was allowed to vary with Reynolds number (see Table 6). For Reynolds numbers of ca. 105 , a value of n = 7 is commonly used. Note that Equation (2.287) gives an infinite velocity gradient at the tube wall. As a result, drag force values cannot be predicted directly from velocity gradients calculations at the wall. Table 6. Characteristics of velocity profile for turbulent flow in a tube [50] Red

n in Equation (2.287)

uavg /umax

4×103 1.1×105 3.24×106

6 7 10

0.791 0.817 0.865

Although Equation (2.281) – (2.283) and (2.287) were developed originally for tube flow, they are also valid for turbulent boundary layers on flat plates. In this case, umax in these equa(t) tions is replaced by u∞ and r 1 by δ m . Also, 2 because τ w = f  uavg /2 for tubes, an expression for uavg /umax must be used to apply the results to flat plates. As illustrated in Table 6, this ratio for tube flows varies with Reynolds number. For Reynolds numbers of ca. 105 , a value of uavg /umax ≈ 0.8 is commonly used.

smoothing Equations (1.65) and (1.73) term by term. Before carrying out these operations, Equation (1.73) can be put in the equivalent form ∂ (u) + ∇ · (uu) = − ∇p − ∇ ·τ + g (2.288) ∂t

where the equation of continuity has been used on the left side of Equation (1.73) and Equation (1.84) on the right side. The key time-smoothing identities are Fϕ = Fϕ

(2.289)

Fϕ ψ = Fϕ ψ+Fϕ ψ 

(2.290)

where F is any linear time or spatial derivative operator (e.g., ∂/∂t, ∇, ∇ ·) and ϕ and ψ represent scalar, vector, or tensor functions, or combinations thereof. By using these identities, the following time-smoothed equations are obtained (incompressible fluids): ∇ ·¯ u= 0

The equations of continuity and motion given previously are perfectly valid for describing turbulent flow; however, because of the unsteady, three-dimensional fluctuations of velocity and other dependent variables, the analysis requires simultaneous solution of the full threedimensional equations of motion along with the equation of continuity. Even if appropriate boundary and initial conditions could be specified, such solutions are beyond current mathematical capabilities. A less ambitious approach is to focus on the time-smoothed quantities and to describe these variables as a function of position and time by using time-smoothed forms of the equations of continuity, motion, and energy. The specific forms of the continuity and motion equations can be obtained by time-

(2.291)

 ∂u ¯ +u ¯ ·∇¯ u = − ∇¯ p − ∇· ∂t  τ¯ −  u u + g 

 

(2.292)

If this equation is compared with the form of Equation (2.288) for incompressible fluids, i.e.,  

2.5.2. Time-Smoothed Momentum and Energy Balances for Turbulent Flow

327



∂u + u ·∇u ∂t

= − ∇p − ∇ ·τ + g

(2.293)

the time-smoothed equation is seen to have the same form, except that u¯ and p¯ replace u and p, and τ (t) ≡ τ¯ − u
u
replaces τ . The components of u
u
are called the Reynolds stresses. Physically, these represent the convective momentum flux components associated with the fluctuating part of the velocity. One could expect that just as τ depends on the deformation rate, the Reynolds stresses might be expressible in terms of the mean velocity and its spatial derivatives. Unfortunately, this does not appear to be possible, and when attempted, the coefficients associated with the mean velocity terms and its derivatives are spatially dependent and specific to each flow field. In general, a proper description of the u
u
contributions requires a more complete statistical analysis of turbulence. Although progress is being made in this area, an approach that can be

328

Transport Phenomena

used in engineering analyses is not yet available. As a result, only forms of the Reynolds stresses that are valid for specific flows can be obtained. This is the approach taken here in restricting attention to flow in tubes and along flat surfaces. Finally, note that the time-smoothed form of the energy balance Equation (2.191) is  ¯    ∂T  cp +u ¯ ·∇T¯ = − ∇ · q¯ +  cp u T  ∂t (2.294)

where q¯ = − λ ∇T¯ . This equation is of the same form as Equation (2.199), except that q (t) ≡ q¯ + cp u
T
replaces q = − λ ∇T . Also, this equation neglects viscous heat dissipation, reversible work, and heat generation. The quantity cp u
T
is the heat flux associated with turbulent fluctuations. As with Reynolds stresses, cp u
T
must be expressed in terms of the timesmoothed quantities and their derivatives before solutions to Equation (2.294) can be obtained. Similar to Reynolds stresses, these relations are not universal and the coefficients in such expressions are spatially dependent and specific to each flow. 2.5.3. Mixing Length Theories One of the common approaches used in relating  u
u
and  cp u
T
to time-smoothed quantities is through a mixing length interpretation of turbulent transport. In particular, this approach assumes a mechanism for turbulent transport which involves irregular movement of discrete fluid elements in the flow in a manner analogous to the kinetic behavior of molecules in a gas. For the case of unidirectional turbulent flow along a surface as illustrated in Figure 20, where the time-smoothed entity M (momentum, energy, mass, etc.) varies with distance y from the surface, the flux arising from the turbulent fluctuations is given by Flux of Min y direction = − εM

¯ dM dy

(2.295)

Here εM is the eddy diffusivity associated with the transfer of M by the turbulent fluctuations. Basically, the fluctuations give rise to the “diffusion” of discrete fluid elements that produce a flux of M down the gradient of decreasing M.

For the exchange of x-momentum between different y layers moving at different velocities u¯ x :  uy ux = εm

d d¯ ux ( u ¯x ) =  εm dy dy

(2.296)

where εm is the eddy momentum diffusivity. Similarly, for the heat flux between different y layers at different temperatures T¯ :  cp uy T  = εT

d dT¯  cp T¯ =  cp εT dy dy

(2.297)

where εT is the eddy thermal diffusivity. From largely physical arguments, Prandtl [46], [51] was able to relate εm and εT to the motion of discrete fluid elements between layers with different mean velocities to obtain  d¯   ux  εm = εT ≡ ε = l2   dy

(2.298)

where l is the Prandtl mixing length. The mixing length is a measure of the distance (in the ydirection) that the discrete fluid elements move before they accommodate to the mean velocity of the surroundings. This quantity is somewhat analogous to the mean free path in the kinetic theory of gases. For one-dimensional flow in tubes and along surfaces (e.g., Fig. 20), the mixing length has been empirically found to increase with distance from the wall. Specifically, the relation l = ky

(2.299)

appears to represent experimental observations with the constant k being ca. 0.4. In summary, for turbulent transport in shear flows where u¯ x = u¯ x (y), the total x-momentum flux in the y-direction can be given by (t)

τyx = −  (ν + ε)

d¯ ux dy

(2.300) (t)

and the total heat flux qy in the y-direction by (t)

qy

= −  cp (α + ε)

dT¯ dy

(2.301)

where ν and α are the molecular momentum and thermal diffusivities, respectively, and ε is the eddy diffusivity. At positions close to the wall in the laminar sublayer, the fluctuations have negligible importance, and ν  ε and α  ε;

Transport Phenomena

329

Figure 20. Turbulent transport in unidirectional flow along a surface

whereas in the bulk turbulent field away from the wall, ε  ν and ε  α for high-intensity turbulent flow. In the buffer region between the laminar sublayer and the turbulent core, the molecular and eddy diffusivities are generally of the same order of magnitude. 2.5.4. Turbulent Heat Transfer in Tubes – Reynolds, Prandtl, and von Karman Analogies Results are now presented for estimating heattransfer coefficients for convective heat transfer in tubes under turbulent flow conditions. The heat transfer is assumed to arise from temperature differences between the heated (or cooled) bounding surface and the bulk fluid. The analysis follows that of Prandtl [51] and is aimed at developing an expression involving the heattransfer coefficient h, the friction factor f , the average velocity uavg , and the properties of the fluid. The key assumptions are (1) the turbulent flow field is divided into two zones, a laminar sublayer that extends to y+ = 5 and a turbulent core that occupies the region beyond this point; (2) the velocity profile in the laminar sublayer is given by Equation (2.281); and (3) the ratio q¯ y /¯ τyx of the heat and momentum fluxes is constant in both the sublayer and the turbulent core. The last assumption implies that the temperature and velocity profiles are functionally similar. In the laminar sublayer where turbulent eddy transport is negligible,

¯

T − cp α ddy q¯y cp α dT¯ = = d¯ u x τ¯yx ν d¯ ux − ν dy

(2.302)

or, because q¯ y /¯ τyx is assumed to be constant, q¯y qw cp dT¯ = = τ¯yx (−τw ) P r d¯ ux

(2.303)

Integration of this equation over the laminar sublayer gives Pr T¯ξ − Tw = − cp



qw τw

 u ¯ξ

(2.304)

where Pr = ν/α. Here the edge of the sublayer is denoted by y = ξ, and T¯ ξ and u¯ ξ are the temperature and velocity at this position. In the turbulent core the eddy momentum and energy transport dominate the molecular transport. As a result, ¯

dT q¯y qw − cp εT dy = − ≈ d¯ u τ¯yx τw − εm dyx

(2.305)

or −

dT¯ qw = cp τw d¯ ux

(2.306)

where εT = εm = ε is assumed as in the Prandtl mixing length theory. Integrating this equation from y = ξ, where u¯ x = u¯ ξ and T¯ = T¯ ξ , to the bulk of the fluid, where u¯ x = uavg and T¯ = T ∞ , gives T∞ − Tξ = −

1 cp



qw τw





u ¯avg − u ¯ξ

(2.307)

Adding Equations (2.304) and (2.307) and using qw = h (Tw − T∞ )

(2.308)

330

Transport Phenomena

τw = f

1  u2avg 2

(2.309)

and, u ¯ξ = 5 τw / = 5 uavg f /2,

(2.310)

give St ≡

h N ud = = Red P r  cp uavg

f /2 1 + 5 f /2 (P r − 1)

(2.311)

where St is the Stanton number. This equation is called the Prandtl analogy and relates the heattransfer coefficient to the friction factor (or drag coefficient). The quantities Nud and Red are the Nusselt and Reynolds numbers based on the tube diameter d and the average velocity, i.e., h d/λ and uavg d/ν, respectively. For Prandtl numbers of 1 (applicable for many gases), Equation (2.311) reduces to h f =  cp uavg 2

(2.312)

which is the Reynolds analogy. This result can be obtained directly from the above analysis by neglecting the sublayer and allowing the turbulent core to occupy the entire flow region. Hence, uξ = 0 and Tξ = T w in Equation (2.307), and the Reynolds analogy follows directly from Equations (2.308) and (2.309). Von Karman [52, pp. 223 – 227] conducted a more complete analysis including all three zones in the turbulent field (i.e., laminar sublayer, buffer zone, and turbulent core). The final result was St ≡ =

h  cp uavg

1+5

(2.313)

f /2 8  ; f /2 P r − 1 + ln 1+ 56 (P r − 1)

This analogy includes the Reynolds and Prandtl analogies as special cases. To use the above analogies to estimate heattransfer coefficients, an expression is needed for the friction factor f in terms of the Reynolds number and appropriate roughness parameters. For smooth tubes, a useful empirical expression for f in the range 4×103 ≤ Red ≤ 105 is the Blasius formula [53] (→ Fluid Mechanics, Chap. 3.3.1.):

f ≡

τw 1  u2avg 2

=

0.0791

(2.314)

1/4

Red

At higher Reynolds numbers (> 105 ), the universal velocity profile can be used to develop an expression for the friction factor. In particular, if the turbulent core is assumed to extend to the wall, Equation (2.283) can be integrated over the tube cross section to obtain the average velocity: .  u r / ∗ 1 uavg = u∗ 1.75 + 2.5 ln ν

(2.315)

Note that u∗ ≡

τw / = uavg f /2

(2.316)

thus, Equation (2.315) can be simplified to the von Karman friction factor relation, i.e.,  -  1 √ = 4.07 log Re f − 0.601 f

(2.317)

This result is quite close to the empirical result of Nikuradse [50] given by  -  1 √ = 4.0 log Re f − 0.40 f

(2.318)

Both of these results are for smooth tubes and are generally used at elevated Reynolds numbers (> 105 ). For rough tubes, the effects of relative roughness e/d (e ≈ height of wall protuberances) must be included. √ For Reynolds numbers above the value Re f = 0.01d/e, the Colebrook equation can be used to determine f [54]: 1 √ = − 4 log f



4.67 e √ + d Re f

 + 2.28

(2.319)

The Colebrook equation reduces to the Nikuradse result when e/d = 0. The degree of roughness of a tube is not always easy to determine accurately. As a result, pressure drops predicted by substituting the above results into Equation (2.80) can often involve considerable error because of uncertainties in estimating e/d. This is particularly true for tubing susceptible to scaling and corrosion and tubing that has been in use for some time. In addition to the above analogies, another result can be obtained from the boundary layer results of Equations (2.235) and (2.255). These latter equations can be combined to give

Transport Phenomena St P r2/3 = f /2

(2.320)

where f has been substituted for C Dx . This equation is identical with the Reynolds analogy (Eq. 2.312) when Pr = 1. Based on this agreement at Pr = 1, Colburn suggested the use of Equation (2.320) at higher Prandtl numbers even for turbulent flow (Colbum analogy). Although it is strictly valid only for laminar flow, this simple relation has been found to be a useful approximation for turbulent flow applications particularly, where there is no form drag and where 0.5 < Pr < 50. The Colburn analogy can be put in another form: jH =

F 2

(2.320a)

where jH ≡ St P r2/3

(2.320b)

is the Colburn j factor for heat transfer. The mass transfer analogy of jH will be discussed later. 2.5.5. Turbulent Momentum and Heat Transferon a Flat Plate For a turbulent boundary layer on a flat plate, the wall shear stress can be related to the free stream velocity u∞ and the turbulent boundary (t) layer thickness δ m by the Blasius expression:  τw = 0.0225  u2∞

1/4

ν

(2.321)

(t)

u∞ δm

This equation follows from Equation (t) (2.314) if d = 2 r 1 → 2 δm , umax → u∞ , and uavg /umax ≈ 0.8. It is valid for flat plate Reynolds numbers up to 107 . In terms of the drag coefficient C Dx , this equation can be written as CDx ≡

τw 0.0450 = 1 1/4  u2∞ Rex 2



x (t)

1/4 (2.322)

δm

To use Equations (2.321) and (2.322), the tur(t) bulent boundary layer thickness δ m must be determined. If Equation (2.287) is used with (t) n = 7, r 1 → δm , and umax → u∞ in the timesmoothed momentum integral balance (the same

331

as Eq. 2.226, but with the dependent variables replaced by time-smoothed quantities), (t)

δm 0.371 = 1/5 x Rex

(2.323)

and CDx =

0.0577

(2.324)

1/5

Rex

which are valid up to a local Reynolds num(t) ber of 107 . In determining δ m , the turbulent boundary layer has been assumed to extend to (t) the leading edge of the plate (δ m = 0 at x = 0). Obviously, this can lead to considerable error, particularly because the laminar boundary layer that exists up to x cr = 2×105 (ν/u0 ) has a squareroot dependence on x compared to the 4/5 power dependence that results from Equation (2.323). Regardless, if Equation (2.324) is used in the Colburn analogy (with C Dx replacing f and Stx replacing St) 4/5

N ux = 0.0289 Rex

P r1/3

(2.325)

is obtained. Given the restrictions on Equation (2.322), this equation is limited to Rex < 107 . Based on the experimental work of Zhukauskas and Ambrazyavichyus [55], Whitaker [16, pp. 333 – 336] suggests the use of 4/5

N ux = 0.029 Rex

 P r0.43

η∞ ηw

1/4 (2.326)

instead of Equation (2.325), where all properties are evaluated at the free stream temperature T ∞ . This equation is similar to Equation (2.325), except for the power dependence on the Prandtl number and the addition of a temperature-dependent viscosity effect. Remember that the origin of the 1/3 power dependence on Pr in Equation (2.325) was from laminar boundary theory. In extending this result to turbulent boundary layers, some modifications would be expected. The 1/4 power dependence on (η ∞ /η w ) is different from that for tube flow where a 0.14 dependence was found. Although such differences are justified on the basis of available data, relatively small experimental errors could cause these differences. As a result, some uncertainty must be associated with the power dependences on Pr and (η ∞ /η w ).

332

Transport Phenomena

For a plate of length L, the overall heat-transfer coefficient can be determined from  x
cr
L  1  hL = hx,lam dx + hx,turb dx  L 

(2.330) (2.327)

Using a modified form of Equation (2.252) for αx,lam [Eq. 2.252 with (η ∞ /η w )1/4 as a multiplier], and Equation (2.326) for hx, turb gives 

1/4

η∞ 4/5 N uL = 0.0361 ReL P r0.43 ηw  ) ( 0.473 −4/5 1− · 1 − 17 400 ReL P r0.1

(2.328)

This equation assumes that the transition Reynolds number is given by Recr ≡ (u∞ x cr /ν) = 2×105 . For sufficiently large values of ReL where Recr /ReL  1, Equation (2.328) can be simplified to 4/5

N uL = 0.0361 ReL P r0.43



η∞ ηw

xcr

0

xcr

0

CDL

 x
cr
L  1  = CDx, lam dx + CDx, turb dx  L 

1/4 (2.329)

As with Equation (2.326), all properties in Equations (2.328) and (2.329) are evaluated at the free stream conditions. These equations are valid for ReL < 107 , Pr between 0.7 and 380, and viscosity ratios between 0.26 and 3.5. Whitaker found that the use of Equation (2.328) versus Equation (2.329) depends on the degree of turbulence in the free stream. When the latter is high, the transition Reynolds number is less than 2×105 and Equation (2.329) tends to be valid. When the free stream turbulence is low, Equation (2.328) represents experimental observations more accurately. In addition to this effect of free stream turbulence, the roughness of the plate also has a significant effect on the transition Reynolds number. In particular, very smooth plates can exhibit transition values considerably above 2×105 . Because such effects are difficult to characterize in many applications, some degree of uncertainty is associated with the use of Equations (2.328) or (2.329). If the fluid properties are evaluated at the film temperature T f ≡ (T w + T ∞ )/2 instead of T ∞ , the (η ∞ /η w ) dependence in Equations (2.328) and (2.329) can be dropped. Such differences are noted in Table 7. To calculate the overall drag coefficient on a plate of length L that extends into the turbulent boundary layer region, we use

Using Equations (2.235) and (2.324) and assuming that Recr = 2×105 give −1/5

CDL = 0.072 ReL



−4/5

1 − 9 200 ReL



(2.331)

7

This equation is valid for ReL < 10 . At higher Reynolds numbers, the result of Prandtl can be used [56]: CDL =

0.455 [log ReL ]2.58

(2.332)

which is valid for ReL from 106 to 109 .

2.6. Summary of Heat-Transfer Relations In Table 7, heat-transfer and friction (or drag) coefficient results are summarized for various geometries. They include the flat plate and tube flow results presented here, as well as results for heat transfer around cylinders and spheres. In addition, free convection results are provided for several common situations.

3. Transport in Multicomponent Systems Multicomponent transport processes require consideration of the fact that to the extent mixture compositions are not uniform, diffusion (the motion of species of the mixture relative to each other) will be nonzero. In binary systems, diffusion processes can be analogous to heat transfer. Multicomponent systems are considerably more complex, however. Useful references on diffusion and general transport references are given in [63–67]. An extensive review of methods for obtaining diffusion coefficients appears in [68].

3.1. Diffusive Mass Transfer in Binary Systems 3.1.1. Species Mass Balances In considerations of single-component (compositionally homogeneous) systems the equation of continuity (conservation of total mass)

Transport Phenomena

333

Table 7. Summary of heat-transfer and friction coefficient results (all properties evaluated at Tf ≡ (Tw − T∞ )/2 unless otherwise noted) Situation

Equation

Restrictions

Equation number ( ) or reference [ ]

Flow over Flat Plates Heat-transfer coefficients Laminar, local

Nux = 0.332 Rex 1/2 Pr 1/3

constant wall temperature; Rex < 2×105 , 0.6 < Pr < 50

(2.252)

Laminar, local

Nux = 0.453 Rex 1/2 Pr 1/3

constant wall heat flux; Rex < 2×105 , 0.6 < Pr < 50

(2.256)

Laminar, overall

NuL = 0.664 ReL 1/2 Pr 1/3

constant wall temperature; ReL < 2×105 , 0.6 < Pr < 50

(2.255)

Turbulent, local

Nux = 0.029 Rex 4/5 Pr 0.43 (η ∞ /η w )1/4

constant wall temperature; 2×105 < Rex < 107 , 0.7 < Pr < 380; properties evaluated at T ∞

(2.326)

Turbulent, overall

N uL =   4/5 −4/5 1− 0.0361 ReL P r 0.43 (η∞ /ηw )1/4 × 1 − 17 400ReL

0.473 P r 0.1



(2.328)

constant wall temperature; 2×105 < ReL < 107 , 0.7 ≤ Pr ≤ 380; relatively low free stream turbulence; properties evaluated at T ∞ Turbulent, overall

NuL = 0.0361 ReL 4/5 Pr 0.43 (η ∞ /η w )1/4

Laminar, local

C Dx = 0.664 Rex −1/2

same as above except relatively high free stream turbulence

(2.329)

Rex < 2×105

(2.235)

ReL < 2×105

(2.236)

Drag coefficients −1/2

Laminar, overall

C DL = 1.328 ReL

Turbulent, local

C Dx = 0.0577 Rex −1/5

2×105 ≤ Rex < 107

(2.324)

Turbulent, overall

C DL = 0.072 ReL −1/5 (1 – 9200 ReL −4/5 )

2×105 ≤ ReL < 107

(2.331)

Flow in Tubes (Properties evaluated at average bulk mean temperature between z = 0 and z = L unless otherwise noted) Heat-transfer coefficients Fully developed laminar

Nud = 3.658

T w = const.; Red < 2100; z > 0.05 d Red Pr

(2.203)

Fully developed laminar

Nud = 4.364

qw = const.; Red < 2400; z > 0.05 d Red Pr

(2.202)

Developing laminar

Nud , lm = 1.86 Red 1/3 Pr 1/3 (L/d)−1/3 (η ∞ /η w )0.14

(2.205) T w = const.; Red < 2400; intermediate-length tubes

Developing laminar

3

ψ N ud,lm = 3.66 + 10.0745 + 0.04 ψ 2

Red < 2400

(2.209) (2.210)

Red > 2400

(2.313)

Red > 4000; 0.4 < Pr < 600

[57]

where ψ ≡ (Red Pr)1/3 (L /d)1/3 (η bm /η w )0.14

Turbulent

N ud Red P r = f /2    √ 1+5 f /2 P r − 1 +ln 1 + 5 (Pr − 1) 6 Nud ,lm = 0.015 Red 0.83 Pr 0.42 (η bm /η w )0.14

Fully developed laminar

f = 16/Red

Red < 2100; z > 0.05 Red d

(2.87)

Turbulent

f = 0.0791 Red −1/4

2100 ≤ Red ≤ 105 ; smooth tubes

(2.314)

Red > 105 ; smooth (e = 0) and rough tubes

(2.319)

Turbulent

St ≡

Friction coefficients

Turbulent

1 √ F

= − 4 log



4.67 √ Red f

+

e d

 + 2.28

334

Transport Phenomena

Table 7. (Continued) Situation

Equation

Restrictions

Equation number ( ) or reference [ ]

Flow Around Bodies Flow normal to cylinder

N ud = 1/2 P r 1/3 d 0.3 +  3/4 1+(0.4/Pr)2/3 0.62 Re

Flow past a sphere

 · 1+



Red 28200

5/8 4/5

Nud = 2 + (0.4 Red 1/2 + 0.06 Red 2/3 ) · Pr 0.4 (η ∞ /η w )1/4

102 < Red < 4×106 ; Red ≡ u∞ d/ν; properties evaluated at film temperature 3.5 < Red < 8×104 ; 0.7 < Pr < 380; properties evaluated at T ∞

[58]

[59]

Free Convection from Various Geometries 1/4 0.67 Ra L 9/16 4/9 0.492 1+ Pr  1/6 0.387 Ra L 0.825 +  8/27 9/16 1 +(0.492/P r)

RaL < 109

[60]

T w = const.; 10−1 < RaL < 1012

[60]

Vertical surface, laminar, Nux = 0.60 Rax 1/5 local

qw = const.; 105 < Grx Nux < 1011

[61]

Vertical surface, laminar, Nux = 0.17 Rax 1/4 local

qw = const.; 2×1013 < Grx Nux Pr < 1016

[61]

T w = const.;1 0−6 ≤ Rad ≤ 109

[61]

10−5 ≤ Rad ≤ 1012

[61]

Horizontal plate, down NuL = 0.58 RaL 1/3 ward facing hot plates or upward facing cold plates; overall

T w = const.; 106 < RaL < 1011

[62]

Horizontal plate, upward NuL = 0.13 RaL 1/3 facing hot plates

T w = const.; RaL < 2×108

[62]

T w = const.; 2×108 ≤ RaL < 1011

[62]

Vertical surface, overall

Vertical surface, overall

N uL = 0.68 + 

N uL =



1/4 0.518 Ra d

2

Horizontal cylinder, laminar, overall

N ud = 0.36 + 

Horizontal cylinder, overall

N ud = 1/6 2    Rad 0.60 + 0.387  16/9 9/16   1 +(0.559/P r)

1 +(0.559/P r)9/16

4/9

1/3

or downward facing cold NuL = 0.16 RaL plates; overall

∂ = − ∇ · (u) ∂t

(3.1)

plays a vital role in establishing relationships between components of velocity. In this form, (∂/∂t)x physically represents the accumulation rate per unit volume of mass; the expression (− ∇ ·  u) represents the net input rate per unit volume to the volume fixed in space as a result of the mass flux  u. An analogous result can be obtained for an individual species (e.g., species A), however, since individual chemical species are not conserved (except for the elements), the net production of A due to reaction must be considered. Accord-

ingly, an equation accounting for the amount of species A is given by ∂A = − ∇ · (A uA ) + rA ∂t

(3.2)

where r A is the rate of production of species A per unit volume in the mixture, A is the mass of species A per unit volume of the mixture, and uA is the average velocity of the molecules of species A in the mixture. This equation can also be expressed in terms of molar concentration by dividing by molecular mass to give ∂cA = − ∇ · (cA uA ) + RA ∂t

(3.3)

Next Page Transport Phenomena where cA is the number of moles per volume of species A in the mixture and RA is the molar rate of production of species A by reaction per unit volume of the mixture. The velocity in this equation is the same as in the preceding equation. An equation of this sort in either mass or molar form can be written for each species in the mixture. When all the equations for all the species of an n-component mixture are added together, the following result is obtained: n

n

n

 ∂i   = − ∇ · (i ui ) + ri ∂t i=1 i=1 i=1

(3.4)

which, with interchange of the summations and derivatives, is n

∂



i

i=1

∂t

 n  n   = − ∇·  i ui  + ri i=1

(3.5)

i=1

Now, by definition, the sum of the individual species’ densities (mass per unit volume of the mixture) is the total mass per unit volume of the mixture, which is customarily defined as  to be consistent with the notation for compositionally homogeneous systems. Furthermore, the sum of all the individual species production rates by reaction must be zero, when expressed in mass units, because total mass is conserved (i.e., no net generation of total mass occurs). Consequently, the equation for the total mixture becomes   ∂ = − ∇· i ui ∂t

(3.7)

i=1

n

u ≡

i ui

i=1 n i=1

n

n

(3.9)

Again the summation and derivatives can be interchanged and the sum of the individual species concentrations (number of moles of species i per unit volume of the mixture) gives the total number of moles of the mixture per unit volume, customarily denoted as c. Accordingly, an accounting equation for the total number of moles in the mixture is   ∂c Ri = − ∇· ci ui + ∂t

(3.10)

In terms of moles, however, the net number of moles produced by reaction is not necessarily zero; the total number of moles due to reaction is not conserved. Consequently, this total molar production rate must be included in the total molar accounting equation for the mixture. However, for convenience, a mixture average velocity is defined such that the summation of the individual species fluxes represents a mixture average flux in terms of the total molar concentration and the mixture molar average velocity. Accordingly, cu ˜ ≡



ci ui

(3.11)

i=1

or, equivalently, n

u ˜ ≡

ci ui

i=1

n

(3.12) ci

i=1

i ui

or

n

 ∂ci   = − ∇ · (ci ui ) + Ri ∂t i=1 i=1 i=1

n

n



Thus, in a mixture, different species may move at different velocities and the mixture velocity is defined as a weighted average of these individual velocities. A parallel set of equations exists for molar concentrations. Accordingly, the sum of the molar concentrations of the individual species gives

(3.6)

Because of the similarity of this equation to the total mass conservation equation, the sum of the individual species fluxes is commonly defined as the product of the total mass density times an average velocity for the mixture. In fact, this becomes a defining relation for what is referred to as the mixture mass average velocity u u ≡

335

(3.8) i

This molar average velocity is not the same as the mass average velocity (Eqs. 3.7 and 3.8) because the individual species are not necessarily of the same molecular mass. The definition of a molar average velocity for the mixture allows

Previous Page 336

Transport Phenomena

the molar accounting equation to be written in the form n  ∂c Ri = − ∇ · (c u ˜) + ∂t i=1

(3.13)

Again, the net number of moles produced by reaction is not necessarily zero as it was for mass, and as a result, this term must remain in the equation. Mass and molar accounting equations are thus obtained for the individual species, along with statements of the conservation of total mass for the mixture. Now diffusion must be defined for the individual species and methods for calculating diffusion fluxes must be introduced. 3.1.2. Species Diffusion Fluxes The preceding discussion recognized that each species may be characterized by its own velocity which, in general, is different from the average velocity of the mixture. Furthermore, the mixture average velocity may be defined in terms of mass or in terms of number of moles, and the two velocities in general are different. As a result, each species has a relative motion with respect to the mixture average motion. This difference in fluxes between each species relative to the mixture average is referred to as diffusion. Because two mixture average velocities were defined, one based on number of moles and one based on mass, two different diffusion fluxes exist for each species, one with respect to the mixture molar average velocity and one with respect to its mass average velocity. Accordingly, the molar diffusion flux J˜ A of species A can be defined as the difference between the total flux of species A and that due to the mixture motion on the average [63]: ˜) = cA uA − cA u ˜ J˜A ≡ cA uA − xA (c u

(3.14)

where x A is the mole fraction of species A. Using this definition of a diffusion flux in the accounting equation for species A gives ∂cA ˜) − ∇ ·J˜A + RA = − ∇ · (cA u ∂t

(3.15)

Here, the accumulation rate of the number of moles of A results from the molar flux of A due to the average flow of the mixture, plus the molar diffusion flux of species A above and beyond

this mixture molar average flow, plus the molar rate of production of A per unit volume. Similarly, a diffusion flux can be defined in terms of mass fluxes, i.e., the difference between the total mass flux of species A and the mass flux of species A that results from the mixture moving at its mass average velocity: j A ≡ A uA − A u

(3.16)

This mass diffusion flux used in the mass accounting equation for species A gives ∂A = − ∇ · (A u) − ∇ ·j A + rA ∂t

(3.17)

Again the accumulation rate of species A at a point in space results from the mass flux of species A due to the bulk flow of the fluid (the mass average velocity of the mixture) plus the diffusion flux of species A plus the mass rate of production (per unit volume) of A by reaction. In addition to the molar diffusion flux of A and the mass diffusion flux of A as defined above, a molar diffusion flux of A could be defined relative to the mass average velocity according to the equation J A ≡ cA uA − cA u

(3.18)

or a mass diffusion flux of A could be defined relative to the molar average velocity: ˜ J˜A ≡ A uA − A u

(3.19)

Neither of these is as commonly used as the first two, so they are not considered further here. The point of this discussion is that when talking about diffusion and diffusion coefficients the velocity used to characterize the mixture must be specified. If the molar average velocity is used, one value for diffusion is appropriate; if the mass average velocity is used, another (different) diffusion is described. 3.1.3. Fick’s First Law of Diffusion Until now the transport equations presented in this section have been truly general for multicomponent (as opposed to binary) systems. The species accounting equations and the mixture mass conservation equations are correct, independent of whether binary or multicomponent

Transport Phenomena systems are involved. Likewise, the definitions of diffusion flux are still valid for multicomponent systems. However, at this point a relationship is defined between diffusion driving forces and species fluxes. The relationship to be discussed is restricted to binary systems and is known as Fick’s first law of diffusion. This law is a linear transport equation relating mass transfer or diffusion driving forces to the diffusion flux. Accordingly, in a binary system the driving force for diffusion (i.e., the reason different species move with different velocities) is the compositional inhomogeneity (the existence of mole fraction gradients). (Pressure and temperature gradients may also induce diffusion [63]. These effects are normally small and are not considered further here.) In the presence of such gradients, species within the mixture move in such a way as to remove these mole fraction gradients, a motion that occurs spontaneously (implying an increase in entropy of the mixture). In accordance with a linear relation between this driving force and diffusion flux, Fick’s first law of diffusion is written as J˜ = − c DAB ∇xA

(3.20)

In this equation, ∇x A is the driving force, and the product of the total mixture molar concentration and the diffusitivity is the proportionality factor required to convert this driving force to a molar flux. The minus sign indicates that the diffusion flux is in the direction opposite to the direction of increasing mole fraction. The diffusivity is a parameter that describes how easily species A moves through the rest of the mixture, species B – consequently, the notation DAB . Similarly, the mass diffusion flux of species A can be written in terms of a mass fraction gradient with the proportionality factor being the product of the mixture mass density and the same diffusion coefficient as for the molar diffusion flux. A priori, the same diffusivity might not be expected to appear in both the molar and mass diffusion flux equations. However, this must be the case for binary systems. Furthermore DAB = DBA in a binary system, because x A = 1 − x B and ∇x A = − ∇x B , and the sum of the two species diffusion fluxes must be zero. Consequently, for binary mixtures a single diffusivity needs to be reported for any given mixture composition of A and B.

337

3.1.4. Special Cases of Diffusion Using the relations for diffusion molar and mass fluxes (Fick’s first law, Section 3.1.3) and the accounting equations presented in Section 3.1.2 yields the general results that account for species A in either moles or mass. Accordingly, in molar units ∂cA ˜) +∇ · (c DAB ∇xA ) + RA (3.21) = − ∇ · (cA u ∂t

or in mass units ∂A = − ∇ · (A u) +∇ · ( DAB ∇wA ) + rA (3.22) ∂t

where wA is the mass fraction of A in the mixture. These equations are quite general within the restrictions of Fick’s first law. They do not assume constancy of total molar concentration or of diffusivities. Three special cases are of particular interest [63]. Case 1. If both the mixture mass density  and DAB are constant, then the following result is obtained: ∂A = − u ·∇A +  DAB ∇2 wA + rA ∂t

(3.23)

which may be divided by molecular mass to yield a molar accounting equation ∂cA = − u ·∇cA + c DAB ∇2 xA + RA ∂t

(3.24)

This result, which assumes that the divergence of the velocity is zero (the mixture mass density is constant), is appropriate for dilute liquid solutions at constant temperature and pressure. This particular form of the equation is also expressed in terms of mixed quantities; the concentration is given in terms of molar concentration, but the mixture average velocity is the mass average velocity. Case 2. If the mixture total molar concentration c and DAB are constant, the mixture continuity equation gives ∇ ·˜ u=

RA + R B c

(3.25)

338

Transport Phenomena

and the accounting equation for species A becomes ∂cA = −u ˜ ·∇cA + c DAB ∇2 xA ∂t + RA (1 − xA ) − xA RB

(3.26)

This result is appropriate for low-density gases at constant temperature and pressure. Case 3. Furthermore, if no chemical reaction occurs, if (for liquids) the mixture mass density is constant, and if the mixture mass average velocity is zero or (for gases) the mixture molar concentration is constant and the mixture molar average velocity is zero (i.e., no bulk motion of the fluid occurs), then Equations (3.25) and (3.26) reduce to the form ∂cA = DAB ∇2 cA ∂t

(3.27)

This result is known as Fick’s second law of diffusion and states that the accumulation of species A at a point in space results solely from the diffusion flux of species A; no convective transfer occurs, nor does any generation or consumption as a result of chemical reactions. This result is also called the diffusion equation and is identical to the partial differential equations (1) that are obtained for heat conduction in solids of constant heat capacity, thermal conductivity, and density; and (2) that result from the equation of motion for a fluid of constant density and viscosity near a wall suddenly set in motion. This type of agreement between different modes of transport provides the basis for analogous correlations between certain heat-, mass-, and momentum-transfer problems. That is, the same mathematical equations can govern these three different modes of transport. Then, with analogous geometrical systems and boundary conditions, exactly the same mathematical problem exists, resulting in indentical mathematical solutions. Consequently, the results of many diffusion problems are available in the heat-transfer literature, e.g., [67], [69]. 3.1.5. Diffusivities of Gases and Liquids Gases. The diffusivities of gases can be estimated quite well by using molecular theory (similar to viscosity). Diffusion calculations are

not complicated by multiple internal energy modes as in the case of thermal conductivity, although the presence of two different molecules must be taken into account. Accordingly, the Chapman – Enskog equation for diffusivity (in square centimeters per second) in a binary mixture of gases is DAB = 1.883 × 10−3 T 3/2 · 

1/2 Mr,A + Mr,B /Mr,A Mr,B 2 Ω p σAB D

(3.28)

where p is in bar, σ AB is in a˚ ngstr¨oms, and T is in kelvin. This result incorporates a molecular potential energy function such as the Lennard – Jones 12 – 6 function, which may be used for mixtures of nonpolar species. For a binary mixture the characteristic separation distance σ AB is an arithmetic average of the values for the two pure species. Also, the collision integral ΩD is a function of k T/εAB , where εAB is the geometric mean of the values for the two species. As for viscosity, values of the collision integral are tabulated or may be calculated by using a relation given by Neufeld et al. [70] and presented by Reid et al. [68]. However, the values for calculating diffusivities are different from those used for viscosity. For binary mixtures with at least one polar species, a different potential function may be used (along with its own values for the molecular parameters) to calculate a different collision integral. Alternatively, the Lennard – Jones collision integral may be modified by an additive contribution due to molecular polarity as manifested by the dipole moment [71]. Diffusivities for low-density gases can also be estimated by using empirical relations built around the basic form of the Chapman – Enskog relation. The relation of Fuller et al. [72] is applicable to both polar and nonpolar molecules and is similar to a group contribution method wherein the different structural units of a molecule are assigned specific diffusion volume increments. The total diffusion volume for each molecule is calculated and used to determine the binary diffusivity. Finally, corresponding states methods also are available for estimating gas diffusivities.

Transport Phenomena Liquids. Theories for estimating liquid diffusivities are not nearly as successful as for gases because of the much smaller separation distances between molecules, which result in multibody interactions and magnified interaction forces. The Wilke – Chang method is probably the most widely used relation for estimating the diffusivity (in square centimeters per second) of a solute A at infinite dilution in a solvent B [73]: 0 DAB = 7.4 × 10−11



1/2 ϕ Mr,B T ηB VA0.6

339

Equimolar Counterdiffusion. The term equimolar counterdiffusion means that the two species in the binary mixture diffuse at exactly the same molar flow rates per unit area but in opposite directions. A diffusion cell schematic that approximates ECD is depicted in Figure 21.

(3.29)

Accordingly, M r,B is the molecular mass of the solvent; η B the solvent viscosity, in pascal seconds, and D0AB the diffusivity when the solute is at infinite dilution in the solvent. Also, an association factor ϕ accounts for the degree of association exhibited by the solvent. It is 1.0 for nonassociated solvents and increases with association: 1.5 for ethanol, 1.9 for methanol, and 2.6 for water. Other methods are available and are summarized and compared in [68]; see also → Estimation of Physical Properties, Chap. 6.3.2. 3.1.6. Theoretical Foundations of Steady-State Measurement of Diffusion To understand the techniques for measurement of diffusion, two situations are addressed with respect to gas-phase diffusion that occur as idealized limiting cases. The first is equimolar counterdiffusion (ECD) in which the two species of a binary mixture have exactly equal and opposite molar fluxes. The second is the diffusion of one species A through a stagnant film of a second species B (ATSB); the flux of one species is not zero whereas the other is exactly zero. Both situations are considered to be occurring at steady state. Since they are gas-phase diffusion problems at constant pressure the equations are given in molar form. In each case analysis involves using a continuity equation for each species (Eq. 3.3), a diffusion flux equation giving the total flux of a species as the sum of that due to the mixture molar average flux plus that due to diffusion (Eq. 3.14), and finally Fick’s first law of diffusion (Eq. 3.20).

Figure 21. Schematic of an equimolar counterdiffusion (ECD) cell

This is a one-dimensional problem for which the mixture molar average velocity (Eq. 3.11) is zero; hence, the flux of either species occurs only because of its diffusion. If NA is defined as the total molar flux of species A, from Equation (3.14) N A = cA uA = cA u ˜ + J˜A = J˜A

(3.30)

or in one dimension using Fick’s first law NAz = − c DAB

dyA dz

(3.31)

Now, an accounting of species A (the continuity equation, Eq. 3.3) with no reactions and at steady state gives ∇ ·N A = 0

(3.32)

which in one-dimensional form is dNAz = 0 dz

(3.33)

and from which the molar flux of species A (and similarly of species B) is seen to be invariant with respect to position in the direction of flow. Consequently, Equation (3.31) can be integrated directly (for constant total concentration and diffusivity) to give a linear relation for the concentration of species A (and hence also of species B) with respect to position in the direction of flow

340

Transport Phenomena 

yA (z) = yA0 +

yAL − yA0 L

 z

(3.34)

where yA0 and yAL are the mole fractions of A at z = 0 and z = L, respectively. The molar flux in terms of diffusivity and concentrations at the ends of the diffusion tube is NAz =

−c DAB ( yAL − yA0 ) L

(3.35)

This result then can be used to measure diffusivities if the tube-end concentrations are known as a function of time. The above derivation assumes a steady state, i.e., the two end-cell concentrations do not change with time. The result can still be used if the changes in concentration are slow compared to the establishment of the concentration profiles in the diffusion cell. By using this pseudosteady-state assumption the changes in concentration at the two cell ends can be related to the flux through the cell by writing a mass accounting balance for species A in each cell. Accordingly,

ATSB Diffusion. For the diffusion of species A through a stagnant film of B (ATSB), the situation is again taken to be constant total pressure and steady state. Diffusion occurs through a tube of uniform diameter and length L. In this case, however, the motion of one of the species is constrained to be zero as in the evaporation of a liquid A from a tube open at one end (Fig. 22). The second gas-phase component is not soluble in A (or at least A is saturated with B), so its flux at the interface of liquid A with the gas phase is zero. The liquid evaporates at the interface and diffuses to the open end of the tube where its concentration is maintained at a reduced level by a steady flow of fresh gas phase across the open end of the tube.

d (cA0 V0 ) = dt NAz Ac = −

Ac DAB (cAL − cA0 ) L

(3.36)

where Ac is the cross-sectional area through which diffusion occurs and d (cAL VL ) = dt Ac DAB (cAL − cA0 ) −NAz Ac = L

(3.37)

from which d (cA0 − cAL ) = dt   Ac DAB 1 1 − + (cAL − cA0 ) L V0 VL

(3.38)

so that [cAL (t) − cA0 (t)] = [cAL (t0 ) − cA0 (t0 )]  ) (  1 Ac DAB 1 (t − t0 ) + · exp − L V0 VL

(3.39)

Measuring the concentration difference between cells as a function of time gives DAB . The diffusion cell itself can consist of a glass frit or even filter paper. In the latter case the cell length is illdefined because of tortuosity and the coefficient Ac (1/V 0 + 1/VL )/L is determined by calibration with a material of known diffusivity [65].

Figure 22. Schematic of “A through stagnant B” (ATSB) diffusion process

For steady state with no reactions in the gas phase, the continuity of species A in the gas phase requires (as in the equimolar counterdiffusion problem) that

Transport Phenomena 0 = − ∇ ·N A = −

dNAz dz

(3.40)

Again, the molar flux of A along the length of the tube is constant. Likewise, a similar result is obtained for species B. Because liquid A is saturated with B, the flux of species B into the liquid at the interface must be zero. Then, because this flux is constant along the entire length of the tube, it must be zero everywhere; species B is stagnant. Now, by writing NA (= cA uA ) and NB (= cB uB ) as the molar fluxes of species A and B, respectively, the combined molar flux for the mixture is given by (see Eq. 3.11) cu ˜ = cA uA + cB uB = N A + N B

(3.41)

which, combined with Equation (3.14), yields N A = yA (N A + N B ) + J˜A

(3.42)

i.e., the total flux of A is that fraction due to the total mixture average flow plus the flow due to the diffusion of species A. With species B stagnant, NB = 0 and NA =

J˜A (1 − yA )

(3.43)

which in one-dimensional form is NAz dz = − c DAB

dyA (1 − yA )

(3.44)

Because of the constancy of NAz along the length of the tube, this result can be integrated directly to give either a result for the molar flux of A (in terms of tube length, total pressure, diffusivity, and concentrations at the end of the tube) NAz =

c DAB ln L



1 − yAL 1 − yA0

 (3.45)

or a result for the concentration profile of species A along the length of the tube  yA (z) = 1 − (1 − yA0 )

1 − yAL 1 − yA0

z/L (3.46)

Note that, contrary to equimolar counterdiffusion, the concentration profile is nonlinear even though the flux of species A is still constant along the tube length.

341

Comparison of ATSB and ECD Results. Comparing the ATSB and ECD results shows that for the same mixture concentration c, diffusivity DAB , and driving force for diffusion dyA /dz, the flux for ATSB is greater than that for ECD due to the fact that the mole fraction of species A is less than unity and hence the driving force is divided by (1 − yA ). This enhancement is perhaps best understood by looking at Equation (3.42) for the total molar flux of A in terms of the bulk flow of the mixture and the diffusion flux. The bulk flow of the mixture is not zero for ATSB, whereas it is zero for ECD. Enhancement occurs in the flux of species A out of the tube as a result of this nonzero bulk flow. Note also that even though B is stagnant, its diffusion flux, through Fick’s law, is nonzero because its mole fraction gradient is nonzero. With species B, however, the diffusion flux down the tube toward the liquid in the bottom of the tube is exactly counterbalanced by its flow up the tube because of the bulk flow of the gas mixture. The net effect is that species B is stagnant (N B = 0) even though it is in fact diffusing. In principle, this experimental situation provides another method for measuring gas-phase diffusivities. In terms of changes in the amount of evaporating liquid in the tube, NAz (t) =

 dL Mr dt

(3.47)

so that Equation (3.45) becomes  dL c DAB = ln Mr dt L



1 − yAL 1 − yA0

 (3.48)

which can be integrated to give L2 =

   2 c DAB Mr 1 − yAL (t − t0 ) + L 20 ln  1 − yA0 (3.49)

Measuring L as a function of time (by weight) can then be used to determine DAB . This result, like the ECD method, rests upon the pseudosteady-state assumption, in this case that the concentration profile and the flux adjust rapidly to changes in L. Further complications in this technique involve maintaining constant temperature in the presence of evaporation and convection due to differences in the molecular mass of the two gases [63].

342

Transport Phenomena

3.1.7. Diffusion with Homogeneous Chemical Reaction A situation is now considered that involves chemical reaction in addition to diffusion: the transfer of a component from a gas phase to a liquid film followed by diffusion away from the gas – liquid interface and reaction within the film. At a distance δ from the gas – liquid interface, the film meets an impermeable solid boundary (Fig. 23). As a boundary condition it is assumed that diffusion of the gas to the liquid film is not limiting so that the concentration of the gaseous material in the liquid film at the gas – liquid interface is that established by equilibrium with the bulk gas-phase concentration. Species A then diffuses away from the gas – liquid interface in an attempt to increase its concentration within the liquid film. However, the chemical reaction serves to deplete the concentration and a nonlinear steady-state concentration profile is achieved. At the solid – liquid interface, the concentration profile must be flat; because of the impermeability of the solid surface, the flux at that surface is zero.

This reaction occurs homogeneously throughout the liquid film in the sense that it takes place everywhere in the film that species A and B exist together. Reaction rate varies with position because of variation of the concentration of A with position z through the film. As usual, we begin with the equation of continuity for A which is ∂cA = − ∇ ·N A + RA ∂t

(3.52)

For the steady-state condition and for the rate of production of A given by chemical kinetics, this equation becomes dNAz = − k1 cA (z) dz

(3.53)

This situation differs in a fundamental way from the previously described situations in that the flux of A in the diffusion direction is not constant because of the depletion of A by the chemical reaction. The concentration of A is a function of position z so that the result cannot be integrated to obtain an immediate conclusion about the flux of A. Instead, the flux equation of A is used in terms of bulk flow in the solution and diffusion: N A = yA (N A + N B + N C ) − c DAB ∇yA (3.54)

Figure 23. Schematic of diffusion and chemical reaction in a liquid film

The objective of the analysis is to obtain a relation for the molar flow rate of A within the film and the concentration profile of A across the film. For reactions of species A with the liquid (species B), consider a situation with the reaction given by A+B →C

(3.50)

which is characterized by a rate constant k 1 such that the rate of production of species A is given by the kinetic first-order equation RA = − k 1 cA (RA < 0 means that A is being consumed)

(3.51)

Because species A reacts as it diffuses into the thin film, a reasonable assumption (depending on the reaction kinetics versus the diffusion rate) may be that the concentration of A in the liquid film is everywhere small so that the bulk flow (convection) term that contributes to the total flux of species A can be neglected relative to the diffusion flux term. In this case and for one dimension, the equation becomes NAz = − c DAB

dyA dz

(3.55)

Combining this with the continuity equation (Eq. 3.53) gives a second-order ordinary differential equation for the concentration of species A d 2 yA k1 − yA = 0 dz 2 DAB

(3.56)

This equation is subject to the boundary conditions at the gas – liquid film interface where the concentration in the liquid film is established by thermodynamic equilibrium (yA = yA0 ) and at the solid surface (z = δ) where N Az = 0 and

Transport Phenomena dyA /dz = 0. Imposing these conditions allows integration of the equation to give .   yA (z) = yA0 cosh z k1 /DAB −   /  tanh δ k1 /DAB sinh z k1 /DAB

(3.57)

The flux of A is a function of position and is given by NAz (z) = c yA0 k1 DAB ·  .  − sinh z k1 /DAB +   /  tanh δ k1 /DAB cosh z k1 /DAB

(3.58)

Note that the concentration profile through the liquid film is exponential in form through the cosh and sinh functions and that the degree of decay depends on the ratio (relative magnitude) of the reaction rate to the diffusion rate. The larger the reaction rate, the more steeply the profile falls as the distance into the film away from the gas interface increases. Conversely, the faster the diffusion (i.e., the higher the diffusion rate relative to the reaction rate), the more slowly the (steady-state) concentration falls or decreases with increasing distance from the interface. A similar problem for an infinitely thick liquid film is approached in a similar way except that instead of the no-flux boundary condition at the solid surface the condition cA → 0 as z → ∞ is used. Such a solution is appropriate even for liquid films of finite thickness if the reaction rate is large enough compared to the diffusion rate so that the diffusing species A in fact never “sees” the solid surface, i.e., it is depleted by reaction before it reaches the solid surface. 3.1.8. Diffusion with Heterogeneous Reaction The next problem addresses diffusion through a gas film to a surface at which a reaction occurs and from which the reaction product must diffuse back through the gas phase. This is the situation in many catalytic conversion reactions where a catalyst pellet may be considered to be surrounded by a thin stagnant film through which a species must diffuse. Because the film is stagnant, diffusion rather than reaction kinetics may be the controlling phenomenon.

343

The situation is depicted schematically in Figure 24 for a spherical catalyst particle of radius R surrounded by a boundary layer gaseous film of thickness δ. Reactant A diffuses through the film to the surface where the reaction A + A → A2 is assumed to occur very quickly compared to the diffusion processes. As a result, at the catalyst surface the concentration of reacting species is assumed to be zero; the reacting species is depleted by reaction as quickly as it is delivered to the surface by diffusion. This is a binary system with A and A2 as the two components. Because the reaction occurs at the solid surface rather than in the gas phase, it is not relevant to the continuity equation of species A within the gas phase. The role of the reaction stoichiometry and kinetics is to provide a boundary condition on species fluxes at the catalyst surface.

Figure 24. Schematic of diffusion and heterogeneous chemical reaction at a catalyst surface

As for the preceding problems, the objective is to establish quantitative relations for the flux of reactant A through the stagnant thin film and for the concentration profile of A across the film. As before, the continuity equation for reacting species A is employed: ∂cA = − ∇ ·N A + RA ∂t

(3.59)

For steady state with no homogeneous reaction, this gives ∇ ·N A = 0

(3.60)

which in spherical coordinates is

1 d 2 r NAr = 0 r 2 dr

(3.61)

which is integrated to give r 2 NAr = constant

(3.62)

344

Transport Phenomena

In this spherical geometry, the flux of a species is not constant but is proportional to 1/r 2 . Now because at any position r the spherical surface area is proportional to r 2 , the molar flow rate of A rather than the flux is constant, independent of position, according to r 2 NA NA r2 NAr = constant = = 4 π r2 4π

(3.63)

Here N A is the number of moles of species A per unit time passing through the entire spherical shell at radius r. This molar flow rate is a constant, independent of radial position. As before, the result for the molar flux of species A is written in terms of the convective bulk flow and diffusion:

N A = yA = N A + N A2 − c DAB ∇yA

(3.64)

which, in one-dimensional form, is

dyA NAr = yA NAr + NA2 r − c DAB dr

(3.66)

These are molar flow rates integrated over the entire surface of a spherical shell and, as shown above, because of continuity they are independent of radial position. The fluxes of each of the species can be written in terms of these molar flow rates so that Equation (3.65) becomes NA = yA r2



NA NA + 22 r2 r



dyA − 4 π c DAB dr

dr 4 π c DAB dyA

= r2 1 − 12 yA

  1 yA (r) = 2 1 − 1 − yAδ 2

NA = 8 π c DAB

As the molar flow rate is a constant, this result can be integrated directly to give a relation for the concentration profile through the film. In doing so, the boundary conditions are that the concentration of reactant A is zero at the surface of the pellet because it is instantaneously depleted by the reaction, and the concentration

γ R+γ



R+δ δ



  (3.69)

R (R + δ) ln δ

  1 1 − yAδ 2

(3.70)

Now this result is written in terms of an unknown film thickness δ. It can be evaluated, however, for two limiting cases to obtain limiting or boundary concentration profiles and molar flow rates. If gas flow across the pellet is slow or essentially stagnant, then the film thickness goes to infinity and the results obtained are 

yA (r) = 2 1 −

1−

(1− R )

r 1 yAδ 2

(3.71)

and  NA = 8 π c DAB R ln

1−

1 yAδ 2

 (3.72)

Alternatively, if the gas velocity is high enough, the stagnant film becomes very thin (δ  R) and results obtained are 



yA (r) = 2 1 −

1−

( γ )

δ 1 yAδ 2

(3.73)

and NA = 8π c DAB

(3.68)



where γ ≡ r − R and

(3.67)

which, because of the relation between the molar flow rates of A and A2 , gives NA



 (3.65)

Because of the reaction stoichiometry, the molar flow rate of species A can be related to that of product A2 according to N A = − 2 N A2

at the outer extent of the film (i.e., at r = R + δ) is a known value (the concentration in the bulk gas phase). The resulting concentration profile is

R2 ln δ

 1−

1 yAδ 2

 (3.74)

In these latter equations, the film thickness remains as a parameter, whereas in the stagnant film situation the velocity profile and molar flow rate can be calculated directly; as the gas film becomes very thick, the molar flow rate and concentration profile become independent of the film thickness. At the other extreme, for a very thin film, the molar flow rate becomes inversely proportional to the film thickness; a doubling of film thickness halves the molar flow rate.

Transport Phenomena 3.1.9. Perspective The aforementioned results are useful in the context of this article; not so much for the specific problems addressed as for the types of approaches that are made to solving diffusion problems. In these problems, steady-state situations have involved either thin films or infinitely thick films, and slowly changing processes have been observed which can be treated in a pseudosteady-state manner by first solving the steady-state problem and then integrating over time to obtain the results as a function of time. Flat and spherical geometries have also been observed, although curved geometries, mathematically, look exactly like flat geometries when the film thickness is small in relation to the radius of curvature. Therefore, flat geometry calculations take on an importance above and beyond their actual application to specific physical problems. Finally, both homogeneous and heterogeneous chemical reactions have been addressed; homogeneous chemical reactions occur throughout the phase through which the diffusion or motion of molecules is occurring, whereas heterogeneous reactions occur at a boundary or surface associated with the diffusion process. As a result, homogeneous reaction kinetics appear directly in the continuity equations for the reacting species within the diffusion phase, whereas heterogeneous reaction kinetics play a role in the fluxes that occur at interfaces or boundaries of the diffusion region.

345

liquid film that moves on a solid surface. A simple example of such a situation is shown in Figure 25, where the gas species A is exchanged between a bulk gas phase and a falling liquid film of component B. This problem is analyzed first as steady-state simple absorption with no chemical reaction. Then, this solution is used to analyze the more general case in which simultaneous absorption and reaction occur.

Figure 25. Absorption and reaction in a falling liquid film

3.2. Convective Mass Transport In the previous examples, motion of the individual species resulted from the diffusive fluxes that occur because of concentration gradients. In this section, examples in which species transport arises from bulk flow (or convection), as well as diffusion, are considered. 3.2.1. Gas Absorption in a Falling Film with Reaction A problem encountered in many industrial scrubbers, fixed-bed gas – liquid reactors, and other mass-transfer devices, is the exchange of a gas species A between a gas mixture and a

In the analysis, A is assumed to be only slightly soluble in Newtonian liquid B, so that the viscosity and density of the liquid film are not changed appreciably. Under such conditions, the film velocity profile is given by (end effects neglected)   y 2  uz = umax 1 − δ

(3.75)

The concentration of A in the film varies with both y and z, and the species mass balance (Eq. 3.24) can be written as uz

∂cA = DAB ∂z



∂ 2 cA ∂ 2 cA + ∂y 2 ∂z 2



with the boundary conditions

(3.76)

346

Transport Phenomena

at z = 0, cA = 0

(3.77)

at z → ∞, cA → cA0

(3.78)

at y = 0, cA = cA0

(3.79)

∂cA = 0 ∂y

(3.80)

at y = δ,

The first boundary condition expresses the fact that pure B enters at z = 0; the second, that the film is saturated with A after a sufficiently long exposure distance. The third condition gives the concentration cA0 at the gas – liquid boundary. Generally, unless significant resistance to mass transfer occurs on the gas side of the interface, the concentration cA0 is the concentration in the liquid that is in equilibrium with the gas. The fourth condition simply indicates that no mass flux exists normal to the solid wall; hence, the concentration gradient ∂cA /∂y is zero at the wall. By assuming that over some film length z ∼ δ the gas penetrates only a fraction of the distance into the film (e.g., y ∼ ε δ), the y and z variables can be scaled as y ≡ ε δ y∗ z ≡ δz

∗

1−ε y

∂c∗A 1 = ∂z ∗ P eδ

(3.81d)



∂ 2 c∗A 1 ∂ 2 c∗A + 2 ∗2 ε ∂y ∂z ∗2

 (3.82)

where P eδ ≡

umax δ DAB

Equation (3.82) can then be written as  1−

1 ∗2 y P eδ



∂ 2 c∗A ∂c∗A 1 ∂ 2 c∗A = + ∂z ∗ ∂y ∗2 P eδ ∂z ∗2

(3.85)

Equation (3.84) indicates that short penetration distances (ε  1) are associated with large P´eclet numbers. For the specific case in which umax ≈ 0.1 cm/s, δ ≈ 0.1 cm, and DAB ≈ 10−5 cm2 /s, the value of Peδ is 103 (or ε ≈ 0.03). Such high P´eclet number, short penetration distance conditions are typical of those encountered in many gas – liquid contact processes. Under such conditions, Equation (3.82) can be reduced to ∂c∗A ∂ 2 c∗A = ∗ ∂z ∂y ∗2

(3.86)

with the boundary conditions at z ∗ = 0,

c∗A = 0

(3.87)

at y ∗ = 0,

c∗A = 1

(3.88)

The solution of Equation (3.86) subject to these boundary conditions is [63, p. 539]

Equation (3.76) can be written as 2 ∗2

(3.84)

1/2

P eδ

(3.81b)

and



1

at y ∗ → ∞, c∗A → 0

(3.81c)

uz = umax u∗z

ε ≡

(3.81a)

where y∗ and z∗ are dimensionless quantities of order 1. Further, if cA and uz are scaled by cA = cA0 c∗A

(the y diffusion term). In particular, this requires that

(3.83)

is the mass-transfer P´eclet number. If convection is important, the left-hand side of Equation (3.82) must be of the same order as the largest diffusion term on the right-hand side

(3.89)

√

y

4DAB z/umax
2 cA 2 ∗ cA = = 1− √ e−ζ dζ cA0 π 0   y = 1 − erf (3.90) 4 DAB z/umax

or cA = erfc cA0



y 4 DAB z/umax

 (3.91)

where erf and erfc are the error and complementary error functions, respectively. For a contact film of length L and width W , the rate of mass transfer (number of moles of A per unit time) to the film is given by WA = NAy |y=0 W dz = 0

(3.92)  ∂cA  −DAB W dz  y=0 ∂y


L 


L

0

Transport Phenomena or, if Equation (3.91) is used, 1 WA = L W cA0

4 DAB umax πL

(3.93)

If the mass-transfer coefficient kc is defined by WA = kc L W cA0

(3.94)

then ShL ≡

kc L 2 2 1/2 1/2 = √ ReL Sc1/2 = √ P eL (3.95) DAB π π

where ShL is the Sherwood number, and ReL and Sc are the Reynolds and Schmidt numbers, respectively, which are defined by ReL

umax L ν ≡ , Sc ≡ ν DAB

(3.96)

where ϕ ( y∗, z∗) is the cA solution from Equation (3.91). Using Equation (3.92) to determine the total gas exchange over a surface of length L and width W gives umax WA = W cA0 k1 /DAB 1    q 1 √ · + q erf q + · exp (−q) 2 π

(3.102)

where q ≡

DaL ReL Sc

(3.103)

In terms of ShL ≡ kc L/DAB , this result can be written as 1/2

Hence, for short penetration distances ε (or short contact times or high P´eclet numbers), the dimensionless form of the mass-transfer coefficient (ShL ) varies as the square root of the product of ReL and Sc. If the chemical reaction

ShL = ReL Sc1/2   (1 + 2 q) 1 √ · erf q + √ exp (−q) √ 2 q π

A+B →C

(3.97)

in the film is now considered and a sufficient excess of B is assumed so that a pseudo-firstorder reaction expression is valid, i.e., RA = k cA cB ≈ k cB0 cA = k1 cA

(3.98)

the mass balance takes the form ∂c∗A ∂ 2 c∗A Daδ c∗A = − ∗ ∗2 ∂z ∂y Reδ Sc

(3.99)

where Daδ is the Damk¨ohler number defined by k1 δ 2 DAB

(3.104)

If q → 0 (no chemical reaction), this equation reduces to Equation (3.95); if q → ∞, 1/2

Daδ ≡

347

ShL = DaL

= (q ReL Sc)1/2

(3.105)

In Figure 26, the results are given for a full range of q values. As q increases (higher reaction rates), mass transfer is significantly enhanced (i.e., higher mass-transfer coefficients) compared to the nonreacting case (q = 0). Many scrubbers and other absorption processes utilize absorption with reaction to obtain the most efficient removal of dilute gases from bulk gas streams. LIVE GRAPH

Click here to view

(3.100)

Here again, in writing Equation (3.99), the penetration distance is assumed to be small (i.e., Peδ ≡ Reδ Sc is large). Also, the boundary conditions are assumed to be the same as given by Equations (3.87) – (3.89). The solution to this problem can be expressed in terms of the solution for the nonreacting case (Eq. 3.91) by [69, pp. 32 – 33] c∗A =

cA = ϕ ( y ∗ , z ∗ ) exp cA0

Daδ + Reδ Sc


z 0

∗

ϕ ( y ∗ , ζ) exp

 

Daδ ∗ z Reδ Sc Daδ ζ Reδ Sc



 dζ

(3.101)

Figure 26. Enhancement of mass transfer with chemical reaction in a falling liquid film

348

Transport Phenomena

3.2.2. Mass Transfer in Laminar and Turbulent Boundary Layers


L WA =

kc (cA0 − cA∞ ) W dx 0

Consider the convective mass transfer of species A from a flat or nearly flat surface into a free stream of species B. Assume that concentration changes of A are restricted to a concentration boundary layer of thickness δc . Conceptually, this is analogous to the problem illustrated in Figure 15, except that δc replaces δT , cA∞ replaces T ∞ , and cA0 replaces T w . For this problem where cA = cA (x, y), the species mass balance takes the form ux

∂cA ∂cA ∂ 2 cA + uy = DAB ∂x ∂y ∂y 2

(3.106)

where the x diffusion term [DAB (∂ 2 cA /∂x 2 )] has been neglected compared to the y diffusion term (right-hand side of above equation). This equation is analogous to Equation (2.239) for heat transfer in thermal boundary layers and can be integrated over the boundary layer thickness to obtain d dx


δc ux (cA − cA∞ ) dy = NAy |y=0 0

= −DAB

∂cA   ∂y y=0

(3.107)

The solution of this equation for δc (x) follows in the same way δT was obtained from the integral form of the energy balance (Eq. 2.240). If a cubic expression is used for cA in terms of y and the constants a, b, c, and d are evaluated with the boundary conditions at y = 0, cA = cA0

(3.108)

at y = δc , cA = cA∞

(3.109)

at y = δc ,

∂nc

A ∂y n

= 0 , n = 1, 2, . . .

= kcL W L (cA0 − cA∞ )

(3.111)

where kcL ≡

1 L


L kc dx

(3.112)

0

is the average mass-transfer coefficient over the plate length L. These results are valid for the laminar part of the boundary layer. In the turbulent part (x ≥ x cr = 2×105 ν/u∞ ), the molar flux in the ydirection can be written as cA ¯Ay = − (DAB + εc ) d¯ N dy

(3.113)

where N¯ Ay and c¯ A are time-smoothed quantities and εc is the eddy mass diffusivity. If a Prandtl mixing length expression is assumed for εc , then  d¯   ux  εc = εm = εT = ε = l2   dy

(3.114)

with the mixing length l given by Equation (2.299). Here again, as with energy transfer in turbulent flow, DAB  ε for the laminar sublayer next to the wall, DAB ≈ ε in the buffer zone, and DAB  ε in the turbulent core. Carrying out an analysis analogous to that of Section 2.5.4 gives the mass-transfer forms of the various analogies: Reynolds analogy: Shx kc = CDx /2 = Rex Sc u∞

(3.115)

Prandtl analogy:

(3.110)

the same results can be obtained as for Equations (2.250) and (2.251), with δc replacing δT , Sc ≡ ν/DAB replacing Pr, and Shx ≡ kc x/DAB replacing Nux . The Blasius solutions (Eqs. 2.252 and 2.255) also have the same forms for the mass-transfer results if the same replacements are made. The mass-transfer rate (number of moles A per unit time) from the surface is then

Shx CDx /2 = Rex Sc 1 + 5 CDx /2 (Sc − 1)

(3.116)

Von Karman analogy: Shx = Rex Sc 1+5

(3.117)

CDx /2 CDx /2 {Sc − 1 + ln [(1 + 5 Sc) /6]}

Further, the Colburn analogy can be written as Shx CDx /2 kc = = Rex Sc u∞ Sc2/3

(3.118)

Transport Phenomena The last result is obtained directly from laminar boundary layer results; however, as noted in Section 2.5.4 it can be used as a first approximation to the Reynolds analogy for turbulent transport at Schmidt numbers different from 1. All of these analogies can be used for turbulent mass transport in tubes if u∞ is replaced by uavg . Equation (3.118) can also be written as jD =

CDx 2

(3.118a)

where jD ≡

kc Sc2/3 2u∞

(3.118b)

is the j factor for mass transfer. When this result is combined with Equation (2.320 a) with C Dx replacing f, we obtain jD =jH =

CDx 2

(3.118c)

This result is called the Chilton – Colburn analogy and connects the mass transfer, heat transfer, and momentum transfer processes in a given flow geometry. Although Equation (3.118 c) is obtained directly from analysis of transport on laminar boundary layers, it can be used as an approximation for other geometries and for turbulent flows as long as there is no form drag. If form drag is present, the first part of Equation (3.118 c), i.e., jD = jH , can still be used to relate mass transfer coefficients to heat transfer coefficients if 0.6 < Sc < 2500 and 0.6 < Pr < 100. Finally, analogous to Equation (2.325), 4/5

Shx = 0.0289 Rex

Sc1/3

(3.119)

for mass transfer in turbulent boundary layers. Combining this with the mass-transfer equivalent of Equation (2.252) for the laminar part of the boundary layer, i.e., 1/2

Shx = 0.332 Rex

Sc1/3

(3.120)

yields   4/5 −4/5 ShL = 0.0361 ReL Sc1/3 1 − 9170 ReL (3.121)

for the overall mass-transfer coefficient relation over a plate of length L where L ≥ x cr . Equation (3.121) is the mass-transfer equivalent of Equation (2.328).

349

3.3. Mass Transfer Across Interfaces In the situations described in the preceding sections, the rate of mass transfer within a single phase is calculated by using fundamental transport relations (mass balances and diffusion flux relations) and transport properties (diffusivities). Such calculations are possible for welldefined geometries and flow situations of sufficient simplicity that can be modeled and calculated a priori and without empiricism. This not always the case, however, because a great many more situations occur in which the flow of the fluid phases or a complex geometry prevents exact modeling or calculation. An example is forced convection transfer across an interface. In this case, the situation is described by using mass-transfer coefficients instead of diffusivities. These coefficients play a role similar to diffusivities in that they describe the transport rate of mass that occurs primarily because of molecular motion, but they also allow for other effects such as the enhancement of mass transfer because of forced or natural convection. Once these coefficients have been determined experimentally for a number of flow and mass-transfer situations, and correlated by dimensionless groups, the coefficients for analogous situations can be estimated and used for process design. The calculations are complicated, however, by the dependence of mass-transfer coefficients on mass flux. For example, in Section 3.1.6 the diffusion of species A through a stagnant film B was enhanced, above and beyond pure diffusion, by the net bulk flow of the mixture. Similarly, mass-transfer coefficients that apply to low mass-transfer rates are affected by the species mass fluxes and should be adjusted accordingly for the most accurate design calculations at high mass-transfer rates. Figure 27 shows how mass-transfer coefficients that are calculated for low mass-transfer rates by using diffusion coefficients or dimensionless group correlations can be adjusted for high mass-transfer rates. Then the coefficients for mass transfer through films on both sides of an interface can be combined (a process requiring thermodynamic equilibrium between the two phases) to give an overall coefficient for transfer between phases.

350

Transport Phenomena

Figure 27. Representation of the calculation of overall mass-transfer coefficients showing corrections of the single-phase coefficients for high mass-transfer rates

The commonly used definitions of masstransfer coefficients for each side of an interface and for the overall coefficients calculated from them are given in Section 3.3.1; their functional forms based upon theoretical considerations for low mass-transfer rates or equimolar counterdiffusion are presented in Section 3.3.2; and correlations for calculating mass-transfer coefficients for low mass-transfer rates and the adjustments that may be made for high masstransfer rates are considered in Sections 3.3.3 and 3.3.4, respectively. General references providing more detailed discussion of mass-transfer coefficient definitions and calculations include texts in the areas of transport phenomena [63], [74] and chemical engineering unit operations [75–79]. 3.3.1. Mass-Transfer Coefficients Mass transfer occurs because of an imbalance of concentrations, a departure from equilibrium. This imbalance provides a driving force for mass

transfer. Uniformity of composition is the equilibrium state in a single phase; if mole fractions are not uniform, then a nonequilibrium condition exists and diffusion occurs until uniformity is reached. Two-phase thermodynamic equilibrium is the equilibrium state across an interface; to the extent that the two phases on opposite sides of an interface are not in equilibrium, mass transfer tends to occur in such a way as to move the system toward equilibrium. The degree of departure from equilibrium directly affects the rate of mass transfer. In a single phase, the degree of departure from equilibrium is represented by the mole fraction (or mass fraction) gradient, and Fick’s first law of diffusion (the most commonly used flux relation) models the diffusion flux as proportional to this driving force. The proportionality factor defines the diffusivity. For mass transfer across interfaces or films, an analogous relationship is normally used to define mass-transfer coefficients. The mass-transfer flux of a species at an interface is modeled as proportional to the driving force (concentra-

Transport Phenomena tion difference) which exists for that transfer, through a thin film next to the interface. This situation is depicted in Figure 28. At the interface the two phases are normally assumed to be in thermodynamic equilibrium. Away from the interface, however, the bulk concentrations of the two phases are not necessarily at equilibrium with each other, and possible concentration or mole fraction profiles are shown as a function of distance from the interface. The majority of the concentration change is modeled to occur over a laminar film region near the interface. The actual concentrations and film depths are not known, however, which makes the definitions quite empirical and dependent on parameters such as fluid flow and turbulence. In Figure 28, concentration profiles are shown in both phases and, for simplicity, one phase is called a gas phase and the other a liquid phase, although this is not a limitation or constraint on the situation. The discussion could just as well be for two liquid phases or for a fluid and a solid phase. The model also normally assumes that concentrations at the interface are at steady state; flux to the interface through one phase equals that away from the interface through the other.

Figure 28. Hypothetical concentration profiles across the gas – liquid interface region with the transfer of A from the gas to the liquid yAb = mole fraction of A in bulk gas phase; yAi = gas-phase mole fraction of A at interface; x Ai = liquid-phase mole fraction of A at interface; x Ab = mole fraction of A in bulk liquid phase

Mass-transfer coefficients, then, are defined for each of the two phases. The definition of a liquid-phase mass-transfer coefficient (based on a liquid-phase mole fraction driving force) is Flux of A = kx (xAi − xAb )

(3.122)

351

Likewise, the defining relation for the gas-phase mass-transfer coefficient for species A based on the gas-phase mole fractions is Flux of A = ky ( yAb − yAi )

(3.123)

In each of these equations, a departure from equilibrium exists that represents the extent to which the interface mole fraction (x Ai or yAi ) differs from that in the bulk fluid (x Ab or yAb ) of the same phase. Whereas the above relations define masstransfer coefficients for a driving force within a single phase at an interface, interphase masstransfer coefficients are also defined according to concentration or mole fraction differences that exist across the two phases, where the average or bulk concentrations are used for each phase. In this case the mass-transfer coefficients Kx and Ky are defined according to the relations Flux of A = Kx (xAe − xAb )

(3.124)

Flux of A = Ky ( yAb − yAe )

(3.125)

and are called overall mass-transfer coefficients. They describe the flux in terms of mole fractions in the bulk phases. Here, instead of defining a driving force that exists within one phase or the other, a driving force that spans the two phases is defined. The mole fractions and driving forces are shown relative to a typical interfacial equilibrium curve in Figure 29. For a mass-transfer coefficient based on liquid-phase mole fractions, the driving force that is used is the difference between the actual mole fraction of A in the bulk liquid phase (x Ab ) and the mole fraction of A that would exist (x Ae ) if the liquid phase were in equilibrium with the mole fraction of A in the bulk gas phase. Likewise, in terms of gas-phase concentrations, mass transfer of A occurs to the extent that the bulk gas-phase mole fraction ( yAb ) differs from the value that would exist ( yAe ) if the gas phase were in equilibrium with the actual bulk liquid-phase mole fraction.

Next Page 352

Transport Phenomena and 1 1 m = + Ky ky kx

Figure 29. Relationships among interface, bulk, and equilibrium concentrations used in mass-transfer rate equations

The slopes of lines that represent the ratios of mass-transfer coefficients are also shown in Figure 29. If species A does not accumulate at the interface, the liquid- and gas-phase relationships for flux in terms of mass-transfer coefficients must be equal. Accordingly, kx (xAi − xAb ) = ky ( yAb − yAi )

(3.126)

which gives −

ky xAb − xAi = kx yAb − yAi

(3.127)

and the ratio of the interphase mass-transfer coefficients is the slope of a tieline connecting the point with composition coordinates equal to the liquid- and gas-phase bulk concentrations to a point with coordinates equal to the equilibrium interface liquid- and gas-phase concentrations. Similarly, a ratio can be obtained for the overall transfer coefficients: Kx yAb − yAe = Ky xAe − xAb

(3.128)

In the limit of small driving forces or for a linear isotherm this ratio is the slope m of a tangent to the equilibrium curve in the concentration region of interest. From the definition of the mass-transfer coefficients and for a locally linear isotherm (slope = m), 1 1 1 = + Kx kx m ky

(3.129)

(3.130)

Hence, the overall or combined resistance to mass transfer through the two phases (1/Kx or 1/Ky ) is equal to the sum of the resistances through each of the phases individually. Before summing, however, one of the individual phase coefficients must be scaled by using the (local) slope of the equilibrium curve in order to be consistent with the resistance offered by the other mass-transfer coefficient. Note that if kx /m  ky , then the gas-phase mass transfer is limiting and Ky ≈ ky . Dimensions of Mass-Transfer Coefficients. Because the flux of A is the number of moles of A per time per (cross-sectional) area, the masstransfer coefficients as defined by these relations must also have the dimensions of number of moles per time per area. Other definitions using different driving force concentration units are employed, however, and the dimensions of the mass-transfer coefficient vary accordingly. For example, number of moles per volume is frequently used for liquid-phase concentrations and partial pressure for gas-phase concentration. In these situations, mass-transfer coefficients may be defined according to Flux of A = kc (cAi − cAb )

(3.131)

Flux of A = kG ( pAb − pAi )

(3.132)

Flux of A = Kc (cAe − cAb )

(3.133)

Flux of A = KG ( pAb − pAe )

(3.134)

Here, k c and K c have the dimensions of volume per time per area (length per time), and k G and K G have the dimension of number of moles per time per area per unit pressure. 3.3.2. Functional Forms of Mass-Transfer Coefficient Relations In the preceding discussion, the mass flux through a film is assumed to be proportional

Previous Page Transport Phenomena to the driving force (the concentration difference) across a film, and this proportionality factor defines the mass-transfer coefficient. Combining this definition with theoretical models for mass transfer in terms of diffusion coefficients allows calculation of the dependence of mass-transfer coefficients on diffusivity. These are only estimates, however, because the situations are too complex for exact modeling, which is why mass-transfer coefficients were defined in the first place. A complete discussion of several models appears in [75]. Two-Film Theory. A particularly simple but useful model for steady-state transfer between two phases, which is frequently used to describe mass-transfer coefficients in terms of diffusion coefficients, is the two-film theory of Whitman [80]. The concentration profile is taken to be flat (independent of position) in the bulk fluid and then to vary linearly across a thin film of thickness δ up to the interface (Fig. 30). For steadystate mass transfer at low mass-transfer rates the results of this model are the same as for equimolar counterdiffusion (see Section 3.1.6); the molar flux is directly proportional to the diffusivity, and the mass-transfer coefficient, in terms of film thickness and diffusivity, is kx =

c DAb δ

(3.135)

Hence, this film theory predicts a mass-transfer coefficient that is directly proportional to diffusivity through the film.

353

and its adaptations [82], [83]. The penetration theory is applicable to a species that is being transferred from a gas to a liquid phase; it gives an estimate of the mass-transfer coefficient in the liquid phase based upon a nonsteady-state situation. The liquid phase is assumed to be a falling film and the velocity profile of the falling film at the gas – liquid interface is essentially flat, i.e., the velocity of the falling liquid is nearly independent of penetration depth from the gas into the liquid as long as the penetration depth is low. This is true if the time that the gas is in contact with the liquid is short. In this situation, any one packet of fluid is in contact with the gas for a fixed amount of exposure time; therefore, the diffusion of a species from the gas to the liquid – from the viewpoint of this packet of fluid – is a nonsteady-state process as opposed to the steady-state diffusion of the film theory. This different type of model gives rise to a mass-transfer coefficient that is proportional to the square root of diffusivity: ! kx = 2 c

DAB π texp

(3.136)

Here, t exp is the exposure time of a given packet or element of fluid to the gas stream and kx is an average value over this time. Exposure time can be calculated in terms of the fluid velocity at the gas film interface and the distance of contact of the gas and liquid L: texp =

L uinterface

(3.137)

A third theory, actually a combination of both, is the film-penetration theory [84]. For some flow situations, the film theory result is recovered, whereas for others, the penetration theory’s D1/2 dependence holds, suggesting that different dependencies might be expected in different flow situations.

Figure 30. Simplified concentration profiles for the twofilm theory

Other Theories. Other theories relating the coefficient to diffusion may also be appropriate. Examples are the penetration theory [81]

3.3.3. Relations for Low Mass-Transfer Rates By using a somewhat different approach, dimensionless group correlations can be postulated to exist for mass-transfer coefficients. For example, the Sherwood number (ShAB = kx d/c DAB )

354

Transport Phenomena

can be expressed in terms of a function of the Reynolds number (Re = d u /η), Schmidt number (Sc = η/ DAB ), and L/d [63]: ShAB = f (Re, Sc, L /d)

(3.138)

The specific functional form depends on the actual flow situation. Furthermore, as discussed in Section 3.2, analogous correlations exist for heat, mass, and momentum transfer because of similarities in the conservation equations that govern all these situations and in the corresponding flux relations. Consequently, correlations exist for mass-transfer coefficients that are very similar to those for heat or momentum transfer (friction loss) in analogous situations of geometry and boundary conditions. Results for some specific flow and mass-transfer situations are summarized in Table 8. Note that in these relations, the dependence of the mass-transfer coefficient on diffusivity ranges from about D0.6 AB to DAB , corresponding to the D0.5 to D AB predicted by the AB penetration and film theories, respectively. 3.3.4. Relations for High Mass-Transfer Rates In Section 3.1.6, two important limiting cases of binary diffusion are considered for a onedimensional diffusion process. In the case of equimolar counterdiffusion the flux of each species is the result of diffusion only and is calculated by using Fick’s first law of diffusion; for each species the flux is directly proportional to the mole fraction gradient of that species. For the case of one species diffusing through a stagnant film of the other, however, the diffusion of each species is the result of the combination of Fick’s law of diffusion and a bulk flow that is induced by the diffusion process. For the species that was defined to be stagnant, the diffusion flux exactly balances its flux due to the mixture bulk flow. For the other species, however, the total flux is that due to Fick’s law of diffusion enhanced by the bulk flow, which results in the Fick’s law flux being divided by (1 − x A ). In the latter case, if the nonstagnant species is present at low molar concentration (infinite dilution or approaching infinite dilution), then obviously the same result for the flux of this

species is obtained as for the equimolar counterdiffusion situation but for a different reason. The other species is still stagnant, and the bulk flow is not reduced by the low concentration but rather by transport of the nonstagnant species that occurs because of reduced bulk flow. These two limiting cases (ECD and ATSB) provide a basis for understanding the corrections to mass-transfer coefficients that are made for high mass-transfer rates. The mass-transfer coefficients described in the correlations of Table 8 are defined for equimolar counterdiffusion-like situations; that is, they apply to true equimolar counterdiffusion situations or to low transfer rate calculations. Evidently, then, if they are to be used for ATSB situations, they must be corrected in a way similar to the comparison between equimolar counterdiffusion and ATSB calculations; the presence of a stagnant film and high mass-transfer rates provides an induced bulk flow that enhances the mass-transfer coefficient. If the low mass-transfer coefficient is used in a high mass-transfer stagnant film situation, the mass-transfer rates may be significantly underestimated. A common correction to the low mass-transfer rate – ECD coefficients is to introduce a drift factor (1 − yA )L analogous to the adjustment of ECD to ATSB. A common, although not unique, notation is to distinguish between these two mass-transfer coefficients with a prime (
); however, the notation may be switched with k
indicating ECD rather than ATSB as is done here: ky =

ky

(3.139)

(1 − yA )L

where (1 − yAi )L is the nonlinear, log-mean average of (1 − yAi ) and (1 − yAb ): (1 − yA )L =

(1 − yAi ) − (1 − yAb ) ln

(1 − yAi ) (1 − yA )

(3.140)

This correction is commonly used for ATSBlike situations and appears in design calculations such as packed gas-absorption towers where one species in the gas phase is absorbed and the others are not. Instead of a simple ATSB correction to the mass-transfer coefficient, however, less restrictive theories for flux ratios may be used to make the adjustment. These are (1) the film theory, (2) the penetration theory, and (3) the boundary

Transport Phenomena Table 8. Mass-transfer coefficients a

355

356

Transport Phenomena

layer theory. Each of these theories provides a means of calculating a correction factor to the mass-transfer coefficient. The situation of ATSB diffusion is actually a specific limiting case of the more complete film theory summarized by Bird et al. [63]. They use the film theory to adjust EDC mass-transfer coefficients for the enhancements which result from diffusion-induced bulk flow. The model allows for relative fluxes of the two species (A and B) that differ from the stagnant B situation. For a given flux ratio, adjustments to the low mass-transfer rate coefficient can be calculated (or, equivalently, the ECD mass-transfer coefficient), which would be calculated or estimated from the correlations of the previous section. Again, ATSB situation falls within this less restrictive film theory. The penetration theory of Higbie [81] has also been used to extend the low mass-transfer rate calculations to high mass-transfer rates due to diffusion-induced enhancement. The results obtained are presented in a similar way to those for the film theory, and correction factors are given for a boundary layer theory approach [63]. Qualitatively, the corrections calculated by these three theories are very similar although the quantitative results are somewhat different. The penetration and boundary layer theories predict an even larger effect of mass transfer on the coefficients than does the film theory.

4. Macroscopic Systems The development of conservation and accounting equations for macroscopic systems parallels the development of those for differential systems that led to the partial differential equations of Table 1. Each law or statement is written for a single extensive property (a property that can be counted, for which the whole is equal to the sum of the parts) for a specified region (system) inside a closed-surface boundary and for a specified time period (either finite or differential) (Fig. 31). In the case of macroscopic (finitesize) systems, either process rate equations are obtained, which are ordinary differential equations, or for a finite time period, an algebraic equation is obtained. For conserved properties, the generation (or consumption) is always zero.

The conservation and accounting equations and their development are summarized below for compositionally homogeneous systems. The purpose is to provide an overall picture relating conservation laws, accounting equations, U, H, A, G, and second law equations of thermodynamics, which provide the basics for quantitatively understanding transport phenomena (and other areas of engineering as well) on a macroscopic scale, rather than to give handbook-type details and examples. General references offering further discussion of the development and application of macroscopic equations include texts in the areas of transport phenomena [98– 100], thermodynamics [101], [102], chemical engineering unit operations [103], and fluid mechanics [104].

4.1. Conservation of Total Mass Total mass is conserved in the absence of nuclear conversions or relativistic effects. This means that for the system and its surroundings together the total amount of mass is a constant. Equivalently, if the amount of mass contained within the system changes, then the surroundings must change to an equal but opposite extent; there is a one-for-one exchange of mass between the system and its surroundings as mass moves across the system boundary. Allowing for multiple flow streams into and out of the system, an equation for the conservation of total mass contained within the system is 

( uavg A)in −

entering streams



( uavg A)out

exiting streams

=

d ( V )system dt

(4.1)

At each area A where mass exchanges occur between the system and surroundings, the fluid density and velocity components normal to the boundary surface u are appropriately averaged values. Likewise, the density of the system is averaged over the system volume V system [98], [100], [105]. In addition to a conservation equation for total mass, accounting statements for individual species within the system can be written. If the system contains n species that together comprise

Transport Phenomena

357

Figure 31. The concept of accounting for an extensive property – the cornerstone of transport phenomena Input and output represent an exact exchange with the surroundings; a conserved property has no generation or consumption term.

the total mass (and could be molecular, ionic, or other species), then n accounting equations exist, one for each species. Because species are not necessarily conserved (except for the chemical elements), these accounting equations must allow for any generation or consumption of individual species through chemical reactions. Accordingly, for each of the n species within the system, 

4.2. Conservation of Linear Momentum

(i uavg,i A)in −

entering streams



(i uavg,i A)out + ri Vsys

exiting streams

=

d (i V )system dt

cause total mass is conserved, the sum of all of the generation and consumption terms in the n species equations must be zero; no net generation or consumption of mass occurs). Note that the total mass of each species entering and leaving the system is obtained by summing over all entering and leaving streams.

(4.2)

The densities and velocities are average values, as discussed above, and ri is an average generation rate of species i per unit volume of the system. An equivalent accounting can be done in molar concentration units (see, e.g., Section 4.9.3). Of these n + 1 equations, however, only n are independent inasmuch as the total mass must be the sum of the individual species mass (be-

Linear momentum is conserved. Given a system, the amount of linear momentum contained therein may change only as a result of exchange of momentum between the system and its surroundings; linear momentum is not generated or consumed. Linear momentum may be exchanged as the result of mass entering or leaving the system and as the result of forces acting on the system by its surroundings. The rate of momentum exchange as mass enters or leaves the system is the momentum per unit mass u multiplied by the rate at which that mass enters or leaves the system. External forces act upon the system either by contact with the system’s surfaces or as body forces, such as gravity, that

358

Transport Phenomena

act individually on each piece of mass within the system. The momentum of the system then changes according to 



u2

avg

uavg  2  u avg uavg +



−

 uavg A in

 uavg A

F external

out

dpsystem = dt

(4.3)

The (u2 )avg term arises because the momentum flow rate, calculated as the momentum per unit mass u times the mass flow rate  u dA, is integrated (averaged) over the cross section [98, Chap. 7], [99, Chap. 7]. If the velocity profile across the area is not flat, then (u2 )avg = (uavg )2 . If αmom ≡ (uavg )2 /(u2 )avg , where αmom is a measure of the extent to which the velocity profile is not flat, this equation becomes

   (uavg )2 A αmom +



in

F external

−



  (uavg )2 A αmom

dpsystem = dt

where Lsystem is the total angular momentum of the system. Again, the entering and leaving momenta are actually appropriate integral averages across the flow cross section, and Lsystem is integrated over the entire volume. The conservation of angular momentum is not considered further here. It is useful, for example, for the analysis of torques on pumps and compressors for design purposes [100].

out

(4.4)

For fully developed laminar flow of a Newtonian fluid, αmom is 3/4; for turbulent flow, it is nearly unity [106, p. 252]. Also psystem is integrated over the entire macroscopic system.

4.3. Conservation of Angular Momentum Angular momentum is also a conserved property. The angular momentum of a system can change only by exchange with its surroundings. A piece of mass possesses angular momentum that is equal to the cross product of the position vector of that piece of mass and its momentum. Consequently, angular momentum enters and leaves with mass. Also, angular momentum enters the system as a result of external forces and is equal to the cross product of the position vector to the point of application of the force with that force: r in ×uin ( uavg A)in − r out ×uout ( uavg A)out  dLsystem + (r × F ext ) = (4.5) dt

4.4. An Accounting of Mechanical Energy Mechanical energy accounting can be made for finite-sized systems and can be obtained by integrating the continuum conservation law of linear momentum. It cannot be obtained from a macroscopic linear momentum law and consequently is an independent equation providing additional insight into flow situations. In this sense, the set of macroscopic equations differs from the continuum equations. The mechanical energy equation is an accounting equation, as opposed to a conservation equation, because mechanical energy is not conserved. Conversion of mechanical energy to thermal or other forms of energy may occur so that the amount of mechanical energy in an isolated system is not necessarily constant. Nevertheless, such an accounting is extremely useful for certain problems, e.g., the design and sizing of piping systems and flow networks. Derivation of the mechanical energy accounting equation is complex and lengthy [107]. The result is stated here for steady-state flow situations with one entrance and one exit. Mechanical energy is accounted for by the following steadystate equation (expressed as energy per unit mass of flow; the symbolˆover a quantity implies “per unit mass”): 1 g (z1 − z2 ) + 2



(u2,avg )2 (u1,avg )2 − αKE,1 αKE,2

−

2




dp ˆ es − Fˆ = 0 ˆ s+ W +W 

(4.6)

1

Here αKE ≡ (uavg )3 /(u3 )avg is another measure of the flatness of the velocity profile, this time with respect to kinetic energy. For fully developed laminar flow of a Newtonian fluid, αKE

Transport Phenomena is 1/2; for turbulent flow it is nearly unity [106, p. 252]. This result holds regardless of heat transfer or other thermal effects that may occur. Equation (4.6) is referred to as the extended Bernoulli equation (→ Fluid Mechanics, Chap. 3.5.) and through Fˆ allows for the irreversible or dissipative conversion of mechanical energy to thermal energy in a manner analogous to that of Equation (1.80), the mechanical energy equation for a continuum. These losses are of the type encountered in flow through pipes where friction losses result in pressure drop in a straight run of horizontal pipe. The potential energy gz and kinetic energy u2 /2 αKE terms are evaluated (as averages over the cross section) at the entrance (1) and exit (2) points of the sysˆ s and the flow work due tem. The shaft work W ˆ es act across the systo the fluid extra stresses W tem boundary, whereas the friction losses occur both within the system (from viscous stresses inside the pipe) and across the system boundary (from viscous stresses at the pipe wall). The pressure term dp/ must be integrated along a path through the process from the entrance to the exit, not necessarily an easy task for a compressible flow. If the flow is incompressible, then the extended Bernoulli equation becomes g (z1 − z2 ) + +

1 2



(u1,avg )2 (u2,avg )2 − αKE,1 αKE,2



(p1 − p2 ) ˆ ˆ es − Fˆ = 0 + Ws + W 

(4.7)

for which the pressure terms also use entrance and exit values only. Mechanical energy in the form of (1) upstream kinetic energy, (2) potential energy, (3) isotropic and extra stress flow work, and (4) shaft work is either redistributed downstream among these forms of mechanical energy, converted to work, or lost as mechanical energy by conversion to thermal energy.

4.5. Conservation of Total Energy Total energy is conserved. Mass possesses energy in the form of kinetic, potential, and internal energy. When mass enters a system, it brings with it energy in these forms, and the rate at which it does so is calculated as the product of the specific energy in each of these forms (i.e., the energy per unit mass) and the rate of mass flow into the system.

359

Heat Q and work W exist at the system boundary but are not energy possessed by mass. Heat enters the system by either conduction or radiation because of a temperature difference across the system boundary. Work is energy transferred when a force acts through a distance upon elements of the system at the boundary. More properly, work may be expressed as
s2 F ·ds

W =

(4.8)

s1

The dot product of the force F with the displacement ds provides that the component of force in the direction of displacement produces work, whereas that which is normal to the displacement does not. The convention here is that both heat and work are positive when energy is added to the system. With energy existing within the system as kinetic, potential, and internal energy, the conservation of total energy for a macroscopic system is expressed in rate forms as   u2 + g z ( uavg A) 2 αKE in   2   u ˆ+ − U + g z ( uavg A) 2 αKE out .  / d ˆ + KE+ < P < ˙ + Q˙ = U E m +W sys dt  

ˆ+ U

The dots over W and Q imply work rate and heattransfer rate, respectively, rather than a derivative rate of change of work or heat. Again, properties of the mass entering and leaving the system are averaged over the flow cross section. Usually, the form of the equation is changed by separating the work required to move mass into and out of the system against the pressure (isotropic stress) at the boundary (expressed per unit mass p/) from all other work of the process, ˙ np . (The term p/ is the “nonpressure” work W commonly referred to as “flow work”. However, because it represents only the flow work due to isotropic stress and does not include that due to extra stress, it is referred here to as pressure work and all other forms as “nonpressure” work.) The statement is rewritten by combining the p/ terms with the other inlet and outlet flow stream terms and replacing the total work term ˙ np . The p/ terms are combined with the with W

360

Transport Phenomena

internal energy to give the enthalpy of the flow streams (Hˆ ≡ Uˆ + p/) and the resulting equation is  

  u2 + g z ( uavg A) 2 αKE in     u2 ˆ − H+ + g z ( uavg A) 2 αKE out .  / d ˙ ˙ ˆ < < + Wnp + Q = (4.10) U + KE+ P E m sys dt ˆ H+

The “nonpressure” work includes work associated with volume changes of the system acting against external forces (a nonsteady-state term), all shaft work (compressors, pumps, turbines, electric motors or generators, etc.), other forms of energy (electrical, magnetic, etc.), plus – if important – any flow work due to extra (nonisotropic) stresses at the system boundaries. The last term is frequently neglected.

4.6. An Accounting of Thermal Energy By subtracting the steady-state mechanical energy equation for one inlet (point 1) and one outlet ( point 2) flow stream (Eq. 4.6) from the corresponding result for total energy conservation (Eq. 4.10), an accounting of thermal energy per unit mass is obtained which, although not independent from the total and mechanical energy equations, can be a useful way of viewing thermal effects:
p2 ˆ1 = ˆ2 − H H p1

dp ˆ + Fˆ +Q 

(4.11)

This result is true regardless of any changes in elevation, pressure, density, or flow velocity or the amount of shaft work, as long as a steady state exists through a single conduit. Furthermore, for a reversible situation (Fˆ = 0), this result is readily identified as the thermodynamic relation dHˆ = T dSˆ + dp/, which is integrated to follow an element of the fluid through the process along the flow path in a sequence of quasiequilibrium steps. For irreversible processes this thermodynamic relation still holds, but Qˆ and Fˆ ˆ both together correspond to the integral of T dS; heat transfer to the system and irreversibilities result in an increase in entropy (see Section 4.7). These thermodynamic relations for dUˆ (and, ˆ and dG) ˆ dA, ˆ may be viewed equivalently, for dH,

as thermal energy accounting equations, expressed in terms of thermodynamic state functions rather than the path functions Qˆ and Fˆ of Equation (4.11). On the one hand, they represent relations between thermodynamic state variables regardless of the path or process used to bring about change; on the other hand, they represent thermal energy conversion for a specific process. Both statements have intrinsic value in their own right. Given a change in enthalpy that occurs in a process and is calculated from such a thermal energy equation, the thermodynamic relations can be used to express this change in terms of changes in observable properties such as heat capacity, temperature, pressure, and volume. Hougen et al. [101, p. 508] recognize this separation of mechanical and thermal energy, and say that these state functions of a body do not depend upon “its external position or motion relative to other bodies.” This result is considered further in Section 4.9.1. The approach outlined in Sections 4.2, 4.3, 4.4, 4.5, and 4.6 for deriving the macroscopic momentum, mechanical energy, total energy, and thermal energy equations is not the most common. More frequently, the extended Bernoulli equation is hypothesized by considering total energy for adiabatic, incompressible situations or by applying the thermodynamic realtions for dUˆ or dHˆ (e.g., in the form of Eq. 4.11) to total energy (Eq. 4.10) to obtain Equation (4.6) or (4.7). The approach given here is preferred because it is more direct and parallels the traditional approach used in continua (Chap. 1). This preference has also been expressed by Whitaker [105].

4.7. The Second Law of Thermodynamics The second law of thermodynamics places constraints on processes that can occur in macroscopic systems. As discussed for continua, the thermodynamic property of entropy quantifies the fact that certain processes are observed to occur in only one direction. In the context of a macroscopic law, the entropy contained within a macroscopic system changes through exchange

Transport Phenomena with the surroundings and also through generation within the system as a result of irreversibilities. On the one hand, accounting for entropy is similar to accounting for other nonconserved properties because the entropy of the system may change as a result of generation within the system, as well as exchange with the surroundings. Unlike other accounting statements (e.g., mechanical energy, electrical energy, individual chemical species), however, entropy cannot be consumed within the system; it can only be generated. Also, unlike other accounting relations, an independent relation exists for the surroundings; the change in entropy of the surroundings due to this process arises because of exchange with the system and also as a result of generation of entropy within the surroundings due to the process. Again, this generation of entropy must be positive; it can never be negative. Consequently, when the two entropy accounting expressions for the system and surroundings are summed, the total entropy change of the system plus surroundings is necessarily positive (or, in the limit of a reversible process, equal to zero). These statements are embodied in an accounting of entropy for a macroscopic system / /  .   .  Sˆ  uavg A − Sˆ  uavg A in out      Q˙ + + S˙ gen system T system .  / d = (4.12) Sˆ m system dt

and its surroundings /  .  Sˆ  uavg A

out

−

/  .  Sˆ  uavg A

   Q˙ + + S˙ gen surr T surr d . ˆ / = S m surr dt 



in

(4.13)

and for the system and surroundings combined     d ˆ  ≥0 + Sˆ m Sm system surr dt

(4.14)

In these equations, the exchange of entropy between the system and surroundings is the result of mass ( possessing entropy per unit mass) transferring between the system and surroundings (values of Sˆ are averaged over the flow cross section).

361

Heat transfer is also a one-for-one-exchange between the system and its surroundings, provided the temperatures of the system and the surroundings at the boundary (T system, bound and T surr, bound , respectively) are the same, this is normally the case (however, see the next paragraph concerning entropy generation). In this case, the rate of entropy added to the system (S˙ in, system ) due to a rate of heat transfer from the surroundings to the system Q˙ is S˙ in, system =

Q˙ Tsystem, bound

(4.15)

and the rate of entropy transfer to the surroundings (S˙ in, surr ) due to the same heat transfer is S˙ in, surr = −

Q˙ Tsurr, bound

(4.16)

If the temperatures of the system and surroundings at the boundary are the same, then these two terms cancel exactly and a true exchange of entropy occurs between the system and surroundings. The generation of entropy in the system and surroundings is frequently difficult or impossible to quantify (although not always) but conceptually is the result of finite driving forces for mass, energy, or momentum transfer. Consequently, if temperature, mole fraction, or velocity gradients within the system or surroundings are dissipated as a result of heat, mass, or momentum transfer within the system, then these dissipation processes cause a movement toward a homogeneous condition which necessarily results in an increase in entropy. Likewise, if heat or momentum transfer at the system boundary results from finite differences in temperature or forces, then an entropy increase is associated with the heat or momentum transfer due to subsequent temperature or momentum equilibriation within the system and surroundings. For the system and surroundings of homogeneous but different temperatures (T system and T surr ), the net total increase in entropy of the universe resulting from the two-step heat transfer and subsequent equilibration process is calculated according to Q Q ≥0 − Tsystem Tsurr

(4.17)

as though a single-step process of transfer to the system (whose boundary temperature

362

Transport Phenomena

is T system ) and from the surroundings (with boundary temperature T surr ) has occurred. Again, because entropy is not conserved, an accounting of entropy for either the system or the surroundings alone is not an adequate statement of the second law of thermodynamics. Furthermore, the entropy of a material is a function of state and consequently can be calculated (with respect to an arbitrarily defined reference value) in terms of heat capacities, equations of state, and energy changes associated with phase changes. Consequently, the entropy exchange associated with mass crossing the system boundary is well defined, and the change of entropy of the system, given its composition and change in state, is also well defined. As mentioned previously, however, the generation of entropy as a result of irreversibilities is not always as well defined. In some cases, such as the diffusive mixing of two ideal gases at equal pressure and temperature, the entropy increase can be calculated from theoretical considerations. In other cases, however, such as the rapid expansion of a gas against an external force, the entropy increase due to irreversibilities cannot be calculated because the expansion is associated with a turbulent velocity profile that is unpredictable and, because of velocity gradients, ultimately results in an increase in entropy.

4.8. Tabular Summary of the Macroscopic Equations The macroscopic equations given in Sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, and 4.7 are summarized in Table 9. Also given are the number of independent scalar equations that are represented in each case and any limitations on the result.

4.9. Some Examples 4.9.1. Pressure Drops and Temperature Changes for Closed Channel Flow Perhaps the most basic problem of macroscopic fluid mechanics is to calculate the pressure drop or pump requirement (or turbine output) associated with fluid flow through a pipe, tube, or other closed conduit. The basic design problem specifies a given amount of material that is to

flow from one location to another at a given design flow rate. This requires either sufficient upstream kinetic, potential, and pressure energy or the addition of energy through a pump. Calculation of the required upstream energy or pump work is done by accounting for mechanical energy through the extended Bernoulli equation (Eq. 4.6 for a compressible fluid and Eq. 4.7 for an incompressible fluid). For example, to move water to an elevated storage tank, a pump must be supplied to overcome the change in potential energy and any frictional losses of mechanical energy due to the flow of water through the pipe. Kinetic energy changes may also be a factor, although these are usually relatively small. One of the primary factors in such flow calculations is determining this mechanical energy loss term. This is not a calculation that can be made solely on theoretical grounds. Instead, it is handled through a dimensionless group correlation for a friction factor f from which the mechanical energy losses for flow at average velocity uavg through a straight section of horizontal pipe of length L and diameter D are calculated according to  Fˆ = 4 f

L D



u2avg 2

 (4.18)

The dimensionless group correlation typically appears in graphic form (the original chart of Moody [108] or its modifications) or in analytical form, convenient for computer use (given by Churchill [109]). [Alternative definitions for the friction factor in the literature differ from each other by a factor of 2 or 4. Care must be taken that the dimensionless group correlation for f versus Re and the friction factor definition (e.g., Eq. 4.18) employed are consistent with each other.] The friction factor depends on the Reynolds number of the flow through the pipe and, for turbulent flow, the roughness of the wall. Additional losses occur in valves, fittings, and pipe bends; relations for calculating these losses also are found in standard texts on fluid flow. The effect of mechanical energy losses on the temperature of the fluid is normally quite small for liquids, although in some instances it may have to be taken into account. The thermal energy equation provides a basis for calculating this effect as discussed in Section 4.6. For

Transport Phenomena

363

Table 9. Summary of the macroscopic equations Equation name

Equation number

Number of independent macroscopic equations

Total mass Species mass (n species) Linear momentum Angular momentum Mechanical energy (compressible) Mechanical energy (incompressible) Total energy Thermal energy Entropy, system Entropy, surroundings Total entropy (second law of thermodynamics)

(4.1) (4.2) (4.4) (4.5) (4.6) (4.7) (4.10) (4.11) (4.12) (4.13) (4.14)

1 n−1 3∗ 3∗ 1∗∗ 1∗∗ 1 0 1 1 0

Comments

steady state, one flow stream steady state, one flow stream steady state, one flow stream

an inequality

∗ One equation for each spatial dimension, e.g., for a two-dimensional situation, only two nontrivial equations exist. ∗∗ Only one of these two equations can be used in any one situation.

a compressible, single-phase fluid the change in enthalpy is written in terms of heat capacity and thermodynamic p –  – T properties by using standard thermodynamic arguments:
T2 ˆ1 = ˆ2 − H H


p2 (1 − T β)

cp (∂T )p1 + p1

T1

(∂p)T2  (4.19)

where the temperature integral is at constant pressure, the pressure integral is at constant temperature, and β is the isobaric coefficient of thermal expansion [ β ≡ − 1/ (∂/∂T )p ]. Therefore, by combining Equations (4.11) and (4.19), in the absence of heat transfer and for a temperature-independent heat capacity cp , the net temperature change is obtained:  
p2
p2 (∂p)T dp 1 ˆ  T2 − T1 = (1 − T β) F+ − cp   p1

p1

(4.20)

where Fˆ is given by Equation (4.18). If the fluid is truly incompressible, then β = 0, the two pressure integrals cancel exactly and the temperature increases in direct proportion to frictional losses. For compressible fluids, however, this heating is counteracted by the expansion of the fluid; if p2 < p1 , then
p2 β T / (∂p)T < 0 p1

which, if large enough, can result in net cooling.

As an example, for the flow of oil through the Trans-Alaska pipeline between pumping stations, the appropriate parameters for this problem are D = 1.19 m, L = 107 km, uavg A = 3.18×105 m3 /d, β = 8.17×10−4 K−1 , f = 0.00335, and cp = 1.8 kJ kg−1 K−1 [110], and the temperature rise is approximately 1 T2 − T1 ≈ cp





4f

L D



u2avg



2

 (1 − β T1 )

= 2.7 K

(4.21)

For comparison, neglecting the fluid compressibility results in a temperature increase of 3.7 K. Mechanical energy conversion to thermal energy in pumps, when present, also contributes to temperature increases. Although the temperature increase can be fairly small for flow of a fluid over a short distance, it can accumulate and become appreciable if distances are long enough as in a transcontinental pipeline or a continuously circulating flow loop. Temperature changes accompanying flow can also contribute to errors in viscosity measurement. 4.9.2. Compressible Flow in a Tube If the flow through a closed channel or tube is compressible, then accurate design and flow calculations can be considerably more complicated. The compressible form of the extended Bernoulli equation must be used (again this is

364

Transport Phenomena

for steady-state flow situations), and the friction factor is calculated in the same way as for incompressible flow by using the friction factor correlations. However, the pressure term is shown as an integral from the entrance of the flow channel to the exit. The difficulty of this calculation is that the density must be known as a function of pressure through the pipe and along the actual flow path of the fluid through the pipe. Knowing the conditions at the entrance and exit is not enough; conditions must be known from one end to the other, along the path taken by the fluid. Normally, these are unknown because of the flow complexity. To handle this situation, idealized approximations to the flow are calculated as limiting or extreme cases. For example, if the flow of an ideal gas ( p1 M r = RT 1 1 , where R is the ideal gas constant and M r the molecular mass) is adiabatic and reversible (no heat transfer occurs to or from the gas, and mechanical energy losses are negligible – isentropic flow), the pressure and density are related by p p1 = const. = κ κ 1

(4.22)

where κ is the ratio of the heat capacities, cp /cv , so that
p2 − p1

  (κ−1)/κ

dp RT1 κ p2 = 1−  (κ − 1) p1

(4.23)

A second situation is isothermal flow of an ideal gas for which p RT = = constant  Mr

(4.24)

and
p2 − p1

dp RT p1 = ln  Mr p2

that obtained for the same pressure difference with isothermal conditions.” Furthermore, they point out that the difference becomes less for long pipes and that “for pipes of length at least one-thousand diameters the difference does not exceed about 5 %.” Thus, isothermal flow and adiabatic reversible flow are commonly assumed to bracket the actual observed situation, and for long pipelines, little difference exists between these two extremes. One other aspect of compressible flow should be pointed out. For large pressure drops and high flow rates, a limit to the flow rate may be obtained even though the pressure drop is allowed to increase by lowering the downstream pressure. This limit is established by the sonic velocity of the gas in the pipe. If the gas reaches sonic velocity, any further reduction of the downstream pressure at constant pipe cross-sectional area results in no additional increase in velocity. The sonic velocity of the gas at the downstream end, based on thermodynamic principles and in terms of the density , pressure p, and specific ˆ is entropy S,

(4.25)

For the flow of a nonideal gas, the equation of state for that gas can be used instead of the ideal gas equation of state to obtain corresponding integrals. For adiabatic, nonreversible flow, Coulson and Richardson [111] give an approach for calculating the pressure integral and conclude that “the rate of flow of gas under adiabatic conditions is never more than 20 % greater than

 usonic =

∂p ∂

 1/2 (4.26) ˆ S

4.9.3. Heterogeneous Reaction in a Fluidized-Bed Reactor Consider now the reaction of component A to product B carried out in a well-mixed fluidizedbed reactor (Fig. 32). Such a conversion might represent the catalytic cracking of heavy hydrocarbons in a refinery process. The reactor stream is fed to the fluidized bed as a gas with sufficient velocity to maintain the catalyst particles in a fluidized state. Once inside the reactor, reacting species A is converted to product B at the catalyst surface. However, before this reaction can occur, the reactant must first migrate to the surface of the catalyst pellet. Then, after reaction at the surface, the product B must migrate to the bulk fluid to be removed from the reactor in the exit flow stream. What complicates this situation is that the catalyst itself may be porous, so that not only does the reactant contact the exterior surface of the catalyst, but part of the reactant species also diffuses into the micropores

Transport Phenomena before reacting. Consequently, some reactant is converted at a reduced concentration inside the pores and, therefore, at a lower rate than if it were converted at the concentration existing at the macroscopic exterior surface of the catalyst. Calculation of the effective reaction rate requires considering each of these mass-transfer and reaction rate processes. To illustrate the concept, the macroscopic material balance of reacting species A is considered. Under the assumption of a well-mixed fluidized bed, the gas-phase concentration is the same everywhere in the reactor and equal to the concentration in the exit flow stream. Of course, this model is an approximation because concentration gradients exist near the catalyst pellets due to the heterogeneous reaction. The model assumes that enough reaction locations of sufficiently small size exist for them to be treated like a homogeneous reaction occurring uniformly throughout the system volume. An accounting of species A within the fluidized bed in terms of its concentration in the feed stream cAin , the concentration in the bulk fluid in the bed and outlet flow stream cA , and the feed and product volumetric flow rates (F in and F out , respectively) is cAin Fin − cA Fout = rA V

(4.27)

The amount of A that enters the reactor at concentration cAin either leaves at concentration cA or is converted to products at an effective reaction rate r A per unit volume. Although complicated by r A , this result provides a clear approach to reactor design. Once an expression for this reaction rate is obtained based on transport and kinetic factors, the problem can be completed to obtain the conversion in the reactor. Then economic and design calculations can be made for the reactor in terms of fractional conversion, space time, and reactor volume and flow rates [112], [113]. The key to reactor design is to determine the overall reaction rate for a given situation. The effect of transport on this effective rate is as follows. The transport of species from the bulk of the fluid in the fluidized bed to the catalyst surface can be estimated by using mass-transfer coefficient correlations for fluidized beds. A j factor correlation has been presented by Chu et al. [114] that is applicable to gas – solid and liquid – solid systems and for both fluidized-bed

365

and fixed-bed reactors; this is presented in Table 8. Graphs of these correlations are given in [113], [114]. For porous catalysts, determination of the reaction rate at the catalyst, although primarily a problem in reaction kinetics, can also be a problem in transport phenomena because of diffusion in the catalyst pores. The chemical kinetics problem must be determined from experimental laboratory reactors. Lin provides extensive literature references to studies of a range of industrial reactions [113]. The transport problem is normally addressed by the use of an effectiveness factor η. The effectiveness factor adjusts the reaction rate to account for the fact that within the pores of a catalyst the reaction occurs at a reduced rate compared to the pellet’s exterior surface. At the catalyst surface, the reaction proceeds at a rate that is dictated by the chemical kinetics and by the concentration of the reacting species at the surface. However, as the reactant becomes depleted with greater depth into the pores, the reaction rate decreases accordingly; the effectiveness of the catalyst surface within the pores is lower than that at the surface. Calculation of an average (averaged over the entire catalyst surface, whether deep inside pores or not) effectiveness factor is based on combined diffusion and reaction inside the catalyst pores and was addressed originally by Thiele [115] and Aris [116] and subsequently in many references on reaction and transport [98], [112], [117]. Such analysis leads to calculation of an average effectiveness factor for a catalyst pellet as a function of a dimensionless variable called the Thiele modulus. This parameter depends on the relative rates of the kinetic reaction to diffusion. If the diffusion rate into the pores is very high compared to the reaction constant, then the fact that some of the catalyst surface is available only in pores makes little difference and the effectiveness factor is close to unity. However, if the kinetic reaction rate constant is large compared to the diffusion rate, then the concentrations, and hence reaction rates, within the pores are reduced considerably, giving a substantially lower effectiveness factor. By using the effectiveness factor η to adjust for the reaction rate at the catalyst, the reaction rate ( per unit volume of catalyst particle) for a reaction that is first-order irreversible in A with rate constant k is given by

366

Transport Phenomena

Figure 32. Modeling of the fluidized-bed reactor: macro-scale for the reactor accounting of mass (A), mesoscale for the mass-transfer rate (B), and microscale for definition and calculation of the effectiveness factor (C) A) Macroscale: fluidized-bed region is modeled as well mixed and uniform in temperature, composition, and reaction rate; B) Mesoscale: mass transfer occurs between the bulk fluid and the catalyst surface because of concentration differences; C) Microscale: diffusion within the catalyst pores reduces the effective reaction rate

rA = η k cAi

(4.28)

If a steady state exists at the surface of the catalyst (i.e., no accumulation of materials occurs at the catalyst surface), the transport rate of materials to the pellet equals the reaction rate at the pellet. For species A then (by using the masstransfer coefficients of Section 3.3), kG Sp (cA − cAi ) = η k Vp cAi

(4.29)

where the external pellet surface area S p and the pellet volume V p are required because k G is per surface area and k is per pellet volume. For a true first-order kinetic reaction, the rate constant is a function of temperature but not of species concentrations. If the reaction is not first order, an additional functionality depending on the interface concentration exists. This equation can be solved for the interface concentration in terms of the rate constants and the bulk fluid concentrations, which can then be used to express the overall reaction rate of species A in terms of the bulk phase concentration r A = K cA

(4.30)

where K =

1 1 η k (Vp /Sp )

+

1 kG

(4.31)

This shows that both the mass-transfer coefficient to the catalyst and the effectiveness factor – rate constant product act as resistances to the conversion of the reactant to the product. When the resistances are added in series, a combined rate constant is obtained. This result leads to the notion of transportlimited versus kinetic-limited reactions. If the mass-transfer coefficient is small enough, mass transfer is the rate limiting step and reaction kinetics play no role in the overall conversion rate. Likewise, if the kinetic rate constant – effectiveness factor product is small enough (compared to the mass-transfer coefficient), it becomes the limiting part of the rate process and the mass-transfer coefficient plays no role in establishing the reaction rate. If both rate constants are of the same order, then the overall rate coefficient depends on both the kinetic and the mass-transfer rate factors.

5. References 1. R. Aris: Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice Hall, Englewood Cliffs, N.J., 1962.

Transport Phenomena 2. L. Brand: Vector and Tensor Analysis, J. Wiley & Sons, New York 1957. 3. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, J. Wiley & Sons, New York 1960. 4. S. Whitaker: Fundamental Principles of Heat Transfer, Pergamon Press, New York 1977. 5. J. C. Slattery: Momentum, Energy and Mass Transfer in Continua, 2nd ed., R. E. Krieger Publishing Co, Huntington, N.Y. 1981. 6. L. E. Malvern: Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, N.J., 1969. 7. G. E. Mase: Theory and Problems of Continuous Mechanics, McGraw-Hill, New York 1970. 8. R. C. Reid, J. M. Prausnitz, T. K. Sherwood: The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York 1977. 9. S. Chapman, T. G. Cowling: The Mathematical Theory of Nonuniform Gases, Cambridge University Press, New York 1939. 10. J. O. Hirschfelder, C. F. Curtiss, R. B. Bird: Molecular Theory of Gases and Liquids, Wiley Interscience, New York 1954. 11. P. D. Neufeld, A. R. Janzen, R. A. Aziz, J. Chem. Phys. 57 (1972) 1100. 12. R. S. Brokaw, Ind. Eng. Chem. Process Des. Dev. 8 (1969) 240. 13. C. Truesdell: The Elements of Continuum Mechanics, Springer Verlag, New York 1966. 14. R. Darby: Viscoelastic Fluids, Marcel Dekker, New York 1976. 15. W. R. Schowalter: Mechanics of Non-Newtonian Fluids, Pergamon Press, New York 1978. 16. S. Whitaker: Fundamental Principles of Heat Transfer, Pergamon Press, New York 1977. 17. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, J. Wiley & Sons, New York 1960. 18. A. J. Chapman: Heat Transfer, Macmillan Publ. Co., New York 1960. 19. H. S. Carslaw, J. C. Jaeger: Conduction of Heat in Solids, 2nd ed., Oxford University Press, London 1959. 20. L. Brand: Vector and Tensor Analysis, J. Wiley & Sons, New York 1947. 21. J. H. Lienhard: A Heat Transfer Textbook, Prentice Hall, Englewood Cliffs, N.J., 1987. 22. H. P. Gurney, J. Lurie, Ind. Eng. Chem. 15 (1923) 1170. 23. H. C. Groeber, VDI-Z. 69 (1925) 705. 24. A. Shack: Industrial Heat Transfer, J. Wiley & Sons, New York 1933.

367

25. M. P. Heisler, Trans. ASME 69 (1947) 227 – 236. 26. J. R. Welty, C. E. Wicks, R. E. Wilson: Fundamentals of Momentum, Heat and Mass Transfer, J. Wiley & Sons, New York 1976. 27. L. E. Sissom, D. R. Pitts: Elements of Transport Phenomena, McGraw-Hill, New York 1972. 28. F. W. Kroesser, S. Middleman, Polym. Eng. Sci. 5 (1965) 1. 29. R. W. Flumerfelt, M. W. Pierick, S. L. Cooper, R. B. Bird, Ind. Eng. Chem. Fundam. 8 (1969) 354. 30. J. Happel, H. Brenner: Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, N.J., 1965. 31. S. M. Selby: Standard Mathematical Tables, 16th ed., The Chemical Rubber Company, Cleveland 1968. 32. R. B. Bird, R. C. Armstrong, O. Hassager: Dynamics of Polymeric Liquids, vol. 1, Fluid Mechanics, J. Wiley & Sons, New York 1977. 33. A. B. Metzner, J. C. Reed, AIChE J. 1 (1955) 434. 34. A. H. P. Skelland: Non-Newtonian Fluid and Heat Transfer, J. Wiley & Sons, New York 1967. 35. D. W. Dodge, A. B. Metzner, AIChE J. 5 (1959) 189. 36. S. Goldstein: Modern Developments in Fluid Dynamics, Oxford University Press, London 1938. 37. S. Prasad: Non-Newtonian Flow Through Constricted Geometries, M. S. Thesis, University of Houston, Houston, TX, 1978. 38. T. B. Drew: Trans. Am. Inst. Chem. Eng. 26 (1931) 26. 39. H. L. Langhaar, J. Appl. Mech. 64 (1942) A-55. 40. W. M. Kays, Trans. ASME 77 (1955) 1265. 41. J. R. Sellars, M. Tribus, J. S. Klein, Trans. ASME 78 (1956) 441. 42. E. N. Sieder, G. E. Tale, Ind. Eng. Chem. 28 (1936) 1429. 43. H. Hausen, Verfahrenstechnik (Berlin) 4 (1943) 91. 44. H. Schlichting: Boundary Layer Theory, 6th ed., McGraw-Hill, New York 1968. 45. L. M. Milne-Thomson: Theoretical Hydrodynamics, 5th ed., Macmillan, New York 1968. 46. J. Schetz: Foundations of Boundary Layer Theory, Prentice-Hall, Englewood Cliffs, N.J., 1984.

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47. E. Pohlhausen, Z. Angew. Math. Mech. 1 (1921) 115. 48. S. W. Churchill, H. H. S. Chu, Int. J. Heat Mass Transfer 18 (1975) 1323. 49. P. S. Klebanoff: “Characteristics of Turbulence on a Boundary Layer with Zero Pressure Gradient,” NACA Report 1247, 1955. 50. J. Nikuradse, Ing. Arch. 1 (1930) 150; VDI-Forschungsh. 361 (1933) 1; Pet. Eng. 11 (1940) 164; 11 (1940) 75; 11 (1940) 124; 11 (1940) 38; 11 (1940) 83. 51. L. Prandtl, Z. Angew. Math. Mech. 5 (1925) 136. 52. J. M. Kay, R. W. Nedderman: An Introduction to Fluid Mechanics and Heat Transfer, 3rd ed., Cambridge University Press, London 1974. 53. H. Blasius, VDI-Forschungsh. 131 (1913). 54. C. F. Colebrook, J. Inst. Civ. Eng. (London) 133 (1938 – 1939). 55. A. A. Zhukauskas, A. B. Ambrazyavichyus: NACA Report 909, 1949. ¨ 56. L. Prandtl: “ Uber den Reibungswiderstand str¨omender Luft,” Reports of the Aerol. Versuchsanst. G¨ottingen, 3rd Series, 1927; see also: “Zur turbulenten Str¨omung in Rohren und L¨angsplatten,” Reports of the Aerol. Versuchsanst. G¨ottingen, 4th Series, 1931. 57. W. L. Friend, A. B. Metzner, AIChE J. 4 (1958) 393. 58. S. W. Churchill, M. Bernstein, J. Heat Transfer 99 (1977) 300. 59. S. Whitaker, AIChE J. 18 (1972) 361. 60. S. W. Churchill, H. H. S. Chu, Int. J. Heat Mass Transfer 18 (1975) 1323. 61. J. P. Holman: Heat Transfer, 6th ed., McGraw-Hill, New York 1986. 62. T. Fujii, H. Imura, Int. J. Heat Mass Transfer 15 (1972) 755. 63. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, J. Wiley & Sons, New York 1960. 64. J. C. Slattery: Momentum, Energy and Mass Transfer in Continua, 2nd ed., R. E. Krieger Publishing Co, Huntington, N.Y. 1981. 65. E. L. Cussler: Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, New York 1984. 66. E. L. Cussler: Multicomponent Diffusion, Elsevier Scientific Publishing Company, New York 1976. 67. J. Crank: The Mathematics of Diffusion, Clarendon Press, Oxford 1956. 68. R. C. Reid, J. M. Prausnitz, T. K. Sherwood: The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York 1977.

69. H. S. Carslaw, J. C. Jaeger: Conduction of Heat in Solids, Oxford University Press, London 1959. 70. P. D. Neufeld, A. R. Janzen, R. A. Aziz, J. Chem. Phys. 57 (1972) 1100. 71. R. S. Brokaw, Ind. Eng. Chem. Process Des. Dev. 8 (1969) 240. 72. E. N. Fuller, P. D. Schettler, J. C. Giddings, Ind. Eng. Chem. 58 (1966) 18. 73. C. R. Wilke, P. Chang, AIChE J. 1 (1955) 264. 74. J. R. Welty, C. E. Wicks, R. E. Wilson: Fundamentals of Momentum, Heat, and Mass Transfer, 2nd ed., J. Wiley & Sons, New York 1976. 75. J. M. Coulson, J. F. Richardson: Chemical Engineering, 3rd ed., vol. 1, Pergamon Press, Oxford 1977. 76. A. S. Foust et al.: Principles of Unit Operations, 2nd ed., J. Wiley & Sons, New York 1980. 77. W. L. McCabe, J. C. Smith, P. Harriott: Unit Operations of Chemical Engineering, 4th ed., McGraw-Hill, New York 1985. 78. R. E. Treybal: Mass-Transfer Operations, 3rd ed., McGraw-Hill, New York 1980. 79. C. O. Bennett, J. E. Myers: Momentum, Heat, and Mass Transfer, 2nd ed., McGraw-Hill, New York 1974. 80. W. G. Whitman, Chem. Metall. Eng. 29 (1923) 147. 81. R. Higbie, Trans. Am. Inst. Chem. Eng. 31 (1935) 365. 82. P. V. Danckwerts, Ind. Eng. Chem. 43 (1951) 1460. 83. P. Harriott, Chem. Eng. Sci. 17 (1962) 149. 84. H. L. Toor, J. M. Marchello, AIChE J. 4 (1958) 97. 85. A. L. Hines, R. N. Maddox: Mass Transfer, Fundamentals and Applications, Prentice Hall, Englewood Cliffs, N.J., 1985. 86. W. H. Linton, T. K. Sherwood, Chem. Eng. Prog. 46 (1950) 258. 87. E. R. Gilliland, T. K. Sherwood, Ind. Eng. Chem. 26 (1934) 516. 88. P. Harriott, R. M. Hamilton, Chem. Eng. Sci. 20 (1965) 1073. 89. M. J. Christian, S. P. Kezibs, AIChE J. 5 (1959) 61. 90. C. H. Bedingfield, I. B. Drew, Ind. Eng. Chem. 42 (1950) 1164. 91. R. L. Steinberger, R. E. Treybal, AIChE J. 6 (1960) 227. 92. F. H. Garner, R. D. Sucking, AIChE J. 4 (1958) 114.

Transport Phenomena 93. E. J. Wilson, C. J. Geankoplis, Ind. Eng. Chem. Fundam. 5 (1966) 9. 94. A. Sen Gupta, G. Thodos, AIChE J. 9 (1963) 751. 95. P. N. Dwivedi, S. N. Upadhyay, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 157. 96. A. Sen Gupta, G. Thodos, AIChE J. 8 (1962) 608. 97. J. C. Chu, J. Kalil, W. A. Wetteroth, Chem. Eng. Prog. 49 (1953) 141. 98. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, J. Wiley & Sons, New York 1960. 99. R. S. Brodkey, H. C. Hershey: Transport Phenomena, A Unified Approach, McGraw-Hill, New York 1988. 100. J. R. Welty, C. E. Wicks, R. E. Wilson: Fundamentals of Momentum, Heat, and Mass Transfer, 2nd ed., J. Wiley & Sons, New York 1969. 101. O. A. Hougen, K. M. Watson, R. A. Ragatz: Chemical Process Principles, part II, Thermodynamics, 2nd ed., J. Wiley & Sons, New York 1959. 102. J. M. Smith, H. C. Van Ness: Introduction to Chemical Engineering Thermodynamics, 3rd ed., McGraw-Hill, New York 1975. 103. W. L. McCabe, J. C. Smith, P. Harriott: Unit Operations of Chemical Engineering, 4th ed., McGraw-Hill, New York 1985. 104. F. M. White: Fluid Mechanics, 2nd ed., McGraw-Hill, New York 1986.

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105. S. Whitaker in N. A. Peppas (ed.): One Hundred Years of Chemical Engineering, Kluwer Academic Publishers, Dordrecht 1989, pp. 47 – 109. 106. J. C. Slattery: Momentum, Energy, and Mass Transfer in Continua, 2nd ed., R. E. Krieger Publishing Co, Huntington, N.Y. 1981. 107. R. B. Bird, Chem. Eng. Sci. 6 (1957) 123. 108. L. W. Moody, Trans. ASME 66 (1944) 672. 109. S. W. Churchill, Chem. Eng. (N.Y.) 87 (1977) Nov. 7, 91. 110. P. R. Hooker, W. E. Brigham, JPT J. Pet. Technol. 30 (1978) 747. 111. J. M. Coulson, J. F. Richardson: Chemical Engineering, 3rd ed., vol. 1, Pergamon Press, Oxford 1977. 112. O. Levenspiel: Chemical Reaction Engineering, J. Wiley & Sons, New York 1962. 113. K.-H. Lin in R. H. Perry, C. H. Chilton (eds.): Chemical Engineer’s Handbook, 5th ed., section 4, McGraw-Hill, New York 1973. 114. J. C. Chu, J. Kalil, W. A. Wetteroth, Chem. Eng. Prog. 49 (1953) 141. 115. E. W. Thiele, Ind. Eng. Chem. 31 (1939) 916. 116. R. Aris, Chem. Eng. Sci. 6 (1957) 262. 117. C. D. Holland, R. G. Anthony: Fundamentals of Chemical Reaction Engineering, Prentice Hall, Englewood Cliffs, N.J., 1979.

Fluid Mechanics

371

Fluid Mechanics See also → Mathematics in Chemical Engineering David V. Boger, Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia Y. Leong Yeow, Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia 1. 2. 2.1. 2.2. 2.3. 2.4. 3. 3.1. 3.2. 3.3. 3.3.1. 3.3.2. 3.3.3. 3.4. 3.4.1. 3.4.2. 3.4.3. 3.5. 3.5.1. 3.5.2. 3.5.3. 4. 4.1.

Introduction . . . . . . . . . . . . . . . Basic Equations of Fluid Mechanics Continuity Equation . . . . . . . . . . Cauchy Equations of Motion . . . . Energy Transport Equation and Bernoulli Equation . . . . . . . . Constitutive Equations and Classification of Fluids . . . . . . . . Newtonian Fluids . . . . . . . . . . . . Deviatoric Stress and Viscosity . . . Navier – Stokes Equations . . . . . . Applications of Navier – Stokes Equations . . . . . . . . . . . . . . . . . Flow in Pipes . . . . . . . . . . . . . . . Concentric Cylinder Flow . . . . . . . Creeping Flow Past a Sphere . . . . . . Some Other Important Flows . . . . Flow Through Granular Beds . . . . . Fluidization . . . . . . . . . . . . . . . . Gas – Liquid Flow . . . . . . . . . . . . Mechanical Energy Balance for Macroscopic Systems . . . . . . . . . Fully Developed Tube Flows . . . . . . Accelerating and Decelerating Flows The Orifice Plate and Similar Flow Rate Measurement Devices . . . . . . Non-Newtonian Fluids . . . . . . . . . Classification of Fluids According to Viscosity Behavior . . . . . . . . . . .

372 374 375 376

4.1.1. 4.1.2. 4.1.3. 4.1.4.

378

4.2. 4.2.1.

379 379 379 381 381 381 383 385 387 387 388 389 391 392 393 394 395 396

Symbols: A C Cd d D De Dhy Dp Dv e

cross-sectional area, surface area, amplitude of oscillation loss coefficient drag coefficient, discharge coefficient distance diameter extrudate diameter hydraulic diameter particle diameter vane diameter surface roughness, end correction

4.2.2. 4.2.3. 4.2.4. 4.3. 4.3.1. 4.3.2. 4.3.3. 4.3.4. 4.3.5. 5. 5.1. 5.2. 5.3. 6.

Bingham Type Behavior . . . . . . . . Shear-Thinning (Pseudoplastic) Fluids Shear-Thickening (Dilatant) Fluids . . Fluids with Time-Dependent Viscosity ( Thixotropic Fluids) . . . . . . . . . . . Fully Developed Tube Flow . . . . . Volumetric Flow Rate – Pressure Drop Relationship . . . . . . . . . . . . . . . . Generalized Treatment . . . . . . . . . Velocity Distribution . . . . . . . . . . . Friction Factor – Reynolds Number Relationships . . . . . . . . . . . . . . . Viscoelastic Fluid Mechanics . . . . Steady Shear Behavior of Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . Behavior of Viscoelastic Fluids in Oscillatory Shear Flow . . . . . . . . . . . Examples of Constitutive Equations for Viscoelastic Fluids . . . . . . . . . . Extensional Behavior of Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . Accelerating and Decelerating Flows of Viscoelastic Fluids . . . . . . . . . . Numerical Methods in Fluid Mechanics . . . . . . . . . . . . . . . . . Finite Difference Method . . . . . . . Finite Element Method . . . . . . . . General Remarks . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

396 397 398 398 399 400 400 401 402 403 405 406 407 409 411 415 415 417 420 420

e rate of strain tensor exx , exy ,eij components of rate of strain tensor Ev frictional loss Ev ex losses in valves, fittings, etc. Ev fd losses in fully developed flows f friction factor F force vector Fd drag force Fi component of force vector G volumetric flow rate in gas phase G
storage modulus G

loss modulus

372 h H Hv i I j JG JL k

Fluid Mechanics

elevation gap between plates vane height unit vector in x direction identity matrix unit vector in y direction superficial gas phase velocity superficial liquid phase velocity thermal conductivity, parameter in the power-law equation k unit vector in z direction K
intercept in the log – log plot of τ w versus 8 V /D L length, volumetric flow rate in liquid phase Le equivalent length L0 initial length n flow-behavior index of power law equation n unit outward pointing normal n
slope of log – log plot of τ w versus 8 V/D N1 first normal stress p isotropic pressure q heat flux vector Q volumetric flow rate r radial coordinate r0 plug radius Rc , Rx , Ry residuals R, R1 , R2 radius Re Reynolds number Re
generalized Reynolds number for timeindependent fluid S path joining two points t time T temperature T stress tensor Tij , Txx , Txy , Txz stress components u velocity vector uf fluidizing velocity ui , ux , uy , ur , ut , u0 velocity components U, U 0 , U∞ velocity V average velocity, volume W mass flow rate W s shaft work x position vector xi components of position vector z elevation α parameter in Meter model γ shear strain γ0 amplitude of sinusoidal shear strain γ˙ shear rate γ˙ 0 amplitude of sinusoidal shear rate

Γ , Γ m torque δij Kronecker delta ε porosity ε˙0 constant extensional rate η viscosity η el extensional viscosity η pl plastic viscosity viscosity parameter of Meter, Maxwell, η0 and Oldroyd-B models η
dynamic viscosity η

component of complex viscosity θ azimuthal coordinate λ bulk viscosity, time constant λel extensional viscosity λ1 relaxation time λ2 retardation time  density p particle density τ deviatoric stress tensor τij component of deviatoric stress tensor τ rz , τ rθ shear stress τw wall shear stress τy yield stress τ 1/2 parameter of Ellis and Meter model Φ interpolation function ψ1 first normal stress coefficient ψ2 second normal stress coefficient Ψ stream function ω 1 , ω 2 , ω angular velocity, vorticity Ω angular velocity

1. Introduction The traditional approach for writing an article or a text book on fluid mechanics has been to deal with the Newtonian fluid and the resulting mechanics associated with such materials. Thus the basic tools (in a sequence of events such as those illustrated in Figure 1) are treated in detail to illustrate how to solve a Newtonian fluid mechanics problem. The laws of conservation of mass (continuity equation) and conservation of momentum (Cauchy momentum equations or equations of motion) are established for a general system and in multidimensional form. From these equations it then becomes clear that the solution to any fluid mechanics problem is not possible unless a relationship is developed between the stress tensor and rate of deformation tensor, or, between the stresses and velocity gradients in the system. Such an equation is termed

Fluid Mechanics the constitutive equation. For Newtonian fluids the relationship between the stress tensor and the rate of deformation is linear, the proportionality constant is the viscosity. Substitution of this relationship into the Cauchy momentum equations leads to the Navier – Stokes equations. Thus the Navier – Stokes equations, in conjunction with the viscosity, form the basis for the solution of classical fluid mechanics problems. Much of the effort in classical fluid mechanics is devoted to establishing methods of solution of the Navier – Stokes equations. These methods can be analytical in nature and in recent times, of course, are numerical. While one whole section of this article is devoted to numerical methods because of their importance in modern fluid mechanics, no attention is directed towards classical analytical methods such as those used in the solution of boundary layer and slow flow (creeping flow) problems. The emphasis is directed towards the industrial chemist, process scientist, and chemical engineer who encounter a vast range of materials in today’s industries where the Newtonian fluid is only one small subset of a vast array of fluid behavior which can be observed. Figure 2 illustrates the range of behavior that can be encountered in terms of the viscosity of fluids and in terms of whether materials are classified as being inelastic or viscoelastic fluids. It is clear from examination of Figure 2 that a vast range of behavior in terms of the viscosity of materials can be and is observed. An industrial chemist or process engineer will encounter low molecular mass liquids and gases (Newtonian materials) but will also be dealing with polymer solutions, polymer melts, rubbers, mineral suspensions, food products, pharmaceuticals, and energy products such as coal oil and coal water fuels, etc. In such systems the viscosity can be constant, but more likely will vary with shear rate or rate of flow. Viscosity can be a function of shear rate and time of shear; or it can be a function of shear rate, time, and thermal history and each of these viscosity behaviors can be observed in inelastic and/or viscoelastic fluids. In addition many materials will not flow until a certain stress is exceeded (e.g., toothpaste). This stress, termed yield stress, can be present in both inelastic and viscoelastic fluids. Therefore, it is important to be aware of the vast range of material behavior present in industry.

373

In Chapter 2 the basic equations for fluid mechanics – the continuity equation, the Cauchy equations of motion, and the energy transport and Bernoulli equations – are established. Chapter 3 deals with Newtonian fluids and Newtonian fluid mechanics where the Navier – Stokes equations are developed. In addition, the mechanical energy balance is established for macroscopic systems in order that pipe flow systems can be designed. The emphasis in the article is directed toward pipe flows because the process engineer is more concerned with flow in conduits than in other geometries. Little emphasis is placed on the details associated with turbulence because process engineers are primarily concerned with turbulent flow in pipes of Newtonian fluids and are not concerned with design of aerofoil structures, etc. Chapter 4 concentrates on non-Newtonian fluids. In this chapter fluids are classified according to their viscosity behavior and are examined in some depth in fully developed tube flow. Viscoelastic fluid mechanics are also summarized – the steady shear and dynamic shear behavior of viscoelastic fluids are reviewed and examples of simple constitutive equations are given. The article is completed with a brief examination of numerical methods in fluid mechanics. The emphasis of this article is shifted toward non-Newtonian fluid mechanics, the sequence of events and tools of importance are illustrated in Figure 3. Here the basic conservation of mass (continuity equation) and the conservation of momentum (Cauchy momentum equations) remain the equations of fundamental importance, but, the relevant constitutive equation in most cases is still unknown. Thus the Navier – Stokes equivalent set of equations in many cases is not known and because any number of constitutive equations are available it is not entirely clear what basic flow property information is required for the most general viscoelastic material. It is only now (1990) that fluid mechanics problems are solved for the first time for viscoelastic fluids. The solutions evolving now for such problems will be extremely important in the future. It is also of considerable importance to keep abreast of the numerical software packages which have been developed in this area particularly in regard to the processing of polymers [1], [2].

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Fluid Mechanics

Figure 1. Newtonian fluid mechanics

Figure 2. Schematic classification of fluid behavior

2. Basic Equations of Fluid Mechanics

Figure 3. Non-Newtonian fluid mechanics TH = Thermal history

The basic equations of fluid mechanics can be derived from a small number of fundamental physical laws such as conservation of mass, of momentum ( Newton’s second law), and of energy. Although very general statements of these laws can be written down (applicable to all substances, solids as well as fluids), in fluid mechanics these laws are formulated in terms of physical variables such as velocity u, pressure p, temperature T , and density . The resulting statements may be less general but they are more directly applicable in analyzing fluid mechanical problems.

Fluid Mechanics

2.1. Continuity Equation The requirement that mass be conserved at every point in a flowing fluid imposes certain restrictions on the velocity u and density . Consider an arbitrary region in space, of volume V , bounded by surface A, through which the fluid flows (see Fig. 4). The rate of increase in mass within this region is given by the volume integral


∂ dV ∂t

(1)

375

at time t and its density  (x + δx, t + δt) at a later time t + δt when it is occupying position x + δx, it can be seen that D/Dt is the rate of change in density experienced by a fluid element as it moves about in space. D/Dt should not be confused with ∂/∂t which is the rate of change in density at a fixed point in space. The substantial derivative operator can also be applied to the velocity u, temperature T , or other properties associated with a fluid element with a similar physical interpretation of the result.

V

The rate at which fluid is leaving the region is given by the surface integral of the outward pointing normal velocity multiplied by the fluid’s density taken over the surface A



∇· (u) dV

u·ndA= A

(2)

V

where n is the unit outward pointing normal at any point on A. The divergence theorem has been used to convert the surface integral into a volume integral. Mass conservation requires the sum of the rate of accumulation in V and the rate of outflow from V to be zero:


 ∂ +∇· (u) dV = 0 ∂t

(3)

V

Since V is arbitrary, the integrand itself must vanish at every point in space: ∂ +∇· (u) = 0 ∂t

(4)

Equation (4) is usually referred to as the continuity equation. It is a statement of mass conservation in terms of fluid velocity and density. Expanding the second term and regrouping, the equation can be written as ∂ +u·∇+∇·u= 0 ∂t

A number of special cases of the continuity equation are frequently encountered. For example, under steady-state conditions the equation reduces to ∇· (u) = 0

(6)

where D/D t is used to represent the operator ∂ ∂t +u·∇. It is known as the substantial derivative operator. By following the change in the density  (x, t) of a fluid element occupying position x

(7)

Much of the subsequent analyses will be restricted to fluids that are incompressible and homogeneous. For such fluids the continuity equation takes on a particularly simple form ∇·u= 0

(5)

or D +∇·u= 0 Dt

Figure 4. Fluids flowing through an arbitrary region in space of volume V and surface area A

(8)

In many practical problems the fluid flow can be regarded as approximately two-dimensional. For such problems, Equation 8 (in cartesian coordinates) takes on the form ∂ux (x,y) ∂uy (x,y) + =0 ∂x ∂y

(9)

Equation (9) and, hence, mass conservation are identically satisfied if the velocity components are given by

376 ux =

Fluid Mechanics ∂Ψ (x,y) ∂Ψ (x,y) and uy = − ∂y ∂x

(10)

where Ψ (x, y), the stream function, is any scalar function of x and y. A simple physical interpretation can be given to Ψ . Consider a line of constant Ψ in the x − y plane, along such a line δΨ =

∂Ψ ∂Ψ δx+ δy= 0 ∂x ∂y

(11)

i.e., −uy δx+ux δy= 0

where g is the acceleration due to gravity. Another external force is that exerted by the fluid outside V . This force acts across the closed surface A and can be expressed in terms of the stress tensor T on the surface. In cartesian coordinates this stress tensor has nine components Txx , Txy , Txz , Tyx , Tyy . . . Tzz , or more concisely Tij ; i, j = 1, 2 or 3. x 1 , x 2 , x 3 are identified with x, y and z, respectively. Tij is the force per unit area in direction j on a surface with normal in the i direction. The force δF acting on an area dA which has a normal n is given by

or dy uy = dx ux

(12)

δF =T ·ndA or δFi =


ΨB −ΨA =

(15)

dΨ dS = dS




Equation (15) is often referred to as the Cauchy stress principle. For the fluids commonly encountered, the stress tensor can be taken to be symmetric, i.e., Tij = Tji for all combinations of i and j. Thus, the external force acting through the closed surface A is given by the surface integral

(13)




i.e., volumetric flow crossing S; un is the velocity component normal to S. Equation (13) shows that the difference in stream function between two streamlines is equal to the volumetric flow rate across any line joining the two streamlines. The continuity equation and the stream function take on different forms when written in noncartesian coordinates. Listing of the equations of fluid mechanics in a number of commonly encountered coordinates can be found in most standard textbooks on this subject, e.g., in [3]. A partial list is given in Table 1.

A

S

un dS

T ·ndA

un

2.2. Cauchy Equations of Motion Before deriving the equations of motion for fluids, it is necessary to examine the forces acting on a fluid element as it moves about in space. Consider again the arbitrary volume V in space (see Fig. 4). One of the external forces acting on the fluid inside V is its weight which is given by the volume integral
V

Tji nj dA

j=1

Since dy/dx is the local slope of the line, Equation (12) shows that the velocity vector is everywhere tangential to lines of constant Ψ . Such lines are referred to as streamlines. They give the instantaneous flow pattern of the fluid. By integrating along any path S joining streamlines where Ψ = Ψ A and Ψ = Ψ B , it can be seen that

(16)

which is equivalent to
∇·T dV

(17)

V

where the divergence theorem is again used to convert the surface integral to a volume integral. The i component of the vector (∇ · T ) is (∇·T )i =

3  ∂Tij ∂xj j=1

(18)

To obtain the equations of motion in terms of T, u, and ( g), it is convenient to start with a statement of conservation of momentum which asserts that: the rate of change of momentum inside the volume V and the net rate at which momentum is being convected out of the volume is equal to the sum of the external forces acting on the volume. The rate of accumulation is given by


g dV

3 

(14) V

∂ (u) dV ∂t

(19)

Table 1. Continuity and Navier – Stokes equations for incompressible homogeneous fluids in cartesian, cylindrical, and spherical coordinates

Fluid Mechanics 377

378

Fluid Mechanics

The rate at which momentum is being convected out is given by the surface integral



∇· (uu) dV

u (u·n) dA= A

(20)

V

where  u is the momentum per unit volume and u · n is the volumetric flux leaving A. Conservation of momentum in V then requires



∂ (u) dV + ∇· (uu) dV ∂t V V

= gdV + ∇·T dV V

(21)

V

Since V is arbitrary, the integrands must be equal ∂ (u) +∇· (uu) =g+∇·T ∂t

(22)

Upon expanding the terms on the left-hand side and regrouping, Equation (22) becomes 

   ∂ ∂u +∇· (u) + +u·∇u ∂t ∂t =g+∇·T

u

(23)

From the continuity equation (Eq. 4) the first group of terms on the left-hand side is zero. The second group of terms can be identified as the product of  and the substantial derivative of velocity. Therefore 

Du =g+∇·T Dt

(24)

This is the equation of motion for fluids. It is often referred to as the Cauchy momentum equation or equation of motion. The left-hand side is referred to as the inertia term. The first term on the right-hand side is the force per unit volume acting on a fluid element arising from gravitational body force. The second term is the force per unit volume acting on the fluid element as a result of the spatial variation of the stress tensor. The form of the Cauchy equations of motion in different coordinate systems can be found in [3].

needed to relate the local temperature T of the fluid to variables such as velocity, velocity gradient, and temperature gradient. Such an equation takes the general form DT = −∇·q − T Dt +dissipation terms

CV



∂ ∂T

 ∇·u V

where q is the heat flux and CV is the heat capacity at constant volume. This equation simplifies to Cp

DT =k∇2 T +dissipation terms Dt

for an incompressible fluid with a heat capacity, Cp , at constant pressure and when the temperature dependence of the thermal conductivity k can be neglected. In arriving at Equation (25), Fourier’s expression for heat conduction (q = − k ∇T ) was used to express q in terms of temperature gradient. Equation (25), the energy transport equation, is a statement of conservation of energy and is a form of the First Law of Thermodynamics. It equates the rate of energy transfer to a fluid element by thermal conduction and the rate of dissipation/conversion of kinetic energy into thermal energy to the accumulation of thermal energy within the element. The dissipation terms arise as a result of the deformation suffered by the fluid element. The derivation of the energy transport equation and the exact form of the dissipation terms are given in [3]. In the following sections only isothermal flows will be discussed. In such flows the dissipation terms are negligible and the energy transport equation is satisfied. This greatly simplifies the analysis. In many flow fields of practical interest, a simple statement of conservation of kinetic and potential energy can be derived. This is obtained by forming the dot product between the velocity vector u, and the equations of motion (24), and integrating the result between any two points on the same streamline. The resulting equation takes the form 1 2 p1 1 p2 u + gh1 + = u21 +gh2 + +losses 2 1  2 

2.3. Energy Transport Equation and Bernoulli Equation In flow problems where a substantial variation in temperature occurs, an additional equation is

(25)

(26)

This is one form of the well-known Bernoulli equation; h, u, and p are the vertical elevation, the velocity, and the isotropic pressure of the fluid respectively. Subscript 1 and 2 denote where these variables are to be evaluated

Fluid Mechanics on the streamline. Losses refer to the conversion of mechanical (kinetic and potential) energy into internal (thermal) energy between the two points. In deriving this equation, the fluid is taken to be incompressible and the flow to be steady. Relaxation of these assumptions will result in a more general form of the Bernoulli equation [1]. The Bernoulli equation finds many practical applications, especially in flows where the losses are small or can be estimated with reasonable accuracy. In many applications the spatial average value of the variables at point 1 and point 2 are used instead of local point values, resulting in further simplification (see Section 3.5).

2.4. Constitutive Equations and Classification of Fluids The equations derived from the conservation laws do not contain any information about the mechanical properties of fluids. An applied force or stress acting on fluids with different properties will result in different flow patterns. Conversely, the same flow pattern will induce different stresses in different fluids. The diverse mechanical properties of fluids are described by equations that relate the local stress tensor to the local flow kinematics. Such equations are known as rheological constitutive equations. One of the most important kinematic variables that appear in rheological constitutive equations is the rate of strain tensor e. This is defined as e=

/ 1. ∇u+ (∇u)T 2

1 2



∂ui ∂uj + ∂xj ∂xi

3. Newtonian Fluids 3.1. Deviatoric Stress and Viscosity In a stagnant fluid, the stress at any point is the hydrostatic pressure. This is an isotropic compressive stress, i.e., the stress has the same magnitude in any direction. Such a stress can be represented by T = −pI

 (28)

The rate of strain tensor is a symmetric tensor. Simple physical interpretations can be given to the components of e. For example, e11 or exx is the rate at which a fluid element is being stretched in the x direction, e12 or exy is the rate at which the fluid element is being sheared in the x – y plane. The form of the rate of strain and other kinematic tensors in different coordinate systems can be found in [4]. It should be

(29)

or in component form (30)

where I is the identity tensor 

or in cartesian coordinates eij =

mentioned that some authors, particularly those working on non-Newtonian fluids, define the rate of strain tensor without the factor of 1/2 as shown in Equation (27). This difference in the definition can lead to considerable confusion and should carefully noted. A convenient way of classifying fluids is according to the form of their constitutive equations. For a large class of fluids, the stress tensor T is a linear function of the rate of strain tensor e. Such fluids are referred to as Newtonian fluids. If T is a non-linear function of e the fluids are said to be non-Newtonian. In chemically more complex fluids, the stress is not only a function of e, it also depends on the deformation or the entire history of deformation suffered by the fluid. Such fluids may also exhibit solidlike elastic behavior and are known as viscoelastic fluids. ( Non-Newtonian and viscoelastic fluids are treated in greater detail in Chap. 4).

Tij = − pδij (27)

379

1  0 0

0 1 0

 0  0 1

and δij is the Kronecker delta δij = 1 i = j δij = 0 i= j

(31)

Unless otherwise stated, i and j take on values 1, 2, or 3 in all the equations in this article. The shear components (when i is not equal to j ) of the isotropic stress are identically zero and the three normal components take on the same value

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Fluid Mechanics

p, the magnitude of the pressure; p is a function of the vertical position in the fluid. It is customary to have a negative sign before p so that for hydrostatic compressive stress, the numerical value associated with p is positive. All these are in agreement with the well known nature of hydrostatic pressure. For a fluid in motion the stress will, in general, no longer be isotropic. At any point the stress is a combination of the hydrostatic pressure and the stresses arising from the deformation experienced by the fluid. It is convenient to decompose the stress tensor Tij into an isotropic part – p δij and a nonisotropic part τij , Tij = −pδij + τij

(32)

τij , usually referred to as the deviatoric stress tensor, is a consequence of the deformation. For a fluid element that has not experienced any deformation, the deviatoric stress vanishes. In a stagnant fluid the deviatoric stress is therefore zero and the resulting isotropic stress can be identified with the hydrostatic pressure. However, because of fluid motion, the isotropic component of the stress tensor will, in general, be different from the hydrostatic pressure. The nature of the relationship between deviatoric stress and deformation depends on the rheological constitutive equation of the fluid. The constitutive equation completely describes the deformation arising from a specified deviatoric stress and, conversely, gives the stresses needed to produce a specified deformation. For a large class of incompressible fluids the deviatoric stress τ is directly proportional to the rate of strain tensor e: τ = 2ηe or τij = 2ηeij

(33)

where η the constant of proportionality, is a property of the fluid and is known as the viscosity. Equation (33) in the Newtonian constitutive equation. It is an example of a rheological constitutive equation used to relate kinematic tensors, such as the rate of strain tensor, to the deviatoric stress. Most commonly encountered low molecular mass liquids and gases follow the Newtonian constitutive equation, they are called Newtonian fluids. Viscosity is the most important flow property of fluids. It is a strong function of temperature. The SI unit of viscosity is pascal – second. It should be pointed out that

when dealing with compressible fluids, such as gases, the Newtonian constitutive equation must be modified to allow for compressibility. This is done by introducing a bulk viscosity λ into the Newtonian constitutive equation τ =λ∇·uI +2ηe

(34)

The bulk viscosity term is zero for incompressible fluids. This article is concerned almost exclusively with incompressible fluids, thus the bulk viscosity term will not appear in all subsequent equations and discussions. For further discussion on compressible flow and bulk viscosity see [5]. Specialized instruments known as viscometers have been developed for measuring viscosity. The basic principles of such instruments can be explained by examining the rate of strain and the deviatoric stress tensors in a fluid contained between two parallel plates, shown in Figure 5. The lower plate is held stationary while the upper is moving parallel to itself, with a constant velocity u0 . Let F denote the force required to maintain the motion of the upper plate, A the area of the plates, and d the spacing between the plates. In this flow, the velocity of the fluid varies linearly from zero at the lower plate to u0 at the upper plate. With reference to the coordinates shown, it can be seen that the only nonzero components of the rate of strain tensor are exy =eyx =

1 dux 1 u0 = 2 dy 2 d

(35)

where u0 /d is the velocity gradient across the plate spacing. This velocity gradient is often referred to as the shear rate and denoted by γ. ˙ In this simple flow field (the steady shear flow) for Newtonian fluids, the only nonzero components of the deviatoric stress are τxy =τyx =

F A

(36)

By definition, the viscosity is given by η=

τxy F  u0  = / 2exy A d

(37)

In a viscometer F and u0 are measured, and A and d are obtained from the dimensions of the instrument. From these data the viscosity can be calculated. A detailed description of viscometric techniques is given in [6–8].

Fluid Mechanics



3 ∂ui ∂ui  + uj ∂t ∂x j j=1

=−

Figure 5. Shearing motion between two parallel plates

Typical values of viscosity (in Pa s at 20 ◦ C or as indicated) for several fluids are given in the following list. These values show the large variation of viscosity that can be encountered in fluid mechanical problems as well as the variation of viscosity with temperature. Steam (373 K) Carbon dioxide Air (273 K) (293 K) (373 K) (473 K) Pentane Hexane Heptane Octane Benzene Mercury Water (273 K) (283 K) (293 K) (303 K) Olive oil Castor oil Glycerol (273 K) (293 K) (298 K) (303 K) Low- density polyethylene at zero shear rate (385 K) (403 K) (443 K) (463 K)

1.28×10−5 1.46×10−5 1.71×10−5 1.81×10−5 2.18×10−5 2.58×10−5 2.34×10−4 3.26×10−4 4.09×10−4 5.45×10−4 6.47×10−4 1.55×10−3 1.79×10−3 1.30×10−3 1.00×10−3 7.98×10−4 8.40×10−2 9.86×10−2 1.21×101 1.49 0.942 0.622

381

3  ∂ 2 ui ∂p +η +gi ∂xi ∂x2j j=1

(39)

The first term on the right-hand side is the isotropic pressure term and the second term is the viscous term. This equation together with the continuity equation forms a set of four equations which can be solved to give the three unknown velocity components and the isotropic pressure for any flow of a Newtonian fluid. Equation (39), known as the Navier – Stokes equation, describes the motion of Newtonian fluids. The equivalent forms in some of the more commonly encountered coordinate systems are listed in Table 1. Viscosity is a strong function of temperature in nonisothermal flow, where the variation of viscosity with temperature cannot be ignored, the Navier – Stokes equations must be modified. Thus, it now takes the form 

3 ∂ui ∂ui  uj + ∂t j=1 ∂xj

=−

  3 ∂ui ∂p  ∂ η (T ) +gi + ∂xi j=1 ∂xj ∂xj

(40)

This equation must now be solved simultaneously with the continuity equation and the energy transport equation for the unknowns – the three velocity components, the isotropic pressure, and the temperature. It is assumed that the variation of viscosity with temperature is a known experimentally measured quantity. Variable viscosity adds to the difficulty of solving the equations describing the flow. The complexity of these equations is such that they often do not have simple analytical solutions and computers are used to generate approximate numerical answers.

2.65×105 1.05×105 2.38×104 1.40×104

3.2. Navier – Stokes Equations When the Newtonian constitutive equation is substituted into the Cauchy equations of motion (Eq. 24), the following equation is obtained: 

Du = −∇p+η∇2 u+g Dt

or in cartesian component form

(38)

3.3. Applications of Navier – Stokes Equations 3.3.1. Flow in Pipes Steady flow of Newtonian fluids in pipes is probably the most commonly encountered flow field. It is also one of the small number of flow fields for which the Navier – Stokes equations have a

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Fluid Mechanics

simple analytical solution. In this flow the only nonvanishing velocity component is that along the axis of the pipe. It is convenient, in this case, to write down the equations of motion in cylindrical coordinates. The axial velocity uz is a function of the radial coordinate only and is independent of the other coordinates and time. Examination of the Navier – Stokes equations in cylindrical coordinates in Table 1 shows that the only nontrivial component is 

0 =−

∆p 1 d +η L r dr

 r

duz dr

 (41)

The driving force of this flow is the applied pressure gradient ∆p/L; the effect of gravity has been left out in this equation. Equation (41) can be integrated to give uz =

1 ∆p 2 r +A lnr+B 4η L

(42)

where A and B are the constants of integration that can be determined by considering the boundary conditions. The velocity along the axis of the tube is finite; this then requires A to be zero. At the pipe wall r = R, it is assumed that the fluid adheres to the wall, i.e., uz = 0. This is the well known no-slip boundary condition that is normally imposed at the interface between a solid wall and a flowing fluid. This boundary condition is satisfied for B= −

1 ∆p 2 R 4η L

Thus the resulting velocity profile in the pipe is given by uz =

R2 4η

 −

∆p L

  r2 1− 2 R

(43)

From Equation (43) the volumetric flow rate Q through the pipe can be obtained by integrating
R 2πruz dr=

Q= 0

πR4 8η

 −

∆p L

 (44)

The average velocity V in the pipe is given by V=

R2 8η

 −

∆p L

 (45)

In terms of V , the velocity profile in the pipe takes the simple form

 uz = 2V

1−

r2 R2

 (46)

Thus the velocity profile is a parabola and the maximum velocity, which occurs at the centre of the pipe, is twice the average velocity. For a given pipeline, the driving force needed to attain a specified average velocity is given by ∆p 8ηV =− 2 L R

(47)

Equation (45) is the Hagen – Poiseuille equation. It can be put in a dimensionless form f= −

D∆p 4L V 2 2

=

16 V D η

=

16 Re

(48)

where D is the diameter of the pipe, Re =  V D/η, and f is as defined by this equation; f is the friction factor and Re is the Reynolds number. Both these quantities are dimensionless. Such dimensionless numbers are widely encountered in fluid mechanical analysis. Equation (48) shows that f is a function of Re only. It will be shown later that shear stress at the pipe wall is given by D ∆ p/4 L, while shear rate at the pipe wall is 8 V/D. Thus the ratio of the two measurable quantities ∆ p D/ 4 L and 8 V/D gives the viscosity of the fluid. Measurement in pipe flow can be used for viscosity measurements. For flow in pipes with Re < 2100, the simple relationship between f and Re in Equation (48) has been confirmed by experimental measurements. Such flows are known as laminar flows. As the Reynolds number is increased above 2100, the flow gradually loses it steady nature and becomes more and more chaotic. The expressions derived for the velocity profile, volumetric flow rate, and friction factor are no longer valid. The chaotic fluid motion associated with flows at high Reynolds number (Re > 4000) is known as turbulent flow. Turbulent flow is characterized by rapid and apparently random fluctuations in fluid velocity and pressure. The same definition of f and Re given in Equation (48) can be used in turbulent flow. However, the relationship between these two dimensionless numbers in turbulent flow can not be determined analytically and a number of empirical and semiempirical correlations have been developed. In the range of 4000 < Re < 105 , experimental data of f versus Re closely follow the empirical Blasius equation

Fluid Mechanics

f = 0.079Re−1/4

(49)

383

LIVE GRAPH

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Another equation that can be used to describe the relationship between f and Re is the von Karman – Nikuradse equation  -  1 √ = 4.0log Re f −0.4 f

(50)

Equation (50) is valid for Re ≥ 4000 [5]. The constants 4.0 and 0.4 are determined from experiment. Analysis of turbulent flow requires special techniques and will not be discussed in this article. For further details see [9]. Equations (48) – (50) which relate the friction factor to the Reynolds number were derived for pipes with a smooth inner surface. In commercial pipes surface irregularities are invariably present – depend on the material of construction and the manufacturing process. A quantitative measure of this surface irregularity is the relative roughness of the pipe which is defined as the ratio of the average surface roughness e to the pipe diameter D. The relative roughness has to be included as an additional factor in the correlations for friction factor. For most commercial pipes the relative roughness is much smaller than unity and surface irregularities have no noticeable effect on pressure drop as long as the flow remains laminar. Thus Equation (48) remains valid for rough pipes. In turbulent flow, the situation is quite different. The friction factor is altered significantly even for small e/D. Empirical equations have been developed to relate the friction factor to the relative roughness and the Reynolds number. One example is the Colebrook equation: 1 √ = −4.0log f



e 4.67 √ + D Re f

Figure 6. Friction factor for flow in circular pipes at different Reynolds number and for different relative pipe roughness (e/D) [1]

3.3.2. Concentric Cylinder Flow

 +2.28

(51)

A disadvantage is that for a given Re and e/D, Equation (51) must be solved iteratively for f . Another difficulty in using the Colebrook equation is the large uncertainty associated with the relative roughness. The friction factor is important in the sizing of pipelines. It is customary to plot f against Re in the form of a log – log plot. An example of such a plot is shown in Figure 6 ( Moody chart). The relative roughness of the pipe is included in this plot as an independent parameter.

Figure 7. Flow between two concentric cylinders

Another exact solution of the Navier – Stokes equation is the one that describes the velocity profile and stresses generated in a fluid contained between the annular gap of two rotating concentric cylinders (see Fig. 7). This flow is of great practical interest because it is the basis of a large class of commercial viscometers. It is also of

384

Fluid Mechanics

considerable theoretical importance; it is one of the few flows for which the transition (via a series of progressively more complicated flows) of the simple laminar solution to turbulent flow has been carefully observed and analyzed theoretically. To simplify the analysis, the length of the cylinders are assumed to be long compared to their radii so that the end effects can be ignored. There is no applied pressure gradient in this flow. Fluid motion is brought about by the rotation of the cylinders. Let ω 1 and ω 2 denote the angular velocity of the outer and inner cylinders respectively. The corresponding linear velocity of the cylinders are ω 1 R1 and ω 2 R2 , where R1 and R2 are the respective radii. The only nonvanishing velocity component, in cylindrical coordinates, is the azimuthal component uθ . The Navier – Stokes equations reduces to 

u2θ



dp =− r dr   d 1 d (ruθ ) 0=η dr r dr

−

(52) (53)

Equation (53) can be integrated to give uθ in terms of the radial coordinate and the boundary conditions on the inner and outer cylinder. Using the newly obtained velocity, Equation (52) can in turn be integrated to give the pressure. The resulting expressions for uθ and p are  

R12 R22 2 1 2 (ω + r R ω −R ω − ω ) 2 1 1 2 2 1 R22 −R12 r 

2 2

2 R2 ω2 − R12 ω1 r −R22  + p=p2 + 2 R2 − R12 2 uθ =

r + 2R12 R22 (ω1 −ω2 ) R22 ω2 − R12 ω1 ln R2   1 1 R14 R24 (ω1 −ω2 )2 − 2 r 2 R2

where p2 is the pressure at the inner cylinder wall. It can easily be verified that the no-slip boundary condition is satisfied at the outer and inner cylindrical walls. In most commercial concentric cylinder viscometers, the outer cylinder is held fixed while the inner one is rotated at a steady angular velocity by an applied torque Γ . For this special case, the velocity in the annular gap is given by uθ =

1 R22 −R12

 rR22 ω2 −

R12 R22 ω2 r

 (54)

The corresponding shear rate and shear stress on the inner wall are γ˙ rθ2 =

τrθ2 =

2ω2

(55)

1− (R2 /R1 )2 Γ 2πR22 L

(56)

where L is the length of the cylinder. In viscometry, these two expressions are combined to yield an explicit expression for the viscosity of the fluid: η=

τrθ2 Γ 1 = γ˙ rθ2 L 4πω2



1 1 − R22 R12

 (57)

With Equation (57) the viscosity of the fluid can be calculated from the applied torque per unit height of the cylinders Γ /L and the measured angular velocity ω 2 . For Newtonian fluids, in principle, only a single reading of the torque together with the corresponding angular velocity is needed to determine the viscosity. For nonNewtonian fluids, the situation is more complex and a series of data points are needed to determine the viscosity as a function of shear rate [6]. According to the solution of the Navier – Stokes equations, the streamlines should form a family of circles concentric with the cylinders. At low Reynolds number this is indeed the observed flow pattern. However, as the Reynolds number is increased (either by increasing the angular velocity or decreasing the viscosity) this simple flow loses its stability and gives way to a new steady flow in which the streamlines are spirals whose axes are concentric with the cylinders. A sketch of the new flow pattern, at sufficiently large Re, is shown in Figure 8. The vortices formed by the fluid are known as Taylor vortices. The transition between the two steady flows can be observed by plotting the torque Γ against the shear rate γ˙ (see Fig. 9). The abrupt change in slope of the Γ – γ˙ plot can be used to locate the transition point accurately. In using Equation (57) to obtain the viscosity from concentric viscometer measurements, it is important to ensure that all the data points are taken prior to the onset of transition flow. At even higher Reynolds number, the Taylor vortices in turn become unstable and are replaced by a series of progressively more complicated flows, which ultimately lead to turbulent flow. The detailed

Fluid Mechanics step by step mapping of the laminar-to-turbulent transition has been investigated experimentally and analyzed theoretically with excellent agreement between the two.

385

In making viscosity measurements, it is common practice to use more than one viscometer so that the experimental data covers as wide a range of shear rate as possible. This step is particularly important if the possibility exists that the fluid may not be Newtonian (i.e., it has a viscosity that is a function of shear rate). Figure 10 shows a log – log plot of shear stress against shear rate obtained using four different viscometers. The data at low shear rates were obtained using a concentric cylinder viscometer. Those at higher shear rates were obtained from the volumetric flow rate and pressure gradient data from three different capillary tubes. (Experimental techniques associated with capillary viscometer are described in Section 4.2). The fact that all the data points, from four different viscometers, fall on the same straight line with slope equal to unity means that the fluid is Newtonian over the shear rates covered and the viscosity can be read off from the intercept of the straight line with the vertical line at γ˙ = 1.0; η is, in this case, ca. 0.91 Pa s. LIVE GRAPH

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Figure 8. Taylor vortices [5] (reproduced with permission of McGraw-Hill)

LIVE GRAPH

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Figure 10. Log – log plot of shear stress versus shear rate for a Newtonian fluid (glycerol at 21.6 ◦ C) obtained using four different viscometers
D = 5.573 mm, L = 300 mm, L/D = 53.8;  D = 5.569 mm, L = 477 mm, L/D = 85.7; ◦ D = 5.595 mm, L = 700 mm; L/D = 125.1;
concentric cylinder viscometer

3.3.3. Creeping Flow Past a Sphere Figure 9. Change in slope in the torque – shear rate plot for flow between concentric cylinders indicating the onset of Taylor vortices [1]

Steady flow of a Newtonian fluid past a sphere has been the subject of numerous investigations.

386

Fluid Mechanics

This flow is of great practical importance in its own right, but also because it illustrates the steps involved in the evaluation of the drag force exerted by a flowing fluid on an object. It is also of considerable theoretical interest because of the part it played in the development of some of the modern mathematical techniques, such as singular perturbation and matched asymptotic analyses, used in solving flow problems. The mathematical analysis is greatly simplified if the inertia terms in the Navier – Stokes equations are assumed to be small compared to the viscous terms and thus are ignored. This assumption is valid if the fluid velocity or density is small, its viscosity large, or if the sphere diameter D is small. These conditions are equivalent to requiring the Reynolds number Re =  U 0 D/η, to be small; U 0 is the undisturbed velocity of the fluid at a large distance from the sphere. Flows with low Reynolds number are often referred to as inertialess or creeping flows. In spherical coordinates, the equations for creeping flow past a sphere are 0=−

   ∂ur 1 ∂ ∂p +η 2 r2 + ∂r r ∂r ∂r

1 r2 sinθ

∂ ∂θ

 sinθ

∂ur ∂θ

 −

2ur − r2

2 ∂uθ 2 − 2 uθ cotθ r2 ∂θ r    ∂uθ 1 ∂p 1 ∂ 0=− +η 2 r2 + r ∂θ r ∂r ∂r  sinθ

∂uθ ∂θ

 +

2 ∂ur uθ − 2 2 r 2 ∂θ r sin θ

 (58)

It is convenient to locate the center of the sphere at the origin of the spherical coordinates and to have the undisturbed upstream velocity of the fluid parallel to the θ = 0 line (see Fig. 11). In this coordinate system, ur and uθ are the only two nonvanishing velocity components. The Newtonian fluid is assumed to adhere to the surface of the sphere (the no-slip boundary condition), i.e., ur =uθ = 0 at r=D/2

(59)

Far away from the sphere, u approaches the uniform velocity U 0 , i.e., ur =U0 cosθ, uθ = −U0 sinθ as r→∞

In addition, the velocity components have to satisfy the continuity equation 1 ∂ 2 1 ∂ r ur + (uθ sinθ) r 2 ∂r rsinθ ∂θ

(61)

Equations (58) and (61) can be solved to give the following expressions for the velocity components and pressure 



D 3 cosθ 2r   

1 D 3 3D uθ = −U0 1− − sinθ 8 r 4 2r   2  D 3ηU0 p=p0 − cosθ D 2r

ur =U0 1−

3D 1 + 4 r 2



(62)

where p0 is the uniform pressure far from the sphere. For mathematical details leading to these expressions see [1]. To calculate the drag force on the sphere, the isotropic pressure and deviatoric stress on the surface of the sphere must be evaluated. These are given, respectively, by



1 ∂ r2 sinθ ∂θ

Figure 11. Definitive sketch for flow past a sphere

(60)

3ηU0 p=p0 − cosθ (63) D     ∂ uθ 3ηU0 1 ∂ur =− + sinθ τrθ =η r ∂r r r ∂θ r=D/2 D

The other components of the deviatoric stress are zero on the surface. The drag force F d is obtained by integrating the resulting Cauchy stress over the entire surface of the sphere. From the symmetry of the problem, it is clear that the drag force is in the same direction as the undisturbed velocity:
π
2π  Fd = ϕ=0 θ=0

−p0 +

3ηU0 cosθ D

 cosθ+

 3ηU0 sin2 θ R2 sinθdθdϕ= 3πηDU0 D

Next Page Fluid Mechanics This result is often referred to as Stokes law. One of the practical applications of Stokes law is the falling ball viscometer. Here the terminal velocity of a small sphere of known diameter that falls through a fluid is measured. When the sphere has attained its terminal velocity, the drag force on the sphere is exactly balanced by it weight, i.e.: 4 Fd = π (D/2)3 ∆g = 3πηDU0 3

(64)

where ∆  is the difference between the density of the solid sphere and the fluid. This then yields a simple expression which allows the fluid viscosity to be calculated from measurable quantities such as the densities of the fluid and the sphere, and the terminal velocity: 1 D2 ∆g η= 18 U0

(65)

Usually the drag force F d is expressed in terms of a dimensionless drag coefficient C d obtained by dividing F d by 12 U02 and by the cross-sectional area of the sphere normal to the velocity U 0 ; thus the drag coefficient is Cd =

24η 3πηDU0 = 1 2 πD 2 /4 U U 0D 0 2

387

shown in Figure 47) The laminar flow eventually gives way to turbulent flow, and C d is expected to change dramatically with Re. Experimentally measured drag coefficients as a function of Re are shown in Figure 12. The validity of Equations (67) and (68) is restricted to Re less than ca. unity with the range of validity of Equation (68) slightly larger than that of Equation (67). For Re in the range 1 to 103 , C d can be approximated by Cd ≈ 18Re−0.6

(69)

And for 103 ≤ Re ≤ 2×105 , C d is approximately constant Cd ≈ 0.44

(70)

For a detailed description of the flow phenomena at increasing Re, including the sudden decrease in C d at Re ≈ 2×105 , see [1]. LIVE GRAPH

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(66)

In terms of C d and the Reynolds number Re =  U 0 D/η, Stokes law takes the form Cd = 24/Re

(67)

This result was obtained for Re much smaller than unity: in this case the error incurred is acceptable for most practical purposes. For larger Re, the inertia terms can no longer be ignored and additional terms must be included in the expression for C d . The result now takes the form of an infinite series in Re, the leading terms are Cd =

  24 3 9 1+ Re+ (lnRe) Re2 +... Re 16 160

Figure 12. Log – log plot of drag coefficient on a sphere as a function of the Reynolds number

3.4. Some Other Important Flows 3.4.1. Flow Through Granular Beds

(68)

The validity of this series extends to Re of the order of unity. The mathematical analysis leading to this series is described in [2]. Expressions (67) and (68) for C d are based on the assumption that the flow remains steady and laminar and Re is small. As Re is increased, the flow pattern around the sphere becomes more and more complicated. (See the numerically generated flow pattern around a cylinder

The flow of fluids through granular beds occurs widely in industry and in nature, some examples are flow of water down a filter bed, seepage of oil through porous underground soil structure, and flow of chemicals in a packed bed reactor. The key variable in the study of flow through a granular bed is the volumetric flow rate per unit cross-sectional area of the bed, Q/A. This flow rate depends on the density  and viscosity η of the fluid, the average size (and shape) of the

Previous Page 388

Fluid Mechanics

particles Dp that make up the bed, the applied pressure gradient ∆ p/L across the bed, and the porosity of the bed ε. The relationship between these variables can be expressed in the general form ∆p/L=f (Q/A,,η,ε,Dp )

The exact form of the function in this equation is expected to be complex and cannot be determined from fundamental considerations only. It must therefore be determined by experimental measurements. A clearer understanding of the influence of each of these variables can be achieved by presenting the experimental results in a dimensionless form. For the present problem, the relevant dimensionless variables are an appropriately defined friction factor f and an appropriately defined Reynolds number: f=

Dp ε3 ∆p 2 (1−ε) L U∞

Re=

Dp U∞ (1−ε) η

(71)

(72)

U ∞ is the superficial velocity and is defined by U ∞ = Q/A. From the accumulated data in the literature, it can be shown that f and Re are reasonably correlated by the following semi- empirical expressions: f = 150/Re for Re τ y . In contrast the viscosity for the Bingham and any yield stress material becomes unbounded as | γ˙ | → 0. When the shear stress – shear rate function is nonlinear the data are often fit by a Herschel – Buckley model:  τ= η=

τy +k γ˙ n−1 |γ| ˙

 γ˙ for |τ |≥τy

τy +k|γ| ˙ n−1 for |τ |≥τy |γ| ˙

(105a)

(105b)

The parameters k and n are obtained by plotting τ − τ y versus γ˙ on logarithmic coordinates. Another equation often used to describe yield stress materials is the Casson equation  τ=

|γ| ˙ 

η=

2

1/2

τy

1/2

+ηpl 1/2

|γ| ˙

(106a)

for |τ |≥τy

(106b)

2

1/2

τy

γ˙ for |τ |≥τy

1/2

+ηpl 1/2

Typical shear stress – shear rate data for a meat extract concentrate and for a tomato soup concentrate are presented in Figures 24 and 25, respectively. The meat extract behaves like a Bingham solid while the tomato soup concentrate exhibits Herschel – Buckley behavior. Figure 23. Typical flow curves

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4.1.1. Bingham Type Behavior Bingham type material exhibits a yield stress. That is, the shear stress τ must exceed a certain yield value τ y before the fluid deforms and flows. Classical Bingham plastic behavior is illustrated in Figure 23 and can be described by  τ=

i.e.,

τy +ηpl |γ| ˙

 γ˙ for τ ≥τy

(104a) Figure 24. Shear stress – shear rate data (flow curve) for a meat extract

Fluid Mechanics LIVE GRAPH

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Figure 26. Viscosity as a function of shear rate for a polyacrylamide solution

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397

less entangled and less resistant to deformation as the deformation rate is increased. The viscosity of pseudoplastic fluids is typically plotted versus shear rate on logarithmic coordinates because the viscosity for such fluids can change by orders of magnitude over several decades of shear rate. The data for the polyacrylamide solution shown in Figure 26 are typical for a macromolecular fluid. The viscosity approaches a constant value at low shear rates called the zero-shear viscosity η 0 and a constant value in the limit of very high shear rates called the infinite shear viscosity η ∞ . Experimental measurements in both regions can be difficult, particularly in the infinite shear region. Thus data are often only available between the two limits. Between the two limits the shear stress (or viscosity) is often linear with shear rate over several decades on logarithmic coordinates. In this restricted region logτ = logk+nlogγ˙

(107a)

logη= logk+ (n−1) logγ˙

(107b)

where n is the slope in the linear region of the plot (n = 0.41 for the data in Fig. 26). Thus Figure 25. Shear stress – shear rate data (flow curve) for a tomato soup concentrate

The yield stress is associated with a threedimensional structure that must deform elastically before flow can occur. It is commonly observed in fine particle suspension systems, especially at high particle concentrations. Common examples are toothpaste, paint, oil well drilling muds, foams, and many food and pharmaceutical products. 4.1.2. Shear-Thinning (Pseudoplastic) Fluids Solutions and melts of flexible macromolecules and many suspensions exhibit pseudoplastic shear stress – shear rate behavior such as that illustrated in Figure 23 and shown specifically for a polymer solution in Figure 26. Here the ratio τ /γ˙ (i.e., η) is a decreasing function of γ. ˙ Such a fluid has a shear-thinning viscosity and is referred to as pseudoplastic. The decrease of viscosity with increasing rate of deformation can usually be attributed to the breakdown of a structure at the colloidal or molecular level. Macromolecules will become more aligned and hence

τ =k|γ| ˙ n−1 γ˙

(108a)

η=k|γ| ˙ n−1

(108b)

Equation (108) is known as the power law and sometimes as the Ostwald – de Waele model. This law is empirical and must fail for both high and low shear rates. The factor k is very temperature sensitive (and concentration sensitive for suspensions), but n is typically insensitive to temperature (and concentration) changes. Many empirical and semi- empirical equations have been proposed to represent the viscosity data for shear-thinning fluids. The Meter model: η=η∞ +

η 0 − η∞  α−1   1 + τ τ 

(109)

1/2

has four parameters, η 0 , η ∞ , τ 1/2 , and α with the following asymptotic behavior τ → 0, η≈η0 τ1/2

(109a)

τ → ∞, η ≈ η∞ τ1/2

(109b)

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Fluid Mechanics

and for    τ α−1 η0   1 L R

(130)

is necessary for the initiation of flow in a pipe for yield stress materials.

The yield stress of a material can be determined by extrapolation of the basic shear stress – shear rate data to zero shear rate. However, in view of the importance of the yield stress special techniques have been developed for its measurement. In one such technique, a vane is inserted into the fluid (see Fig. 30). The minimum torque Γ m required to initiate a rotation motion in the vane is recorded. The yield stress is then related to this torque by Γm =

πDv3 2



Hv 1 + Dv 3

 τy

(131)

In deriving Equation (131) it is assumed that the yielding surface is the cylindrical surface and the two end circular disks traced out by the vane. For short vanes, end effects may become important and must be taken into consideration [21].

4.2.4. Friction Factor – Reynolds Number Relationships General procedures applicable to all types of fluids are of course desirable because of the difficulty in classifying a fluid as one type or another, and because of the tendency of some fluids to change from one type to another with changing shear rate, temperature, or composition. At present generalizations are established only for time-independent fluids. Laminar Flow. For laminar flow of Newtonian fluids in circular pipes, the friction factor satisfies the relation (see Section 3.3.1 and Equation (48)) f=

16 Re

with the Reynolds number defined as Re = (D V )/η. For the laminar flow of any timeindependent fluid in a tube, the wall shear stress is given by τw =K 

Figure 30. The vane used for yield stress measurement Dv = vane diameter; H v = vane height



8V D

n (132)

where, in general, K
and n
are not constant, but depend on 8 V/D. When Equation (132) is substituted into the defining equation for the friction factor one obtains

Fluid Mechanics 16 f=  Re

(133)

where 

Re =



Dn V 2−n K  8n −1

(134)

which is a generalized Reynolds number valid for all time-independent fluids. In the special case where n
= 1 and K
= η, Equation (134) reduces to the Reynolds number for a Newtonian fluid. For a power-law fluid n
= n and Equation (122 a) is valid. LIVE GRAPH

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Figure 31. Friction factor design chart for non-Newtonian fluids —— Experimental regions; – – – extrapolated regions

Transition Flow. So far only the laminar flow of non-Newtonian fluids in tubes has been considered and the important question, when is the flow laminar, has been avoided. For Newtonian fluids, the question is answered by calculating the Reynolds number and comparing it to the empirically determined value of 2100 below which the flow is laminar. A similar criterion has been sought for non-Newtonian fluids by Metzner and Reed [22] and Dodge and Metzner [23]. Metzner and Reed first considered that stable laminar flow in tubes usually ends when the friction factor becomes < 0.008, which corresponded to a generalized Reynolds number of 2100. In the later work by Dodge and Metzner, which led to the generalized f – Re chart shown in Figure 31, laminar flow was found to end at generalized Reynolds numbers which increase slightly as n
decreases. At n
values of 1.0, 0.726, and 0.38, the transition region

403

appeared to begin at Re
of about 2100, 2700 and 3100, respectively. Others have examined the flow transition question for non-Newtonian fluids but the conservative conclusion remains: Re ≤ 2100

(135)

has been accepted and is used to establish whether the flow is laminar or turbulent for a non-Newtonian fluid. Turbulent Flow. The friction factor chart for time-independent inelastic fluids is reproduced in Figure 31. The friction factor is defined by !

/ .  1 0.4 4.0 log Re f 1−n /2 − 1.2 = 0.75 f (n ) n

(136)

Equation (136) reduces to the von Karman – Nikuradse equation (50) for Newtonian fluids when n
= 1. The solid lines in Figure 31 are based on experimental results, and the dashed lines are extrapolations made using Equation (136). The constants in Equation (136) were evaluated from experimental data. If the fluid does not obey the power law with an n
and K
which vary with 8 V/D, then a trial-and-error procedure is required to calculate f, τ w , or ∆p. First a value of ∆p is assumed, then the approximate values of n
and K
are found from the laminar flow curve (τ w versus 8 V/D) at the assumed τ w . From these parameters, Re
is calculated and f is obtained from Figure 31. If the pressure drop calculated from the friction factor does not agree with the original assumed value, another iteration is carried out. The process converges quickly.

4.3. Viscoelastic Fluid Mechanics Materials such as molten plastics, solutions of synthetic macromolecules and many biological fluids – apart from possessing the properties normally associated with purely viscous fluids – also exhibit a range of behavior which is normally associated with elastic solids. For this reason such materials have been given the name viscoelastic fluids. In a viscoelastic fluid the stress generated by deformation is no longer a simple function of the instantaneous rate of strain. It also depends, as in solids, on the strain and, for

Next Page 404

Fluid Mechanics

Figure 32. Rod-climbing by a viscoelastic fluid

large strains, on the strain history experienced by the fluid. As discussed earlier, in a purely viscous fluid, all the energy required to produce the deformation is dissipated as heat while in a perfectly elastic solid this energy is stored and is completely recovered when the forces acting on the solid are removed. In a viscoelastic fluid, this energy is partly dissipated and partly stored, the ratio of dissipation to storage depends on the fluid, the nature of the deformation, and the history of the deformation. Thus, depending on the flow, a viscoelastic fluid may exhibit, in varying degrees, the elastic properties of solids and the viscous properties of fluids. It is therefore not surprising that some of the flow behaviors of viscoelastic fluids have no parallel in purely viscous fluids. A number of these viscoelastic phenomena are reproduced in Figures 32, 33, 34, and 35. Figure 32 shows a rod slowly rotating in a beaker filled with a viscoelastic fluid. The rotation causes the viscoelastic fluid to climb up the rod. Under similar conditions, a purely viscous fluid is undisturbed by the slow rotation and its surface remains essentially flat. Rod climbing by a viscoelastic fluid is

Figure 33. Extrudate swell by a viscoelastic fluid

often referred to as the Weissenberg effect. When a viscoelastic fluid is extruded from a capillary (see Fig. 33) the resulting fluid stream may swell up to several times the capillary diameter. This phenomenon is known as extrudate swell. Newtonian fluids also exhibit extrudate swell at small Reynolds number, but the increase in diameter is only 13 %. Large extrudate swell is observed in polymer extrusion and clearly has to be taken into account in the design of extrusion shaping dies. Figure 34 shows a tubeless syphon, where a viscoelastic fluid is being sucked vertically upwards into a tube. The suction tube can be lifted to a large height (well over four to five tube diameters in the case shown) above the viscoelastic fluid without the syphoning action being interrupted. In comparison, for a purely viscous fluid, syphoning ceases as soon as the tube is lifted out of the fluid. The tubeless syphon illustrates the large tensile stress that can be sustained in a viscoelastic fluid. Figure 35 is a comparison of the streamline patterns formed by a Newtonian fluid and a viscoelastic fluid as they flow through a pipe contraction. Figure 35A shows a Newtonian fluid; Figures 35B – D show a viscoelastic

Previous Page Fluid Mechanics fluid at increasing flow rate. For the Newtonian fluid, two small recirculating vortices can be observed at the contraction plane. The size of the Newtonian vortex, shown at low Reynolds number, is not very sensitive to changes in flow rate. For viscoelastic fluids, the situation is quite different. At low flow rates the viscoelastic vortices are approximately of the same size as those for the Newtonian fluid, but they rapidly increase in size, up to four times the size of Newtonian vortex, as the flow rate is increased. The flow then becomes unstable and the vortices begin to pulsate, growing and decreasing in size in some erratic manner. Although the streamline pattern of the Newtonian fluid can be obtained by solving the Navier – Stokes equations, attempts to understand the viscoelastic flow patterns have so far been, at most, only partially successful [24]. The most likely source of the problem here is the present inability to describe, with the necessary degree of accuracy, the complex rheological behavior of the viscoelastic fluid when it undergoes shearing and stretching deformation as it flows through the contraction.

405

Some of the more fundamental aspects of viscoelastic fluid mechanics will be discussed in the next section. The treatment is, however, of a more descriptive nature. 4.3.1. Steady Shear Behavior of Viscoelastic Fluids Under steady shear the shear stress Txy generated in a viscoelastic fluid is related to the shear rate by Txy =η (γ) ˙ γ˙

where η is the shear-rate dependent viscosity of the fluid. Figure 26 is the plot of the viscosity of a solution of polyacrylamide, a typical viscoelastic fluid. Like most non-Newtonian fluids, most viscoelastic fluids exhibit shear thinning, and, over a range of shear rates, behave like a powerlaw fluid. However, if the normal components of the stress tensor in the x direction, Txx , and in the y direction, Tyy , are measured, it will be found that they are, in general, not equal. This is not observed in Newtonian and generalized Newtonian fluids. The difference in Txx and Tyy is referred to as the first normal stress difference N 1 . Similarly, it will be found that Tyy and Tzz are unequal, and the difference Tyy − Tzz is known as the second normal stress difference. The first and second normal stress difference are related to the shear rate by Txx − Tyy =ψ1 (γ) ˙ γ˙ 2 Tyy − Tzz =ψ2 (γ) ˙ γ˙ 2

Figure 34. Viscoelastic fluid being sucked up (tubeless syphon)

The functions ψ 1 and ψ 2 are known as the first and second normal stress coefficients, respectively. They are a function of the shear rate γ: ˙ η, ψ 1 and ψ 2 are known as the viscometric functions of the fluid which together describe the response of the viscoelastic fluid under steady shear. ψ 1 is measured in special viscometers, which measure the normal stress, as well as the shear stress, exerted by the test fluid when it is being sheared [8]. Figure 36 is a plot of the first normal stress difference exhibited by a polyacrylamide solution. For comparison, the shear stress is also included in Figure 36. It can be seen that the normal stress is very much larger than the shear stress which means that this viscoelastic fluid can be expected to behave quite differently from a purely viscous shear-thinning

406

Fluid Mechanics

Figure 35. Comparison of the flow patterns exhibited by a Newtonian fluid and by a viscoelastic fluid A) Newtonian fluid; B) – D) Viscoelastic fluid at increasing flow rates

fluid with similar shear stress versus shear rate behavior. The second normal stress difference is usually much smaller than the first normal stress difference and is very difficult to measure accurately. It is not normally reported in the literature. LIVE GRAPH

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ciated with viscoelastic fluids. Boger fluids are used extensively in the comparison of experimentally observed elastic response of fluids with that predicted by viscoelastic constitutive equations. They play an important role in the development and selection of such constitutive equations. 4.3.2. Behavior of Viscoelastic Fluids in Oscillatory Shear Flow

Figure 36. Normal stress versus shear rate for a viscoelastic fluid a) Shear stress τ ; b) First normal stress difference N 1

In laboratory investigations of viscoelasticity, it is very useful to be able to isolate the effects of shear thinning and fluid inertia from that of fluid elasticity so as to concentrate on the latter. For this purpose, a class of special test fluids have been developed. They are commonly referred to as Boger fluids or ideal elastic fluids. Typically, they are prepared by dissolving a high molecular mass solute in a highly viscous Newtonian solvent. The resulting solutions are characterized by their near constant viscosity, over a wide range of shear rates, and they exhibit all the elastic properties normally asso-

In steady shear motion the viscoelastic fluid is subjected to a constant rate of strain which does not reveal the dependence of the current stress on the strain history. Viscoelastic behavior under a time-dependent rate of strain is frequently studied experimentally by putting the fluid through an oscillatory shear flow, such as that generated when the fluid is sheared between two parallel plates: one of the plates is held stationary and the other oscillates sinusoidally with a frequency ω. The resulting shear rate and shear stress both have the same frequency ω, but there is a phase difference between them. If the oscillatory shear strain is represented by γ =γ0 sin (ωt)

(137)

where γ 0 is the amplitude of the sinusoidal shear strain, then the shear rate experienced by the fluid is given by γ˙ =γ0 ωcos (ωt) = γ˙ 0 cos (ωt)

(138)

γ˙ 0 (= ω γ 0 ) is used to denote the amplitude of the sinusoidal shear rate.

Fluid Mechanics It is customary to express the resulting shear stress in the form   Txy (ω) =γ0 G (ω) sinωt+G (ω) cosωt


(139)





The terms associated with G (ω) and G (ω) are the in-phase and the out-phase component of the shear stress. G
(ω) is usually referred to as the storage modulus of the fluid, it is related to the storage of energy. G

(ω) is the loss modulus of the fluid and is connected with energy dissipation. As indicated above, G
(ω) and G

(ω) are functions of ω, and for small amplitude oscillatory shear, they are independent of γ 0 . For purely viscous fluids G
(ω) is zero. G
(ω) and G

(ω) are material properties of a viscoelastic fluid. Two other material functions, related to G
(ω) and G

(ω) are also in common use. These are the complex viscosity of the viscoelastic fluid, defined by η  (ω) =G (ω) /ω

(140a)

. /1/2   η (γ) ˙ = η  (ω)2 +η  (ω) 

ω=γ˙

(143)

The Cox – Merz rule is often used to extend the steady shear viscosity data to large shear rates where only oscillatory measurements have been carried out. Further details and the limitations of this and other similar relationships are given in [4]. G
(ω) and G

(ω) or equivalently η
(ω) and

η (ω) are measured with special viscometers in which one of the shearing surfaces can be driven sinusoidally [8]. A typical set of oscillatory data is shown in Figure 37. These data were extracted from a comprehensive study carried out on polyethylene [25]. Oscillatory measurements are now routinely carried out by polymer manufacturers as a means of characterizing their products. Click here to view

η  (ω) =G (ω) /ω

(140b)

The function η
(ω) is called the dynamic viscosity of the fluid; η

(ω) does not seem to have a generally accepted name. For a Newtonian fluid η
(ω) is equal to the viscosity of the fluid and η

(ω) is zero. It is evident that a large number of material functions are needed just to describe the behavior of viscoelastic fluids in shear. At low shear rates and low angular frequencies, the material functions for steady-shear and oscillatory shear are related to one another. The dynamic viscosity at very low frequencies is related to the steady shear viscosity at very low shear rates by lim η  (ω) = lim η (γ) ˙

ω→0

(141)

γ→0 ˙

Furthermore, the storage modulus at very low frequencies is related to the first normal stress coefficient ψ 1 (γ) ˙ at very low shear rates by  ω→0

the steady and oscillatory data. An example of these is the Cox – Merz rule which relates η (γ) ˙ to η
(ω) and η

(ω):

LIVE GRAPH

and

lim

407

2G (ω) ω2



 = lim

γ→0 ˙

˙ N1 (γ) γ˙ 2

 = lim ψ1 (γ) ˙ γ→0 ˙

(142)

These limiting relationships are very useful for checking the consistency of experimental data on viscoelastic fluids. A number of other empirical relationships have been developed to relate

Figure 37. Storage and loss modulus of a low density polyethylene [25]
Storage modulus G ;
Loss modulus G

4.3.3. Examples of Constitutive Equations for Viscoelastic Fluids Numerous constitutive equations have been proposed to describe the diverse relationship, observed in viscoelastic fluids, between the stresses and the various kinematic quantities used to measure the strain and the rate of strain experienced by the fluid. One of the earliest viscoelastic constitutive equations is the linear Maxwell equation. In shear flow, the stress Txy and the shear rate γ˙ are related by a constitutive equation of the form

408 λ

Fluid Mechanics

dTxy +Txy =η0 γ˙ dt

(144)

λ and η 0 are the two material parameters of the linear Maxwell equation where λ is the time constant of the fluid and η 0 is its viscosity. Under steady shear, the stress is not a function of time, hence the first term on the left-hand side vanishes and the linear Maxwell equation reduces to the Newtonian constitutive equation where the stress is related to the shear rate via the viscosity η 0 . However, the introduction of the time constant λ means that, in general, the stress is no longer a unique function of the instantaneous shear rate. For example, during the start up of the shear flow or in oscillatory shear test, the stress is out of phase with the instantaneous shear rate. In particular, in oscillatory shear test the oscillation of the top plate is given by A sin ω t. The velocity of the top plate is u0 =Aωcosωt

For most viscoelastic fluids of practical interest, experimental conditions are such that the inertia terms in the Cauchy equations of motion, Equation (24), are small and can be ignored. As a result, the velocity variation across the gap separating the plates is linear and the velocity profile is given by u (y,t) =

Aωy cos (ωt) H

(145)

where H is the gap between plates. The shear rate is given by Aω γ˙ (t) = cos (ωt) =γ0 ωcos (ωt) H

(146)

where γ 0 = A/H is used to denote the amplitude of the shear strain. According to the linear Maxwell constitutive equation, the stress exerted by the fluid on the oscillating top plate is given by λ

dTxy +Txy =γ0 ωcos (ωt) dt

(147)

This equation can be solved to give  η0 λω 2 η0 ωcosωt + sinωt 1+λ2 ω 2 1+λ2 ω 2 +transient terms 

Txy =γ0

(148)

The transient terms decay away exponentially. As oscillatory measurements are taken at large

t, the transient terms can be ignored. The resulting sinusoidally fluctuating stress can be written in the form 

Tyx =γ0

 η0 ω η0 λω 2 cosωt+ sinωt 2 2 2 2 1+λ ω 1+λ ω

(149)

From the definition of G
(ω) and G

(ω) and of η
(ω) and η

(ω) it can then be seen that for a linear Maxwell fluid η0 λω 2 1+λ2 ω 2

G (ω) =

η0 1+λ2 ω 2

η  (ω) =

G (ω) = η  (ω) =

η0 ω 1+λ2 ω 2

η0 λω 1+λ2 ω 2

(150a) (150b)

By choosing the appropriate value for the viscosity η 0 and time constant λ, these functions can provide an adequate description of the observed behavior of real viscoelastic fluids. The linear Maxwell equation does not satisfy all the requirements of modern continuum mechanics for a theoretically sound constitutive equation. Numerous modifications of the Maxwell equations can be found in the literature. Some of the modifications were made so as to satisfy the requirements of continuum mechanics, others resulted from improved understanding of the molecular structure and dynamics of the polymer constituents of viscoelastic fluids [4], [26]. A constitutive equation that has been very popular is the Oldroyd-B equation. It relates the deviatoric stress tensor τ to the strain tensor γ by an equation of the form

λ1 τ (1) +τ =η0 γ (1) +λ2 γ (2)

(151)

where λ1 and λ2 are the relaxation and retardation time of the Oldroyd-B fluid, respectively and η 0 is, as before, the viscosity of the fluid. Subscripts (1) and (2) are used to denote the first and second convected derivatives of the stress and the strain tensor. The definition of the convected and other derivatives can be found in [4]. The introduction of these derivatives is required by the theory of constitutive equations in continuum mechanics. The derivative γ (1) is equal to the rate of strain tensor e. The Oldroyd-B equation satisfies all the requirements of continuum mechanics. In terms of the three material parameters, λ1 , λ2 , and η 0 the viscometric functions (i.e., the steady shear properties) and the oscillatory behavior of the Oldroyd-B equation can be shown to be

Fluid Mechanics η=η0 ψ1 = 2η0 (λ1 − λ2 ) ψ2 = 0 G (ω) =

η0 (λ1 − λ2 ) ω 2

G (ω) = η  (ω) =

1+ (λ1 ω)2

η0 ω 1+λ1 λ2 ω 2

1+ (λ1 ω)2

η0 1+λ1 λ2 ω 2

η  (ω) =

(152a)

1+ (λ1 ω)2 η0 (λ1 − λ2 ) ω 1+ (λ1 ω)2

(152b)

409

agreement between experimental observations and theoretical predictions could be observed for the first time. It has also been possible to relate the parameters in the constitutive equations to the physical properties of the constituents of Boger fluids. LIVE GRAPH

Click here to view

(152c)

As with the linear Maxwell equation, the parameters of the Oldroyd-B equations are chosen so that these functions give an approximate description of the measured viscometric and oscillatory properties of real fluids. It is interesting to note that both linear Maxwell and the Oldroyd-B equations give a constant viscosity. This means that these equations are particularly suitable for describing the rheological behavior of viscoelastic fluids that are not shear-thinning. In Figure 38 the measured G
(ω) and η
(ω) for a silicone oil are plotted against ω. This fluid can be regarded as a non-shear-thinning viscoelastic fluid ( Boger fluid). The expression for these two quantities according to the Maxwell and the Oldroyd-B equations are also plotted on the figure. Standard procedures have been applied to extract the fluid parameters of these two constitutive equations [27]. The Oldroyd-B equation provides a better description of the rheological behavior of this viscoelastic fluid than the simpler Maxwell equation. For comparison, the steady shear data for this silicone oil are also included in Figure 38. The experimental data confirm the limiting relationship between N 1 and 2 G
and between η and η
at low γ˙ and ω. The linear Maxwell and the Oldroyd-B equations are just two examples of the large number of viscoelastic constitutive equations in the literature. The construction of constitutive equations is guided by the principles of continuum mechanics and by the understanding of the dynamical properties of the macromolecules and the solvent that make up the viscoelastic fluids. For more details on these and related topics see [28], [29]. One of the recent developments in viscoelastic fluid mechanics is the use of Boger fluids to gain a physical understanding of constitutive equations. With these ideal test fluids good

Figure 38. Comparison of the observed normal stress and storage modulus of a silicone oil against that predicted by the Maxwell and the Oldroyd-B constitutive equations, fluid: silicone oil ◦ N 1 , Pa;  2 G , Pa; • η, Pa s;  η  , Pa s

4.3.4. Extensional Behavior of Viscoelastic Fluids In many of the flow fields encountered in industrial processes the fluid undergoes a stretching deformation. A specific example is the deformation suffered by a fluid element when it is forced to flow through a channel or tube of gradually or abruptly changing cross-sectional area. This kind of deformation is usually referred to as extensional deformation. In extensional deformation the macromolecules in the viscoelastic fluid are being stretched and may generate large tensile stresses as a result of this. The behavior of a viscoelastic fluid in extensional deformation can be quite different from its behavior in shear deformation. An idealized flow field that can be used to investigate the extensional behavior of a viscoelas-

410

Fluid Mechanics

tic fluid is shown in Figure 39. The tensile stress Tzz applied to stretch the specimen of test fluid, in the form of a circular cylinder, is measured. The applied stress Tzz is varied so that the axial velocity gradient is uniform within the fluid sample and remains constant in time, i.e., ezz =

duz 1 dL = ε˙0 = dz L dt

(153)

where ε˙0 is the constant extensional rate, which can be identified with the ratio of the instantaneous rate of increase in length of the fluid sample dL (t)/dt to its instantaneous length L (t). It can be assumed that the azimuthal velocity component in the test specimen is zero. The continuity equation for an incompressible fluid then requires the velocity components in the axial and radial directions to be given by uz = ε˙0 z 1 ur = − ε˙0 r 2 uθ = 0

(154)

The rate of strain tensor, in cylindrical coordinates, is given by   ε˙  0  e=  0  0

0 − 12 ε˙0 0

  0   0   − 12 ε˙0 

(155)

A characteristic of extensional flow is the vanishing off-diagonal elements in the rate of strain tensor. This idealized constant-rate fluid stretching experiment, known as the steady axisymmetric extensional flow, is the extensional equivalent of steady shear flow. It is a difficult experiment to carry out and consequently reliable extensional properties of fluids are difficult to obtain. One of the difficulties becomes apparent when Equation (153) is integrated to give ε˙0 t

L=L 0 e

(156)

where L 0 is the initial length of the cylindrical fluid test specimen. The exponential increase in length of the test specimen makes it very difficult to sustain measurement for a period long enough for the effects of initial conditions to be ignored. Specialized rheometers have been designed to overcome this and other related problems [8]. However, good elongational flow measurements are rare.

Figure 39. Schematic diagram of extensional flow

An important extensional property of a viscoelastic fluid is its extensional viscosity η el . In terms of the idealized extensional measurement described above, η el is defined by ηel =

Tzz − Trr ε˙0

(157)

The difference between the normal stress in the axial direction Tzz and that in the radial direction Trr gives the deviatoric tensile stress arising from the extensional deformation. The definition of extensional viscosity is analogous to the definition of shear viscosity. Like its shear counterpart, the extensional viscosity is in general a function of the extensional rate ε˙0 . The extensional viscosity predicted by different constitutive equations can be found in the literature [30]. The extensional viscosity of a Newtonian fluid with constant viscosity η 0 is relatively simple to obtain. The total normal stress in the axial and the radial directions are given by duz = −p+2η0 ε˙0 dz dur Trr = −p+2η = −p − η0 ε˙0 dr Tzz = −p+2η

Hence ηel =

Tzz −Trr = 3η0 ε˙0

(158)

According to this result, the extensional viscosity of a Newtonian fluid is three times its shear viscosity. This result has been verified experimentally by Trouton in 1906 and extensional viscosity is often referred to as Trouton viscosity [31]. The extensional viscosity of viscoelastic fluids is, in general, much larger than three times it shear viscosity. The extensional viscosity obtained for the Oldroyd-B equation can be shown to be  ηel (ε˙0 ) = 3η0

(1−λ2 ε˙0 ) (1 + 2λ1 ε˙0 ) (1+λ1 ε˙0 ) (1 − 2λ1 ε˙0 )

 (159)

Fluid Mechanics At low extensional rate, the Newtonian result of 3 η 0 is again obtained. As the extensional rate is increased, the extensional viscosity increases rapidly. This large extensional viscosity is in agreement with the observed extensional behavior of viscoelastic fluids. However, according to the Oldroyd-B equation, the extensional viscosity grows without bound as the extensional rate approaches λ1 /2. This singularity in the relationship between extensional rate and extensional viscosity is physically unrealistic and is an indication that the Oldroyd-B equation must be modified. Comparison of the measured extensional viscosity with that predicted by a constitutive equation provides an additional test of the validity of the constitutive equation. It is clear that fluid elasticity has greatly increased the complexity of fluid motion. There are a large number of flow phenomena which are only observed in viscoelastic fluids. Many of these have practical implications for industrial processes. For further details on the mechanics of viscoelastic fluids see [4], [19], [26]. 4.3.5. Accelerating and Decelerating Flows of Viscoelastic Fluids Fluid elasticity does not affect the energy requirements for fully- developed/laminar tube flow. However, in accelerating and decelerating flows, such as in the entrance and exit of a tube, the influence of fluid elasticity becomes quite pronounced. Entrance and exit effects are conveniently discussed with reference to the capillary rheometer, an important instrument for the measurement of fundamental flow properties of fluids. One essential feature of the capillary viscometer is that the wall shear stress can be directly determined from the measured fullydeveloped flow pressure drop (see Eq. 87). However, for a laboratory-scale capillary viscometer it is more practical to measure the overall pressure drop from the upstream fluid reservoir to the exit of the tube, rather than the pressure drop in the tube itself. Schematics of a pressuredriven and a ram-driven capillary rheometer are shown in Figure 40. In the pressure-driven instrument, the independent variable is the shear stress, whereas in the ram-driven instrument the independent variable is the shear rate. Shear

411

rates of > 103 s−1 are easily obtained in the capillary rheometer. This is one of its main advantages over conventional rotation instruments. Assuming that the flow is laminar and that the fluid is a time-independent inelastic or viscoelastic fluid, corrections to the measured pressure drop ( pgas − patm ) for pressure- driven rheometers and ( papp − patm ) for ram- driven rheometers may have to be made due to the following effects: 1) 2) 3) 4)

Head of fluid above the tube exit Kinetic energy effects Tube entrance and exit losses Weight of filament after exit (this will be ignored here)

pgas −patm = ∆pfd1 +∆pen +∆pfd2 +∆pex +∆KE +∆P E

(160)

Equation (160) is a mechanical energy balance written between surface 0 and 2 as shown in Figure 40: ∆pfd1 and ∆pfd2 are the fully- developed flow pressure drops in the reservoir and tube, respectively; ∆pen is the entry loss over and above the fully-developed flow loss for the flow into the tube; and ∆pex is the exit loss over and above the fully-developed flow loss for flow out of the tube to the atmosphere; ∆KE and ∆PE are the kinetic and potential energy losses, respectively. Kinetic and potential energy effects can normally be neglected for polymer melts and for concentrated solutions and suspensions. In addition, the reservoir diameter is usually much greater than that of the downstream tube, so that the fully- developed pressure drop in the upstream tube or reservoir can also be neglected. Thus, Equation (160) becomes pgas −patm = ∆pen +∆pfd2 +∆pex

(161)

where ∆pfd2 =

2τw L R

If high L/R capillary tubes are used for the pressure drop – flow rate measurements (L/R ≥ 200), fundamental shear stress – shear rate data can be determined directly with the capillary rheometer because 2τw L >> ∆pen +∆pex R

(162)

412

Fluid Mechanics

Figure 40. Schematic diagram of (A) pressure and (B) ram driven capillary rheometers

Therefore τw =

R (pgas −patm ) 2L

(163)

i.e., the wall shear stress can be directly determined from the measured pressure drop, and the wall shear rate is specified by the Rabinowitsch – Mooney equation  γ˙ w =

3n +1 4n



8V D

 (164)

where n =

dlnτw

(165)

dln 8V D

However, for many materials in tubes with a high L/R ratio, the pressures required to obtain shear rates of interest are prohibitive, and methods have been derived to correct for entry and exit effects when low L/R capillary tubes are used. The method most commonly used is that first suggested by Bagley in 1957 [32]: ∆pen +∆pex =

2τw L e = 2τw e R

(166)

where e is the dimensionless extra length of tube which defines the exit and entry losses in excess of the fully-developed flow losses. pgas −patm = 2τw (L/R+e)

and

(167)

τw =

pgas −patm 2 (L/R+e)

(168)

The end correction e is determined by first plotting ( pgas − patm )/(2 L/R) versus 8 V/D on log – log coordinates. Different lines or curves such as those illustrated in Figure 41 will be obtained for different L/R tubes. From such a graph pgas − patm can be determined as a function of L/R for various values of 8 V/D. A linear plot of pgas − patm versus L/R is then made for various values of 8 V/D. The end correction e is the intercept on the abscissa of this plot (see Fig. 42). The end correction is then a known function of 8 V/D and hence the corrected value of τ w can be computed as a function of 8 V/D from Equation (168) whereas the true shear rate can be computed with the aid of Equation (164) and the slope of a log – log plot of τ w versus 8 V/D (Eq. 165). The capillary rheometer is an important instrument for the measurement of the shear stress as a function of shear rate, particularly for molten polymers where measurements can be made at processing shear rates. It is also an instrument for determining quantitative measurement of fluid elasticity. For instance, the end correction is strongly influenced by the elasticity of the fluid. Figure 43 shows end correction results for molten polymers while Table 2 lists the end

Fluid Mechanics

413

Figure 41. First plot of capillary rheometer data for determination of the end effect

Figure 42. Determination of the end correction

correction for inelastic power-law fluids as determined from the numerical solution of the tube entry and exit flow problem [34].

also highly elastic. Even higher end correction values than those illustrated for molten polymers in Figure 43 have been observed for concentrated polymer solutions. The end correction is used to differentiate between different polymers and in fact to distinguish between different grades of the same polymer (see curves d and e in Fig. 43). Significantly different behavior is indeed observed in inlet and exit flows of inelastic and viscoelastic fluids. Figure 44 shows streamline photographs obtained for an inelastic Newtonian fluid (Fig. 44A) and for an elastic fluid which shows no shear thinning (Fig. 44B). Both fluids have identical viscosities. Flow is from left to right in the photographs and represents flow from the reservoir into the tube of a capillary rheometer. For the inelastic Newtonian fluid a small secondary flow is present in the corner of the reservoir. This cell remains essentially constant in size with increasing flow rate and ultimately disappears for Reynolds numbers > ca. 0.1 when the fluid inertia starts to become important. For the viscoelastic fluid of the same viscosity (2000 Pa s) the secondary flow is much larger. The size of the secondary flow grows with increasing flow rate for Reynolds numbers < 0.1 and continues to grow until the flow becomes unstable. For a molten polymer being extruded through a die, the flow instability results in a distorted extrudate. The flow phenomena, called melt fracture, represents an upper limit on the rate at which a molten polymer can be

Table 2. The end correction for inelastic power-law fluids Power-law index, ∆pen /2 τ w n

∆pex /2 τ w

e

1.0 0.5 0.3 0.167

0.246 0.26 0.28 0.59

0.834 1.60 2.04 2.92

0.588 1.34 1.76 2.33

LIVE GRAPH

Click here to view

Figure 43. Typical values of the end correction as a function of shear rate [33] a) Polystyrene; b) Poly(methyl methacrylate); C) Lowdensity polyethylene MFI 2.0; d) High- density polyethylene MFI 0.25; e) High- density polyethylene; f ) Polyacetal

In the absence of fluid elasticity, the end correction increases as a result of the shear-thinning characteristics of the fluid, but not to the extent of the values observed for many commercial polymers which in general are shear-thinning and

414

Fluid Mechanics

extruded. Similar flow instabilities are not observed in tubular inlet flows for inelastic fluids.

where the subscript w indicates that the stresses are to be evaluated at the wall shear rate for fullydeveloped flow. Die swell ratios of 2 or more are not unusual in the processing of molten polymers. Since the die swell depends not only on the particular polymer but also on the operating conditions such as temperature and flow rate, the industrial problems related to extrudate swell are particularly complex and challenging. Fluid elasticity does not effect the energy requirements for fully-developed laminar tube flow, that is fviscoelastic =finelastic

Figure 44. Comparison of entry flow patterns for a Newtonian and a non-shear thinning elastic fluid with the same viscosity Reservoir to tube diameter ratio is 7.67 A) Newtonian; B) Non-shear thinning elastic

For a viscoelastic fluid the exit flow from a capillary tube also differs significantly from that of an inelastic fluid. For fully-developed flow of a viscoelastic fluid in a tube, a tension along the streamlines associated with the deviatoric normal stresses is present. When the fluid passes through the exit of the tube to the atmosphere, it will relax the tension along the streamlines by contracting in the longitudinal direction. For an incompressible fluid, this results in a lateral expansion of the fluid. This relaxation phenomena results in extrudate swell (see Fig. 33), where the diameter of the extrudate De is significantly greater than the internal tube diameter. For large tube length to diameter ratios the extrudate swell can be estimated as follows [35]:   1/6  De 1 τ11 −τ22 2 = 0.1+ 1+ 2 2τ12 D w

(169)

The equivalent conclusion, however, is not applicable for viscoelastic fluids in turbulent flow. Here considerable reduction in the friction factor below the expected inelastic value is observed. This drag reduction phenomenon, first observed by Toms in 1948, has received considerable attention in the literature because of its possible commercial significance. Parts per million of certain polymers dissolved in water can reduce the friction factor considerably; drag reduction by as much as 90 % has been observed. Figure 45 shows some friction factor – Reynolds number data for a very dilute aqueous polymer solution. The data agrees with the laminar flow prediction but deviates from the solvent line characterization for turbulent flow for Reynolds numbers > 104 . LIVE GRAPH

Click here to view

Figure 45. Friction factor – Reynolds number data for Boger fluids, distilled water, and a solution of 296 ppm by weight of polyethylene oxide in distilled water in an 8.46 mm-diameter pipe [36]. (Reproduced by permission of the American Inc 1975 AIChE) stitute of Chemical Engineers   Boger fluid; ◦ Distilled water; • 296 ppm by weight polyethylene oxide

Fluid Mechanics

5. Numerical Methods in Fluid Mechanics Many of the flows of practical importance have irregular geometry and their boundary conditions are often mathematically difficult to handle. Furthermore, the fluid properties may vary rapidly with flow and thermal conditions. These complications mean that the set of equations that describe the fluid motion are unlikely to have an analytic solution. Even in those rare cases where an analytic solution can be found, it may be in a form, that is not convenient for practical use, e.g., as a slowly converging infinite series. For such flows, it is necessary to obtain an approximate numerical solution to the governing equations. Numerical methods have always been used, but their development has been greatly accelerated due to the availability of modern digital computers that perform the repetitive calculations. Computational fluid mechanics is now a well established subject capable of producing highly accurate predictions of the fluid motion at great speed. The ever increasing computing power combined with the graphical capabilities of modern computers has established numerical solution of fluid mechanical problems as an extremely effective way of generating and presenting information about fluid flows. In this short section, the principles of two general numerical methods for solving the steady-state Navier – Stokes equations and the continuity equation in two dimensions will be described. Further details are given in [37–41].

5.1. Finite Difference Method (→ Mathematics in Chemical Engineering, Chap. 7.3.) In the finite difference method, the partial derivatives in the governing equations are approximated by finite differences. To do this, the flow field of interest is divided into a regular grid as shown in Figure 46. At any grid point (i, j ), the first derivative ∂ux /∂x in the continuity equation can be approximated by 

∂ux ∂x

 = i,j

uxi+1,j −uxi−1,j 2∆x

(170)

415

Subscripts are used to indicate the grid point at which the variable is to be evaluated. The error incurred in this approximation is dependent on the size of the grid ∆x. It approaches zero as ∆x approaches zero. A similar approximation can be obtained for ∂uy /∂y. It is common practice to make ∆y equal to ∆x. The finite difference approximation of the incompressible continuity equation at the grid point (i, j ) is uxi+1,j +uyi,j+1 −uxi−1,j −uyi,j−1 = 0

(171)

This is an algebraic equation for the unknowns uxi+1, j , uxi−1, j etc. A similar equation can be written down for each of the grid points, 1 ≤ i ≤ Ni and 1 ≤ j ≤ Nj , where Ni and Nj are the number of grid points in the x and y directions, respectively.

Figure 46. Finite difference grid

At the grid point (i, j ), ∂ 2 ux /∂x 2 , a typical second derivative, can be approximated by the finite difference: 

∂ 2 ux ∂x2

 = i,j

uxi+1,j +uxi−1,j −2uxi,j ∆x2

(172)

Replacing all the partial derivatives by their finite difference approximations, the following finite difference equivalent of the x- component of the Navier – Stokes equation is obtained:

416

Fluid Mechanics the unknown velocity components. These equations are not easy to solve directly. In one of the numerical procedures developed to deal with the nonlinear terms, the typical inertia term ux ∂ux /∂x is approximated by

   uxi+1,j −uxi−1,j  uxi,j + 2∆x  uyi,j

uxi,j+1 −uxi,j−1



2∆y   uxi+1,j +uxi−1,j −2uxi,j pi+1,j −pi−1,j +η = 2∆x ∆x2   uxi,j+1 +uxi,j−1 −2uxi,j (173) +η ∆y 2

A similar finite difference equation can be written for the y- component of the Navier – Stokes equation. The finite differences have converted the differential equations describing the fluid flow to a set of simultaneous algebraic equations for the unknown velocity components and pressure at the grid points. After the incorporation of the boundary conditions, they can, in principle, be solved for the unknown variables. However, because of the complexity of the Navier – Stokes equations, the success of the finite difference method depends critically on how the finite difference equations are handled. They often have to be completely reorganized and rewritten in various forms that are numerically and computationally more manageable. For example, in this description of the finite difference method, the physical variables ux , uy and p appear explicitly. The finite difference method can be reformulated in terms of the stream function Ψ , or a combination of stream function and the vorticity ω. For two-dimensional flow ω is defined as the z component of curl of the velocity vector: ω= −

∂ux ∂uy + ∂y ∂x

(174)

In two-dimensional flows, the stream function or stream function-vorticity formulations are, in fact, used in preference to the velocity – pressure formulation. Further details, particularly regarding the choice of numerical schemes for handling the resulting equations, can be found in the references listed. In this introductory description of the finite difference method, the partial derivatives are approximated by central differences. Other finite difference approximation schemes can be applied. Irrespective of the finite difference scheme employed (because of the inertia terms on the left-hand side of the Navier – Stokes equations) the resulting algebraic equations are nonlinear in

  (n) (n) ∂ux (n−1)  uxi+1,j −uxi−1,j  ux ≈ uxi,j ∂x 2∆x

(175)

Here the superscript n is used to denote the values of the unknowns at the nth iteration. At the nth iteration the values of the (n − 1)th iteration are already known. This way the nonlinear terms are approximated as linear ones and the finite difference equations become a set of linear algebraic equations for which many computationally efficient methods of solution are available. The calculation is terminated when the difference between two successive iterations is less than some preassigned small tolerance. To start the iterative procedure, it is necessary to supply the zeroth iteration of the unknown vari(0) ables ui, j , etc. In the absence of any information, these zeroth iteration values can be taken to be zeros. Convergence of this straightforward iterative scheme is not guaranteed and even if it converges, it may require a large number of iterations. Convergence, for example, becomes slower and slower as the Reynolds number of the flow is increased and may fail completely when the Reynolds number becomes too large. Many numerical procedures have been developed to improve the convergence performance. Iterative procedures have also been developed to deal with the nonlinearity introduced by variable fluid properties. Figure 47 shows the streamlines of the flow of a Newtonian fluid past a cylinder at increasing Reynolds number obtained by finite difference computation using the stream function – vorticity formulation [42]. Numerical convergence at a Reynolds number as high as 100 was achieved by a specially developed iteration scheme. The effects of increasing Reynolds number show up clearly in the increasing size of the recirculating wake behind the cylinder. These numerical results are in excellent agreement with the available experimental data.

Fluid Mechanics

417

Figure 47. Streamlines around a cylinder at increasing Reynolds number obtained by finite difference computation [42] (reproduced with permission of Cambridge University Press) A) Re = 5; B) Re = 7; C) Re = 10; D) Re = 20; E) Re = 40; F) Re = 70; G) Re = 100.

5.2. Finite Element Method (→ Mathematics in Chemical Engineering, Chap. 7.5., Chap. 7.6.) The finite element technique was originally developed by structural engineers for calculating the stresses and strains in structures of complex shapes. It has been generalized and developed as a numerical technique for solving differential equations, particularly partial differential equations. As with the finite difference method, the differential equations for the unknown variables are converted into a set of algebraic equations which can then be solved for the unknown variables at discrete points. The application of finite element computation to fluid mechanics started in the early 1960s and has since been greatly refined. A brief description of the finite element technique based on the Galerkin approach will be presented here. The mathematical principles and numerical details of the Galerkin finite element method, particularly the handling of boundary conditions, can be found in [37–41].

In finite element approximation, the flow field of interest is again subdivided by a grid into a large number of small connected domains of different shapes and sizes known as elements. (Following the general practice in finite element computation, the grid points, referred to as the nodal points, are identified by a single subscript i instead of the double subscripts i, j in finite difference.) The unknown variables, ux , uy , and p are approximated by series of the form ux (x,y) =

N 

uxi Φi (x,y)

i=1

uy (x,y) =

N 

uyi Φi (x,y)

(176)

i=1 

p (x,y) =

N 

pi Ψi (x,y)

i=1

where N (and N
, see below) is the total number of nodal points in the grid; uxi , uyi , and pi are unknown numerical coefficients to be determined; Φi (x, y), for i = 1 to N, and Ψi (x, y) for

418

Fluid Mechanics

i = 1 to N
are two sets of known functions defined for each of the nodal points. They are usually referred to as the interpolation functions. While one is free to choose the form of these functions, Φi (x, y) and Ψi (x, y) are usually restricted to simple functions such as low-order polynomials. In many finite element computations Φi (x, y) are taken to be quadratic functions of x and y and Ψi (x, y) to be linear functions of x and y. Both Φi (x, y) and Ψi (x, y) take the value of unity at the nodal point i and decrease to zero at all the neighboring nodes surrounding node i. These functions are also defined to be identically zero beyond these neighboring nodes, i.e., Φi and Ψi are nonzero only in those elements which have node i as one of its nodal points. The quadratic interpolating function Φi (x, y) is also defined to be zero at the midpoints (the mid-side nodes) of each side of the elements that have node i as one of its nodes. The total number of nodes N for the quadratic interpolation functions includes the mid-side nodes. Thus N
, the total number of nodes for the linear interpolation function, is < N. In order that the series representation approaches the true solution as N and N
become large, the interpolation functions have to satisfy a number of mathematical requirements. For example, all the interpolation functions have to be continuous across the common boundary of two adjacent elements. Typical examples of the interpolating functions for quadrilateral and triangular elements are shown in Figure 48. In the finite element method, elements of different sizes and shapes can be mixed relatively easily. This greatly simplifies the discretization of irregularly shaped flow fields and is the major advantage of the finite element method over the finite difference method. When the series representations (Eq. 176) are substituted in the equations of motion and the continuity equation, the following are obtained: 

N 



uxi Φi

i=1 N 

 uyi Φi

N  i=1



N  i=1

uxi

i=1

i=1

+

N 

pi

uxi

∂Φi ∂x

∂Φi ∂y



+





N N  ∂Ψi ∂ 2 Φi  ∂ 2 Φi uxi + u −η xi ∂x ∂x2 ∂y 2 i=1 i=1

=Rx (x,y)

(177a)

 

N 

 uxi Φi

i=1 N 

 uyi Φi

N  i=1



N 

uyi

i=1

i=1

+

N 

pi

i=1

uyi

∂Φi ∂x

∂Φi ∂y

 +





N N  ∂Ψi ∂ 2 Φi  ∂ 2 Φi uyi + u −η yi ∂y ∂x2 ∂y 2 i=1 i=1

=Ry (x,y) N  i=1

uxi

N ∂Φi  ∂Φi uyi + =Rc (x,y) ∂x ∂y i=1

(177b) (177c)

In general the series representations do not satisfy the governing equations and leave behind residuals Rx (x, y), Ry (x, y), and Rc (x, y). These residuals are functions of the unknown coefficients uxi , uyi , pi , as well as x and y. An approximate solution is obtained by finding the set of the unknown coefficients that minimize, in some sense, these residuals. In the case of Galerkin finite element method, this is done by requiring the Rx (x, y) and Ry (x, y) to be orthogonal to all the Φi (x, y) and Rc (x, y) to be orthogonal to all the Ψi (x, y) over the region of interest, i.e.,
Rx (x,y) Φk (x,y) dA= 0 k= 1toN

(178a)

Ry (x,y) Φk (x,y) dA= 0 k= 1toN

(178b)

Rc (x,y) Ψk (x,y) dA= 0 k= 1toN 

(178c)

A




A




A

A is the area occupied by the flow field. After performing the integration, these become a set of algebraic equations which can be solved for the unknown coefficients uxi , uyi , pi . It will be found that the number of independent algebraic equations, after taking into consideration the boundary conditions, is exactly equal to the number of unknown numerical coefficients in the series representation. From the definition of the interpolation functions, it is clear that at node j, x = xj and y = yj , the only nonzero terms in the series representation are Φj (xj , yj ) and Ψj (xj , yj ) where they take on the value of unity. Hence ux (xj ,yj ) =uxj Φj (xj ,yj ) =uxj

(179a)

Fluid Mechanics

419

Figure 48. Typical finite element linear and quadratic interpolation functions

uy (xj ,yj ) =uyj Φj (xj ,yj ) =uyj

(179b)

p (xj ,yj ) =pj Ψj (xj ,yj ) =pj

(179c)

Thus the numerical coefficients in the series representation are also the values of the unknown velocity components and pressure at the nodal points. A considerable amount of algebraic manipulation and numerical integration must be carried out to set up and solve these algebraic equations. Some of these steps require considerable ingenuity in computer programming. In a well designed finite element computer package most of these are taken care of by the computer requiring a minimal amount of human intervention. Because of the inertia terms and variable fluid properties, the algebraic equations are again nonlinear in the unknown coefficients. A procedure similar to that outlined above for handling the set of nonlinear algebraic equations arising from the finite difference method can also be applied here, the details for implementing the iterative procedure may, of course, be quite different. In the finite element method ux , uy , and p appear explicitly in the computation. It is again possible to formulate a finite element scheme where the unknown variable is the stream function or a combination of stream function and vorticity. In the case of viscoelastic fluids, it is also common practice to have the deviatoric stress components τxy , τxx , and τyy as well as ux , uy ,

and p appearing explicitly in the finite element computation scheme.

Figure 49. Finite element mesh and streamlines in a branching channel at increasing Reynolds number obtained by finite element computation A) Finite element mesh; B) Streamlines for Re = 10; C) Streamlines for Re = 100

Figure 49 shows the streamlines obtained by finite element computation for the flow in a branching channel. The finite element grid employed is shown in Figure 49A. The use of a small number of triangular elements together with rectangular elements has made the discretization of the irregularly shaped flow field a relatively simple task. A Newtonian fluid, traveling vertically upwards, enters the channel as a single stream and leaves as two separate streams, generally of unequal size. The effects of increasing Reynolds number, based on upstream

420

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channel width, now show up as a recirculating zone in the branch channel. At low Reynolds number, the flow is divided equally between the two branches (Fig. 49B). As the Reynolds number is increased (Fig. 49C), the flow through the straight channel increases at the expense of the branch channel.

5.3. General Remarks Both the finite difference method and the finite element method are now routinely used to solve fluid flow problems. A detailed discussion of the relative merits of these two methods would not serve a very useful purpose and is certainly out of place here. It is, however, generally agreed that the principles of the finite difference method are easier to understand and simpler to implement on computers. In contrast the finite element method can cope with irregular flow fields very efficiently, better than the finite difference method. The development of a computer program to solve a nontrivial flow problem numerically is a major undertaking that requires expertise not only in fluid mechanics, but also in numerical analysis, computer programming, and organization. It is not a task that can normally be carried out by a single person working in isolation. A number of commercial software packages specially designed for solving fluid flow problems are now available. Some of these packages are very well designed and simple to use. Most of them incorporate special procedures for dealing with highly nonlinear flow problems. They usually have built-in computer graphics for efficient presentation of the large amount of numerical results generated by these packages. Intelligent use of these computing tools coupled with a sound understanding of the physics and mathematics of fluid mechanics has lead rapidly to advances in the solution of flow problems that would otherwise be intractable.

6. References 1. M. M. Denn: Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs 1980. 2. S. W. Churchill: Viscous Flow the Practical Use of Theory, Butterworths, Boston 1988.

3. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, J. Wiley & Sons, New York 1960. 4. R. B. Bird, R. C. Armstrong, O. Hassager: Dynamics of Polymeric Liquids, 2nd ed., vol. 1, J. Wiley & Sons, New York 1987. 5. H. Schlichting: Boundary-Layer Theory, 7th ed., McGraw-Hill, New York 1977. 6. K. Walters: Rheometry, Chapman and Hall, London 1975. 7. R. W. Whorlow: Rheological Techniques, Ellis Horwood, London 1980. 8. J. M. Dealy: Rheometers for Molten Plastics, Van Nostrand, New York 1982. 9. J. O. Hinze: Turbulence, 2nd ed., McGraw-Hill, New York 1975. 10. D. Kunii, O. Levenspiel: Fluidization Engineering, R. E. Krieger Pub. Co., Huntington 1977. 11. J. M. Kay, R. M. Nedderman: Fluid Mechanics and Transfer Processes, Cambridge University Press, Cambridge 1985. 12. Y. Taitel, A. E. Dukler, AIChE J. 26 (1980) 345. 13. G. E. Alves, Chem. Eng. Progr. 50 (1954) 449. 14. J. M. Mandhane, G. H. Gregory, K. Aziz, Int. J. Multiphase Flow 1 (1974) 537. 15. P. Griffith: “Two-Phase Flow,” in W. M. Rohsenow, J. P. Hartnett, E. N. Ganic (eds.): Handbook of Heat Transfer Fundamentals, 2nd ed., McGraw-Hill, New York 1985. 16. F. A. Holland: Fluid Flow for Chemical Engineers, E. Arnold, London 1973. 17. R. H. Perry, D. W. Green: Perry’s Chemical Engineers Handbook, 6th ed., McGraw-Hill, New York 1984. 18. A. S. Foust et al.: Principles of Unit Operations, 2nd ed., J. Wiley & Sons, New York 1980. 19. H. A. Barnes, J. F. Hutton, K. Walters: An Introduction to Rheology, Elsevier, Amsterdam 1989. 20. A. B. Metzner, M. Whitlock, Trans. Soc. Rheol. 2 (1958) 239. 21. Q. D. Nguyen, D. V. Boger, J. Rheol. (N.Y.) 29 (1985) 335. 22. A. B. Metzner, J. C. Reed, AIChE J. 1 (1957) 434. 23. D. W. Dodge, A. B. Metzner, AIChE J. 5 (1955) 189. 24. D. V. Boger: “Viscoelastic Flows Through Contractions,” in J. L. Lumley, M. Van Dyke, H. L. Reed (eds.): Annual Review of Fluid Mechanics, vol. 19, Ann. Reviews Inc., Palo Alto 1987.

Fluid Mechanics 25. J. Meissner, Pure Appl. Chem. 42 (1975) 553. 26. R. B. Bird, C. F. Curtiss, R. C. Armstrong, O. Hassager: Dynamics of Polymeric Liquids, 2nd ed., vol. 2, J. Wiley & Sons, New York 1987. 27. G. Prilutski, R. K. Gupta, T. Sridhar, M. E. Ryan, J. Non-Newtonian Fluid Mech. 12 (1983) 233. 28. M. Doi, S. F. Edwards: The Theory of Polymer Dynamics, Oxford University Press, Oxford 1986. 29. R. L. Larson: Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston 1988. 30. C. J. S. Petrie: Elongational Flows, Pitman, London 1979. 31. F. T. Trouton, Proc. R. Soc. London A 77 (1906) 426. 32. E. B. Bagley, J. Appl. Phys. 28 (1957) 624. 33. J. A. Brydson: Flow Properties of Polymer Melts, Butterworth, London 1970. 34. D. V. Boger, R. K. Gupta, R. I. Tanner, J. Non-Newtonian Fluid Mech. 4 (1978) 239.

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35. R. I. Tanner, J. Polym. Sci. Part A-2 8 (1970) 2067. 36. P. S. Virk, AIChE J. 21 (1975). 625. 37. T. J. Chung: Finite Element Analysis in Fluid Dynamics, McGraw-Hill, New York 1978. 38. M. M. Gupta: “Numerical Methods for Viscous Flow Problems,” in A. S. Mujumdar, R. A. Mashelkar (eds.): Advances in Transport Processes, vol. 1, Wiley, New York 1980. 39. A. J. Baker: Finite Element Computational Fluid Mechanics, McGraw-Hill, New York 1983. 40. O. C. Zienkiewicz, K. Morgan: Finite Elements and Approximation, Wiley & Sons, London 1983. 41. M. J. Crochet, A. R. Davies, K. Walters: Numerical Simulation of Non-Newtonian Flow, Elsevier, Amsterdam 1984. 42. S. C. R. Dennis, G. Z. Chang, J. Fluid Mech. 42 (1970) 471.

Design of Experiments

423

Design of Experiments Sergio Soravia, Process Technology, Degussa AG, Hanau, Germany (Chaps. 1 – 3) Andreas Orth, University of Applied Sciences, Frankfurt am Main, Germany; Umesoft GmbH, Eschborn, Germany (Chaps. 4 – 7)

1. 1.1. 1.2. 1.3. 1.4. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 3. 3.1. 3.2. 3.3. 3.4. 4. 4.1. 4.2. 4.3.

Introduction . . . . . . . . . . . . . . . . . General Remarks . . . . . . . . . . . . . Application in Industry . . . . . . . . . Historical Sidelights . . . . . . . . . . . . Aim and Scope . . . . . . . . . . . . . . . Procedure for Conducting Experimental Investigations: Basic Principles . . System Analysis and Clear Definition of Objectives . . . . . . . . . . . . . . . . Response Variables and Experimental Factors . . . . . . . . . . . . . . . . . . . . Replication, Blocking, and Randomization . . . . . . . . . . . . . . . Interactions . . . . . . . . . . . . . . . . . Different Experimental Strategies . . Drawback of the One-Factor-at-a-Time Method . . . . . Factorial Designs . . . . . . . . . . . . . . Basic Concepts . . . . . . . . . . . . . . . The 22 Factorial Design . . . . . . . . . The 23 Factorial Design . . . . . . . . . Fractional Factorial Designs . . . . . . Response Surface Designs . . . . . . . . The Idea of Using Basic Empirical Models . . . . . . . . . . . . . . . . . . . . The Class of Models Used in DoE . . . Standard DoE Models and Corresponding Designs . . . . . . . . .

423 423 424 425 425 425 426 426 427 428 428 429 430 430 432 435 436 441 441 442 443

1. Introduction 1.1. General Remarks Research and development in the academic or industrial context makes extensive use of experimentation to gain a better understanding of a process or system under study. The methodology of Design of Experiments (DoE) provides proven strategies and methods of experimental design for performing and analyzing test series in a systematic and efficient way. All experimen-

4.4. Using Regression Analysis to Fit Models to Experimental Data . . . . . 5. Methods for Assessing, Improving, and Visualizing Models . . . . . . . . . 5.1. R2 Regression Measure and Q2 Prediction Measure . . . . . . . . . . 5.2. ANOVA (Analysis of Variance) and Lack-of-Fit Test . . . . . . . . . . . . . . 5.3. Analysis of Observations and Residuals . . . . . . . . . . . . . . . . . . . 5.4. Heuristics for Improving Model Performance . . . . . . . . . . . . . . . . . 5.5. Graphical Visualization of Response Surfaces . . . . . . . . . . . . . . . . . . . . 6. Optimization Methods . . . . . . . . . . 6.1. Basic EVOP Approach Using Factorial Designs . . . . . . . . . . . . . . . . . . . . 6.2. Model-Based Approach . . . . . . . . . 6.3. Multi-Response Optimization with Desirability Functions . . . . . . . . . . 6.4. Validation of Predicted Optima . . . . 7. Designs for Special Purposes . . . . . . 7.1. Mixture Designs . . . . . . . . . . . . . . 7.2. Designs for Categorical Factors . . . . 7.3. Optimal Designs . . . . . . . . . . . . . . 7.4. Robust Design as a Tool for Quality Engineering . . . . . . . . . . . . . . . . . 8. Software . . . . . . . . . . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . .

444 444 445 446 448 449 450 451 451 451 452 453 453 454 456 457 458 459 459

tal parameters are varied in an intelligent and balanced fashion so that a maximum of information is gained from the analysis of the experimental results. In most cases, the time and money spent on the experimental investigation will be greatly reduced. In all cases, an optimal ratio between the number of experimental trials and the information content of the results will be achieved. DoE is a powerful target-oriented tool. If it is properly employed, creative minds with a scientific and technical background will best

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Design of Experiments

deploy their resources to reach a well-defined goal of their studies. In contrast to what researchers sometimes fear, experimenters will not be hampered in their creativity, but will be empowered for structuring their innovative ideas. Of course, adopting DoE requires discipline from the user, and it has proved very helpful to take the initial steps together with an expert with experience in the field. The rewards of this systematic approach are useful, reliable, and well-documented results in a clear time and cost frame. A comprehensible presentation and documentation of experimental investigations is gratefully acknowledged by colleagues or successors in research and development teams. The application of DoE is particularly essential and indispensable when processes involving many factors or parameters are the subject of empirical investigations of cause – effect relationships. DoE is a scientific approach to experimentation which incorporates statistical principles. This ensures an objective investigation, so that valid and convincing conclusions can be drawn from an experimental study. In particular, an honest approach to dealing with process and measurement errors is encouraged, since experiments that are repeated under identical conditions will seldom lead to the same results. This may be caused by the measuring equipment, the experimenter, changes in ambient conditions, or the natural variability of the object under study. Note that this inherent experimental error, in general, comprises more than the bare repeatability and reproducibility of a measurement system. DoE provides basic principles to distinguish between experimental error and a real effect caused by consciously changing experimental conditions. This prevents experimenters from drawing erroneous conclusions and, as a consequence, from making wrong decisions.

1.2. Application in Industry In industry, increasingly harsh market conditions force companies to make every effort to reach and maintain their competitive edge. This applies, in particular, in view of the following goals: – Quality of products and services in conformance to market requirements

– Low costs to ensure adequate profits – Short development periods for new or improved products and production processes (time to market) Quality engineering techniques are powerful elements of modern quality management systems and make it possible to reach these goals. One important challenge in this context is not to ensure quality downstream at the end of the production line, but to ensure product quality by a stable and capable production process which is under control. By this means, ongoing tests and checks to prove that the product conforms to specification requirements are avoided. This can be realized by knowing the important and critical parameters or factors governing the system and through the implementation of intelligent process management strategies. A methodical approach, sometimes referred to as off-line quality engineering [24], [31], focuses even further upstream. By considering quality-relevant aspects in the early stages of product and process development, quality is ensured preventively in terms of fault prevention [30], [39]. Naturally, there is considerable costsaving potential in the early stages of product and process design, where manufacturing costs are fixed to a large extent. The losses incurred for each design modification of a product or process gradually increase with time. In addition, design errors with the most serious consequences are known to be committed in these early stages. As an outstanding quality engineering tool, DoE occupies a key position in this context. The emphasis is on engineering quality into the products and processes. At the same time, DoE opens up great economic potential during the entire development and improvement period. It is well-known that the implementation and use of corresponding methods increases competitiveness [4], [23], [29], [37], [40]. DoE is applied successfully in all high-technology branches of industry. In the process industry, it makes essential contributions to optimizing a large variety of procedures and covers the entire lifecycle of a product, starting from product design in chemical research (e.g., screening of raw materials, finding the best mixture or formulation, optimizing a chemical reaction), via process development in process engineering (e.g., test novel technological solutions, determine best oper-

Design of Experiments ating conditions, optimize the performance of processes), up to production (e.g., start up a plant smoothly, find an operating window which meets customer requirements at low cost, high capacity, and under stable conditions) and application technology (give competent advice concerning the application of the product, customize the properties of products to specific customer needs). In particular, technologies like highthroughput screening or combinatorial synthesis with automated workstations require DoE to employ resources reasonably.

1.3. Historical Sidelights The foundations of modern statistical experimental design methods were laid in the 1920s by R. A. Fisher [16] in Great Britain and were first used in areas such as agricultural science, biology, and medicine. By the 1950s and 1960s, some of these methods had already spread into chemical research and process development where they were successfully employed [3], [13], [34]. During this period and later, G. E. P. Box et al. made essential contributions to the advancement and application of this methodology [4–9]. In 1960, J. Kiefer and J. Wolfowitz initiated a profound research of the mathematical theory behind optimal designs, and in the early 1970s the first efficient algorithms for so-called D-optimal designs were developed. Around the same time, G. Taguchi integrated DoE methods into the product and process development of numerous Japanese companies [38]. One of his key ideas is the concept of robust design [24], [31], [39]. It involves designing products and processes with suitable parameters and parameter settings so that their functionality remains as insensitive as possible to unavoidable disturbing influences. Taguchi’s ideas were discussed fruitfully in the United States during the 1980s [4], [18], [22], [24] and caused an increasing interest in this subject in Western industries. At the same time, another set of tools, which in general is not suitable for chemical or chemical engineering applications, became popular under the name of D. Shainin [2], [26]. The 1980s also saw the advent and spread of software for DoE. Various powerful software tools have been commercially available for sev-

425

eral years now. They essentially support the generation, analysis, and documentation of experimental designs. Experience has shown that such software can be used by experimenters once the basic principles of the methods applied have been learned. A list of software tools is given in Chapter 8. Despite these stimulating developments, the majority of scientists and engineers in industry and academic research have still not yet used DoE.

1.4. Aim and Scope The target group of this article consists of scientists and engineers in industry or academia. The intention is – To give an insight into the basic principles and strategies of experimental design – To convey an understanding of the important design and analysis methods – To give an overview of optimization techniques and some advanced methods – To demonstrate basically the power of proven methods of DoE and to encourage their use in practice References for further reading and detailed study are given on all subjects. In particular, [7], [27], and [33] may be regarded as standard monographs on DoE in their respective languages.

2. Procedure for Conducting Experimental Investigations: Basic Principles DoE should become an integral part of the regular working tools of all experimenters. The methods provided by this approach may be used in experimental investigations on a smaller scale as well as in large projects which, depending on their importance and scope, may involve putting together an interdisciplinary team of appropriate size and composition. It proved to be advantageous to also include staff who take care of the equipment or facility on-site. They often bring in aspects and experiences with which

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Design of Experiments

Figure 1. Three essential phases characterize the basic structure of a DoE project

the decision-makers involved are largely unfamiliar. Moreover, involving, for instance, laboratory staff in the planning phase of experiments has a positive effect on their motivation. A DoE project essentially subdivides into three characteristic phases: the design or planning phase, the actual experimental or realization phase, and the analysis or evaluation phase (Fig. 1). It should be stressed that the design phase is decisive because it is this phase that determines the level of information attainable by the analysis of the experimental results. For more information on this subject and on basic principles of DoE, see [11], [19], [21].

2.1. System Analysis and Clear Definition of Objectives A system analysis involves collecting all existing information about the system to be examined and describing the current situation. A precise formulation of the problem and a clear definition of the objectives are very important prerequisites for a successful procedure and for attaining useful results from an experimental investigation. Experiments should never be conducted for their own sake but are to provide objective and reliable information, particularly, as a sound basis for decisions to be made. A clear statement of objectives is crucial, since the experimental strategy and hence the design of the experiments is essentially influenced by the goals to be reached (see Section 2.5). This sounds trivial but is frequently not handled carefully enough in practice. The actual planning or even the performance of experimental trials should not start until all of the aforementioned points have been settled satisfactorily.

2.2. Response Variables and Experimental Factors Each experiment can be regarded as an inquiry addressed to a process or system (see Fig. 2). Naturally, it must be possible to record its answer or result in terms of measurable or quantifiable response variables (dependent variables). Response variables must be selected such that the characterization of the interesting properties of the system is as complete and simple as possible. Let us consider a batch reaction, for instance. In this case it is certainly not only the conversion of starting material that is of interest as a test result; other potential response variables could be the concentration of undesired byproducts or the duration of the reaction (capacity). For some system characteristics, such as foaming during stirring operations or the visual impression of a pigment, there are often no measurable quantities. In these cases, it may be helpful to assess the results by a subjective rating (e.g., 0: no foam or very beautiful pigment, 1: some foam or beautiful pigment, up to about 7: very much foam or very ugly pigment). When the response variables are selected, it is important to ask questions about the reliability of the corresponding values: How much do the results vary if the experimental runs are conducted under conditions that are as identical as possible? Ideally, a detailed analysis of the measurement system is available which, in particular, provides information on repeatability and reproducibility. The results of experimental runs, i.e., the values of response variables, are affected by various factors. In practically all applications there are disturbing environmental factors, which may cause undesired variations in the response variables. Some of them are hard to control or uncontrollable, others simply may not even be known. Examples of such variables are different batches of a starting material, various items of equipment

Design of Experiments with the same functionality, a change of experimenters, atmospheric conditions, and – last but not least – time-related trends such as warmingup of a machine, fouling of a heat exchanger, clogging of a filter, or drifting of a measuring instrument.

Figure 2. Input – output model of a process or system

However, besides collecting and discussing the uncontrollable and disturbing variables, it is essential to collect and weigh up those factors or parameters that can be controlled or adjusted (independent variables), such as temperature, pressure, or the amount of catalyst used. The decision as to which of these experimental factors are to be kept constant during a test series and which are to be purposefully and systematically varied must also be carefully weighed up. The entire scientific and technical system know-how available by then from literature and experience, as well as intuition, must decisively influence not only the choice of the factors to be varied (one should focus on the important ones here according to the latest knowledge) but also the determination of the experimental region, i.e., of the specific range over which each factor will be varied (e.g., temperature between 120 and 180 ◦ C and pressure between 1200 and 1800 mbar). A good experimental design will efficiently cover this experimental domain such that the questions related to the objectives may be answered when the experimental results are analyzed.

2.3. Replication, Blocking, and Randomization To take the effects of disturbing environmental variables into account and decrease their impact to a large extent, the principles of replication, blocking, and randomization are employed. Moreover, by considering these principles, the

427

risk of misleading interpretation of the results is minimized. Replicates serve to make results more reliable and to obtain information about their variability. A genuine replicate should consider all possible environmental influences leading to variations within the response variables, i.e., the whole experiment with its corresponding factor-level combination should be repeated from the beginning and with some time delay in between. The results of data analyses must always be seen in the light of this inherent experimental error. Hence, replication does not mean, for instance, analyzing a sample of a product several times. This variability of a response variable is solely a measure of the precision of the analytical test procedure (laboratory assistant, measuring instrument). It is of crucial importance that environmental factors and experimental factors of interest do not vary together such that changes in a response variable cannot be unambiguously attributed to the factors varied. For example, if two catalyst types were to be compared at various reaction temperatures and if two differently sized reactors were available for this purpose, it would be unwise to conduct all experiments with one catalyst in the smaller reactor and all experiments with the other catalyst in the larger reactor. If, in this case, the results differed from each other, it would be impossible to decide whether the catalyst type or the reactor type or both caused the deviations. The objective of blocking is to predetermine relatively similar blocks – in this case, the two reactors – in which test conditions are more homogeneous and which allow a more detailed study of the experimental factors of interest. Regarding the selection of the catalyst type and reaction temperature, the experiments to be conducted in each of the two reactors must be similar in the sense that variations in the values of a response variable can be interpreted correctly. It is not always possible, however, to clearly identify unwanted influences and to take them into account, as is the case when blocking is used. Yet these side effects can be counterbalanced by a general use of randomization. Here, in contrast to systematically determining which experiments are to be conducted, the order of the experiments is randomized. In particular, false assignments of time-related trends are avoided.

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Let us consider a rectifying column, for instance, in which the effects of operating pressure, reflux ratio, and reboiler duty on the purity of the top product are to be examined. Let us assume that the unit is started up in the morning and that the whole test series could be realized within one day. Now, if one conducted all experiments involving a low reflux ratio before those involving a high reflux ratio, the effect of the reflux ratio could be falsified more or less by the unit’s warming up, depending on how strong this influence is (poor design in Fig. 3). Such an uncontrolled mixing of effects is prevented by choosing the order of the experimental runs at random (good design in Fig. 3).

about the joint effects of experimental factors, that is, their interactions. Two variables are said to interact if, by changing one, the extent of impact on a third, namely, a response variable, depends on the setting of the other variable. In other words, interaction between two experimental factors measures how much the effect of a factor variation on a response variable depends on the level of the other factor. Interactions are often not heeded in practice, or they are studied at the price of spending large amounts of time and money on the associated experimental investigation. In addition, what interaction actually means is often not clearly understood. In particular, interaction is not to be confused with correlation. Two variables are said to be correlated if an increase of one variable tends to be associated with an increase or decrease of the other. Especially factorial experimental designs (see Chap. 3) allow, among other things, a quantitative determination of interactions between varied experimental factors.

2.5. Different Experimental Strategies

Figure 3. Time-related effects caused, e.g., by instrument drifts may falsify the analysis of the results when factor settings are not changed randomly (poor design). If environmental conditions vary during the course of an experiment, their effect will be damped or averaged out by randomizing the sequence of the experiments (good design)

The decisive reason for employing the principles of replication, blocking, and randomization is therefore to prevent the analysis of systematically varied experimental factors from being unnecessarily contaminated by the influences of unwanted and often hidden factors. While blocking and randomization basically do not involve additional experimental runs, each replicate is a completely new realization of a combination of factor settings. An appropriate relation between the number of experimental runs and the reliability of its results must be established here on an individual basis.

2.4. Interactions To avoid an overly limited view on the behavior of systems, it is of great importance to know

When processes are to be improved or novel technical solutions are to be tested, but also when plants are started up, several factors are often varied, in the hope of meeting with shortterm success, by using an unsystematic iterative trial-and-error approach until satisfactory results are eventually produced. The expenditure involved quickly takes on unforeseeable proportions without affording important insights into the cause-and-effect relationships of the system. The result of this procedure is that, in the end, a comprehensible documentation is not available, and objective, reliable reasons for process operations or factor settings are missing. Furthermore, very little is known in most cases about the impact of factor variations. Experimenting in this way might be acceptable for orientation purposes in a kind of pre-experimental phase. However, one should switch to a judicious program as soon as possible. To study causal relationships systematically, the experimental factors or parameters are usually varied separately and successively, and the values of a response variable (product, process, or quality characteristic), such as the yield of a chemical product, are shown in a diagram

Design of Experiments (see Fig. 4). This one-factor-at-a-time method (see Section 2.6), however, provides only few insights into the subject under study because the effect of a particular factor is only known at a single factor-level combination of the other factors. The response variable may have quite another shape if the levels of the remaining factors are set differently. If the experimental factors in their effect on a response variable do not act additively according to the superposition principle, i.e., if the factors influence each other in their effect on the response variable by existing interactions, a misinterpretation of the results is easily possible, particularly when optimum conditions are to be attained. When statistical experimental design methods are used, all considered factors are varied in a systematic and balanced way so that a maximum of information is gained from the analysis of the corresponding experiments. This may comprise the statistically sound quantitative determination of the effects of factor variations on one or several response variables (see Chap. 3) or a systematic optimization of factor settings (see Chap. 6). Depending on the experimenter’s intention, the following questions can be answered: – What are the most important factors of the system under investigation? – To what extent and in which direction does a response variable of interest change when an experimental factor is varied? – To what extent is the size and direction of the effect of a factor variation dependent on the settings of other experimental factors (interactions)? – With which factor settings does one obtain a desired state of a response variable (maximum, minimum, nominal value)? – How can this state be made insensitive to disturbing environmental factors or how can an undesired variability of a response variable be reduced (robust design)? The question of which experimental strategy should be chosen from a comprehensive range of methods will be governed by the objectives to be achieved in each individual case, taking, e.g., system-inherent, financial, and time-related boundary conditions into account. Every project has its peculiarities. Carefully planned experiments cover the experimental region to be in-

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vestigated as evenly as possible, while ensuring to the largest possible extent that changing values in a response variable can be attributed unambiguously to the right causes. Information of crucial importance is frequently obtained by a simple graphical analysis of the data without having to employ sophisticated statistical analysis methods, such as variance analysis or regression analysis. On the other hand, the best statistical analysis is not capable of retrieving useful information from a badly designed series of experiments. It is therefore decisive to consider basic DoE principles right from the beginning, above all, however, before conducting any experiments.

2.6. Drawback of the One-Factor-at-a-Time Method A crystallization process is used in the following to illustrate the deficiency of the frequently used one-factor-at-a-time method. Factors influencing this system are, for instance, crystallization conditions such as geometry of the crystallizer, type and speed of the agitator, temperature, residence time, and concentrations of additives like crystallization and filter aids, as well as of two presumed additives A and B. Possible response variables may be bulk density, abrasion, hardness, and pourability of the crystallization product. Let us assume the simple case that the effects of the two experimental factors – additive A and additive B – on the material’s bulk density are to be systematically examined with the aim of obtaining a maximum bulk density. As mentioned before, the experimental factors are usually examined and/or optimized separately and successively. In the example considered here, one would therefore begin by keeping factor B constant and varying A over a certain range until A has been optimally adjusted in terms of a maximum bulk density and enter the result in a diagram (see Fig. 4). The optimal value for A would then be selected and kept constant. The same procedure would then be employed for B. The result of this is a presumably optimal setting for A and B, and hasty experimenters would jump to the conclusion that, in this case, after varying A between 15 and 40 g/L and B between 5 and 17.5 g/L, the highest bulk density is obtained by setting A to 33 g/L and B to 8.5 g/L and that its

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value is approximately 825 g/L. However, the response surface in Figure 4, which shows the complete relationship between both experimental factors and the response variable, reveals how misleading such a conclusion can be. This drastic misinterpretation is based on the, in this case, false assumption that the effect of varying one factor is independent of the settings of the other factor. For instance, by using the response surface, one can show that the bulk density values take on a decidedly different shape compared to the first diagram when varying A for B = 15 g/L. The following should be noted: If there are interactions between experimental factors, the one-factor-at-a-time method is an unsuitable tool for a systematic analysis of these factors, which holds true in particular when factor settings are to be optimized. If such interactions can definitely be ruled out, it might well be used. In chemistry, however, it is rather the rule that interactions occur.

3. Factorial Designs 3.1. Basic Concepts The statistical experimental designs most frequently used in practice are the two-level factorial designs. These designs are called two-level because there are only two levels of settings, a lower (−) and an upper level (+), for each of the experimental factors. A full two-level factorial design specifies all combinations of the lower and upper levels of the factors as settings of the experimental runs (2n design, where n denotes the number of factors). Their principle is illustrated by a simple example of a chemical reaction for which the influence of 2 (22 design) and 3 (23 design) experimental factors on the product yield is to be examined (Sections 3.2 and 3.3). For a growing number of factors, the number of runs of a full factorial design increases exponentially, and it provides much more information than is generally needed. Particularly for n > 4, the number of experimental settings can be reduced by selecting a well-defined subgroup of all 2n possible settings of the factors without losing important information. This leads to the fractional factorial designs (Section 3.4). The restriction of initially using just two levels for each experimental factor often causes

some uneasiness for experimenters using this method for the first time. But by using two-level factorial designs, a balanced coverage of the interesting experimental region is achieved very economically. Moreover – owing to the special combination of factor levels – it is also possible to gain deeper insights from the associated individual values of the response variables. A decisive advantage of the two-level factorial designs is that they allow the effects of factor variations to be systematically and reliably analyzed and quantified and that they provide information on how these effects depend on the settings of the other experimental factors. These insights are gained by calculating so-called main effects and interaction effects. In the calculation of these effects, all experimental results can be used and are included to form well-defined differences of corresponding averages (see Figs. 6 and 9), thereby increasing the degree of reliability. The essential results of this effect analysis can be visualized by simple diagrams (see, e.g., Fig. 7). In a factorial design, not only continuous experimental factors, such as temperature, pressure, and concentration, which can be set to any intermediate value, but also discrete or categorical factors, such as equipment or solvent type, may be involved. If at least one categorical variable with more than two levels is involved or if curvatures in the response variables are expected and to be explored, factorial designs with more than two levels may be used, e.g., 3n designs, in which all factors are studied at three levels each, or hybrid factorial designs with mixed factor levels like the 2 × 32 design, in which one factor is varied at two levels, and two factors at three levels [27]. However, especially in the case of continuous factors, other so-called response surface designs are more efficient (see Chap. 4). In the following, the expression “factorial designs” always refers to two-level factorial designs. For the sake of simplicity, replicates are neglected in the following examples, and variability in the process and in measurement are assumed to be very small. Note, however, that being aware of the impact of experimental error on the reliability or significance of calculated effects is an essential principle of DoE and crucial to drawing valid conclusions. Variability within individual runs having the same settings of the experimental factors will propagate and cause variability in each calculated variable,

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Figure 4. The one-factor-at-a-time method can lead to misinterpretations in systems that are subject to interactions

Figure 5. Example of a 22 factorial design, including a graphical and a tabular representation of the experimental settings and results

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e.g., main effect or interaction effect, deduced from these single results. Experimental designs, particularly the factorial designs, minimize error propagation.

which contain the settings of the experimental factors form the so-called design matrix. The resultant values of the response variable y obtained for the settings A−B−, A+B−, A−B+, and A+B+ are referred to as yA−B− , yA+B− , yA−B+ , and yA+B+ respectively. They are entered in the column of the response variable and in the corresponding positions of the graph. Due to the special constellation of the experimental runs, it is possible to see how y changes when factor A is varied at the two levels of B and what happens when factor B is varied at the two levels of A. Figure 5 reveals that, at the lower temperature of 70 ◦ C (B−), an increase of the catalyst quantity from 100 g (A−) to 150 g (A+) increases the response variable yield by 2 %, while at the higher temperature of 90 ◦ C (B+), increasing the catalyst quantity enlarges the value of the response variable by 18 %, i.e., Effect of AforB− = EB− (A) = yA+B− − yA−B− = +2% Effect of AforB+ = EB+ (A) = yA+B+ − yA−B+ = +18%

Accordingly, the individual effects of a temperature increase can be determined for the different catalyst quantities: Effect of BforA− = EA− (B) Figure 6. Calculation of the two main effects and the interaction effect in a 22 factorial design

= yA−B+ − yA−B− = +14% Effect of BforA+

Factorial designs are treated in most textbooks on DoE, e.g., [7], [25], [27], [33], [36].

= EA+ (B) = yA+B+ − yA+B− = +30%

3.2. The

22

Factorial Design

Let us suppose the influence of two factors – catalyst quantity A and temperature B – on yield y of a product in a stirred tank reactor is to be examined. Figure 5 shows the two levels of both factors involved, as well as a tabular and a graphical representation of the associated experimental 22 factorial design. The – in this case two – columns

As mentioned before, key results of a factorial design are obtained by the calculation of main effects and interaction effects (Fig. 6). The main effect of a factor is a measure of the extent to which a response variable changes on average when this factor is varied from its lower to its upper level. To calculate this main effect, all experimental results obtained at the upper (+)

Design of Experiments and lower (−) level of the factor are averaged, and the result at the lower setting is subsequently subtracted from that at the upper. This leads to the following equations: Main effect of A = ME (A) yA−B− + yA−B+ yA+B− + yA+B+ − = 2 2 (yA+B− − yA−B− ) + (yA+B+ − yA−B+ ) = 2 EB− (A) + EB+ (A) = 2 = 10% Main effect of B = ME (B) yA−B− + yA+B− yA−B+ + yA+B+ − = 2 2 (yA−B+ − yA−B− ) + (yA+B+ − yA+B− ) = 2 EA− (B) + EA+ (B) = 2 = 22%

The expressions preceding the numerical results in the calculations above show that the main effect of a factor can also be calculated from the mean of the individual effects involved. The two calculated results can now be read as follows: When the catalyst quantity A is increased from 100 to 150 g, the yield increases on average by 74 92 10 % from 60 + = 67% to 62 + = 77%. 2 2 The temperature B has a stronger impact on the response variable yield within the range of 70 – 90 ◦ C. An increase in temperature leads to an average increase in yield of 22 %. The maineffect diagrams in Figure 7 illustrate these relations graphically. The effect of a factor variation is strongly dependent on the respective setting of the other factor. This was already shown in Figure 5 and is further illustrated by the two interaction diagrams in Figure 7. The four lines in these diagrams correspond to the four edges of the response surface, also shown in Figure 7. This interaction is a typical case of a synergetic interaction. A simultaneous increase of A and B has a clearly higher impact on the response variable than the additive superposition of the individual effects, EB− (A) and EA− (B), would lead one to expect. If the value of yA+B+

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were not 92 but 76 %, there would be no interaction between A and B. The effect of a variation of one of the two factors on the response variable would be independent of the adjustment of the other factor. In this case, the two individual effects would be identical for each factor, i.e., EB− (A) = EB+ (A) = + 2 % and EA− (B) = EA+ (B) = + 14 %. The corresponding lines in the interaction diagrams would then be parallel. It seems reasonable now to determine a quantitative measure for the interaction of two factors as the difference of these individual effects, i.e., Interaction effect between AandB = IE (AB) 1 = [EB+ (A) − EB− (A)] 2 (yA+B+ − yA−B+ ) − (yA+B− − yA−B− ) = 2 yA+B+ + yA−B− yA−B+ + yA+B− = − 2 2 (yA+B+ − yA+B− ) − (yA−B+ − yA−B− ) = 2 1 = [EA+ (B) − EA− (B)] 2 = IE (BA) = +8%

From these sequences of expressions it can be seen that the interaction between A and B could be equally defined through the difference of the individual effects of A or the individual effects of B, giving the same numerical result each time. Moreover, the expression in the middle shows that the interaction between A and B – as is the case for the main effects – is nothing but the difference of two averages (this is basically the reason why a factor of 12 is introduced in the definition). This corresponds to the difference of the averages of the results located on the diagonals in Figure 6. The calculation of the two main effects and of the interaction effect is also represented geometrically in Figure 6. The corresponding analysis table contains columns of signs, which allow calculation of these effects. Each effect may be computed as the sum of signed response values divided by half the number of experiments, where the signs are taken from the column of the desired effect. Note that the signs of the interaction column AB can be generated by a row-wise

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Figure 7. Diagrams of main effects, interactions, and response surfaces illustrate conspicuously important relations governing the system

multiplication of the main-effect columns A and B. Today, however, the effects do not need to be computed like this anymore. Specific DoE software tools (see Chap. 8) use methods such as those described in Section 4.4 and yield numerical and graphical results more easily. A final interpretation of the experiments could read as follows: The catalyst shows a not yet satisfactory activity at the lower temperature (70 ◦ C). An increase of the catalyst quantity from 100 to 150 g gives a slight improvement but does not yet yield satisfactory values. The situation is a different at the higher temperature (90 ◦ C), where the catalyst clearly performs better. In addition, an increase in the catalyst quantity at this temperature has a strong impact on yield. If all experimental factors are continuous, it will be possible and useful to perform experimental runs at the center point. It is obtained by setting each factor to the midpoint of its fac-

tor range, in our example, to 125 g of catalyst and 80 ◦ C. This isolated additional experimental point gives a rough impression of the behavior of response variables inside the experimental region of interest. If the result in the center point does not correspond to the mean of the results obtained in the corner points of the factorial design, then the response surface of the response variable will have a more or less pronounced curvature that depends on the magnitude of this deviation. The graph in Figure 5 shows a result of 70 % at the center point, which is slightly below the 72 % obtained by averaging the four results of the factorial design. Thus, the response surface must be imagined as slightly sagging in its middle. Of course, with this single additional experiment, it is impossible to determine which of the experimental factors is (are) ultimately responsible for the curvature. This question can only be settled by a response surface design (see Chap. 4).

Design of Experiments The analysis shown for one response variable is performed for each response variable so that the effects of factor variations on every response variable are finally known.

3.3. The

23

Factorial Design

The concepts and notions introduced in the Section 3.2 can be generalized to three or more factors. Let us suppose that, in addition to the example in Section 3.2, not only the effects of temperature and catalyst quantity but also the impact of changing the agitator type on the yield are to be examined. In analogy to Figure 5, the experimental factors with their respective two settings as well as the 23 factorial design, which is obtained by realizing all possible combinations of factor settings, are represented in graphical and tabular form in Figure 8. By comparing the values at the ends of the various edges of the cube in Figure 8, it is possible to perform a very elementary analysis. For each factor, the effect of its variation can be studied for the four different constellations of the other two factors. For example, the change from the current to the new agitator type does not lead to the presumed improvement in yield. This may be verified by looking at the four edges going from the front face to the back face of the cube. The deterioration is particularly severe at high temperature where the yield decreases from 74 to 67 % (for the lower level of catalyst) and from 92 to 82 % (for the upper level of catalyst). More detailed information about the specific and joint effects of the factors is obtained by calculation of the main effects and interaction effects introduced in Section 3.2. The main effect thereby indicates how much a response variable is affected on average by a variation of a factor and is measured as the difference in the average response for the two factor levels. Figure 9 illustrates this for the factors A (catalyst) and B (temperature). The main effect for C (agitator type) is obtained analogously by calculating the difference of the two averages at the back face and the front face of the cube. The interaction effect of two factors, more precisely, the two-factor interaction was introduced in Section 3.2. Generally, there will be 

n 2



=

n· (n − 1) 2

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two-factor interactions, where n denotes the number of factors. For three factors there are   3 =3 2

two-factor interactions, namely, AB, AC, and BC. The calculation of a two-factor interaction in a 23 factorial design will be demonstrated for AB. This interaction is obtained by calculating the two-factor interaction IEC− (AB) at the lower level of factor C and the two-factor interaction IEC+ (AB) at the upper level of factor C, and then by averaging these two values: Interaction effect between AandB = IE (AB) = =

IEC− (AB) + IEC+ (AB) 2

57 + 82 60 + 92 62 + 74 + − − 2 2 2

59 + 67 2

2 62 + 74 + 59 + 67 60 + 92 + 57 + 82 − = 4 4 = +7.25%

A two-factor interaction in a factorial design with more than two factors is obtained by taking the average of all individual two-factor interactions at the different constellations of the other factors. The calculation of IE(AB) is also illustrated in Figure 9, where it is seen to be the difference of averages between results on two diagonal planes. Due to the inherent symmetry of the design, the calculation of the other twofactor interactions, IE(AC) and IE(BC), can be performed in a similar way. This leads to the respective columns of signs within the analysis table in Figure 9 and to the corresponding diagonal planes within the cube. Now, if the interaction between A and B at the lower level of C differs from their interaction at the upper level of C, there will be a three-factor interaction, which is defined by the difference of these two individual two-factor interactions: IE (ABC) 1 = [IEC+ (AB) − IEC− (AB)] 2   1 57 + 82 59 + 67 = − 2 2 2   60 + 92 62 + 74 − − 2 2 60 + 92 + 59 + 67 62 + 74 + 57 + 82 − = 4 4 = −0.75%

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Figure 8. Example of a 23 factorial design, including a graphical and a tabular representation of the experimental settings and results. The experimental settings for agitator type “current” correspond to the 22 factorial design in Figure 5

Note that this result can also be obtained by using the column of signs corresponding to the threefactor interaction ABC in Figure 9. As in the case of two-factor interactions, the signs within this column are obtained by a row-wise multiplication of the signs of the main effect columns A, B, and C. Obviously, the three-factor interaction is also the difference of two averages (as in the case of the two-factor interaction presented in Section 3.2, this is the reason why the factor of 12 is introduced in the definition). They are obtained by averaging the results located at the vertices of the respective two tetrahedra which make up the cube. The numerical results of all effects obtainable from the 23 factorical design are summarized in the following: ME(A) ME(B) ME(C) IE(AB) IE(AC) IE(BC) IE(ABC)

= = = = = = =

9.25 % 19.25 % − 5.75 % 7.25 % − 0.75 % − 2.75 % − 0.75 %

Note again that numerical results like these and graphical results like those of Figure 7 are easily obtained by using specific DoE software (see Chap. 8). Diagrams of main effects and interaction effects may also be generated for factorial designs with more than two factors.

Even though three-factor interactions may really exist in some cases, it is seldom that they play an essential role. So, if their absolute value is clearly higher than most of the main effects and two-factor interactions, it seems reasonable to conclude that the experimental error is of a magnitude that does not allow the reliable estimation of most of the effects (perhaps because they are very small) and/or that at least one response value has been corrupted by a gross systematic error. If, on the whole, systematic errors can be excluded, it is a legitimate practice to neglect higher-order interactions, such as three- and four-factor interactions, because main effects tend to be larger than two-factor interactions, which in turn tend to be larger than three-factor interactions, and so on. Moreover, if no information about the magnitude of the experimental error is available, it will be possible to obtain a rough estimate of this error by using higherorder interactions, like five-, four-, and even three-factor interactions.

3.4. Fractional Factorial Designs For a growing number n of experimental factors, the number of experimental settings 2n increases exponentially in a full factorial design. Simul-

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Figure 9. Calculation of main effects and interaction effects in a 23 factorial design shown for the main effects of A and B and their two-factor interaction AB

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taneously, the proportion of higher-order interactions increases rapidly. For instance, if n = 5, there are 5 main effects, 10 two-factor interactions, and 16 interactions of higher order. Obviously, if n is not small, there is some redundancy in a full factorial design, since higher-order interactions are not likely to have appreciable magnitudes. At this point, the following questions arise: – Is it possible to reduce the amount of experimental effort in a sophisticated way so that the most important information can still be obtained by analysis of the data? – Is it possible to study more experimental factors instead of higher-order interactions with the same number of experimental settings? The answer in both cases is “yes”. It leads to the 2n−k fractional factorial designs, where 2n−k denotes the number of experimental settings, n the number of experimental factors, and k the number of times by which the number of settings has been halved compared to the corresponding complete 2n design with the same number of experimental factors (1/2k · 2n = 2n−k ). The application of fractional factorial designs yields a reduction in experimental effort, which is adapted to the complexity of the system under investigation and to the information required. Once the experimental runs have been performed, fractional factorial designs are analyzed like full factorial designs, except that fewer values are available. As introduced in Sections 3.1, 3.2, and 3.3, effects are obtained by calculating corresponding differences of averages. However, by doing so, one will discover that effects are confounded. Confounding is defined as a situation where an effect cannot unambiguously be attributed to a single main effect or interaction. Let us consider the 23−1 design, in which the effects of three factors are studied with four experimental settings (see Figure 10). This is an example of a half-fraction factorial design (see Figure 11). By calculating the main effect of C [ME(C) on the left-hand side of Fig. 10] the outcome is not the difference between the averages of four (as in the case of the 23 design) but the averages of only two results which are calculated. A similar situation occurs when calculating the interaction between A and B [IE(AB) on the right-hand side of Fig. 10]. Moreover, calcu-

lating these effects leads to the same expression. It is actually the sum ME(C)+IE(AB) of both effects. It can also be verified that ME(A) is confounded with IE(BC), and ME(B) with IE(AC). This is an example of a so-called resolution III design. A slightly different situation occurs when considering the 24−1 design (see Fig. 11). Here the main effects are confounded with threefactor interactions, e.g. ME(A) with IE(BCD), and the two-factor interactions are confounded with each other, e.g., IE(AC) with IE(BD). This is an example of a resolution IV design. The 25−1 design also shown in Figure 11 is an example of a very efficient resolution V design. The resolution of a fractional factorial design largely characterizes the information content obtainable by analyzing the results of a fractional factorial design: – A fractional factorial design of resolution III does not confound main effects with each other but does confound main effects with two-factor interactions. – A fractional factorial design of resolution IV does not confound main effects with twofactor interactions but does confound twofactor interactions with other two-factor interactions. – A fractional factorial design of resolution V does not confound main effects and twofactor interactions with each other but does confound two-factor interactions with threefactor interactions. The resolution of the most important fractional factorial designs can be seen in Table 1. Designs of resolution III are used in the early stages of experimental investigations to gain a first insight into the possibilities and behaviors of systems. Particularly beneficial are saturated designs for 2n − 1 factors in which all degrees of freedom of a 2n design are exploited. The above-mentioned 23−1 design is an example of such a design in which three factors are studied with four experimental settings. It is “generated” by replacing all columns of signs in the analysis table of Figure 6 by experimental factors, i.e., the settings of factor C at the top of Figure 11 are determined by the column of the two-factor interaction AB in the analysis table of Figure 6. A further important example of a saturated design is the 27−4 design, which allows seven factors

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Figure 10. Example of a 23−1 fractional factorial design. In the top section the experimental settings and results are shown in geometrical and tabular form. In the bottom section the problem of confounding is illustrated: Calculating ME(C) and IE(AB) leads to the same expression, which is actually the sum of both. It is not possible to determine whether the calculated value is caused by the main effect of C, by the interaction effect of A and B, or by both Table 1. The 2n−k (fractional) factorial designs with a maximum of 32 experimental settings and their resolution. 2n−k denotes the number of experimental settings, n the number of experimental factors, and 1/2k the factor by which the number of settings has been reduced compared to the corresponding full factorial design with the same number of experimental factors. Resolution

III IV V VI Full

Number of experimental settings 4

8

16

32

23−1

25−2 – 27−4 24−1

29−5 – 215−11 26−2 – 28−4 25−1

217−12 – 231−26 27−2 – 216−11

22

23

24

to be studied with eight experimental settings. It is generated by replacing AB by D, AC by E, BC by F, and ABC by G in the heading of the columns of signs used for the calculation of effects in Figure 9. With resolution III designs special care must be taken when interpreting

26−1 25

the analysis results. Calculated main effects may also be two-factor interactions of other factors. Resolution IV designs are employed to gain unambiguous information about the individual impact of the experimental factors, while unambiguous information about their two-factor interactions is not yet required. Designs of resolu-

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Figure 11. Geometric representation of the most important half-fraction factorial designs and their respective tabular representation, i.e., their design matrices

tion III and IV are mostly used to find out which factors play an important role. This technique of isolating the important factors is sometimes referred to as screening. Using designs of resolution V or higher does not lead to any loss of decisive information, since the main effects and two-factor interactions are

only confounded with higher-order interactions, which in most cases can be neglected. Particularly when n > 4, experimental settings can be halved to achieve designs of at least resolution V. Fractional factorial designs support the iterative nature of experimentation. The experimen-

Design of Experiments tal runs of two or more fractional factorial designs conducted sequentially may be combined to form a larger design with higher resolution. In this way it is possible to resolve ambiguities by the addition of further experimental runs. Half-fraction designs possess an interesting and useful projection property: the omission of one arbitrary column in the designs always leads to a full factorial design with respect to the remaining columns or factors. So, if a factor proves to have no significant effect on a response variable, the remaining factors can be analyzed as in the full factorial design. For example, the omission of factor B in the 24−1 design of Figure 11 leads to a 23 design for the factors A, C, and D. This may be verified by examining the three remaining columns in the table but also by pushing the upper faces of the two cubes into the lower faces.

4. Response Surface Designs In some situations factorial designs in which all factors are varied by only using two settings are not adequate for describing the behavior of an experimental system, because a more detailed insight is needed to predict its responses or to find optimal factor settings. In this case, it is often necessary to extend factorial designs and to do additional experiments at other points in the experimental domain. To decide which points to use, response surface designs are used [5], [28]. These designs are based on mathematical models that describe how responses depend on experimental factors.

4.1. The Idea of Using Basic Empirical Models A model is a way to describe a part of reality; ideally it is much simpler and more manageable than that which it describes, but it should nevertheless adequately fulfill a predefined modeling purpose. Since a model can only be an approximate description of the original, it is important to be aware of this purpose when constructing models (see Chap. 2 for typical goals in conjunction with DoE). Models used in DoE are polynomials in several variables, in which the y

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variable is a response, and the x variables are experimental factors. A set of coefficients is used to describe how the y variables depend on the x variables. Often a process or experimental system can only be adequately described by more than one response, in which case there will be one model for each response, and each model will have its own set of coefficients. Coefficients are estimated from experimental data which are collected in a corresponding experimental design. Estimating model coefficients from experimental data is called model fitting and represents the principle task of statistical analysis to be performed as soon as response values from experiments are available. An equally important task consists of assessing the quality of a fitted model, as an important step towards qualifying it for use in prediction and optimization or whatever other purpose. In fact it is often possible to improve the performance of a model by taking small corrective measures such as those described in Section 5.4. The important ideas behind empirical modeling are: – The model must describe the entire behavior of the process or experimental system that is relevant to answering the questions of the experimenter. – The experimental design is based on the model and determines which information can be extracted from the experimental results. Statistical analysis is only the tool for extracting this information. – Neglecting experimental design essentially means missing out on finding all the relevant answers. Picking the wrong design also means missing out on relevant answers. – Enlarging the scope of the questions always means extending the model and adding experiments to the design. Using models and setting up designs in this fashion requires that one proceed in the systematic way that has been described in Chapter 2. There are several additional aspects that play an important role in modeling: – Definition of the experimental domain in which the model should be useful for prediction – Selection of the correct model type

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– Choice of the experimental design that corresponds optimally to the experimental domain and to the model type chosen – Estimation of coefficients by regression analysis – Qualification and refinement of the model by continued statistical analysis – Validation of model predictions by confirmatory experiments – Use of the model for the purpose of finding optimal factor settings

4.2. The Class of Models Used in DoE The class of models that is normally used in DoE contains only models which are linear with regard to the unknown coefficients. This is why, in order to estimate coefficients, linear regression methods can efficiently be used [15]. Nonlinear or first-principle models, such as mechanical models, reaction-kinetic models, and more general dynamic models are only rarely used directly when designing experiments; they are commonly approximated by simple polynomial models at the cost of restricting the domain of validity of the model. A direct generation of optimal designs for nonlinear models, i.e., models that are nonlinear in the parameters that are to be estimated, is sometimes possible. However, it is particularly important that such models are very accurate, that experimental errors are small and that initial estimates of the parameters are already available. It is this last point that often makes setting up the correct design very difficult [8]. Polynomial models used in DoE are built up as a sum of so-called model terms: y = b0 + bA xA + bB xB + bAB xA xB + bAA x2A + bBB x2b + ε

This is an example of a quadratic model for two factors A and B, containing a constant term b0 , linear terms bA x A and bB x B , an interaction term bAB x A x B , quadratic terms bAA x 2A and bBB x 2B , and an error term ε. The x represent the settings of the factors in the experimental domain. In factorial designs, they are coded as − 1 and + 1. When factors have continuous scales, like temperature or pressure,

this coding can be understood as a simple linear transformation of the factor range onto the interval [− 1, + 1]. This transformation is called scaling and centering; the corresponding equation is: xcentered & scaled = 2 (x − xcenter ) / (x max − xmin ) .

Since x center = (x max + x min )/2, x max transforms into + 1 and x min transforms into − 1 (or simply + and − respectively, when using factorial designs). Centering and scaling allows the influences of different factors with different scales to be compared. In the following discussion it is assumed that all factors are either coded or scaled and centered in this fashion. The b are the coefficients that are estimated by regression after the experiments have been completed. A question that arises is: How do calculated effects in a factorial design as described in Chapter 3 compare to estimated coefficients of linear or interaction terms in a fitted model? The answer to this question is quite interesting. Estimating coefficients by multiple linear regression and calculating effects for factorial designs as described in Chapter 3 are mathematically equivalent. In fact, coefficients are simply half of the corresponding effects: Calculating a main effect of a factor ME(A) means estimating the difference ∆y in the response that has been provoked by changing the factor A from its lower to its upper level. In contrast, the corresponding coefficient, bA , is the geometrical slope of the curve describing the dependency of y upon x, i.e., ∆y/∆x. Since DoE is based on scaled and centered variables, ∆x is exactly 2. So bA can be estimated by ME(A)/2. This is also true for interaction effects: bAB can be estimated by IE(AB)/2. Estimators are often denoted by ˆ, so ˆbA = IE (AB) /2. The constant b0 can be estimated by calculating the mean of all response values. For the example that was discussed in Section 3.2, the model equation is y = b0 + bA x A + bB x B + bAB x A x B and the estimated coefficients are: ˆb0 = 1 (74 + 92 + 70 + 60 + 62) = 71.6% 5 ˆbA = ME (A) /2 = 5% ˆbb = ME (B) /2 = 11% ˆbAB = IE (AB) /2 = 4%

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Figure 12. Examples of response surfaces of standard models for two factors: linear (A), interaction (B), and quadratic (C) models

The benefit of using coefficients lies in the greater generality of the response surface models. These allow: – Prediction of response variables within the experimental region – Use of quadratic models for modeling maxima and minima – Use of mixture models for modeling formulations (see Section 7.1) – Use of dummy variables for modeling categorical factors (see Section 7.2) – Correcting factor settings when prescribed settings cannot be exactly met in the experiment – Nonstandard domains for the model (and the design) that are subject to additional constraints

4.3. Standard DoE Models and Corresponding Designs For standard DoE models the designs can be chosen off the peg. This means that the design structure is predefined and available in the form of a design table (see, e.g., Fig. 11 for half-fraction factorial designs). For nonstandard models an optimal design has to be generated by using a mathematical algorithm (see Section 7.3). Standard DoE models are – Linear models (i.e., linear with regard to the factor variables), containing the constant term and linear terms for all factors involved – Interaction models, which additionally contain interaction terms of the factors involved – Quadratic models, which, in addition to all interaction terms, contain quadratic terms

Standard models for two and three factors are shown in Table 2, examples of response surfaces are shown in Figure 12. Adding isolated interaction terms to linear models, taking away interaction terms from interaction models, taking away square terms from quadratic models, or even adding cubic terms like bAAB x 2A xB , bABC x A x B xC , or bAAA x3A , to quadratic models gives nonstandard models which can also be used for DoE. Nonstandard models are also obtained when mixture components are investigated together with normal factors, or when so-called dummy variables (indicator variables) are used to code categorical factors that have three or more settings. Optimal designs for standard and nonstandard models are: – Linear models: resolution III factorial designs or so-called Plackett – Burman designs (which are very similar to factorial designs) if no interactions are present – Interaction models with only some interaction terms: resolution IV factorial designs or so-called D-optimal designs (see Section 7.3); resolution IV designs are also used for linear models when interactions may be present but are assumed to be unimportant – Interaction models with all interaction terms: resolution V (or higher) factorial designs or D-optimal designs – Quadratic models: central composite designs (CCD; see below) or so-called Box – Behnken designs – All nonstandard models: D-optimal designs (see Section 7.3)

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Table 2. Standard models for two and three factors

Two factors Three factors

Linear model

Interaction model

Quadratic model

y = b0 + bA x A + bB x B y = b0 + bA x A + bB x B + bC x C

. . .+ bAB x A x B . . .+ bAB x A x B + bAC x A x C + bBC x B x C

. . .+ bAA x 2A + bBB x 2B . . .+ bAA x 2A + bBB x 2B + bCC x 2C

Factorial designs are discussed in detail in Chapter 3. Central composite designs (CCD) are extensions of factorial designs, in which socalled star points and additional replicates at the center points are added to allow estimation of quadratic coefficients (see Fig. 13 and Table 3). The number of star points is simply twice the number of factors and, ideally, the number of replicates at the center is roughly equal to the number of factors. The distance α from the center point to the star points should be greater than one, i.e., the star points should be outside the domain defined by the factorial design. A star distance α = 1 is sometimes used; then star points lie in the faces of the factorial design. A good alternative to a CCD is the Box – Behnken design, in which no design points leave the factorial domain [5]. Box – Behnken designs do not contain a classical factorial design as a basis. Table 3. Design matrix of a CCD for two factors. The run Nos. 1 to 4 form the factorial design, Nos. 5 to 8 the star points, and No. 9 the center point. No.

A

B

1 2 3 4 5 6 7 8 9

−1 1 −1 1 − 1.41 1.41 0 0 0

−1 −1 1 1 0 0 − 1.41 1.41 0

y

CCDs are normally used with full factorial designs, and sometimes with resolution V designs. Another class of designs, called Hartley designs [20], are similar to CCDs and are based on resolution III factorial designs. When, as is sometimes done in industrial practice, only some quadratic terms are added to an existing interaction model, it is useful to include star points only for those factors for which quadratic terms have been added.

4.4. Using Regression Analysis to Fit Models to Experimental Data Factorial designs are orthogonal for linear and interaction models which means that coefficients can be estimated independently of each other. This is why calculating effects as described in Chapter 3 is so easy. For more complex models and designs, the analysis will be based on multiple linear regression (MLR) and its variants, such as stepwise regression, variable subset selection (VSS), ridge regression (RR), and partial least squares (PLS) [17]. All of these methods represent ways of fitting the model to the data, in the sense of minimizing the sum of squared distances from measured response values to model values (Fig. 14). They differ in that the minimization procedure is subject to different constraints, and their performance differs only in the case of badly conditioned designs, i.e., when the design is not really adequate for estimating all model coefficients. If yi stands for the observed response value at experiment i and yˆi represents the predicted value at that point, then the least-squares estimates for the coefficients (b0 , bA , bB , bAB , ...) 2 are those for which Σ (yi − yˆi ) is minimized (remember: yˆi depends on the ˆb values).

5. Methods for Assessing, Improving, and Visualizing Models There are many statistical tools that allow a basic judgement of whether a fitted model is sound. A useful selection of these is: – The regression measure R2 to check the quality of fit – The prediction measure Q2 to check the potential for prediction and to prevent so-called over-fit, which means that the model is so close to the data that it models experimental errors

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Figure 13. Central composite design for two factors (left) and three factors (right)

Figure 14. How least-squares regression works: squared distances from the model to the observed values are minimized

– Analysis of variance (ANOVA), to compare the variance explained by the model with variance attributed to experimental errors and to check for significance – Lack-of-fit test (LoF) to assess the adequacy of the model – Analysis of the residuals to find structural weaknesses in the model and outliers in the collected data These methods are used both to qualify models for prediction and optimization purposes and also to find indications of how to improve the models in the sense of increasing their reliability. The different statistical methods are explained and ways to interpret and use them toward model improvement are discussed.

5.1. R2 Regression Measure and Q2 Prediction Measure The regression measure R2 is the quotient of squared deviations due to the model, SSreg = 2 Σ (ˆ yi − y¯) , and the total sum of squared deviations of the measured data about the mean 2 value y¯, SStot = Σ (yi − y¯) , i.e., R2 = SSreg /SStot .

If ordinary least squares is used for fitting the model, R2 = 1 − SSres /SStot , where SSres = 2 Σ (yi − yˆi ) , because SSres = SStot − SSreg , the sum of squared residuals. An example for the calculation of R2 is given in Figure 15 and Table 4. R2 is always between 0 and 1 and should be as close as possible to 1; how close it should be depends upon the context of the application. When a measuring device is calibrated, R2

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should be above 0.99, whereas when the output of a chemical reaction with several influencing factors is examined, R2 may well be around 0.7 and still belong to a very useful model.

Figure 15. From the example in Section 3.2: observed values yi and predicted values yˆi from a model with coefficients ˆb0 = y¯ = 71.6, ˆbA = 5, ˆbB = 11, ˆbAB = 4 (see Section 4.2), so that yˆi , yi − yˆi andyi − y¯ can be calculated 2

R is a deceptive measure because it is prone to manipulation: by adding terms to a model, it will always be possible to get a value of R2 that is almost one, without really improving the quality of the model. On the contrary, many models with a very high R2 tend to “over-fit” the data, i.e., they model the experimental errors. This is a common and unwanted phenomenon because it decreases the prediction strength of a model. Especially when a design is not orthogonal, one should be wary of over-fit. To counteract overfit and improve the reliability of model predictions it is useful to consider the prediction measure Q2 . The calculation of Q2 is similar to that of R2 , except that in the second equation, the prediction error sum of squares (PRESS) is used instead of SSres : Q2 = 1 − PRESS/SStot

2
 Here, PRESS = yi − yˆˆi , where yˆˆi is the prediction for the ith experiment from a model that has been fitted by using all experiments except this ith one. In a sense yi − yˆˆi is a fair measure of prediction errors. PRESS is always greater then SSres , and therefore Q2 is always smaller then R2 . The relationship between Q2 and R2 and an example for the calculation of Q2 are given in Figure 16 and Table 5. In good models, Q2 and R2 lie close together, and Q2 should at least be greater than 0.5. This

may not be the case if the design is saturated or almost saturated, which means that the number of experiments that have been carried out equals or only slightly exceeds the number of terms in the model. In this case, Q2 may underestimate the quality of the model, because leaving out single measurements may destroy the structure of the design. It is more dangerous, however, to overestimate the quality of a model, which may happen if many experiments are replicated. In these experiments yˆˆi will be very close to yˆi because only one of the measurements is left out in the calculation of yˆˆi . This means that PRESS may be unduly close to SSres , and Q2 unduly close to R2 . Nevertheless, Q2 is normally quite a useful measure to prevent overfit.

5.2. ANOVA (Analysis of Variance) and Lack-of-Fit Test Analysis of variance, usually abbreviated as ANOVA, is a general statistical tool which is used to analyze and compare variability in different data sets. It becomes a powerful tool that can be used for significance testing when assumptions about the error structure underlying the data can be made. In the context of DoE, ANOVA can be used to complement regression analysis and to compare the variability caused by the factors with the variability due to experimental error. Strictly speaking, “analysis of variance” should be referred to as “analysis of sum of squares” or, even more correctly, “analysis of the sum of squared deviations”, because it is actually this sum of squares that is decomposed (Fig. 17). However, the aim of ANOVA is to see to what extent the variability in the measured data is explainable by the model and to judge whether the model is statistically significant. To make this comparison, SSreg and SSres must first be made comparable by considering the degrees of freedom. Already when R2 is calculated, it is arguable that the number of terms in the model p and the number of runs in the design N should be considered in the calculation, and that R2adjusted = 1 − [(N − 1)SSres ]/[(N − p)SStot ]

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Table 4. Calculation of R2 for the example above: SSres = 3.2, SSreg = 648, SStot = 651.2, hence R2 = 1 − SSres /SStot = 0.995. This is a very good value for the regression measure R2 .

1 2 3 4 5 Squared sum

A

B

yi

yˆi

yi − yˆi

− + − + 0

− − + + 0

60 62 74 92 70

59.6 61.6 73.6 91.6 71.6

0.4 0.4 0.4 0.4 − 1.6 SSres = 3.2

yi − y¯ − 11.6 − 9.6 2.4 20.4 − 1.6 SStot = 651.2

ˆ Figure 16. How Q2 relates to R2 : PRESS ≥ SSres and Q2 ≤ R2 ; yˆi is usually between yi and yˆ i Table 5. Calculation of Q2 for the example above: PRESS = 260 and Q2 = 1 − 260/651.2 = 0.601. This is quite a reasonable Q2 value. Hence, there is no indication of over-fit.

1 2 3 4 5 Squared sum

A

B

yi

yi

y i − yi

yi − y¯

− + − + 0

− − + + 0

60 62 74 92 70

52 54 66 84 72

8 8 8 8 −2 260

− 11.6 − 9.6 2.4 20.4 − 1.6 651.2

should be used instead of R2 = 1 − SSres /SStot . But since R2adjusted normally lies somewhere between R2 and Q2 , these two are quite sufficient for a first model assessment. In any case, N − 1 is the total number of degrees of freedom (of variation with respect to the mean value), p − 1 is the number of degrees of freedom of the model (not counting the constant), and N − p is the so-called residual number of degrees of freedom.

Figure 17. Decomposition of SStot into SSreg and SSres

ANOVA compares the model to the residuals and tells us whether the model is statistically sig-

nificant under the assumption that the residuals can be used to estimate the size of random experimental error. MSreg = SSreg /(p − 1) is compared to MSres = SSres /(N − p), also called the mean square error (MSE), by subjecting them to a so-called F-test for significance. If the quotient, F emp = MSreg /MSres , is greater than 1, then there is reason to suspect that the model is needed in order to explain the variability in the experimental data. If this has to be proven “beyond a reasonable doubt” (i.e., for statistical significance), F emp must be greater than the theoretical value of F crit (p − 1,N − p,γ), which is always greater than 1, where γ is the desired level of confidence. Taking the square root of MSres furnishes a reasonable estimate of the standard deviation of a single measurement, provided N − p is not too small and MSres does not change dramatically when terms with small coefficients are added

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to or removed from the model. This number is called the residual standard deviation (RSD or SDres ). In the example that has been used SSreg = 648, p − 1 = 3, hence MSreg = 216. SSres = 3.2, n − p = 1, hence MSres = 3.2. F emp MS can be calculated as Femp = MSreg , but a comres parison to F crit (3, 1, 95 %) is not meaningful, because there is just one √degree of freedom. Also the estimate RSD = MSres = 1.789 should not be taken too seriously in this example. The lack-of-fit test addresses the question of whether the model may have missed out on some of the systematic variability in the data. This test can only be performed if some of the experimental runs have been replicated: When replicates are present, SSres can be further decomposed into a pure error part SSp.e. and a remaining lack-of-fit part SSlof (Fig. 18).

Figure 18. Decomposition of SSres into SSlof and SSp.e.

The corresponding significance test compares MSlof = SSlof /(N − p − r) to MSp.e. = SSp.e. /r, where r is the total number of replicates of experimental runs in the design. (A replicate count does not include the original run: if there are five runs at the center point, then one of these runs is counted as the original and the four others as replicates; hence, r = 4; if a design consisting of eight runs is completely replicated, there are eight runs that have each been replicated once, hence r = 8.) As in ANOVA, a quotient MSlof /MSp.e. greater than 1 means that there may be a lack of fit although the evidence is still weak, whereas a quotient greater than F crit (N − p − r, r, 1 − α) > 1 means a significant lack of fit at the error probability level α. The lack-of-fit test method can be regarded as a good complement to the Q2 prediction measure because the former works well when many replicates are involved, while the latter makes sense when there are only few replicates. Of course, the latter situation is more common in DoE.

5.3. Analysis of Observations and Residuals Ideally, when models are fitted to data, the model describes all of the deterministic part of the measured values and the part due to experimental error is reflected by the residuals. It is a presumption in linear modeling, i.e., when using least squares fitting as a criterion, that the experimental error is – Identically (i.e., evenly) distributed over all measurements – Statistically independent of the measured value, the order of the experiments, the preceding measurement, the settings of the factor variables, etc. To verify this assumption and to detect outliers, a rudimentary examination of residuals should always be performed as a step toward qualifying the model. The most important tests for residual structure are: – Test for influence of single observations on the model – Test for normal distribution and test for outliers – Test for uniform variance – Test for independency of measurement errors These tests can be performed formally as described above for ANOVA and lack-of-fit testing. However, formal testing of hypotheses of the type necessary here usually requires a very high number of residual degrees of freedom N − p. Since this number is actively and consciously kept low in DoE, these tests do not often yield useful results. This is why it is common practice to plot observations and residuals in different types of graphs in order to detect structural weaknesses and to find hints on what the problem might be and on how to avoid it. Common plots are: – Observed values versus predicted values to see whether there are observations that had undue influence on the model – Ordered residuals in a normal probability plot to detect outliers and indications of a possibly nonnormal distribution of residuals – Residuals versus predicted values in order to spot inhomogeneities in variance

Design of Experiments – Residuals versus different factor variables to detect weaknesses in the model – Residuals versus run order to check latent influences of time and autocorrelation There is a further systematic approach to coping with possible inhomogeneities in variance that was proposed by G. E. P. Box and D. R. Cox in 1964 and that can be summarized in the so-called Box – Cox plot: In addition to fitting a model to the response y, they suggest fitting models to the transformed response (yλ − 1)/λ for different λ and then to choose λmax such that residuals are closest to normally distributed. The Box – Cox plot, displays performance of a transformation against λ. It is a practical way of finding indications of what type of transformation to the response data may be useful.

–

–

5.4. Heuristics for Improving Model Performance An interesting although somewhat dangerous aspect of DoE and statistical analysis is that of “pruning” models. What is meant by this is the following: When a model displays some weaknesses in the analysis phase explained above, there is often the possibility to improve the performance of a model by simple measures that do not require additional experiments: – Excluding model terms with insignificant coefficients – Introducing a transformation – Excluding observations that seem to unduly dominate the model or that lie far away from the model – Include one or two terms in the model without overstressing the design, i.e., without having to perform further experiments Of course there are always situations in which a model remains inadequate and further experiments must be performed to reach the objectives. The following is a set of heuristics, which may help to improve a model in one of several possible situations that may arise during the analysis phase: – R2 and Q2 are both small (i.e., there is just bad fit): check for outliers using a normal probability plot; check that the response values correspond to the factor variables; check

–

–

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pure error if experiments have been repeated; check that no important factors and interactions are missing in the model. R2 is high but Q2 is very low (below 0.4, i.e., tendency for over-fit): remove very small and insignificant terms from the model (they may reduce the predictive power of the model); check for dominating outliers (by comparing observed and predicted values) and try fitting the model without them (remember, however, that for screening designs with few residual degrees of freedom, Q2 may be low although the model is good). There are clear outliers in the normal probability plot: usually these outliers have not had much influence on the model (otherwise they would be seen when comparing observed and predicted values), however, they often lead to low R2 values; check what happens when they are removed from the model; check the records, repeat the experiment; mistrust predictions of the model in the vicinity of such outliers; consider that the outlier may contain important information (maybe this is a new and better product). There is some structure in a plot showing residuals versus a factor variable (Fig. 19): this is a sign that the model is too weak and should be expanded; this can usually not be done without enlarging the design. In rare cases of analysis it may be observed in a plot displaying residuals versus predicted values that residuals are not homogeneous; this may be the case when a response varies over several orders of magnitude and the size of errors is either proportional to the response values or satisfies some other relation. Indications of this can also be seen in the Box – Cox plot if the optimal exponent λmax deviates significantly from unity. A transformation of the response, e.g., taking logarithms or square roots, may be the correct measure.

Sometimes, the measures suggested above do not lead to a stable model; outliers seem be present in spite of all attempts to improve the situation, there are bends and curves in the normal probability plot, coefficients are not significant but also not negligible, and so on. This is usually an indication that not enough experiments have

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Figure 19. Standardized residuals are residuals divided by RSD = SDres (see Section 5.2). When plotted against a factor variable they may give an indication of how a model can be improved

been done. It is the authors’ recommendation in this case to either – Reduce all “manipulation” to a minimum (eliminate only obvious outliers from the data and very small coefficients from the model), and use the model knowing that it is weak but may still be useful, or to – Strip the model down to linear, find the optimal conditions and repeat a larger design here. In any case the heuristics above cannot repair a model that is simply incorrect, and care must be taken not to succumb to the temptation of systematically perfecting such an incorrect model. This is usually not the purpose of modeling.

Figure 20. Contour plot of yield, showing the dependency on catalyst and temperature (for the data from Fig. 6 and Fig. 15). It can be seen that there is an interaction between the two factors

5.5. Graphical Visualization of Response Surfaces When statistical tests such as those described above detect no serious flaws in the model, it is useful to plot the model by reducing it to two or three dimensions and by representing it as a contour diagram (see Fig. 20) or a response surface plot (see Fig. 12). When the model consists of more than two factors, surplus factors are set constant, usually to their optimum level, i.e., where responses are optimal. Setting a factor constant corresponds to slicing through the experimental domain and reducing its dimension by one. An interesting possibility is setting a third factor constant at three levels, and placing the three contours side by side.

Figure 21. Superimposed contour plot for two responses used for finding specification domains for the two factors x A and x B , given the specifications for the responses Y (1) and Y (2)

Visualizing models is particularly interesting when the modeling involves several responses, because contour diagrams can be used to find specification domains for the factors when specification ranges for the response variables have

Design of Experiments been imposed (e.g., by the customer of the chemical product to be produced; see Fig. 21).

6. Optimization Methods A major purpose of using response surface modeling techniques is optimization. There are several approaches to optimization depending on the complexity of the situation. Single-response optimizations are easier to handle and are treated in Sections 6.1 and 6.2. Multi-response optimization requires weighting the different responses according to their importance. This can be quite difficult, and the relevant techniques are treated in Section 6.3.

6.1. Basic EVOP Approach Using Factorial Designs A very basic approach to optimization was proposed in the 1960s by G. E. P. Box and N. R. Draper [6]. It is known as Evolutionary Operation, or EVOP (and has nothing to do with evolution strategy or genetic optimization algorithms). When performing optimization experiments in a running production unit, it is not possible to vary factors over very large domains. Hence, effects will be small and very hard to detect against an underlying natural variability of the response. The idea is to simply repeat a small factorial design for, for instance, two factors, as often as is necessary so that the random noise can be averaged out and the effects become apparent.

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The direction of maximal improvement of a response can be deduced from these effects and can subsequently be used to find a new position for a second factorial design. At this new position the second factorial is repeated until significant effects or a satisfactory result is obtained. This procedure can be repeated until a stable optimum is found (Fig. 22).

6.2. Model-Based Approach When it is less difficult to establish the significance of a model, as is usually the case on a laboratory scale or in a pilot plant, the model can directly be used for optimization. Gradients of the polynomial models are calculated from the model coefficients because they point to the directions of maximum change in the responses. These directions are then used in an iterative search for a maximum or minimum, as was proposed by Box and Wilson [9]. For quadratic and interaction models, it makes sense to use conjugate gradients, which correct the direction of maximum change by the curvature of the model function (Fig. 23). In this way the search for an optimum can often be accelerated. When a predicted optimum lies within the experimental domain, it is usually quite a good estimate of the real optimum, particularly when a response surface model has been used. When it is outside of the domain, where, by construction, the model is most probably not adequate in describing the process or system under study, it should be validated in the fashion described in Section 6.4.

Figure 23. How the conjugate gradient compares to the gradient Figure 22. Example of a moving EVOP design (22 factorial design with a center point) that has found a stable optimum on the right

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6.3. Multi-Response Optimization with Desirability Functions In an industrial environment, it is usually not sufficient to maximize or minimize one response only. A typical goal is to maximize yields, minimize costs, and reach specification intervals or target values for quality characteristics. But how should one proceed when these goals are contradictory? This question arises quite frequently in practice. For example, yield, cost, and the quality characteristics are used as responses in an experimental design. The design should be chosen to allow adequate models to be fitted to all responses. For each response, a model will be fitted to its corresponding measured data. Now optimization can be started. Before being able to apply any mathematical optimization algorithm, the goal of the optimization has to be translated from a multidimensional target to a simple maximization or minimization problem. This can effectively be done by using desirability functions. A desirability function dj (y(j) ) for one response y(j) , measures how desirable each value of this response is. A high value of dj (y(j) ) indicates a high desirability of y(j) . Many types of desirability functions have been proposed in literature (e.g., [14]). Two-sided desirability functions are used if a target value or a value within specification limits is to be reached (Fig. 24, left), whereas one-sided desirability functions are used if the corresponding response is to be maximized (Fig. 24, right) or minimized. An example of a two-sided desirability function is:   (j) (j) dj y (j) = 0 f or y (j) > ymax or y (j) y (j) > ymin ,    t  t (j) (j) (j) dj y (j) = ymax − y (j) / ymax − ytarget (j)

(j)

f or ymax > y (j) > ytarget . (j )

This function depends on a target value ytarget , a (j ) minimum acceptable value ymin , a maximum ac(j ) ceptable value ymax , and two exponents s,t > 0

(Fig. 24, left). High values of s and t emphasize the importance of reaching the target; smaller

(j )

values leave more room to move within ymin (j ) and ymax . To build one desirability function for all responses, the geometric mean is usually taken:     /1/k .  D (y) = d1 y (1) ·d2 y (2) ·...·dk y (k)

By substituting the y by the predicted values of the model functions, desirability D becomes a function of the factor variables, D = D(x). To find the best possible factor settings, this function D should be maximized by using any efficient mathematical optimization procedure. In many typical applications, production cost and yield tend to increase simultaneously, and the optimizer, in trying to raise yield and lower cost, reaches an optimum somewhere within the experimental domain. Then it is essential to have good information about the quality of predictions in the interior of the domain. It may be necessary to fit a quadratic model to obtain such good predictions. In a similar fashion, quality characteristics are often contradictory, e.g., more textile strength means less skin and body comfort; again, the optimizer will find an optimal compromise, which means suboptimal settings for the individual responses within the experimental domain. Often, the predicted optimum is at the border of the experimental region. It is in fact quite typical that, in cases where linear or interaction models are used and no compromises as described above are necessary, two or more corners of the experimental domain are found to be locally optimal. From a mathematical point of view this seems to present a problem, but in fact it is a very promising situation for further improvement of the product or the process under study. Although the models were made to work inside the experimental region, they often also give very useful information just outside the experimental domain. To be able to make a compromise between different y, it is sometimes more practical to use desirability functions dj that are not exactly 0 for y(j) > y(j) max or y(j) < y(j) min , but only close to 0. This allows the comparison of factor settings for which desirability would otherwise be zero,

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Figure 24. Two responses with corresponding desirability functions: a two-sided desirability function for y(1) (target value (1) ytarget desired) and two different one-sided desirability functions for y(2) (maximization of y(2) desired)

because some of the y are outside of specification. In any case, constructing desirability functions is quite a delicate task, and it is useful to play around with the desirability function when looking for an optimum. When extrapolating outside the domain of x, remember that predictions using the model deteriorate when the experimental region is left. This means that predicted optima outside the region must always be validated by further experiments.

6.4. Validation of Predicted Optima There are several reasons why predicted optima from empirical models should always be validated by further experiments: – Errors in observed responses propagate into predictions of the model and hence into the position of the predicted optima – The model may be insufficient to describe the relevant behavior of the experimental system under study and lead to bad predictions of optima – The desirability function used in multiresponse optimization may not correctly depict all aspects of the real goal – Predicted optima may lie outside the initial experimental domain, where the quality of the model is doubtful – Predicted optima may be unsatisfying in that not all responses lie near the desired values or within specified ranges. It is possible within the scope of linear modeling to calculate confidence intervals for all responses, and hence to get a good idea of the quality of prediction. This calculation of confidence

intervals is only correct for qualified models as explained in Chapter 5. Confidence intervals of this type must be mistrusted if there is a significant lack of fit or if Q2 is very low. They should also be mistrusted if residuals of observed responses in the vicinity of the predicted optima are very large. In this case and when predicted optima are outside the experimental domain, further experiments are inevitable. A good strategy for this is to find a sequence of experimental settings for which the predictions gradually improve (according to the model). This sequence will start within the experimental domain and will typically leave it at some point (see Fig. 25). Experiments in this sequence should be carefully performed and checked against predicted results. This procedure will usually lead to very good results. It should be emphasized, however, that these experiments are purely confirmatory in nature, they may lead to better products and better processes but they will not lead to better (fitted) models.

7. Designs for Special Purposes The design methods that have been discussed until now cover many situations in which the cause and effect relationship between several factors and one or more responses of an experimental system are investigated. However there are cases where further methods are necessary to solve modeling problems. Four characteristic situations are treated in this chapter: – Designs for modeling and optimizing mixtures are used in product optimization

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Figure 25. Predicting optima based on the model for increasing domains: A strategy in optimization consists of predicting a series of optima while slowly increasing the domain. Experiments should be performed at these predicted optima for validation purposes. In this way either a satisfactory optimum is reached or a further design is employed

– Designs for categorical factors may be used for raw product screening and for implementing blocking (as described in Section 2.3) – So-called optimal or D-optimal designs which are implemented for advanced modeling when nonstandard models or irregular experimental domains are to be investigated – Robust design techniques have the aim of not only optimizing response values but also response variability. Further interesting topics, such as – Nested designs, where the levels of factor B depend on the setting of factor A [7], [27] – The field of QSAR (quantitative structure activity relationships), which is becoming more and more interesting in conjunction with the screening of active ingredients in medicines – The whole field of DoE for nonlinear dynamic models involving differential algebraic equations and others, cannot be treated here. The advanced reader is invited to research on his own, particularly on the latter two themes, which are still in constant motion (both of them incidently make extensive use of D-optimal designs to be shortly discussed in the following).

7.1. Mixture Designs All designs described above assume that all factor variables can be set and varied independently of each other in an arbitrary manner. This is the typical situation in process optimizations, where factors are technical parameters such as temperatures, pressures, flow rates, and concentrations. However, when the goal is to model the properties of a mixture in order to find optimal ratios for its components, then this factor independence may no longer be presumed. There are many branches of the chemical industry and rubber industries where mixtures are investigated, and there is a whole range of characteristic problems: – Optimizing the melting temperature of a metal alloy – Reducing cost in producing paints while keeping quality at least constant – Optimizing the taste of a fruit punch or the consistency of a yogurt – Increasing adhesion of an adhesive – Finding the right consistency of a rubber mixture for car tires These are cases where experimental design methods must be modified to cope with the fact that all components of the mixture must add up to unity, i.e., for three components, x A + x B + x C = 1. Nevertheless there are useful models and designs for mixtures. They are based on the so-called mixture triangle or the mixture

Design of Experiments

455

Figure 26. For three-component mixtures the cube is no longer the appropriate geometrical object for modeling the experimental domain. The equation x A + x B + x C = 1 defines a triangle. Unfolding this triangle leads to the mixture triangle for three factors (left). Mixtures with four components can be modeled by a simplex; setting one of the factors constant again leads to a triangle for the others, e.g., x D = 0.5, i.e., x A + x B + x C = 0.5 (right)

simplex when four or more components are involved (Fig. 26). All simplices have the following properties, which facilitate analysis and the visualization of results: – Response plots can be generated based on the mixture triangle (as contour plots). – A simplex is very similar to the cube in that its boundaries and also its sections parallel to its boundaries are again simplices in a lower dimension. This allows contour plots to be used in the mixture triangle even in situations where more than four components are involved; just set the surplus factors constant and put the three important ones into the mixture triangle. – Lower and upper levels of the mixture factors can be visualized geometrically as cutting off parts of the simplex. An active upper level means cutting off a corner, an active lower level means cutting off the base (or side) of a simplex. When upper levels are active, the experimental domain is no longer a regular simplex, but only a subset thereof, a so-called irregular mixture region. The best experimental designs for regular mixture regions are so-called simplex lattice designs. Simplex lattice designs place experiments at the corners and along edges and axes of a simplex. Depending on the complexity of the model that is to be fitted, different strategies in picking out the points are used. The most typical design for a linear model is the axial simplex design

which consists of experiments at the corners, at the centroid (equal amounts of all components) and midpoints of the axes from the corners to the centroid (Fig. 27, Table 6). Details can be found in [12].

Figure 27. Geometrical view of an axial simplex design for three-component mixtures Table 6. Design matrix for an axial simplex design for three mixturecomponents Run no.

Comp. A

Comp. B

Comp. C

1 2 3 4 5 6 7

1 0 0 0.6666 0.1667 0.1667 0.3333

0 1 0 0.1667 0.6666 0.1667 0.3333

0 0 1 0.1667 0.1667 0.6666 0.3334

When investigating mixtures, it is important to keep in mind that, although it is possible to correlate changes in the responses to changes in

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the factor settings, it is principally impossible to tell which factors have caused these changes in the responses, because whenever one factor has changed, at least one other factor has also changed. So which one should be made responsible? Example 1: A fruit punch containing pineapple, grapefruit, and orange juice seems to become more tasty when pineapple juice is added. Example 2: A long drink containing pineapple and orange juice and vodka seems to become less bitter when pineapple juice is added. Is it possible, in the two examples, to attribute the change in taste to the change in the amount of pineapple? When mixtures are modeled and optimized, it is not always necessary to use simplex designs and corresponding models. There are principally three ways to proceed, only one of which involves using simplex designs: – Define component ratios as factors and use the classical factorial design or a CCD (instead of ratios, any other transformation leading to independent pseudocomponents can be used). – Use a simplex design, as described above, or D-optimal design, as described in Section 7.3. – Identify a filler component, for example, x C , which does not have an effect on any of the responses, use a factorial design, a CCD, or a D-optimal design for the remaining components (together with the constraint x A + x B ≤ 1, if necessary), and regression analysis for a model in which the filler variable does not appear. Analyzing mixture designs requires some care, because due to the dependencies amongst the mixture factors the models must be adapted to the situation. H. Scheff´e and D. R. Cox were the pioneers who developed ways to use models correctly for the mixture problem. Details cannot be included here. The reader is referred to the literature [10], [12], [33], [35].

7.2. Designs for Categorical Factors Not all factors that influence a product or a process can be quantified. Examples of nonquantifiable or so-called categorical (or discrete) factors are:

– The supplier of a raw material, who may influence some of the quality characteristics of an end product – The date or the time of the year may influence how well a process will perform – Different persons doing the same experiments – The type of catalyst or solvent – Mutants of a strain of bacteria in a fermentation process All these are possible influencing factors that cannot be quantified in a satisfactory manner. (In the case of investigating solvents it would be a very good idea to consider polarity as a quantifiable factor instead of just the type of solvent.) Hence, particularly when more than two instances of a categorical variable must be considered, new design techniques are necessary. Typical questions that arise in conjunction with categorical or discrete factors are: – How large is the impact of a qualitative factor? Is it significant? – If so, which instance yields the best results? – Does the categorical factor interact with other factors? Are there interactions between different categorical factors? – Is a possible optimum for the other (continuous) factors robust with respect to varying the categorical factor? If so, where does it lie? If there are just two instances or categories of a qualitative factor, it is easy to encode them as “−” and “+” and to use factorial designs. The categorical or qualitative factor can then be used in calculations like a continuous quantitative factor. Effects and coefficients can be calculated in the usual way, and they measure how large the influence of changing categories is. The same is true for interactions with other factors; they measure how the influence of these factors on the responses changes when changing categories. Treating categorical factors with three or more instances is much more delicate. Within generalized linear modeling, it is possible to treat factors with three or more instances by increasing the number of dimensions of the problem. Dummy variables are used to differentiate between instances: Let p1 , p2 be two dummy variables and encode instances i1 , i2 , i3 in the fashion shown in Table 7.

Design of Experiments Table 7. How to encode three instances of a categorical factor by using two dummy variables Coding for

p1

p2

Instance i1 Instance i2 Instance i3

1 0 −1

0 1 −1

Geometrically the three instances become three points in the two-dimensional coordinate system of the dummy variables. The coded design with example values for a response y as well as the corresponding geometrical visualization is shown in Figure 28.

457

By additionally setting c3 = − c1 − c2 the same measure has been found for instance i3 . For the example shown in Figure 28 the coefficients are c0 = 9, c1 = − 3, c2 = − 1, c3 = 4. Designs for categorical variables will normally be adapted to the problem at hand and generated by a computer algorithm. In fact optimal designs, as described in Section 7.3, should be used. It is interesting to note that dummy variables can be used to model interactions: Just use these variables in interaction terms of a model. The coefficients then measure by how much the effect of another factor varies from a given instance to its mean effect taken over all instances (i.e., its main effect). It would go beyond the scope of this article to go into details about this, in particular about calculation of degrees of freedom. A short digression on dummy variables and their use for modeling blocking can be found in [15].

7.3. Optimal Designs

Figure 28. Geometrical and tabular representation of the design and the response values

Now the advantage of using dummy variables becomes apparent: Analysis of data can be done using the same regression methods as described in Section 4.4, and the interpretation of results, although being somewhat different, is still relatively straightforward: The model has the form y = c0 + c1 p1 + c2 p2

where c1 and c2 are coefficients pertaining to the dummy variables p1 and p2 . Model predictions for the instances i1 , i2 , i3 become: yˆ (i1 ) = c0 + c1 yˆ (i2 ) = c0 + c2 yˆ (i3 ) = c0 − c1 − c2

So the coefficients c1 , c2 are actually a measure of the extent to which instances i1 and i2 differ from the mean c0 .

The usual procedure in planning an experimental design is to identify factors and responses, to determine ranges for the factors, and then to choose a standard design amongst those that have been discussed. This design may be factorial, Plackett – Burman, CCD, Box – Behnken, Simplex, or the like. However, in some situations none of the standard designs are suitable. Such situations typically are: – Use of nonstandard models – Use of nonstandard experimental domains (i.e., constraints involving more than one factor) – Special restrictions on the number of runs or tests that can be performed – Use of mixture designs when either upper factor levels are active (this means that the regular simplex structure is corrupted) or when normal continuous factors are to be investigated in the same design – One or more categorical factors with three or more discrete settings are present. In these cases it is common practice to generate a design by using a mathematical algorithm that maximizes the information that the design will contain. Such designs depend on the model to be used and on the experimental domain to be covered.

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Design of Experiments

Criteria involved in optimizing designs are based on the extended design matrix, usually denoted by X, which is built up of the design matrix and extended by a column for each term in the model. The basic idea behind optimal designs is to ensure that there are no correlations between the columns of this X matrix. For if there are correlations here, then the influence of the terms involved cannot be resolved (this situation is similar to that described in Section 7.1 in conjunction with mixture components, which are always correlated). Working with optimal designs (Fig. 29) involves: – Defining the experimental domain (including possible constraints) – Choosing the appropriate model – Specifying experiments that the experimenter explicitly wants to perform or has already performed – Selecting the optimization criteria for the design – Specifying approximately how many experimental runs can be performed. Typically not just one design is generated, but a whole number of designs of differing size (test or run number). This makes it possible to evaluate designs by comparing the optimality criteria for different designs. Typical criteria are D-optimality, G-optimality, A-optimality, and E-optimality. They are essentially varianceminimizing design criteria in the sense that the variances of model predictions or model coefficients are minimized. The most commonly used criterion is D-optimality, which leads to a maximized determinant of the squared X matrix. For more details, in particular, for further optimality criteria, the reader is referred to special literature on optimal designs [1], [32]. Optimal designs will have a high quality if a sufficient number of experimental runs or tests has been allowed for. They are analyzed by regression analysis like other designs, and the corresponding models can be used for prediction and optimization purposes.

7.4. Robust Design as a Tool for Quality Engineering Sometimes the goal of investigation goes beyond estimating the effects of influencing fac-

tors and predicting process behavior or quantifiable product characteristics. A typical application of DoE techniques is toward finding conditions, i.e., factor settings, for which not only quantifiable responses are optimal (usually in a multivariate way as described in Section 6.3) but also the variability of response values is minimal with respect to disturbing or environmental factors that act during production or field use of the product. Typically, these influences can not all be controlled or are too expensive to be controlled even during the experimentation phase. Doing DoE with the goal of reducing variability is known as robust design. In a robust design, control factors or design factors are varied in a design as described above, typically in a screening design, and experiments are repeated at each trial run so as to estimate the standard deviation (or variance) at each trial. This dispersion measure is employed as a new response value. Statistical analysis and optimization tools such as those described above are then used to quantify and finally to minimize variation, while at the same time improving product characteristics. Taguchi [18], [24], [38] introduced an additional idea into robust design, namely, the concept of so-called noise factors or environmental factors [3], [24]. These are introduced as perturbing factors in a designed experiment with the idea that they simulate possible external influences which may effect the quality of products after they have left the plant. These factors, which will in fact increase the standard deviation, are varied in a second experimental design which is performed for each of the trial runs in the design for the design factors. To distinguish between the two designs, the design for the environmental factors is sometimes called outer array and that for the design factors is called inner array. The use of outer arrays is recommended, when noise factors can rather cheaply be varied independently of the controlled factors. Example: During the production of CDs a transparent coating is applied to protect the optical layer. CDs are placed onto a turntable, a droplet of lacquer is applied, and the CD is spun to spread the lacquer. Parameters to be varied are the size of the droplet, the speed and the acceleration of the turntable and the temperature in the room. CDs are to be subjected to different

Design of Experiments

459

Figure 29. An optimal design with constraints and inclusion of some old experiments Table 8. A design for a quality engineering problem, using a 24−1 factorial with center point as inner array and a 22 factorial as an outer array. Run

1 2 3 4 5 6 7 8 9 10 11

Design factors; A, B, C, D

Environmental factors: H, T

A

B

C

D

− + − + − + − + 0 0 0

− − + + − − + + 0 0 0

− − − − + + + + 0 0 0

− + + − + − − + 0 0 0

H=− T=−

H=− T=+

H=+ T=−

H=+ T=+

y1

y2

y3

y4

extreme climatic conditions in order to simulate their performance in the field. This is to be done in a climate chamber. Factors to be varied in the corresponding outer array are humidity and temperature in the climate chamber. For the inner array a 24−1 factorial design with 3 realizations at the center point is chosen, for the outer array a simple 22 full factorial (Table 8). Results from the four experiments in the outer array are taken together, and the mean and the variance are calculated and subjected to effect calculation or regression analysis.

8. Software Selected DoE software and suppliers are listed in Table 9.

9. References 1. A. C. Atkinson, A. N. Donev: Optimum Experimental Designs, Oxford University Press, Oxford 1992.

Mean

Variance

y-mean

y-var

2. K. R. Bhote: Qualit¨at – Der Weg zur Weltspitze (World Class Quality), IQM, Großbottwar 1990 (American Management Association, New York 1988). 3. S. Bisgaard: Industrial Use of Statistically Designed Experiments: Case Study References and Some Historical Anecdotes, Quality Engineering 4 (1992) no. 4, 547 – 562. 4. G. E. P. Box, S. Bisgaard: The Scientific Context of Quality Improvement, Quality Progress June (1987) 54 – 61. 5. G. E. P. Box, N. R. Draper: Empirical Model-Building and Response Surfaces, John Wiley & Sons, New York 1986. 6. G. E. P. Box, N. R. Draper: Evolutionary Operation – A Statistical Method for Process Improvement, John Wiley & Sons, New York 1969. 7. G. E. P. Box, W. G. Hunter, J. S. Hunter: Statistics for Experimenters – An Introduction to Design, Data Analysis, and Model Building, John Wiley & Sons, New York 1978. 8. G. E. P. Box, H. L. Lucas: Design of Experiments in Nonlinear Situations, Biometrika 46 (1959) 77 – 90.

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Design of Experiments

Table 9. A selection of DoE software tools and providers (as of March 2002). Tool

Company

WEB link

Design-Expert D.o.E. FUSION ECHIP JMP, SAS/QC MINITAB MODDE Starfire, RS/Series STATGRAPHICS Plus STATISTICA STAVEX

Stat-Ease Inc. S-Matrix Corp. ECHIP Inc. SAS Institute Inc. Minitab Inc. Umetrics Brooks Automation Inc. Manugistics Inc. StatSoft Inc. AICOS Technologies AG

http://www.statease.com http://www.s-matrix-corp.com http://www.echip.com http://www.sas.com http://www.minitab.com http://www.umetrics.com http://www.brooks.com http://www.statgraphics.com http://www.statsoft.com http://www.aicos.com

9. G. E. P. Box, K. B. Wilson: On the Experimental Attainment of Optimum Conditions, J. Roy. Statist. Soc. B 13 (1951) 1 – 45. 10. J. Bracht, E. Spenhoff: Mischungsexperimente in Theorie und Praxis (Teil 1 und 2) Qualit¨at und Zuverl¨assigkeit (QZ) 39 (1994) no. 12, 1352 – 1360, Qualit¨at und Zuverl¨assigkeit (QZ) 40 (1995) no. 1, 86 – 90. 11. D. E. Coleman, D. C. Montgomery: A systematic approach to planning for a designed industrial experiment (with discussion), Technometrics 35 (1993) no. 1, 1 – 27. 12. J. A. Cornell: Experiments with Mixtures – Designs, Models, and the Analysis of Mixture Data, 2nd ed., John Wiley & Sons, New York 1990. 13. O. L. Davies (ed.): The Design and Analysis of Industrial Experiments, 2nd ed., Oliver & Boyd, London 1956. 14. G. Derringer, R. Suich: Simultaneous Optimization of Several Response Variables, Journal of Quality Technology 12 (Oktober 1980) no. 4, 214 – 219. 15. N. R. Draper, H. Smith: Applied Regression Analysis, 3rd ed., John Wiley & Sons, New York 1998. 16. R. A. Fisher: The Design of Experiments, 8th ed., Oliver & Boyd, London 1966. 17. I. E. Frank, J. H. Friedman: A Statistical View of Some Chemometrics Regression Tools, Technometrics 35 (May 1993) no. 2, 109 – 148. 18. B. Gunter: A Perspective on the Taguchi Methods, Quality Progress (June 1987) 44 – 52. 19. G. J. Hahn: Some Things Engineers Should Know About Experimental Design, Journal of Quality Technology 9 (January 1977) no. 1, 13 – 20.

20. H. O. Hartley: Smallest composite designs for quadratic response surfaces, Biometrics 15 (1959) 611 – 624. 21. C. D. Hendrix: What every technologist should know about experimental design, CHEMTECH March (1979) 167 – 174. 22. J. S. Hunter: Statistical Design Applied to Product Design, Journal of Quality Technology 17 (October 1985) no. 4, 210 – 221. 23. ISO 3534-3: 1999 (E/F): Statistics – Vocabulary and symbols – Part 3: Design of experiments. 24. R. N. Kacker: Off-Line Quality Control, Parameter Design, and the Taguchi Method (with discussion), Journal of Quality Technology 17 (October 1985) no. 4, 176 – 209. 25. R. L. Mason, R. F. Gunst, J. L. Hess: Statistical Design and Analysis of Experiments with Applications to Engineering and Science, John Wiley & Sons, New York 1989. 26. B. Mittmann: Qualit¨atsplanung mit den Methoden von Shainin, Qualit¨at und Zuverl¨assigkeit (QZ) 35 (1990) no. 4, 209 – 212. 27. D. C. Montgomery: Design and Analysis of Experiments, 5th ed., John Wiley & Sons, New York 2000. 28. R. H. Myers, D. C. Montgomery: Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York 1995. 29. A. Orth, M. Schottler, O. Wabersky: Statistische Versuchsplanung, Serie: Qualit¨at bei Hoechst, Hoechst AG 1993. 30. T. Pfeifer: Qualit¨atsmanagement: Strategien, Methoden, Techniken, Hanser, M¨unchen 1993. 31. M. S. Phadke: Robuste Prozesse durch Quality Engineering (Quality Engineering Using Robust Design), gfmt, M¨unchen, 1990 (Prentice Hall, London, 1989).

Design of Experiments 32. F. Pukelsheim: Optimal Design of Experiments, John Wiley & Sons, New York 1993. 33. E. Scheffler: Statistische Versuchsplanung und -auswertung – Eine Einf¨uhrung f¨ur Praktiker, 3., neu bearbeitete und erweiterte Auflage von “Einf¨uhrung in die Praxis der statistischen Versuchsplanung”, Deutscher Verlag f¨ur Grundstoffindustrie, Stuttgart 1997. 34. K. H. Simmrock: Beispiele f¨ur das Auswerten und Planen von Versuchen, Chem.-Ing.-Tech. 40 (1968) no. 18, 875 – 883. 35. R. D. Snee: Experimenting with mixtures, CHEMTECH (November 1979) 702 – 710. 36. E. Spenhoff: Prozeßsicherheit durch statistische Versuchsplanung in Forschung, Entwicklung und Produktion, gfmt, M¨unchen 1991.

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37. S. Soravia: Quality Engineering mit statistischer Versuchsmethodik, Chem.-Ing.-Tech. 68 (1996) no. 1 + 2, 71 – 82. 38. G. Taguchi: System of Experimental Design, vol. I and II, Kraus International Publications, New York 1987. 39. J. Wallacher: Einsatz von Methoden der statistischen Versuchsplanung zur Bestimmung von robusten Faktorkombinationen in der pr¨aventiven Qualit¨atssicherung, Fortschr.-Ber. VDI Reihe 16 Nr. 70, VDI-Verlag, D¨usseldorf 1994. 40. DuPont Quality Management & Technology: Design of Experiments – A Competitive Advantage, E. I. du Pont de Nemours and Company 1993.

Computational Fluid Dynamics

463

Computational Fluid Dynamics Anja R. Paschedag, Technische Universität Berlin, Germany

1. 2. 3. 3.1. 3.2. 3.3. 3.4. 4.

Introduction . . . . . . . . . . . . . . Procedure . . . . . . . . . . . . . . . . Modeling . . . . . . . . . . . . . . . . Transport Equations . . . . . . . . Initial and Boundary Conditions Turbulent Flow . . . . . . . . . . . . Multiphase Approaches . . . . . . Numerics . . . . . . . . . . . . . . . .

. . . . . . . .

464 464 465 466 467 469 472 475

Cµ , Cε1 , Cε2 c mol/m3 cp J /(kgK) C D m2 /s F N F f G g I Jφ k m n N Pˆ

m/s 2 kg · m2 m2 /s 2 kg m2 /s 3

p N/m2 q˙ J /(m2 s) r Re Sc Shr Su

Basics . . . . . . . . . . . . . . . . . Finite Volume Method . . . . . Pressure Correction Methods Lattice Boltzmann Method . . Interpretation . . . . . . . . . . . Industrial Application . . . . . References . . . . . . . . . . . . .

Sφ

Symbols A m2 CBC1 . . . CBC4

4.1. 4.2. 4.3. 4.4. 5. 6. 7.

cell surface (FVM) constants of mathematical expressions for boundary conditions constants in k − ε model concentration specific heat capacity Courant number diffusion coefficient force general function weighting factor in time discretization filter function (LES) gravitational acceleration moment of inertia molecular flux turbulent kinetic energy mass normal vector number of nodes source term in transport equation for k pressure heat flux density residual Reynolds number source in concentration equation source in temperature equation source in energy equation

T T t u V v x

N ·m K s J /m3 m3 m/s m

αi 

m2 /s

δ ε κ λ µ µt ρ τ τs τt

m2 /s 3 Pa · s J /(msK) Pa · s Pa · s kg/m3 N/m2 N/m2 N/m2

φ 

m/s

. . . . . . .

. . . . . . .

. . . . . . .

475 476 480 480 480 482 484

source in general transport equation torque temperature time specific internal energy volume velocity position vector volume fraction of phase i molecular transport coefficient unit tensor energy dissipation rate dilatational viscosity heat conductivity dynamic viscosity turbulent viscosity (RANS) density Newtonian shear stress turbulent stress tensor (LES) turbulent stress tensor (RANS) general transport quantity rotational speed of particles

Indices B, E, N, P , S, T , W b, e, n, s, t, w Pi

nodes of the FVM grid cell faces of the FVM grid i th particle parcel

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Computational Fluid Dynamics

Computational Fluid Dynamics (CFD) is a numerically based tool for the prediction of flow field, concentration and temperature distribution. Its main parts are mathematical modeling, discretization, numerical solution of the discretized equations and the interpretation of numerical results. Basic equations in all mathematical models for CFD are balances for momentum and total mass determining velocity, pressure and density field. Depending on the case considered they are supplemented by mass balances for single species and a heat balance. Additional models are required to describe e.g. turbulence, multiphase flows, chemically reactive systems and other special cases. Basis for the discretization of the balance equation is the discretization of a space – the grid generation. Most codes can handle unstructured grids. Nevertheless, certain requirements concerning grid structure have to be fulfilled to get stable convergence and an accurate solution. Traditionally, most CFD codes use finite volume discretization for the balance equations, even if finite element algorithms are of increasing relevance for simulations with adaptively moving grids and for coupling CFD with structural dynamics simulations. A new approach for simulations with high resolution in space and time is the lattice Boltzmann method. Finally, the numerical results have to be graphically presented and interpreted. Because of the huge amount of numerical data provided by each simulation this cannot be done with a general method. It must always be analysed in reference to a certain research question. Errors caused by the model formulation and by the numerical scheme have to be analyzed in order to judge the accuracy of a simulation. Quantitative estimates are required for an adequate interpretation of the results.

1. Introduction The increasing pressure on development time and product quality in industrial process development stimulates the application of numerical methods in addition to traditional experimental ones. In many cases the numerical tools can save time and costs if they are used complementary to experimental methods. The simulation tool for

modeling fluid dynamic processes is called computational fluid dynamics (CFD). A high complexity of the mathematical models is needed for a sufficient description of the relevant interactions in reactive flows occurring on different time scales. Even if not all aspects can be considered in detail to keep the equation system in a manageable frame, the complex mathematical models are far from being able to be solved analytically. Therefore, numeric methods play an important role in CFD and many applicants associate CFD mainly with its numerical aspects. But the subject is defined much wider: formulation of the mathematical model, visualization and interpretation are all essential parts of the tool.

2. Procedure This chapter sketches the general steps of a CFD simulation. It gives an overview of the procedure and the relation of the individual parts. The points mentioned here are described in more detail in the following chapters. As with all investigation methods, the formulation of the exact question is a crucial point for the entire CFD procedure. The question includes the target value and its accuracy, the dependencies of importance and possible approximations and assumptions which can be made. Also the range in space and time to be considered is a part of it. All significant physical parameters or the range of their variation have to be specified together with the physical state at the boundaries and at the initial time. The formulation of the question determines the effort and the time required to solve the problem and should be therefore as strict as necessary and as weak as possible. The mathematical model can be formulated based on the question posed. Its central elements are balance equations for mass, momentum and energy. These are supplemented by additional model equations if necessary. All partial differential equations require initial and boundary conditions for their solution. Parameters and constants in the equations have to be specified in a way that the model is mathematically closed. For this purpose, additional laws, like an equation of state or temperature dependencies of physical properties (e.g. density, viscosity)

Computational Fluid Dynamics might be required. Not always can the most exact mathematical formulation be solved with reasonable effort. In such cases, additional model assumptions such as averaging procedures, reduction of the number of variables or restriction of the range in space and time considered have to be introduced. The resulting system of differential, integral and algebraic equations is too complex to be solved analytically. Therefore, numerical algorithms are applied to derive a system of linear algebraic equations which provides an approximative solution of the original mathematical model. In most cased this discretization is performed using the finite volume method or the finite element method. Specialized methods exist for certain applications. A relatively new alternative to classic discretization algorithms are statistical methods like the Lattice Boltzmann method. As a result of the great complexity of the mathematical model, the numerical solution of the governing equations must be post-processed to extract the information necessary to answer the original question. Most often this information can best be presented in a graphic plot, but the data to be plotted are usually not the direct solution of the model. Relevant data have to be selected, linked, transformed, and in some cases statistically treated and normalized to create a data set which allows the user to answer the question. Last but not least, the accuracy of the numerical result has to be judged, even if the simulation is not erroneous in an obvious sense. Starting from the setup of the mathematical model over all numerical steps up to the postprocessing, inaccuracies are introduced or at least accepted. Only a rough quantitative error estimation can prove if the solution provided is accurate enough for the question considered.

3. Modeling The central CFD modeling approach is to balance momentum, mass and heat. This approach is based on the continuity hypothesis which states that all balance quantities and physical parameters change continuously within each phase. Consequently, limiting considerations for infinitesimally smallvolumes never consider

465

changes at the molecular level. Discontinuous jumps are found only at interfaces. Therefore, in some points, modeling of multiphase systems requires different approaches than modeling of single-phase systems. For an easier understanding of the following discussion a few terms shall be defined: – Balance quantities Quantities governed by laws of conservation can be balanced. In general, mass, momentum and energy are such balance quantities. For detailed considerations masses of different chemical species or different kinds of energy can be balanced separately. In such balances source terms for possible conversions appear. Balance quantities are extensive values. – Transport quantities While the basic step of balancing is an integral consideration of the balance quantities, differential transport equations can be derived from the original balances. Transport equations describe the distribution and the transport of intensive values, either mass based (mass fraction, specific energy) or volume based (density, concentration), which are the so-called transport quantities. – Independent variables Independent variables in transport equations are normally the three spacial directions and time. In special cases symmetry can be used to reduce the number of spacial directions, in the case of stationary considerations, time is not considered. For certain model approaches additional independent variables have to be included. In probability density models, e.g. for micromixing, the species composition is an additional independent variable, in population balances particle properties such as the characteristic diameter take this functionality. – Parameters All known values in the transport equations – besides transport quantities and independent variables – are called parameters. ’Known’ has to be understood in a mathematical sense: the solution of the equations is only possible if values for the parameters are given. In the physical sense the determination of these values might be problematic or only possible with help of an additional equation.

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3.1. Transport Equations Transport equations are derived based on balances. For the general transport quantity φ the transport equation reads: ∂φ = −∇(vφ) − ∇Jφ + Sφ ∂t (a) =

(b)

(1)

+ (c) + (d)

Term (a), the time derivative, codes transient changes. Term (b) models the convective transport with v being the convective velocity. Term (c) in this general form is the molecular transport term including the molecular flux vector Jφ . In certain cases, in particular in turbulence modeling, structurally similar terms can occur which are numerically treated in the same way but have different physical meanings. Finally, term (d) is the source term. The kinds of sources which occur depend strongly on the transport quantity φ. In the momentum balance these are forces, in the energy balance these are exchange rates to other types of energy and in mass balances for single species these are conversion rates due to chemical reactions. Detailed derivations and discussions of the transport equations can be found in [1], see also → Transport Phenomena. Continuity equation. A basic physical law is the conservation of mass. From this the continuity equation can be derived: ∂ρ = − ∇ (ρv) ∂t

(2)

As there is no molecular mass transfer relative to the convective velocity (with its mass based definition) and no source of mass in a nonrelativistic system, this equation contains only the terms (a) and (b) of Eq. (1) with density ρ being the transport value. Equation of motion. CFD considers transport phenomena in fluids, so the convective velocity is a central quantity. This becomes obvious also from Eq. (1) where velocity is the only general, φ-independent value contained. The velocity is determined by the equation of motion: ∂ρv = −∇ (ρvv) − ∇ τ − ∇p + ρg ∂t

(3)

The velocity itself is not the transport quantity here but rather the volume based momentum ρv. The stress tensor τ plays the part of the molecular flux. It is a function of the velocity gradient, but this functionality can not be expressed in a general form because it depends on the rheologic properties of the fluid. In many applications a Newtonian fluid can be assumed and τ can be expressed in the following way:
   2 τ = −µ ∇v + (∇v)T − κ − µ (∇ · v)δ 3

(4)

µ is the viscosity which is independent of shear stress and time in Newtonian fluids. κ is the dilatational viscosity, which in most real cases has no influence on the flow. The last two terms of Eq. (3) are the source terms to be considered in most cases. The pressure term is of highest importance. It has a dual role in the structure of the equation. As mentioned, it can be seen as a source term, but it also could be counted as a special type of stress and included together with τ in a general stress term. The first concept is applied here because it provides a more suitable structure for the discussion of numerical methods. Gravity acts in all systems and is therefore also written here. In special cases other forces acting on a fluid like electromagnetic forces or centrifugal forces also have to be considered. Two things complicate the numerical handling of the equation of motion. The first is the nonlinear structure of the convective terms. While in other transport equations the velocity can be handled like a parameter, in the equation of motion it is the unknown. The second problem occurs from the pressure term. There is no transport equation available for pressure. In compressible fluids an equation of state is used additionally which allows one to calculate the pressure if the density field is known from the equation of continuity. The solution of these three coupled equations (one of which has three components) is numerically not handsome, but possible without significant problems. In incompressible systems the density is constant and an equation of state cannot be used to determine the pressure. The remaining equation for the pressure is the equation of continuity, but it does not contain the pressure. The relation can be described in the way that only the velocities which

Computational Fluid Dynamics satisfy the equation of motion with the right pressure gradient also satisfy the equation of continuity. Navier-Stokes equations. A special and commonly used version of the equation of continuity and the equation of motion are obtained for Newtonian fluids with constant density and constant viscosity. The equations simplify to: Equation of continuity: ∇v = 0

(5)

∂v = −ρv∇v + µ∇ 2 v − ∇p + ρg ∂t

(6)

This equations are called Navier-Stokes equations. They are of restricted validity because changes in temperature as well as changes in the chemical composition effect density and velocity. Nevertheless, they are a reasonable approach in many cases. Concentration equation. The equation of continuity results from the overall mass balance. If a multi-component system is considered and the concentration of the different species is of interest for the simulation, separate transport equations have to be solved for them. As the global mass balance is solved in all cases, the number of additional equations needed for a complete description is one smaller than the total number of species in the system. The transport equation for a concentration c reads: ∂c = −v∇c + ∇(D∇c) + Sc ∂t

equation for the temperature field. For the application of turbulence modeling, a certain part of the kinetic energy is of interest. This second case is discussed in Chapter 3.3, only the internal energy is considered here. Internal energy covers the kinetic energy of atomic and molecular motion, potential energy of intermolecular interactions, energy of chemical bonds and nuclear energy. Not all aspects of this energy can be determined absolutely. Therefore, only differences in the internal energy between two states are discussed. Balancing of the internal energy gives: ∂ρu = −∇(ρuv) − ∇ q˙ − (ττ : ∇v) − p(∇ · v) + Su ∂t

Equation of motion: ρ

467

(7)

D is the molecular diffusion coefficient and Sc is the source term resulting from a chemical reaction. It is expressed via the chemical reaction kinetics which in many cases include nonlinear expressions and dependencies between concentrations of different chemical species. Energy equation. Formulating of energy equations is a large area because there are many different forms of energy which can be converted into each other. For CFD the equation of the internal energy is usually considered to obtain an

(a) =

(b)

+ (c) +

(d)

+

(e)

+ (f ) (8)

with (a) temporal change (b) convective transport (c) heat conduction (d) increase by viscous dissipation (e) increase by compression (f ) conversion from other types of energy If an incompressible fluid is considered, in addition viscous dissipation, radiation and other kinds of energy conversion can be neglected and the mixing enthalpy is also negligible the only potential source is the enthalpy of a chemical reaction Shr . Density, heat conductivity and heat capacity values are calculated for the mixture in the case of a multi-component system. With these assumptions the temperature equation can be derived to the form of a commonly used equation: ∂ρcp T = −v∇(ρcp T ) + ∇(λ∇ρcp T ) + Shr ∂t

(9)

cp is the heat capacity and λ the heat conductivity. If the assumptions cannot be upheld as described above, extended forms of the temperature equations must be derived.

3.2. Initial and Boundary Conditions Partial differential equations cannot be solved without initial and boundary conditions. In a

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mathematical sense this can be interpreted as such that the solving of a differential equation requires its integration and to perform this the integration constants must be determined. Consequently, the number of additional conditions needed for each independent variable is equal to the number of derivatives. Therefore, in time only one condition is needed, which in most cases is given at the initial time while for the space coordinates two conditions are required which are realized normally by one condition at each boundary point. For the initial conditions values of the transport quantities φ must always be given. These values may vary in space but cannot be a function of φ. Three different types of boundary conditions are possible: Dirichlet condition φ|BC = CBC1

Neumann condition  ∂φ  = CBC2 ∂n BC

fixed value of φ given at the boundary

derivative of φ normal to the boundary is given

Cauchy condition  ∂φ  +CBC3 φ|BC = CBC4 ∂n BC

combination of value and derivative of φ is given

In a finite region of the boundary a Dirichlet condition is required. The use of these boundary conditions can be segregated in relation to the physical nature of boundaries. The ones most often used are presented here, but for special flow situations other types can also be defined. It is not always easy to find the expression required and in some cases a setup is more easily obtained by moving the boundary location than by constructing a complex formulation at the original location. An example is shown in Figure 1. – Inlet It is assumed that the conditions at the inlet are known. Therefore, fixed values are given at the inlet for all transport quantities. These fixed values may vary along the inlet, e.g. a parabolic velocity profile can be given for a laminar flow in a tube.

area relevant for simulation

unsuitable location for boundary conditions suitable location for boundary conditions

Figure 1. Location of boundary conditions

– Wall Most model systems are bounded at least partly by solid walls. They can be fixed or moving (e.g. wall and stirrer of a stirred vessel). Walls considered here are impermeable for mass fluxes but may possibly conduct heat. To prevent convective fluxes through the wall the normal velocity component is set to zero or equal to the normal velocity of the moving wall. Assuming no slip conditions, the tangential velocity is equal to the wall velocity, which is zero with fixed walls. If slip is considered, the gradient of the tangential velocity components normal to the wall is zero. To avoid diffusive fluxes through the wall the concentration gradient normal to the wall is set to zero. For heat transfer different types of walls have to be considered. The definition of temperature boundary conditions at walls is an approximation in most cases, because neither the physical conditions are well known there nor are they easy to control. Adiabatic walls are modeled using a zero temperature gradient normal to the wall. At walls of fixed temperature this temperature is given as boundary condition. Heat fluxes across a wall can be described using Neumann or Cauchy conditions. – Outlet No assumptions about the quantities of the transport values can be made for outlet conditions. However, it is assumed that the flow is uniform and no further sources occur. In that case velocities tangential to the outlet are zero. For all other unknowns (normal velocity, con-

Computational Fluid Dynamics centrations, temperature) zero gradients normal to the outlet are used. Furthermore, for incompressible flows the mass flow at each outlet has to be specified to preserve the total mass. It is not always easy to find a location at which outlet boundary conditions can be specified with sufficient accuracy. Special problems occur in turbulent flows if vertex structures are not averaged but resolved and the assumptions for the velocity field do not hold. In such a case, boundary conditions for the pressure are stated instead of velocity conditions. The strong coupling between pressure and velocity described in context of the equation of motion allows one to solve these equations with this type of boundary conditions. – Symmetry Some systems show symmetries which allow one to reduce the model space and consequently the numerical effort. Symmetry can be utilized if it holds not only for the geometry, but also for the flow and for all transport variables. Symmetry is given when all gradients normal to the symmetry plane are zero and the velocity component normal to the symmetry plane is zero, too.

3.3. Turbulent Flow While laminar flow is characterized by parallel stream lines in a turbulent flow, vertex structures of a large size range are found causing highly frequent fluctuations in all transport variables. All the equations introduced up to now are valid in laminar as well as in turbulent flows, as long as the conditions mentioned hold. The greatest problem in the simulations of turbulent flows is that the resolution of the high frequent fluctuations requires a fine numerical grid and small time steps. The number of grid cells N needed for a three-dimensional simulation can be estimated by N = 53 Re9/4

(10)

with Re being the Reynolds number of the flow. Such simulations of turbulent flows, called direct numerical simulations (DNS) are possible only for small academic cases and relatively low

469

Reynolds numbers with the present computing power. In most cases the exact structure of the turbulent flow is of minor interest and information on local averaged values is sufficient. Unfortunately, such an averaging can not be done in a simple way because the coarse scale structure of the flow interacts strongly with the small scale structure. Special attention has to be given to chemically reactive flows. Chemical reactions can take place only if the species are mixed on the molecular scale. A model averaging this scale brings the danger of overestimating the rate of fast chemical reactions. Two general modeling approaches for turbulent flows are used nowadays: Reynolds averaging (Reynolds averaged Navier-Stokes – RANS) and large eddy simulations (LES). RANS modeling is based on averaging all transport quantities over all scales of turbulent fluctuations. This allows one to use relatively coarse grids and can be applied on geometrically large systems with reasonable effort. The influence of turbulent structures is included in additional models. Large eddy simulations work with a grid size which allows the direct resolution of the large scale part of the turbulence structures. Since the large eddies contain the major amount of turbulent energy, the return in accuracy is superproportional to the increase in effort with this approach. The filter width is a function of the grid spacing and can therefore be influenced by the user. Nevertheless, sub-grid scale models (SGS models) are required also for LES. The numerical effort for large eddy simulations is significantly larger than for RANS simulations and can be handled at the moment only on high performance parallel computers. Figure 2 gives an impression of the resolution of turbulent eddy structures by the different methods. LES and RANS are illustrated in more detail for the equation of motion. Concentration and energy equations can be handled in a similar way, but all turbulence models are developed in the first step for momentum transport. Thus the mass and energy transport models use proportionality approaches in most cases based on the parameters computed for momentum transport. Reynolds averaging. Reynolds averaging is based on the assumption that scales of the main

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Computational Fluid Dynamics

a) Direct numerical simulation

b) Large eddy simulation

c) Reynolds averaged simulation

Figure 2. Resolution of turbulent eddy structures by different modeling approaches

flow and of turbulent fluctuations differ significantly. Therefore, each transport quantity can be split up into a time averaged value φ and a fluctuating value φ  in such a way that possible macroscale fluctuations of φ are included in φ while turbulent fluctuations are covered by φ  : (11)

In accordance with the goal of RANS to obtain averaged values, the transport equations are averaged. This yields for the Navier-Stokes equations:

ρ

0

∂v = −ρv∇v +∇ττ t + µ∇ 2 v − ∇p + ρg ∂t

(12)

where an additional term including the Reynolds stress tensor τ t occurs. 

(13)

With this term the averaged equation of motion is not closed because the velocity fluctuations are unknowns. There is a large number of approaches to model the Reynolds stress tensor. To explain all of them is beyond the scope of this article. A detailed presentation is given by Pope [2]. Therefore, only the most commonly used concept of the eddy viscosity approach together with the k − ε model will be sketched. Boussinesq [3] interprets the effect of the Reynolds stress tensor in analogy to the Newton stress tensor. Therefore, it can be expressed proportionally to the gradient of the averaged velocity with a proportionality factor µt called eddy viscosity:

(14)

1   v ·v 2

(15)

In contrast to the molecular viscosity, µt is the eddy viscosity, which is not a material constant but depends on the flow structure. The k − ε model expresses this dependence as a function of the turbulent kinetic energy k and its dissipation rate ε: µt = Cµ ρ

k2 ε

(16)

with ε=



ρvx vx ρvx vy ρvx vz   −ττ t = ρv × v = ρvy vx ρvy vy ρvy vz  ρvz vx ρvz vy ρvz vz

2 ρkδ 3

δ is the unit tensor and k is the turbulent kinetic energy defined by k=

φ = φ + φ

∇v =

τ t = µt [∇v + (∇v)T ] −

µ |∇v + ∇vT |2 2ρ

(17)

k and ε computed by their own transport equations. The original model by Launder and Spalding [4] uses the following form:   µt ∂ρk = −∇(vρk) + ∇ ∇k + ∂t σk

Pˆ

−

ρε (18)



∂ρε µt ∇ε = −∇(vρε) + ∇ ∂t σε



ε2 ε + Cε1 Pˆ − Cε2 ρ k k

with the production term Pˆ defined as µt |∇v + ∇vT |2 Pˆ = − 2

(19)

The constants of the model are listed in Table 1.

Computational Fluid Dynamics Table 1. Constants of the k − ε model

Cµ 0.09

Cε1 1.44

Cε2 1.92

σk 1.0

σε 1.3

The k − ε model is valid for fully developed isotropic turbulence. If this condition is not given, either extended RANS models or large eddy simulations are required for an acceptable flow prediction. Large eddy simulations. A high accuracy alternative to Reynolds averaging is given by large eddy simulations. In this approach the transport values are filtered with a filter function G(x, x ): (x) = φ

G(x, x )φ(x )dx

(20)

G describes the influence of the value of φ in at position x. position x  on the filtered value φ Several forms of G are possible, but it is always a function of the grid size. Filtering the equation of motion gives: ρ

∂ v v − ∇p + ρg = −ρ v∇ v − ∇ττ s + µ∇ 2 ∂t

(21)

In this equation the sub-grid scale Reynolds stress tensor τ s occurs. Its meaning is analogous to the Reynolds stress tensor in RANS but it considers only the effect of unresolved sub-grid scale structures. Eq. (21) can be solved only if τ s is expressed as a function of filtered values. For this purpose so-called sub-grid scale models (SGS models) are used. SGS models of the Smagorinsky type are widespread [5]. They are similar to the eddy viscosity models in RANS, apart from the fact that they consider only sub-grid scale effects. In their structure they do not use information about the eddy structure of the resolved scales. Similarity models like the one from Bardina et al. [6] are based on the assumption that there is a similarity in the eddy structures of different scales. Therefore, the analysis of the resolved structure can be used to model the sub-grid scale. This models fail to describe the dissipation at the smallest eddy scales because this effect is not found at larger scales. So they have to be combined with other models to cover this aspect. Dynamic models like the ones from Germano et al. [7]

471

and Meneveau et al. [8] are based on this idea and extend upon it. For this approach two simulations on different grids are carried out. The sub-gid scale model is adapted to the differences in the predictions of both models. In this way parameters of the sub-grid scale model can be varied in time and space and dissipation effects are described. Such models are interesting for cases in which constant parameters are only a rough approximation. The major drawback of this method is its numerical effort which is significantly higher than for the other methods. Micromixing models. If systems with fast chemical reactions are to be simulated, a description of the mixing state on the microscopic level is required. Neither Reynolds averaging nor large eddy simulation provide such an description because they average or filter small scale fluctuations. A remedy is the use of micromixing models which provide information about the micromixing quality as a function of turbulence quantities, like eddy viscosity or turbulent kinetic energy dissipation. A widespread micromixing model is the probability density function approach (PDF) [9], [10]. It considers the possible mixing states of a fluid of certain composition and gives for each computational cell a probability density function for each state in a way that the integral composition in this cell is satisfied. For a mixture of only two species A and B this is shown in Figure 3. The local probability density functions are computed from transport equations for the PDF. Each mixing state is related to a certain chemical reaction rate. The integral of the reaction rates over the whole composition space gives the total reaction rate in the appropriate numerical cell. Alternative micromixing models are – Eddy break up model [11] This is a simple empirical model which assumes binary chemical reaction with only one product. It is mainly applied for the modeling of combustion. – Flamelet model [12], [13] This model assumes the chemical reaction in a two-dimensional laminar layer. It was developed for premixed and non premixed flames with a different definition of the reaction layer for both cases.

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Computational Fluid Dynamics p

0

A a)

Distribution of species A (black) and B (white) in a turbulent flow with incomplete micromixing

1 cA cA +cB Probability density function of mixing between A and B xA =

B b)

Figure 3. Modeling micromixing in a computational cell with probability density function approach

– Engulfment model [14] This model assumes that a fluid is mixed into another one of different composition by the formation of a rotational cylinder. Within the cylinder diffusion is the dominant mixing process and the composition changes proportionally to the change of cylinder size.

3.4. Multiphase Approaches Multiphase systems are characterized by interfaces at which the fluid properties change discontinuously. This requires extensions of the model approaches introduced up to now for their application to multiphase systems. At the beginning of this chapter multiphase systems will be classified to describe the appropriate model for each type. One type of classification refers to the geometric structure of the system (see Figure 4). If one phase consists of solid or fluid particles with a small size compared to the total system the system is called a disperse system. The particles form the disperse phase while the other phase is called the continuous one. For modeling disperse systems, an extended form of the continuum hypotheses is applied. Properties of both phases are considered to be continuously distributed over the whole system; specific properties of single particles are not relevant. The central property in that case is the volume fraction α which for phase i is defined as the ratio of the volume of this phase, Vi and the total volume V :

αi =

Vi V

(22)

The sum of all volume fractions must be one: 

αi = 1

(23)

i

The assumption that the continuum hypotheses holds for disperse systems implies that changes of properties in a particle are not resolved. This means for example in a temperature distribution that each particle is characterized by a mean temperature, but the different mean temperatures of all particles determine a particle temperature field of the system. If this approach is insufficient for the description of a system, e.g. because the temperature gradient at the interface is needed to determine the heat transfer between the phases exactly, it cannot be modeled with the approach for disperse systems. An important property of disperse systems is the size distribution. Many phase interactions like buoyancy force or heat transfer rate depend on particle size and therefore the particle size distribution is needed to determine these interactions exactly. The easiest case from the point of modeling are monodisperse systems, which means that all particles have the same size. For some applications the differences in the interactions do not need to be resolved, and this approach can be used. If the size distribution has to be considered the particle size becomes an additional independent variable in the system, which increases the complexity of the mathematical

Computational Fluid Dynamics coarse disperse

fine disperse

with size distribution

monodisperse

discontinuous

473

Figure 4. Geometric structure of multiphase systems

system significantly. Size distributions are relevant for all cases where the size range of the particles is very wide or where the size distribution changes in the course of the process e.g. by nucleation, evaporation, agglomeration or similar processes. The computation of particle size distributions is based on the solution of population balances. This model approach is not discussed here further but details can be found e.g. in [15–17]. Another characteristic of a disperse system is the relative velocity between the phases. If the differences in gravitational forces are small compared to the momentum transfer between the phases, both phases have nearly the same velocity. This is the case for phases with similar densities or for systems with very small particles. Another description refers to the interactions during momentum transfer. If particles are moved with the continuous phase but do not significantly influence the flow field of this phase it is called a one-way interaction. Systems with one-way interactions do not have to be modeled with multiphase approaches. The flow field is modeled with the single phase equations and the volume fraction can be handled like a concentration. The term two-way interaction is used, if there is a relative velocity between both phases which causes a reciprocal momentum transfer. If in addition the volume fraction of the disperse phase is so high that interactions between the

particles and also between particles and walls have to be considered, one speaks of four-way interactions. If in a multiphase system a disperse phase cannot be defined clearly because the extension of all phases is too large to neglect gradients of the transport values in it, the system is called a discontinuous multiphase system. The way to model such a system is to use the approaches introduced in the previous chapter for single-phase systems and additionally to describe the condition at the interface. This works well if the extent of each single-phase region covers a significant part of the total domain. The conditions at the interface act as boundary conditions for the balances in both phases. Therefore, two conditions have to be defined for each transport variable in each point. One is the equilibrium of fluxes, the other is the relation between the values of the transport quantity. As neither momentum nor mass or energy can be accumulated at the interface, the fluxes must be equal on both sides. The ratio of the values depends on the quantity considered. As a result of the non-slip condition the tangential velocities are equal. Unless there is a significant mass transfer over the interface (e.g. melting or evaporation) this holds also for the normal velocity, otherwise source terms must be considered in the stress balance at the interface. Thermal equilibrium at the interface states that the interfacial

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Computational Fluid Dynamics

temperature for both phases is the same. The distribution of chemical species is not equal in equilibrium but described by a distribution coefficient. Detailed explanations and equations for such systems can be found e.g. in Deen [18] and Slattery [19]. Euler/Euler. The Euler/Euler method is used for modeling disperse systems. It utilizes continuous transport equations for each phase. These equations are based on single phase equations but weighted with the volume fraction and appended with source terms describing the phase interaction [20]. For the equation of motion the interactions between phases are forces. In all systems with a relative velocity between the phases the drag force plays a major role. It is the main force which prevents the continuous increase in the relative velocity between the phases. Further forces which have to be considered depending on the properties of the phases are the virtual mass force, the Basset force or other forces caused by different relative velocities at different points of the particle surface, such as the Saffman force and the Magnus force. Mass and heat transfer between the phases determine the interaction terms in the concentration and temperature equations. For systems with a significant mass transfer over the interface it must be considered that heat and momentum are also transferred together with that mass. The Euler/Euler approach is preferably used for systems with high volume fractions of the disperse phase or for relatively large particle sizes because there are no formal restrictions for these properties. Models for the changes in turbulence caused by phase interaction are not well developed yet. Size distributions can be described only with a high numerical effort because most forces change as a function of the particle size and therefore the size distribution must be divided into size classes whereby each class is handled as an individual phase. Euler/Lagrange. Alternatively, the Euler/ Lagrange approach can be used for disperse systems. In this approach the continuous transport equations are also used for the continuous phase and extended by interaction terms. The disperse phase is modeled differently: particle parcels are defined and distributed in a way that they rep-

resent the particle phase with its volume fraction and properties. For each of this parcels P i the path and the velocities (convective and rotational) are determined by ordinary differential equations: dxP i dt mP i IP i

= vP i

 dvP i = Fi dt

(24)

ωP i dω = Ti dt

xP i is the position of the parcel, vP i and ω P i its convective and rotational velocity, Fi the forces acting between the phases, IP i the moment of inertia and Ti the torque. The velocity of the continuous phase and the particle phases are corrected in an iterative procedure considering interactions based on current particle paths. The particle parcels only give a statistical representation of the particle phase if their characteristics are averaged over a reasonable time interval. The time interval for averaging has no limits in the stationary case aside from giving statistically reliable results. For transient investigations the time interval of averaging determines the time resolution of the simulation. If statistical reliability is insufficient, the time steps of the particle simulation must be reduced. The forces for phase interaction in the second equation of (24) are the same as those mentioned for the Euler/Euler approach. Mass and energy transfer also have to be considered along the particle paths and statistically evaluated. Consideration of the particle paths reduces the volume fraction and particle sizes which can be handled by this method. The volume of a parcel can not be larger than the smallest grid cell. For high volume fractions there is a danger of having more than one parcel in a cell at a time which further restricts the volume ratio between a cell and a parcel. Reasonable results are therefore only obtained for very low volume fractions, usually in the range of 1%, exceptionally up to 5%. The numerical effort of the method increases with the volume fraction, because additional parcels have to be introduced. On the other hand, this method provides better models of the influence of particle on turbulence than the Euler/Euler approach. Defining parcels with different properties allows one to

Computational Fluid Dynamics model size distributions with only a small additional effort. A detailed description of the Euler/Lagrange method is given by Crowe et al. [21] and by Sommerfeld and Krebs [22]. Grid adaptation to interface. The best way to model a discontinuous multiphase system is to define the reference system in a way that the interface has a fixed position in space. In that case the grid can be constructed so that the interface is located at grid lines. The interfacial conditions can then be applied easily. In most cases interfaces move and the reference system cannot be defined as described above. In this case the location of the interface is a part of the solution of the model. To incorporate the interfacial conditions as with boundary conditions the grid can be adapted to the new location of the interface. As long as the dislocation is small, this affects only the cell layer close to the interface and the numerical error introduced is small. For large dislocations the cells near the interface would be deformed severely and it is necessary to add or delete cells. Detailed descriptions of adaptive methods are given by Ferziger and Peri˘c [23]. Volume of fluid. If there is a strong movement of the interface, the grid adaptation method is numerically laborious and erroneous. An alternative is to locate the interface at an arbitrary position of the grid. Different methods act in this way (segment method [24], marker and cell method [25]). The most popular method is the volume of fluid method first presented by Hirt and Nichols [26]. A volume fraction is defined and a transport equation is solved for it similarly to the Euler/Euler method. The volume fraction in most cells is either zero or one and the interface is located in the cells where the volume fraction lies between these values. The exact location of the interface is determined by the volume fraction in the cell. The direction of the interface is normal to the gradient of the volume fraction.

4. Numerics 4.1. Basics In the previous section mathematical models for the description of transport processes in fluids

475

have been introduced. It is obvious that the coupled system of integrodifferential equations and algebraic equations can be solved analytically only for a very small number of limiting cases. For all practically relevant situations only a numerical, computer based solution is possible. The basic idea of a numerical solution is to replace the partial differential equation which is continuously defined over the range in space and time with a system of algebraic equations which gives the solution only at defined discrete points or for discrete intervals. For the finite difference method (FDM) values at single points are computed, for the finite volume method (FVM) characteristic values are determined, which are constant for a numerical cell, and for the finite element method (FEM) parameters of a polynomial function over a cell yield the solution. All these are available for separate times. A completely different approach used with the lattice Boltzmann method will be described separately. As the solution of the algebraic equations, called the discretized equations, is fixed to certain points or elements is space and time, the discretization of space and time is the first step to be performed. Time is a one-dimensional coordinate, and so it can simply be split into intervals, i.e. time steps. The time step size can vary over the total time period considered. Stability and accuracy of the procedure on the one side and computing time on the other side depend on the time step size chosen in relation to the space discretization (see Chapter 4.2). The discretization of the three-dimensional space offers much more possibilities. The two discretization methods mainly used in CFD, FVM and FEM, require the subdivision of the total model space into non overlapping cells. In the ideal case the grids formed out of this cells meet the following requirements: – Geometrical requirements The outer border of the grid should fit as closely as possible to the physical borders of the system to introduce the boundary conditions at the correct location. In the case of simple geometrical structures this is not a problem but it might become one for jagged structures. The more variable cell shapes are accepted by a CFD code the easier such structures can be described without extensive refinement close to the boundaries.

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– Physical requirements To achieve sufficient accuracy in the solution the grid size should increase in areas of steep gradients and especially where they are varying. This criterion has the major problem that it can be fulfilled precisely only if the solution is known. In many cases general physical understanding of the flow investigated is sufficient to estimate where local grid refinement is necessary. In other situations preliminary computations on a rough grid give an orientation for grid improvements. The simulation of turbulent flows requires a few grid cell layers parallel to the boundary for an appropriate modeling of the laminar sub-layer. – Numerical requirements The numerical requirements depend mainly on the numerical method used and the way it is implemented in the code. The crux of the matter is whether a structured grid is needed or an unstructured one is allowed. Especially for unstructured grids a wide variety of cell shapes can be imagined but not every one is supported by every code. Nearly all unstructured solvers can handle tetrahedral and hexahedral cells separately or in combination in one grid. In some cases more general formulations of polyhedral cells are available but are not standard. From a numerical point of view serious jumps in the size of adjacent grid cells should be avoided. Furthermore, cells should not tend to degenerate, which means that the ratio of volume to surface gets too small. In many cases very large numbers of cells would be needed to meet all requirements sufficiently. This might push the computational effort to an unrealistic level. Therefore, a reasonable compromise has to be found. The major distinction of grid types is made between structured and unstructured ones. A structured grid is defined by three bands of grid faces, where the faces of one band do not intersect with each other but with the faces from the other bands. There is no restriction on the shape of the faces or the coordinate system in which they are defined. The cell shape in structured grids are hexahedral, sometimes with one collapsed cell face. Structured grids have only restricted applicability to complex geometries

but for simple geometries they can be generated easily. In unstructured grids no global grid faces or lines are defined, but the domain may be divided in any way. This offers many more possibilities for grid fitting, but it normally cannot be done without an automated grid generator. For some simulations it is necessary to perform grid adaptation in the course of the simulation. Reasons might be, for example, moving boundaries (as in stirred vessels), moving of interfaces which have to be tracked or strong transient changes in the solution which requires local changes in refinement. If the new structure of the grid depends on the numerical solution, the numerical effort for the adaptation strategies is relatively high. Nevertheless, in many cases with grid adaptation a significant improvement in accuracy and efficiency of a simulation can be gained. The model equations can be discretized based on the discretization of space and time. Different methods are available varying in their mathematical approach and in the range of application where they are used. An insight shall be given here for two methods: the finite volume method and the lattice Boltzmann method with stress on the first of the two. In accordance with the overview character of this contribution, the methods cannot be discussed in detail, but references for further reading will be given. A third widespread method is the finite element method. It is extensively explained in → Mathematics in Chemical Engineering and → Fluid Mechanics and therefore not discussed here.

4.2. Finite Volume Method Historically, the most common numerical method in CFD is the finite volume method (FVM). It is also used by the presently widest distributed commercial CFD codes FLUENT, CFX and Star-CD. Its success in CFD is related to physical equivalents in the approach. The fundamental concept is the consideration of grid cells (also called control volumes) in which all physical parameters and system characteristics are assumed to be constant. The balance equations are integrated over each of this

Computational Fluid Dynamics cells. For volume sources these integrals can be solved directly considering these sources as constant over the cell. The volume integrals of the transport terms are transformed into surface integrals using the theorem of Gauss. As the physical properties are constant within each cell, they can also considered to be constant along each cell face, which permits the surface integral to be resolved. The value or the derivative of the solution at the cell face is contained in the expressions gained. While the values are constant within each cell, they jump unsteadily at the cell faces and the values and derivatives at the faces have to be expressed by the values in the cells which are fixed at certain reference points, i.e. the nodes. This approach is, in general terms, the reverse of the derivation of the transport equations, where a finite model volume is considered initially and fluxes over the boundaries and volume sources are balanced. In the second step the limiting expression for the model volume approaching zero is determined. This analogy reveals another advantage of the method in a physical sense: as the surface integral of the transport terms expresses the fluxes over the cell surface, a numerical error in the determination of these fluxes does not violate the integral conservation of the balanced property. The laws of conservation of mass, momentum and energy are the fundamentals of modeling for CFD, and therefore a method upholding this laws inherently has been most widely accepted by engineers. Nowadays, this advantage of FVM is no longer significant, firstly, because the accuracy of all methods has been increased so that for well designed grids the error in the computation of fluxes is small compared to other errors in the course of simulations and secondly, because advanced grid techniques, like certain types of unstructured grids or sliding grids do not uphold this advantage exactly. Nevertheless, it is still the most widely distributed CFD method and shall therefore be described here as an example. More details can be found e.g. in [15],[27–29]. Discretization in space. We first consider the general shape of a transport equation for the quantity φ. For purposes of simplicity only the two-dimensional version is used, but the extension to three dimensions is straightforward. The discretization in time will be mentioned later, so

477

we consider here the stationary form. The velocity field is assumed to be known as well as the molecular transport coefficient  which might vary in a known way over the analyzed space.     ∂φ ∂φ ∂ ∂φ ∂ ∂φ − vy +  +  + Sφ ∂x ∂y ∂x ∂x ∂y ∂y (25)

0 = −vx

For reasons of simplicity the integration is shown here for a equidistantly structured grid. But there is no general restriction – finite volume discretization can also be carried out for nonequidistant and unstructured grids. Note that in the two-dimensional case the ’volume’ for integration is a face and the ’volume surface’ is a line. For the grid the compass notation as shown in Figure 5 is normally used [27]. The nodes are marked with capital letters, P being the one of the cell considered. The points at cell faces are marked by lower case letters. The final linear equation system can contain only values with capital letter indices, values with lower letter indices are not defined and have to be replaced by the former ones. Integration of Eq. (25) over the control volume gives

0=−

vx

V

+

∂φ dV − ∂x

vy

∂φ dV + ∂y

V

 

∂ ∂φ  dV + Sφ dV ∂y ∂y

V

  ∂ ∂φ  dV ∂x ∂x

V

(26)

V

Conversion of volume integrals of the transport terms and resolution lead to:

0=−

nvx φdA −

A

+

  

 ∂φ ∂φ n  dA + n  dA ∂x ∂y

A

A

+

nvy φdA A

Sφ dV V

0 = − (vx Aφ)e + (vx Aφ)w − (vy Aφ)n + (vy Aφ)s       ∂φ ∂φ ∂φ + A − A + A ∂x e ∂x w ∂y n   ∂φ − A + SV ∂y s

(27)

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Computational Fluid Dynamics NW

N

NE

n WW ww

W

P w

E e

EE ee

s SW

S

ss

SE

SS

Figure 5. Nomenclature in a two-dimensional grid for finite volume method

Ae , Aw , An and As are the cell faces crossing e, w, n and s and n is the face normal. Finally, the values and the derivatives of φ at the cell faces have to be replaced as a function of the values at the nodes. The derivatives are approximated using the central difference which is accurate to the second order, e.g. 



A



∂φ φ − φW = w Aw P ∂x w δxW P

 (28)

where δxW P is the distance between the points W and P . To express the values of φ various approaches are used which differ in their stability and accuracy. Most of them are asymmetric and depend therefore on the flow direction. We discuss them for cell face w and a velocity vx,w from W to P . – Upwind Differencing Scheme – UD φw = φW

(29)

This scheme is unconditionally stable, but it is only accurate for the first order. It is the only scheme which does not produce any overshots or undershots without additional limiter and therefore guarantees the solution to be in the physically meaningful range (no negative concentrations etc.). On the other hand, the numerical error is relatively large. It has the effect of smoothing strong changes in the gradients – a property called numerical diffusion. The error can be minimized if the local veloc-

ity vector is nearly parallel to the face normal vector. – Central Differencing Scheme – CD φw =

δxW w φW + δxwP φP δxW P

(30)

This scheme is only conditionally stable but it is second order. Central schemes give normally a better approximation of steep slopes but they show significant non-physical oscillations of the solution. In the worst case these oscillations can build up and lead to a diverging solution. CD schemes are stable and advisable for diffusion-dominated processes. – Quadratic Upstream Interpolation for Convective Kinetics – QUICK 1 6 3 φw = − φW W + φW + φP 8 8 8

(31)

QUICK is the basis for most third-order schemes used today. With a combination of central and upwind aspects it joints properties of both schemes discusses above. Oscillations occur, but to a lesser extent than with the CD scheme. QUICK has a much wider range of stability than CD but restrictions remain for systems without significant diffusion terms. To improve this third-order scheme various advancements have been developed. The main advantage is gained from the combination with so-called limiters. Limiters are extensions of the numerical scheme which prevent

Computational Fluid Dynamics t

t

479

t

t+ t

t

x WW W P E a) explicit

EE

x WW W P E b) implicit

EE

x WW W P E EE c) Crank-Nicolson

Figure 6. Discretization in time – coupling of values according to different methods

the formation and pronunciation of local extrema but keep the numerical accuracy of a scheme. For the source term a linear approach in φ is used: SV = S0 + S1 φ

(32)

Discretization in time. For transient cases a time derivative of the transport quantity also occurs in the equation. If the two-dimensional Eq. (25) is extended in this way one obtains: ∂φ = (33) ∂t     ∂φ ∂φ ∂ ∂φ ∂ ∂φ − vy +  +  + Sφ −vx ∂x ∂y ∂x ∂x ∂y ∂y

The philosophy for the time discretization is the same as for the discretization in space: The equation is integrated over the grid cell as well as over the time interval. The integration of the left hand side can be performed easily:

t+t

t

V

∂φ dtdV = V (φPt+t − φPt ) ∂t

(34)

For the time integral of the right hand side a linear relation to the values of φ at the old and the new time step is assumed: t+t

φdt = [f φ t+t + (1 − f )φ t ]t

(35)

t

f is a weighting factor between zero and one. For a space discretization which needs only the values of φ in P and its direct neighbor points the discretized equation reads:

aPt+t φPt+t = t+t t ] + a [f φ t+t + (1 − f )φ t ] aE [f φE + (1 − f )φE W W W t+t t ] + a [f φ t+t + (1 − f )φ t ] + (1 − f )φN +aN [f φN S S S

+aPt φPt

(36)

in which the coefficients aPt+t and aPt depend on f linearly. ai are the coefficients resulting from space discretization and from Eq. (34). For different values of f different methods can be derived. The most common ones are shown in Figure 6. – f = 0 – explicit method This method is numerically simple to use because only independent linear equations are derived. It is of first-order accuracy. On the other hand, it is only conditionally stable. For applications in which the flux is dominated by the convective part (and this is the major part of all CFD applications) the CourantFriedrich-Lewy criterion determines the stability range: 1 ≥ C =

|v|t x

(37)

C is the so-called Courant number. – f = 1 – implicit method This method is unconditionally stable and also first-order accurate. If a final state of a process is more interesting than the transient behavior, this method can be used with relatively large time steps. – f = 0.5 – Crank-Nicolson method This method is second-order accurate. It is conditionally stable but larger Courant numbers than in the explicit method can be handled. It is used if a high accuracy in time is required.

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4.3. Pressure Correction Methods One of the crucial points of fluid dynamic modeling are the nonlinearities in the equation of motion and the lack of an explicit pressure equation for incompressible flow. This has been outlined in connection with the equation of motion. There are different approaches to deal with this problem. The directly coupled solution of the equation of motion and the equation of continuity is possible but is mathematically instable, especially for the finite volume method and is seldom used. Also, the approach of an artificial compressibility is only successful and efficient in a few cases. The most commonly used alternatives are pressure correction schemes. They breakup the velocity and pressure into an estimated value and a corrective part. Starting with an estimated pressure field, they compute an estimated velocity field, or vice versa, using the equation of motion. For the velocities in front of the derivatives in the convective terms (see Eq. (6)) estimates are used. Reorganization of the discretized momentum balance gives an equation for the pressure correction as a function of velocity correction. With the help of this equation the discretized continuity equation can be transformed into a pressure correction equation. Iterative solution of all these equations lead to a stepwise approach of velocity and pressure. It is important to update the ’parameter-velocity’ in the convective term and other dependent parameters in each iterative cycle. If necessary, coupled equations (e.g., turbulence models) have to be solved at the end of each step. One of the most basic pressure correction algorithms is the SIMPLE algorithm from Patankar [27]. Because it does not converge stably, a significant under- relaxation has to be used. Reversed algorithms have been developed since then, but the underlying basic principles are the same.

4.4. Lattice Boltzmann Method A numerical method which has undergone a rapid development during recent years is the lattice Boltzmann method (LBM). It differs greatly from FVM because it is not a discretization of the transport equations but a statistical method which gives an average result for the solution of

the equations. To provide sufficient statistics the method requires a relatively fine grid and small time steps, on the other hand it performs much faster on a fine grid than FVM does on the same grid. The grid used is completely regularly structured but the location of wall boundaries is not restricted to the grid lines. Therefore, the method is suited for geometries with sophisticated wall shapes like porous media. While some years ago the grid requirements caused a impassable numerical effort, such a method can be handled today on parallel computers. It is especially suited for simulations which require a fine grid to handle the models, like large eddy simulations of turbulent flows. For lattice Boltzmann simulations a number of numerical particles with properties like velocity, temperature and concentrations are distributed over the grid in a way that averaging of their properties gives the initial conditions of the flow. The simulation consists of a first step in which the particles move due to their individual velocities. In a second step collisions between particles and also between particles and walls are interpreted according to collision rules. The collision rules describe the change in the properties of each of the particles in the collision. They have to be formulated in a way that they cause statistical changes so that the averaged property field fulfills the transport equations [30], [31]. As the method is still relatively new, not all models can be used yet but the current development is rapid.

5. Interpretation As mentioned in Chapter 2, the volume of computed data of a CFD simulation is too large to interpret the numbers directly. In a graphic visualization the data are much easier to comprehend. A basic step of a suitable visualization is an appropriate reduction of the data. Threedimensional transient data cannot be visualized at once, and the visualization of the velocity vector field can not be realized effectively in a two-dimensional plane. But not all aspects of the data field are relevant to answering the initial question. Relevant data, however, can be extracted, data can be averaged or combined to derive meaningful results. To transfer information into a graphical presentation requires that

Computational Fluid Dynamics the figure be clearly readable and quantitative. Aspects of data reduction (location of a slice, chosen velocity components) have to be marked and a legend must be given which allows quantitative interpretation of colored figures. Visualization is not only useful to find the result of the simulation. It also helps to detect errors. For this purpose the whole data field should at least be scanned roughly. As the aim of this procedure is to find irregularities in the solution, the layout of the presentation is of minor interest, but a quantitative interpretation of the data must be possible. To judge the results all errors of the simulation must be analyzed and as well as possible and quantified. The main categories are: – Modeling errors Modeling errors refer to the differences between the values of all quantities in a real physical system and the values resulting from an exact solution of the mathematical model. Not all physical relations can be reproduced exactly in a mathematical formulation. Furthermore, not all mathematical models can be solved numerically with reasonable effort. Therefore, additional assumptions and restrictions are introduced which increase the modeling error. Examples are the application of turbulence models, the neglect of temperature dependencies of parameters or simplified approaches for boundary conditions. A quantitative estimate of the modeling errors is very difficult. If the mathematical model can be solved analytically (at least for a limiting case) and the corresponding quantities can be measured, a direct comparison is possible. Even in this case the measurement error from the experiment has to be regarded. In most cases such a direct comparison is not possible and one is restricted to experience from similar cases or to comparison of different model approaches for the same situation. – Discretization errors Discretization errors refer to the difference between the solution of the continuous mathematical model and the discretized equations. The discrete solution is computed only for the nodes (which are representative for a certain volume according to FVM) and for time steps. For the locations between these points

481

in space and time only approximations based on the values at the nodes can be used. The solution of the discrete equations at the nodes differs from the solution of the continuous equations at the same points. This is caused by the assumptions and approaches used in the course of discretization. While this is valid for all discretization schemes, it can be most easily explained for the finite difference method (FDM) (see also → Fluid Mechanics Chap. 5 and → Mathematics in Chemical Engineering or [32]). According to this method, the approximation is derived from the Taylor series where only the first terms are considered. The neglected terms give the discretization error. The procedure is based on the assumption that the following terms in the Taylor series are decreasing. Therefore, the discretization error can be approximated by the first neglected term. This term is proportional to the grid size with a certain power. This power is the so-called order of accuracy. The more terms considered the higher the order of the scheme. The order of a scheme is also used for other discretization methods. It states by which power the discretization error is reduced in relation to the reduction of the grid size. This means that a scheme of higher order is not always more accurate than a scheme of lower order if the schemes are derived in a completely different manner. But the influence of grid size reduction is greater for the scheme of higher order. – Truncation of iterative procedures Iterative procedures are used on different levels for the solution of the discrete equations. Normally the linear algebraic equation systems are solved iteratively. Also, the coupling between the equations is often handled iteratively. The most important example of this are the pressure correction schemes (see Chapter 4.3). Iterative procedures approach the solution asymptotically. They are terminated if a certain accuracy is reached. Criterion for termination is the residual. It can be defined in two different ways. The first alternative is the difference in the solution between two iterations. This value is easily determined, but it is dangerous because for slow convergence the differences between two iterations is small, ever far away from the solution. The second

482

Computational Fluid Dynamics

alternative can be used if the problem can be formulated as: F(x) = 0

(38)

The residual of the nth iteration rn is then defined as: F(xn ) = rn

(39)

Both definitions of the residual give values for each node. For a truncation criterion different statistical methods are used to derive a characteristic value. This can be either the maximum, the average or a weighted average of the value itself or of a normalized one. – Representation errors In the computer all numbers are represented with a limited number of digits. This might result in significant errors, e.g., if a small difference of large numbers is computed. Normally, this type of error is of minor importance in CFD simulations. The different types of errors are usually additive. Therefore, it does not help to improve the accuracy of one aspect if the error in this regard is smaller than other errors. In most cases for simulation the limiting factor for error reduction is the modeling error. All other errors are usually small by comparison. A quantitative judgment has a higher priority than indifferent formulations such as ’grid independence’.

6. Industrial Application CFD is a tool of increasing importance in industrial applications. The expertise of industrial users is mainly in the field of fluid dynamics or chemical engineering, not in numerics or programming. Therefore, industrial problems are solved with commercial codes which are available at high level with regard to modeling and numerics. Furthermore, such codes have a user friendly interface and support which allows even inexperienced users to obtain first results quite fast. Commercial codes are continuously being developed and achive more or less the state of the art. Nevertheless, the different codes are specialized in different fields of application and therefore it is necessary to check the available models

before a new kind of applications is tackled. Often additional models can be implemented by user coding, but the effort to do so is great and it should be checked before beginning if another code provides a better foundation for the case considered. To illustrate different modeling requirements, some examples for applications in chemical engineering will be presented. This cannot give a complete overview of the most current usage of CFD in chemical engineering but gives an impression of the variety of applications. – Tubular reactor Tubes without internals have a simple geometric structure. They are a part of nearly all technical apparatuses as inlet and outlet pipes and in some cases they are used as reactors. Problems in modeling occur from the large ratio between length and diameter, which requires a compromise in meshing between number of cells and shape of cells. In some cases the number of cells can be reduced by utilizing symmetry properties of the system. Tubular reactors are often used to mix nonNewtonian fluids or, in a jet configuration, to mix Newtonian fluids very fast down to microscopic levels. For non-Newtonian fluids, e.g. in polymerization processes, chaotic mixers [33] consisting of tubular loops or static mixers [34], [35] within the tubes are used. In both cases the geometry of the flow becomes much more complicated. Especially for static mixers sufficient flexibility is only provided by unstructured grids. In jet flows the modeling of turbulence is of high importance. As tubular reactors are relatively narrow, wall effects have a significant influence on the flow. Therefore, in most cases special wall models have to be used which increases the modeling and numerical effort. Furthermore, turbulence in tubes is often nonisotropic. This reduces the applicability of RANS simulations. – Stirred vessel Stirred vessels are widely used in chemical engineering for single phase and multi-phase flows. The flow field is markedly inhomogeneous and transient. Characteristic of all stirred vessel configurations is the combination of static and moving elements. In the

Computational Fluid Dynamics simplest case the static parts are rotationally symmetric. Then they can be modeled as a moving wall in a rotating reference frame [36]. If there are baffles, the rotational symmetry is broken and extended approaches are used. A good overview about this approaches is given by Brucato et al. [37]. If only the stationary solution averaged over a high number of stirrer evolutions is interesting, the power input from the stirrer can be included in source terms. This system can be solved for the stationary case on a fixed grid. For more detailed simulations the grid is divided in two overlapping or non-overlapping parts. One of them is moving with the stirrer, the other is static. As the stirrer motion has to be resolved sufficiently in time, only small time steps can be used and the simulation becomes relatively extensive. RANS simulations often do not give a sufficient description, especially in presence of chemical reactions, because the flow field is inhomogeneous. As the fine resolution required for large eddy simulations is numerically extensive, a remedy is to apply the lattice Boltzmann method [38]. – Precipitation reactor Precipitation is usually simulated to predict the particle size distribution in the system [39], [40]. The initial supersaturation in precipitation processes is high and therefore nucleation and particle growth start at a high rate. This requires appropriate turbulence modeling, including micromixing models [41]. The particles are very small, so only one-way coupling is relevant and the simulation can be carried out as pseudo-single-phase. On the other hand, the solution of the population balance must be integrated into the simulation [15]. – Bubble column Bubble columns are a typical example for two phase flows. The difference in density between gas and liquid causes a relevant relative velocity between the phases. Depending on bubble size and volume fraction, either Euler/Euler [42], [43] or Euler/Lagrange [44], [45] approaches are used. In most practical applications the requirements for the Lagrangian approach are not met because the volume fraction is too high. But to describe the non-uniform bubble size population, balances have to be solved. This is much easier

483

in connection with the Lagrangian approach than with the Eulerian approach. – Membrane module One of the newest apparatuses used in chemical engineering is the membrane module. Its numerical simulation is at the very beginning because of the variety of aspects which have to be considered. These are for instance: Flow through a porous medium Coupling of two flow regions Multi-phase flow Non-Newtonian fluids Fouling Presently the modeling approaches do not consider all of this aspects. The main stress is on modeling the two flow regions divided by the porous membrane under certain pressure conditions [46], [47]. It is obvious from this listing that the qualified application of CFD can not be performed without well founded knowledge of applicable models and some insight into the numerical procedures. The development of refined products like fine chemicals and active agents will be of growing importance in the chemical industry and CFD will play an important part in the design of the necessary apparatuses. This requires the considerations of parallel and consecutive chemical reactions of different rates and simulations of multiphase systems including the prediction of particle size distributions. Consequently, emphasis in CFD development for this field will be put on models and solution methods for fine spatial and temporal resolution and for a detailed description of physics. The advances in CFD interact strongly with the hardware development. On one hand the available computing power determines which mathematical models and which numerical methods can be used with reasonable numerical effort, on the other hand it affects the direction of development of new models and methods. Due to the fast pace at which new powerful compute hardware evolves and becomes available, model development does not restrict itself to quantitative refinement but regularly includes new fields of research. Latest improvements are Large eddy simulations of turbulent flows instead of Reynolds averaged approaches

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and models for multiphase systems. A new trend in the field of numerical methods is the application of the lattice Boltzmann method. Further innovations can be expected.

7. References 1. R.B. Bird, W.E. Steward, E.N. Lightfoot: Transport Phenomena, 2nd ed., John Wiley & Sons, New York 2002. 2. Pope S.: Turbulent flows, Cambridge University Press, Cambridge 2000 3. J.V. Boussinesq: “Essa sur la theories des eaux courantes”, Mém. prés. par div. savants à l’acad. sci. de Paris 23 (1877) 1. 4. B.E. Launder, Spalding D.B.: “The numerical computation of turbulent flows”, Computational Methods in applied mechanics and Engineering 3 (1974) 269. 5. J. Smagorinsky (1963): “General circulation experiments with the primitive equations, part I: the basic experiment”, Monthly Weather Rev. 91 (1963) 99. 6. J. Bardina, J.H. Ferziger, W.C Reynolds: Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows, Technical Report TF-19, Thermal sciences div., Dept. of Mech. Engg., Stanford Univ., Stanford, CA. 1980. 7. M. Germano, U. Piomelli, P. Moin, W.H. Cabot: “A dynamic subgrid scale eddy viscosity model”, in: Proc. Summer workshop, Center for turbulent research, Stanford CA 1990. 8. C. Meneveau, T.S. Lund, W.H. Cabot: “A Lagrangian dynamic subgrid-scale model of turbulence”, J. Fluid Mech. 319 (1996) 353. 9. J. Bałdyga: “A Closure Model for Homogeneous Chemical Reactions”, Chem. Eng. Sci. 49 (1994) 1985. 10. R.O. Fox: “On the Relationship between Lagrangian Micromixing Models and Computational Fluid Dynamics”, Chem. Eng. Process 37 (1998) 521. 11. B.F. Magnussen, B.W. Hjertager: “On the structure of turbulence and a generalised eddy dissipation concept for chemical reaction in turbulent flow”, in: 19th AIAA Aerospace Meeting, St. Louis, USA. 1981 12. A. Linan: On the internal structure of laminar diffusion flames, Technical Note, Inst. nac. de tec. aeron., Esteban Terradas, Madrid, Spain 1961.

13. N. Peters: “A spectral slosure for premixed turbulent combustion in the flamelet regime”, J. Fluid Mech. 242 (1992) 611. 14. J. Bałdyga, J.R. Bourne: Turbulent Mixing and Chemical Reactions, John Wiley & Sons Ltd., Chinchester 1999. 15. A.R. Paschedag: CFD in der Verfahrenstechnik: Allgemeine Grundlagen und mehrphasige Anwendungen, Wiley-VCH, Weinheim 2004. 16. D. Ramkrishna: Population Balances. Theory and Application to Particulate Systems in Engineering, Academic Press, San Diego 2000. 17. A. Gerstlauer, A. Mitrovi´c, S. Motz, E.-D. Gilles: “A population balance model for crystallization processes using two independent particle properties”, Chem. Eng. Sci. 56 (2001) 2553. 18. W.M. Deen: Analysis of Transport Phenomena, Oxford University Press, New York 1998. 19. J.C. Slattery: Advanced Transport Phenomena, Cambridge University Press, New York 1999. 20. D.A. Drew: “Mathematical modeling of twophase flow”, Ann. Rev. Fluid Mech. 15 (1983) 261. 21. C.T Crowe., M. Sommerfeld, Y. Tsuji: Multiphase flows with droplets and particles, CRC Press, Boca Rato 1998. 22. M. Sommerfeld, W. Krebs: “Particle dispersion in a swirling confined jet flow”, Part. and Part. Syst. Characterization 7 (1990) 16. 23. J.H. Ferziger, M. Peri˘c: Computational Methods for Fluid Dynamics, 2nd rev. ed., Springer Berlin 1999. 24. B.D. Nichols, C.W. Hirt: “Calculating ThreeDimensional Free Surface Flows in the Vicinity of Submerged and Exposed Structures”, J. Comput. Physics 8 (1971) 434. 25. F.H. Harlow, J.E. Welch: “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface”, Phys. Fluids 8 (1965) 2182. 26. C.W. Hirt, B.D. Nichols: “Volume of fluid (VOF) method for the Dynamics of Free Boundaries”, J. Comput. Physics 39 (1981) 201. 27. S.V. Patankar: Numerical Heat Transfer and Fluid Flow, Hemisphere Publ. Co. Washington 1980. 28. H.K. Versteeg, W. Malalasekera: An intriduction to computational fluid dynamics. The finite volume method, Longman, Harlow 1995. 29. C.A.J. Fletcher: Computational Techniques for Fluid Dynamics, vol 1+2, Springer, New York 1988. 30. Frisch U., D. d’Humieres, B. Hasslacher: “Lattice Gas Hydrodynamics in Two and Three Dimensions”, Complex Systems 1 (1987) 649

Computational Fluid Dynamics 31. Succi S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, Oxford 2001 32. J.D.Anderson Jr.: Computational Fluid Dynamics: the basics with applications, McGraw-Hill, New York 1998. 33. A. Birtigh, G. Lauschke, W.F. Schierholz, D. Beck, Ch. Maul, N. Gilbert, H.- G. Wagner, C.Y. Werninger: “CFD in der chemischen Verfahrenstechnik aus industrieller Sicht”, Chem. Ing. Tech. 72/3 (2000) 175. 34. Th. Avalosse, M.J. Crochet: “Finite Element simulation of mixing: 2. Three-dimensional flow through a Kenics mixer”, AIChE J. 43/3 (1997) 563. 35. E.S. Mickaily-Huber, F. Bertrand, P. Tanguy, T. Meyer, A. Renken, F.S. Rys, M. Wehrli: “Numerical simulations of mixing in an SMRX static mixer”, Chem. Eng. J. 63 (1996) 117. 36. F. Bertrand, P.A. Tanguy, E. Brito de la Fuente, P. Carreau: “Numerical modelling of the mixing flow of second-order fluids with helical ribbon impellers”, Comp. Methods Appl. Mech. Engng. 180 (1999) 267. 37. A. Brucato, M. Ciofalo, F. Grisafi, G. Micale: “Numerical prediction of flow fields in baffled sirred vessels: A comparison of alternative modelling approaches”, Chem. Eng. Sci. 53/21 (1998) 3653. 38. J. Derksen, H.E.A. Van den Akker: “Large eddy simulations on the flow driven by a Rushton turbine”, AIChE J. 45/2 (1999) 209. 39. D.L. Marchisio, A.A. Barresi, R.O. Fox: “Simulation of Turbulent Precipitation in a Semi-batch

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41. 42.

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47.

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Taylor-Couette Reactor Using CFD”, AIChE J. 47/3 (2001) 664. Paschedag A.R.: “Modelling of Mixing and Precipitation Using CFD and Population Balances”, Chem. Eng. Tech. 27/3 (2004) 232. D.L. Marchisio: Precipitation in Turbulent Fluids, PhD thesis, Politecnico di Torino 2002. A. Sokolichin, G. Eigenberger: “Applicability of the standard k − ε turbulence model on the dynamic simulation of bubble columns: Part I Detailed numerical simulations”, Chem. Eng. Sci. 54 (1999) 2273. L.I. Zaichik, V.M. Alipchenkov: “A kinetic model for the transport of arbitrarydensity particles in turbulent shear flows”, in: Proc. of Turbulence and Shear Flow Phenomena 1, Santa Barbara 1999. A. Lapin, A. Lübbert: “Numerical simulation of the dynamics of two-phase gas- liquid flows in bubble columns”, Chem. Eng. Sci. 49 (1994) 3661. L. Sanyal, S. Vásquez, S. Roy, M.P. Dudukovic: “Numerical simulation of gas-liquid dynamics in cylindrical bubble column reactors”, Chem. Eng. Sci. 54 (1999) 5071. D.E. Wiley, D.F. Fletcher: “Techniques for computational fluid dynamics modelling of flow in membrane channels”, J. Membrane Sci. 211 (2003) 127. C.A. Serra, M.R. Wiesner: “A comparison of rotating and stationary membrane disk filters using computational fluid dynamics”, J. Membrane Sci. 165 (2000) 19.

Model Reactors and Their Design Equations

487

Model Reactors and Their Design Equations Vladimir Hlavacek, Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Jan A. Puszynski, Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Hendrik J. Viljoen, Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Jorge E. Gatica, Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States

1. 2. 2.1. 2.1.1. 2.1.2. 2.2. 2.2.1. 2.2.2. 2.2.3. 3. 3.1. 3.2. 3.3. 3.4. 4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.1.4. 4.2. 4.2.1. 4.2.2. 4.2.3. 4.3. 4.4.

Introduction . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . Homogeneous Systems . . . . . . . . . Isothermal Reactors . . . . . . . . . . . Nonisothermal Reactors . . . . . . . . . Nonhomogeneous Systems . . . . . . Gas – Liquid Reactions . . . . . . . . . Solid – Solid Reactions . . . . . . . . . Solid – Gas Reactions . . . . . . . . . . Continuous Stirred-Tank Reactors (CSTR) . . . . . . . . . . . . . . . . . . . Isothermal Homogeneous System . Isothermal Heterogeneous Reactors Nonisothermal Continuous StirredTank Reactors . . . . . . . . . . . . . . Cascade of Tank Reactors . . . . . . Packed-Bed Reactors . . . . . . . . . Mass and Energy Balances . . . . . . The Single-Particle Model . . . . . . . The Two-Phase Model . . . . . . . . . . The One-Phase Model . . . . . . . . . . Boundary Conditions . . . . . . . . . . Values of the Parameters . . . . . . . Average Bed Porosity . . . . . . . . . . Effective Transport Coefficients . . . . Wall Heat-Transfer Coefficient . . . . Further Simplifications . . . . . . . . Parametric Sensitivity . . . . . . . . .

489 490 490 490 490 493 493 494 496 498 498 502 502 505 507 507 508 509 509 510 511 511 512 513 513 515

Symbols a a a A B, B∗

activity thermal diffusivity, m2 /s interfacial area per unit volume, m2 /m3 heat transfer area, m2 dimensionless adiabatic temperature rise

4.5. 4.5.1. 4.5.2. 4.5.3. 4.5.4. 4.6. 4.7. 4.7.1. 4.7.2. 5. 5.1. 5.2. 5.2.1. 5.2.2. 5.2.3. 5.3. 5.4. 5.5.

6.

c cJ cp , cv

Flow Field Description . . . . . . . . . Boundary Conditions . . . . . . . . . . Permeability and Inertia Coefficient . Thermal-Expansion Coefficient . . . . Mass-Expansion Coefficient . . . . . . Thermomechanical Effects in the Reaction System . . . . . . . . . . . . . Numerical Simulation . . . . . . . . . Discretization of the Physical Domain Discretization of the Governing Equations . . . . . . . . . . . . . . . . . . . . . Optimization of Chemical Reactors The Objective Function for Chemical Reactors . . . . . . . . . . . . . . . . Elementary Optimization Problems Optimization of a Batch System . . . . Optimization of a Continuous System (CSTR) . . . . . . . . . . . . . . . . . . . Optimization of Complex Systems: Illustrative Example . . . . . . . . . . . Optimization of Reactors by the Search Method . . . . . . . . . . . . . . Optimum Temperature Profile and Optimization of Multistage Reactors Optimization of a Multibed Adiabatic Reactor with Heat Exchange Between Catalytic Stages . . . . . . . References . . . . . . . . . . . . . . . . .

517 518 518 518 519 519 520 520 521 522 522 523 523 524 524 524 527

528 529

concentration, kmol/m3 molar concentration of species J, kmol/m3 heat capacity per unit mass at constant pressure and volume, respectively, J kg−1 ◦ C−1

488 cpm

Model Reactors and Their Design Equations

specific heat capacity of the cooling/heating medium C reduced concentration, c/c0 Cp , Cv molar heat capacity at constant pressure and volume, respectively, J kmol−1 ◦ C−1 d characteristic dimension, diameter, m diameter of impeller dI D diameter of reactor, m D coefficient of molecular diffusion, m2 /s e particle emissivity Ea activation energy E (t) residence time distribution function or frequency function, s−1 F molar flow rate g gravitational acceleration, m/s2 h overall heat-transfer coefficient, W m−2 K−1 H enthalpy, J HJ molar enthalpy of species J, J/kmol ∆H r heat of reaction at constant pressure as associated with the stoichiometric equation (∆H r )J heat of reaction at constant pressure for the conversion of or to one molar unit of J, J/kmol J component J molar flux, kmol m−2 s−1 JJ molar flux of species J, kmol m−2 s−1 k homogeneous reaction velocity constant, dimension depends on the kinetics; for nth order: m3(n−1) /(kmol(n−1) s) K chemical equilibrium constant KD ratio between two concentrations KL mass-transfer coefficient, m/s L length, m m total mass of a system, kg mJ mass of species J in a system, kg MJ molar mass of species J, kg/kmol n sequence number of a tank reactor in a cascade or of a bed in a multibed reactor n outward unit vector on S N number of revolutions (impeller) per unit time N QI dimensionless constant Nu αd/λ; Nusselt number p total pressure, Pa P permeability ∆Pr pressure drop across the reactor Pe total pressure in the bulk stream

vL /D1 ; P´eclet number for longitudinal dispersion Peh vL  cp /λ; P´eclet number for heat dispersion Pet vd t /Dt ; P´eclet number for transverse dispersion Pr µ cp /λ; Prandtl number Q amount of heat, J Q˙ heat flow, W r cylindrical or spherical coordinate, m R rate of reaction R gas constant, 8.314 J mol−1 K−1 RJ molar rate of production of J per unit volume of the reaction phase, kmol m−2 s−1 | R |J molar rate of conversion per unit volume of the reaction phase, kmol m−3 s−1 Re v d/ν; Reynolds number Rep v d p /ν; Reynolds number in packed bed related to particle diameter s ratio of reaction rate constant S surface, m2 Sc ν/D; Schmidt number St α/ cp v; Stanton number for heat transfer St K G /v or K L /v; Stanton number for mass transfer St
UA/ cp Φv ; modified Stanton number t time, s t
off-line time, s t∗ reaction time, s T temperature, K or ◦ C ∗ T optimum reaction temperature, K or ◦ C T  average radial temperature for a two dimensional model ∆T ad adiabatic temperature rise of a reaction mixture after complete conversion, ◦ C e T equilibrium temperature u internal energy per unit mass, J/kg u average internal energy (over the reactor volume), J/kg u linear velocity, m/s us control volume velocity, m/s UJ molar internal energy of species J, J/kmol ∆U r heat of reaction at constant volume as associated with a stoichiometric equation, J (∆U r )J heat of reaction at constant reaction volume for the conversion of or to one molar unit of J, J/kmol Pe

Model Reactors and Their Design Equations v vF vr V Vr VR wJ W W ˙ W x xJ x yJ z Z α β¯ γ γ γ¯ δ¯ δ ε ε¯ ϑ Θ Θc ζ ζ¯ κ λe λ λ µ µ ν ν ξJ η  σ

velocity, m/s front velocity, m/s relative velocity; velocity of one phase relative to another phase, m/s volume, m3 volume of reaction mixture, m3 reactor volume, m3 mass fraction of species J mass of the catalyst amount of work, J rate of work done on the surroundings, W direction of propagation, m mole fraction of species J in any phase coordinate, m mole fraction of species J in any gas phase only used in connection with x J for the liquid phase flow direction, m dimensionless coordinate in the direction of flow, z/L heat-transfer coefficient, W m−2 K−1 dimensionless heat-transfer coefficient ratio between kinetic and mass-transfer rates temperature factor E/RT a ; index a varies, depending on situation dimensionless activation energy dimensionless parameter displacement vector porosity dimensionless parameter dimensionless temperature dimensionless time dimensionless cooling temperature relative degree of conversion, fraction of reactant converted average conversion ratio between reaction rate constant effective thermal conductivity, W m−1 K−1 thermal conductivity, W m−1 K−1 Lam´e constant Lam´e constant dynamic viscosity, Ns/m2 kinematic viscosity, m2 /s stoichiometric coefficient degree of conversion of species J effectiveness factor density, specific mass, kg/m3 Stefan – Boltzmann constant, 5.67×10−8 J m−2 K−4 s−1

τ Φ ΦP ΦmJ Φc Φv ψ

489

average residence time in reactor system, s flow rate production rate, kg/s; kmol/s mass flow rate for species J, kg/s circulation rate volumetric flow rate, m3 /s shape factor

Subcripts 0 av A, B c e ext f fus F g G i ign J l L L m n r s S t v w z

initial average reactants coolant, continuous equilibrium extern fluid fusion fluid phase gas gas phase interface ignition arbitrary species liquid, longitudinal liquid phase at the outlet of a tubular reactor of length L medium, mass output of the nth tank reactor in a cascade, or of a segment n reaction, radial direction solid solid phase transverse volumetric wall axial direction

1. Introduction Reactor modeling represents a simultaneous solution and analysis of mass, energy, and momentum transfer equations along with reaction rate and equilibrium data. Sometimes additional information (e.g., of economic nature in reactor optimization calculations) must be also included in the reactor model. Important goals of reactor modeling are [1]:

490

Model Reactors and Their Design Equations

1) Scale-up of experimental laboratory units to pilot-plant size 2) Scale-up of pilot-plant units to full scale size 3) Prediction of behavior of different reactor feedstocks and new catalysts 4) Prediction of transients in reactors for better control 5) Optimization of steady-state operating conditions 6) Better understanding of the system that may lead to process and design improvements Reactor modeling in the hands of a practitioner is a powerful tool and can result in faster, more economical process design. It should always be coupled with pilot-plant experiments and process development. The most important parts of reactor modeling are described in this article; namely, simulation of batch and continuous stirred-tank reactors, tubular systems, and packed beds. Some elements of optimization of chemical reactors are presented as well.

2. Batch Reactors


ξJ t= 0

2.1.1. Isothermal Reactors Batch reactors are usually operated as closed systems with no net inflow or outflow of mass. The systems are also well-mixed and no spatial gradients exist. The material balance for a component J in a batch reactor which meets these conditions, is:

t=

(2.2)

t=

(1 −ξJ )1−n − 1 k (n − 1) cJn−1 0

, n = 1.

(2.3)

ln (1 −ξJ ) ,n= 1 −k

(2.4)

The rate equations of reversible first-order reactions with initial concentrations cJ0 and cP0 and rate constants k 1 and k 2 for the forward and backward reactions respectively, can also be integrated: . / cJ +cP0 ln (1 −ξJ ) (k1 +k2 ) −k2 0c J0

− (k1 +k2 )

(2.5)

The general problem in reactor design is to calculate the volume of a reactor for a certain average rate of production. The volume can be obtained from the required degree of conversion and the corresponding reaction times. In batch operation, a certain time is used for filling, heating, cooling, and cleaning and during this time no production occurs. If t
denotes this off-line time and ΦP denotes the desired production of product P, the reactor volume is

(2.1)

where V r is the volume of the reaction mixture, m the total mass of the system (kg), mJ the mass of component J in the system (kg), ξ J the degree of conversion of J, M J the molar mass of component J (kg/kmol), and RJ the molar rate of production of J per unit volume of the reaction phase (kmol m−3 s−1 ). In the isothermal case, RJ is only a function of the reactants for an irreversible reaction or a function of reactants and products for a reversible reaction. In terms of the density of the reaction mixture,  = m/V r , the material balance can be written in integral

 dξJ M J RJ

If the density of the reaction mixture remains constant during the course of the reaction, the integration is simplified. In the case of irreversible nth-order reactions integration gives

t=

2.1. Homogeneous Systems

dmJ dξJ =m =MJ RJ Vr dt dt

form (Eq. 2.2) and the reaction time t that is required for a certain degree of conversion can be obtained by integrating Equation (2.1).

VR =

νA ΦP (t +t ) νP MP cA0 ξA

(2.6)

where ν A and ν P are the stoichiometric coefficients of reactant A and product P, respectively. 2.1.2. Nonisothermal Reactors All chemical reactions are accompanied by evolution or absorption of heat. The most important considerations which determine the choice of temperature in the reactor are:

Model Reactors and Their Design Equations 1) Chemical equilibrium and the conversion rate of the desired reaction; the maximum conversion of a reversible exothermic reaction is reduced when the temperature is increased, but the kinetic rate increases, therefore an optimum temperature exists. 2) Undesirable side reactions (which also depend on temperature and their role in the overall equilibrium) can influence the maximum yield of product. 3) Phase transformations of reactants or products; there are examples where it is better to carry out the reaction in the liquid instead of the gaseous phase to achieve maximum conversion and mass transfer. An example is the formation of methyl tert-butyl ether from isobutene and methanol. 4) Cost implications of heating reactants to the reacting temperature; energy cost must be compared with the gain in yield at increased temperature as maximum yield and maximum profitability will not always be at the same temperature [2]. Constant Pressure. At constant pressure, the energy balance in a well-mixed batch reactor is: cp

dT Q˙ + (∆Hr )J RJ = dt Vr

(2.7)

where cp denotes the specific heat capacity, (∆H r )J the reaction enthalpy for the conversion of or to one molar unit of J, and Q˙ denotes the rate of heat flow to or from the reactor. In the case of adiabatic operation, Q˙ = 0. Otherwise, Q˙ can be determined as follows:

Q˙ =hA T −Tc, H

(2.8)

where h is the overall heat-transfer coefficient and A the heat-transfer area. The external temperature can be cooler (T c ) or hotter (T H ) than T , depending on whether cooling or heating must be accomplished. The overall heat-transfer coefficient h is calculated from correlations [3] and it depends on the temperature of the cooling or heating medium. Thus, a heat-transfer coefficient hav is calculated at the average medium temperature. If Φm is the mass flow rate and cpm is the specific heat capacity of the cooling or heating medium, then h is corrected according to:

h=

hav 1 + 2 ΦhavcA

491 (2.9)

m pm

Example. In a batch reactor with a volume of 5 m3 , an exothermic reaction A → P is carried out in the liquid phase. The density of the mixture does not change during the reaction. The rate of reaction is given by RA = − kcA

with k = 4×106 exp (− 7500/T ) s−1 . The heat of reaction is (− ∆H r )A = 1.67×106 J/kg, the initial concentration of A is cA0 = 1 kmol/m3 , M A = 100 kg/kmol and Cp = 4.2×106 J m−3 K−1 . The initial temperature of the reaction mixture is 298 K. The reactor is fitted with a heat-exchange coil with a surface area of 3.3 m2 . The heat exchanger is operated with steam (T H = 393 K, h = 1000 W m−2 K−1 ) and with cooling water (T c = 290 K, h = 1400 W m−2 K−1 ). Filling and emptying of the reactor take 600 s and 900 s, respectively. For a required conversion ξ A ≥ 0.95, the temperature profile over one reaction cycle and the duration of a reaction cycle shall be determined when the following policy of operation is followed: heat up to 350 K, the reaction then proceds under isothermal conditions until ξ A = 0.95, followed by cooling down to 310 K. Solution. The following equations must be solved simultaneously for the first part of the reaction cycle: dξA =k (1 −ξA ) dt dT = (− ∆Hr )A cA0 MA k (1 −ξA ) cp dt hAs [Ts −T ] + Vr

Starting from the initial concentration and temperature, a small temperature interval ∆T is selected. Average values for T and k over this interval are then calculated and used together with the initial value for ξ in the second equation to solve for ∆t. This value is then used in the first equation to solve for ξ. An average value for ξ over this interval can then be calculated. With the average values of T , k, and ξ, an improved ∆t value can be calculated from the second equation and the process is repeated until the required degree

492

Model Reactors and Their Design Equations

of accuracy is reached. In Figure 1 the temperature profile is shown. Once the conversion is 0.95, cooling commences. The cooling period can be calculated by integrating the following energy equation from T = 350 K to T = 310 K: dT = (− ∆Hr )A cA0 MA k (1 −ξA ) dt hAc [Tc − T ] + Vr

cp

Example. A closed tank with an internal volume of 250 m3 contains a mixture of air with 4 % butane at 18 ◦ C and a pressure of 100 kPa. The mixture is ignited and the reaction proceeds according to: C4 H10 + 6.5 O2 → 4 CO2 + 5 H2 O

Integrating the first two equations from the initial conditions until T is 350 K gives a preheating period of 2384 s. The temperature is then kept constant and only the first equation is integrated until the conversion reaches 0.95; this period takes 1000 s. For the cooling-off period the third equation is integrated and this gives 5426 s. Adding 1500 s for filling and emptying gives a total period of one cycle of 10 310 s. LIVE GRAPH

The heat of combustion at 18 ◦ C and 100 kPa is (− ∆H r ) = 2880 kJ/mol. The pressure in the vessel shall be determined after the explosion. Solution. Since the reaction proceeds adiabatically and at constant volume, the total internal energy of the reaction mixture does not change. The temperature rise is calculated from:
T


ξB cv dT +

T0

ξB0

(∆Ur )B dξB = 0 MB

where the index B denotes butane. To calculate (∆U r )B , the relationship between U and H and the ideal gas law will be used:

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∆Hr = ∆Ur +R T ∆n

The total change in number of moles is (4 + 5 − 7.5 = 1.5), hence: RT ∆n = 1.5 × 8.314 J mol−1 K−1 ×291 K = 3.63 × 103 J/mol

Figure 1. Temperature profile over a reaction cycle

Constant Volume. The energy balance for a constant volume process is cv

dT Q˙ + (∆Ur )J RJ = dt Vr

(2.10)

Since ∆U r is the change of internal energy at constant volume and temperature, for the case of an ideal gas ∆H r and ∆U r are related as follows: ∆Hr = ∆Ur +RT ∆n

(2.11)

This relation will be different for reaction mixtures with another equation of state.

Since O2 is the limiting reactant (i.e., O2 is present in substoichiometric amounts), not all butane will react. The composition of the final mixture can then be calculated. In Table 1 the initial and final compositions are given. The specific heat capacity at constant volume (cv ) of the product mixture is a function of temperature and can be found in thermodynamic tables. In this example the average value of cv is 1.06 kJ/kg. The mole fractions of butane before and after the reaction were 0.04 and 0.0089, respectively. These values can be converted to mass fractions by multiplying the mole fraction of each component in Table 1 by its molecular mass and normalizing. The mass fractions of butane before and after reaction are 0.077 and 0.017, respectively.

Model Reactors and Their Design Equations Table 1. Initial and final gas compositions Compound

Mole fraction before reaction

Mole fraction after reaction

Butane Nitrogen Oxygen Carbon dioxide Water

0.04 0.758 0.2028

0.0089 0.708 0.182 0.093

The temperature rise can now be determined:  cv × (T − T0 ) +

 (− ∆Hr )B − RT ∆n × (∆ξB ) = 0 MB

so: 1.06 kJ/kg× (T −291 K)   −2880 kJ/mol − 3.63 kJ/mol + 58 g/mol × (0.077 − 0.017) = 0 T = 3103 K = 2830 ◦ C

Applying the ideal gas law, the pressure can be determined: P ×250 m3 = 1.046×n0 ×8.314 J mol−1 K−1 × (3103 K) = 1115.4 kPa

where 1.046×n0 is the total number of moles present after the explosion. The last column in Table 1 is the mole fraction based on kmol/kmol original mixture and adding this column will give the relative change in the number of moles in the mixture.

2.2. Nonhomogeneous Systems Batch systems are spatially homogeneous when the reactant mixture is well-mixed during the course of the reaction. Nonideal mixing can cause settling or separation of phases in liquid – gas mixtures, mixtures of nonmiscible liquids or liquid – solid mixtures. However, even reactants which are well-mixed, do not necessarily react in a homogeneous way. Even if a single phase is present, the system can still be spatially nonhomogeneous. 2.2.1. Gas – Liquid Reactions In the following example, the design of a batch reactor, where both a liquid and a gas phase are present, will be demonstrated.

493

Example. A batch reactor must be designed for the production of 1×105 kg/day fats containing monounsaturated fatty acids by hydrogenation of cottonseed oil. The reaction is carried out on a nickel catalyst that is dispersed in hard stearin. Hydrogen is bubbled through the oil, thus two phases are present. The following equations describe the conversion and formation of acid moieties with diunsaturated (B), cismonounsaturated (R1 ), trans-monounsaturated (R2 ), and saturated (S) aliphatic chains: dcB = − (k1 +k2 ) c0.5 Hs cB dt dcR1 0.5 =k1 c0.5 H s c B − k 3 c Hs c R 1 dt +k4 c0.5 Hs c R 2 − k 5 c H s c R 1 dcR2 0.5 =k2 c0.5 Hs cB +k3 cHs cR1 dt − k4 c0.5 H s c R 2 − k 6 c H s cR 2 dcs dt

=k5 cHs cR1 +k6 cHs cR2

The hydrogen concentration in the gas phase (cHi ) is in equilibrium with the hydrogen on the catalyst surface, cHs . Since the gas is essentially pure hydrogen, cHi is constant. Furthermore it is assumed that the rate of consumption of adsorbed hydrogen on the catalyst surface is balanced by the rate of mass transfer from the gas phase. Let K D = cHs /cHi denote the ratio between the two concentrations, then the following relation exists: 0.5 KD =

. / − 1 + s1 [1 −ξB ] 2  +  YR YR 2 γ1 + s 1 + s 2 5

6

 2  0.5 YR YR 1 + s1 (1 −ξB )2 + γ4 γ1 + s 1 + s 2 2 5 6   YR YR 2 γ1 + s 1 + s 2 5

6

where Y R1 = cR1 /cB0 , Y R2 = cR2 /cB0 , s2 = k 1 /k 2 , s3 = k 1 /k 3 , γ = k 1 cB0 /K L av c0.5 Hi , 0.5 s4 = k 1 /k 4 , s5 = k 1 /k 5 c0.5 Hi , s6 = k 1 /k 6 cHi . The product of the mass-transfer coefficient and the bubble area av was determined experimentally as K L av = 0.022 s−1 . Whereas the si values denote the ratios of reaction rate constants, γ is a ratio between kinetic and mass transfer rates.

494

Model Reactors and Their Design Equations

Experiments on the hydrogenation of cottonseed oil using a Rufert nickel catalyst gave the following values for  the reaction rate constants [4]: k1 = 0.254 L/mol/min, s2 = 2.89, s3 = 5.05, s4 = 15.1, s5 = 3.38, s6 = 3.38. The hydrogen concentration in the gas phase is cHi = 0.0129 mol/L. The feedstock composition (wt %) consists of 25.6 % S, 27 % R1 , 0.4 % R2 and 47 % B. The molar concentration of B, cB is 1.45 mol/L. The value for γ can now be calculated:

density of cottonseed oil. Using a height to diameter ratio of two, the specification of the vessel is 2.92×5.84 m. LIVE GRAPH

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L/mol/min × 1.45 mol/L (60 × 0.022) min−1 ×0.1136 mol/L/min = 2.45

γ=

0.254

The reactions are irreversible, hence cs does not appear in the rate expressions for components B, R1 , and R2 and they can be integrated independently from the balancing equation for S. Time can be substituted in terms of the conversion of B: YR

YR

YR

0.5 1 2 1 dYR1 (1 −ξB ) − s3 + s4 − s5 KD   = 1 dξB 1+ s (1 −ξB ) 2   YR1 YR YR 1 −ξB 0.5 − s + s 2 − s 1 KD dYR2 s2 3 5   4 = dξB 1+ 1 (1 −ξB ) s2

These two equations, together with the algebraic equation for K D can now be integrated, using the initial conditions Y R1 = 27/47, Y R2 = 0.4/47, and ξ B = 0. In Figure 2 A, the total yield of monounsaturates is shown as a function of ξ B . A maximum is obtained at ξ B = 0.84 and the time it takes to obtain 84 % conversion of B is 115 min (see Fig. 2 B). The processing time, excluding the reaction time, is best ascerted from plant studies; for this example 2.5 h are taken. Hence the total time (conversion plus processing) is 4.42 h. Thus, five batches can be processed per day, each batch producing 20×103 kg. At 84 % conversion the product mixture consists of 67 kg monounsaturates per 100 kg of mixture, hence the reactor volume is: Vr = 1.05×

100 kg 1 ×20 000 kg× 67 kg 800 kg/m3

= 39.2 m3

where the factor 1.05 is introduced because 5 % was added for head space and 800 kg/m3 is the

Figure 2. Total yield of monounsaturates in the hydrogenation of cottonseed oil (A) and conversion time of diunsaturates (B)

2.2.2. Solid – Solid Reactions Even when only one phase is present, the reaction can progress in a nonhomogeneous way. Reactions with large activation energies require a certain amount of preheating before they be-

Model Reactors and Their Design Equations gin. If the temperature profile in a batch system is not uniform, the reaction rate will also be spatially distributed and if the heat of reaction is large, it will further contribute to the concentration of high temperatures in local areas inside the reactor. Solid – solid reactions that are associated with large activation energies and large heats of reaction often take place in the combustion mode. The combustion mode is associated with strong spatial gradients and modeling these type of reactions requires special techniques. Examples of solid – solid reactions which occur in the combustion mode are those between Ti and C, Mo and Si, and Ta and C. The well-mixed reactants are usually prepared as a preform that is ignited at one end of the sample (see Fig. 3). Once the combustion wave has traversed the length of the sample, the reaction is complete. Conversion is high in these combustion reactions and under ideal conditions complete conversion can be accomplished. Two factors determine the degree of conversion. If the system loses a lot of heat during the reaction, the degree of conversion drops. Thus, the closer the system can be operated to adiabatic conditions, the higher will be the degree of conversion. The second factor is incomplete mixing of the reactants. In these type of reactions it is important to have intimate contact between the different species on the micro-level. Therefore, mixing of the reactants is one of the most time- consuming and expensive steps in the process.

cp

∂T = ∇ ·λ ∇T + (− ∆Hr )J RJ ∂t

Maximum Temperature Rise. For T ad < T fus the adiabatic temperature T ad rise of a system where reactant A is transformed to product P, is given by: Tad =T0 +

(− ∆Hr ) cA cp

∂cJ = − RJ ∂t

(2.12)

(2.14)

When T ad > T fus and (−∆Hr )
5 − 10 Φm

(3.4)

where Φv is the volumetric flow rate; ζ J the relative degree of conversion; cJ0 the molar concentration of species J; and RJ the molar rate of production of J per unit volume of the reaction phase. The mass balance equations for various reaction orders are listed in Table 2.

The impeller discharge rate depends on the type of impeller, the tank geometry, and operating conditions. This discharge rate can be estimated from:

Table 2. Ratio of outlet to inlet concentrations in a CISTR for reactions of different order

where d I is the impeller diameter, N the number of impeller revolutions per unit time, and N QI is a dimensionless constant. The value of this constant depends on the axial blade width of the impeller, the number of blades, curvature or pitch, and the ratio of impeller velocity to fluid rotational velocity [8]. The following values are recommended for agitators in baffled tanks which are commonly used in industry: Marine-Type Impeller:

Reaction order, n

Reaction rate equation, RA

−1

k c−1 A

0

k

1 2

k

1

k cA

2

k c2A

√ cA

Concentration ratio, cA /cA0 "  # # 1  1+ $ 1− 2 1 − ck τ ' A0 1+ 1 1 +k τ 1 2k cA τ 0

k2 τ 2 4cA 0

 

4k τ c2 A0

 kτ − 2√ c

A0

( ( ) ( ( 1 + 4k cA0 −1( (

Most important in designing a homogeneous CISTR system is to select equipment and stirring procedures that will assure essentially perfect mixing. Choice of a suitable type of impeller and reactor arrangement, the required power,

Φc =NQI N d3I

(3.5)

NQI = 0.5

Turbine (with six blades and width-to-diameter ratio of 1 : 5): NQI = 0.93D/dI for Reimp > 104

where Reimp =  N d 2I /µ is the impeller Reynolds number, and D the diameter of the reactor.

500

Model Reactors and Their Design Equations

Many reactions require rapid dispersal of reactants in order to avoid even momentary buildup of high concentrations in the system. In homogeneous reacting systems, the most rapid dispersal can be accomplished by discharging the feed as close as possible to the center of the impeller. Discharge ports for product removal should be located at the opposite end of the flow pattern from the inlet nozzle. In general, the flow of a reacting fluid through a reactor is a very complex process. The interaction of flow pattern and chemical reaction cannot be analyzed rigorously even in the simple case of a single isothermal reaction. Therefore simplified and/or idealized models are necessary.

CISTR. When segregated flow takes place, the situation is different. Each aggregate in the outlet stream can be considered as a batch reactor with a reaction time equal to its residence time. The fraction of aggregates having a residence time between t and t + ∆t is E ∆t. If the remaining reactant concentration after conversion in a batch reactor with residence time t is cBatch (t), A the average concentration at the outlet of the segregated continuous flow reactor CSSTR can be calculated: c¯A 1 = c A0 c A 0


∞ cBatch (t) Edt A

(3.7)

0

or Residence-Time Distribution (RTD). The frequency function often used in statistics (called the E diagram in RTD description) can be derived for a continuous stirred-tank reactor from a mass balance of the reactor with an ideal pulse perturbation at the inlet stream. The pulse is immediately distributed over the reactor, moreover the product concentration remains the same everywhere and equals the outlet concentration. The E (t) diagram reads: 1 E (t) = e−t/τ τ

(3.6)

For a continuously operated mixing vessel which shows a RTD of an ideal mixer (Eq. 3.6), the different states of micromixing are: 1) CISTR: the ideal mixer (mixing on molecular level) 2) CSSTR: the completely segregated mixer where the fluid consists of lumps which move through the reactor without exchange of matter with their surroundings 3) CSTR: the partially segregated mixer A detailed description of the amount of micromixing with one parameter is generally not possible. For non-first-order reactions, apart from the reaction rate not only the intensity of mass exchange between fluid elements is important, but also the residence time of a molecule in the reactor. In the case of maximum micromixing and constant density of the reacting system, the conversion is identical to that calculated for the

c¯A 1 = c A0 c A 0


∞ cBatch (t) A 0

1 −t/τ e dt τ

(3.8)

The average outlet concentrations for different reaction kinetic expressions for both ideal micromixing and segregated flow are listed in Table 3 [9]. Outlet concentrations are identical for both cases if a first-order reaction takes place. Example [9]. Reactant A (M A = 104 kg/kmol) is to be polymerized in the dispersed phase in a continuous stirred-tank reactor. Based on batchscale experiments, the following reaction rate expression was found: RA = − k c1.5 A with k = 2.5 × 10−4 s−1 kmol−0.5 m1.5

The required size of the tank reactor shall be calculated for a production of 10 t/d of polymer with a relative degree of conversion ζ¯A = 0.9, under the assumption that coalescence between the liquid drops does not occur. The feed of the dispersed phase consists of pure A, and the density of this phase remains constant at a value of 832 kg/m3 during the reaction. The volume fraction of the dispersed phase in the reactor is 0.165. Solution. The average conversion ζ¯A at the outlet of the CSSTR is obtained by rearrangement of Equation (3.8):
∞ ζ¯A =

Batch ζA (t) 0

1 −t/τ dt e τ

(3.8a)

Model Reactors and Their Design Equations Table 3. Design equations for ideally mixed CISTR and for completely segregated fluids CSSTR

501

By integrating Equation (3.8 a) a value of √ kτ cA0 = 15.4 is found for ζ¯A = 0.9. The calculated residence time τ is 6 h and the required volume of the dispersed liquid Φm τ /0.9 A is equal to 3.34 m3 . The total required volume in the reactor is 3.34 m3 /0.165 = 20.2 m3 . Start-Up of a CISTR. For non-steady-state operations, the accumulation term in Equation (3.1) cannot be neglected. Two different periods of non-steady-state operations in a CISTR can be distinguished: 1) Filling a reactor with reactants (variable reactor volume) 2) Non-steady-state operation at a constant reactor volume In the case of an irreversible liquid reaction (A → B) which takes place in a CISTR under isothermal and constant density conditions, the instant concentration in the reactor can be calculated as follows: For t ≤ τ , no liquid leaves the reactor and the material balance leads to: − d (cA V ) = −Φv cA0 +kcA V dt

(3.9)

in which the reaction volume is V = Φv t. Integration of Equation (3.9) gives: cA =

 c A0  V for t ≤ 1 − e−kt kt Φv

(3.10)

For t > τ the reaction volume V is constant and equals V r and the product flow out of the reactor equals the feed rate Φv . The material balance for reactant A now becomes: − d (cA Vr ) dcA = − Vr dt dt

= −Φv cA0 −cA +kcA Vr

(3.11)

Integration yields: cA0 − (1 +kτ ) cA = e−(1+kτ )(t−τ )/τ cA0 − (1 +kτ ) cAτ for t >τ cA0 kτ

The material balance for a batch system together with the rate expression yields: 

= 1− ζ¯ Batch A

2 √ 2 +k t cA0

2



−kτ



(3.12)

1−e where cA = After a long time (t → ∞) the right-hand side of Equation (3.12) approaches zero, so that ultimately the concentration cA in the reactor reaches the value cA∞ . A conservative estimate for the time required to reach steady state can be made by substituting cAτ = cA0 in Equation (3.12) for the required degree of conversion.

502

Model Reactors and Their Design Equations

3.2. Isothermal Heterogeneous Reactors Contrary to reactions in a single phase where mixing takes place on a molecular scale, in a multiphase system not all reactive molecules present may be available for chemical reaction due to mass transfer limitations. For a reacting system consisting of two phases L and G, with a reaction taking place in phase L only, the mass balance equations are: dmJL = − ∆ (ΦmL wJL ) dt +JJL MJ a Vr +RJ MJ (1 −ε) Vr

(3.13)

3.3. Nonisothermal Continuous Stirred-Tank Reactors

(3.14)

All chemical reactions are in principle accompanied by evolution or absorption of heat and these effects often have to be taken into account. For flow processes, the energy balance equation is

and: dmJG = − ∆ (ΦmG wJG ) +JJG MJ aVr dt

The term RJ M J (1 − ε) V r is the production rate of J by a homogeneous chemical reaction in phase L where (1 − ε) is the volume fraction of the reaction phase per unit volume of a reactor. The terms (−∆Φm wJ ) and (J J M J a) represent the net supply of component J by convection and the mass flow of J from the interface to the bulk of the phase, respectively. When a reaction takes place in a system of two immiscible or partially miscible liquid phases, the overall process can be controlled by chemical reaction and/or mass transfer between both phases. In the case of a chemically controlled process, the overall rate expression in Equation (3.1) is the sum of the reaction rates in the individual phases. When mass transfer is the rate-determining step, phase equilibrium cannot be achieved and the reaction rate is fast compared to the rate of mass transfer. Thus, the reaction proceeds as the reactants diffuse through the interface into the bulk of the other phase. If reactant A is a major component of phase I while phase II consists of reactant B and if equilibrium is attained at the interface, the rate of mass transfer of A from phase I to phase II would be equal to the rate of disappearance of A by chemical reaction in phase II. For an irreversible, second-order reaction between A and B, mass transfer of A from phase I to phase II can be described as:  JAII =kII a

cAI −c0AII KA

 (VI +VII )

a is the interfacial area per unit volume of both phases (m2 /m3 ), cAI is the concentration of reactant A in phase I, K A is the distribution coefficient of A between both phases, and c0AII is the steady-state concentration of A in phase II. The value of the product of mass-transfer coefficient and interfacial area strongly depends on the reactor arrangement and empirical correlations must be used to scale-up such reacting systems [10], [11].

(3.15)

where k II is the mass-transfer coefficient in phase II per unit volume of both phases (m/s),

d

9 : u m dt

˙ = − ∆ (hΦm ) +Q˙ −W

(3.16)

where h is the enthalpy per unit mass of the reaction mixture and u is the average internal energy over the reactor volume. For the steadystate operation of a continuous tank reactor provided with a heat exchange area A, the energy balance becomes: Φm (h1 −h0 ) = U A (Tc −T1 )

(3.17)

where the subscripts 1 and 0 refer to the conditions at the outlet and inlet, respectively; T c is the temperature of the cooling or heating medium. The difference in enthalpy (h1 − h0 ) can be calculated from:


T1 ξJ,1

h1 −h0 =



cp dT + (∆hr )J dξJ



(3.18)

T0,O

where cp is the specific heat capacity at constant pressure and (∆hr ) the enthalpy of reaction at constant pressure per unit mass of J converted. If the density and heat capacity of the system are constant, the energy balance for reactant A gives:

− (∆Hr )A cA0 −cA1 = cp  (T1 −T0 ) +

UA (T1 −Tc ) Φv

(3.19)

Model Reactors and Their Design Equations where (∆H r )A is the heat of reaction per mole of A converted. In the case of a first-order irreversible reaction Equations (3.1) and (3.19) can be rewritten to: kτ 1 +kτ UA (T1 −Tc ) = cp (T1 −T0 ) + Φv − (∆Hr )A cA0

(3.20)

In order to find the maximum temperature T 1 , Equation (3.20) must be solved by trial and error, either numerically or graphically, because k is an exponential function of temperature. The left-hand side of Equation (3.20) represents the heat produced per unit volume of a reaction mixture. The right-hand side represents the heat removed per unit volume of the reaction mixture as a result of the heat absorbed by the cold feed and the heat transferred to the cooling medium. The maximum temperature rise can be reached when the system is adiabatic (no heat transfer to the surroundings) and complete conversion is achieved: ∆Tad =

− (∆Hr )A cA0 cp

503

is higher than that generated in the system. The intermediate intersection point is unstable. Since the slope of the heat production curve at the intermediate point is greater than that of the heat removal line, any positive temperature perturbation will be amplified until the reactor reaches the upper stable operating point, and negative temperature deviation leads to extinction (lower stable operating point). If the feed temperature T 0 and the cooling capacity are such that the heat removal line 3 prevails (see Fig. 8), the reactor operates in an upper stable state and no special measures need be taken to start up a process.

(3.21)

Autothermal Reactor Operation. The heat released during an exothermic reaction is often used for preheating inlet reactant streams. A reactor system, in which such a feedback of reaction heat to the incoming reactant stream takes place is said to operate under autothermal conditions. The theory of steady-state behavior of autothermal reactors is described in [12], [13]. Multiplicity. In Figure 8 both right- (heat removal) and left-hand sides (heat production) are plotted for different inlet and operating conditions. At the intersection of the heat production (solid line) and heat removal curves (dashed lines), Equation (3.20) is obeyed. For a certain range of inlet parameters several solutions may exist (multiplicity of steady states). If the heat removal is represented by line 1 in Figure 8, the reactor is cooled so effectively that steady-state operation is possible only at a low temperature and a very low degree of conversion. If the cooling rate is lowered (line 2), three points of intersection occur with the heat generation curve. The two intersection points, the lowest and highest reaction temperature, represent stable conditions due to the fact that the rate of heat removal

Figure 8. Heat production (—-) and heat removal (– – –) in a nonisothermal CISTR for an irreversible exothermic reaction A → R

LIVE GRAPH

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Figure 9. Regions of multiplicity in a CISTR [5] T = Trifurcation point; ——– τ = l s; – – – τ = 0.5 s

504

Model Reactors and Their Design Equations

In Figure 9 a multiplicity region is presented as a function of the maximum  temperature in∆Tad crease in the reactor, 1 +St . In the area enveloped by the curves multiplicity will occur. Outside, only one (stable) operating point is possible. Operating points located above the multiplicity region are distinguished by a high degree of conversion, those below the multiplicity region by a low degree of conversion. The point ad with the lowest value of 1∆T +St where multiplicity occurs is often called the trifurcation point, and the curves departing from it, the bifurcation lines. Hlavacek et al. derived the criteria for existence of multiplicity of steady states for firstorder reactions [14]. Multiplicity occurs when the dimensionless adiabatic temperature rise (− ∆H ) c E B = cp Tr 0 A0 RT exceeds a critical value of 0 ∗ B : 1) Adiabatic case ( β¯ = 0) B∗ =

4¯ γ γ ¯ −4

2) Adiabatic case ( β¯ = 0); γ¯ → ∞ B∗ = 4

3) Nonadiabatic case; γ¯ → ∞ B ∗ = 4δ¯

4) Nonadiabatic case

¯γ 2 4 ε¯+δ¯ B∗ = ¯ γ ¯ δ (¯ γ − 4) − 4¯ ε

where γ¯ = E/RT 0 is a dimensionless activation energy, β¯ = UA/Φv  cp is a dimen¯ sionless heattransfer coefficient, δ¯ = 1 + β, E ¯ ¯ ε¯ = β Θc , and Θc = RT 2 (Tc −T0 ) is a di0 mensionless cooling temperature. Hysteresis in Autothermal Systems. If the feed temperature T 0 is increased at a constant feed rate, the reaction will be ignited at a certain feed temperature (T 0 )ign and the reactor will operate at the upper stable steady state. If T 0 is then lowered again, the reactor continues operating at a high conversion level until extinction takes place at a feed temperature (T 0 )ext .

Stability of Autothermal CISTR. The transient behavior of strongly exothermic reacting systems may lead to unexpected, undesired changes in a reactor. This is a result of coupling between the material and energy balances (Eqs. 3.1 and 3.16). Detailed information on dynamic behavior of autothermal systems can be found in [15–17]. The mass and energy balance equations Equations (3.1) and (3.16) can be rewritten to: dζ τ RA = −ζ + dΘ c A0

(3.22)

and: dT RA τ +St (Tc −T ) =T0 −T − ∆Tad dΘ c A0

(3.23)

where τ =  V r /Φm is the average residence time, Θ = t/τ is a dimensionless time, and St
a modified Stanton number (St
= UA/cp Φv ). In the case of a first-order reaction Equations (3.22) and (3.23) can be rewritten to: dζ = −ζ +ks τ (1 −ζ) e∆ϑ dΘ

(3.24)

dϑ = − 1 +St ∆ϑ + ∆ϑad ks τ (1 −ζ) e∆ϑ dΘ + ∆ϑ0 +St ∆ϑc (3.25)

where: E RT 2s E ∆ϑc = (Tc −Ts ) RT 2s E ∆ϑad = ∆Tad RT 2s ∆ϑ0 = (T0 −Ts )

and k s is the reaction rate constant at temperature T s. When a small perturbation is imposed on a steady state, Equations (3.24) and (3.25) can be linearized as follows: d ∆ζ 1 ∆ζ +ζs ∆ϑ≡a11 ∆ζ +a12 ∆ϑ (3.26) = − dΘ 1 −ζs

and:

  d ∆ϑ ∆ϑad ζs ∆ζ + ∆ϑad ζs − 1 +St ∆ϑ = 1 −ζs dΘ ≡ a21 ∆ζ + a22 ∆ϑ (3.27)

Model Reactors and Their Design Equations where ∆ζ = ζ − ζ s and ∆ϑ = ϑ − ϑs ; ζ s and ϑs are the conversion and the dimensionless temperature at steady state. The solution of Equation (3.26) and (3.27) has the form: ∆ζ =b11 ep1 Θ +b12 ep2 Θ

(3.28)

p1 Θ

(3.29)

∆ϑ =b21 e

+b22 e

p2 Θ

where: (a11 +a22 ) p1, 2 = 2 !  

4 (a11 a22 −a12 a21 ) 1− 1± (a11 +a22 )2 1 + St >ζs (1 −ζs ) ∆ϑad

(3.30)

and: 1 − ∆ϑad ζs + 1 +St > 0 1 − ζs

(3.31)

where ζ s = k τ /(1 + k τ ) in the case of first-order reaction. If the second condition (Eq. 3.31) is met, the reactor is dynamically stable and operates at its steady state (ζ s , T s ). However, if this condition is not satisfied the reactor is dynamically unstable and operates in a fixed limit cycle around the steady-state point (ζ s , T s ) (Fig. 10). LIVE GRAPH

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Figure 10. The limit cycle in CSTR [18]

3.4. Cascade of Tank Reactors [21] Continuous stirred-tank reactors in series (Fig. 11) are used when the required residence time is long and more than one vessel is necessary for efficient, economical mixing. Conversions, yields, and residence times of the individual stirred-tank reactors are calculated analogously to those for a single CISTR. The steadystate material balance over the nth reactor for a component J reads: ξJn −ξJ (n−1) =

The chemical process is stable when:

505

Vrn MJ RJn Φm

(3.32)

In most cases, the set of equations describing the degree of conversion in a cascade cannot be solved analytically. For a system consisting of n equal-sized CISTRs in series an analytical solution can be obtained for first-order irreversible reactions [19]: cAn =cA0 (1 +kτ )−n

(3.33)

and for second-order reaction: cAn = 14 k τi cAn =

1  −2 + 2· 4kτi

!

1 −1...+

 2 n − 1 + 2 1 + 4k cA0 τi · s

(3.34)

In other cases a solution has to be found by an algebraic stepwise method or by a graphical technique [9], [20]. Example. The second-order, reversible reaction 2 A
C + D is carried out in a single CISTR with total volume of 2.665 m3 . The volumetric flow rate of reactant A is 7.866× 10−4 m3 /s, and its inlet and outlet concentrations are 24.03 and 6.94 kmol/m3 , respectively. The forward rate constant at the reaction temperature is 1.156 × 10−4 m3 kmol−1 s−1 and the equilibrium constant K is 16.0. It now becomes necessary to replace the working reactor with a series of identical CISTRs. The volume of each reactor in a new reactor arrangement should be one-tenth of the current one. How many CISTRs are required?

506

Model Reactors and Their Design Equations

Figure 11. Continuous stirred-tank reactors in series (cascade)

Solution. The reaction rate can be expressed as follows:  RA =k1 c2A −

2

cA0 −cA 4K

The relation between RA and cA described by the above equation is shown in Figure 12. In addition, the balance equation RA =

 Φv  cAn − cA(n−1) Vr

is also valid for each reactor in a series. The slope Φv /V r in the case of a new arrangement is 2.95 × 10−3 s−1 . A straight line through cA0 = 24.03 kmol/m3 with this slope can be drawn to the intersection with the reaction rate curve. This process has to be repeated until cAn = 6.94 kmol/mol is reached. From Figure 12, the number of stages required is 3.8. Hence four CSTRs should be used in this case.

Figure 12. Graphical estimation of required number of CSTRs in series

Figure 13. Different configurations for packed-bed reactors A) Multitube heat exchanger; B) Multistage adiabatic reactor According to “Chemical Reactor Analysis and Design” G. F. Froment, K. B. Bischoff, John Wiley & Sons, New York 1990.

Model Reactors and Their Design Equations

507

4. Packed-Bed Reactors The packed-bed reactor constitutes the keystone in the production of several important chemicals. Several reactor configurations are encountered in practice (e.g., Fig. 13). The heat-exchanger type of reactor (Fig. 13 A) is used to carry out highly exothermic reactions. The adiabatic multistage reactor (Fig. 13 B) with cooling between stages is used to carry out exothermic equilibrium reactions. The autothermal configuration, an interesting alternative, is a heat-exchanger type of reactor in which the reactants circulate alternatively through shell and tube arrangements with the double purpose of being preheated and keeping the reactor operating within a prespecified thermal regime by cooling it down. The analysis of transport phenomena for systems with chemical reaction is usually based on the transport equations resulting from the differential balance laws. To predict global effects, detailed information about the velocity profiles, and temperature and concentration fields is required. This information is extracted from the solution of the associated transport equations, subject to pertinent boundary conditions. When flow through a complex structure such as a porous medium is involved, these governing equations are valid even inside the pores. However, the geometric complexity of a randomly ordered particulate system prevents any general solution of detailed concentration, temperature, and flow fields. Instead, some form of macroscopic balances based on the average over a small volumetric element must be employed. A common practice is, with the help of some empirical relations, to replace the microscopic momentum, mass, and energy balances with corresponding macroscopic equations. Due to the distributed nature of the system, a dynamic model contains two or more independent variables (time and spatial coordinates). Simultaneously, it is important in principle to include all relevant transport resistance (see Fig. 14) to avoid unrealistic results. Since a very complex model may be too difficult to solve, a number of simplifying assumptions are often adopted in practice. However, care must be taken when choosing the adequate degree of complexity for the reactor model, so as to guarantee sufficient accuracy for a given practical application.

Figure 14. Energy transport mechanisms in packed beds

4.1. Mass and Energy Balances Mathematical models for adiabatic and nonadiabatic packed-bed reactors have been systematically listed in reviews by Froment [22], Hlavacek [23], and Hofmann [24]. In these reviews an attempt is made to classify the models according to their complexity. Basically, the mathematical modeling of fixed-bed reactors can be performed either through the finite-stage or the continuum approach [25]. The continuum approach leads to a system of partial differential equations in which mixing effects are accounted for in the form of mass diffusion and heat conduction terms. This approach is being increasingly favored, mainly due to the continuous progress in computer software and numerical modeling. In general, for the mathematical treatment of the heat and mass transfer processes in packed beds, with or without chemical reaction, three types of continuous models have been developed:

508

Model Reactors and Their Design Equations

1) The single-particle model, where intra particle phenomena are accounted for separately 2) The two-phase (heterogeneous) model in which both phases, solid and fluid, exchange heat and/or mass, and such transport processes are accounted for 3) The one-phase (pseudohomogeneous) model in which the reactor is approximated as a quasi- continuum model

4.1.1. The Single-Particle Model Basically the single-particle model is also a heterogeneous model. However, it differs from the two-phase model in that the solid phase is not regarded as a continuum since the concentration and temperature distribution inside the single pellet are accounted for. Consider a catalyst pellet positioned in a flowing stream (Fig. 15). Three different regions can be distinguished: the bulk fluid phase, the boundary layer surrounding the particle, and the particle itself.

Figure 15. Schematic of a single particle in a flowing stream

Bulk Fluid Phase. The bulk fluid phase can be modeled as a constant temperature and constant concentration field. Perturbations at the inlet of the reactor propagate through the fluid and the conditions at the outer edge of the boundary layer change accordingly; this dynamic evolution of the boundary condition can be solved by using a heterogeneous model (see Section 4.1.2). Boundary Layer. Temperature and concentration conditions in the boundary layer are frequently assumed to be identical to those of the bulk fluid phase. This is not always the case and,

particularly if it is desired to trace the dynamic behavior of the concentration and temperature fields inside the catalyst pellets, the temperature and concentration in the boundary layer must be obtained by solving the corresponding mass and energy balances. The first step in solving these fields is to determine the extent of the boundary layer. This can be accomplished by performing overall energy and mass balances across the boundary layer. Once the thickness of the boundary layer has been found (e.g., ∆r), the governing equations and their boundary conditions will be defined by: Mass balance: ∂Cf =Df ∇2 Cf ∂t

(4.1)

Energy balance: g Cp, f

∂Tf =λf ∇2 Tf ∂t

(4.2)

subject to:

where η is the effectiveness factor, the subscript f denotes fluid, and R (C s , T s ) = k ∗0 C s exp (− E a /RT ) stands for the equation for the reaction rate [26]. Catalyst Pellet. Similar to the analysis of the boundary layer, the pellet itself is modeled in terms of mass and energy transport. The pellet can be either porous or nonporous; in the latter case there is no mass diffusion of species into the pellet and only an energy balance is necessary. For certain classes of reactions, the assumption of nonporous catalyst is justifiable, e.g., for reactions whose high exothermicity prevents the reaction front from penetrating deep into the pellet even if it were porous. Under these assumptions the energy balance for the catalyst pellet gives:

Model Reactors and Their Design Equations

∂Ts =λs ∇2 Ts s Cp, s ∂t

(4.5)

subject to: ∂Ts = 0 at r =0 ∂τ ∂Tf  ∂Ts  ks − λf   ∂r r→R− ∂r r→R+ =η (− ∆Hr ) R (Cs ,Ts ) at r = R

(4.6)

4.1.2. The Two-Phase Model Of the two remaining models, the two-phase model is not only more realistic but it also constitutes the basis on which the derivation of the pseudohomogeneous approximation relies. The most important feature of this model is that temperature and concentration differences are considered between solid and fluid phases. The mass and energy transport processes are governed by the following equations: Fluid phase: ε

∂ Ci, f

= − ε ∇ · uCi, f +ε ∇ · Df ·∇Ci, f ∂t

(4.7) +av λf, s Ci, s − Ci, f

∂Tf = − εf Cp, f u ·∇Tf +ε ∇ · λf ·∇Tf ∂t +av hf, s (Ts −Tf ) (4.8)

εf Cp, f

Solid phase: (1 − ε) εp

∂ Ci,s

= ∂t

− η (1 −ε) s R C1,s , . . . ,Ci,s , . . . ,CS,s ,Ts

+av λf,s Ci,f −Ci,s

(4.9)

∂Ts = ∂t

η (1 −ε) s (−∆Hr ) R C1,s , . . . ,Ci,s , . . . ,CS,s , Ts (1 − ε) s Cp,s

+ (1 − ε) ∇ · λs ·∇Ts +av hf,s (Tf −Ts )

(4.10)

4.1.3. The One-Phase Model By introduction of the effective transport concept heterogeneous fluid – solid systems can be treated as quasi- continuum media [27]. In the

509

pseudohomogeneous model, additional transfer mechanisms are superimposed on the contributions that originate from the global movement of the fluid. These additional mechanisms are based upon the observation that the fluid traveling between two points of the packed bed undergoes a tortuous journey that is composed of a large number of random steps. It is assumed that the transfer mechanisms resulting from this process can be treated as diffusion- and conductionlike phenomena. The fact that the modeled system is actually heterogeneous is accounted for by defining transport coefficients where both contributions due to the solid and fluid phases are considered. Although the two-phase models fulfill the obvious physical requirements of considering a system where two phases exchange mass and/or energy, they also contain a number of approximations. For instance, separation of thermal dispersion effects, which in fact are interconnected, is very difficult. An even more difficult problem arises when heat losses must be accounted for; in this case the wall heat transfer coefficient has to be resolved into its solid- and fluid-phase components. Ramkrishna and Arce discussed the validity of the pseudohomogeneous model to represent the heterogeneous nature of packed-beds [28], [29]. The simplest, most obvious pseudohomogeneous model was derived by Yagi et al. [30]. These authors made use of the fact that, at low Reynolds number, temperature differences between the solid and fluid phases are negligible and both phases can be assumed to have the same temperature. This approach has been generalized, and the one-phase model is adopted, even for high Reynolds number, based on the assumption of equal temperatures in both phases (e.g., [31]). The idea of equivalence was extended by Vortmeyer et al. [32], [33] for arbitrary Reynolds numbers. These authors combined the energy balances for the fluid and solid phases (Eqs. 4.8 and 4.10) by assuming that the driving forces for heat conduction were similar in both phases. The main difference to the approach followed by Yagi and coworkers is that Vortmeyer and coworkers derived a dispersion term which reflects the exchange of energy between both phases; this term is added to the thermal

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Model Reactors and Their Design Equations

conductivity of the solid phase to form a quantity which is known as the effective thermal conductivity. An additional assumption concerns the mass transport between the solid and fluid phases. The resistance to mass transfer of the film adjacent to the solid particles can often be considered negligible; thus, the surface reactant concentration can be considered similar to that in the bulk of the fluid phase. Under these assumptions and by combining both energy balances (Eqs. 4.8 and 4.10), the transport processes in the packed-bed reactor can be considered as governed by the following set of equations: Mass balance: ε

∂ (Ci ) = −ε∇ · (uCi ) +ε∇ · D e ·∇Ci ∂t −η (1 −ε) s R (C1 , . . . ,Ci , . . . ,CS ,T )

(4.11)

Energy balance: ∂T (4.12) = − ε f Cp,f u ·∇T +∇ · λe ·∇T ∂t +η (1 −e) s (−∆Hr ) R (C1 , . . . ,Ci , . . . ,CS ,T )

 Cp

where: Cp =εf Cp, f + (1−ε) s Cp,s

An alternative two-phase model was proposed by Carberry and White [34]. This model combines the pseudohomogeneous and heterogeneous models; it is based on the catalyst particle but uses effective transport properties. As a hybrid between the two previous models it has been termed the “pseudoheterogeneous” model. By considering the solid phase to be discontinuous, this model appears to perform better when matching models. A thorough analysis of this model is given in [35]. 4.1.4. Boundary Conditions Since the classical works by Hulburt [36], Danckwerts [37], and Wehner and Wilhelm [38] appeared, boundary conditions in packedbed reactors have received much attention. The formulation of the boundary conditions proposed by Danckwerts is the most often followed. This formulation meets the limiting mixing conditions of the continuous stirred-tank reactor (CSTR) and the plug-flow reactor (PFR).

This was proven for isothermal [39] and nonisothermal [40] systems. These boundary conditions have also been shown to be adequate for systems with a multiplicity of steady states [41]. They also predict the behavior of nonadiabatic systems satisfactorily [42] even though the number of possible steady states in these systems could change if the Wehner and Wilhelm approach is followed [43]. Even though the formulation of the boundary conditions as suggested by Wehner and Wilhelm [38] appears more logical from a physical point of view, their correspondence with Danckwerts’ suggestion for steady-state operation of adiabatic systems has been proven [44]. Modifications have been proposed when radial dispersion is also taken into account [45], and the boundary conditions have also been modified to treat the transient situation [46], [47]. For selection of the proper boundary and initial conditions for flow systems see [48][49][50]. A simple pseudohomogeneous model without radial dispersion is now considered. The reactor is visualized as a tube with three distinguishable sections (see Fig. 16): the reaction zone packed with the catalyst (0 ≤ z ≤ L ), and a fore (z < 0) and aft (z > L ) sections of inert packing. A general formulation of the inlet and exit boundary conditions can be written as: Inlet (z = 0):  De,z

∂Ci − uz Ci ∂z

 0−

  ∂Ci = De,z − uz Ci ∂z 0+ (4.13)

 λe,z  λe,z

∂T − uz f Cp, f T ∂z ∂T − uz f Cp, f T ∂z

 

0−

= (4.14)

0+

Exit (z = L ): 

   ∂Ci ∂Ci = De,z ∂z L− ∂z L+     ∂T ∂T λe,z = λe,z ∂z L− ∂z L+ De,z

(4.15) (4.16)

where a given approach is followed by selecting the diffusivities/conductivities as indicated in Table 4 (“yes” means parameter is included in the model, “no” means not included).

Previous Page Model Reactors and Their Design Equations Table 4. Dispersion coefficients for different boundary conditions Parameter

Hulburt

Danckwerts

Wehner and Wilhelm

D/λe, z | 0− D/λe, z | 0+ D/λe, z | L − D/λe, z | L +

no no yes no

no yes yes no

yes yes yes yes

The conditions for the mass balance at the lateral walls are such that it is assumed that no mass flux occurs because the lateral walls are often impermeable (radial reactors are an exception). The boundary conditions for the energy balance, however, vary depending on whether the reactor is operated adiabatically or nonadiabatically. For a pseudohomogeneous model with radial dispersion these boundary conditions are: Reactor wall (r=R): ∂Ci = 0 ∂r

λe,r

∂T =hw (Tw −T ) ∂r

511

nonuniform distribution of the porosity causes the flow to be nonuniformly distributed, particularly near the reactor walls [58], [59]. The inclusion of radial porosity and velocity profiles improves the agreement between experimental and theoretical results [60], [61]. It has been found [62], however, that the predictions obtained by assuming constant average porosity and mainflow velocity for the entire packed bed will be qualitatively correct in most cases.

(4.17)

(4.18)

Reactor centerline (r=0): ∂Ci = 0 ∂r

(4.19)

∂T = 0 ∂r

(4.20)

4.2. Values of the Parameters Various steps have been taken towards the derivation of an integral model for the packedbed reactor by considering the individual steps involved in the transport of a given property in a packed bed. In this section only few correlations for typical applications will be mentioned, for more details, see [25], [51–54]. 4.2.1. Average Bed Porosity The existence of oscillatory radial variations of the void fraction in packed beds has been recognized since the early work of Roblee et al. [55] and Benenati and Brosilow [56], [57]. This

Figure 16. Schematic of the flow system in a packed-bed reactor

In order to use a constant average porosity, the following integral relation must be fulfilled: ε=

1 πR2


R ε (r) r dr 0

There is no unique functional relation between the radius and the porosity, but many functionalities have been proposed in the literature. To find a realistic value for the average porosity, the approach suggested by Martin can be followed [63]. The bed is considered to be formed by two concentric rings with their own properties; a central core with an average porosity ε0 and a one-particle-diameter-thick concentric shell with porosity εw . The average porosity, as defined above, must satisfy the relation:

512

Model Reactors and Their Design Equations A more heuristic approach is to assume a constant value for the axial and radial Bodenstein (or mass P´eclet) numbers, i.e.,

ε0 A0 +εw Aw =εA

Then, from geometric considerations, the following equation results: ε =εw − (εw −ε0 ) (1−d/D)2

where the porosity for the outer shell (εw ) is estimated as [63]: εw = 1− (1−εmin )

2 3



D/d−7/8 D/d−1/2



where εmin is the minimum value observed for the void fraction (εmin = 0.23 and ε0 = 0.39 are taken from [56]). 4.2.2. Effective Transport Coefficients Since the effective transport coefficients are mainly determined by the flow characteristics, in general, packed beds are anisotropic for effective transport, so that the radial transport component is different from its analogue in the axial direction. For the sake of simplicity, effective transport coefficients are often considered to be unaffected by secondary flow and dependent on the mainflow (forced) only. Mass Transport. A correlation for the mass transport in both spatial directions is [64]:   9 : C1 0.7    + De(r,z) = u z d × C2 Re Sc 1 + Re Sc

where the constants C 1 and C 2 depend on the geometry of the packing and the fluid flowing through the packed bed as listed in Table 5. Table 5. Constants for the effective mass-transport coefficients [64] Constant

C1 (longitudinal) C 1 (radial) C2 (longitudinal) C 2 (radial)

Liquids

Gases

Spherical Irregular packing packing

Spherical packing

Irregular packing

2.5

2.5

0.7

4

0.08 8.8

0.08 7.7

0.12 5.8

0.12 5.1

78 ± 20

P em,z/r =

9 : u zd De,z/r

= constant

where z/r denotes that the expression is valid for both axial and radial transport. Experimental evidence has shown that axial dispersion in packed beds is typically characterized by Pem, z ≈ 2. In other words, CSTR conditions (small Pem, z ) are reached in the void within each void cell, and the packed-bed reactor can be viewed as (L /d) CSTRs in series [65]. Therefore, axial mass diffusion can be neglected for ratios L /d > 50. This criterion applies to systems with particle Reynolds number larger than unity (i.e., Rep = f uz d/µf > 1). Radial mass dispersion is characterized by a constant radial mass P´eclet number: P em,r =

9 : u zd De,r

≈ 8 − 10

Energy Transport. Energy transport is supplemented by radiation and conduction in the solid phase. Therefore, it is more difficult to define constant values for the heat P´eclet numbers. The problem of equivalence between the heterogeneous and pseudohomogeneous models has been addressed in [51], [66]. In general the effective thermal conductivity λe(r , z ) is formed by two contributions: the quiescent bed conductivity λ0s and a contribution due to the flow, i.e., λe(r,z) =λ0s +Cλf ReP r

where the conductivity for the quiescent bed λ0s is obtained from the correlation proposed by ¨ Zehner and Schlunder [67], while the contribution due to the flow can be found from the analysis by Vortmeyer and Berninger [66]. The constant in the above equation is chosen as C = 0.1 for the radial (r) direction and C = 0.1 – 0.2 for the axial (z) direction. The equivalence between the heterogeneous and pseudohomogeneous models has been proved for nonreactive conditions [51]. For further information on energy transport in packed beds see [51], [52], [68–71].

Model Reactors and Their Design Equations Effect of Radiation. Transport by radiation is frequently ignored when estimating heat transfer coefficients. This transport mechanism depends on the temperature range for the reactor operation; for a temperature of 600 K at the reactor wall, radiation has been found to contribute up to 20 % to the overall radial heat flux [72]. The contribution of radiation to effective energy transport in packed beds has been analyzed [72], [73]. In the model presented the heat transfer is considered to be controlled by two parallel mechanisms and the overall effective thermal conductivity is assumed as λe,r =λ0e,r +λr a de,r

where the contribution due to radiation λr ad e, r is λr ade,r = 4ψσ dT 3   8 σ dT 3 =Cr T 3 = 2/e−0.264

1 ∂ r ∂r

 r λ0e,r

∂T ∂r

the radial temperature profiles predicted for the solid and fluid phases. Based on the physical restriction that both profiles can never cross, most of them were discarded and only a few could be considered reliable [53]. The most often accepted correlation has been presented by Dixon et al. [74]. A thorough analysis of the estimation of this coefficient has been presented [75]. The recommended correlation has the form: αf, w d N uf, w = kf / . = 1 − 1.5 (D/d)−1.5 P r1/3 Re0.6

(4.21)

where αf, w is the fluid-to-wall heat transfer coefficient.

4.3. Further Simplifications

where σ is the Stefan – Boltzmann constant (5.67 × 10−8 J m−2 K−4 s−1 ), ψ is a shape factor, and e is the particle emissivity. The treatment of this contribution is carried out by adding an additional term to the conduction term in the energy balance, i.e., ∇ ·λe · ∇T =

513

 +

 ∂T +... ∂r     1 ∂ ∂T 2 ∂T =λe,r r + 3Cr T 2 +... r ∂r ∂r ∂r 

r Cr T 3

It is apparent that this contribution can become important with increasing particle emissivity and its omission would lead to conservative temperature predictions. 4.2.3. Wall Heat-Transfer Coefficient The wall heat-transfer coefficient can be estimated from many correlations. The correlations available, however, predict heat transfer coefficients which are largely scattered. For instance, for a particle Reynolds number Re = 100, the heat transfer coefficient can differ by almost one order of magnitude [53]. Odendaal et al. analyzed the available correlations and compared

Despite its simplicity, the pseudohomogeneous model has proven to be a very reliable representation for transport processes in packed beds [66]. If care is taken when evaluating the transport coefficients, predictions obtained from a one-phase model will not differ significantly from those yielded by a two-phase representation. Every model has its deficiencies, thus, the heterogeneous model cannot capture the radial temperature and concentration gradients present in the pellets under reactive conditions, and the pseudohomogeneous model lacks an adequate representation of the solid phase. Windes et al. [54] have suggested an approach to achieve equivalence between the pseudohomogeneous and heterogeneous models. The questions concerning equivalence stem from the fact that packed beds behave differently under reactive and nonreactive conditions. This was initially interpreted as a change in the heat transport coefficients under reactive conditions [76]. Wijngaarden and Westerterp [77] have addressed the subject in detail and provided a comprehensive explanation for the difference between reactive and nonreactive situations. For the mixing coefficients independence of chemical reaction was demonstrated by Gunn and Vortmeyer [78].

514

Model Reactors and Their Design Equations

One-Dimensional Models versus TwoDimensional Models Although radial temperature profiles can be minimized in the design of the reactor, it is not necessary to suppress radial dependence of temperature to obtain reliable predictions with one-dimensional models. In general, the sophistication of the model should never exceed that of the experimental data. For this reason, one-dimensional models have become increasingly useful. Beek and Singer [79] addressed this point by assuming a weak parabolic radial dependence of the temperature profiles on the radial dimension. Then it can be shown that predictions from a one-dimensional model equal the average radial temperature T  for a two-dimensional model [25]. The radial temperature distribution is then obtained as [25]: T (r/R) ≈ 

9 : 1− (r/R)2 + (2/Bi ) Tw + 2 T − Tw 1 + (4/Bi )

The equivalence between the one-dimensional model and the radial average temperature for a two-dimensional model is obtained by a proper computation of the overall heat-transfer coefficient. For a parabolic temperature profile: 1 1 D = + U hw 8ke,r

A more refined approach was followed by Crider and Foss [80], who solved the heat transfer problem analytically under nonreactive conditions. The equivalence of one/twodimensional models was achieved by combining the radial heat-transfer parameters as follows: 1 D 1 + = U hw 6.133λe,r

This was later confirmed by Hlavacek et al. [81] who solved the linearized problem and found identical results. An important theoretical conclusion can be drawn from this last observation; as long as only small radial temperature gradients exist, the chemical reaction only seems to have a significant effect on the axial temperature dependence. Under these circumstances, the radial temperature dependence can be described via a radially lumped model.

Example. The above analysis is illustrated by solving for the steady-state temperature and concentration profiles in a typical industrial reactor with the partial oxidation of o-xylene to phthalic anhydride as an example. This highly exothermic process is carried out in heat-exchangertype reactors. The reactor consists of a bundle of tubes with a cooling fluid circulating around the tube arrangement. The process (of German origin, but used worldwide) employs V2 O5 on silica gel pellets as catalyst. The geometric description and kinetic information have been taken from [82]. If the pseudohomogeneous, two-dimensional model is lumped radially, i.e., if the equations are integrated across the cross-sectional area, the following steady-state, one-dimensional model results: ∂ 2 CX ∂ (uz CX ) +De,z ∂z ∂z 2 9 : − (1−ε) s R CX , T

(4.22)

9 : ∂ f Cp, f uz T ∂z 9 :

∂2 T 4U 9 : − T −Tw +λe,z 2 ∂z D 9 : + (−∆Hr ) (1−ε) s R CX , T

(4.23)

0 = −

0 = −

where R (C X , T ) = k 0 pO pX exp (−E a /Ru T ) represents the rate of reaction, and pX and pO represent the partial pressures of oxylene and oxygen, respectively. The reaction is performed in excess oxygen (i.e., pO /ptot = 0.21 ≈ constant). Although the reaction mechanism for this process is completed by parallel and consecutive reactions, only the main reaction is addressed here. The results obtained with the onedimensional (Eqs. 4.22 and 4.23) and twodimensional models are presented in Figure 17 for two different operating conditions. The onedimensional model predicts reactor behavior satisfactorily for the conditions of Figure 17 A (Tw = 630 K). For the conditions of Figure 17 B (T w = 645 K), however, the differences between this model and the two-dimensional models are significant. This can be better understood by analyzing the results presented in Figure 18 where the radial temperature profile is shown as a function of radial position r/R at different

Model Reactors and Their Design Equations axial locations. The high-temperature situation (Fig. 18 B) shows the nonlinear effect of the reaction rate and the temperature profile deviates markedly from a parabolic dependence(as shown Fig. 18 A). These results show that, for mild radial dependence of the temperature distribution, one-dimensional models can be used as a reliable design tool. LIVE GRAPH

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LIVE GRAPH

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Figure 17. Comparison between one- and two-dimensional models A) Tw = 630 K, x x, 0 = 0.00924; B) Tw = 645 K, x x, 0 = 0.00924 a) Two-dimensional model; b) Crider and Foss model (onedimensional; c) Beek and Singer model (one-dimensional) Open circles in Figures 17 A and 17 B denote axial locations for curves shown in Figures 18 A and 18 B, respectively.

4.4. Parametric Sensitivity Since the 1960s, great improvements have been made in the design and operation of catalytic reactors. In industrial conditions, however, due to the high exothermicity of the chemical reac-

515

tion, the temperature rises sharply in the catalytic bed towards a maximum or hot spot (usually located near the reactor inlet). These sharp axial temperature gradients can cause poor reaction selectivity and extreme temperatures can be responsible for rapid deactivation or even deterioration of the catalyst. Therefore, the hot spot must be kept within permissible bounds. Furthermore, the attainable conversion frequently has an upper safety limit at levels where the temperature profile becomes extremely sensitive to changes in operational and/or physicochemical parameters. Bilous and Amundson [83] termed such a reactor condition as “parametric sensitivity”. Many studies have been reported on the prediction of this phenomenon, which in turn can lead the reactor to runaway operation [23], [84]. The danger inherent in running such systems is widely recognized, and the final reactor design must guarantee a safe mode of operation. Most studies on parametric sensitivity have assumed constant temperature for the cooling medium [85–91]. This is a suitable approach for perfectly mixed coolant, or reactors employing a boiling liquid or a fluid of abnormally large heat capacity as cooling medium. However, in the more common case of molten salt circulation, the thermal gradients in the shell side cannot be neglected in the reactor model. Few papers have studied the nonisothermal situation. For more information; see [92–106]. Example. Consider again, the production of phthalic anhydride via partial oxidation of oxylene in a multitube heat-exchanger-type reactor. The steady-state operation will be described by the one-dimensional pseudohomogeneous model (Eqs. 4.22 and 4.23). Parametric sensitivity can be estimated as originally proposed by Bilous and Amundson [83]. It is computed as the derivative of the variable to be analyzed with respect to one of the physicochemical parameters of the process. Thus, the parametric sensitivity (Sij ) of the variable χi with respect to the parameter πj is defined as: Sij =

∂χi ∂πj

In what follows the effect of the reactor wall temperature on the reactor thermal condition is analyzed, i.e., π = Tw and χ = T . This is a typical problem found in industrial practice where

516

Model Reactors and Their Design Equations LIVE GRAPH

LIVE GRAPH

Click here to view

Click here to view

Figure 18. Radial temperature profiles A) Tw = 630 K, x x, 0 = 0.00924; B) T w = 645 K, x x, 0 = 0.00924

the thermal state of the reactor is monitored (i.e., the observed variable would be the readouts of the reactor thermocouples) and the operator acts on the cooling fluid (i.e., the manipulated variable would be the temperature or coolant flow rate) to keep the reactor operating near the desired state. In order to obtain numerical values for the parametric sensitivity of the reactant concentration (SC ) and the bed temperature (ST ), the governing equations are differentiated with respect to the wall temperature, i.e., ∂ 2 SC ∂ (uz SC ) +De,z ∂z ∂z 2 ∂R − (1−ε) s ∂Tw 0 = −

(4.24)

∂ f Cp,f uz ST ∂z ∂ 2 ST 4U (ST −1) − +ke,z ∂z 2 D ∂R + (−∆Hr ) (1−ε) s ∂Tw

0 = −

(4.25)

where:   ∂R Ea =k0 pO SC +pX 9 :2 ST ∂Tω Ru T 9 : exp −Ea /Ru T

If Equations (4.24) and (4.25) are integrated simultaneously with the reactor model (i.e., Eqs. 4.22 and 4.23), the sensitivity of the reactant conversion and bed temperature to changes in

Model Reactors and Their Design Equations the coolant temperature is obtained at any point of the reactor length. Of these two sensitivity profiles, only ST (z) guarantees a safe mode of operation. The four additional boundary conditions needed are derived from the original boundary conditions, i.e., Inlet, z = 0: 0 = uz SC −De,z

∂SC ∂z

(4.26)

f Cp, f uz =f Cp, f uz ST −ke,z

∂ST ∂z

(4.27)

Exit, z = L: 0 =

∂SC ∂ST = ∂z ∂z

(4.28)

LIVE GRAPH

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517

parametric sensitivity; a 1 K rise in the inlet temperature is sufficient to drive the reactor away from a safe operation regime. Figure 19 A, on the other hand, only shows a small difference in the reactor thermal state after a change of 5 K in the inlet temperature. Indeed the 5 K temperature increase is transmitted almost linearly along the reactor. These two situations can be better understood by inspecting the normalized temperature sensitivity associated with the lower inlet temperature operating condition. In Figure 19 A the temperature sensitivity remains almost constant along the reactor, i.e., the temperature profile should experience an almost constant increase for a temperature rise at the inlet of the reactor. In Figure 19 B, on the other hand, the temperature is predicted to grow dramatically in the vicinity of the hot spot. Furthermore, the sensitivity crosses to negative values after the hot spot axial location; this suggests a temperature overshoot promoting a high degree of conversion and mostly cooling afterwards. Indeed, the temperature profiles corresponding to the higher inlet temperature confirm the predictions extracted from analyzing the temperature sensitivity.

4.5. Flow Field Description

LIVE GRAPH

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Usually the flow distribution in unbounded porous media is assumed to be well represented by a linear relation between the pressure drop and the fluid velocity ( Darcy’s law). When using such an approximation, however, two main points are disregarded: 1) Effect of the boundaries on the flow field 2) Increasing importance of the inertial effects as the flow speed increases If the fluid obeys the Boussinesq approximation, the flow field will be governed by [110]: ∂u µf = −∇p − u ∂t κ   1 −f β1 (T − T0 ) +β2 (C0 −C) g C0

f Figure 19. Parametric sensitivity in packed-bed reactors for different inlet temperatures and x x, 0 = 0.019 A) T w = 620 K and 615 K; B) T w = 626 K and 625 K

In Figure 19 the axial temperature profiles for the previous example are shown for two different inlet temperatures and the normalized temperature sensitivity associated with the lower inlet temperature. Figure 19 B illustrates the concept of

∇·u= 0

(4.29)

(4.30)

which is known as the Darcy – Oberbeck – Boussinesq model for flow through a porous medium [111].

518

Model Reactors and Their Design Equations

As the flow speed increases, the inertial terms become important and the results of Darcy’s model are inefficient in describing the momentum transfer in packed beds. The flow through porous media is now said to be described by a “modified” Darcy’s law:  µf ∂u 1 + u · ∇u = −∇p − u ∂t e κ   1 (C0 −C) g −f β1 (T −T0 ) +β2 C0 

u = [0,uz ]

(4.34)

Inlet (z = 0): u = [0,uz (r)]

(4.35)

Exit (z = L): u = [0,uz (r)]

(4.36)

f

(4.31)

where the inertial forces are represented by the term u · ∇u. Even though this term arises from a formal volume averaging in the point field equations [112], its inclusion may lead to inconsistencies between boundary conditions and governing equations [113]. Forchheimer [114] first proposed addition of higher order inertial terms to the relation between pressure drop and fluid velocity and modified Darcy’s equation to: dp +a1 uz +a2 (uz )2 = 0 dz

Later he added a third-order term to fit experimental data. Forchheimer’s modification for the one-dimensional flow can be formalized for two and three dimensions as [115], [116]: ∇p − f g +

µf f u + bu |u| = 0 κ κ

f

(4.32)

Only two boundary conditions are necessary to solve the momentum equations. The classical nonslip boundary conditions are imposed on the solid walls and a given velocity profile is imposed at the entrance and exit of the reactor, i.e., Reactor wall (r = R):

Reactor centerline (r = 0):

∆p 150 (1−ε)2 1.75 (1−ε) 2 µuz + = uz H ε3 d2 ε3 d

However, for a certain range of fluid velocities, a linear dependence of the pressure drop on the fluid velocity (namely, the Carman – Kozeny term) provides a reliable representation description. Thus, as Darcy’s law proposes a linear dependence between pressure drop and fluid velocity, the above equation can be used to estimate the packing permeability κ and the Forchheimer correction term b as:

b=

d2 ε3 150 (1−ε)2

1.75d 150 (1−ε)

4.5.3. Thermal-Expansion Coefficient

4.5.1. Boundary Conditions

u = [0, 0]

The pressure drop in a porous medium can be approximated by the Ergun correlation, a linear combination of the Carman – Kozeny and Burke – Plummer equations [117]:

κ=

where b can be considered a structure property associated with inertia effects in the porous matrix. Then the momentum equation becomes: ∂u µf f = −∇p − u − bu |u| ∂t κ κ   1 −f β1 (T − T0 ) +β2 (C0 −C) g C0

4.5.2. Permeability and Inertia Coefficient

(4.33)

The thermal-expansion coefficients are tabulated for most of the fluids of interest. When not available, a satisfactory estimation can be obtained for gaseous fluids by resorting to the ideal gas law. For instance, the expansion coefficient for temperature variations can be approximated as: =

pM RT

 ∂ = T ∂T

then: β1 ≈

1 ≈ 10−3 T0

Model Reactors and Their Design Equations 4.5.4. Mass-Expansion Coefficient The expansion coefficient for concentration changes is not as readily obtained as its temperature analogue. A satisfactory estimation can still be obtained by assuming ideal gas behavior. For instance, for a binary mixture A + B where the reaction A → B takes place, the fluid density can be written as:   =0

1 1 +cCA y



where c stands for a combination of the initial and final molar masses: c=

Mi −Mf Mf

If c < 0, then a lower molar mass product is formed in the chemical reaction (for instance, a decomposition reaction). In contrast c > 0 would indicate the formation of a product with a higher molar mass (for instance, a polymerization reaction). If a perturbation is considered in a situation where no conversion has been achieved (i.e., ξ → 0), the gas density can be represented in a linear approximation as:  =0 (1−β2 CA ξ)

Taking into account that c < 1 and ξ < 1, the dependence of the density on the conversion can be written as:  ≈0 (1−cCA ξ)

then: β2 ≈ c

In the case of a reaction in the liquid phase, the density must be determined from an appropriate equation of state, and the expansion coefficients should be found experimentally.

4.6. Thermomechanical Effects in the Reaction System Several phenomena can be observed when analyzing the dynamic behavior of catalytic and noncatalytic packed-bed reactors, e.g., “parametric sensitivity”, “wrong-way” behavior, travelling reaction fronts, and ignition and extinction

519

phenomena. All these temperature excursions may disappear or be too small to drive the system out of control. However, the development of thermal stresses during these temperature excursions adds a new factor to the reactor design and operation. The mechanical behavior of the catalyst pellet can be analyzed following the basic theory of elastic behavior of solids (see [118]). For a general situation the equations can be written as: Energy balance: ∂T =ks ∇2 T ∂t ∂ (∇ · u) − (3λ + 2µ) αT0 ∂t

s Cp, s

(4.37)

Thermomechanical equations: ∂2 δ =µ∇2 δ + (λ +µ) ∇ (∇ · δ) ∂t2 − (3λ + 2µ) α∇T

s

(4.38)

subject to: δr = 0 at r = 0 σrr = 0 at r = R

(4.39)

where λ and µ are the Lam´e constants, α is the coefficient of linear expansion, T 0 the temperature of stressless condition, δ is the displacement vector, and σrr are the radial stresses. Thus, the energy balance has been augmented from its previous form (i.e., Eq. 4.5 in Section 4.1.1), by an additional term. This term represents what is known as thermoelastic dissipation. Nevertheless, the temperature fluctuations due to deformations are usually small and, for most situations, they can be neglected. A more systematic criterion is to neglect the thermoelastic dissipation term whenever the inequality below holds: (3λ + 2µ)2 α2 T0 CH2 >CH− >C< =CH2 =CH− =C< =C= ≡CH ≡C− Ring increments −CH2 − >CH− >C< =CH− =C< Halogen increments −F −Cl −Br −I Oxygen increments −OH (alcohol) −OH (phenol) −O− (nonring) −O− (ring) >C=O (nonring) >C=O (ring) O=CH− (aldehyde) −COOH (acid) −COO− (ester) =O (except as above) Nitrogen increments −NH2 >NH (nonring) >NH (ring) >N− (nonring) −N= (nonring) −N= (ring) −CN −NO2 Sulfur increments −SH −S− (nonring) −S− (ring)

ITc

I pc

0.0141 0.0189 0.0164 0.0067 0.0113 0.0129 0.0117 0.0026 0.0027 0.0020

−0.0012 0 0.0020 0.0043 −0.0028 −0.0006 0.0011 0.0028 −0.0008 0.0016

65 56 41 27 56 46 38 36 46 37

23.58 22.88 21.74 18.25 18.18 24.96 24.14 26.15 9.20 27.38

−5.10 11.27 12.64 46.43 −4.32 8.73 11.14 17.78 −11.18 64.32

0.0100 0.0122 0.0042 0.0082 0.0143

0.0025 0.0004 0.0061 0.0011 0.0008

48 38 27 41 32

27.15 21.78 21.32 26.73 31.01

7.75 19.88 60.15 8.13 37.02

0.0111 0.0105 0.0133 0.0068

−0.0057 −0.0049 0.0057 −0.0034

27 58 71 97

−0.03 38.13 66.86 93.84

−15.78 13.55 43.43 41.69

0.0741 0.0240 0.0168 0.0098 0.0380 0.0284 0.0379 0.0791 0.0481 0.0143

0.0112 0.0184 0.0015 0.0048 0.0031 0.0028 0.0030 0.0077 0.0005 0.0101

28 −25 18 13 62 55 82 89 82 36

92.88 76.34 22.42 31.22 76.75 94.97 72.24 169.09 81.10 −10.50

44.45 82.83 22.23 23.05 61.20 75.97 36.90 155.50 53.60 2.08

0.0243 0.0295 0.0130 0.0169 0.0255 0.0085 0.0496 0.0437

0.0109 0.0077 0.0114 0.0074 −0.0099 0.0076 −0.0101 0.0064

38 35 29 9

66.89 52.66 101.51 48.84

34 91 91

73.23 50.17 52.82 11.74 74.60 57.55 125.66 152.54

0.0031 0.0119 0.0019

0.0084 0.0049 0.0051

63 54 38

63.56 68.78 52.10

20.09 34.40 79.93

Comparison of the methods listed in Table 4 shows the considerable improvement of Lydersen’s method by Joback. Somewhat more accurate but more complicated than Joback’s method is the method of Ambrose [94], [95]. With Fedor’s method [96] only critical temperatures can be predicted. Normal boiling point data are not needed, but the results are less reliable than those obtained by the methods of Joback and Ambrose.

IV c

ITb

ITm

68.40 59.89 127.24

When the properties of molecules with strongly interacting groups are estimated, the application of classical group contribution methods leads to systematic errors. Moreover, these methods cannot distinguish between certain isomers, e.g., o-, m-, and p-derivatives of benzene. To avoid these shortcomings Constantinou and Gani [97] proposed a new incremental method which does not require any information other than the molecular structure of the com-

546

Estimation of Physical Properties

Table 4. Selected methods for the prediction of critical data of organic compounds Method

Applicability

Information required

Error

Lydersen

T c , pc , V c with increments for 41 functional groups; >Si< increments for T c and pc ; –B< increment for T c

T b for T c

T c : 1.5 %

M for pc

pc : 10 % V c: 4 % T c : 0.8 % pc : 6 % V c : 2.5 % T c : 0.7 %

1955 [90] Joback 1984 [91]

T c , pc , V c with increments for 40 functional groups

T b for T c

Ambrose

T c (52), pc (52), V c (35) with increments for functional groups (number of functional groups given in parentheses)

T b for T c

1978 [94] 1979 [95] Fedors 1982 [96] Constantinou, Gani

M for pc T c with increments for 48 functional groups

T b not required

pc : 5 % V c: 3 % T c: 5 %

T c , pc , V c with increments for 78 functional groups and for structural effects (e.g., proximity, isomerism, ring structure)

T b not required

T c : 0.9 % pc : 3 % V c: 2 %

1994 [97]

pound. To allow for secondary structural effects, so-called second-order groups are introduced in addition to the 78 first-order groups. Including the increments of second-order groups reduces the error in estimated critical data by a factor of about 1.5 on average. Comprehensive compilations of critical data have been published for organic compounds [98], [99] and inorganic compounds [100].

2.4. Other Characteristic Constants of Pure Compounds The Acentric Factor ω. In order to extend corresponding states methods to compounds and mixtures consisting of nonspherical and polar molecules, the Pitzer acentric factor ω was defined as a third parameter:  ω=−log lv

lv 

p pc

(2.16)

Tr =0.7

where p is the vapor pressure of the pure component. The Pitzer acentric factor is intended to give a measure of nonspherity or acentricity of a molecule’s potential force field. For an ideal spherical molecule it should be zero. Acentric factors for some compounds are given below [89]:

Compound Water Ammonia Hydrogen Phenol Aniline

ω 0.344 0.250 − 0.216 0.438 0.384

Knowing the critical temperature and pressure of a substance and its vapor pressure plv (i.e., with the Antoine coefficients A, B, and C, Eq. 4.5), the acentric factor can be calculated as follows: 

ω=log 



pc

 exp A −

B 0.7Tc +C

  −1

(2.17)

If only the normal boiling point T b is known the Equation of Edmister [101] gives a reasonable estimate (pc in atmospheres): ω=

−1

ω 0.008 0.445 0.212 0.212 0.556 0.587

Compound Methane Nonane Cyclohexane Benzene Methanol 1-Octanol

3 7



log pc Tc /Tb −1



−1

(2.18)

The average uncertainty of this approximation is 5 % [102]. Example. Calculation of the acentric factor for benzoic acid T c = 752 K; pc = 45.6 bar; T b = 523 K; ω exp = 0.620 [88]; Antoine constants [89] A = 10.5432, B = 4190.70, C = −125.2 Substituting into Equation (2.17) 



ω

cal

= log 

 exp 10.5432 −

= 0.617

pc 4190.7 (0.7×752)−125.2

 − 1

Estimation of Physical Properties Relative deviation = 0.6 %. With Equation (2.18) ω cal =

3 7



1 (752/523) −1



 log

45.6 1.013

 −1 = 0.618

Relative deviation = 0.3 %. An incremental method for the estimation of the acentric factor has been developed by Constantinou et al. [103]. It is based on the same structural groups as the procedure of Constantinou and Gani [97] for the prediction of critical properties. No experimental data are needed for this method. The average error is 3 %; however, for 31 of the 78 structural groups (e.g., nitrogen and sulfur groups), increments are not available. The Stiel Polar Factor χ. This factor relates the actual reduced vapor pressure to the value calculated by the Pitzer equation (Eq. 2.16) at the reduced temperature T r = 0.6:  χ=log

plv(exp) plv(Pitzer)

 (2.19) Tr =0.6

It is therefore also a measure of polarity and is related to the acentric factor by  χ=log

plv pc



547

principles and easy-to-handle methods. More detailed information is given in [104–106]. For n moles of an ideal gas consisting of rigid molecules of zero volume and zero interaction potential energy the following equation holds pV =RT

(3.1)

where V is the molar volume. The ideal gas law is a universal limiting law valid for all substances when the density  approaches 0. Equation (3.1) can usually be used with sufficient accuracy for gases at low pressure p CH2 =CH− 23.58 22.88 24.96 T b = 368.7 K Deviation: − 4.3 K

=CH2 18.18

q 29.0

−COO− 81.10

A new and more highly refined incremental method for predicting T b and T m has been developed by Constantinou and Gani [157] (see also Section 2.4) with average errors of 5.4 K for T b and 14 K for T m .

melting point Tm = 122.0+



ITm [K]

(4.19)

Increments IT b and IT m are given in Table 3.

4.1.2. Enthalpy of Vaporization Determination from Vapor Pressure Data. As has been shown in Section 1.3 the vaporiza-

Estimation of Physical Properties tion enthalpy of a pure liquid can be determined from vapor pressure data with the rigorous equation of Clausius and Clapeyron (Eq. 1.1). Its approximate version (Eq. 1.2) yields ∆H lv =RT 2

dlnplv dT

(4.20)

With the Antoine Equation (4.5) for the differential quotient d ln plv /d T the following relation is obtained: ∆H lv =RT

B

(4.21)

(T +C)2

Equation (4.21) can be readily used to calculate ∆H lv with the aid of tabulated Antoine constants [149]. For moderate to high pressures ( plv > 0.1 MPa) the simplifying assumptions leading to Equation (1.2) are not justified. This means that the molar volumes of the liquid and the vapor must be taken into account. In Equation (1.4) this has been done only for the vapor phase. A convenient way to consider both vapor and liquid molar volume is described by Haggenmacher [158]. He employs the difference in the compressibility factors of the vapor and the liquid:  p  v V −V l ∆z lv =z v −zl = RT

(4.22)

559

Other Methods. Vapor pressure data are often unavailable or of insufficient accuracy for the application of the Clausius – Clapeyron equation. Other estimation methods then have to be used for the estimation of the vaporization enthalpies. Two of these are described in Section 1.3 (Eqs. 1.5 and 1.6). Vetere [159] suggested the following set of correlations (∆H lv , J/mol; M, g/mol; T b , K) [147]: 1) Hydrocarbons ∆HTlv

b

RTb +

=7.0054 + 1.647log M . /−1.037 0.7806 Tb − (263M )0.581 M

(4.24)

2) Alcohols, organic acids, methylamine ∆HTlv

b

RTb

=9.7643 + 1.5748log 3.1018Tb 0.0176375Tb2 + M M 2.57131 × 10−5 Tb3

Tb − −

M

(4.25)

3) Esters

which can be approximated by the relation  lv

∆z =

plv 1− r3 Tr

∆HTlv

1/2

b

(4.23)

Use of Equations (4.22) and (4.23) allows determination of V v − V l . Introduction of this term into Equation (4.1) improves the calculation of the enthalpy of vaporization except in the region close to the critical point. Correction factors for acetone and trichloromethane are shown in Figure 7. LIVE GRAPH

Click here to view

RTb

=5.501 + 1.901log

0.04852Tb 0.53689 × 10−3 Tb2 + M M −6 0.6977 × 10 Tb3 (4.26) − M Tb +

4) Other polar compounds ∆HTlv

b

RTb

=5.3405 + 1.8453log

0.047109Tb 0.52125 × 10−3 Tb2 + M M 0.67738 × 10−6 Tb3 (4.27) − M Tb +

Corresponding states methods (e.g., Pitzer’s correlation) should also be mentioned. These methods are discussed in [160].

Figure 7. Influence of the Haggenmacher correction on the prediction of the heats of vaporization

560

Estimation of Physical Properties

Table 10. Selected methods for the prediction of the enthalpy of vaporization ∆ H lv of pure substances Method

Range of application Substances

Riedel 1954 [162] Chen 1965 [163] Vetere 1973 [159] Antoine equation, (Haggenmacher 1946 [158]) Wagner 1973 [164] Kobayashi 1984 [165] Watson 1943 [166] Lee – Kesler – Pitzer 1975 [150]

nonpolar, weakly polar nonpolar, weakly polar nonpolar, weakly polar

Information required ∗

Error, %

T = Tb T = Tb T = Tb 0.4 < T r < 0.8

T b , T c , pc T b , T c , pc plv (T ), T c , pc plv (T ), T c , pc , l

T vls < T < 0.95 T c T r < 0.98 0.5 < T r < 0.75 0.5 < T r < 0.98

plv (T ), (T c , pc ) T c, ω T c , ∆H lv ref T c , pc , ω

2 2 2 ∗∗ for T r > 0.5 with inclusion of Eq. (4.23) 1 ∗∗ 1.5 0.05 (T − T ref ) 1.5

Temperature

∗ In addition to the corresponding parameters and increments. ∗∗ Depends on the accuracy of vapor pressure data.

A summary of estimation methods for ∆H lv appears in Table 10. Figure 8 shows a comparison of different estimation methods. Experimental data can be found in [161]. LIVE GRAPH

Click here to view

Temperature Dependence. The temperature dependence of the vapor pressure can be represented satisfactorily using Watson’s equation [166]:  ∆HTlv = ∆HTlvb

1−Tr 1−Tbr

n (4.28)

A value of 0.38 is recommended for the exponent n, but if other values of ∆H lv are available this equation can be used efficiently for the correlation of the temperature dependence of ∆ H lv by adjusting the exponent n.

4.2. Mixtures Calculation of phase equilibria is based on the thermodynamic conditions for equilibrium, i.e., equality of temperature T , pressure p, and fugacities f i of each component i in the coexisting phases. This means that for the mathematical description of phase equilibria, expressions representing fugacities as a function of composition are required. Two types of formulations are used which differ in the state of reference. With the ideal gas as state of reference, the fugacity fi of component i in a mixture is given by Figure 8. Comparison of estimation methods for the prediction of heat of vaporization of chloromethanes a) Dichloromethane; b) Trichloromethane; c) Tetrachloromethane Open symbols denote experimental data, lines denote predicted values

fi =xi ϕi P

(4.29)

where xi is the concentration of i (usually as mole fraction) and ϕi the fugacity coefficient (with ϕi = 1 for ideal gases). With an arbitrary state of reference, which is very often the pure component i at its vapor pressure, the following relation is defined:

Estimation of Physical Properties fi =xi γi fi0

(4.30)

with γi as activity coefficient of i in the mixture and f 0i as fugacity of i at the state of reference. 4.2.1. Vapor – Liquid Equilibria with Liquid-Phase Activity Coefficients 4.2.1.1. Basic Considerations For the correlation and prediction of vapor – liquid equilibria at low to moderate pressure (< ca. 1.0 MPa) Equation (4.30) is usually applied to the liquid phase. The vapor phase is either considered to be ideal or a correction for real gas behavior is applied (e.g., by estimated second virial coefficients). The reference state in the liquid phase for each component i in the mixture is the pure liquid i with a vapor pressure plv 0i at temperature T . With this reference state Equation (4.30) for component i in a liquid mixture is given by   1 fil =xi γi plv 0i ϕ0i exp  RT


p

  l V0i dp

(4.31)

p0i

which includes two corrections, i.e., the fugacity coefficient ϕ0i for the pure liquid at pressure p0i and the exponential term called the Poynting factor which corrects for the difference of fugacity between plv 0i and system pressure p. If the liquid volume V l0i is assumed to be constant (independent of pressure), the Poynting factor Πi is simply given by  Πi =exp

l V0i



 p − plv 0i RT

(4.32)

Equality of fugacities in the vapor and liquid phases yields with Equations (4.29) and (4.31) the following general relation for vapor – liquid equilibria: yi ϕi p=xi γi plv 0i ϕ0i Πi

561

This means that a simplified version of Equation (4.33) can be used for the majority of vapor – liquid equilibria without much loss in accuracy: yi p=xi γi plv 0i

(4.34)

With Equation (4.34) the vapor phase is considered ideal and all nonideality effects are attributed to the liquid-phase activity coefficient γi . For γi = 1, Equation (4.34) transforms into Raoult’s law for the partial pressure pi in ideal mixtures: pi =yi p=xi plv 0i

(4.35)

The activity coefficients γi in liquid mixtures are directly related to an important thermodynamic quantity characterizing nonideal mixing behavior, i.e., the molar excess Gibbs energy of mixing ∆GE which is defined as the difference in the molar Gibbs energy of mixing between real and ideal mixtures: ∆GE = ∆G−∆Gid

(4.36)

This quantity is related to the activity coefficients γi of all components i in a mixture by the following equation: ∆GE =RT



xi lnγi

(4.37)

Conversely, the activity coefficients γi in a mixture can be obtained from ∆ G E when this quantity is known as a function of composition (i.e., of all xi ) according to:  RT lnγi =

∂n·∆GE ∂ni

 (4.38) T ,p,nj

where n is the total number of moles and ni the number of moles of component i. 4.2.1.2. Data Correlation: Binary and Multicomponent Mixtures of Nonelectrolytes

(4.33)

with yi and xi as the mole fractions of i in the vapor and liquid, respectively. At pressures up to several hundred kilopascal (several bar) the fugacity coefficients ϕi and ϕ0i and the Poynting factor Πi are usually not significantly different (< 2.3×10−3 ) from unity.

Numerous correlation methods have been published. They include flexible polynomials (e.g., Margules, van Laar, Redlich – Kister series) and other more sophisticated semitheoretical models, especially those based on the concept of local composition [Wilson, Non-Random

562

Estimation of Physical Properties

Table 11. Model equations for calculating the molar excess Gibbs energy of mixing ∆GE in liquid mixtures Model

Parameter

Equation

Margules-3 (binary systems)

A B

Wilson [167]

Λij , Λji (Λii = Λjj = 1)

ln γ 1 = x 22 [A + 2 (B − A) x 1 ] ln γ 2 = x 21 [B + 2 (A − B) 

xr2 ] r



∆GE = − x ln xj Λij i RT i i  r r



lnγi = − ln xj Λij +1− i

with Λij = NRTL [168]

τij , τji , αij = αji

∆GE RT

=−

lnγi =

r

xi

i

j

Gki xk

τji Gji xj

r

+ Gki xk

x Λ

k ki xj Λkj j

τji Gji xj

r

k

k

τij , τji

r

 λ −λ  exp − ijRT ii

i r

UNIQUAC [169]

V l0j Vl 0i

k

r

j



r

 τij −

xj Gij r

Gkj xk k

k

 xk τkj Gkj

r

k

Gkj xk

 

with τij = g −giiij and RT Gij = exp {− αij τij }





E A ∆G xi lnSi + z2 qi xi ln S i − qi xi ln xj Aj τji RT = i

i

i

R ln γi = ln γ C i + ln γ i



i

S

i

S



lnγiC = 1−Si +lnSi − z2 qi 1+ Ai +ln Ai i i

 qj xj τji

qj xj τij j

− lnγiR =qi 1 − ln q x q x τ j

j

j

−∆u

j

k

k



k kj

with τij =exp T ij , ∆uii =∆ujj =0, z = 10 Si , Ai , see Equations (4.42) and (4.43)

Two-Liquid ( NRTL), and UNIversal QUAsiChemical ( UNIQUAC) equations]. Relations for some of these model equations are given in Table 11. If reliable experimental data are available, equations with two parameters per binary system are usually suitable for correlating and extrapolating available data with respect to composition and, to a limited degree, to temperature. Numerous papers have been published comparing different ∆G E models. But no essential advantages for any of these correlation methods could be found with respect to their ability to fit data. Another important aspect has to be considered. The majority of published vapor – liquid equilibrium data are concerned with binary systems, only a few with ternary systems, and very few with systems of four and more components. In practical chemical engineering problems, however, systems with at least five or more components often have to be treated. Here the models based on the concept of local composition show the advantage that only binary parameters are required for the representation of multicomponent systems. The local composi-

tion concept was introduced by Wilson [167]; the essential feature of this idea is that it considers only binary interactions between neighboring molecules in the liquid mixture. Other models using the local composition concept are the NRTL [168] and UNIQUAC [169] equations. In contrast to the Wilson equation these models are able to describe the coexistence of two liquid phases, i.e., liquid – liquid phase splitting. 4.2.1.3. Prediction of Equilibrium Data Solution-of-Groups Concept. Measurement of vapor – liquid equilibria is difficult, time-consuming, and costly. Thus reliable and widely applicable estimation methods are of utmost interest. Numerous attempts to derive methods for the prediction of the properties of nonelectrolyte liquid mixtures using only pure component information (e.g., dipole moment or parachor) have failed. In contrast, methods based on structural groups have turned out to be quite successful. The starting point for these concepts is the assumption that a mixture is com-

Estimation of Physical Properties posed of structural groups and not of molecules. Consequently interactions between structural groups are considered instead of interactions between molecules. The large variety of molecular mixtures resulting from the huge number of compounds is reduced to a limited number of combinations of structural groups (e.g., −CH3 , > CH2 , −OH, −CHO, > C=O). This “solutionof-groups” concept was originally introduced to phase equilibrium calculations by Redlich et al. [167] and Wilson and Deal [171]. The first method of practical importance based on this concept was the Analytical Solution Of Groups (ASOG) method of Derr and Deal [172]. ASOG Method. This method employs an athermal Flory – Huggins term to consider size effects and the Wilson equation for the interactions of the structural groups. Therefore the activity coefficient γi is split into two parts: lnγi = lnγiC +lnγiR

(4.39)

where the combinatorial part ln γ C i considers size effects and the residual part ln γ R i the group interactions. Equation (4.39) is the basic relationship of the solution-of-groups concept. The residual part ln γ R i is related to the pure liquid i according to lnγiR =



.

(i)

νki lnΓk − lnΓk

/ (4.40)

k

where νki is the number of groups of type k in ( component i; Γk and Γ k i) are the corresponding group activity coefficients in the mixture and in the pure liquid, respectively. They are calculated from interaction parameters that have been determined from reliable vapor – liquid equilibrium data. The latest version of ASOG gives interaction parameters for 43 structural groups [173]. UNIFAC Method. More recently Fredenslund et al. [174] derived the UNIversal Functional group Activity Coefficients (UNIFAC) method. This group contribution model applies the UNIQUAC equation [169] to the solution-of-groups concept (Eq. 4.39). Since this model is most widely accepted and applied, it is described here in more detail. The combinatorial part contains pure component parameters

563

only and is the Staverman expression taken from the UNIQUAC model: z lnγiC = 1−Si +lnSi + qi 2

 1−

Si Si +ln Ai Ai

 (4.41)

where xi is the mole fraction. The volume fraction Si and surface fraction Ai are defined as follows: Si =

ri r j xj

(4.42)

qi q j xj

(4.43)

j

Ai = j

where the parameters ri and qi represent the relative volume and relative surface area of the component i; z is the coordination number and is usually fixed as z = 10. These size parameters have been tabulated for many groups [175] or can be calculated [176]. The residual part considers group – group interactions via the group activity coefficient Γi following the UNIQUAC model for the main groups m and n: lnΓk =  Qk 1 − ln



 Θm Ψmk

  Θm Ψkm  − (4.44) Θn Ψnm m n

with the area fraction Θm of group m defined by Qm Xm Θm = Qn Xn

(4.45)

where Xm is the group mole fraction and Qm < the group surface area. The group energy term Ψnm for the interaction between the groups m and n is defined by  Ψnm =exp

−amn T

 (4.46)

At present, interaction parameters amn are known for 50 main groups, some of which comprise subgroups (e.g., main group CH2 with subgroups C, CH, CH2 , and CH3 ). Several smaller molecules (e.g., water, methanol) and some molecules with special structures (e.g., pyridine, N-methylpyrrolidone) have been defined as main groups. The interaction parameters have been determined from reliable, thermodynamically consistent experimental vapor – liquid equilibrium

564

Estimation of Physical Properties

Figure 9. Status (1991) of the UNIFAC parameter matrix [178] Abbreviations: A = aromatic; (C)3 N = tertiary amine; DMF = dimethylformamide; DMSO = dimethyl sulfoxide; DOH = 1,2ethanediol; NMP = N-methylpyrrolidone

data [175], [177], [178]. The present state can be seen in Figure 9. The predictive ability of the UNIFAC method is quite remarkable. Provided accurate purecomponent vapor pressures are known, errors in vapor phase composition are usually ca. 1 mol%. Typical results of UNIFAC predictions are displayed in Figure 10, which demonstrates the predictive power of the UNIFAC method. In cases where only little or unreliable primary data are available for the determination of group interaction parameters (e.g., for combinations with aldehydes) serious deviations may be expected. This original UNIFAC method has additional limitations: 1) Limited temperature dependence leading to unsatisfactory prediction of heat of mixing data 2) In many cases isomers cannot be distinguished from one another

3) Neighboring effects (i.e., interaction between “strong” functional groups within the same molecule) are not considered. Much effort has been and still is devoted to overcoming these shortcomings, e.g., [179]. Gmehling et al. [180], [181] have included experimental data for both heats of mixing and activity coefficients at infinite dilution into their data base from which they determined temperature-dependent group interaction parameters. They showed that this modified UNIFAC method can be used to predict temperaturedependent activity coefficients at both bulk composition and high dilution and also mixing enthalpies with satisfactory accuracy. A different approach in applying group contributions for the prediction of activity coefficients was proposed by Kehaian et al. [182]. Their method is based on Barker’s lattice theory; interaction parameters have only been published for a limited number of group combinations.

Estimation of Physical Properties

565

Figure 10. Results of UNIFAC predictions (curves) compared with experimental data (dots) for 16 ketone – alkane mixtures x 1 = mole fraction of underlined component in the liquid phase; y = mole fraction of underlined component in the vapor phase Only two interaction parameters (aCH2 , CH2 CO = 476.4 K, aCH2 CO, CH2 = 26.76 K) have been used for all diagrams [177].

For practical purposes the classical UNIFAC method with its large interaction parameter matrix provides a useful tool for the prediction of vapor – liquid equilibria of unknown systems and also for the selection of solvents in extractive and azeotropic distillation [183]. The matrix of UNIFAC interaction parameters is given in [178]. Extensions of the UNIFAC Method for Polymer Solutions and for Gas Solubilities. In order to apply the UNIFAC method to poly-

mer solutions Oishi and Prausnitz proposed the inclusion of a free volume term ln γ FV in i Equation (4.39) [184]: lnγi = lnγiC +lnγiR +lnγiFV

(4.47)

For gas – liquid equilibria the activity coefficient approach (Eq. 4.30) employing the pure liquid as reference state cannot apparently be applied when the system temperature exceeds the critical temperature of one of the components, as is the case with solubilities of so-called permanent gases (e.g., nitrogen, oxygen, methane,

566

Estimation of Physical Properties

ethylene) in liquids at ambient temperature. A hypothetical liquid state of the gaseous component i may then be defined by extrapolating the standard fugacity f 0i beyond the critical temperature. Using this hypothetical liquid state Nocon et al. [185], [186] obtained satisfactory predictions of solubilities of nitrogen, oxygen, carbon dioxide, methane, and ethylene in hydrocarbons and alcohols with a UNIFAC method based on Equation (4.47) (i.e., by including a free volume term). 4.2.2. Vapor – Liquid and Gas – Liquid Equilibria with Equations of State 4.2.2.1. Basic Considerations Taking the ideal gas as standard state for the fugacities (Eq. 4.29), the equilibrium condition of equal fugacities in the vapor and liquid phases yields for component i: yi ϕvi =xi ϕli

(4.48)

With this relation the same thermodynamic model can be used for both phases. The fugacity coefficient can be calculated from the rigorous relation lnϕi =

1 RT


p  0

∂V ∂ni

 − T ,p,nj=i

RT p

dp

a consequence, the molecular methods have not yet become established for engineering calculations. 4.2.2.2. Data Correlation An extensive compilation of high-pressure vapor – liquid equilibria and gas – liquid equilibria data has been presented by Knapp et al. [189]. They tested the capability of four common equations of state (Lee – Kesler – Pl¨ocker, LKP; Benedict – Webb – Rubin – Starling, BWRS; Redlich – Kwong – Soave, RKS; Peng – Robinson, PR; see Section 3.1.1) in representing experimental data for a large number of binary systems. The calculations of Knapp et al. show that there is no principal advantage of any of these equations. They yield satisfactory results when the components involved are not highly polar or do not differ too much in size. More generally applicable are the so called ∆G E mixing rules. Huron and Vidal [190] proposed using the following definition of the excess Gibbs energy:    ∆GE = RT lnϕ − xi lnϕi0

where the fugacity coefficient ϕ accounts for the real behavior of the mixture:

(4.49)

employing an equation of state for V . The advantage of this approach is that no standard-state fugacity is necessary. This is particularly useful when the mixture consists of both sub- and supercritical components. A further advantage is the possibility of calculating other mixture properties such as densities, enthalpies, and entropies of the two phases. Numerous equations of state have been published but accurate prediction of phase equilibria of mixtures is not yet possible from pure-component data only. Hence, binary interaction parameters (see Section 3.1.3.1) must be obtained from experimental data for mixtures. Modern statistical mechanics have achieved remarkable progress in the last two decades [187], [188]. Nevertheless, computation time is still considerable. Adequate force models are available only for rather simple molecules. As

(4.50)

lnϕi =

1 RT


p  V − 0

RT p

 dp

(4.51)

The fugacity coefficients ϕi0 of the pure components are given by Equation (4.49). Substitution of ∆G E in Equation (4.50) by Equation (4.37) leads to an expression from which mixing rules for the parameter a of cubic equations of state (Section 3.1.1) can be derived. When infinite pressure (p → ∞) is chosen as reference state, it can be assumed that the parameter b is equal to the molar volume V . With this assumption and the linear mixing rule for bm (Eq. 3.39) the Huron – Vidal (HV) mixing rule for the RKS equation of state (Eq. 3.11) can be derived: am = bm RT



xi ·

ai − bi RT



xi lnγi 0.6931

 (4.52)

Other reference states may be chosen. For example with p = 0 the modified Huron – Vidal (MHV) mixing rules have been derived [191].

Estimation of Physical Properties With a constant value for the ratio V /b the MHV1 mixing rule [192] for the RKS equation of state is obtained:

(Eq. 4.54) of several hydrocarbon gases (1) in the liquid solvent (2) eicosane. H12 = lim

am =

(4.53) 

 xi ln bbm xi lnγi +  ai i   bm RT xi · + bi RT A

Depending on the value of V /b different values for A will result. In the PSRK method (predictive Soave – Redlich – Kwong [193], Section 4.2.2.3) with V /b = 1.1 the value of A is − 0.64663. Equation (4.53) allows any G E model for the liquid phase to be incorporated (see Section 4.2.1.2) into the RKS equation of state. Based on the ∆G E mixing rules several other models have been proposed; they differ mainly in the reference pressure and in the V /b ratio. They are all well suited for the representation of vapor – liquid and gas – liquid equilibria, also at high pressures and for strongly polar systems. For the latter systems they should be preferred to equations of state combined with densitydependent mixing rules (e.g., [194]) or with association models (e.g., [195], [196]).

567

xi →0

f1 x1

(4.54)

where f 1 is the fugacity of gaseous component (1) and x 1 the mole fraction of gaseous component (1) in the liquid. LIVE GRAPH Click here to view

4.2.2.3. Prediction of Vapor-Liquid and Gas-Liquid Equilibria Since any type of GE model can be introduced into Equation (4.53), group contribution methods can be applied, too. The PSRK group contribution equation of state [193] combines the UNIFAC method (Section 4.2.1.3) with the RKS equation of state. Thus all group interaction parameters tabulated for UNIFAC can be used. Moreover new groups for gaseous components have been included: NH3 , CO2 , CH4 , O2 , Ar, N2 , H2 S, H2 , CO, SO2 , NO, N2 O, SF6 , He, Ne, Kr, Xe, HCl, HBr [193], [197]. Compared to other group contribution equations of state proposed for estimating vapor – liquid equilibria up to the critical region [198], [199], [200] the PSRK method can be applied over a wider range of temperature and pressure. In addition, many more groups and their parameters have been defined for the PSRK method. As an example of its predictive power, a comparison of predicted gas solubilities is given in Figure 11. It shows the Henry coefficients H 12

Figure 11. Results of PSRK predictions (lines) compared with experimental data (•) from the Dortmund Data Bank) for Henry coefficients of hydrocarbon gases in eicosane (nC20 H42 )

4.2.3. Liquid – Liquid Equilibria Phase splitting of a liquid mixture into two liquid phases occurs when a single liquid phase is thermodynamically unstable. For the computation of liquid – liquid equilibria, equality of fugacities for each component in the two phases is usually applied. Since this equilibrium condition is only necessary, but not sufficient, it should

568

Estimation of Physical Properties

be combined with an analysis of thermodynamic stability in order to find the true equilibrium. Using Equation (4.30) for the fugacities fi
and fi

in phases I and II with the same reference state for f 0i yields the following relation:  xi γi =x i γi

(4.55)

The products x
i γi
and x
i γ
i are the so-called activities. Since Equation (4.55) holds for each component of a liquid – liquid system, it should be possible to predict liquid – liquid equilibria when the ∆G E function (Eq. 4.37) and consequently the activity coefficients of the individual components in the multicomponent system are known (e.g., from vapor – liquid equilibria or from prediction methods developed for these phase equilibria, such as UNIFAC). This procedure entails several problems, however: 1) Whereas in vapor – liquid equilibrium calculations activity coefficients serve as a correction of Raoult’s law (cf. Eqs. 4.34 and 4.35), they are the sole numerical input into calculations of liquid – liquid equilibria. Therefore, the accuracy of activity coefficients γi for modeling and predicting liquid – liquid equilibria has to be considerably higher than for vapor – liquid equilibria. 2) Systems with phase splitting often show rather high temperature dependence of activity coefficients, which is in general predicted less reliably. 3) The majority of vapor – liquid equilibrium data and the associated activity coefficient data have been determined at other, usually higher temperatures than are of interest for liquid – liquid equilibria. Modifications of the UNIFAC method, either by creating a separate UNIFAC parameter matrix for liquid – liquid equilibrium prediction [201] or by including experimental liquid – liquid equilibria data into the determination of interaction parameters [181], [202], have achieved limited success. The use of equations of state has been tested with promising results for the calculation of liquid – liquid equilibria, for example, extrapolation of liquid – liquid equilibria to higher pressures and representation of vapor – liquid – liquid equilibria [203].

Octanol – Water Distribution Coefficients. The distribution coefficient of toxic substances between octanol and water is an important property in toxicology and environmental science since it correlates with the distribution of substances between a lipid and an aqueous phase. For the estimation of octanol – water distribution coefficients, Hansch and Leo [204] developed a much used group contribution method. Another such method has been proposed by Wienke and Gmehling [205]. It is based on the UNIFAC method and has the advantage that other equilibrium constants relevant to the distribution of chemicals in the environment, i.e., Henry constants and water solubilities, can be predicted with the same parameters. 4.2.4. Solid – Liquid Equilibria Solid – liquid equilibria may be roughly classified into systems without miscibility of the components in their solid states (eutectic systems) and systems that form mixed crystals. With nonelectrolytes the first type of behavior is much more frequent. For these systems the following relation between the solubility of pure solid i (expressed as liquid mole fraction x li ) and temperature T holds:     ∆Hisl T 1− ln xli γil = − RT Tm,0i

(4.56)

where ∆H sl i is the enthalpy of melting and T m, 0i the melting point of pure component i (terms of higher order have been neglected in Eq. 4.56). The liquid-phase activity coefficient γ li can be estimated with group contribution methods such as UNIFAC [206], [207]. At high concentrations of i the equilibrium temperature T (i.e., the freezing point of a liquid solution or melt) can be predicted quite reliably without knowing the value of the liquid-phase activity coefficient γ li , because γ li generally approaches unity at these concentrations. With this assumption, rearrangement of Equation (4.56) (with some simplifications) yields the well-known Raoult equation for the lowering of the freezing point of solutions at low solute concentrations: Tm,i − T =

RTm,0i (1−xi ) ∆Hisl

(4.57)

Estimation of Physical Properties

5. Thermodynamic Data of Chemical Reactions 5.1. Definitions Thermodynamic data of chemical reactions are highly important in process design for two reasons: 1) Molar reaction enthalpies (heats of reaction) ∆H r are required in the design of chemical reactors, since the heat effect connected with the reaction must be known to control the reaction temperature. 2) The molar Gibbs energy ∆G r (also called the free enthalpy) of a chemical reaction is directly related to the reaction equilibrium; knowledge of ∆G r allows prediction of equilibrium concentrations.

569

is still in use. Generally the standard formation enthalpy ∆H 0f298 of a compound is given for its most stable state at standard pressure (1.01325 bar at 298.15 K) . For the elements ∆H 0f298 is zero by definition in their most stable state under standard conditions (for gases: the ideal gas state). The standard reaction enthalpy ∆H 0r298 for Equation (5.1) is thus given by 0 0 0 ∆Hr298 =γ ∆Hf298,C +δ∆Hf298,D 0 0 −α∆Hf298,A −β ∆Hf298,B

(5.4)

or in general 0 ∆Hr298 =



0 νi ·∆Hf298,i

(5.5)

At temperature T the heat of reaction ∆H 0rT is calculated from ∆H 0r298 with the aid of the molar heat capacities (ideal gas state) c0pi of the reactants:
T 0 ∆HrT

5.1.1. Reaction Enthalpy

0 =∆Hr298 +

∆cpr dT

(5.6)

298.15

Reaction enthalpies can be determined from the enthalpies of formation ∆Hf of the compounds taking part in the reaction. Thus, for the reaction αA + βBγc + δD

(5.1)

the reaction enthalpy ∆H r is given by ∆Hr =γ ∆HfC +δ∆HfD −α∆HfA −β ∆HfB

(5.2)



νi ∆Hfi


T 0 ∆HrT

or by the general form ∆Hr =


with ∆cpr = νi cpi . This is in fact a heat balance. Therefore in the case of phase changes between standard-state and reaction conditions (T, p) for one or several reactants the respective phase change enthalpies ∆H trs also have to be included yielding the following general expression:

(5.3)

where ν i is the stoichiometric coefficient of compound i (according to the rules of chemical thermodynamics, ν i is positive for products and negative for educts) and ∆H fi is the enthalpy of formation of compound i. For the calculation of reaction enthalpies the formation enthalpies of many compounds have been tabulated for a standard state of 1.01325 bar = 1 atm at 298.15 K [208–215]. They are designated as ∆H 0f298 (with superscript 0 for standard pressure and subscript 298 for standard temperature, 298 K). Since 1982 0.1 MPa = 1 bar has been recommended as standard pressure, but in chemical thermodynamics and relevant data compilations 1.01325 bar

=

+

0 ∆Hr298 +



∆cpr dT

298.15

νi ∆Hitrs

(5.7)

5.1.2. Gibbs Energy of Reaction and Chemical Equilibrium Gibbs energies of reaction are calculated analogously to reaction enthalpies. For the reaction of Equation (5.1) the Gibbs energy ∆Gr is equal to the sum of the Gibbs energies of formation ∆G f of the reactants multiplied by their stoichiometric coefficients: ∆Gr =γ ∆GfC +δ∆GfD −α∆GfA −β ∆GfB

The general form of Equation (5.8) is

(5.8)

570

Estimation of Physical Properties

∆Gr =



νi ∆Gfi

(5.9)

In order to determine the standard Gibbs energy of reaction ∆G 0r298 with Equation (5.8) or (5.9) the relevant standard quantities of formation ∆G 0f298 have to be used. Another way of calculating ∆G 0r298 uses the definition ∆Gr = ∆Hr −T ∆Sr

(5.10)

Application of Equation (5.10) requires ∆H 0r298 from Equation (5.5) and ∆S 0r298 obtained from standard entropies S 0298 according to Equation (5.11): 0 ∆Sr298 =



0 νi S298

(5.11)

Data for ∆G 0f298 and S 0298 can be found for many compounds in data collections, e.g., in [208], [211], [213], [214]. Conditions for equilibrium at constant pressure and temperature yield the following relation between the Gibbs energy of reaction ∆G0rT (ideal gas at standard pressure = 1.01325 bar = 101.325 kPa) and equilibrium constant K: ∆G0rT = −RT lnK (T )

(5.12a)

For the chemical reaction of Equation (5.1) K is defined as K=

γ δ 0α f 0β fC fD f A · 0γ B β 0δ αf fC fD fA B

pγC pδD β pα A pB

≈ K· (1atm)

∆G0r298 298.15R

lnK298.15 = −

0 dlnK ∆HrT = dT RT 2

(5.15)

Integration of Equation (5.15) with ∆H 0rT as a function of temperature according to Equation (5.6) yields a general relation for calculating K at temperature T when it is known at temperature T 0 (e.g., 298.15 K): lnK (T ) = lnKT0 − −

1 RT

0 ∆HrT 0

R

νi

(5.14)



1 1 − T T0




T ∆cpr dT T0

+

1 R


T

∆cpr dT T

(5.16)

T0

For a first approximation the integrals in Equation (5.16) can be neglected which means that the reaction enthalpy ∆H r is considered to be independent of temperature. With K known at T 0 = 298.15 K this first approximation is lnK (T ) = lnK298.15 +



(5.12b)

The dependence of K on temperature is given by the Van’t Hoff Equation:

(5.13)

with the standard fugacity f 0i of the reactants A, B, C, D (equal to the standard pressure of 1 atm). K is a function of temperature only, but not of pressure. At low pressure where real gases do not show much difference from ideal gas phase behavior, the fugacities fi can be considered as ideal gas partial pressures. The “thermodynamic” (true) equilibrium constant K is then practically equal to the numerical value of the equilibrium constant K p expressed in partial pressures pi = yi p ( yi is the mole fraction of component i in the gas phase) of the reactants as: Kp =

For all those reactions for which values of either ∆G 0f298 or ∆H 0f298 and S 0298 are available for each reactant the value of K at 25 ◦ C (298.15 K) can easily be determined:

0 ∆Hr298 R



 1 1 − (5.17) 298.15 T

In a second approximation the sum of the molar heat capacities ∆cpr is considered to be constant [∆cpr (T ) = ∆cpr ]:  1 1 − 298.15 T   T 298.15 ln + −1 (5.18) 298.15 T

lnK (T ) = lnK298.15 + +

∆cpr R

0 ∆Hr298 R



For approximate estimations Equations (5.17) and (5.18) are sufficient; accurate predictions, especially at high temperatures, require evaluation of the integrals in Equation (5.16). Concerning the effect of pressure on equilibrium concentrations the influence of real gas behavior on K p has to be considered. At

Estimation of Physical Properties low to moderate pressures (up to several hundred kilopascal) K p is approximately equal to

vi (Eq. 5.14). But with higher presK(1 atm) sure the difference between the numerical values of K and K p increases and cannot be neglected at pressures > ca. 1 MPa. This effect can be predicted when the pVT behavior of the reaction mixture is known. Using the relation fi =ϕi yi p=ϕi pi

(5.19)

where ϕi is the fugacity coefficient for component i, Equation (5.13) is transformed as follows:   νi pγ pδ ϕγC ϕδD 1 K= C D · · β α β atm pα A pB ϕA ϕB  νi  1 =Kp Kϕ · atm

molecules can be used as the basis for estimating thermodynamic reaction quantities, in particular of organic substances. Various such group contribution methods have been proposed for the estimation of standard enthalpies, Gibbs energies of formation, and standard entropies (Table 13). The method of Joback [216], [217] is explained in more detail. In the Joback method ∆H 0f298 , ∆G 0f298 , and the molar heat capacities c0p are calculated via Equation (5.21), (5.21a), and (3.46): 0 ∆Hf298 = 68.29+

∆G0f298 = 53.88+ (5.20)

The parameter Kϕ represents the effect of real gas behavior on chemical equilibrium. The magnitude of this effect on high-pressure equilibria can be seen from Table 12, where data for the methanol synthesis equilibrium are given. Table 12. Effect of pressure on equilibrium of methanol synthesis at 300 ◦ C (CO + 2 H2
CH3 OH) K = 2.316×10−4 p, bar (MPa)

Kϕ

K p , atm−2

10 (1) 25 (2.5) 50 (5) 100 (10) 200 (20) 300 (30)

0.96 0.90 0.80 0.61 0.38 0.27

2.41×10−4 2.57×10−4 2.90×10−4 3.80×10−4 6.09×10−4 8.58×10−4

5.2. Group Contribution Methods for the Estimation of Enthalpies and Gibbs Energies of Formation Especially in processes for new products, thermodynamic data (∆H 0f , ∆G 0f ) are often not available for all compounds and their estimation is extremely helpful. Chemical reactions can be considered as processes in which molecular bonds are broken and others are formed. The thermodynamic functions for a chemical reaction can therefore be considered as the sum of the contributions of those chemical bonds which are altered (i.e., broken or formed) by the reaction. In a similar approach the structural groups of the reacting

571

 

si IHi [kJ/mol]

(5.21)

si IGi [kJ/mol]

(5.21a)

where si is the number of structural groups of type i and IHi and IGi are the increments of structural groups i for enthalpy and Gibbs energy of formation, respectively. The structural groups are the same as in the Joback method for the estimation of critical data (Chap. 2). The values of the group contributions IHi for ∆H 0f298 ; IGi for ∆G 0f298 ; and I a , I b , I c , and I d for the coefficients of the polynomial for c0p are given in Table 14. As is evident from Equations (5.21) and (5.21a), standard formation enthalpies ∆H 0f and standard Gibbs energy of formation ∆G 0f can be obtained directly for 298.15 K only. In order to calculate these quantities and the equilibrium constant K for other temperatures, the molar heat capacities c0p have to be evaluated as a function of temperature with Equation (3.46); ∆H 0f and K can be then determined with Equations (5.6) and (5.16), respectively. Deviations of ∆H 0f298 and ∆G 0f298 calculated with the Joback method from literature data are mostly below 10 kJ/mol; in a few cases, however, differences of more than 20 kJ/mol were found [218]. With the methods of Benson [219], [220] and of Yoneda [221] errors are somewhat smaller because they use larger numbers of groups which allow higher differentiation in structure. Application of these methods is, however, rather tedious. In both methods ∆G 0f298 is obtained via Equation (5.10), formulated for the reaction of formation for which ∆H 0f298 and the standard entropy S 0298 are estimated with group contributions. In the same way, molar heat capacities c0p

572

Estimation of Physical Properties

Table 13. Group contribution methods for the estimation of ∆H 0f , ∆G 0f , and S 0 Authors

Estimated parameters

Form in which contributions are given

Average error, kJ/mol

Secondary sources for increments; remarks

Benson [219], [220] Constantinou and Gani [225] Franklin [222] Joback [216], [217] van Krevelen and Chermin [224] Verma and Doraiswamy [223] Yoneda [221]

∆H 0f , S 0 ∆H 0f , ∆G 0f ∆H 0f ∆H 0f , ∆G 0f ∆G0f

IH (298), IS (298); Ic (T ) IH (298); IG (298) IH (T ) IH (298); IG (298); Ic = f (T ) IG = A + B T

10 10 15 15 15

[218] ∆H 0f and ∆G 0f at 298 K only [227]; in kcal/mol this article (Table 14); [218] [226], [227]; in kcal/mol

∆H 0f ∆H 0f , S 0

IH = A + B T IH (298), IS (298); Ic = f (T )

15 10

[227]; in kcal/mol [218]

Table 14. Increments for the calculation of enthalpies and Gibbs energies of formation by Joback’s method [216], [217] Type of increment

Nonring increments −CH3 >CH2 >CH− >C< =CH2 =CH− =C< =C= ≡CH ≡C− Ring increments −CH2 − >CH− >C< =CH− =C< Halogen increments −F −Cl −Br −I Oxygen increments −OH (alcohol) −OH (phenol) −O− (nonring) −O− (ring) >C=O (nonring) >C=O (ring) O=CH− (aldehyde) −COOH (acid) −COO− (ester) =O (except as above) Nitrogen increments −NH2 >NH (nonring) >NH (ring) >N− (nonring) −N= (nonring) −N= (ring) =NH −CN −NO2 Sulfur increments −SH −S− (nonring) −S− (ring

IH , kJ/mol

Molar heat capacities ∗, J mol−1 K−1

IG , kJ/mol ∆a

∆b

∆c

∆d

−76.45 −20.64 29.89 82.23 −9.63 37.97 83.99 142.14 79.30 115.51

−43.96 8.42 58.36 116.02 3.77 48.53 92.36 136.70 77.71 109.82

1.95 E + 1 −9.09 E −1 −2.30 E + 1 −6.62 E + 1 2.36 E + 1 −8.00 −2.81 E + 1 2.74 E + 1 2.45 E + 1 7.87

−8.08 E −3 9.50 E −2 2.04 E −1 4.27 E −1 −3.81 E −2 1.05 E −1 2.08 E −1 −5.57 E −2 −2.71 E −2 2.01 E −2

1.53 E −4 −5.44 E −5 −2.65 E −4 −6.41 E −4 1.72 E −4 −9.63 E −5 −3.06 E −4 1.01 E −4 1.11 E −4 −8.33 E −6

−9.67 E −8 1.19 E −8 1.20 E −7 3.01 E −7 −1.03 E −7 3.56 E −8 1.46 E −7 −5.02 E −8 −6.78 E −8 1.39 E −9

−26.80 8.67 79.72 2.09 46.43

−3.68 40.99 87.88 11.30 54.05

−6.03 −2.05 E + 1 −9.09 E + 1 −2.14 −8.25

8.54 E −2 1.62 E −1 5.57 E −1 5.74 E −2 1.01 E −1

−8.00 E −6 −1.60 E −4 −9.00 E −4 −1.64 E −6 −1.42 E −4

−1.80 E −8 6.24 E −8 4.69 E −7 −1.59 E −8 6.78 E −8

−251.92 −71.55 −29.48 21.06

−247.19 −64.31 −38.06 5.74

2.65 E + 1 3.33 E + 1 2.86 E + 1 3.21 E + 1

−9.13 E −2 −9.63 E −2 −6.49 E −2 −6.41 E −2

1.91 E −4 1.87 E −4 1.36 E −4 1.26 E −4

−1.03 E −7 −9.96 E −8 −7.45 E −8 −6.87 E −8

−208.04 −221.65 −132.22 −138.16 −133.22 −164.50 −162.03 −426.72 −337.92 −247.61

−189.20 −197.37 −105.00 −98.22 −120.50 −126.27 −143.48 −387.87 −301.95 −250.83

2.57 E + 1 −2.81 2.55 E + 1 1.22 E + 1 6.45 3.04 E + 1 3.09 E + 1 2.41 E + 1 2.45 E + 1 6.82

−6.91 E −2 1.11 E −1 −6.32 E −2 −1.26 E −2 6.70 E −2 −8.29 E −2 −3.36 E −2 4.27 E −2 4.02 E −2 1.96 E −2

1.77 E −4 −1.16 E −4 1.11 E −4 6.03 E −5 −3.57 E −5 2.36 E −4 1.60 E −4 8.04 E −5 4.02 E −5 1.27 E −5

−9.88 E −9 4.94 E −8 −5.48 E −8 −3.86 E −8 2.86 E −9 −1.31 E −7 −9.88 E −8 −6.87 E −8 −4.52 E −8 −1.78 E −8

−22.02 53.47 31.65 123.34 23.61 55.52 93.70 88.43 −66.57

14.07 89.39 75.61 163.16

2.69 E + 1 −1.21 1.18 E + 1 −3.11 E + 1

−4.12 E −2 7.62 E −2 −2.30 E −2 2.27 E −1

1.64 E −4 −4.86 E −5 1.07 E −4 −3.20 E −4

−9.76 E − 8 1.05 E −8 −6.28 E −8 1.46 E −7

79.93 119.66 89.22 −16.83

8.83 5.69 3.65 E + 1 2.59 E + 1

−3.84 E −3 −4.12 E −3 −7.33 E −2 −3.74 E −3

4.35 E −5 1.28 E −4 1.84 E −4 1.29 E −4

−2.60 E −8 −8.88 E −8 −1.03 E −7 −8.88 E −8

−17.33 41.87 39.10

−22.99 33.12 27.76

3.53 E + 1 1.96 E + 1 1.67 E + 1

−7.58 E −2 −5.61 E −3 4.81 E −3

1.85 E −4 4.02 E −5 2.77 E −5

−1.03 E −7 −2.76 E −8 −2.11 E −8

∗ E = exponent of 10: E + 1 = 101 , E − 2 = 10−2 etc.

Estimation of Physical Properties are determined when Equations (5.6) and (5.16) are applied for ∆H 0r and K at other temperatures. Straightforward estimation of ∆H 0f and ∆G 0f at different temperatures is also possible [222– 224]. Application of these older methods is convenient because they do not require estimation of heat capacity data. Results from these methods are, however, in kilocalories per mole.

Group i

si

IHi

si IHi

−CH3 −CH2 − −OH −O−

1 5 1 1

− 76.45 − 20.64 −208.04 −132.22

− 76.45 −103.20 −208.04 −132.22

si IHi = − 519.91 kJ/mol

Equation (5.21): ∆H 0f298 = 68.29 − 519.91 = − 451.62 kJ/mol

5.3. Applications 5.3.1. Reaction Enthalpy In the determination of reaction enthalpies with Equation (5.3) the enthalpy of formation is often not known for all reactants. The following procedure can then be adopted: Example. Calculation of reaction enthalpy for ethoxylation of n-butanol at 390 K. Monoethers of ethylene glycol (e.g., nbutoxyethanol) are produced by ethoxylation of alcohols, i.e., by addition of ethylene oxide:

Equation (5.5): ∆H 0r298 = − 451.62 + 274.9 + 52.67 = − 124.05 kJ/mol Step 2. ∆H 0r390 with Equation (5.6) Sum of molar heat capacities:

νi −1

n-Butanol

Ethylene −1 oxide nButoxyetha- 1 nol

a

The design of the reactor for the ethoxylation of n-butanol at 390 K is dependent on the enthalpy of this reaction (∆H r390 ). Determination of ∆H r390 consists of three steps: 1) Calculation of ∆H 0r298 from the standard formation enthalpies ∆H 0f298 with Equation (5.5) 2) Calculation of ∆H 0r390 with Equation (5.6) 3) Calculation of the ∆H r390 with n-butanol and n-butoxyethanol as liquids and ethylene oxide in the gaseous state, using Equation (5.7) Step 1.

with Equation (5.5).

∆H 0f298 n-butanol ∆H 0f298 ethylene oxide ∆H 0f298 n-butoxyethanol:

− 274.9 kJ/mol − 52.67 kJ/mol no data available

Estimation with Joback’s method [216], [217]

b

c

d

4.1782 4.543 ×10−1 −37.79 3.682 ×10−1 5.4462 28.255 ×10−1

−2.242 ×10−4 −3.467 ×10−4 − 2.220 ×10−4

4.62 ×10−8 20.86 ×10−8 1.52 ×10−8

−2.414 ×10−1 = ∆b

3.489 ×10−4 = ∆c

− 23.96 ×10−8 = ∆d

61.502 = ∆a

∆H 0r298

573


T ∆cpr dT T0


T =



a+bT +cT 2 +dT 3 dT

T0

∆b 2 T −T02 2

∆d 4

∆c 3 T −T03 + T −T04 + 2 4

=∆a (T − T0 ) +

390


∆cpr dT =924.0Jmol−1

298.15 0 ∆Hr390 = −124.05 + 0.924 = −123.13kJ/mol

Step 3. ∆H r390 with Equation (5.7) 

lv lv −∆Hn−butanol νi ∆Hitrs = ∆Hn−butoxyethanol

= 48.517 − 43.781 = 4.736kJ/mol ∆Hr390 = −123.13 + 4.74 = −118.4kJ/mol

574

Estimation of Physical Properties

5.3.2. Chemical Equilibrium With Equation (5.12 a) the equilibrium constant K of a chemical reaction can be determined. The necessary thermodynamic data have been tabulated for many inorganic and organic compounds. However, in using results of such calculations, inaccuracies in the original data must be considered. Thus enthalpies and Gibbs energies of formation are derived from calorimetric measurements: formation enthalpies of some organic compounds are usually determined from measured heats of combustion. With combustion enthalpies of ca. ≥ 1000 kJ/mol, inaccuracies of about ± 1 kJ/mol in the final values of ∆G 0r and ∆H 0r have to be expected independent of the absolute value of the resulting quantities. If the Gibbs energy of formation has to be estimated for one or more of the reactants (cf. Section 5.2), the error in ∆G 0r is ≥ 10 kJ/mol. In Table 15 an example of the effect of errors in ∆G 0r on K and on equilibrium concentrations is given. Obviously, K values obtained in this way are too inaccurate to be used for exact process calculations without experimental verification. Table 15. Deviations of equilibrium constant K caused by errors in ∆G0r (∆G0r = 0.0 kJ/mol at 500 K) Error in ∆G 0r , kJ/mol

K

+ 1.0 (max.) − 1.0 (min.) + 10.0 (max.) − 10.0 (min.)

0.786 1.272 0.090 11.08

For preliminary investigations and for feasibility studies, however, predicted Gibbs energies of reaction with limited accuracy are helpful. For | ∆G 0r | values > 40 kJ/mol, the equilibrium is either highly favorable for the reaction (for negative ∆G 0r ) or the reaction is practically impossible (for positive ∆G 0r ). With knowledge of ∆G 0r it is possible to predict whether a reaction is impossible or thermodynamically feasible, i.e., whether the reaction equilibrium may yield the desired products if equilibrium can be achieved. Equilibrium may not, however, be reached in practice. For example, the velocity of the equilibrium reaction may be so low that no detectable conversion occurs. In other cases consecutive reactions of the equi-

librium reaction may be so fast that product recovery becomes practically impossible. The application of ∆G 0r for the selection of processes and process conditions will now be demonstrated for hydrocarbon pyrolysis. Figure 12 shows the temperature dependence of the Gibbs energy of formation for several hydrocarbons. All of the hydrocarbons apart from acetylene show increasing instability with higher temperatures, i.e., increasing decomposition into the elements carbon (as graphite) and hydrogen. The much lower temperature dependence of ethylene and benzene means that these compounds are less stable than paraffinic hydrocarbons (e.g., C6 H14 and C20 H42 ) at low temperature and more stable at high temperature. Therefore, in pyrolysis of naphtha (C5 – C10 paraffins) for the production of ethylene (steam cracking), temperatures of 500 – 900 ◦ C are employed. Apparently it is pointless to try to develop a catalyst for the synthesis of ethylene from naphtha at lower temperatures (e.g., 300 – 400 ◦ C). On the other hand naphtha pyrolysis yielding acetylene requires higher temperatures, preferably > 1150 ◦ C; with methane as raw material even higher temperatures are necessary. LIVE GRAPH

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Figure 12. Temperature dependence of the Gibbs energy of formation for selected hydrocarbons a) Methane; b) Ethane; c) Hexane; d) Eicosane (n-C20 H42 ); e) Acetylene; f ) Ethylene; g) Benzene

Estimation of Physical Properties Due to increasing instability at higher temperature, reaction time has to be kept short to prevent decomposition; fast cooling of the reaction mixture after leaving the reactor is also necessary. Example. Calculation of equilibrium constant for the reaction of 2-propanol to diisopropyl ether 2-Propanol is produced by reacting propene with water. In a subsequent reaction diisopropyl ether is formed at ca. 150 – 250 ◦ C:

The loss in 2-propanol due to this reaction might be reduced by altering the reaction temperature. The temperature dependency of the equilibrium constant must therefore be determined. This can be done in three steps: 1) Determination of ∆G 0r298 (with Eq. 5.8) and of the equilibrium constant K 298 (with Eq. 5.12 b) 2) First approximation for K T at temperature T (Eq. 5.17) with ∆H 0r298 by using Equation (5.5). 3) Second approximation for K T with Equation (5.18) Step 1. ∆G 0r298 with Equation (5.8) and K298 with Equation (5.12 b) Sum of the standard Gibbs energies and standard enthalpies of formation; data from [214]. νi

∆G 0f298 , kJ/mol

∆H 0f298 , kJ/mol

2-Propanol − 2 − 173.71 Diisopropyl ether 1 − 121.96 Water 1 − 228.77 ∆G0r298 = − 3.31

− 272.77 − 319.03 − 242.00 ∆H 0r298 = − 15.96

−3310 = 1.3353 8.314 × 298.15 K298 = 3.80 lnK298 = −

Step 2. First approximation of ln K = f (T ) with Equation (5.17). lnK (T ) =1.3353+

−15960 8.314



1 1 − 298.15 T



575

T , ◦C

ln K (T )

K (T )

0 25 50 100 150 200 250

1.9246 1.3353 0.8372 0.0412 −0.5667 −1.0461 −1.4338

6.852 3.801 2.310 1.042 0.567 0.351 0.238

Step 3. Higher accuracy for ln K = f (T ) with Equation (5.18) Example: Reaction temperature
T = 473.15 K (= 200 ◦ C), calculate ∆cpr = νi cpi (385.65K) with Joback’s method [216], [217].

2-Propanol Diisopropyl ether Water

0 385 ,

J mol−1 K−1

νi

cpi

−2 1 1

106.79 191.71 34.35 ∆cpr = 12.48

Evaluation of the last term of the sum in Equation (5.18):  T 298.15 + −1 298.15 T   12.48 473.15 298.15 ln + −1 = 0.138 = 8.314 298.15 473.15

∆cpr R



ln

exp (0.138) = 1.148; K (473.15 K ) = 1.148 0.351 = 0.403 Comparison with the result from Step 2 (K = 0.351) shows that the contribution of the last term of Equation (5.18) is only minor. Although formation of diisopropyl ether is less favored at higher temperature (250 ◦ C), the equilibrium concentration of this product is still considerable. Moreover, the velocity of the sequential reaction increases with temperature. Therefore, a decrease in diisopropyl ether formation is not to be expected when the reaction temperature is increased. In the above calculation tabulated data of ∆G 0f298 are used, in the following K 298 is determined from ∆G 0f298 estimated with Joback’s method. Calculation of K 298 in Step 1 using ∆G 0r298 determined with Joback’s method

576

Estimation of Physical Properties The kinematic viscosity ν [m2 /s] is defined by

2-Propanol Diisopropyl ether Water

νi

∆G 0f298 , kJ/mol

−2 1 1

− 164.88 − 110.24 − 228.56 ∆G0r298 = − 9.04

ν=

−9040 = 3.6469 8.314 × 298.15 K298 = 38.4 lnK =−

The value of K obtained with Joback’s method is ten times higher than that determined with tabulated ∆G0f298 data.

η 

(6.1)

and is still often expressed in Stokes [cm2 /s]. A review on methods for predicting and correlating viscosity data has been given by Monnery et al. [228]. The following discussion is restricted to Newtonian fluids and thus includes most organic liquids and solvents but excludes many polymer solutions, melts, and slurries. Typical viscosity behavior of a pure substance ( ethanol) is shown in Figure 13. LIVE GRAPH

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6. Transport Properties of Pure Compounds and Mixtures Transport properties largely determine the type and dimensions of equipment in the process industries. They are therefore of great importance in the engineering design of many processes – not just those involved in the chemical industry. Although transport properties are macroscopic nonequilibrium quantities, they may, to a certain extent, be related to static equilibrium quantities. In particular, kinetic theory has been very useful in explaining the mechanisms of heat, mass, and momentum transport in dilute gases. For a detailed description of transport phenomena, see → Transport Phenomena. From the ratios of the transport coefficients of the liquid to the gas phase given below it is evident that transport processes in the two phases differ considerably: Density ratio Viscosity ratio Thermal conductivity ratio Diffusion coefficient ratio

l

g

 / η l /η g λl /λg Dl /Dg

3

ca. 10 10 – 100 10 – 100 ca. 10−4

6.1. Viscosity The dynamic viscosity η of a fluid is defined as the ratio of the shear stress to velocity gradient. The SI unit of dynamic viscosity is Pa · s = N · s/m2 but poise is also still in use where 1 P = 1 g s−1 cm−1 = 10−1 N · s/m2 and 1 cP = 1 mPa · s

Figure 13. Viscosity behavior of ethanol [232]

6.1.1. Viscosity of Pure Gases Gases at Low Pressure (< 1 MPa). Kinetic theory yields the following relation for gases consisting of rigid, noninteracting, elastic spherical molecules with a Maxwellian velocity distribution: η=C 

M 1/2 1/2 T σ2

(6.2)

where η is expressed in µPa · s and σ is the colli˚ M is the molecular sion diameter in 10−10 m [A]; mass in g/mol and C
a constant (C
= 2.6693). Chapman and Enskog described the effect of intermolecular forces by considering the potential energy of atoms and molecules [229]. They found that viscosity can be expressed in the general form η=C 

M 1/2 1/2 T σ 2 Ωv

(6.3)

where Ωv , the collision integral, can be calculated from an appropriate potential energy function, such as that of Lennard – Jones (see Section

Estimation of Physical Properties

577

LIVE GRAPH

2.1). Despite the simplifying assumptions used when deriving Equation (6.3), the accuracy of the viscosity of nonpolar gases is quite acceptable (ca. 1 %). Information on the potential energy function and the molecular parameters for more complex molecules is, however, scarce. Values for the molecular parameters and solutions for Ωv for various compounds are tabulated in [230], [231]. Figure 14 shows viscosity data of some organic gases and vapors. In polar gases dipole – dipole interactions must be taken into account. Consequently the collision integral is calculated employing suitable potential functions, e.g., the Stockmayer potential [233]. The accuracy in the estimation of viscosity of polar gases by Equation (6.3) is usually better than 2 %.

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Figure 15. Generalized reduced viscosities as a function of reduced temperature [235] Values on the curve denote the reduced pressure pr = p/pc .  η = C

M 3 p4c Tc

1/6 (6.4)

Figure 15 shows the generalized reduced viscosities as a function of reduced temperature. In the accurate method proposed by Lucas [236], [238] the basic relation for the reduced viscosity η r is ηr =

η ηc

0 FP0 FQ ·  −0.357exp (−0.449Tr ) = 0.807T 0.618 r

+0.34exp (−4.058Tr ) +0.018] ·

(6.5)

where Figure 14. Viscosity data of some gases and vapors [232] a) Helium; b) Carbon dioxide; c) Nitrogen; d) Hydrogen chloride; e) Hydrogen sulfide; f ) Water; g) Ammonia; h) Methane; i) Hydrogen cyanide

Corresponding States Methods. Several methods employ the principle of corresponding states by defining a critical viscosity η c . 1/3 With the relation σ = const. × V c , Equation (6.3) transforms into:

1/6  1 = 1.76 10−4 Tc M −3 p−4 c ηc

(6.6)

η is in µPa · s and pc is in MPa. The correction factors F 0P and F 0Q account for polarity and quantum effects, respectively, and can be obtained from critical data. In terms of the reduced dipole moment µr µr = 524.6

pc 2 µ Tc2

(6.7)

three different equations have been formulated for F 0P :

Next Page 578

Estimation of Physical Properties

FP0 = 1 0 ≤µr 20 : To = 8.164N + +238.59

(6.12b)

and N + ≤ 20 :

2 Ba = 24.79 + 66.885N + −1.3173 N +

3 −0.00377 N + (6.13a)

properties as well as the acentric factor must be known. An overview of selected estimation methods for the viscosity of liquids is given in Table 18. In addition, viscosities of some liquids are presented as a function of temperature in Figure 16. LIVE GRAPH

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N + > 20 : Ba = 530.59 + 13.740N +

(6.13b)

The structure parameters are given in Table 17. Since the method is only fairly accurate (errors 10 – 15 %) it cannot easily be applied to complex molecules [234]. Moreover, the predictions of viscosity for the first members of homologous series often yield higher errors. Example. Calculation of viscosity of liquid diethylamine (CH3 CH2 )2 NH, using the method of Van Velzen et al. [248]. With N = 4: For secondary amines: From Equation (6.13 a): From Equation (6.12 a):

N + = 4 + 1.390 + 0.461×4 = 7.234 I B = 25.39 + 8.744 N+ = 88.64 Ba = 438.27 T 0 = 236.66 K

At 283.2 K η lcal = 0.4306 mPa · s, η lexp = 0.3878 mPa · s [249]; rel. deviation: 11.0 % At 310.8 K η lcal = 0.2944 mPa · s, η lexp = 0.2732 mPa · s [249]; rel. deviation: 7.8 % At higher temperatures (T r > 0.75) the method of Letsou and Stiel [254] should be used. Their method is based upon the principle of corresponding states and valid in the range 0.76 < T r < 0.98. Only liquids at saturation pressure can be considered. The critical

Figure 16. Dynamic viscosity η l of some liquids [232] a) Hydrogen chloride; b) Acetone; c) Trichloromethane; d) Methanol; e) Benzene; f ) Water; g) Decane; h) Formic acid; i) Aniline

6.1.4. Viscosity of Liquid Mixtures The viscosity of a liquid mixture is not usually a linear function of composition and is very difficult to predict. A first estimate of the viscosity of liquid mixtures can be obtained by

580

Estimation of Physical Properties

Table 17. Parameters of structural groups in the method of Van Velzen et al. [248] for estimating viscosities of liquids Group n-Alkanes Isoalkanes Saturated hydrocarbons with two methyl groups in iso position n-Alkenes n-Alkadienes Isoalkenes Isoalkadienes Hydrocarbon with one double bond and two methyl groups in iso position Hydrocarbon with two double bonds and two methyl groups in iso position Cyclopentanes N < 16 N ≥ 16 Cyclohexanes N < 17 N ≥ 17 Alkyl benzenes N < 16 N ≥ 16 Polyphenyls Alcohols: Primary Secondary Tertiary Diols (correction) Phenols (correction) −OH on side chain to aromatic ring (correction) Acids 3 ≤ N < 11 N ≥ 11 Iso acids 3 ≤ N < 11 N ≥ 11 Acids with aromatic nucleus in structure (correction) Esters Esters with aromatic nucleus in structure (correction) Ketones Ketones with aromatic nucleus in structure (correction) Ethers Aromatic ethers Halogenated compounds: Fluoride Chloride Bromide Iodide Special configurations (corrections): C(Cl)x −CCl−CCl− −C(Br)x − −CBr−CBr− CF3 , in alcohols CF3 , in other compounds Aldehydes Aldehydes with an aromatic nucleus in structure (correction) Anhydrides Anhydrides with an aromatic nucleus in structure (correction) Amides Amides with an aromatic nucleus in structure (correction)

IN 0 1.389 – 0.238 N 2.319 – 0.238 N

IB 0 15.51 15.51

−0.152 – 0.042 N −0.304 – 0.084 N 1.237 – 0.280 N 1.085 – 0.322 N 2.626 – 0.518 N

−44.94 + 5.410 N + −44.94 + 5.410 N + −36.01 + 5.410 N + −36.01 + 5.410 N + −36.01 + 5.410 N +

2.474 – 0.560 N

−36.01 + 5.410 N +

0.205 + 0.069 N 3.971 – 0.172 N 1.48 6.517 – 0.311 N 0.60 3.055 – 0.161 N −5.340 + 0.815 N

−45.96 + 2.224 N + −339.67 + 23.135 N + −272.85 + 25.041 N + −272.85 + 25.041 N + −140.04 + 13.869 N + −140.04 + 13.869 N + −188.40 + 9.558 N +

10.606 – 0.276 N 11.200 – 0.605 N 11.200 – 0.605 N

−589.44 + 70.519 N + 497.58 928.83 557.77 213.68 213.68

16.17 – N −0.16

6.795 + 0.365 N −249.12 + 22.449 N + 10.71 −249.12 + 22.449 N + 6.795 + 0.365 N − 0.24 nic −249.12 + 22.449 N + 10.71 − 0.24 nic −249.12 + 22.449 N + (nic = number of methyl groups in iso position) 4.81 −188.40 + 9.558 N + 4.337 – 0.230 N −1.174 + 0.376 N

−149.13 + 18.695 N + −140.04 + 13.869 N +

3.265 – 0.122 N 2.70

−117.21 + 15.781 N + −760.65 + 50.478 N +

0.298 + 0.209 N 11.5 – N

−9.39 + 2.848 N + −140.04 + 13.869 N +

1.43 3.21 4.39 5.76

5.75 −17.03 −101.97 + 5.954 N + −85.32

1.91 – 1.459x 0.96 0.50 1.60 −3.93 −3.93 3.38 2.70

−26.38 0 81.34 – 86.850x −57.73 341.68 25.55 146.45 – 25.11 N + −760.65 + 50.478 N +

7.97 – 0.50 N 2.70

−33.50 −760.65 + 50.478 N +

13.12 + 1.49 N 2.70

524.63 – 20.72 N + −760.65 + 50.478 N +

Estimation of Physical Properties

581

Table 17. (Continued) Group

IN

Amines: Primary Primary amine in side chain of aromatic compound (correction) Secondary Tertiary Primary amines with NH2 group on aromatic nucleus Nitro compounds: 1-nitro 2-nitro 3-nitro 4-nitro; 5-nitro Aromatic nitro-compounds

IB

3.581 + 0.325 N −0.16

25.39 + 8.744 N + 0

1.390 + 0.461 N 3.27 15.04 – N

25.39 + 8.744 N + 25.39 + 8.744 N + 0

7.812 – 0.236 N 5.84 5.56 5.36 7.812 – 0.236 N

−213.14 + 18.330 N + −213.14 + 18.330 N + −338.01 + 25.086 N + −338.01 + 25.086 N + −213.14 + 18.330 N +

Table 18. Selected methods for estimating the viscosity of pure liquids Method

Range of application Substances

Thomas 1946 [250]

Morris 1964 [251] Van Velzen et al. 1972 [248] Orrick – Erbar 1974 [252] Przezdziecki – Sridhar 1985 [253] Letsou – Stiel 1973 [254] Joback – Reid 1987 [255] Tatevskii et al. 1987 [256]

Information required ∗

Error, % ∗∗

T c , l

20

Tc

14

Temperature

no strongly polar and naphthenic compounds but sulfur-containing substances

T > T m+ 5 K T < Tb T r < 0.75 T > T m+ 5 K T r < 0.75 T r < 0.75

no sulfur-containing compounds, no highly branched substances no sulfur-containing compounds, not for the first members of homologous series no sulfur- or nitrogen-containing compounds T r < 0.75 no sulfur- or nitrogen-containing T r < 0.75 compounds, no alcohols nonpolar, weakly polar compounds at 0.76 < T r < 0.98 saturation hydrocarbons T > T m+ 5 K T r < 0.75 T > T m+ 5 K no sulfur-, nitrogen-, or fluorine-containing compounds T r < 0.75

10 – 15 l

T c , M,  15 T c , pc , V c , M, ω,  l 15 T c , pc , M, ω

10 10 18

∗ In addition to the corresponding increments. ∗∗ In some cases errors may be considerably higher.

lnη=



xi lnηi

(6.14)

Methods to date do not allow confident predictions, particularly when dealing with highly polar compounds or components differing greatly in size. On the basis of Eyring’s theory, McAllister published a relation for the kinematic viscosity of binary and ternary mixtures which allows estimations for chemically similar compounds with average errors of 15 % [257]. For details and recommendations see [258]. Viscosity data can be found, e.g., in [259]. The pressure – temperature dependence of vis-

cosity for 50 pure fluids together with a comprehensive discussion of theories for dense fluids are given in [260].

6.2. Thermal Conductivity The thermal conductivity λ is defined as the proportionality constant between heat flux and temperature gradient. It is usually expressed in terms of W m−1 K−1 .

582

Estimation of Physical Properties

6.2.1. Thermal Conductivity of Gases at Low Pressure This chapter discusses methods for the prediction of thermal conductivities at pressures up to 1 MPa. The lower boundary is the region of extremely low pressures (< 10−2 kPa) where the mean free path of molecules becomes large compared to the macroscopic dimensions of the apparatus (Knudsen domain). Thermal conductivities of some organic gases and vapors are shown as a function of temperature in Figure 17. In the limiting case of noninteracting monatomic hard spheres, kinetic theory leads to the basic relation λg = 2.63 × 10−23

T 1/2 M 1/2 σ 2 Ωv

(6.15)

where T is the absolute temperature, M the molecular mass, σ the characteristic dimension of the molecule, and Ωv the collison integral. Equation (6.15) gives the temperature dependence of λg at low pressure.

diction methods (e.g., the Eucken model and the Mason and Moschick approach) have been proposed based on the dimensionless Eucken factor E E=

λM ηcv

(6.16)

where cv is the molar heat capacity at constant volume. For further details see, e.g., [234]. Table 19. Coefficients A, B, and C for the Roy and Thodos method for predicting thermal conductivities of liquids (Eq. 6.20) [234] A T r + B T 2r + C T 3r

Substance Saturated hydrocarbons ∗ Olefins Acetylenes Naphthalenes and aromatics Alcohols Aldehydes, ketones, ethers, esters Amines and nitriles Halides Cyclic compounds (e.g., pyridine, thiophene, ethylene oxide, dioxane, piperidine)

− 0.152 T r + 1.191 T 2r − 0.039 T 3r − 0.255 T r + 1.065 T 2r + 0.190 T 3r − 0.068 T r + 1.251 T 2r − 0.183 T 3r − 0.354 T r + 1.501 T 2r − 0.147 T 3r 1.000 T 2r − 0.082 T r + 1.045 T 2r + 0.037 T 3r 0.633 T 2r + 0.367 T 3r − 0.107 T r + 1.330 T 2r − 0.223 T 3r − 0.354 T r + 1.501 T 2r − 0.147 T 3r

∗ Not recommended for methane.

For the estimation of the thermal conductivity of gases, Roy and Thodos [262] used an approach based upon the reduced thermal conductivity λr λr

λ= . 210

(M 3 Tc /p4c )1/6

/

(6.17)

with the pressure pc expressed in bar. As proposed by Eucken, the reduced thermal conductivity is separated into two terms λr =λtrans +λint r r

LIVE GRAPH

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Figure 17. Thermal conductivities λg of some organic gases and vapors (data from [261]) a) Methane; b) Ethane; c) Dimethylamine; d) Water; e) Ethanol; f ) Dimethyl ether; g) Chloromethane; h) Trichloromethane

The thermal conductivity λ and the viscosity η of a gas are closely related. Successful pre-

(6.18)

The term λtrans considers the translational enr ergy of the molecules and is determined by correlating dilute gas properties with temperature. The second term λint r takes internal (i.e., rotational and vibrational) energy contributions into account and employs specific polynomial coefficients for different classes of compounds. λtrans = r 8.757 [exp (0.0464Tr ) −exp (−0.2412Tr )]

(6.19)

  2 3 λint r =β ATr +B Tr +C Tr

(6.20)

Estimation of Physical Properties The coefficients A, B, and C are tabulated for nine classes of organic compounds (e.g., paraffins, olefins, alcohols). The additional parameter β is determined by employing group increments Ii β=



Ii

(6.21)

The method of Roy and Thodos yields estimates for both polar and nonpolar organic substances at ambient pressure. The constants for this method are given in Tables 19 and 20. The only specific data required are the molecular mass and the critical temperature. The average error is ca. 5 %. Example. Estimation of the thermal conductivity of 2-butanol, CH3 CH2 CH(OH)CH3 , at 480 K by the method of Roy and Thodos. M = 74.122 g/mol, T b = 372.66 K, T c = 535.95 K, pc = 41.95 bar [249].

583

extended their equation for low-pressure thermal conductivity data: 31.2θη 0 M

λg =



 1 1/2 +B6 y +qB7 y 2 T r G2 (6.22) G2

with λ expressed in terms of the reduced density function y (V , V c in cm3 /mol) y=

Vc 6V

(6.23)

and the parameters q= 3.586 × 10−3

(Tc /M )1/2 2/3

Vc

viscosity at low pressure, Pa · s generalized functions of reduced density parameters which are functions of the acentric factor ω, the reduced dipole moment µ (Eq. 6.31), and the association factor k (Eq. 6.24) function of T r , cv /R, and ω (Eq. 6.27)

η0 G1 , G2 B1 – B7

θ

β = (–CH3 ) + (–CH2 –) + (–CH2 –) + (–CH2 ) + (–OH) = 0.73 + 2.0 + 3.18 + 3.68 + 4.62 = 14.21

with A = 0, B = 1, C = 0 from Table 19 From Equation (6.19): From Equation (6.20): From Equation (6.21): Experimental: Relative deviation: − 0.8 %

λtrans = 2.073 λint = 11.398 λ = 0.0316 W m−1 K−1 λ = 0.0324 W m−1 K−1 [263]

Other more sophisticated methods have been suggested based upon the principle of corresponding states, e.g., [264]. These methods employ the Eucken factor and require much more information on the properties of the substances considered, i.e., the critical properties (T c , pc , and V c ), the heat capacity at constant volume cv , and the viscosity. 6.2.2. Thermal Conductivity of Gases at High Pressure Typical p  λ behavior of fluids is displayed in Figure 18. Some methods used for gases at low pressures, particularly the corresponding states methods using the Eucken factor, have been extended to correlate high-pressure thermal conductivity data. For the region close to the critical point, however, other methods must be chosen. The method of Chung et al. [266] is typical and will be briefly described here. These authors

Additional equations: Bi =ai +bi ω+ci µr +di k

(6.24)

The parameters ai , bi , ci and d i are listed in Table 21. G1 =

1 − 0.5y (1−y)3

(6.25)

(6.26) G2 = B1 [1 − exp (−B4 y)] +B2 G1 exp (B5 y) +B3 G1 y (B1 B4 +B2 +B3 ) θ= 1+α



(6.27)  0.215 + 0.28288α − 1.061β +0.26665Z 0.6366+β Z +1.061αβ

α=cv /R − 1.5

(6.28)

β =0.7862 − 0.7109ω+1.3168ω 2

(6.29)

Z =2.0 + 10.5Tr

(6.30)

µr =

131.3µ (Tc Vc )1/2

(6.31)

Data for the associaton factor k are given, e.g., in [234, p. 396]. Since no appropriate correlation for the polar factor is available, the application of the Chung method is limited to nonpolar and a few polar substances. Errors are usually < 8 % [234].

584

Estimation of Physical Properties

Table 20. Parameters for the Roy and Thodos method for predicting thermal conductivities of gases [234]

Estimation of Physical Properties

585

Table 21. Constants (Eq. 6.24) for Chung’s method , p. 522 [234] ∗ i

ai

bi

ci

di

1 2 3 4 5 6 7

2.4166 E + 0 −5.0924 E − 1 6.6107 E + 0 1.4543 E + 1 7.9274 E − 1 −5.8634 E + 0 9.1089 E + 1

7.4824 E − 1 −1.5094 E + 0 5.6207 E + 0 −8.9139 E + 0 8.2019 E − 1 1.2801 E + 1 1.2811 E + 2

−9.1858 E − 1 −4.9991 E + 1 6.4760 E + 1 −5.6379 E + 0 −6.9369 E − 1 9.5893 E + 0 −5.4217 E + 1

1.2172 E + 2 6.9983 E + 1 2.7039 E + 1 7.4344 E + 1 6.3173 E + 0 6.5529 E + 1 5.2381 E + 2

∗ E = exponent of 10: E + 1 = 101 , E + 2 = 102 etc.

LIVE GRAPH

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erties, the acentric factor, the constant-volume heat capacity cv at low pressure, and low-pressure viscosity data are required for the prediction. The accuracy of estimation methods is usually < 10 % [234]. An overview of estimation methods is given in Table 22. 6.2.3. Thermal Conductivity of Gas Mixtures As a first approximation the thermal conductivity of a gas mixture can be described as a linear function of composition. Deviations increase with increasing polarity of the components and increasing difference in size and structure. Methods for the prediction of mixture data are based upon the theoretically derived Wassiljewa equation:

Figure 18. Isotherms of the thermal conductivity of carbon tetrachloride [265]

Example. Calculation of thermal conductivity of carbon dioxide at 473.15 K and 30 MPa M = 44.01 g/mol, ω = 0.239, T c = 304.1 K, V c = 93.9 cm3 / mol [234], µ = 0 Debye; hence from Equation (6.31): µr = 0, V = 112.95 cm3 /mol,  = 8.8537×10−3 mol/cm3 , cv = 39.5 J mol−1 K−1 [267], η 0 = 23.04×10−6 (at 473.15 K, 0.1 MPa). Calculation:

Experimental: Relative deviation

G 1 = 1.4560, G2 = 0.6489, α = 3.251, β = 0.692, θ = 1.8687, z = 18.34 λ = 0.04968 W m−1 K−1 λ = 0.05181 W m−1 K−1 [238] = − 4.1 %

The reliable estimation of thermal conductivities in dense gases is only possible for nonpolar substances; information about the critical prop-

yi λi λm = yj Aij

(6.32)

The binary interaction parameters Aij can be determined employing the methods of Mason – Saxena or Lindsay [234]. Errors are usually < 8 % according to [234], but greater deviations are possible for components differing greatly in size and polarity. Nevertheless, reasonable estimates of thermal conductivities of mixtures can be obtained from pure component data using a linear combination of λi in mole fraction. 6.2.4. Thermal Conductivity of Liquids Thermal conductivities of liquids are significantly higher than those of gases due to their much higher density. Liquid thermal conductivities λl are usually in the range

586

Estimation of Physical Properties

Table 22. Selected methods for estimating the thermal conductivity of gases at low pressure Method

Range of application Substances

Eucken and modifications Stiel – Thodos 1964 [270]

Information required ∗

Error, %

M, cv , η

10 – 15

Temperature and pressure

polar and nonpolar compounds, no associating gases no associating gases

Misic – Thodos 1961 [268] Roy – Thodos 1968, 1970 [262], [269] Ely – Hanley 1983 [264]

hydrocarbons only preferably for polar compounds

Chung et al. 1980 [266]

weakly polar but preferably nonpolar compounds

T r < 10 10−4 < pr < 0.2

low- and high-pressure version T r< 5

M, T c , pc , cv , η

11

M, T c , pc , cp M, T c , pc

8 8

M, T c , V c , zc , ω, cv

5–7

M, T c , ω, cv (η)

5–7

∗ In addition to the corresponding increments.

0.1 – 0.17 W m−1 K−1 (Fig. 19) and are, like liquid densities, only a weak function of pressure. Comprehensive compilations of thermal conductivity data are given in [271], [272]. Since the numerical variation of λl values is small, estimation is comparatively easy, except for strongly polar or associating substances. Empirical methods usually provide estimates of considerable accuracy, often comparable to experimental uncertainty.

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Sato’s Method. Sato found that a simple empirical relationship exits between λl at the normal boiling point T b and the molecular mass M [273]: λl (Tb ) =

1.11 M 1/2

(6.33)

In order to extend this relation the equation [274] can be employed to give / . λl =B 3 + 20 (1−Tr )2/3

(6.34)

where B is a substance-specific parameter [279]. Combination of Equations (6.33) and (6.34) yields 3 + 20 (1−Tr )2/3 1.11 λl = √ · M 3 + 20 (1−Tb /Tc )2/3

Figure 19. Thermal conductivities λl of some organic liquids (data from [261]) a) Methanol; b) Ethanol; c) 1-Butanol; d) 1-Octanol; e) Trichloromethane; f ) Chlorodifluoromethane; g) Aniline; h) Pyridine; i) Nitrobenzene

(6.35)

The only information required is the critical temperature and the normal boiling point. Errors are usually < 15 %, but are generally higher for strongly polar substances.

Method of Latini and Baroncini [275], [276]. This estimation method is based upon the relation λl =

A∗ Tbα (1−Tr )0.38 1/6 M β Tcγ Tr

(6.36)

Specific data required for the application of the method are the normal boiling point T b , the criti-

Estimation of Physical Properties cal temperature T c , and the molecular mass. For chlorofluorohydrocarbons the method is applicable up to T r < 0.9. The exponents α, β, γ and the parameter A∗ are given in Table 23 for various organic compounds. According to [234] the errors in estimating liquid thermal conductivity are usually < 10 %. Table 23. Constants for the Latini – Baroncini method (from [234]) Compounds

A∗

α

β

γ

Saturated hydrocarbons Olefins Cycloparaffins Aromatics Alcohols Organic acids Ketones Esters Ethers Chlorofluorohydrocarbons: R 20, R 21, R 23 Others

0.0035 0.0361 0.0310 0.0346 0.00339 0.00319 0.00383 0.0415 0.0385

1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

0.5 1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0

0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167

0.562 0.494

0 0

0.5 0.5

−0.167 −0.167

Both estimation methods can be employed reliably up to T r < 0.65. Numerous other methods have been published, but are either of limited applicability or require much more specific information without offering significant improvement in reliability and accuracy. An overview of methods for estimating the thermal conductivities of liquids is given in Table 24. Example. Estimation of thermal conductivity of liquid isopropylbenzene, C9 H12 , at 323 K. M = 120.194 kg/kmol, T b = 425.56 K, T c = 631.13 K [249] Experimental: Calculation: Sato’s method (Eq. 6.35): Latini’s method (Eq. 6.36):

λl = 0.1187 W m−1 K−1 [276] λl = 0.125 W m−1 K−1 Relative deviation=5.1 % λl = 0.119 W m−1 K−1 Relative deviation=0.2 %

587

are recommended, preferably with (1 − T r ) as an independent variable. Effect of Pressure on Thermal Conductivity. The thermal conductivity increases significantly at pressures above pr > 0.5. Missenard suggested an empirical approach for the dimensionless thermal conductivity normalized by the thermal conductivity at low pressure (p0 ) [281]: λ1 (pr ) = 1+C p0.7 r λ1 (p0 )

(6.38)

Values of the generalized parameter C = f (T r , pr ) are reported by the author for several compounds. 6.2.5. Thermal Conductivity of Liquid Mixtures The thermal conductivity of organic liquid mixtures λm can usually be approximated with sufficient accuracy by linear combination of the thermal conductivities of the pure components. For binary systems Jamieson et al. recommended the following relation in terms of the weight fraction wi [272]: λm =w1 λ1 +w2 λ2   1/2 w2 −α (λ2 −λ1 ) 1−w2

(6.39)

which employs an adjustable binary parameter α. If no experimental mixture data are available, α is set equal to unity. For multicomponent mixtures Li suggested an equation with volume fractions [282]: λm = 2

r r   i

j

Φ i Φj

1 (1/λi +1/λj )

(6.40)

where the volume fraction Φi of component i is defined by Effect of Temperature on Thermal Conductivity. At low to moderate pressures (< 1 MPa) and over a limited range of temperature, the thermal conductivity decreases linearly with temperature λl =A − B T

(6.37)

For more accurate description over wider temperature ranges, higher order polynomial series

xi Vi Φi = ( xj Vj )

(6.41)

6.3. Diffusion Coefficients The diffusion coefficient D is defined as the proportionality factor between mass flux and the

588

Estimation of Physical Properties

Table 24. Selected methods for estimating the thermal conductivities of liquids Method

Range of application Substances

Robbins – Kingrea 1962 [277]

Information required

Error, %

Temperature 0.4 < T r < 0.9

no nitrogen- or sulfur-containing compounds

T b , T c , cp , l , ∆H lv b T b , clp ,

Missenard 1965 [278] Sato – Riedel 1973 [279]

no low molecular mass hydrocarbons, no highly polar compounds Baroncini – Latini 1978, 1984 [275], no nitrogen- or sulfur-containing [276] compounds, no aldehydes 50 ≤ M ≤ 250 Nagvekar – Daubert 1987 [280] no sulfur-containing compounds

0.5 < T r < 0.75

0.3 < T r < 0.8

M, M, T b , T c

lb

5 8 15

M, T b , T c

8

Tc

6

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concentration gradient; it is usually given in [cm2 /s]. In this discussion only molecular diffusion is considered with concentration gradients (i.e., primarily gradients in chemical potential) as driving force in systems which are free of external force fields and are homogeneous with respect to temperature and pressure. Diffusion is treated comprehensively in [233], [283]. 6.3.1. Diffusion Coefficients of Gases at Low and Moderate Pressures Molecular Theory. Statistical mechanics lead, via integration of the Boltzmann equation, to the Chapman and Enskog equation for the mutual diffusion of the components i and j: Dij =3/16 (2πkT )1/2

(1/Mi +1/Mj )1/2 fD (6.42) 2 Ω πN σij D

where k is the Boltzmann constant; Mi , Mj the molecular masses of components i, j respectively; N the number of molecules per unit volume; σij a characteristic length; and f D a correction term. Introducing the ideal gas law into this equation gives a basic relation that is valid for spherical molecules in dilute gases: Dij =0.01333

T 1.5 [1/Mi +1/Mj ]0.5 1 2 Ω p σij D

(6.43)

This expression ( p in MPa) contains the characteristic length σij (in 10−10 m) and the dimensionless collison integral ΩD , which can be determined from an appropriate intermolecular potential. For monatomic gases the potential function of Lennard and Jones may be chosen (values of σij and ΩD for the Lennard – Jones potential are given for some components, e.g., in [234]).

Figure 20. Diffusion coefficients of selected gases in air at ambient pressure a) Hydrogen; b) Helium; c) Water; d) Ammonia; e) Methane; f ) Methanol; g) Ethanol; h) Benzene; i) Hexane

Employing simple combination rules, i.e., σij =

σi +σj and εij = (εi εj )1/2 2

(6.44)

diffusion coefficients can be determined with reasonable accuracy without considering concentration effects, even for complicated molecules. For mixtures containing highly polar compounds more sophisticated intermolecular potential functions must be applied. Diffusion coefficients of some components in air are

Estimation of Physical Properties shown as a function of temperature in Figure 20.

Method of Fuller et al. Fuller and coworkers suggested an incremental method for Dij [cm2 /s] based on Equation (6.43) [283], [284]: Dij =

1.013 [1/Mi +1/Mj ]1/2 · 104 p

.

T 1.75 1/3

(ID )i

/ 1/3 2

(6.45)

with p in MPa where (Σ I D )i and (Σ I D )j are the diffusion volumes of components i and j, respectively which are given by Fuller et al. for simple molecules such as the noble gases, nitrogen, and oxygen [284]. Some values for diffusion volumes (Σ I D ) follow: He Ne Ar Kr Xe H2 D2 N2 O2 Air

2.67 5.98 16.2 24.5 32.7 6.12 6.84 18.5 16.3 19.7

CO CO2 N2 O NH3 H2 O SF6 Cl2 Br2 SO2

18.0 26.9 35.9 20.7 13.1 71.3 38.4 69.0 41.8

For other molecules (Σ I D ) is calculated by summing up the increments I D for the elements and structural groups in the molecule. Values for these increments follow [284]: C H O N Ring, aromatic Ring, heterocyclic

15.9 2.31 6.11 4.54 −18.3 −18.3

Experimental: D12 = 0.236 cm2 /s (at 328.4 K) [234] Relative deviation: 2.7 % Temperature Dependence of Binary Diffusion Coefficients. Molecular theory (Eq. 6.42) predicts the following form for the temperature dependence of binary diffusion coefficients: D ∼ T 1.5

+ (ΣID )j

F Cl Br I S

14.7 21.0 21.9 29.8 22.9

The authors claim an average accuracy in the prediction of diffusion coefficients of ca. 4 %. Example. Calculation of D12 of ethylene – water using Fuller’s method. 1) C2 H4 : M = 28.05 g/mol; (Σ I D )C2 H4 = 2 C + 4 H = 2×15.9 + 4×2.31 = 41.04 2) H2 O: M = 18.02 g/mol; (Σ I D )H2 O = 2 H +1 O = 2×2.31 + 1×6.11 = 10.73 From Equation (6.45): D12 = 0.242 cm2 /s (at 0.1 MPa, 328.4 K)

589

(6.46)

[neglecting ΩD (T )]. For real systems the exponent is often about 1.8. Effect of Pressure on Binary Diffusion Coefficients of Gases. In general, diffusion coefficients decrease with increasing pressure according to Equation (6.43). At higher pressures ( p >> pc ) the influence of pressure becomes more complex and corresponding states methods must be employed [285], [286]. Effect of Concentration on Diffusion Coefficients of Gases. Binary diffusion coefficients can usually be assumed to be independent of composition. A discussion of the theory of diffusion in multicomponent gas mixtures is given, e.g., in [233], [283], [287]. A procedure for predicting multicomponent diffusion phenomena is given in [288]. 6.3.2. Diffusion Coefficients of Liquids The Stokes – Einstein equation provides the basis for many empirical correlations of diffusion coefficients in liquids. This relation describes the movement of a large hypothetical spherical particle through a solvent consisting of infinitely small molecules. With these assumptions binary diffusion coefficients are given as a function of solvent viscosity ηj and the “molecular” radius of the solute ri : Dij =

kT 6πηj ri

(6.47)

The effect of solvent viscosity on binary diffusion coefficients is shown in Figure 21.

590

Estimation of Physical Properties In many cases the surface tensions σi and σj are similar and their ratio can be set equal to unity, making this expression very convenient. 2) n-Paraffin solutions: ∞ Dij = 1.33 × 10−7

T 1.47 c ηj V 0.71 0i

(6.50)

where the exponent c = (10.2/V 0i ) − 0.791. According to the authors errors are < 4 %. 3) Organic solutes in water:   −0.19 ∞ Dij = 1.25 × 10−8 V0i −0.292 T 1.52 ηjδ (6.51) Figure 21. Diffusivity of carbon tetrachloride in various solvents [289] a) Hexane; b) Heptane; c) Isooctane; d) Methanol; e) Toluene; f ) Benzene; g) Carbon tetrachloride; h) Cyclohexane; i) Ethanol; j) Dioxane; k) Kerosine; l) Decalin

Since Equation (6.47) only holds for simple systems, a variety of empirical correlations have been published. They are usually applicable to solvent – solute systems with low to moderate viscosity (< 20 – 40 mPa · s). Diffusion Coefficients at Infinite Dilution. The diffusion coefficient D∞ ij for solute i at infinite dilution in solvent j is a useful reference value for the determination of binary diffusion coefficients of liquid mixtures. The estimation technique of Hayduk and Minhas comprises several equations for different types of solutions [290]: 1) Nonelectrolyte systems: ∞ Dij =

Pj0.5 1.55 × 10−8 T 1.29 0.23 η 0.92 Pi0.42 V0j j

Example. Calculation of infinite dilution diffusion coefficients using the Hayduk – Minhas method. 1) Acetone in ethyl acetate [249]:

M, g/mol V l 0i , cm3 /mol ηi , mPa · s σi , N/m

Acetone

Ethyl acetate

58.08 73.52

88.106 97.83 0.4508 0.02375

0.02332

From Equation (6.49): D∞ 12 = 2.59 × 10−5 cm2 /s −5 Experimental: D∞ cm2 /s at 12 = 3.18×10 293 K [235] Relative deviation: − 18.6 % 2) Toluene in n-hexane [249]:

(6.48)

where V 0j [cm3 /mol] is the liquid molar volume of solvent j at its normal boiling point, η j its viscosity [mPa · s], and P the parachor (see Section 7.1) [cm3 g0.25 s−0.5 mol−1 ]. From the definition of the parachor as P ≡ σ 0.25 V 0i (Eq. 7.2 where σi is the surface i tension of component i ), Equation (6.48) can be rewritten as ∞ Dij = 1.55 × 10−8

where the parameter δ = (9.58/V l0i ) − 1.12. According to the authors the errors are usually < 10 %.

0.420.27 0.125 V0j T 1.29 σj (6.49) V0i ηj0.92 σi0.105

M, g/mol V l0i , cm3 /mol ηi , mPa · s

Toluene

n-Hexane

92.14 106.9

86.177 131.60 0.2942

Substituting c = − 0.6956 in Equation (6.50): −5 D∞ cm2 /s 12 = 4.90×10 −5 Experimental: D∞ cm2 /s at 12 = 4.12×10 298 K [235] Relative deviation: 18.9 %

Estimation of Physical Properties 3) Aniline in water [249]:

591

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M, g/mol V l0i , cm3 /mol ηi , mPa · s

Aniline

Water

93.128 77.42

18.02 1.002

Substituting δ = − 0.996 in Equation (6.51): −5 D∞ cm2 /s 12 = 1.021×10 ∞ Experimental: D12 = 0.92×10−5 cm2 /s at 293 K [235] Relative deviation: 11.0 %

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The application of the method of Hayduk et al. is somewhat more complicated for systems with associating compounds (e.g., organic acids). Table 25 gives an overview of estimation methods. Effect of Composition on Binary Diffusion Coefficients in Liquids. The dependence of binary diffusion coefficients on concentration is shown in Figure 22. In mixtures with compounds differing greatly in size and polarity the effect of concentration on diffusion coefficients cannot be neglected. Even a linear combination of diffusion coefficients at infinite dilution does not yield accurate results. The thermodynamic correction term (∂ ln ai / ∂ ln xi ) is common to many estimation methods as in the Darken equation [291] for a binary mixture of components i and j

∞ ∞ ∂ lnai Dij = xi Dij +xj Dji ∂ lnxi

(6.52)

where x is the mole fraction and a the activity; ai (xi ) can be obtained from vapor – liquid equilibrium data. From the Gibbs – Duhem equation it follows that ∂ lnai ∂ lnaj = ∂ lnxi ∂ lnxj

(6.53)

Application of the NRTL equation (Section 4.2.1.2) yields for a binary mixture: ∂ lna1 = 1 − 2x1 x2 ∂ lnx1   τ21 G221 τ12 G212 (6.54) × + 3 3 (x1 +x2 G21 ) (x2 +x1 G12 )

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Figure 22. Effect of composition on binary diffusion coefficients [294] A) Benzene – carbon tetrachloride at 298.41 K; B) Chlorobenzene – bromobenzene at 299.93 K; C) Toluene – chlorobenzene at 300.11 K

For systems showing ideal mixing behavior the differential quotient ∂ ln ai / ∂ ln xi is equal to zero. In view of the fact that Equation (6.53) in some cases overcorrects for nonideal behavior, Rathburn and Babb recommended the introduction of an exponential constant f for the thermodynamic correction [292]:

∞ ∞ Dij = xi Dij +xj Dji



∂ lnai ∂ lnxi

f (6.55)

The values for f originally proposed by Rathburn and Babb (in parentheses) have been reevaluated by Siddiqi and Lucas [293]: f = 0.6 (0.6) for positive deviations from Raoult’s law and f = 0.4 (0.3) for negative deviations. Another correlation has been proposed by Vignes [295]: Dij =

.

∞ Dij

xj ∞ xi / ∂ lnai Dji ∂ lnxi

(6.56)

Siddiqi and Lucas found that the average error was less than 5 % for the prediction of 79 data sets. For systems with polar components they report deviations of 11 %.

592

Estimation of Physical Properties

Table 25. Selected methods for estimating diffusion coefficients in liquid solutions at infinite dilution Method

Range of applicability

Wilke – Chang 1955 [296]

not reliable for water as solute

Scheibel 1954 [297]

usually appropriate for aqueous solutions preferably for aqueous solutions

Othmer – Thakar 1953 [298] Reddy – Draiswamy 1967 [299] Lusis – Ratcliff 1968 [300] Tyn – Calus 1975 [301] Nakanishi 1978 [302] Hayduk – Minhas 1982 [290] Siddiqi – Lucas 1986 [293]

preferably for alkane – alkane systems, not for aqueous solutions different correlations for different types of systems not for highly viscous solvents (η < 20 – 30 mPa · s) different correlations for different types of systems including dissolved gases

Information required ∗ l 1 , (T b ),

Error, % ∗∗

M 2 , η2 , V association parameters l M 2 , η 2 , V 1 , (T b )

11 (w), 20 (o)

V l01 , η water M 2 , V l01 , V l02 η 2 V l01 , V l02 , η 2

5 (w) 22 (o), 25 (w) 29 (o)

M 2 , V l01 , V l02 , P1 , P2 , η 2

11 (w), 20 (o)

9

V l01 , V l02 , η 2

14

M 2 , V l02 , V l01 , η 2 , P1 , P2 , (σi )

10

V

l 01 ,

V

l 02 ,

η2

13 (o), 20 (w)

∗ Subscripts: 1 = solute; 2 = solvent. ∗∗ w = aqueous solutions; o = organic solutions.

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( p >> pc ); experimental data can be found in the compilation of Jasper [303]. Water exhibits an extraordinarily high surface tension of 72 mN/m at room temperature. The variation of surface tension with temperature of some organic compounds is shown in Figure 23. Salt melts typically have surface tensions of ca. 100 – 200 mN/m and metal melts of 300 – 1000 mN/m. At low to moderate pressures ( p < pc ) the influence of inert gases can be neglected.

7.1. Surface Tension of Pure Liquids

Figure 23. Variation of surface tension with temperature [304] a) Acetic acid; b) Diethyl ether; c) Ethyl acetate

7. Surface Tension of Liquids

Two basically empirical categories of methods are used for the estimation of surface tension. One type uses the parachor P and the other employs the principle of corresponding states. Parachor Methods. The parachor is a characteristic parameter which, according to its definition in Equation (7.1), is a function of surface tension. It has been defined by Sudgen [305] as 1/4

Surface and interfacial tension influence mass and heat transfer processes considerably. In this chapter only the liquid – vapor interface is discussed. Surface tension σ is expressed in units of [N/m] or preferably in [mN/m] (≡ [dyn/cm]). Surface tensions of organic liquids vary between 20 and 50 mN/m at moderate pressure

Pi =σi

Mi li − vi

(7.1)

where li and vi are the liquid and saturated vapor densities of pure compound i, respectively. At temperatures considerably lower than the critical temperature the density of the vapor vi can be neglected compared to the density of the liquid li , thus simplifying Equation (7.1) to

Estimation of Physical Properties 1/4 Mi li

Pi =σi

1/4

=σi

V li

(7.2)

The parachor is virtually independent of temperature; it is approximately an additive quantity which can be determined by summing up increments. Several such methods have been proposed. The method of Quale [306] is recommended (for increments see Table 26). Surface tensions predicted with this method usually show errors of less than 5 % for nonpolar and weakly polar compounds and 5 – 10 % for compounds with hydrogen bounds [307]. Example. Calculation of surface tension of acetamide, CH3 CONH2 , at 378.2 K from parachor estimated with the method of Quale [306] Structural groups: 1 (CH3 −), 1 (−CONH2 ) (see Table 26) P = 55.5 + 91.7 = 147.2 cm3 g0.25 s−0.5 mol−1 With M = 59.068 g/mol, l = 977.05 kg/m3 , v lv plv 0i = 12.4 hPa [308] and  = p0i M/(RT ) = 3 0.023 kg/m from Equation (7.1): σ = 0.0352 N/m Experimental: σ = 0.03696 N/m at 378.2 K [308] Relative deviation: − 4.9 % (Deviation between Eqs. 7.1 and 7.2 = 0.01 %) Corresponding States Methods. These methods are based upon a relation proposed by Van der Waals [309]: 1/3 1/32/3

σ=kTc

p

(1−Tr )m

(7.3)

where k and m are constants, T c is the critical temperature, pc the critical pressure, and T r the reduced temperature. Employing the Stiel polar factor χ (see Section 2.4), Hakim et al. [310] transformed Equation (7.3) into a relation which is particularly useful for alcohols ( pc in bar, σ in N/m):  σ=p1/32/3 T 1/31/3

1−Tr 0.4

m

Qp (χ,ω) 1000

(7.4)

with the functions Qp and m in terms of the Stiel polar factor χ and the acentric factor ω Qp = 0.156 + 0.365ω−1.754χ −13.57χ2 −0.506ω 2 +1.287ωχ

Errors are usually between 5 and 10 %.

Table 26. Quale’s increments for the estimation of the parachor Pi [cm3 g0.25 s−0.5 mol−1 ] [306] Group Carbon – hydrogen C H CH3 CH2 in −(CH2 )n − n < 12 n > 12 Alkyl groups 1-Methylethyl 1-Methylpropyl 1-Methylbutyl 2-Methylpropyl 1-Ethylpropyl 1,1-Dimethylethyl 1,1-Dimethylpropyl 1,2-Dimethylpropyl 1,1,2-Trimethylpropyl C6 H5 Special groups −COO− −COOH −OH −NH2 −O− −NO2 −NO3 (nitrate) −CO(NH2 ) R−[−CO−]−R (ketone) Number of C atoms in R + R : R + R = 2 R + R = 3 R + R = 4 R + R = 5 R + R = 6 R + R = 7 −CHO O (not noted above) N (not noted above) S P F Cl Br I Ethylenic bonds Terminal 2,3-position 3,4-position Triple bond Ring closure Three-membered Four-membered Five-membered Six-membered

Increment 9.0 15.5 55.5 40.0 40.3 133.3 171.9 211.7 173.3 209.5 170.4 207.5 207.9 243.5 189.6 63.8 73.8 29.8 42.5 20.0 74 93 91.7

51.3 49.0 47.5 46.3 45.3 44.1 66 20 17.5 49.1 40.5 26.1 55.2 68.0 90.3 19.1 17.7 16.3 40.6 12 6.0 3.0 0.8

(7.5)

m= 1.21 + 0.5385ω−14.61χ−32.07χ2 −1.656ω 2 +22.03ωχ

593

(7.6)

A simpler relation based on Equation (7.3) was proposed by Brock and Bird [311]; it should only be used for nonpolar or weakly polar liquids.

594

Estimation of Physical Properties

Example. Calculation of surface tension of 1-butanol, CH3 CH2 CH2 CH2 OH, at 303.2 K according to Hakim et al. [310]. M = 74.122 g/mol, pc = 44.13 bar, T c = 563.0 K, χ = − 0.07, ω = 0.593 [308]. From Equation (7.6): m = 0.8981. From Equation (7.5): Qp = 0.1974. From Equation (7.4): σ = 0.02314 N/m. Experimental: σ = 0.02378 N/m [308]. Relative deviation: − 2.7 %.

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7.2. Surface Tension of Liquid Mixtures Organic Systems. Surface tensions of mixtures of organic compounds can be estimated by the simple mixing rule: σm =



xi σi

(7.7)

where xi is the mole fraction of component i or by the more general relation β σm =



xi σiβ

(7.8)

with the exponent β. An exponent β = − 1 is sometimes used: σm =



xi σi

Figure 24. Surface tension of binary organic mixtures at 298 K [311] a) N itrobenzene – carbon tetrachloride; b) Acetophenone – benzene; c) N itrobenzene – benzene; d) Diethylether – benzene The mole fraction refers to the underlined component.

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−1 (7.9)

Equation (7.2) can be extended for nonaqueous mixtures: σm =



 Pi

lm xi M

4 (7.10)

where xi and yi are the mole fractions of compound i in the liquid and vapor phases, respectively, and   1 = xi /i m

(7.11)

Again, the parachor of the pure compounds is employed (see Section 7.1). When experimental surface tension data for the mixture are available, Pi can also be used as a correlation parameter. The accuracy of Equation (7.10) is usually < 10 %. If the parachor is applied as an adjustable parameter using experimental information, precision is of course much higher. The concentration dependence of the surface tension of some binary organic solutions is shown in Figure 24.

Figure 25. Surface tension of aqueous solutions of organic compounds (data from [312]) a) Sucrose; b) Methanol; c) Acetonitrile; d) Ethanol

A large number of more sophisticated methods have been published based on the principle of corresponding states or on phenomenological or statistical thermodynamics. These methods are discussed in detail in [307].

Estimation of Physical Properties Aqueous Systems. The surface tension of water is strongly affected by small amounts of dissolved organic compounds because hydrophobic molecules are repelled by the water molecules and are therefore enriched at the surface. The surface tension of aqueous solutions of organics therefore depends on the hydrophobicity of the organic solute with a highly nonlinear influence of solute concentration. In general, the surface tension of water is greatly decreased by hydrophobic solutes (Fig. 25). For estimating the effect of organic solutes on the surface tension of water Meissner et al. [313] have proposed a method which is based on the following equation: .  xi / σm =σ0w 1 − 0.411log 1+ A

(7.12)

where σ 0w is the surface tension of pure water, xi the molar concentration of the organic component, and A is a substance-specific constant. Values of A (×104 ) for a number of organic substances follow: Propionic acid 1-Propanol 2-Propanol Methyl acetate

26 26 26 26

Propylamine 19 Methyl ethyl 19 ketone Butyric acid 7 Isobutyric acid 7 Butanol 7 Isobutanol 7 Propyl formate 8.5 Ethyl acetate 8.5 Methyl propionate 8.5 Diethyl ketone 8.5

Ethyl propionate Propyl acetate

3.1 3.1

Valeric acid Isovaleric acid Pentanol Isopentanol

1.7 1.7 1.7 1.7

Propyl propionate Caproic acid Heptanoic acid Octanoic acid Decanoic acid

1.0 0.75 0.17 0.034 0.0025

The accuracy of this method is reported to be better than 15 %. Another more precise but more complicated estimation method was suggested by Tamura et al. [314]. This method is applicable to binary mixtures only and is discussed in detail in [307].

8. References General References Methods 1. S. Bretsznajder: Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford 1971.

595

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210. M. W. Chase et al.: Joint Army Navy Air Force (JANAF) Thermochemical Tables, 3rd ed., Journal of Physical and Chemical Reference Data, Suppl. 14,1, American Chemical Society, American Institute of Physics, Washington, D.C., 1986. 211. J. D. Cox, G. Pilcher: Thermochemistry of Organic and Organometallic Compounds, Academic Press, London 1970. 212. F. D. Rossini: Selected Values of Chemical Thermodynamic Properties, National Bureau of Standards, Circular 500, Washington 1952. 213. D. R. Stull, G. C. Sinke: “Thermodynamic Properties of the Elements,” in: Advances in Chemistry Series, vol. 18, American Chemical Society, Washington, D.C., 1956. 214. D. R. Stull, E. F. Westrum, G. C. Sinke: The Chemical Thermodynamics of Organic Compounds, Wiley, New York 1969. 215. Thermodynamics Research Centre (TRC): Selected Values of Properties of Chemical Compounds, Texas A & M University, College Station, Texas, 1982 Selected Values of Properties of Hydrocarbons and Related Compounds, vols. 1, Texas A & M University, College Station, Texas, 1977; TRC Thermodynamic Tables – Non-Hydrocarbons, vols. 1 – 8, Texas A & M University, College Station, Texas, 1955; TRC Thermodynamic Tables – Hydrocarbons, vols. 1 – 11, Texas A & M University, College Station, Texas, 1942. 216. K. G. Joback: MS thesis, Massachusetts Institute of Techn., Cambridge, Mass., 1984. 217. K. G. Joback, R. C. Reid, Chem. Eng. Commun. 57 (1987) 233. 218. R. C. Reid, J. M. Prausnitz, B. E. Poling: The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York 1987. 219. S. W. Benson: Thermochemical Kinetics, Wiley, New York 1968. 220. S. W. Benson et al., Chem. Rev. 69 (1969) 279. 221. Y. Yoneda, Bull. Chem. Soc. Jpn. 52 (1979) 1297. 222. J. L. Franklin, Ind. Eng. Chem. 41 (1949) 1070; J. Chem. Phys. 21 (1953) 2029. 223. K. K. Verma, L. K. Doraiswamy, Ind. Eng. Chem. Fundam. 4 (1965) 389. 224. D. W. Van Krevelen, H. A. G. Chermin, Chem. Eng. Sci. 1 (1951) 66; Chem. Eng. Sci. 1 (1952) 238. 225. L. Constantinou, R. Gani, AIChE J. 40 (1994) 1697. 226. R. C. Reid, J. M. Prausnitz, T. K. Sherwood: The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York 1977.

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227. R. C. Reid, T. K. Sherwood: The Properties of Gases and Liquids, 2nd ed., McGraw-Hill, New York 1966. 228. W. D. Monnery, W. Y. Svrcek, A. K. Mehrota, Can. J. Chem. Eng. 73 (1975) 3. General References References for Chapter 6 229. S. Chapman, T. G. Cowling: The Mathematical Theory of Nonuniform Gases, Cambridge, NY, 1939. 230. S. Weiß et al. (eds.): Verfahrenstechnische Berechnungsmethoden, part 7: Stoffwerte, VCH Verlagsgesellschaft, Weinheim 1986. 231. R. C. Reid, J. M. Prausnitz, T. K. Sherwood: The Properties of Gases and Liquids, 3d ed., McGraw-Hill, New York 1977. 232. VDI W¨armeatlas, 7th ed., VDI Verlag, D¨usseldorf 1994. 233. J. O. Hirschfelder, C. F. Curtiss, R. B. Bird: Molecular Theory of Gases and Liquids, 4th print, Wiley, New York 1967. 234. R. C. Reid, J. M. Prausnitz, B. E. Poling: The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York 1987. 235. O. A. Uyehara, K. M. Watson, Natl. Pet. News 36 (1944) R 714. 236. K. Lucas: Phase Equilibria and Fluid Properties in the Chemical Industry, Dechema, Frankfurt 1980, p. 573. 237. K. Lucas, Chem. Ing. Tech. 53 (1981) 959. 238. VDI-W¨armeatlas, 7th ed., VDI-Verlag, D¨usseldorf 1994. 239. C. R. Wilke, J. Chem. Phys. 18 (1950) 517. 240. R. S. Brokaw, NASA Tech. Note D-4496, Apr. 1968. 241. R. S. Brokaw, Ind. Eng. Chem. Process Des. Dev. 8 (1969) 240. 242. D. Reichenberg, Symp. Transp. Prop. Fluids and Fluid Mixtures, Natl. Eng. Lab., Glasgow, UK, 1979; see also [234], p. 404. 243. T.-H. Chung, L. L. Lee, K. E. Starling, Ind. Eng. Chem. Fundam. 23 (1984) 8. 244. T. H. Chung, L. L. Lee, K. E. Starling, Ind. Eng. Chem. Res. 27 (1988) 671. 245. L. A. Bromley, C. R. Wilke, Ind. Eng. Chem. 43 (1951) 1641. 246. D. Reichenberg, AIChE J. 16 (1970) 854; AIChE J. 21 (1975) 181. 247. E. N. da C. Andrade, Endeavour 13 (1954) 117. 248. D. Van Velzen, R. L. Cardozo, H. Langenkamp, Ind. Eng. Chem. Fundam. 11 (1972) 20.

249. J. A. Riddick, W. B. Bunger: Organic Solvents, Physical Properties and Methods of Purification, 3rd ed., vol. II, 1970, 4th ed., 1986, J. Wiley, New York. 250. L. H. Thomas, J. Chem. Soc. 1946, 573. 251. P. S. Morris, MS thesis, Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1964. 252. C. Orrick, J. H. Erbar in [234], p. 456. 253. J. W. Przezdziecki, T. Sridhar, AIChE J. 31 (1985) 333. 254. A. Letsou, L. I. Stiel, AIChE J. 19 (1973) 409. 255. K. G. Joback, R. C. Reid, Chem. Eng. Commun. 57 (1987) 233. 256. V. M. Tatevski, A. V. Abramenkov, O. E. Grikina, Russ. J. Phys. Chem. 61 (1987) 1454. 257. R. A. McAllister, AIChE J. 6 (1960) 427. 258. J. B. Irving: “Viscosities of Binary Liquid Mixtures: A Survey of Mixture Equations,” NEL Rep. GB 630 (1977); “Viscosities of Binary Liquid Mixtures: The Effectiveness of Mixture Equations,” NEL Rep. GB 631 (1977). 259. Y. S. Touloukian, C. Y. Ho: “Viscosity. Thermophysical Properties of Matter,” The TPRC Data Ser., vol. 2, Plenum Press, New York 1975. 260. K. Stephan, K. Lucas: Viscosity of Pure Dense Fluids, Plenum Press, New York 1979. 261. W. Blanke (ed.): Thermophysikalische Stoffgr¨oßen, Springer Verlag, Berlin 1989. 262. D. Roy, G. Thodos, Ind. Eng. Chem. Fundam. 7 (1968) 529. 263. N. B. Vargaftig: Tables on the Thermophysical Properties of Gases and Liquids, 2nd ed., Hemisphere, Washington 1975. 264. J. F. Ely, J. M. Hanley, Ind. Eng. Chem. Fundam. 22 (1983) 90. 265. L. A. Guildner, Proc. Natl. Acad. Sci. USA 44 (1958) 1149. 266. T. H. Chung, L. L. Lee, K. E. Starling, Ind. Eng. Chem. Fundam. 19 (1980) 186. 267. S. Angus et al. (eds.): IUPAC: International Thermodynamic Tables of the Fluid State: Carbon Dioxide, Pergamon Press, Oxford 1973. 268. D. Misic, G. Thodos, AIChE J. 7 (1961) 264; J. Chem. Eng. Data 9 (1963) 540. 269. D. Roy, G. Thodos, Ind. Eng. Chem. Fundam. 9 (1970) 71. 270. L. I. Stiel, G. Thodos, AIChE J. 7 (1961) 611; AIChE J. 10 (1964) 26. 271. Y. S. Touloukian, C. Y. Ho: “Thermal Conductivity. Thermophysical Properties of Matter,” The TPRC Data Ser., vol. 3, Plenum Press, New York 1972.

Estimation of Physical Properties 272. D. T. Jamieson, J. B. Irving, J. S. Tudhope: Liquid Thermal Conductivity: A Data Survey to 1973, H. M. Stationery Office, Edinburgh 1975. 273. T. Maejima in [234], p. 550. 274. L. Riedel, Chem. Ing. Tech. 21 (1949) 349; 23 (1951) 59, 321, 465. 275. G. Latini, M. Pacetti, Therm. Conduct. 15 (1978) 245. 276. C. Baroncini, G. Latini, P. Pierpaoli, Int. J. Thermophys. 5 (1984) no. 4, 387. 277. L. A. Robbins, C. L. Kingrea, Hydrocarbon Process. Pet. Refiner 41 (1962) 133. 278. A. Missenard, Comptes Rendus 260 (1965) 5521. 279. K. Sato in [234], p. 550 280. M. Nagvekar, T. E. Daubert, Ind. Eng. Chem. Res. 26 (1987) 1362. 281. A. Missenard, Rev. Gen. Therm. 5 (1970) no. 101, 649. 282. C. C. Li, AIChE J. 22 (1976) 927. 283. W. Jost: Diffusion in Solids, Liquids, Gases, Academic Press, New York 1960. 284. E. N. Fuller, J. C. Giddings, J. Gas Chromatogr. 3 (1965) 222. E. N. Fuller, K. Ensley, J. C. Giddings, Ind. Eng. Chem. 58 (1966) no. 5, 18. 285. L. S. Tee, G. R. Kuether, R. C. Robinson, W. E. Stewart, Am. Pet. Inst. Div. Refin. May 1966. 286. S. Takanishi, J. Chem. Eng. Jpn. 7 (1974) 417. 287. E. L. Cussler: Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge 1984, Chaps. 3, 7. 288. D. Reinhardt, K. Dialer, Chem. Eng. Sci. 36 (1981) 1557. 289. W. Hayduk, S. C. Cheng, Chem. Eng. Sci. 26 (1971) 635. 290. W. Hayduk, B. S. Minhas, Can. J. Chem. Eng. 60 (1982) 295. 291. L. S. Darken, Trans. Am. Inst. Min. Metall. Eng. 175 (1948) 184. 292. R. E. Rathburn, A. L. Babb, Ind. Eng. Chem. Process Des. Dev. 5 (1966) 273. 293. M. A. Siddiqi, K. Lucas, Can. J. Chem. Eng. 64 (1986) 839.

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294. H. R. Kamal, L. N. Canjar, AIChE J. 8 (1962) no. 3, 329. 295. A. Vignes, Ind. Eng. Chem. Fundam. 5 (1966) 189. 296. C. R. Wilke, P. Chang, AIChE J. 1 (1955) 264. 297. E. G. Scheibel, Ind. Eng. Chem. 46 (1954) 2007. 298. D. F. Othmer, M. S. Thakar, Ind. Eng. Chem. 45 (1953) 589. 299. K. A. Reddy, L. K. Draiswamy, Ind. Eng. Chem. Fundam. 6 (1967) 77. 300. M. A. Lusis, G. A. Ratcliff, Can. J. Chem. Eng. 46 (1968) 385. 301. M. T. Tyn, W. F. Calus, J. Chem. Eng. Data 20 (1975) 106. 302. K. Nakanishi, Ind. Eng. Chem. Fundam. 17 (1978) 253. General References References for Chapter 7 303. J. J. Jasper, J. Phys. Chem. Ref. Data 1 (1972) 841. 304. D. B. Macleod, Trans. Faraday Soc. 19 (1923) 38. 305. S. Sugden, J. Chem. Soc. 125 (1924) 1177. 306. Q. R. Quale, Chem. Ref. 53 (1953) 439. 307. R. C. Reid, J. Prausnitz, B. E. Poling: The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York 1987. 308. J. A. Riddick, W. B. Bunger, T. K. Sakano: Organic Solvents. Physical Properties and Methods of Purification, 4th ed., vol. II, Wiley, New York 1986. 309. J. D. Van der Waals, Z. Phys. Chem. 13 (1894) 716. 310. D. I. Hakim, D. Steinberg, L. I. Stiel, Ind. Eng. Chem. Fundam. 10 (1971) 174. 311. J. R. Brock, R. B. Bird: AIChE J. 1 (1955) 174. 312. W. Blanke (ed.): Thermophysikalische Stoffgr¨oßen, Springer Verlag, Berlin 1989. 313. H. P. Meissner, A. S. Michaelis, Ind. Eng. Chem. Ind. Ed. 41 (1949) 2782. 314. M. Tamura, M. Kurata, H. Odani, Bull. Chem. Soc. Jpn. 28 (1955) 83.

Construction Materials in Chemical Industry

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Construction Materials in Chemical Industry ¨ Hubert Grafen, Bayer AG, Leverkusen, Federal Republic of Germany

1. 2. 2.1. 2.2. 2.3. 2.4. 3. 4.

5. 5.1. 5.1.1. 5.1.2. 5.1.3. 5.1.4. 5.1.5. 5.1.6. 5.1.6.1. 5.1.6.2.

Introduction . . . . . . . . . . . . . . Material Requirements . . . . . . . Processability and Joining . . . . . Mechanical Stability and its Dependence on Temperature . . . Corrosion Resistance . . . . . . . . Resistance to Wear . . . . . . . . . . Choice of Materials . . . . . . . . . Quality Assurance through Material Tests and Checking of Fabrication and Functioning . Properties and Applications of Materials . . . . . . . . . . . . . . Steels . . . . . . . . . . . . . . . . . . . Unalloyed and Low-Alloy Steels for Vessels and Pipelines . . . . . . . . . Steels with High-Temperature Strength . . . . . . . . . . . . . . . . . Heat-Resistant Steels . . . . . . . . . Steels for Low Temperatures . . . . Steels Resistant to Pressurized Hydrogen . . . . . . . . . . . . . . . . . . Stainless Steels . . . . . . . . . . . . . Technical Properties . . . . . . . . . . Chemical Properties . . . . . . . . . .

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1. Introduction Components of chemical plant are generally subjected to thermal, chemical, and mechanical stresses. The combination of these stresses places very heavy demands on plant materials, especially with regard to corrosion. Thus even unalloyed and low-alloy steels have to meet very strict quality requirements. They must have high purity, for example, and their nonmetallic inclusions must be finely dispersed, since they affect the ability of plant to withstand corrosion cracking, such as stress corrosion cracking and damage by hydrogen. By far the most important materials are the highly alloyed stainless steels and nickel-based alloys, though aluminum, copper and their alloys, and refractory metals, and organic and inorganic materials are also important. In this article the applications of materials

5.1.6.3. 5.2. 5.3. 5.3.1. 5.3.2. 5.3.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.10.1. 5.10.2. 5.10.3. 5.11. 5.11.1. 5.11.2. 5.11.3. 5.11.4. 6.

Development State of Stainless Cr – Ni Steels . . . . . . . . . . . . . . . . . Cast Iron . . . . . . . . . . . . . . . . Nickel and Nickel Alloys . . . . . . Nickel – Copper Alloys . . . . . . . . Nickel – Chromium Alloys . . . . . . Nickel – Molybdenum and Nickel – Molybdenum – Chromium Alloys . Aluminum and Aluminum Alloys Copper and Copper Alloys . . . . Lead and Lead Alloys . . . . . . . . Zinc and Zinc Alloys . . . . . . . . Tin and Tin Alloys . . . . . . . . . . Titanium, Zirconium, Niobium, and Tantalum . . . . . . . . . . . . . Organic Materials . . . . . . . . . . Selection Criteria . . . . . . . . . . . Properties and Application Criteria Thermosetting Plastics . . . . . . . . Inorganic Nonmetallic Materials Glass . . . . . . . . . . . . . . . . . . . Graphite . . . . . . . . . . . . . . . . . Refractory and Acid-Resistant Bricks . . . . . . . . . . . . . . . . . . Engineering Ceramics . . . . . . . . References . . . . . . . . . . . . . . .

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in process plant manufacture are described and development trends are discussed.

2. Material Requirements In the choice of materials for production plant in the chemical industry, there are three basic considerations: 1) The processibility of the material in its commercially available form (sheet, piping, profiles, etc.) 2) The ability of materials to withstand production processes. This is a complex property that includes mechanical stability and its dependence on temperature, resistance to corrosion, and possibly resistance to wear also. 3) The costs of materials, of their processing,

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and of the inspections of the chemical apparatus during its useful lifetime. As so many factors have to be taken into consideration, choosing materials is not easy, especially if the plant is to be used for a recently developed chemical process. The prospect of a suitable choice is best where the material scientist, chemical engineer, plant designer, and plant engineer have worked closely together [1], [2].

2.2. Mechanical Stability and its Dependence on Temperature [9] Chemical apparatus in use is subjected to widely differing mechanical stresses. The possible variations of mechanical stress with time are shown schematically in Figure 1. These stresses may be monoaxial, biaxial, or triaxial.

2.1. Processability and Joining Shaping (e.g., bending, rounding, and flanging), separating (e.g., cutting and machining) and joining (e.g., welding, bonding, and pipe rolling) are particularly important processes in the fabrication of chemical plant. Coating and processes that modify the properties of materials (e.g., quenching, tempering, nitriding, age hardening) are also widely used. The choice of fabrication processes depends on the properties of the material concerned, e.g., on its suitability for cold shaping or welding. Fabrication must not affect materials so drastically that their resistance to the conditions of subsequent use is significantly impaired. Joint welding is by far the most important joining process in chemical plant fabrication, while build-up welding is used widely both for coating in original plant fabrication, and for repairs [3–5]. The weldability of a component depends on the weldability of the material and on the design and fabrication of the part [6]. Although each factor may be decisive on its own, the interplay of factors must never be overlooked. It is thus pointless to choose a steel with the highest possible yield strength unless one has checked that dangerous peak stress will not occur (through deficiencies of design) and that welding defects will not be caused by the use of welding techniques unsuited to the material. Bonding is used for materials that cannot be welded or whose properties are changed excessively by welding [7], [8]. As adhesive bonds cannot be exposed to elevated temperatures, this method of joining parts has acquired little importance for chemical apparatus.

Figure 1. Forms of the time function of mechanical stress

The behavior of materials under polyaxial stress – the states of stress in plant components are almost always of this kind – is seldom known as such, because strength data are usually available only as yield strength and tensile strength for loads exerted monoaxially in a tensile test. Therefore, where a particular exposure is concerned, the material’s mechanical stability must be assessed by comparing its known mechanical strength values with a stress calculated according to the appropriate theory of strength [10]. In the fabrication of chemical apparatus preference is given to materials that are easily shaped, since they react to the application of excessive force by undergoing energy-consuming changes of shape instead of simply breaking. Temperature has an important influence on the mechanical stability of materials. As the temperature rises, strength decreases, ease of shaping increases, and creep behavior becomes the main factor determining mechanical stability. Thus, above a limit temperature, which depends on the material, the yield strength or tensile strength at the envisaged operating temperature can no longer serve as a characteristic value for calculating the stress of a chemical apparatus (Fig. 2, see next page) [9]. The strength of metals increases with decreasing temperature. Therefore, components intended for use at room temperature might be expected to be more resistant to loads exerted at lower temperatures. That this is only partly true is explained by the deformation behavior of materials. The characteristic strain values of metals tend to drop suddenly as the tempera-

Construction Materials in Chemical Industry ture decreases, and this is accompanied by an increase in the tendency to undergo brittle fracture. The inherent risk of brittle fracture is not accounted for directly by strength calculations.

Figure 2. Dependence of characteristic strength data on temperature (schematic)

Susceptibility to brittle fracture is represented by values for elongation at rupture and reduction of area in the tensile test, and also by the transition temperature obtained from a plot of the notched-bar impact energy versus temperature (Fig. 3) (see also → Mechanical Properties and Testing of Metallic Materials). In choosing a material to suit a particular set of requirements, adequate safety must be ensured by taking this transition temperature into consideration, as well as the influences exerted on the material during its production and processing. In the case of highstrength materials, it is also advisable to perform fracture toughness testing [11].

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will enable a part to be designed so that it satisfies safety requirements is often difficult, because the combinations of loads exerted in practical use cannot be accounted for fully by measuring the alternating stress endurance limit (e.g., by the W¨ohler method). The shape and surface condition of the part must also be considered (Fig. 4). More realistic criteria can be obtained by determining the resistance to service conditions of components themselves. The AD (Arbeitsgemeinschaft Druckbeh¨alter) leaflets [12] on calculations for pressure vessels or their components are based on the assumption that these vessels are normally subjected to static loads. If the pressure fluctuates, the resultant additional stresses can be taken into account according to AD leaflet S 1, in which allowance for them is made by reducing the permissible stresses (Fig. 5). Prediction of useful lifetime is particularly difficult when alternating stresses and creep stresses are superimposed. LIVE GRAPH

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LIVE GRAPH

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Figure 4. Influence of the surface quality on alternating stress tests on high-presure tubes (nominal bore 6 mm; nominal pressure 250 MPa) made of alloy 30CrMoV9 (material no. 1.7707; HB30 ≈ 3000 MPa; tensile strength = 900 – 1100 MPa) a) Cold-worked; b) Nitrided; c) Electropolished; d) Nitrided; e) Polished; f) and g) As-delivered

Figure 3. Transition temperature T t of the notched-bar impact energy a) High level, fracture on working; b) Transition range, mixed fracture; c) Low level, brittle fracture

Where alternating stresses occur, determining suitable characteristic material values that

In such cases there is still no accepted method for prediction of useful life. It would also appear unrealistic to seek “universally valid” theories and better to simulate typical stress processes in test specimens similar to plant components, thus providing the design engineer with useful life data for particular kinds of materials and problems. According to the available experience it is best to design plant and components so that they

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can withstand the stress that gives the shortest useful life. LIVE GRAPH

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iron-based and nickel-based alloys, this interrelation affects the static load fatigue resistance of parts exposed to corrosion processes at high temperatures. LIVE GRAPH

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Figure 5. Admissible number of cycles for pressure vessels as a function of stress a) Range of alternating compressive stress = 90 %; b) Range of alternating compressive stress = 100 %

2.3. Corrosion Resistance (→ Corrosion) The longevity of most apparatus depends not only on mechanical loads but also on the nature and spectrum of the corrosive ambient medium. As a rule the characteristic symptom of corrosion damage is not the loss of material but an impairment of function and load-bearing capacity. Uniform corrosion, for example, may cause a considerable loss of mass before the serviceability of a part begins to suffer. Localized or selective corrosion proceeds rapidly towards the inside of a part, such that a notching effect is exerted on parts that bear mechanical loads. Corrosion of this kind may rapidly cause vessels to leak, or parts to fail through low-ductility fracture. If the surface becomes creviced or pitted, the stress becomes nonuniform, and the operational stability of the component under alternating stresses is impaired. This also reduces the static fatigue resistance under constant load, as shown by the example in Figure 6 [13]. The static fatigue resistance of copper samples exposed simultaneously to corrosion and mechanical loading is impaired to a much greater extent that of precorroded samples that have already undergone a considerably greater loss of mass and are then stressed mechanically. This exemplifies the interaction between corrosion and creep. For

Figure 6. Creep rupture behavior of copper at 95 ◦ C a) 0.5 M H2 SO4 (air saturated); b) 0.5 M H2 SO4 (air); c) Air (precorroded in 0.5 M H2 SO4 ); d) Air ∆m = Mass loss, wt %

Basically it may be assumed that the behavior of a part exposed to corrosive influences depends largely on the stability of the material, which, in turn, depends mainly on the film which forms at the surface. Efforts are therefore being made to find alloys that can improve the conditions for surface film formation. It is also necessary to ensure that the microstructures of alloys remain as stable as possible, so that demixing (which, for example, could impair the passivation capability of alloys) cannot occur in welding, hot forming, etc. This is particularly important because precipitation may strengthen the tendency towards local activation (local removal of the passive layer), whereby local corrosion may be initiated. Highly localized corrosion that results from simultaneous chemical attack and mechanical stress is particularly serious. The criterion of failure here is the occurrence of lowductility fracture. Stress corrosion cracking, fatigue cracking, and water-induced cracking are phenomena of this kind. They are particularly important with regard to serviceability, useful life, and operational reliability. The special danger of anodic stress corrosion cracking and hydrogen-induced cracking is that they can rarely be detected while the cracks are spreading and before the apparatus starts to leak

Construction Materials in Chemical Industry or a part breaks. Although anodic stress corrosion cracks outwardly resemble brittle fractures, the metal itself retains its ductility. The initiation of cracking by the various stress corrosion cracking mechanisms that result from the action of a given corrodent on a given material depends very much on the mechanical stresses involved. In the case of fatigue cracking the loss of fatigue strength for finite life and loss of fatigue strength have caused considerable difficulties in the dimensioning of parts.

2.4. Resistance to Wear (see also → Abrasion and Erosion) Since all production processes, except chemical reactions, involve physical operations such as comminution, conveyence, and separation of phases, the wear resistance of materials is important for many types of chemical apparatus. Models of the various kinds of wear have been developed [14]; in practice, however, the conditions are complex, for not only may severals kinds of wear occur simultaneously, but overlap and interaction of wear and corrosion processes can also arise. Thus, as a rule, only practical trials reveal the actual stress conditions involved. Compared to corrosion, however, wear is less problematic because it does not cause cracklike damage, and its effects are more easily repaired, e.g., by build-up welding.

3. Choice of Materials Designers of chemical apparatus must pay careful attention to materials as well as the purpose for which the apparatus is intended. This is important at all stages, from initial planning to the detailed drawing of the finalized design. For a selection process, however, not just the function and properties needed by the components, but also the operations by which the components are fabricated from the material, must be taken into consideration. Not only do these operations call for definite properties on the part of materials, they may also affect the behavior of components under service conditions.

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The demands made on chemical plant in use are becoming increasingly strict, and the variety of fabrication processes and range of materials available are growing. These factors, together with the need to use economical fabrication techniques, have given the choice of materials a complexity that calls for systematic consideration. A basic procedure for the selection of materials for chemical plant is shown in Figure 7 (see next page). Clearly the demands of the application and those of the fabrication process must be taken into account as fully as possible so that the overall requirements are correctly formulated and properly interpreted in the fixing of verifiable property data. At the same time the behavior of materials under operating conditions and those of fabrication must be known so that one can judge reliably whether or not a given material is suitable for a given purpose. Often, these preconditions cannot be met entirely. In many cases, therefore, model experiments and serial testing are necessary as empirical aids to selection. Establishing and then interpreting the combination of mechanical and corrosion-chemical exposures, all of which are often supplemented by wear, is particularly difficult for chemical plant. Frequently, corrosion behavior is treated in the selection procedure as the decisive criterion. In fact, experience in the operation of chemical plant has shown that much of the damage that occurs to chemical apparatus arises from the use of materials that are insufficiently resistant to corrosion under practical conditions, and that damage of purely mechanical origin is generally much less frequent [15]. Therefore, technical rules exist for designing and dimensioning apparatus to meet combined mechanical and thermal stresses. The standardized calculation procedures based on strength and ductility data are intended to provide the designer with a proven basis on which to work. However, where wear processes (corrosion and abrasion) are concerned, that is not possible; here, instead, special design criteria must be worked out in each individual case. A shortlist of materials can be drawn up from corrosion tables, which give the corrosion behavior of specific materials [16], [17]. Nevertheless, the most extensive tables can never take into consideration all the conditions of a given practical case. Almost always there is more than one corrosive agent, because chemi-

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Figure 7. Basic procedure for the choice of materials for chemical apparatus

cal processes normally involve starting materials and end products, intermediates and byproducts, often with unknown properties, and solids that promote wear. Furthermore, reactions are influenced by pressure and temperature. Thus the corrosion behavior of materials cannot be given in tables with any degree of accuracy, and terms such as resistant or nonresistant may be inappropriate in an actual application. The results of laboratory tests are consideraly more useful. The most reliable predictions are those based on plant tests performed either in a plant engaged in practical production or in a pilot plant. Often, however, such tests are very laborious. Even in cases where tests have been carried out, corrosion may still occur, possibly because the stability of a material has been impaired by excessive cold forming or by excessive heat input in welding, or because the states of the material in the test plant and in the subsequently constructed chemical apparatus were not identical.

In view of this complexity, close cooperation between material scientists, chemical engineers, chemists, designers, apparatus manufacturers, and – where chemical apparatus subject to compulsory testing, e.g., pressure vessels, is concerned – the representatives of the officially recognized supervisory authorities, is particularly desirable. Because corrosion reactions are so complex and depend on so many factors, an assessment of the corrosion behavior of a given material – corrodent system will differ greatly according to the method of investigation. False interpretations can be avoided and comparison improved if one has a good knowledge of the kinds of information provided by the chosen test method as well as knowing its limitations. It is therefore understandable that standardized corrosion tests are few in number or contain more general instructions. Frequently, they cover only the states of materials that are most favorable from the aspect of corrosion chemistry (that is to say the ho-

Construction Materials in Chemical Industry mogeneous structural states existing at the time of delivery), describing, for example, the testing of stainless steels for resistance to intergranular corrosion (DIN 50 914) or the testing of stainless steels in boiling nitric acid (DIN 50 921 and ASTM A 262-70). A comparison of the sensitivities of unalloyed and low-alloyed steels to intergranular stress corrosion cracking is given in DIN 50 915. A common feature of all investigation methods is that they refer to the material, and not directly to the stresses that occur in practice. Supplementary chemical or electrochemical tests simulating the conditions that occur in practice are therefore necessary in the selection of materials. Chemical tests are performed on samples exposed to gaseous and liquid corrodents at the operating pressure and temperature, whereas in electrochemical corrosion tests, the dependence of corrosion on potential is investigated and provides information on the effects of variables which alter the potential. The results of chemical corrosion tests and electrochemical corrosion tests may differ fundamentally. The differences arise because in chemical tests the corrosion potential may vary with time. In electrochemical tests the potential is fixed, so that, though they are more informative in some respects, their practical value may be limited in others. The characteristic stability data needed in the choice of materials can be obtained in two ways: 1) By inserting suitable material specimens in pilot plants or in existing, but not yet entirely satisfactory, production plants, 2) By performing laboratory corrosion tests. With regard to the corrosive medium in corrosion tests, careful attention must be paid to a number of important factors, such concentration, temperature, dissolved gases, impurities, solid matter, and rate of flow. As there are so many parameters, tests should use suitable specimens in a plant and under conditions as close as possible to those of practical use. The specimens should be placed at several representative locations. In a distillation column, for example, they should be placed in the pit, in the vicinity of the feed point, and in the head. The shape of a specimen used depends on the types of corrosion expected. Where general corrosion predominates, it is sufficient to use

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welded sheets that have the surface quality of the parts to be used in practice. If stress corrosion cracking is involved, plastically and elastically stressed specimens or tuning fork specimens are used. Material specimens are generally attached to interior parts, such as an agitator or a thermometer protection tube. To prevent polarization of specimens through contact with plant components, and consequent falsification of measurements, the fixing screws are placed in insulating sheaths. In addition to the exposure of test specimens, the parts of a pilot plant itself should be examined as a source of further information on corrosion behavior. For critical plant units such as heat exchangers, it has been found advisable to remove a tube from time to time, to cut it open, and to examine the inside and outside surfaces for corrosion symptoms. If stability tests cannot be performed in pilot installations, laboratory tests under conditions as close as possible to those of practice must be carried out. They should also be carried out as a supplementary measure in cases where plant tests have not clearly revealed the time dependency of the corrosion processes, and severer conditions of attack may therefore help to complete the picture. General guidelines on the conduct of corrosion tests are given in DIN 50 905. These guidelines should be followed as a route to reciprocally comparable results that can be transferred to the plant component in question. For further details on chemical and electrochemical corrosion tests, see → Corrosion.

4. Quality Assurance through Material Tests and Checking of Fabrication and Functioning To ensure that a chemical apparatus will be properly fabricated and that its functioning free from disturbances, many tests must be carried out before and during fabrication, before the apparatus is put into operation, and while it is in actual use. At the stage of testing of materials when they leave the factory and arrive at the plant manufacturer’s establishment it is normally fairly easy to ascertain whether or not they are of the required

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quality, as the required property data will have been agreed on the basis of quality standards or similar specifications. The same cannot be said of fabrication checks, which include, in particular, checking the construction for compliance with the blueprints and examining the welding work. Here the test engineer has more latitude, especially in interpreting the results of nondestructive tests. It must be emphasized that the first step in fabrication testing should be a thorough visual inspection; this gives the tester an overall impression of the care taken by the manufacturer. Checking compliance with legislation and official regulations is mainly the responsibility of the Technical Control Associations and the plant safety inspection departments of the large chemical firms. Functioning tests check the behavior of the apparatus under conditions close to those of practice. Of exceptional importance are the regular inspections of plant in use that are intended to reveal incipient damage early enough to ensure that plant shut-down can be avoided and any necessary repairs need not be undertaken in haste. Once again, visual inspection takes precedence. Much emphasis is also placed on nondestructive methods – thickness measurement, for example, and tests that reveal initial cracking [18]. Testing of plant in service also provides knowledge that may be useful in selecting materials for new chemical apparatus by revealing weaknesses of design, materials, or fabrication.

5. Properties and Applications of Materials 5.1. Steels Steels are still the materials most commonly used for chemical plant. Their variety of alloy compositions and the range of variation of their properties permit an exceptional degree of adaptation to practical requirements. 5.1.1. Unalloyed and Low-Alloy Steels for Vessels and Pipelines Sheet and piping made of unalloyed and lowalloyed steels with carbon contents up to ca. 0.25 wt % are used extensively in chemical plant for vessels (pressure vessels, storage vessels, etc.) and pipelines that are not exposed to particularly severe corrosion. The main standards and designations of several typical steels are compiled in Table 1. Comparable steels with comparable properties are described in the standard specifications of other countries – see, for example, SAE, AISI, ASTM (United States), BS (Great Britain), NF (France), SS (Sweden), UNI (Italy) and NBN (Belgium). The steels are normally processed in the normalized state. Owing to the introduction of weldable finegrained structuralsteels with yield strengths of

Table 1. Quality standards for unalloyed and low-alloy steels widely used in chemical plant Standard

Short title

Typical steels Short name

Material number

DIN 17 100

Steels for general structural purposes

RSt 37 – 2 St 44 – 2

1.0038 1.0044

DIN 1629

Seamless unalloyed circular steel tubes for special requirements

St 37.0 St 44.0 St 52.0

1.0254 1.0256 1.0421

DIN 17 155

Steel plates and strips for pressure purposes

HI H II 15 Mo 3

1.0345 1.0425 1.5415

DIN 17 102

Weldable, finegrained construction steels (normalized)

StE 285 StE 355 WStE 460

1.0486 1.0562 1.8935

DIN 17 175

Seamless tubes of heat-resistant steels

St 35.8 St 45.8 15 Mo 3

1.0305 1.0405 1.5415

Stahl-Eisen-Werkstoffblatt 087 – 81

Structural steels resistant to weathering

WTSt 37 – 2 WTSt 37 – 3

1.8960 1.8961

Construction Materials in Chemical Industry ≥ 360 MPa, boiler plate consisting of unalloyed steels, whose yield strengths are considerably lower, is used on a smaller scale than it was formerly. The use of high-strength heattreated structural steels with yield strengths of ≥ 700 MPa [19–23] will presumably increase. Pipes consisting of welded steel strip will be used increasingly instead of seamless pipes [24]. Steels are suitable for fusion welding by all processes, but the welding filler metals must be suited to the base material. Specific welding guidelines must be adhered to increasingly as the alloy content and yield strength rise; these guidelines can be found by consulting either the standards or manufacturers of steels and welding filler materials. Processing guidelines for weldable fine-grained structural steels are compiled in DIN standards. Normalized fine-grained structural steels are now very popular. They combine the excellent strength and toughness imparted by fine-grain hardening [25] with the advantages offered by modern ladle metallurgy in respect of purity and microalloying. The blast or shot injection of calcium compounds to reduce the sulfide content of steel (TN process) and the addition of cerium and zirconium to produce globular, finely distributed sulfide inclusions should also be mentioned. These processes improve purity, block segregation, weldability, and toughness. In special cases, these desulfurization methods enable the sulfur content to be reduced to below 0.005 %. Consequently, and through the influence exerted on the nature of the precipitation, the anisotropy of the steel’s properties is reduced considerably, as may be seen from the percentage reductions of area at fracture of fine-grained structural steels in the thickness direction of the sheet (Fig. 8) [26]. Figure 9 [26] shows that the fatigue resistance is also improved. Where damage occurs, the leak-beforefailure behavior imparted by high toughness is exceptionally favorable because the risk of sudden failure as a result of unstable crack propagation is practically eliminated. The high purity of fine-grained steels, together with the fine dispersion of their nonmetallic inclusions, notably the sulfides, is very favorable with regard to corrosion resistance. Coarse sulfides promote the formation and growth of cracks both in hydrogen-

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LIVE GRAPH

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Figure 8. Influence of sulfur content on the reduction of area at rupture in the thickness direction of StE 355 steel plates a) 40 – 55 mm plate, 0.015 – 0.054 % S; b) 40 mm plate, ≤ 0.010 % S; c) ≥ 50 mm plate, ≤ 0.006 % S; d) 20 – 50 mm plate, TN treated, 0.002 % S

Figure 9. Influence of the TN process on the dynamic strength of steel StE 355 (stress ratio R = 0, thickness of plate: 50 mm) A) Untreated; B) Calcium treated (TN process); a) Transverse; b) Longitudinal; c) Normal

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induced crack formation and in stress corrosion cracking [27]. Upon chemical exposure to aqueous solutions, the sulfides, especially under the influence of the hydrolytic acidification that occurs in cracks, may be converted to H2 S, which is a vigorous promoter of hydrogen cracking. It is now believed that crack growth in anodic stress corrosion cracking is also promoted by hydrogen. That would explain why steel StE 355, in particular, behaves favorably towards such crack-initiating media as alkalis, nitrates, and liquid ammonia. This steel is now a standard material for these media. For greater safety, the finished apparatus should be subjected to stress relief annealing. Considerable progress has been made through efforts to improve the economy of structural steels by raising their strength. Starting from steel St 52-3, it has been possible to raise the yield strength from 355 to ca. 900 MPa. The first step in this direction consisted in raising the alloy contents of normalized structural steels. But the scope for improvement in this way was limited by the fact that increasing the alloy content impairs the cold cracking resistance in welding. Agreement has therefore been reached that the yield strength of normalized steels should not exceed 500 MPa. Considerably higher yield strengths, even at low alloy contents, are obtainable by quenching and tempering. In the case of heat-resistant structural steels, air tempering is preferred. Water quenching followed by tempering, on the other hand, is more favorable for steel needing high toughness. By far the most important representative of this group of steels in terms of quantity is StE 690, whose minimum yield strength is 690 MPa (Euronorm 137). Steels with comparable yield strengths are also used on a large scale for pressure vessels and other chemical apparatus, such as the pressure-bearing parts of multistory vessels. StE 890 has been used to an increasing extent since the mid 1980s [28]. 5.1.2. Steels with High-Temperature Strength High-temperature structural steels are used in the chemical industry mainly for heavily stressed parts of steam boilers (e.g., collectors, drums, piping) and for pressure vessels and tubes

operating at up to ca 800 ◦ C [29]. These materials are also very important in reactor technology. According to their maximum service temperatures that can be maintained for long periods, the following groups are distinguished: 1) For temperatures up to ca. 400 ◦ C: unalloyed steels, in some cases melted as fine-grained structural steels, whose main strength-related property is the high-temperature yield strength. These steels are standardized in DIN 17 155 (boiler plate types), DIN 17 175 (pipes) and DIN 17 102 (finegrained structural steels) and are suitable for all fusion welding methods. Typical steels belonging to this group are the grades HI (1.0345), St 35.8 (1.0305), and WStE 355 (1.0565). 2) For temperatures of 400 – 550 ◦ C: unalloyed and low-alloy steels having good long-term, high-temperature strength-related property values. The main alloying elements are manganese (up to 1.3 wt %), chromium (up to 2.5 wt %), and molybdenum (up to 1.2 wt %), vanadium also being used occasionally (up to 0.5 wt %). These materials are standardized in DIN 17 155, DIN 17 175, DIN 17 240 (nuts and bolts), and DIN 17 245 (steel castings). These steels are suitable for fusion welding, though preheating and post-welding annealing may be necessary, details of which can be found in the standards. 3) For temperatures of 550 – 600 ◦ C: martensitic steels containing up to ca. 12 wt % chromium and additions of molybdenum, vanadium, nickel, and tungsten [30]. These are standardized in DIN 17 459 (sheet, pipes, forgings) and in DIN 17 245 (steel castings). These steels are supplied in the quenched and tempered state; a typical representative is X 20 CrMoV 12 1 (1.4922). Being airhardenable, these steels must be kept at 250 to 450 ◦ C during welding, after which they must be cooled to ca. 120 ◦ C and then immediately annealed. 4) Temperatures of 600 – 800 ◦ C: austenitic steels containing 16 – 21 wt % chromium, 11 – 32 wt % nickel, and additions of molybdenum, tungsten, niobium, tantalum, aluminum, and other elements, e.g., X 8 CrNiNb 16 13 (material no. 1.4961), X 8 CrNiMoNb 16 16 (material no. 1.4981),

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Table 2. Approximate temperature limits for the use of steels in a weakly oxidizing flue-gas atmosphere Steel

Material number

Standard

Approximate temperature, ◦ C

St 35.8 St 45.8 15 Mo 3 13 CrMo 44 10 CrMo 9 10 X 20 CrMoV 12 1 X 8 CrNiNb 16 13 X 8 CrNiMoNb 16 16

1.0305 1.0405 1.5415 1.7335 1.7380 1.4922 1.4961 1.4981

DIN 17 175 DIN 17 175 DIN 17 155 DIN 17 155 DIN 17 175 DIN 17 175 DIN 17 459 DIN 17 459

500 500 530 560 590 600 750 750

and X 8 NiCrAlTi 32 21 (material no. 1.4959), which are standardized in DIN 17 459. These steels are welded with filler metals of the same composition, and at the lowest possible heat input to avoid hot cracking of the weld metal. As austenitic steels are very expensive, they are used only in the zones which reach the highest temperatures, outside of which lowalloy steels are used. Special techniques have to be used to form welds between ferritic and austenitic steels because of the diffusion processes that occur in welding and because of the differences in thermal expansion [31]. For temperatures above 800 ◦ C only iron – chromium – nickel alloys and nonferrous alloys based on cobalt or nickel are suitable [32], [33]; see also Section 5.3. Table 2 gives approximate figures for the highest temperatures at which a number of steels with high strength at elevated temperature can be used in a weakly oxidizing flue-gas atmosphere. Long-term heat stability values are compiled in [34]; the importance of these values in calculating the useful lives of components subjected to this exposure is discussed in [35]. 5.1.3. Heat-Resistant Steels Heat-resistant steels are those that have good strength-related property data and are distinguished by exceptional resistance to exposure for short or long periods to hot gases or combustion products at temperatures exceeding 550 ◦ C, and which are thus resistant to scaling [36]. Rotary furnaces, cracking units, and muffle furnaces are examples of chemical plant in which these conditions arise.

Resistance to scaling is obtained mainly by alloying with chromium, but further improvements are possible if silicon and aluminum are added. The most commonly used heat-resistant steels are standardized in Stahl-Eisen-Werkstoffblatt 470-76. Distinctions are made between the ferritic steels, such as X 10 CrAl 7 (1.4713), X 10 CrAl 13 (1.4724) and X 10 CrAl 24 (1.4762); the ferriticaustenitic steel X 20 CrNiSi25 4 (1.4828); and austenitic steels such as X 15 CrNiSi 20 12 (1.4828), X 15 CrNiSi 25 20 (1.4841) and X 10 CrNiAlTi 32 20 (1.4876). The above-mentioned publication also gives long-term heat resistance values for periods of up to 105 h. The highest service temperatures listed, which extend to 1200 ◦ C, apply to air. They may be reduced greatly by admixtures to the air, e.g., of water vapor or sulfur-containing or carburizing matter, because the reaction products do not form sufficiently thick surface layers. Within certain temperature ranges, heatresistant steels tend to become brittle. Ferritic steels with ≥ 12 wt % chromium do so between 400 and 530 ◦ C. This so-called 475 ◦ C embrittlement can be eliminated by annealing briefly above 600 ◦ C. In the case of ferritic steels with ≥ 17 wt % chromium and austenitic steels, an intermetallic iron – chromium σ-phase, which causes severe embrittlement, is formed between 600 and 900 ◦ C. It endangers particularly the weld interfaces. Within the same temperature range, austenitic steels precipitate chromium carbides, which further reduce the toughness. The steel most severely affected is X 15 CrNiSi 25 20 (1.4841), which should be used only above 900 ◦ C. The σ-phase and the carbides can be redissolved by annealing above 1000 ◦ C, followed by quenching.

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The heat-resistant steels can be welded by the usual methods, provided the guidelines for alloyed steels are followed. Suitable filler metals are listed in Stahl-Eisen-Werkstoffblatt 47076. It should be noted that in ferritic steels with ≥ 12 wt % chromium, coarse grains, which can no longer be removed by heat treatment, are formed at temperatures above 900 ◦ C. Heat-resistant cast steel, as used in the manufacture of pipes by centrifugal casting, for example, is standardized in Stahl-EisenWerkstoffblatt 471-76. 5.1.4. Steels for Low Temperatures Refrigeration is very important in the chemical industry, e.g., in the fractional distillation of hydrocarbons or storage and transportation of liquid gases. Steels for the pressure vessels must still have sufficient toughness at the lowest operating temperature. The degree of low-temperature toughness depends particularly on the steel’s composition and heat treatment. Notched-bar impact energies, measured on DVM (Deutscher Verband f¨ur Materialforschung und -pr¨ufung e. V., Berlin) longitudinal specimens, of ≥ 40 and ≥ 60 J/cm2 , for cast steel and other steels, respectively, are regarded as evidence of sufficient low-temperature toughness. Hence the lowest service temperature of a steel with good low-temperature tenacity is that at which the notch impact energy is still above this limit. Steels for low temperatures can be divided into four groups: 1) Unalloyed aluminum-killed steels for service temperatures down to − 50 ◦ C in the normalized state and − 80 ◦ C in the heat-treated state; e.g., TTSt 35 (1.1101). 2) Unalloyed and low-alloy weldable finegrained structural steels in the normalized state for service temperatures down to − 60 ◦ C, e.g., TStE 380 (1.8910). 3) Nickel-alloyed heat-treatable steels with 1.5 – 9 wt % Ni for service temperatures of − 100 to − 190 ◦ C, e.g., 12 Ni 19 (1.5680). 4) Austenitic chromium – nickel steels for service temperatures extending close to absolute zero; e.g., the steels specified in DIN 17 440 and DIN 17 441. The relevant quality standards are DIN 17 173 and DIN 17 174 for groups 1 and 3

and DIN 17 102 for group 2, and Stahl-EisenWerkstoffblatt 685 – 82 for cast steel with good low-temperature toughness. In the welding of steels for low temperatures it is necessary to use filler materials that give a weld metal whose strength and toughness are equal to those of the base material [37–39]. 5.1.5. Steels Resistant to Pressurized Hydrogen [40], [41] Steels for high-pressure plant in the chemical industry can divided into the following two groups: 1) Steels for components subjected to purely mechanical loads or to pressurized hydrogen, in both cases at temperatures of ≤ 200 ◦ C, 2) Steels for components exposed to pressurized hydrogen at temperatures > 200 ◦ C, For the conditions of group 1, as present in the production of low-density polyethylene (pressures of up to about 400 MPa), mainly unalloyed and low-alloy heat-treatable steels are used; these are standardized in DIN 17 200 and in the case of large forgings, in Stahl-Eisen-Werkstoffblatt 550-76. If relatively good corrosion resistance is required, stainless heat-treatable chromium and chromium – nickel steels according to DIN 17 440, and occasionally hardenable stainless steels, are used instead [42]. The periodic alternation of the pressure subjects the components to a severe fatigueinducing stress. Their shape stability therefore depends decisively not just on the choice of materials, but also on their being designed to suit the conditions of exposure as well as on satisfactory fabrication and installation [43]. Hot pressurized hydrogen, which is needed in many high-pressure syntheses (e.g., ammonia synthesis and pressure hydrogenation) dissociates on the surface of many steels, diffuses into the metal, and reacts with carbon to form methane, thus causing decarburization, embrittlement, and cracking: Fe3 C + 2 H2 → 3 Fe + CH4

This reaction between hydrogen and cementite occurs mainly at the grain boundaries, the

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Table 3. Composition of pressurized-hydrogen-resistant steels Steel type

25 CrMo 4 16 CrMo 9 3 26 CrMo 7 24 CrMo 10 10 CrMo 11 10 CrMoV 10 20 CrMoV 13 5 X 20 CrMoV 12 1 X 8 CrNiMoVNb 16 13

Material number

1.7218 1.7281 1.7259 1.7273 1.7276 1.7766 1.7779 1.4922 1.4988

Composition, wt % C

Cr

Mo

V

0.22/0.29 0.12/0.20 0.22/0.30 0.20/0.28 0.08/0.12 0.15/0.20 0.17/0.23 0.17/0.23 0.04/0.10

0.90/1.20 2.0/2.5 1.5/1.8 2.3/2.6 2.7/3.0 2.7/3.0 3.0/3.3 11.0/12.5 15.5/17.5

0.15/0.25 0.30/0.40 0.20/0.25 0.20/0.30 0.20/0.30 0.20/0.30 0.50/0.60 0.80/1.20 1.10/1.50

0.10/0.20 0.45/0.55 0.25/0.35 0.60/0.85

grains thus losing their cohesion. Hydrogen and methane accumulate, giving rise to high pressure whose cleaving action leads to internal microcracks. Together with the stresses exerted on the component and the loss of mechanical strength, this results finally in pronounced brittle fracture. The resistance of carbon steels to attack by pressurized hydrogen depends partly on the ambient conditions, but also on the microstructure, cold working, effects of welding, impurities, and heat treatment. Elements that form stable carbides, such as chromium, molybdenum, tungsten, vanadium, titanium, and niobium, can be added to the steel to prevent the reaction between hydrogen and cementite. The pressurized-hydrogen-resistant steels now in use contain up to 16 wt % Cr, and also in many cases 0.2 – 1.5 wt % Mo. Some types are additionally alloyed with up to 0.85 wt % V (Table 3). The low-alloy ferritic steels are suitable for fusion welding, provided that they are preheated. The chromium steels have to be kept at 250 – 450 ◦ C throughout welding, and partially annealed immediately afterwards. The austenitic steel (1.4988) tends to suffer hot cracking on welding. Short-term tests give very little indication of the limits to the use of steels of this kind, because the time factor is much too important and the attack of pressurized hydrogen on steels has a significant incubation period. As an illustration, Figure 10 gives stability limits for various steels, as compiled by Nelson [44]. It can be seen that even at low hydrogen partial pressures (e.g., 2 MPa) an unalloyed steel should be used only below 300 ◦ C. Austenitic steels with 18 % Cr and ca. 9 % Ni have good resistance to pressurized hydrogen; they can be exposed to

Others

Ni: 12.5/14.5 N: 0.07/0.13

hydrogen throughout the range of temperatures used in normal high-pressure processes.

Figure 10. Resistance diagram for the attack of pressurized hydrogen on steels (Nelson diagram) a) 1.25 Cr 0.5 Mo steel; b) 5.0 Cr 0.5 Mo steel; c) 8.0 Cr 0.5 Mo steel; d) 2.25 Cr 1.0 Mo steel; e) 2.0 Cr 0.5 Mo steel; f ) 1.25 Cr 0.5 Mo steel; g) 1.0 Cr 0.5 Mo steel; h) 0.5 Mo steel; i) Mild steel

5.1.6. Stainless Steels [45] Among the highly alloyed steels, the most important group comprises those that are chemically resistant. They have chromium contents ≥ 14 wt %. As the steels of this group resist heat as well as chemicals, this group also includes a considerable number of the heatresistant steels discussed in Section 5.1.3. Often they contain nickel in addition to chromium. The more highly alloyed corrosion-resistant (acid-resistant) steels must additionally withstand general corrosion and localized corrosion in relatively aggressive corrodents (salt solutions; acids, even at fairly high concentrations). Characteristic alloying components, apart from chromium and nickel, are molybdenum, copper, and in some cases silicon.

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There are many kinds of steels and alloys, each being represented by a material number (DIN 17 007) or by chemical composition (DIN 17 006). The mechanical properties of steels are determined by the microstructure, which depends on composition and heat treatment. Stainless steels are divided according to microstructure into four groups: 1) Martensitic (hardenable) steels with > 0.12 wt % C and ≤ 15wt % Cr. They are hardened at temperatures > 1000 ◦ C (e.g. cutting steels). After being hardened they can be improved by tempering at 500 to 600 ◦ C – their strength thus being reduced to a desired lower level – and henceforth combine high strength with good ductility. In the heat treatment, a ferritic structure, with precipitated carbides of high chromium content of type M23 C6 or M7 C3 , is formed. Fixation of chromium and the formation of chromium-depleted zones reduces the resistance to corrosion. To equalize the chromium content of the ferritic matrix again it is necessary to anneal the steel for a fairly long time at elevated temperature. Although this restores the corrosion resistance to some extent, it impairs the strength. As some of the chromium is still bound as carbide, the corrosion resistance after annealling procedure is still considerably poorer than that of the martensite. 2) Ferritic steels with body-centered cubic lattice (α-phase). Those of greatest importance are the ferritic chromium steels with ca. 17 wt % chromium. Their mechanical and technical properties, however, are unsatisfactory; when welded, they tend to become coarse-grained and brittle. Nevertheless they have the advantage of resisting stress corrosion cracking in chloride-containing media. 3) Austenitic steels with face-centered cubic lattice (γ-phase). Particularly important are the austenitic chromium – nickel steels with ca. 18 wt % Cr and 10 wt % Ni, but without addition of molybdenum. Their strength is relatively low and their ductility very good. In chloride-containing corrodents, however, they undergo transgranular stress corrosion cracking. 4) Ferritic – austenitic steels combine good mechanical properties with improved re-

sistance to stress corrosion cracking. They are therefore used mainly under conditions which may cause fatigue corrosion and stress corrosion cracking in austenitic steels. Pure iron has a ferritic structure. The alloying elements that may be added to iron are either ferrite-forming (chromium, molybdenum, silicon, titanium) or austenite-forming (carbon, nitrogen, nickel, manganese). The austenitic chromium – nickel steels, with or without molybdenum, are used in the solution-heat-treated state (solution heat treatment temperature 1000 – 1100 ◦ C). At solution heat treatment temperatures these steels are close to the boundary of the α + γ field in the iron – chromium – nickel phase diagram (Fig. 11). As the content of chromium increases, more nickel or nitrogen must be added to maintain the austenitic structure. If the content of molybdenum is raised, then it is also necessary to increase the content of austenitizers and/or to reduce the chromium content.

Figure 11. Ternary phase diagram of Fe – Cr – Ni (section at 1100 ◦ C)

With increasing chromium content, and especially increasing molybdenum content, the tendency to segregate intermetallic compounds (σ-phase, χ-phase, Laves phase, Fe2 Mo), increases greatly (see Fig. 12). If the precipitation tendency is strong enough, intermetallic compounds may already be precipitated during the

Construction Materials in Chemical Industry relatively short-lived heat exposure of welding. Hence the addition of molybdenum, which is desirable for reasons of corrosion chemistry, has an upper limit. In the case of austenitic chromium – nickel steels it is about 6 wt % Mo.

Figure 12. Precipitation fields of chromium-rich carbides M23 C6 , σ- and χ-phase, molybdenum-rich Laves phase (Fe2 Mo), and of a complex nitride (Z-phase) for the steel X 2 CrNiMo 18 14 3

Addition of nitrogen retards the precipitation of the intermetallic phases and compounds considerably, as well as improving the mechanical properties by introducing nitrogen atoms into interstices in the metal lattice. Nitrogenalloyed austenitic and ferritic – austenitic steels are therefore of industrial importance. In ferritic – austenitic steels, the ferritizers chromium and molybdenum are enriched in the ferrite phase, whose nickel content is correspondingly depleted. The austenitic phase contains, conversely, more nickel and less chromium and molybdenum. These differences of concentration are relatively slight, however, and are only important for the use of ferritic – austenitic steels in borderline cases. The alloy composition of these steels is also chosen such that the ferritic and austenitic phases are present in the structure in approximately equal proportions and the chromium content of the austenite is not less than 17 wt %. Annealing within certain temperature ranges may cause carbides, nitrides, and intermetallic compounds to be precipitated in the microstructure of stainless steels. Owing to the special importance of chromium and molybdenum in imparting corrosion resistance to stainless steels, the precipitation of phases and compounds that contain these elements exert strong effects. The segregation of chromium- and molybdenumrich phases depletes the matrix in these elements.

619

The matrix therefore loses its resistance to corrosion. In ferritic chromium steels and austenitic chromium – nickel steels, the chromium-rich σphase may precipitate on annealing at 600 – 900 ◦ C. For ferritic chromium steels with up to ca. 18 wt % chromium this precipitation has no importance, while in the case of molybdenum-free austenitic chromium – nickel steels it is, at the most, important only in connection with the welded material. For reasons of welding technique (prevention of hot cracking) the welded material of normal austenitic steels almost always contains some δ-ferrite, which decomposes when the steel is annealed. In a corrosion exposure this decomposed δ-ferrite may be selectively dissolved. Damage must be expected where a coherent δ-ferrite network is present (proportion of δ-ferrite in the structure > 10 %). The designations of the most important stainless steels for chemical plant are compiled in DIN 17 440 (Table 4, see next page). Chemical engineering also uses a number of other special materials notable for their high resistance to pitting corrosion and stress corrosion cracking, as well as to mineral acids (Table 5, see next page). Particularly high molybdenum contents are present in the steels listed in Table 6 (see next page). The stability of the austenite is due to nickel and nitrogen. Unlike ferritic and austenitic steels, ferritic – austenitic steels have a two-phase structure, which, in contrast to the composition of austenitic steels, is obtained by raising the contents of ferrite-stabilizing elements, such as chromium and silicon, and reducing the austenite-stabilizing nickel content (Table 7, see page after next). By virtue of their high yield strength of at least 450 MPa, steels of this kind are used for components which, while exposed to corrosive media, are additionally subjected to wear (cavitation, erosion) and vibration. Being more resistant than commercial austenitic steels to stress corrosion cracking in neutral chloride solutions, they are also being used increasingly to handle aggressive cooling water. A commonly used material of this group is X 2 CrNiMoN 22 5 (1.4462), which, in addition to about 22 wt % Cr, has a nickel content of about 5.5 wt %, an Mo content of about 3 wt %, and an austenitic structure proportion of about 60 % [46], [47].

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Construction Materials in Chemical Industry

Table 4. Composition of stainless steels Short name (DIN)

US Standard (AISI)

Ferritic and martensitic steels X 7 Cr 13 X 7 CrAl 13 X 10 Cr 13 X 15 Cr 13 X 20 Cr 13 X 40 Cr 13 X 45 CrMoV 15 X 8 Cr 17 X 8 CrTi 17 X 8 CrNb 17 X 6 CrMo 17 X 12 CrMoS 17 X 22 CrNi 17 Austenitic steels X 12 CrNiS 18 8 X 5 CrNi 18 9 X 5 CrNi 19 11 X 2 CrNi 18 9 X 10 CrNiTi 18 9 X 10 CrNiNb 18 9 X 5 CrNiMo 18 10 X 2 CrNiMo 18 10 X 10 CrNiMoTi 18 10 X 10 CrNiMoNb 18 10 X 5 CrNiMo 18 12 X 2 CrNiMo 18 12 X 2 CrNiMo 18 16 X 2 CrNiN 18 10 X 2 CrNiMoN 18 12 X 2 CrNiMoN 18 13

Material number

Composition, wt %

C

Cr

434 430 F 431

1.4000 1.4002 1.4006 1.4024 1.4021 1.4034 1.4116 1.4016 1.4510 1.4511 1.4113 1.4104 1.4057

10 cm), field inspection becomes difficult because high-energy sources are needed. Ultrasonic methods, especially with shear waves, are widely used for weld inspection. Thickness measurement using the normal beam is commonly practiced for corrosion monitoring. Those areas having a high corrosion rate or history of attack are inspected thoroughly.

3.2. Composite Pressure Vessels and Piping The performance of fiber-reinforced plastic vessels and piping has been poor, and many failures have been recorded. Apart from acoustic emission, no satisfactory test exists for determining the structural adequacy of fiber-reinforced plastic equipment. Radiography is difficult because resins have low absorption coefficients. Fiber-reinforced plastics are anisotropic, and ultrasonic attenuation is very high, making pulseecho techniques hard to use. In addition, the various types of life-limiting flaws are quite different and some of them are not amenable to conventional ultrasonic inspection. Fiber fracture and fiber – matrix debonding are difficult to detect with ultrasonics. The use of acoustic emission for testing fiberreinforced plastic vessels and piping has been highly successful, and standard test procedures have been established [9, E 1067], [28], [35], [36]. Test vessels and piping are pressurized up to 150 % of the maximum allowable working pressure. The procedures are designed to locate substantial flaws, which are then evaluated by other techniques such as ultrasonic testing (delamination) or visual (resin loss) and penetrant (matrix cracking) inspection. A vessel (or piping) to be tested can be new, in-service, or repaired, and different steps are specified for atmospheric, vacuum, and pressurized vessels. A vessel that has been in service must be preconditioned by reducing the operating pressure; for

this, the maximum operating pressure within the previous year must be known. Acoustic emission instrumentation should have a sufficient number of channels to localize sources by using the zone location method; that is, many AE sensors are installed to completely cover the vessel with the corresponding zone marked around each sensor. Acoustic emission activities of flaws within each zone are detected, and the zone represents the approximate position of these flaws. High-frequency (100 – 200 kHz) sensors are used for zone location. Two or more low-frequency (25 – 75 kHz) sensors are used to evaluate the adequacy of coverage of the highfrequency sensors. If a low-frequency sensor detects acoustic emission whereas none of the high-frequency sensors do, the latter must be relocated. Sensors are positioned to detect structural flaws at critical sections of the test vessel, such as high-stress areas, geometrical discontinuities, nozzles, manways, repaired regions, support rings, and visible flaws. Pressurization of a vessel during AE testing proceeds in steps, with pressure hold periods. For atmospheric vessels, the pressure is held at 50, 75, 87.5, and 100 % of the test pressure. Pressure vessels are stressed with 10 % increments, with depressure increments (also 10 %) above 30 % of the test pressure. A test is terminated whenever a rapid increase in AE activity indicates an impending failure. Acoustic emission data from the high-frequency sensors are used for evaluation, whereas the low-frequency sensors generally detect acoustic emission from significant flaws. Detected flaws are graded according to using several criteria, including emissions during pressure hold periods, felicity ratio (the ratio of the load at the onset of significant emissions to the maximum prior pressure), total AE counts, high-amplitude events, and longduration events. Emissions during hold indicate continuing permanent damage and lack of structural integrity. For in-service vessels, the felicity ratio criterion (when it is less than 0.95) is an important measure of previous damage. Highamplitude events indicate structural (fiber) damage, especially in new vessels. Long-duration events are characterized by measured area of the rectified signal envelope (MARSE), which is an indicator of combined signal and amplitude duration. Large MARSE values result from

Nondestructive Testing delamination, adhesive bond failure, and crack growth.

3.3. Weldments Nondestructive testing methods used for completed fusion weldments include (1) visual inspection, (2) radiography, (3) ultrasonic pulse echo, (4) magnetic particle and leakage field testing, (5) liquid penetrant testing, (6) leak testing, and (7) acoustic emission testing. For many noncritical welds, integrity is assured mainly by visual inspection to look for cracks, bead thickness, bead contour, undercut, overlap, and spatter. For critical welds, both faces and root surfaces are examined, especially for cracks, undercut, root penetration, and unfilled craters. These are tested further by radiography and ultrasonics for internal flaws. Radiography is commonly used for detecting porosity, slag entrapment, and inclusions. A round or oval dark spot represents the image of a pore. Slag inclusions appear along the weld edge as irregular or continuous dark lines, whereas tungsten inclusions give rise to single or clustered light spots. Cracks are sometimes visible in radiographs as dark narrow irregular lines, but the lack of any radiographic image of cracks does not assure their absence. Radiography may be unable to detect incomplete fusion and incomplete penetration because of their small effects on X-ray absorption. They appear as very narrow dark lines. Ultrasonic pulse-echo techniques are used effectively for the detection and location of planar defects such as laminations, unbonded areas, cracks, hidden surfaces, and other flaws. They are also used to reveal a lack of root penetration, porosity, and unbonded sidewalls of fusion zones. Ultrasonic testing and radiography are complementary techniques, and both are used for inspection of critical welds. When no access to the opposite side of the weldment is available, ultrasonic testing is the only option for internal flaw inspection. Shear wave beams are generally used in weld inspection. To detect longitudinal flaws (along the weld), the search unit is moved along a zigzag scanning path either with sharp changes in direction or with right-angle changes. To detect transverse flaws in welds, the search unit is

817

placed on the base metal surface at the edge of the weld. The sound beam is directed into the weld by angling the search unit at ca. 15◦ . The scanning of the search unit is parallel to the weld. Magnetic particle testing and penetrant testing are used mainly for surface-breaking flaws (also for subsurface flaws in magnetic particle testing). These methods are relatively inexpensive but can reveal even fine cracks with clarity. In magnetic particle testing, the magnetic field is applied in two mutually perpendicular directions. Leak testing of welded vessels uses a tracer gas under pressure or vacuum. Welds are tested for leak location or for leak rate. Acoustic emission tests for large-scale structures can narrow the areas of inspection by locating sources of acoustic emission. Typically, the structure is stressed by pressurization while AE sensors are mounted in arrays on the surface of the structure. When the applied stress exceeds the previously applied stress level, acoustic emission activities increase rapidly. The positions of such activities are identified by triangulation or the zonelocation method. Often, the source of acoustic emission is localized to specific weld regions which are inspected further by ultrasonic and radiographic methods.

3.4. Other Uses Polymeric materials are adversely affected by improper curing, inclusions, porosity, environmental degradation, machining and impact damage, and fretting. These are detected by specialized application of various nondestructive testing methods. Ultrasonic propagation characteristics are used to reveal anisotropy of elastic moduli, film and fiber orientation, structural relaxation, glass transition, degree of crosslinking, and state of cure in polymers. Dielectric measurements are used to monitor the curing process [37]. In situ radio-frequency impedance is determined for this purpose. Microwave transmission and reflection methods are used for flaw detection (cracks, voids, inclusions) and for monitoring moisture, thickness, and cure. Optical techniques are employed to measure the state of mechanical strain in transparent polymers (photoelasticity). This is used

818

Nondestructive Testing

widely in studying stress concentration effects in model components. Thermographic techniques are useful in testing polymers because they have low thermal conductivities. Infrared spectroscopy provides information on molecular structure and has also been used to measure stress concentration effects near crack tips. Radiography of polymers requires lowenergy (ca. 20 kV) radiation because polymers have low absorbance. To enhance contrast, high atomic number penetrants are used. Test pieces are immersed in fluids such as tetrabromoethane and diiodobutane. This is only beneficial for surface-breaking flaws. Although difficult to use, neutron radiography is effective for polymers. Paper and paperboard products require special testing methods during production. Radiation thickness gauging with beta emitters (85 Kr and 90 Sr) is used for the measurement of basis weight (areal weight). For determining ash in paper due to fillers and coating minerals, the attenuation and fluorescence of low-energy photons (4 – 5 keV) are used. Various on-line optical methods are employed in determining opacity, surface roughness, color, and reflectance. Infrared absorption in the 1.95-µm region is used for moisture measurement. For heavyweight paper, moisture is measured by microwave methods. The mechanical properties of a paper can be probed by means of ultrasonic wave propagation. An online caliper system obtains the thickness and ultrasonic propagation data and the results are correlated to the mechanical strength of the paper.

4. References 1. Metals Handbook, “Inspection and Quality Control,” 8th ed., vol. 11, Amer. Soc. Metals, Metals Park, Ohio 1976. 2. Metals Handbook, “Nondestructive Evaluation and Quality Control,” 9th ed., vol. 17, ASM International, Metals Park, Ohio 1989. 3. D. E. Bray, R. K. Stanley: Nondestructive Evaluation, McGraw-Hill, New York 1989. 4. R. Halmshaw: Nondestructive Testing, Arnold, London 1987.

5. Nondestructive Testing,“A Survey,” NASA SP-5113, NASA, Washington D.C. 1973. 6. Nondestructive Testing Study Guides, “Basic Magnetic Particle Testing, Penetrant Testing, Eddy-Current Testing and Radiography (1983),” General Dynamics, Convair Div., 1977 – 1983. 7. Nondestructive Testing Handbook, 2nd ed., vol. 1, “Leak Testing,” 1982; vol. 2, “Liquid Penetrant Tests,” 1982; vol. 3, “Radiography,” 1985; vol. 4, “Electromagnetic Methods,” 1987; vol. 5, “Acoustic Emission Testing,” 1987; vol. 6, “Magnetic Particle Testing,” 1989, ASNT, Columbus, Ohio. 8. ASME Boiler and Pressure Vessel Code, ASME, New York 1989. 9. 1989 Annual Book of ASTM Standards, vol. 3.03, “Nondestructive Testing,” ASTM, Philadelphia 1989. 10. Redi-Reference Guide, “An Issue of Materials Evaluation,” published yearly by ASNT, Columbus, Ohio. 11. R. S. Sharpe (ed.): Research Techniques in Nondestructive Testing, vol. 1 1970, vol. 2 1973, vol. 3 1977, vol. 4 1980, vol. 5 1981, vol. 6 1982, vol. 7 1984, vol. 8 1985, Academic Press, London. 12. D. O. Thomson, D. E. Chimenti (eds.): Quantitative NDE, vol. 1 – 7, Plenum Publishing, New York(1982 – 1988). 13. Proc. World Conference on Nondestructive Evaluation, 11th, Las Vegas, Nov. 1985. J. Boogard, G. M. van Dijk (eds.): Non-Destructive Testing (Proc. World Conference on Nondestructive Evaluation), Elsevier, Amsterdam 1989. 14. 17th Symposium on Nondestructive Evaluation, San Antonio, April 1989. 15. J. M. Farley, R. W. Nichols (eds.): Nondestructive Testing, vol. 1 – 4, Pergamon Press, Oxford 1988. 16. Materials Evaluation; NDT International; Research in Nondestructive Evaluation; Brit. J. Nondestructive Testing; Soviet J. Nondestructive Testing; NTIAC Newsletter. 17. F. Feigl, U. Anger: Spot Tests in Inorganic Analysis, 6th ed., Elsevier, Amsterdam 1972. 18. G. Tschorn: Spark Atlas of Steels, Macmillan Publ. Co., New York 1963. 19. A. R. Fee, R. Sagabache, E. L. Tabolski, Metals Handbook, “Hardness Testing,” 9th ed., vol. 8, ASM International, Metals Park 1985, pp. 69 – 113.

Nondestructive Testing 20. R. Halmshaw: Industrial Radiography-Theory and Practice, Applied Science Pub., Englewood, NJ 1982. 21. R. A. Armistead, R. N. Yancey, Mater. Eval. 47 (1989) 487 – 491. B. D. Hansche, Mater. Eval. 47 (1989) 741 – 745. 22. J. Krautkr¨amer, H. Krautkr¨amer: Ultrasonic Testing of Materials, 3rd ed., Springer Verlag, Berlin 1983. 23. M. G. Silk: Ultrasonic Transducers for Nondestructive Testing, A. Hilger, Bristol 1984. 24. C. Broere et al., in [15], vol. 4, pp. 2424 – 2432. 25. Progress in Acoustic Emission, (a) I (1982), (b) II (1984), (c) III (1986), (d) IV (1988), Japan Soc. Non-Destructive Inspection, Tokyo. 26. Many articles in J. Acoustic Emission, vols. 1 – 9 (1982 – 1990). 27. R. K. Miller et al., J. Acoustic Emission 8 (1989) 25 – 29. 28. R. Davies, in [25, (c)], pp. 9 – 25. 29. K. G¨obbels, G. Ferrano, in [15] vol. 4, p. 2763. 30. W. Lord (ed.): Electromagnetic Methods of Nondestructive Testing, Gordon and Breach, New York 1985. 31. D. C. Jiles, NDT Int. 21 (1988) 311 – 319. 32. K. Ono, in [25, (c)]pp. 200 – 212. 33. V. S. Cecco, F. L. Sharp, NDT Int. 22 (1989) 217 – 221. 34. T. J. Fowler, in [25, (c)]pp. 150 – 162.

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35. Recommended Practice for Acoustic Emission Testing of Fiberglass Reinforced Plastic Tanks/Vessels, Soc. Plastics Industry, New York 1982. 36. T. J. Fowler, Chem. Proc. (Chicago) March 1984, 24 – 27. 37. F. I. Mopsik et al., Mater. Eval. 47 (1989) 448 – 453. 38. C. A. Salvado, Proc. Design and Manufacturing of Advanced Composites, ASM International, Metals Park 1989, pp. 111 – 119. 39. T. J. Fowler, J. A. Blessing, P. J. Conlisk, AECM-3 (Third Int. Symp. Acoustic Emission from Composite Materials, Paris 1989), ASNT, Columbus 1989, pp. 16 – 27. 40. S. S. Russell, E. G. Henneke, NDT Int. 17 (1984) 19 – 25. 41. J. W. Wagner, in [22], pp. 405 – 431. 42. R. C. Anderson, Inspection of Methods: Visual Examination, vol. 1, ASM, Metals Park, Ohio 1983. 43. E. P. Chiang in C. P. Grover (ed.): Optical Testing and Metrology, vol. 661, SPIE International Soc. Optical Eng., Bellingham 1986, pp. 249 – 261. 44. R. Jones, C. Wykes, Holographic and Speckle Interferometry, Cambridge University Press, Cambridge 1983. 45. In [7, vol. 1], pp. 346 – 368.

On-Line Monitoring of Chemical Reactions

821

On-Line Monitoring of Chemical Reactions Wolf-Dieter Hergeth, Wacker Polymer Systems, Burghausen, Federal Republic of Germany

1. 2. 2.1. 2.2. 2.2.1. 2.2.2. 2.3. 3. 3.1. 3.2. 3.3. 4. 4.1. 4.2. 5. 5.1. 5.2. 5.3. 5.3.1. 5.3.2. 5.3.3. 5.3.4. 5.3.5.

Introduction . . . . . . . . . . . . . . . Reaction Calorimetry . . . . . . . . . Introduction . . . . . . . . . . . . . . . Heat Flow Balance and Principles of Measurement . . . . . . . . . . . . . . . Heat Flow Balance . . . . . . . . . . . . Basic Modes of Operation . . . . . . . Applications and Instrumentation . Ultrasonic Methods . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . Dielectric Spectroscopy . . . . . . . . Theory and Mechanisms . . . . . . . Instrumentation and Applications . Optical Spectroscopy . . . . . . . . . . Introduction . . . . . . . . . . . . . . . Instrumentation for Reaction Monitoring . . . . . . . . . . . . . . . . Applications of Optical Spectroscopy . . . . . . . . . . . . . . . UV – VIS Spectrocopy . . . . . . . . . NIR Spectroscopy . . . . . . . . . . . . IR Spectroscopy . . . . . . . . . . . . . Raman Spectroscopy . . . . . . . . . . Fluorescence . . . . . . . . . . . . . . . .

823 825 825 826 826 828 829 830 830 830 832 833 833 835 837 837 837

7.4. 8. 8.1. 8.2. 8.3. 8.4.

839 839 839 842 844 847

9. 9.1. 9.2. 9.3. 9.4. 10.

Abbreviations AOTF APM ATR CCD CHDF CID DGEBA DGEBF DDM DLS DWS EWA FFF FID FIR FOCS FODLS

6. 6.1. 6.1.1. 6.1.2. 6.1.3. 6.1.4. 6.2. 7. 7.1. 7.2. 7.3.

acousto-optic tunable filter acoustic plate mode attenuated total reflection charge-coupled device capillary hydrodynamic fractionation charge-injection device diglicidyl ether of bisphenol A diglicidyl ether of bisphenol F diaminodiphenylmethane dynamic light scattering diffusing wave spectroscopy evanescent wave absorption field flow fractionation free induction decay far infrared fiber-optic chemical sensor fiber-optic dynamic light scattering

Particle Size Analysis . . . . . . . . . Scattering Techniques . . . . . . . . . Turbidimetry . . . . . . . . . . . . . . . . Angular Static Light Scattering . . . . Dynamic Light Scattering . . . . . . . Other Optical Techniques . . . . . . . . Separation Techniques . . . . . . . . Chromatography . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . GC Hardware Components . . . . . Applications of Gas Chromatography . . . . . . . . . . . . Other Chromatographic Techniques Electroanalytical Methods . . . . . . Introduction . . . . . . . . . . . . . . . Conductometry . . . . . . . . . . . . . Potentiometry . . . . . . . . . . . . . . Amperometry, Voltammetry, and Coulometry . . . . . . . . . . . . . . . . Miscellaneous Methods . . . . . . . . Mass Spectrometry . . . . . . . . . . . Densimetry and Dilatometry . . . . . Rheometry . . . . . . . . . . . . . . . . NMR Spectroscopy . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

848 848 848 849 849 850 850 851 851 852 854 855 855 855 855 856 857 858 858 859 861 862 862

FOQELS fiber-optic quasi elastic light scattering FRA frequency response analyzer FT Fourier transform FTIR Fourier transform infrared HT Hadamard transform HDC hydrodynamic chromatography IR infrared LEC liquid exclusion chromatography LED light-emitting diode NIR near infrared NMR nuclear magnetic resonance PCS photon correlation spectroscopy PS power source QELS quasi-elastic light scattering QCM quartz crystal microbalance RIM reaction injection molding SAW surface acoustic wave SEC size exclusion chromatography

822 SERS SH SLS SRS UV

On-Line Monitoring of Chemical Reactions surface-enhanced Raman scattering shear-horizontal static light scattering stimulated Raman scattering ultraviolet

Symbols absorbance (extinction), instrument constant AR reactor wall heat exchange area aη Kuhn – Mark – Houwink exponent B instrument constant c concentration C0 equivalent capacitance of free space ci concentration of component i cp,cool specific heat capacity of coolant cp,k specific heat capacity of kth reactor feed cp,liq specific heat capacity of jacket liquid Cp,mixture heat capacity of reaction mixture Cp,tot heat capacity of filled reactor d distance, path length D diffusion coefficient, dielectric displacement dp particle diameter d stir stirrer diameter E electric field f res resonance frequency f frequency, particle size distribution function g contraction factor G shear modulus, distribution function g1 electric field autocorrelation function g2 light intensity autocorrelation function G conductance h depth ∆H r,j reaction enthalpy of jth reaction I electrical current, light intensity I0 incident light (radiation) intensity Id detected light intensity Il light intensity at distance l K bulk modulus, scattering coefficient Kη Kuhn – Mark – Houwink parameter l (optical) path length, distance kQ heat transfer coefficient Boltzmann constant Kb m refractive index ratio mk mass of component k mcool mass of coolant mliq mass of jacket liquid Mw weight-average molecular mass M w,crit critical weight-average molecular mass A

n refractive index nm refractive index of medium np refractive index of particle Nν stirrer speed p pressure P polarization p0 pressure amplitude pbubble bubble pressure pgel gel point ph hydrostatic pressure px pressure amplitude at distance x P0 power number q scattering vector Qacc accumulated heat Qcal heat generated by calibration heaters Qel heat generated by electrical heaters Qfeed heat generated by material feed Qi sources and sinks of heat Qloss heat losses Qosc heat of forced temperature oscillations Qr heat of reaction Qreflux heat flow over reflux condensor Qstir heat input due to stirring Qtrans heat transfer through reactor wall R reflection coefficient, tube diameter r radius rh hydrodynamic radius reaction rate of jth reaction Rj t time T transmission coefficient, temperature δT amplitude of temperature oscillations T1 outer reactor wall temperature T2 inner reactor wall temperature T cool,in coolant temperature (inflow) T cool,out coolant temperature (outflow) Tg glass transition temperature reactor jacket temperature TJ T J,in jacket temperature (inflow) T J,out jacket temperature (outflow) Tk temperature of kth reactor feed T osc period of tube oscillations T osc,cal period of tube oscillations filled with calibration liquid TR reactor temperature u sound velocity U electrical voltage u0 sound velocity of reference component U0 initial electrical voltage (∆u/∆c)i sound velocity-concentration coefficient i ul,liq longitudinal sound velocity of liquids ul,sol longitudinal sound velocity of solids

On-Line Monitoring of Chemical Reactions umix utr,liq utr,sol Ux V V excess Vi Vr V total x xr Xr Y Z

sound velocity of a mixture transversal sound velocity of liquids transversal sound velocity of solids electrical voltage at distance x volume excess volume volume of component i volume of reaction mixture total volume distance monomer conversion calorimetric conversion complex admittance acoustic impedance, complex impedance Z
real component of complex impedance Z

imaginary component of complex impedance α sound attenuation αliq sound attenuation in liquids β adiabatic compressibility δ loss angle ε decadic extinction coefficient ε∗ dielectric permittivity ε0 free space dielectric permittivity ε∞ high frequency dielectric permittivity ε
storage component of dielectric permittivity ε

loss component of dielectric permittivity ε∗ dipolar dipolar reorientation contribution to ε∗ γ surface tension Γ decay constant η (bulk) viscosity η0 viscosity of dispersion medium shear viscosity ηs ηv volume viscosity ϕ wave phase, volume fraction ϕr phase shift between temperature oscillations Λ conductivity λ wave length λexc excitation wave length λm medium light wave length λQ heat conductivity ν frequency (of light), average velocity θ scattering angle  density cal density of calibration liquid end density at reaction completion mix density of reaction mixture

start σ dc τ ω ω max

823

density at reaction start dc conductivity oscillation period, relaxation time, optical transmittance (turbidity) oscillation (angular) frequency frequency at loss maximum

1. Introduction The terms “on-line monitoring” and “process analytical chemistry” are often used synonymously. They are gaining increasing attention and importance in both industry and academia. The extent of agreement between the two terms is indeed very broad in that they provide sufficiently accurate and immediate information on variables that describe the state of a chemical reaction. On-line monitoring of reactions encompasses on-stream and on-reactor application of analytical methods to monitor the chemical composition of a reaction mixture, to identify processrelated chemical species, and to quantify the concentration of reaction ingredients, products and byproducts. In addition to revealing the state of the reactor, on-line analysis of physical parameters (temperature, pressure, level, density, viscosity, etc.) may also reflect the extent of a chemical reaction. Process analytical chemistry comprises applications which supply relevant process information of interest “in-time”: The time for sampling and analysis is very short compared to the overall reaction time and thus allows adequate monitoring and efficient control of the reaction. The utilization of the analytical data for control strategies makes process analytical chemistry an essential integral part of process engineering and control systems (see also → Process Development). Process analytical methods may be classified as off-line, at-line, on-line, or in-line with respect to sampling, sample transport, and analysis itself (Table 1) [1]. There is no clear-cut line between the different classes, and the boundaries are even moving: “Some of today’s offline techniques may become tomorrow’s on-line techniques” [2]. An increasing number of offline techniques have been converted into online methods by automated, robot-assisted with-

824

On-Line Monitoring of Chemical Reactions

Table 1. Classes of process analyzers (adapted from [1]) Process analyzer

Sampling

Sample transport

Analysis

Off-line At-line On-line In-line Noninvasive

manual discontinuous/manual automated integrated no contact

to remote or centralized laboratory to local analytical equipment integrated no transport no transport

automated/manual automated/manual quick check automated automated automated

drawal of samples from the reactor or from bypass or process streams and feeding them into off-line instruments. On-line monitoring and control of chemical reactions contributes to: – Guaranteeing and improving product quality and consistancy (i.e., repeatability of product properties within narrow specification ranges) – Increasing the efficiency of the process – Ensuring safe reactor operation by monitoring process and reactor parameters – Understanding fundamentals of the reaction itself – Saving time for analysis and sample transport – Reducing emissions by avoiding sample withdrawal and transport – Reducing costs for labor, raw materials, offspec products, and process waste

Figure 1. Total costs for on-line and off-line process analyses (including instrument costs, labor costs, energy, material, etc.) A) Costs for purchase of off-line instruments; B) Costs for purchase and installation of line equipment

In most cases, purchasing and installation costs of on-line and in-line analytical instruments exceed those of off-line equipment, but as soon as the measurements required exceed a certain (relatively small) number per day, online analysis becomes superior to off-line analysis (Fig. 1). However, the real savings of on-line reaction monitoring are due to improved process efficiency, lower raw materials consumption and waste generation, and, most important, the ability to manufacture high-quality products. Several issues have to be addressed when designing an on-line analysis application [1], [3– 8]: – Information necessary to monitor and control the process (physical parameters, chemical composition, etc.) – Frequency of measurements with respect to the time scale of the reaction – Average values, typical fluctuations, dynamic ranges and expected extremes of properties – Type, precision, and response time of sensors – Robustness of sensors and simplicity of installation – Full automation and minimum maintenance of equipment – Number (combination) of sensors and location of measurement – Proper sampling and, if required, sample conditioning – Form of data output and further handling of information – Compatibility with process control system – Safety precautions and possible hazards – Costs of instrumentation and availability of trained personnel Demands on performance characteristics of on-line analytical techniques differ for academic research applications, process development and pilot plant operation, and monitoring of indus-

On-Line Monitoring of Chemical Reactions

825

Table 2. Relative importance of various performance characteristics of on-line analyses in academic research, process development, and industry [9] Characteristics

Academic research

Process development

Manufacturing

High accuracy Good reproducibility Good selectivity Good sensitivity Extended linearity Good stability Robustness High analysis frequency Short time delay of result Low price Multi-analyte analysis Ease of use Ease of validation High flexibility Ease of implementation Low maintenance

+++ +++ +++ +++ ++ ++ ++ ++ ++ +++ +++ + + + + +

+++ +++ +++ +++ +++ +++ +++ +++ ++ ++ +++ +++ ++ +++ +++ +++

+++ +++ +++ +++ +++ +++ +++ +/+++∗ +++ + + +++ +++ + ++ +++

∗

Process-dependent. +++ Very important, ++ important, + not very important.

trial manufacturing processes, as summarized in Table 2 (see also → Plant and Process Safety; → Process Development).

2. Reaction Calorimetry 2.1. Introduction The majority of chemical processes are accompanied by temperature changes of the reaction mixture owing to release or consumption of heat in the course of the reaction. Heat evolution is a definite, reproducible, and directly measurable characteristic of a chemical reaction. This enables one to monitor the extent of a reaction by measuring temperature or heat flux changes (see [10–21] for introduction, overviews, applications, and related subjects). Reaction calorimetry is noninvasive, rapid, accurate, robust, and, as it is based on temperature measurements, relatively easy to carry out. Temperatures can be measured directly within the reaction mixture (reactor interior), at the reactor/exterior interface (reactor wall), or in the heating/cooling liquid of the reactor jacket. An advantage of reaction calorimetry is that the heat of reaction Qr derived from temperature measurements is directly proportionalto the

rate of reaction. This allows easy access to basic kinetic and thermodynamic data of chemical reactions. With reaction calorimetry, rates of reaction or conversion of reactants can be determined quasi-instantaneously and continuously with a high degree of resolution. This makes reaction calorimetry an ideal tool for real-time feedback control of chemical composition during the course of reaction. Reaction calorimetry not only provides quantitative information on the chemical process itself (e.g., heat of reaction, reaction rate, conversion) but also on reactor parameters necessary for safe reactor operation and process design: – – – – –

Global reaction kinetics Heat production rates Necessary cooling power Reactant accumulation Adiabatic temperature rise to avoid runaway reactions – Heat-transfer coefficients for scale-up However, reaction calorimetry is a nonselective method. It is impossible to distinguish between parallel chemical reactions with heat generation and simultaneous enthalpic processes within the system such as phase transitions, crystallization, mixing, and dissolution.

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On-Line Monitoring of Chemical Reactions

2.2. Heat Flow Balance and Principles of Measurement 2.2.1. Heat Flow Balance The basis for reaction calorimetry is the energy balance around the reactor. Sources and sinks of heat Qi which contribute to the overall heat flux balance (Eq. 1) are shown schematically in Figure 2:

Feeding of material into the reactor with a mass flow rate dmk / dt (and also withdrawal of material from the reactor) leads to a heat flux dQfeed / dt Q˙ feed =



m ˙ k cp,k (Tk − TR )

where cp,k is the specific heat capacity of the kth reactor feed and T k its temperature. The heat input by stirring the (viscous) reaction mixture Qstir is given by Q˙ stir = P0 Nν3 d5stir mix

Figure 2. Schematic of a reaction calorimeter 

Q˙ i = 0

(1)

i

The quantity of interest in reaction calorimetry is the heat flux Qr generated by chemical reactions within the reactor Q˙ r = VR



rj (−∆Hr,j )

(2)

j

where V R is the volume of the reaction mixture, rj the rate of reaction j, and ∆H r,j the reaction enthalpy of reaction j. Different rates of heat generation within the reactor and heat flow out of or into the reactor lead to temperature changes of the reaction mixture. This accumulated heat Qacc is a major cause of uncertainty and error in quantitative analysis of reaction calorimetry data because it depends on the derivative of the reactor temperature T R Q˙ acc = Cp,tot T˙R

(3)

where Cp,tot is the heat capacity of the filled reactor (reactor plus reaction mixture).

(4)

k

(5)

where P0 is the power number, N v the stirrer speed, d stir the diameter of the stirrer, and mix the density of the reaction mixture. The energy balance of the reactor is also influenced by heat losses to the surroundings Qloss (e.g., by conduction or radiation). It is important to identify and quantify all possible sources of heat losses around a reactor because heat losses might be significantly larger (2 – 5 times) than Qacc , Qstir , or Qfeed . In most cases, heat losses must be determined by separate experiments prior to the reaction or without a running reaction. Additional heat can be removed by the insertion of reflux condensers into the reactor. The heat flow over the reflux condenser can be calculated according to Equation (6)

˙ cool cp,cool Tcool,in − Tcool,out Q˙ reflux = m

(6)

˙ cool is the coolant mass flow through the where m condenser, cp,cool is the specific heat capacity of the coolant, and T cool,in − T cool,out the temperature difference across the condenser. For calibration purposes (see below), heat may also be generated by additional heaters Q˙ cal = U I

(7)

where U and I are the voltage and current of the electrical heater, respectively. The quantity that is actually measured in a calorimetric experiment is the heat flux through the reactor wall to the cooling/heating system Qtrans . There are two main methods to determine Qtrans : heat flux calorimetry and heat balance calorimetry.

On-Line Monitoring of Chemical Reactions Heat Flux Calorimetry. In heat flux calorimetry, the generated heat is measured by means of the temperature difference between reactor and jacket. The heat transfer between reactor interior and jacket medium Qtrans depends on the overall heat transfer coefficient k Q through the wall, the heat exchange area of the reactor wall AR that is actually in contact with the reaction mixture (“wet area”), and the difference between reactor temperature and mean jacket fluid temperature T¯ J

Q˙ trans = kQ · AR T¯J − TR

(8)

Typically, the temperature difference is sufficiently large for heat flux calorimetry to be very sensitive. Advantages of heat flux calorimetry are its (i) high sensitivity because of the (relatively large) temperature difference between reaction medium and jacket fluid, (ii) fast response and high accuracy, and (iii) its independence on the flow rate of the jacket medium. The reactor area for heat exchange with the jacket AR is almost constant for batch reactions. There can be some variation in AR as a result of density changes of the medium during the course of the reaction. In semibatch reactions, AR is no longer a constant. Feeding of reaction components into the reactor leads to an increase of its filling level. If the feeding rates are known their influence on AR can be taken into account. A major problem in heat flux calorimetry is the determination of the exact k Q value. The overall heat transfer coefficient may change considerably during the course of the reaction because it depends on, for example, the mixture’s viscosity and density, the stirring rate (hydrodynamics of the reactor), and heat transfer through the reactor/jacket interface (influence of film formation or reactor fouling). Hence, k Q AR must be calibrated separately in a typical heat flux calorimetry experiment. Several variations are proposed in the literature and offered by instrument manufacturers: Continuous calibration during the reaction can be achieved by application of well-defined heat pulses produced by a calibration heater. The reactor heat measured with these pulses enables one to quantify the heat of reaction. However, the hot surface of the calibration heater may also cause problems with temperature-sensitive samples (e.g., product degradation, film formation).

827

Reichert [22], [23] developed an elegant method to overcome the problem of variations in k Q AR , known as temperature oscillation calorimetry. A small sinusoidal temperature change is added to the overall temperature/time characteristic of the reactor via the jacket liquid or a calibration heater. These forced temperature oscillations Q˙ osc ∼ (1 + sin (ωt))

(9)

create temperature oscillations of the reactor with amplitude δT R and frequency ω = 2π/τ , where τ is the oscillation period. Decoupling and separate determination of oscillating and nonoscillating terms in the energy balance equation allows the simultaneous on-line calculation of the rate of reaction as well as the heat transfer value of the reactor k Q AR : kQ · AR = − ω Cp,mixture

δTR sin ϕr δTJ − δTR cos ϕr

(10)

where ϕr is the phase shift between the oscillations, and δT J the amplitude of temperature oscillations of the jacket liquid. One- or Two-Point Calibration. In the case of a constant heat transfer coefficient during the reaction the heat transfer value k Q AR can be determined by calibration before the start of the reaction. Several commercially available calorimeters apply a two-point calibration. The first calibration must be carried out before the reaction run starts to determine an initial value of k Q AR . A second calibration after completion of the reaction in combination with the assumption of a linear behavior between start and end calibration allows one to back-interpolate the reaction data after the end of the reaction. However, the reactor/jacket interlayer is prone to film formation and precipitation of reaction products or byproducts, and the viscosity of the reaction mixture may change dramatically in a short time interval, all of which can alter k Q by orders of magnitude at a certain time (not continuously). Hence, interpolation is inaccurate, and even pre- and post-calibration of k Q by separate experiments may lead to erroneous results. Absolute Heat Flux Calibration. Absolute heat flux calorimetry (ChemiSens AB, Lund, Sweden) relys on the fact that the thermal conductivity λQ within the reactor wall is constant

828

On-Line Monitoring of Chemical Reactions

throughout the reaction, in contrast to the overall heat transfer coefficient k Q through the wall (wall + interlayers). Calorimeter calibration can be carried out by utilizing the temperature gradient (T 1 − T 2 )/d (see Fig. 3). Thus, the heat flux can be calculated according to kQ · AR (TJ − TR ) =

λQ AR (T1 − T2 ) d

(11)

Figure 3. Absolut Heat Flux Calorimetry: Schematic of reactor wall temperature gradient (reproduced by permission of ChemiSens AB, Lund, Sweden)

Note that the immediate vicinity of the reactor wall to both sides (i.e., reaction mixture and jacket liquid) may behave like an additional thin film characterized by other distinct heat transfer resistances. Heat Balance Calorimetry. An alternative approach for measuring the heat generated within the reactor is to determine the total heat balance of the cooling/heating liquid circulating through the jacket:

˙ liq cp,liq TJ,in − TJ,out Q˙ trans = m

(12)

where cp,liq is the specific heat capacity of the jacket liquid. In this case, Qtrans is independent of both heat exchange area and heat transfer coefficient. However, it depends on the circulation ˙ liq of the jacket liquid. Heat losses at the rate m outer jacket wall directly influence the heat balance of the circulating liquid. Heat balance calorimetry is relatively slow and insensitive compared to heat flux calorimetry because of the small temperature difference T J,in − T J,out . This also leads to stricter requirements with respect to the accuracy of temperature measurements. To achieve 0.5 W resolution, it is necessary to measure the temperature with an accuracy of about 0.001 K (compared to

0.05 K in heat flux calorimetry for the same resolution). Under typical conditions, energy resolution in heat balance calorimetry is on the order of 2 W, compared with 0.5 W in heat flux calorimetry. Heat Compensation Calorimetry. In heat compensation calorimetry, both the vessel temperature T R and the jacket-fluid temperature T J are fixed with T J < T R . Heat from an additional electrical heater Qel is supplied to the mixture to maintain the temperature T R of the reactants. After the reaction has been started and a heat Qr released, the electrical power Qel is reduced to hold T R constant. This reduction in Qel is equal to the reaction heat Qr under the condition that k Q AR and the heat losses Qloss are constant throughout the reaction. As in heat flux calorimetry, the unknown and changing quantity k Q AR requires calibration of the calorimeter. The hot surface of the electrical heater is prone to the formation of films, coatings, precipitates, and hot spots. 2.2.2. Basic Modes of Operation From a thermodynamic point of view, three basic modes of operation in heat flux calorimetry can be distinguished: adiabatic, isoperibolic, and isothermal calorimetry. Adiabatic Reaction Calorimetry. An adiabatic reaction calorimeter is characterized by thermal insulation of the reaction mixture from the surroundings. As a result, the heat released during the reaction is stored within the reaction mixture. Thus, the temperature gradient of the reaction mixture directly reflects the ongoing reaction, and the total heat balance is simply given by (cf. Eq. 13): Q˙ r = CR,tot T˙R

(13)

In practice, thermal insulation of the reaction mixture can be achieved either by means of an infinitely large thermal resistance between mixture and surroundings (e.g., Dewar calorimeter with evacuated reactor jacket) or by continuously matching the jacket temperature to that of the mixture such that there is no heat transfer through the reactor/jacket wall (T J (t) = T R (t)).

On-Line Monitoring of Chemical Reactions A major disadvantage of adiabatic reaction calorimetry is the interdependence of mass balance and heat balance via the reaction rate, which requires an adequate kinetic model for data analysis. Isothermal Reaction Calorimetry. In isothermal reaction calorimetry, the reaction mixture temperature T R is held constant, for example, by adjusting the temperature of the jacket fluid T J or by controlling the power of a compensation heater. In practice, it is very difficult to keep the reactor strictly isothermal because of the nonzero heat transfer resistance of the reactor wall and deviations and delays in power control. Thus, heat accumulation must be taken into account in the isothermal heat balance

Q˙ r = kQ AR TR − T¯J + Cp,total T˙R

(14)

Isoperibolic reaction calorimetry is opposite to isothermal calorimetry in the sense that the jacket temperature T J is held constant instead of the reactor temperature T R . As a consequence, T R changes during the reaction. Heat is both accumulated in the reaction mixture and transferred to the jacket liquid. Hence, the heat balance can be described as in isothermal calorimetry (see Eq. 14).

829

surface area to volume compared to bench-scale reactors and calorimeters; hence, their heatremoval capacity is limited. To achieve a maximum production rate in industrial-scale reactors, the reaction rate should be matched to the heat-removal capacity throughout the reaction. On-line reaction calorimetry provides some of the data necessary for a desired reaction run and safe reactor operation (e.g., heat release rates, heat losses, reaction rate, conversion). Reaction calorimetry has been used to monitor polymerization reactions and crystallization processes [28–30]. Strategies for on-line control of reactions on the basis of calorimetric measurements during the course of the reaction have been described [31–33]. Reaction calorimetry is the method of choice for studying thermal runaway reactions [34]. The application of calorimetry to investigating the kinetics of reactive blending and reactive extrusion is discussed in [35], and the application of adiabatic temperature rise for monitoring reaction injection molding (RIM) in [36].

2.3. Applications and Instrumentation The heat of reaction Qr (t) can be calculated according to Equation 1 if all contributions Qi to the overall heat balance of the reactor are measured or known. If only one exothermic chemical reaction proceeds in the vessel, the reaction rate is given by Equation 2, and the actual relative calorimetric conversion X r (t) is given by Equation 15 t Xr (t) =

Qr (t) dt

0 t,final 

(15) Qr (t) dt

0

On-line calorimetry has been used by chemical companies to monitor the extent of reactions for several decades [24–27]. Industrial-scale reactors have a relatively low ratio of heat-exchange

Figure 4. Reaction calorimeter RM-2S for heat flow and heat balance measurements (reproduced by permission of ChemiSens AB, Lund, Sweden)

Most instrument suppliers (e.g., MettlerToledo, ChemiSens, HEL) offer computercontrolled calorimeters that can be run in different modes (Figure 4). The nonselectivity of re-

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On-Line Monitoring of Chemical Reactions

action calorimetry in monitoring complex reactions (e.g., reactions in which parallel processes occur, copolymerizations, etc.) can be overcome by combining calorimetry with other online methods to obtain additional information. These methods include density, pressure, gas chromatography, dielectric spectroscopy, and infrared and Raman spectroscopy (Fig. 5).

In most applications, ultrasound is generated by piezoelectric transducers. Because of the reversibility of the piezoelectric effect, the transducers can act both as emitters and receivers of ultrasonic waves. Depending on the requirements of the application, a wide variety of readily available and cost effective piezoelectric materials can be utilized as transducers (e.g., quartz, lithium niobate, barium titanate, poly(vinylidene fluoride), zinc oxide). Ultrasonic sensors and ultrasonic systems for materials characterization, process monitoring, and applications in chemistry are reviewed in [41–48].

3.2. Theory A planar elastic ultrasonic wave can be described by its complex alternating pressure p(x,t) as a function of time t and distance x (Eq. 16) .  x / p (x, t) = p0 exp iω t − · exp [−αx] u

Figure 5. Reaction calorimeter RC1 equipped with React IR infrared Spectrometer (ASI Applied Systems) for simultaneous on-line calorimetry and infrared analysis (courtery of Mettler-Toledo GmbH, Analytical)

Calorimetric data of chemical reactions have also been determined by differential scanning calorimetry [37–40].

3. Ultrasonic Methods 3.1. Introduction The propagation of ultrasound in matter is a physical effect that is receiving increasing attention for on-line monitoring of chemical processes. Ultrasonic methods are easy to use, safe, nondestructive, and noninvasive. Signal response times on the order of milliseconds allow real-time monitoring of fast reactions.

(16)

where ω = 2πf (f = frequency), p0 is the pressure amplitude, and u and α are the velocity and attenuation of sound in matter. The frequency of ultrasound waves is in the range of 20 kHz to about 1 GHz with corresponding wavelengths on the order of 1.6 cm to 0.3 µm in air (u ≈ 330 m/s), 6 cm to 1.2 µm in liquids (u ≈ 1200 m/s), and 20 cm to 4 µm in solids (u ≈ 4000 m/s). Both u and α strongly depend on the material’s state and properties. In solids, the longitudinal and transverse sound velocities ul,sol and utr,sol are related to the bulk modulus K, the shear modulus G, and the density  according to u2l,sol =

K + 

4 G 3

and u2tr,sol =

G 

(17)

The longitudinal ultrasonic velocity in liquids ul,liq depends on both the adiabatic compressibility β of the liquid and its density (Eq. 18). u2l,liq =

1 β

(18)

Ultrasonic shear waves in liquids (utr,liq ) do not propagate significantly. They are strongly dampened, and therefore not of technical importance. The sound attenuation α of solids and liquids depends on several loss mechanisms. Intrinsic, viscous, and thermal losses may contribute to

On-Line Monitoring of Chemical Reactions α, as well as scattering and reflection effects at interfaces in heterogeneous samples, relaxation processes in polymers, electrokinetic losses in disperse systems, and structural losses in concentrated systems (e.g., pseudoplasticity). Thus, α is influenced by material properties such as density, viscosity, thermal conductivity, expansion, and capacity; by the morphology of the sample (grain size and shape); and by temperature and pressure. It is difficult to derive analytical expressions for α because of its complex dependence on these parameters. However, in simple homogeneous liquid systems, α can be described as αliq 2 · π2 = 2 f  · u3



4 ηs + ηv 3

Z1 − Z2 = 1−T Z1 + Z2

(20)

(21)

where Z 1 and Z 2 are the acoustic impedances of the materials on both sides of an interface, and T is the transmission coefficient. A common method for determining acoustic properties is to monitor (compressional) sound pulses travelling over a well-defined distance (Fig. 6). The emitting transducer E generates a sound burst of initial amplitude p0 (proportional to the applied voltage U 0 ) which travels a distance x in a time ∆t. In multiple-echo mode, ∆t is the time delay between successive echoes, and x is twice the distance between transducers. The dampened (and broadened) pulse reaching the receiving transducer R generates an electrical voltage Ux proportional to the pressure amplitude px at the receiver. By using this method, sound velocity and attenuation can simply be calculated according to Equations 22 and 23. u =

x ∆t

(23)

(19)

It is typically measured in reflection experiments at interfaces by means of the reflection coefficient R (Eq. 21). R =

1 p0 1 U0 = ln ln x px x Ux



where η s and η v are the shear and the volume viscosities of the liquid. Sound is strongly absorbed at high frequencies (f 2 dependence) and cannot penetrate deeply into the material. A third acoustic parameter of importance with respect to reaction monitoring is the acoustic impedance Z of the material (Eq. 20). Z = ·u

α =

831

(22)

Figure 6. Schematic of ultrasonic pulse travelling measurements. (The transducer arrangement can also be used in multiple reflection echo mode.)

Measurements of pulse travelling time are highly accurate and relatively easy to carry out compared to sound attenuation measurements. The frequency of ultrasonic sensors for reaction monitoring is generally in the 1 – 10 MHz range. In principle, the accuracy of sound velocity measurements is poor for very low frequencies (< 100 kHz) and increases with increasing frequency. However, attenuation does also increase for f  10 MHz, and the distance between transducers must be shortened for the signal to reach the receiver. As a result, the accuracy of pulse travelling time measurements also deteriorates at very high frequencies. Typical demands on the accuracy of ultrasonic sensors for reaction monitoring are ∆u ∆α ∆Z ≤ 10−3 ≈ 0.5 % ≤ 10−3 u α Z

These conditions imply time measurements in the nanosecond and picosecond range, and of amplitude resolutions of 12 bit and higher [41]. Additionally, temperature control has to be better than ± 0.1 K. In addition to u and α, other sound wave characteristics (e.g., wave phase ϕ, resonance frequency f res and shift ∆f res ) can be measured in certain ultrasonic sensor applications for materials characterization and chemical processes monitoring [41].

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On-Line Monitoring of Chemical Reactions

3.3. Applications The basic condition for monitoring chemical processes by means of ultrasound is that physical properties that determine u, α, or Z change during the course of the reaction (see Eqs. 17 – 20). In polymer production and modification, curing and cross-linking reactions, mechanical moduli as well as the mixture’s density, compressibility, viscosity, thermal conductivity, etc. alter dramatically, thus influencing sound attenuation and velocity (see Table 3). Hence, those processes are ideal for application of ultrasound to detect the extent of reaction. Even electrochemical processes can be monitored because of the sensitivity of ultrasound to the type of ion and concentration.

tic impedance Z via reflection coefficient measurements may reduce the bubble disturbances. However, so far, no application of Z determination in reaction monitoring has been described in the literature. LIVE GRAPH

Click here to view

Table 3. Ultrasound velocities of monomers and polymers at 20 ◦ C Substance

Monomer

Polymer

Butyl acrylate Styrene Vinyl acetate Vinyl chloride

1233 m/s 1354 m/s 1150 m/s 897 m/s

1375 m/s 2120 m/s 1853 m/s 2260 m/s

LIVE GRAPH

Click here to view

LIVE GRAPH

Click here to view

Figure 7. Influence of gas bubbles on ultrasound velocity measurements during emulsion polymerization

A major advantage of ultrasound (velocity) measurements in industrial applications is the capability to plug the sensor directly into the reacting mixture. They do not necessarily require a sampling loop. Care should be taken because plug-in transducers may cause fouling at their edges or lead to formation of coagula and films. The disturbing influence of gas bubbles in the reaction mixture between the transducers is demonstrated in Figure 7. Monitoring of acous-

Figure 8. Ultrasonic velocity (A) and relative ultrasonic absorption (B), both measured at 1 MHz, as a function of the extent of epoxide ring reaction P for a DGEBA/DDM system at 70 ◦ C; gel point Pgel = 0.58 (reprint from [62], with permission from Elsevier Science)

Previous Page On-Line Monitoring of Chemical Reactions A simple empirical but useful approach for describing the sound velocity of multicomponent reaction mixtures is based on the assumption that each component i of the mixture contributes independently to the sound velocity of the mixture umix according to its concentration ci umix = u0 +

  ∆u  i

∆c

ci

(24)

i

where u0 is the sound velocity of a reference component (e.g., solvent, dispersion medium) whose ultrasonic velocity is either constant throughout the process or changes in a known manner, and ∆u/∆c is the (temperaturedependent) sound velocity – concentration coefficient, which itself may depend on concentration. However, higher order terms are neglected in Equation 24. The influence of component i on the sound velocity of component j is also neglected. Equation 24 is the basis for on-line concentration measurements in numerous applications in chemistry, biochemistry, and the foods industry. Ultrasound has been used to monitor chemical reactions for more than twenty years [49–52]. Ultrasonic on-line monitoring of polymerization reactions was reported in [53–57]. Crosslinking reactions (curing, gelation) were investigated with ultrasound in [52], [58–64]. As an example, the influence of the extent of reaction of the epoxide ring in a system consisting of the diglycidyl ether of bisphenol A (DGEBA) and 4,4
diaminodiphenylmethane (DDM) on the ultrasonic velocity and relative ultrasonic absorption is shown in Figure 8. Viscoelastic properties of polymer melts in extruders can be monitored by in-line determination of ultrasonic parameters, along with temperature and pressure measurements [35], [51], [65–67]. Ultrasound is sensitive to the local concentration of oil in emulsions and thus enables monitoring of creaming and flocculation processes [68], [69]. It is applicable to both dilute and concentrated emulsions. Ultrasound is not only a tool for monitoring chemical reactions but also one for initiating them. The origin of the enhancement of chemical reactions by ultrasonication is the intensity of the cavitation bubbles produced in the sonicated medium at higher sound intensities [47], [48], [70–72].

833

The dependence of ultrasonic parameters on grain sizes in heterogeneous media enables one to determine particle sizes and particle size distributions in latices, emulsions, suspensions, etc. even at very high concentrations of the disperse phase [73–77]. Thus, acoustic spectroscopy (as well as electroacoustic spectroscopy [75], [78]) is developing rapidly as an alternative to lightscattering methods for particle size analysis in dense media. Ultrasonic microsensors are used for materials characterization and reaction monitoring [41], [43]. Ultrasonic microsensors based on the quartz crystal microbalance (QCM) principle are very sensitive for a wide variety of chemical applications. The high precision and ease of measurement of the QCM frequency has made it a useful tool for measuring slight changes in mass on the QCM electrode surface. By coating the surface with a sensitive layer, the QCM can be used as selective gas or even under-liquid sensor. The determination of contact angles and surface tensions with a QCM is described in [79]. Detergency processes have been monitored in real-time by using a QCM [80]. The use of shearhorizontal (SH) acoustic plate mode (APM) devices to monitor cross-linking reactions in polymer films in real time is described in [81]. Photocross-linking of diacetylene thiol-based monolayers on surface acoustic wave sensors (SAW) has been monitored in situ [82]. Recent developments include the application of fiber-optic interferometric ultrasound sensors for monitoring the cure of epoxy resins [83], [84].

4. Dielectric Spectroscopy 4.1. Theory and Mechanisms Dielectric spectroscopy deals with the electrical response of the polarization P (ω) of matter to the application of an oscillating electric field E (ω) D (ω) = εo E (ω) + P (ω) = ε∗ (ω) ε0 E (ω) (25)

where D is the dielectric displacement vector, ω is the angular frequency, and ε∗ and ε0 are the dielectric permittivity and the permittivity of free space, respectively. Several atomic and molecular mechanisms contribute to the dielectric behavior of matter [85–88]:

834

On-Line Monitoring of Chemical Reactions LIVE GRAPH

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Figure 9. Dielectric relaxation: Dielectric permittivity and dielectric loss as a function of frequency

1) Electronic polarization (i.e., induced dipole moment due to the displacement of electrons from their equilibrium position with respect to the nucleus) 2) Displacement polarization (atomic, ionic, lattice polarization; e.g., induced dipole moment due to the deflection of ions from their equilibrium positions in molecules) 3) Orientation polarization of permanent dipoles present in the system (i.e., alignment to the direction of the external field) 4) Motion of intrinsic and extrinsic charge carriers (ions, particles) 5) Charge transfer and polarization at electrodes 6) Interfacial polarization and distortion of the electrical double layers at interfaces in heterogeneous systems A certain mechanism contributes to the overall polarization of the sample if its characteristic relaxation time is faster than the time scale of the external field. Hence, the various polarization mechanisms can be distinguished on the basis of their different frequency dependencies. Time scales for resonances of electronic and displacement polarization of atoms and ions are in the femtosecond (10−14 to 10−15 s; UV region) and picosecond range (10−12 to 10−13 s; IR region), respectively. Thus, these induced dipoles contribute instantaneously to the dielectric properties of matter typically measured in the frequency range 10−5 to 1011 Hz. Key phenomena determining the change of dielectric properties during the course of a chem-

ical reaction are orientation polarization of permanent dipoles, migration of charges, and interfacial effects in heterogeneous systems. In contrast to electronic and atomic polarization, the orientation of permanent dipoles is time-dependent and not always instantaneous. It is governed by the (micro) viscosity of the immediate surroundings. In polymeric systems, it requires cooperative motion of chains or chain sequences, too. Dipole relaxation is the strongly temperature dependent reorientation of dipoles due to thermal fluctuations after the removal of the external field. As the frequency is scanned, various characteristic dipole relaxations accompanied by corresponding losses are observed. The migration of charges mainly influences the dielectric losses. The (ionic) conductivity contribution to the loss factor is inversely proportional to the frequency of the external field. Ionic conductivity can be correlated with viscosity, since fluidity is indicated by the ease with which charge carriers can migrate through the sample. The complex dielectric permittivity ε∗ (ω) is composed of two parts, a real (or storage) component ε
(ω) in phase, and an imaginary (or loss) component ε

(ω) out of phase with the applied oscillating electric field (Eq. 26). 

ε∗ (ω) = ε (ω) + iε (ω)



(26)

where i is (−1). ε
(ω) is proportional to the capacitance of the sample, and ε

(ω) is influenced both by the material’s conductivity and

On-Line Monitoring of Chemical Reactions by dipolar losses. A sigmoidal change in ε
(ω) and the appearance of a peak in ε

(ω) is indicative of a relaxation process (Fig. 9) 

ε (ω) = 

ε0

−

ε0 − ε∞ ω 2 τ 2

1 + ω2 τ 2

ε0 − ε∞ ωτ 1+

ω2 τ 2

4.2. Instrumentation and Applications (28)

where τ is the relaxation time. The loss factor tan δ is defined by tanδ (ω) =

ε (ω) ε (ω)

formalism (e.g., Z

vs. Z
plots) is particularly helpful in separating bulk and surface phenomena (e.g., polarization in the bulk and electrode polarization).

(27)



ε (ω) =

835

(29)

Instrumentation. According to the frequency of the external electric field, several frequency regimes for dielectric experiments can be distinguished which require different measurement principles [90–92] (Fig. 10).

As indicated above, the dielectric permittivity is not a constant but a function of frequency. Additionally, ε∗ is strongly influenced by temperature since both the transitional mobility of charge carriers as well as the reorientational motion of dipoles are temperature-dependent. For some cases in the absence of chemical reactions, the temperature dependence of dielectric properties can be described in terms of Arrhenius and Williams – Landel – Ferry (Vogel – Fulcher) type relations, depending on the nature of the dipolar motions (single activation and cooperative process, respectively). In addition to the complex dielectric permittivity, there are several other quantities that can be alternatively derived from dielectric spectra [89], [90]. These include the impedance and admittance (or conductivity), the dielectric modulus, and the susceptibility. Admittance spectra Y ∗ (ω), the inverse of impedance spectra Z ∗ (ω) = Z
(ω) − iZ

(ω) = (Y ∗ (ω))−1 for linear response, can be calculated from permittivity data according to Equation 30. ε∗ (ω) =

Y ∗ (ω) iωC0

(30)

where C 0 is the equivalent capacitance of free space. Transformations from one formalism to another can help resolve particular aspects of the relaxation processes, but no new information can be extracted in the linear response regime of the material. The impedance spectrum Z ∗ (ω) is often used for discussion because dielectric data can be analyzed in terms of equivalent electrical circuits consisting of resistors and capacitors connected in parallel and series. The Z ∗ (ω)

Figure 10. Principles of dielectric measurements: A) Frequency response analysis technique; B) Resonance circuit; C) standing wave in hollow conductor PS: Power source

The common instrumentation in the impedance regime from about 10−5 to 108 Hz includes current – voltage measurements in electrode/sample/electrode sandwich arrangements

836

On-Line Monitoring of Chemical Reactions

of electrical (ratio-arm) bridges, and resonance (oscillating) circuits. The main techniques for dielectric reaction monitoring are (1) the direct ac measurement of amplitude and phase between voltage and current over a certain frequency range by using a frequency response analyzer (FRA), and (2) step voltage application (or removal) with rapid acquisition of the current response over time and transformation of the signal into the frequency domain. Minimum time intervals for dielectric measurements are on the order of tenths of seconds, so dielectric spectroscopy in the impedance frequency regime is suitable for on-line monitoring of chemical reactions. Typical accuracies of measurements achievable in the impedance frequency regime are about ± 0.05 % for ε
, ± 0.2 % for ε

, and better than 10−4 for tan δ. Dielectric cells for liquids are generally simple and consist of two concentric cylinders with the fluid in the space between them. These measurements are of very high precision because of the good dimensional stability of the cells. Solid samples are typically sandwiched between metal plate electrodes as in a capacitor. Compared to liquid cells, solid sample set-ups are quite complicated because they must take into account for the sample’s thermal coefficient of expansion. The radio regime from about 106 to 1010 Hz requires resonance circuits (low ω) or guidedwave techniques (high ω; e.g., standing waves in hollow conductors). The typical equipment for the microwave region from 109 to several 1011 Hz is coaxial or rectangular waveguides. Propagating wave techniques are utilized in the IR and UV – VIS regimes above a few 1011 Hz with Fourier transform technique as the method of choice. Applications. Chemical reactions are accessible to reaction monitoring by dielectric spectroscopy if they lead to changes in the rate of motion of molecules contributing to the dielectric response of the mixture (e.g., rotational or vibrational motion of dipoles and translational motion of charge carriers). Many applications of dielectric spectroscopy published so far are for monitoring the cure of resins. Dielectric spectroscopy can detect changes in the polymer cross-link density during the course of polymerization (Fig. 11).

When van der Waals or other weak interactions between molecules are replaced by covalent bonds during the conversion of low-viscosity monomers into polymers, the dipole moments of molecules and the mechanisms of their Brownian diffusion change, as does the size of the diffusing moieties. The restrictions on the diffusion of dipoles and charge carriers increase [93–98]. LIVE GRAPH

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Figure 11. Real and imaginary impedance as a function of time with frequency as parameter during the reaction of the diglycidyl ether of bisphenol F (DGEBF) with 4,4 methylenedianiline (MDA) at 36 ◦ C [96] a) Z  = 500 Hz; b) Z  = 500 Hz; c) Z  = 2000 Hz; d) Z  = 2000 Hz; e) Z  = 10 000 Hz; f) Z  = 10 000 Hz

Both the real and the imaginary parts of the dielectric permittivity can have dipolar and ioniccharge polarization components. At lower frequencies, ε

is typically dominated by the mobility of ions in the sample, whereas dipolar reorientation mobility contributes to ε

at higher frequencies (Eq. 31). ε∗ (ω) = ε∗dipolar − i

σdc ωε0

(31)

Here σ dc is the dc conductivity due to everpresent ionic impurities [99], [100]. The ionic mobility (i.e., σ dc ) is a molecular probe for quantitatively monitoring viscosity changes during cure. At short curing times, ε

is dominated by the dc conductivity. When gelation occurs, the dc conductivity decreases strongly since the mobility of the charge carriers decreases due to network formation. Peaks in ε

indicate when dipolar reorientation processes contribute to ε

. Dielectric spectroscopy in the 10 to 105 Hz frequency range with a ratio-arm bridge has been used to monitor the UV polymerization of a polymer-stabilized liquid crystal cell containing a mixture of nematic and chiral liquid crystals

On-Line Monitoring of Chemical Reactions together with DSM monomer and photoinitiator [101]. The authors discuss their results in terms of capacitance units. They have shown that the decreasing loss serves as an excellent monitor for changing molecular processes and the state of photopolymerization, whereas the capacitance itself does not change dramatically during the course of the polymerization. Lowfrequency losses observed at the beginning of the reaction are mainly caused by ionic impurities. After polymerization, the liquid crystal cell has the dielectric characteristic of a relatively low loss dielectric. In particular, dielectric spectroscopy is the method of choice for following the curing of polar coatings. In these thin-layer systems, very high signal-to-noise ratios can be achieved because their capacitance is inversely proportional to their thickness, like the capacitance in a plate capacitor as a function of interplate distance. Dielectric spectroscopy has been also applied to disperse systems [102–105]. The relaxation pattern is the dielectric fingerprint of a colloidal system [106]. Correlations between relaxation frequencies and suspension properties such as particle volume fraction, electrolyte concentration, Zeta potential, pH, surface conductivity, particle radius, Debye screening length, degree of ionization, and diffusion coefficient have been developed [107], [108]. These characteristics change during reactions in colloidal systems, and on-line monitoring is possible. Moreover, dielectric spectroscopy is especially valuable for studying concentrated and opaque dispersions. For a latex exhibiting a low-frequency relaxation process, the relaxation time at the loss maximum τ = 1/ω max is proportional to the square of the particle radius, and inversely proportional to the diffusion coefficient of the electrical double layer of counterions up to 30 % total solids [109].

5. Optical Spectroscopy 5.1. Introduction Optical spectroscopy covers the range of the electromagnetic spectrum from the ultraviolet (UV, λ  10 nm) to the infrared (IR, λ < 1 mm). Experimental equipment for optical

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spectroscopy in this wavelength range is constructed with typical optical elements such as mirrors, lenses, gratings, and optical filters. The incident energy is delivered by light sources. The long-wavelength end of the electromagnetic spectrum (i.e., millimeter range) requires completely different experimental techniques (e.g., hollow conductors for microwave spectrometry) as does the short-wavelength end (λ  100 nm) with special X-ray and γ-ray equipment. Electronic absorptions determine the spectral features observable in the UV – VIS region. Molecular vibrations, rotations, and combined rotational-vibrational transitions of molecules can be detected in the near (NIR) to the far (FIR) infrared wavelength range. Spectra can either be recorded in absorption (UV, VIS, NIR, IR, FIR) or emission (IR, Raman, fluorescence).

5.2. Instrumentation for Reaction Monitoring Spectroscopy over the entire spectral range of optical wavelengths has been applied to online process monitoring for several decades. The vast majority of applications is based on nondispersive instruments, and light absorption or emission is typically measured either at a single wavelength or by integration over a certain (rather small) wavelength range. In absorption spectroscopy, the light is generated by monochromatic light sources (including lasers) or polychromatic light sources combined with optical filters. Tuning of the wavelength to characteristic absorptions of certain molecules enables one to detect substances and to observe the variation of their concentration as a function of time with high selectivity and sensitivity. In transmission measurements, the fundamental relation between the incident light intensity I 0 , the transmitted light intensity I l , and the concentration of the analyte c in a cell of optical length l is given by the Lambert – Beer law: log

I0 (ν) 1 = log = A (ν) = ε (ν) cl Il (ν) τ (ν)

(32)

where ε(ν) is the (decadic) extinction coefficient of the sample at frequency ν, l is the length of the measurement cell, A is the absorbance (extinction), and τ is the transmittance of the sample. Monitoring procedures based on this classical

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photometric method are still extensively used because of its simplicity, ruggedness, sensitivity, and low cost and maintenance requirements. It works best for well-characterized continuous material streams but is less advantageous for dynamic processes (e.g., some batch operations). The number of substances that can be detected simultaneously by nondispersive instruments in absorption and emission modes is limited, and detailed molecular information on the species undergoing a reaction is not available. These restrictions can be overcome by parallel installation of several nondispersive instruments operating at different wavelengths, or utilization of dispersive instruments (spectrometers) to exploit the whole spectral information. With classical dispersive wavelengthscan spectrometers, data collection is timeconsuming because such an instrument records the data points of the spectrum consecutively. Therefore, dispersive scanning spectrometers are not suitable for process monitoring. However, the substitution of point detectors by array detectors in dispersive instruments offers the opportunity to acquire large amounts of spectrally and/or spatially resolved data in extremely short periods of time. Semiconductor array detectors are characterized by high quantum efficiency, wide dynamic range, fast response, low noise, low power requirement, and continuously decreasing costs. Array detectors suffer from either limited resolution or limited bandwidth. However, the two-dimensionality of some types of array detectors in combination with cross-dispersion enables the complete spectrum to be obtained with high resolution. Since the whole spectrum is available with one shot of light on the detector, chemical reaction monitoring on the sub-second timescale is possible. Another advantage of two-dimensional array detectors is the ability to record several spectra simultaneously in different lines of the array thus enabling multiplexing. Fundamentals and applications of array detectors [charge-transfer devices (CCD, CID), photodiode arrays, and photoconductor arrays] in both absorption and emission spectroscopy are reviewed in [110]. With Fourier transform (FT) spectrometers, the spectral information is first recorded in the time domain as an interferogram and then Fourier transformed into the frequency (wavenumber) domain. The optical throughput

of an FT instrument is very high because light of all frequencies is recorded simultaneously. Depending on the spectral resolution and the signal-to-noise ratio required, FT spectrometers are suitable for monitoring very fast chemical processes on the timescale of seconds to minutes. However, they are less rugged than photometers or array-detector instruments because the interferometer is susceptible to vibrations, for instance, in a production-plant environment. The introduction of fiber optics has been a major step forward in reaction monitoring by optical spectroscopy. Nowadays, most instrument suppliers offer their spectrometers with optional fiber optics. Optical fibers connect the remote instrument with the reactor or process line for online monitoring with the following advantages: – Reduced risk of fire or explosion in production units – Reduced interaction of the delicate spectrometer with the hostile process environment – Enhanced accessibility of various measurement points in a process stream or reactor – Simplified multiplexing Table 4. Fiber types and typical optical wavelength range [111] Fiber type

Approximate optical range

High-purity fused silica Plastics Glass Low-OH fused silica Zirconium fluoride Chalcogenide Silver halides

200 – 2000 nm 350 – 1100 nm 350 – 1800 nm 250 – 2700 nm 2 – 5.5 µm 3 – 11 µm 5 – 20 µm

Light may be delivered and collected through separate single fibers or fiber arrays; alternatively, fibers can be arranged in integrated fiber bundles in which a light source fiber is surrounded by detector fibers, or in which source and detection fibers are randomly distributed within the bundle. A review of the developments in fiber optic applications in molecular spectroscopy is given in [111]. Commonly used fiber types are listed in Table 4. Recently, the development of fiber-optic chemical sensors (FOCS) based on evanescent wave absorption (EWA) spectroscopy has experienced much progress and improvement.

On-Line Monitoring of Chemical Reactions

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FOCSs can be employed for real-time in situ monitoring by simply immersing them in the mixture and establishing a connection between the fiber, the light source, and the detection system. They offer fast response times on the order of seconds with all the advantages of ATR spectroscopy, and considerably improved sensitivity. Figure 12 shows various instrument/probe/process vessel configurations for the application of optical spectroscopy to the monitoring of chemical reactions. Spectra can either be recorded in transmission, reflection, transflection, or attenuated total reflection (ATR). Probes can be directly immersed into the reaction mixture, or the spectra can be recorded through windows in the reactor or pipe wall.

The versatility of UV – VIS spectroscopy in process monitoring has been extended by the development of UV array spectrometers [112]. Optical fibers for use in the VIS region are readily available. Transmission of visible light through optical cables is possible over long distances for remote process monitoring. In the short-wavelength UV, however, the optical absorption of the fiber core and/or cladding material restricts light throughput. Recently, evanescent-wave fiber optic absorption sensors for use in the UV and VIS regions have been described [113], [114]. UV reflection spectroscopy with a fiber optic accessory has been used to characterize the cure of bis(maleimide) resins [115], [116].

5.3. Applications of Optical Spectroscopy

5.3.2. NIR Spectroscopy

5.3.1. UV – VIS Spectrocopy UV – VIS spectroscopy is highly sensitive, but the spectra are less informative because they typically consist of a few broad absorption peaks. Therefore, UV – VIS spectroscopic reaction monitoring is less commonly used than other spectroscopic techniques. Typical applications of UV – VIS spectroscopy include photometric gas analysis in transmission gas cells for the detection of sulfur dioxide, nitrogen oxides, chlorine, mercury vapor, toluene, benzene, methane, ethanol, etc. Detection limits are in the order of tens of ppm in air [8]. A major advantage of the UV – VIS spectral region is that water does not absorb considerably in the entire VIS region up to the mid-UV, thus making it attractive for water and wastewater analysis. UV – VIS absorption measurements in aqueous or other fluid process streams are typically performed in transmission or ATR modes. With ATR probes, even highly opaque mixtures and liquid systems containing strongly scattering particulate solids are accessible to process analysis. In disperse systems, light scattering by particles accounts for much of the attenuation of the incident light. Thus particle size measurement by UV – VIS scattering methods is common practice in colloid laboratories and at production sites.

Absorption bands in the near infrared originate from electronic transitions of the heaviest atoms and from overtones and combinations of fundamental vibrations in the IR. In organic substances, C – H, N – H, and O – H vibrations give rise to dominant absorptions in the nearIR. Therefore, the detection of water was one of the first on-line applications of NIR spectroscopy. However, water interference may also complicate NIR spectroscopy of aqueous samples [117]. NIR-active vibration overtones and combinations are characterized by low molar absorptivities and high scattering efficiencies, and the NIR bands are broad and overlapped. The peculiarities of NIR spectrometry lead to important consequences in practice: 1) Near IR wavelengths are able to penetrate into and sometimes even through a sample, thus enabling measurements on real world samples in the millimeter or centimeter thickness range. Spectra can be recorded on most types of liquid, solid, powder, or gaseous samples without sample pretreatment. Sample size may vary from tiny bits to large chunks. 2) NIR spectra contain all the spectral information necessary for fingerprinting, albeit buried under broad and strongly overlapped absorption features. In most cases, the spectra cannot be interpreted directly, unlike the individual peaks of IR spectra, and calibrations and chemometric methods have to be

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On-Line Monitoring of Chemical Reactions

Figure 12. Instrument probe process vessel configurations for the application of optical spectroscopy to monitoring chemical reactions A) Transmission; B) Transflection; C) Reflection; D) Scattering (90 ◦ ); E) Attenuated total reflection (crystal); F) Attenuated total reflection (single-bounce tip); G) Evanescent-wave fiber optic FO = fiber optic; L = light source; D = detector

On-Line Monitoring of Chemical Reactions

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Figure 13. Ceramic transmission NIR probe with variable optical pathlength and diffuse reflectance dipper probe for insertion into extruder (courtesy of IPF Dresden)

exploited for quantitative analysis. This may cause problems in NIR reaction monitoring of batch processes with frequently changing recipes and reaction regimes because of the necessity to provide an extensive training data set for each process [118]. Typical analysis time is on the order of seconds or even shorter, with sensitivities down to tenths of a percent; hence, NIR is an ideal tool for observing fast reactions. Spectra of chemically changing mixtures can be recorded through quartz glass windows mounted in reactors and pipes or by fiber optic immersion probes for diffuse reflectance, attenuated total reflectance (ATR), transmission, and transflection measurements. The application of cheap silica fibers allows truly remote sensing with distances up to kilometers between process and spectrometer because of the extremely low attenuation of quartz in the VIS to NIR range. NIR equipment suitable for reaction monitoring includes both holographic grating/semiconductor array detector instruments and Fourier transform spectrometers. Lightemitting diode (LED) based instruments have no moving parts because each wavelength band is produced by a different diode with precise wavelength tunability and monochromaticity, thus making the instrument rugged to harsh

process environments. LED spectrometers are highly sensitive and selective, with fast response and exceptional dynamic range. So far, they have been applied to on-line monitoring of HF and H2 S traces in refineries. However, their high costs are prohibitive for most on-line applications. Acousto-optic tunable filter (AOTF) based instruments have been occasionally used for on-line analysis. Hadamard transform (HT) spectrometers have not been described for NIR on-line applications so far. Typical problems with NIR measurements are the need to check the calibration of the instrument, since the instrumental response may change after some time [119], and the sensitivity of NIR spectra to the sample temperature, which complicates training procedures. NIR equipment and the application of NIR to reaction monitoring are reviewed in [120–124]. NIR spectroscopy has been widely used for the characterization of epoxy curing [125] with conventional fiber optics [126], [127], evanescent wave high-index fiber optic sensors [128], and the NIR ATR technique [129]. Other fiber optic applications include reaction monitoring in explorative organic synthesis [130], real-time analysis of light alkenes in high-pressure reactors at elevated temperatures [131], remote monitoring of styrene emulsion polymerization in the short-wavelength NIR region [132], and deter-

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On-Line Monitoring of Chemical Reactions

Figure 14. On-line NIR spectra of a polypropylene/ethylene – vinyl acetate copolymer during blend formation (courtesy of IPF Dresden)

mination of octane in refinery processes [133]. A fiber optic coupled AOTF NIR spectrometer has been described for dryer effluent monitoring in a chemical plant [134]. NIR spectroscopy has been shown to be suitable for online detection of reaction completion in a closedloop hydrogenator in pharmaceutical production [135]. Diffuse reflectance NIR spectroscopy has been evaluated as an on-line technique for the monitoring of powder blending [136–138]. Gasphase real-time monitoring by transmission NIR spectroscopy is described in [139]. A promising application of remote-sensing NIR spectroscopy is the on-line identification and classification of plastics material [140–142]. Near-infrared spectroscopy is one of the few analytical methods applicable to real-time monitoring of chemical processes in polymer melts during extrusion. Figure 13 shows a ceramic transmission NIR probe with variable optical pathlength that is adapted to a corotating twin-screw extruder and interfaced to an NIR spectrometer via fiber optic cables. Online NIR spectra, as shown in Figure 14, provide the quantitative composition of a polypropylene/ethylene – vinyl acetate copolymer mix-

ture during blend formation with an accuracy of about 1 %. With the help of ATR (Fig. 15) and diffuse reflectance probes the formation of styrene – maleimide copolymer (Fig. 16), and the filler concentration in molten, opaque polypropylene/chalk powder composites have been monitored. In the latter application, the inhomogeneous distribution of filler particles may lead to substantially different spectra during the course of extrusion. The scattering contribution of the particles to the NIR spectra has to be corrected before chemometric analysis. Appropriate chemometric calibrations models are required in both cases [143– 145]. Hansen et al. [146–149] were even able to derive rheological properties of ethylene – vinyl acetate copolymers from NIR spectra recorded in-line in an extruder via fiber optics. 5.3.3. IR Spectroscopy Infrared spectroscopy has a long tradition of use in chemistry, materials science, physics, and even reaction monitoring [150]. It is one of the standard techniques in most analytical laboratories. The IR region of the spectrum is highly spe-

On-Line Monitoring of Chemical Reactions cific and very sensitive to molecular structures. Individual peaks can be assigned unequivocally to chemical species and calibrated for analyte concentration. The timescale for FTIR spectroscopic analysis is on the order of seconds. All of this makes IR spectroscopy attractive for on-line process monitoring.

Figure 15. ATR immersion probe adapted to a corotation twin-screw extruder (courtesy of Carl Hanser Verlag) a) ATR immersion probe; b) ZnSe crystal; c) Intermediate plate; d) Melt in the itermeshing zone; e) screw

High molar absorptivities in the IR are advantageous for gas analysis but may present challenges for liquid analysis because sample layer thicknesses (optical pathlengths) would have to be in the micrometer range to transmit IR radiation. This distance is too short for circulation of the reaction mixture, so transmission and transflection measurements cannot be used for process monitoring. The technique of choice is ATR spectroscopy. The short penetration depth of the evanescent wave is both the strength and a limitation of ATR spectroscopy. ATR crystals are prone to surface fouling and layer build-up, which leads to interference with the process signals. Additional issues to be considered with respect to on-line applications of ATR are the scratch resistance of the crystal; the relatively low light throughput, which requires sensitive detectors; and the linearity (or nonlinearity) of calibration.

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A major obstacle to industrial IR on-line applications is the lack of appropriate fibers transparent to IR radiation. The chalcogenide fibers currently used are very expensive and have only limited light throughput and hence do not enable remote monitoring. Infrared spectrometers should be installed close to the process to allow the use of automated sampling devices, short fibers, or optical systems that transmit the light via mirrors and tubes to the analyzer. The latter option is shown in Figure 5, in which a “mirrorarm” is interfaced to an FTIR spectrometer. In Figure 17, a waterfall spectrum and the corresponding line intensities as a function of time are shown for the reaction of secondary amines with paraformaldehyde. In this case, a linear calibration model makes it possible to examine the reaction kinetics on-line on a quantitative basis. In aqueous systems, the strong water absorption in the IR region may interfere with spectral features of interest for reaction characterization. Nevertheless, on-line monitoring of 2-ethylhexyl acrylate/styrene emulsion copolymerization with an ZnSe ATR fiber optic FTIR device has been demonstrated successfully [151]. Applications of FTIR spectroscopy to polymerization reactions include kinetic studies of laser-induced photopolymerization directly in a spectrometer [152], multiacrylate polymerization studies with a quartz crystal ATR device [153], and monitoring of carbocationic polymerization with ATR probes [154], [155]. A water-cooled mid-infrared chalcogenide fiberoptic probe with an ZnSe crystal was described for in situ monitoring of an acid-catalyzed esterification reaction in toluene at 110 ◦ C [156]. Cross-linking reactions in various resins have been studied by IR spectroscopy [157–159], and FTIR imaging with focal-plane array detection [160]. The melt crystallization process of isotactic polystyrene was studied by means of in situ FTIR spectroscopy [161]. On-line gas process stream analysis was carried out by piping the gas through a transmission gas cell [162] or by means of photoacoustic gas sensors [163]. Chemical processes in thin films and coatings have been investigated by photoacoustic FTIR spectroscopy [164] and in situ IR ellipsometry [165].

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On-Line Monitoring of Chemical Reactions

Figure 16. Formation of styrene – maleimide copolymer monitored with a diffuse reflectance dipper probe (courtesy of Carl Hanser Verlag) FA = C-18 fatty amine; PS = polystyrene; SMA = styrene – maleic anhydride copolymer; SMI = styrene – maleimide copolymer

5.3.4. Raman Spectroscopy

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Figure 17. Waterfall spectrum (a) and the corresponding line intensities (b) as a function of time for the reaction of secondary amines with paraformaldehyde (courtesy of Mettler-Toledo GmbH, Analytical)

Fundamentals. Raman spectroscopy combines some of the advantages of (FT)IR and NIR spectroscopy without the limitations associated with these techniques. Nowadays, it is one of the fastest growing areas of analytical chemistry. Raman spectroscopy produces well-resolved spectra of fundamental vibrations comparable to IR spectroscopy. The interaction probabilities of light with the molecule for the two techniques are quite different. Infrared absorption is favored when a molecule has a permanent dipole which is modulated by the vibration. Raman scattering occurs when the molecule is polarizable, with the polarizability modulated by the vibration. Thus the intensities of IR and Raman bands of the same substance are fundamentally different, and the two methods provide complementary information about the molecule. Some of the bands can even be absent in one or the other spectrum. The Raman line intensity is directly proportional to the number of corresponding oscillators in the scattering volume and the intensity of the illuminating radiation. Line intensity changes primarily reflect concentration changes within the sample, thus making calibration of the spectra rather simple and straightforward. Raman

On-Line Monitoring of Chemical Reactions spectroscopy does not require extensive sample preparation, and Raman spectra can be taken from almost all samples without pretreatment. The Raman effect is an extremely weak, inelastic scattering process. Typically, only 10−6 to 10−9 of incident photons undergo Raman scattering, and low-loss optical devices are required for proper recording of spectra. In Raman spectroscopy, there is always a competitive mechanism for light emission present, namely, fluorescence of the sample. In some cases, the quantum yield of fluorescence outweighs the Raman scattering efficiency by several orders of magnitude, and the fluorescence of the sample completely obscures the Raman spectrum. The most convenient way to avoid sample fluorescence is excitation at longer wavelengths. However, as the excitation wavelength increases, the light scattering intensity decreases according to a λ−4 power law. Raman signals of some species are much more sensitive than the corresponding IR absorptions. The ability to measure Raman spectra of symmetrical diatomic molecules (for instance O2 , N2 , H2 , F2 , Cl2 , Br2 , I2 are not active in IR absorption) opens up opportunities in gas analysis. Raman spectroscopy is sensitive to nonpolar molecular vibrations. Hence, double or triple bonds in monomeric or polymeric molecules are strong Raman scatterers. Vinyl or diene monomers can easily be identified, and their concentrations determined by means of their double bonds. Therefore, Raman spectroscopy is an ideal tool for monitoring the disappearance of monomers during the course of polymerization, and for detecting residual monomers in final products. Metal carbonyls in catalysts and cyanides in plating baths give rise to strong Raman signals due to their triple bonds. Raman spectroscopy can be used to monitor the quality of diamond or diamond-like carbon and to distinguish between isomers [166]. Water, with a strong molecular dipole, has an intense IR absorption, but a very weak Raman response. Thus water obscures IR spectra, whereas Raman scattering is more or less oblivious to the presence of water. Hence, Raman spectroscopy can easily be applied to aqueous solutions, emulsions, latices, and suspensions. Disperse systems scatter light because of refractive index differences between particles and the surrounding continuous phase (e.g., elastic Rayleigh or Mie

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scattering of latices, emulsions, aerosols). Multiple elastic light scattering reduces severely the scattering volume from which the light for Raman spectroscopy is collected. Hence, confocal illumination and collection optics are essential for efficient sample excitation and signal detection. Light scattering affects spectral intensities but not band shapes. With confocal optics, interference of both the window material and films built up on window surfaces with the spectrum is much reduced, and spectra can be obtained from samples through colored or opaque sample container walls. Instruments. Dispersive scanning Raman spectrometers operating in the UV or VIS region are not very suitable for process monitoring because of the time required to record a spectrum. Additionally, the short wavelength excitation may cause severe sample fluorescence. The main advantage of dispersive scanning instruments is the very high resolution achievable by using double or triple monochromators. As in IR spectroscopy, FT Raman spectrometers are suitable for monitoring even very fast chemical processes on the timescale of seconds. For practical and cost reasons, most instrument manufacturers use the same type of interferometers in their IR and FT Raman spectrometers. Hence, excitation in FT Raman spectroscopy is typically at 1064 nm with Nd:YAG lasers. Therefore, sample fluorescence is very much reduced, but FT Raman spectra of aqueous samples are modified by self-absorption of water in the NIR (i.e., for wavenumbers above 2000 cm−1 with excitation at 1064 nm). Pros and cons of FT Raman spectroscopy are reviewed in [167–169]. The application of holographic transmission gratings and semiconductor array detectors (CCD or CID) has led to the rediscovery of dispersive Raman spectrometers. The major advantage of these instruments is they do not contain any moving parts and are thus rugged and compact for on-line applications in chemical plants. However, the insensitivity of a CCD at wavelengths above 1050 nm restricts the wavelength of the incident light. For a CCD-based instrument, excitation of the sample has to be done below 800 nm to obtain the entire spectrum up to 3000 cm−1 . Most instruments for reaction monitoring are equipped with recently

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On-Line Monitoring of Chemical Reactions

developed inexpensive and compact NIR solidstate diode lasers operating between 780 and 850 nm, whereby 785 nm or 840 nm are commonly used. A review on diode lasers for Raman spectroscopy is given in [170]. In this wavelength range, low-cost silica optical fibers deliver the laser light to the measurement site and return the signal efficiently for remote monitoring. Holographic filters integrated into the probe head remove the Raman signal generated within the excitation optical fiber and prevent reinjection of elastically scattered laser light into the signal return fiber. Raman spectrometry with fiber-optic sampling is described in [171–174]. For trends in instrumentation and applications, see [175–179]. Applications. Prior to the 1990s, only a few publications deal with Raman reaction monitoring. The validity of the method has been demonstrated for the suspension polymerization of styrene [180] and vinyl chloride [181], the thermal polymerization of styrene [182– 184] and methyl methacrylate [184], the solution polymerization of methyl methacrylate [185], the γ-initiated polymerization of diacetylene [186], and the microemulsion polymerization of styrene and methyl methacrylate [187]. The decreasing intensity of the ν(C=C) Raman lines of the monomer during the course of the reaction was monitored as a measure of the extent of conversion to polymer. The positions of the Raman double bond stretching vibration of several monomers are listed in Table 5. Raman spectra recorded during emulsion copolymerization of styrene and butyl acrylate are shown in Figure 18. The ν(C=C) double bond peaks of the two monomers cannot be resolved since the spectral resolution of the instrument was on the same order of magnitude as the difference between their peak maxima (ca. 7 cm−1 ). The double bond peak disappears almost completely during the reaction. Based on the spectral changes as a result of polymerization, the conversion of the two monomers as a function of time can be calculated with an accuracy of about ± 1 %. The combination of Raman on-line monitoring and automated spectra evaluation enables chemical reactions to be controlled efficiently.

Table 5. Double bond Raman lines Monomer

ν ˜(C=C), cm−1

Vinyl chloride Acrylonitrile Vinyl sulfonate N-Methyl acryl amide Styrene Methyl acrylate 2-Ethylhexyl acrylate Butyl acrylate Diisopropyl fumarate Ethyl acrylate Butadiene Glycidyl methacrylate Methyl methacrylate Allyl methacrylate Veo Va 9/10∗ Vinyl acetate Vinyl propionate Crotonic acid

1607 1610 1619 1629 1631 1635 1637 1638 1638 1638 1639 1640 1641 1641/1648 1646 1648 1648 1658

∗

Vinyl esters of versatic acids.

In the 1990s, a wealth of publications appeared on in situ Raman investigations, for instance, of polymerization reactions [188– 193], epoxy resin cure [194–196], crystallinestate photoreactions [197], fast high-pressure decomposition reactions [198], hydrolysis of acetic anhydride to acetic acid in a hydrothermal/supercritical water reactor [199], liquidphase chemistry of aliphatic organic peroxides [200], crystallization of polymer melts [201], a PCl3 production process [202], petroleum distillate quality in pipelines [203], and chlorosilane stream composition [204]. Chemical reactions have even been monitored in aerosol particles by means of Raman spectroscopy [205–208]. The study of Raman spectra of adsorbed molecules on surfaces is one of the most promising areas in Raman spectroscopy. Molecules adsorbed on metal surfaces show enhancement of the scattering efficiency by up to seven orders of magnitude [209], [210]. This surfaceenhanced Raman scattering (SERS) is ultrasensitive for detecting numerous adsorbed compounds by means of their vibrational spectra. Originally, SERS experiments were carried out with either bare metal surfaces or metal colloids mixed with solutions of the sample. Covering the substrates with extremely thin modifying layers enables the construction of chemical sensors for remote monitoring [211–214].

On-Line Monitoring of Chemical Reactions

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Figure 18. Raman spectra recorded during emulsion copolymerization of styrene and butyl acrylate. The spectra were recorded with a fiber-based instrument equipped with a CCD detector. Laser excitation at 785 nm with an output power of about 90 mW was performed through a glass window in the reactor wall. The accumulation time for each spectrum was about 15 s. Spectra were recorded at regular intervals of 1 min. Hence, the instrument meets the requirements for an on-line process monitoring system.

A nonlinear Raman technique, stimulated Raman scattering (SRS), has been used to monitor the bulk polymerization of styrene and methyl methacrylate [215]. In contrast to spontaneous Raman scattering, most compounds show only a few Stokes lines in their SRS spectrum. As a result, spectral interference of SRS-active compounds in a reaction mixture is minimized. 5.3.5. Fluorescence Fluorescence spectral signals are quite intense because of the efficient electronic excitation and fluorescent emission process. This enables measurements down to the nanosecond timescale on small samples with remarkable accuracy. The sensitivity is high over a wide concentration range. The spectral features are rather broad compared to those of other spectroscopic methods. Fluorescent techniques provide information on molecular arrangement and dynamics of micro-heterogeneous systems, and on micropolarity or microvicosity. Steady-state fluorescence spectroscopy and fluorescence lifetime measurements have been used for on-line reaction monitoring. Sources of fluorescence can be the reaction components themselves or a fluorescent probe that is added to enhance the selectivity of the measurement. Re-

action monitoring with fiber-optic fluorescence sensors coated with fluorescent probe molecules is an emerging technology. In contrast, fluorescence signals in Raman spectroscopy are a source of disturbance because they originate from largely unknown impurities and mask the more specific Raman signal. The fluorescence technique is potentially much cheaper than Raman spectroscopy. Fluorescence methods have been demonstrated to be useful in academic research for in situ monitoring of curing [216–218] and polymerization [219–225]. Polymer colloids have been characterized by fluorescence quenching techniques in order to draw conclusions about the internal particle structure, transport phenomena, and particle flocculation of latices [226], [227]. A traditional field for the application of various fluorescence techniques are micellization behavior, molecular interactions, aggregation and clouding phenomena in surfactant solutions [228–231]. Optical fluorescence microscopy is capable of in situ chemical analysis of phase-separating polymer blends [232]. Various aspects of latex coalescence and film formation were studied by fluorescence techniques [233–236]. Various monitoring applications of fluorescence-based sensors and fluorescent probes are described in [237–242].

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Fluorescence is not commonly used for online monitoring of chemical processes outside academia. Nevertheless, fuel in the cylinder wall oil film of a combustion engine has been detected in situ with a fiber-optic laser-induced fluorescence system [243].

6. Particle Size Analysis An important characteristic of disperse systems (latices, emulsions, aerosols, suspensions, powders) is their particle size distribution. Particle sizes affect almost all properties of a disperse system (electrical, optical, rheological, etc.), as well as its stability. The evolution of the particle size distribution is a sensitive indicator of the progress of heterophase reactions, and it determines the application properties of final reaction products. In this chapter, the main focus is on process particle size measurements in suspensions.

6.1. Scattering Techniques The scattering of electromagnetic waves (light, X-rays) and neutrons by particulate matter is a powerful tool for studying particle sizes and shapes, internal particle morphologies, particle dynamics (even under shear), the structure and dynamics of concentrated disperse systems, and particle charges. Wave interference yields information on particle sizes and shapes whenever the wavelength is of a comparable order of magnitude to the size of the scatterer. Neutron and X-ray scattering both rely on the availability of corresponding neutron and radiation sources. Hence, their use for on-line size monitoring is extremely limited. Particle size determination by light scattering techniques is a well established broad field of experimental methods. In addition to inelastically scattered light (Raman and Brillouin scattering), there are two basic approaches that exploit light scattering data for on-line particle size analysis: Static light scattering (SLS) and dynamic light scattering (DLS). SLS methods involve measurement of the time-averaged angular scattering as well as turbidity measurements (i.e., elastically scattered light). DLS measures the very small frequency shift of scattered light caused by translation and rotation

of scatterers, or, in the FT time domain, fluctuations of the light intensity by correlation techniques. Thus, quasi-elastic light scattering (QELS), Rayleigh linewidth/spectroscopy, intensity correlation spectroscopy, and photon correlation spectroscopy (PCS) are synonymous with DLS. Most of the light scattering techniques require extensive dilution of the extracted sample to avoid multiple-scattering effects. Automatic dilution devices are interfaced to the light scattering instrument for on-line applications. Direct optical access to the (undiluted) reaction mixture became possible with the development of fiberoptic probes and instruments. For general reading see [244–246]. 6.1.1. Turbidimetry The turbidimetric experiment is rapid, precise, reproducible, and absolute (i.e., no calibration is required). Spectrophotometers are readily available. Turbidity gives a measure of the attenuation of light traversing a suspension or aerosol of nonabsorbing particles. Turbidimetry may require dilution of the mixture to avoid multiple scattering effects. The wavelength-dependent turbidity τ of a (diluted) sample provides information on particle size and concentration. Turbidimetric (and nephelometric) measurements at single wavelengths can be used to determine the concentration of particulate matter in liquid and gaseous process streams. Fraunhofer diffraction is used for particles sizes  1 µm. Several publications describe the determination of particle size during emulsion polymerization by specific turbidity τ /ϕ or turbidity ratio measurements [247–249]. For a polydisperse suspension, the turbidity is related to the polymer volume fraction ϕ according to ∞ 

τ =

3 ϕ 2

0

d2p · K



∞  0

dp np , λm nm

 f (dp ) ddp (33)

d3p f (dp ) ddp

where K, f , λm , and d p are the scattering coefficient, the size distribution function, the wavelength of light in the medium, and the particle diameter, respectively. The quantities K and τ

On-Line Monitoring of Chemical Reactions are functions of both the refractive indices of the particles np and of the medium nm , and of the size of particles relative to the wavelength of the light d p /λm . K can be calculated from the general Mie theory. If the size distribution function f is known, the particle size distribution can be estimated from specific turbidity measurements at several wavelengths. Despite the complicated dependence of τ on K and its direct dependence on the distributional form f (which is sometimes unknown), the specific turbidity can yield (1) the turbidity average particle diameter and the volume-surface average (i.e., D3,2 ) diameter for small and large particles, respectively, for any value of m = np /nm , (2) the weight average diameter for m < 1.15 and particles that are smaller than the wavelength of the light, and (3) an estimate of the weight average particle diameter in all other (monomodal) cases if a log-normal particle size distribution is assumed [248], [249]. Turbidimetry has been used to study the coagulation kinetics of aqueous dispersions [250], [251]. Efforts were made to extend the theoretical basis of turbidity to higher concentrations [252–254]. 6.1.2. Angular Static Light Scattering Angle-dependent static light scattering measurements with goniometers are not very suitable for on-line monitoring because of the time required and on other specific demands. A fastresponse multichannel photometer capable of on-line monitoring even at moderate concentrations is described in [255]. The scattered light is simultaneously measured at 168 angles. This static light scattering instrument has a response time of 100 ms and an angular resolution of 1◦ . Static light scattering at a fixed angle is valuable for studying the formation and aggregation of latex particles during emulsion polymerizations [256]. Several instruments are on the market which automatically dilute aqueous dispersions to a desired (extremely low) concentration and perform static multiple-angle light scattering. Different modes of light scattering (including Frauenhofer diffraction) are typically combined within one instrument to enable particle size analysis over a wide range of sizes.

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6.1.3. Dynamic Light Scattering Particles suspended in a continuous medium undergo random Brownian motion. Hence, the phase of light waves scattered by Brownian particles also fluctuates randomly in a time-resolved light scattering experiment. Their mutual interference leads to a net randomly fluctuating scattered light intensity I(t) at the detector. The normalized autocorrelation function g2 (t
) of the light intensity as a function of time delay t
is given by g2 t =

9

I (t) · I (t + t ) 9 :2 I

: (34)

with I(t) · I(t + t
) (t
→ ∞) = I2 , the square of the average scattered light intensity. The field autocorrelation function g1 (t
) can be calculated if the Siegert relation holds  2   g2 t = A + B · g1 t 

(35)

where A and B are instrument-related constants. For a monodisperse system of Brownian particles, their diffusion coefficient D can simply be derived from g1 (t
) according to Equation 36



g1 t = exp −q 2 Dt = exp −Γ t

(36)

where Γ is the decay constant. The scattering vector q is defined as q =

4πn sin λ

  θ 2

(37)

where n, λ, and θ are the refractive index, the wavelength of light, and the scattering angle, respectively. The Stokes – Einstein equation relates the diffusion coefficient D to the hydrodynamic radius of the particles r h rh =

kb T 6πηD

(38)

where kb , T , and η are the Boltzmann constant, temperature, and viscosity of the dispersion medium, respectively. For polydisperse systems, the autocorrelation function becomes a sum of exponentials with a distribution function G(Γ )

850 g1 t =

On-Line Monitoring of Chemical Reactions
∞

G (Γ ) exp −Γ t dΓ

(39)

0

Particle size characteristics and distribution functions can be derived from measured autocorrelation functions by (1) directly inverting the Laplace integral equation (Eq. 39, a mathematically ill-conditioned problem), or (2) parameter fits, exponential sampling, regularization techniques, and histogram analysis. Some advantages make dynamic light scattering attractive for on-line applications: 1) DLS is an absolute method; no calibration over time is necessary. With conventional DLS systems, accuracies of ± 1 % for the mean particle diameter can be achieved on a routine basis within a few minutes of measurement time. Care must be taken in reacting systems, where either temperature or viscosity change during the course of the reaction (see Eq. 38). 2) Particle properties such as density or refractive index do not affect the time behavior of the intensity fluctuations. Thus, the derived particle sizes are independent of the chemical composition of the particles. 3) In the single-scattering regime, diffusion coefficients and particles sizes are independent of particle concentration. This is an advantage over turbidity measurements, for which exact knowledge of the particle concentration is crucial for size calculations. 4) The use of inexpensive fiber-optics allows remote monitoring. Two approaches are possible for on-line DLS measurements: 1) A device interfaced with the process line or reactor automatically captures a certain amount of the mixture and dilutes it sufficiently to avoid multiple scattering and allow noise-free measurements [257]. 2) A fiber-optic probe (optode) is directly immersed into the mixture and both illuminates the colloidal system and collects the scattered light (FOQELS: fiber optic quasi elastic light scattering; FODLS: fiber optic dynamic light scattering [258– 260]. This backscattering arrangement shortens the light path within the dispersion

and makes the method applicable to turbid and concentrated systems. Monomode fibers further suppress multiple scattering effects [261]. Particle growth during emulsion polymerization has been studied by conventional DLS [262–266] and fiber-optic DLS [267], [268]. 6.1.4. Other Optical Techniques The effect of multiple scattering can be useful for studying highly concentrated dispersions. The path of multiply scattered light in the dispersion is similar to the path of particles or molecules under the influence of Brownian diffusion [269– 272]. The concept of photon migration or diffusing wave spectroscopy (DWS) has already been applied to particle size determination. Colloidal refractometry is an optical method that measures the refractive index n of a dispersion and analyzes it by using Mie theory. This method allows the structure of concentrated dispersions to be probed in the undiluted state. The measured values of n can provide a measure of the volume average particle size [273].

6.2. Separation Techniques Recently, particle separation techniques (chromatography, fractionation) have been developed that are capable of on-line monitoring of particle size. These methods require fully automated removal of samples from the reaction mixture, dilution, and injection into the instrument. Size exclusion chromatography (SEC) is applicable with porous (liquid exclusion chromatography LEC) and nonporous packing of the column (hydrodynamic chromatography HDC). Particles are separated because of the dependence of the rate of particle flow through gaps between packing beads on particle size. Larger particles are eluted first since they travel through larger gaps with higher flow rates. The main advantage of SEC methods is that particle size distributions can be obtained directly without any assumption regarding the mathematical form of the distribution. Calibration of the instruments with particle size standards is necessary but simple. Disadvantages are the limited resolution because of radial dispersion, and relatively long elution times of up to half an hour [274], [275].

Next Page On-Line Monitoring of Chemical Reactions Much higher resolution than with SEC methods can be achieved by capillary hydrodynamic fractionation (CHDF) [276] and field-flow fractionation (FFF). The fluid in a CHDF capillary has a parabolic velocity profile with the greatest fluid velocity at the center of the tube and zero velocity at the wall. Particles move radially due to their Brownian motion. Larger particles are unable to approach the wall as closely as smaller particles. Hence, larger particles travel through the tube faster than smaller particles. This separation effect is exclusively a function of particle size; it is independent of particle density. The efficiency of capillaries for separating particles depends on the eluent viscosity, the flow rate, and the capillary diameter. The optimum particle size for CHDF analysis is < 1 µm. The main disadvantage of this method is the possibility of capillary clogging. The CHDF method has been used to monitor the evolution of the particle size distribution during emulsion [277] and miniemulsion polymerization [278]. Field-flow fractionation (FFF) is a family of chromatography-like elution techniques based on influencing the rate of particle flow through a narrow channel by applying an external field perpendicular to the flow direction [279–283]. The external field separates particles of different sizes by driving them into different localized laminas (< 10 µm thick) of the parabolic velocity profile of the eluent within the fractionation channel thus causing their separation. In contrast to CHDF, FFF does not necessarily require calibration of the instrument. Retention times for particles of different sizes can be calculated directly from first principles. FFF can be classified according to the type of applied field into sedimentation, thermal, electrical, cross-flow, and steric. The most suitable external fields for particle size analysis are crossflow and sedimentation fields. The particle separation in cross-flow occurs according to the hydrodynamic radius of the particles, whereas their effective weight separates particles in a sedimentation field. The particle size ranges for crossflow and sedimentation field FFF are 10 nm to 100 µm and 50 nm to 100 µm, respectively. Recently, FFF (including thermal FFF) has been used to characterize the size and composition of core/shell latices [284]. On-line coupling of FFF with multi-angle laser light scattering is described in [285], [286].

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7. Chromatography 7.1. Introduction Chromatographic methods (especially gas chromatography) are the most widely applied family of techniques for determining the chemical composition of reaction mixtures. A comprehensive discussion of (gas) chromatography for on-line monitoring of chemical reactions has been published [287]. Process applications can also be found in [288–291]. The term “process chromatography” almost exclusively refers to the application of gas chromatography (GC) to monitor chemical reactions or process streams. Advantages of GC are its simplicity, high sensitivity and selectivity, and ease of automation. Liquid chromatography (LC, including HPLC), supercritical fluid chromatography (SFC), electrophoresis (EP), and ion chromatography (IC) are less common in process monitoring because of their higher maintenance requirements, greater complexity, and higher costs for purchase and maintenance. Because GC (like other chromatographic methods) is an inherently discontinuous batchlike analytical technique which is applied to continuously monitor a chemical system in a nonbenign environment, there are distinct differences between laboratory and process instruments, although the basic components of a gas chromatograph can be found in instruments of both categories (Fig. 19) and the basic principles of separation are the same. In contrast to laboratory instruments, process analyzers have to meet stringent safety and explosion-proof requirements because of their close vicinity to the product line. The typical GC process instrument is designed to detect and monitor only one or a few (previously known) components of the reaction mixture so as to control a chemical reaction or process stream. This requires special injection valves and columnswitching devices. As in conventional GC, the amount of sample required for process GC is extremely small. Hence, the sampling system must ensure that the sample is representative of the process to be monitored. Often, a sample conditioning system is necessary to further process

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Figure 19. Schematic of a process GC analyzer GS: gas supply for carrier, auxiliary, and purge gases; V: valves for preseparation, column switching, and injection; VP/LP: vapor-phase/liquid-phase sampling; SHC: sample handling and conditioning system; R: reactor; S: separation unit; C: column(s); D: detector; O: thermostatically controlled oven; PC/C/NW: personal computer/controler/network; W: waste gases

the sample, which may still be reactive, for GC analysis. The nonspecific character of detection based exclusively on retention time (i.e., no positive identification) makes GC preferably suitable for process monitoring of continuous processes with small variations in composition. Different molecules may have the same retention time, confusing quantification of the chromatograms and leading to misinterpretation of overlapping peaks. Additionally, GC analysis requires time for separation and detection. This time interval must be reasonably short compared to both the run time of the reaction and the time required to feed the analytical result into a process controler, and to influence the reaction.

7.2. GC Hardware Components Carrier Gas. In principle, the same carrier and auxiliary gases can be used in both laboratory and process GC analyzers (usually hydrogen, helium, argon, synthetic air, or nitrogen, depending on the application). Most high-purity gases are supplied in containers (e.g., high-pressure gas cylinders). For an on-line operation of

the GC analyzer, the gas storage and delivery systems should be equipped with valves that switch atomatically between empty and full containers. Sampling. The quality of a GC analysis of a reacting mixture is only as good as the quality of the samples withdrawn from it. Because of the very small amounts of sample necessary for GC, it is crucial to withdraw samples that are representative of the mixture. The sampling point (sample probe) should be located at a point in the bulk of the medium where it is homogeneous and well mixed so as to obtain the “true” bulk composition of the mixture. Close to the walls, adsorption and deposition phenomena and stagnant flow may locally change the overall composition. From an analytical point of view, the best location for the analyzer would be right next to the process line or sampling loop. However, a number of factors and restrictions have to be taken into consideration for placing the chromatograph relative to the sampling point, e.g., 1) space available for installation, 2) maintenance

On-Line Monitoring of Chemical Reactions requirements, 3) safety requirements, 4) accessability of sampling point and chromatograph, 5) connection to utilities and supplies (carrier gas, electricity, compressed air), 6) optimization of instrument lag time versus control cycle time, 7) minimization of sample transport, 8) reactivity/stability of the analyte. In most applications, a sample transfer line bridges the distance between sampling point and analyzer. Length and diameter of the transfer line must be optimized for (high) analyte flow. Pressure drop and temperature changes along the line must be controlled. When the analysis is complete, the sample is returned to the process by a separate line or disposed of along with the carrier/purge gas. In many processes, GC samples are taken at elevated temperature and/or pressure. These samples may contain unconverted components, other reactive ingredients, water vapor, liquids, or solid impurities. Hence, process samples must be appropriately conditioned before feeding them into the GC analyzer. The sample conditioning system peforms depressurizing and temperature reduction as well as filtration and removal of liquids (e.g., by vaporization). Its different elements may be close to the sample point (pressure reducer, filter) or close to the analyzer (temperature control, flow control). The conditioning system ensures that the temperature of gaseous samples is held above the dew point to avoid condensation and that the temperature of liquid samples is well below the boiling point to avoid bubble formation in the line. Sample transfer and conditioning are the most time-consuming part of an on-line GC analysis. The lag time between process and analysis is on the order of a few minutes and must be taken into account in designing reaction control systems based on conventional GC. With microsensor GC systems, timescales on the order of (milli) seconds for on-line analysis are possible. Analyzer Oven. In contrast to most laboratory instruments, the actual process chromatograph (i.e., injection system and valves, column(s), detector(s), controls) is placed in a single temperature-controlled oven. Temperature control of better than ± 0.05 K suppresses fluctuations in temperature-sensitive processes such as gas flow, mobile-phase diffusion, adsorption/desorption, and chemical inter-

853

actions. Since detection is solely based on retention time, all processes that can influence retention times must proceed unaltered to avoid loss of calibration under automated operating conditions. Constant temperature is usually guaranteed by circulating heated air within the process analyzer oven or by using high-mass metal heat reservoirs. In some applications, tighter temperature control can be circumvented by an independent delayed injection of a standard or by adding an internal standard to the carrier gas. Pressure control and programming is dominant in process chromatographs, although some temperature programmable analyzers and gradient systems have been recently described in the literature. Valves and Columns. In contrast to laboratory instruments, a process chromatograph is usually designed to monitor only one or just a few components of the reacting system for control purposes. Exceptions include some petrochemical applications, in which up to several hundreds of components are monitored simultaneously. Additionally, the process GC analyzer must operate unattended in a rather harsh environment over extended periods of time, but still provide reproducible and stable data. Of special importance is a constant sample volume/mass for injection (and volume of standard, if necessary). Therefore, dedicated valve systems for sample injection and column switching have been developed [287]. For gaseous samples, the sample size is normally defined by the volume of an external sampling loop connected to different valve ports. The injection volume of liquid samples can be defined by the size of internal grooves within the valve or by external sampling loops. Sliding, rotary, and diaphragm valves with a wide variety of internal pathways and port configurations are commonly used in process chromatography of gaseous samples. For liquid samples, piston (syringe) valves may also be applied. Valves are heated externally or internally. Depending on the component(s) of interest within the complex reaction mixture, cutting (mainly heart-cut technique) and other column-switching techniques (e.g., backflushing) or multicolumn analysis are often combined with precolumn separation. The main reason for using column-switching techniques in on-line process GC is to drastically reduce the total anal-

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On-Line Monitoring of Chemical Reactions

ysis time and to fully separate the components of interest from the mixture. In general, the same types of valves are used for injection and column switching. Traditionally, the use of packed columns predominates in process GC. As in laboratory GC, packed columns are being replaced by capillaries in an increasing number of on-line applications. With capillary columns, chromatographic separation is improved, resolution is higher, and analysis time is shorter than with packed columns because of the much simpler flow and diffusion pattern of the analyte. However, capillaries are normally less sensitive, more expensive, and not so easy to use as packed columns. A major drawback of capillary columns in process applications is their sensitivity to flow, pressure, and especially temperature fluctuations. Columns, their supports, coatings, and packing materials have to withstand possible corrosive action of ingredients of the reaction gas mixture and should be inert to reactions with solute molecules (e.g., by silylation of the column packing material or capillary coatings). The presence of water vapor and oxygen traces in process gases may also be of relevance to column selection. Detectors in process chromatographs have to comply with the overall safety, robustness, and reliability standards of an on-line instrument. The overwhelming majority of GC detectors utilized in process analyzers are of the thermal conductivity (TCD) and flame ionization (FID) types because of their ease of use and sensitivity to the process components of interest. TCDs are sensitive to changes in the bulk thermal conductivity of the carrier gas in the presence of the analyte, which gives a concentration-dependent signal, whereas FIDs sense the number of certain ionized molecules entering the flame with a specific flow rate. In some rare cases, photoionization (PID) and flame photometric (FPD) detectors are applied. Several methods have been utilized in process GC to convert detector output signals (peak areas) into desired analyte concentration values, such as the separate injection of calibration standards, the use of internal standards, and calculations based on relative response for the different detector types.

7.3. Applications of Gas Chromatography GC has a long tradition of use for controlling processes and process streams in industry. Online GC is one of the most important techniques for monitoring distillation, rectification, and conversion processes in refineries and chemical and pharmaceutical plant. It is extensively applied in catalysis research, development, and production to evaluate catalysts and study reaction mechanisms and kinetics. Another field of use is in metallurgical smelters for gas and sulfur analysis in metals. It is widely utilized as a part of workplace safety systems, e.g., to ensure air quality in production units, pilot plants, and laboratories. On-line GC has proved useful for determining organic substances in water [292], [293], wastewater analysis [294], and air monitoring [295–297]. Reports have been published on the on-line analysis of petroleum [298], [299] and gasoline [300–302], the determination of octane numbers with n-alkanes as reference [303], the on-line monitoring of methyl formate conversion to methanol and carbon monoxide [304], Fischer – Tropsch synthesis [305] and thermolysis/decomposition of organic compounds [306], [307]. The application of gas chromatography (among other techniques) for on-line monitoring of polymerization reactions is reviewed in [308]. Gas chromatography of the reaction mixture [309], [310] and the head space of the reactor [311], [312] can be applied for online determination of residual monomer composition in emulsion polymerization. Head-space GC in liquid reaction mixtures is not straightforward for direct reaction monitoring because it requires knowledge of the partitioning of each component between the gas phase above the reaction mixture and the liquid mixture, and, especially in multiphase reactions, between the different phases of the reaction mixture (e.g., aqueous phase, monomer droplets, and polymer particles in an emulsion polymerization). Additionally, the head space should be in thermodynamical equilibrium with the fluid reaction mixture. The utilization of (mass) sensors for gas and liquid analysis has been reviewed in [313–315],

On-Line Monitoring of Chemical Reactions In-line applications of chemical sensors to monitor polymerization reactions has not been described in the literature so far. Such sensors should have a very high selectivity so as to detect components well below the percent range in a matrix of solvent or solvent vapor (including water) containing a large number of ingredients, some of them with a chemical structure comparable to that of the analyte. Additionally, the sensors would have to deal with precipitates on their surfaces.

7.4. Other Chromatographic Techniques Applications of liquid chromatographic systems to monitoring chemical and biochemical processes are described in [316–322]. On-line ion chromatography has been used to monitor adipic acid production [323], trace metal ions [324] and cations [325], and oxides in flue gases [326]. Reviews on process ion chromatography can be found in [327] and [328]. In some rare cases, capillary electrophoresis [329], [330] and supercritical fluid chromatography [331], [332] have been described as on-line reaction monitoring techniques.

8. Electroanalytical Methods 8.1. Introduction Electroanalytical methods are based on the interaction of electrical fields with chemical processes or on chemically induced electrical signals on electrodes. The major advantage of electroanalytical methods is the electrical output of the measurement (i.e., electrical current, potential, resistance, or charge), which can directly be processed electronically without any further transformation, in contrast to other monitoring techniques. Although the main application for electroanalytical techniques is process monitoring in aqueous solutions, they can also be applied to nonaqueous and nonliquid systems. In most electroanalytical measurement systems, the sensing electrodes are in direct contact with the reaction medium and all of its components, not only with the species of interest. This can lead to cross-sensitivity and poisoning, and

855

to abrasion, deactivation, and contamination of the active electrode surfaces. Additionally, the analytical information of interest must be extracted from the integral signal. Calibration of the sensing system is typically necessary at the beginning of a measurement cycle and subsequently in regular intervals. Accumulation of static electricity within the sensing system and on its body must be avoided.

8.2. Conductometry Conductometric measurements rely on the existence of charges in the medium that are mobile in an applied electrical field (e.g., ions in electrolyte solutions or melts, electrons in metals). In conductometry, no reaction takes place at the electrodes. Electrical conductance of the medium or electrical capacitance of a measurement cell is the measured quantity. Conductometry in solutions and melts is a nonspecific method because all anions and cations contribute to the medium’s conductivity. However, the method is very robust, precise, sensitive, fast, and does not require extensive maintenance. Concentration ci and µi of all the ions i present in a medium determine its conductivity Λ (and conductance G) Λ = kG = kF



ci ni µi

(40)

i

where ni is the ion valency, F is Faraday’s constant, and k a cell constant. This linear relationship between concentration and conductivity is the basis for concentration measurements in the lower concentration range (≤ 0.1 Ω−1 cm−1 ). For higher electrolyte concentrations, Equation (1) becomes nonlinear because of ionic interactions. Additionally, polarization effects may occur at the electrodes. The temperature dependence of the ionic mobility µ leads to an almost (quasi) linear dependence of Λ on temperature in the temperature range common for conductivity measurements in electrolyte solutions (0 – 100 ◦ C). Two different principles of conductivity measurements are in industrial use: 1) two- or fourelectrode conductive cells with direct electrode contact to the medium, and 2) inductive, electrodeless, and contactless devices. Contact electrode cells are applied in nonaggressive media. The cell itself is typically part

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On-Line Monitoring of Chemical Reactions

of a Wheatstone bridge operating with lowfrequency alternating current to avoid electrode polarization. Two-electrode cells are in use in the conductance range 10−8 to 10−1 Ω−1 cm−1 , whereas four-electrode cells are suitable for the higher conductance range of 10−3 to 10 Ω−1 cm−1 . Electrodeless inductive measurements are the method of choice for highly conductive systems; for aggressive, abrasive, or corrosive media; and for high-temperature applications. The medium to be measured is circulated through a loop; two transformer coils are attached to the inert wall material of the loop pipe (Fig. 20). A generator supplies alternating voltage to the primary coil. The induced alternating voltage at the secondary coil depends on the conductivity of the solution circulating through the loop. For contactless measurements, the medium’s conductance is typically much higher than 10−1 Ω−1 cm−1 . Inductive devices are less sensitive than electrode cells.

mation, disappearance of particles, phase inversions, and particle swelling in heterophase reactions such as crystallization and emulsion and suspension polymerization [334–337]. Most applications of conductometry deal with monitoring of process streams rather than direct reaction monitoring. Typical application fields are drinking water purification and tap water monitoring, wastewater treatment, cleanness of ultrapure water in the electronics industry, distillation processes, neutralization and precipitation reactions, concentration monitoring of acids (e.g., in production of concentrated sulfuric acid), lyes, and salt solutions, and monitoring of process streams in dairies and breweries.

8.3. Potentiometry Potentiometric techniques are characterized by the fact that electrode reactions at equilibrium are involved in the potential measurement. In potentiometry, the actual electrical current through the electrode is zero. The electrode potential cannot be measured directly but only as a difference with respect to the potential of another electrode (reference electrode). Electrode potentials and the potential difference between two electrodes can be calculated according to Nernst’s equation (Eq. 42) Uelectrode ∼

Figure 20. Schematic of an electrodeless conductivity measurement device G: generator; D: detector; P: primary coil; S: secondary coil; ≈: alternating current

The elctrical conductivity Λd of a disperse system is a function of both the volume fraction of the disperse phase ϕ (ϕ  1) and the electrical double layer properties of the particles (surface conductivity Λs , Zeta potential ζ) Λd =

1 Λ0 + ΛS f

(41)

where ΛS is the conductivity of the dispersion medium, Λ0 the conductivity of the dispersion medium and f a form factor accounting for the shape and the volume of the (nonconducting) particles [333]. Thus, conductivity measurements provide information on particle for-

RT ln F



ai ai,ref



(42)

where R is the gas and F is Faraday’s constant. The ion activities ai can be activities of two different ions within the same solution (redox pair), activities measured with two different electrodes, or activities at two different concentrations or partial pressures. Ion-Selective Electrodes. The swelling glass layer of ion-selective electrodes is designed to be sensitive to a certain type of ions and insensitive to others. This can be achieved by using membranes containing inorganic salts of the ions of interest (e.g., Na+ , K+ , Cu2+ , NH4+ , Cl− , F− , I− ). Conditioning of the mixture to a certain conductivity and pH level is a prerequisite for ensuring precise concentration measurements with ion-selective electrodes. Hence, the use of this type of electrodes for direct (in-line) insertion into a reaction mixture is limited [338], [339].

On-Line Monitoring of Chemical Reactions pH measurement is the most important and most widely used potentiometric technique in aqueous solutions or water-containing systems. Measurements of pH are crucial for monitoring processes in the chemical and pharmaceutical industries, the food and beverages industry, wastetreatment plants, environmental protection, and biochemical operations [340]. In particulate systems, the pH has a dramatic influence on colloidal stability (e.g., cosmetic products, emulsion and suspension polymerization). Care must be taken to take into account temperature variations of the reacting system because of the strong temperature dependence of electrode potentials. pH measurements are easily performed with standard glass electrodes. For process applications, the most common pH-sensitive membranes consist of lithium glasses. Other alkali metal glasses are prone to increased hydrolysis and instability, especially in alkaline solution. Several types of glasses are available for different process environments (standard, highly alkaline, highly acidic, high temperature). Problems may arise with glass electrodes due to contamination of the surface by precipitation or film formation. Cleaning and maintenance of glass electrodes on a regular basis are critical for successful on-line pH measurements. Glass electrodes with counterpressure are applicable for on-line monitoring of reactions up to temperatures of about 80 ◦ C and pressures up to 60 bar. Mechanical instability is a major drawback of glass pH electrodes in high-pressure applications. Solid-state ion-sensitive field effect transistor (ISFET) sensors show faster response, lower alkaline errors, and improved mechanical stability than glass electrodes. However, they also suffer from surface contamination. The applicability of ISFET pH sensors is limited to pressures well below 50 bar and temperatures up to 85 ◦ C [341]. There is no solid-state sensor on the market applicable for pressures above 60 bar. An as-yet not commercially available ZrO2 sensor for pH measurements under high pressure (83 bar) and high temperature (285 ◦ C) is described in [342]. A number of different optical pH sensors have been developed [343–345]. Redox Potential. The oxidizing or reducing character of a solution can be measured with (inert) precious metal electrodes (Pt, Au) relative to

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a reference electrode. Redox potential measurements are highly nonspecific because all redox couples of a mixture contribute to the potential. The redox potential also strongly depends on pH. Thus, the applicability of redox potential measurements is mainly limited to determining qualitatively, e.g., nitrites in wastewater (denitrification), cyanides, copper, and chromium in galvanic wastes, or chlorine in public swimming pools. Potentiometric Oxygen Analysis. Solidstate electrolytes that exhibit ionic conductivity at elevated temperatures (> 500 ◦ C) can also be used for potentiometric measurements. Porous ZrO2 membranes covered with platinum grids have been used to determine oxygen in gases potentiometrically. Different oxygen concentrations on the two sides lead to a potential difference across the membrane in proportion to the difference in oxygen partial pressure.

8.4. Amperometry, Voltammetry, and Coulometry Amperometric, voltammetric, and coulometric techniques are not so widely used as other electroanalytical methods for reaction monitoring. They are characterized by electrode reactions at polarized electrodes that are detected by current – voltage measurements. Because of the electrode reactions the electrodes have to be cleaned on a regular basis, and a certain fraction of the mixture under investigation is consumed during the measurement. Applications include the amperometric determination of chlorine, dissolved oxygen, hydrazine, and ozone in pulp-bleaching baths; tap, industrial, and wastewater, amperometric monitoring of hydrogen peroxide in textile bleaching baths [346], the amperometric monitoring of gases and air at work places, the voltammetric determination of heavy metals in waste streams, and the coulometric analysis of sulfur compounds.

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On-Line Monitoring of Chemical Reactions

9. Miscellaneous Methods 9.1. Mass Spectrometry In the 1990s, mass spectrometry (MS) emerged as technique for on-line analytical purposes. The method is highly sensitive and fast with an extraordinary molecular selectivity. The detection principle of MS is based on an ionization of chemical species followed by a sparation of the fragments according to their mass-to-charge ratio. Due to the wide variety of ionization techniques, most types of sample can be analyzed by MS. Ionization techniques for process applications include electron impact (EI), resonanceenhanced multiphoton ionization (REMPI), microwave plasma ionization, thermal ionization, and electrospray ionization (ES). Direct capillary and membrane inlets are most commonly used for sample introduction. MS process applications are reviewed in [347–350]. In an increasing number of process applications MS is interfaced to pre-separation techniques such as GC or LC to improve resolution and selectivity in the analysis of complex (reaction) mixtures. Reduced costs, size, and complexity of the instruments have even made portable MS devices possible. Efforts have been undertaken within the last couple of years to miniaturize MS instruments (“MS on a chip”). The traditional on-line fields of use of MS are process gas stream and environmental gas analysis, environmental and wastewater monitoring, and monitoring of fermentation processes. Trace amounts of aromatics and chlorinated compounds in flue gases of industrial incinerators have been detected by REMPI-MS [351]. MS has also been applied to monitoring liquid-phase and gas-phase chemical reactions. On-line MS is useful for monitoring alkanes and alkenes in rubber production [352], monomers and solvents in polyethylene production [353], or electrochemical polymerization of aniline [354]. The modern semiconductor industry relys heavily on MS process monitoring [355], [356].

MS can be used to monitor gaseous impurities in high-purity gases, to study plasma etching processes [357], molecular beam epitaxy, and chemical vapor deposition processes [358].

Figure 21. U-shaped tube for density measurements (courtesy of Anton Paar KG)

On-Line Monitoring of Chemical Reactions

9.2. Densimetry and Dilatometry

Xr (t) =

The physical basis of densimetry and dilatometry is the mass density difference (if any) between initial reaction mixture and the final reaction product. As the reaction proceeds, this density difference leads to an increase or decrease in the density of the mixture (t) and to an overall shrinkage or expansion, respectively, of the volume of the reaction mixture V . Density changes can be measured continuously with responses on the order of one measurement per second, while the repetition time for measurements of volume changes is in the millisecond range. Therefore, the extent of reaction can be calculated at any time if the amount of ingredients initially charged to the reactor and/or continuously fed to the reaction mixture is known. The following factors must be considered in applying these techniques: 1) For a quantitative analysis of density changes, it has to be checked carefully whether the reaction mixture behaves as an ideal mixture or not; that is, does Equation 43 Vtotal =



Vi + Vexcess

(43)

i

with V excess = 0 hold, where V total , Vi , and V excess are the volume of the reaction mixture, volume of component i, and excess volume, respectively. 2) Accurate densimetric and dilatometric measurements rely substantially on the accuracy of temperature control to be better than ± 0.1 K (in some cases even better than ± 0.01 K). 3) As with all other “integral” methods, a single variable (density, volume, temperature, pressure) is insufficient to describe multicomponent reaction mixtures. Density measurements are suitable for reaction monitoring of polymerizations (in bulk, solution, emulsion, suspension) because of the considerable density differences between monomers and polymers (Table 6). In a twocomponent system, the extent of reaction (e.g., conversion x r (t) in polymerizations) can simply be calculated according to Equation 44 by assuming ideal mixing behavior

1 start 1 start

− −

1 (t) 1 end

859 (44)

where start and end are the density of the mixture at the beginning and end of the reaction. Most of the common principles to perform density measurements (pyknometer, buoyancy) are unsuitable for reaction monitoring. Methods that have been applied to process monitoring include hydrostatic pressure measurement, the maximum bubble pressure technique, oscillations of U-shaped tubes, cylinders or plates, and absorption of radioactive radiation or X-rays. Table 6. Densities of monomers and polymers at 20 ◦ C Density, g/cm3

Monomer

Acrylic acid Acrylonitrile 1,3-Butadiene Butyl acrylate n-Butyl methacrylate Ethyl acrylate 2-Ethylhexyl acrylate Ethyl methacrylate Methyl acrylate Methyl methacrylate o-Methylstyrene Styrene Vinyl acetate Vinyl chloride Vinyl methyl ether

Monomer

Polymer

1.051 0.806 0.621 0.899 0.886 0.924 0.887 0.914 0.954 0.944 0.916 0.906 0.932 0.911 0.750

1.37 1.17 0.97 1.08 1.06 1.12 0.99 1.13 1.22 1.19 1.01 1.04 1.18 1.39 1.06

A rather robust technique is the measurement of the resonance frequency of a U-shaped tube (Fig. 21). This resonance frequency is a very sensitive function of the mass of fluid within the tube, and, hence, of its density. The densimeter has to be calibrated with a liquid of known density cal that gives a period T osc,cal of the U-tube oscillations. The density of the mixture (t) can be derived from the change of the period of oscillations T osc (t) provided the temperature T of the mixture is held constant (Eq. 45)   2 2  (t) − cal = const. Tosc (t) − Tosc,cal

(45)

To obtain noise-free results, gas bubbles should be avoided while circulating the reaction mixture through the tube densimeter. In lowviscosity media, bubble traps can prevent bubbles entering the densimeter tube. A prerequisite for the application of Equation (45) is that the viscosity of the calibration liquid is roughly

860

On-Line Monitoring of Chemical Reactions

equal to the viscosity of the mixture because of its influence on the tube attenuation. The accuracy of density measurements with commercially available oscillating tubes can reach the range 10−6 to 10−7 g/cm3 . Applications of Ushaped tube density measurements to monitoring reactions can be found in [359–361]. In addition to gas bubbles and temperature uncertainties, film formation on the inner surface of the tube, formation of sediments, and clogging of the pipe may cause problems with this method. These problems can be partially overcome by the inverse arrangement: a tuning fork inserted into the mixture. Again, the change of resonance frequency of the fork is related to the density of the surrounding liquid, and the amplitude with which the fork oscillates (i.e., the attenuation) is a function of the viscosity of the medium. The maximum bubble pressure technique is based on the pressure pbubble of a spherical gas bubble of radius r at a capillary tip immersed in the reaction mixture at depth h (Fig. 22). This pressure is the sum of the hydrostatic pressure of the liquid above the capillary tip, and the pressure generated by the liquid – bubble interfacial tension γ pbubble = gh +

2γ r

(46)

where g is the acceleration due to gravity [362], [363]. Thus, pbubble depends on both the immersion depth of the capillary tip into the liquid, and the capillary and, therefore, bubble diameter.

For density measurements, two capillaries of equal diameter are inserted into the mixture at different depths hi beneath the surface. This leads to a difference of pressure readings ∆pbubble of the two capillaries, and  =

(47)

The internal bubble pressure due to surface tension is identical for the two capillaries. In addition, the two-capillary principle can also be used to measure the surface tension of the liquid. In this case, two capillaries of different diameters r 1 and r 2 are immersed into the liquid at the same depth and experience the same hydrostatic pressure but different internal pressures due to different bubble curvatures. Thus, γ = 2

∆p  bubble  1 − r1 r 1

(48)

2

Corrections must be made to account for the influence of capillary forces and nonspherical bubble shapes [362]. In stirred or turbulent media, bubble shear-off may cause problems as well as coagulum formation or clogging of capillaries in film-forming media. The accuracy of bubble densimeters is on the order of 10−3 g/cm3 . The hydrostatic pressure ph of a liquid measured at a depth h is ph = gh

(49)

The hydrostatic pressure difference ∆ph between two pressure sensors at different depths of distance ∆h = h1 - h2 is directly proportional to the liquid density  =

Figure 22. Schematic of maximum bubble pressure method

∆pbubble (h1 − h2 ) g

∆ph ∆hg

(50)

The physical basis of Equations (49) and (50) is essentially identical. In stirred or flowing media, the hydrodynamics may add considerable noise to the pressure readings of the sensor. The absorption of radioactive radiation (including X-rays) depends on the molecular/atomic absorption coefficient, the density (concentration) of the ingredients, and the thickness of the material through which the radiation passes (path length d). After calibration of the absorption of both the source/cell/detector arrangement and the mixture, the density can be calculated according to

On-Line Monitoring of Chemical Reactions Id ∞I0 exp (−d)

(51)

where I 0 and I d are the intensity of the radiation source and the detected radiation, respectively. Radiometric density determination is advantageous in systems under very high pressure and in corrosive media because neither the source nor the detector is in direct contact with the mixture. The application of dilatometry to reaction monitoring is discussed in detail in [364]. In general, dilatometry is a very useful tool for academic research but only of limited value for reaction monitoring in (continuously stirred) industrial reactors.

9.3. Rheometry The theory of rheology, instrumentation and applications are extensively described in [365– 367] (see also → Fluid Mechanics). Various processes in the plastics industry are accompanied by viscosity changes of the reaction mixture: 1) Most of plastics manufacturing processes lead to a substantial increase of viscosity because of growing chains, branching, and cross-linking. Rheological characterization of plastics is essential for understanding and influencing their processability. 2) Curing reactions transform liquids into gels and solids. Thus, the resistance of the material to flow changes by orders of magnitude. 3) Melt flow index is an important parameter for monitoring and controlling reaction extrusion, reaction blending, and reaction injection molding [35], [36]. Increasing viscosity in noncross-linked systems can be related to the molecular mass and concentration of growing chains. For linear polymers, the bulk viscosity η is a function of the weight average molecular mass M w similar to the Kuhn – Mark – Houwink equation for the intrinsic viscosity [η] of polymers in solution a

η = Kη Mwη

(52)

where Kη is a parameter that depends on interchain friction. The exponent aη is about 1 – 2 for short chains, and increases to 3.4 above a

861

critical molecular weight M w,crit [368]. Equation 52 holds also for branched polymers if M w is replaced by gM w , where the contraction factor g is defined as the ratio of the radius of gyration of the branched molecule to that of a linear molecule of the same weight average molecular mass. In cross-linking reactions, the viscosity of the mixture increases dramatically while approaching the gel point [369], [370]. After reaching the gel point, the cross-linked reaction mixture behaves like a soft solid. For temperatures above the glass transition temperature T g the mechanical properties of the material may be treated by rubber elasticity theory. The viscosity of disperse systems depends on the volume fraction ϕ of disperse material, particle shape and size distribution, interparticular interaction, etc. For monodisperse spherical particles at low ϕ, viscosity is proportional to the volume fraction of the disperse phase η = η0 (1 + 2.5 ϕ)

(53)

where η 0 is the viscosity of the dispersion medium. For higher volume fractions and systems with dominant interparticle interaction, numerous theoretically based, semi-empirical, and empirical η – ϕ relations can be found in the literature [371], [372]. Thus, viscosity reflects changes of particle number and size [373] in heterophase reactions (e.g. emulsion formation, emulsion polymerization, suspension polymerization, dispersion polymerization). Principles and methods for on-line rheometry include rotational and capillary process viscometers, as well as the application of ultrasound (see Chap. 3), and tube or fork oscillations (see Section 9.2). Widely used techniques for measuring melt viscosity in extruders are capillary viscometry and pressure drop analysis [35], [36], [374], [375]. The pressure drop ∆p of melt flow through a nozzle is directly proportional to the melt vicosity according to the Hagen – Poiseuille law ∆p = −

8η¯ vl R2

(54)

where υ¯, l, and R are the average velocity in the pipe, its length, and diameter, respectively.

862

On-Line Monitoring of Chemical Reactions

9.4. NMR Spectroscopy Nuclear magnetic resonance (NMR) spectroscopy is probably the most important spectroscopic technique for studying chemical structures. Signals of molecules and sub-units of molecules, and the signals of their various spatial arrangements and configurations are much more pronounced in NMR than in any other spectroscopical method. Nevertheless, NMR is still mainly an off-line technique because of elaborate sample preparation requirements, the instruments sensibility to temperature variations, the necessity of instrument tuning (magnet shimming, drift compensation), and the time required to run spectra. However, some recent developments in instrumentation (magnets, radio frequency source and detector circuitry), and the adaption to flow cells have made NMR spectroscopy applicable to on-line analysis. A major advantage of NMR spectroscopy is the noninvasive character of the technique. Most of the process measurements so far utilize pulsed NMR in which the free induction decay (FID) signal of magnetic-field aligned molecular spins after the application of a perpendicular radio frequency pulse is analyzed as a function of time. Network formation during the radiation-induced polymerization of Nisopropylacrylamide has been monitored by T 2 spin – spin relaxation time measurements [376]. The solids content of rubber latex can be determined without any special sample preparation with a total measurement time of ca. 10 s by quasi-on-line application of pulsed NMR [377]. Other applications include determination of hydrogen in various materials, and several solvents in gasoline. FT-NMR spectroscopy has been rarely applied to on-line analysis. Because of its natural abundance, 1 H NMR is superior to all other nuclei for process measurements. Applications are monitoring processes in refineries [378] and the emulsion polymerization of butyl acrylate [379].

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298. C. J. Venkatramani, J. B. Phillips, J. Microcolumn Sep. 6 (1994) 229. 299. R. G. Mathews, J. Torres, R. D. Schwartz, J. High Resol. Chromatogr. Chromatogr. Commun. 1 (1978) 139. 300. M. G. Bloch, R. B. Callen, J. H. Stockinger, J. Chromatogr. Sci. 15 (1977) 504. 301. J. Clemmons, Adv. Instrum. Control 50 (1995) 97. 302. J. Durand, J. Beboulene, A. Ducrozet, Analusis 23 (1995) 481. 303. J. P. Durand, Y. Boscher, M. Dorbon, J. Chromatogr. 509 (1990) 47. 304. F.-Q. Ma, D.-S. Lu, Z.-Y. Guo, J. Chromatgr. A 740 (1996) 65. 305. K. Snavely, B. Subramaniam, Ind. Eng. Chem. Res. 36 (1997) 4413. 306. E. Dorrestijn, P. Mulder, J. Anal. Appl. Pyrolysis 44 (1998) 167. 307. V. Pacakova, V. Kozlik, Chromatographia 11 (1978) 266. 308. O. Kammona, E. G. Chatzi, C. Kiparissides, J. Macromol. Sci. - Rev. Macromol. Chem. Phys. C39 (1999) 57. 309. A. Guyot, J. Guillot, C. Pichot, L. Rios Guerrero in D. R. Basset, A. E. Hamielec (eds.): Emulsion Polymers and Emulsion Polymerization, ACS Symp. Ser. 165 (1981) 415. 310. A. Urretabizkaia, J. R. Leiza, J. M. Asua, AIChE Journal 40 (1994) 1850. 311. M. Alonso, M. Recasens, L. Puigjaner, Chem. Eng. Sci. 41 (1986) 1039. 312. M. Alonso, M. Alivers, L. Puigjaner, M. Recasens, Ind. Eng. Chem. Res. 26 (1987) 65. 313. J. Janata, M. Josowicz, D. M. DeVaney, Anal. Chem. 66 (1994) 207R. 314. J. Janata, M. Josowicz, P. Vanysek, D. M. DeVaney, Anal. Chem. 70 (1998) 179R. 315. P. H. M¨uller, D. Werner, Chemie Technik 26 (1997) 48. 316. C. L. Guillemin, Process Control Qual. 3 (1992) 153. 317. R. E. Cooley, C. E. Stevenson, Process Control Qual. 2 (1992) 43. 318. N. C. van de Merbel, I. M. Kool, H. Lingeman, U. A. T. Brinkman, A. Kolhorn, L. C. de Rijke, Chromatographia 33 (1992) 525. 319. F. J. Sauter, Am. Lab. 10 (1992) 33. 320. E. Verette, F. Qian, F. Mangani, J. Chromatogr. A 705 (1995) 195. 321. Z. Ge, R. Thompson, D. Detora, T. Maher, P. Mckenzie, D. Ellison, J. Crutchfield, J. Proc. Anal. Chem. 3 (1997) 1.

On-Line Monitoring of Chemical Reactions 322. M. G. Musolino, G. Neri, C. Milone, S. Minico, S. Galvagno, J. Chromatogr. A 818 (1998) 123. 323. J. C. Thompson, R. H. Smith, Proc. Control. Qual. 2 (1992) 55. 324. L. Ebdon, H. W. Handley, P. Jones, N. W. Barnett, Microchim. Acta 2 (1991) 39. 325. M. A. Rey, J. M. Riviello, C. A. Pohl, J. Chromatogr. A 789 (1997) 149. 326. M. Nonomura, T. Hobo, J. Chromatogr. A 808 (1998) 151. 327. G. J. Lynch, Proc. Control. Qual. 1 (1991) 249. 328. R. M. Montgomery, R. Saari-Nordhaus, L. M. Nair, J. M. Anderson, J. Chromatogr. A 804 (1998) 55. 329. H. B. Wan, J. Liu, K. C. Ang, S. F. Y. Li, Talanta 45 (1998) 663. 330. B. A. P. Buscher, U. R. Tjaden, J. van der Greef, J. Chromatogr. A 788 (1997) 165. 331. B. E. Richter, B. A. Jones, N. L. Porter, J. Chromatogr. Sci. 36 (1998) 444. 332. Y. Shen, M. L. Lee, J. Chromatogr. A 778 (1997) 31. 333. M. H. Wright, A. M. James, Kolloid-Z. Z. Polym. 251 (1973) 745. 334. S. Bisal, P. K. Bhattacharya, S. P. Moulik, J. Phys. Chem. 94 (1990) 350. 335. L. F. J. No¨el, R. Q. F. Janssen, W. J. M. van Well, A. M. van Herk, A. L. German, J. Colloid Interface Sci. 175 (1995) 461. 336. J. L. Reimers, F. J. Schork, J. Appl. Polym. Sci. 60 (1996) 251. 337. K. Voloudakis, P. Vrahliotis, E. G. Kastrinakis, S. G. Nychas, Meas. Sci. Technol. 10 (1999) 100. 338. A. M. Santos, P. Vindevoghel, C. Graillat, A. Guyot, J. Guillot, J. Appl. Polym. Sci. 34 (1996) 1271. 339. M. Patek, S. Bildstein, Z. Flegelova, Tetrahedron Lett. 39 (1998) 753. 340. S. Zoll, Chemie Technik 27 (1998) 34. 341. N. Le Bris, D. Birot, Anal. Chim. Acta 356 (1997) 205. 342. L. W. Niedrach, Angew. Chem. 99 (1987) 183. 343. T. Werner et al., Fresenius J. Anal. Chem. 359 (1997) 150. 344. E. Pringsheim, E. Terpetschnig, O. S. Wolfbeis, Anal. Chim. Acta 357 (1997) 247. 345. U.-W. Grummt, A. Pron, M. Zagorska, S. Lefrant, Anal. Chim. Acta 357 (1997) 253. 346. P. Westbroek, E. Temmerman, P. Kiens, Anal. Comm. 355 (1998) 21. 347. R. G. Cooks, S. H. Hoke, K. L. Morand, S. A. Lammert, Int. J. Mass Spectrom. Ion Processes 118 (1992) 1.

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348. S. Bohatka, Adv. Mass. Spectrom. 13 (1995) 199. 349. M. A. DesJardin, S. J. Doherty, J. R. Gilbert, M. A. LaPack, J. Shao, Proc. Control Qual. 6 (1995) 219. 350. K. D. Cook, K. H. Bennet, M. L. Haddix, Ind. Eng. Chem. Res. 38 (1999) 1192. 351. R. Zimmermann, H. J. Heger, A. Kettrup, Fresenius J. Anal. Chem. 363 (1999) 720. 352. C. Didden, J. Duisings, Proc. Control Qual. 3 (1993) 263. 353. A. M. Gregory, L. A. Kephart, Adv. Instrum. Control 50 (1995) 933. 354. H. Deng, G. J. van Berkel, Anal. Chem. 71 (1999) 4284. 355. P. Fiorani, G. Margutti, G. Mariana, G. Moccia, Proc. SPIE-Int. Soc. Opt. Eng. 3509 (1998) 221. 356. J. J. Chambers, K. Min, G. N. Parsons, J. Vac. Sci. Technol. B 16 (1998) 2996. 357. V. M. Donelly, Mater. Res. Soc. Symp. Proc. 406 (1996) 3. 358. D. R. McKenzie, W. D. McFall, W. G. Sainty, Y. Yin, A. Durandet, R. W. Boswell, Surf. Coat. Technol. 82 (1996) 326. 359. S. Canegallo, G. Storti, M. Morbidelli, S. Carra, J. Appl. Polym. Sci. 47 (1993) 961. 360. F. Bleger, A. K. Murthy, F. Pla, E. W. Kaler, Macromolecules 27 (1994) 2559. 361. A. Penlidis, J. F. MacGregor, A. E. Hamielec, J. Appl. Polym. Sci. 35 (1988) 2023. 362. F. J. Schork, W. H. Ray, J. Appl. Polym. Sci. 28 (1983) 407. 363. F. J. Schork, W. H. Ray, J. Appl. Polym. Sci. 34 (1987) 1259. 364. R. G. Gilbert: Emulsion Polymerization – A Mechanistic Approach, Academic Press, San Diego 1995. 365. W. R. Schowalter: Mechanics of Non-Newtonian Fluids, Pergamon, Oxford 1978. 366. H. A. Barnes, J. F. Hutton, K. Walters: An Introduction to Rheology, Elsevier, Amsterdam 1989. 367. C. W. Macosko: Rheology: Principles, Measurements and Applications, VCH Publishers, New York 1994. 368. K.-F. Arndt, G. M¨uller: Polymercharakterisierung, Carl Hanser Verlag, M¨unchen – Wien 1996. 369. P. J. Halley, M. Mackay, Polym. Eng. Sci. 36 (1996) 593. 370. D. Hesekamp, H. C. Broecker, M. H. Pahl, Chem. Ing. Tech. 70 (1998) 286.

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371. R. M. Fitch: Polymer Colloids: A Comprehensive Introduction, Academic Press, San Diego 1997. 372. E. J. Schaller in P. A. Lovell, M. S. El-Aasser (eds.): Emulsion Polymerization and Emulsion Polymers, John Wiley & Sons, Chichester 1997. 373. R. Pal, AIChE J. 42 (1996) 3181. 374. R. Trim in Polymer Testing  97 Design and Rheological Data in the Plastics Industry, RAPRA, Paper 1, 1.

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Plant and Process Design

Process Development

873

Process Development Herbert Vogel, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany

1. 1.1. 1.2. 1.3. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 3. 3.1. 3.2. 3.2.1. 3.2.1.1. 3.2.1.2. 3.2.1.3. 3.3. 3.4. 3.4.1. 3.4.1.1. 3.4.1.2. 3.4.1.3.

Introduction . . . . . . . . . . . . . . The Objective of Industrial Research and Development . . . . The Production Structure of the Chemical Industry . . . . . . . . . . The Task of Process Development Process Data . . . . . . . . . . . . . . Chemical Mechanism . . . . . . . . Physicochemical Data . . . . . . . . Processing . . . . . . . . . . . . . . . Patenting and Licensing Situation Development Costs . . . . . . . . . . Location . . . . . . . . . . . . . . . . . Market Situation . . . . . . . . . . . Raw Materials . . . . . . . . . . . . . Plant Capacity . . . . . . . . . . . . Waste-Disposal Situation . . . . . . End Product . . . . . . . . . . . . . . The Standard Course of Process Development . . . . . . . . . . . . . . The Iterative Nature of Process Development . . . . . . . . . . . . . . Drawing up an Initial Version of the Process . . . . . . . . . . . . . . . Tools used in Drawing up the Initial Version of the Process . . . . . . . . Databases . . . . . . . . . . . . . . . . Simulation Programs . . . . . . . . . Expert Systems . . . . . . . . . . . . . Checking the Individual Steps . . Testing the Entire Process on a Small Scale . . . . . . . . . . . . . . . The Miniplant Technique . . . . . . Introduction . . . . . . . . . . . . . . . Construction . . . . . . . . . . . . . . The Limits of Miniaturization . . . .

873

3.4.1.4.

873

3.4.2. 4. 4.1. 4.1.1. 4.1.2. 4.1.3.

874 876 876 876 877 878 878 879 879 879 880 880 882 882 883 883 885 887 887 887 888 889 890 890 890 891 893

1. Introduction 1.1. The Objective of Industrial Research and Development In the chemical industry, ca. 6 % of turnover is spent on research and development [1, pp. 282,

4.1.4. 4.1.5. 4.1.5.1. 4.1.5.2. 4.1.5.3. 4.1.5.4. 4.1.6. 4.1.6.1. 4.1.6.2. 4.1.6.3. 4.1.6.4. 4.1.6.5. 4.1.6.6. 4.1.6.7. 4.1.7. 4.1.8. 4.2. 4.2.1. 4.2.2. 4.3. 4.3.1. 4.3.2. 5. 6.

The Limitations of the Miniplant Technique . . . . . . . . . . . . . . . Pilot Plant . . . . . . . . . . . . . . . Process Evaluation . . . . . . . . . Preparation of Study Reports . The Summary . . . . . . . . . . . . . Basic Flow Diagram . . . . . . . . Process Description and Flow Diagram . . . . . . . . . . . . . . . . Waste-Disposal Flow Diagram . . Estimation of Capital Expenditure Introduction . . . . . . . . . . . . . . ISBL Investment Costs . . . . . . . OSBL Investment Costs . . . . . . Infrastructure Costs . . . . . . . . . Calculation of Production Costs . Feedstock Costs . . . . . . . . . . . Utilities . . . . . . . . . . . . . . . . Waste-Disposal Costs . . . . . . . . Staff Costs . . . . . . . . . . . . . . . Maintenance Costs . . . . . . . . . Overheads . . . . . . . . . . . . . . . Capital-Dependent Costs (Depreciation) . . . . . . . . . . . . Technology Evaluation . . . . . . . Assessment of the Experimental Work . . . . . . . . . . . . . . . . . . Return on Investment . . . . . . . Static Return on Investment . . . . Dynamic Return on Investment . . Economic Risk . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . Amortization Time . . . . . . . . . Trends in Process Development References . . . . . . . . . . . . . .

. . . . . .

894 894 895 895 895 895

. . . . . . . . . . . . . .

895 896 896 896 896 898 898 898 899 899 902 902 903 903

. 903 . 903 . . . . . . . . .

907 907 908 908 909 909 909 909 910

302], an amount which is often of the same order of magnitude as the profit of the undertaking. It is the job of research management to use these resources to achieve competitive advantages. After all, for the chemical industry the halcyon days of the sellers’ market (demand > supply) have receded into the distant

874

Process Development

past; it is now the customer who calls the tune (buyers’ market). Unlike consumer goods such as cars and clothes, most commercial chemical products are faceless (e.g., hydrochloric acid), and as a rule the customer is therefore only interested in sales incentives such as price, quality, and availability. All the research activities of an industrial enterprise must therefore ultimately boil down to two basic competitive advantages, namely those of being “cheaper” and “better” than the competitor. At the same time, “better” now refers not only to availability and product quality, but also environmental compatibility of the process, and the quality assurance approach of the supplier, etc.

1.2. The Production Structure of the Chemical Industry If the production structure of the chemical industry is examined, it is seen that there are only a few hundred major basic products and intermediates that are produced on a scale of at least a few thousand to several million tonnes per annum worldwide. This relatively small group of key products, which are in turn produced from only about ten raw materials, are the stable foundation on which the many branches of refining chemistry (dyes, pharmaceuticals, etc.), with their many thousands of often only short-lived end products, are based [2, p. 57 ff]. This has resulted in the wellknown family tree (see Fig. 1), which can also be regarded as being synonymous with an integrated production system, with synergies that are often of critical importance for success. A special characteristic of the major basic products and intermediates is their longevity. They are statistically so well protected by their large number of secondary products and their wide range of possible uses that they are hardly affected by the continuous changes in the range of products on sale. Unlike many end products, which are replaced by better ones in the course of time, they do not themselves have a so-called life cycle. However, the processes for producing them are subject to change. This is initiated by new technical possibilities and advances opened up by research but is also dictated by the current raw material situation (see Fig. 2).

Figure 1. Product family tree of the chemical industry: starting from raw materials and progressing through the basic products and intermediates, to the refined chemicals and final consumer products [3], [4]

Figure 2. How the raw material base of the chemical industry has changed with time

Process Development

875

Figure 3. Life cycles of the acrylic acid processes a) Cyanohydrin,acrylonitrile,and propiolactone processes; b) Reppe process; c) Propylene oxidation; d) New process?[4]

Here it is not the individual chemical product, but the production process or technology which has a life cycle. Figure 3 shows the life cycles of the acrylic acid processes [4]. Enterprises which already have competitive advantages must take account of this technology S curve [5] in their research and development strategy (see Fig. 4). The curve shows that as the research and development expenditure on a particular technology increases, the productivity of such expenditure decreases with time [6]. If enterprises are approaching the limit of a particular technology, they must accept disproportionately high research and development expenditure, with the result that the contribution made by these efforts to the research objectives of “cheaper” and/or “better” becomes increasingly small, thereby always giving the competitor the opportunity of catching up on the technical advantage. Once an enterprise has reached the upper region of the product or technology S curve, the question arises whether it is necessary to switch from the standard technology to a new trendsetting technology in order to gain a new and sufficient competitive advantage [7]. Figure 4 depicts this switch to a new technology schematically and shows that on switching from a basic

technology to a new trend-setting technology, the productivity in the research and development sector increases appreciably, thus enabling substantial competitive advantages to be achieved. It is precisely on this innovative activity that the prosperity of highly developed countries with limited raw material sources such as Germany and Japan is based.

Figure 4. The technology S curve [8]: The productivity of the research and development expenditure increases considerably by switching from the basic technology to a new trend-setting technology

876

Process Development

To assess whether a “better” and/or “cheaper” research and development strategy is still acceptable in the long term for a given product or production process, the R & D management must develop an “early warning system” that determines the optimum time for switching to a new product or a new technology [9].

1.3. The Task of Process Development The task of process development is to extrapolate a chemical reaction discovered and researched in the laboratory to an industrial scale, taking into consideration the economic, safety, and ecological boundary conditions. The starting point is the laboratory equipment and the outcome of development is the production plant; in between, process development is required. The following account shows how this task is generally handled. Although the sequence of steps in the development process described are typical, they are by no means obligatory, and it is only possible to outline the basic framework.

Thermodynamic Equilibrium. Many reactions, such as esterifications and dimerizations, proceed to chemical equilibrium. The thermodynamic equilibrium provides information about the maximum possible conversion, and to assess the potential conversion it is essential to know how far the intended reactions are from chemical equilibrium. Example 2.1. The formation of trioxane from formaldehyde

in the gas phase is a typical equilibrium reaction. The equilibrium constant, which is given by Kp =

pT 4 p0F −p =

3 p3F 3p−p0

(2.2)

F

where p = total pressure at equilibrium p0F = initial partial pressure of formaldehyde pT , pF = equilibrium partial pressure of trioxane or formaldehyde, respectively, can be expressed according to [10] as:

2. Process Data

 7350  log Kp /bar −2 = −19.8 T /K

When the laboratory phase has been completed and before the actual process development is started, further information must be obtained since the latter stage is normally associated with high costs.

It follows that if the process conditions assumed are T = 353 K and p0 F = 0.2 bar, a maximum formaldehyde conversion of:

2.1. Chemical Mechanism The core of a chemical plant is the reactor. Its input and output decide the structure of the entire plant built up around it. Therefore, detailed knowledge of the chemical reaction must be available at the earliest possible stage. In particular, the following questions must be clarified: 1) Thermodynamic equilibria 2) Kinetics of the main, secondary, and side reactions 3) Dependence of selectivity and conversion on the process parameters 4) Heats of reaction

U=

3·pT ·100 = 35% p0F

(2.3)

(2.4)

can be achieved in one throughput. The conversion actually achieved will be < 35 % and depends on the catalyst and reactor systems. The thermodynamic data (standard enthalpy and entropy of formation, specific heat of the reactants) required to calculate the equilibrium constant can be found in published tables or can be estimated by approximation methods (see → Estimation of Physical Properties ) [11]. Kinetics. In order to specify the type of reactor to be used later, information must be available on the potential reaction routes to the main and secondary products and the byproducts. The rate of formation and its dependence on process parameters such as temperature, pressure, and catalyst concentration should, if possible, be known quantitatively.

Process Development Example 2.2 [12, pp. 173 – 175]. The starting material A reacts not only to form product P, but also (1) to form an undesirable byproduct Y in a parallel reaction or (2) to form an undesirable secondary product X in a sequential reaction:

k1 ·c0A k1 −k2 · {exp (−k2 ·t) −exp (−k1 ·t)}

877

cP =

( cX =c0A ·

1+

(2.14)

k1 ·exp (−k2 ·t) − k2 −k1 )

k2 ·exp (−k1 ·t) k2 −k1

1) First-order parallel reaction For the starting material A: dcA = − (k1 +k3 ) ·cA dt

(2.6)

with cA = c0A at time t = 0, integration results in: cA =c0A ·exp {− (k1 +k3 ) ·t}

(2.7)

For the useful product P: dcP =k1 ·cA dt

(2.8)

with cP = c0P for t = 0, integration gives cp = c0p +

k1 ·c0p

(k1 +k3 ) · {1 − exp [− (k1 +k3 ) ·t]}

(2.9)

The same applies to cY . 2) First-order sequential reaction This is covered by three simultaneous differential equations: dcA = −k1 ·cA dt

(2.10)

dcP =k1 ·cA −k2 ·cP dt

(2.11)

dcX =k2 ·cP dt

(2.12)

If c 0P = c0X = 0, integration of these equations yields: cA =c0A ·exp (−k1 ·t)

(2.13)

(2.15)

2.2. Physicochemical Data A knowledge of exact physicochemical data has become more important for a number of reasons: 1) The increasing use of simulation programs has made the requirements for exact physicochemical data increasingly important. The result of a simulation calculation can only be as good as the quality of the physicochemical data. 2) To approve chemical plants, the authorities demand information on the toxicity, degradability, and safety of the materials involved [13], [14]. 3) The public are demanding more information on the effect of the materials being handled on the environment. At the start of process development, material data files for pure substances as well as binary and ternary mixtures are started. As development advances, these grow and must be continually updated. They subsequently form a document which is passed to the planning and plant construction departments. The layout shown in Table 1 has proved useful as a model for the collection of puresubstance data. Only data which have been reliably evaluated are kept in the files and these are the only values fed into the simulation programs. At the start of process development, the first step is to collect all available literature data on a material (see→ Estimation of Physical Properties).

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Process Development

Many companies now maintain their own material databases. The data have been evaluated and should have been found satisfactory when used in practice. The times when a single company had 10 different sets of Antoine parameters for water between 0 and 100 ◦ C ought now to be over. If no material values can be found in the literature, which is often the case for binary and ternary data, they should initially be estimated by using empirical formulae or realistic values that have proved themselves for many substances (e.g., heat capacity≈2 kJ kg−1 K−1 for most organic liquids). However, experimental determination or confirmation of the most important values on which the plant design is based is unavoidable. It is sheer negligence to rely on physicochemical data originating from a single literature reference.

2.3. Processing Processing is intimately linked to the chemical reaction. While the chemical aspects are still subject to extensive modification (for instance, in the choice of catalyst, solvents, etc.), there is little point in paying a great deal of attention to processing. It is only when the reaction mixture is being produced in a representative manner that an initial processing procedure can be devised (see Section 3.2). There is no generally accepted way of going about this. It is more an art than craft, and reliance must be placed on inspiration and experience. It has, however, always been helpful to gather as many experienced specialists as possible and to discuss the problems regularly. Although tentative efforts are now being made to compile collective know-how in expert systems for processing strategies, these are still far from being universally applicable. The above deliberations culminate in a separation concept which can be broken down into individual unit operations. Initially, these operations can be examined in the laboratory either on a batch basis, or in a continuous process to determine whether they are feasible in principle (e.g., are azeotropes formed?, how difficult is separation?, what is the dissolution rate?, can the phases be separated?, etc.). This gives the initial process concept, which can be used as a basis for starting the actual process development.

Table 1. Important pure-substance data frequently required in process development using acrylic acid as an example [15] Product name CAS no. Formula Molar mass, kg/kmol

acrylic acid [79-10-7] C3 H4 O2 72.06

Melting point, ◦ C Boiling point, ◦ C Antoine parameters ∗ A B C Vapor pressure at 20◦ C, mbar Heat of evaporation at bp, kJ/kg Heat capacity, kJ kg−1 K−1 Density, kg/m3 Viscosity, mPa · s Heat of formation, kJ/mol Gross calorific value, kJ/kg Heat of transformation, kJ/kg Heat of fusion, kJ/kg Solubility in H2 O Solubility of H2 O in acrylic acid MAK Toxicity (LD50 , rat, oral) Water hazard class Odor threshold Flash point, ◦ C Ignition temperature, ◦ C Lower explosion limit, vol % Upper explosion limit, vol %

13.5 141.0 9.135 −3 245 216.4 10 633 1.93 (liquid) 1 040 at 30◦ C 1.149 at 25◦ C 19 095 1 075 (polymerization) 154 at 13◦ C miscible miscible 340 mg/kg 1 (“low water hazard”) pungent 54 390 2.4 (47.5◦ C) 16 (88.5◦ C)

∗ ln (p/bar) = A + B/(C + T /◦ C).

2.4. Patenting and Licensing Situation If a new idea is to be used in developing a process, a careful check should be made to determine whether any third-party proprietary rights would be infringed. The best situation is one in which the ideas are completely new and the company can file its own patents to ensure it can use the ideas without restriction. However, patent applications not only provide protection but also represent a source of information for competitors [16]. Careful consideration must therefore be given to the question of whether it is desirable to keep an idea secret or whether it should be patented. The risk involved in deciding against patenting in-house knowledge can be reduced by eliminating the patentable novelty of that information on a worldwide basis through publication in a suitable form, e.g., in an in-house journal or a technical bulletin, etc.

Process Development If the idea is restricted by third-party proprietary rights, the following questions should be answered: 1) When do those rights expire? 2) How can they be rendered void? 3) How can they be circumvented by modifications? If the idea is completely covered by third-party proprietary rights, the only way out is to enter into licensing negotiations. Normally, a process should only be developed after the legal situation has been clarified.

2.5. Development Costs Development costs often constitute a considerable proportion of the total cost of a project. If the process is completely new, these costs are often around 50 % of the investment costs for the industrial plant, and perhaps more if a pilot plant must be built. Development costs can be quoted with sufficient accuracy if the following items are assessed: Man years: This involves determining how many chemists and engineers will be employed on the project and for how long. Since the costs associated with each of the specialists working on the individual investigations are known, the man-year costs can then be calculated. Setting up costs: A rough estimate of the costs of setting up a small-scale plant is sufficient since this item is normally the smallest in the development costs. Once some notion of the process is available (a rough process flow diagram), these costs can be determined with reasonable accuracy. Operating costs: The operating costs can be estimated from the assumed development time and the supervisory staff (shift personnel) required. A rule of thumb is that the annual operating costs of an experimental plant will be at least equal to those required to set it up. More difficult to estimate are the amounts required annually for modifications and repairs to the experimental plant. These depend very much on whether new or already proven technologies are being used.

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The development costs determined in this way do not contain any contributions arising from personnel not directly involved in development, such as management or patent specialists. Nor is account taken of unsuccessful developments. It nevertheless seems reasonable to apportion the direct development costs to the product and to allow for the other research and development costs in some other way, for example by means of a general cost provision (see Section 4.1.6.6). Although this method only accounts for the costs of successful development projects, the financial burdens on the product will be relatively high if it must bear all the development costs of the first plant.

2.6. Location The main competitive advantages such as price and availability are nowadays often dependent on the location of the production plant. Thus, there is now an increasing tendency to move the production of major basic chemicals such as methanol to the oil-producing countries. However, products whose production requires considerable know-how are less location-dependent and will continue to be produced in the industrialized countries, even in the future. Extreme cases are those in which legal requirements decide the location, as is the case in the field of gene technology. In deciding on location, consideration should be given to the following aspects: 1) Proximity to the sales market 2) Proximity to the sources of raw materials or precursors 3) Transport facilities (ship, rail, road) 4) Quality of the available labor 5) Availability of cooling water 6) Waste-disposal management (sewage treatment plant, incineration plant) 7) Political considerations (tax, investment aid, approval situation)

2.7. Market Situation The competitive advantages generated by process development should ultimately benefit the

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market and its customers. It is therefore impossible to target process development correctly without a precise knowledge of that market. The following information is required if the basic data needed to plan production (plant size, maximum production costs) are to be obtained 1) Market price (how it will develop and vary with time) 2) Demand, broken down into in-house and third-party demand 3) Market growth 4) Utilization of capacity in existing plants, with a breakdown in terms of location (United States, Western Europe, Japan) and production process 5) Competitive situation (who is the largest competitor?) 6) Customer situation (are there many small customers or a few large ones?) The question of whether a process development or improvement will be worthwhile can be analyzed by using a portfolio representation (Fig. 5). Portfolio analysis is a strategic planning tool, used to concentrate research investment on products for which market prospects appear favorable and competitive advantages can be exploited [17].

2.8. Raw Materials Raw materials, their availability, and their price structure have always been crucial factors responsible for shaping the technological base of the chemical industry and, consequently, its growth and expansion (see Fig. 2). The chemical industry produces a wide variety of products from only a few inorganic and organic raw materials. Thus, raw materials such as natural gas, petroleum, air, and water are used to produce basic chemicals such as synthesis gas (CO – H2 mixture), acetylene, ethylene, propene, benzene, ammonia, etc. These are used in turn to manufacture intermediates such as methanol, styrene, urea, ethylene oxide, acetic acid, acrylic acid, cyclohexane, etc. (see Fig. 1). Any change in the price and availability of raw materials will therefore have a substantial effect on the production processes for secondary products.

Since the exact purchase prices (transfer prices) are an important source of information in assessing manufacturing costs, they must always be available and kept up to date in a chemical company. Figure 6 shows the variation in the price of some basic chemicals. It is not only the price of potential raw materials which must be determined but also their availability and quality (purity, state as delivered). It is never possible to reach a generally applicable decision on a particular raw material since much depends on its availability at a particular location and variation with time.

2.9. Plant Capacity Optimum plant capacity, i.e., the size of plant which yields the maximum return (see Section 4.2), will be achieved if a degree of capacity utilization of 100 % is reached 3 – 5 years after start-up. However, since the plant capacity must be decided about 4 – 5 years before start-up for planning and legal approval reasons, and plants which are not fully utilized over a period of time increase the specific production costs, it is essential to be able to look 7 – 10 years into the future in determining the optimum capacity. Therefore, precise knowledge of the market potential and of market growth trends is required. The relationship between production price and degree of capacity utilization is as follows [19] P 1+F/V = P0 1+F/V ·Keff /K0

(2.16)

where P P0 F V K eff

= production price in monetary units per kilogram, = production price at rated capacity, = fixed costs, = variable costs, = effective plant capacity, in tonnes per annum,

K0 = rated capacity, in tonnes per annum. Thus, as the degree of capacity utilization drops, the production price increases hyperbolically and this dependence is the greater, the

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Figure 5. Portfolio matrix [17] A) Sleeping dog products These are problem products whose competitive situation is weak because of their low market share. In such cases process development would not be worthwhile. B) Cash cow products These are products which, although their market growth is low, have a high market share. To exploit the cost advantages to the full, there is little point in heavily investing in process improvements. Instead, these products should be “milked” and the resulting cash flow used to support the star products. C) Star products All the efforts should be concentrated on these products since they will safeguard the survival of the company in the long term. D) Question mark products In these cases the company must decide whether to increase the market share (turn them into star products) or to abandon them if the prospects are slim.

Figure 6. Variation in the price of some basic chemicals with time [18]

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higher the contribution made by the fixed costs (see Fig. 7).

be undecided and the process should therefore be designed to produce only waste products which can be readily incinerated. Processes which produce salts or large amounts of dilute aqueous solutions containing materials that are difficult to degrade biologically are inherently locationdependent.

2.11. End Product

Figure 7. Dependence of the relative production price on degree of capacity utilization for different fixed cost/variable cost ratios F/V

The greater the uncertainty in the market studies and the newer the technology used, the greater will be the preference for choosing a lower capacity. Many Japanese chemical companies adopt this approach and prefer to set up several plants with smaller capacities. However, lower capacities result in higher specific plant costs and therefore in higher production costs. This is represented by the following empirical relationship: I2 =I1 · (K2 /K1 )χ

(2.17)

where I K χ

= capital expenditure (investment), = capacity, = degression exponent.

For individual items of equipment, χ = 0.6, and for plants containing the same number of items of equipment χ = 2/3 [20].

2.10. Waste-Disposal Situation (see Section 4.1.6.3) Waste-disposal facilities (incineration, sewage treatment plant, waste-disposal site) depend on the location. However, at the beginning of process development the location will generally still

It is essential to determine the demand for the end product as well as its specification and achievable price. A strongly competitive situation will mean particularly severe changes with time, and in this case it is necessary to adopt a particularly critical approach to forecasting. The product specification is of particular significance for process development. In the simplest case, it consists of a minimum purity, for example, that the end product should contain at least 99 % of a particular chemical compound. The permissible impurity content may vary considerably from one product to another. Thus while it may be a few percent in the case of dyes, the limit is often only a few parts per million for monomers. However, in most cases the end product cannot be specified so simply since its subsequent use may be affected in different ways by individual impurities. In such cases, it is not sufficient to specify an upper limit for the sum of impurities; instead, limits must be specified for important individual components. Often, the chemical analysis is not sufficient or is not sufficient on its own for specifying the end product, a case in point being, for example, polymers. In such cases chemical analysis is supplemented by process engineering properties which, in the simplest case, can be reduced to simple physical properties. The process development engineer endeavours to obtain as detailed a specification as possible for the end product before he starts work, but usually the specification will only be finalized during process development. Statements such as “as pure as possible” are inadequate since the removal of the final traces of impurities is time consuming and expensive. Individual specifications which are of no significance in characterizing the end product should be avoided. If possible, a composition which can be determined analytically is preferable to a

Process Development purely physical characterization since it is more closely related to the process sequence.

3. The Standard Course of Process Development 3.1. The Iterative Nature of Process Development The development of chemical processes is a complex procedure. The first hurdle in establishing a new process is overcome when a promising synthetic route, usually with associated catalysts, is discovered, but many issues must be clarified and many problems solved before industrial implementation is feasible and all the documents necessary to design and operate a chemical plant have been assembled [21], [22]. How are these planning documents arrived at? The conventional procedure is carried out in three stages (see Fig. 8): An initial process concept is developed on the basis of an optimized laboratory synthesis. This is not usually continuous (see Section 3.2), and the individual process steps are examined independently of each other in the laboratory (see Section 3.3). A continuous laboratory plant (the so-called miniplant) is then designed, set up, and operated (see Section 3.4.1). This is a small but complete plant handling production quantities of ca. 100 g/h and consisting of a synthesis section, working-up, and all recycling streams. Once the process concept has been confirmed in the miniplant, the next step is to design and set up a trial plant with a much higher capacity, the scale of this pilot plant being between that of the miniplant and that of the industrial plant. The quantities produced are a few kilograms per hour or tonnes per annum and enable, for example, application tests to be carried out on the product or large-scale deliveries to be made to customers. Operation of the pilot plant makes it possible to complete and verify data and documentary information obtained at an earlier stage of process development (see Section 3.4.2). The scale-up factor from one stage to the next is always limited by the “minimum principle”, i.e., the process stage or piece of equipment with the lowest scale-up feasibility determines

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the maximum capacity of the next larger plant whose operating performance can be calculated. It is here that the process development engineer has an opportunity to save time and money. If reliable documents can be drawn up which enable the maximum scale-up of four powers of ten (100 g/h×104 = 1 t/h) to be spanned in designing a production plant, omitting the intermediate stage also eliminates the costs of the pilot plant and 3 – 4 years’ development time [23], providing an important advantage in terms of cost effectiveness and marketing. Consequently, efforts are now made to extrapolate directly from the miniplant to the production scale. The following list summarizes some of the important process steps and typical maximum scale-up values above which reliable scale-up is no longer possible [24]: Reactors Multitubular reactor, homogeneous tube, homogeneous stirred tank Bubble column Fluidized-bed reactor

>10 000 < 1000 50 – 100

Separation processes Distillation and rectification Absorption Extraction Drying Crystallization

1000 – 50 000 1000 – 50 000 500 – 1 000 20 – 50 20 – 30

Larger scale-up factors are possible for gasphase reactors such as multitubular reactors. An overview of the main types of reactor and a discussion of the criteria for the choice of a particular reactor and reactor scale-up during process development can be found in Parts A and B of this volume. The use of dimensional analysis in scale-up is discussed in→ Scale-Up in the Chemical Industry. The unit operations of distillation, rectification, and absorption can be scaled up without an intermediate stage. This explains, for example, why much effort is expended on obtaining operating conditions that allow the reaction and processing of gases and liquids. The behavior of gases and gas – liquid equilibria can be explained well in physical terms and calculation is therefore straightforward.

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Figure 8. The stages involved in developing a process The time intervals specified for design, installation, and operation are only tentative and, in specific cases, the real values may differ greatly from those given

Example 3.1. Vapor – liquid equilibria can readily be measured experimentally and can be described well in mathematical form. The general relationship for the equilibrium between a liquid phase and an ideal gas phase takes the form: γi (xi ) ·Pi0 (T ) yi (xi ,T ) = ·xi P

(3.1)

where yi

= mole fraction of component i in the gas phase, xi = mole fraction of component i in the liquid phase, P0i (T ) = saturation pressure of pure component i at the system temperature T , P = vapor pressure of the mixture, γi = activity coefficient of component i.

The vapor pressure of the pure component i, P0i (T ), is often represented by a three-parameter Antoine expression: Pi0 (T ) =Ai +

Bi Ci +T

(3.2)

Several models are available for describing the activity coefficient γ i . Examples are the Wilson model, which can be applied to completely miscible mixtures, and the NRTL model, which is suitable for systems with a miscibility gap [25], [26]. For solids, scale-up factors are several orders of magnitude lower because of complications such as deposits, encrustations, and abrasion. This is true not only for synthesis (e.g., in fluidized-bed reactors), but also for processing (e.g., drying and crystallization). Process development does not, however, take place as a one-way street. Assumptions are made for the individual development stages which are

Process Development only confirmed or refuted when the next stage is being worked on. It may be necessary therefore to go through the individual stages several times with modified assumptions, resulting in a cyclic pattern (see Fig. 9).

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possible so that consideration of a larger number of possibilities is restricted to the laboratory. Prolonged investigation of two variants on a trial-plant scale should be avoided. Most mistakes are made at the beginning of the activity, but it is still relatively easy and cheap to eliminate them at the miniplant stage. However, the further process development advances, the more expensive it becomes to eliminate mistakes (see Fig. 10). In the final production plant, corrections can only be made with an enormous expenditure of time and money. As shown in Figure 9, each development stage is followed by an evaluation to decide whether development should be continued, stopped, or started again at an earlier development level (see Section 4).

3.2. Drawing up an Initial Version of the Process

Figure 9. The cyclic pattern of process development

The most important task is to find the weak points and subject them to particularly close scrutiny. The entire process will then be examined again with the improved data obtained in this way, and so on. The fact that many decisions must be taken with incomplete knowledge is a fundamental but inevitable difficulty. To delay development until all uncertainties have been eliminated would be just as wrong as starting industrial development on the basis of laboratory discoveries alone. A start should be made on choosing between various processes or process variants as early as

Once all the information has been assembled, an initial version of the process is drawn up [27]. In general, it has been found useful to adopt the following procedure: After the initial versions of the process, which should be the most reasonable ones possible at that stage, have been drawn up, they should be discussed by the project team. This discussion should take account of all the information gathered hitherto and, after all the advantages and disadvantages have been reviewed, only one version of the process should remain. There is at present no method which will result in the “best” version of the process being invariably chosen at the outset and there will presumably not be one in the future either [28], [29]. Finding the optimum version of a process requires not only knowledge and experience but also a considerable creativity, since the number of possible ways of carrying out a given task is almost infinite and many of the rules can still not be quantified [30]. Example 3.2 : The esterification of an impure organic acid with methanol yields a reaction mixture containing at least 5 components (N = 5): unreacted acid, unreacted alcohol, ester, volatiles such as H2 O, lower esters, dimethyl ether, and high-boiling components such as higher esters. According to Stephanopolous [31] the possible number of arrangements for the separation

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Figure 10. The costs of eliminating mistakes and the investment costs increase by a factor of 10 from one development stage to the next

operations can be calculated from the relationship: Z=

[2· (N −1)] ! (N −1) ·S N !· (N −1) !

(3.3)

where Z N S

= number of alternative arrangements, = number of components, = number of separation processes.

If all components can be separated by distillation, (N − 1) = 4 separation columns are necessary, which can be arranged in 14 different ways. If a second separation process – for example, an extraction, if the ester, alcohol, and water form azeotropes – must be added, this already results in 224 possible arrangements. If the number of possible reaction procedures (e.g., batch/continuous; gas/liquid; ion exchanger/mineral acid) is also taken into consideration, there will be an almost infinite number of ways of carrying out the process (a socalled combinatorial explosion). To find the best process among all these variants is quite impossible simply because the ideal solution is unknown. To calculate all possible

variants, to compare them, and then select one is not feasible because of the large number. Many of the variants can of course be eliminated at the start for trivial reasons (industrial, economic, legal reasons, etc.). However, even the number of potential variants left is still so large that it is impossible to calculate investment and production costs for all of them. Furthermore, at this point the information available for doing this is often inadequate and there is consequently a risk of eliminating promising variants. Perhaps this example reveals why inventing processes is still an art rather than a craft. However, the tools available for performing this art are being continuously improved [30], [32], [33] (see Section 3.2.1). On the basis of this initial version, an industrial plant is designed. The individual unit operations (reactor, absorber, distillation, etc.) are designed on the basis of the existing information (approximate size and diameter of columns, etc.). By scaling down this hypothetical large-scale plant a trial plant is designed, which is nowadays generally a miniplant (see Section 3.4.1). At the same time, the industrial plant is simulated by

Process Development using a computer program and an initial complete mass and energy balance sheet is drawn up. 3.2.1. Tools used in Drawing up the Initial Version of the Process Requirements which must be met by modern EDP tools are as follows: 1) Fast access to the information 2) Information which can be conveniently handled at the workplace 3) The information must be continuously updated

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Databases differ as to whether they contain the information itself (primary information or factual data) or whether they refer to other information sources (secondary information or literature data). Some types of data which are important for process development are [36]: 1) 2) 3) 4)

Material properties [37] Physicochemical data Ecological and toxicological data Costs of raw materials, intermediates, and end products 5) Energy and equipment costs 6) On-line literature searches

Table 2. A selection of simulation programs for steady-state processes Program

Supplier

ASPEN PLUS

Aspen Technology Inc., 251 Vassar Street Cambridge, MA 02139 United States BASF AG, Ludwigshafen, Germany Chemstations Engineering Software COADE 10375 Richmond Ave., Suite 1225 Houston, TX 77042 United States Chemshare ICI Hydrotech. Ltd. 400, 119 14th Street N.W. Calgary, Alberta T2N1Z6 Canada Simulation Science Inc., 1 051 W Bastanchury Road, Fullerton, CA 92633 United States Bayer AG, Leverkusen, Germany

CHEMASIM

CHEMCAD

DESIGN II FLOWPACK HYSIM

PROCESS PRO/II

Figure 11. On-line service components [35]

VTPLAN (CONTI)

3.2.1.1. Databases [34] To carry out process development as efficiently as possible, it is important to avoid duplicate activities. This includes taking account of existing knowledge or exploiting it. For this reason, chemical companies not only build up internal databases but also utilize external databases. Figure 11 provides an overview of the structure of an on-line data bank.

3.2.1.2. Simulation Programs [38] Simulation using computer programs furnishes the process development engineer with a very effective tool. The development of very fast, large computers and effective physical models has now made it possible to simulate individual units (e.g., distillation), networks of units of the

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same type (e.g., a stirred tank cascade, a heat exchanger network) and networks of dissimilar units (e.g., chemical plants). In practice, the simulation program used depends on the objective; there is, after all, little point in trying to solve detailed problems with a simulation program which, although it is allembracing, is slow if there are very much more efficient programs available which are designed for the specific requirement. Programs for solving specific problems are usually developed by the user himself and are not generally available or are available only in rare cases. Numerous programs for dealing with flow sheet problems are available from universities, industry, or the commercial market (see Tables 2 and 3) [33]. Table 3. A selection of simulation programs for dynamic systems [39–42] Program

Supplier

CHEMADYN

BASF AG, Ludwigshafen, Germany Prof. Dr. Gilles, University of Stuttgart, Germany H¨ochst AG, Frankfurt, Germany The Dow Company Midland, Mi 48674, United States Prosys Technology Ltd., Cambridge, England, (since 15.10.1991 Aspen Technology Inc.)

DIVA

SATU

SIMUSOLV

SPEEDUP

3.2.1.3. Expert Systems The difference between simulation programs and expert systems is that in the former the modeling is rigidly embodied by an algorithm, while in the latter, knowledge of the model is stored in the knowledge base independent of the deductive mechanism [43–45]. An expert system is a computer program equipped with knowledge and ability which can carry out complex tasks by imitating human intelligence. Such systems are therefore particularly suitable for use in process development where complex problems are solved by draw-

ing conclusions and using experimental knowledge. In this case, however, the knowledge does not only consist of facts which can be stored in databases but is acquired by carrying out process development, experiencing failures, being successful, repeating the same process, and learning at the same time; in short, by acquiring a feel for a problem. Knowledge is gained as to when it is necessary to stick to the rules and when they can be broken; a stock of practically proven, heuristic knowledge is built up [30]. Some examples of heuristic rules in process development are: 1) Carry out each reaction or separation only once (e.g., volatiles should only be removed once) 2) Remove substances which can be readily isolated immediately after they are produced and do not spread them over the entire working-up process 3) Design recycling paths so that, if possible, they only result in small feedbacks 4) If possible, use substances which are in any case present in the reaction output as auxiliary solvents since this may reduce the working-up costs considerably Combinatorial problems such as drawing up an initial version of a process or developing a complex catalyst system result in a combinatorial explosion (see Section 3.2). Time does not allow all the possible solutions to be tested. The human expert copes with such a combinatorial explosion by eliminating those possibilities which seem to him to be unfruitful, concentrating on those which seem to be feasible and using heuristic knowledge which steers him towards the optimum solution but does not guarantee it. The main components of an expert system are as follows (see also Fig. 12): 1) Knowledge base: heuristic knowledge is represented in this case by a system of symbols 2) Knowledge editor: a module which assists in changing, adding, or removing rules and formalizes their evaluation 3) The deductive mechanism, which combines knowledge and problem data by deriving further data from the rules stored in the knowledge base 4) input – output systems

Process Development

Figure 12. Structure of an expert system [46]

This technique was first applied in 1969 for describing and designing composite heat exchanger systems [47]. It was rule-oriented, i.e., the expert system involved a series of rules whose combination led to different results. The system was controlled by weighting factors, which were in turn dictated by success. The system “learned” through this method of changing weights and was therefore capable of yielding increasingly better results.

Figure 13. Checking the individual steps in the laboratory using pure feedstock with the absorber stage as an example (see Fig. 14)

The first expert systems were laboriously written in FORTRAN, but the technique acquired a new impetus in the 1980s when pro-

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gramming languages such as LISP (List Processor) were introduced. These are able to process symbols and symbol structures using a computer [48]. In addition a wide range of computer tools became available. Although these have made the development and handling of expert systems easier, they have contributed almost nothing to an understanding of the phenomena [49]. It is now beyond dispute that expert systems have justified themselves, although the euphoria of the 1970s has given way to a more sober assessment. The belief that a lack of knowledge and understanding can be overcome by an expert system has been found to be completely erroneous. On the contrary, an expert system can only be used efficiently by experts.

3.3. Checking the Individual Steps Like the reaction step or steps, the chosen preparatory and separation steps are first checked individually and independently of one another on a laboratory scale. Even at this stage, new requirements may be imposed on the reaction stage. For example, difficulty in removing a byproduct may make it necessary to alter the reaction conditions or modify the catalyst. Along with the experimental work, e.g., testing an absorber (see Fig. 13), thermodynamic simulation calculations are carried out for the unit concerned. Laboratory experiments are generally carried out with pure, well-defined materials. Thus, the solvent selected for the absorption is used as received from the chemical stores. This differs, of course, from the subsequent situation, where the solvent must be recycled for cost reasons and inevitably becomes enriched with byproducts, some of which are unknown, however, at this point in time. The feedstock used is a synthetic mixture of gases prepared from pure materials which also does not contain byproducts. The experiment shows whether it is physically possible to absorb the useful product from the gas stream with reasonable amounts of solvent. If the absorber can be simulated with the available material data in a calculation run in parallel, work on this step will be complete. The unit is then another piece which can be fitted into the jigsaw puzzle of the

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entire process, which is the next item which must be considered in an integrated trial plant.

tion, the solvent circuit may become enriched in byproducts and this may lower product purity. In principle, these problems can only be solved by mathematical simulation, but since many of the quantities that are required for a mathematical description are unknown, an experimental approach must be adopted. Example 3.3. When a plant which includes recycling streams (see Fig. 15) is operated, the solvent circuit becomes enriched in a highboiling substance whose existence was not previously known because it was below the analytical limit of detection in the reaction product. This high-boiling substance also has the troublesome property of depositing when it reaches a certain concentration in the solvent and forms a coating in the column. Since the high-boiling substance was not known before, it was not possible to include it in the mathematical simulation. Now that it has been discovered as a result of operating an integrated plant, it can be characterized and incorporated in the simulation.

Figure 14. An initial version of a process for oxidizing a starting material with air and then isolating the product by absorption and distillation

3.4.1. The Miniplant Technique 3.4.1.1. Introduction

Once all the individual steps have been successfully tested, it is possible to draw up a reliable flow sheet of the entire process (see Fig. 15). However, if any subsidiary step is found not to be feasible at this stage, the process concept must be changed. This preliminary flow sheet, which has not as yet been fully tested as a whole, can be used to draw up an initial rough cost estimate (see Section 4).

3.4. Testing the Entire Process on a Small Scale As Figure 15 shows, when the individual steps are put together, recycling streams are created (in this case one gas and one solvent recycling path). These recycling streams or feedbacks are an economic necessity, but they raise new process engineering problems. Thus, material recycled to the reactor may drastically affect the activity and service life of the catalyst. In addi-

Improvement in computer programs for modeling processes has resulted in the integrated miniplant technology acquiring ever increasing importance because the synergism between the miniplant technique and mathematical simulation means that the margin of safety in scaling up is as good as that obtained by setting up a pilot plant [50–52], [53, pp. 179 –182]. The miniplant technique has the following characteristics: 1) The trial plant includes all the recycling paths (it is a so-called integrated miniplant) and it can consequently be extrapolated with a high degree of reliability. 2) The components used (columns, pumps, condensers, pipelines) are often the same as those used in the laboratory. In addition, they are largely reusable standardized components which have been tried and tested in continuous operation. For these reasons, investment costs are low and flexibility is high.

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Figure 15. Example of a simplified process flow chart produced after testing the individual steps

3) The plant is operated round the clock on a shift basis for weeks and it therefore must be as fully automated as possible to keep operating costs low, but not, however, at the expense of flexibility. The measuring and control instruments used are standard components which are nowadays generally connected to a small process control system [54, pp. 183 – 186], thus making it possible to carry out modifications in the measurement and control rapidly. 4) The entire plant is normally set up in an extracted chamber so that explosion-proof operation is possible and safety-at-work requirements can be met more easily.

3.4.1.2. Construction The following design documents are required as a minimum for constructing a miniplant: 1) A process flow sheet showing all the quantity flows in the miniplant and the associated temperature and pressure conditions 2) Engineering flow diagram Required information: a) All equipment and machines b) Internal diameters and pressure ratings of pipelines and construction material

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c) The objectives of the instrumentation d) Information on the insulation of the equipment and pipelines 3) List of measuring points (measuring point number, specification, parts list, point of installation) 4) Safety concept, i.e., how the sensors (e.g., for temperature, pressure, flow) are linked to the actuators (e.g., valves, pumps) by the safety logic

efficiently (because of excessive temperature differences, foaming, etc.). Example 3.4. A miniplant produces a highboiling residue at a rate of 1 g/h. This residue is removed from the bottom of a distillation column with a distillation boiler capacity of 200 mL, so that the mean residence time is 200 h. The time required to reach the steady state is consequently about 25 d (about 3 times the mean residence time). Only then will it be possible to assess this bottom residue and draw conclusions about its fouling behavior, composition, properties, etc. Specification of Material Balance. In Section 3.2 it was shown that a miniplant should be designed by scaling down a hypothetical largescale plant. The scale-down factor is determined by finding the points in the process where the smallest and largest quantity flows occur. The levels at which these flows can be handled determine the scale-down factor. Normally only flows of 1 g/h and 10 kg/h can be handled in a miniplant. At the same time, the minimum flow is primarily determined by the time required to reach a steady state (see Example 3.4), while the upper limit of the flow is set by the maximum hydrodynamic load of the laboratory column used. Example 3.5. In a hypothetical large-scale plant producing 5 t/h, the minimum flow is 50 kg/h of a high-boiling substance which must be removed and incinerated, while the maximum flow is 100 t/h of the reaction throughput, which contains the useful product as a 5 % aqueous solution.

Figure 16. Example of a miniplant

Specification of Equipment Size. In a miniplant, the hold-up for a given throughput is almost always greater than in an industrial-scale plant. The size of the vessels and column bottoms should therefore be minimized, otherwise it will take too long for the miniplant to reach a steady state because of long residence times. In the column bottoms, which are particularly critical, a hold up of 100 – 200 mL is normally the lower limit. Below this level, a laboratory evaporator can usually no longer be operated

Scale-down factor: for the minimum flow: 50 kg/h ÷ 1 g/h = 50 000, for the maximum flow: 100 t/h ÷ 10 kg/h = 10 000. Thus the miniplant should be designed to produce not less than 100 g or not more than 500 g of useful product per hour. Specification of Type of Equipment. Unlike normal laboratory experiments in which the equipment operates only during the daytime, miniplants must run continuously over a period of weeks to be able to fulfil their task. This is due to the fact that the time required to reach

Previous Page Process Development a steady state is often long and because it is only possible to draw reliable conclusions about long-term effects such as corrosion, fouling, catalyst deactivation after a long time (1 week to 6 months). Unlike laboratory plants, the requirement imposed on the reliability of miniplant components are therefore very high and almost the same as for a large-scale plant. Occasionally this sets very tight restraints on choosing equipment and in some cases necessitates in-house development if nothing suitable is available on the market.

monitor liquid levels, deposits, foam formation, etc., than in the large-scale plant. Schulz-Walz has given examples of measuring and metering techniques for miniplants [55]. In the last two decades, important advances have been made in sensor technology and, along with electronics and processor technology, this has considerably simplified the miniaturization of sensors for miniplants (see Table 5).

Table 5. Established miniaturization limits for instrumentation components [51] Measured quantity

Principle of measurement

Established minimum measurement range

Gas flow

thermal volumetric thermal Coriolis force magnetic inductance volumetric

0.02 – 0.6 L/h 2 – 200 L/h 2 – 30 g/h 0.07 – 1.5 kg/h 0.6 – 6 L/h 0.1 – 1 L/h for piston meters 2 – 40 L/h for oval disk meters 1 mL/h upwards for metering pumps 2 – 50 g/h (balance range 0 – 5000 g) T min range 30◦ C (maximum range −200 to 600◦ C) T min range 50◦ C (maximum range 0 – 1000◦ C) >50 mm >50 mm >4 mbar 0 – 1.25 mbar

3.4.1.3. The Limits of Miniaturization Table 4 lists the minimum dimensions for which it is possible to scale-up a process.

Liquid flow

Table 4. Examples of tried and tested miniaturization limits [51] Example

Tried and tested limit

Columns

30 mm diameter packed columns 35 mm diameter columns with structured packing, dual-flow plate columns 50 mm diameter bubble-cap columns 1 mL/h injection pumps 10 mL/h piston pumps 100 L/h rotary pumps 50 mm belt width delivery 5 L/h 1.5 mm diameter, metal or Teflon

Pumps

Belt filter Centrifuges Pipelines

Columns can still be operated effectively down to 35 mm diameter when filled with an ordered packing. For plate columns having one bubble cap per plate, the limit is ca. 50 mm. At still smaller dimensions, wall effects are difficult to suppress. Experimental equipment for handling solids or dealing with extreme operating conditions is often not available. This often necessitates inhouse developments and cooperation with specialist firms, university institutes, and research companies. Expenditure on instrumentation is often very much lower for a miniplant than for the corresponding large-scale plant. For this to be the case, however, most of the miniplant components must be made of glass and therefore operate at low-pressure, thus making it much easier to

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gravimetric Temperature

resistance change

thermoelectric voltage Level

conductive capacitive hydrostatic Differential pressure strain gauge piezoresistive capacitive

There are four stages in the automation of a miniplant: Stage 1. At this stage, the electrical instrumentation of the miniplant consists of compact, self-contained individual components, e.g., balances, metering controls, metering pumps, thermostats, temperature controllers, pH sensors, individual controllers, control valves, pressure sensors, and plotters. Monitoring and operation is the responsibility of the shift personnel. Stage 2. At this stage the structure of the sensors, actuators, and individual components is the same as in Stage 1, but the individual components may be connected to a data processing

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computer via a data interface. Monitoring and operation is again the responsibility of the shift personnel. Stage 3. Here the system for logging the measured data is the same as in Stage 2, but the lower-ranking self-contained individual components are controlled and provided with set points by a higher-ranking automation software, so that shift personnel are able to concentrate on monitoring tasks. Stage 4. At this stage the miniplant is equipped with a compact process control system. While the actuator and sensor system is the same as at Stage 1, all the other functions such as controlling, measuring, regulating, calculating, processing data, optimizing, keeping records, alerting, logging, operating, and observing are performed by the process control system. The choice of automation stage is decided during the miniplant design phase on the basis of the specific plant requirements and boundary conditions as well as on the results of trials. The higher automation Stages 3 and 4 are only justified if a miniplant is to be operated over a long period of time. 3.4.1.4. The Limitations of the Miniplant Technique A disadvantage of the miniplant approach emerges if individual steps in the process are subject to an unduly large risk in extrapolating from the miniplant to the industrial-scale plant. For example, scale-up factors of ca. 104 are still not feasible for extraction and crystallization steps even today. This disadvantage can often be circumvented by isolating critical sections of the process and working on them in an intermediate stage (pilot stage). If enough of the unmodified product can then be collected from an integrated miniplant at an acceptable cost for it to be possible to operate, e.g., an extraction column or a crystallizing plant on a pilot scale with a representative feed for a sufficiently long time, the entire plant can again be extrapolated upwards with a calculable risk. This may obviate the need for constructing a cost-intensive and time-consuming pilot plant for the entire process [56].

3.4.2. Pilot Plant It may be necessary to construct a pilot plant if one of the following conditions applies: 1) The scaling-up risk is too large to proceed directly from the miniplant to the industrialscale plant. The reasons for this may be: a) The process involves several critical stages (e.g., handling of solids) which cannot be described by physical models b) a difficult or completely novel technology is being used 2) It is necessary to provide representative product quantities, e.g., for the market launch, and these cannot be produced by the miniplant in a reasonable time The operation of the pilot plant should clarify all the issues which have not been fully dealt with in the miniplant. These may include, for example [56], [57]: 1) Checking design calculations 2) Solving scale-up problems 3) Checking experimental results obtained with the miniplant 4) Measuring the true temperature profiles in the reactors and columns under adiabatic conditions 5) Gaining process know-how (dynamic behavior of the plant, start-up and shut-down procedures) 6) Producing representative sample quantities in fairly large amounts 7) Precisely assessing fairly small flows (e.g., residues) 8) Training the personnel who are to run the plant 9) Improving the estimate of the expected service life 10) Carrying out material tests under realistic conditions A pilot plant should be designed as a scaleddown version of the industrial-scale plant and not as a larger copy of the existing miniplant [58]. Nowadays it costs almost as much to design, construct, and operate a pilot plant as an industrial-scale plant. The decision to build a pilot plant results in considerable costs. As a rule of thumb, the initial investment is at least 10 % of the investment costs of the subsequent industrial-scale

Process Development plant. Moreover, if they use particularly toxic substances, pilot plants now often require approval, and this may considerably lengthen the time required to put them into operation. Once process development on a pilot scale has been successfully concluded, the pilot plant must be kept on stand-by until the industrialscale plant is running satisfactorily. Normally, when the larger plant is started up, the pilot plant is operated simultaneously so that any problems which occur in the former can be dealt with rapidly.

4. Process Evaluation After each development stage (see Section 3.1) the existing knowledge should be documented and the status of the process evaluated. This involves preparing study reports and is now standard in many enterprises. At this stage it is necessary to answer the question posed in Section 1 as to whether the process is “better” and “cheaper” than that of the competitors.

4.1. Preparation of Study Reports The study report documents knowledge about the status of the development of a process after specified time intervals or specified development stages to provide the basis for making a decision. The objective of the study report is to answer three important questions: 1) Can the production process be implemented in this way in principle? 2) What is the return on investment? 3) How big is the risk in economic and technological terms?

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1) A summary 2) A basic flow diagram 3) A process flow diagram and a description of the process 4) A waste-disposal flow diagram 5) An estimate of the capital expenditure 6) A calculation of the production costs 7) A technology evaluation 8) The level to which the experimental details have been worked out 4.1.1. The Summary The summary must start with a precise definition of the objective. This is followed by a short description of the process and of the most important process stages. The chemistry of the process should be represented by overall reaction equations, with the assumed conversion and selectivity being specified. Details of the result of the study, the production costs, the capital expenditure for a specified annual production output with an indication of the accuracy of the estimate and the planned location of the plant, if known, are other important points. The summary concludes with information on the state of knowledge and on the major risks. 4.1.2. Basic Flow Diagram The basic flow diagram provides a quick overview of the total process. It shows the input, output, and recycling streams and identifies the individual process stages (see Fig. 14). 4.1.3. Process Description and Flow Diagram

1) One at the beginning of process development 2) One in the course of working out the details of the process 3) One at the end of process development

All the individual steps of the process should be described as accurately as possible on the basis of all the information available at the time, reference being made to the process flow diagram (see Fig. 15). The latter should clearly show the process in detail and the composition of the flows. The following information should be included in the process flow diagram:

A study report should include the following items:

1) All equipment and machines (columns, vessels, heat exchangers, reactors, etc.)

At least three project study reports must be completed in the course of the development of a process:

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Table 6. Example of a material balance table

2) All product streams (with numbering) Beneath the process flow diagram the material balance should be presented in a clearly laid out table (see Table 6). When development is at an advanced stage or is complete, the process flow diagram should show the following additional information: 1) Identification of the energy carriers (type of steam, cooling water, compressed air, electric power, nitrogen, deionized water, heating gas, refrigerants, etc.) 2) Details on the construction materials, size and power of the equipment and machines 3) Typical operating conditions (e.g., pressure and temperature in columns and pipelines) 4) Important fittings 5) Purpose of instrumentation

4.1.4. Waste-Disposal Flow Diagram The waste-disposal flow diagram illustrates the waste streams for which there is no further economic use in the plant (see Section 1.3) and indicate their flow numbers from the process flow diagram. Precise information should be given on the toxicity, degradability, water danger, flash point, ignition temperature, MAK, odor threshold, etc. of the individual components, their mass flows, and the state of aggregation of the flow (viscosity, sediment content, etc.). This information makes it possible to specify the type of waste disposal required, for example: 1) Sewage treatment plant

2) Incineration in a power station or an incineration plant 3) Central hazardous waste incineration 4) Waste-disposal site

4.1.5. Estimation of Capital Expenditure 4.1.5.1. Introduction An important reason for preparing study reports is to determine the capital expenditure (investment) of a project [59], [60]. This is made up of the independent location ISBL (inside battery limits) process plant costs such as investment in the production plant and the associated control rooms, laboratories, employee facilities, tank farms, and loading and unloading stations, as well as the location-dependent OSBL (outside battery limits) costs of all the service facilities, such as investment in storage buildings, cooling towers, and waste-disposal facilities. Infrastructure costs are also to be included (see Fig. 17). The accuracy of the investment cost estimate depends critically on the level of maturity the process has reached [62]. 4.1.5.2. ISBL Investment Costs Of the many methods for determining the capital required to set up a plant, a method of estimation which derives the costs of a new plant from the known costs of existing plants with an accuracy of ± 30 % is now firmly established (the additional-cost calculation method). Such a method is only possible because the structure of chemical plants is always very similar, i.e.,

Process Development

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Figure 17. Subdivision of a project into ISBL, OSBL, and infrastructure activities [61]

they are made up of interlinked pieces of equipment of similar construction whose number does not vary widely. For this reason, the overall investment cost of the entire plant when installed exceeds the average value of the machines and equipment (= total machine and equipment costs divided by the number of machines and pieces of equipment) by a factor which depends on the capacity. The total machine and equipment costs are determined from the parts list, while the costs of individual units can be obtained from in-house databases or directly from the manufacturers. Example 4.1 [59]. The overall factor for the chemical plant construction projects handled by BASF in 1984 was on average 3.9; i.e., the direct plant costs of an average chemical plant is 3.9 times the machine and equipment costs. The overall factor is essentially determined by the following items: 1) Operating conditions (pressure, temperature) 2) Type of construction materials used (steel, stainless steel, special materials) 3) Size of machines and pieces of equipment According to Miller [63], the effect these items have on the overall factor can be deter-

mined from the average value of the machines and equipment (see Fig. 18). This results in the following simple relationship: directplantcosts = overallfactor× Σ (valuesof machinesanditemsof equipment) . (4.1)

Since direct plant costs are always estimated for some date in the future, it is necessary to determine the effect of increases in price. Use is therefore made of price indexes which are published for chemical plants in the individual countries (see Table 7). The price index i determined by extrapolation is divided by the index i0 at the date of the cost determination. The ratio i/i0 is then a measure of the increase in investment costs to be expected. The direct plant costs make no allowance for the cost of engineers (usually 10 – 20 %) or for contingencies (ca. 5 – 10 %). This method of estimation has the disadvantage that while the machine and equipment costs can be determined fairly precisely, the average value of these items is multiplied by a comparatively inexact overall factor. The accuracy can be considerably increased by breaking the overall factor down into individual factors.

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Figure 18. Overall factor for the direct plant costs as a function of the average value of the machines and pieces of equipment in 1984 [59]: a) Best-fit curve; b) Confidence limits for a statistical confidence interval of 95 %

Table 7. Price indexes i for chemical plants in Germany , p. 280[64] Year

i

1976 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

100 125.3 132.9 140.5 144.4 147.2 150.4 154.0 159.6 164.7 170.9

4.1.5.3. OSBL Investment Costs The OSBL investment depends on the location of the plant. It includes, for example, the cost of storage buildings, cooling towers, wastedisposal facilities, linking the OSBL facilities to the location in question (e.g., water and electricity supply), etc. Since the question of location is generally still unresolved at an early stage in development, it is not possible to specify the OSBL costs explicitly but they can be tentatively taken as 20 – 30 % of the ISBL costs. For small plants the percentage tends to lie near the upper limit,

while for larger plants it is likely to be close to the lower limit. If the location is known, the OSBL costs can be estimated very accurately. 4.1.5.4. Infrastructure Costs The infrastructure costs depend on the location of the plant. They are not, however, related exclusively to the project since the infrastructure, such as workshops, the central store, and the canal system, is also used by other production facilities. Obviously, this type of cost can only be specified if the precise position of the plant at a location is known. 4.1.6. Calculation of Production Costs The cost effectiveness of a new process depends on the production costs of a product (unit of currency divided by unit of quantity) (see Section 4.2). The following information is required to determine these costs: 1) Material balance

Process Development 2) Waste-disposal flow diagram 3) Utilities 4) Investment The production costs can be calculated by adding up the following items: 1) Feedstock costs (raw materials and auxiliaries) 2) Manufacturing costs, which can be subdivided into: a) Utility costs b) Waste-disposal costs c) Staff costs (wages and salaries) d) Maintainance e) Miscellaneous costs (concern overheads) 3) Capital-dependent costs (depreciation, interest, insurance taxes)

Figure 19. Temperature – enthalpy diagram

4.1.6.1. Feedstock Costs The raw material requirements are given by the material balance for the chosen capacity, but to determine the feedstock costs the prices of the raw materials are required. The change in raw material prices while the process is being developed can be estimated, but it is more difficult to allow for the fact that the erection of the plant will have an impact on the raw material market and consequently alter the price structure. Therefore, in some cases it is necessary to enter into negotiations with the raw material suppliers at an early stage. If the raw materials originate from another plant in the same concern, market prices are replaced by transfer prices [65], but

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this does not overcome the problem of the impact of the proposed plant on the market – it only shifts it. The significance of raw material costs in cost estimation varies widely. Thus, in processes involving a high level of material refinement (as is the case for some pharmaceuticals) or in those with high specific energy costs (e.g., chlorine production), precise determination is not as important as in the case of cheap, mass-produced products, such as petrochemicals, manufactured in large plants. Credits for byproducts can be set against the raw material costs from the outset. 4.1.6.2. Utilities For large-scale processes, the cost effectiveness depends appreciably on the cost of the utilities. The combination of utilities required in the plant therefore acquires considerable importance and consideration must be given to the ideal thermal coupling even at an early stage in the process development since it can have a considerable impact on the configuration of the process [66]. For example, whether the energy derived from condensing vapors can be utilized depends on the behavior of the materials. Clean and stable condensates usually present no problem in using the heat of condensation to raise the temperature of cold streams, but in the case of products which tend to form deposits (by decomposition, polymerization, etc.) optimum thermal utilization of this type of energy is not possible since the vapors must be quenched to prevent deposits and this results in loss of energy. This type of question should be resolved in the miniplant phase. A suitable aid for systematizing such considerations is Linhoff analysis [67], [68]. The Linhoff analysis determines the process configuration for which the energy supplied from or lost to the surroundings is a minimum. The problem which must be solved is how to couple the heat sinks and heat sources within the plant such that minimum energy consumption is achieved. The initial step in the calculation is to determine the so-called composite curves and pinch temperature, which is the temperature above which heat can be lost and below which heat can be absorbed (see Fig. 19). For a given interconnection system, the Linhoff analysis shows how far the heat utilization

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is from being ideal. However, there are often reasons for not choosing an ideal interconnection: 1) Start-up operations often require start-up heat exchangers, and this means higher investment costs 2) Energy utilization only becomes possible as a result of increasing the column pressure, and this means higher investment costs and may result in material problems (decomposition, side reactions, etc.) 3) Problems associated with the formation of deposits during heat transfer, which necessitates direct heat removal (quenching) Generally, as the plant capacity increases, reducing the operating utilities (i.e., optimizing the thermal combination) rather than increasing the investment will have a more beneficial effect on production costs. The energy analysis is used to find the amounts of energy required or released and to calculate the energy per unit quantity. The determination of energy costs requires information on the prices of the different types of energy, which depend on the quantities required and the location. Steam is the energy carrier most widely used for heat columns because it offers the following advantages: 1) It is inexpensive since it can often be generated directly in the synthesis section of the plant 2) There are no explosion hazards provided the hot pipe surfaces are properly insulated 3) The temperature range covered is large (see Table 8) Table 8. Enthalpy and temperature of some frequently used steam pressures [69] Gauge pressure, p, bar

Temperature, ◦ C

Heat of vaporization ∗, kWh/t

0 1.5 4 16 25 40 100

100 127 153 204 226 252 312

627 605 585 534 508 477 361

∗ With indirect heating, only the difference between the incoming stream and the discharged condensate can be utilized.

An initial estimate of the steam requirement of a column is given by: t/hsteam = 1/5 (reflux + headproduction) t/h (4.2)

This is based on the assumption that: 1) The thermal combination is optimum (e.g., the hot discharge from the bottom is used to heat the feed) 2) Normal organic liquids are involved (heats of evaporation ca. 110 kWh/t) Electrical energy is more than twice as expensive as steam, and because of the explosion hazard it is normally not suitable for heating columns. Its most important applications are in feeding material (pumps, impellers, compressors) and in carrying out operations such as size reduction, mixing, and separation. The pump delivery N required can be estimated using the following formula [70]: N /kW =

V˙ ·H ··g 3600·103 ·ηtot

(4.3)

where V˙ H  g η tot

= the useful feed flow in m3 /h, = delivery head in m, = density in kg/m3 , = acceleration due to gravity (10 m/s2 ), = overall efficiency.

The overall efficiency is made up of the mechanical and the electrical efficiencies: ηtot =ηmech ·ηelectr.

(4.4)

For rotary pumps, the overall efficiency is ca. 0.7 – 0.8 while for piston pumps it is 0.8 – 0.9. For small pumps (< 5 m3 /h), the overall efficiency may, however, be much lower. Another application for electrical energy in chemical plants is in auxiliary electrical heating systems (e.g., frost protection). Fuels are only used in chemical plants if streams must be heated to such high temperatures that steam is no longer economical (the limit is ca. 100 bar of steam, i.e., 310◦ C). However, the use of fuels in chemical plants always entails a risk from the ignition source.

Process Development Only the net calorific value of the fuel can be used since the steam formed can generally not be condensed. A few typical values of the standard enthalpy of combustion are given in Table 9. Table 9. Standard enthalpy of combustion at 25◦ C [71] Substance

kWh/t

Acetaldehyde Acetic acid Carbon Carbon monoxide Ethane Ethanol Formaldehyde Formic acid Hydrogen Methane Methanol

7 360 4 040 9 110 2 810 14 440 8 250 5 190 1 590 39 740 15 460 6 310

Cooling water can be drawn from a body of surface water or from the groundwater [72]. Compared with surface water, brackish water or seawater has the advantage of a more constant temperature and a lower infestation with algae, but requires special measures to be taken against corrosion. A recirculating procedure (Fig. 20) is preferred to a once-through system, with only sufficient cold river water being used for topping up as is required to maintain the specified temperature (the advantages of this are that economies in river water are possible in the cold season, the minimum temperature can be kept constant, and high flow rates in the circuit prevent corrosion and fouling).

901

The cooling water required to remove a given quantity of heat depends on the temperature of the water and its quality. If more detailed information is not available, it is sufficient as a first approximation to determine the amount of cooling water required by assuming its temperature rises by 20◦ C. For high product temperatures, an alternative to water cooling is to use air coolers. Although these require a higher investment than water-cooled condensers, they are often cheaper to operate and are less prone to fouling. However, unless there is a shortage of cooling water at a known location, it is generally sufficient to assume water cooling in an initial cost effectiveness calculation. Refrigeration Energy. If temperatures below approximately 25 – 30◦ C must be achieved in a process, energy must be expended on refrigeration. Refrigeration is generally achieved by compressing, cooling, and adiabatically expanding coolants in the plant itself; i.e., the refrigeration requirement can be expressed as a requirement for electrical energy and cooling water. To remove a given quantity of heat, about 20 – 50 % of that quantity, depending on the temperature level required, must additionally be expended in the form of electrical energy, thereby increasing the amount of heat which must be removed to about 120 – 150 %. Compressed Air. Small quantities of compressed air are drawn from pipelines. An example is the control air required by closed-loop control instruments (the rule of thumb is 1 m3 /h (STP) of control air per instrument). Larger amounts are produced by an air compressor in the plant and the requirement can then be expressed in electrical energy [73]. The power N expended on compressing gases is given by: Ntheo =P1 ·V˙ 1 ·

n · n−1



P2 P1

 n−1 n

−1

(4.5)

where

Figure 20. Example of a recirculation system for cooling a chemical plant with river water

n = cp /cv , P1 or P2 = initial or final pressure, respectively, cp or cv = heat capacity of the gas at constant pressure or constant volume, V˙ 1 = initial volumetric flow.

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For air compression (n = 1.4, overall efficiency η ≈ 0.7):

N/kW = 0.14·V˙ 1 / m3 /h · 6

(P2 /bar)0.286 −1

7 (4.6)

4.1.6.3. Waste-Disposal Costs The waste-disposal situation has become much more difficult for chemical plants, a trend which is continuing. The delivery of effluent and wastes to central sewage plants, waste incineration plants, or waste-disposal sites is restricted by official regulations, and as a result waste-disposal costs have risen dramatically in Germany in recent years (see Fig. 21).

Figure 21. Expenditure on environmental protection investment in the chemical industry in Germany [74, p. 303]

This poses a challenge to process development. The slogan “built-in rather than add-on environmental protection” expresses the current demand for environmental pollution to be reduced in the manufacturing process itself and for efforts to be made to put even the byproducts of a chemical process to good use if possible [75– 78]. Sewage Treatment Plant. Effluents discharged into a sewage treatment plant must be readily capable of biological degradation and must not be toxic to the bacteria in the plant. In addition, each substance in an effluent stream

should have a sufficiently high biological degradation rate, regardless of its total content in the stream. This requirement prevents individual undegradable substances being lost in an otherwise readily degradable mixture. To assess whether disposal of a stream via a sewage treatment plant is possible, the following information must be collected for each substances it contains: 1) 2) 3) 4)

BOD5 and COD value [79] Zahn – Wellens test [80] Bacteria and fish toxicity [81–84] Water hazard class [85]

Incineration Plants [86, pp. 213 – 217]. Waste streams which consist only of compounds containing carbon, hydrogen, and oxygen as aqueous solutions with a concentration >10 % are ideally suitable for disposal in an incineration plant or a power station (if the content is >10 %, the heat of incineration is roughly equal to the heat of evaporation of the water). Disposing of waste by incineration is more difficult if other elements such as nitrogen, chlorine, sulfur, and metals are present. Such wastes can only be disposed of economically in a central waste incineration plant equipped with suitable absorption systems for NOx , HCl, SO2 , etc. [87], [88]. Waste-Disposal Site. The restrictions imposed on wastes which can be dumped are continuously increasing, and new dumping space is no longer available. Consequently, this wastedisposal option is virtually no longer available for new processes. 4.1.6.4. Staff Costs Operating staff costs in chemical plants can vary widely. For large automated plants, they may be less than 3 % but for small batch processes they may amount to 25 % of the manufacturing costs. Often they are 5 – 10 % of the manufacturing costs. In new plants, whether batch-operated or of the single-train type, the trend is towards the fullest possible automation. The introduction of process control engineering has made it possible to operate and monitor virtually every aspect of a plant using pictorial displays [89, pp. 158 –162].

Process Development Almost regardless of the size of the plant, a wellplanned and automated single-train installation requires the following minimum staff: 1) 2) 3) 4)

One shift foreman or his deputy Two shift employees for the control room One shift employee for outside operations One shift employee as a reserve

This minimum staff of five should be multiplied by a factor, specific to the company, which reflects the type of shift model, the sickness statistics, and vacations to give the total number of employees required. Special operating requirements such as frequent changes in production schedule, periodic start-up and shut-down of the plant, loading and unloading of tankers may make additional staff necessary. 4.1.6.5. Maintenance Costs Annual repair and maintenance costs generally amount to 3 – 6 % of the invested capital [90]. About half of this amount is material costs and the rest wages. Their long-term increase can therefore be assumed to be between that of the investment costs (see Table 7) and that of the wage rates (see Table 10). Experience shows that high repair costs are incurred shortly after a plant is put into operation, in eliminating start-up difficulties, and after a prolonged operation because of the need to carry out major repairs. Depending on how new a process is, annual costs of 5 – 10 % of the estimated total investment should be allowed for in an initial estimate.

Table 10. Gross hourly earnings of an average worker in the chemical industry in Germany , p. 286[91] Year

Earnings, ¤/h

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990

7.22 7.68 8.10 8.47 8.72 8.99 9.30 9.69 10.06 10.42 11.11

903

4.1.6.6. Overheads The overheads of the company consist of the total costs of the central facilities (research, personnel, energy-supply department), works management, staff facilities, internal plant traffic, etc. These costs are very dependent on the structure of the company. As a rule they are ca. 5 – 20 % of the combined energy, staff, and depreciation costs. 4.1.6.7. Capital-Dependent Costs (Depreciation) The plant capital can be depreciated by various methods [92]. For a cost efficiency calculation it is generally sufficient to assume linear depreciation over a ten-year utilization period. The annual depreciation is then 10 % of the plant capital: Depreciation =

I 10·N

(4.7)

where I N

= total investment in currency units = nominal capacity in kg/a

4.1.7. Technology Evaluation Technology evaluation should provide information on the technical risk associated with a process. This risk is equivalent to the economic damage which would occur in the extreme case if the process did not work at all or could only be rendered operational by carrying out subsequent improvements. Technology evaluation should also reveal whether the individual unit operations or the equipment and machines used are of technically established types and whether particular risks are incurred by breaking new ground (size, construction, material, scale-up factor, etc.). It is also necessary to include a statement as to whether the process is based on a technology with which the company is familiar (e.g., high pressure, gas-phase oxidation, phosgenation technology, etc.) or whether fundamental innovations are involved. If limits are exceeded (e.g., if the dimensions of the largest extraction

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column previously operated by the company are considerably exceeded in the new project), the limits which have been tested should be specified. This information should be elaborated in detail as early as possible in order to recognize, consider, and evaluate potential risks and causes of failure. During operation of the miniplant, close attention should be paid to deposits, frequently occurring malfunctions, etc., since such problems often subsequently lead to insuperable difficulties in the industrial-scale plant (e.g., foaming, emulsion formation, aerosol formation, etc.). The technology evaluation gives a list of weak points arranged in order of technical risk (e.g., catalyst service life, corrosion, etc.). This list can then be used to take steps which reduce such risks. In principle, two steps are possible: 1) Expenditure on R & D can be increased at the weak point (e.g., the unit concerned may be checked on a pilot scale or a well-established solution may be sought). This option should always be chosen if, in an extreme case, failure of the unit concerned would be accompanied by a total investment loss. 2) Failure scenarios can be developed, i.e., what can be done later if problems occur or if a unit fails completely?

train plant increases as the number of process steps and the probability of failure of individual steps increase: a single-train plant can only function if each step is operational. If partial streams are fed back to earlier stages, the coupling between them is particularly strong. This is also true of the energy crosslinks within a process (see Section 4.1.6.2). The increase in risk as the number of process steps increases is independent of whether the process is continuous or batch. In practice, various options are now adopted in resolving this dilemma (see Figs. 22 and 23): 1) Decoupling by inserting tanks for intermediates (the greater the buffer tank, the greater the independence). For larger quantities of product or gaseous intermediates, this approach is virtually impracticable. 2) Decoupling by installing parallel arrangements of equipment which is particularly susceptible. For example, filters and fittings that are susceptible to failure should be provided with a by-pass if safety considerations permit (e.g., heat exchangers in the main flow in which severe fouling is to be expected).

Normally, these two options have to be weighed against each other, i.e., it is necessary to clarify whether it is more economic to minimize the risk by increasing R & D expenditure (e.g., by building a pilot plant), or to reduce or eliminate it at a later date by technical measures (e.g., additional instrumentation or backup units). Some general aspects, which may be useful in assessing the technical reliability, are summarized below. From the estimate of the risk and the measures it suggests, the economic effect can be quantified very exactly (surcharges on the development costs or on the investment and repair costs). This transforms technical reliability into cost effectiveness considerations and results in the possibility of expressing the risk of a development project in monetary units. Improving Technical Reliability Decoupling Process Steps. The susceptibility to failure of a process carried out in a single-

Figure 22. Example of decoupling a single-train plant: a) Intermediate tank to smooth out variations in feed flow; b) Recycling to smooth out variations in feed flow

Process Development

Figure 23. Decoupling of a single-train plant by including redundant equipment A) Filter station with single by-pass; B) Pumping station with A and B pump; C) Preheater with flushing facilitya) Auxiliary preheater; b) Double block and bleed; c) Preheater

Each measure which is aimed at increasing the uptime of a single-train plant results in an increase in investment costs. In practice it is often the case that such steps are only taken after the plant is put into operation for the first time. Choice of Process Steps which can be Safely Scaled-Up. An example of the dependence of technical reliability on the operational method employed is mass transfer. While simple standard equipment can be used for rectification, a wide variety of equipment in which phases are mixed in several stages and then separated has been developed for extractions. Such equipment generally requires more expenditure for safe operation than distillation columns. The complexity of an operation increases as the number of substances and phases it involves increases and as the operating conditions become more extreme. Thus, the presence of two liquid phases requires special precautions to be taken. The same applies if particularly high or low temperatures or pressures are involved.

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If solids are processed, the process must be assessed with particular care. The scaling-up of equipment for separating solids should be subjected to particularly careful scrutiny. Even the bunkering and transportation of solids present difficulties which are often specific to the materials. If new technology is employed, the operational safety can be expected to be lower than that of well-established processes. It can be increased if similar types of equipment are used in the experimental plant as in the industrial-scale plant. This allows scaling-up to be possible but in the case of reaction equipment, it is not always the case. Materials to be Processed. If toxic, inflammable, caustic, or otherwise hazardous substances are to be handled, a higher degree of operational safety is required from the outset than in the case of harmless substances. Corrosion is an ever-present risk and necessitates the use of special materials. In making an initial selection, use may be made of published tables (see → Corrosion), but in most cases it is necessary to carry out corrosion tests. Trace impurities can lead to considerable difficulties if they accumulate at particular points in the equipment. The most reliable information in this connection is provided by operating an integrated miniplant. If the flow rate of mixed phases is high, erosion of the wall material can be expected. The extent of erosion can often be lowered by the choice of material and by effective design. Increases in wall thickness are generally of no help because the erosion usually takes the form of pits. Miniplant experiments are of no assistance in this connection since they are usually not representative in hydrodynamic terms. In the case of moving solids, not only the walls of the equipment but also the solids themselves are subject to abrasion. If a process is intended to give a particular size, the abrasion loss may have a considerable effect and, moreover, an increase in this abrasion is always expected on scaling up a process. Solids often deposit inside equipment and result in fouling. Valuable information about this is provided by miniplant results. In heat exchangers, allowance is made for fouling by increasing the heat transfer resistance by empirically determined amounts (fouling factors) (see Fig. 24).

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Figure 24. Effect of the fouling factor Fi on the heat transfer coefficient k F 1 = 7×10−4 m2 K W−1 (typical value for river water); k 0 = 1000 W m−2 K; Q˙ = k A ∆T ; 1/k = 1/k 0 + Fi ; Fi = Si /λs where Si = thickness of fouling layer; λs = thermal conductivity of fouling layer; Q˙ = heat flux; A = surface area of heat exchanger

If the expected operating time is inadequate, a back-up heat exchanger must be installed in parallel. In larger plants, heat exchangers are usually subdivided into several units for design reasons. Thus, if suitable fittings (double block and bleed) are provided to ensure that each heat exchanger can be cleaned while the plant is operating, the effect on the availability of the entire installation is small (see Fig. 23). However, this is only true provided the cleaning frequency is technically feasible. The design of heat exchangers and the cleaning operations should be carefully matched to the type of fouling, and extensive automation is always worthwhile in the case of large pieces of equipment. If fouling is expected in columns, larger columns should be designed for whole-body access to each plate, while in small columns a manual-access hole should be provided for each plate to allow cleaning. Furthermore, designs should be used whose operation is relatively insensitive to fouling and which can readily be cleaned. The choice of reaction equipment can be considerably influenced by fouling. As fouling increases, the pressure drop in fixed-bed reactors increases, whereas fluidized-bed reactors are virtually unaffected. Moreover, fixedbed reactors must be opened and emptied in or-

der to regenerate the contaminated catalyst, but fluidized-bed catalysts can usually be regenerated in the reactor. Often it is also possible to remove, regenerate, and replace the catalyst in a sidestream while the plant is operating. Fouling that arises both from the material being processed and from the material of which the equipment is made is more troublesome. For example, in many polymerization reactors, wall deposits can be suppressed by suitable choice of the wall material and processing method. In such cases, steps should be taken to ensure that the same wall material is used in the trial plant and in the industrial-scale plant. If wall erosion or deposit growth occurs at a constant rate, the difficulties can be expected to decrease in bulk-storage equipment and pipelines with increasing scale-up but to remain unchanged in columns and heat exchangers (equipment with large surface areas). Machines. Since machines have moving parts, they are always a potential source of malfunction and the reliability of a process decreases as the proportion of machines it employs increases. In general, machines having reciprocating parts are regarded as less reliable than those employing only rotating parts. For this reason it is particularly important to provide a reasonable level of back-up. Because of their relatively low cost, duplicate full-capacity pumps are usually installed, but occasionally two pumps having different functions can be provided with a common backup. In the case of reciprocating machines, in addition to a complete backup (2×100 % capacity), 2×66 % capacity, or in exceptional cases 3×50 % capacity, may also be considered. Turbo machines are usually installed at 1×100 % capacity, but provision is made for rapid replaceability of parts which are subject to wear. Alternatively, two machines with 50 % capacity are often installed. Instrumentation. The narrower the limits in which quantity flows and operating conditions have to be kept for reliable operation, the higher will be the requirements imposed on the instrumentation. To increase plant availability, provision is made for n measurements instead of one measurement in the case of critical circuits and control loops. The complete unit will operate as specified provided m of the n measurements

Process Development (m 100 DIN A2 sheets. The P & I diagram contains all essential information developed by the individual disciplines: process engineering; equipment, machinery, piping engineering; engineering for electricals and control systems. It must also include data provided by the manufacturers of equipment and machinery (e.g., the control system of a compressor). This information becomes available over a prolonged span of time. The P & I diagram is therefore revised several times in the engineering stage (Fig. 25 is a

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Figure 24. General arrangement drawing

portion of a P & I diagram). An attempt should be made to review the P & I diagram with all responsible persons (and, if possible, representatives of the client). After this review, the P & I diagram is “frozen” and only essential changes should be subsequently allowed (e.g., changes concerned with plant safety). Minor changes are often made during commissioning; these should be incorporated into the P & I diagram so that

the document reflects the “as built” state of the plant. 5.4.3. Apparatus and Machinery All important process engineering data for apparatus (e.g., heat exchangers, reactors, towers, vessels, tanks) and machinery (e.g., pumps, blowers, compressors, turbines) are specified

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Figure 25. Section of a piping and instrumentation (P & I) diagram (e.g., LIC = level control; TE = local temperature indicator; TI = temperature indicator in the control room; TIC = temperature control)

Previous Page Chemical Plant Design and Construction during basic engineering. Data sheets prepared in this stage contain essential information on overall dimensions, pressures, temperatures, quantities, materials of construction, etc., of each piece of apparatus and machinery. In detail engineering the apparatus and machinery engineers complete this information. The result of this work is a set of specifications in the form of drawings and descriptions, which enable qualified manufacturers to submit bids for apparatus and machinery. The equipment engineers prepare so- called guide drawings, which are scaled drawings indicating all dimensions dictated by process engineering (e.g., number and diameter of trays in a tower, spacing of trays, and tower height). All dimensions of importance for shipping are also shown. Relevant legal provisions must be taken into account. Nozzle tabulation and other important design data are attached to the guide drawing (see Fig. 26). The wall thickness is estimated so that the weight of the apparatus can be calculated The number and dimensions of the nozzles, and frequently their elevations, are stated. The horizontal orientation of the nozzles is determined later when the exact position can be ascertained from the piping design. The guide drawings and supplementary information form the technical portion of the bid invitation, which is sent to selected manufacturers. The information in the bid invitation must be presented in such a way that the bidders can submit comparable bids (see also Section 5.5.1). The design office of the manufacturer prepares detailed workshop drawings and calculates the final wall thicknesses. The workshop drawings are checked by the engineering firm who also fixes the position of the nozzles and informs the manufacturer of any changes. Once the drawings have been approved, production can begin. The manufacturer is responsible for compliance with legal provisions. Specialist engineers periodically inspect complicated equipment, even during its production. Such inspections are independent of those per-formed by a third party (e.g., one of the Ger¨ organizations or Lloyds) when manman TUV dated by law (e.g., for pressure vessels). The equipment is cleared for shipping only after final acceptance by the same specialist engineers. The specification and procurement of machinery are similar to that for apparatus. In con-

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trast to apparatus, which is usually custom-built and thus individually designed and drawn, an attempt is made to use off-the-shelf machinery. There are two reasons for doing so: to minimize engineering costs and to hold down the purchase price. The machinery engineers prepare specifications for every machine to be procured. The specifications are based on the data sheets compiled by the process engineers, which contain all information relevant to the process (operating conditions, materials of construction). As an example, Figure 27 shows a data sheet for centrifugal pumps taken from the bid specification. The machinery manufacturer supplements the data sheet with further information on the model he has selected for the bid. An important element of order handling is the time schedule according to which the machinery manufacturer is to submit information about the machine (e.g., dimensions, weight, vibratory behavior). It is important for the engineering firm to obtain this information as early as possible to avoid delays in the design of footings and foundations, buildings, and piping. Such information should also be finalized as soon as possible to avoid duplication of work. For noise-abatement design see Section 3.3.2.2. Spare parts required for plant startup and the first two years of on-stream operation are commonly ordered at the same time as the machinery. The manufacturer recommends the type and quantity of spares. The subsequent operator of the plant makes the final decision once the specialist engineer has checked the bid. 5.4.4. Piping The objective of piping design is to prepare all drawings and specifications needed for procurement and installation of the piping components. The engineering of piping systems is closely linked with the engineering of all other disciplines. In the initial phase of engineering, information is incomplete and often preliminary. The data become more complete and exact as work progresses. Because the need for on-time availability of piping data requires early information from the other engineering disciplines, a stepby-step procedure is employed. Often the first

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Figure 26. Guide drawing for a vessel

Chemical Plant Design and Construction

Figure 27. Data sheet for a centrifugal pump

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steps are based on assumptions, so that frequent corrections are required later. Piping accounts for a relatively high proportion of chemical plant costs and piping engineering may represent as much as 20 – 40 % of total engineering. Refineries and petrochemical plants lie at the upper end of this range. Piping engineering can begin once the following information (at least in preliminary form) is available: 1) Standards and codes (of the investor or the engineering firm) 2) P & I diagram 3) Layout model 4) Plot plan 5) Guide drawings for apparatus 6) System drawings for machinery 7) Preliminary civil and structural steel drawings 8) Data on electricals and control systems Piping Specification. For the sake of efficiency in engineering, procurement, and piping installation, and in view of the wide variety of piping components and design, a piping specification is first prepared. This document is based on the standards and codes applicable to the plant and relevant engineering regulations. The piping specification also contains special piping design guidelines for the project.

Figure 28. Piping classes (pressure – temperature diagram)

An important element in the piping specification is the piping classification which minimizes the different types of piping components required. All piping components (pipe, fittings, flanges, bolts, seals, etc.) are classified on the

basis of flowing media, pressures, and temperatures that occur in the plant. A piping class comprises the expected dimensions of components and their materials of construction for a given set of media, pressures, and temperatures. The classification is based on pressure – temperature diagrams from DIN 2401 or ANSI B 16.5 (see Fig. 28). Each of the areas represents one piping class. All components within one such area are uniformly sized. Fittings and flanges are standardized according to pressure level. Wall thicknesses are calculated from the pressure and temperature. Once the piping classes have been worked out, the figures are stored in a data base and can be retrieved as needed. The use of such a data base greatly reduces the amount of work to be done in specifying the piping system for a plant. Piping List. All pipe runs are identified by a code number and a piping class, and are compiled in a piping list. The associated data are stored in a data file. The piping list is prepared at the same time as the P & I diagram. Isometric Piping Drawings and Piping Model. Drawings that show both the geometry of the run and its location in the plant are needed for the prefabrication and installation of piping. It used to be common to plot every pipe run and every fitting in a piping diagram with plan, elevation, and section views. This method has been largely replaced by a diagram of a single pipe run and its components, along with measurement and control devices and piping supports. The initial piping studies and the final isometric drawing of a pipe run are done in parallel with the construction of a piping model. A bill of materials containing all piping components is drawn up for every pipe run. Figure 29 gives an example of such an isometric drawing. Computer-aided design (CAD) techniques are used increasingly in preparing isometrics. The isometric drawings are supplemented by plans for pipe bridges and underground pipe runs. The model shop uses the plot plan to make a basic model at 1 : 33 1/3 or 1: 25 scale that includes all apparatus, buildings, frameworks for equipment, stairs, ladders, platforms, and pipe bridges (Fig. 30). The piping model forms

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Figure 29. Isometric piping drawing Symbols and designations mark piping components (e. g., valves, flanges, reducers), dimensions, position of the pipe run

the center for coordination of all detail designs [127]. Installation work carried out on the model in parallel with piping includes air conditioning and cable ducts, control panels, hoists, and cranes. A piping model has the following advantages: 1) Piping routes are easy to check for collisions with other equipment 2) The plant operator can check ease of operation and maintenance 3) The model can serve as a training facility for operating personnel and as a form of instruction during installation The completed model should be thoroughly assessed by all discipline engineers, the later maintenance engineer, and the operator. Any necessary changes made at this stage are much less costly than if they are made later on the construction site.

Piping Calculations. Piping calculations cover strength calculations for individual elements (wall thicknesses, flanged joints) and stress analysis of the piping system. Wall thicknesses are calculated from the pressure and temperature ratings of the piping class. The safe functioning of a piping system depends on correct sizing and proper layout. Special attention should be given to the elasticity of the piping and the use of supports and anchors. Temperature changes give rise to stresses in piping systems, which in turn generate forces and moments at connection and support points. Computer-aided elasticity calculations are performed to make certain that the strains resulting from stresses in a given piping layout are within the elastic range. If the stresses are too high, a different configuration must be selected or compensators must be inserted. For small-diameter pipes and moderate temperatures, these expen-

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Figure 30. Piping model (fluid catalytic cracker complex, courtesy of Lurgi AG)

sive calculations are often superfluous, since experienced piping engineers lay out such piping with adequate elasticity. Material Takeoff and Procurement. A first rough material takeoff can be performed when the P & I diagram has reached a certain level of completeness and the plot plan has been drawn up so that fittings can be counted and the lengths of the main pipe runs can be estimated. The objective of this preliminary takeoff step is to invite bids and place orders for piping components with long delivery times. Sufficient material can thus be made available on site when piping installation begins. As piping design advances, isometrics and piping plans with bills of material are generated. Another (not yet final) material takeoff is then prepared and has a higher degree of accuracy than the preliminary takeoff. Further orders are then placed. At this stage, bid invitations can be sent for the installation work, and the piping installation contractor can be selected. The final

material takeoff is worked out after all isometrics and piping plans are complete. The preparation of the material takeoffs is computer-aided. Sorting and condensing programs calculate the quantities that have to be ordered [128]. An integrated materials management system allows the print out of lists and calculations for every step (e.g., bid invitation, ordering, expediting, and material handling on site). The efficient use of such a system requires a consistent high-order data structure as well as the unambiguous definition of piping components in terms of piping classes. Insulation and Coatings. The thickness of insulation needed on equipment and piping must be established at a fairly early stage since this value may influence other parameters such as the length of nozzles and the width of pipe bridges. Insulation thicknesses are entered on the P & I diagram and in the piping list. Thermal insulation for pipes carrying hot media generally consists of mineral wool enclosed in galvanized or aluminum jackets. Polyurethane foam enclosed

Chemical Plant Design and Construction in sheet metal is widely used for pipes carrying cold media to prevent icing and cold bridges. Uninsulated surfaces of tanks, piping, and steel structures must be painted to protect them against corrosion. As a rule, machinery is delivered with the specified prime and topcoats. Rust must be removed from the surface before application of the prime coat. In many cases, a second prime coat is needed before the first and second topcoats are applied. Underground pipe is either coated with asphalt or jacketed in plastic. 5.4.5. Control Systems [129–133] Automation of chemical plants is increasing. Rapid progress in microprocessor technology has led to the development of distributed control systems (DCS) that can meet the increasingly stringent requirements of modern process operation. The objectives are to improve the availability of operating process plants, enhance their reliability, and optimize their operation. The distributed functions facilitate the engineering, operation, and maintenance when broken down into levels (Fig. 31). Centralization of process control systems means that plants are chiefly fitted with electronic devices, since this kind of equipment with its reliable signals is suitable, even for explosion hazard areas. Pneumatic instrumentation is limited to pneumatically actuated controllers and low-order local control loops. With the help of process monitoring and control systems, advanced control strategies can be built up in modular form. The modules perform both computing and dynamic functions, so that a variety of signal processing algorithms can be selected for optimal control strategy. The control of material streams, plant optimization, and balancing is implemented by process control computers at a level above the process control systems. These computers may have an on-line function or a data management function. Software vendors offer appropriate modular software. Such programs are linked together to permit control of products by centralized programs. Optimization programs did not gain wide use until stable on-line analytical instruments with short response times were developed. The addi-

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tional investment is amortized after as little as one to two years. Complicated analytical systems, however, have higher costs for maintenance, which is performed by specially trained personnel. Analytical instruments are built in prefabricated enclosures and tested at the vendor’s workshop. Installation simply involves connecting the process loop lines, utilities, and data cables. Plant Safety and Availability. Regulations on plant safety and environmental protection have rapidly become more stringent, influencing the choice of automation hardware and system structure. Safety control requires the use of redundant systems approved by the regulatory authorities. Interfaces connect these systems to the process control system, special attention should be paid to transmission time between the different systems. Process Monitoring and Control System. In petroleum refineries and petrochemical plants, it is often necessary to operate 3000 – 4000 loops and give the plant operator access to these in a meaningful order. Sensor signals relating to the process and control functions, along with signals to motor control centers (MCC) and valve actuators, are handled in the processing stations, which perform configured tasks such as signal conditioning, control, and signal processing. The processing stations are assembled from modules and tailored to individual functions. Process information is transmitted to the operating and monitoring system via serial busses. The chemical plant is controlled with extended software functions for graphical display, along with process graphics overlays. The hierarchical information structure of data representation leads the operator to the proper level in the information structure. The alarm functions notify the operator directly of the inititating measuring point in the process. The automation structure is governed by the following important criteria: 1) Size of process plant 2) Continuous or batch process 3) Behavior of process over time – Steady and stable – Product output or quality strongly affected by load variations and/or perturbations

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Figure 31. Levels of distributed control systems (DCS)

4) Complexity of process – Simple control strategies – Complex interdependences – Frequently changing recipes 5) Local or central process control 6) Startup and shutdown strategies 7) Upgradability 8) Amenability to changes 9) Capability of linking with automation systems at other plants, other monitoring/control levels, or information systems 10) Type of reporting 11) Safety, availability 12) Standardization 13) Environmental restrictions 14) Maker and service capabilities 15) Personnel considerations (crew size, qualifications) 16) Economic and management aspects Only close collaboration between process engineering and process control specialists belonging to the staff of the engineering contractor and the owner can ensure proper decisions. In the central control room, the operating and monitoring devices collect all needed process information. All signals from the plant (flow, pressure, temperature, etc.) are dynamically displayed on a screen. Two to three screens per workstation, with the necessary operating features (touch screen, light pen, keyboard), have proved optimal with regard to cost and volume of information. The process control system includes reporting features that maintain a contin-

uously updated record of alarm and condition reports, series of measurements, balances, and operator actions. Trend displays of the process variables are replacing conventional chart recorders. The latter are needed only as required by the regulatory authorities (e.g., emission measurements) or to record guaranteed values (e.g., temperatures in a catalyst bed). The workstations and peripherals should be arranged so that the operator sees the whole working field as a closed area and operators can exchange information in order to coordinate their actions in case of abnormal occurrences (Fig. 32). Engineering of Control Systems. It is useful to break the engineering of modern measurement and control equipment into field devices and central control rooms (process control systems). Different levels of detailed knowledge are needed for these two areas. The high rate of innovation in process monitoring and control systems demands continuous retraining of the design engineers. The main activities involved in engineering follow and make use of computer-aided engineering (CAE) systems with different software requirements: 1) Preparation of basic documents such as coding system, power supply and distribution, materials of construction, P & I diagrams, control strategy, functional diagrams for process and device control systems, instrument

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Figure 32. Control room, courtesy of Lurgi AG

2)

3)

4) 5)

list, quantity structure list, requirements for process measurement and control system Preparation of instrument specifications such as data sheets for all instruments to be installed, with information on designation of measurement location, process data, manufacturer’s data, materials of construction Engineering of process measurement and control system: configuration documentation, process graphics, loop sheets, description of process measurement and control system Planning of central facilities such as power distribution, instrument cabinets, monitoring rooms Preparation of installation documents such as cable run plans, cable lists, hookups, and list of installation materials

The pareparation of as-built documents once the plant has been commissioned and the maintenance of important documents thereafter fa-

cilitate plant maintenance and the remedying of malfunctions. Information on reliability, maintenance cost, spare parts management, experience, availability over an extended time, and service are important factors in the selection of instruments and systems. Ease of access to instruments and systems greatly reduces the number of plant malfunctions and thus increases profits. 5.4.6. Electrical Design The objective of electrical design is to supply electric power reliably and economically to all consumers. The designer does not create isolated solutions component by component but must find the optimal solution for the system as a whole. Design begins where high-voltage power enters the plant, and may include mediumand low-voltage switchgear, transformers, generators, emergency backup systems, lighting,

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grounding, and communications. It covers three areas: 1) Planning of power generation and distribution 2) Planning of electric utilities 3) Installation planning Planning of Power Generation and Distribution. The operator wants to optimize use of the electric power and to insure that the system can handle short- circuit loads, that the power grid can handle short- circuit loads, that the power grid offers the necessary reliability, that investment costs are minimized, and that operating costs are held down. The following points therefore have to be examined carefully in the design of the distribution grid: 1) 2) 3) 4)

Selection of voltage levels Determination of transformer ratings Location of load centers Location of distribution stations (with allowance for danger zones) 5) Reliability of supply from electric utilities and/or in-plant generating capacity 6) Materials of construction The elements of the electrical grid must be selected and sized. The use of powerful computer programs is indispensable (to keep the consumer list, perform short- circuit/load – flow calculations, determine the run up behavior of motors, and carry out the sizing of cables). The results of these steps are entered in the block diagram. The list of electrical consumers gives a detailed description of energy consumption and provides the basis for the energy balance, in which the installed power and net power demand are calculated. Planning of Electric Utilities. Once the results of the above activities are available, documents needed for specifying the electrical utilities are prepared. These include technical specifications, engineering data sheets, circuit diagrams, cable lists, terminal diagrams, and mimic diagrams. These technical procurement documents form the basis for bid comparisons and order specifications. The plans are prepared by CAD methods.

Installation Planning. Electrical installation accounts for a significant fraction of investment costs, so detailed planning is a prerequisite for economic execution. At this stage access to a plant model is extremely helpful. The installation plans are drawn up by CAD methods on the basis of layout plans and comprise: 1) 2) 3) 4) 5) 6) 7)

Position plan for electrical consumers Cable run plan Cable run sections Grounding position plan Lighting layout plan Hazardous area classification Layout plan for communication systems

On the basis of the designs and documents prepared, the quantities needed are determined and bid invitations are prepared for bulk materials and installation. The ordered equipment is inspected at the workshops to check for compliance with the specifications. The cost of electrical equipment and materials makes up 6 – 10 % of the total chemical plant costs. The high end of the range applies to grass-roots plants where a new infrastructure must be created. For further information, see [134], [135].

5.5. Procurement Procurement activities consist of three main tasks: 1) The purchase of plant components and services 2) Expediting during the fabrication of plant components (i.e., supervising the fabrication sequence of plant components according to an agreed time schedule) 3) The shipping of plant components to the construction site In U.S. oriented regions, workshop inspection of plant components also comes under this heading. Depending on the terms of the contract, the engineering firm procures plant equipment in its own name or in the name and on behalf of the investor. The procurement department of an engineering firm is acquainted with the world market

Chemical Plant Design and Construction and carefully observes trends. The procurement and engineering activities are closely linked together. The purchasing and shipping agents as well as the expediters are also members of the project team, and thus subordinate to the project manager. Procurement man-hours make up 8 – 12 % of total engineering hours spent on project execution. 5.5.1. Purchase of Equipment and Services The main steps in purchasing are: 1) Preparation of a vendor list, possibly in collaboration with the investor 2) Preparation and dispatch of bid invitations based on requisitions written by the specialist engineers 3) Handling queries from bidders, checking ontime receipt of bids, checking received bids for completeness 4) Checking bids for comparability by the engineer 5) Preparation of bid comparison and order recommendation by the purchasing department 6) Checking the order recommendation (by cost engineers, specialist engineers, and possibly the project manager) 7) Negotiations with the assistance of the specialist engineer and possibly the project manager 8) Preparation of order documents 9) Checking order confirmation 10) Approval of bill payments after confirmation from the specialist engineer 11) Compilation of lists of bid invitations, bid comparisons, and orders Bid Invitations and Comparisons. The engineer responsible for a particular discipline prepares specifications for the components he needs in the form of data sheets, guide drawings, descriptions, and information on when each item will be needed. These documents are sent to the purchasing department, which adds relevant business conditions and sends the packages to selected bidders. The potential bidders are chosen by the responsible engineers and purchasing agents when the vendor list is drawn up. If goods are purchased in the name and on behalf of the investor, the investor commonly has a say in the process. Good definition of equipment items

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covered by the bid invitations is important so that bids from competing vendors are comparable as to content and thus price. The above procedure is also followed in the procurement of services. The technical documentation (e.g., for installation of piping) is prepared by the specialist engineers responsible for piping, in close cooperation with the erection planning department. This documentation includes the piping material takeoff, plant layout, specifications for piping installation, information on material storage capabilities, and schedules. The received bids for plant components and services are examined by the responsible engineers to insure that they are comparable and conform to the requirements stated in the bid invitations. Bidders may often be asked to correct their scope of delivery and services. The bid price may be adjusted as a result. Technical bid comparison is followed by commercial bid comparison and an order recommendation. Often, the final decision is only made after verbal negotiations with two or three bidders. Decision criteria include not only lowest price but also the technical reliability of the plant component covered by the bid, the experience and reliability of the manufacturer, and the vendor’s solvency and workshop capacity utilization. Orders. Especially in the case of complicated plant components and large service packages, the order often goes out in abbreviated form by Telex, to make the best possible use of the agreed-on lead time. The detailed order document follows immediately. The order is prepared by computer-aided techniques so that, for example, the value of the order is transferred directly into the computer-aided cost control system. If the plant components are clearly defined, the specification used for the bid invitation can also be employed for ordering. For larger and more complicated items, the scope of delivery or service must be unambiguously described with a statement of exclusions. Imprecision at this point can result in unpleasant confrontations with the vendor. Receipt of the order is confirmed in writing by the vendor.

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5.5.2. Expediting The engineering firm has an obligation to the investor to erect the plant within a certain time. The time schedule agreed between the engineering firm and the vendor must therefore also be complied with. The engineering firm establish a production schedule monitoring system for this purpose. Supervision begins as soon as stocks of material are ordered, it covers the manufacturer’s design work (workshop drawings) and other production steps. Expediters from the engineering firm carry out their checks by telephone calls and regular visits to material vendors and the manufacturers of plant equipment. They report regularly to the expediting engineer and the project manager. When delays are expected, corrective measures must be instituted in collaboration with the manufacturer (e.g., changing material vendors, weekend work, night shifts).

5.5.3. Shipping The packaging and shipping of plant components to the construction site may be included in the order given to the vendor, or may be the responsibility of the engineering firm. In larger projects, the engineering firm should take responsibility for coordinating and supervising packaging and shipping. The principal activities of the shipping department are: 1) Checking the specifications for packaging, shipping, payments, and duties (these are usually part of the contract with the investor) 2) Drawing up plans for the delivery of equipment 3) Issuing standard invoices for import licenses 4) Obtaining packaging bids and issuing orders 5) Checking container lists and obtaining transport approval for large and heavy containers Further activities include booking freight space, procuring insurance, and supervising loading and transport. Shipping, customs, and bank documents (including invoices) must also be prepared. Finally, damage and faults must be taken care of.

5.6. Planning and Execution of Civil Work and Erection The main execution phases of a project up to mechanical completion (engineering, procurement, civil work, erection) overlap one another in time. The sequence of engineering work should guarantee that 1) Plant equipment with long delivery times (e.g., compressors, complicated apparatus) can be ordered as early as possible 2) Civil work (e.g., foundations, cable ducts, buildings) is begun early so that equipment erection is not delayed Civil work should be begun as soon as the engineering work is 25 – 30 % complete. By way of example, Figure 33 shows the project master schedule for the design and construction of the expansion of a refinery complex including the progress curves for engineering and construction. The time to mechanical completion is 30 months if basic engineering (which must be performed by the licenser) is complete at the outset. Usually the time required for basic engineering is four to six months. Figure 33 also shows that engineering office work is only complete by the time of “mechanical completion.” The remaining work includes the preparation of final documentation. 5.6.1. Planning of Civil Work and Erection The planning of civil work and erection is part of detail engineering. 5.6.1.1. Planning of Civil Work (Including Structural Steel Work) As a rule, engineering for civil work and structural steel is done by engineers in the civil engineering department assigned to the project team. Often, however, their activity is limited to basic civil design, while detailed civil design is assigned to engineering firms in the country where the plant will be built. These firms are familiar with local conditions, know the local regulations, and have short lines of communication to the construction site and the firm performing the civil work. Important information required at the start of basic civil design includes:

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Figure 33. Project master schedule (refinery expansion project)

1) Soil evaluation report that contains, data on the subsurface soil conditions at the plant site, water table, water analysis, and soil bearing capacity. It should also give settlement calculations so that plant components subjected to severe dynamic loads can be calculated and designed. 2) Data on the proposed wastewater system. 3) An approved layout. 4) Geological and climatic figures such as earthquake factor, prevailing wind direction, and severe snow conditions. 5) Static and dynamic load data for the foundations of machinery, equipment, furnaces, and steel structures. 6) Footprint dimensions of machinery and equipment: piping and cable cutouts in floors, platforms, and walls. On the basis of this information, preliminary plans are drawn for foundations, buildings, steel frames, traffic routes, underground

piping, sewers, wastewater systems, and heating/ventilation/air conditioning systems. They are the basis for detailed planning by the civil engineering subcontractor or in the office. Together with the estimated quantities determined for steel and concrete, these documents are used in inviting bids. The detail engineering for structural steel is usually carried out by the structural steel supplier. After the submitted bids have been evaluated, the contractors for civil work and structural steel are selected on the basis of qualitative and price aspects. 5.6.1.2. Erection Planning Planning the installation of plant equipment starts at a relatively early stage in the engineering process. The sequence of installation activities can strongly influence the detailed scheduling of engineering and procurement. The sched-

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ule for overall project execution should be developed backward from the agreed mechanical completion date and specifically for equipment with long delivery times. In large plants, separate schedules are worked out for each plant section. Erection schedules are revised at intervals throughout detail engineering and procurement as the agreed equipment delivery times are incorporated into the schedules. Drawing up the plot plan calls for the cooperation of an experienced erection engineer, who must consider in particular the erection requirements for heavy components (space requirement, accessibility). The land on which the plant will later be built must be prepared prior to construction work. The planning of temporary facilities is generally the responsibility of the engineering firm. Besides surveying and leveling, it is necessary to plan for the delivery of utilities and the removal of runoff and sewage. Other facilities include construction offices, stores, open-air storage areas, site roads, site fencing, piping prefabrication shops, communication facilities, toilets, first-aid station, guardroom, changing rooms, and accommodation for subcontractor personnel. The engineering firm subcontracts installation work to qualified firms specializing in the erection of structural steel, apparatus, machinery, and installation of piping, electricals, and control systems. Subcontracting takes place in the detail engineering stage as soon as sufficiently exact information is available on the plant components and bulk materials. This is especially important when installation is covered by a fixed-price contract. 5.6.2. Execution of Construction The construction work performed by specialist subcontractors is generally directed by the engineering firm. If the contract provides for the investor or a third party to do the construction, the engineering firm only supplies technical advisory services. 5.6.2.1. Construction-Site Organization and Management The construction manager and his team supervise, coordinate, and direct construction. Re-

sponsibility for all site activities belongs to the construction manager, as specified in the contract, relevant codes, and regulations. The construction manager is answerable to the project manager and is the engineering firm’s principal representative to the investor on the site. The organization of the construction team must take account of the size and complexity of the project, time schedule, local conditions, and contractual obligations. Figure 34 shows a typical site organization for a large project. Commonly the investor maintains a similar, but smaller organization so that discussions can be carried on at all technical levels. The main tasks of the construction site team are: 1) To plan, coordinate, and manage all site activities 2) To arrange for site offices and establish site security systems 3) To organize and oversee materials management 4) To perform scheduling and progress control 5) To define working methods 6) To prepare and carry out quality control 7) To establish and supervise work safety procedures 8) To coordinate and oversee the work of construction subcontractors 9) To manage and clarify the construction documentation 10) To implement a cost control system and arrange payments 11) To prepare construction-site orders 12) To prepare deficiency reports and control insurance cases 13) To prepare as-built drawings 14) To implement a change order management system 15) To submit reports to the investor and the project manager 16) To direct and oversee functional tests 17) To initiate and direct the final plant inspection and, when the plant is mechanically complete, to pass it on to the commissioning manager or turn it over to the custody of the future plant operator

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Figure 34. Organization of a construction team

5.6.2.2. Time Scheduling and Progress Control Time Schedules. An overall construction schedule is created during detail engineering. This plan is continually refined as new information becomes available. Construction work calls for more detailed, individual schedules for on-site activities, which allow better monitoring of individual jobs and fast response to schedule changes. Experience shows that detailed construction network diagrams are unwieldy because of the large quantity of data. Individual schedules for use at the construction site are therefore usually prepared as bar charts that also show the interdependencies between the various activities. Two types of bar charts are usually used: 1) Detail schedules for plant sections, subdivided according to functional disciplines. These charts form the basis for assessing progress of work. 2) Detailed schedules for each functional discipline that include all plant sections. These charts are used for capacity planning for construction personnel and their tools, equipment, materials, and consumables.

Progress of Work. Regular evaluation of progress in construction provides reliable information about the current status of the project. Schedule changes show up and measures can be planned and carried out early enough to insure on-time completion of the plant. Progress planning is based on a detailed schedule, list of plant components, material takeoffs, and specific rating factors. A specific rating factor is an empirical number of hours required for a specified activity (e.g., hours per tonne, m3 , or piece). If there are no rating factors, the hours required for stated activities are estimated in advance. The total number of hours thus found for each activity is allotted to the planned execution time for individual activities. Expressing such allotments in percent allows a target progress curve to be determined for each functional discipline in each plant section; this curve serves as a reference for monitoring construction progress. Progress in each special discipline is evaluated every two to four weeks. Activities not completed at the time of progress assessment must be included. The activities of the special disciplines

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are therefore divided into steps and evaluated. A breakdown for above-ground piping installation serves as an example: Prefabrication: Pick up material Prepare and tack parts Weld Fit up and weld small parts Installation: Transport Install, tack

2% 16 % 16 % 6% 5% 20 %

Weld 11 % Attach clamps, supports 9% Inspection: Do preliminary tests, remedy 6% deficiencies Pressure test, flush 7% Do final test and prepare 2% report Total 100 %

Assessments of individual operations result from years of experience. The progress values found for each discipline are then summarized for each plant section, yielding the progress for each plant section or the plant as a whole. These values are compared with the planned target progress values. If there are deviations, the causes are analyzed and appropriate measures taken. The progress report contains the above information and is an essential part of the regularly updated construction-site report. 5.6.2.3. Construction Work [123] Work at the construction site begins with preparation of the terrain. The plot must be surveyed, graded, and terraced if necessary. Access roads must be laid out and old structures demolished. Utilities for construction must be brought in. Construction-site offices, storage areas, and workshops must be built. Accomodation (containers) for construction personnel must be provided. The construction work begins with excavation and foundation work. If the soil quality necessitates driving piles, this must be done first. The sequence of pouring foundations depends on the order in which equipment is to be delivered and installed. The first step in erection is the erection of heavy equipment (reactors, towers) and steel structures (pipe bridges and equipment supporting frames). Very large process equipment often

cannot be shipped in one piece. Tall towers are divided into sections, while large tanks are delivered in the form of prefabricated pieces. The tools needed for assembly (e.g., welding and cutting machines) and facilities for stress-relief annealing of the welds must be provided at the site. If possible, the delivery of heavy units should be scheduled so that they can be placed on their foundations or supported in their frames immediately. Specialists provided by the manufacturer usually assist in the installation of pumps, compressors, and turbines, as well as “package units” such as refrigeration systems and complicated conveyors; these experts later commission the components installed. Piping installation at a chemical plant is often the most labor intensive and longest phase of installation. It starts with the placement of underground pipelines,the mounting of straight piping on bridges, and the prefabrication of piping. When the number of connection points to apparatus and machinery is great enough (i.e., when the devices have been delivered and put in place), the prefabricated piping sections are connected. Satisfactory progress in prefabrication and smooth installation of piping depends on skillful scheduling of the preparation of isometric drawings and the observance of this schedule. Furthermore, the procurement of piping material should be scheduled so that the material for prefabrication arrives at the site on time. Computeraided integrated material management systems are a great help in the handling of bulk material. Weld inspection is carried out by X-ray methods. Which piping is to be inspected depends on the quality assurance specifications. After a pipe run has been installed, it is pressurized with water to reveal any leaks. The pressure test is documented in a report and the piping is approved for painting or insulation. Insulation work starts at vessels, towers, and reactors and often requires the construction of complex scaffolds. Pipes should not be insulated until a given plant section has a sufficiently large number of pipe runs that have been approved for insulation. The installation of electronic devices and control systems takes place after a section of piping has been completed. Devices in control rooms and substations can, however, be installed independent of other work as soon as the buildings have been completed. Underground electri-

Chemical Plant Design and Construction cal cables are layed after piping. Electrical and pneumatic cables for measurement and control are installed in cable ducts after the completion of underground work. The laying of cables on cable trays is put off until as late as possible to prevent damage during simultaneous piping installation. The same applies to the installation and junctions of field instruments. Furnaces are lined with refractories before shipping to the site or on site. Trays are installed in towers after access to the towers has been provided by platforms and ladders. Lightning protection and grounding wires are installed at an early stage, during the fill work of foundation excavations. Functional tests of the installed equipment mark the end of erection work. These tests are done with the plant in the cold condition and with no product. The contract must precisely define “mechanical completion,” since the contractual obligations of the engineering contractor to the investor often end at this point. Responsibility for the commissioning of the plant may lie with the investor, the licenser, or the engineering contractor. Generally speaking, a plant is mechanically completed when subsequent commissioning will not be delayed or disrupted by installation work and that the safety of the plant is fully guaranteed. The certificate of mechanical completion is usually accompanied by a “punch list” that defines all installation work that is still outstanding and to be done in the commissioning phase. The certificate is generally granted when the following activities have been performed: 1) Pressure testing of equipment and vessels with air, water, or nitrogen 2) Purging and, if necessary, chemical cleaning and pressure testing of piping 3) Testing of stress-free piping connections to machinery and checking of rotation direction and coupling seating 4) Brief trial run of pumps (with water) and of machinery and motors (as possible without product) 5) Calibration of measuring instruments, alarms, interlocks, and cutoff points 6) Functional checking of electrical equipment and control systems

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Figure 35 shows a three-train reformer plant under construction.

5.7. Commissioning 5.7.1. Plant Design and Commissioning Commissioning must be considered even during basic and detail engineering in the design of equipment, piping, and control systems. Faulty process design can have serious effects on the time required for commissioning and the amount of corrective work needed. The start of production may be significantly delayed and the owner may suffer a substantial loss of production and revenue. Difficulties in commissioning and causes of delays have been identified as [136]: 26 – 29 % 56 – 61 % 13 – 15 %

faulty design failure of plant components errors by operating personnel

Commissioning costs as a percentage of total plant investment are: 5 – 10 % 10 – 15 % 15 – 20 %

for established processes for relatively new processes for novel processes

Commissioning should be a special consideration when the piping and instrumentation diagram is designed. A commissioning engineer with relevant experience should be brought in during the planning work. This engineer should prepare the complete operating manual which should be available before the final version of the piping and instrumentation diagram. The experienced commissioning engineer along with specialists (and maintenance engineers) working for the future plant operator should also be involved in checking the piping model. Errors in pipe routing and poor access for the servicing, installation, and removal of equipment can thus be remedied at an early stage. 5.7.2. Operating Manual The operating manual is a condensed “reference book” for the entire plant. It should contain all important details about the design and operation. The typical contents of an operating manual for a chemical plant follow [123], [136], [137]:

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Figure 35. Reformer plant during construction, courtesy of Lurgi AG

Part I. Operating Instructions 1. Design Principles Statement of the type, purpose, and capacity of the plant; specification of quality and quantities of feedstocks and products (including waste streams); utilities and consumables. 2. Description of the Process and the Plant 2.1. Description of the process with its principles (e.g., chemical and physical principles of the process stages). The process itself is demonstrated by process flow diagrams showing equipment, machinery, and instruments and important process conditions. Auxiliary systems such as refrigeration, steam, or slop (wastewater) systems are similarly shown. The delivery and disposal of utilities and consumables is also discussed. 2.2. The plant description covers the designs of the individual plant sections, the functions of their components, and the control of the installed unit operations. 2.3. Material balances.

2.4. Process principles and guidelines for plant operation explain the theoretical basis of the process, as well as the process variables and their effects on product quality or composition. Diagrams, formulas, nomographs, and tables allow estimation of these variables (e.g., raw-materials composition, cooling water temperatures). 3. Special Equipment This chapter contains an in-depth description of special-purpose or critical equipment (e.g., reactors, compressors, turbines). A subchapter deals with particularly important or complex control loops or interlockings and emergency shutdown systems. 4. Preparation of the Plant for Commissioning This chapter lists the preparatory steps required for commissioning (e.g., flushing, cleaning, and neutralizing of piping, equipment, and plant sections; inspections; pressure and leak tests; inspection of safety devices; mechanical tests of machinery; drying

Chemical Plant Design and Construction of furnace refractory or reactor linings; and specifications for charging catalysts and consumables). 5. Plant Startup An outline of the overall plan for starting up the plant is first given. All startup operations are then described in detail step-bystep. Special precautions and unusual design conditions are highlighted. The startup instructions are broken down as follows: a) Initial startup after installation is complete. b) Startup after a prolonged shutdown. c) Restart after a brief shutdown when the plant is still warm. d) Procedures for catalyst regeneration or replacement. e) Measures to be taken after abnormal occurrences. Possible disturbances are listed together with their effects and countermeasures. This section discusses how to remedy problems during operation; how to keep plant sections in operation while problems are being remedied; and how to perform restart afterward. 6. Plant Shutdown This chapter describes procedures for planned and unplanned shutdown of the plant. It is subdivided as follows: a) Partial shutdown for periodic catalyst regeneration or removal of cracking deposits from furnace tubes b) Procedure for brief shutdown c) Procedure for extended shutdown d) Shutdown on utilities outage e) Emergency shutdown and special precautions 7. Analysis Specifications Analysis specifications: required or recommended number of analyses during commissioning under steady-state operating conditions and exceptional operating conditions. 8. Operating Report This chapter describes which data are to be recorded during plant operation. A standard form for reporting during steady-state operation or commissioning is recommended. 9. Safety Practices The safety regulations are summarized. Potential hazards are discussed, and the be-

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havior of the operating personnel is recommended or prescribed. Information about safety facilities and the locations of first-aid stations is given. 10. Miscellaneous List of blinds, setpoints for alarm and switching functions. Part II. Drawings and Equipment Specifications 1. Drawings This chapter contains drawings that relate to the process or to the plant as a whole: a) Process flow diagrams and P & I diagram b) Plant layout plan c) Underground plan d) Selected overall drawings of important equipment (e.g., reactors) 2. Specifications Equipment design specifications are compiled. Relevant drawing numbers, technical procurement specifications, and other documents containing supplementary information are also noted. 3. Equipment Manufacturers Operating Instructions Part III. Technical Documentation and Drawings Manuals are prepared for each engineering discipline. These manuals show all specifications and drawings relating to plant components and their operation and maintenance. 5.7.3. Responsibility and Organization Responsibility for commissioning generally lies with the party granting the process license: the investor/owner, licenser, or the engineering firm. The commissioning team is led by the commissioning manager. The key positions are occupied by experienced startup engineers. Startup operation goes on around the clock, so that an adequate number of startup engineers must be available for shift work. In large plants involving two or more process steps, it is desirable to break the plant down into sections and assign responsibility for each to a smaller startup team. On mechanical completion of the plant most of the installation personnel leave. Some specialist engineers remain on site, however, especially

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those involved with piping, electricals, control systems, and machinery, who solve problems that arise during commissioning. The actual manual commissioning activities are generally performed by personnel of the operator who have already gone through classroom training. The operating personnel can often be trained in similar plants operated by associated firms. 5.7.4. Preparation for Commissioning The commissioning manager must ensure that the plant is supplied with the necessary quantities and qualities of feedstocks, utilities, consumables, and energy in time for the planned start of commissioning. Spare parts and a fully equipped repair shop must also be available. The cooperation of specialists provided by the manufacturers is essential for the commissioning of complicated equipment (e.g., compressors, refrigeration plants). The plant laboratory has a vital function during commissioning. Analytical data are an important input to process control. Sampling and analysis programs must be prepared and discussed with the laboratory. The commissioning team should be present on site during the final installation phase. While the installation team performs function tests, the commissioning engineers perform a detailed check of the plant, focusing on process design and operation (e.g., inspection of towers, internals, and control systems). Changes requested by the commissioning team can then be carried out by the installation team. When the commissioning manager is convinced that the plant is ready for operation, he takes over responsibility for further activities from the erection management.

time, lined furnaces are dried and heated in accordance with the vendor’s specifications. The catalyst is charged, reduced when necessary, and brought up to reaction temperature. Plants in which combustible media circulate must be purged with inert gas so that they are oxygenfree before charging. Steam lines must be carefully dewatered. All measurements are recorded and balances are run so that incorrect behavior of the plant can be quickly detected and corrective measures instituted. Initial disorders are almost always encountered: these result, for example, from utilities outages, mechanical damage, and hot running of bearings or stuffing boxes. An attempt should be made, however, to get the plant running first and start up all systems, provided the safety of personnel and equipment is not endangered. The defects can then be remedied during the first scheduled shutdown of the plant. After operation has stabilized, conditions are optimized. When the planned values of product quantity and quality, utilities consumption etc., have been attained, the guarantee test is carried out. Guarantee values and conditions for performing the guarantee test are stipulated in the contract. If the test results are satisfactory, a report for handover of the plant to the owner is signed. Responsibility for the plant and its operation now shifts to the owner. Any defects still to be remedied are entered in a punch list and a deadline for corrective action is established. Generally the service life of plant equipment is guaranteed for a further, contractually agreed period (parts subject to wear are usually exempt from this guarantee). Whether this guarantee is the responsibility of the manufacturers or the engineering firm depends on the contract. Figure 36 shows part of a plant complex for olefin production.

5.7.5. Plant Startup The measures described in Sections 5.7.3 and 5.7.4 apply to chemical plants in general, whereas activities during the initial startup of a plant depend on the type of process. Commissioning takes place step by step as specified in the operating manual. The first units to be started are utilities and off-sites (e.g., cooling water loop, steam generation). At the same

6. Computer Support Most engineering contractors have invested heavily in computerization, with the emphasis on computer-aided design (CAD), computeraided engineering (CAE), design calculations, data-base management,and office communication sys-tems [138]. Decisions on the use of such systems, in particular CAD and CAE, are

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Figure 36. Section of an olefin complex, courtesy of Lurgi AG

driven by benefits, chiefly reductions in costs and turnaround times, gains in the transparency of methods used, and systematic support for project procedures.

6.1. Role of Computers in Project Execution A variety of systems based on discrete and closed mathematical models are employed in process engineering for the simulation and design of processes. They can be accessed from mainframe computers, workstations, or personal computers (PCs). For special processes, firms use internally developed programs and modules based on standard PC software. Internationally recognized simulation programs are employed to prepare mass and energy balances and to optimize heatexchanger systems. Powerful systems are also available for computer-aided process analysis in

large plants, dynamic simulation, and plant optimization. Project management uses computer software with a high degree of integration for scheduling, cost planning, and project control. These systems are generally accessible on both mainframes and PCs. The specific applications of computer systems vary. Graphical documents (e.g., process flow diagrams, P & I diagrams, loop diagrams) are prepared with CAD systems which are being increasingly linked to engineering data bases. Two-dimensional design instruments are in widespread use for site layout planning and plant design; three-dimensional design systems are occasionally used in special piping-intensive projects [139]. The advantages of graphics in plant design are the consistent and systematic use of models and overlays and the reuse and evaluation of graphical elements with variable intelligence [140].

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CAD use improves collaboration between individual disciplines. For example, if plant and civil engineering both use the same CAD system, plant design can be optimized “at the source.” Engineering calculations can be performed with PC programs developed in-house or with internationally recognized standard software products. Standard systems are predominant in direct daily use by engineers. In subfields, such as finite-elements design, tasks are delegated to specialists who can perform optimizations with special computer tools. Every step in the procurement of equipment is computer-aided. All data and functions are integrated into a system that performs bid invitation, bid comparison, ordering, expediting, and shipping. Relevant data are available not only to the cost control system but also to accounting. As a rule, the entire accounting process is also computer-aided. Specialized relational data-base systems are widely used in the procurement of plant components and bulk materials. These systems are linked by interfaces. Administrative functions are supported by integrated office communications systems. Clerical staff work with PC support connected to laser printers.

sign and engineering of plant equipment. The following general-purpose programs are also commercially available: PROCESS, ASPEN PLUS SDC HTRI ROHR2 PROBAD/FEZEN ANSYS, STRUDL

simulation programs, flow sheeting material data compiler design of heat exchangers strength calculations for piping strength calculations for apparatus finite-element method (FEM) programs for structural analysis and material and heat flow

Data-Base Management Systems. The following list includes systems developed by Lurgi from relational data-base systems: ANSY LUPREA MASY MVS/LUROMAK DISPO MOSY BISAM VERONA ATERM KAPAZ KOKO DOSY ¨ REPRU

mechanical equipment electrical equipment control systems piping disposition of bulk materials management of materials on-site bid invitation and ordering shipping expediting capacity planning cost control documentation and archiving accounting

Graphics. Standard systems for graphics applications include: CADAM/AEC INTERGRAPH/IGDS, CADEX AUTOCAD

two- and three-dimensional plant layout and design preparation of P & I diagrams preparation of P & I diagrams in special fields

6.2. EDP Infrastructure and Systems Progressive engineering companies strive for complete system integration so that engineering, commercial, and management data can be utilized in interlinked modules. In addition to mainframe computers, PCs are increasingly used for individual support at engineering workstations. The growing demand for “distributed intelligence” and advances in computer capabilities are now leading to the use of interconnected, decentralized workstations with alphanumeric applications (data bases, calculations) and graphical ones (CAD). Extensive standardization of the data-processing infrastructure is desirable. The variety of software systems used in project execution are illustrated by the following examples: Design Calculations. Many software systems are developed in-house for the process de-

6.3. Coordination and Interfaces Growing importance is now attached to the fast, error-free transfer of information generated in computer-aided operations of the engineering firm during project execution, to the owner and to engineering partners (and vice versa). Accordingly, the partners must arrive at a good understanding as to the content of documents and data files to be transmitted; standards and codes (data structures, nomenclature, symbols); and data formats and EDP procedures. The definition of interfaces between two different EDP systems is an important factor in data transfer. Standardization of interfaces to a reliable extent does not yet exist for plant design and construction. Various engineering companies have devised interfaces for two-dimensional plant layout and design so that drawing data can be flexibly transmitted to any computer system.

Chemical Plant Design and Construction

7. Quality Assurance The primary goal in the design and construction of chemical plants is to satisfy quality requirements. These are defined by agreement between the investor and the engineering contractor, by legal regulations, and by objectives set by the engineering firm. The quality requirements are generally specified in the contract between the investor and the engineering firm. Quality assurance is ensured by installation of relevant systems. These quality assurance systems cover all technical and organizational practices needed to achieve the desired quality. Requirements for quality assurance systems are defined in standards. The international standards ISO 9000 – 9004 have already been incorporated into most national standards systems [141]. Increasing numbers of quality assurance systems have been developed, introduced, and documented in production and service companies. Certification of these quality assurance systems by neutral organizations is in development. The idea behind the creation of quality assurance systems is that the quality of a product should not only be established after its production, but rather that the entire production process should be subjected to appropriate quality assurance practices on a phase-by-phase basis. Quality assurance practices must be defined and implemented for all services performed by the engineering firm itself (e.g., project management, engineering, procurement, supervision of construction, commissioning). The company quality assurance system is usually documented in a “quality assurance manual” which contains information on organizational structure and processes, as well as the procedures, means, and methods used to assure quality. It may also include references to internal procedures and work instructions which are not part of the manual. The manual gives the owner and third parties a summary of the company’s quality policy and quality system. It is also an instrument for communicating the company’s quality policy to management and employees. A typical table of contents follows: 1) Quality policy of the company 2) Brief description of the company 3) Elements of quality assurance:

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Management tasks Company quality assurance system Marketing Research and development, engineering, project management Documentation Procurement Fabrication, civil work, erection, commissioning Measuring and test equipment, inspection status, corrective action Quality records Internal quality audits Education and training The quality assurance system must not be regarded as a fixture that schematizes all procedures – it needs to be continuously improved in the light of practical knowledge. An engineering firm does not usually have its own fabrication capacity or perform construction services. Equipment as well as construction and installation services must therefore be procured. Suppliers of equipment and firms performing construction work must demonstrate their own quality assurance systems to the engineering firm and allow them to be verified. In addition to this, employees of the engineering firm monitor the fabrication of equipment and the construction work in accordance with established rues to ensure that delivery and performance are in accordance with quality requirements and planned schedules. Such supervision does not, however, release a manufacturer or a construction contractor from its contract obligations. In the normal case, a project can be executed and meet the quality requirements if the provisions of the quality system are satisfied. In complicated projects or those involving a high degree of risk or innovation, a quality assurance plan must be drawn up [142]. This sets forth in detail the quality practices to be followed during execution of the project. A quality assurance manager is designated for the project who, after consultations with the project manager and the heads of the functional divisions, directs the quality assurance activities. He is independent of the project team, reports directly to company management, and confers with the investor on all questions of quality assurance.

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Many quality assurance practices are involved in the execution of a project, a few examples follow: 1) Checking contracts against checklists 2) Defining the degree of checking of technical specifications and drawings for equipment 3) Implementing design change control 4) Instituting design reviews to check, for example, process flow diagrams, layout plans, piping and instrumentation diagrams, and piping models 5) Selecting competent manufacturers for critical equipment 6) Identifying “hold points” and intensity of inspection for equipment during fabrication Quality obtained on the basis of an assurance system “tailored” to the company results in several important benefits: 1) After consultation of quality assurance documents, the customer of a production or service company can convince himself of the company’s ability to achieve the agreed-on quality of the product or service 2) A company’s quality assurance systems, examined and certified by a competent neutral organization, can mean a competitive advantage 3) If a quality assurance system is organized in a meaningful and expedient manner, agreedon quality of a product can be obtained at low cost and with little expenditure of time Quality assurance practices must already be applied, for example, during the design stage for a piece of plant equipment. Expensive postfabrication corrections on a wrongly specified device are then avoided.

8. Training of Plant Personnel Preliminary Planning. The people who are to operate and maintain a chemical plant must have the necessary theoretical background, practical training, and know-how. This applies in particular to personnel in developing countries, who should participate in specially developed know-how transfer programs. Many owners and operators of chemical plants write into their contracts with engineering

firms the transfer of operating and maintenance knowledge to their specialists, supervisors, engineers, and technicians. Training covers technical and commercial jobs as well as middle and upper management. The following questions should be explored at the feasibility study stage: 1) What level of education exists in the region? 2) Will skilled labor be available in the region? 3) Is the project a newly-built plant or an expansion of an existing facility? 4) If it is an expansion, can skilled operating personnel be found? 5) Is the plant a labor-intensive production facility or one that can run automatically? 6) Can the plant operate autonomously (e.g., in a virgin forest area or on an island) or does it require an industrial infrastructure? Training Plan. The training plan must be adapted to the needs of the plant and its surroundings and should be workable regardless of the initial qualifications of the workers being trained. It should also convey state-of-the-art knowledge. The training plan comprises organizational charts, job descriptions, definition of minimum qualifications of future jobholders, training schedules, and identification of facilities for practical training. It may also include administrative topics (e.g., how to obtain a visa and residence and work permits, assistance in finding accomo-dation, where to get work and safety clothing, personal insurance and medical care during the training period). The planning documents provide a basis for hiring tests and results are compared with the profile of requirements in the job descriptions. The tests should pose and evaluate technical and management skills questions. If necessary, an institute or an industrial psychologist can be brought in. Execution of Training. Every participant should receive a training schedule in which the subject, day/time, and discussion partner or instructor are listed. Training should take place as late as possible so that there is no lag between training and job assignment (fluctuation danger). However, training activities should be started early enough so

Chemical Plant Design and Construction that the future operator’s personnel can see their own plant demonstrated in the final installation phase and can perform some functions themselves. This improves their sense of responsibility. The training program is usually broken down as follows: Phase I. Phase II.

Phase III.

Phase IV.

Presentation of basic information about the plant. Presentation of generally important instructions on plant operation, maintenance of machinery, safety practices, and organization. The trainees are divided into operating personnel, maintenance personnel, and administrative and management personnel. When possible, these groups are trained on the same or similar facilities for the jobs they will later perform. During the final phase of installation, the plant personnel familiarize themselves in depth with their own plant. The training period ends with active participation in plant commissioning.

Training Costs. Training costs comprise personnel, nonpersonnel, and incidental costs. Personnel costs are incurred for the people who prepare, execute, and coordinate training, as well as the trainees salaries. Nonpersonnel costs include payments to the operators of facilities where training takes place and the outfitting of training rooms on the construction site. Incidental costs comprise costs for accomodation, travel, work clothing, insurance, and utilities. These may make up a significant fraction of total costs if the trainees are sent abroad.

9. References 1. H. Popper: Modern Cost Engineering Techniques, McGraw-Hill, New York 1970. 2. H. K¨olbel, J. Schulze: Der Absatz in der chemischen Industrie, Springer-Verlag, Berlin 1982. 3. J. T. Thorngren: “Probability Technique Improves Investment Analysis,” Chem. Eng. Cost File, vol. 9, McGraw-Hill, New York 1967. 4. D. J. Massey, I. H. Black: “Predicting Chemical Prices,” Chem. Eng. Cost File, vol. 11, McGraw-Hill, New York 1969. 5. J. Jung: “Globale Vorausberechnung von Investitionskosten,” in GVT-Hochschulkurs: Angewandte Kosten- und Wirtschaftlichkeitsberechnungen bei der Projektierung verfahrenstechnischer Anlagen, University of Dortmund 1984, 90 – 114.

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6. V. D. Herbert, A. Bisio: “The Risk and the Benefit,” part 2, CHEMTECH 6 (1976) no. 7, 422 – 429. 7. H. K¨olbel, J. Schulze: Projektierung und Vorkalkulation in der Chemischen Industrie, Springer-Verlag, Berlin 1982. 8. J. E. Haselbarth: “Updated Investment Cost for 60 Types of Chemical Plants,” Chem. Eng. (N. Y.) 74 (1967) Dec. 4, 214. 9. K. M. Guthrie: “Capital and Operating Cost for 54 Chemical Processes,” Chem. Eng. (N. Y.) 77 (1970) June 15, 140 – 156. 10. J. Jung: “Detaillierte Vorausberechnung von Investitionskosten,” in GVT-Hochschulkurs: Angewandte Kosten- und Wirtschaftlichkeitsberechnungen bei der Projektierung verfahrenstechnischer Anlagen, University of Dortmund 1984, pp. 115 – 137. 11. F. Strailk: “Das Verfahrensschema als Grundlage bei der Planung in der chemischen Technik,” Chem. Appar. 92 (1968) 419 – 434. 12. H. C. Lang, Cost Engineering in the Process Industries, McGraw-Hill, New York, 1960, pp. 7 – 14. 13. C. H. Chilton, Chem. Eng. (N. Y.) 73 (1966) 184 – 190; C. H. Chilton: Cost Engineering in the Process Industries, McGraw-Hill, New York 1960. 14. W. E. Hand, “From Flowsheet to Cost Estimate,” Pet. Refin. 37 (1958) 331. 15. W. Burgert “Kostensch¨atzung mit Hilfe von Kostenstrukturanalysen,” Chem-Ing.-Techn. 51 (1979) 484 –487. 16. C. A. Miller: “Factor Estimating Refined for the Appropriation of Funds,” Chem. Eng. Cost File, vol. 7, McGraw-Hill, New York 1965. 17. K. M. Guthrie: “Date Techniques for Preliminary Capital Cost Estimating,” Chem. Eng. (N. Y.) 76 (1969) Jan. 13, 138;76 (1969) March 24, 114 –142;76 (1969) April 14, 201 – 216. Chem. Eng. Cost File, vol. 11, McGraw-Hill, New York 1969. 18. Chemical Engineering, McGraw-Hill, New York –D¨usseldorf, published monthly. 19. Chemische Industrie, Verlag Handelsblatt, Frankfurt am Main (published monthly). 20. P. M. Kohn: “CE Cost Indexes Maintain 13-Year Ascent,” in: Modern Cost Engineering: Methods and Data, McGraw-Hill, New York 1979. 21. J. Schulze: “Modernisierter Preisindex f¨ur Chemieanlagen,” Chem. Ind. (D¨usseldorf) 32 (1980) no. 10, 657 – 663.

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22. J. Jung: “Globale Vorausberechnung von Betriebskosten,” in GVT-Hochschulkurs: Angewandte Kosten- und Wirtschaftlichkeitsberechnungen bei der Projektierung verfahrenstechnischer Anlagen, University of Dortmund 1984, pp. 90 – 114. 23. J. T. Sommerfeld, C. T. Lenk, Chem. Eng. 77 (1970) May 4, 136 – 138. 24. H. Gaensslen, Chem. Ing. Tech. 48 (1976) no. 12, 1193 – 1195. 25. F. C. Vilbrandt, C. E. Dryden: Chemical Engineering Plant Design, McGraw-Hill, New York 1959. 26. ICI “Factest,” Eur. Chem. News 21 (1972) March 17, 32. 27. ASPEN: TECHNOLOGY ASPEN plus Costing Manual, Cambridge 1984. 28. M. S. Peters, K. D. Timmenhaus: Plant Design and Economics for Chemical Engineers, McGraw-Hill, New York 1980. 29. Verband der chemischen Industrie: Kostenrechnung in der Chemischen Industrie, Gabler, Wiesbaden 1962. 30. R. I. Reul: “Which Investment Appraisal Technique Should You Use?” Chem. Eng. Cost File, vol. 10, McGraw-Hill, New York 1968. W. R. Park: “The Investment Profit-Prophet,” Chem. Eng. Cost File, vol. 10, McGraw-Hill, New York 1968. 31. D. Spethmann: “Flexibilit¨ats¨uberlegungen bei Investitionsprogrammen,” Vortrag auf dem 22. Deutschen Betriebswirtschaftler Tag, Deutsche Gesellschaft f¨ur Betriebswirtschaft, Berlin 1971. 32. H. Jonas: Investitionsrechnung, De Gruyter, Berlin 1964. 33. J. Leibson, C. A. Trischmann: “When and How to Apply Discounted Cash Flow and Present Worth,” Chem. Eng. (N. Y.) 78 (1971) Dec. 13, 97 – 106. 34. A. Fletscher, G. Clarke: Mathematische Hilfsmittel der Unternehmensf¨uhrung, Verlag W. Dummer & Co., M¨unchen 1966. 35. H. M¨uller-Merbach: “Operations Research/Rev. 1,” Methoden und Modelle der Optimalplanung, 3rd ed., M¨unchen 1973. 36. H. F. Rase, M. H. Barrow: Project Engineering of Process Plants, J. Wiley, New York 1957. 37. B. Aggteleky: Fabrikplanung, Hanser Verlag, M¨unchen 1987, 1990. 38. E. Mach: Planung und Errichtung chemischer Fabriken, Sauerl¨ander, Frankfurt 1971. 39. A. Franke: “Risk Analysis in Project Management,” Int. J. Project Management 5 (1987) no. 1, 29 – 34.

40. A. Jaafari: “Management Know-how for Project Feasibility Studies,” Int. J. Project Management 8 (1990) no. 3, 167 – 172. 41. W. T. Nichols, Ind. Eng. Chem. 43 (1951) no. 19, 2295. 42. American Association of Cost Engineers: A Guide to Capital Estimating (1988) 14. 43. R. J. Loring: “Cost of Preparing Proposals,” Chem. Eng. (N. Y.) 77 (1970) Nov. 16, 126. 44. P. Grassmann: Physikalische Grundlagen der Chemie-Ingenieur-Technik, Sauerl¨ander, Frankfurt 1983. 45. E. U. Schl¨under: Heat Exchanger Design Handbook, Hemisphere Publ. Corp., Washington 1983. 46. J. H. Perry: Chemical Engineers Handbook, McGraw-Hill, New York 1984. 47. McGraw-Hill, New York: L. Clarke, R. L. Davidson: Manual for Process Engineering Calculations, 1962; E. W. Comings: High Pressure Technology, 1956; D. L. Katz: Handbook of Natural Gas Engineering, 1959; A. L. Kohl, F. C. Riesenfeld: Gas Purification, 1960; W. L. Nelson: Petroleum Refinery Engineering, 1958; M. van Winkle: Distillation, 1967. H. Harnisch, R. Steiner, K. Winnacker: Chemische Technologie, C. Hanser Verlag, M¨unchen 1984. 48. Gulf Publ. Co., Houston: R. N. Watkins: Petroleum Refinery Destillation, 1979; E. E. Ludwig: Applied Process Design for Chemical and Petrochemical Plants, 1964. 49. R. H. Perry: Perry’s Chemical Engineers’ Handbook, McGraw-Hill, New York 1984. 50. K. Weissermel, H.-J. Arpe: Industrial Organic Chemistry, VCH Verlagsgesellschaft, Weinheim 1993. 51. L. B. Evans: Foundations of Computer-Aided Chemical Process Design, vol. 1, Engineering Foundation, New York 1981, 425 – 469. 52. R. Raman: Chemical Process Computations, Elsevier Appl. Sci. Publ., London 1985. 53. DECHEMA, DETHERM-SDC (Stoffdaten-Compiler), Frankfurt, updated regularly. Design Institute of Physical Property Data (DIPR), American Institute of Chemical Engineers (AIChE), DIPPR Data Compilation, uptated regularly. Thermodynamics Research Center (TRC), Vapor Pressure Datafile, updated regularly. 54. Landolt-B¨ornstein: Zahlenwerte und Funktionen aus Naturwissenschaft und Technik, Springer-Verlag, Berlin 1985. 55. D’Ans-Lax: Taschenbuch f¨ur Chemiker und Physiker, Springer-Verlag, Berlin 1970.

Chemical Plant Design and Construction 56. D. Behrens, R. Eckermann: Chemistry Data Series, DECHEMA, Frankfurt 1977 ff. 57. M. Munsch, T. Mohr, E. Futterer, Chem. Ing. Tech. 62 (1990) 995 – 1002. 58. D. W. Townsend, B. Linnhoff, AIChE J. 29 (1983) 742 – 771. 59. M. A. Duran, I. E. Grossmann, AIChE J. 32 (1986) 123 – 138. 60. B. Linnhoff: “New Concepts in Thermodynamics for Better Chemical Process Design,” Chem. Eng. Res. Des. 61 (1983) 207 – 223. 61. B. Linnhoff: “W¨arme-Integration und Prozeßoptimierung,” Chem. Ing. Tech. 59 (1987) no. 11, 851 –857. 62. A. W. Westerberg, Comput. Chem. Eng. 13 (1989) 365 – 376. 63. V. Hlavacek, Comput. Chem. Eng. 2 (1978) 67 – 75. 64. N. Nishida, G. Stephanopoulos, A. W. Westerberg, ALChE J. 27 (1981) 321 – 351. 65. A. W. Westerberg, Comput. Chem. Eng. 9 (1985) 421. 66. G. Hosemann: “Methodisches Grundkonzept technischer Sicherheit,” DIN-Mitteilung 70, no. 3, Beuth Verlag, Berlin 1991. 67. E. Tr¨oster: “Sicherheitsbetrachtungen bei der Planung von Chemieanlagen,” Chem. Ing. Tech. 57 (1985) no. 1, 15 – 19. 68. Bundes-Immissionsschutzgesetz vom May 14, 1990 (BGBl I, 880) 69. Zw¨olfte Verordnung zur Durchf¨uhrung des Bundes-Immissionsschutzgesetzes (St¨orfallverordnung) vom September 20, 1991 (BGBl I 1991, 1891) 70. Gewerbeordnung, Jan. 1, 1987 (BGBl. I, 1987, ¨ 925). Amended by Gesetz zur Anderung der Gewerbeordnung, April 5, 1990 (BGBl. I, ¨ 1990, 706) and Gesetz zur Anderung der Gewerbeordnung Nov. 9, 1990 (BGBl. I, 1990, 2442) 71. Bundesnaturschutzgesetz (Mar. 12, 1987) Bundesgesetzblatt (BGBl) I (1987), 889. 72. G. Hohe: “Sicherheitsabst¨ande – Definition und Bedeutung in Rechtsvorschriften und Technischen Regelwerken,” Berichtsband zum BPU-Seminar: “Sicherheitsabst¨ande bei Planung und Betrieb industrieller Anlagen,” BPU GmbH, Berlin 1990. 73. Gesetz zur Umsetzung der Richtlinie des Rates, June 27, 1985 Concerning environmental compatibility testing in public and private projects (85/337/EWG). (BGBl. I, 1990, 205).

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74. Gesetz u¨ ber die Umwelthaftung, Bundesgesetzblatt (BGBl) F (1990), 2634. 75. Abst¨ande zwischen Industrie- bzw. Gewerbegebieten und Wohngebieten im Rahmen der Bauleitplanung (Abstandserlaß), Minister f¨ur Umwelt, Raumordnung und Landwirtschaft, March 21 1980 (MBl.NW, 504 / SM Bl. NW, 283). 76. Erste Allgemeine Verwaltungsvorschrift zum Bundes-Immissionsschutzgesetz (Technische Anleitung zur Reinhaltung der Luft – TA Luft), Feb. 27, 1986. 77. DIN 52 900,Feb. 1983, DIN Safety Data Sheet for Chemical Substances and Preparations, Beuth Verlag, Berlin 1983. 78. T. Redeker, W. M¨oller: “Chemsafe-Datenbank f¨ur sicherheitstechnische Kenngr¨oßen,” TU 29 (1988) no. 5, 174, VDI Verlag, D¨usseldorf. 79. V. Pilz: “Sicherheitsanalysen zur ¨ systematischen Uberpr¨ ufung von Verfahren und Anlagen – Methoden, Nutzen und Grenzen,” Chem. Ing. Tech. 57 (1985) no. 4, 289 – 307. 80. A Guide to Hazard and Operability Studies, Chemical Industry Safety and Health Council of the Chemical Industries Association, Alensbie House, 93 Albert Embankment, London SE 1 7TU. Reprinted 1985. 81. Risikobegrenzung in der Chemie, PAAG-Verfahren (HAZOP), ISSA Prevention Series No. 2002, May 1990. Published by: IVSS Internationale Vereinigung f¨ur soziale Sicherheit, Berufsgenossenschaft der chemischen Industrie, Geisbergstraße 11, D-6900 Heidelberg. 82. Wasserhaushaltsgesetz Sept. 23, 1986 (BGBl. I, 1986, 1529) 83. Allgemeine Rahmen-Verwaltungsvorschrift u¨ ber Mindestanforderungen an das Einleiten von Abwasser in Gew¨asser (Rahmen Abwasser VwV) vom Sept. 8, 1989 (GMBI, 1989, 518). 84. H.-O. Braubach, E. Sommer, K. Zellmann: “Anforderungen an verfahrenstechnische Anlagen durch die Weiterentwicklung der Gew¨asserschutz-Vorschriften,” Chem. Ing. Tech. 57 (1985) no. 3, 113 –118. 85. Allgemeine Verwaltungsvorschrift u¨ ber die n¨ahere Bestimmung wassergef¨ahrdender Stoffe und ihre Einstufung entsprechend ihrer Gef¨ahrlichkeit (VwV wassergef¨ahrdende Stoffe), Mar. 9, 1990 (GMBI, 1990, 114). 86. Gesetz u¨ ber die Vermeidung und Entsorgung von Abf¨allen (Abfallgesetz – AbfG), Aug. 27, 1986 (BGBl I, 1986, 1410).

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87. Verordnung zur Bestimmung von Abf¨allen nach § 2 Abs. 2 des Abfallgesetzes (Abfallsbestimmung-Verordnung – AbfBestV) April 3, 1990 (BGBl. I, 1990, 614). 88. Allgemeine Abfallverwaltungsvorschrift u¨ ber Anforderungen zum Schutz des Grundwassers bei der Lagerung und Ablagerung von Abf¨allen, Jan. 31, 1990 (GMBI, 1990, 74). 89. MAGS-L¨armschutz beim Raffineriebau, Minister f¨ur Arbeit, Gesundheit und Soziales des Landes Nordrhein-Westfalen, 1977. 90. Forschungsbericht 79-105-03-302, Stand der Technik bei der L¨armminderung in der Petrochemie, commissioned by the Bundesumweltamtes, Dec. 1979. 91. H. M. Bohuy: L¨armschutz in der Praxis, R. Oldenbourg Verlag, M¨unchen 1986. 92. M. Heckl, H. A. M¨uller: Taschenbuch der Technischen Akustik, Springer-Verlag, Berlin 1975. 93. Allgemeine Verwaltungsvorschrift u¨ ber genehmigungsbed¨urftige Anlagen nach § 16 der Gewerbeordnung GewO, Technische Anleitung zum Schutz gegen L¨arm (TA L¨arm), July 16, 1968 (Beilage zum BAnz Nr. 137, July 26, 1968) 94. Verordnung u¨ ber Arbeitsst¨atten, March 20, 1975 (BGBl. I, 729). Modified by decrees passed on Jan. 2, 1982 (BGBl. I, 1) and Aug. 1, 1983 (BGBl. I, 1057) 95. Unfallverh¨utungsvorschrift L¨arm (UVV-L¨arm) Oct. 10, 1990, Hauptverband der gewerblichen Berufsgenossenschaft. 96. Gesetz u¨ ber Betriebs¨arzte, Sicherheitsingenieure und andere Fachkr¨afte f¨ur Arbeitssicherheit, Dec. 12, 1973 (BGBl. I, 965). Modified by law passed on April 12, 1976 (BGBl. I, 965) 97. Verordnung u¨ ber Druckbeh¨alter, Druckgasbeh¨alter und F¨ullanlagen (Druckbeh¨alterverordnung), Feb. 27, 1980 (BGBl. I, 173). Modified by decree passed on April 21, 1989 (BGBl. I, 830) 98. M. P¨utz: Die Genehmigungsverfahren nach dem Bundes-Immissionsschutzgesetz. Handbuch f¨ur Antragsteller und Genehmigungsbeh¨orden, Erich Schmidt, Berlin 1986. 99. Neunte Verordnung zur Durchf¨uhrung des Bundes-Immissionsschutzgesetzes (Grunds¨atze des Genehmigungsverfahrens – 9. BImSchV), Feb. 18, 1977 (BGBl. I, 274) and May 19, 1988 (BGBl. I, 608, 623) 100. Ber¨ucksichtigung von Emissionen und Immissionen bei der Bauleitplanung sowie bei

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der Genehmigung von Vorhaben (Planungserlaß), Minister f¨ur Landes- und Stadtentwicklung, Minister f¨ur Arbeit, Gesundheit und Soziales und Minister f¨ur Wirtschaft, Mittelstand und Verkehr, July 8, 1982 (MBl. NW, 1366 /SM Bl. NW, 2311). Die Beteiligung an der Bauleitplanung (Beteiligungserlaß), Minister f¨ur Landes- und Stadtentwicklung, July 16, 1982 (MBl. NW, 1375 / SM Bl. NW, 2311). A. Rahmel, W. Schenk: Korrosion und Korrosionsschutz von St¨ahlen, Verlag Chemie, Weinheim 1977. U. Heubner et al.: Nickellegierungen und hochlegierte Sonderst¨ahle; Kontakt und Studium, vol. 153, Expert-Verlag, Sindelfingen 1985. H. Gr¨afen et al.: Die Praxis des Korrosionsschutzes; Kontakt und Studium, vol. 64, Expert-Verlag, Grafenau 1981. U. R. Evans: Einf¨uhrung in die Korrosion der Metalle, Verlag Chemie, Weinheim 1965. DIN Taschenbuch 219: Korrosion und Korrosionsschutz, Beuth-Verlag, Berlin 1987. H. Gr¨afen et al.: Kleine Stahlkunde f¨ur den Chemieapparatebau, VDI Verlag, D¨usseldorf 1978. VDI-Berichte 385: Schadensverh¨utung durch Qualit¨atssicherung von Werkstoffen und Bauteile-Optimierung des Bauteilhaltbarkeit. Tagung, M¨unchen 1980, VDI-Verlag, D¨usseldorf. D. Behrens: DECHEMA Corrosion Handbook, VCH Verlagsgesellschaft mbH, Weinheim 1988 ff. J. R. Brauweiler: “Economics of Longlife – Shortlife Material,” Chem. Eng. (N. Y.) 70 (1963) no. 2, 128. R. E. Meissner, D. C. Shelton, Chem. Eng. (N.Y.), April 1992, 81 – 85, 89 – 94. R. S. Aries, R. D. Newton: Chem. Eng. Cost Estimation, New York (1955) 2, 77. J. T. Gallagher: “A Fresh Look at Engineering Construction Contracts,” Chem. Eng. Cost File, vol. 9, McGraw-Hill, New York 1967, p. 23. J. in ’t Veld, W. A. Peters: “Keeping large Projects under Control: The Importance of Control Type Selections,” Int. J. Project Management 7 (1989) no. 3. A. Morna, N. J. Smith: “Project Managers and the Use of Turnkey Contracts,” Int. J. Project Management 8 (1990) no. 3, 183 – 189. J. Madauss: Projektmanagement, 3rd ed., C. E. Poeschel Verlag, Stuttgart 1989.

Chemical Plant Design and Construction 117. E. Mosberger: “Projektmanagement im Anlagenbau,” Chem. Ing. Tech. 63 (1991) no. 9, 921 – 925. 118. D. Arnold: “Projektmanagment im internationalen Großanlagenbau und Konsequenzen f¨ur die Aus- und Weiterbildung von F¨uhrungskr¨aften,” Zfbf Sonderheft 24 (1989) 55 – 77 (Verlagsgruppe Handelsblatt, D¨usseldorf-Frankfurt). 119. D. Kumar: “Developing Strategies and Philosophies Early for Successful Project Implementations,” Int. J. Projekt Management 7 (1989) no. 3. 120. L. F. Rolstad: “Project Start-Up in Tough Practice,” Int. J. Project Management 9 (1991) no. 1, 10 – 14. 121. H. Reschke, H. Schelle, R. Schnoop: Handbuch Projektmanagement, vol. 1, Verlag ¨ Rheinland, K¨oln 1989. TUV 122. N. Thumb: Grundlagen und Praxis der Netzplantechnik, Verlag Moderne Industrie, M¨unchen 1969. 123. G. Bernecker: Planung und Bau verfahrenstechnischer Anlagen, 3rd ed., VDI Verlag, D¨usseldorf 1984. 124. H. Groh, R. Gutsch: Netzplantechnik, 3rd ed., VDI Verlag, D¨usseldorf 1982. 125. A. Franke: “Risikobewußtes Projektcontrolling, Risikomanagement als Element eines integrierten Projectcontrollings,” Dissertation, Universit¨at Bremen 1991. 126. S. C. Ward, C. B. Chapman, B. Curtis: “On the Allocation of Risk in Construction Projects,” Int. J. Project Management 9 (1991) no. 3, 140 – 147. 127. E. Diegelmann, H. Benesch: “Industriemodelle – Methodik, Abwicklung,” 3 R International (1983) no. 11, 535 – 541. 128. E. Diegelmann: “Erfassung, Beschaffung und Montage von Rohrleitungsteilen,” Chem. Ing. Tech. 54 (1982) no. 4, 303 – 313. 129. J. Hengstenberg, B. Sturm, O. Winkler: Messen, Steuern, Regeln in der Chemischen

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Technik, 3rd ed., vol. V, Springer-Verlag, Berlin 1985. VDI 3546 – Konstruktive Gestaltung von Prozeßleitwarten, Beuth-Verlag, Berlin 1987 – 90. VDI 3683 – Beschreibung von Steuerungsaufgaben, Pflichtenheft, Beuth-Verlag, Berlin 1986. VDI 3693 – Verteilte Prozeßleitsysteme, Pr¨ufliste f¨ur den Einsatz, Beuth-Verlag, Berlin 1985. VDI 2180 – Sicherung von Anlagen in der Verfahrenstechnik, Bl. 1 bis 5, Beuth-Verlag, Berlin 1984 – 88. ABB (ed.): Schaltanlagen, 8th ed., Cornelsen-Verlag, D¨usseldorf 1988. Siemens (ed.): Handbuch der Elektrotechnik, 3rd ed., Verlag Girardet, Essen 1971. J. Matley: “Keys to Successful Plant Start-Ups,” Chem. Eng. (N.Y.) 76 (1969) Sept. 8, 110 – 130. W. Langhoff: “Inbetriebnahme,” Beitrag im Kursushandbuch “Planung und Bau von Großanlagen der Chemischen Industrie,” DECHEMA, Erfahrungsaustausch, D. Behrens, K. Fischbeck, Frankfurt 1969. VDI-Richtlinien, Datenverarbeitung in der Konstruktion, Einf¨uhrungsstrategien und Wirtschaftlichkeit von CAD-Systemen, VDI 2216 (October 1990, draft). W. Hilkert, R. Pfohl: Anlagenkonstruktion mit CAD/CAE-Werkzeugen – ein Erfahrungsbericht – VDI-Berichte Nr. 861.5, 1990. H. Schmidt-Traub: “Integrierte Informationsverarbeitung im Anlagenbau,” Chem. Ing. Tech. 62 (1990) no. 5, 373 – 380. R. Gareis: Projektmanagement im Maschinenund Anlagenbau, Manz-Verlag, Wien 1991. S. J. Heisler: “Project Quality and the Project Manager,” Int. J. Project Management 8 (1990) no. 3, 133 – 137.

Pinch Technology

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Pinch Technology Bodo Linnhoff, University of Manchester Institute of Science and Technology, Manchester, United Kingdom Vimal Sahdev, Linnhoff March Ltd., Manchester, United Kingdom

Introduction . . . . . . . . . . . . . . . . . Composite Curves and Energy Targets Pinch Principle . . . . . . . . . . . . . . . Energy – Capital Tradeoff . . . . . . . . Process Modifications and Applications . . . . . . . . . . . . . . . . . 5.1. Batch Processes . . . . . . . . . . . . . . .

1. 2. 3. 4. 5.

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1. Introduction The discovery of the pinch principle by Linnhoff has provided engineers with a scientific design methodology which has achieved outstanding results across the range of process industries [1]. The way in which industrial processes are designed and retrofitted has been revolutionized. Using the pinch principle, engineers can obtain a better understanding of the fundamentals of a particular process. Some of the original applications in ICI [2] and Union Carbide [3] showed energy savings and capital savings in new designs. The technology has evolved to incorporate cogeneration schemes [4], to improve plant flexibility [5], [6], increase production capacity [5], and reduce the effect of fouling [7] in both new designs and existing plants. The early applications were in the oilrefining, petrochemical, and bulk chemical plants, but the technology has also been proved across a wide range of process industries, including pharmaceuticals, food, pulp and paper, cement, brewing, coffee making, and ice-cream and dairy products. Pinch technology has been used in various kinds of processes including batch, semicontinuous, and continuous operations incorporating various operating parameters, such as different feedstocks, seasonal demand fluctuations, multiple utilities, quality constraints, and environmental constraints.

5.2. 5.3. 6. 6.1. 6.2. 7.

Cogeneration . . . . . . . . . . . . . . . . Flexibility . . . . . . . . . . . . . . . . . . Specific Examples . . . . . . . . . . . . . Modification of a Continuous Process Integration of a Batch Operation . . . References . . . . . . . . . . . . . . . . . .

1078 1078 1079 1079 1080 1080

2. Composite Curves and Energy Targets All processes consist of hot and cold streams. A hot stream is defined as one that requires cooling, and a cold stream as one that requires heating. For any process, a single line can be drawn on a temperature – enthalpy plot which represents either all the hot streams or all the cold streams of the process. A single line representing all the hot streams and a single line representing all the cold streams are called the hot composite curve and the cold composite curve, respectively. The construction of a composite curve is illustrated in Figure 1. Two hot streams are shown on a temperature – enthalpy diagram. Stream 1 is cooled from 200 ◦ C to 100 ◦ C. It has a CP (i.e., mass flow rate times specific heat capacity) of 1; therefore, it loses 100 kW of heat. Stream 2 is cooled from 150 ◦ C to 50 ◦ C. It has a CP of 2; therefore, it loses 200 kW of heat. The hot composite curve is produced by simple addition of heat contents over temperature ranges. Between 200 ◦ C and 150 ◦ C, only one stream exists and it has a CP of 1. Therefore, the heat loss across that temperature range is 50 kW. Between 150 ◦ C and 100 ◦ C, two hot streams exist, with a total CP of 3. The total heat loss from 150 ◦ C to 100 ◦ C is 150 kW. Since the total CP from 150 ◦ C to 100 ◦ C is greater than the CP from 200 ◦ C to 150 ◦ C, that portion of the hot composite curve becomes flatter in the second temperature range from 150 ◦ C to 100 ◦ C. Be-

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tween 100 ◦ C and 50 ◦ C, only one stream exists, with a CP of 2. Therefore, the total heat loss is 100 kW. LIVE GRAPH Click here to view

The point at which the curves come closest to touching is known as the pinch. At the pinch the curves are separated by the minimum approach temperature ∆ T min . For that value of ∆ T min , the region of overlap shows the maximum possible amount of process-to-process heat exchange. Furthermore, QH, min and QC ,min are the minimum utility requirements.

Figure 1. Two hot streams plotted on a temperature – enthalpy diagram

Figure 2 shows the hot composite curve for the problem. The cold composite curve is constructed in the same way. In practical applications the number of streams is generally much greater, but these streams are constructed in exactly the same way. Figure 3 shows the hot and cold composite curves plotted on the same temperature – enthalpy diagram. The diagram represents the total heating and cooling requirements of the process. LIVE GRAPH Click here to view

Figure 2. Hot composite curve

Along the enthalpy axis the curves overlap. The hot composite curve can be used to heat up the cold composite curve by process-toprocess heat exchange. However, at either end an overhang exists such that the top of the cold composite curve needs an external heat source (QH , min ) and the bottom of the hot composite curve needs external cooling (QC , min ). These are known as the hot and cold utility targets.

Figure 3. Composite curves showing the pinch and the energy targets

3. Pinch Principle Once the pinch and utility targets of a process have been identified, the three “golden rules” of pinch technology can be applied. The process can be considered as two separate systems (Fig. 4) – a system above the pinch and a system below the pinch. The system above the pinch requires only residual heat and is, therefore, a heat sink, whereas the system below the pinch has heat to reject and is, therefore, a heat source. The three rules are as follows: 1) Heat must not be transferred across the pinch. 2) There must be no outside cooling above the pinch. 3) There must be no outside heating below the pinch. If the amount of heat traveling across the pinch is α, then an extra amount (α) of hot utility must be supplied and an extra amount of cold utility α is required (Fig. 5). Similarly, any outside cooling of the heat sink and any outside heating of the heat source increases the energy requirements. Thus:

Pinch Technology T =A−α

where T A α

= target energy consumption, = actual energy consumption = cross-pinch heat flow

To achieve the energy targets, cross-pinch heat flows must be eliminated.

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simply by shifting one or both of the composite curves horizontally relative to the other. Vertical movement of the composite curves is not allowed since temperatures are governed by the process. If ∆ T min is increased, the overlap decreases and the energy target increases. However, capital cost decreases. If ∆ T min decreases, capital cost increases but energy cost decreases. Therefore, a tradeoff between energy and capital and, thus, a cost optimum exists. Figure 6 shows the effect of ∆ T min on energy, capital, and total cost. Pinch technology provides the value of ∆ T min with the minimum total cost prior to design. This is determined by a procedure known as supertargeting [8].

Figure 4. Schematic representation of the systems above and below the pinch

Figure 6. Energy cost, capital cost, and total cost plotted against ∆ T min

5. Process Modifications and Applications Figure 5. Heat transfer across the pinch from heat sink to heat source

4. Energy – Capital Tradeoff The composite curves can be moved anywhere parallel to the enthalpy axis, because the composite curves represent changes in enthalpy between certain temperatures, and not absolute values. Thus, ∆ T min can be increased or decreased

The insight into the thermodynamics of the process provided by pinch technology enables an experienced practitioner to identify potentially advantageous process changes. For example, by alteration of the temperatures or pressures, it may be possible to reduce substantially the energy bill without an adverse effect on the product or process. Proper modification of the process also leads to other benefits, such as the removal of bottlenecks and the improvement of process operability.

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A systematic search of the data for such opportunities is an essential element in any pinch technology survey. This can be an iterative procedure: once any beneficial changes are highlighted, the composite curves must be redrawn and the minimum energy needs recalculated, which may then open the way to further process changes [3], [9]. Table 1. Savings from some applications of pinch technology ∗ Process description

Savings

Crude oil unit

savings of ca. $ 1.75×106 at 1.6 year payback savings over ca. $ 7.00×106 with paybacks from 12 to 20 months

Large petrochemical complex manufacturing ethylene, butadiene, HDPE, LDPE, and polypropylene Tailor-made chemicals, batch process with 30 reactors and over 300 products Sulfur-based speciality chemicals, batch and continuous Edible oil refinery, batch operation, wide range of feedstocks

savings of ca. $ 0.45×106 at paybacks of 3 months to 3 years

30 % savings to total site energy bill (worth ca. $ 0.18×106 at paybacks of 9 – 16 months savings of 70 % of process energy equivalent to ca. $ 0.79×106 with paybacks from 12 to 18 months and debottlenecking equivalent to 15 % increased capacity Batch processing of dairy savings of 30 % (equivalent to ca. $ products and dried 0.20×106 ) with payback of less than 1 year beverages Brewery savings from 12 % to 25 % of energy costs with paybacks from 9 months to 2 years “State-of-the-art” whisky significant debottlenecking and distillery savings of ca. $ 0.35×106 with paybacks from 18 months to 2 years Paper mill savings of 8 – 20 % of energy bill at paybacks from 1 to 3 years Continuous cellulose savings of ca. $ 0.28 × 106 at 1 year payback acetate processing Continuous dry cement large energy savings process ∗ Savings mentioned above are concerned primarily with energy costs. The majority of the companies also benefited from increased throughput and improved process flexibility and operability; the economic value of these benefits is not included in the table above.

Since 1983, pinch technology has achieved energy savings in the range of 10 – 70 %, reductions in the capital cost of new plants by up to 25 %, increased process capacity by the removal of bottlenecks, and greater flexibility and operability. The projects summarized in Table 1 indicate the variety of industries in which pinch technology has been successfully applied: food, paper, dairy products, whisky, cement, speciality and bulk chemicals, and oil refining. To date,

the projects identified by pinch technology have been implemented, in whole or part, in all instances mentioned in Table 1.

5.1. Batch Processes In the early days pinch technology achieved its most impressive results in continuous processes, but Table 1 includes a number of processes with exceptionally successful studies in batch operations. The key to applying pinch technology in noncontinuous processes lies in the data extraction. There are no shortcuts; detailed measurements and timings of all the process streams are essential if cost-saving opportunities are to be found. Where the technology has been applied to existing operations, there have frequently been wider process benefits, as well as lower energy bills. By improved integration of the plant, capacity has often been increased, without adversely affecting operability [10].

5.2. Cogeneration The principles governing the correct siting of cogeneration systems were established several years ago [11], [12]. A pinch study uncovers opportunities for introducing such equipment or optimizing existing units. In the pulp and paper industries, for example, power generation is an important element, and pinch technology has been applied successfully to improve performance and to evaluate alternatives. Some of the examples in Table 1 feature the use of cogeneration schemes.

5.3. Flexibility A well-integrated plant need not be less flexible [5], [6]. In fact, the opposite is the case, with techniques having been developed to ensure that flexibility is incorporated into the process design in the most cost-effective way. For example, it is possible to allow for changes in feedstock and product specifications, turndown requirements, and seasonal variations. In a study for Shell, UK, where flexibility was of paramount importance, an integrated design was produced that included twelve operating modes [13]. Throughput was also increased

Pinch Technology

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Figure 7. Example of a process flow diagram a) Fractionation column; b) Furnace; c) Reactor; d) Stabilizer column; e) Benzene column; f) Toluene column

and energy savings of ca. $ 2×106 were achieved within a 1.6 year payback period.

which are mixed with a preheated hydrogen-rich stream before entering the main reactor. The reactor product is cooled and partially separated before being fed to the first of a series of three distillation columns, each fitted with a reboiler and a condenser. The pinch analysis without process modifications showed that the utility energy target was 22 % lower than the actual use. The pinch point occurred at 150 ◦ C on the hot curve.

Figure 8. Original flow diagram of distillery cooker system a) Water tank; b) Cooker; c) Fermenter; d) Ejector

6. Specific Examples The two examples that follow give some indication of the way in which pinch technology is applied to specific problems and help to show why it has had such an impact on every industry that has used it.

6.1. Modification of a Continuous Process Figure 7 shows a simplified flow sheet of an organic chemicals plant. The crude feed is separated into residue and light aromatic fractions,

When the aspect of process modification was included, it was obvious that on all four of the columns (the initial fractionation column and the three distillation columns), the reboiler temperature was above the pinch and the condenser temperature was below it. If the possibility existed to reduce the reboiler temperature to below 150 ◦ C, where excess heat is readily available, then the requirement for heating above the pinch would be correspondingly reduced, thus offering significant savings. In fact, practical considerations limited the scope for this change, but change was achieved on one of the columns by a reduction of the operating pressure. The next step was to examine the furnace, which supplied most of the process heating. The flue gas exit temperature was well above 150 ◦ C; therefore, recovery of this heat above 150 ◦ C would reduce the heating target. After composite curves had been redrawn and process modifications incorporated, the revised energy target was less than half the former energy use.

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The implementation of these proposals produced an integrated design that used 45 % less energy than before, with the investment that was involved being paid for in less than one year.

need for steam to be brought in. The resulting energy savings were only part of the benefit; the cooking cycle was shortened because the cooling was faster and, thus, a process bottleneck was removed and the production capacity of the plant increased. The revised design of the batch cooker is shown in Figure 9.

7. References

Figure 9. Distillery cooker system after integration a) Water tank; b) Cooker, c) Buffer tank; d) Interchanger; e) Fermenter

6.2. Integration of a Batch Operation The batch cooking operation in a distillery is shown in Figure 8. Preheated water and grain are cooked by steam injection; cooling begins by venting steam at above atmospheric pressure to the water preheat tank and is then speeded by a steam ejector that evaporates water from the cooker under vacuum. The cooled batch then moves to the fermentation stage. The pinch analysis of the distillery as a whole had identified the pinch point at 90 ◦ C, which meant that a substantial cross-pinch heat transfer existed within the batch cooking process. Vent steam hotter than the pinch was heating the process water below it, and the ejector exhaust steam was being condensed by cooling water. The cross-pinch heat transfer was eliminated by a series of modifications. By introduction of a buffer tank between the cooker and the fermenter, the process water could be preheated from the cooling cooked grain, which was below the 90 ◦ C pinch, thus eliminating the inappropriate steam ejector. The vent steam, no longer needed for preheating the process water, was utilized elsewhere on the site to reduce the

1. B. Linnhoff., D. W. Townsend, D. Boland, G. F. Hewitt, B. E. A. Thomas, A. R. Guy, R. H. Marsland: User Guide on Process Integration for the Efficient Use of Energy, IChemE (Rugby, UK) 1982. 2. B. Linnhoff, J. A. Turner: “Heat Recovery Networks – New Insights Yield Big Savings,” Chem. Eng. November 2, 1981 pp, 56 – 70. 3. B. Linnhoff, D. R. Vredeveld: “Pinch Technology Has Come of Age,” Chem. Eng. Prog. July 1984, pp. 33 – 40. 4. B. Linnhoff, A. R. Eastwood: Combined Heat and Power and Process Integration, Institution of Gas Engineers, 51st Autumn Meeting, London, November 1985. 5. E. Kotjabasakis, B. Linnhoff: “Sensitivity Tables for the Design of Flexible Processes,” Chem. Eng. Res. Des. 64 (1986) 197 – 211. 6. E. Kotjabasakis, B. Linnhoff: Flexible Heat Exchanger Network Design: Comments on the Problem Definition and on Suitable Solution Techniques, paper presented at IChemE Symposium “Innovation in Process Energy Utilisation” Bath, 16 – 18 September 1987. 7. E. Kotjabasakis, B. Linnhoff: “Better System Design Reduces Heat Exchanger Fouling Costs,” Oil Gas J. September 1987, pp. 49 – 56. 8. B. Linnhoff, S. Ahmad: Supertarget: Optimal Synthesis of Energy Management Systems, presented at the ASME Winter Meeting, Anaheim, December 1986. 9. B. Linnhoff, H. Dunford, R. Smith: “Heat Integration of Distillation Columns into Overall Process,” Chem. Eng. Sci. 38 (1983) no. 8, 1175 – 1188. 10. B. Linnhoff, G. J. Ashton, E. D. A. Obeng. Process Integration of Batch Processes, paper presented at the 79th AIChE Annual Meeting, New York November 1987, pp. 15 – 20. 11. D. W. Townsend, B. Linnhoff: “The Preliminary Design of Networks in Process Design,” Part 1: ‘Criteria for placement of heat

Pinch Technology and engines and heat pumps in process networks’, AIChE J. 29 (1983) pp. 742 – 771. 12. D. W. Townsend, B. Linnhoff: “Heat and Power Networks in Process Design,” Part 2: ‘Design Procedure for Equipment Selection

Piping Systems

→

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and Process Matching’, AIChE J. 29 (1983) no. 5, 748 – 771. 13. T. N. Tjoe, B. Linnhoff: “Using Pinch Technology for Process Retrofit,” Chem. Eng. April 28 (1986) pp. 47 – 60.

Chemical Plant Design and Construction

Pilot Plants

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Pilot Plants Richard P. Palluzi, Exxon Research and Engineering Company, Annandale, NJ 08801, United States

1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . Classification of Pilot Plants . . . . . . Scale-Up . . . . . . . . . . . . . . . . . . . Planned Pilot Plant Experimentation Pilot Plant Design . . . . . . . . . . . . . Methods of Estimating Pilot Plant Costs . . . . . . . . . . . . . . . . . . . . . Pilot Plant Space . . . . . . . . . . . . .

. . . . .

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. 1087 . 1087

1. Introduction While advances in modeling and predictive methods have greatly expanded the range and utility of theoretical research, there still can remain a significant degree of uncertainty in the final results before a major investment is justified. Small-scale laboratory tests help minimize this uncertainty but are fraught with risk due to the small scale and manual, batch simulation of most of the process steps. Hence the concept of piloting the process in a miniature unit in a way which more closely simulates the actual process plant. Pilot plants have evolved from multistory semi-works units, designed for one-tenth of final process scale, to microunits, which fit in larger laboratory hoods. Their cost and complexity often belie their small size and great utility. The piloting process often occurs several times over the life of a research program, Table 1 shows a sequence of this piloting that might be part of a research program. Depending on the complexity of the process, the results of subsequent steps, and the organization’s experience and comfort with the process and results, some of these steps may be eliminated. Recycle and feedback between each step is also possible when problems arise or concerns refuse to be resolved. As each step along this path involves an exponential increase in the resources, time, and money required, there is a strong incentive to progress to the next stage as soon as is practical. Conversely, there is a need to minimize the amount of recycling as the project progresses through the various steps to the commercial unit.

8. 9. 10. 11. 12. 13.

Scheduling . . . . . . . . . . . . . . Pilot Plant Control System and Instrumentation . . . . . . . . . . . Feed and Product Handling . . . Pilot Plant Start-Up . . . . . . . . Safety In Pilot Plant Design And Operation . . . . . . . . . . . . . . . References . . . . . . . . . . . . . .

. . . . 1088 . . . . 1088 . . . . 1090 . . . . 1090 . . . . 1091 . . . . 1091

While this rapid progress is desirable, the consequences of skipping a necessary preliminary stage or failing to fully understand the results of the previous stage can be devastating. Significant time and resources can be wasted at the next stage if it is focused along an incorrect or at least nonoptimum path. A substantial amount of time and money has progressively been invested, and any desire to change the process to make minor improvements must be resisted. At some point the decision must be made that the remaining unanswered process questions are acceptable risks. Table 1. Typical research program steps Step

Intention

Concept

Initiate ideas/concepts to be investigated Basic research Determine basic feasibility of concept Preliminary economic Determine if economic incentive is evaluation sufficient to proceed with investigation Laboratory research Develop basic data Product evaluation Evaluate product suitability Process Produce preliminary design of actual development/preliminary commercial plant, resolving process engineering questions as they arise Pilot plant study Prove basic reliability of proposed process or process design Demonstration unit Demonstrate process feasibility on small commercial scale Prototype unit Demonstrate process feasibility on small commercial scale Commercial plant Produce and sell a product at a profit

In today’s climate of rapid change, timeliness can be as important or more important than minimizing the risks associated with a new product or process. Examples include securing a new mar-

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ket with a totally novel product or attempting to secure a patent position before a competitor. In this situation, the decision may be made to proceed to commercialization earlier than desirable, prior to satisfactorily resolving all design concerns. This often usually requires a more conservative (and more expensive) design approach, results in a significantly longer start-up, or produces a first unit plagued with operational difficulties. Resolving significant problems during start-up or in an operating unit, while feasible, is risky, expensive, and time-consuming. Rarely is it more effective than additional research. A pilot plant is a collection of equipment designed and constructed to investigate some critical aspect(s) of a process operation or to perform basic research. It is a tool rather than an end in itself. A pilot plant can range in size from a laboratory bench-top unit to a facility approaching the size of a commercial unit. The purposes for its construction and operation can vary widely; they include: – Confirming the feasibility of a proposed process – Providing design data – Determining the economic feasibility or a new process – Determining optimum construction materials – Testing the operability of a control scheme – Determining the extent of plant maintenance – Producing sufficient quantities of product for market evaluation – Obtaining kinetic data – Screening catalysts – Proving areas of advanced technology – Providing data for solutions to scale-up problems – Providing technical support to an existing process or product – Assessing process hazards – Determining operating costs The need for a pilot plant is a measure of the degree of uncertainty in developing a process from the research stage to a full commercial plant.

2. Classification of Pilot Plants Pilot plants can be categorized by a number of different criteria. Size is the most common classification as it is the most uniformly proportional to construction and operating costs; a common classification is shown in Table 2. While a common classification, this approach shows little but a range of sizes and costs. The pilot plant’s degree of automation is another classification as instrumentation typically represents large portion of the initial construction and annual operating costs; a common classification is shown in Table 3. While more useful in some respects, this classification is also primarily an indication of cost and complexity. In addition, the high cost of operating labor and the difficulties in manually a process while acquiring accurate data has forced the level of automation of virtually all but the simplest new pilot plants to be increased to such an extent that most fall into the highest automation class. Other classifications based on strategic purpose and/or design philosophy exist [1]. These are often more useful as shorthand methods of communicating useful concepts about pilot units in general. All these classifications are elastic.

3. Scale-Up Scale-up is the process of developing a plant design from experimental data obtained from a unit orders of magnitude smaller (see also → Scale-Up in Chemical Engineering, Chap. 4.; → Process Development, Chap. 3.). This activity is considered successful if the commercial plant produces the product at design capacity, cost, and quality. This step from pilot plant to full-scale operation is the most potentially risky of all the phases of new process development, as the highest expenses are committed when the greatest risks occur. Process plant design has come a long way from the early 1930s, when process designers were limited to rules-of-thumb that a process facility should not be scaled-up more than tenfold [2].

Pilot Plants Table 2. Typical classification of pilot plants by size Class

Characteristics

Small size (< 2 m2 ), typically found in a hood or on a bench top Small tubing used throughput (3 – 6 mm diameter) Small feeds (Small size (< 4 L) Integrated pilot plant Larger size (usually 10 – 25 m2 ), located in open bay, walk in hood, or containment cell Tubing and small (13 – 25 mm diameter) pipe used Fully automated Feed and product systems with limited capacity usually included Demonstration or Very large size (> 250 m2 ), usually in a dedicated Prototype Unit building or area Pipe used extensively Fully automated usually with a dedicated computer system Feed and product systems, including significant storage capacity included

Micro-unit or bench-top unit

Cost range $ 10 000 to $ 100 000

$ 50 000 to $ 500 000

$ 1 000 000 and up

American Oil’s Ultracracking unit at Texas City, for example, was designed with data from a small pilot plant with a scale-up factor of 80 000 [3]. However, many scale-ups still require smaller ratios, albeit greater than 10 : 1, particularly for new processes. Scale-up problems exist for many reasons, including basic equations unsolvable by known mathematical techniques, interconnected physical and chemical process aspects resulting in coupled basic equations, solutions used on the pilot plant that are not suitable on the commercial scale, and unknown equipment performance at sizes never used before. Discussions of scaleup problems and solutions are available in [2– 9]. While successful pilot plant operation does not guarantee successful commercial plant operation, it considerably decreases the risk. All stages of a research program should confirm good scale-up practices are being followed.

4. Planned Pilot Plant Experimentation Planning the experimental program is an important step in determining the type of pilot plant

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required. This step should be performed as soon as the research program objectives are defined, as the type of experimental program required sets, to a large degree, the total program cost, the type of pilot plant required, and the required pilot plant operation. Hence planning the experimental program is one of the major effects on the economic justification of research programs. Planned experimentation – Reduces the number of tests required – a key factor in controlling annual costs as it is not unusual for the annual operating costs of a pilot plant to be two to three times the initial construction costs. – May alter the design or show that a pilot plant is not the best place to obtain some of the desired information. – May highlight the need for more laboratory work to support the pilot plant operation. Laboratory work is less costly and frequently will give insight to correlations or models for the process to be studied in the pilot plant. – Is instrumental in determining the overall project schedule. Hence, the early inclusion of planned experimentation in any research program is vital. Pilot plant design specifications should be established only after careful experimental program development, as decisions on the accuracy of instruments, analyzers, and other equipment should be based on the planned experimental requirements, not haphazard assumptions. Flexibility and versatility are important but costly; when provided unnecessarily or too profusely, they can result in a unit that is expensive and timeconsuming to construct and start up, as well as difficult or impossible to operate successfully. There is always a tendency to ensure the design covers any possible eventuality as well as to try to maximize the unit’s use across future programs. Both tendencies often result in overly complex units, which often make achieving the original goal more difficult, costly, and timeconsuming. Statistical designs for experiments maximize information and reduce research time and costs. These techniques are less likely to miss synergistic factors affecting performance or product quality, minimize the element of human bias, eliminate less productive avenues of experimentation by taking advantage of previous data, and

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reduce the number of pilot plant runs needed to define the effects of variables. Overall, statistical designs increase the confidence level in the experimental results. Additional information is available in [9–17], [70]. Table 3. Typical classification of pilot plants by degree of automation Degree of automation

Characteristic

Cost of instrumentation ∗

Manual

]All process conditions set and monitored by operators Poor repeatability Loose process control Most important variables monitored by dedicated controllers Tighter process control Possibly some data-gathering instrumentation All variables monitored by controllers (may be local or distributed) with some computer-controlled or -monitored Usually data gathering provided via “smart” data logger or computer Usually older system upgraded to more modern standards All or majority of systems computer-controlled Tightest process control May be supervisory or direct digital control High repeability Lowest operating labor Most batch operations may be automated Data gathering integral part of control system

< $ 2000

Local control

Mixed systems

Automated

$ 10 000 – $ 20 000

$ 15 000 – $ 30 000

$ 20 000 – $ 60 000

∗ Costs include installation of equipment but not field wiring and are based on a typical medium-sized pilot plant of 10 – 20 control loops. For smaller units (less than 10 control loops), reduce the above by 25 – 40 %; for larger units (20 – 40 control loops) increase costs by 50 – 100 %.

5. Pilot Plant Design A pilot plant’s design affects all aspects of its life and performance including its cost, operability, and effectiveness. While all operating ranges may not yet be fully defined at the initial design stage, a realistic preliminary range is required before the design is commenced, as is a clear definition of the pilot plant’s purpose. The first step in pilot plant design is to determine whether to design a pilot plant for process modeling or problem investigation. Modeling the process involves reproducing the specific unit operations

on a smaller scale. This promotes safe scale-up, minimizes design time, and reproduces all process operations of interest. It is usually expensive since all operations are reproduced, not just the most important ones. Investigating the problem involves designing a pilot plant to look at a specific area of interest. In this case, the pilot plant may not resemble the commercial operation. While this approach is usually cheaper and quicker, it carries the inherent risk of missing the real problem and producing nothing of value. Several references have addressed each type of approach [1], [18–21]. Pilot plants may be designed with conventional design techniques that mimic commercial process design. This generally provides for a safe and operable design, as the basic methodology is known and the effectiveness of the final results have been proven previously. While safer, this approach is not always feasible: the proposed scale of the pilot plant operation may be completely outside the range of all commercial design techniques, the design suggested may not be economical on the scale envisioned or in the location proposed, or copying the commercial design may carry some inherent limitations that adversely impact operations at ranges or conditions which are of interest in the pilot plant but not in the commercial plant. An alternative approach is to use a design methodology oriented to the pilot plant, by using many conventional design techniques, but trying to maximize the advantages of the pilot plant operation with regard to scale, techniques, and operation. Advantages include economic savings due to reduced equipment, construction, and operating costs; economical and efficient ways around otherwise expensive problems; increased simplicity of operation, resulting in reduced maintenance and operating staff, and improved process understanding through use of specialized equipment. The lack of published design information, however, raises the risk factor; hence, the quality of the final design is very dependent on the skill level and experiences of the design organization/engineer. A significant problem is the limited number of in-house pilot-plant-specific design organizations and truly pilot-plant-specific contractors. This makes using the alternative approach sometimes riskier, except for larger organizations with the requisite in-house resources

Pilot Plants or after a diligent and often expensive contractor evaluation and qualification.

6. Methods of Estimating Pilot Plant Costs Pilot plants costs range from US $ 10 000 to 10 × 106 , but most are commonly in the $ 50 000 to $ 250 000 range when an existing facility is available to house them. There are three basic methods for estimating the costs to design and construct a pilot plant; similarity, cost ratios, and detailed labor and materials. Similarity involves estimating the cost of the pilot plant on the basis of costs to design and construct a similar unit. While the fasted method, it is the least accurate, with errors of ± 100 % not uncommon. As pilot plants are generally built for new processes or process improvements, most are more or less unique. Few units are ever similar enough for good cost estimation. Cost ratios develop the cost estimate by relating the overall cost of the pilot plant or a part of the pilot plant to a known factor such as the cost of major process equipment, the number of control loops, the size of the equipment, or a variety of similar factors. The cost estimate is built up by using the ratios to develop the cost of the entire unit, individual equipment, or separate subsystems, depending on how detailed a cost ratio estimate is made. Unfortunately, cost ratio information, while widespread as a plant-estimating tool, is rarely available for pilot-plant-scale equipment. The lack of a large data pool also renders much of what is available suspect with regard to its overall accuracy. Depending on the type of cost ratios used and the experience of the personnel making the estimate, accuracies of ± 25 – 50 % are typical. Cost ratios can be used more accurately if they are restricted to small subsystems, as numerous information is available in this area [22]. This, however, usually requires a substantial increase in effort. Detailed labor and material estimation involves breaking the pilot plant construction down into a detailed series of small tasks and estimating the labor and materials required for each separate task. This estimation may be performed by in-house or contract engineers, obtained from contractors, or solicited, or solicited via bids or consulting. This method can produce estimates with accuracies of ± (10 – 20) % but requires more effort

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than either of the previous methods. See Table 4 and [1], [22–28], [71] for further information. In general, similarity or general cost ratios estimates are used for developing screening estimates when an estimate is required quickly and a lower accuracy is acceptable. Detailed cost ratios or detailed labor and materials estimates are generally developed prior to actual appropriation of funds so as to have a more accurate estimate for budgeting and cost control. The costs for operating a pilot plant are a summary of the costs of the feedstock, product disposal, utilities, operating labor, spare parts, maintenance, and support services. Major efforts to reduce operating costs have focused on reducing the operating labor through automation and unattended operation [1], [29–37], as well as controlling waste-disposal costs by careful design and planning [1], [38]. While often overlooked or, at best, cursorily estimated in advance, operating costs outweigh construction costs within one to three years. Hence, investments in reducing these costs, while adding to initial unit costs, are generally prudent.

7. Pilot Plant Space Pilot plant space can be divided into several basic types: separate buildings, containment cells or barricades, open bays, walk-in hoods, and laboratory areas. Space is often selected more for availability than suitability or long-term operability. This can lead to significant problems later. For a more detailed summary of the advantages and disadvantages of each type of space and a full discussion of the impacts, see [1]. The space required for a pilot plant will vary tremendously with its size and type. A small unit may require only part of a laboratory (perhaps 5 – 10 m2 ) while an “average” pilot plant may require a large room or building (perhaps 500 – 2000 m2 ), excluding extended feed or product storage. Pilot plant space often requires significant exhaust ventilation, large numbers of utilities, and special power and climate control. Hence, it is expensive to construct and even more expensive to operate.

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Table 4. Pilot plant cost estimating techniques Method

Accuracy range

Information required

Time to develop ∗ (weeks)

Similarity

± (50 – 100) %

50 (vessel with baffles) or Re > 5 × 104 (unbaffled vessel) a constant Newton number Ne ≡ P/(ρn3 d 5 ) is found. In this case, viscosity is irrelevant, we are dealing with a turbulent flow region.

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3) Understandably, the baffles do not influence the power characteristics within the laminar flow region. However, their influence is extremely strong at Re > 5 × 104 . Here, the installation of baffles under otherwise unchanged operating conditions increases the power consumption of the stirrer by a factor of 20! 4) When liquid is stirred in an unbaffled vessel, it begins to rotate and a vortex is formed. Gravitational acceleration g, and hence the Froude number Fr ≡ n2 d/g, plays no role under these circumstances as determined experimentally (Fig. 3). This is confirmed by the points on the lower Ne(Re) curve where the same Re value was set for fluids with different viscosities. This was only possible by a proportional alteration of the stirrer speed. At Re = idem (= identical value), Fr clearly was not idem, but this had no influence on Ne, g is therefore irrelevant!

4. Fundamentals of the Theory of Models and Scale-Up 4.1. Theory of Models The results in Figure 3 could have also been acquired by changing the stirrer diameter: It does not matter, by which means a relevant number is changed. This gives clear evidence that the representation of a physico-technological problem in a dimensionless form is independent of scale (“scale-invariant”), and this presents the basis for a reliable scale-up: Two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe them have the same numerical value (Πi = identical or idem). Clearly, the scale-up of a desired process condition from a model to industrial scale can be accomplished reliably only if the problem was formulated and dealt with according to the dimensional analysis.

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Scale-Up in Chemical Engineering

Figure 3. Power characteristics of a leaf stirrer of a given geometry and given installation conditions [3]

4.2. Model Experiments and Scale-Up In Example 2 process characteristics were evaluated, that presented a comprehensive description of the process. This often expensive and timeconsuming method is certainly not necessary if a given process condition only has to be scaled up from the model to the industrial plant (or vice versa). With Example 2, and assuming that the Ne(Re) characteristics like that in Figure 3 is not known, the task is to predict the power consumption of a leaf stirrer of d = 2 m, installed in a baffled vessel of D = 4 m (D/d = 2) and rotating with n = 10 min−1 . The kinematic viscosity of the liquid is ν = 1 × 10−4 m2 /s. It only has to be known – and this is essential – that the hydrodynamics in this case are governed solely by the Reynolds number and that the process is described by an unknown dependence Ne(Re). Then Re of the industrial plant can be calculated:

Ne ≡ P/(ρ n3 d 5 ) calculated. We find Ne = 8.0. Because Re = idem results in Ne = idem, the power consumption PT of the industrial stirrer can be obtained: N e = idem → N eT = N eM →     P P = 3 5 3 5 n d T n d M

From Ne = 8.0 found in laboratory measurement, it follows for the leaf stirrer having d = 2 m and stirrer speed n = 10 min−1 : P = N en3 n5 =



3 8.0 × 1 × 103 kg/m3 × 10/60s−1 × 25 m5 ∼ 1.2kW = 1185W =

We realize that in scale-up, knowledge of the functional dependency f (Πi ) = 0 is not necessary. All we need is to know which pi space describes the process.

Re ≡ n d 2 /ν = 6.7 × 103

Let us assume that we have a geometrically similar laboratory device of D = 0.2 m and d = 0.1 m and that the stirrer speed can be arbitrarily chosen. Which rotational speed must be chosen to obtain Re = idem using water (ν = 1 × 10−6 m2 /s) as model liquid? The answer is n d 2 /ν = 6.7 × 103 → n = 40 min−1

Under these conditions the stirrer power must be measured and the power number

5. Further Procedures to Establish a Relevance List 5.1. Consideration of the Acceleration due to Gravity g If a natural or universal physical constant has an impact on the process, it has to be incorporated into the relevant list, whether it will be altered or not. In this context the greatest mistakes are made with regard to the gravitational constant g.

Scale-Up in Chemical Engineering This is all the more surprising in view of the fact that the relevance of this quantity is easy enough to recognize if one asks the following question: Would the process function differently if it took place on the moon instead of on earth? If the answer to this question is affirmative, g is a relevant variable. The gravitational acceleration g can be effective solely in connection with the density as gravity gρ. When inertial forces play a role, the density ρ has to be listed additionally. Thus it follows that: 1) In cases involving the ballistic movement of bodies, the formation of vortices in stirring, the bow wave of a ship, the movement of a pendulum and other processes affected by the earth’s gravity, the relevance list comprises gρ and ρ. 2) Creeping flow in a gravitational field is governed by the gravity gρ alone. 3) In heterogeneous physical systems with density differences (sedimentation or buoyancy), the difference in gravity g∆ρ and ρ play a decisive role. In Example 3 the problem is discussed which belongs to the mentioned second class. Example 3: Which Pi Number Describes the Creeping Flow of a very Viscous Liquid in a Gravitational Field (see Fig. 4)? Physical quantity Flow velocity Pipe diameter Dynamic viscosity Liquid weight

Symbol v d µ ρg

Dimension L T−1 L M L−1 T−1 M L−2 T−2

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liquid motion is creeping flow, because without the fluid density the acceleration due to gravity would not work. Therefore, the liquid density has to be combined with the gravitational acceleration as liquid weight, gρ. In this case, the dimensional analysis is very easy: 1) Elimination of the base dimension [M] leads to µ/ρg [LT] 2) Elimination of the base dimension [T] leads to vµ/ρg [L2 ] 3) Elimination of the base dimension [L] leads to vµ/(ρg d 2 ) [–] The resulting pi number, which governs the process, can be interpreted as the quotient of the Reynolds number Re and the Froude number Fr: Fr vdr vµ v2 = and F r ≡ with Re ≡ Re grd2 µ dg

(3)

5.2. Introduction of Intermediate Quantities Relevance lists of some problems contain a whole host of parameters. This makes the elaboration of the process characteristics a difficult endeavor. In some cases a closer look at a problem (or previous experience) facilitates a reduction of the number of physical quantities in the relevance list. This is the case when some relevant variables affect the process by way of a socalled intermediate quantity. Assuming that this intermediate quantity can be measured experimentally, it should be included in the problem relevance list, if this facilitates the removal of more than one quantity from the list. If, e.g., the target quantity, y, depends on five parameters y = f 1 (x 1 , x 2 , x 3 , x 4 , x 5 )

and three of them can be replaced by an intermediate quantity z z = f 2 (x 2 , x 3 , x 4 ) Figure 4. Sketch of creeping flow of a liquid in a gravitational field

then the introduction of z reduces the functional dependence by two parameters to

Here, the density has to be incorporated into the relevance list in spite of the fact that the

y = f 3 (x 1 , z, x 5 )

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In the following, a few examples are presented. A universally known intermediate quantity is the flow velocity, v, in pipes or the so-called superficial gas velocity, vG , in stirring vessels at gassing or in bubble columns: volumethroughputq q v= ∝ 2 S = πD2 /4 cross − sectionalareaS D

Its introduction into the relevance list replaces two quantities: throughput, q, and diameter, D. A settling process (sedimentation) depends on the parameters of the disperse phase. They can be substituted by the sedimentation velocity vs : vs = f (d p , g∆ρ, ϕp , µ)

nΘ = f (Re) where Re ≡ n d 2 /ν

Example 4: Mixing Characteristics for Liquid Mixtures with Density and Viscosity Differences. The mixing time Θ necessary to achieve a molecular homogeneity of a liquid mixture – normally measured by decolorization methods – depends in material systems without differences in density and viscosity on only four parameters: stirrer diameter d, density ρ, kinematic viscosity ν, and stirrer speed n: (4)

(5)

In material systems with differences in density and viscosity, the relevance list, Equation (4), enlarges by the physical properties of the second mixing component, by the volume ratio of both phases ϕ = V 2 /V 1 and, because of the density differences, inevitably by the gravity difference g∆ρ, to nine parameters: {Θ; d; ρ1 ; ν 1 ; ρ2 ; ν 2 ; ϕ; g∆ρ; n}

(6)

This results in a mixing time characteristic incorporating six numbers: nΘ = f (Re, Ar, ρ2 /ρ1 , ν 2 /ν 1 , ϕ)

where d p is the mean particle diameter, g∆ρ the weight difference between solid and liquid phase, ϕπ the volume portion of the solid phase, and µ the dynamic viscosity of the liquid. In some cases, e.g., in the flotation process, many influencing parameters can be replaced by only one or two intermediate quantities. Then, it is legitimate to speak of intermediate quantities as of “lumped parameters”. For details see [4, Example 31]. A tableting process depends on the physical parameters of the powder, as for instance mean particle diameter d p , bulk density (porosity), and flow behavior (lubricating ability). The flow behavior depends also on humidity. These physical properties can be replaced by the lumped parameter powder compressibility κ [5]. The advantage gained by the introduction of intermediate quantities is demonstrated by an elegant example.

{Θ; d; ρ; ν; n}

This five-parametric dimensional space leads to the two-parametric mixing time characteristic:

(7)

where Ar ≡ g∆ρ d 3 /(ρ1 ν 21 ) is the Archimedes number. Meticulous visual observation of this mixing process (the slow disappearance of the Schlieren patterns as result of the disappearance of density differences), reveals that macromixing is quickly accomplished as compared with micromixing. The time-consuming micromixing process already takes place in a material system which can be fully described by the physical properties of the mixture: ν ∗ = f (ν 1 , ν 2 , ϕ) and ρ∗ = f (ρ1 , ρ2 , ϕ)

By introducing the intermediate quantities ν ∗ and ρ∗ , the relevance list (Eq. 6), can be reduced by three parameters to {Θ; d; ρ∗ ; ν ∗ ; g∆ρ; n}

(8)

to obtain a mixing characteristic composed of only three numbers: nΘ = f (Re, Ar)

(9)

(In this case, Re and Ar have to be formed by ρ∗ and ν ∗ !) The process characteristics of a cross-beam stirrer was established in this pi space by evaluation of corresponding measurements [3, pp. 110 – 111], in two differently sized mixing vessels (D = 0.3 and 0.6 m), using different liquid mixtures (∆ρ/ρ∗ = 0.01 – 0.29 and ν 2 /ν 1 = 1 – 5300). The characteristics read:

Scale-Up in Chemical Engineering √

  nJ = 51.6Re−1 Ar1/3 + 3

Re = 101 − 105 ; Ar = 102 − 1011

This example clearly shows the big advantages achieved by the introduction of intermediate quantities.

5.3. Scale-up Procedure at Unavailability of Model Material Systems When model material systems are not available (e.g., with non-Newtonian fluids) or the relevant material parameters are unknown (e.g., with foams, slimes, and sludges), model measurements must be carried out in models of various sizes. The unavailability of the model material systems can sometimes limit the application of the dimensional analysis. In such cases it is, of course, absolutely wrong to speak of “limits of the dimensional analysis”. The following example shows how design and scale-up data can be obtained by model measurements with the same material system in differently sized laboratory devices.

1103

centrifuge: The centrifugal acceleration n2 d exceeds the gravitational one (g) by far! However, the water content of the foam entering the centrifuge depends very much on the gravitational acceleration: On the moon the water drainage would be by far less effective! In contrast to the dimensional analysis presented elsewhere [6] we are well advised to add g to the relevance list: {nmin ; d; type of foamer, cf ; qG , g}

(10)

For the sake of simplicity, in the following nmin is replaced by n and qG by q. For each type of foamer we obtain the following pi space: (

nd3 q 2 , , cf q d5 g

) or, abbreviated {Y, F r, cf }

(11)

To prove this pi space, measurements must be made in model equipment of different sizes, to produce a reliable process characteristics. Figure 5 gives results for a commercial foamer (Mersolat H, Bayer, Germany), the pi space of Equation (11) is seen to be fully satisfied. The straight line in Figure 5 corresponds to the analytic expression Y = Fr −0.4 c0.32 f

(12)

which can be reduced after remodeling to Example 5: Scale-up of Mechanical Foam Breakers. To obtain reliable information on dimensioning and then scale-up of a given type of mechanical foam breaker (foam centrifuge, see sketch in Fig. 5), we wish to ascertain the mode of performing and to evaluate the model measurements of the device. Preliminary experiments have shown that for each foam emergence – proportional to the gas throughput qG – for each foam breaker of diameter d a minimum rotational speed nmin exists that is necessary to control foam formation. The dynamic properties of the foam (density and viscosity, elasticity of the foam lamella, etc.) cannot be named or measured. We have to content ourselves by listing them wholesale as material properties S i . In our model experiments we will of course be able to replace S i by the known type of surfactant (foamer) and its concentration cf , expressed in ppm. In discerning the process parameters, we realize that the gravitational acceleration g has no impact on the foam breaking within the foam

n d = const q0.2 g0.4 f (cf )

(13)

Here, the foam breaker will be scaled-up according to its tip speed u = πn d in model experiments, which will also moderately depend on the foam yield (q). In all other foamers examined [6], the relationshipe Y ∼ Fr −0.45 was found. If the correlation Y ∼ Fr −0.5 f (cf )

proves to be true, then it can be deduced to n2 d/g = cb f

In this case the centrifugal acceleration (n2 d) would present the scale-up criterion and would depend only on the foamer concentration and not on foam yield (q).

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Figure 5. Process characteristics of a foam centrifuge (sketch) for a given foaming agent (Mersolat H) in two concentrations (cf in ppm) [6]

5.4. Partial Similarity In the scale-up from a model (subscript M) to the industrial scale (subscript T), not only must the geometric similarity be ensured, but also all dimensionless numbers describing the problem retain the same numerical values (Πi = idem). This means, for example, that in the scale-up of boats or ships the dimensionless numbers governing the hydrodynamics Fr ≡

vL v2 and Re ≡ Lg ν

must retain their numerical values: Fr T = Fr M and ReT = ReM . It can easily be shown that this requirement cannot be fulfilled here! Since the gravitational acceleration g cannot be varied on earth, the Froude number of the model can be adjusted to that of the full-scale vessel only by its velocity vM . Subsequently, Re = idem can be achieved only by the adjustment of the viscosity of the model fluid. If the model size is only 10 % of the full size (scale factor µ = L T /L M = 10), Fr = idem is achieved in the model at vM = 0.32 vT . To fulfill Re = idem, for the kinematic viscosity of the model fluid, ν M , it follows: νM vM LM = = 0.32 × 0.1 = 0.032 νT vT LT

No liquid exists, whose viscosity would be only 3 % of that of water!

Some requirements concerning physical properties of model materials cannot be implemented. In such cases only a partial similarity can be realized. For this, essentially only two procedures are available (for details see [4, Examples 9, 10 and 41]). One consists of a wellplanned experimental strategy, in which the process is divided into parts, which are then investigated separately under conditions of complete similarity. This approach was first applied by William Froude (1810 – 1879) in his efforts to scale-up the drag resistance of the ship’s hull. The second approach consists in deliberately abandoning certain similarity criteria and checking the effect on the entire process. This ¨ technique was used by Gerhard Damkohler (1908 – 1944) in his attempts to treat a chemical reaction in a catalytic fixed-bed reactor by means of dimensional analysis. Here the problem of a simultaneous mass and heat transfer arises, two processes that obey to completely different fundamental principles. It is seldom realized that many “rules of thumb” utilized for scale-up of different types of equipment are represented by quantities which fulfill only a partial similarity. Examples are the volume-related mixing power P/V (widely used for scaling up mixing vessels) and the superficial velocity vG (normally used for scale-up of bubble columns). The volume-related mixing power P/V presents an adequate scale-up criterion only in

Scale-Up in Chemical Engineering liquid-liquid dispersion processes and can be deduced from the pertinent process characteristics d p /d ∼ We−0.6. (d p , droplet diameter; We, Weber number). In the most common mixing operation, the homogenization of miscible liquids, where macro- and backmixing are required, this criterion fails completely [3]. Similarly, the superficial velocity vG of the gas throughput as an intensity quantity is a reliable scale-up criterion only in mass transfer in gas-liquid systems in bubble columns. In mixing operations in bubble columns, requiring the whole liquid content being backmixed (e.g., in homogenization), this criterion completely loses its validity [4]. Thus, a particular scale-up criterion that is valid in a given type of apparatus for a particular process is not necessarily applicable to other processes occurring in the same device.

6. Short Summary of the Essentials of the Dimensional Analysis and Scale-Up 6.1. Advantages of Dimensional Analysis The advantages made possible by correct and timely use of dimensional analysis are as follows: 1) Reduction of the number of parameters required to define the problem. The Π theorem states that a physical problem can always be described in dimensionless terms. Thus the number of dimensionless groups that fully describe it is much smaller than the number of dimensional physical quantities and is generally equal to the number of physical quantities minus the number of base dimensions contained in their dimensions. 2) Reliable scale-up of the desired operating conditions from the model to the full-scale plant. According to the theory of models, two processes may be considered to be similar if they take place under geometrically similar conditions and all dimensionless numbers which describe the process have the same numerical value. 3) A deeper insight into the physical nature of the process. When experimental data are pre-

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sented in a dimensionless form, distinct physical states can be isolated from one another (e.g., turbulent or laminar flow regions), and the effect of individual physical variables can be identified. 4) Flexibility in the choice of parameters and their reliable extrapolation within the range covered by the dimensionless numbers. These advantages become clear when considering the Reynolds number, Re ≡ v l/ν , which can be varied by altering the characteristic velocity v, or a characteristic length l, or the kinematic viscosity v. By choosing appropriate model fluids, the viscosity can be very easily altered by several orders of magnitude. Once the effect of the Reynolds number is known, extrapolation of both v and l is allowed within the examined range of Re.

6.2. Area of Applicability of Dimensional Analysis The application of dimensional analysis is indeed heavily dependent on the available knowledge. The following five steps (see Fig. 6) can be outlined: 1) The physics of the basic phenomenon is unknown. → Dimensional analysis cannot be applied. 2) Enough is known about the physics of the basic phenomenon to compile a first, tentative relevance list. → The resultant pi set is unreliable. 3) All the relevant physical variables describing the problem are known. → The application of dimensional analysis is unproblematic. 4) The problem can be expressed in terms of a mathematical equation. → A closer insight into the pi relationship is feasible and may facilitate a reduction of the set of dimensionless numbers. 5) A mathematical solution of the problem exists. → The application of dimensional analysis is superfluous. It must, of course, be said that approaching a problem from the point of view of dimensional analysis also remains useful even if all the variables relevant to the problem are not yet known:

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The timely application of dimensional analysis may often lead to the discovery of forgotten variables or the exclusion of artifacts.

or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. When must model experiments be carried out exclusively with the original material system? Where the material model system is unavailable (e.g., in the case of non-Newtonian fluids) or where the relevant physical properties are unknown (e.g., foams, sludges, slimes), the model experiments must be carried out with the original material system. In this case measurements must be performed in models of various sizes (cf. Example 5).

7. Treatment of Variable Physical Properties by Dimensional Analysis

Figure 6. Applicability of dimensional analysis, as dependent on the knowledge available; after J. Pawlowski

6.3. Experimental Methods for Scale-Up In Chapter 1 a number of questions were posed which are often asked in connection with model experiments. How small can a model be? The size of a model depends on the scale factor L T /L M , and on the experimental precision of measurement. Where L T /L M = 10, a ± 10 % margin of error may already be excessive. A larger scale for the model will therefore have to be chosen to reduce the error. Is one model scale sufficient or should tests be carried out in models of different sizes? One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called “process point” in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters

When using dimensional analysis to tackle engineering problems, it is generally assumed that the physical properties of the material system remain unaltered in the course of the process. Relationships such as the heat transfer characteristic of a mixing vessel Nu = f (Re, Pr) are valid for any material system with Newtonian viscosity and for any constant process temperature, i.e., for constant physical properties. However, constancy of physical properties cannot be assumed in every physical process. A temperature field may well generate a viscosity field or even a density field in the material system treated. In non-Newtonian (pseudoplastic or viscoelastic) liquids, a shear rate can also produce a viscosity field. Although most physical properties (e.g., viscosity, density, heat conductivity and capacity, surface tension) must be regarded as variable, it is particularly the value of viscosity that can be varied by many orders of magnitude under certain process conditions. Besides this, it shows the highest temperature coefficients. Why is this topic given so much space in a treatise on dimensional analysis? An explanation is easy to give: A complete similarity requires a geometrical, material, and processrelated similarity, but the material similarity often cannot be easily obtained. A problem arises, e.g., when model (laboratory, bench-scale) measurements are to be per-

Scale-Up in Chemical Engineering formed in a so-called “cold model”, but the industrial plant operates at high temperatures (petrochemicals; T ≈ 800 – 1000 ◦ C). How can we ascertain that the laboratory model system behaves hydrodynamically similarly to that in the industrial plant? Here, different temperature dependence of physical properties (viscosity, density) may cause problems. A problem also arises when laboratory measurements are to be performed with cheap and easy to handle model fluids in order to gain information about the scaling-up of an apparatus for treatment of cell cultures in biotechnology (mammal and plant cells, aerobic cultures, yeasts), the rheological behavior of which is very complex (non-Newtonian: pseudoplastic and viscoelastic). Which model system may we choose? The answer is clear and unambiguous: We may choose any model material system whose dimensionless material function in question is similar to that of the original material system. In this chapter the necessary procedures to obtain this information will be shown.

7.1. Dimensionless Representation of the Material Function µ(T)

To achieve this standard representation, the original dependence µ(T ) had to be enlarged by three additional parameters to µ(T , µ0 , T 0 , γ 0 ). Figures 7A and 7 B depicts the dramatic effect of this standard transformation. They show that all liquids presented in this diagram behave similarly to one another with respect to µ(T ). This is the more surprising in view of the fact that their viscosities cover a range of five orders of magnitude and also their temperature coefficients of the viscosity γ 0 differ widely. T 0 stands for the mean process temperature. In fact, any temperature could have been taken in the case of these substances as the reference temperature in the range covered by experiments. The presented evaluation of µ(T ) is independent of the reference point T 0 , it is invariant of it. Engineers prefer the representation µ/µ0 = exp [−γ 0 (T − T 0 )]

µ/µ0 = f {−γ0 (T − T 0 )}

(14)

meets the requirement f (0) = f
(0) = 1.  γ0 ≡

1 ∂µ µ ∂T

 T0

is the temperature coefficient of viscosity and µ0 ≡ µ(T 0 )

where T 0 is any reference temperature.

(15)

which is a special case of Equation (14) – f being an e function – , although this is not the best possible approximation (represented by the dotted line in Fig. 7B). A slightly better approximation of the material function µ(T ) is provided by the well known Arrhenius relationship: ( µ/µ0 = exp

Similar behavior of a certain physical property common to different material systems can only be visualized by a dimensionless representation of the material function of that property. It is furthermore desirable to formulate this function as uniformly as possible. This can be achieved by the “standard representation” of the material function µ(T ) in which a standardizing transformation is defined in such a way that the expression produced

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−

E0 RT0



)

T0 −1 T

(16)

see [4, Chap. 8]. Because of the relation between µ/µ0 and γ 0 ∆T in Equation (15), the engineering literature prefers the incorporation of the quotient µ/µ0 , e.g., µw /µ (w = wall), instead of γ 0 ∆T in the heat transfer characteristics. Therefore they normally read Nu = f (Re, Pr, µw /µ)

(17)

Were the function µ(T ) not independent (invariant) of the reference point T 0 , an additional number, ∆T /T 0 , had necessarily be incorporated in Equation (17) to fully describe the µ(T ) behavior. In order to describe the process by dimensional analysis, it is advisable to formulate the reference temperature T 0 in a process-related manner, using a characteristic, possibly mean process temperature as reference. A class of functions exists, however, which are independent (invariant) of the reference point T 0 . The

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Figure 7. Temperature dependence of the viscosity (A) and its standard representation (B) which proves to be invariant with respect to the reference point T 0 .

solid line in Figure 7B shows a function of this type. (Note: Functions given in Eqs. 14, 15, and 16 are special cases of invariant functions.) For details, see [4, 7].

7.2. Dimensionless Representation of the Material Function ρ(T) The dependence ρ(T ) for some organic fluids and water is represented in Figure 8, whereas Figure 9 shows the standard transformation of this dependence. The temperature coefficient of the density, β 0 , is given by the expression  β0 ≡

1 ∂ 0 ∂ T

 (18) T0

As shown in Figure 9 the standard transformation of ρ(T ) is invariant of the reference point T 0 only for the organic fluids considered, but not for water (see the signs o and •). This indicates that water cannot be taken as a model fluid in the so-called “cold model”, in order to obtain a reliable information about the behavior of organic fluids at high temperatures in an industrial plant. Only in the range close to the reference point, β 0 ∆T ≈ 0, water behaves similarly to these organic fluids with respect to ρ(T ).

7.3. Pi Set for Variable Physical Properties The type of the dimensionless representation of the material function affects the extended pi set

Scale-Up in Chemical Engineering

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Figure 8. Dependence ρ(T ) for some organic fluids and water

within which the process equation is formulated. In case of the standard representation of µ(T ), the relevance list must include two additional parameters γ0 , T 0 . This leads to two additional dimensionless numbers in the process characteristics. With regard to the heat transfer characteristics of a mixing vessel, it followed that Nu = f (Re0 , Pr 0 , γ 0 ∆T , ∆T /T 0 )

(19)

where the subscript 0 in Re and Pr denotes that these two dimensionless numbers are to be formed with µ0 (which is the numerical value of µ at T 0 ).

antly with regard to the reference temperature T 0 , see Fig. 7B), the relevance list is extended by only one additional parameter γ0 . This, in turn, leads to only one additional dimensionless number. For the above problem it then follows that Nu = f (Re0 , Pr 0 , γ 0 ∆T ) or Nu = f (Re0 , Pr 0 , µw /µ0 ) (20)

7.4. Material Function in Non-Newtonian Liquids In Newtonian liquids the shear stress τ (Pa) is proportional to the shear rate γ(s ˙ −1 ) τ = µ γ˙ → µ = τ /γ˙

(21)

and the proportionality constant is the dynamic viscosity µ (→ Fluid Mechanics, Chap. 3.1.). In non-Newtonian liquids µ actually depends on the effective shear rate γ˙ and occasionally on its history as well. Such liquids are subdivided accordingly to their flow behavior into three classes (→ Fluid Mechanics, Chap. 4.):

Figure 9. Standard transformation of the dependence ρ(T ) in Figure 8, which proves that for water this dependence is not invariant with respect to the reference point T 0

Considering that the standard transformation of the material function can be expressed invari-

1) The viscosity does not depend upon the duration of the shear 2) The viscosity depends upon the duration of the shear 3) The liquid partly behaves like a solid body These interrelations are recorded in the rheological constitutive equations.

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7.4.1. Pseudoplastic Flow Behavior (→ Fluid Mechanics, Chap. 4.1.) The pseudoplastic or shear-thinning fluids represent the most important group within the first class of materials. Under shear stress aggregates of a liquid – solid or liquid – liquid (e.g., dyes) dispersion disintegrate into single particles, which then orientate themselves in the direction of flow. For instance, entangled chains of macromolecules of a polymer solution or melt are stretched, spherical erythrocytes of blood are elongated. In these cases viscosity is degraded by shear. In most pseudoplastic liquids, Newtonian flow behavior is observed at sufficiently low and at high shear rates γ, ˙ see Figure 10. Viscosity approaches a constant value with low shear rates, which is called the zero-shear viscosity, µ0 , and its constant value at very high shear rates is called the infinite shear viscosity, µ∞ .

Figure 10. Typical course of µeff (γ) ˙ in pseudoplastic fluids

In a dimensional-analytical discussion of the rheological constitutive equations, Pawlowski [7] furnished the proof that the rank of the dimensional matrix is always two. By a convenient association of the quantities contained in them, two dimensional material parameters can be produced (see Fig. 10): H – a characteristic viscosity constant, e.g., µ0 , and (22) Θ – a characteristic time constant, e.g., 1/γ˙ 0 or 1/γ˙ ∞ (23)

The relevance list of a process in which a rheological material participates is therefore extended by only two dimensional quantities. All the other material parameters can be transferred into dimensionless pi numbers Πrheol . Thus the rheological constitutive equation reads

µ/H = f (γΘ, ˙ Πrheol ) → µ/µ0 = f (γ/ ˙ γ˙ 0 , Πrheol )

(24)

If the viscosity curve is plotted in a log – log scale and the transition range between µ0 and µ∞ is represented by a straight line with the slope m < 1 (see Fig. 10), this is called an Ostwald – de Waele (power law) fluid (→ Fluid Mechanics, Chap. 4.1.2.) µeff = K γ˙ (m − 1)

(25)

where µeff is the effective viscosity, K the consistency index, and m the flow index. Shear rates normally appearing in mixing vessels are in the range of γ˙ = 50 – 500 s−1 , therefore many liquids behave like Ostwald – de Waele fluids. This explains why the power law is so often used to describe rheological behavior. Many years ago, Pawlowski [2, p. 124] pointed out that Equation (25) injures the principle of consistency of physical quantities, because the dimension of [K] = M L−1 Tm−2 depends on the power m. If the viscosity is temperature-dependent this entails that in a temperature field, for example, the quantity K changes its dimension from point to point and therefore neither grad K nor K/K 0 can be formed. The flow index m is already a dimensionless parameter of the set Πrheol . With the aid of parameters H and Θ (Eqs. 22, 23), the power law can be transformed into a dimensionless form. The rheological constitutive equation of an Ostwald – de Waele fluid reads: µeff K = (γ˙ J)m−1 → H H J m−1 µeff K = (γ/ ˙ γ˙ 0 )m−1 µ0 µ0 (1/γ) ˙ m−1

(26)

According to Equation (24), the material function of pseudoplastic fluids (Πrheol = m), whose viscosity obeys the power law of Ostwald – de Waele, can be represented in the pi space as follows: {µeff /µ0 , γ/ ˙ γ˙ 0 , m}

Henzler correlated the viscosity behavior of aqueous carboxymethyl cellulose (CMC) and xanthan solutions in this pi space with good success, as can be seen in Figure 11 [9]. The fitting line corresponds to the process equation µeff /µ0 = (1 + (γ/ ˙ γ˙ 0 )2(1−m) )−1/2

(27)

Scale-Up in Chemical Engineering

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Figure 11. Dimensionless, standardized material function of some pseudoplastic fluids [9]; for the meaning of γ˙ 0 , see Figure 10

7.4.2. Viscoelastic Flow Behavior (→ Fluid Mechanics, Chap. 4.3.)

The isotropic pressure, p, can be eliminated by the formation of normal stress differences:

The viscoelastic fluids represent the third class of non-Newtonian fluids. Many liquids also possess elastic properties in addition to viscous properties. This means that the distortion work resulting from a stress is not completely irreversibly converted into frictional heat, but is stored partly elastically and reversibly. In this sense, viscoelastic fluids are similar to solid bodies. The liquid strains give way to the mechanical shear stress as do elastic bonds by contracting. This is shown in shear experiments as a restoring force acting against the shear force which, at the sudden ending of the effect of force, moves back the plate to a certain extent. (The procedure is fully described under → Fluid Mechanics, Chap. 4.) In viscoelastic fluids at steady-state laminar flow, besides shear stress τ = σ 21 = µ γ, ˙ normal stresses are observed in all three directions:

1.normalstressdifference N1 = σ11 − σ22 2.normalstressdifference N2 = σ22 − σ33

inthedirectionof flow

N 2 values are always lower than N 1 values, see, e.g., [9]. Therefore, for many processes considering only N 1 will suffice. The normal stress differences are independent of the direction of flow and, in laminar flow (low γ) ˙ are proportional to γ˙ 2 . In following µ = τ /γ˙ for a Newtonian fluid, normal stress coefficients Ψ 1 ≡ N 1 /γ˙ 2 and Ψ 2 ≡ N 2 /γ˙ 2 are occasionally used. Their dependence on the shear rate Ψ (γ) ˙ describes the non-linear viscoelastic behavior of the fluid. For a correct dimensional-analytical representation of the viscoelastic behavior of a fluid, the ratio of normal stress to shear stress is used. The so-called Weissenberg number Wi is defined as Wi1 ≡ N 1 /τ

(31)

In Figure 12 Wi1 values as a function of γ˙ are presented for aqueous CMC and xanthan solutions.

σ11 + p perpendiculartothedirectionof flow σ22 + pandσ33 + p

(29) (30)

(28)

1112

Scale-Up in Chemical Engineering

Figure 12. (Wi1 , γ) ˙ dependence for CMC and polyacrylamide (PAA) solutions [9]

If, according to the concept of Metzner and Otto, γ˙ is replaced by the stirrer speed n, then Equation (32) can also be written as ∧ De ≡ λ n = λ γ˙ = Wi1 /2

(33)

The product n λ is named Deborah number, De (→ Rheology, Chap. 1.2.2.). Figure 13 shows the dimensionless, standardized material function of two viscoelastic fluids, whose dependences Wi1 (γ) ˙ were given in Figure 10. The fitting line corresponds to the process equation Wieff /Wi0 = (γ/ ˙ γ˙ 2 )a + (γ/ ˙ γ˙ 3 )b

(34)

(Exponents a and b have different values depending on the respective substance; see Fig. 13).

7.5. Pi Space in Processes with Non-Newtonian Fluids Figure 13. Dimensionless standardized material function of some viscoelastic fluids [9]

Frequently, a characteristic relaxation time, λ, is used to describe viscoelastic behavior. It is a measure for the time needed to transform the reversibly-elastically stored energy into friction heat: λ ≡ N 1 /(2 τ γ) ˙ = Wi1 /(2 γ) ˙

(32)

As mentioned in Section 7.4.1, the transition from Newtonian to non-Newtonian fluids has the following consequences with respect to the enlargement of the pi space: 1) All pi numbers of the Newtonian case also appear in the non-Newtonian case, whereby µ is replaced by the quantity H (usually µ0 ) with the same dimension (see Eq. 22). 2) An additional pi number is included which contains Q (usually 1/γ˙ 0 ). 3) The pure material numbers are increased by Πrheol

Scale-Up in Chemical Engineering This will be demonstrated on the heat-transfer characteristics of a smooth straight pipe. Case “a” is valid for the temperature independent viscosity and “b” for the temperature dependent viscosity (Eq. 35): Newtonian fluid a Nu, Re, Pr b Nu, Re0 , Pr 0 , γ 0 ∆T

Geometry: Physical properties:

Stirrer diameter density viscosity stirrer speed gravitational constant

Process parameters:

1113 d ρ µ n g

non-Newtonian fluid Nu , ReH , Pr H , νΘ/L, Πrheol Nu, ReHo , Pr Ho , νΘ 0 /L, γHo ∆T , γ Q /γ Q , Πrheol

In case (b) µw /µ and γ Ho ∆T , respectively, as well as γΘ /γH have to be added (γΘ ≡ δlnΘ /δT ). In addition, it can happen that in the nonNewtonian case completely new phenomena take place (e.g., shaft climbing by a viscoelastic fluid against the acceleration due to gravity, the so-called Weissenberg effect, → Fluid Mechanics, Chap. 4.3.), calling for additional parameters (in this case g). Figure 14. Sketch of the pipe stirrer

8. Optimization of Process Conditions by Combining Process Characteristics In this chapter, one example is presented to demonstrate the possibility of optimizing a process with respect to the desirable objective by the appropriate combination of process characteristics. Example 6: Process Characteristics of a Self-Aspirating Hollow Stirrer and the Determination of its Optimum Process Conditions. As a result of their form, hollow stirrers utilize the suction generated behind their edges (Bernoulli effect) to suck in gas from the head space above the liquid. As “rotating ejectors”, they are stirrers and gas pumps in one and are therefore particularly suitable for laboratory use (especially in high pressure autoclaves) because they achieve intensive gas-liquid contacting via internal gas recycling without a separate gas pump [3]. A particularly effective type of this stirrer, the pipe stirrer, is depicted in Figure 14. In operating a hollow stirrer, both target quantities – self-aspirated gas throughput, q, and stirrer power, P, – adapt themselves simultaneously. Both target quantities depend on the following parameters:

Thus we obtain two separate relevance lists: {q; d, ρ, µ; n, g}

(36)

{P; d, ρ, µ; n, g}

(37)

The first relevance list (36) leads to the following matrix ρ

d

n

q

µ

g

M L T

1 −3 0

0 1 0

0 0 −1

0 3 −1

1 −1 −1

0 1 −2

M 3M + L −T

1 0 0

0 1 0

0 0 1

0 3 1

1 2 1

0 1 2

which results in the following three pi numbers: q µ ≡ Q Π2 ≡ ≡ Re−1 nd3 rd2 n g Π3 ≡ ≡ F r−1 dn2

Π1 ≡

(38)

Π1 is named the gas throughput number, Q, Π2 is the inverse Reynolds number, Re, and Π3 is the inverse Froude number, Fr. The gas throughput characteristics of a hollow stirrer then reads: q = f1 nd3



nd2 r n2 d , µ µ

 → Q = f1 (Re, F r)

(39)

1114

Scale-Up in Chemical Engineering

Figure 15. Gas throughput characteristic of a three-edged stirrer [3]

Figure 16. Power characteristic of a three-edged stirrer [3]

The power characteristics, obtained from the relevance list (Eq. 37) by a similar dimensional matrix containing P instead of q leads to: P = f2 rn3 d5



nd2 r n2 d , µ µ

 → N e = f2 (Re, F r) (40)

The pi number containing P is termed the Newton number, Ne. Model experiments with another type of hollow stirrer (three-edged stirrer, see sketch in Fig. 16) were performed in water-air under defined experimental conditions and the scale factor µ = d T /d M was changed in the range of µ = 1 : 2 : 3 : 4 : 5. The results demonstrate that the Reynolds number is irrelevant in the turbulent flow range (Re > 104 ) and the process is exclusively governed by the Froude number. Both process characteristics can therefore be represented as Q = f 1 (Fr) and Ne = f2 (Fr) (see Figs. 15 and 16). These results impressively

demonstrate to which extent information can be compressed by dimensional analysis. These process characteristics present a reliable basis for the scale-up of this hollow stirrer under the given geometric conditions. But they also allow a further optimization of this process, for example with respect to the process conditions under which this stirrer will achieve a given gas throughput with the minimum power, P/q = min. These conditions can be easily found by combining the above characteristics in Figures 15 and 16 in such a way that a dimensionless expression for P/q is produced. This is the pi number combination N eF r P ≡ = f3 (F r) Q qdrg

(41)

This new pi number is, as are its constituents, also a function of the Froude number. This dependence is shown in Figure 17. It can be seen that two sections exist:

Scale-Up in Chemical Engineering Fr ≤ 10 : Ne Fr/Q = const → P/q ∝ d

Here, P/q diminishes directly proportionally to scale (d). Fr > 10 : Ne Fr/Q ∝ Fr → P/q ∝ d 2

In this range, the hollow stirrer is less effective, because the power per unit gas throughput increases with the square of the scale (d 2 ). Under these circumstances, which can be described by “small is beautiful”, it can be clearly shown that hollow stirrers are not suitable for sucking in large amounts of gas on a full-scale. In this case the gas throughput and the stirrer power have to be separated by choosing an appropriate stirrer (Rushton turbine is a good choice!) and providing the gas from below the stirrer by a blower.

Figure 17. Dimensionless interdependence of P/q and d under different process conditions (Fr)

In transport-limited reactions in gas – liquid systems, mass transfer is usually dimensioned according to P/V = idem and v = q/S = idem; (S = πD2 /4). In scaling up, these conditions also speak in favor of decoupling the gas supply and stirrer speed, because processes with two mutually independent process parameters can be

1115

more easily optimized than those having only one process parameter. However, there are many chemical reactions in the gas – liquid system in which the gas throughput does not play a role because microkinetics is rate determining. For such cases, the hollow stirrers, due to their dual role as stirrers and gas conveyers, are especially suited, particularly in high pressure chemical engineering. For details see [3, Section 4.12.1].

9. References 1. M. Zlokarnik: Dimensional Analysis and Scale-up in Chemical Engineering, Springer-Verlag Berlin, Heidelberg, New York 1991. ¨ 2. J. Pawlowski: Die Ahnlichkeitstheorie in der physikalisch-technischen Forschung, Springer-Verlag, Berlin-Heidelberg-New York 1971. 3. M. Zlokarnik: R¨uhrtechnik – Theorie und Praxis, Springer-Verlag, Berlin-Heidelberg-New York 1999. M. Zlokarnik: Stirring – Theory and Practice, Wiley-VCH, Weinheim 2001. 4. M. Zlokarnik: Scale-up – Modell¨ubertragung in der Verfahrenstechnik, Wiley-VCH, Weinheim 2000. M. Zlokarnik: Scale-up in Chemical Engineering, Wiley-VCH, Weinheim 2002. 5. M. Levin, M. Zlokarnik, in Pharmaceutical Process Scale-Up, Marcel Dekker, Inc., New York 2002. 6. M. Zlokarnik, Ger. Chem. Eng. 9 (1986) 314 – 320. 7. J. Pawlowski, AIChE J. 15 (1969) 303 – 305. 8. J. Pawlowski: Ver¨anderliche Stoffgr¨oßen in ¨ der Ahnlichkeitstheorie, Salle + Sauerl¨ander, Aarau -Frankfurt/M. 1991. 9. H.-J. Henzler, Chem.-Ing.-Tech. 60 (1988) 1 – 8.

Biochemical Engineering

1117

Biochemical Engineering Martin Krahe, Bioengineering AG, 8636 Wald, Switzerland

1. 2. 2.1. 2.1.1. 2.1.2. 2.1.2.1. 2.1.2.2. 2.1.2.3. 2.1.3. 2.1.3.1. 2.1.3.2. 2.1.3.3. 2.2. 2.2.1. 2.2.2. 2.2.2.1. 2.2.2.2. 2.2.2.3. 2.2.2.4. 2.2.3. 2.2.4. 2.3. 2.3.1. 2.3.2. 2.4. 2.4.1. 2.4.1.1. 2.4.1.2. 2.4.1.3. 2.4.2. 2.5. 2.5.1. 2.5.2. 2.5.2.1. 2.5.2.2. 2.5.3. 2.5.3.1. 2.5.3.2.

Introduction . . . . . . . . . . . . . . Requirements . . . . . . . . . . . . . Microorganisms: Growth and Bioreaction . . . . . . . . . . . . . . . Biomass Concentration – Cell Density . . . . . . . . . . . . . . . . . . Unstructured Model of Growth . . . Monod Model . . . . . . . . . . . . . Inhibition Kinetics . . . . . . . . . . . Nutrient Uptake Rate . . . . . . . . . Aerobic Bioreactions . . . . . . . . . Stoichiometry of an Aerobic Growth without Product Formation . . . . . Oxygen Uptake Rate (OUR) . . . . . Generated Heat . . . . . . . . . . . . . Sterility . . . . . . . . . . . . . . . . . Thermal Microbial Death Rates . . Steam Sterilization Processes . . . . Sterilization in the Autoclave . . . . “Full” Sterilization In Place (SIP) . “Empty” Sterilization In Place (SIP) Continuous Water/Steam Sterilization . . . . . . . . . . . . . . . Sterile Filtration . . . . . . . . . . . . Incinerators . . . . . . . . . . . . . . . Cleaning . . . . . . . . . . . . . . . . Manual Cleaning . . . . . . . . . . . . Cleaning in Place (CIP) . . . . . . . Homogenous Conditions due to Mixing . . . . . . . . . . . . . . . . . . Mechanical Mixing . . . . . . . . . . Stirred Tank Reactors – Energy Dissipation . . . . . . . . . . . . . . . Vibromixer . . . . . . . . . . . . . . . Hydraulic Mixing . . . . . . . . . . . Pneumatic Mixing . . . . . . . . . . . Gas – Liquid Mass Transfer . . . . Solubility of Oxygen and Other Gases . . . . . . . . . . . . . . . . . . . Oxygen Transfer Rate for Stirred Tank Reactors . . . . . . . . . . . . . Liquid Mass-Transfer Coefficient k L Specific Interfacial Area a . . . . . . Aeration of Stirred Tank Reactors . Gas Hold-up and Foaming in Stirred Reactors . . . . . . . . . . . . . . . . . Volumetric Mass-Transfer Coefficient in Stirred Reactors . . .

1120 1121 1121 1121 1121 1122 1122 1122 1122

2.5.4. 2.6. 2.6.1. 2.6.2. 2.6.3. 2.6.4. 2.6.5. 2.6.6.

1123 1124 1124 1125 1125 1126 1126 1127 1127 1127 1127 1127 1128 1128 1128 1130 1131 1131 1132 1132 1132 1133 1133 1133 1133 1134 1134 1134 1134

2.6.7. 2.6.8. 2.7. 3. 3.1. 3.1.1. 3.1.2. 3.1.3. 3.2. 3.3. 3.4. 3.5. 3.6. 3.6.1. 3.6.1.1. 3.6.1.2. 3.6.2. 3.6.2.1. 3.6.2.2. 3.6.2.3. 3.7. 3.8. 4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.1.3.1.

Volumetric Mass-Transfer Coefficient in Airlift Reactors . . . . Measurement and Control Loops Temperature Measurement and Control . . . . . . . . . . . . . . . . . . Agitation Control . . . . . . . . . . . pH Measurement and Control . . . . Measurement and Control of Dissolved Oxygen . . . . . . . . . . . Measurement and Control of the Volume or the Level . . . . . . . . . . Measurement and Control of Gas-Flow Rate . . . . . . . . . . . . . Measurement and Control of Liquid Flows . . . . . . . . . . . . . . . . . . . Other Measurement Systems . . . . Containment . . . . . . . . . . . . . . Processes and Related Equipment Media Preparation and Sterilization . . . . . . . . . . . . . . Media Preparation Techniques . . . General Procedure of a “Full” SIP . General Procedure of an “Empty” SIP . . . . . . . . . . . . . . . . . . . . Inoculation . . . . . . . . . . . . . . . Batch Cultivation . . . . . . . . . . . Fed-Batch Cultivation . . . . . . . Continuous Cultivation . . . . . . . Continuous Cultivation with Cell Retention . . . . . . . . . . . . . . . . Filtration Systems for Suspended Organisms . . . . . . . . . . . . . . . . Cross-Flow Filtration . . . . . . . . . Rotor Filter . . . . . . . . . . . . . . . Immobilized Organisms . . . . . . . Pellets, Porous Matrices, or Beads . Fluidized-Bed Reactors . . . . . . . . Fixed-Bed Reactors . . . . . . . . . . Dialysis Cultivation . . . . . . . . . Selection of Equipment Related to Specific Processes or Products . . Materials . . . . . . . . . . . . . . . . Austenitic Steel . . . . . . . . . . . . Different Alloys of Austenitic Steels Welding . . . . . . . . . . . . . . . . . Surfaces of Austenitic Steels . . . . Definition and Measurement of Surface Roughness . . . . . . . . . .

1135 1135 1135 1136 1137 1137 1138 1139 1139 1140 1140 1140 1140 1140 1141 1143 1143 1143 1144 1145 1146 1146 1146 1146 1147 1147 1148 1148 1149 1149 1150 1150 1150 1152 1154 1154

1118

Biochemical Engineering

4.1.3.2. 4.1.3.3. 4.1.3.4. 4.1.4. 4.2. 4.3. 4.3.1. 4.3.2. 5.

Mechanical Surface Treatment . . . Chemical Surface Treatment . . . . Electrochemical Surface Treatment Storage and General Working Rules Polymers . . . . . . . . . . . . . . . . Other Materials . . . . . . . . . . . . Glass . . . . . . . . . . . . . . . . . . . Grease and Lubricants . . . . . . . . System Components and Detailed Engineering . . . . . . . . . . . . . . The Vessel . . . . . . . . . . . . . . . Components Mounted or Welded on the Vessel . . . . . . . . . . . . . . . . Seals . . . . . . . . . . . . . . . . . . . Ports . . . . . . . . . . . . . . . . . . . Viewing Glasses . . . . . . . . . . . . Manways . . . . . . . . . . . . . . . . Heating/Cooling Jackets . . . . . . . Heating/Cooling Coils . . . . . . . . Pressure Relief and Safety Devices Standard Stirred Tank Reactor . . . Standard Airlift Reactor . . . . . . . Glass Vessels . . . . . . . . . . . . . . Special Polyamide Film Sleeve Bioreactors . . . . . . . . . . . . . . . Agitation System Design . . . . . . Top or Bottom Drive? . . . . . . . . . Mechanical Seals versus Magnetic Coupling . . . . . . . . . . . . . . . . . Shafts with a Mechanical Seal . . . Magnetic Coupled Shafts . . . . . . Transfer Lines . . . . . . . . . . . . . Needle Connection . . . . . . . . . . Steam-Sterilizable Transfer Line . .

5.1. 5.1.1. 5.1.1.1. 5.1.1.2. 5.1.1.3. 5.1.1.4. 5.1.1.5. 5.1.1.6. 5.1.1.7. 5.1.2. 5.1.3. 5.1.4. 5.1.5. 5.2. 5.2.1. 5.2.2. 5.2.2.1. 5.2.2.2. 5.3. 5.3.1. 5.3.2.

1157 1157 1158 1158 1158 1159 1159 1159 1159 1160 1160 1160 1162 1162 1162 1162 1163 1163 1163 1164 1165 1165 1166 1166 1167 1167 1169 1169 1169 1169

Abbreviations AF AISI ASME BgVV BPVC cdw CFR CHO CIP cww ELISA

Anti-foam American Iron and Steel Institute American Society of Mechanical Engineers Bundesinstitut f¨ur gesundheitlichen Verbraucherschutz und Veterin¨armedizin Boiler and Pressure Vessel Code Cell dry weight Code of Federal Regulations (USA) Chinese hamster kidney Cleaning in place Cell wet weight Enzyme linked immunosorbent assay

5.3.3. 5.3.4. 5.3.5. 5.4. 5.4.1. 5.4.1.1. 5.4.1.2. 5.4.2. 5.4.3. 5.4.4. 5.4.5. 5.4.6. 5.4.7. 5.5. 5.5.1. 5.5.2. 5.5.3. 5.5.4. 5.6. 5.6.1. 5.6.2. 5.6.3. 5.6.4. 5.7. 5.8. 5.8.1. 5.8.2. 5.9. 6.

EPDM FIC GILSP GMP GTAW HE HEPA IIW MDBK MV NC NO OD P

Steam-Sterilizable Transfer Line with CIP . . . . . . . . . . . . . . . . . Manually Operated Transfer Panel . Ring Transfer Panel . . . . . . . . . . Valves . . . . . . . . . . . . . . . . . . Diaphragm Valves . . . . . . . . . . . The Valve Body . . . . . . . . . . . . Diaphragms . . . . . . . . . . . . . . . Bellows-Type Sealed Valves . . . . Piston Valves . . . . . . . . . . . . . . Ball and Butterfly Valves . . . . . . . The Actuators . . . . . . . . . . . . . . Relieve Valves, Safety Valves . . . . Check Valves (Nonreturn Valves) . Pipes . . . . . . . . . . . . . . . . . . . Pipe and Tube Sizes . . . . . . . . . . Pipe Applications and Sloping of Lines . . . . . . . . . . . . . . . . . . . Bends, Reducing-Fittings, T-Piece Connectors . . . . . . . . . . . . . . . Sterile Coupling of Pipes . . . . . . . Gassing Devices for Bioreactors . Tubes and Ring Spargers . . . . . . . Sinter Metal and Porous Membrane Spargers . . . . . . . . . . . . . . . . . Surface Gassing . . . . . . . . . . . . Bubble-Free Membrane Gassing . . Mechanical Foam Separators . . . Sterile Filtration . . . . . . . . . . . Depth Filters . . . . . . . . . . . . . . Membrane Filters . . . . . . . . . . . Hydraulic Pumps . . . . . . . . . . . References . . . . . . . . . . . . . . .

1169 1170 1171 1171 1171 1173 1173 1173 1174 1174 1175 1176 1176 1176 1176 1176 1177 1178 1178 1178 1178 1179 1180 1180 1181 1181 1181 1182 1184

Ethylene-propylene-diene-rubber Flow indicator and controller Good Industrial Large Scale Practice Good Manufacturing Practice Gas tungsten arc-welding Heat-exchanger High efficiency particulate air (filter) International Institute of Welding Madin darby bovine kidney Measured value Normally closed Normally open Optical density Controller with a proportional algorithm

Biochemical Engineering PED PI PI PIC PID PLC PTFE pO2 Pt-100 SCBS SIC SIP SP STR TIC TIS TOC TS VERO vvm WFI

Pressure Equipment Directive Controller with a proportional and an integration algorithm Pressure indicator (pressure gauge) Pressure indicator and controller Controller with a proportional an integration and a derivative algorithm Programmable logic controller Polytetrafluoroethylene Partial pressure of dissolved oxygen Temperature measurement sensor Swiss Interdisciplinary Commitee for Biosafety in Research and Technology. Z¨urich / Switzerland Stirrer speed indicator and control Sterilisation in place Set point Stirred tank reactor Temperature indicator and controller Temperature indicator and switch Total organic carbon Temperature switch African green monkey kidney Gas volume per liquid reactor volume per minute Water for injection

Symbols A area, m2 a interfacial area per unit volume, m−1 , number of moles H atoms in C mole biomass, mol/C-mol b number of moles O atoms in C mole biomass, mol/C-mol c number of moles N atoms in C mole biomass, mol/C-mol c∗o2 O2 concentration in equilibrium with the gas phase, mol/m3 c¯ o2 mean O2 concentration in the broth, mol/m3 de stable bubble diameter in bubble column, mm di diameter of impeller, m D dilution rate, h−1 Dd diameter of draft tube, m Di diffusivity of compound i in the liquid, cm2 /s Dj diffusivity of compound j in the liquid, cm2 /s Ds diameter of the settler, m

Dt Dztemp F Fb Fg Fp F ztemp H Hb Hl HL Hi Ht I k kL kLa Ks Kp Li l m

ms mO 2 M bio n N No Ns OTR OUR OURmax

1119

diameter of the tank, m decimal reduction time at “temp” for z, min volumetric liquid flow rate, m3 /h bleed, volumetric liquid flow rate, m3 /h volumetric gas flow rate, m3 /s volumetric product flow rate, m3 /h lethal time at given temperature, min total height of the tank, m length of the baffle, m liquid height below lower impeller, m total liquid height in tank, m liquid height in between two impellers, m height of liquid above top impeller, m volume of inoculum per working volume, % specific death rate of microorganisms, min−1 liquid side mass transfer coefficient, m/h volumetric liquid side mass transfer coefficient, h−1 Michaelis-Menten constant of substrate, g/L inhibition constant of a product, g/L blade length of the turbine impeller, m number of moles H atoms in C mole substrate, mol/C-mol number of moles O atoms in C mole substrate, mol/C-mol, multiplying factor for k L a in pure water maintenance factor for substrate, g substrate · (g biomass)−1 · h−1 maintenance factor for oxygen, g oxygen · (g biomass)−1 · h−1 molar mass of biomass, g/C-mol agitation speed, s−1 number of viable microorganisms, CFU initial number of viable microorganisms, CFU number of viable microorganism after sterilization, CFU oxygen transfer rate, mmol · L−1 · h−1 oxygen uptake rate, mmol · L−1 · h−1 maximal oxygen uptake rate, mmol · L−1 · h−1

1120 p P Po Pg Qg qsx q o2 x Ra Rq Rz Rmax s so tD us VL Wb Wi x Y Y os Y’xo 2 Y xo 2 ,obs Y xo 2 ,max Y”x s Y x s,obs Y x s,max x z α β γ γ’s

Biochemical Engineering product concentration, g/L biomass production, g · L−1 · h−1 ungassed power input of one impeller, W gassed power input of one impeller, W generated heat due to bioreaction, kJ · L−1 · h−1 rate of organic substrate consumption, g substrate · (g biomass)−1 · h−1 rate of oxygen consumption, g oxygen · (g biomass)−1 · h−1 arithmetical average roughness, µm root mean square deviation of profile, µm average peak to valley height, µm maximum peak to valley height, µm substrate concentration, g/L initial substrate concentration, g/L doubling time, generation time, h linear gas velocity, m/s liquid reactor volume, m3 width of the baffle, m blade width of the turbine impeller, m biomass concentration, g/L sorption number oxygen per substrate requirement, g oxygen/g substrate or mL oxygen/g substrate biomass yield coefficient per oxygen, C-mol biomass/mol O2 observed biomass yield coefficient per oxygen, g biomass/g O2 max. biomass yield coefficient per oxygen, g biomass/g O2 biomass yield coefficient, C-mol biomass/C-mol substrate observed biomass yield coefficient per substrate, g biomass/g substrate maximal biomass yield coefficient per substrate, g biomass/g substrate biomass concentration, g/L cdw z-value, ◦ C molar flow of substrate, mol · L−1 · h−1 molar flow of ammonia, mol · L−1 · h−1 molar flow of oxygen, mol · L−1 · h−1 degree of reduction of substrate (NH3 ), –

γ’x δ ε κ µ η o2 ρ ν

degree of reduction of biomass (NH3 ), – molar flow of biomass, C-mol · L−1 · h−1 molar flow of water, mol · L−1 · h−1 molar flow of carbon-dioxide, mol · L−1 · h−1 growth rate, h−1 oxygen efficiency factor of growth density of fluid, kg/m3 kinematic viscosity, m2 /s

Indices max

maximum value of the parameter

Dimensionless numbers Ne

Power or Newton number

Re Q

Reynold Number for stirrer Flow-rate or aeration number

Y

Sorption number

Po r · n3 · d5 i n · d2 Re ≡ u i Fg Q = n · d3 i  2 k a Y ≡ uLs · ug

Ne ≡

1/3

1. Introduction Biochemical reactions involve biocatalysts, such as microorganisms, plant cells, animal cells or enzymes, and result in the transformation and production of biochemical substances. Biochemical engineering covers the design of vessels and apparatus suitable for performing biochemical reactions. This article will treat the design of apparatus for the cultivation of microorganisms, plant and animal cells. Apparatus for enzymatic reactions, the waste-water treatment and for solid-state fermentation are not discussed. The article focuses on sterilizable reactors for suspended cultivation. In American English a differentiation is made between bioreactors and fermentors, the first characterizing reactors used to cultivate animal cells and the latter to cultivate bacteria or fungi [1]. Generally when running a bioprocess in a reactor, the periphery and the reactor are first sterilized. Organisms are then inoculated to a liquid medium and start growing and metabolizing substrates. Such processes are widely used in life science industry to produce vaccines, vitamins, starter cultures, monoclonal antibodies, aroma,

Biochemical Engineering enzymes, antibiotics and many others. Reactor volumes from 1 to 500 000 L are treated. Reactors of smaller scale such as petri dishes, roller flasks, and shakers are not discussed.

2. Requirements Fermentors and bioreactors should provide optimal growth conditions for microorganisms to achieve conversion and/or production of biological products. Generally: only one specific form of life has to grow and produce the desired product (axenic culture). To prevent contamination with other microorganisms the sterility of the reactor is an indispensable requirement. Cleaning is important to make sure that impurities will not affect the bioconversion or spoil the final product. Homogeneous conditions with respect to temperature, pH, dissolved oxygen, substrate, and product concentration have to be maintained in the reactor. The bioprocess needs to be controlled and all available measurements should be logged to enable quality assurance. Safety regulations have to be adhered in order to prevent accidents and the release of toxic products. Documents describing the parts, instruments, control loops, the operation and the maintenance of the equipment, are required.

2.1. Microorganisms: Growth and Bioreaction Microorganisms, plant and animal cells are described in [2], [3]. Generally, organisms with larger cell size, such as animal cells compared to bacteria, have a complex demand of nutrients and a lower growth rate. On the other hand their ability to produce complicated proteins is increased [4]. 2.1.1. Biomass Concentration – Cell Density For a better understanding of the bioprocess it is important to measure or calculate the biomass concentration x. Several units and measurement methods are in use. The recommended unit to employ if a balance of substrates and products (usually in g/L or mol/L) has to be calculated is the cell dry weight

1121

(cdw, in g/L). This is the most widely used unit for bacteria and fungi density analysis. The cell wet weight (cww, in g/L) can be measured far more quickly but not as precisely. The cdw corresponds to 20 to 25 % of the cww [5]. The measurement of the optical density (OD) of the culture medium is a fast and relatively easy process used to determine the cell density (the wavelength of the measurement should always be indicated, e.g., 548, 550, 556, 600 nm). Nevertheless, the density values depend very much on the procedure and on the photometer. For a medium containing E. coli an OD of 1 (at 548 nm) was found to correspond to 0.25 g/L cdw [5]. The density of cells of larger organisms, such as insect or animal cells, or very low densities of bacteria can be given in number of cells per milliliter. The cells are counted in counting chambers (Neubauer or Thoma chamber) under the microscope. When staining methods are applied, viable cells can be differentiated from dead ones. Most of these determinations are performed off-line. The optical density can be measured in situ. 2.1.2. Unstructured Model of Growth Growth is defined as the increase of viable biomass. This increase can be due to an increase in size of the microorganisms or due to proliferation. During the exponential growth phase, cells divide according to a defined doubling time (generation time) t D [6]. The growth during this phase follows the equation: dx = µ·x dt

(2.1)

with µ, the growth rate, being µ=

ln2 tD

(2.2)

The equation of biomass growth and the following equations in this chapter show that high bioreaction rates will be generated at high cell density with high growth rate. The consideration of high bioreaction rates allows the calculation of the maximal required mass flow rates of substrates, products and the generated heat of exothermic bioreactions. The considered “high” bioreaction rate has a direct impact on the design of the fermentor.

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2.1.2.1. Monod Model The Monod model allows the mathematical correlation of substrate concentration with growth kinetics [7]. The model is based on enzymatic Michaelis–Menten kinetics: µ (s) = µmax ·

s Ks + s

(2.3)

where µmax is the maximum specific growth rate of the organisms, s the substrate concentration, and K s the Michaelis-Menten constant. K s varies with the type of substrate and the diameter of the organism. Examples for µmax and K s are given in Table 1. The values correspond to the efficiency of microorganisms in substrate uptake. E. coli, for example, reaches 95 % of its maximal growth rate with a glucose concentration of only 0.076 g/L. With a dissolved oxygen concentration of only 0.22 % of air saturation (air saturation corresponds to a dissolved oxygen concentration of ≈ 6 mg O2 /L) 95 % of maximal growth is reached. In practice when cultivating freely suspended organisms the dissolved oxygen concentration is kept above 5 to 20 % air saturation. The examples demonstrate that when homogeneous conditions are achieved in a reactor and the organisms are freely suspended in the broth, only relatively low substrate concentrations are required to maintain maximal growth.

At high concentrations, substrates may inhibit growth. The extended Monod model includes the growth inhibition by substrates or products: s Kp · Ks + s Kp + s

(2.4)

where K p is the inhibition constant. For E. coli growing on glycerol the constants were found to be Ks = 0.68 g/L, and Kp = 87.4 g/L [10]. In the case of acids usually only the undissociated part will inhibit growth, so the pH of the medium has to be taken into account. 2.1.2.3. Nutrient Uptake Rate Biomass increase requires a certain amount of substrate, expressed in the observed biomass yield coefficient per substrate Y xs,obs :

(2.5)

Besides substrate uptake for growth, organisms require a certain amount of energy for their maintenance, ms . This energy is needed to maintain the correct ionic composition and turnover of cellular constituents and for several other vital processes. An equation relates growth yields, maintenance factor, and specific growth rate to the rate of substrate or oxygen consumption per biomass (qsx , qo2 x ) [11]: ds = qsx ·x = dt µ ·x = Yxs,obs



do2 = qo2 x ·x = dt µ ·x = Yxo2 ,obs

ms +

µ

 ·x

Yxs,max

(2.6)  mo2 +

µ Yxo2 ,max

 ·x (2.7)

The equations show that cell maintenance requires a constant amount of substrate and oxygen, independent of the growth rate. Therefore at low growth rates the observed biomass yield coefficients on substrate will be much smaller than at a high growth rate [11]. Yxs,obs =

µ qsx

1 1 ms = + Yxs,obs Yxs,max µ

2.1.2.2. Inhibition Kinetics

µ = µmax ·

ds dx 1 1 · ·µ·x = = dt Yxs,obs dt Yxs,obs

(2.8) (2.9)

Several examples of maintenance and yield factors are given in Table 2. 2.1.3. Aerobic Bioreactions The majority of organisms are chemoheterotrophs, requiring an organic carbon source and a chemical energy source. The stoichiometry of a bioreaction can be used to evaluate the mass-flow rates of nutrients, the demand of oxygen, the accumulation of products and the heat generated during growth. The model also indicates how many mass flows need to be measured to estimate the remaining mass flows [12].

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Table 1. Maximum growth rates µmax and Michaelis–Menten constant values K s for selected substrates and microorganisms (data from [8]) Microorganism

Substrate

µmax , h−1

K s , g/L

Escherichia coli Escherichia coli Escherichia coli Escherichia coli Saccharomyces cerevisiae Baker’s yeast Candida tropicalis Klebsiella aerogans Aerobacter aerogenes

glucose glycerol lactose oxygen glucose oxygen glucose glycerol glucose

0.8 – 1.4 0.87 0.8

2 – 4 × 10−3 2 × 10−3 20 × 10−3 7.1 × 10−7 ∗ 25 × 10−3 2.06 × 10−5 ∗ 25 – 75 × 10−3 9 × 10−3 1 – 10 × 10−3

∗

0.5 – 0.6 0.5 0.85 1.22

Data from [9].

Table 2. Maintenance m and true yield coefficients Y on the basis of glucose and oxygen for selected microorganisms (from [11]) Microorganism

ms , g · g−1 · h−1

Yxs, max , g/g

mo , g · g−1 · h−1

Yxo,max , g/g

Escherichia coli Escherichia coli Saccharomyces cerevisiae Aerobacter aerogenes

0.072 0.090 0.018 0.054

0.35 0.53 0.51 0.380

0.6 3.0 0.6 1.4

24.7 42 34.5 44

2.1.3.1. Stoichiometry of an Aerobic Growth without Product Formation Based on the elemental composition of the cell and assuming constant cell composition and growth only on ammonia, oxygen, and a carbon source to form biomass, water, and carbon dioxide, the equation of the bioreaction can be written:

  δ  Yxs =   α

(2.10)

with: CHa Ob Nc elemental composition of the cell (C mole of biomass) CHl Om elemental composition of the carbon source (C mole of substrate) [13], [8] The elemental composition of bacteria and yeast cells does not vary strongly. An average composition was estimated to be CH1.64 O0.52 N0.16 [8]. Based on a general ash content of biomass of 5 %, this will result in a molar mass on C basis of M bio ≈ 25.5 g/C mol. The elemental composition of several carbon sources is given in Table 3. Six unknown flows of molecules α, β, γ, δ, ε, κ (mol L−1 h−1 ) have to be determined. According to the atomic balance four equations can be formulated. Following rules of linear algebra two flows are therefore free. When assuming or measuring two flows the other can be estimated [12]. In the following, the flows of biomass (δ) and substrate (α) are considered to be known.

(2.11)

The other mass flow rates are calculated accordingly: Substrate flow α=

α · CHl Om + β · NH3 + γ · O2 −→ δ · CHa Ob Nc + ε · H2 O + κ · CO2

A macroscopic yield factor Y

xs for the number of C moles of biomass produced per C mole of substrate consumed is considered:

δ  Yxs

(2.12)

Ammonia flow β β = c·δ

(2.13)

Oxygen flow γ γ =

1 · 4



γs − γx  Yxs

 ·δ

(2.14)

Biomass flow δ =δ

(2.15)

Water flow ε ε=

1 · 2





l − a + 3c  Yxs

·δ

(2.16)

CO2 flow κ  κ=

1 −1  Yxs

 ·δ

(2.17)

where c is the number of moles of N atoms in C mole biomass, and a the number of moles of

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Table 3. Elemental composition of averaged biomass and selected substrates (from [8]) Microorganism substrate

Molecular formula

Biomass

CH1.64 O0.52 N0.16

Methane Pentadecane Methanol Ethanol Glycerol Mannitol Acetic acid Lactic acid Glucose Formaldehyde Succinic acid Citric acid Formic acid Oxalic acid

CH4 C15 H32 CH4 O C2 H6 O C3 H8 O3 C6 H14 O6 C2 H4 O2 C3 H6 O3 C6 H12 O6 CH2 O C4 H6 O4 C6 H8 O7 CH2 O2 C2 H2 O4

∗

Molar mass, g/mol

16 212 32 46 92 182 60 90 180 30 118 192 46 90

Molar mass on C mole basis, g/mol

Degree of reduction γ’ (NH3 )

25.5∗∗

4.17 (s = ± 3 %)∗

16 14.1 32 23 30.7 30.33 30 30 30 30 29.5 32 46.0 45.0

8 6.13 6 6 4.67 4.33 4 4 4 4 3.50 3.0 2.0 1.0

s = standard error expressed in %. 5 % ash is considered in the calculation of the molar mass.

∗∗

H atoms in C mole biomass. To simplify Equation (2.14), the so-called generalized degree of reduction of biomass γ
x and of substrate γ
s are introduced [14]: γx

= 4 + a − 2b − 3c

γs = 4 + l − 2m

yield of C mole biomass per consumed mole oxygen, Y
x o2 , can be calculated by introducing the oxygen efficiency factor of growth, η o2 : (2.20)

 Yxo 2

(2.21)

(2.18)

(2.19)

where b is the number of moles of O atoms in C mole biomass, l the number of moles of H atoms in C mole substrate, and m the number of moles of O atoms in C mole substrate. The rationale behind the degree of reduction is the concept of the number of electrons available for transfer to oxygen during combustion of a compound. However, this concept must be used carefully, since the degree of reduction depends on the nitrogen source introduced in the bioreaction. The equations for the degree of reduction of biomass and substrate are in fact linear combinations of the equations of the atomic balances. Therefore these equations can be seen as a simplification but do not give any further information about the bioreaction. An excellent review of the calculation, considering also products or other nitrogen sources in the bioreaction, is given in [13], [15].

 ·γ  Yxs x γs   4 δ ηo2 =  · = γ γx 1 − ηo2

η o2 =

Since the degree of reduction of biomass γ
x can be assumed to be constant, the yield of biomass growth per mole oxygen will only depend on η o2 . Table 4 shows the yield and efficiency factor of several microorganisms growing on minimal media. With the indication given in Tables 1, 2, 3, and 4 the maximal oxygen requirement OURmax at maximal growth rate and cell density can be calculated: OU Rmax =

µmax ·xmax  ·m Yxo bio 2

(2.22)

The OUR of mammalian cells can be estimated by indicating the oxygen demand of a single cell per hour. Respiration parameter in the range of 2 to 8 × 10−9 mg O2 · cell−1 · h−1 are reported [16]. Typical cell densities are in the range of 1 – 3 × 106 cells/mL at the end of a batch cultivation and 0.8 – 2 × 107 cells/mL for continuous cultivation with cell recycle. 2.1.3.3. Generated Heat

2.1.3.2. Oxygen Uptake Rate (OUR) The theoretical oxygen uptake rate OUR can be calculated using Equation (2.14) [13], [15]. The

Cooney et al. measured that during aerobic metabolism the rate of heat production Qg correlates with the rate of oxygen consumption

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Table 4. Selected microorganisms with indication of the substrate (minimal medium) and yield coefficients [11], [8] Microorganism

Carbon source

Y xs , g/g

Calculated∗ Y”xs , C-mol/C-mol

Calculated∗ ηo 2

Calculated∗ Y’xo 2 , C-mol/mol

Candida utilis Candida utilis Candida utilis Saccharomyces cerevisiae Penicillium chrysogenum Escherichia coli Escherichia coli Aerobacter aerogenes Aerobacter aerogenes Bacillus subtilis

glucose glucose ethanol glucose glucose glucose glucose glucose mannitol glucose

0.504 0.515 0.687 0.556 0.483 0.377 0.522 0.404 0.524 0.4517

0.593 0.606 0.620 0.654 0.568 0.444 0.614 0.475 0.623 0.531

0.618 0.632 0.431 0.682 0.592 0.462 0.640 0.495 0.600 0.554

1.553 1.645 0.726 2.056 1.394 0.825 1.707 0.942 1.437 1.191

∗

For calculation the values of Table 3 were used.

(OUR) [17]. Luong and Volesky found the correlation factor to be [18]: Qg = (0.465 ± 0.084) ·OU R

(2.23)

This correlation is predicted by the theory of the degree of reduction and the thermodynamic treatment of the energetics of growth (see Section 2.1.3.1). The correlation is independent of growth rate, substrate, or microorganism. The theoretical correlation factor was calculated to be 0.460 kJ/mmol [13], [14], [19]. The theoretical value is very consistent with the measurement performed by Luong and Volesky. The measurement from Cooney et al. is slightly higher (0.519 ± 0.013 kJ/mmol).

2.2. Sterility Sterility is defined as a state being free from microorganisms or spores which could reproduce themselves (CEN-prEN 13092, 1998). This absolute statement regarding the absence of viable microorganisms cannot be proven. It can only be said that no organisms were present that were capable of growing under the specific test conditions [20]. In practice processes will be called sterile, when established and recognized methods of sterilization are used. The process of inactivation of viable microorganisms during a sterilization procedure is usually described by an exponential function. Mathematically the results of such functions can be reduced to very low fractions, but never to zero. A degree of inactivation or a certain probability of contamination can be calculated. Different administrations [21] accept a probability of

contamination for samples (infusion flasks, ampoules) of 10−6 . It means that one out of 1 million samples might be contaminated. The sterilization procedure used for the bioprocess should guarantee a contamination probability of < 10−3 . A sterile system inoculated with only one desired form of life, or a pure culture, is called “axenic”. 2.2.1. Thermal Microbial Death Rates When organisms (microorganisms, viruses, plant or animal cells) are exposed to dry heat or saturated pressurized steam their death rate is proportional to their number: dN = −k ·N dt   N (t) ln = −k ·t No

(2.24) (2.25)

The proportionality factor k depends on the temperature and the type of microorganisms. For more convenience the decimal reduction time Dtemp was defined. It is the time required to reduce the amount of microorganisms by 90 % of the initial value (for a given temperature expressed in ◦ C).  ln

1 10



Dtemp =

= −k ·Dtemp 2.303 k

(2.26) (2.27)

The relation between the decimal reduction time and the temperature is represented in the thermal killing time curve. The microorganism specific z

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value indicates the temperature increase (in ◦ C) required to decrease the D value by a factor of 10.  log

z DT 1 z DT 2

 =

T2 − T1 z

(2.28)

The F ztemp value is the lethal time in minutes of a sterilization process. This calculation assumes a specific organism (therefore indicating z), an initial number of these organisms, N o , and a final number of organisms after sterilization, N s .  z Ftemp = log

No Ns

 z ·Dtemp

(2.29)

The advantage of the calculation is the quantification of the lethal effect in terms of the F ztemp . Based on this theory, equivalent processes at other temperatures and sterilization times can be calculated and compared (Table 5): FTz1 FTz2



= 10

T2 − T1 z



(2.30)

Several examples are given in Table 5. Table 5. Lethal time and decimal reduction time at different temperatures of a steam sterilisation. A sterilisation at 121 ◦ C for 20 min is regarded as safe. A microorganism with D6121 = 1.5 min is considered. Values from [22] ◦

∗

Temperature T , C Lethal time for a constant lethal effect F 6T , min

D6T ,

100 105 110 115 120 121 125 130 135 140

4743 696 102 15.0 2.20 1.50 0.323 0.047 0.007 0.001

63 246 9283 1363 200 29 20 4.309 0.632 0.093 0.014

min

∗ Required lethal time to achieve the same lethal effect as with F 6121 = 20 min.

The decimal reduction time depends strongly on the organism and on the sterilization procedure. Dry heat (e.g., Bacillus subtilis: D23 121 = 30 min) is much less effective than saturated steam (e.g., Bacillus stearothermophilis: D6121 = 1.5 min) [22]. When considering an initial concentration of microorganisms (specified with D6121 = 1.5 min) of N 0 /V L = 103 CFU/mL, a working volume V L = 100 m3 and a final number of organisms of N s = 10−3 the required lethal time F 6121 can

be calculated to be 21 min. The probability of contamination is 1 to 1000. However the difficulty of sterilizing a bioreactor is not the determination of the required lethal time for sterilization or the determination of the initial cell concentration, but to make sure that saturated steam and temperature prevail in all dead-ends, crevices and lowest points of the reactor. 2.2.2. Steam Sterilization Processes When performing steam sterilization, it is most important to evacuate all air and other gases from the system to be sterilized. First because the lethal effect is dramatically reduced by air in the system and second because when controlling the temperature of a system with an airsteam mixture a higher pressure will be obtained compared to a system filled with saturated steam only. A system with 50 % air will have an absolute pressure of 3 bar at 121 ◦ C while a system with saturated steam has 2 bar at 121 ◦ C. 2.2.2.1. Sterilization in the Autoclave Small bioreactors made of glass (3 to 15 L) are often sterilized in an autoclave. Stainless steel vessels of several hundred liters on wheels are seldom autoclaved. The advantage of an autoclavable system is that the reactor and all connections to transfer lines are sterilized together. The complete system is therefore relatively simple with a small number of valves and clamps. The disadvantages are multiple: • The transfer in between the production place and the autoclave is always accompanied with the risk to damage the fragile glass vessel. • It is not always possible to guarantee a complete removal of air out of the tubes, lines, and headspace of the reactor. To overcome this disadvantage some autoclaves have the possibility to apply a vacuum to the system prior to steam sterilize it. • Due to lack of mixing of the media, the heatup times in the autoclave are long and true temperatures difficult to check. The inhomogeneous condition lead to long sterilization times, which might harm the media.

Biochemical Engineering 2.2.2.2. “Full” Sterilization In Place (SIP) A “full” SIP procedure heats up the liquid in the reactor (media, parts of the media, or water) to generate the steam necessary for steam sterilization. Heating jackets, coils, electrical heat fingers or direct steam injection can be used to heat up the liquid. The temperature of the liquid is controlled and homogeneous conditions achieved by mixing. Air or other gases must be evacuated from the reactor. When leaving the upper point of a reactor open, the generated steam will expel the air at around 100 ◦ C. After this step the reactor is closed to pressurize it. “Full” SIP procedures are fast, do not require an autoclave, and enable good quality assurance compared to sterilization performed in an autoclave. These advantages are paid with increased complexity. Transfer lines and filters require several valves and need to be sterilized separately in most cases (see Section 5.3). 2.2.2.3. “Empty” Sterilization In Place (SIP) Empty SIP is performed by injecting clean steam in the empty vessel. Heat-up time might be supported by heating the liquid in the temperature jacket (or coils) or by draining the liquid in the jacket prior to starting the sterilization. Air in the vessel has to be evacuated through the top and/or bottom, while condensate is drained at the lowest point of the vessel. The main steps are the same as for a “full” SIP. When cooling down, high flow rates of sterile air are required to break the vacuum at around 100 ◦ C. Empty SIP is preferred when heat sensitive media are used (cell culture) or to reduce the heat and cooling times (especially large vessels 10 to 500 m3 ). Media require a separate sterilization procedure before being added to the sterile reactor. A disadvantage is that pO2 and pH probes suffer more during an empty SIP compared to a full SIP. 2.2.2.4. Continuous Water/Steam Sterilization [23], [24] Instead of being sterilized batchwise in the reactor or the media tank, medium can be sterilized continuously. Generally shorter holding times (≈ 90 s) at higher temperatures (> 130 ◦ C) are used (Table 5).

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The process requires a pump to pressurize the medium; a heat exchanger to heat up the medium to the sterilization temperature; a holding section; a heat exchanger to cool the medium; a pressure relief valve (Fig. 9). Before starting the sterilization procedure, the equipment itself must be sterilized. The liquid flow of an ideal continuous sterilization process has a plug-flow characteristic. Plug-flow or low axial dispersion is achieved with a turbulent liquid flow (Re > 2 × 104 , high P´eclet number: Pe > 1000) [23], [24]. 2.2.3. Sterile Filtration (→ Filtration) During cultivation several flows enter and escape the reactor. The entering flows must be sterilized to avoid contamination of the reactor, while the escaping flows should be sterilized in order not to contaminate the environment. This is done by filtration. Liquids and gases can be filter sterilized to withdraw microorganisms and viruses (see Section 5.8). A filter is characterized by the removal of particles of a stated size at a stated efficiency. Such ratings can be absolute removal (100 % for membrane filter), or percent removal of a specific particle size fraction (depth filter). The absolute filter rating is the diameter of the largest hard spherical particle that passes through the filter under specified test conditions (e.g., flow rate, temperature, pressure drop). A filter or filter cartridge is usually located in a filter housing which requires to be sterilized with all involved tubing before operation. This is usually performed by steam sterilization (Fig. 11). Absolute filters (membrane filters) have in comparison to depth filters a low capacity of retaining particles before clogging. To take advantage of both types of filters first a depth filter immediately followed by an absolute filter can be installed. While the depth filter retains most of the particles without clogging, the absolute filter ensures that all particles are retained (Fig. 8). 2.2.4. Incinerators In bioreactor systems incinerators are applied for sterilization of exhaust gases. Processes involving pathogenic microorganisms might use

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such a procedure as a second barrier in the exhaust-gas line (Fig. 1). An incinerator operates continuously between 350 to 450 ◦ C (CEN, prEN13095, 1998). The exhaust gas flows either directly through a flame or is heated by electrical heaters.

2.3. Cleaning After running a process, the reactor must be cleaned. Deposits of sugar, fats, proteins and salts must be removed to avoid contamination and pollution of the following process. Residues of fungi, baking media in the upper region of the heating jacket (see Section 5.1.1.5), and the headspace of a tank or the exhaust gas line (foam) are among the most difficult parts to clean. The cleaning process is influenced by four parameters: • Physical shear stress (e.g., mixing, fluid velocity) • Temperature • Chemical influence, efficiency of the chemical agent • Time required for cleaning Generally smooth surfaces without crevices, edges, and fissures are easier to clean. Validated cleaning processes require the measurement of selected “key” components occurring in the biological and in the cleaning process. The measured values must be below a critical value, which is considered tolerable. Extremely accurate measurements and detectors are therefore required (e.g., determination of total organic carbon (TOC), DNA test, enzyme-lined immunosorbent assay ELISA test, Lowry’s nonspecific protein detection). However the definition of “clean”, the required tolerances and the procedures often lead to controversial discussions. 2.3.1. Manual Cleaning Manual cleaning is performed in small and in very large reactors. Often the dirty reactor is first rinsed with normal tap water. The vessel is afterwards filled with cleaning solution, the solution in the reactor powerfully mixed and temperature regulated. Different cleaning steps can follow each other (DIN 11483):

• 0.1 – 1 M NaOH 60 to 140 ◦ C for several hours • 0.1 – 0.5 M H3 PO4 or HNO3 up to 90 ◦ C for less than 1 h • Protease solutions for several hours At the end and in between the steps the vessel is rinsed with tap water, demineralized water, or water for injection (WFI), depending on the required quality of cleaning. Laboratory- and pilot-scale reactors are partly dismounted for in-depth cleaning. Sensors, baffles, shaft plugs, or other parts in the vessel should be removed to allow separate cleaning. Soft sponges, brushes, or tissues are used to physically remove the dirt. It must be avoided to scratch or damage the surface of stainless steel parts. Thorough procedures imply the removal and greasing of O-rings and O-ring grooves. Manual cleaning in larger reactors requires a crane to remove the lid or a manway to get into the vessel. The reactor can be cleaned with a high-pressure water injector, either from the top or by a person inside the vessel. 2.3.2. Cleaning in Place (CIP) Dismantling a reactor for manual cleaning is always time consuming. Cleaning of reactors in place (CIP), especially when being partly or completely automated is fast and more convenient. In general for CIP the reactor needs to be equipped with CIP supply and return lines and spraying nozzles. Cleaning solutions are prepared in a “CIP kitchen”. A CIP-kitchen allows the preparation (concentration and temperature adjustment) of cleaning solutions (Fig. 2). To adjust the concentration conductivity probes or scales might be used. For temperature adjustment usually an external heat exchanger with a recirculation pump is installed. CIP kitchen with multiple tanks have higher investment costs but require less time for cleaning, since cleaning solutions are stored ready to use in holding tanks. The reactor to be cleaned in place requires additional valves to supply the cleaning solutions through transfer lines into the vessel. Spraying nozzles must be placed in the top area of the

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Figure 1. Incinerator: continuous sterilization in the exhaust gas-line

Figure 2. Example of a CIP kitchen preparing and supplying the cleaning solutions to a plant consisting of four vessels (R1 to R4). A preparation and a cleaning step are indicated.

vessel in order to cover all surfaces with cleaning solution. To avoid shadows on parts installed in the vessel, several spray nozzles or special spray valves need to be installed (Fig. 3). To clean pipes, flow velocities of 1.5 to 2.5 m/s are required. Vessels should be covered with at least 30 – 40 L · min−1 · (m circumference of

vessel)−1 . To avoid aerosols the pressure drop on the nozzles should be below 2.5 bar. Typical cleaning blocks consist in: (a) flushing with tap water, (b) flushing or recirculation with cleaning solution 1 (usually an alkali solution), (c) flushing or recirculation with cleaning solution 2 (usually an acid solution), and (d) final

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Figure 3. Spray nozzles and application examples to clean a stirred tank [25] A) spray ball with action radius of 360 ◦ and flow rates from 1 to 15 m3 /h; B) Stirred vessel requiring a minimum of two spray balls; C) Spray valve; D) Application to sparge directly on critical parts a) Spray nozzle part; b) Flange on vessel; c) Condensate out; d) Pneumatic actuator; e) Cleaning detergent or steam; f) Cleaning detergent outlet (spray direction)

rinse with WFI or deionized water. In between and at the end of those blocks the equipment is blown-out with air. The quality of the cleaning procedure can be estimated by measuring the conductivity of the rinsing liquid in the return line. When WFI solution is used to rinse the equipment the criteria for efficient rinsing is below 1 to 5 µS/cm. Conductivity probes without temperature compensation should be used for this purpose [26]. Good CIP systems are characterized by efficient cleaning, short cleaning times and low water consumption.

2.4. Homogenous Conditions due to Mixing Efficient mixing (→ Mixing, Introduction) in fermentors and bioreactors is one of the main objectives to reach: • Homogenization: equalization of differences in concentration, temperature, and pH • Intensification of heat transfer between the heating surface and the liquid • Dispersion of gas in liquid and thereby intensification of the gas-liquid mass transfer • Suspension of solids such as microcarriers or pellets which must be kept in suspension

Biochemical Engineering Mixing in reactors is limited by: • The difficulty in removing the dissipated energy generated by mixing • The intensive foaming of the medium due to mixing and gassing. • The gas hold-up requiring additional volume in the bioreactor • Shear sensitivity of the microorganisms Two main categories of mixing are usually distinguished, the mechanical and the pneumatic mixing. When considering sterile application, about 95 % of the reactors are mechanically and 5 % are pneumatically mixed. Mixing in almost all mechanical mixed reactors is achieved by stirring. The turbulence in the media is generated by rotating impellers and in most cases promoted by static baffles. 2.4.1. Mechanical Mixing Three categories of mechanical mixing can be distinguished: • Stirring by a rotating shaft with impellers, (largest category, about 98 % of the cases). • Vibromixing, based on a translatory vibrating instead of a rotating motion of a shaft. • Hydraulic mixing, usually using a rotating impeller inducing a liquid flow, which is afterwards dissipated to create turbulence.

2.4.1.1. Stirred Tank Reactors – Energy Dissipation A broad range of impellers were investigated for stirring. Mainly two types are used for reactors: The six blade turbine impeller (also called turbo or Rushton impeller) for aerobic growth of microorganisms requiring high OTR and the marine impeller (propeller) for gentle mixing at low shear rate (e.g., animal cell cultures). The Rushton impeller is chosen in the following to calculate the dissipated energy in a stirred tank reactor (STR). When mixing is investigated the rheology of the fluid has to be considered. Broth of most microorganisms and animal cells have a Newtonian fluid characteristic and keep it over the cultivation time [24].

1131

The general procedure to calculate the power input of stirrers is widely described in literature [27], [24], [8] and consists in calculating the power input of a nongassed system Po and adjusting the calculation to the gassed system Pg . To determine power input Po the dimensionless power or Newton number Ne was defined by Rushton et al. (1950) [28]: Ne ≡

Po r ·n3 ·d5i

(2.32)

The “power characteristic of a stirrer” is given in the graph correlating the dimensionless Reynold number for stirrers to the power number: Re ≡

n·d2i v

(2.33)

with n (s−1 ) the rotational speed, d i (m) the diameter of the impeller, ν the kinematic viscosity (m2 /s) of the broth and ρ (kg/m3 ) the density of the broth. Since reliable kinematic viscosities and densities of the cultivation media are lacking, the viscosity and density of water at 30 ◦ C is used for calculation (ν = 0.8 × 10−6 m2 /s, ρ ≈ 1000 kg/m3 ). The “power characteristic of a stirrer” can be separated into three types of flow regimes: laminar, transient, and turbulent. A Rushton impeller with baffles in the vessel reaches turbulent mixing when Re > 5 × 103 . Under these conditions the power number is constant (Ne = 5, for other flow regimes or stirrers. Thus the ungassed power input of one impeller Po under turbulent flow regime is: Po = 5·r ·n3 ·d5i

(2.34)

The power input of a gassed impeller Pg is reduced compared to a nongassed system. Consequences can be quite dramatic and even result in a flooding of the impeller. At that stage the impeller is completely surrounded with gas bubbles and its mixing efficiency dramatically reduced. To estimate the gassed power input again a dimensionless factor, the aeration- or flow-rate number Q is introduced. Q=

Fg n·d3i

(2.35)

with F g (m3 /s) the volumetric aeration rate. A graph showing the gassed power number Ne as

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a function of the aeration-rate number Q is used for the estimation of Po and Pg of single impellers. With aeration rate numbers Q > 0.05 the gassed power Pg is reduced by 50 % compared to the ungassed power Po . The impeller can be considered to be flooded. If multiple impellers are used and if these are properly spaced (distance in between the impellers 1 to 2 times the diameter of the impeller), Po can be multiplied by the number of impellers to obtain the total ungassed power absorbed in the broth. 2.4.1.2. Vibromixer A vibromixer consists of a disk with conical perforation, perpendicularly attached to a shaft moving up and down with a controlled frequency and amplitude. By increasing the amplitude and the frequency of the translatory movement the turbulence in the reactor is increased. The sealing of the shaft is achieved with a membrane or bellows. Vibromixer do not require a dynamic sealing, are a closed system and are therefore considered as very safe (containment). The mixing is efficient and gentle (low shear forces). Vibromixers find application in production of tetanus toxin and animal cell culture. 2.4.1.3. Hydraulic Mixing In hydraulic mixing turbulence is created with a hydraulic pump. Two-phase nozzle injectors generate small gas bubbles by dissipating the kinetic energy of the liquid [29]. Liquids with low coalescence can efficiently be mixed and aerated with injectors. However aseptic design of pumps in a reactor system is always a problem and probably the reason why until now injectors are mainly found in wastewater treatment. In fluidized-bed (Figs. 15 and 16) or fixedbed reactors (Fig. 17), hydraulic pumps might be used to circulate the media. These two processes require an additional gassing system to allow gas-liquid mass transfer. 2.4.2. Pneumatic Mixing Pneumatic mixing is achieved by sparging gas into liquid. Bubbles rising in the liquid will lo-

cally vary the density of the broth and thereby generate turbulences and liquid flow. Bubble sizes will depend on the type of the sparger and the coalescence of the media. No mechanical impeller or other device is used to crush the bubbles. When bubbles freely rise, the bioreactor is called bubble column. Airlift reactors might be built with a concentric draft tube. In such units bubbles usually rise outside the draft tube and create thereby a liquid upflow (riser), on the inner side of the tube the liquid flows downwards (downcomer). Downcomer and riser can also be built in separate tubes (loop reactors). Draft tubes, riser, and downcomer are guiding the bubbles and thereby the liquid flow. A better control of the flow is thereby achieved, whether this results in improved mass transfer or homogeneity is questionable [16]. Large bubble column bioreactors (> 100 m3 ) produce excellent results, when using spargers that efficiently disperse the air bubbles. Characteristics and main advantages of pneumatically mixed reactors include: • Simplified design and low maintenance costs compared to mechanically agitated systems (mechanically moving parts have not to be introduced and sealed in the aseptic area) • Mixing and gassing is performed in a single process • Gentle mixing at low shear forces When related to the necessary power input, heat and mass transfer are excellent; absolute values will be lower than in the STR (Rushton impeller) • Airlift or bubble columns are usually long and narrow and therefore require only a small base area This type of reactors is used for large bulk productions and animal cell cultures. The main parameters influencing the mixing and the gas – liquid mass transfer are the geometry of the gas sparger, the gas flow rate, the length of the reactor, the coalescence of the bubbles, and the rheology of the broth. Higher gas flow rates will increase the turbulence and the mixing. Small bubble size will improve the gas – liquid mass transfer.

Biochemical Engineering

2.5. Gas – Liquid Mass Transfer The oxygen demand of the cells (OUR, see Section 2.1.3.2) and the low solubility of oxygen in water, require a continuous transfer of oxygen from the gas phase into the liquid phase. This mass transfer is called oxygen transfer rate (OTR). On the other hand product gases, such as CO2 or H2 S from aerobic and anaerobic cultivations, might accumulate in the broth and inhibit growth. To prevent inhibition, these gases have to be stripped from the medium. With a high gas – liquid mass transfer, the stripping effect is more effective. 2.5.1. Solubility of Oxygen and Other Gases The solubility of gases in liquid media is described by Henry’s law. These solubilities vary as a function of the temperature and the concentration of salts in the solution (Table 6). The solubility of CO2 in water is about 30 times higher than that of oxygen. −3 Table 6. Oxygen solubility c∗ · bar−1 ) for various o2 (mol · m temperatures in demineralized water (unless otherwise described)

Temperature T , ◦ C 15 20 25 30 30 35 37 40

−3 c∗ · bar−1 ] o [mol · m 2

the partial pressure of oxygen in the gas phase (or saturation concentration) and c¯ o2 (mol/m3 ) the mean oxygen concentration in the broth. From this equation, reliable guidelines regarding the design of a STR for aerobic processes are extracted. In order to increase the oxygen transfer rate, either the mass transfer coefficient k L or the specific interfacial area a have to be increased. The interfacial area a increases, when the intensity of the dissipated energy (power input per reactor volume) or when the gas flow rate increases. Another possibility consists in increasing the saturation concentration c∗ , by either increasing the percentage of oxygen in the inlet gas or the overall pressure applied on the broth. The relation further indicates, what will happen during an aerobic cultivation. At constant k L a and c∗o2 and with increasing oxygen demand of the cultivation, the OTR is increased by a decrease of the oxygen concentration in the broth. If c¯ o2 drops below a critical concentration (Table 1), growth will be oxygen limited. When considering a biological process with maximal OUR (from Eq. 2.22) the maximal required k L a can be calculated for a maximal allowed driving force (c∗o2 − c¯ o2 ) using Equation (2.36), assuming that the OUR equals the OTR: kL amax =

1.54 1.38 1.26 1.16 0.75 to 0.875 ∗ 1.09 0.96 ∗∗ 1.03

∗

In a broth used to cultivate microorganisms. ∗∗ In a cell culture medium.

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µmax ·xmax

 ·m ∗ Yxo ¯o2 max bio · co2 − c 2

(2.37)

When designing a STR the geometry and the rotation speed must meet the maximal required k L a value. Several correlations exist between the gassed power input, the aeration rate and the mass-transfer coefficient (see Section 2.5.3.2). 2.5.2.1. Liquid Mass-Transfer Coefficient kL

2.5.2. Oxygen Transfer Rate for Stirred Tank Reactors The OTR (mol · m−3 · h−1 ) per unit of reactor volume V L in an ideally STR is given by:

OT R = kL a· c∗o2 − c¯o2

(2.36)

where k L (m/h) is the liquid mass transfer coefficient, a (m2 /m3 ) the specific interfacial area (gas to liquid phase) of mass transfer per liquid reactor volume, c∗o2 (mol/m3 ) the hypothetical oxygen concentration in the broth in equilibrium to

The reciprocal of k L is the resistance to gas – liquid mass transfer in bubble aeration (two-film theory, → Transport Phenomena, Chap. 3.3.2.). The value depends on the diffusivity Di of the gas i in the liquid. For stagnant media the correlation k L ∼ Di exists. For moving bubbles with mobile surfaces the√following correlation was demonstrated: kL ∼ Di (penetration theory). These correlations are useful, when the masstransfer coefficient of another gas or at another

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temperature has to be estimated. The mass-transfer coefficient k L,i of a gas with Di in a given system can be estimated if the k L,j for a gas with the diffusivity Dj is known: ! kLi = kLj ·

Di Dj

(2.38)

2.5.2.2. Specific Interfacial Area a The interfacial area a is the surface area A of the gas – liquid interface divided by the reactor working volume V L . To increase the interfacial area the gas-flow rate or the gas hold-up in the gas-liquid mixture can be increased. Another possibility, for a given gas hold-up, consists in decreasing the bubble size. Many small bubbles will have a larger interfacial area than some few large bubbles. Parameters such as the dissipated energy in the gassed liquid, the geometry of the gas sparger, the coalescence of the bubbles and surfactants in the media therefore influence the interfacial area. The equations correlating parameters with the mass-transfer coefficients usually are indicated with tolerances of 20 to 40 %. The reason for this large tolerance is found in the difficulty to estimate the coalescence and thus the a of the k L a coefficient. Zlokarnik formulated the “multiplying factor” m, relating the k L a of deionized water to solutions of different salt concentrations [29]. m=

(kL a)solution (kL a)pure water

(2.39)

Multiplying factors of up to 6 were found for solutions with high salt concentrations. Surfactants used as antifoam agents considerably reduce m. Low viscous Newtonian-type media used to cultivate bacteria or mammalian cell cultures often have multiplying factors in the range of 1.5 to 2.

working volume V L . The volume increase per total working volume V L is the gas hold-up. The gas hold-up is influenced by the foaming properties or the coalescence of a medium. The gas hold-up of a nonfoaming, bubble coalescing medium will be in the range of 3 to 12 % compared to 5 to 25 % for a foaming medium [30]. Calculations of the gas hold-up are based on the power dissipated per reactor volume Pg /VL and the superficial gas velocity us [8]. Since the gas hold-up is related to the interfacial area a, foaming media have higher OTR values than nonfoaming media. When an antifoam agent is added to a foaming medium, foam will immediately disappear and the gassed liquid level be lowered. At the same time the measurement of oxygen partial pressure in the broth drops. This demonstrates that the driving force (c∗o2 − c¯ o2 ) is increased to compensate the effect of a lower k L a value. To achieve high OTR values in foaming media, mechanical foam breaker instead of chemical antifoam agents should be used. Fermentors should be designed to allow high gas holdups (25 %). 2.5.3.2. Volumetric Mass-Transfer Coefficient in Stirred Reactors Van’t Riet compared data on mass transfer measurements in stirred tanks with volumes in the range of 0.0025 to 5.1 m3 and power per reactor volumes Pg /V L in the range of 0.5 to 10 kW/m3 [31]. According to these measurements he proposed a correlation for “totally” coalescing media (air sparged in pure water):  kL a = 93.6·

2.5.3.1. Gas Hold-up and Foaming in Stirred Reactors When aerating a stirred tank reactor the gassed volume will be higher compared to the ungassed

0.4 ·u0.5 ± 20 − 40% s

(2.40)

where us (m/s) corresponds to the linear gas velocity (us = F g /(π · D2t )). For media with salts repressing the coalescence: 

2.5.3. Aeration of Stirred Tank Reactors

Pg VL

kL a = 7.2·

Pg VL

0.7 ·u0.2 ± 20 − 40% s

(2.41)

These high tolerances and differences in the exponent indicate the difficulty to predict the masstransfer coefficient. Generally, when designing a STR the Pg /VL should not exceed 5 kW/m3 . When using higher

Biochemical Engineering Pg /V L the mass transfer efficiency (mass-transfer coefficient per power input) is considerably decreased. To further increase the OTR not an increase in the k L a value but alteration of other parameters such as higher oxygen partial pressures or reduced usage of antifoam agent (using, e.g., mechanical foam separator instead) should be considered. The air-flow rate indicated in the correlation as us is another important design criterion. For bioprocesses requiring high OTRs an often found rule of thumb is the aeration with 1 to 2 vvm [(gas volume) · (liquid reactor volume)−1 · (minute)−1 ]. When using this rule of thumb, one should be aware that with higher reactor volumes, us will considerably increase [32]. By increasing us , the gas hold-up is increased; the gassed power input Pg and the heat transfer are reduced. High us might also create excessive foaming. 2.5.4. Volumetric Mass-Transfer Coefficient in Airlift Reactors In bubble columns due to oxygen transfer, bubbles reduce their oxygen partial pressure when rising to the surface. Smaller bubbles might flow downwards and decrease even more their oxygen partial pressure. This situation complicates the modeling of the system because it is questionable whether really an ideally mixed STR can here be considered. Other models for long and narrow reactors [33] or loop reactors [34] might be more appropriate. However for a first approach the model of the STR (Eq. 2.36) is used here. For bubble columns the mass-transfer coefficient k L a is proportional to the linear gas velocity [33]: kL a ∝ us

(2.42)

Combining this proportionality with Equation (2.36) shows that the OTR is directly proportional to the: • Concentration difference (c∗o2 − c¯ o2 ) • Liquid height in the reactor H L • Gas flow rate F g This correlation is valid for noncoalescing media and for us values below 200 m/h [33].

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Maximal gassing rates for standard airlift reactors are indicated with 720 m/h [35]. To calculate the mass transfer coefficient the dimensionless sorption number Y has been introduced: Y ≡

kL a · us



v2 g

1/3 (2.43)

The sorption number is a function of the final stable bubble diameter d e in noncoalescing systems (Table 7), it decreases with increasing bubble diameter. The kinematic viscosity of water is ν = 0.8 × 10−6 m2 /s; the acceleration due to gravity g = 9.81 m/s2 . Table 7. Sorption number Y for bubble columns as a function of the final stable gas bubble diameter d e or for gassing devices. Only noncoalescing media are considered [33] Stable bubble diameter d e or gassing device

Sorption number, 105 · Y

0.5 mm 0.6 mm 0.7 mm 1.0 mm 2.0 mm Sparger with frit and (us ≤ 180 m/s) Bubble column, sparger with 1 mm Ø holes

129 107 92 64 32 14 9

2.6. Measurement and Control Loops To maintain optimal growth of biomass, several parameters such as the temperature, pH, pO2 , and a sufficient substrate concentration must be maintained in the reactor. Sterilizable sensors in the reactor can measure on-line. By taking samples of the broth, off-line measurements can be performed. 2.6.1. Temperature Measurement and Control The temperature of the liquid is usually measured with a Pt-100 sensor (“Pt 100” because it provides a resistance of 100 Ω at 0 ◦ C). Usually a PID (Proportional Integral Differential) controller activates control elements (valves) for heating or cooling (Fig. 4) An accuracy of ± 0.5 ◦ C should be achieved. In small reactors heating is achieved by activating an electrical heater immersed in the broth. To cool the reactor coolant is circulated through a coil in contact with the broth.

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Larger reactors (10 to 3500 L) usually have a jacket with a temperature circuit. A pump circulates a liquid (usually tap water) through the jacket. A multitude of possibilities exist to heatup the fluid. For relatively small units (10 to 150 L), and when steam is not available, an electrical heater can be mounted in the loop. Steam can also directly be injected in the loop through a steam injector. A popular solution consists in mounting a heat exchanger in the loop, heatedup with industrial pressurized steam (Fig. 4).

complexity (O-rings, crevices, edges) to the system is questionable. The second method does not change the internal pipework, is a flexible system, but measures the temperature on the outer side of the pipe. Temperature differences between the outer measurement and the inner temperature of less than 3 ◦ C can be reached with a device shown in Figure 5.

Figure 4. Temperature control of a jacketed bioreactor. The temperating fluid is heated and cooled by two heat exchanger and circulates through the jacket.

To cool down the reactor, tap water is either directly injected into the loop or supplied into a heat exchanger. Heat exchanger for cooling allow the use of closed chilled water systems such as glycerol – water (Fig. 4). Very large vessels (> 1500 L) might have several parallel temperating loops either flowing through different sections of the jacket or into separate coils. Temperature measurement might also be used to trace the sterilization procedure at several “critical spots” (Section 3.1.2). Either special fittings, allowing temperature probes to be mounted, are welded into the pipework or the measurement is performed on the outer side of the pipe. With the first method the exact temperature is measured inside the pipe but the added

Figure 5. Measurement of the temperature on the outer side of the pipe to control the sterilization process. The temperature difference between the inner and outer side is less than 3 ◦ C (at 121 ◦ C internal temperature). The example shows a measurement on a sterile cross.

2.6.2. Agitation Control To guarantee homogeneity of the parameters the agitation speed of the stirrer should be continuously controlled. A minimal speed is normally set at about 10 % of the maximal rotation speed

Biochemical Engineering of the motor. Gears or pulleys with belts are used to reduce the rotating speed of the electrical motor. Compared to d.c. motors a.c. motors do not require any maintenance. Usually an accuracy of ± 3 % of the maximum rotation speed of the motor is acceptable. 2.6.3. pH Measurement and Control For pH measurement and control in-situ sterilizable pH sensors measuring a potential difference (+ 500 to − 500 mV) between the broth and a reference electrolyte of pH 7 are in use. Before installation the sensor has to be calibrated in buffer solutions of known pH. The sensor is in-situ sterilized. Probes with liquid electrolyte (3 M aqueous KCl) require to be pressurized during sterilization to prevent penetration of broth or steam into the electrolyte (through the diaphragm or the membrane). pH electrodes with gel electrolyte need not be pressurized, their electrolyte however can not be exchanged for maintenance purposes. Membrane and diaphragm of the sensor must be cleaned on a regular basis (pepsin solution to clean the diaphragm, the glass membrane must be etched). A proportional band controller with deadband should be used for pH control. The output of the controller can for example activate a liquid pump or valve in the transfer line supplying acid (e.g., 1 – 5 M H2 SO4 ) or alkali solution (e.g., 1 – 5 M NaOH, 20 % NH3 ) to the media. The supply of gaseous ammonia raises the pH and balances the nitrogen demand of a culture (e.g., E. coli) without diluting the media. To reduce the pH of an animal cell culture, CO2 can be supplied to the inlet gas. Due to its high solubility it is often sufficient to supply the CO2 to the headspace of the reactor thereby avoiding to produce bubbles (shear stress of “exploding” gas bubbles at the liquid surface might harm the cells [36]). To increase pH, CO2 has to be stripped out of the media by sparging with air or N2 . pH should be controlled with an accuracy of ± 0.1. A severe problem might result from drifting of the pH electrode. While media like those used for animal cell culture usually do not tend to induce drifting of probes, others such as those

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containing sulfur compounds affect the measurement considerably. Depending on the bioprocess it might be useful to recalibrate the pH probe during cultivation. The online measurement can for example be adjusted to the measurement obtained from a sample by a single point calibration. This method is often criticized, since due to gassing out of CO2 from the sample and changed conditions (pressure, temperature) the measured value might be severely different. When accurate recalibration of the pH sensor is required during cultivation, interchangeable probe devices should be used (Fig. 47). 2.6.4. Measurement and Control of Dissolved Oxygen Dissolved oxygen is usually measured with an in-situ sterilizable Clark-type electrode immersed in the broth. A current proportional to the oxygen partial pressure (pO2 ) or the fugacity of oxygen in the media is measured. The probe should be calibrated in-situ after sterilization and before inoculation. Probes used to monitor aerobic growth are usually one-point calibrated at air saturation. When no current is measured it is assumed that the probe is not exposed to oxygen. To control the concentration of dissolved oxygen in the broth, parameters influencing the OTR must be activated (see Section 2.5): • • • •

Agitation speed of the impeller Oxygen or air flow-rate Oxygen partial pressure of the inlet gas Overall pressure of the tank

One or several of these parameters can be used to control the pO2 . A PI type controller should be used for control. The output signal of this controller can be used to change the set points of one or several of the above parameters. Such a system of two controllers in series is called cascade controller (Fig. 6). The “pO2 controlled substrate limited fedbatch strategy”, is another method to control the pO2 , which consists in feeding a controlled amount of substrate to the cultivation (see Section 3.4).

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Figure 6. O2 cascade control loops SIC = Stirrer indicator and controller; PIC = Pressure indicator and controller; FIC = (Air) flow rate indicator and controller; SP = Set point; MV = Measured value

2.6.5. Measurement and Control of the Volume or the Level In fed-batch or continuous cultivation the broth volume is an important parameter for the bioprocess. A popular but expensive method to evaluate the volume of the broth is to weigh the entire reactor. Platform scales or load scales have a precision of 0.05 to 0.1 % of the maximal load. The load of the vessel itself must therefore be considered. A vessel of a total volume of 150 L will for example weigh around 500 kg. The maximum range of the scale must therefore be above 650 kg. A platform scale for 1000 kg is for example chosen. A theoretical accuracy of around ± 1 kg results. In fact the main difficulty is to withdraw all cross influences of transfer lines connected to the vessel. These lines must

be flexible and/or bent so that they do not exert a force (or only a stable force) on the vessel to be scaled. Lateral forces on the reactor must also be avoided. The temperature dependent dilatation of the vessel during sterilization must be considered. Because of these influences the load measured at varying temperatures is often affected with an accuracy far above the ± 1 kg for 150 kg liquid. During operation it must further be considered that any manipulation on the vessel will influence the measurement. Far less expensive than weighing, are measurement principles, which determine the liquid level in the reactor. To convert the level height into a reactor volume, the cross-sectional area as a function of height should be known or better be constant. If the level is measured with the capacitive measurement principle the gas hold-

Next Page Biochemical Engineering up and the foam are parameters influencing the measurement. Another possibility consists in measuring the pressure difference in between the headspace of the reactor and the bottom, the height of the water column is thereby measured (Fig. 7). An accuracy of ± 6 mm water column can be achieved. For a 150-L tank with standard dimensions (H : Dt = 1 : 3) this corresponds to an accuracy of ± 0.8 kg. A possible cross influence is the pressure fluctuation due to turbulences (agitation, aeration) in the broth. The main advantage of this measurement principle is that it is independent of the gas hold-up of the liquid and external forces on the vessel. To control the liquid volumes, pumps or valves are activated to supply or withdraw the liquids. Media tanks and transfer-lines are required.

Figure 7. Weighing probe, the measurement of the pressure difference in between the headspace and the reactor bottom allows to quantify the water column and thereby the volume of the bioreactor

2.6.6. Measurement and Control of Gas-Flow Rate The measurement of gas-flow rates is a nonsterile operation. Sterilization occurs usually downstream of the flow meter. A cheap and reliable method to measure gasflow rates consists in using float meters (rotameters). Here, a float hovers in the upward stream of gas passing through a vertical tube that is slightly tapered, the top of the tube being wider than the

1139

bottom. The reading is affected by the gas pressure and temperature but can be corrected by calculation. Accuracy up to 2 % can be achieved. Thermal mass flow meters measure the heat transferred to the gas flow. The heat transfer is directly proportional to the mass flow. The measurement is not affected by the pressure. The measurement is very reliable for clean gases but difficult to service and recalibrate. To control the gas flow, often a thermal mass flow meter is used, followed by a continuous valve. The combination is a stand-alone unit supplied with a controller. The units often have pressure drops of 1 to 2 bar. 2.6.7. Measurement and Control of Liquid Flows Liquid flow meters are used in transfer lines and must therefore be cleanable and sterilizable. If a good accuracy is achieved the integration of the measurement is used to determine the transferred volume. To control the flow a throttle valve is activated by a flow controller. Magnetic or electromagnetic flow meters measure the voltage created by a conducting liquid flowing through a magnetic field. The voltage is directly proportional to the flow rate. Temperature, solids in the liquid, and pressure changes do not affect the measurement. Gas bubbles must be avoided. The measurement requires a conductivity of the liquid above 10 µS/cm, WFI can therefore not be measured. The Coriolis flow meter consists of a bent tube where the medium flows through. The tube is activated to vibrate at its normal frequency. When measuring the amplitude at two different points of the bent tube, a phase shift can be detected, which is due to Coriolis forces. This phase shift is proportional to the mass flow. External forces or mechanical stress on the bent tube must be avoided. Vibration of the plant might affect the measurement. If these parameters are well controlled, an accuracy of 0.5 % can be expected. Liquid flow might also be controlled by calibrated metering pumps. Piston or diaphragm pumps generating flows independently of the back pressure are suitable. The system is efficient but has the disadvantage of not measuring the flow (no feed back control).

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2.6.8. Other Measurement Systems The inlet and exhaust gas may be analyzed to calculate the OTR. Carbon dioxide is measured using infrared spectrometry. The gas must be dried prior to measurement. Oxygen is measured with a paramagnetic analyzer. In-situ sterilizable probes measuring on-line parameters such as the concentration of alcohol, the optical density, or the redox potential exist. The torque of the agitator is measured to indicate the gassed power input to the bioreactor. The pressure is measured and controlled to increase the OTR (see Section 2.5.4).

water (WFI, demineralized, or tap water) and the media components are added in form of powders or concentrated buffer solutions. The components are dissolved or homogeneously dispersed by mixing and heating the media in the reactor. Media sterilization is combined with the “full” SIP of the reactor (see Section 2.2.2.2).

2.7. Containment The release of biological material from the bioreactor has to be limited for the following reasons: • Larger amounts of proteins, which are foreign to the human body, might produce an allergic reaction in humans. • Microorganisms might be pathogenic to humans, animals or plants due to their toxicity or the toxicity of their metabolic products. Bacteria, viruses and fungi are classified in four risk classes: harmless, low risk, medium risk, and high risk (e.g., [37]). To each risk class corresponds a safety level with defined precautions [38]. As an example the minimal precautions for the exhaust gas line are listed in Table 8 (see also CEN, prEN 13095, 1998). Internal standards of large companies are often much higher than those required by the official regulations. The official or the company internal regulations have a strong impact on the design of the bioreactor (e.g., sampling system, transfer line, mechanical seal of the agitator [39]). Generally the regulations differentiate between, laboratory/pilot, and production scale processes [40].

3. Processes and Related Equipment 3.1. Media Preparation and Sterilization 3.1.1. Media Preparation Techniques The simplest way to prepare media for a cultivation, is to use the reactor directly. It is filled with

Figure 8. Combination of prefilter (depth filter) and an absolute filter (membrane filter) for the continuous filtration of liquids. Two prefilter are mounted in parallel to allow the exchange, cleaning and sterilization of one of them, while the filtration process is on-going with the other one.

Media components such as vitamins, certain amino acids, or proteins are heat-labile and might be destroyed by steam sterilization (121 ◦ C, 20 min). In order to prevent this, only heat-resistant compounds will be sterilized in the reactor and heat-labile compounds are added afterwards by sterile filtration. It is also reported that a sterilization at higher temperatures for shorter times is less harmful to certain heatlabile compounds, such as thiamin [27]. Media for animal cell cultures are usually prepared separately and added by sterile filtration into the sterile reactor (Fig. 8). Nowadays, completely sterile media are readily available for delivery in bags. The filtration, as well as the bag

Biochemical Engineering

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Table 8. Risk classes with corresponding safety precautions. Example of minimal required precautions to be taken in the exhaust gas line (pilot and production scale) Risk class

Safety precautions

Minimal required precautions for the exhaust gas line (pilot and production scale [37])

Harmless microorganisms

no treatment of exhaust gas is required

Low risk microorganisms

GILSP (Good Industrial Large Scale Practices) containment category 1

Medium risk microorganisms

containment category 2

High risk microorganisms

containment category 3

the exhaust gas is to be treated to limit the release of microorganisms to a minimum (e.g., mechanical foam separator) the exhaust gas must be sterilized by either passing it through a HEPA filter system or by heat treatment (incinerator) the exhaust gas must be passed through two HEPA filter systems. The filter must be exchanged under aseptic conditions

Figure 9. Continuous sterilization with constant holding time and variable length for different flow rates FIC = Flow indicator and controller; HE1 = Heat exchanger to heat up the medium; TIC = Temperature indicator and controller; TIS = Temperature indicator and switch; HE2 = Heat exchanger to cool down the medium

method require a sterile transfer into the reactor. Large reactors are often sterilized empty to reduce the heat-up time. Media may be sterilized continuously (Fig. 9) to fill the sterile reactor. Apart from the degradation of heat-labile components, certain reactions between components may contribute to the loss of nutrient quality during steam sterilization. A common phenomenon is the occurrence of Maillard-type browning reactions of carbonyl groups (usually from reducing sugars) with amino acids and proteins [41], [42]. To prevent the degradation a separate sterilization of the carbohydrate components (e.g., glucose, glycerol etc.) from the rest of the media is necessary. A sterilizable media preparation tank with agitation and temperature control and a sterilizable transfer line is often used to prepare the carbo-

hydrate part of the media (Fig. 10). Such a preparation tank prepares, sterilizes, and stores media used in fed-batch and continuous cultivation. 3.1.2. General Procedure of a “Full” SIP Before applying a SIP procedure to a reactor often the gas supply and the exhaust gas filter must be sterilized first. When two sections are separately steam sterilized, these are usually connected by a sterile cross (Fig. 11). The general SIP procedure of a “full” reactor can be described as follows: 1) Sensors requiring off-line calibration (e.g., pH sensors) are calibrated and mounted; the reactor is equipped with all fittings, connections, and plugs.

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2) The reactor is filled with media or another liquid. 3) The medium is heated up while being homogeneously mixed. 4) At around 100 ◦ C the liquid boils, the generated steam expels the air from the reactor. 5) Above 100 ◦ C, the reactor must be heated and pressurized to reach the sterilization temperature. Therefore the vents must be closed. Out- and incoming pipes require a flow of steam to heat them up during the sterilization process. Steam-traps or pulsed valves can be used for this purpose (Fig. 11). The temperature of such “critical spots” might be monitored to control the lethal effect (e.g., calculation of Fztemp ). 6) Depending on the sterilization strategy the temperature measured in the reactor and on all “critical spots” has to be above a critical sterilization temperature (e.g., 121 ◦ C) for a defined time (e.g., 20 min). 7) When cooling down, the pressure of the reactor drops. All valves or steam traps must be closed. To avoid an underpressure in the reactor, sterile gas should be injected to keep the pressure above ambient pressure. The exhaust gas line should be closed to avoid accumulation of condensate. 8) The reactor cools down and once the cultivation temperature is reached the air flow through the reactor should be operated to dry the inlet and outlet gas filter. Tubes dipping into medium, such as spargers, require special care during full sterilization. When simply connecting the dipped tube to a pulsing valve, steam trap, or other vent, media would flow out of the reactor, as soon it is steam pressurized (above 100 ◦ C). This must be prevented by opening a bypass in between the head space of the reactor and the line connected to the dipped tube (Fig. 11) or by directly supplying clean steam through the dipped tube. The filter is separated from the tank by a sterile cross (valves 4, 5 and 6). In a first step the filter is steam sterilized, steam is supplied via valve 1 and condensate drained through valve 3 and 6. The filter cartridge, the pipe in the direction of valve 4 and the diaphragm seat of valve 4 are sterilized and the procedure is monitored by measuring the sterilization temperature (TIR). In a second step the filled tank is steril-

ized. The medium is heated by heating the jacket of the tank. To prevent draining of the medium through the dip tube, the bypass valve 7 needs to be opened. Valve 4 is closed, valve 5 and 6 are open to drain the condensate. Again the sterilization is monitored with the TIR.

Figure 10. Media preparation tank for separate sterilization of carbohydrates or concentrated feed, steam-sterilizable, temperature-controlled, and agitated.

Figure 11. Example of a design for a gas supply filter. Separate sterilization of the filter, temperature monitoring and integrity test are the main characteristics of the design.

Valves 8 and 9 (always manual valves) are required for the integrity testing of the filter cartridge (absolute filter). Since the valves are mounted on the non-sterile side of the filter, the

Biochemical Engineering integrity test can be performed before or after sterilization of the filter. Usually the pressure decay test is performed [39]. 3.1.3. General Procedure of an “Empty” SIP The general “empty” sterilization procedure is very similar to the “full” SIP. Steam injected to the reactor directly drives the air off. The tube of the sparger or of a CIP line can for example be used to supply the steam to the reactor. Condensate accumulating on the bottom must be drained. To check the sterilization process, the temperature on the lowest point of the reactor must be traced. The design of the reactor must consider that larger quantities of air need to be removed in the heat-up phase and a higher flow rate of sterile air is required to break the vacuum during cooling (compared to the full SIP).

3.2. Inoculation The inoculation volume required for the total working volume is an important parameter when designing a plant of several reactors. Depending on the bioprocess the inoculation volume is 2 to 20 % of the working volume. Larger inoculation volumes will shorten the cultivation time, but require additional or larger reactors to produce the inoculum. If the initial amount of biomass is supposed to be proportional to the inoculation percentage (I = inoculation volume per working volume), the time savings t when starting the cultivation with a higher inoculation percentage (I h compared to the lower I l ) can be calculated as: t=

1 µmax

 ·ln

Ih Il

 (3.1)

For a given microorganism growing at µmax = 0.5 h−1 the time saving when inoculating at 10 instead of 2 % will be 3.2 h. When considering a 1500 L production reactor (1000 L working volume) the total volume of the seed reactor used to produce the inoculum will be 150 (10 %) instead of 30 L (2 %). Each project requires an analysis of the resulting process times and investment costs for the vessels. The timing must be integrated in the upstream and downstream process. Small reactors (about 2 to 30 L

1143

total volume) are inoculated with cultures grown in Erlenmeyer flasks. A fermentation plant at production scale often consists of one large production reactor, which may be inoculated via two seed reactors. This results in a higher security level of the production process, if one seed reactor is contaminated or organisms are not well growing, the other remains as inoculum.

3.3. Batch Cultivation A batch cultivation is characterized by the growth of microorganisms without supply of additional substrate after inoculation. While the substrate is metabolized, biomass and products are formed during cultivation (Fig. 12 A). The batch is stopped and harvested when the desired product concentration is maximal. Thereafter the reactor is cleaned and sterilized for the next batch. During batch cultivation several components other than substrate might be added. In aerobic cultivation air or O2 is continuously added through a sparger. To maintain constant pH, alkali or acid components are added to the broth. Antifoam agents are required for foam control. Agents might be added to the broth to induce the production of a desired protein. These examples show that batch processes require several sterilizable transfer lines between the holding tanks and the reactor. The “repeated batch” operation mode consists first of a batch cultivation. Instead of harvesting the total volume some broth remains in the reactor as an inoculum. Fresh sterile media are added to immediately restart a batch. This procedure can be continued until the production rate declines. The time-consuming cleaning and sterilization in between two batches is avoided. The problem here consists in the validation of the process. The starting conditions are not clearly defined, because residues from former batches might influence the cultivation. Batch processes are simple, reliable and therefore widely used in biotechnology. The limit of a batch process is set by the maximal initial substrate concentration. As described in Section 2.1.2.2, high concentrations of substrates might inhibit growth.

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3.4. Fed-Batch Cultivation To overcome the growth limitation due to the initial substrate concentration (see Section 3.3), substrates can be added to the broth during cultivation. This mode of operation is called fedbatch cultivation. Substrates are either supplied stepwise during the cultivation or continuously (Fig. 12 B). Depending on the bioprocess different fed-batch techniques are in use:

Figure 12. Several standard bioprocesses A) Batch cultivation; B) Fed-batch cultivation; C) Continuous cultivation; D) Continuous cultivation with cell recycle V L = Working volume; F = Flow of fresh substrate; F p = Permeate flow, cell free flow of product; F b = Bleed, flow of broth with cells out of the bioreactor; s = Concentration of substrate; x = Concentration of biomass; p = Concentration of product; so : Concentration of substrate in the flow F; D = F/V L Dilution rate

1) Constant or Predefined Growth Rate. To maintain a steady growth rate or exponential growth, the flow of substrate must, to a certain extent, increase exponentially. Linear growth of biomass is achieved by a constant flow of substrate. 2) Bioprocess Requiring a Constant Substrate Concentration. To maintain a constant substrate concentration in the media, the substrate concentration must be measured or estimated. The substrate feed-rate is adjusted according to the measurement. The feeding strategy can be automated if an on-line estimation or measurement of the substrate concentration and a controller for the feed are installed. However, measurement (estimation) of the substrate concentration is complex and difficult. The controller requires measurement kinetics, which are about 10 times faster than the growth kinetics. 3) Substrate-Limited Fed-Batch. “Manual adjustment”. Several bioprocesses require low substrate concentration for optimal production. For this method the substrate must be supplied continuously with a pump. Defined cultivation media (without complex nutrients such as yeast extract) are required. A minimal substrate concentration can be maintained, if the substrate feed rate is kept below the maximum amount, which could be metabolized by the organisms. To check the substrate limitation the supply is stopped from time to time. When in the following seconds the dissolved oxygen concentration is sharply rising, this indicates, that the process is substrate limited. The observations are used to adjust the feed rate, which is immediately started after the check.

Biochemical Engineering 4) pO2 Controlled feeding is a variation of the substrate-limited feeding strategy mentioned above. pO2 Controlled feeding is used when low substrate concentration are promoting the production. The control strategy requires a constant k L a and a constant hypothetical saturation concentration c∗O2 (stirrer speed, aeration flow rate, O2 partial pressure of the inflowing gas and pressure in the vessel must be kept constant). Under these conditions and if the fed-batch is substrate limited, the dissolved pO2 will rise if the feed rate is reduced and vice versa (see Section 2.6.4). This behavior is used to automate the controlled feeding. The fed-batch technique is the state of the art for the efficient production of bacteria and fungi. Modern biotechnology processes with genetically engineered microorganisms, often use fed-batch techniques to reach high cell densities before adding the inducing agent to start the production of the desired product. Two reasons may limit the growth of fedbatch cultivation: The accumulation of inhibiting products in the broth or lack of reactor volume for substrate addition. Optimizing a fedbatch consists in concentrating the added substrates and starting the batch with a low working volume. The minimal working volume of a reactor is therefore an important criterion. It depends on the location of the sensors which must all dip into the broth, the jacket or heat exchanger which must ensure a safe temperature control and the position of the lowest impeller (STR) to ensure good mixing. This mode of operation requires a separate tank where the concentrated feed is prepared and sterilized before being transferred.

3.5. Continuous Cultivation Continuous cultivation operation is characterized by the continuous addition of fresh media and the withdrawal of broth at constant volume [24]. Fresh nutrients promoting the growth are added, while growth inhibiting metabolic products are withdrawn from the system. The parameter characterizing the continuous cultivation is the dilution rate D: D =

F VL

(3.2)

1145

with F representing the flow rate of fresh medium and V L the working volume of the cultivation. When formulating a mass balance for the substrate and biomass, without considering the product formation or the substrate requirement for maintenance, the following equations can be written (see Section 2.1.2 and Fig. 12 C): dx = (µ (s) − D) ·x dt ds µ (s) ·x = D · (so − s) − dt Yxs,obs

(3.3) (3.4)

Of special interest are the steady states of a continuous cultivation. It shows that the specific growth rate µ(s) equals the dilution rate D. From Equations (3.3) and (3.4), and when considering the Monod kinetics (Eq. 2.3) the substrate and biomass concentration can be calculated as: D ·Ks µmax −  D x = Yxs,obs · so −

s=

D ·Ks µmax − D



(3.5) (3.6)

The maximal dilution rate Dmax before washing microorganisms out of an ideally mixed reactor is: Dmax =

µmax ·so so + Ks

(3.7)

When the biomass is the desired product the productivity P is: P = D ·x

(3.8)

The main advantage of continuous processes is that productivity is theoretically maintained for indefinitely. The turn-down time of a plant is kept low. For slowly growing organisms such as animal cells with product formation proportional to the biomass growth continuous processes are often chosen. There are two standard types of continuous cultivation. The chemostat cultivation is characterized by a continuous substrate feed, where at least one nutrient is limiting. The steady states of the cultivation are of interest and are used to characterize the growth of the organism. In a turbidostat cultivation, the biomass concentration is measured and kept constant by varying the dilution rate. In both cases a sterile media supply tank and a harvesting tank are required. Level controller, scales or metering pumps are used to control the feed rate (Section 2.6.7). The productivity of a continuous bioprocess is limited by the washing out of the cells (cell density x) and the limited dilution rate D.

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Biochemical Engineering

3.6. Continuous Cultivation with Cell Retention To overcome the production limitation of a continuous cultivation, organisms must somehow be retained in the reactor. Such a process would allow the increase of the cell density and of the dilution rate, both factors increasing the productivity (see Section 3.5). The process would uncouple the supply of nutrients and the withdrawal of inhibiting products from the cell density in the reactor. Most of the applications of continuous cultivation with cell retention are found in animal cell cultivations. Several techniques retain suspended cells in the broth while other immobilize the cells on matrices. 3.6.1. Filtration Systems for Suspended Organisms To retain suspended cells in reactors a filtration process can be used. The biomass is concentrated by withdrawing cell-free culture broth. If the desired product is dissolved in the broth (extracellular production), the procedure enables the continuous harvest in the cell-free permeate (Fig. 12 D). The production rate compared to a classical continuous cultivation is increased, because of higher cell densities and dilution rates. To continuously withdraw dead cells from the cultivation and control the cell density a small flow rate, called bleed (F b ) is taken from the reactor. Filtration units are either directly incorporated in the reactor or placed in a bypass. The first design has the advantage of being a compact system. The second is more flexible, because the unit can be dismounted, cleaned, sterilized and reconnected during cultivation. Filtration or membrane systems are characterized by the membrane material, flow/pressure characteristics across the membrane and the required membrane surface-to-volume ratio. When scaling-up a membrane system, the surface-to-volume ratio should be constant. It is the factor, that limits the possibilities of scaleup. Another difficulty is the fouling or clogging of the membrane. The transmembrane flux has the tendency to settle particles or organisms on

or within the membrane. Thus the pressure drop across the membrane is increased or the permeate flow decreased [43]. The filtration methods described in the following try to overcome this fouling using special techniques. 3.6.1.1. Cross-Flow Filtration To keep the membrane surface free of organisms, cross-flow filtration units maintain a tangential flow of the feed stream parallel to the separation surface. The aim is to provide sufficient shear at the membrane surface to keep solids and other particles from settling [44]. At a laboratory scale, filtration units built into the reactor [45] or in a bypass [46] were very successful. However the difficulty to maintain a constant ratio of membrane surface per reactor volume of about 20 m2 /m3 limits the possibilities of scale-up. 3.6.1.2. Rotor Filter The rotor filter (also called rotating sieve or spin filter) represents another method to avoid the fouling of the membrane (Fig. 13). When built in a narrow cylinder, Taylor vortices create the necessary turbulence in the vicinity of the sieve preventing solids from settling. Centrifugal forces might also have a positive influence [47]. The rotor filter is very popular for animal cell cultivation and is found in many production processes and design variations [48]. The membrane consists of a cylindrical stainless steel sieve available with various mesh sizes (2 to 60 µm). It is interesting that cells of about 5 to10 µm in diameter are retained by up to 90 % by a sieve with mesh size of 20 µm, rotating at 1000 rpm (cylinder diameter 100 mm). The permeate is not completely cell-free, but dead cells are thereby directly withdrawn from the system (bleed flow). Stable permeate flow rates of up to 40 to 150 L · m2 · h−1 can be achieved over several weeks with animal cell culture densities of up to 107 cells per milliliter. Such cell densities are reached by exchanging the working volume one to two times per day (D = 0.041 to 0.083 h−1 ). This results in reasonable sieve areas per reactor volume of about 0.3 to 2 m2 · m−3 . When placing the rotor filter within the reactor, the system does not require a pump for recirculation of broth. The bypass system can be

Biochemical Engineering

1147

Figure 13. Rotor-filter build into the bioreactor or in a bypass circuit. The rotating speed can be adjusted separately from the stirrer speed.

separated, cleaned, sterilized, and reconnected to the reactor (Fig. 13). Twin filter systems enable to switch over to a second rotor filter and clean the first in the meantime (Fig. 14). Magnetic drive gear pumps, diaphragm pumps, or peristaltic pumps are suitable to circulate the broth without cell damage. The circulation flow should be about 10 to 20 times higher than the permeate flow. 3.6.2. Immobilized Organisms Organisms are immobilized when they form cell aggregates (pellets), adhere on porous matrices, or are entrapped in beads. The possible technique depends on the considered organism and its characteristics. Immobilization is mainly applied to animal cells and mycelia type fungi. Immobilized organisms are cultivated in fixed-bed or fluidized-bed reactors. 3.6.2.1. Pellets, Porous Matrices, or Beads Mycelia type fungi can spontaneously form pellets if the media or the environment are manipulated to a certain extent (pH, addition of components). An immobilization on porous microcarrier such as celite beads (porous diatomaceous earth) is also possible. A positive side effect is a higher OTR compared to the viscous broth of mycelia [49].

Although relevant industrial mammalian cell lines, such as Chinese hamster ovary (CHO) or Baby hamster kidney (BHK) have been successfully adapted to grow in suspension, similar to “classical” suspendable cell lines like hybridoma, most primary cells (i.e., hepatocytes, epidermal cells) and most cell lines derived from tissues still need a surface for adherence in order to proliferate. Among these anchorage-dependent cells are cell lines (i.e., VERO, MDBK) for the production of viral particles used as veterinary and human vaccines. Immobilized animal cells are protected from shear forces. This allows higher mixer speed and aeration rates. Because of this advantage even non-adherent cell lines are often immobilized for cultivation with cell retention. Anchorage-dependent cells can be immobilized on porous carriers such as glass, cellulose, dextran, collagen, or gelatin. Macrocarriers have diameters of 0.4 to 5 mm and allow the cells to grow inside the porous structure. Microcarriers have diameters of only 0.1 to 0.2 mm. Here, cells grow mainly on the surface in monolayers and therefore are not so well protected. Cell densities on or in the carriers of 0.5 to 5 × 108 cells per milliliter carrier were reported. The homogeneity of parameters and the mass transfer of nutrients or metabolic products in or out of carriers might become critical for larger carriers. To immobilize nonadherent cells, carriers with modified surfaces enabling an adherence

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Biochemical Engineering

can be used. Another possibility consists in encapsulating the cells into beads or perform an entrapment in a matrix (e.g., calcium alginate, agarose, carrageenan [50], [51]).

nique to withdraw the permeate. A simple stainless steel mesh might already be sufficient for small-scale reactors (< 30 L). At larger scale a rotor filter (Section 3.6.1.2) can be used for separation.

Figure 14. Twin rotor-filter for cell retention in continuous cultivation. Flexible separate unit connected by steam sterilizable transfer-lines to the bioreactor.

3.6.2.2. Fluidized-Bed Reactors Reactors keeping the solids (carriers, pellets, or beads) in suspension are called fluidized-bed reactors. Usually three phases (solid, liquid, and gaseous) are involved. To keep the solids in suspension all types of mixing procedures can be used provided the solids are not damaged. When a settler is used the density of the solids should be slightly higher than that of the media. A good settling is achieved, using beads with densities of 1.03 to 1.045 kg/L. Reactors with settling zones allow direct withdrawal of the cell- and particle-free broth (Figs. 15 and 16). A settler in an external loop (e.g., lamella clarifier) can also be used. Stirred tank reactors, which usually do not have any settling zone require a separation tech-

Figure 15. Three phase fluidized-bed reactor for immobilized cultures. The separation system in the headspace of the reactor allows the particle-free withdrawal of media. a) Air inlet; b) Gas disperser, e.g., frit or jet; c) Sampling port; d) Jacket for temperature control; e) Draft tube; f) Circulation chamber; g) Degassing chamber; h) Separation chamber; i) Viewing glass

3.6.2.3. Fixed-Bed Reactors Instead of keeping the immobilized cells in suspension (fluidized-bed reactor) they can be trapped in a fixed bed. The system therefore does not require a separation technique to withdraw permeate. The medium is circulated through the fixed bed to supply nutrients and withdraw metabolites or products from the cells. The

Biochemical Engineering medium has to be oxygen enriched before being circulated through the fixed bed. The length of the fixed bed is the critical parameter, because the dissolved oxygen concentration in the medium decreases when flowing through the fixed bed with metabolizing cells [52]. A radial flow through a cylindrical fixed bed is proposed to allow a scale-up into length (Fig. 17 and 18) [53].

Figure 16. Fluidized-bed reactor. The medium is circulated with a peristaltic pump. Upflowing media keep carrier in suspension. In the upper part, due to the larger diameter, the media flow is reduced to allow sedimentation of carrier. a) Glass vessel with double jacket; b) Medium circulation inlet; c) Medium circulation outlet; d) Temperature control fluid inlet; e) Temperature control fluid outlet; f) Storage vessels base/acid; g) Storage vessels harvest substrate; h) Frame; i) Carrier material, pellet; j) Steel ball; k) Hydraulic pump

3.7. Dialysis Cultivation High cell densities or high production rates can be achieved by performing a dialysis process during cultivation. Extremely high cell densities of E. coli up to 174 g/L cdw were for example reached [54]. The driving force of the diffusive mass transfer across a membrane for a given compound is its concentration difference between both sides of the membrane. Mass transfer in both directions across the membrane is therefore possible [55], [57]. Nutrients can be supplied across

1149

the membrane to the cultivation while inhibiting products are withdrawn. Depending on the cut-off of the membrane compounds of higher molecular mass (e.g., > 10 000) will be retained by the membrane. That way a concentration of products can be achieved.

Figure 17. Radial flow fixed-bed bioreactor; media circulation is performed with an external hydraulic pump

Figures 19 and 20 show a reactor with an integrated dialysis membrane separating two reaction chambers. A coculture of two different strains separated by the membrane can be studied in such a laboratory system. Generally fast growing bacteria require a dialysis membrane surface ratio per reactor volume of at least 10 to 20 m2 /m3 . Reactors with a volume of more than 30 L therefore require the dialysis membrane to be mounted in an external loop [56]. Batch, fed-batch, continuous, or continuous cultivation with cell recycle can be performed in dialysis membrane reactors.

3.8. Selection of Equipment Related to Specific Processes or Products Tables 9, 10, 11 are intended to give an overview on suitable equipment for different types of or-

1150

Biochemical Engineering

ganisms, products, and the type of production. Such tables can only give a rough idea of what kind of equipment might be selected for a specific process, because each project has its special features. The huge variety in biotechnology makes it very difficult to standardize equipment for bioreactors and fermentors.

used for cultivation of extremophiles or anaerobic microorganisms. These reactors contain corrosive media (pH 1.5 – 10, NaCl concentration up to 100 g/L, sulfur compounds). Parts in the periphery of the main reactor such as storage tanks for acid (hydrochloric, nitric, phosphoric, or citric acids) or alkali (ammonia, sodium hydroxide) and transfer pipes are also exposed to highly corrosive media. Sterilizable reactors must resist temperatures up to 125 ◦ C and pressures of 1.5 bar for a short time. Temperature circuits are exposed to 140 ◦ C and 2.5 bar. Aggressive agents are used to clean the inside of vessels and pipes. In some cases ethanol, formaldehyde, or other aggressive agents are also applied to the outer side of the equipment. In that case painted ferritic steels and thermoplastics should not be used, preference should be given to stainless steel. The surfaces of materials must be smooth and easy to clean. Generally it is differentiated between materials in contact and those not in contact with the product (culture broth).

4.1. Austenitic Steel With a few exceptions, austenitic steels (with a maximum of 0.08 % carbon) are the material of choice in biotechnology. Compared to the nonaustenites, these steels show improved corrosion and heat resistance, and they are nonmagnetic. Other advantages are good tenacity and workability [58], [59]. 4.1.1. Different Alloys of Austenitic Steels

Figure 18. Photograph of a radial flow fixed-bed bioreactor

4. Materials Materials used in the construction of fermentors and bioreactors (vessel, pipes, fittings, instruments) are usually long-term exposed to moderate conditions, i.e., temperatures between 28 and 40 ◦ C, pH values between 6 and 8 and a salt concentration below 5 g/L. Exceptions are reactors

The characteristics, composition and name of the different austenitic steels used in biotechnology are summarized in Table 12. In general, vessels used in biological processes are manufactured from 316 or 316L steel. The expensive 316L has both excellent electropolishing and welding characteristics. The cheaper and less corrosion-resistant steel grades (304, 304L) are mostly used in food technology or for certain noncritical harvest or storage tanks such as waste tanks. Parts of the vessel not in contact with the product, such as the heating circuit the heat-transfer jacket or the platform,

Biochemical Engineering

1151

Table 9. Suitable equipment for specific processes Type of organism Bacteria aerobic

Equipment, mixing STR ∗∗ Marine impeller Rushton ∗∗ impeller Baffles ∗∗ Airlift reactor ∗ Vibromixer Hydraulic ∗ mixing Mechanical ∗∗ seal Magnetic ∗ coupling Top drive ∗ Bottom drive ∗∗ Jacket for heat ∗ transfer Coils for heat ∗ transfer Equipment, other Stainless steel ∗∗ vessel Glass vessel ∗ Plastic film ∗ sleeve reactor Depth filter ∗ (gas) Membrane ∗ filter (gas) Ring sparger, ∗∗ fish tail for gassing Sinter metal sparger Surface gassing Bubble free membrane gassing Process Batch Fed-batch Continuous cultivation Rotor-filter Fluidized bed Fixed-bed Dialysis CIP SIP Sterilization in the autoclave Containment

∗∗ ∗∗ ∗

E.coli

Bacteria Yeast anaerobic

Pichia pastoris

Animal cells

Hybridoma Insect cells Adherent cells cells on carriers

Algae (phototroph)

∗∗

∗∗ ∗

∗∗

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∗

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∗∗ = Very suitable. ∗ = Suitable, possible, but depending on other parameters. = usually not suitable, / = not applicable.

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Biochemical Engineering

Table 9. (Continued) Type of organism Bacteria aerobic

E.coli

Bacteria Yeast anaerobic

Pichia pastoris

Animal cells

Hybridoma Insect cells Adherent cells cells on carriers

Type of production GMP ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ production Bulk product ∗ ∗ ∗ ∗ ∗ Product for ∗ ∗ ∗ ∗ ∗ food industry ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Large scale ∗∗ > 3.5 m3 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 0.3 < pilot scale < 3.5 m3 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Laboratory scale 3 < 0.3 m mm ∗∗ = Very suitable. ∗ = Suitable, possible, but depending on other parameters. = usually not suitable, / = not applicable.

are usually made of 304, 304 L or non-stainless steel materials.

Algae (phototroph)

∗ ∗ ∗ ∗∗ ∗∗ ∗∗

4.1.2. Welding Gas tungsten arc welding (GTAW) is the most common technique to weld vessels and pipes in biotechnology. Fillet weld joints mostly applied to vessels are polished on the inside of the vessels. Polishing the partly overlapping fusion points of such fillet welds on the outside of a vessel gives a smooth surface but a visual control of the welds is no longer possible. Automatic welding such as orbital welding (butt welds) is used to connect pipes wherever the geometry of the pipework allows this technique. A considerable effort is put into inspection and labeling of weld seams. To control the welding each weld seam needs to be labeled on labeled pipes and components. Drawings or isometrics of pipes may be provided to indicate the location of the weld seam. The date, the name of the welder or parameters of the welding machine, and the label of the pipe are reported for each weld seam in a log file. Classes of acceptance and acceptance criteria for weld seams are defined by the International Institute of Welding (IIW, classes 1 – 5; 5: highest quality) or by CEN 25817 (class B, C and D; B: highest quality). Examples of inspections include:

Figure 19. Schematic of a dialysis membrane reactor. A dialysis membrane separates two reaction chambers from each other. Low molecular compounds, driven by the concentration difference in between the two chambers, permeate through the membrane. One or both chambers can be used for the cultivation of microorganisms.

1) Spot check of 10 – 100 % of the welds with a boroscope to detect cracks, incomplete fusions or alignment deviations. Photographs, or video of this inspection might be stored. 2) Surface and color of the seam have to satisfy acceptance criteria.

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Table 10. Suitable equipment for specific products Products Amino Organic acids, acids, vitamins ethanol Equipment, mixing STR ∗∗ Marine impeller Rushton ∗∗ impeller Baffles ∗∗ Airlift ∗ reactor Vibromixer Hydraulic ∗ mixing Mechanical ∗∗ seal Magnetic coupling Top drive ∗∗ Bottom ∗ drive Jacket for ∗ heat transfer Coils for ∗∗ heat transfer Equipment, other Stainless ∗∗ steel vessel Glass vessel Plastic film sleeve reactor Depth filter ∗ (gas) Membrane ∗∗ filter (gas) Ring ∗∗ sparger, fish tail for gassing Sinter metal sparger Surface gassing Bubble free membrane gassing Process Batch ∗∗ Fed-batch ∗∗ Continuous cultivation Rotor-filter Fluidized bed Fixed-bed Dialysis ∗ CIP ∗ SIP ∗∗ Sterilization in the autoclave Containment

Technical enzymes

Single Baker cell yeast protein

Antibiotics Vaccines Tetanus vaccine

Protein Monoclonal Virus (pharma antibodies (VERO industry) cells)

Viruses from bacteria

∗∗

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∗

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∗

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∗∗ = Very suitable. ∗ = Suitable, possible, but depending on other parameters. = usually not suitable, / = not applicable.

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Table 10. (Continued) Products Amino Organic acids, acids, vitamins ethanol Type of production GMP production Bulk ∗∗ product Product for ∗∗ food industry Large scale ∗∗ > 3.5 m3 0.3 < pilot ∗ scale < 3.5 m3 Laboratory ∗ scale < 0.3 m3

Technical enzymes

Single Baker cell yeast protein

Antibiotics Vaccines Tetanus vaccine

Protein Monoclonal Virus (pharma antibodies (VERO industry) cells)

Viruses from bacteria

∗

∗∗

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∗

∗

∗

∗

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∗∗

∗

∗

∗∗

∗

∗∗ = Very suitable. ∗ = Suitable, possible, but depending on other parameters. = usually not suitable, / = not applicable.

3) Spot check (e.g., 10 %) with X-ray inspection to detect pores or slag intrusions. 4) Welders certificates need to be provided. 5) Parameters of automatic welding machines must be logged. Tests of weld seams may be performed and kept for later inspection. The time required for those inspections and the data logging can be estimated with 12 – 1 h per weld seam. Generally the surface criteria (corrosion, hygienic constraint) is more important than the detection of pores/inclusions (affecting the stability) in the inside of a weld seam. Therefore boroscope inspection makes more sense than X-ray inspection. In addition, X-ray inspection is by far more expensive than the boroscope method. 4.1.3. Surfaces of Austenitic Steels Surface treatments of austenitic steels in biotechnology are important for two reasons. First, the cleaning of the equipment largely depends on the surface roughness. The smoother the surface is, the easier the cleaning will be, and less product is retained in the pores. However, there is a lower limit of surface roughness of Ra = 0.4 µm, under which cleaning time will not further be decreased [35], [60]. Second, on polished surfaces a homogeneous microscopically thin chromium oxide layer will be formed.

This layer increases the corrosion resistance of stainless steels. In general, both the inside and the outside of a reactor have a defined surface finish. Different methods such as cauterization, ball blasting (mainly for outside vessel), polishing, electropolishing, and passivation are used for surface treatment, the final result of the treatment being directly related to the steel quality and the previous working steps [58]. For the inside of vessels a standard surface roughness of Ra ≤ 0.8 µm and for the inside of pipes of Ra ≤ 5 µm is suggested [35]. Straight pipes are available with surface roughness of Ra ≤ 0.5 – 0.8 µm. However when bending such pipes the surface roughness will become 2.5 to 4 times higher. For the outside specification of Ra ≤ 2 µm are common. High (Ra ≤ 0.5 µm) and very high qualities of (Ra ≤ 0.25 µm) surface finishes are sometimes required for the inside of vessels. The increasing demands for surface quality are often difficult to achieve, expensive, and extremely timeconsuming. It is questionable whether this expense is made up with improved cleaning or biological results. 4.1.3.1. Definition and Measurement of Surface Roughness In the past the roughness was defined by the grit size (P60, P100 . . . P320, P400) of the abrasive

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Table 11. Suitable equipment for specific types of organisms Type of production

Equipment, mixing STR Marine impeller Rushton impeller Baffles Airlift reactor Vibromixer Hydraulic mixing Mechanical seal Magnetic coupling Top drive Bottom drive Jacket for heat transfer Coils for heat transfer Equipment, other Stainless steel vessel Glass vessel Plastic film sleeve reactor Depth filter (gas) Membrane filter (gas) Ring sparger, fish tail for gassing Sinter metal sparger Surface gassing Bubble free membrane gassing Process Batch Fed-batch Continuous cultivation Rotor-filter Fluidized bed Fixed-bed Dialysis CIP SIP Sterilization in the autoclave Containment Type of production GMP production Bulk product Product for food industry Large scale > 3.5 m3 0.3 < pilot scale < 3.5 m3 Laboratory scale < 0.3 m3

GMP production Bulk product

Product for food industry

Large scale > 3.5 m3

0.3 < pilot scale < 0.3 m3

Laboratory scale < 0.3 m3

∗ ∗ ∗ ∗ ∗ ∗

∗

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∗∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗∗

∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗

∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗

∗

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/

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∗ ∗

/ /

/ /

/ /

/

/

/

∗ / / ∗ ∗ ∗

∗ ∗

∗∗ = Very suitable. ∗ = Suitable, possible, but depending on other parameters. = usually not suitable, / = not applicable.

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Table 12. Austenitic steels with application in biotechnology (based on DIN 17440/41) Alloy component according to DIN W N◦ , wt %

Name/indication DIN, W N◦

AISI

Other

C

Cr

1.4301 1.4306 1.4541 1.4401 1.4404 1.4571 1.4439

304 304L 321 316 316L 316Ti 316L

V2A V2A V2A V4A V4A V4A V4A

≤ 0.07 ≤ 0.030 ≤ 0.08 ≤ 0.07 ≤ 0.030 ≤ 0.08 ≤ 0.030

1.4435

316L

V4A

≤ 0.030

Mo

Ni

17.0 – 19.0 18.0 – 20.0 17.0 – 19.0 16.5 – 18.5 16.5 – 18.5 16.5 – 18.5 16.5 – 18.5

2.0 – 2.5 2.0 – 2.5 2.0 – 2.5 4.0 – 5.0

8.5 – 10.5 19.0 – 12.5 9.0 – 12.0 10.5 – 13.5 11.0 – 14.0 10.5 – 13.5 12.5 – 14.5

17.0 – 18.5

2.5 – 3.0

12.5 – 15.0

Other

Ti = (5 × %C) – 0.80 Ti = (5 × %C) – 0.80 N = 0.12 – 0.22, S ≤ 0.025 S ≤ 0.025

Figure 20. Dialysis membrane reactor

belts. The roughness achieved by polishing with a defined grit size can widely vary since it is related to the belt quality. Therefore not the grit size but an unambiguous measurement as, e.g., with the electric stylus instrument should be used to evaluate the roughness of a surface (Fig. 21, [61], [62]). Several parameters characterizing the roughness profile are calculated. 1) The arithmetical average of the absolute values of the distances y from mean line to roughness profile within the sampling length (DIN 4768): Ra =

1 lm

x=l
m

ydx x=0

(4.1)

2) The average peak to valley height Rz of five adjoining single sampling lengths (Fig 22). RZ =

Z1 + Z2 + Z3 + Z4 + Z5 5

(4.3)

3) The maximum peak to valley height Rmax is the highest of the 5 peak to valley heights Z i (Fig. 22). In practice surface definitions based on the mechanical treatment with different grain or grit sizes of polish belts are still in use. Manufacturer and customer should define a smoothness such as Ra , which is easily measured and unambiguous. Table 13 gives a rough correlation between the grit and the average roughness Ra .

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Figure 21. The roughness profile measured with the electrical stylus instrument Table 13. Approximative correspondence of the arithmetic average roughness Ra , the average peak to valley height Rz and the maximum peak to valley height Rmax to the grit size Grit size

60

≈ Ra , µm ≈ Rz , µm ≈ Rmax , µm

2.2 13.4 18.2

100 1.8 12.8 17.1

180 1.3 9.7 11.6

220

280

1.1 7.3 8.6

0.5 4.4 6.5

320 0.45 3.2 4.7

400 0.4 2.8 3.5

factory. Often, finer polishing leads to poorer smoothness values after mirror polishing • Steel used for vessels should be cold-rolled sheet material. It is extremely difficult and time-consuming to polish warm rolled sheets

Figure 22. Determination of the average peak to valley height Rz and Rmax (DIN) from the roughness profile

Ball blasting is used for special surface finishing and gives various possibilities for optical surface effects with a dull-finished or dullpolished appearance. The ball blasting decreases the risk of tension cracks and corrosion. It is used mostly for the vessel exterior.

4.1.3.2. Mechanical Surface Treatment

4.1.3.3. Chemical Surface Treatment

The surface quality of steels can be greatly improved by mechanical treatment.

Cauterization. Impurities and surface imperfections prevent the formation of a perfect passive layer. The purpose of cauterization is to dissolve these flaws with suitable acid mixtures, such as 15 – 25 vol % aqueous HNO3 and 1 – 8 vol % aqueous HF for steel with chromium content above 15.5 %. The impurities that are cauterized may be oxidation tint, welding slag residues, surface impurities containing unalloyed steel, and fine scales or overlaps generated during the working process.

Polishing. Polish belts of increasing grit sizes are used (Table 13) to obtain specified smoothness values. The surface quality obtained with mechanical treatment depends on and is limited by the steel quality. Some examples are: • Titanium-stabilized chrome-nickel or molybdenum steels, 1.4541 (AISI 321) and 1.4571 (AISI 316Ti), are not suited for high surface qualities • Best possible smoothness values (Ra = 0.25 to 0.5 µm) are reached with steel grades 1.4306 (AISI 304L) and 1.4406 (316L) • For later mirror polishing, a preceding polish to Ra ≈ 0.6 µm (P220) is completely satis-

Passivation. The natural passive layer is formed when stainless steel is exposed to air. An artificial passivation with dilute nitric acid is frequently used for corrosion resistance at critical conditions. The oxidizing effect of the nitric

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acid accelerates the formation of a dense passive layer. Passivation is either used after cauterization or as a final treatment of ground, brushed, or polished steels with special surface structures that would be destroyed when cauterized. New studies on passivation show that alternatives to nitric acids, such as organic acid chelating agents can produce a long-lasting passive layer [63].

material are not suited for storage. Also, locations near railways and tramways (which may produce ferritic brake dust) and pavements in winter (dry sand and salt) should be avoided. For long storage periods, it is best to cover and wrap up delicate parts, as well as workpieces, in polyethylene film.

4.2. Polymers 4.1.3.4. Electrochemical Surface Treatment Mirror polishing (electropolishing) is used to smoothen and polish rough and dull surfaces. The quality of the mirror polishing is only as good as the source material and its mechanical pretreatment. Advantages of mirror polishing are: • Despite good Ra values, the surface may have roughness in the microscopic range, which is smoothed by the electrolytic process • Corrosion resistance is increased due to exposure to high oxygen concentration, which enables and increases the formation of a dense chromium oxide layer Disadvantages of electrolytic polishing are: • Formerly existing imperfections on the material surface are more accentuated after mirror polishing (Ra increasing after electropolishing) • The difficulty to mirror polish rolled plate steel • It is impossible to work titaniumstabilized steels (1.4541-AISI 321; 1.4571AISI 316Ti) After mirror polishing, the parts have to be thoroughly cleaned to remove remaining electrolyte. 4.1.4. Storage and General Working Rules Tools such as saws and files must only be in contact with one type of austenitic steel. Never use these tools with ferritic steels. Workpieces have to be stored on a clean and non-absorbing base support, protected from dust and water spray. Workshops handling ferritic

Polymers are used for O-rings, gaskets, diaphragms, or filters. Those materials have to withstand temperatures between 0 and 125 ◦ C, in some cases up to 135 ◦ C. An optimal grade of hardness for sealing material is “medium” to “firm” between 65 and 75 Shore A. Polymers must be resistant to acid and alkali solutions (see Chap. 4, Introduction). Polymers, which are in contact with the product (culture broth), must be recognized as safe for use in food or food packaging. The German BgVV issues recommendations of types of materials and the allowed migration, five categories are declared. Only quite safe materials are admitted. The FDA standard relating to elastomers is the FDA Code of Federal Regulations (CFR), Chapter 1, Title 21, part 177 (ref. 21 CFR 177.2600 in section 2600 “Rubber articles intended for repeated use” [64]). An alternative is the USP 23 approval for elastomers. The applicable standard in this case is laid down in section Biological Reactivity Tests in Vivo Plastic Classes I to VI. This standard relates to results of animal testing by exposure to elastomers or plastics by injections of extracts or implants from test material. For sanitary process application elastomers must meet the specification of class V or VI [65]. Several synthetic polymers are recommended [35], [66] and widespread in biotechnology because of their excellent temperature, acid and alkali resistance. Their main advantages and disadvantages are shown in Table 14. EPDM has versatile qualities and is therefore widely used for O-rings, diaphragms of valves and gaskets. The material is appropriate to resist to flows of hot water and steam. When PFTE is used an increased resistance to chemicals is required. The disadvantages of PFTE are that

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Table 14. Characteristics of synthetic polymers used in reactors Synthetic polymers

Ethylene-propylene-diene Fluoroelastomer rubber

Silicone or fluorosilicone elastomer

Fluoropolymer

ASTM code

EPDM

Registered trade name, e.g.

Nordel, Buna AP, Keltan

Temp. min Max. dry temp. Max. steam temp. Resistance to conc. acids Resistance to dilute acids Resistance to alkali Resistance to steam

− 40 ◦ C 135 ◦ C 140 ◦ C reasonable good very good good

Q, MQ, VMQ, FVMQ Silopren, SE, Blensil, Silastic − 55 ◦ C 200 ◦ C 125 ◦ C good very good reasonable poor

PTFE Teflon Fluon, Hostaflon − 25 ◦ C 260 ◦ C 150 ◦ C very good very good very good very good

FPM, FKM Viton, Tecnoflon, Fluorel − 20 ◦ C 210 ◦ C 120 ◦ C very good excellent not recommended poor

it cannot be stretched, is hard and has coldflow properties under pressure. Higher temperatures dramatically increase this cold flow behavior. PFTE is used to coat EPDM diaphragms of valves, when these have to resist specified chemicals.

product is given under “Indirect Food Additives: Adjuvant, production aids, and sanitizers” of the CFR (21CFR178.3570) [64].

4.3. Other Materials

Cleaning and asepsis are the two most important points of consideration when designing components for bioreactors and fermentors. Good design practice follows several cardinal rules:

4.3.1. Glass Because of its excellent temperature and chemical resistance and because it enables transparent sight into the reaction chamber, glass is a frequently used material for reactor vessels of volumes below 15 L. A disadvantage of glass is its brittleness. When bouncing glass, this may break or even worse have micro-fissures, which are not directly discovered. This may lead to accidents during sterilization. To obtain smooth glass surface (sealing purposes) glass must be polished. The design parameters for pressure glass vessels are summarized in [75]. Glass vessels are used for viewing glasses built into stainless steel vessels. 4.3.2. Grease and Lubricants Lubricants are used for O-rings and gaskets, to prevent their damage during installation and prolong service life. Lubricants have incidental contact with the product and are subject to regulations such as USDA H1. A positive list of allowed substances and their limitations in the

5. System Components and Detailed Engineering (Fig. 23)

1) All parts must be self-draining, puddles must be avoided or minimized. 2) Dead-ends, resulting in poor heat transfer and air pockets must be avoided or minimized. 3) Corners, crevices, and edges (screws) must be avoided. The design principle of maximizing the crevice should be applied to enhance cleaning (see Fig. 24 A). 4) “Noncontrolled” closed spaces (e.g., in between two seals, see Fig. 24 B) must be avoided. The design must consider that organisms have to be killed during sterilization and kept outside the cultivation system to avoid contamination. Higher design criteria are necessary, when the release of organisms or products to the environment must be avoided. In the case of pathogenic organisms (see Section 2.7) the contact to operators and the direct surroundings of the reactor must be avoided. These additional requirements have a considerable impact on the design of components.

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Figure 23. Components for a bioreactor

Figure 24. Inclined 25 mm port (Ingold type) A) Principle of maximizing unavoidable crevices. B) Negative example of a port with crevice difficult to clean, double O-ring creating an uncontrolled space

5.1. The Vessel A reactor system is often composed of several vessels. The core is the cultivation vessel, which might be surrounded by acid, alkali, antifoam agent, fresh media, storage, or harvest tanks. Up to a size of 1500 L, stainless steel vessels are usually built with a removable lid sealed by an axial positioned O-Ring. For vessels above this volume the costs to manufacture. Such large sealing surfaces dramatically increase. Therefore non-removable dished tops with a manway are designed [24]. A vessel and its jacket (if any) are designed to withstand a certain pressure. On an inscription plate the maximal allowed pressure and the volume of the tank respectively the jacket must be indicated. Further the manufacturer, the year of

construction and a serial number must be indicated. Steam sterilized vessels must be inspected according to the national pressure vessel regulation (e.g., ASME, PED 97/23, BPVC). 5.1.1. Components Mounted or Welded on the Vessel 5.1.1.1. Seals Sealing a reactor correctly according to the hygienic and sterile design rules is one of the main challenges. Seals can be divided in two groups: static seals which are stressed statically and dynamic seals (see Section 5.2.2.1), which have a sliding contact to another surface. Wherever removable components are mounted static seals are required. Static seals

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are divided in two subgroups: round shaped seals like O-rings, which seal along a line and flat seals (gaskets, tri-clamp type seals), which seal a surface. Common materials are EPDM or fluoroelastomers such as Viton. O-Rings. Using O-rings is the state of the art. The design of grooves and its tolerances for the O-rings are described in DIN 3771 or ISO 3601/I. O-rings must be elastic and incompressible. The dimension of the O-ring groove should be designed to enable the O-ring contact with all four surfaces during sealing to minimize the space where dirt or broth residues could accumulate or remain. On the other hand, to prevent damage of the incompressible O-ring, the groove should leave enough space for the Oring to react to compression. The swallow-tail groove, often used for O-rings sealing the lid minimizes the “dirty” space (Fig. 25). Large diameters (> 200 mm) such as those for manways and lids are sealed with an axial O-ring. Radial O-rings are preferred for smaller diameters (< 200 mm) to avoid dead-ends (Fig. 26). The O-ring should always be positioned closed to the end to minimize the space where “dirt” could accumulate.

Figure 25. Swallow-tail groove with compressed O-ring, the large groove allows the expansion of the compressed O-ring

High containment design is sometimes performed by designing seals with two O-rings being steam sterilized in between. However in practice the lifetime of O-rings often exposed to steam is dramatically reduced. When considering the additional pipework and the complexity of the system it is questionable whether the system will really be safer [66].

Figure 26. Axial and radial position of an O-ring A) If h > d there is an increased risk of dead-end. Axial positioned O-rings will therefore be used for large diameter (d > 200 mm) B) Radial positioned O-rings require a high precision of the sealing parts, which can only be achieved at reasonable costs for small diameter (d < 200 mm)

Gaskets and Tri-Clamps. The sealing surface of a gasket is large compared to an O-Ring. This surface must be clean and even to properly seal. To reach a comparable pressure on the sealing surface, the force applied on a gasket, compared to an O-ring, must be higher. Despite these disadvantages several application require a flat sealing: 1) Noncircular openings such as longitudinal viewing glasses. 2) The sealing of stainless steel with another brittle material such as bursting disks or glass. Gaskets avoid strain peaks compared to O-rings. 3) Depth filters, which must be sealed over a surface. 4) Tri-clamp connections for pipes are widespread in North America and accepted by the FDA. While adepts of tri-clamps emphasize the easy handling (no tools required) others argue that when the pipe is pressurized the operator has no chance to feel the danger while opening the connection, further three hands are required to open and close a tri-clamp.

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5.1.1.2. Ports Ports on vessels are required for probes and transfer lines. In Europe more or less standardized ports with inner diameter of 12, 19, 25 (Ingold type) and 32 mm are in use. Stanardization encompasses dimension and tolerance of the inner diameter. Probe manufacturer can therefore supply probes with an adaptor sealing with a radial O-ring (Fig. 24). However different lengths of the Ingold type port are in use and make it therefore difficult to place the O-ring on the optimal location [67]. Lateral ports on the vessel can be mounted horizontally since an O-ring is placed next to the vessel wall (Figs. 44 and 45). Probes containing electrolyte (e.g., pH, pO2 ) need to be installed in inclined ports (usually 15 ◦ , Fig. 24). In North America a different standardization is employed. Not the diameter of the port is standardized but the end, which should always be a tri-clamp. The seal is located on the tri-clamp and not next to the vessel wall, this situation always creates a small dead-leg, which is difficult to clean. Lateral ports need to be slightly inclined to allow draining of the small dead-leg. 5.1.1.3. Viewing Glasses As a matter of fact almost all stainless steel reactors are equipped with lamps and viewing glasses, although the opinions about viewing glasses on the lid or the vessel wall are controversial. A first group estimating that every gasket/O-ring is a source of risk will not agree to install a viewing glass. Further it is argued that longitudinal viewing glasses on the vessel wall interfere with the liquid circulation in the jacket, reduce the surface of heat transfer and the strength of the pressure vessel. A second group would not like to miss the quick look into the vessel to control the broth/liquid level and the actual condition of foam. The viewing glass flange should be flush with the internal wall of the vessel and have rounded edges to promote drainage. Glass flanges are usually sealed with gaskets. Construction details such as frames, size and geometry are defined in the national pressure vessel guidelines (e.g., DIN 7080, 7081, 28210, 28121). During cultivation foam residues and condensate accumulates on lid viewing glasses. For

a transparent sight into the reactor the glass must be cleaned. There are two cleaning systems available: a manually operated wiper blade or a steam jet (Fig. 27). The disadvantage of the wiper blade is that it is designed with a dynamic O-ring.

Figure 27. Round viewing glass with steam jet cleaning unit

5.1.1.4. Manways Vessels with a volume above 1500 L seldom have removable lids and require manways for cleaning and servicing the inside of the vessel. Baffles, impellers, and the shaft are also introduced or removed through the manway. The manway is closed with a lid and sealed with an O-ring on a flange. Nuts or kind of butterfly nuts are used to lock the manway. Usually a hinge is designed to open the lid of the manway. The design of the manway must avoid dead spaces and consider that all surfaces are accessible for cleaning fluids (CIP) but drain readily. 5.1.1.5. Heating/Cooling Jackets Jackets are responsible for heat transfer in and out of the vessel. Three general concepts predominate in the layout: the double jacket, the full pipe, and the half pipe (see Table 15). Double jackets are the most common solutions for vessels up to about 3 m3 . A tangential inlet and outlet on the jacket should be designed to create a fluid circulation around the vessel. This allows the total surface of the jacket to be used for heat transfer. However, ports and longitudinal viewing glasses interfere with this circulation (inhomogeneous conditions and pressure drop). Jacketing the bottom area of the tank is difficult to manufacture, but advantageous when small working volumes are used or the tank is poorly mixed. It is recommended to jacket up to the unaerated maximal working

Previous Page Biochemical Engineering volume (about 2/3 of the total volume). This choice is a tightrope walk between lower jackets, which prevent baking of media during sterilization of low working volumes, and higher jackets, which increase the surface area of the jacket and thereby improve the heat transfer of aerated full working volumes. 5.1.1.6. Heating/Cooling Coils Large fermentors or those with extremely high power input per volume, high OTR, and thereby strong exothermic reactions must remove large amounts of heat and are therefore equipped with internal cooling coils. A spiral coil is more effective if an axial flow pattern is created. Tube bundles arranged vertically along the vessel wall are preferred for stirred reactors with Rushton impellers and replace baffles. Vertical coils are also preferred to spiral coils since they have less horizontal surfaces, on which residues might remain, when the vessel is drained. 5.1.1.7. Pressure Relief and Safety Devices Vessels and jackets resist a certain pressure. Pressure may rise above the limit, if the temperature control of the vessel or the pressure reducing valve of the inlet gas is defect. For safety reasons vessels are therefore equipped with a pressure relief valve or a bursting disk. The advantage of a pressure relief valve is that it closes after a release as soon the pressure drops below the maximal allowed pressure. Such an event would not contaminate an aseptic vessel. Usually such valves are used for small vessels of up to 100 L. Vessels above 100 L are usually equipped with bursting disks, made of carbon or synthetic polymers. The disadvantage of a bursting disk is the fact that it is broken after an overpressure. Furthermore it should be exchanged on a regular base. Vent lines from a pressure relief valve or a bursting disk are directed to the floor or connected to a kill tank (contained drain). Safety devices are not required on the vessel if all lines providing pressure to the vessel are equipped with safety devices rated below or at the maximal pressure of vessel.

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5.1.2. Standard Stirred Tank Reactor About 95 % of the reactors are stirred tank reactors, which are available from laboratory scale (2 L) up to 250 m3 . Often encountered volume sizes due to standardized components are: 30, 42, 300, 500, 750, 1500, 3000, 5000, 7500, 15 000, and 30 000 L [35], [66]. As a rule of thumb about 2/3 of the total vessel volume is the maximal unaerated working volume of a stirred tank. The liquid broth volume depends mainly on the aeration rate (gas hold-up) and the foam development during the process. The minimal working volume is defined by the location of the necessary sensors and the lowest impeller. It requires a sufficient heating/cooling surface area of the jacket to ensure a proper temperature control. Baffles, absolutely necessary in reactors stirred with Rushton type impellers, improve mixing by eliminating the vortex and promoting turbulence. Usually four baffles are either permanently welded to the vessel or designed as removable units [28]. Storage tanks or bioreactors for animal cells requiring low mixing can do without baffles. Impellers generating an axial flow pattern (e.g., marine impeller) situated off center and/or at an angle to the axis are used here. Top and bottom drives are available. Figure 28 illustrates the most important vessel dimensions and components. Several dimensions are related to each other and rules of thumb for ratios can be formulated for the most important dimensions. Especially the H/D ratio is an often discussed parameter. Reactors with a high H/D ratio (≈ 3) generally have the following advantages over lower ratios: • Higher surface/volume ratio resulting in better heat transfer via jackets. • Higher hydrostatic pressure at the sparger, longer bubble residence time in the reactor resulting in higher OTR. On the other hand reactors with a low H/D ratio (≈ 2) feature also some advantages: • Lower overall height (fit better into rooms with low ceiling).

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Table 15. Design of Jackets

Type Double jacket

Full pipe

Half pipe

Advantages – Max. contact surface – Min. pressure drop – Few welding seams

– Totally independent from the vessel, no pressure on the vessel wall

– Good contact surface

– High pressure on the vessel, vessel wall has to be reinforced-reduced energy transfer

– High pressure drop – Min. contact surface – Many welding seams

– High pressure drop – Many welding seams

Disadvantages

• Higher reactor cross-section-to-volume ratio enables better surface gas-liquid mass transfers when gassing the headspace (e.g., bubble-free gassing of O2 , CO2 for animal cell culture). • Better axial mixing (only of importance at large scale > 3 m3 ). • Shorter shafts, with usually only two impellers, having better mechanical stability (only of importance at large scale > 3 m3 ).

Figure 28. Standard stirred tank reactor, schematic drawing with ratios of main dimensions d i /Dt = 0.3 – 0.5; H l /d i = 0.3 – 0.5; H/Dt = 2 – 3; H i /d i = 1 – 2; 4 baffles L b /Dt = 0.08 – 0.1; H b  × H; L c /Dt = 0.02

Reactors requiring a high OTR and an efficient heat transfer (e.g., aerobic cultivation of prokaryotic cells and fungi) will often have a H/D ratio of around 3, on the other hand reac-

tors requiring homogeneous mixing at low shear forces (e.g., animal cell culture) will have ratios of about 2. However, the importance of the dimensions must not be overestimated, most processes will run equally with both extremities of dimensions. Therefore often the available space in the building will define the overall dimensions of the reactor. 5.1.3. Standard Airlift Reactor Good mixing, an efficient OTR, low shear forces and a relative simple design (no agitation shaft to be inserted in the vessel) are the main advantages of an airlift reactor. However, the high OTR values reached in a STR are not achieved. The height to diameter ratio of an airlift reactor varies between 5 and 10 [34], [35]. Compared to STR (H/D ratio 2 to 3), jackets along the vessel cylinder of airlift reactors (Fig. 29) have therefore much better heat transfer rates. Airlift reactors are available from laboratory to very large production scale. Due to their relatively simple design these reactors are well suited for very large applications (100 to 500 m3 ). Standard total volumes of airlift reactors are 16.8, 26, 50 and 1500 L, resulting in a working volume of 12, 21.5, 30 and 1200 L, respectively [35]. Gas distribution is achieved by perforated pipes, sintered bodies, hydrophobic tubes or static mixers. The OTR depends on the aeration flow-rate, the reactor height, and the oxygen concentration difference between the gas and liquid

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Figure 29. Standard airlift reactor with draft tube

phase (see Section 2.5.4). As shown in Figure 29 airlift reactors often have larger diameters on the top end to act as gas separators. 5.1.4. Glass Vessels Glass vessels are used up to total volumes of around 15 L. Glass is appreciated in R&D for an overall view inside the reactor. When cultivating phototrophic organisms light is transmitted through the glass. Glass vessels are often sterilized in an autoclave and designed with a jacket for heating and cooling (Fig. 30). All transfer lines with bottles (antifoam agent, alkali solution, etc.) can be sterilized together with the glass reactor. The components of such systems are therefore quite simple and are easy to maintain. On the other hand transport and handling of the fragile glass vessel into the autoclave and back to the laboratory bench might be delicate. Long heat-up times during sterilization in the autoclave, due to inhomogeneous conditions, must be accepted. Heat-up times are improved when the jacket of the vessel is filled with water. In-situ sterilizable (SIP) glass vessels consist usually of a glass cylinder, which is closed at the bottom and the top by stainless steel plates. The construction must absolutely avoid strain peaks on the glass. Heat transfer is achieved by jacket-

ing the bottom of the reactor or by introducing heat exchangers (electrical heater, cooling coils) into the reactor. The cylinder must be handled with care and inspected before each sterilization. The explosion of a damaged glass vessel filled with liquid at 121 ◦ C at 1 bar overpressure represents a major risk in the laboratory. To protect the surroundings it is necessary to position a cylindrical metal shield around the glass vessel during sterilization. This shield must withstand a possible explosion of the glass reactor. Tests have shown that slightly conical cylinders allowing the emission of vapor and glass splitter into a vertical direction are best suited. 5.1.5. Special Polyamide Film Sleeve Bioreactors A safe alternative to the in-situ sterilizable glass vessel is the in-situ sterilizable STR made of a special polyamide film sleeve. The film sleeve is supported by a stainless steel base and lid (Fig. 31). Metal shells are used to provide support for the film during sterilization (overpressure!) and are removed afterwards. This construction principle is the only feasible and safe solution for larger in-situ sterilizable reactors (15 to 100 L), which need external illumination (phototrophic culture). Long airlift reactors were designed with a polyamide film sleeve (Fig. 32).

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Figure 30. Autoclavable stirred tank reactor with glass jacket (5 L); stirrer with axial magnetic coupling, bottom driven a) Glass vessel with a glass jacket; b) Bottom drive; c) Upper magnetic coupling; d) Lower magnetic coupling; e) Plug-in support; f) Temperature control fluid; g) Inlets/outlets and probes; h) Stirrer shaft with magnetic coupling

The construction is limited by the strength of the plastic film sleeve exposed to the hydrostatic pressure of the liquid. Main design parameters are reported in [55].

5.2. Agitation System Design 5.2.1. Top or Bottom Drive? The advantages of a bottom driven shaft are: • A shorter length of the shaft (less vibration, reduced design dimensions, more stability)

• Space on the top head of the reactor for transfer connections and sensors (especially laboratory reactors) • Lubrication of the primary mechanical seal or of the bushing type bearing of magnetic coupled shafts by the medium/broth The advantages of a top driven shaft are: • Abrasive media can be cultivated without damage of the bearing seal • No media leakage in case of a failure of the mechanical seal Reactors equipped with a bottom drive are therefore suitable for most applications.

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pathogenic organisms), the magnetic coupling should be chosen.

Figure 31. Visual safety bioreactor, 2.4 L total volume. Insitu sterilizable stirred tank reactor designed with a special polyamide film sleeve supported by the stainless steel base and lid, metal shells are used to provide support for the film during sterilization and are removed afterwards.

5.2.2. Mechanical Seals versus Magnetic Coupling The difficulty consists in introducing a rotating shaft into a “closed system”. When using a mechanical seal, the rotating shaft is held by bearings located outside the reactor and the mechanical seal seals the rotating shaft against the static vessel. The design allows good cleaning, it must, however tolerate minor leakage of broth in case of failure of the mechanical seal. When using a magnetic coupling the vessel can be considered closed. Torque is transmitted through the reactor wall with magnets. The disadvantage is that the shaft is held inside the vessel by bushing type bearings, which are difficult to clean. If the most important requirement to the reactor is very high torque transmission and/or efficient cleaning (CIP) the mechanical seal should be used. If containment is most important (use of

Figure 32. Visual safety fermentor; the outer cylinder of the airlift reactor is a transparent plastic-film sleeve; the heat exchanger is integrated in the draft tube

5.2.2.1. Shafts with a Mechanical Seal The mechanical seal consists of the static seat ring on the vessel and the rotating ring on the shaft. The polished surfaces of both rings must be perfectly parallel to slide one on each other. This is achieved by positioning one of the rings on springs to form a floating bearing (Fig. 33). The springs are compressed to produce a defined force necessary to seal both parts. Reactors often have a double mechanical seal. The primary mechanical seal is in contact with the medium, the secondary with the environment (Fig. 33) [32]. The materials mostly used in biotechnology are carbon for the rotating ring and ceramics for the seat ring. More resistant but also more expensive materials are silicon or tungsten carbide (static and rotating ring same material). Lubrication of the seal is very important. The primary mechanical seal of a bottom drive is automatically lubricated by the medium/broth. A top drive or a secondary mechanical seal requires

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an external lubricant. Either steam condensate or glycerol is used.

Figure 33. Double mechanical seal of a bottom drive

When using steam condensate as a lubricant, the space in between the two mechanical seals can be sterilized. For sterilization, the condensate is evacuated and the space in between the two mechanical seals steamed (Fig. 34). On the end of the sterilization, to again create conden-

sate, cold water is circulated through a heat exchanger in the condensate reservoir. In cultivation mode, the condensate is pressurized by sterile air at a pressure above the pressure in the tank. A leakage from a defect primary or secondary seal would therefore result in a level decrease in the condensate reservoir. By mounting a float switch in the reservoir an alarm can be send out. Broth will not leak out of the reactor. When using glycerol the space in between the mechanical seals is not sterilized. The design thus requires far less valves and less maintenance. The system is reliable since glycerol is a better lubricant than steam condensate. Glycerol is therefore an excellent alternative for processes where a small leakage of broth is tolerated [68]. Steam condensate as well as glycerol are used for top or bottom drives. Shafts mounted with external bearings are called overhung shafts. When reaching a critical length shafts need not be supported by a steady bearing inside the reactor. Bottom drives have shorter shafts than top drives, and do not need a steady bearing up to volumes of about 3500 L. Larger reactors will require a steady bearing, usually a bushing type bearing, to hold the end of the shaft. Materials such as synthetic coals, met-

Figure 34. Lubrication of a double mechanical seal with steam condensate. Steam condensate is created in a reservoir connected to the mechanical seal. During operation the reservoir and therefore the lubricating condensate is pressurized with sterile air. A float or level switch in the reservoir detects a leak of the lubricant.

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als, or metal carbides such as silicon and tungsten carbide are used. To ensure lubrication and allow cleaning, the bushing often has a groove through which the lubricant (medium/broth) can flow.

quired. These lines must be cleanable and sterilizable. Depending on the containment class, the size, the length and the slope of the line, the design can considerably differ.

5.2.2.2. Magnetic Coupled Shafts

5.3.1. Needle Connection

A magnetic coupling provides torque transmission from the motor shaft to the agitator shaft. The magnetic torque is transmitted through the reactor wall. It is therefore the ideal technique when complete containment is required. Generally it is differentiated between axial (Fig. 30) and radial torque transmission (Fig. 35). Radial couplings can transmit a higher torque than axial couplings. The torque transmission is not the limiting factor up to vessel volumes of 750 L. The stability of the shaft is, however, not so easy to achieve, because the shaft is hollow and mounted on an arbor by bushind type bearings. The stability is given by the arbor, which has a smaller diameter. The stability compared to a shaft directly introduced into a reactor (mechanical seal) is therefore reduced.

Figure 35. Radial magnetic torque transmission, bushing type bearing of the shaft in contact with the product

The bushing type bearings require lubrication and cleaning. To meet this requirement holes are drilled on the top and bottom of the shaft. Broth and cleaning solution can thereby penetrate into the narrow gaps.

5.3. Transfer Lines To inoculate, feed, supply or withdraw a liquid from or to a reactor, sterile transfer lines are re-

A bottle can be connected via a flexible hose to a sterile reactor with a needle connection (Fig. 36). The bottle, the hose, and the needle are protected from the nonsterile environment by a tube with filter and are sterilized in an autoclave. By a burning few drops of ethanol on the septum the surface of the septum is sterilized. The reactor is sterile and needs to be pressureless when connecting the needle to it. The needle is taken off from its protecting tube and pierced through the flame and septum. These types of transfer lines are widely used on a laboratory scale. 5.3.2. Steam-Sterilizable Transfer Line Figure 37 shows a transfer line from the harvest valve of vessel 1 to a sterile cross connection on the top of vessel 2. The harvest valve is mounted flush to the bottom of an in-situ sterilizable tank; the steam line is mounted next to the sealing. The sterile cross on the end of the transfer line, next to vessel 2, is connected to a steam trap. The transfer-line can be sterilized by supplying steam to the harvest valve and draining condensate through the sterile cross, while the two vessels remain in operation. The configuration shown in Figure 37 is only valid for transfer lines of small nominal diameters (< DN 10). In pipes of low diameters condensate accumulating at the lowest point of the line is chased by the steam flow. Larger lines require a steam trap at the lowest point of the transfer-line in order to drain condensate. 5.3.3. Steam-Sterilizable Transfer Line with CIP Figure 38 shows a steam-sterilizable transfer line, which can be cleaned in place. The CIP supply line, connected to the sterile cross of vessel 2, provides cleaning solution or WFI. The connection is used to either clean the transfer line or the

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Figure 36. Transfer line with needle connection. Separate sterilization of the needle, bottle, and the flexible hose in the autoclave.

nozzle of vessel 2. On the other hand the CIP return line is used to drain the cleaning solutions, when the transfer line or vessel 1 is cleaned. Instead of using steam traps as shown in Figure 37, pneumatic actuated pulsed valves with nozzles are used to drain the condensate. When briefly opening, those valves drain the accumulating condensate. The pulsation is controlled by the programmable logic controller (PLC). 5.3.4. Manually Operated Transfer Panel Figure 37. Steam sterilizable transfer line (SIP) for tubes or hoses withstanding 2 bar pressure and diameters below DN 10

The previous example show that a large number of valves are required to design a sterilizable

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Figure 38. Steam sterilizable transfer line with CIP supply and return line to clean the transfer. Pulsed valves with nozzles are used instead of steam traps to drain the condensate.

and cleanable transfer line. When multiple transfer-lines in between several tanks are needed, the number of valves increases tremendously. In order to reduce the number of valves so-called transfer panels manually operated with pivoting elbows, bends, or connecting pipes can be designed (Fig. 39). Thus the number of valves is strongly reduced, but skilled operators, are required to connect the elbows. Manually operated transfer panels are open systems, since operators might be in contact with the product.

rier when closed. Valves should have a minimal number of joints and connections to reduce the maintenance costs and downtime of a plant. Parts requiring maintenance (e.g., actuators, diaphragms) must be removable. Ball, gate, butterfly, and piston valves have cavities, which restrict their use for hygienic/ sterile applications. Thus, diaphragm valves and bellows type valves are widely used valves in biotechnology [35], [39], [69], [70]. 5.4.1. Diaphragm Valves

5.3.5. Ring Transfer Panel Ring transfer panels have similar functions as manual transfer panels but have the additional advantage of being a closed system. Special Tand corner-type diaphragm valves installed in a ring configuration allow the transfer of any inlet to any outlet valve (Fig. 40).

5.4. Valves Valves used in equipment for sterile and hygiene application, must constitute a sterile bar-

Diaphragm valves are best suited for sterile/hygienic applications in stainless steel pipework. The main advantages of a diaphragm valve are: 1) Pocketless and cavity-free design 2) Good draining in horizontal pipes 3) In- and outlet of the diaphragm valve is inline with the pipe 4) Maintenance can be performed on build-in valve bodies, therefore valve bodies can be welded in the pipework

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Figure 39. Transfer panel to connect several tanks to each other via pivoting elbows or bent pipes.

Figure 40. Ring transfer panel with SIP and CIP. Special T- and corner-type diaphragm valves allow a ring design without dead-spaces. Any inlet can be connected to any outlet.

Figure 41. Diaphragm valve, valve body and body with manual actuator

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5) Manual and powered actuators (electrical or pneumatic) are available A diaphragm valve consists of a valve body, the diaphragm, and the actuator (Fig. 41). Complete closure of the valve is achieved by contact of the elastic diaphragm with the metal weir of the valve body. Beside this classical diaphragm valve there are models with a built-in drain line connection adjacent to the weir (Fig. 42). Combinations to form sterile crosses, and rings are also available. To ensure free draining of horizontal pipes, diaphragm valves need to be positioned in a lateral or inclined position.

Figure 43. Picture and drawing of a welded diaphragm valve body (DN 100)

Valve bodies of diaphragm valves offer the choice of all type of end connections (tri-clamp, butt-welded, flare-type fitting, etc.). Whenever possible, valves should be welded into the pipework to eliminate unnecessary joints. For automated orbital welding the butt tube of the body requires a minimal length, depending of the tube size (e.g., 11 mm for DN 10). Figure 42. Diaphragm valve with drain line adjacent to the weir

5.4.1.1. The Valve Body Valve bodies can be cast, forged, welded, or be machined out of bar material. In the past most valve bodies were cast. The surface of such bodies is of poor quality and a big effort is required to reach an acceptable roughness (by welding, polishing). Further the ferrite content is high and cavities are possible. Forged bodies, where cavities can be excluded and the amount of ferrite be reduced below 1 %, are therefore preferred [35]. Excellent quality and flexibility is reached when bar material is machined. The quality of stainless steel (e.g., 316L) is chosen and the pipe connections are milled according to the specification of each customer. Surface roughness of Ra ≤ 0.8 µm is standard, but even higher demands down to Ra ≤ 0.25 µm can be reached. Bodies of bar material and those, which are forged, are usually available for pipe sizes in between DN 08 to DN 32. Tube sizes with higher nominal diameter, up to DN 150, require cast or welded (Fig. 43) valve bodies, which are remachined.

5.4.1.2. Diaphragms The weakest point of a diaphragm valve is the diaphragm itself. It is a wear and tear part which has to be checked on a regular basis. Usually those diaphragms that are in direct contact with steam or abrasive media first begin to show deficiencies. In a plant diaphragms are therefore differently stressed and have different lifetimes. When specially stressed diaphragms are identified in a plant, these can be used as an indicator to define the time to change all the diaphragms of a plant. Usually EPDM or EPDM covered with PTFE is used. 5.4.2. Bellows-Type Sealed Valves Bellows-type sealed valves can be designed in order to seal flush with a tank wall. The closed valve can be sterilized separately from the tank. Therefore this type of valves finds applications as harvesting (Fig. 44) and sampling valves (Fig. 45). Bellows-type valves find also application as control and process valves in pipework.

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Figure 44. Pneumatic actuated harvest valve with bellows

Bellows materials are stainless steel or PTFE. No dynamic O-rings are used in this type of design.

Figure 46. Flush mounted sampling valve with dynamic O-ring

A special application of a piston valve is found in an interchangeable probe armature (Fig. 47). The armature enables external probe maintenance during cultivation. The probe is sterilized before being reinserted. 5.4.4. Ball and Butterfly Valves Figure 45. Flush mounted sampling valve with static Orings and bellows in closed position. When turning the wheel the bellows contracts and the valve opens.

The weak point of this valve are the bellows. To control the integrity of the bellows a leakage pipe can be used as an indicator. Double bellows with internal pressure indicator are also available. Cleaning of the bellows might be difficult in special applications involving sticky media or mycelia-forming fungi.

Ball and butterfly valves can be mounted in any position, when open they are always selfdraining and have almost no pressure drop. To open and close the valves a swivel drive needs to be turned by 90 ◦ . To a limited extent these valves can be used as control valves. Problematic is the packing of the ball or the sealing of the shaft. Cavities at these locations cannot be excluded, this limits their use for sterile and hygienic applications [35]. Since they are cheaper than diaphragm valves they are usually found in the nonsterile pipework of the plant (cooling water circuit, air line before sterile filter, etc.).

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Figure 47. Interchangeable probe in contact with the medium (measurement position). When being retracted the probe is separated from the medium, the probe can be steam sterilized or dismounted to be recalibrated and inspected. Before being reinserted to the media the sensor is steam sterilized. a) Bioreactor wall; b) Port 25 mm; c) Handle; d) Housing; e) Sliding part; f) Sensor

An adapted butterfly valve for sterile conditions has a steam-sterilizable dual shaft sealing. Such rather complicated units are gaining interest for lines larger than 100 mm in diameter [70]. 5.4.5. The Actuators Actuators of valves can be divided into two groups, those describing an axial and others describing a rotating movement. An axial movement is used to actuate diaphragm, bellows-type, and piston valves. The rotating movement is used to actuate ball, gate, and butterfly valves. Actuators are manually, electrically, or pneumatically powered. Generally the pneumatic actuators for axial movement are bigger than the valve body. The actuator of a butterfly valve has the advantage to be smaller than that of the diaphragm valve. This is a further reason why butterfly valves become interesting for lines larger than 100 mm in diameter. Process, harvesting, and sampling valves have an open and a closed position. Control valves require a continuous movement of the valve to control a flow rate or a pressure. Axial Actuators. Manual axial actuators for diaphragm and bellows-type valves are usually designed with an insert screw. Several turns of a hand wheel are necessary to open and close the valve. To not over-tighten the valve and thereby damage the seal the actuator should be mechanically blocked in the closed and open position.

A visual indication showing the position of the actuator is advantageous. The insert screw of the actuator makes it possible to use the valve as a control valve. Manual axial actuators for piston valves can be designed with a hand lever, which directly acts on the piston. A spring ensures that the valve is automatically in the closed position when not activated (Fig. 46). Automated actuators for process valves are often pneumatically and seldom electrically powered. So called NO (normally open) or NC (normally closed) actuators are available. This indication defines the position of the actuator during pressure or power loss (Fig. 48). Used materials for pneumatic actuators are aluminum, synthetic polymers, or stainless steel. Pneumatic actuators for control valves also exist.

Figure 48. Normally closed (NC) and a normally open (NO) pneumatic actuators

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Quarter Turn Actuators. The swivel drive of butterfly and ball valves requires to be rotated by 90 ◦ . Manual actuators are open when the hand lever is directed into pipe direction. The pneumatic actuator is available as single-action or double-action. The single-action actuator uses a spring to close and air pressure to open the valve. The double-action actuator is opened and closed with air pressure. Position Feedback. To provide a position feedback of actuators and valves, limit switches are used. By use of micro-switches or by the principle of induction the closed and open position of a valve can be detected and transmitted to a PLC.

• All valves and instruments have to be located so as to be both easily operable and maintainable. • All pipes should be drainable, pockets need to be minimized if any. • The amount of reducers, elbows, adapters and connections have to be minimized, this criteria requires a correspondence of the armatures to the tubing. • Welded connections will always be preferred to dismountable connections with joints.

5.5.1. Pipe and Tube Sizes

Check valves allow the gas or fluid to flow only into one direction. They find many applications in the piping of utilities. Check valves are designed with seals, balls, and springs, have a pressure drop and are seldom self-draining. This limits their use in sanitary piping or pumps. Cleaning of a sanitary pipe with a built-in check valve in countercurrent is for example not possible. Check valves in sanitary piping are therefore being replaced by process-controlled diaphragm valves.

Pipes or tubes in contact with the product are made of stainless steel (standard 1.4435/316L) as discussed in Section 4.1. Many standards, defining the outer diameter and the wall thickness of a pipe, are in use in biotechnology [67]. To avoid tremendous welding (to connect pieces of one standard to the other) it is important to have a consistent standard within one plant. Three types of standardization are the most used. ISO tubes (in fact ISO 1127, corresponding to DIN 2462/3) are frequently used in Europe. This standardization does not define very small tube sizes and therefore the metrical standardization DIN 11850 is used for inner diameters of 6, 8, 10, and 15 mm. In the US mainly “imperial” pipes (also called O.D. tubing), based on an inch measurement of the outer diameter, are used. It is differentiated between seamless pipes and welded tubing. The inner surface of a seamless pipe is not as good as that of a welded pipe made out of a cold rolled plate. Seamless tubes therefore might require polishing to achieve higher inner surface qualities. Cold rolled welded pipes can have excellent inner surface finish without extra polishing and are therefore often preferred. Standard inner surface roughness of straight tubes are available in the range of Ra ≤ 0.2 to 0.8 µm.

5.5. Pipes

5.5.2. Pipe Applications and Sloping of Lines

Generally the following criteria can be postulated for pipework [70]:

Pipes must be routed in order to be self-draining. A slope of 0.3 to 0.5 % is generally sufficient to allow low viscous media to be drained.

5.4.6. Relieve Valves, Safety Valves To prevent pipes and vessels from bursting due to overpressure, relieve or safety valves are installed (see Section 5.1.1.7). An often found design principle consists of an O-ring sealed piston valve held by a spring. The preloading of the spring defines the critical pressure. The released fluid needs to be drained to the floor or into a kill-tank. 5.4.7. Check Valves (Nonreturn Valves)

• The number of lines and their length has to be kept to a minimum.

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Figure 49. Distribution piping for WFI or deionized water A) Negative example: piping with dead-end to the point of use; B) Good example: piping with a loop down to the point of use. By using a 3/2 way (T) valve dead-ends can be avoided.

Fluid distribution piping with T-piece connectors should avoid dead-ends longer than 6 times the diameter of the pipe (6 d rule) to the point of use. In fact such dead-ends can normally be avoided by using special T-valves and clever routing of pipes (Fig. 49). Distribution piping for steam should be taken from the top of the horizontal main line, to avoid taking condensate to the point of use (Fig. 50).

5.5.3. Bends, Reducing-Fittings, T-Piece Connectors Pipes can be bent but the surface quality of bent pipe is reduced compared to the original surface. Generally the surface roughness Ra of a bent pipe is about 2.5 to 4 times higher than the one of the original straight tube. Prefabricated 90 ◦ bends achieve lower surface roughness by polishing the internal surface after bending the pipe. However, it must be considered that with each prefabricated bend

Figure 50. Pipe routing for steam lines. Lowest points and end of pipes require a valve – steam-trap combination. Tconnections from horizontal pipes to sub-users must start from the top of the pipe.

two welds are added to the plant. Bends of 60 ◦ or lower are not considered to be very critical. When bending welded tubes, the weld seam should be on the lateral side of the bend, because this line is neither condensed nor stretched. Reducing fittings exist in eccentric and concentric design. In vertical lines concentric reducing fittings will be used while in horizontal lines

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eccentric fittings are build in to allow the fluid to be drained (Fig. 51). T-piece connectors exist in a broad range of variations adapting all kind of tube sizes.

5.6. Gassing Devices for Bioreactors

Figure 51. A) Eccentric reducing fittings can be used in horizontal, slightly inclined pipes; B) Concentric reducing fittings are only used in vertical pipes

The formation and size of bubbles depends on the gas sparging device, the coalescence of the media and the turbulence in the vessel. Small gas bubbles have a higher interfacial surface area a than large bubbles. Therefore the OTR can be increased when smaller bubbles are created (see Section 2.5). Shear stress generated by “exploding” gas bubbles at the liquid surface may damage suspended animal cells. The number of gas-bubbles must therefore be minimized or completely be avoided (bubble-free aeration). The bubble size also has an influence on the structure of the foam. Foam created by small bubbles in a protein rich-medium (e.g., serum) is more stable than the one created by bigger bubbles.

5.5.4. Sterile Coupling of Pipes

5.6.1. Tubes and Ring Spargers

Wherever possible, pipes should be welded for a maintenance-free coupling. On the other hand the possibility to dismount parts of piping to perform maintenance or exchange parts of a plant is often required. Several criteria can be formulated to pipe couplings:

Tubes, fish tail and ring spargers are made of simple stainless steel pipes. The sparger can end with a tube opening, a fish tail (squashed pipe end) or with a perforated ring (ring sparger). In all cases large bubbles are created, which are usually smashed and dispersed by a Rushton impeller rotating at high velocity just above the sparger (Fig. 53). The sparger pipe must be selfdraining to avoid a puddle when emptying the reactor. The ring sparger requires holes on the bottom of the ring to be drained.

– A static seal is required. For hygienic design reasons, seals with an O-ring are preferred compared to a gasket. – The compression and position of the seal should be defined by the design of the coupling – The seal should be clamped on one pipe end to avoid failure when being assembled. – Pipe-ends should be identical. Male/female ends should be avoided. – The in-line position of the two pipe ends must be defined by the shape of the coupling. – When using a screwed pipe connection threads with a broad, steep and round pitch should be used. Such threads, originally used in milk industry, are very robust. – Easy and safe handling. Figure 52 shows different types of pipe couplings with their advantages and drawbacks.

5.6.2. Sinter Metal and Porous Membrane Spargers When small bubbles are required and high stirrer speed is not applicable, sinter metal or porous hydrophobic membranes are mounted on the end of the aeration tube. Such spargers are often used for animal cell cultivation or for airlift reactors. The final bubble diameter size in the upper region of the reactor will much depend on the coalescence capability of the medium (see Section 2.5.2.2). Such sparger need to be dismounted for separate cleaning.

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Figure 52. Sterile coupling of pipes A) Flared tube connection with O-ring for small pipes (DN 6-10); B) Sterile coupling designed with a clamped O-ring, robust round thread and a shape aligning the pipes, for DN 15 to DN 50 pipes; C) Sterile flange connection with clamped O-ring, shape aligning the pipes, for large pipes > DN 50; D) Tri-clamp connection (gasket, clamped on pipe-end, identical tube ends)

Figure 53. Tube type sparger below Rushton impeller A) Tube sparger; B) Ring-sparger with sparging holes on the top and few holes on the bottom to drain the sparger when emptying the reactor

5.6.3. Surface Gassing To avoid bubbles in the broth the reactor headspace can be gassed. Of course the supply of oxygen is limited by the liquid surface. Aeration through the surface is therefore reserved to small reactor volumes or to cell cultures with low densities. The gas – liquid mass transfer is increased

by mixing the media (the liquid boundary layer at the phase surface is reduced). Surface gassing is performed in animal cell cultivation to control the pH. CO2 is stripped from the media by supplying air to the headspace, or solved in the media by supplying CO2 . The first will increase and the second reduce the pH.

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Figure 54. Support with winded membrane hose for bubble free aeration. The membrane is kept in a translatory movement [71]

5.6.4. Bubble-Free Membrane Gassing Bubble-free membrane gassing is used for shearsensitive animal cell cultivation (e.g., insect cells) or in few applications where highly volatile products should not be stripped out of the media. The main advantage of the method is that foam formation is completely avoided, thus foaming media such as those containing serum can be used for cultivation. The technique uses a membrane separating the liquid from the gas phase. Mass transfer of gases from one phase to the other is achieved through the membrane. Usually thinwalled membrane hoses of silicone or porous hydrophobic materials such as polypropylene are used. The gas phase is pressurized, but the pressure drop across the membrane must be controlled to prevent the formation of gas bubbles on the liquid side of the membrane. By creating a turbulence in the liquid phase the mass transfer can be increased and gas bubbles be avoided. One possible method consists in winding the membrane hose on a support, which is kept in a translatory movement (Fig. 54) [71]. Another method consists in using reinforced silicone hoses to be able to increase the pressure of the gas phase and thus increase the gas transfer [72].

The possibilities to scale-up membrane gassed reactors is limited by the scale-up criterion membrane area to reactor volume ratio. Realistic are ratios of 10 to 25 m2 /m3 in tanks up to 150 L. The OTR will usually be high enough for animal cell cultures cultivated without cell retention (see Section 2.1.3.2).

5.7. Mechanical Foam Separators Foaming can be one of the major problems during aerobic growth of bacteria. Antifoam agents might severely reduce the OTR (see Section 2.5.3.2) or have a detrimental impact on the product recovery later in the downstream processing. Their use can be avoided when a mechanical foam separator working on the principle of a cyclone is used. Due to centrifugal forces the foam is separated into liquid and gas phases. The exhaust gas is withdrawn through the center of the separator. Mechanical foam separators are mounted in the headspace of the reactor or in the exhaust gas line (Fig. 55). The liquid phase is usually recirculated to the broth. The shaft of the rotor must be sealed. Usually single or double mechanical seals, which require lubrication, are used.

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Typical filter materials are ceramics, sinter metal, polypropylene, cellulose acetate and microglass. The advantages of materials such as ceramics and sinter metal lie in their resistance to acids and superheated steam. Depth filter are therefore robust and can easily be cleaned or regenerated. Gases in the laboratory as well as in large scale production are often sterilized with depth filter. For liquid filtration they are often used as a prefilter because of their ability to retain large amounts of particles without clogging (Fig. 8). Depth filter cannot be integrity tested without destroying them, this limits their use for GMP production. 5.8.2. Membrane Filters

Figure 55. Mechanical foam separator mounted in the the exhaust gas line

5.8. Sterile Filtration [73]

Membrane filters, also called absolute filters, retain particles on their surface similar to a sieve. Often filter cartridges are designed with a pleated membrane in a polypropylene support cage. The cartridge has O-rings, which fit into a filter housing (Fig. 56).

In biotechnology filters are used for sterile filtration of gases and liquids. Typical particles sizes to be separated by filters are bacteria (0.3 – 1 µm), mycoplasma (0.1 – 0.2 µm) or even viruses (0.04 µm). Filters are often characterized by their absolute filter rating expressed in diameter length of the largest hard spherical particle that passes through the filter under specified test conditions. 5.8.1. Depth Filters Depth filters used in biotechnology consist of a porous structure or fibers, which form an irregular three-dimensional net. Particles are not only retained on the surface but also inside the filter. Electrostatic forces, adsorbing forces, diffusion and the impact of the particles on the filter material are responsible for the retention. Depth filters can retain a high load of particles but the gas velocity across the filter must be adapted (≈ 0.3 m/s). Depth filters do not have an absolute rating since their retention rate will be proportional to their depth length, they therefore have to be sealed with gaskets over their depth.

Figure 56. Filter housing for a pleated membrane filter in cartridge

Membrane filters can be used to sterilize liquids (hydrophilic membranes) as well as gases (hydrophobic membranes). Typical wet filter ratings are:

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• 0.45 µm: in most cases sufficient to retain bacteria • 0.2 µm: absolutely secure against bacterial contamination • 0.1 µm: secure against mycoplasma contamination • 0.04 µm: secure against virus contamination Dry removal filter rates are smaller than the indicated wet ones. The main advantage of membrane filters is their ability to be integrity tested after sterilization without contaminating the process. Standard test methods consist in wetting the membrane, applying a certain pressure below the bubble point and measuring the diffusion flow (forward flow test). By measuring the pressure decay the diffusion flow can be calculated (pressure hold test). Those nondestructive test methods have to be validated by the filter vendor by comparing them to destructive test methods. Automated integrity test instruments are available to perform, monitor, and log the test. A GMP production usually require the test to be performed after the sterilization, before-, and immediately after the production process. Since the sterilization of membrane filters requires clean steam, the filter housing is often designed to be sterilized separately from the reactor. To perform a filter integrity test two supplementary connections on the non-sterile side of the filter cartridge are required (Fig. 11).

5.9. Hydraulic Pumps Hydraulic pumps can be grouped in centrifugal and positive displacement pumps. In the following a selection of pumps are presented and their advantages, disadvantages, and application fields for equipment used in biotechnology are discussed [74]. The following characteristics should be considered when choosing a pump: • Total dynamic pressure of the pump (usually expressed in height of water column) • Flow rate • Ability to be self-draining • Hygiene, sterilizable design • Materials in contact with the product • Ability to be self-priming • Steady continuous flow versus pulsating flow • In-line design of connections

• • • • • •

Ability to handle abrasive media Ability to handle viscous media Ability to handle shear-sensitive cells Ability to run dry Noise Costs (price per flow rate)

Centrifugal pumps Centrifugal pumps are often found in CIP units. To meet the hygienic design requirements centrifugal pumps should be equipped with a mechanical seal. The pump is not self priming and difficult to drain. When taking into account these disadvantages in the process design, the pump can, however, be integrated in the equipment. An advantage is that a broad range of centrifugal pumps with different flow-rates and pressures are available on the market. Peristaltic pumps depend on progressive squeezing of flexible tubing by rollers mounted on a rotating plate. The liquid is pushed towards the discharge as the roller travels its circular path. The major advantage of such principle is that nothing but the tubing itself is ever in contact with the media. Typical tubing selections are silicone and Viton. The pump can run dry, is self-priming up to a certain degree, can handle abrasive media and provides constant flow as long as the tubing is well adapted to the rollers and not too stressed. Usually the tube is sterilized in the autoclave and afterwards inserted in the pump-head. Large reinforced tubes can also be sterilized in-situ. Those advantages make it to be the pump par excellence in small to middle scale. Peristaltic pumps are operating at low shear forces and are therefore suitable to pump media with animal cells. Their usage is just limited by the pressure. Depending on the spring forces of the rollers, the wall thickness, the tubing diameter and material, pressures of 0.5 to 2 bar can be achieved. Piston pumps are self-priming, metering pump that provide accurate low flow rates at high or even very high pressure (up to 500 bar). For safety reasons a pressure relief valve should therefore always be installed behind the pump. Often frequency and stroke can be regulated to control the flow rate. Pulsation can be reduced when using two or three pistons or when building in dampeners.

Biochemical Engineering The sealing of the piston and the check valves are the weak points. Usually only clean nonviscous media can be delivered, solids would score the piston. A piston pump therefore requires further design tricks, such as a steam barrier sealing of the piston or the usage of a membrane (Fig. 57) to meet the standards of sanitary design.

Figure 57. Piston-type pump with diaphragm. The axial movement of the piston lubricated by oil, is separated from the product by the diaphragm. Cleaning in counter-current is not possible due to the check valves. a) Check valve; b) Diaphragm; c) Pump head; d) Oil chamber; e) Piston

Diaphragm pumps are very reliable. A common model used for sanitary transfers in bet-

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ween vessels is the pneumatically driven double diaphragm pump. Two diaphragms, each separating a wetted chamber, are connected by a shaft inside the air motor, so that when one is on the pressure stroke pushing forward the other is correspondingly pulling back to create vacuum. The advantages of double diaphragm pumps are their ability to stall if run against too great a head pressure, their ability to prime abrasive or shear sensitive media, their self-priming and dry running capability and the inherent variable flow rate capability (regulated by the air flow). Disadvantages are the need for clean, dry, compressed air, the air turbulence created around the pump (usually to be avoided in clean rooms), the fundamental inefficiency of a compressor/pump system, the noise of the check valves and the high repair and maintenance costs those pumps in practice usually demand. The check valves of a double diaphragm pump can be replaced by diaphragm valves (Fig. 58). Cleaning (CIP) in counter-current to the transfer can thereby be achieved. Other Pumps with Rotating Shaft. Gear, lobe, piston, circumferential piston, flexible impeller, sliding vane, and progressing cavity pumps are all positive displacement pumps with a rotating shaft. The difficulty is to seal the insertion of the shaft, which might be accomplished by using mechanical seals or magnetic couplings (Fig. 59). The pumps are usually self-priming.

Figure 58. Air-driven double diaphragm pump with diaphragm valves instead of check valves A) The opening and closing sequence of all diaphragm valves is computer controlled; B) Example of usage as a transfer-pump from reactor 1 to 2. CIP usually performed in counter-current is possible with such pump.

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Figure 59. Gear-pump with magnetic coupling to the motordrive. The pump-head is either sterilizable in place or separately in the autoclave. a) Pump housing; b) Gear driving part; c) Gear receiving part; d) Inside magnet; e) Outside magnet; f) Magnet cap

6. References 1. E. L. Stadler (1998) Dual purpose fermentor and bioreactor? A capital quandary! Pharmaceutical Engineering 18. 3. 2. H.-J. Rehm (1980) Industrielle Mikrobiologie. Springer Verlag 3. H. G. Schlegel: Allgemeine Mikrobiologie, Thieme Verlag, Stuttgart 1992. 4. H. Dellweg: Biotechnologie. Grundlagen und Verfahren. VCH, Weinheim 1987. 5. H. M¨arkl et al.: “Cultivation of Escherichia coli to high cell densities in a dialysis reactor”. Appl. Microbiol Biotechnol. 39 (1993) 48 – 52. 6. W. Crueger, A. Crueger (1989) Biotechnologie – Lehrbuch der angewandten Mikrobiologie. M¨unchen; Wien; R. Oldenburg Verlag. 7. J. Monod (1942) Recherches sur la Croissance des Cultures Bact´eriennes. Hermann & Cie. Paris 8. B. Atkinson, F. Mavituna: Biochemical Engineering and Biotechnology Handbook, Nature Press, 1991. 9. I. S. Longmuir (1954) Respiration rate of bacteria as a function of oxygen concentration. Biochem 57. 81 – 87. 10. A. Ch. Dubach, H. M¨arkl (1992) Aplication of an extended Kalman filter method for monitoring high density cultivation of Escherichia coli, J. of Ferment Bioeng. 73, 5, 396 – 402. 11. B. O. Solomon, L. E. Erickson (1981) Biomass yields and maintenance requirements for growth on carbohydrates, Process Biochemistry Feb/March 44 – 49.

12. Stephanopoulos et al. Studies on On-Line Bioreactor Identification I to IV. Biotechnol. Bioeng. 26, 1176 – 1218. 13. J. A. Roels (1983) Energetics and kinetics in biotechnology, Elsevier biomedical press. 14. I. G. Minkevich, V. K. Eroshin (1973) Productivity and Heat Generation of Fermentation Under Oxygen Limitation. Folia Microbiol. 18, 376 – 385. 15. H. W. Blanch, D. S. Clark (1996) Biochemical engineering, ed. M. Dekker cop. New York, Basel. 16. J. G. Aunins, H.-J. Henzler: Aeration in cell culture bioreactors. In Biotechnology Vol.3 ed. Stephanopoulos. VCH Weinheim, 1993. 17. C. L. Cooney, D. I. C. Wang, R. I. Mateles (1969) Measurement of heat Evolution and correlation with oxygen consumption during microbial growth. Biotechnol. Bioeng. 11, 3, 269 – 281. 18. J. H. T. Luong, B. Volesky “Determination of Heat of some Aerobic Fermentations”.The Canadian Journal of Chemical Engineering 58 (1980) 497 – 504. 19. M. S. Kharasch, “Heats of cumbustion of organic compounds”. Bur. Stand. J. Res. 2 (1929) 359. 20. F. Bader in A. L. Demain, N. A. Salomon (eds.): “Sterilization: Prevention of Contamination”, Manual of Industrial Microbiology and Biotechnology, Chap. 25, American Society for Microbiology, Washigton DC 1986. 21. K. H. Wallh¨auser (1985) Sterilization in: Biotechnology Vol. 2 Fundamentals of Biochemical Engineering. Ed. H.-J. Rehm and G. Reed. VCH, Weinheim. 22. K. H. Wallh¨auser (1995) Praxis der Sterilisation - Desinfektion – Konservierung. Georg Thieme Verlag Stuttgart – New York. 23. G. K. Raju, C. L. Cooney (1993) Media and Air Sterilization. In: Biotechnology Vol.3 ed. G. Stephanopoulos. VCH Weinheim. 24. Sh. Aiba, A. E. Humphrey, N. F. Millis: Biochemical Engineering, Academic Press Inc., 1973. 25. F. Wilde (1997) Sterilization and cleaning. Chemical Plants and Processing, 3. 26. H. Schunn (1998) Hohe Massst¨abe, Leitf¨ahigkeitsmessung nach USP 23. Process 11. 27. D. I. C. Wang et al. (1979) Fermentation and Enzyme Technology J. Wiley & Sons. 28. Rushton et al. (1950) Power characteristics of mixing impellers II. Chem. Eng. Prog. 46: 467 – 476.

Biochemical Engineering 29. M. Zlokarnik (1985) Tower-shaped reactors for aerobic biological waste-water treatment in Biotechnology Vol. 2 ed. H. Brauer VCH Weinheim. 30. M. Yasukawa et al. (1991) Gas Holdup, Power Consumption, and Oxygen Absorption Coefficient in a Stirred-Tank Fermentor under Foam Control. Biotechnol and Bioengineering, Vol. 38, 629 – 636. 31. K. Van’t Riet (1979) Review of Measuring Methods and Results in Nonviscous Gas-Liquid Mass Transfer in Stirred Vessels. Ind. Eng. Chem. Process Des. Dev. 18. 357 – 364. 32. M. Charles, J. Wilson (1994) fermentor Design; In: Bioprocess Engineering Edited by B. K. Lydersen et al.; J. Wiley & Sons, Inc. New York, 3 – 67. 33. M. Zlokarnik (1999) R¨uhrtechnik Theorie und Praxis. Springer Verlag. 34. H. Blenke (1987) Process Engineering Contributions to bioreactor design and operation. In: Biochemical engineering. Eds: Chmiel et al. G. Fischer Verlag. 69 – 91. 35. Dechema (1991) Standardisierungs- und Ausr¨ustungsempfehlungen f¨ur Bioreaktoren und periphere Einrichtungen. Dechema Frankfurt a. M. 36. I. J¨obses et al. (1991) Lethal events during gas sparging in animal cell culture. Biotechnol. Bioeng. 37. 484 – 490. 37. SCBS (1995) Guidelines for work with genetically modified organisms. Z¨urich Switzerland. 38. Belgian Biosafety Server http://biosafety.ihe.be. 39. ISSA (2000) Control of Risks in Work with Biological Agents. Part 3: Production. Prevention Series N◦ 2039. 40. M. M. Meagher (1992) Design of a University-Based. Biosafety Level 2, large-Scale Fermentation facility. BioPharm 10. 30 – 35. 41. J. Mauron (1981) The Maillard reaction in food. Progress In Food and Nutrition Science. 5, 5 – 35. 42. M. Krahe, G. Antranikian, H. M¨arkl (1996) Fermentation of extremophilic microorganisms. FEMS Microbiol. Rev. 18, 271 – 285. C. Kroll (1992) Neuere apparative Konzepte in der Bioreaktortechnik. Chemie-Technik 8 p 14 – 18. 43. R. Rautenbach, R. Albrecht (1981) Membrantrennverfahren Ultrafiltration und

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Umkehrosmose. Ed. Otto Salle Verlag Frankfurt a. M. R. J. Datar, C-G. Ros´en (1993) Cell and cell debris removal: Centrifugation and Crosslow filtration. In Biotechnology Vol. 3 ed. Stephanopoulos. VCH Weinheim. T. Suzuki (1996) A dense culture system for microorganisms using a stirred ceramic membrane reactor incorporating asymmetric porous ceramic filters. J. ferment. Bioeng. 82/ 3/ 264 – 271. Hayakawa et al. (1990) High density culture of Lactobacillus casei by cross-flow culture method based on kinetic properties of the microorganism. J. ferment. Bioeng. 70/ 6/ 404 – 408. A. Sambanis, W.-S. Hu (1993) Cell culture bioreactors. In Biotechnology Vol.3 ed. Stephanopoulos. VCH Weinheim. J. D. Macmillan et al. (1987) Monoclonal antibody production in stirred reactors. In: Large scale cell culture technology. Ed. B. K. Lydersen. Hanser Publishers Munich. L. A. Behie et al. (1987) The application of continuous three phase fluidized bed bioreactors to the production of pharmaceuticals. In: Biochemical Engineering. Ed. Chmiel et. al. G. Fischer Verlag Stuttgart. K. Nilsson (1987) Entrapment of cultured cells in agarose beads. In: Large scale cell culture technology. Ed. B. K. Lydersen Hanser Publishers Munich. R. Rupp et al. (1987) Cellular microencapsulation for large-scale production of monoclonal antibodies. In: Large scale cell culture technology. Ed. B. K. Lydersen Hanser Publishers Munich. D. Fassnacht, R. P¨ortner, (1999) Experimental and theoretical considerations on the oxygen supply of animal fixed-bed cultures. J. Biotechnol. 72. 169 – 184. R. P¨ortner et al.: “Immobilization of Mammalian Cells in Fixed Bed Reactors”.Bioforum International, G.I.T. Verlag Publishing Ltd. 1999. R. P¨ortner, H. M¨arkl (1998) Dialysis cultures. Appl. Microbiol. Biotechnol. 50. 403 – 414. H. M¨arkl (1989) Folien und Membranen als neue Elemente im fermentorbau. forum mikrobiologie. GIT Verlag GmbH 12 234 – 237. J. Ogbonna, H. M¨arkl (1993) Nutrient-Split Feeding Strategy for Dialysis Cultivation of Escherichia coli. 41. 1092 – 1100.

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57. H. M¨arkl et al. (1990) A new dialysis fermentor for the production of high concentrations of extracellular enzymes. J. Ferment. Bioeng. 69. 244 – 249. 58. C. Dillon, D. Rahoi, A. Tuthill (1992) Stainless steel for Bioprocessing, in BioPharm 5: Part 1: Materials selection, (April), 38 – 42; Part 2: Classes of Alloys, (May), 32 – 35; Part 3: Corrosion phenomena, (June), 40 – 44. 59. P. Meyer in B. K. Lydersen et al. (eds.): “Vessels for Biotechnology: Design and Materials”.Bioprocess Engineering, J. wiley & Sons Inc., New York 1994, pp. 189 – 214. 60. R. Schmidt (1999) Nicht nur sauber sondern steril, PROCESS 11.1999 p. 46 – 49. 61. W. K¨uppers (1976) Die Oberfl¨ache von nichtrostenden Feinblechen – Anlieferung und Weiterverarbeitung. Thyssen Edelstahlwerke AG, Technische Ver¨offentlichung 6327/6 Krefeld, Germany. 62. G. Graf (1987) Die Oberfl¨achenbehandlung von Nichtrostenden Chromnickelst¨ahlen, Metall. 9. 63. D. Coleman, R. Evans (1990) Fundamentals of Passivation and Passivity in the Pharmaceutical Industry, Pharm. Eng. 10, 43 – 49. 64. http://www.acessdata.fda.gov/scripts/cdrh/ cfdocs/cfcfr/showcfr.cfm 65. Ph. Bellow, J. Larsen: (1999) Elastomers in aseptic processing. In: Pharmaceutical processing. (Sept.), 106 – 107. 66. F. Menkel (1992) Einf¨uhrung in die Technik von Bioreaktoren, R. Oldenbourg Verlag M¨unchen.

67. P. Meyer (2000) Handlungsbedarf: Standards international vereinheitlichen. Pharma + Food 5. 66 – 69. 68. E. Breuker, U. Drees 1984) Pilotfermentor in der Industrie: Kriterien und konzipierung eines praxisnahen Systems. In: forum mikrobiologie 7 p. 12 – 18. 69. W. Storhas (1994) Bioreaktoren und periphere Einrichtungen. Vieweg-Verlag Braunschweig. 70. H. Adey, M. S. Polan (1994) Piping and Valves for Biotechnology.; In: Bioprocess Engineering Edited by B. K. Lydersen et al.; J. Wiley & Sons, Inc. New York, 215 – 252. 71. B¨untemeyer, Lehmann (1987) Device and process for bubble-free gasification of liquids, in particular culture media for tissue culture reproduction. PCT/EP86/00566. 72. I. Sekoulov, H.-J. Br¨autigam: (1987) Process for supplying oxygen from fermentation plant and device for implementation of the process. Patent PCT/EP86/00744; WO87/03615. 73. J. M. Martin et al. (1994) Cartridge filtration for Biotechnology. In: Bioprocess Engineering. Edited by B. K. Lydersen et al.; J. Wiley & Sons, Inc. New York, 189 – 214. 74. R. Stover in B. L. Lydersen et al. (eds.): “Pumps”. Bioprocess Engineering, J. wiley & Sons Inc., New York 1994, 189 – 214. 75. AD-Merkblatt N4. (1983) Druckbeh¨alter aus Glas. Beuth Verlag GmbH, Berlin (Sept).

Energy Management in Chemical Industry

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Energy Management in Chemical Industry Colin D. Grant, University of Strathclyde, Glasgow, United Kingdom 1. 2. 2.1. 2.2. 3. 4. 4.1. 4.2. 4.3. 4.4. 4.5. 5. 5.1. 5.2. 5.3. 5.4. 6. 6.1.

Introduction . . . . . . . . . . . . . . . . . Areas and Levels for Energy Management . . . . . . . . . . . . . . . . Areas . . . . . . . . . . . . . . . . . . . . . Levels . . . . . . . . . . . . . . . . . . . . . Good Housekeeping . . . . . . . . . . . . Audits, Monitoring, and Control . . . Energy Audits . . . . . . . . . . . . . . . . Monitoring and Control . . . . . . . . . Energy Usage and Production Levels Batch Processes . . . . . . . . . . . . . . . Advanced Monitoring and Control Systems . . . . . . . . . . . . . . . . . . . . Heat Recovery . . . . . . . . . . . . . . . Process Design for Heat Recovery . . Types of Heat Exchanger . . . . . . . . Equipment Design . . . . . . . . . . . . . Example: Optimum Duty of a Waste-Heat Boiler . . . . . . . . . . . . . Distillation . . . . . . . . . . . . . . . . . . Column Design . . . . . . . . . . . . . . .

1187 1188 1188 1189 1189 1190 1190 1190 1191 1192 1192 1192 1193 1193 1193 1194 1195 1196

1. Introduction Objectives of Energy Management. The chemical industry uses energy for fuel to provide primary heat and power for the operation of various processes. Energy management is concerned with the efficient use of this energy; its primary objective is to reduce operating costs, rather than simply to reduce energy consumption. The purpose of energy management is, therefore, to reduce the total energy requirement of a process per unit of output while reducing the total cost of production (or at least holding it constant). Energy usage for space heating and lighting is insignificant compared to process energy usage, except for some sectors of the fine-chemical industry where all operations are carried out indoors. Many methods for improving the energy efficiency of processes require an expenditure of either capital investment or manpower or both. The return on such investment through reduced operating costs must be acceptable and worthwhile to the company, and investment in energysaving measures will normally compete with

6.2. Methods of Increasing Energy Efficiency . . . . . . . . . . . . . . . . . . . 6.3. Control and Operation of Columns . 6.4. Column Improvements–Internals . . . 6.5. Column Improvements–Control Systems . . . . . . . . . . . . . . . . . . . . 6.6. New Column Designs . . . . . . . . . . . 7. Drying . . . . . . . . . . . . . . . . . . . . . 7.1. Energy Efficiency of Dryers . . . . . . 7.2. Operation and Control of Convective Dryers . . . . . . . . . . . . . . . . . . . . . 7.3. Heat Recovery from Convective Dryers . . . . . . . . . . . . . . . . . . . . . 7.4. Microwave and Radio-Frequency Drying . . . . . . . . 7.5. Hybrid Dryers for Batch Processes . . 8. Process Design . . . . . . . . . . . . . . . 8.1. Less Energy-Intensive Process Conditions . . . . . . . . . . . . . . . . . . 8.2. Improved Conversions . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . .

1196 1197 1197 1198 1198 1199 1200 1200 1200 1201 1201 1201 1202 1202 1202

other investment opportunities. The payback period is a simple criterion (although it has many limitations) widely used for a preliminary analysis of investment in energy-saving measures. Payback period is defined as: Paybackperiod =

Capitalrequired Annualsavings

Most of the methods discussed in this article for improving the energy efficiency of chemical processes are based on established technologies and are realistic in terms of being financially attractive and yielding an acceptable return on investment. Whether or not they should be implemented depends on a detailed technical and economic study of each potential application. Energy for Feedstocks. The chemical industry uses substantial quantities of energy, mainly in the form of oil and gas, as feedstock for conversion to other products. The efficient chemical conversion of energy feedstocks to higher value products is of great importance, but it is normally considered separately from the

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efficient management of energy used to operate processes. Energy usage in the form of feedstocks is not considered in this article. Energy Supply and Distribution to Processes. The energy used in processes, either as heat or as power, can be supplied and distributed in several ways: Heat Primary Steam systems Direct firing Electric heating Secondary Heat recovery

Power Electricity Direct drives (steam turbines)

Energy-Using Operations. Every operation in a chemical process requires energy in some form or other, but for the purposes of energy management, the most important energy-using and energy-intensive operations must be identified. An approximate analysis of the main energy-using operations for the chemical industry as a whole was made by Grant [1]. A more recent and detailed analysis for the chemical industry in the United Kingdom [2] has identified the main energy-using operations in each sector of the industry and for the industry as a whole; the results are given in Table 1. A wide range of processes was analyzed in each sector, and a similar breakdown of energy usage could be expected for the same sectors in the chemical industry worldwide.

For the industry as a whole, around 71 % of the energy is used to provide heat, and around 29 % to provide power. Heat usage is dominated by three operations: (1) heating of process streams, (2) distillation, and (3) drying. Methods for improving the energy efficiency of these operations are considered in this article. The main power-using operations involve fluid flow through pumps, compressors, fans, and blowers. The energy efficiency of these operations can be improved with the use of more efficient equipment, better matching of equipment capacity to throughput, and more energyefficient flow control such as variable-speed drives. These techniques are not considered in detail in this article. The methods are more applicable to large pumps, motors, and other items of equipment. In many plants, energy usage is so widely distributed that the actual use in each item of equipment is small, and in such cases, energy saving through improved equipment is difficult to achieve. Significant power savings are possible if processes are improved to reduce the flow of process streams (for example, by reducing recycles) or to operate at reduced pressure. An example of the latter approach is given (see Section 8.2).

2. Areas and Levels for Energy Management 2.1. Areas Energy management is involved in two main areas:

Table 1. Energy use in the chemical industry in the United Kingdom [2] Operation

Process heating Evaporation Distillation Drying Space heating Fans and blowers Drive motors Refrigeration Compression Mixing Pumping Comminution Total

Energy use in each sector, % Pharmaceuticals Dyestuffs and Pigments

Plastics and rubber

Inorganic

Organic

Miscellaneous

Total, all sectors, %

36 0 15 3 23 2 1 6 6 6 2 0 100

42 3 20 16 5 3 2 2 3 2 2 0 100

52 7 2 17 1 3 4 2 3 1 7 1 100

35 3 23 4 1 1 1 8 16 1 7 0 100

38 3 5 14 9 2 2 2 18 2 4 1 100

40 4 13 10 4 2 2 5 10 2 6 2 100

40 1 4 14 3 2 3 3 4 5 3 18 100

Energy Management in Chemical Industry 1) energy production and distribution, and 2) energy usage within processes. The area of energy production and distribution deals with the operation of boilers, furnaces, and electrical generators; with compressed air and nitrogen generation; and with the distribution of steam, electricity, and other services. These matters are important, and good energy management should ensure that services are generated and distributed as efficiently as possible. The methods for this are well-known and are not discussed here. Once the maximum practical efficiency has been achieved in the supply and distribution of services, the only remaining area for energy saving involves reducing the energy requirements of the process. The greatest potential for more efficient use of energy in the chemical industry will almost always lie in decreasing the process energy requirements and usage, rather than in supplying energy more efficiently to processes. Energy usage within processes should, therefore, be the most important consideration for energy management.

2.2. Levels Energy management is involved at three main levels: 1) Efficient operation of the existing plant (good housekeeping measures) 2) Major improvements to the existing plant (retrofits or revamps) 3) New plant or process designs Level 1 is primarily the concern of plant management, and the energy management measures involved in operating plants efficiently are often referred to as good housekeeping measures. Level 2 involves plant management as well as process and plant design. Level 3 involves principally research and development, together with process and plant design. The potential for substantial improvement in energy efficiency increases in going from Level 1 to Level 3, but the necessary capital costs and the time scale to achieve the improvements also increase.

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3. Good Housekeeping Good housekeeping measures cover a wide range of relatively minor measures which generally represent good plant management and operating practice. The difference between good and bad housekeeping typically represents around 10 % of the total process energy usage, but it might be more in extreme cases. A checklist of the most important measures follows: 1) Steam Systems Reduce leaks Improve operation of steam traps Increase condensate recovery Increase flash steam recovery Use lower steam pressure if possible Use direct steam for heating if possible Improve maintenance of steam ejectors 2) Insulation Check optimum thickness; increase if necessary Ensure flanges and manholes are lagged Improve protection from rain and water Replace after maintenance 3) Heat Exchangers Monitor fouling; reduce if possible Increase cleaning frequencies 4) Pumping and Compression Avoid oversizing pumps Consider replacing undersize lines and valves Consider if flows can be reduced Switch off when not required 5) Electrical Systems Improve power factor; minimize other losses 6) Compressed Air Systems Use minimum operating pressure Use minimum inlet air temperature Reduce pressure drops in pipework Reduce leaks 7) Cooling Water Systems Minimize flow Consider scope for “cascade” flow through units in series Switch off when not required 8) Vacuum Systems Reduce leaks Replace steam ejectors by liquid ring pumps Although these measures may require relatively little capital expenditure, they can involve significant investments of time and manpower. Most of the measures are not “once-and-for-all,” in the

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sense, for example, that installing a new heat exchanger for heat recovery is a once-and-for-all measure. Good housekeeping measures, therefore, require regular attention if energy savings are not to be eroded.

4. Audits, Monitoring, and Control The measurement and control of energy use within processes are interrelated, essential aspects of good energy management.

the reasons should be investigated because they will often indicate possible methods of improving energy efficiency. Figure 1 shows two approaches to establishing targets for improving energy efficiency. In the first, the top-down approach, targets are often set in an arbitrary manner based on current energy use. For example, a target might be set as a 10 % reduction in energy usage within two years. This is an easy way to set targets, but it provides little insight into the real potential for energy savings or the manner in which the targeted reduction might be achieved.

4.1. Energy Audits An energy audit is an analysis of energy usage which identifies the main areas, operations, and types of energy used in a process. It should be based on actual plant measurements and data, rather than on whatever process design data might be available. Plants are frequently modified so that the operation is very different from the original design. Also, design data may be inaccurate or optimistic with regard to energy requirements. In many cases, an audit may involve primarily obtaining accurate heat and mass balances for the process, and the major problems that arise are usually concerned with inadequate instrumentation, particularly for steam flow. For example, electrical power consumption is often better metered and easier to quantify than heat consumption. An energy audit, normally based on a single set of typical operating conditions, is the essential first step in identifying methods of improving the energy efficiency of a process. The results of the audit should, therefore, be analyzed carefully in order to: 1) develop a better understanding of the process, 2) establish a measure of energy efficiency for the process, and 3) establish targets for improving energy efficiency. If the results of an energy audit are in close agreement with the process design data, they may provide little further understanding of the process. However, if substantial differences exist,

Figure 1. Targets for energy savings A) Top-down approach; B) Bottom-up approach

The second method, the bottom-up approach, is to try to identify the minimum practicable energy requirement for the process. This provides an absolute target for improving energy efficiency. This approach requires a thermodynamic analysis to identify first the minimum theoretical energy required for the process route (which may be negative for an exothermic reaction). Then the unavoidable additional energy required to operate a real process with the best available technology must be identified. This may require a mixture of thermodynamics and process engineering techniques. The latter, more rigorous approach is not yet widely applied to complete chemical processes. However, the rigorous thermodynamic approach, in the form of process integration or pinch technology [3], is now being used with considerable success for identifying targets in heat recovery systems (→ Pinch Technology).

4.2. Monitoring and Control Monitoring refers to the regular, systematic measurement of energy use in relation to production

Energy Management in Chemical Industry (rather than the one-shot analysis of usage in an energy audit). A good monitoring and control system requires: 1) good instrumentation, 2) a short time period between measurements (otherwise excessive energy use continues for too long before remedial action is taken), 3) clear norms and targets with which to compare the measured energy usage, 4) knowledge of control action to be taken if energy use exceeds the norm, and 5) control systems to implement the action.

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as shown in Figure 3. Operation of the plant at high output levels, therefore, usually represents an important method of improving energy efficiency. If required production levels are significantly lower than capacity, then a variety of operating strategies, including campaign operation, should be considered [4].

4.3. Energy Usage and Production Levels In establishing norms and targets with which to compare the measured energy usage, problems may arise in quantifying the effects of (1) variations in product mix, (2) variations in weather, and (3) variations in throughput. Of these, the effect of variations in throughput is often the most important. A typical relationship between energy consumption and production level is shown in Figure 2. Many chemical processes have relationships of this form. In many cases, plants cannot operate over a wide range of throughputs– particulary single-stream continuous processes– and operation at very low production levels is not possible. The data can show considerable scatter, and the temptation often arises to fit a regression straight line to all the data points to establish a norm. This should be avoided if possible; a detailed analysis of the data may reveal a more complex and accurate relationship between energy use and production levels. With relationships between energy usage and production of the form shown in Figure 2, both fixed and variable, or output-related, components exist. This can be seen clearly by extrapolating the energy use to zero production. The fixed component can range from 30 % of the total energy consumption at full output to as to as much as 60 % for some batch processes. With energy consumption vs. output relationships of this form, the energy consumption per unit of output–the normal measure of energy efficiency–decreases as output increases,

Figure 2. Energy consumption versus output for a typical process

Figure 3. Energy consumption per unit of output versus output for a typical process

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4.4. Batch Processes Increasing the energy efficiency of batchwise processes through improved technology or process modifications can often be more difficult than for continuous processes. As a simple example, the potential for heat recovery is often more limited in batch processes. The control and operation of batchwise processes is, therefore, particularly important for good energy management. Improved sequencing and control of operations, which reduce batch times and increase throughputs, usually significantly reduce the energy consumption per unit of output [5].

4.5. Advanced Monitoring and Control Systems The monitoring methods discussed above can be applied by using conventional and existing control systems. The use of modern computerbased monitoring and control systems, which are designed to operate on a plant-wide basis, can yield further major improvements in energy efficiency [6–8]. These systems are designed to supervise the individual programmable controllers that operate the control loops on the plant. The system will extract data from the programmable controller networks and then output set points and controller configuration information. Systems are available which are independent of programmable controller manufacture and can link up with most types of instruments and computers. The data collected can be manipulated and presented in a form useful to plant managers and operators. For example, the data could be production rates, or energy use and waste. These can be projected on a real or historic time basis, and displayed to show how a plant is performing against targets. Data can be presented in the form of bar charts, graphs, and tables or as values superimposed on a mimic diagram. The system can be used in a passive role as a monitoring and management tool to assist skilled plant operation by bringing together groups of readings, calculating totals and averages, and indicating trends and optimum operating conditions. The system can also be used as a supervisory control system, for example,

by changing set points on programmable controllers.

5. Heat Recovery Heat recovery is the single most widely used technology for improving energy efficiency in chemical industries. This is to be expected because process heating is by far the most significant energy-consuming operation, accounting for 40 % of the energy used by the industry as a whole and as much as 52 % in the inorganic chemical sector (Table 1). In addition, evaporation, distillation, and drying together account for another 27 % of the energy used by the industry as a whole, and heat recovery is an established method of improving the energy efficiency of these operations (Chaps. 6 and 7). Sometimes, in attempting to consider heat recovery in a process, the potential for reducing the heat requirements of the process is ignored. Improving a process to use less heat is almost always better than modifying it to recover more heat, and the possibility of reducing heat requirements should always be considered first. Two interrelated aspects of applying heat recovery to chemical processes are (1) process design and (2) equipment design and selection. Process design involves determining the process streams between which heat is to be transferred and the amount of heat that is to be transferred; this is also called the “duty” of the heat exchanger. Equipment design involves the design or selection of the best heat exchanger for the duty. The criteria for the best exchanger involve capital costs, installation costs, and considerations of all aspects of operability. The basic design equation for any heat exchanger is an overall heat balance of the form Q= rateof heattransferfromhotstream = rateof heattransfertocoldstream =U A∆Tm

(1)

where Q is duty (W), U is the overall heat-transfer coefficient (W m−2 K−1 ), A is the heat transfer area (m2 ), ∆T m is the mean temperature difference between the fluids (K).

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5.1. Process Design for Heat Recovery

5.2. Types of Heat Exchanger

The first stage in the design of a heat recovery system is to identify potential heat sources and heat sinks. Heat sources can be either process streams (e.g., the outlet stream from a reactor) or waste streams (e.g., a flue-gas stream). Heat sinks can be process streams (e.g., a reactor feed) or other streams such as combustion air or boiler feedwater. The stream data necessary to specify heat sources or heat sinks include the following:

When a feasible match between hot and cold streams has been identified, the most appropriate type of heat exchanger for the required duty must be selected. The main types of exchanger are listed in the following according to type of construction. Tubular heat exchangers:

1) 2) 3) 4) 5) 6) 7)

stream phase (vapor, liquid, mixture), mass flow rate, temperature, pressure, contaminants and fouling characteristics, corrosion and hazardous characteristics, and variations in flowrate and temperature.

The last parameter is important for continuous processes with significant variations in throughput and for any attempt at heat recovery in batch or semibatch processes. The next stage in process design is to consider matching heat sources and heat sinks, while taking account of the above parameters and the physical proximity of the streams. (Installed costs of heat exchangers can be many times greater than the purchase costs of the unit, and long piping runs between units are undesirable.) If a number of hot and cold streams exist between which heat exchange is possible, then the resulting heat-exchanger network can be designed in many different ways. In such cases, the techniques of process integration (or pinch technology) should be applied to achieve the optimum heat recovery network design, by taking into account the balance between energy savings and capital costs, as well as operability considerations for the network (→ Pinch Technology) [3]. Even for a heat recovery system with only a single heat source and sink, consideration must be given to the optimum economic duty for the exchanger. A preliminary economic analysis should, therefore, be carried out before any detailed design calculations are made.

Double pipe∗ Shell-and-tube∗ Plate heat exchangers: Gasketed+ Spiral+ Extended surface heat exchangers: Plate – fin+ Tube – fin∗ Regenerative heat exchangers: Rotary+ Fixed matrix∗ A comprehensive guide to heat-exchanger selection and costs can be found in [3]. The types of heat transfer that can occur in heat recovery and a guide to the most appropriate types of exchanger are given in Table 2. Table 2. Exchanger selection for types of heat transfer Type of heat transfer

Type of heat exchanger

Liquid – liquid Gas – liquid Gas – boiling liquid Gas – gas

tubular, plate extended surface shell-and-tube (waste-heat boiler) tubular, plate, regenerative

5.3. Equipment Design Once a heat recovery duty has been specified and the type of exchanger selected, two interrelated aspects of the detailed equipment design must be considered: (1) thermal design and (2) mechanical design. Some types of exchanger are normally designed by specialist manufacturers for a duty specified by the purchaser (marked by + in preceeding list); other types are more normally designed by the user to standard methods and codes (marked by ∗ in preceeding list).

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Thermal design involves calculating mean temperature differences and heat-transfer coefficients, and hence the required heat-transfer area. The pressure drop for each stream is usually also calculated at this stage. Mechanical design involves specifying the required heat-transfer area in terms of equipment details, at least to the extent of providing sufficient detail for exchanger fabrication to appropriate codes and standards. For example, for a shell-and-tube exchanger, this would involve specifying the diameter, length, layout, and number of tubes; the diameter of the shell; the type and spacing of the shell-side baffles; etc. The two aspects of thermal and mechanical design are interrelated. Heat-transfer coefficients (or pressure drops) cannot be calculated without a knowledge of the basic mechanical features that affect the geometry of the heat transfer area and the velocity of the fluids. An iterative design procedure is, therefore, necessary. A preliminary mechanical design is established, based on assumed values of the overall heat-transfer coefficient. The thermal design of this exchanger is then checked by detailed calculation. If the thermal design is not suitable, the mechanical design is amended and the procedure repeated. This is known as a rating procedure for heat exchanger design. Comprehensive design procedures can be found in [9].

5.4. Example: Optimum Duty of a Waste-Heat Boiler Process Description. An effluent gas stream from a reactor at a temperature of 550 ◦ C can be used to generate steam at 0.5 MPa (5 bar) in a

waste-heat boiler. The waste-heat boiler can be a vertical shell-and-tube exchanger with the hot gases on the tube side and the steam on the shell side. The following data pertain: Gas flow rate 10 t/h Gas specific heat 1.63 kJ kg−1 K−1 Estimated overall heat-transfer coefficient in the waste-heat boiler 0.12 kW m−2 K−1 For steam at 0.5 MPa (5 bar) raised from boiler feed water at 15 ◦ C: saturation temperature, T sat = 150 ◦ C and increase in enthalpy hfg = 2686 kJ/kg Operating period 4000 h/a Cost. The value of the steam raised in the waste-heat boiler is ca. 26 $/t. Approximate purchased costs of shell-and-tube exchangers are available, e.g., in [3]. The installed cost of the exchanger is approximately twice the purchased cost. Method of Analysis. The optimum economic duty depends on the value of the steam raised in relation to the installed cost of the exchanger. Both of these depend on the gas outlet temperature T . The payback period can be used as a first guide for evaluating the investment. A range of values of T is chosen, starting with a “base case” in which a short payback period might be expected. Successively larger exchangers are then priced, and the payback period for each incremental investment is calculated until the return on the additional investment is unattractive. Calculations of Heat-Transfer Areas. The temperature profile in the waste-heat boiler is shown in Figure 4 (for explanation of symbols, see Table 3). From Equation (1):

Table 3. Heat transfer areas for waste-heat boiler∗ T, ◦ C

Q, kW

M, kg/s

A, m2

200 180 170 160

1590 1680 1720 1770

0.589 0.623 0.640 0.657

79 98 113 139

∗ T = gas outlet temperature; Q = duty (rate of heat transfer); M = rate of steam generation; A = heat-transfer area.

Energy Management in Chemical Industry Q= 4.53 (550−T ) = 2.69 × 103 M = 0.12A∆Tm

The values Mand A have been calculated for a range of T , as shown in Table 3.

Figure 4. Temperature profile in a waste-heat boiler

Calculations of Costs. Based on one year’s operation (4000 h), the value of the steam generated is ca. 3.81×105 · M (in $/a). The value of steam generated and the installed cost of the exchanger have been calculated for all four values of T . The payback period for the base-case exchanger, area 79 m2 , was calculated first and then the payback period for the incremental investment in the next three sizes, as shown in Table 4. Comments. Preliminary analysis showed that heat recovery with a waste-heat boiler is economically attractive. The base-case exchanger has a payback period of only 0.3 year. If, for example, the company criterion for investment in energy saving measures required a payback period of about two years (a fairly common requirement), then the optimum economic gas outlet temperature would be between 170

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and 160 ◦ C. Detailed design calculations would be concentrated on conditions in this region.

6. Distillation Distillation is the most widely used operation in the chemical industry for the separation of mixtures of chemical species. It accounts for around 13 % of the energy usage of the industry as whole, and up to 23 % of the energy usage in some sectors such as organic chemicals (Table 1). A summary of 130 of the most important separations in the chemical industry which are carried out by distillation is given in [10]. The basic process of distillation is the equilibrium flash where a mixture of components is separated into a vapor phase and a liquid phase, with the components partitioned selectively between them. The exact equilibrium compositions of the phases are determined by temperature, pressure, initial composition, and relative volatility of the components. Because the equilibrium concentrations for a single flash are not usually pure enough, many successive equilibria are required to achieve good separation. Thus, distillation is carried out in columns having multiple vapor – liquid stages in succession. A typical distillation column is shown in Figure 5. Feed enters at some intermediate position. Vapor rising up the column comes in contact with liquid falling down the column. At the bottom, a portion of the liquid is vaporized in a reboiler. Vapor leaving the top of the column is condensed, and a portion is returned to the column as reflux. A temperature gradient is created in the column by the addition of heat in the reboiler and the removal of heat in the condenser – reflux system. The concentration of the components tends to follow the temperature profile, with the lighter components concentrating in the upper (cooler) section of the column and the heavier components concentrating in the lower (hotter) section.

Table 4. Costs and payback period for waste-heat boiler Exchanger area, m2

Exchanger cost, 103 $

Value of steam, 103 $ Payback period, a

79 98 113 139

66.2 77.7 85.2 99.4

224.5 237.5 243.9 250.4

66.2/224.5 11.5/13.0 7.5/6.4 14.2/6.5

= 0.29 = 0.88 = 1.17 = 2.18

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Energy Management in Chemical Industry mum number of theoretical plates at total reflux and the minimum reflux with an infinite number of theoretical plates, as shown in Figure 6. Column design represents an optimum balance between the relative cost of energy (operating costs) and the cost of the column and internals (capital cost). Current practice is to design for reflux ratios between 10 and 20 % above the minimum.

Figure 5. Heat flow in a simple distillation column

The column is filled with either trays or packing material to promote intimate vapor – liquid contact. The effectiveness of the stage contacting equipment is measured by tray efficiency or by the height of a transfer unit for packed columns. These can be regarded as the fractional approach to equilibrium that occurs in any one physical tray or packing length. Distillation is a heat-driven separation process in which the heat input at the reboiler (often steam-heated) flows up the column and is rejected at a lower temperature in the overhead condenser. Methods of improving the energy efficiency of distillation are almost always directed toward reducing the reboiler heat demand, because this represents the major energy requirement. Electrical energy is also necessary for pumping cooling water through condensers and for reflux pumps, but this is usually of minor significance compared with the heat requirements.

6.1. Column Design For a given separation, the reflux ratio and hence the required heat input decrease as the number of theoretical plates (equilibrium stages) in a column increases. The bounding limits are the mini-

Figure 6. Relationship between reflux ratio and number of plates

Regardless of the design reflux ratio, columns are frequently operated at much higher reflux ratios, with a corresponding increase in energy consumption. Control of the reflux ratio represents a major operating parameter.

6.2. Methods of Increasing Energy Efficiency The methods that can be used to increase the energy efficiency of distillation may be divided into three levels: 1) Better operation and control of existing columns (requiring relatively low costs) 2) Major improvements to existing columns (e.g., new internals, advanced control systems) 3) New designs (e.g., split-tower (multi-effect) distillation)

Energy Management in Chemical Industry The first two levels are generally applicable to the existing plant, while the third level is usually worth considering only for new plants. In addition to these methods which are specific to the distillation column itself, heat recovery from the associated hot streams is an important energy-efficient measure. For example, preheating feed streams with both product streams is common practice. To evaluate and analyze methods for energy efficiency in distillation, detailed calculations of column operation, including stage-to-stage calculations, are usually necessary. Computer programs are now available for the design and simulation of distillation columns, and these greatly facilitate the detailed calculations.

6.3. Control and Operation of Columns The main variable controlling the energy consumption on an existing column is the reflux ratio. A number of other variables can be manipulated to allow a reduction in reflux:

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40 ◦ C, relatively high pressures may be necessary to condense some of the more volatile components. Opportunities may exist for reducing column pressure when cooling capacity is high due to low ambient temperature. Advanced control systems for such pressure reduction are available, but action can also be taken by alert process operators.

6.4. Column Improvements–Internals New columns are designed with reflux ratios close to the minimum, but many columns still in use were originally designed for higher reflux ratios. In such cases, energy consumption can be reduced by revamping the column to increase the number of stages. Table 5 is a very rough guideline to the economic feasibility of this approach. Table 5. Economic feasibility of column improvement Ratio of actual to minimum reflux

Potential cost savings

product specification feed plate location operating pressure

> 1.3 1.3 to 1.1 < 1.1

good possible, worth studying doubtful

Product specification should be examined critically. The column product specification is often set slightly higher than the ultimate specification for sale or further processing. This minimizes production of off-specification material during normal operating conditions, at the expense of increased energy usage. The correct feed-plate location should be used. The feed should enter the column at the point where the composition on the tray closely matches the composition of the feed. If it enters anywhere else, efficiency is lost through blending. Many columns are designed with a number of optimal feed points, and if necessary the feed point can easily be relocated. For many mixtures, separation is easier at lower pressure because the relative volatility increases with decreasing pressure. Lower pressure operation can, therefore, be used to achieve the same separation at lower reflux ratios. The operating pressure of most columns is set to allow condensation of the overhead vapors with cooling water (or air). Because this limits overhead temperatures to a minimum of ca. 35 –

The performance of a distillation column depends on the actual number of stages and the stage efficiency (or the total height of packing and the height of the transfer unit for packed columns). Therefore, for stage columns the number of theoretical stages can be increased by (1) increasing the number of trays by increasing the column height or reducing the tray spacing or (2) replacing existing trays with a more efficient type. Increasing column height can be very expensive. Reducing tray spacing may be expensive because of the cost of modifying tray supports, and problems due to entrainment may result. In many cases, retraying a column with more efficient contacting devices is the most attractive approach [11]. For packed columns, changing to a packing with improved liquid distribution may be attractive. Generalizations about tray efficiency (or height of a transfer unit for packing) are difficult to make because of the dependence on the specific application. A preliminary analysis can be

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made of the energy savings available at various levels of increased tray efficiency. If the results look attractive, tray and packing suppliers can be approached to determine the maximum feasible increase in efficiency, and a detailed economic analysis is then possible.

6.5. Column Improvements–Control Systems The control system for a distillation column has two basic functions: to stabilize and to optimize the operation. When applied properly, the control system should ensure the production of products to specification at the minimum cost. Conventional distillation control systems are designed only for stabilization, relying on operator intervention for any optimization. A conventional system for a two-component fractionator is shown in Figure 7. Product flows are level-controlled. The reflux is set independently at a fixed rate, which is usually conservatively high. Product quality is controlled inferentially by varying the reboiler heat input to hold a constant temperature on some critical tray. This type of system is susceptible to variations in heat balance, which lead to changes in product quality. Advanced control systems are available that use direct material-balance control, feed-for-

Figure 7. Conventional control system for a distillation column TC = Temperature control; LC = Level control; FC = Flow control

ward control, on-stream product analyzers, and digital computers for optimization. Such systems can give more stable operation, control the column at the optimum operating point, and reduce the need for operator action. They offer considerable potential for reducing energy requirements and operating costs [8, 10].

6.6. New Column Designs In conventional distillation, each column has its own reboiler and its own condenser, normally water- or air-cooled. Heat supplied to the reboiler is rejected without further use in the condenser. Many methods of making better use of these heat flows are possible, involving new approaches to column design; these include: heat pumping split-tower operation A heat pump system can upgrade the heat content of the overhead vapors so that they may be used in the reboiler of the same column. A system based on vapor recompression, using the overhead vapors as the heat pump working fluid, is shown in Figure 8. Although heat pumping can produce significant energy savings, only a limited range of applications offer any prospect of economic viability. The major criterion is the temperature range in the column, and potential

Energy Management in Chemical Industry applications are restricted to relatively closeboiling components.

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but each at a different (decreasing) pressure. All the heat is applied to the first column, and its overhead vapors are condensed by reboiling the next column, and so on. The energy requirement for the series is theoretically about 1/N times the single-column energy requirement. Although this method offers substantial energy savings, it has many practical constraints and has not yet been applied widely.

7. Drying Thermal drying of solids accounts for around 10 % of the energy usage in the chemical industry as a whole, and up to 17 % of the energy used in sectors such as inorganics, dyes and pigments, and polymers (Table 1). The three main classes of dryers, depending on the method of heat transfer are Figure 8. Distillation with vapor recompression

The basic principle of split-tower distillation is to increase the thermodynamic efficiency by repeated use of a given quantity of energy (analogous to the repeated use of heat in multi-effect evaporation). One possible application is shown in Figure 9. A series of N towers is operated in parallel with approximately equal feed rates,

Figure 9. Multi-effect distillation

1) Direct contact (convective) dryers: heat transfer by direct contact of the wet solid and a hot gas stream 2) Indirect contact (conduction) dryers: heat transfer to the wet solid through a solid wall 3) Microwave or radio-frequency dryers: heat transfer throughout the wet solid by the action of a high-frequency electric field. Convective dryers are the most widely used type in the chemical industry. The most common ap-

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plication is the evaporation of water vapor, although dryers that evaporate solvents are used in some pharmaceutical and fine-chemical processes. Dryers evaporating organic solvents are less significant energy users because of the lower latent heat of vaporization of organic solvents compared to water.

7.1. Energy Efficiency of Dryers The thermal efficiency of dryers is usually defined as the ratio of the heat required for vaporization of the water removed to the heat actually used. For indirect-contact dryers, the efficiency can be close to 100 %, but the efficiency of many convective dryers may be as low as 30 %. With this definition, an efficiency of 100 % does not represent the minimum energy use because efficiencies greater than 100 % are theoretically possible with heat recovery from the exhaust gases. The primary methods of improving the thermal efficiency of convective dryers are 1) improved operation and control 2) heat recovery from exhaust gases

7.2. Operation and Control of Convective Dryers The flow rate and inlet temperature of the hot gas stream (normally air) are the main parameters in dryer operation. The heat required for a given duty can be reduced by controlling these parameters, i.e., reducing the air flow rate and increasing the inlet temperature, to minimize the exhaust gas enthalpy per unit mass of water vapor [12, 13]. Dryers that operate at a range of throughputs should have correct conditions throughout the range. When running at less than full capacity, many dryers are operated at excessive air flow rates. Spray dryers, fluidized-bed dryers, and rotary dryers may offer little scope for altering the air flow rate without affecting particle transport characteristics within the dryer, and the air inlet temperature may be the only variable that can be controlled. The minimum level of instrumentation and control for convective dryers should involve

temperature measurement of inlet and outlet air, as well as the use of dampers for flow control. A better approach to optimum operation can be achieved with on-line measurement of exhaust air humidity, and hygrometers are now available which can operate up to 100 ◦ C in dust-free gases. Further improvements in operation, including the avoidance of overdrying, may be possible with direct on-line measurement of product moisture content. Several techniques are available for this, such as microwave absorption [14], but reliable methods are not yet possible for all applications.

7.3. Heat Recovery from Convective Dryers Recirculation of a fraction of the exhaust gas stream to the inlet air for the heater represents a form of heat recovery that can yield significant energy savings. While this is a simple concept, the increased moisture content of the hot air entering the dryer means that a higher temperature is necessary to achieve the same drying rate. The capital costs of the ducting and control for air recirculation may be high in some cases. Heat recovery from the exhaust gas stream by means of heat exchangers can result in substantial energy savings. Preheating the inlet air is the most common use of recovered heat, although in principle the heat can be used for other process streams. Low outlet gas temperatures and heatexchanger fouling are practical limitations (apart from the cost of the heat exchanger and associated ductwork) which may restrict heat recovery via heat exchangers. Fouling of heat-exchanger surfaces by particles in the gas stream can often be a major restriction to heat recovery. For example, exhausts from spray, fluidized bed, and rotary dryers may contain too much dust for heat recovery if a cyclone is the only method of particle removal from the gas stream. More efficient gas cleaning (e.g., by filtration) may be uneconomical if required only for heat recovery, but may be justifiable for high-value products if significant additional product recovery occurs. The conventional approach to heat recovery from dryer exhaust gases is aimed primarily at

Energy Management in Chemical Industry recovering sensible heat, i.e., the temperature of the hot gas stream is decreased without condensing any vapor. The potential for heat recovery is greatly increased if the latent heat of the vapor can also be recovered by cooling the exhaust gas below its dew point. This can be achieved in heat pump systems where the recovered latent heat is upgraded to higher temperatures and used to preheat the inlet air. Although the application of heat pumps to drying is the subject of much research and development work, as yet no evidence exists of proven applications in the chemical industry.

7.4. Microwave and Radio-Frequency Drying When a product containing water is treated with microwave or radio-frequency electromagnetic radiation, the energy is absorbed mainly by the water molecules. Drying is, therefore, possible without the use of large quantities of heated air, which suggests that energy savings might be possible in many applications. However, power for microwave and radio-frequency drying can be generated at an efficiency of only ca. 17 % on a primary fuel basis [14], so that applications with net energy savings are very limited. The capital cost of microwave or radio-frequency drying is high and has not been justified so far on the basis of energy savings alone. Factors that justify microwave or radio-frequency drying are more likely to involve improved product quality, ease of operation, and increased throughput on existing convective dryers.

7.5. Hybrid Dryers for Batch Processes The drying of pharmaceuticals and fine chemicals is affected by some special factors, compared to the drying of bulk chemicals. These may include the following: 1) 2) 3) 4) 5)

High product value Low throughput Difficulty of handling wet sticky material Moisture to be evaporated includes solvents Use of plant to process different materials

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For these duties, the trend is toward the use of hybrid solids-processing equipment, in which more than one operation apart from drying can be performed [15]. Examples are combined filter – dryers and equipment in which crystallization, filtration, washing, and drying operations can all be carried out. The aim of hybrid equipment is to reduce material losses (e.g., in dryer exhausts), handling problems, and hazards, and to increase throughput and flexibility of plant use. Improved energy efficiency is also a factor, but it is not the only consideration. However, even if energy efficiency is not a primary consideration, changes to equipment which increase the throughput of batch processes usually also increase the energy efficiency of the process as a whole (discussed in Section 4.4).

8. Process Design For many chemical processes, changes in overall process design or operating conditions offer the greatest potential for reducing energy requirements. Even in “mature” processes, which are well-established and have undergone extensive development, substantial improvements in energy efficiency may still be possible. A good example of this is the ammonia process, which by 1960 was regarded as a mature process. However, since then, considerable efforts in process development have substantially reduced the energy requirements from ca. 70 GJ per ton of ammonia to 30 GJ per ton at present (Fig. 10, see next page). The energy requirements are now beginning to approach the thermodynamic minimum for this process route, which is an absolute limitation on further energy savings. Most other chemical processes still have energy requirements that are far in excess of the theoretical minimum. Processes can be improved in many ways, and generalizations about how to achieve this are not very meaningful. Each process must be considered in detail to identify potential improvements. However, the following examples are given to show the improvements that can be achieved through (1) less energy-intensive process conditions and (2) improved conversions in the reactor.

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Figure 10. Energy requirements for the ammonia process

The reaction has two stages: (1) alkylation to give ethylbenzene, followed by (2) dehydrogenation to styrene. The dehydrogenation does not go to completion, and the products from the reactor are condensed to give a mixture of ethylbenzene, styrene, and other hydrocarbons. Separation of this mixture by distillation–in particular, the styrene – ethylbenzene column–requires most of the energy for the entire process. Improvements in operating conditions and catalysts have increased the conversion to styrene per pass from around 40 % to over 75 %. The effect of this on the styrene – ethylbenzene separation is to reduce column steam requirement from ca. 1.8 kg of steam per kilogram of styrene produced to less than 0.5 kg of steam per kilogram of styrene [18].

9. References 8.1. Less Energy-Intensive Process Conditions If processes can be operated at less extreme conditions of temperature or pressure, then energy requirements may be substantially reduced. An example of this is the production of lowdensity polyethylene. For many years, standard processes involved polymerization at pressures in the range of 100 – 300 MPa (1000 – 3000 bar), with very high power requirements for compression. Typical total energy requirements for the process were around 15 MJ per kilogram of polymer [16]. Modern processes now use lowpressure gas-phase polymerization at about 0.7 – 2 MPa (7 – 20 bar). The total energy requirement for these processes is ca. 3 MJ per kilogram of polymer [17].

8.2. Improved Conversions In many processes, the major energy requirement is for separation. Separation also frequently dominates the capital costs of the process plant. Increased conversions in the reactor, for example, by better and more selective catalysts, may substantially reduce energy requirements for downstream separations. An example of this is the production of styrene monomer from ethylene and benzene.

1. C. D. Grant: Energy Conservation in the Chemical and Process Industries, IChemE/George Godwin Ltd., Rugby 1979. 2. Energy Management Focus, no. 9, Focus on the Chemicals Industry, Dept. of Energy, London 1986. 3. B. Linnhoff et al.: User Guide on Process Integration for the Efficient Use of Energy, IChemE, Rugby 1982. 4. J. Whiston, G. R. Taylor, Proc. Eurochem. ’83, IChemE Symp. Ser. no. 79, Rugby 1983, pp. 335 – 343. 5. M. G. Kemp, P. Smith, in [4] pp. 344 – 353. 6. J. Springell, Proc. Conf. Energy and the Process Industries, IMechE, London 1985. 7. J. Redmond, Chem. Eng. (1987, Jan.) 10. 8. H. Bozenhardt, M. Dybeck, Chem. Eng. (1986, Feb. 3) 99 – 102. 9. Heat Exchanger Design Handbook, Hemisphere Publishing Corporation, 1983. 10. Technology Applications Manual: “Energy Conservation in Distillation,” DOIE/CS/4431-T2, US Department of Energy, 1980. 11. A. Jenkins, A. Challinor, J. Jones, Proc. Eurochem. ’83, IChemE Symp. Ser. no. 79, Rugby 1983, pp. 364 – 375. 12. D. Reay, Chem. Eng. (1976, Jul/Aug.) 507. 13. J. C. Ashworth, Processing (1985, May) 12. 14. Drying, Evaporation and Distillation, Energy Technology Support Unit Market Study no. 3, Energy Efficiency Office, London 1985.

Energy Management in Chemical Industry 15. J. Ashworth, Chem. Eng. (1986, Sept.) 30 – 34. 16. “Process Flowsheets,” Hydrocarbon Process., Nov. 1979. 17. Eur. Chem. News (1980, Sept. 8) 8.

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18. L. F. Albright: Processes for Major Addition Type Plastics and their Monomers, McGraw-Hill, 1974.

Plant and Process Safety

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Plant and Process Safety Volker Pilz, Bayer AG, Leverkusen, Federal Republic of Germany (Chap. 1, Sections 3.1 – 3.3) Herbert Bender, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany (Section 2.1) ¨ Michael Muller, Bayer AG, Leverkusen, Federal Republic of Germany (Section 2.2.1, Section 3.5.3 in part) Dietrich Conrad, Berlin, Federal Republic of Germany (Section 2.2.2) Claus-Diether Walther, Bayer AG, Leverkusen, Federal Republic of Germany (Section 2.2.2) Werner Berthold, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany(Section 2.2.3) Martin Glor, Ciba-Geigy AG, Basel, Switzerland (Section 2.2.3) Peter-Andreas Wandrey, Bundesanstalt f¨ur Materialforschung und –pr¨ufung, Berlin, Federal Republic of Germany (Section 2.2.4) Karl-Heinz Mix, Bayer AG, Leverkusen, Federal Republic of Germany (Section 2.2.4) ¨ Steinbach, Schering AG, Berlin, Federal Republic of Germany(Section 2.3) Jorg Albert Eberz, Bayer AG, Leverkusen, Federal Republic of Germany(Section 2.3) Francis Stoessel, Ciba-Geigy AG, Basel, Switzerland(Section 2.3) Hans Hagen, Bayer AG, Leverkusen, Federal Republic of Germany (Sections 3.4.1, Section 5.1 in part) Helmut Schacke, Bayer AG, Leverkusen, Federal Republic of Germany (Section 3.4.2) Richard Viard, Bayer AG, Leverkusen, Federal Republic of Germany (Section 3.4.2) ¨ Bernd Schrors, Bayer AG, Leverkusen, Federal Republic of Germany (Section 3.4.3) Stephan Weidlich, Hoechst Aktiengesellschaft, Frankfurt, Federal Republic of Germany (Section 3.4.3) Stefan Drees, Bayer AG, Leverkusen, Federal Republic of Germany (Section 3.5.1) ¨ Gunter Hesse, Bayer AG, Brunsb¨uttel, Federal Republic of Germany (Section 3.5.2) ¨ Hans Forster, Physikalisch-Technische Bundesanstalt, Braunschweig, Federal Republic of Germany (Section 3.5.3 in part) Klaus Bartels, Berufsgenossenschaft der chemischen Industrie, Heidelberg, Federal Republic ofGermany (Section 4.1) Ulrich Widmer, Sandoz, Basel, Switzerland (Section 4.2.1) Adrian Geiger, Sandoz, Basel, Switzerland (Section 4.2.1) Klaus Noha, Hoechst Aktiengesellschaft, Frankfurt, Federal Republic of Germany (Section 4.2.2) ¨ Edmund Muller, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany (Section 4.3) Dieter Grenner, Bayer AG, Dormagen, Federal Republic of Germany (Sections 4.4, 4.5) Nikolaus Schulz, Bayer AG, Dormagen, Federal Republic of Germany (Sections 4.4, 4.5) ¨ Jurgen Zimmermann, Bayer AG, Dormagen, Federal Republic of Germany (Sections 4.4, 4.5) ¨ Jurgen Harbordt, Bayer AG, Leverkusen, Federal Republic of Germany (Section 5.1 in part) Matthias Walper, Bayer AG, Brunsb¨uttel, Federal Republic of Germany (Sections 5.2, 5.3)

1206 1. 1.1. 1.1.1. 1.1.2. 1.2. 1.3. 1.4. 1.4.1. 1.4.2. 1.4.3. 1.5. 2.

2.1. 2.2. 2.2.1. 2.2.1.1. 2.2.1.2. 2.2.1.3. 2.2.1.4. 2.2.2. 2.2.2.1. 2.2.2.2. 2.2.2.3. 2.2.2.4. 2.2.2.5. 2.2.3. 2.2.3.1. 2.2.3.2. 2.2.3.3. 2.2.4. 2.2.4.1. 2.2.4.2. 2.2.4.3. 2.2.4.4. 2.2.4.5. 2.2.4.6.

Plant and Process Safety Safety Problems in Chemical Plants . . . . . . . . . . . . . . . . . . Types and Sources of Hazards . . Fundamentals . . . . . . . . . . . . . . Causes and Effects . . . . . . . . . . . Requirements for Safe Chemical Plants . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . Magnitude of Hazard Potentials and Risks . . . . . . . . . . . . . . . . Risk According to Previous Experience . . . . . . . . . . . . . . . Hazard Potential from Release of a Volatile Substance . . . . . . . . . . . Hazard Potential from an Explosion Conclusions . . . . . . . . . . . . . . Determination and Evaluation of Hazardous Properties of Substances (Safety Ratings) . . . . Harmful Effects of Substances . . Ratings of Flammable and Explosive Substances . . . . . . . . Flammability Ratings . . . . . . . . . Introduction . . . . . . . . . . . . . . . Combustibility/Flammability of Chemicals . . . . . . . . . . . . . . Oxidizing Properties of Chemicals . Flammability of Construction Materials . . . . . . . . . . . . . . . . . Explosion Data for Gas Mixtures . Introduction . . . . . . . . . . . . . . . Explosion Limits; Limiting Oxygen Concentration . . . . . . . . . . . . . . Maximum Explosion Pressure and Maximum Rate of Pressure Rise . . Pressure Limit of Stability for Unstable Gases . . . . . . . . . . . . . Ignition Temperature, Ignition Energies . . . . . . . . . . . . . . . . . Explosion Indices of Dust-Air Mixtures . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . Ignition Sensitivity of Dust Clouds Dust Explosibility . . . . . . . . . . . Characterization of Explosive Condensed Substances . . . . . . . . Introduction . . . . . . . . . . . . . . . Chemical Nature and Definition of an Explosion . . . . . . . . . . . . . . Grouping of Explosive Substances . Explosion Mechanisms . . . . . . . . Testing of Explosive Substances . . Conclusions . . . . . . . . . . . . . . .

2.3. 1207 1207 1207 1208

2.3.1. 2.3.1.1. 2.3.1.2.

1209 1210 2.3.1.3. 1211 1211

2.3.1.4.

1212 1213 1215

2.3.2.

1215 1215 1219 1219 1219 1219 1221

2.3.2.1. 2.3.2.2. 2.3.2.3. 2.3.2.4. 2.3.3. 2.3.3.1. 2.3.3.2. 2.3.3.3. 2.3.3.4. 2.3.4. 3.

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3.1.

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3.2.

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3.3. 3.4. 3.4.1. 3.4.1.1. 3.4.1.2. 3.4.2. 3.4.2.1. 3.4.2.2. 3.4.2.3. 3.4.2.4. 3.4.2.5. 3.4.3.

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3.4.3.1. 1235 1236 1236 1237 1238

3.4.3.2. 3.4.3.3.

Exothermic and PressureInducing Chemical Reactions . . . Introduction . . . . . . . . . . . . . . . Exothermic Reactions: Runaway Potential . . . . . . . . . . . Causes and Consequences of Overpressure-Inducing Exothermic Reactions . . . . . . . . . . . . . . . . Processes Hazard Assessment and Safety Evaluation for Exothermic Reactions . . . . . . . . . . . . . . . . Hazard Characteristics of Exothermic Processes due to Process Design . . . . . . . . . . . . . . . . . . Methods of Investigation and their Systematic Application . . . . . . . . Screening Methods . . . . . . . . . . Thermal Aging and Heat Accumulation Storage Tests . . . . . Calorimetry . . . . . . . . . . . . . . . Systematic Testing . . . . . . . . . . . Assessment Criteria . . . . . . . . . . Runaway Scenario . . . . . . . . . . . Severity: Adiabatic Temperature Increase . . . . . . . . . . . . . . . . . Probability and Kinetics of a Runaway . . . . . . . . . . . . . . . . . Assessment of Criticality . . . . . . Measures . . . . . . . . . . . . . . . . Development, Design, and Construction of Safe Plants . . . . Objectives, Regulations, and Concerns . . . . . . . . . . . . . . . . Procedure for Designing and Constructing Safe Plants . . . . . . Methodological Aids . . . . . . . . . Safe Processing: Strategies . . . . Fire Protection . . . . . . . . . . . . . Fire Prevention . . . . . . . . . . . . . Limitation of Fire Effects . . . . . . Explosion Prevention and Protection Introduction . . . . . . . . . . . . . . . Hazard Identification . . . . . . . . . Explosion Risk . . . . . . . . . . . . . Explosion Risk Reduction . . . . . . Legal Aspects . . . . . . . . . . . . . . Safety Techniques Based on Process Control . . . . . . . . . . . . . . . . . . Integration of PCE into the Safety Concept . . . . . . . . . . . . . . . . . Classification of PCE Systems . . . Requirements for PCE Equipment for Process Plant Safety and Design Principles . . . . . . . . . . . . . . . .

1238 1238 1238

1240

1241

1241 1243 1243 1244 1245 1246 1247 1247 1247 1248 1248 1250 1250 1250 1252 1254 1259 1259 1259 1260 1262 1262 1263 1263 1270 1272 1273 1273 1274

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Plant and Process Safety 3.4.3.4. 3.4.3.5. 3.5. 3.5.1. 3.5.1.1. 3.5.1.2. 3.5.1.3. 3.5.1.4. 3.5.2. 3.5.2.1. 3.5.2.2. 3.5.3. 3.5.3.1. 3.5.3.2. 3.5.3.3. 4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.1.4. 4.1.5.

4.2. 4.2.1. 4.2.1.1. 4.2.1.2. 4.2.1.3. 4.2.1.4. 4.2.1.5. 4.2.2. 4.2.2.1. 4.2.2.2.

Operation of PCE Safety System . . Summary . . . . . . . . . . . . . . . . Special Safety Equipment . . . . . Pressure-Relief Devices . . . . . . . Introduction . . . . . . . . . . . . . . . Safety Valves . . . . . . . . . . . . . . Bursting Disks . . . . . . . . . . . . . Sizing of Safety Valves . . . . . . . . Blowdown Systems . . . . . . . . . . Procedure . . . . . . . . . . . . . . . . Alternatives . . . . . . . . . . . . . . . Flame Arresters and Explosion Barriers . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . Flame Arresters for Mixtures of Vapors (Gases) and Air . . . . . . . . Flame Arresters for Dust – Air Mixtures . . . . . . . . . . . . . . . . . Safe Plant Operation . . . . . . . . Safe Handling of Chemicals . . . . Introduction . . . . . . . . . . . . . . . Normal Operation . . . . . . . . . . . Sampling . . . . . . . . . . . . . . . . Cleaning of Vessels . . . . . . . . . . Safety Practices when Working with Hazardous Substances Under Abnormal Operation . . . . . . . . . Safety in Batch and Continuous Processes . . . . . . . . . . . . . . . . Safety in Batch Processes . . . . . . Introduction . . . . . . . . . . . . . . . The Operating Manual . . . . . . . . The Human Aspect of Safety . . . . Internal Organization and Policies . Safety in Production Practice . . . . Safety in Continuous Processes . . . Introduction . . . . . . . . . . . . . . . Analogies with Batch Processes . .

1279 1280 1280 1280 1280 1281 1283 1283 1285 1285 1285 1286 1286 1286 1288 1289 1289 1289 1289 1290 1291

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The entire topic was coordinated by Volker Pilz

1. Safety Problems in Chemical Plants 1.1. Types and Sources of Hazards 1.1.1. Fundamentals Large-scale chemical production is one of the most important industrial activities. Its products

4.2.2.3. 4.3. 4.4. 4.4.1. 4.4.2. 4.4.2.1. 4.4.2.2. 4.4.3. 4.5. 4.5.1. 4.5.2. 4.5.3. 5. 5.1. 5.1.1. 5.1.2. 5.1.3. 5.1.4. 5.1.5. 5.1.6. 5.1.7. 5.2. 5.2.1. 5.2.1.1. 5.2.1.2. 5.2.1.3. 5.2.2. 5.3. 6.

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Special Features of Continuous Processes . . . . . . . . . . . . . . . . Technical Inspection . . . . . . . . . Maintenance . . . . . . . . . . . . . . Actions During Plant Design and Construction . . . . . . . . . . . . . . Actions During Plant Operation . . Maintenance Activity . . . . . . . . . Performance of Maintenance . . . . Continuing Plant Development (Analysis of Weak Points) . . . . . . Modification of Plants . . . . . . . . Reasons for Modifications . . . . . . Procedure for Modification . . . . . Summary of Plant Maintenance and Process Modification . . . . . . . . . Hazard Control . . . . . . . . . . . . Means of Limiting Accident Impacts . . . . . . . . . . . . . . . . . Retention Systems, Catch Wells . . Rapid-Closing Valves, Emergency Compartmentalization Systems . . . Emergency Drain and Collection Systems . . . . . . . . . . . . . . . . . Blowoff and Disposal Systems . . . Spray-Curtain (Drench) Systems . . Partial and Complete Containment . Fire Protection; Retention of Water Contaminated by Fire Fighting . . . Hazard Control Plans at Plant Level and Beyond . . . . . . . . . . Hazard Control Plans . . . . . . . . . Plant Hazard Control Plan . . . . . . Works Hazard Control Plan . . . . . Required Contents of the Plan . . . Off-Battery Hazard Control Plans . Public Awareness and Responsible Care . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

1299 1301 1303 1303 1305 1305 1309 1312 1312 1312 1313 1314 1314 1314 1315 1316 1316 1316 1316 1316 1317 1317 1317 1318 1318 1318 1319 1319 1320

play key roles in human nutrition, health, quality of life, and welfare. Nevertheless, many people regard chemical production as dangerous, even though chemical processes are associated with accident frequencies which are much lower than in other industries – and very few such accidents have anything to do with chemistry [2]. Despite the good accident statistics of chemical plants, grave accidents have occurred, endangering human life and the environment, and causing considerable damage. The names

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Flixborough (1974) [3], Seveso (1976) [4], Bhopal (1984) [5], and Basel (1986) [6] call spectacular events to mind. Three types of event are traditionally associated with the chemical industry: Releases and spills (Seveso, Bhopal) Fires (Basel) Explosions (Flixborough) “Explosion” as used here refers to an event in which large amounts of energy are released very rapidly (ultimately, in the form of expansion energy), producing a pressure wave that is (at least) audible and propagates at high velocity, decaying over distance. Damage caused by the above mentioned three types of event comes about through two distinct processes: Direct action of chemicals on humans and the environment Indirect action of liberated energy In the Bhopal accident, many people died after inhaling a dangerous gas that had escaped from a tank storage facility [5]. Most of the damage from the Basel warehouse fire was due to fire fighting water carrying starting material, decomposition products, and combustion products out of the stored material, harming the ecosystem of the Rhine [6]. The consequences of the Flixborough explosion occurred largely through the destructive action of the pressure wave, initiated by ignition of a cloud of escaped flammable gases [3]. Generally, chemical reactions underlie such harmful effects. When substances act directly on living matter, a chemical reaction may alter or destroy cellular tissue (toxic effect). In an explosion, energy is liberated by an exothermic chemical reaction. It is obvious that the dangers arising from chemical plants have much to do with the nature of the substances being processed, the way they are treated in the plant, and their tendency to take part in chemical reactions under these conditions. A conflict arises between process needs and chemical safety. On the one hand, chemistry requires reactive substances; on the other hand, this reactivity of the substances is a key aspect of the danger they pose.

A chemical process is impossible without substances that show hazardous properties and effects. The substances must therefore be reliably contained in the process equipment, and their reactivity must be governed so that uncontrolled chemical reactions cannot take place. 1.1.2. Causes and Effects Releases and Spills. An uncontrolled release from a closed facility presupposes an undesired opening of the containment (e.g., of a valve), or damage to the walls of vessels, process equipment, fittings, or piping (e.g., as a consequence of corrosion, seal failure, or rupture). If liquids and solids are released, people outside the plant are, as a rule, not directly endangered (or at worst after some time delay) and they are thus more easily protected. The material, on the other hand, can penetrate into the soil, contaminating the groundwater or passing through sewers into surface waters, posing a danger to the environment. Precautions inside the facility include erection of barriers (liquidproof floors, seals on sewers, and retention systems; see Section 5.1). If the substances are gaseous or finely dispersed (aerosols), they are transported by the motion of the ambient air after their release, so that they can be spread over large areas, diluted by convection and diffusion, separated from the air by sorption and possibly sedimentation, and gradually degraded through chemical reactions with the environment. Above certain concentration levels in air, pollutants can harm people, fauna, and flora in downstream regions. The amount of damage depends on the exposure time (dose). Usually, such a drifting gas or aerosol cloud causes a shortterm burden on individuals exposed to varying concentrations. Toxicology must provide information about these harmful effects (see Section 2.1). Explosions. An explosion involves the rapid, almost instantaneous release of heat and expansion energy resulting in the formation of large gas volumes and a (destructive) pressure wave, which propagates at high velocity. At a fixed point, the wave causes a sudden rise in pressure

Plant and Process Safety followed by a pressure drop, all within a fraction of a second. The pressure wave arises because the volumes of gas formed cannot escape quickly enough by mass motion. The destructive action of the wave results from the pressure force and the rapid transfer of momentum to objects in the region of wave propagation. Energy-supplying processes in an explosion can be chemical or purely physical. There are three different types of chemical explosion. The best-known is the instantaneous decomposition of sensitive chemical compounds to gaseous species occupying a large volume, because it figures in the use of explosives. The process is initiated by external excitation. The volume required for the gaseous products is markedly increased by the heat of decomposition evolved in the process. Decomposition can be excited mechanically by shear forces (friction) or impact; thermally by heating; or by the chemically generated shock wave from an igniter (see Section 2.2.4). The second chemical explosion process is also fairly well known: rapid oxidation of gases or finely dispersed particles in oxidizing gases (see Sections 2.2.2, 2.2.3). A highly exothermic reaction (usually combustion) heats the gases to high temperature so quickly that the pressure in confined systems generally increases by a factor of 6 – 10 (depending on the situation); in unconfined systems, the increase in volume can bring about a pressure wave with a peak overpressure of up to 100 kPa, with a positive pressure phase lasting for a few to a few hundred milliseconds (depending on the quantity of material that takes part in the spontaneous reaction, how well the fuel gas is mixed with air, how much turbulence there is to accelerate the combustion process, and how well the reacting mixture is confined). This type of explosion can be prevented “classical” anti-explosion measures (see Section 3.4.2). The third type of chemical explosion, less well known, at least among the general public, is the “thermal runaway” explosion, initiated by a homogeneous exothermic reaction that goes out of control. If the heat of an exothermic reaction cannot be removed, it further raises the temperature of the system. The reaction velocity increases, more heat is produced per unit time, and the system continues to heat up more and more rapidly

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until the maximum possible reaction velocity has been reached. The accompanying temperature rise can lead to a dangerous pressure buildup in two ways: by raising the vapor pressure of reactants and solvents, or by initiating another reaction (e.g., a decomposition) that liberates gases (see Section 2.3). If the pressure increase leads to overstressing of the reactor, it may burst and the expansion of its contents may generate a pressure wave. In principle, an explosion can also be brought about by a purely physical process. A sudden liberation of expansion energy is possible after gases have been compressed or when vaporization is abruptly initiated, e.g., if a hot, condensed substance is quickly mixed with another substance that vaporizes readily. So far, this phenomenon does not seem to have played a major role in chemical plant explosions, and it is not discussed here; but its possibility must be considered in relation to plant safety. Fires. Fires in chemical plants can occur if flammable substances are released and come in contact with an ignition source. Section 2.2.1 deals with the substance properties and ambient conditions that must be present for fires to start; Section 3.4 discusses measures to control fires. The effects of fires relate to the action of heat (energy evolved mainly as thermal radiation) and the release of pollutants (combustion gases and decomposition products).

1.2. Requirements for Safe Chemical Plants The phenomenological discussion in Section 1.1 showed that most of the dangers in the operation of chemical plants have to do with the reactivities of the substances present. The principal hazards are caused by releases and spills of substances outside the plant, and by uncontrolled chemical reactions between substances. The danger is greater, the larger the quantities of substances and energy released. Plant and process safety efforts must have as their essential goals: 1) To minimize the quantities of substances (Section 3.1) 2) To control the potential risks that remain

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Objective (2) requires: Reliably confining hazardous substances in process equipment that can withstand the anticipated stresses (due to pressure, temperature, corrosive attack, etc.). Insuring that process parameters (e.g., pressure, temperature, concentration) do not take on values such that the substances can undergo uncontrolled reactions. Some help is gained from the fact that substances, as a rule, react spontaneously with one another only if they have been suitably prepared (e.g., by treatment to increase their surface area); mixed and concentrated; and excited (e.g., by heat). Experimental laboratory tests are performed, e.g., to determine the temperature and concentration conditions under which reactions can get so far out of control that they can lead to an explosion, or to establish the initiation conditions that can cause dangerous reactions. A point of special interest is where the regions begin in which uncontrolled reactions cannot occur. The characteristic (critical) parameters found in such experiments are called safety ratings. They include such figures as the ignition temperature and the explosion limits. Safety ratings form the basis for the safety measures in chemical plants. All of Section 2 concerns how safety ratings are determined; the account is broken down by the event classes of Section 1.1. At the planning and design stage, plant and process safety requires taking steps so that critical concentrations, temperatures, and pressures are not reached. This is achieved by appropriate process design and process control engineering (Chap. 3). Safety also entails preventing the occurrence of potential ignition sources that could cause an undesired reaction (e.g., hot surfaces or sparks generated by mechanical or electrical equipment); great care needs to be taken in the layout of machinery and process equipment (Section 3.4). Critical process parameters and hazardous potential ignition sources must not occur in the plant as a result of process upsets or human error. These become an additional concern of plant safety, requiring a painstaking cause-and-effect analysis of all possible errors and malfunctions, and the institution of measures to prevent or neutralize situations that could lead to an unsafe condition. Such measures may be technical or

organizational; the second group includes operating instructions, inspections, etc. Chapters 3 and 4 present a fuller treatment of these points. The complete absence of all possible hazards – absolute safety – is not possible, for several reasons. 1) First, it cannot be ruled out that several safety measures will fail simultaneously, so that a potential hazard may become an actual one 2) People make errors from time to time, and they can misjudge things, assess them wrongly, even fail to notice them at all The fact that absolute safety is impossible is important; two inferences follow from it. Knowledge has to be continually advanced, and precautions have to be taken to avert risks in case of failure (Chap. 5).

1.3. Definitions The fact that absolute safety is not possible leads to two key questions: 1) What does plant safety mean in the first place? 2) How safe is safe enough? These intimately related questions can be answered in two different ways. In the first approach, safety is defined in terms of risk; a plant is said to be safe if the risk created by it is acceptable. Here “risk” means the possibility of harm, defined in terms of the probability of the harm during the lifetime of the plant (or the frequency of the harm) and its anticipated severity. In the second approach, a plant is said to be safe if it complies with the appropriate regulations and codes. Then it is also said to be safe enough. While the first method is based on a probabilistic concept, risk, the second employs a deterministic principle. In both cases, what is safe is clearly a matter of convention. The German standard DIN 31 000, Part 2, defines safety as “a state of affairs in which the risk is not greater than the greatest acceptable risk due to the technical process or condition under consideration” [7]. The standard states that this risk is generally not quantifiable, since only in rare cases can it

Plant and Process Safety be expressed as R, the product of a frequency F and a measure of severity S: R=F S

(1)

The standard treats danger as the diametric opposite of safety, where the risk of a process is greater than the acceptable limiting risk (Fig. 1). A useful notion in plant and process safety is the hazard potential, a measure of the greatest harm that can occur in the worst possible event in a plant or plant subdivision. It is reasonable to use this concept in assessing safety measures in a plant: the greater the hazard potential, the more and better safety measures are needed to lower the probability of occurrence of the undesired event to the point that the level of risk is at or below the acceptable risk level. Safety measures may include intrinsic measures and conditions [8], which insure a priori that a hazard potential can become real only in the event of a relatively improbable combination of multiple independent failures. Where product quality considerations make it necessary to design the process and process control system so as to prevent an exothermic reaction getting out of control, safety or protective measures can be built up on the basis of this intrinsic safety in order to lower the risk to the acceptable level.

1.4. Magnitude of Hazard Potentials and Risks The risks created by chemical plants are often overestimated if their principles are not known. 1.4.1. Risk According to Previous Experience An anticipated value for the risk posed by chemical plants to employees or uninvolved third parties can be derived in a relatively simple way by statistical analysis of historical data. Consider the risk of death incurred by a chemical worker due to a typical chemical accident (poisoning, chemical burn, explosion). In the Federal Republic of Germany, this risk can be determined by analysis of the annual reports

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of the mutual accident insurance association of chemical industry, the so- called Berufsgenossenschaft der chemischen Industrie [2]. When the number of persons per year suffering death from poisoning, chemical burn, fire, or explosion is divided by the total number of persons employed in the chemical industry, the annual individual lethal risk averaged over the period 1983 – 1992 is ca. 7×10−6 a−1 , i.e., statistically, 7 persons in 106 die every year owing to an onthe-job chemical accident. This risk is a factor of 20 less than the risk of dying in a traffic accident in Germany (ca. 1.5×10−4 a−1 [9]) and is comparable to the risk of drowning (ca. 8×10−6 a−1 [9]).

Figure 1. Risk chart [7]

Figure 2 illustrates the situation. The vertical axis shows the probability of any employee dying in the next year from one of the causes cited (individual lethal risk). For comparison, values for a range of risks, [9–12], are also plotted. In terms of a statistical average, a chemical worker is much more likely to be killed in a traffic or domestic accident than in an on-the-job accident. The worker is also more likely to be a victim of violent crime than to die in an industrial chemical accident. His on-the-job risk is low compared with other risks threatening his life. Those living nearby and others outside the chemical plant are even safer from chemical effects, because the effects of the infrequent incidents in chemical plants fall off quite rapidly with distance. It can be assumed that this risk is, at most, of the same order as the risks due to natural catastrophes [10]. In Germany, the past 50 years have seen no identifiable serious per-

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sonal injuries or deaths outside a chemical plant site resulting from accidents inside. This shows that the German chemical industry, like those in many other industrialized countries, operates very safely.

Consider a vessel that contains a mass M L of a substance in the liquid state under its vapor pressure pv at a temperature Tα , above the boiling point T b . The mass of vapor in the vessel is M v . If the containment fails, the contents of the vessel expand to ambient pressure; the liquid continues to vaporize and cool until the remaining liquid reaches its boiling point T b at atmospheric pressure. Because this process is very fast, it approximates to an adiabatic expansion. The mass of substance vaporized plus the original mass of vapor present are released into the atmosphere; this is the quantity that determines the hazard potential. A material and energy balance, with some simplifications, gives, for the mass vaporized: ∆Mv =MF(α) [1 − exp (−cL ∆T /∆hv )]

Figure 2. Comparison of lethal risks to humans (per person per year) in the Federal Republic of Germany [2], [9–11]

However, it is true that a low risk may well conceal high hazard potentials, when the probability of occurrence is low. It is therefore advantageous to consider the size of hazard potential in chemical plants. Analysis of hazard potentials is broken down by type of accident: Spills and releases Explosions

1.4.2. Hazard Potential from Release of a Volatile Substance When a volatile substance posing a health hazard is released, the hazard arises because the liberated gas is taken up by air currents, and propagates in the atmosphere as it is steadily diluted. The size of the impact area, the region in which concentration levels of the substance rise above a critical value ϕH , is one measure of the hazard potential; ultimately, however, the hazard potential is governed by the number of persons who stay in this impact area without protection. To assess the hazard potential, it is necessary to determine the greatest mass of gas that can be released (in accordance with the model).

(2)

where ∆T = Tα − T b , cL is the specific heat of the liquid, and ∆hv is the heat of vaporization. If the temperature change is small, i.e., (cL /∆hv )∆T 1, a series expansion leads to the approximation: ∆Mv ≈MF(α) (cL ∆T /∆hv )

(3)

A typical value for cL /∆hv is 5×10−3 K−1 . For ∆T , a value of ca. 50 ◦ C, corresponding to an overpressure of 0.5 – 1.0 MPa, is not unusual for the storage of liquefied gases. Calculation shows that with these assumptions 20 – 30 % of the liquid contents of the vessel vaporizes spontaneously. A small additional amount vaporizes because of heat supplied from the surroundings, especially if liquid escapes from the vessel in finely divided form entrained with the vapor. In any case, this simple analysis shows that only a fraction of liquefied mass under its vapor pressure in a vessel can generally be spontaneously released into the atmosphere on failure of confinement, even when substances are kept under high pressure well above their boiling point. Only if the contents are superheated by some hundreds of degrees will the released fraction approach 100 %. To evaluate the acute hazard created by such a process, atmospheric propagation and dilution of the gas cloud have to be calculated, but these processes show an extremely strong dependence on the motion and turbulence of the atmosphere. Results can differ by orders of magnitude; what

Plant and Process Safety is more, initial dilution (e.g., dilution due to momentum of released vapor), height of release, and gas density all play roles. Exact prediction is impossible, but a rough estimate is possible [13]: under unfavorable conditions, ca. 1000 m3 (reduced to standard temperature and pressure) or several tonnes of gas have to be released to establish maximum concentrations of ca. 10 ppm out to a range of several kilometers from the point of release. Under propagation conditions of higher probability, the concentrations drop below such levels at a range of less than 1 km [13], [19]. As a rule of thumb, a hazard to humans due to a volatile toxic gas release at a range of more than a few hundred meters from the source presupposes catastrophic failure of a vessel, and the vessel must contain several tonnes of the gas in releasable form (compressed or liquefied). Such apparatus can be found in chemical plants, but the kind of catastrophic damage under discussion is extremely improbable because such vessels are designed with safety factors against stress, and, they are subject to regular inspection. From the safety standpoint, it is still important to assess hazard potentials relative to one another and to know how they can be influenced by process parameters. The following argument can be of help. The simple equations describing gas propagation show [13] that the maximum concentration in a gas cloud (plume) at distance x downwind of the source is directly proportional to the quantity released M rel and inversely proportional to the nth power of x: ϕ= const (Mrel /xn )

(4)

where 2 < n < 3 in general. In the case of a release near the surface, the downwind impact area, the region swept over by a concentration higher than ϕH , is commonly cigar-shaped. It becomes broader and shorter, the more turbulent the atmosphere. For a given state of atmospheric turbulence (i.e., fixed n and constant of proportionality), the (maximum) width of the impact region is always proportional to its length. Therefore, the impact region, as a measure of the hazard potential (HP), is proportional to the square of the distance x from the source to the point where the concentration falls below ϕH : HP ∼ x2 = const (Mrel /ϕH )2/n

(5)

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For the case under discussion (spontaneous vaporization and escape of a substance posing a health hazard and having a critical concentration level ϕH ) the hazard potential is: ( HP =K

cL ∆T MF(α) ∆hv ϕH

)2/n (6)

Nevertheless, this is not a well-defined quantity, because K and n depend on the ambient conditions (state of motion of the atmosphere) and can only be derived from the known gas dispersion formula if a special case is assumed [13], [19]. The hazard potential is also a function of the quantity of substance, its properties, and its thermodynamic state inside the vessel. The hazard potential increases with the mass of substance present and its superheating (above boiling point), i.e., with pressure. From this, it is possible to devise ways to reduce the hazard potential, even to the point of inherent safety. 1.4.3. Hazard Potential from an Explosion The hazard potential of an explosion is due to the pressure wave, and is determined by the size of the (ideally circular) area around the explosion center in which there is significant damage, e.g., collapsing walls or broken windows. Window glass typically breaks at a peak overpressure in the 1 kPa range and masonry walls begin to fail at overpressures ≥ 30 kPa [14], [15]. Brasie and Simpson speak of serious damage at peak overpressures > 15 kPa [15]. The task of evaluating the hazard potential can thus be reduced to determining at what distance from the explosion center the peak pressure just falls below a critical value. Suppose the mass M of substance being handled is such that its sudden decomposition liberates an enthalpy of reaction ∆hr . The total energy evolved is: E =M ∆hr

(7)

Hopkinson and Cranz independently found that for two distinct “charges” (releasing energies E 1 and E 2 ) to produce equal effects, the radii (distances R1 and R2 from the source point)

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must be related as the cube roots of the energies [16]: 1/3 R1 E1 = 1/3 R2 E 2

(8)

This result led Sachs to postulate that the dimensionless peak overpressure and the dimensionless momentum of a pressure wave (the quantities that govern the destructive action of the wave) can each be represented as a unique function of a dimensionless distance from the origin [16], [17]. The dimensionless peak overpressure p∗ is obtained by dividing by the ambient pressure; the dimensionless range R∗, from an expression involving the energy and ambient pressure: p∗ =

pp −p0 p0

R∗ =

∆p, kPa

R∗

30 15 1

ca. 1.5 ca. 2 ca. 20

Thus, Equation (11) implies that an explosion corresponding to the detonation of 1 t of TNT would break windowpanes (∆p=1 kPa) out to a range of several hundred meters (Rp = (35 m)×R∗ = 700 m) and cause serious damage (∆p = 15 kPa) in a circle radius 50 – 100 m (Rp = 70 m).

(9)

Rp

(10)

(E/p0 )1/3

The dimensional formula corresponding to Equation (10) is: 

m Nm·m2 N

m 1/3  = m

A diagram presented by Baker and co-workers (Fig. 3) shows that these parameters make possible a relatively accurate dimensionless representation of the characteristic pressure-wave parameters, at least for many explosives [16]. For a provisional estimate, consider 1 t of a typical explosive, TNT (∆hr = 4230 kJ/kg). The energy liberated in an explosion is:

Figure 3. Dimensionless peak overpressure in an explosion pressure wave as a function of dimensionless distance from the source [16] The expression for the dimensionless distance involves the explosion energy and the initial pressure

E = (4230kJ/kg) ×1000kg = 4.23 × 106 kJ = 4.23 × 109 Nm

If p0 = 100 kPa, Equation (10) yields:  Rp =

E p0

1/3

R∗ =



4.23 × 109 Nm

1/3 ∗ Rp = 42300m3 R Rp = (35m) R∗

105 N/m2

1/3

R∗

(11)

Figure 3 yields the following dimensionless ranges:

Explosives and similar materials are not commonly used in chemical plants. Reactions that can lead to explosion-like phenomena (Sections 1.1.2) are governed by other laws and differ strongly in kinetics, enthalpies of reaction, and efficiencies (in terms of energy conversion). For this reason, no direct comparison with the explosion just described is possible. Even so, a number of workers have sought correlations [4], [18]. Marshall [1.3, p. 256], working with event analyses for explosions of unconfined gas clouds, arrives at the conclusion that for less than ca. 1 t of fuel gas forming an

Plant and Process Safety unconfined explosive cloud, peak overpressures on ignition are negligibly low. He estimates that for an unconfined cloud with > 10 t fuel gas, about 5 % of the liberated energy (of combustion) can contribute to a TNT-equivalent pressure wave, but with significantly lower pressures (< 100 kPa) in the near region. Hence, 2 t fuel gas reacting to completion would produce the same effect as 1 t TNT (with allowance for the fact that the enthalpy of combustion of a fuel gas is a factor of 10 greater than the energy of TNT decomposition). In a real unconfined gas cloud formed through damage to a vessel or in a plant, only a fraction of the fuel gas is ever in a region where the concentration is high enough for sudden combustion – in the worst case. It must therefore be supposed that the effect produced by the explosion of 1 t TNT can be matched only by the spontaneous release and ignition of ca. 10 t fuel gas. This could not happen without serious damage to a large vessel. The hazard potential of an explosion is directly proportional to the area within which a certain level of damage occurs. According to Equations (8) or (10) and (7): HP ∼ R2 ∼E 2/3 = (M ∆hr )2/3

reduced in the most efficient way, and safety measures and equipment can be designed to handle the remaining potentials (Chap. 3). Experience has led to the low risk values cited for chemical plants and shown that major chemical plant accidents have been infrequent, especially in view of the scope of industrial activity in this field. The advanced chemical industry has obviously had much success in controlling its hazard potentials.

2. Determination and Evaluation of Hazardous Properties of Substances (Safety Ratings) 2.1. Harmful Effects of Substances According to the EC classification criteria [20], substances are described in terms of 15 hazardrelated characteristics, which can be broken down into four groups: 1)

(13)

i.e., the hazard potential increase is less than directly proportional to the energy release and the mass involved.

1.5. Conclusions The analyses given here show, as does experience, that chemical plants may occasionally represent high hazard potentials, especially when volatile substances of a flammable or healthendangering nature are processed in large quantities, and when exothermic (decomposition and combustion) reactions are possible. It is also clear that, while hazards can be identified and analyzed, the magnitude of hazard potentials cannot be quantified precisely. For the design of safe processes and plants, a qualitative order-of-magnitude estimate, a relative comparison with other hazard potentials, and an identification of key independent variables is sufficient, because such considerations give a point of reference such that hazard potentials can be

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2)

3)

4)

Acute Toxicity: Very toxic substances Toxic substances Harmful substances Corrosive substances Irritant substances Specific Toxic Properties: Sensitizing substances Carcinogens Substances with effects on reproduction Substances with heritable effects Physicochemical Properties: Extremely flammable substances Highly flammable substances Flammable substances Oxidizing substances Explosive substances Environmental Impacts: Substances harmful to the environment

While acute and specific toxic properties can cause direct harm to health, physicochemical properties lead to indirect harm. Substances can take any of three routes into the body: Oral: by mouth directly into the stomach Dermal: through the skin Inhalation: via the respiratory organs

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The actions of a substance can differ greatly depending on the route of exposure. Because of the acidic conditions in some regions of the gastrointestinal tract (pH 1 – 5), hydrolyzable substances can be broken down after oral intake. Chemical reactions can lead to both more toxic substances (metabolic activation) and less toxic ones (detoxification). While acidic compounds are preferentially absorbed in the stomach, basic and lipophilic substances are preferentially taken up in the intestinal tract. Chemicals not resorbed in either the stomach or the intestines can be excreted directly without realizing their toxic potential (e.g., mercury, barium sulfate). The skin protects the body against external effects. It performs very effectively against substances readily soluble in water. Fat-soluble (lipophilic) substances, on the other hand, commonly diffuse well through the skin into the body. Dermal resorption is extremely effective for polar molecules with lipophilic and hydrophilic groups (e.g., many organic solvents such as dimethyl formamide, dimethyl sulfoxide, ethyl acetate, toluene, chloroform). Corrosive chemicals resorbed through the skin (e.g., phenol) are taken up very quickly and effectively by this route. Highly corrosive substances are those that destroy the skin (necrosis) if allowed to act for a maximum of 3 min; corrosive substances have the same effect within 4 h. Examples of highly corrosive substances are inorganic acids and alkalies. In general, the corrosive action of alkaline substances is stronger than that of acids. Irritant substances are characterized by inflammation of the skin after a maximum of 4 h exposure (e.g., dilute acids and alkalies, weak organic acids, acrylates, alcohols, amines). While the irritant action of substances is reversible, corrosive action implies irreversible injury. The mucosa of the eyes and respiratory passages are particularly sensitive to irritation. Highly water-soluble substances, when taken up by inhalation, are generally absorbed by the mucosa in the upper part of the trachea, and do not reach the deeper layers of the lung. Because many receptors are present, irritation reactions such as coughing and sneezing may be initiated by agents such as ammonia, hydrogen chloride, sulfur dioxide, and vapors of acids and alkalies. Compounds that are less water-soluble can readily penetrate into the bronchi. Partial diffu-

sion through the thin bronchial tissue is possible and can lead to lung injury (e.g., chlorine, bromine, iodine, ozone, phosphorus chlorides). If the tissue of the lung suffers local corrosion, liquid can seep into the alveoli (pulmonary edema) and the oxygen transfer required for life can be hindered for hours to days after exposure (latent action; e.g., phosgene, nitrogen dioxide, ozone, some isocyanates). Lipophilic compounds, in particular, can cross over via the alveoli when respired air is exchanged with the blood. Substances with systemic action, after inhalation or oral or dermal exposure, are distributed throughout the body by the circulatory system. In this way they can reach all organs. If injury takes place only at the point of exposure, the substance is said to have a local action. With regard to acute toxic effects, substances are classed as very toxic, toxic, or harmful. This description is based on the median lethal dose LD50 : the dose at which half of the experimental animals die. Toxicity is reported as oral, dermal, or by inhalation. Oral and dermal LD50 values are given in milligrams of the substance per kilogram of the experimental animal for onetime administration; LD50 values by inhalation are given in milligrams of the substance per liter of respired air (exposure time 4 h). Because oral toxicity values are more common, they alone are discussed here. The lethal dose represents only the end point of toxic effects. The toxic action, however, begins to appear even when much smaller quantities are present in the body. The dose at which no biologically relevant effect can be detected is called the no adverse effect level (NOAEL). Table 1 shows how substances are classified in accordance with EU guidelines [20]. Substances are arranged by acute toxicity properties; the table also gives danger symbols and indications along with the “R phrases” (which specify the principal risk more closely). In some nonEuropean countries, the boundaries between classes differ slightly from the values shown. Table 1. LD50 values (oral) Attribute

LD50 , mg/kg

Symbol

R phrases

Very toxic Toxic Harmful

100 MPa m s−1 . Substances are classified in accordance with their explosion violence, as expressed by K max , tabulated as follows: Kmax , MPa m s−1 30

Dust explosion class St 1 St 2 St 3

A condensed-phase explosion is defined as an exothermic chemical reaction, during which gases and vapors are formed at such a high rate as to have destructive effects on their surroundings. Fragments, projectiles, ground and air shock waves are produced by the high temperature, pressures, and rates of pressure rise. The extent of damage depends on the type and the mass of the explosive substance as well as on the nature of the surroundings. An explosive reaction can be initiated only in substances which have a high positive heat of formation or a high negative heat of decomposition. Such substances usually have a defined chemical structure. Being organic in nature and possessing, in most cases, reactive groups

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with available oxygen, they can undergo intramolecular oxidation of the combustible part. Examples of oxidizing reactive groups are nitro, nitroso, and peroxy groups, but organic compounds containing azide, acetylenic, or diazonium groups may also be powerfully explosive substances. Explosive compositions may also be produced by mixing inorganic oxidizing substances with combustible materials; well-known examples are: mixtures of potassium nitrate, sulfur, and carbon (black powder); ammonium nitrate and fuel oil (ANFO); nitration mixtures containing nitric acid; and oxidation mixtures with hydrogen peroxide. Bretherick [75] and King [76] give guidance for identifying substances or mixtures as potentially explosive because of the presence of reactive groups. When a substance or mixture contains reactive groups, it depends on the molecular mass and the type and number of reactive groups, or on the oxygen balance, whether and with what intensity explosive properties are exhibited. An important difference between gas-phase and condensed-phase explosions is that in the latter the participation of oxygen from the air is not necessary. The power of an explosion is closely related to the dynamics of gas production and energy release, as well as to the amount of gas produced and the total heat of decomposition. The energy release when a substance explodes is difficult to determine because it depends on the resulting explosion products. There may be considerable uncertainty in reported heats of decomposition of explosive substances. A slow reaction at low temperature, due to a low degree of adiabatic efficiency under the test conditions, may result in a less energetic decomposition measurement. It is known [77] that the decomposition energy measured can be an underestimate of the energy actually available. On the other hand, computer programs such as CHETAH (chemical thermodynamic and hazard evaluation [78]) can be used to predict the maximum heat of decomposition for any substance or mixture of known chemical formula and structure. As a rough guide, substances with decomposition energy > 500 J/g may have explosive properties, and those > 800 J/g may be detonable. Although this is often a good qualitative prediction of explosivity, reliable assessment of an explosive substance necessarily requires experimentally determined data.

2.2.4.3. Grouping of Explosive Substances Explosive substances are grouped according to their intended use and/or the type of hazard connected with handling them. When substances, because of their energy content, are used to produce explosive, initiating, propellant, or pyrotechnic effects, they are called explosives. They may be used, e.g., for blasting in rocks, coal, or salt, as initiating explosives in blasting caps (detonators), as propellant powders in ammunition, or as components of pyrotechnic articles. All explosives are regulated internationally with respect to handling, storage, and transport. Except for international transport, this is not so for explosive substances used for chemical, scientific, or technical purposes. According to the type of associated hazard, explosive substances (especially explosives and articles with explosive properties) are divided into groups representing: 1) The hazard of a mass explosion (mass detonation) 2) The hazard of explosions (detonations) together with the production of fragments, without the hazard of mass explosion 3) The hazard of a mass fire 4) Ordinary fire hazard This grouping is the basis for all modern regulations and safety measures during transport and storage (e.g., safe distances). 2.2.4.4. Explosion Mechanisms Commonly, “explosion” is used collectively for the three distinguishable mechanisms by which a substance can explode. The thermal explosion proceeds, usually in the liquid phase and more or less homogeneously, as a temperaturecontrolled (Arrhenius law) self-accelerating reaction (see Section 2.3). When, during the homogeneous decomposition, a temperature gradient develops, a deflagration may start at the hottest spot. It passes through the substance as a reaction front in which the heat of reaction and the reaction products are liberated. The deflagration is propagated by conductive, radiative, or convective heat transfer into the unre-

Plant and Process Safety acted materials. If a substance is liable to deflagrate, this reaction can be initiated locally, e.g., by flame, heat, impact, or friction. The deflagration velocity increases with the energy content, temperature, and porosity of the substance, and exponentially with pressure. Typical values are 0.003 – 100 m/s. The pressure dependence of the reaction velocity explains why deflagration of a small quantity of material may proceed slowly, whereas under confinement by pressure buildup, or in even larger masses by self- confinement, it occurs with explosive violence. When the speed of the gaseous deflagration products reaches sonic velocity, a shock wave develops which propagates supersonically into the unreacted substance; a deflagration to detonation transition (DDT) has taken place. Compression combined with strong heating initiates chemical reaction of the substance. Typical detonation velocities are 1000 –8000 m/s; detonation pressures reach values up to several thousand mega pascales. Detonation velocity increases with the energy content of the substance and its density. The detonation reaction may be initiated locally by heat, impact, friction, DDT, or a shock wave from other detonating substances, leading to mass explosion, and producing disastrous damage. Primary (initiating) explosives give rise to direct immediate detonation by flame, impact, or friction in quantities of a few milligrams. Secondary explosives may be detonated by primary explosives or by DDT in larger quantities. The nature of detonation is further described in the thermohydrodynamic theory of detonation [79–81]. 2.2.4.5. Testing of Explosive Substances To determine the risks associated with handling an explosive substance, experimental investigation into its individual explosive properties is imperative. Explosive properties refer to the mechanisms by which an explosive reaction can proceed, to the types of stresses and the ease with which explosions can be initiated (sensitivity), and to the power of the explosion once it takes place. Understanding the initiation, propagation, and the possibility of terminating explosive reactions, and recognition of any destructive potential form the basis for safe handling of an explosive substance [82]. The sensitivity of an explo-

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sive substance is of special importance since, if it is too high, it may preclude some or all modes of technical handling. For several decades, explosive properties have been described by test data determined by standardized test methods [83], [84]. The latest and most comprehensive collection of tests, covering all relevant properties together with evaluation criteria for classifying explosive substances for the purpose of transport, is given in the UN Test Manual [85]. In Part I, the Manual contains test methods and criteria, together with a classification flowchart, for deciding whether a substance or mixture is an explosive substance or article of Class 1 of the UN Recommendations on the Transport of Dangerous Goods [86], and which division of Class 1 best reflects its risks during transport. Part II of the Manual presents tests methods, criteria, and a flowchart for evaluating organic peroxides and self-reactive substances (Divisions 5.2 and 4.1 of the UN Recommendations) for explosive properties. The Manual includes ca. 70 test methods on laboratory and field scale for the substances (packed as for transport). The test methods are related to the properties to be identified: – Detonability by shock wave, including sensitivity to detonation shock of variable strength, detonation velocity, and sensitization by cavitation (gas bubbles) – Deflagration after ignition in an open vessel or under confinement, determining the linear deflagration velocity or the rate of pressure rise – DDT after ignition under confinement – Thermal sensitivity to heating under variable defined degrees of confinement, the sensitivity being characterized by the limiting diameter of a pressure-relief vent, or by the maximum pressure and rate of pressure rise – Sensitivity to mechanical stresses (impact and friction), determination of the sensitivity limits – Explosive power after initiation with a blasting cap or thermal decomposition, measuring the work performed or the specific energy – Thermal stability of explosives on storage at 75 ◦ C, or of organic peroxides and selfreactive substances on storage under adiabatic, isothermal, or heat-accumulation conditions, determining the self-accelerating

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decomposition temperature (SADT) related to the packed substances The European Community has provisionally agreed on three methods for evaluating a substance as explosive or not. These screening methods include testing the sensitivity of substances to impact, friction, and heating under partial confinement [87]. The EC regulates the classification, packaging, and labeling of such substances for all modes of handling except transport. 2.2.4.6. Conclusions Use of standardized test methods resulting in criteria for assessing the safety of chemical procedures and installations involving explosive substances should take special note of certain factors limiting their applicability. It requires considerable experience to perform the tests, to modify them if necessary, and to evaluate observations correctly. Each test result should be assessed for plausibility in connection with all other results. Sensitivity does not depend on the energy content of the substance, as does the explosive power. To increase the reproducibility of test results, standardized test procedures often reflect idealized conditions. The heating rate, degree of filling of the apparatus, and the strength of confinement may greatly influence the results of thermal tests. Problems of scaling must also be addressed. Any test result corresponds to the conditions under which it was obtained, and its validity for substances in a particular plant should be thoroughly checked. Special care should be taken before neglecting a given detonability of a substance. With respect to production in a chemical plant (excluding the explosives industry), it is imperative that initiation of a detonable substance under the conditions of handling should be impossible.

2.3. Exothermic and Pressure-Inducing Chemical Reactions 2.3.1. Introduction The overwhelming majority of chemical reactions are accompanied by heat release, i.e., the

overall change in enthalpy between the starting materials and products is negative (exothermic). The heat produced heats up the material itself, the container (e.g., the reaction vessel), and/or the surroundings. According to the law of conservation of energy, these heat flows equilibrate in a heat balance (Fig. 24). The degree of heat dissipation depends on the heat capacity of the system itself and the heat removal capacity provided by the reaction vessel design and/or the latent heat of phase transitions. If the heat production rate of a chemical process exceeds the heat removal capacity of the system, and if there are no self-stabilizing boundary conditions, e.g., reaching of a boiling point, self-acceleration will occur, resulting in a “runaway” reaction. As runaway reactions are rather hazardous, it is necessary to understand the underlying mechanism, as well as the systematic experimental test procedures to assess the hazard potential for a runaway, and to know the design criteria for safe processes in order to prevent runaway scenarios from occurring.

Figure 24. Schematic representation of heat balance [88]

2.3.1.1. Exothermic Reactions: Runaway Potential To assess the runaway potential of a chemical system it is necessary to understand the basic physicochemistry and its interaction with the transport phenomena of heat, mass, and momentum, which determine the behavior of the system, including the reactor. The hazardous character of a chemical reaction is determined by the

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Table 8. Typical heats of reaction for common chemical processes [89] Reaction type

∆H, kJ/mol

Reaction type

∆H, kJ/mol

Neutralization (HCl) Neutralization (H2 SO4 ) Diazotization Sulfonation Nitration Epoxidation

− 55 −105 − 65 −105 −130 − 96

Hydrogenation (nitroaromatics) Amination Combustion (hydrocarbons) Diazo-decomposition Nitro-decomposition

−560 −120 −900 −140 −400

overall heat of reaction (thermodynamics), and the rate of heat production (strongly influenced by the kinetics). Typical values for heats of reaction are summarized in Table 8. The total heat output of a real system can only be influenced by the concentration of the reacting materials. In the sense of a primary method to prevent runaway reactions it can be stated that the more dilute a system the smaller is its runaway potential. The disadvantage of this rule, besides its effect on the economics of the process, is the environmental consequence of a higher demand for auxiliary materials and energy consumption for their recovery. Consequently, an inherent runaway potential has to be accepted, but must be adequately controlled, either kinetically (e.g., by choice of process temperature), or technically (by reactor design).

the case in well-mixed gases or Newtonian liquid, heat transfer is governed by the boundary, e.g., across a reactor jacket. This situation was originally discussed by Semenov [90]. The other extreme, nonuniform temperature distribution and heat loss governed by conduction through the bulk, is a good model for large, unstirred masses, solids, and powders and was first discussed by Frank-Kamenetskii [91]. Both cases are shown in Figure 25. For more detailed discussion of self-heat models see [92]. For phenomenological discussion of runaway reaction systems the Semenov model [90] is the more adequate. It assumes a chemical reaction A → B with an indefinite amount of initial material, so that its consumption may be neglected. It is further assumed that the temperature dependence follows the Arrhenius relationship. Based on this pseudo-zero-order kinetic approach, the heat production rate Q˙ R is governed by an exponential dependence on temperature:  Q˙ R =V (−∆R H) k∞ cA,∞ exp

−

E RT

 (18)

The rate of heat removal Q˙ C , assuming the coolant has Newtonian properties, depends linearly on the driving temperature difference between the uniform reaction mass temperature and ambient ( jacket) temperature: Figure 25. Semenov (uniform) and Frank-Kamenetskii (nonuniform) temperature profiles

As already mentioned, control of chemical heat release depends on the ability of the system to dissipate this energy. Mathematically this is formulated as a heat balance, including an accumulation term as well as the heat generation and dissipation terms. Appropriate description of the dissipation terms depends strongly on the heat transport mechanism (conductive or convective, free or forced). For a uniform temperature distribution within the reaction mass, which is usually

Q˙ C =U A (TR −TA )

(19)

The controllability of the heat production rate can best be explained by plotting the two heat flows (Eq. 18 and 19) as a function of temperature (Fig. 26). Three different cases can be discussed: with two, one, or no intersections. These three cases can be obtained either by varying the ambient temperature (Fig. 26) or by varying the slope of the heat removal lines (equal to the product of heat transfer area and overall heat transfer coefficient). The intersections represent pseudo-steady-state conditions. In case

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1 (low T A ), small deviations from the steady state, represented by the lower of the two intersections, automatically result in a return to the origin, as can easily be deduced from a comparison of the relative magnitudes of the two heat flow terms. This operating point is rated “stable”. With respect to the upper steady state, once a temperature deviation occurs, the original operating conditions are never reached again. In the case of a temperature decrease, the process quickly approaches the lower steady state; for a temperature increase, the heat production rate always exceeds the heat removal capacity of the system. This leads to an unhindered selfacceleration of the reaction rate and thereby of the heat production rate (thermal explosion or runaway reaction). The same is true for all operating conditions of case 3 (high T A ). Case 2 (medium T A ) represents the limiting case of the first occurrence of an unstable operating point, characterized by the equality of rates and their temperature derivatives. From these characteristics, it can be shown mathematically that, as long as the following relation holds: 2 RTR >TR −TA E

(20)

runaway conditions need not be expected. This may be regarded as a first rule of thumb for the safety assessment of chemical processes. More sophisticated design criteria, presented in 2.3.1.4 and Section 2.3.3.1, extend the stability discussion of chemical processes from this pseudo-steady-state approach to a discussion of dynamic stability.

Figure 26. Semenov plot 1) Low T A ; 2) Medium T A ; 3) High T A

2.3.1.2. Causes and Consequences of Overpressure-Inducing Exothermic Reactions If the defined operating conditions do not include extensive overpressure, as is common for a number of petrochemical processes and hydrogenations, the sudden and unexpected occurrence of overpressure must be regarded as hazardous. The discussion of causes and consequences of a pressure increase must begin with a distinction between effects related to gasgenerating chemistry and those related to process design. For chemistry-related effects, the prime cause of overpressure is the formation of noncondensable gases. These may be reaction products of the desired process or of secondary reactions initiated at the elevated temperatures due to a runaway. For the first category, the basis of a suitable safety assessment is a fundamental understanding of the chemical process. Many common chemical reactions produce large amounts of gases, e.g., reductions with LiAlH4 or substitutions of NH2 groups by halogens via diazonium intermediates. Less obvious processes are those where the gas is formed in parallel reactions, e.g., the formation of gaseous ethylene oxide from 2- chloro-1-ethanol in the presence of a strong base. Provided that the reaction has been properly identified, these kind of gasgenerating processes are controllable by conventional means, as their generation rate is kinetically controlled. The most important class of gas-generating processes are decomposition reactions. These are initiated, e.g., when a runaway reaction raises the operating temperature so that the decomposition reaction overwhelms the desired reaction, or when cooling medium comes into contact with the reaction mixture, owing to a jacket or pipe rupture. Many of such undesired reactions stoichiometrically produce more than 1 mol gas per 1 mol initial material, which accelerates the pressure increase dramatically. In these cases, the process cannot be controlled and only mitigating measures can help reduce the consequences. It is therefore important to assess the process in all conceivable fault conditions, in order to initiate appropriate process or plant modifications to make the process inherently safe,

Plant and Process Safety or to define suitable preventive measures as the second best choice.

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2) It is not performed once and for all, but should be used repeatedly during the lifetime of a plant or process PHASE has to be applied to differing degrees, depending on the status of development of a process. Preliminary assessments should be performed on laboratory scale. The first fully comprehensive application of PHASE is recommended before the first run, on pilot plant scale. Later it should be repeated whenever a process scale-up or process/plant modifications are intended. PHASE is a stepwise iterative procedure (Fig. 27). It starts with an assessment of all chemicals, together with operating and plant conditions, for the desired process. If hazardous situations are identified at this stage, process or plant modifications have to be initiated. Experimental methods and evaluation criteria for this purpose are the subject of Sections 2.3.2 and 2.3.3. If the desired process is rated as safe, possible fault conditions and process deviations have to be identified and evaluated. If the implementation of technical or organizational preventive measures cannot assure safe operation, further process or plant modifications become mandatory, and assessment of the normal operating conditions has to be repeated, accounting for the changed parameters. This is the iterative part of PHASE. For the second step in PHASE, tools such as HAZOP, fault trees, or FMEA are recommended (see Section 3.3).

Figure 27. Process hazard and safety evaluation (PHASE)

2.3.1.3. Processes Hazard Assessment and Safety Evaluation for Exothermic Reactions Process hazard assessment and safety evaluation (PHASE) [93] is a mandatory procedure to assure safe manufacturing of chemicals. The PHASE procedure has two characteristics: 1) It has to follow predefined systematic pathways to optimize its efficiency to identify as many hazardous situations as possible, and to assure the implementation of recommended safety measures

2.3.1.4. Hazard Characteristics of Exothermic Processes due to Process Design Chemical reaction engineering principles form the basis of an understanding of reaction hazards originating from process design. Basically three different designs can be distinguished: continuous, semicontinuous, and batch. A detailed discussion of reactor design principles is given in the standard textbooks [94], [95]. This section reviews those characteristics which have a direct impact on the hazard assessment of exothermic processes. Design and safety criteria depend strongly on whether the system is homogeneous or heterogeneous. Heterogeneous processes depend on a much larger number of parameters (e.g., particle size distribution, surface area, and porosity for

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solid – liquid systems, or mass transfer parameters for liquid – liquid systems). For a detailed discussion of heterogeneous processes see [94], [95]. The following discussion is for homogeneous systems only. Continuous processes are performed either in continuously stirred tank reactors (CSTR) or plug-flow tube reactors (PFTR). Phenomenologically, the longitudinal spatial coordinate of the PFTR is equivalent to the time coordinate of a batch process. Therefore, with respect to continuous processes, the following discussion focuses on CSTR. Continuous processes have two major advantages: the reaction rate can be controlled by the reactant feed stream; the reaction products and the solvent immediately dilute the initial materials entering the reactor. Both characteristics influence the effective concentration of reacting materials directly and reduce the thermal potential. The problem areas with CSTR are the correct design of the start-up procedure and the avoidance of dynamic instabilities. Additionally (but not of primary importance to hazard assessment), continuous processes require a larger reaction volume than discontinuous processes to achieve the same space – time yield. It is recommended to start up a CSTR as a semicontinuous process, up to the point where the expected stationary extent of reaction is achieved, and then to start the second feed stream. To avoid ignition/extinction and oscillatory phenomena, design criteria are available in [94], [95], which can easily be applied, provided the thermal kinetics of the overall process have been determined appropriately, e.g., by reaction calorimetry. The true batch process represents the other extreme of reactor design. Batch reactors are often preferred in the fine chemicals and pharmaceutical industries because of the smaller amounts of substance handled. From a hazard evaluation viewpoint, batch processes are the most difficult, as all reacting materials are charged initially, so that the total hazard potential is present in the reactor right from the beginning. Depending on plant or process design, batch reactors (BR) can be operated either isothermally, under computer control, or isoperibolically, i.e., with a coolant kept at constant temperature (Fig. 28). In the first case, the assessment must focus on the maximum heat rate developed, ideally at t = 0. If the coolant temperature can be rapidly lowered by the con-

trol system, quasi-isothermal conditions can be achieved. On the other hand, sensitivity analyses by Semenov and others suggest, that the cooling temperature difference should not be too great [96]. There are four engineering numbers, which can be used to give guidance on safe operability: the adiabatic temperature increase ∆T ad ; the thermal reaction number B; the dimensionless reaction rate Da (Damkoehler number); and the modified Stanton number St (dimensionless cooling capacity of the system): (−∆R H) cA0 (−νA ) rcP E ∆Tad B = RT 2

∆Tad =

(21) (22)

where ν A is the stoichiometric coefficient of the limiting component. (−νA ) r0 t cA0 U At St = V rcP

Da=

(23) (24)

Figure 28. Batch reactor operating modes (A) Isothermal; B) Isoperibolic Full lines, internal temperature; dotted lines, jacket temperature.

If ∆T ad < 50 K and B < 5, isothermal batch process can be operated safely. Such processes

Plant and Process Safety may be regarded as inherently safe, acknowledging that possible plant hazards are excluded from this assessment. Under isoperibolic operating conditions, the hazard assessment focuses on the maximum temperature difference, which, under these conditions, occurs some time into the reaction. Additionally it has to be assessed whether or not the maximum reaction temperature exceeds a limiting value, which represents the temperature level at which undesired reactions become dominant. It is also necessary to assess the sensitivity of this operating point to the normal variability of other operating parameters, e.g., coolant temperature or overall heat transfer coefficient. If Da/St 1. Provided this is assured by the process design, accumulation of the added component does not pose a hazard [97]. 2.3.2. Methods of Investigation and their Systematic Application It is the aim of safety studies either to determine whether or not a chemical process may be carried out under predefined conditions (e.g., optimal conditions with regard to yield and time), or to establish safe operating conditions with respect to temperature, pressure, concentrations, and time. For this purpose, it is necessary to make experimental investigations by appropriate test methods. Theoretical considerations may be helpful (e.g., estimation of the hazard potential of a process from the chemical equation and molecular structures), but they cannot replace experiments. Several screening methods and more detailed test methods are used in the

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chemical industry to obtain information necessary for the assessment of exothermic or pressurizing chemical processes [98].

Figure 29. Differential thermal analysis (DTA) a) Oven; b) Sample; c) Reference; d) Temperature difference signal; e) Oven temperature signal

2.3.2.1. Screening Methods Thermal Analysis. Differential thermal analysis (DTA) is a common method for the investigation of chemical reaction hazards. A sample of the material to be tested, which may be a pure compound, a reaction mixture, a residue, or a waste sample, is exposed to thermal stress. For this purpose, the sample and a thermally inert reference material (e.g., Al2 O3 ) is placed in an oven. Thermal stress can be applied either dynamically by ramping up the oven temperature at a fixed rate, or isothermally in thermal aging at elevated temperatures. The difference between sample and reference temperature is recorded as a function of oven temperature or time. The recorded signal is converted to a heat flow by a calibration function. For screening purposes, linear heating rates of 1 – 10 K/min, from room temperature up to 300 – 500 ◦ C, are widely used. To suppress the endothermic effect of vaporization and catalytic effects of the vessel material, sealed glass ampoules are often used. A more sophisticated method is differential scanning calorimetry (DSC), where the heat flux between the sample and its surroundings is recorded directly. The resulting thermogram shows exothermic and/or endothermic transitions, originating from physical or chemical effects. Integration of the signal peaks provide

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the heat content of the observed transitions. The measurable beginning of the transition is called the onset temperature (Figs. 29, 30).

Figure 30. Typical thermogram of a solid a) Melting (endothermic); b) Decomposition (exothermic) Heating rate 3 K/min

The advantages of DTA/DSC are the use of small sample masses and the short time (1 – 3 h) required for a run. Onset temperature, peak area, and peak shape give hints on the hazard potential of a sample. It is not permitted to perform a direct scale-up of the measured values to the true chemical process or unit operation conditions. The recorded onset temperature depends critically on the measuring conditions, e.g., heating rate and sensitivity of the apparatus. Furthermore, there is no analytical control on the extent of reaction. So the area of a peak due to a chemical reaction cannot be interpreted as equivalent to the molar heat of a defined reaction. As the sample mass cannot be stirred during the experiment, the measuring conditions are not representative for heterogeneous systems. In particular, thermograms of such samples must be interpreted carefully. Other Screening Methods. Isoperibolic calorimetry on the scale of several grams is another screening method for testing the thermostability of chemical systems, and may be used as an alternative or in addition to DTA/DSC. The calorimeter consists of a jacketed sample vessel (the jacket temperature may be programmed). The temperature difference between the sample mass and the jacket is measured and recorded as a function of jacket temperature or time. The recorded curves are evaluated similarly to DTA/DSC curves. The use of a larger sample mass (compared with DTA/DSC) and the option of stirring allows the investigation of heterogeneous systems with

more success. The low mechanical stability of the sample vessel may be a disadvantage. For a comprehensive assessment of the hazard potential of chemical systems, not only the thermal behavior, but also pressure effects in a closed vessel, or the rate of material loss in an open system, are of interest. Both effects can be measured with the help of these screening tools as a function of the sample temperature (programmed) for sample masses of ca. 50 – 1000 mg. For the interpretation of results, restrictions similar to those for thermograms apply. The results must not be scaled up directly to the true process and plant conditions. 2.3.2.2. Thermal Aging and Heat Accumulation Storage Tests Data from thermal aging tests can be used directly for plant-scale assessment, provided the test conditions are representative of actual process conditions (e.g., quality of solid or liquid sample identical with the actual plant material, temperature and testing time identical with the process temperature and residence time). By varying the sample amount it is possible to extrapolate the experimental results for larger quantities. A sample vessel of defined size (e.g., a glass cylinder 100 – 500 mL) is stored in a drying oven at fixed temperature. Temperature sensors inside the sample detect the onset of an exothermic reaction. It is possible to eliminate the dependence of the exothermic onset on the sample mass by storing the test sample under adiabatic conditions, using so- called heat accumulation storage tests (open or under pressure). For adiabatic tests, the sample is stored in a Dewar flask with an insulating lid. A higher degree of adiabaticity is achieved by reducing heat losses to the surroundings. For this purpose, the Dewar flask is placed in a controllable oven, thereby assuring an ambient temperature equal to the sample temperature (Fig. 31). The heat accumulation storage test provides the sample temperature – time profile under adiabatic conditions, starting at a fixed storage temperature. In evaluating the curves, it is necessary to account for the “thermal inertia” of the experimental system. Thermal inertia expresses the fact that the heat produced by the chemical reaction system is partially absorbed

Plant and Process Safety by the sample container and internal components (temperature sensor, stirrer, etc.). From a heat balance viewpoint, the extent of this effect depends on the ratio of the heat capacity of the complete measuring system to that of the reaction mixture, the so- called Φ-factor: Φ=

mR cPR +mA cPA mR cPR

(25)

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tests when a decomposition reaction occurs. The problem can be reduced by using smaller sample masses (< 10 g) in metallic bombs with high pressure stability. As such apparatus has a high heat-loss factor, it is necessary to use an efficient oven with accurate temperature control to minimize this heat loss. With modern equipment, it is not necessary to predefine a fixed starting temperature for the adiabatic test. The measuring system detects the temperature at which a threshold value of the heat production rate (e.g., 1 – 2 W/kg) is exceeded, by a “search-and-wait” procedure. Evaluation of the test run yields the temperature and pressure rates as a function of temperature, as well as a thermal kinetic interpretation of the exothermic reaction under adiabatic conditions, and the adiabatic temperature increase. The temperature/pressure rate – temperature data of such an experiment (Fig. 33) can be used for designing a rupture disk or a safety relief valve. LIVE GRAPH

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Figure 31. Adiabatic Dewar test setup a) Oven; b) Autoclave; c) Dewar; d) Sample; e) Temperature signal; f ) Pressure signal

Generally, Φ approaches unity with increasing mass of the reaction mixture. This is important as this is both demand and justification for model based mathematical correction to data obtained in small experimental tools. Figure 32 shows typical temperature – time curves for the decomposition of a substance, starting the test at different initial temperatures.

Figure 33. Experimental self-heat and pressure rate data from an adiabatic study of an exothermic reaction

LIVE GRAPH

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Figure 32. Typical recording of decomposition of a substance from adiabatic Dewar experiments

The handling of toxic reaction products and the repair problems are the disadvantages of the thermal aging and heat accumulation storage

2.3.2.3. Calorimetry The methods described usually provide information on the thermal and/or adiabatic runaway potential of batch systems. They are not suitable for studying reactions under their predefined, normal or upset operating conditions. For this purpose, reaction calorimetry is the recommended method. In a reaction calorimeter, batch, semibatch, and continuous processes can be studied, and complex reaction procedures can be simulated (e.g., pH control, temperature or pressure control, reflux, distillation). Moreover, the effect of deviating operating conditions can be tested

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(e.g., by varying the amounts of reaction components, catalyst, or solvent, temperature, loss of agitation, simulation of adiabatic conditions). The equipment consists of a stirred glass reactor with a cooling jacket. For the investigation of reactions under pressure, special test vessels are also available. The reactor lid accommodates several measuring instruments and input/output devices. The experiment is controlled by a data processing/acquisition unit. The calorimeter can be operated isothermally (reaction temperature kept constant throughout the run), in an isoperibolic mode ( jacket temperature is kept constant), in a free-programmable mode (linear temperature ramps for the internal or the jacket temperature can be defined), or in an adiabatic mode. Calorimetric evaluation is based on temperature measurements and calibration runs, to characterize the heat transfer. The evaluation yields the overall heat production rate over time of the reaction system as investigated, as well as the heat of reaction. Special evaluation techniques allow determination of the thermal accumulation potential in the reactor due to unreacted material. Evaluation can be based on heat-flow or heatbalance algorithms. In the first case, the heat flux between the inside of the reactor and the surrounding jacket is determined by measuring the temperature difference between the reaction mass and the coolant. This kind of evaluation may be significantly in error if the overall heat transfer coefficient changes markedly during the reaction in a nonlinear way. This is especially the case for polymerization reactions, as they are usually accompanied by major changes in viscosity. This systematic error can be eliminated by running the calorimeter in the isoperibolic mode, and evaluating the heat balance for the jacket independently. However, this evaluation method is rather complex. If the calorimetric experiment is evaluated according to the heat-balance algorithm, the liquid flow rate of the coolant and its heat capacity must be well defined and measured; otherwise, significant errors may occur. If reactions are investigated under reflux or distillation, it is necessary that all heat flows related to the vapor phase are accounted for appropriately. Figure 34 shows a typical apparatus for reaction calorimetry. The information obtainable from reaction calorimetry is of great use,

not only for process safety, but also for process development and optimization.

Figure 34. Typical reaction calorimetry setup a) Thermostat; b) Controlling unit; c) Reaction temperature; d) Jacket temperature; e) Quality measurement

2.3.2.4. Systematic Testing Some 25 different experimental techniques are commercially available for the investigation and characterization of exothermic reactions. It has to be emphasized that there is no comprehensive method; it is sensible to apply more than one method to obtain the necessary PHASE data. On the other hand, PHASE demands a systematic procedure, so a certain ranking of the different techniques is required. This can be achieved by considering the life cycle of a chemical process. Researchers usually carry out their experiments on the 1-g scale; thorough experimental hazard characterization is not required, as possible consequences of events are limited to the fume cupboard, provided occupational health and safety recommendations have been strictly followed. Process development on the 1-kg scale represents the first milestone of the process life cycle, where the preliminary PHASE should be performed, based on screening test data. Before the first run of a process in a pilot plant, the first comprehensive PHASE is required. This can only be performed reasonably if it is based on data which are not subject to large modelbased, scale-up corrections; reaction calorimetry and Dewar tests are the recommended techniques [99]. For intended plant or process modifications, selection of the appropriate experimental test methods depends on each individual case, and no general recommendations can be given.

Plant and Process Safety 2.3.3. Assessment Criteria The incident risk is defined as the product of incident severity and its probability of occurrence. Hence, risk assessment includes both severity and probability. In the following, a generalized procedure for risk assessment of runaway reactions with special emphasis on evaluation criteria is presented. 2.3.3.1. Runaway Scenario It is assumed that a batch reactor is being operated under normal operating conditions, and a cooling failure occurs. Provided that unconverted material is still present in the reactor, the temperature increases, owing to the heat output related to completion of the reaction. This temperature increase is proportional to the amount of unreacted material. The temperature reached at the end of this period is called the maximum temperature of the synthesis reaction (MTSR). At this level, a secondary decomposition reaction may be initiated (Fig. 35). The heat produced by this undesired reaction leads to a further increase in temperature [100].

Figure 35. Runaway scenario a) Runaway of desired process; b) Runaway of undesired process 1 = time to cooling failure; 2 = ∆T ad due to unreacted material (for desired reaction only); 3 = ∆T ad due to decomposition; 4 = T p − T initial ; 5 = time to reach MTSR; 6 = adiabatic induction period

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The following questions help to characterize the runaway scenario and to provide data for risk assessment: 1) Can the process temperature be controlled by the cooling system? 2) What temperature can be reached after runaway of the desired reaction? 3) What temperature can be reached due to runaway of the decomposition reaction? 4) Which moment of a cooling failure will have the worst consequences? 5) How fast is runaway of the desired reaction? 6) How fast is runaway of the decomposition at MTSR ? This type of scenario should be worked out for the different possible process deviations, e.g., charging errors or higher accumulation due to loss of agitation. It allows assessment of the sensitivity of the process to these deviations. 2.3.3.2. Severity: Adiabatic Temperature Increase Most reactions in the fine chemicals industry are exothermic. In the case of a cooling failure, or if heat removal is not sufficient to compensate for the heat production, the temperature increases proportionally to the heat of reaction. Thus, the reaction energy is a direct measure of the severity of a runaway, i.e., the destructive potential. It is usually expressed as the adiabatic temperature increase ∆T ad .

Figure 36. Adiabatic runaway temperature profiles a) – d) increasing rates of temperature increase with increasing energy

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Independent of its meaning for characterization of the hazard potential and determination of attainable temperature levels, the adiabatic temperature increase assists estimation of the dynamics of a runaway. As a general rule, high energies result in fast runaway or thermal explosion, while lower energies (∆T ad < 100 K) result in slower temperature increase rates (Fig. 36), given the same activation energy and initial heat release rate [89]. The heats of common industrial synthesis reactions are often of the order of − 100 kJ/mol, whereas decomposition reactions may reach far higher levels, e.g., −400 kJ/mol for mononitrated aromatic compounds. In many cases, the desired reactions are not themselves inherently dangerous, but decomposition reactions may lead to dramatic effects. The consequences of a runaway event can be manifold. One possibility is solvent evaporation with the subsequent possibility of a vapor cloud explosion, if the boiling point of the system is reached. Another possibility is formation of gaseous products from a decomposition reaction, which leads to a pressure increase and the risk of vessel rupture. In practice, three levels – low, medium, and high – are sufficient for assessment the severity: Low: Medium: High:

∆T ad < 50 K, no pressure buildup 50 K < ∆T ad < 200 K ∆T ad > 200 K

2.3.3.3. Probability and Kinetics of a Runaway There is no direct quantitative measure for the probability of occurrence of an incident, or in the case of thermal process safety, for the occurrence of a runaway reaction. But, if a runaway, anticipated to occur within several minutes, is compared with one expected to occur over several hours, it is obvious that in the first case there is very little time to take preventive measures, whereas in the second case there is time to regain control of the reaction. The probability of runaway is higher with a fast than with a slow temperature increase. While quantification of probabilities is not easy, at least a comparison is feasible. To assess the probability of occurrence of a decomposition reaction, it is necessary to char-

acterize its kinetics. The concept of time to explosion or TMRad (time to maximum rate under adiabatic conditions) is of great help [98]. It can be estimated by the following equation: TMRad =

cp RT 20 (s) q0 Ea

(26)

where cp is the heat capacity of reaction mixture (J kg−1 K−1 ); R is the gas constant, 8.31431 J mol−1 K−1 ; T 0 is the initial temperature of the runaway (K); q0 is the heat release rate at T 0 (W/kg); and E a is the activation energy (J/mol). This equation was derived for zero-order reaction kinetics, but it can be used for reactions of higher order, provided the influence of concentration on the reaction rate can be neglected. This approximation is particularly valid for fast, highly exothermic reactions. The isothermal mode of DSC provides an easy way to measure the kinetic parameters in the TMRad equation [89]. A set of isothermal experiments is run at different temperatures. The natural logarithm of maximum heat release rate, determined on each thermogram, is plotted against the reciprocal temperature in an Arrhenius diagram (Fig. 37). Runaway curves and the corresponding time to maximum rate can also be measured by adiabatic calorimetry. In practice, three levels are sufficient for the assessment of the probability. For discontinuous chemical reactions on an industrial scale, a probability can be considered low if the time to maximum rate of a runaway reaction under adiabatic conditions is > 24 h. The probability becomes high if the time to maximum rate is < 8 h (a working shift). These timescales are only orders of magnitude, and depend on a lot of organizational and plant design factors, e.g., degree of automation, operator training, frequency of electrical power failures, reactor size. This scaling of probabilities is valid only if safety measures in proportion to the known severity are implemented. 2.3.3.4. Assessment of Criticality For reactions presenting a thermal potential, criticality ranking can be based on the relative values of four temperatures.

Plant and Process Safety LIVE GRAPH

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LIVE GRAPH

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Figure 37. Determination of TMRad by differential scanning calorimetry (DSC) A) Isothermal DTA traces: a) T = 100 ◦ C; b) T = 110 ◦ C; c) T = 120 ◦ C B) Pseudo-Arrhenius diagram for evaluation of the isothermal runs

– T process (process temperature) is the initial temperature in the cooling failure scenario. In nonisothermal processes, it is defined as the temperature level at which a cooling failure produces the most severe consequences (worst case). – MTSR (maximum temperature of synthesis reaction) depends essentially on the degree of accumulation of unconverted reactants, and therefore depends strongly on the process design [101]. – ADT 24 (temperature at which TMRad = 24 h) is defined by the thermal stability of the reaction mixture. – MTT (maximum temperature of technical reasons), in an open system, is the boiling point. For a closed system, it is the temperature at which the pressure reaches the maximum permissible value, i.e., the set pressure of a safety valve or rupture disk.

not be triggered. The safety of the process depends on the heat production rate of the desired reaction at the boiling point [103]. 4) In the case of loss of control of the desired reaction, the boiling point is reached, and the decomposition reaction can, theoretically, be triggered. The safety of the process depends on the heat production rate of both the desired and decomposition reactions at the boiling point. Evaporative cooling may serve as a safety barrier in open systems. 5) In the case of a loss of control of the desired reaction, the decomposition reaction is triggered and the boiling point is reached during runaway of the decomposition reaction. It is very unlikely that evaporative cooling can serve as a safety barrier. The heat production rate of decomposition at the boiling point determines the thermal safety of the process. It is the most critical of all scenarios.

These four temperatures permit classification of the scenarios into five different classes, from the least critical (1) to the most critical (5) [102] (Fig. 38). They are defined as follows: 1) In the case of loss of control of the desired reaction, the boiling point cannot be reached and the decomposition reaction cannot be triggered. 2) The situation is very similar to scenario (1), but if the reaction mass is kept under heat accumulation conditions for a longer time, the decomposition reaction can be triggered and the boiling point of the system can be reached. 3) In the case of loss of control of the desired reaction, the boiling point of the system is reached, but the decomposition reaction can-

Figure 38. Assessment of criticality a = T p ; b = MTSR; c = range where decomposition becomes relevant; d = bp

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2.3.4. Measures Three categories of measures, with different applications, are distinguished: 1) Hazard-preventing measures, which suppress the potential, e.g., dilution, change of synthesis route. These measures render the process inherently safe [101]. 2) Hazard- controlling measures, e.g., change from batch to semibatch or continuous operation, interlocks in the process control system (Section 3.4.3) etc. These measures render the process fail-safe. 3) Consequence-mitigating measures, e.g., dumping, quenching, venting (Sections 3.5, 5.1). Depending on the criticality, different measures can be applied to prevent or control the runaway or to mitigate its consequences: 1) No special measures are required, but the reaction mass should not be kept under heat accumulation conditions for long periods. Evaporative cooling serves as an additional safety barrier. 2) No special measures are required, but the reaction mass should not be kept under heat accumulation conditions for long periods. 3) The measure of choice is to use the evaporative cooling to keep the reaction mass under control. A distillation system must be designed for that purpose, and has to work effectively, even in cases of utility failures. A backup cooling system, dumping of the reaction mass, or quenching can also be used. These measures must be designed adequately and must be available immediately after the failure occurs. 4) Similar to (3). The same measures apply, but the additional heat production rate due to the secondary reaction has also to be taken into account. 5) In this case, the boiling point is very unlikely to serve as a safety barrier. Therefore only quenching or dumping can be used. Since, in most cases, the decomposition reactions release very high energies, particular attention has to be paid to the design of safety measures. It is worth considering an alternative process design, in order to reduce the severity or at least the probability. The follow-

ing possibilities should be considered: reduction of concentration, change from batch to semi-batch, optimizing semibatch operating conditions to minimize accumulation [89], [104], changing to continuous operation, etc.

3. Development, Design, and Construction of Safe Plants 3.1. Objectives, Regulations, and Concerns The goal of all plant safety effort is to eliminate or reduce the possible hazards described in Chapters 1 and 2. According to Section 1.3, this means limiting the risk created by the facility to an acceptable level. At the same time, the relevant regulations and engineering codes must be complied with. In many areas, these implicitly establish the required level of safety (or acceptable risk). All the industrialized countries have adopted regulations to protect workers, uninvolved third parties, and the environment [105]. These regulations employ various combinations of the following approaches: 1) Setting standards for the quality of engineering facilities and their safe operation 2) Establishing approval or licensing procedures for the erection and operation of plants and for substantial modifications to them 3) Requiring regular inspections of the plant and its technical equipment by the operators, government agencies, and (sometimes) independent third parties 4) Assigning civil liability and legal responsibility for damage caused Because there was initially little experience relating to the hazards of technology, the first regulations were written for individual types of apparatus regarded as dangerous (steam boilers, pressure vessels, machinery with moving parts) and aimed chiefly to insure occupational safety and to protect third parties. Later, regulations for other types of equipment were added, attempts were made to extend protection to new areas such as surface waters, and new regulatory principles such as the principle of precautionary action were introduced. It is not unusual that regulations are created by many independent bodies,

Plant and Process Safety with very different jurisdictions. As a result, in Germany (for example) there is a rather complicated system of regulations for chemical plants, with overlaps and multiple rules, and the density and the degree of detailing of regulation differ widely from one regulation to another. The safe design and safe operation of chemical plants in Germany are governed by regulations stemming from the following fields of law [106]: 1) Building and construction law 2) Occupational safety law (including the Equipment Safety Act, regulations defining facilities subject to supervision, and the Accident Prevention Regulations) 3) Hazardous substances law (including the Chemicals Act and the Hazardous Materials Regulation) 4) Water pollution control law (including the Regulation on Facilities with WaterContaminating Substances) 5) Air pollution control law (including the Major Hazards Directive [107]) This complicated situation is not likely to be fundamentally changed by harmonization efforts in the European Single Market, for two reasons: First, under the EEC Treaty (Article 100 a) [108], only requirements on the quality of technical devices are subject to harmonization. Second, regulations in the European Union (EU) continue to be created, adopted, and enacted into law by the member countries according to the old pattern (e.g., for specific types of facilities). As a rule, the regulatory system is structured vertically [109]. Legislation at high level sets forth the scope and objectives as well as the solution approaches in general form, while engineering codes and standards at lower levels are largely concerned with technical details. Regulations, especially at sublegislative levels, are designed deterministically in that the required safety is deemed to be attained if the plant satisfies certain requirements on design, sizing and equipment. An exception can be found in a Dutch regulation under which a chemical plant, to be approved, must operate at a risk below a value described in terms of fatal accident probabilities [110]. Plant builders and operators view plant safety in terms of the types of hazard arising from the process. For them and their dealings with the

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authorities, it would be simpler if there were a single consistent set of regulations, broken down by hazard type (releases and spills, fires, explosions) and – instead of setting fixed technical solutions for protective tasks – such a code would use fixed solutions merely as examples of ways to achieve the stated objective. This approach would have the advantage that the required safety could be achieved by the most advantageous technical resources in each instance. In addition to that, the “plant” orientated view would no longer require multiple specific regulations covering occupational safety and health, air and water pollution control, since one criterion for the plant (e.g., tightness) would simultaneously take care of related problems in the other areas. The main safety tasks for the process designer and the designer and builder of process plants are: 1) To identify and correctly assess all types of hazard 2) To take appropriate steps to reduce and control these hazards The key hazard types are those listed in Section 1.1: Releases and spills Fires Explosions The safety tasks can be broken down into the process itself and into the safe design and operation of the technical facility required for the process. The specific tasks can then be listed as follows: 1) Achieve safe process design by: a) identifying all types of hazards; b) assessing their hazard potentials; c) minimizing the hazard potentials; d) deactivating the hazard potentials 2) Achieve safe plant design and operation by: a) systematically analyzing danger sources; b) evaluating their probabilities of occurrence (usually qualitatively); c) minimizing sources of trouble and error; d) employing fault-tolerant design In detail (see also Section 1.2), this means:

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1 a) For all substances to be processed, safety ratings and toxicologically and ecologically relevant data must be acquired (see Chapter 2). A comparison of process parameters and design data with safety ratings, step by step through the process, reveals where danger sources exist, or may arise. 1 b) The magnitude of each hazard potential is evaluated through analyses, e.g., those described in Sections 1.4.2 and 1.4.3. 1 c) It must be determined whether the hazard potentials can be reduced through suitable process design. The practices recommended here lead to inherent safety [111], [112]. To achieve this goal the planner should replace hazardous substances by less hazardous ones wherever possible. Large inventories should be avoided as far as possible, e.g., by using process steps that can be carried out quickly in a small volume, by introducing continuous operations in place of batch operations, and by eliminating large buffer volumes. 1 d) Any hazard potentials that remain must be deactivated in such a way that they cannot manifest themselves in the process. Lowering the temperature far below critical values or diluting substances and handling them in solution (to lower the vapor pressure) are the approved methods. They also help make the process inherently safe [111], [112]. If the plant is to be conceived for a process that has been safety optimized in this way, two analytical tasks, followed by two design tasks, must still be performed: 2 a) The system plant must be systematically searched for danger sources, i.e., possible defects and failures that can activate the deactivated hazard potential. 2 b) When possible faults are identified, their frequencies or probabilities of occurrence must be evaluated so that appropriate safety measures can be taken. The last two tasks can now be undertaken with an eye to the magnitude of each hazard potential (task 1 b) and a (qualitative) measure of the probability of a defect that would activate the hazard potential: 2 c) All possibilities for minimizing sources of trouble and error must be exhausted [112].

2 d) As far as possible, the facility must be designed and equipped so that faults are “forgiven” without resulting in harm. One way to accomplish this is to use redundant (multiple) safety devices [112] (see Section 3.4.3). In carrying out these tasks, some of the steps (2 a – 2 d) may have to be done more than once, recursively, because any change in the system due to the new measures can introduce new danger sources. For this reason, and because the development of a process and the associated plant is done step by step, with concomitant advances in understanding a “holistic” procedure, segmented by time, technical specialty, and logical relationships, must be adopted in the development, design, construction, and operation of a chemical plant. Examples of such methods are presented in the Section 3.2.

3.2. Procedure for Designing and Constructing Safe Plants Chemical processes and their associated technical facilities are developed in steps. Process development in the laboratory is followed by testing of the process on a pilot scale before the project goes through the various planning stages (preliminary, draft, and detailed design). The planning process culminates in the purchase of equipment and erection of the plant. After a test and commissioning phase, the facility is put into production. The left-hand side of Figure 39 shows these stages [113]. Each phase involves questions as to the safety of the process and the plant, each of which must be answered immediately or, at the latest, before the process goes on to the next phase. The most expedient way of creating a safe plant is thus to plan for safety studies. At each step in process and plant development, safety analyses must be done (ideally integrated into the development) in order to pose the right questions and immediately seek solutions to the problems identified. These ideas are the basis for the procedure chosen by Bayer, a large German chemical company (Fig. 39). Safety analyses are broken down into four sections, each concluding with a certificate prepared by safety experts. This certificate

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Figure 39. Technical safety audit scheme for new plants or changes to existing plants (Bayer) [113]

states that the required studies have been performed, and the correct conclusions drawn from them. The four phases of safety analysis can be listed as follows: A 1) To create safety principles by: compiling and determining safety, toxicological, and ecological data; identifying sources of danger in the process; examining possible safety solutions; establishing the safety concept for the process A 2) Defining the safety concept for the plant by: performing systematic analysis; identifying technical protective measures A 3) Performing a detailed safety analysis by: analyzing all plausible forms of trouble as to cause, effect, and corrective measures; adopting the final detailed safety concept A 4) Conducting the safety acceptance of the plant by: doing a nominal/actual compari-

son (concept/implementation); carrying out functional tests This procedure is organized in an obvious way, and has the advantage that a completion certificate must be prepared at the end of each phase, before the next phase begins. The project is not released unless the planning certificate is ready, and the plant does not go on stream unless there is a safety acceptance certificate. These rules are set forth in an internal directive [114]. Similar procedures are common in other companies. They are not always so formal, but they embody the same principles and use the same resources. BASF, for example, uses a three-phase approach for these studies [115]. Phase 1 generates a basic safety concept with provisional solutions for the main possible hazards in the process. Phase 2 involves elaboration of the safety concept, so that it can be submitted to the author-

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ities along with the approval application (safety analysis, safety report). Detailed safety verification takes place in phase 3. Hoechst utilizes the principle that certain safety analysis and design results, once created, become important documents that must accompany the process and the plant throughout their lives [116]. A substance data report and transmission report have been developed for this purpose; they function to gather key safety information and communicate it within the company; a checklist is used in the safety assessment of production facilities. A number of publications on the performance of safety analyses have been issued by ESCIS, the Expert Commission on Safety in the Swiss Chemical Industry [117], [118]. These stress principles and organization for this kind of work, establish key points to be analyzed at various times, and include methodological aids. Essential in all these procedures is that nothing is overlooked and the proper specialists are brought in at the proper time. All such procedures can therefore be said to regulate three things – who does what and when. The methods employed by Degussa [119] and by Lurgi, a plant construction company [120], assign great importance to these aspects. Over the plant’s economic life, all the results of safety work are used again and again for guidance, e.g., during maintenance and plant modifications, or when new personnel are being trained. Close attention must be paid to clear and complete documentation of the safety analyses; the recommended form for this is a concise, uniform, computerized database keyed to process, process step, plant, and plant section, and covering all danger sources with causes, effects, and corrective measures (including brief justifications). It can be used directly in the writing of safety reports pursuant to the Major Hazards Regulation (Seveso Directive) [107] and can easily be consulted from the plant operating personnel whenever necessary. By way of summary: the design and construction of safe plants calls for a highly structured and organized procedure clearly setting forth what has to be done by whom and in what way, and focusing on the creation and routing of documents. This is called a safety management system.

In accordance with this principle, it has recently been suggested in both the United States [121] and Germany [122] that “safety management” should be implemented on the model of quality management following the well known international standards ISO 9000 ff. This idea makes sense, provided the procedures recommended for quality assurance are reinterpreted for this new purpose. This tends toward the procedures discussed above, which are better suited to the legal situation, at least in Germany. In order that nothing should be overlooked in the analyses, it is desirable to provide methodological aids to the specialists as they deal with various problems. Section 3.3 concerns such aids.

3.3. Methodological Aids The safety concept for a chemical plant must be complete, with an answer ready for any safety question. All necessary questions which can help reveal potential sources of danger must be posed ahead of time. The problem of a comprehensive and complete analysis comes up twice in the course of process and plant development. First, for the process (task 1 a of Section 3.1) the safety figures for the various substances (toxicity, flammability, explosibility) must be determined. Second, for the plant (task 2 a of Section 3.1) defects and malfunctions in the system that can activate the hazard potentials must be identified. Appropriate safety engineering involves assessing the hazards as to both possible scope (task 1 b of Section 3.1) and probability of occurrence (task 2 b of Section 3.1). Methodological aids available for use in these tasks [123], [124] are all characterized by clear and easily understood structures and systematic procedures. The specific methods differ in: Objective of the analysis Functional principle employed Basic knowledge required Appropriate time of use Aids needed Results achievable Documentation Cost/benefit ratio

Plant and Process Safety If we go back to the four chief tasks of safety analysis 1) Completely identifying hazards due to substances (task 1 a) 2) Assessing hazards as to potential scope (determining the hazard potential, task 1 b) 3) Identifying danger sources in the plant (identifying defects and possiblities for malfunctions, task 2 a) 4) Evaluating the possible hazards with respect to probability of occurrence (task 2 b) the methods can be placed in four groups, two relating to the identification of problems and two to their assessment. Table 9 summarizes the most important methods. The literature refers to many more methods than those in the table. In most cases, the additional names do not refer to formal methods, e.g., “safety audit”, “what-if-method”, and “preliminary hazard analysis”, as defined in [124]; or they are synonyms for known methods: “human error analysis”, “action error analysis”, and “human reliability analysis”, all concerned with human errors and using checklists, keywords, tables, or event trees to study their effects; or they are combinations of two methods, e.g., “cause – consequence analysis”, a combination of incident sequence analysis and fault tree analysis [123–125]. The methods of Table 9 increase in difficulty and cost from top to bottom; with the exception of consequence analysis, they are also arranged roughly in chronological order to use in process and plant development. In accordance with this ordering, the amount of input information required, the level of complexity, and the amount of special knowledge required also increase from top to bottom. Therefore assessment methods must generally be carried out by specialists, while the identification methods should be among the tools used by every chemist and engineer in the process industries, particularly the chemical industry. The identification methods used in safety analysis are summarized below. Checklists and Relationship Charts. A checklist enumerates points that, according to experience, are associated with hazards in the handling of substances and mixtures of substances, or in the performance of a technical

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process. It can be as detailed as desired, and must be suitable for the kind of analysis being carried out. For example, checklists can be developed and used for thorough analysis of all safety-relevant material properties. If the concern is whether explosions can be initiated by undesired reactions, the checklist comprises such questions as: – Are the substances stable? – What happens if foreign substances are admitted? – What happens if the process parameters (temperature, concentration) change? The formal method is to work through such lists of questions and to determine whether: 1) A hazard is possible (yes or no) 2) Further study is needed 3) Safety measures are called for The results of the analysis can be documented directly on the checklist or on a summary form reflecting its organization.

Figure 40. Reaction matrix showing all possible pairings of substances (tests for unwanted reactions)

Checklists can be used to insure the completeness of the safety concept in later phases of a plant project, e.g., in the design phase, to insure that possible events such as power outage, cooling outage, or stirrer outage are included in the safety concept. The creation and use of checklists is also customary in plant operation. They have the obvious advantage that they can

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Table 9. Safety analysis methods Task

Aim

Working principle

1) Identification of hazards

1) To complete the safety concept

1) Memory-jogging

2) Evaluation of hazards according to probability of occurrence

3) Evaluation of hazards according to consequences

Method

1) Checklists 2) Relationship charts 2) Use of search aids and tables 3) Failure modes and effects analysis 4) Action error analysis 5) Hazard and operability studies 2) To optimize safety systems with 3) Representation of 6) Incident sequence analysis regard to reliability and availability interconnections of failures in (inductive) graphical form and evaluation of probability 7) Fault tree analysis (deductive) 3) To minimize the hazard potential 4) Mathematical analysis of 8) Hazard consequence analysis and devise optimum protective physicochemical processes measures

be adapted to any problem; their drawback is that things not included may not always be recognized and dealt with. The same limitation applies to the second “memory-jogging” method, the use of compatibility charts [124] or relationship charts [125]. One use for this technique is to check the compatibility of all the substances that can come into contact in a process. If, for example, there are substances A to G, then half of a twodimensional matrix (Fig. 40) gives all possible pairings of substances. By systematically scanning over the interaction fields, it is possible to check whether mixing two substances creates a hazard, and whether further experimental study is called for. The disadvantage of this technique is obvious: mixtures of three or more components do not appear. Caution is therefore indicated, and matrices are more often used simply to illustrate safety situations, e.g., causes and effects or effects and corrective actions [117]. Hazard and Operability Studies (HAZOP) and Similar Methods. The second group of problem-identifying methods includes: – Failure modes and effects analysis [123], [124] – Action error analysis [123], [124] – HAZOP studies (called PAAG methods in Germany) [127], [128] These methods are so similar that they can be described together. All can be characterized as “deviation analysis” in that they look for possible hazards that can arise in the process, in the

plant, or in plant operation if an error occurs, or the state or sequence of actions deviates from the prescribed state of sequence. (The assumption is that there are no hazards in the prescribed state or sequence.) In this analysis of deviations, similar forms are employed for all three cases. As a rule, the form covers the following aspects: – – – –

Deviation (error, failure) Cause Effect Corrective action

In many analysis, the following are also included: – Error detection (how?) – Frequency (often/seldom) – Severity of effects The third trait common to all three methods is the use of search aids. It is here and in the objects of study that the methods differ. Failure modes and effects analysis (FMEA) is oriented to items of equipment and machinery, each having a certain function in the plant. It employs the working hypothesis that this function is not performed, i.e., each piece of equipment is examined for the effect its failure has, and what corrective action may be required. The search aids in action error analysis and HAZOP studies are keywords describing deviations from nominal conditions or a nominal sequence. Keywords used to characterize human errors include: – Too early

Plant and Process Safety – – – –

Too late Not Wrong object Wrong sequence

Typical keywords for the HAZOP method [128] include: – – – –

More Less No Different from

Much has been written about this and the other methods [123–128]; Table 10 presents a few examples of analyses by all three methods for a stirred-tank reactor. All that remains is to indicate the proper time for applying these methods. In the design of a new plant, they should come in at the detailed safety design phase, when most of the piping and instrumentation diagrams are ready. These methods are also useful in the inspection of plants already built and on stream. In the author’s view, the possibility of creating suitable keywords for all parameters, functions, and action sequences in a plant makes the HAZOP method so flexible that it can cover the full range of applications of the other techniques. Just two identification methods are therefore of any practical importance in the development of a complete safety concept: checklist methods and the HAZOP method. Assessment of Hazards by Consequences. The concept of the hazard potential was introduced in Section 1.3 as a way of evaluating the magnitude or severity of a safety problem. Section 1.4 described how the hazard potential can be determined and what difficulties and imponderable factors must be taken into account. If the scope and quality of safety practices are to be suited to the hazard potential so that a generally comparable and acceptable risk level is achieved, it is necessary to get at least a rough idea of the magnitude of the hazard potential (see also Section 3.1). On the other hand, if effect-limiting safety devices are to be custom designed, e.g., an emergency pressure-relief system or a scrubber to handle off-gases in case of an emergency, it is necessary to model the physical and chemical processes taking place during the accident and

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to investigate their effects. Protective systems can be tailormade, and hazard potentials can be minimized (tasks 1 b and 1 c of Section 3.1). Evaluation of Hazards by Probability of Occurrence: Incident Sequence Analysis and Fault Tree Analysis. Both methods in this class examine links between faults, represent them graphically, and (in principle) can assign probabilities to them. Incident sequence analysis starts with a single fault, and observes how it may develop. The working hypothesis is that every safety measure can succeed or fail with a certain probability. Figure 41 shows a simple event tree for a temperature rise in an exothermic semibatch chemical reaction where three safety measures (heating shut-down, feed cut-off, and emergency cooling) have been instituted to prevent the reaction getting out of control. The figure illustrates the three-stage safety concept. The probability of the undesired event can be calculated if the probability of the initial event and the failure probabilities of the several safety measures are known. More careful analysis of the sequence of events reveals that the model is far too simple for the process under discussion. Whether the safety actions succeed or fail depends heavily on the intensity and rate of temperature rise, as well as the dynamic response of each action, and these are quantities that cannot be simply described in probability terms. Therefore, the only utility of event tree analysis for such processes is that it can make relationships easy to understand in simple cases. Similar limitations apply to fault tree analysis, at least when used for typical processes in chemical systems (Fig. 42). The analysis goes in the opposite direction to that in incident sequence analysis: deductively from an undesired event. Faults are connected from top to bottom in chains that can lead to the top event. There are two main kinds of link: 1) The blocking AND junction: more than one event must occur in order to open the path upward 2) The passing OR junction: one event from a set of several events is sufficient to lead to the next higher event

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Figure 41. Event tree diagram showing success or failure of safety measures

Figure 42. Fault tree diagram

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Table 10. Examples of safety analysis methods applied to a stirred-tank reactor Method

Cause

Consequences

Safety measure

Failure modes and effects stirrer: does not turn, turns analysis (failure of wrong way: equipment) steam valve: does not close

no electricity, motor defective, wrongly installed jammed

heat transfer impeded, solids precipitate

revolution counter, feed shut-off, control unit feed shut-off

Action error analysis (human error)

“too late”: steam off, cooling water in

operator not paying attention, overworked

temperature increase in reactor temperature increase, danger of uncontrolled reaction

“not”: steam off

operator not paying attention mix-up unknown reaction pressure reduction fails, no temperature increase, cooling danger of uncontrolled reaction

Hazard and operability studies (deviations in functions and process variables)

Deviation

“incorrect”: product “more”: steam, heat

“none”: cooling water, stirring “reverse”: direction of stirrer

valve does not open, driving mechanism defective wrongly installed

Both fault tree analysis and event tree analysis pose stringent requirements on correct application and interpretation. Both call for experts. The methods have been fully described in the literature [123–126] and have been standardized in Germany [129], [130]; with respect to chemical safety, however, they have so far not played a significant role, for the reasons discussed earlier. Their main area of application is in the assessment and optimization of complicated technical systems in which the components display a yes/no failure behavior with assignable probabilities, and where the failure rates are known (e.g., in aviation and space flight). For further discussion see [123], [124], [131]. It can be maintained that these methods do not provide a satisfactory basis for assessing hazards in chemical processes as to probability or frequency of occurrence. The process variables in the analyses are also functions of time and intensity, and there is not a sufficiently large number of comparable states for statistical analysis. Estimates of probabilities therefore have to be established qualitatively or semiquantitatively on the basis of experience and the number and quality of safety measures. For the same reasons, attempts to evaluate risks from chemical plants on the basis of occurrence probabilities of single events and process and component failure rates have not yielded useful results [131]; but this is not a fatal draw-

alarm, feed shut-off, emergency cooling automatic shut-off labeling analysis steam shut-off by hand, feed shut-off, emergency cooling

heat dissipation impeded, feed shut-off, emergency temperature increase cooling solids precipitate

direction indicator, control unit

back because the risk due to chemical production (Section 1.4.1) can easily be derived from historical data of long-time experience and has proved low in the past.

3.4. Safe Processing: Strategies 3.4.1. Fire Protection 3.4.1.1. Fire Prevention The key factors governing the occurrence and impact of fires are the quantity of combustible substances and possible ignition sources present in the plant. Both factors must be determined and taken into consideration for effective design of fire-protection measures. First, it is necessary to ascertain the form and quantity of combustible, flammable, and explosive substances located inside battery limits. As well as products, auxiliary substances, and packaging materials, thus evaluation must also include energies, construction materials, and combustible parts of the plant. Next, these combustible substances are rated according to their key fire parameters (see Section 2.2.1). The analysis must include the state of aggregation in which these substances are held in the plant under normal or process conditions, as well as the extent to which a possible fire, by raising the temperature, can initiate further critical reactions. A crucial point is whether fire loads

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are unprotected or are protected from the direct action of ignition sources by noncombustible packaging or enclosures, e.g., steel tanks or reactors [132–134]. The first important measure to prevent fires is to minimize the fire loads. In industrial plants, this can be done only to a limited extent, so possible ignition sources must also be identified, eliminated, or isolated from the existing combustible substances by bulkheads. Some examples of ignition sources are [135]: – – – – – – – – –

Open flames Sparks Compensating currents Spontaneous ignition Hot surfaces Electrostatic charges Chemical reactions Vibrations, shock waves Thermal manifestations of mechanical energy (e.g., friction) – Thermal manifestations of electrical energy (e.g., overheating) Ignition sources present during maintenance and repair operations also merit special attention. Sparks from welding, burning, and abrasive cutting have repeatedly caused fires. It is therefore desirable to prescribe the necessary safety precautions in safety certificates (see Section 4.4). Smoking must be expressly forbidden in plants and warehouses. The creation of special smoking areas is preferable to concealed smoking in no-smoking areas. Appropriate containers, e.g., self-extinguishing wastepaper baskets, must be furnished for the disposal (daily if possible) of combustible wastes. To avoid ignition by lightning, all buildings must have lightning protection. Potential compensation between large-area metal structures should be kept in mind, as should the use of overvoltage protection. Critical effects can also follow from reactions between combustible substance and a fireextinguishing agent. If incompatibilities lead to restrictions on extinguishing agents, these must be identified and integrated into the planning of organizational and technical fire-fighting procedures. For example, the reaction of fire-fighting water with a combustible substance may release not only flammable gases (from potassium, al-

kali metal compounds, etc.) but also corrosive vapors (HCl from chlorosilanes, acetyl chloride, etc.). Ignition by static discharge has caused accidents where CO2 or medium-expansion foam has been improperly used [133], [136]. 3.4.1.2. Limitation of Fire Effects The principal effects of fire have to do with the toxic action of the smoke, and the high temperatures. To minimize harm to humans and animals, the environment, and property, a fire-protection concept must be developed for every industrial facility. The following sections identify the measures that must be included in such a concept. Design and Construction. Access to the plant must be guaranteed by linking it to roads, streets, and fire service entries. Staircases – their number and placement dictated by the permissible lengths of escape and rescue routes – must give access to interior areas of the building. Escape and rescue routes simultaneously serve as lines of attack for firefighters, and must therefore be built to remain safely usable for an extended time. In modern industrial buildings, this is achieved by erecting massive stair towers (turrets) or administrative service wings, which are more or less independent of the supporting structure in the production building proper [137– 139]. The availability of fire-fighting water is insured by a main supply designed to serve the needs of the entire developed area, and a plant supply designed to meet the special needs of the facility. Fireproofing of the supporting beams of the building should be suited to the possible thermal loads in the anticipated fire. Simple approximations can be used for rough estimates (e.g., DIN 18 230), or more exact thermal-balance calculations can be performed. Regardless of the result, the use of noncombustible structural materials, or at least materials that burn only with difficulty or are difficult to ignite, is preferable. An effective way to reduce temperature stress is to build in smoke and heat exhausts. Smoke extraction devices, which have to be opened mechanically, serve chiefly to remove cold smoke, and thus protect escape and rescue

Plant and Process Safety routes; heat exhaust systems are intended to remove hot fumes, and thus reduce the thermal stress on the supporting members of the building. Materials that fuse in a short time can be employed effectively and economically in these devices [140], [141]. The subdivision of the plant building into fire compartments or fire-fighting compartments should prevent unacceptable spreading of the fire. This is accomplished by walls and ceilings rated for a specified fire endurance (commonly 90 min). Openings in these members must be closed off appropriately; this includes doors, windows, ventilation ducts, cable and pipe penetrations, openings for tracked conveying systems, and other weak points [139], [142]. When a building is compartmentalized in this way, care should be taken to separate areas allocated to uses, e.g., administration, production, technical zones, and storage. Storage of starting materials and end products in production areas should be avoided. Retention of contaminated fire-fighting water must form part of the design concept for fire protection. Retention is implemented, appropriate to the purpose of the building, the water contamination class of the substances, and the protective measures employed. If facilities are available for intercepting and removing contaminated fire-fighting water, normal water-retention facilities can be erected for more than one plant or storage area [143]. Fire-Protection Equipment. When fire compartments must be very large, for reasons dictated by production aspects, or when very large quantities of combustible substances or high fire risks are present, fire-protection equipment must be installed. The selection of such equipment is governed by the fire risk. The fundamental distinction is between room-protective equipment, which covers the entire enclosed region or fire compartment, and local protective equipment, whose action is restricted to a particular unit [144], [145]: Early fire-detection devices are connected to a continuously staffed central office (that of the fire-fighting service if possible) and give a prompt alarm to the personnel assigned to damage control, so that the time allowed for the fire to develop undisturbed is cut short. Depending

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on the combustile material present, smoke, heat, and flame detectors can be used. Automatic fire-extinguishing systems involve more than mere early detection. They are connected to a continuously staffed central office, and not only send an alarm to fire-fighting personnel, but also take immediate fire-fighting action. A properly designed and constructed system can suppress a nascent fire or block its progress, so that the fire-fighting service has a better chance of success. The design of the fire-extinguishing system must include the proper extinguishing agents carefully matched to the type and quantity of combustible substances. Restrictions on extinguishing agents, e.g., because of critical reactions of stored substances with water, must be observed. A sprinkler system consists of fixed piping with spray nozzles, closed off by heatsensitive elements (liquid-filled glass vessels or fusible closures, with triggering temperatures ca. 50 – 260 ◦ C). If a fire starts, the elements above the fire site experience a thermal load, which opens the outlets of the spray nozzles so that extinguishing agent is supplied to the fire site. Because only the nozzles above the actual site of the fire open, damage by the extinguishing agent is limited. Applications include piece-goods storage areas and production areas with large quantities of combustible substances. Water sprinkler systems are suitable mainly for Class A solid fuels. Sprinkler systems with a mixture of water and foaming medium can be employed with Class A solid fuels and with combustible liquids (Class B). A water-spray or deluge system consists of a fixed pipe grid with open nozzles. In contrast to the sprinkler system, extinguishing agent is delivered through all the nozzles at once, over the entire area of the system. The extinguishing action is stronger than that of sprinklers, but the possibility of secondary damage is greater. This type of system is generally used only where the fire spreading rate is extremely high, e.g., transformer installations or warehouses containing substances that burn very rapidly or explosively. Water and mixtures of water with foaming media can be used. Carbon dioxide fire-extinguishing systems deliver CO2 through piping systems with spray or “snow” nozzles. The high concentration of ex-

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tinguishing agent required means that the compartment must be isolated by bulkheads in case of fire, so systems of this type are preferred for smaller spaces. Their use is essentially restricted to combustible liquids and electrical equipment. Carbon dioxide does not penetrate to a great depth, and so there are problems in putting out solid (smoldering) fires by this method. CO2 is useless against metal and carbon fires, because the high temperatures that may occur in such fires can break down CO2 to carbon monoxide and oxygen, and the liberated oxygen can promote the development of the fire. Certain metals such as magnesium can reduce CO2 to carbon, e.g.: CO2 + 2 Mg −→ 2 MgO + C

so these metals can continue to burn, even after CO2 has displaced oxygen. The toxicity of the extinguishing agent means that special measures must be taken for personnel safety (redundant warning systems). CO2 fire-fighting installations are increasingly used for local protection, e.g., in laboratory fume hoods or computer centers. Powder extinguishing systems can be used primarily as local protective devices for special applications, e.g., to extinguish compressed gases or organometallic liquids. This type of system has the advantage that its use is largely independent of temperature (−50 to +60 ◦ C). Powder extinguishing agents are based on sodium hydrogen carbonate, potassium hydrogen carbonate, potassium sulfate, and other chemicals. Sodium chloride or potassium chloride powder can be used against metal fires. Halon extinguishing systems can be used only with special approval because of current bans on these halogenated hydrocarbons; their application is limited to military uses, and on aircraft, spacecraft, and ships. Semistationary, nonautomatic fireextinguishing systems have recently come into increasing use alongside automatic systems. In such a device, the extinguishing agent is delivered through fixed piping from mobile supply units operated by the fire-fighting service. This kind of system is effective only when it is carefully matched to the fire service’s equipment; it is used chiefly where the plant has its own firefighting unit. The same extinguishing media can be used as in automatic systems.

Dry or wet standpipes are almost always required because they markedly improve the performance of the fire-fighting service. Fire-Fighting Methods. Industrial firefighting includes both the training of employees to carry out immediate response, and the installation of fire extinguishers or wall hydrants to deliver suitable extinguishing agents. Establishing a properly trained in-plant firefighting service improves efficiency; training and equipment should be similar to those of the municipal fire services. Professional industrial fire-fighting services differ from in-plant services in that they are recognized by government agencies, and are trained, certified, and equipped to the same level as municipal fire departments. They represent the highest level of fire protection in industrial plants. Their value stems from local knowledge of the personnel, special equipment tailored to the plant, and the fact that they can work continually to provide high-quality fireprevention and fire-protection services. An essential aspect of industrial fire precautions is continuous firefighter training, together with combined exercises with plant personnel [145]. 3.4.2. Explosion Prevention and Protection 3.4.2.1. Introduction Flammable substances (gases, liquids, and solids), if in the form of gas, mist, or dust and mixed with a gaseous oxidizer (most commonly atmospheric oxygen, but other substances such as pure oxygen and chlorine can serve), can form an explosive mixture. The explosion of such systems is generally a fast, exothermic oxidation – a gas-phase chain reaction with chain branching [146]. Special aspects are presented by substances that are not only oxidizable in the gas phase, but can experience exothermic, explosive decomposition reactions without other reactants; examples are ethylene, ethylene oxide, and acetylene. A gas-phase explosion is initiated by an ignition source, delivering sufficient energy and having a suitable energy distribution. The reaction initiated propagates spontaneously through

Plant and Process Safety the mixture. Such combustion reactions are accompanied by the release of a large quantity of energy with increases in temperature and pressure (and often also by the formation of dangerous reaction products). The hazards associated with an explosion are thus governed by three factors : Occurrence of an explosive mixture Presence of an effective ignition source Effects of an explosion These factors provide a logical structure for explosion prevention and protection in the following order: Hazard identification Risk assessment Identification of countermeasures, risk reduction

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process and plant. Since substance data are obtained under standard laboratory conditions, it is important to know how they depend on factors such as temperature, pressure, volume, and volume of the process and plant under consideration. The basic transformation rules are qualitatively summarized in Table 11 [147]. It is necessary to compare substance properties with parameters of the chemical process and plant, not only for normal operating conditions, but also for failure states of the process and of the plant. Such analysis is particularly important for identification of process and plant conditions that may lead to the release of energy (i.e., creation of an ignition source) into the chemical system being handled (see the discussion of ignition hazards in Section 3.4.2.3). 3.4.2.3. Explosion Risk

3.4.2.2. Hazard Identification (see also Sections 2.2.2, 2.2.3) In gas-phase explosions, it is not the flammable material on its own that represents the potential hazard, but its contact or mixing with an oxidizer. The first step in identifying the hazard is therefore to define the combustion properties of the substances. The result indicates whether, and under what conditions, the substances can give rise to an explosive gas mixture. These conditions are characterized by data such as flashpoint, lower and upper explosion limits, and limiting oxygen concentration. The second step is to determine the (minimum) requirements for the explosion hazard to be activated, i.e., the ignition characteristics of the system. These include the minimum ignition temperature for a dust layer, the ignition temperature of a flammable gas or liquid, the minimum ignition temperature of a dust cloud, and the minimum ignition energy. The third step, focusing on the behavior of the explosive system after ignition, provides information on the expected physical explosion effects. Of interest are the heat of combustion, the explosion pressure, the rate of pressure rise, and the maximum experimental safe gap. After the characteristics of the substances are known, this information has to be linked to the dimensions and parameters of the real chemical

The necessary condition for an explosion in the gas phase is the simultaneous occurrence of an explosive mixture and an effective ignition source. The likelihood of an explosion may therefore be considered as the product of the probability of the occurrence of an explosive mixture and the probability of the presence of an effective ignition source. The risk is defined as the combination of the (relative) frequency of an event occurring and the expected extent of damage; this means that explosion effects must also be considered. Likelihood of Explosive Mixtures. Even when the chemical and physical relationships are understood in detail and years of practical experience have been accumulated, the likelihood that explosive mixtures may arise cannot be quantitatively evaluated, and it is doubtful whether such an evaluation would be of any use. It is more to the point to define a few qualitative categories that take in the limiting cases. In these categories, the probability of the formation or presence of a hazardous explosive mixture is described verbally [148]. The following zone definitions have been worked out in the course of European harmonization of explosion prevention and protection regulations [149]: 1) The explosive atmosphere (i.e., mixture under normal conditions) is present continuously, for an extended time, or frequently

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Table 11. Transformation rules Substance characteristics ∗

Real plant parameters Temperature

T f (flashpoint) LEL (lower explosion limit) UEL (upper explosion limit) LOC (limiting oxidator concentration) MIT (minimum ignition temperature) MIE (minimum ignition energy) pmax (maximum explosion pressure) (dp/dt)max (maximum rate of explosion pressure rise) MESG (maximum experimental safe gap)

− + − − − + −

Pressure

Dispersion

Volume

+ − + − − − + + −

− (+) − − − 0 + (−)

0 0 0 − 0 0 − (0)

∗ Laboratory data +, −, 0 indicate parallel, inverse, or neutral effects; i.e., increase in value of parameter results in increase, decrease, or no change in substance property compared with laboratory data.

(i.e., for most of the time in the process under consideration). This zone is, as a rule, limited to the interior of apparatus: Zone 0 for flammable gases, vapors, and mists; Zone 20 for flammable dusts. 2) The explosive atmosphere may be present occasionally during normal operation: Zone 1 for flammable gases, vapors, and mists; Zone 21 for flammable dusts. 3) The explosive atmosphere is present only rarely and for a short time only (e.g., in the case of an infrequent malfunction or upset): Zone 2 for flammable gases, vapors and mists; Zone 22 for flammable dusts. (Zone 22 also covers deposits of dust that can lead to an explosive atmosphere in the infrequent event of the dust becoming agitated in air.) 4) The “zone-free area” where the occurrence of explosive mixtures is not anticipated, even in infrequent upsets (nonhazardous area). In countries whose codes are based on United States practice, the classification differs slightly, but the rating arrived at is similar [150]. A mixture of flammable substances with air is explosive only if the concentration of flammable substances lies between the lower and upper explosion limits (see Sections 2.2.2 and 2.2.3 for definitions). With flammable liquids, if the processing temperature is above the flashpoint, an explosive mixture can be produced above the free liquid surface. If a flammable liquid is atomized (to form a mist or aerosol), the degree of dispersion can be sufficient to generate an explosion hazard, even at temperatures below the

lower explosion point. The probability of occurrence of explosive mixtures for a given process in a given plant is not constant for all points in the system: 1) Inside the apparatus, this likelihood depends on the substances present, the prevailing process conditions, and the possibility of upsets 2) Flammable gases, vapors, liquids, or dusts can enter the outside atmosphere at openings and leaks (release points) and form explosive mixtures with air (explosive atmosphere) The frequency of a release in the vicinity of apparatus depends on the procedures used; it is important to retain flammable substances in closed vessels, not just on grounds of explosion prevention, but also for reasons of exposure and environmental protection. Accordingly, portions of the plant containing flammable substances, as well as equipment and piping associated with these plant sections, are designed to be technically tight in relation to the mechanical, chemical, and thermal stresses anticipated under the intended operating conditions. The following examples illustrate the differences between various forms of release [151]: – Continuous sources (continuous release) include the permanent free surface of a flammable liquid, open to the atmosphere (open vessel or separator, immersion bath, conveyor belts carrying products moist with solvent) at or above the flashpoint. – Primary sources (occasional release) include expected leaks from flanges, shaft seals, apparatus and connections opened in normal

Plant and Process Safety operation, venting to the atmosphere during filling or as a result of overpressure, occasional upsets such as the failure of an unmonitored cooling system, sampling, and drainage. – Secondary sources (infrequent release) include cracks in apparatus walls or piping, defective gaskets in flange joints, defective shaft seals, and emergency pressure-relief devices (e.g., safety valves). Dilution in the air causes the concentration of flammable substances to decline with distance from the release point. The size of an explosion hazard region thus depends on the rate of release and the propagation conditions. The rate of release depends in turn on the geometry of the source and the substance and process parameters, all of which control the mechanism of release (gas release, flash evaporation, spray jet, pond evaporation). A central problem is to determine the maximum anticipated leak rate [151]. Catastrophic leaks, e.g., a large leak in a pipe or complete pipe rupture, do not fall within the scope of current explosion regulations.

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seal (leak size in the mm2 range) results in a release rate up to ca. 10 kg/h for gases or 100 kg/h for liquids, but in liquids it may be that only a fraction of this amount contributes to the explosion hazard. The main factors governing propagation of released substances are the density of the fluid, the momentum of the release, and the climatic conditions and topography of the surroundings (see Section 1.4.2). Ventilation conditions can contribute greatly to reducing explosion hazards: – Local ventilation. If the atmosphere is exhausted from the release point (e.g., around the margins of open vessels), flammable substances can be kept from spreading into the compartment. – Dilution. The concentration of flammable substances is lowered by dilution in air so that the concentration may fall below the lower explosion limit.

Figure 44. Ventilation model V 0 volume considered; G˙ max release rate of flammables (mass/time); V˙ 0 fresh airflow (volume/time); C = V˙ 0 /V 0 air change (1/time); f ventilation quality factor (1 – 5); Vz = f V˙ min /C hypothetical residual sphere of explosive mixture with radius rz on dilution by fresh airflow; V˙ min = G˙ max /k · LEL required minimum fresh airflow to dilute below LEL with safety factor k Figure 43. Hypothetical extent of an explosive cloud as a function of release rate of flammable substances for different ventilation conditions Enclosed areas: normal, C/f = 5; unfavorable, C/f = 0.2 Open-air situation: normal, C/f = 100; unfavorable, C/f = 4 Extent = 2rz (see Fig. 44 for definitions); assumed typical LEL ca. 50 g/m3

Other expected malfunctions generally lead to leaks up to ca. 0.01 – 1 kg/h. The failure of a

In buildings above ground with no special air intake or exhaust openings, an airflow of at least 1 h−1 is normally maintained. In typical chemical plants with natural ventilation, the temperature differences commonly present between different process apparatus units produce an airflow > 3 h−1 . Below ground, ventilation conditions are generally less favorable; most flamma-

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ble substances are denser than air, so they tend to collect in places such as drains or underfloor spaces. Artificial ventilation moves larger amounts of air, increasing dilution. Air movement can be designed in an artificial ventilation system, with allowance for explosive mixtures with density different from that of air. In an outdoor installation, air movement produces a much more rapid turnover. Figure 43, based on a simple ventilation model (Fig. 44) [148], [151], shows the characteristic size of an explosive cloud as a function of release rate for various airflows. For a given release rate, an explosive cloud in a closed building is a factor of 3 – 6 larger than in an outdoor plant. Initially, a hazardous area is created around any potential release source. Figure 45 shows the size of the hazardous areas around a stirred tank opened for charging in a ventilated room. The areas where an explosion hazard exists are identified in the ground plan. Where these areas overlap, the zone of higher probability prevails. For clarity, the hazardous areas have been smoothed to simple geometric forms containing the actual regions. The result is an area classification document. Figure 46 shows such a document for a simplified facility in a compartment with artificial ventilation. In practice, especially in smaller compartments, it is normal to extend, say, Zone 2 out to the boundaries of the room [151], [152].

Figure 45. Type, shape, and extent of hazardous areas for a stirred vessel opened during normal operation [151]

Ignition Hazards. In view of the various ways energy can be introduced into an explo-

sive system, 13 types of ignition sources have been defined [148]: Hot surfaces Flames, hot gases Mechanical sparks Electrical installations Transient currents Lightning Static electricity Electromagnetic waves (high frequency) Electromagnetic waves (3×1011 – 3×1015 Hz) Ionizing radiation Ultrasonics Adiabatic compression, shock waves Chemical reactions The conditions for such ignition sources to become effective can be generalized in only a few cases. The effectiveness of ignition sources involved in process operation or the working of apparatus depends strongly on the properties of the substances present (see Section 3.4.2.2). Hot surfaces may be present in normal operation (heated piping) or may arise through malfunctions (increased friction between moving parts). A large hot surface can initiate an explosion if its temperature exceeds the ignition temperature of the fuel that is present. With regard to small (cm2 scale) hot surfaces, studies have shown that these must be at temperatures much higher than the ignition temperature before they can initiate an explosion [153]. Flames and hot gases are among the most effective ignition sources. Visible sparks produced mechanically (friction, impact, and grinding) can be attributed to the combustion of metal particles heated to 1000 ◦ C on separation from solid materials. The effectiveness of such sparks as ignition sources depends on the material as well as the ignition temperature of the fuel. For most optimal mixtures of common flammable gases and vapors with air, sparks from friction against steel are effective ignition sources; however, these sparks are effective on dust – air mixtures only if the minimum ignition energy and ignition temperature are low. At relative speeds of ca. 1 m/s, visible frictional sparks no longer occur (Fig. 47). Impact sparks may be effective especially with the combination of light metal and non-stainless steel.

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Figure 46. Plan view of the area classification of an in-house chemical plant Zone 1 enclosed by boundary; Zone 2 by boundary.

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Figure 47. Generation of mechanical sparks Typical curve for steel construction material [153]

Electrical sparks and hot surfaces may occur as ignition sources in electrical installations. Explosive mixtures can be ignited, even at low voltages. Electrical equipment is considered a priori safe if none of the following values is exceeded: voltage 1.2 V, current 0.1 A, energy 20 µJ, power 25 mW. The use of electrical devices in hazardous areas is strictly (and internationally) regulated (Fig. 48), and as a rule such equipment is rated on the basis of type approval tests or comparable technical documentation, e.g., a manufacturer’s certificate [155]. Static electricity as an ignition source is a typical example of how a realistic assessment demands an analysis of the interaction of substance properties and process/plant parameters [156], [157]. Electrostatic discharges occur primarily with substances of very low electrical conductivity; discharges result from fast separation processes at interfaces. Solids with surface resistances < 109 Ω and liquids with conductivities > 108 S/m cannot be charged to dangerously high potentials, provided they are grounded. In the case of a liquid capable of holding a charge, charging (e.g., in stirring) can be significantly increased if the liquid is a multiphase system. Dusts often acquire an electrostatic charge during handling. Gases do not become charged, but solid or liquid contaminants or solid or liquid species formed by condensation in flowing gases can lead to an electrostatic charge. If, say, a pipe wall is sufficiently highly charged, the dielectric strength of the ambient air is exceeded and a gas discharge ensues. The effectiveness of the discharge as an ignition source

Figure 48. Explosion-proof electrical equipment, tested and certified, with the required explosion-resistant marking and data of testing house (with kind permission of the manufacturers STAHL and ABB CEAG) A) Limit switch, flame-resistant enclosure “d”, suitable for use in zone 1, explosion hazard due to products of Group II C and Temperature Class T6 (e.g., CS2 ); B) Electronic device for PCI, intrinsic safety “i(a)”, suitable for use even with zone 0, explosion hazard due to products of Group II C and Temperature Class T 6

Spark discharges take place when charged, conductive objects are brought close together and a discharge channel forms between these “electrodes.” Such discharges are effective ignition sources if the energy stored capacitively is greater than the minimum ignition energy of the mixture, as is commonly the case with large objects. Conductive substances (resistivity

Plant and Process Safety < 104 Ω · m), if isolated, represent an increased ignition hazard. Brush discharges can occur from charged nonconductive substances. The counterelectrode can be a conductive, grounded piece of equipment or a (grounded) person. The effectiveness of bruhs discharges as ignition sources depends on the charged area; they can ignite gaseous mixtures with a minimum ignition energy of up to ca. 3 mJ (dust – air mixtures cannot generally be ignited by brush discharges). At pointed electrodes, brush discharges give way to corona discharges; this weakest form of discharge, in its pure form, is not usually an effective ignition source. If a grounded, conductive body has a thin, chargeable coating, charge can be drained off, and the danger of ignition by brush discharge may be greatly reduced. If rapid charging takes place, as in the pneumatic conveying of dusts, the charge density in the coating can become so high that propagating brush discharges with high energy content can occur (Fig. 49).

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active systems, this can happen through spontaneous exothermic reactions; such systems are said to be auto-igniting or unstable. Self-heating does not occur in less reactive systems under normal operating conditions. If heat removal is blocked, e.g., by (oxidation-sensitive) solid deposits, the temperature can rise to the point that the conditions for ignition are attained. Other conceivable types of ignition source are less important in practice and cannot be discussed here. Ignition sources can be categorized in a similar way to explosion hazards (the rating of explosion prevention measures is discussed later in this section). Explosion Effects. Structures that may rupture in an explosion present a danger to the surroundings, owing to overpressure, underpressure, flames, and expelled fragments. In the field of explosion protection, design data for apparatus are compared with the anticipated maximum explosion pressure. If the possible explosion effects exceed the design limits of the apparatus, and product releases or flying debris from ruptured equipment cannot be ruled out, substantial consequences have to be reckoned with. In general, severe effects will not extend beyond a few tens of meters (see Section 1.4.3). Rating of Explosion Prevention Measures. In the chemical industry, a distinction is made between cases involving appreciable impacts (i.e., severe personal injury or environmental harm) and those where damage to plant and equipment is the most that can be expected. If no appreciable impacts are anticipated, then economic considerations alone (plant availability, product loss, damage) dictate that an analysis be done to determine how tolerable is the occurrence probability of an explosion. If appreciable impacts cannot be eliminated, the occurrence probability of events must be made low enough that such events are regarded as positively prevented. The following commonly accepted rating system should be applied to such cases:

Figure 49. Propagating brush discharge from a chargeable surface [154]

Chemical reactions, by producing heat, can raise the temperature of any substances present, and thus become ignition sources. In highly re-

A) Where an explosive atmosphere is anticipated most of the time (Zone 0 or Zone 20), the occurrence of an ignition source is never tolerable, not even under extreme fault conditions.

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B) Where an explosive atmosphere occurs occasionally (Zone 1 or Zone 21) the only ignition sources that are tolerable are those that result from a very infrequent malfunction, e.g., during run-up of a mill rotor (mechanical sparks) or a short- circuit (electrical sparks). C) Where an explosive atmosphere can occur only infrequently (Zone 2 or Zone 22) continuous ignition sources or those present for long periods or for most of the time, cannot be tolerated. Sources occurring only as a result of a (simple) upset, e.g., a hot surface due to abnormal overheating of a heat exchanger, can be tolerated. D) In an area where an explosive atmosphere cannot occur at any time, even under extreme upset conditions (i.e., nonhazardous area), the question of possible ignition sources is irrelevant. Even continuous ignition sources remain ineffective, and can be tolerated (e.g., the flame in an off-gas flare system). For these four categories, Table 12 uses keywords to summarize complementary measures to restrict the formation of explosive mixtures and to avoid ignition sources. The prerequisite for such a rating (categories B and C) is that the occurrence of an explosive atmosphere and the occurrence of an ignition source are mutually independent; i.e., both do not result from a common malfunction in the process or plant (common mode failure). The explosion rating of a system states what zone exists for the situation under consideration, and in what zone the ignition sources found would still be tolerable. This safety balance can take different forms, depending on the specific question posed: 1) The ignition source appears to be tolerable in the ascertained category of occurrence of explosive mixtures. The system properties are so favorable that the probability of occurrence of an explosion is sufficiently low. Such a system is said to be intrinsically safe. No further precautions are necessary [158]. 2) There is a gap between the category in which the ignition source would still be tolerable and the ascertained zone. Precautions are necessary, but they need only complement the existing intrinsic safety. Three cases can be observed:

a) For a reduction by one level (e.g., explosion hazard lowered from Zone 0 to Zone 1), a “simple” measure is sufficient b) For reduction by two levels (e.g., from Zone 0 to Zone 2), the measure must satisfy extensive requirements c) For a three-level reduction (e.g., from Zone 0 to “zone-free”), the reliability of the measure(s) taken must be enhanced even more, e.g., by redundancy

3.4.2.4. Explosion Risk Reduction There are two options for lowering the explosion risk: measures to prevent an explosion, or measures to limit its effects. If explosion prevention is chosen, measures are instituted to reduce the explosion hazard (preferable) or the ignition hazard; combinations of both approaches are possible. Limitation of the explosion hazard establishes a zone. If the limitation of the ignition danger meets the requirements for this zone, prevention of an explosion is insured. Reduction of Explosion Hazard. The avoidance or restriction of explosive mixtures should always be the first priority. Especially outside closed systems, this rule also applies on grounds of occupational safety and health, as well as environmental protection. A simple way is to replace flammable substances with nonflammable ones (e.g., a flammable solvent by water). Inside apparatus and equipment, the formation of explosive mixtures can be prevented by: 1) Keeping the concentrations of flammable substances in the gas phase outside the explosion range. Gas systems can be operated at overpressure, to prevent entry of air. Suitable purging is needed when such a system is placed in service [159]. In waste-gas streams, the total concentration of flammable components can be held above the upper explosive limit, e.g., by adding flammable gases (enrichment). Conversely, a stream can be diluted with air to reduce the concentration of flammable gases below the lower explosion limit. With liquids, the formation of

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Table 12. Keywords for explosion risk assessment Category

Occurrence of explosive mixtures

Keyword

Tolerance of ignition sources

Keyword

A B C D

continuously/for long periods/frequently occasionally in normal operation in rare situations and with short duration not even in very rare situations

permanently occasionally rarely never

not even in very rare situations not even in rare situations not in normal operation in normal operation

never rarely occasionally permanently

explosive mixtures can be avoided by keeping the temperature sufficiently far below the flashpoint. Substances with low flashpoints can often be replaced by substances whose flashpoints are a safe margin above the ambient and process temperatures. 2) Displacing or replacing air (or other gaseous oxidizing agent) with an inert gas until the oxidizer concentration drops below a casespecific critical level (see Section 2.2.2). A dryer for flammable dusts can be operated with combustion off-gases no longer having oxidizing properties, or a solvent tank can be purged with nitrogen. In the vicinity of process equipment, the occurrence of explosive mixtures can be largely controlled by: – Using closed systems to avoid release of flammable substances during normal operation. Examples include gas equilization lines for storage tanks, and the use of lock systems for tank filling and emptying. – Installing apparatus that has more integrity in terms of possible leaks. Examples include: welded piping to avoid flange joints; O-ring seals instead of stuffing boxes; and materials less susceptible to fracture, such as metal instead of glass. – “Good housekeeping,” i.e., the immediate removal of all flammable liquids and dusts released in the vicinity of plant equipment. – Creating suitable ventilation to insure the instant dilution of any gas mixtures. This practice can be established throughout the plant (e.g., open-air plant). Selective efforts can also be made to avoid formation of explosive mixtures in areas where sparks can occur, e.g., the installation of an open cage on stirrers and pumps insures that vapors escaping at the shaft seal are immediately diluted with air.

Reduction of Ignition Hazard. The ignition hazard from each type of ignition source listed in Section 3.4.2.3 can be reduced by specific practices. Hot surfaces can be avoided by matching the cooling capacity to the energy dissipated in the apparatus, and providing overtemperature protection. If necessary, insulation can be installed as a form of shielding. Possible ignition sources involving frictional heat or sparks can be dealt with by suitable design: – Use of adequately sized components with sufficiently long service lives – Avoidance of improper material combinations, use of materials with favorable dry friction properties – Monitoring of temperatures, and of lubrication and cooling systems – Maintenance of adequate clearance between moving parts Electrical ignition sources can be avoided by limiting electrical characteristics such as energy and power so as not to produce sparks capable of initiating an explosion (“intrinsic” safety, as in instrumentation circuits) or by preventing extreme conditions and overload in equipment that is free of ignition sources under normal conditions (“enhanced” safety, as in electric power distribution and lighting). The chief way of reducing ignition hazards due to electrostatic discharges is to use (grounded) conductive construction materials, and to ground conductive objects, liquids, and personal (resistance to ground < 106 Ω). The use of objects made of chargeable, nonconductive substances is generally permitted in Zone 2, but not in Zones 0 and 1. Ignition hazards can be avoided by limiting the dimensions of such objects, depending on the zone, and the ignition sensitivity of substances present; e.g., chargeable surfaces can be kept below 100 cm2 for substances classified in explosion groups II A and II B in Zone 1. Similarly, the thickness of

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chargeable coatings on conductive substrates can be restricted (e.g., < 2 mm for substances belonging to explosion groups II A and II B) and ignition thus avoided. When working with chargeable liquids, dangerous charging can be avoided by limitations on filling rates (e.g., < 7 m/s for hydrocarbons in a pipe with nominal diameter 50 mm) and by systematic under-level filling. Ignition hazards due to chemical reactions are avoided by excluding pyrophoric systems from hazardous areas, and maintaining safe process conditions (e.g., layer thickness, temperature, residence time, and thermal load in drying processes). Limitation of Explosion Effects. Design practices can insure that, if an explosion does take place in an apparatus, it does not have appreciable impacts. Maximum possible pressures generated by an explosion can be determined to acceptable accuracy by thermodynamic calculations, or measured experimentally. Explosion-pressure resistant design consists in designing apparatus to withstand the anticipated explosion pressure, generally less than 10 times the initial pressure for fuel – air mixtures. If the safety factor relative to yield strength is smaller, the term used is explosion pressure shock resistant design. In piping where detonations are possible, the standard design for PN 10 includes significant hidden reserves that are adequate for nominal diameters ≤ 200 mm; for nominal diameters > 200 mm, the piping should be designed to PN 16 at least. In interconnected vessels, an explosion in one portion can raise the pressure in the remainder; in this way, a (consequent) explosion can be induced, with a higher initial pressure. Explosion protection in such a case may entail decoupling of volumes. Pressure-relief devices (rupture disks or explosion valves) open as the explosion pressure rises and reaches their response pressure. They insure that the apparatus experiences only a reduced explosion pressure. Such systems are designed especially for protection against dust explosions, where the rate of pressure rise is generally fairly low [160], [153]. Naturally, pressure relief must not create new hazards (see Section 3.5.1). In an explosion suppression system, a complex of detectors respond at a very early stage

in the explosion, and an extinguishing agent (e.g., ammonium-phosphate-based powder) is injected on a scale of milliseconds in order to quench the explosion. The protected apparatus can then be designed for a lower pressure, e.g., 200 kPa. The amount of extinguishing agent needed depends on the violence with which the substance explodes, and the volume of apparatus to be protected (typically 50 kg per 25 m3 for St 1 dusts; see also Section 2.2.3). Isolation (“decoupling”) in the context of explosion protection means separation of an explosion-resistant section (where an explosion can take place without danger) and a nonexplosion-resistant section (where explosions must be positively prevented). A number of devices have been created to prevent flame-front breakthrough; these employ several principles: – Fast mechanical closure and lock-type features – Extinguishing of flames in narrow gaps – Use of flame-extinguishing agents (as in explosion suppression) – Flame arresting by high counterflow velocity – Flame arresting by liquid barriers Equipment selection must take into account substance properties; key aspects include sensitivity to ignition and propagation velocity (see Section 3.5.3). Narrow gaps are also employed in the design of explosion-proof equipment, e.g., fans, gas pumps for recycle systems, submerged pumps, and internal combustion engines [161]. This principle, when used in electrical equipment design, results in the rating “flame-proof enclosure.” 3.4.2.5. Legal Aspects In the domain of European legislation, EU directives set the framework for explosion protection at continental and national levels. The principal directives are ATEX 100 a [162], ATEX 118 a [163] and portions of the Machinery Directive [164]. The ATEX 100 a directive, aimed at manufacturers of “devices and protective systems intended for use in explosion hazard regions,” governs the quality and certification of such devices and systems. (For electrical equipment intended

Previous Page Plant and Process Safety for use in Zones 0 and 1, type testing and a certificate of conformity issued by an authorized agency have long been required [165], according to earlier harmonized European standards.) For Zone 2 equipment, the manufacturer’s declaration of conformity is sufficient. Similar rules apply for equipment to be used in areas where there is a dust explosion hazard (this equipment is still classified according to earlier zone definitions with just two types, i.e., Zone 10 and Zone 11, substantially comparable to the “new” zones 20 and 21 or zone 22, respectively). The ATEX 100 a directive, in contrast to its predecessor, now includes equipment and devices as well as protective systems “which are intended to halt incipient explosions immediately and/or to limit the effective range of explosion flames and explosion pressures,” and makes such equipment and systems subject to certification. The pertinent standards are being drawn up or are under revision by the European standards institutions CEN (e.g., committees TC 305 and TC 114) and CENELEC (e.g., TC 31), partly in collaboration with international standards bodies such as ISO and IEC. The ATEX 118 a directive (in preparation) applies to “the minimum requirements for improving the safety and health protection of workers potentially at risk from explosive atmospheres.” It is aimed at plant and process operation and represents a synthesis of earlier national regulations, reflecting the basic principles of explosion protection. 3.4.3. Safety Techniques Based on Process Control Safety objectives in chemical process plants can be achieved with systems based on process engineering and process control engineering (PCE). Technical or organizational measures, or a combination, can be employed. Alternative solutions that are equivalent from the safety standpoint can be found, so that the most appropriate solution from the process engineering standpoint can be selected. Safety concepts for specific safety objectives set down in technical regulations must take first priority. This section deals with PCE equipment for safety in process plants. The discussion is based

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on NAMUR Recommendation NE 31 [166], VDI/VDE 2180 [167], and provisional standard DIN V 19250 [168], with some points from the book Praxis der Sicherheitstechnik (Practical Safety Engineering), vol. 1, Anlagensicherung mit Mitteln der PLT-Technik (Safety Techniques Based on Process Control Engineering) [169]. The PCE safety measures suggested in this section are tailored to the risk component that would arise from a process plant in the absence of such measures. This way of assessing risks is only qualitative, and must be carried out by case. Other procedures leading to a comparable safety level are possible. As used in this section, “safety objective” refers to preventing injuries to persons, major environmental damage, and major equipment damage to property. Safety precautions for electrical installations and equipment, occupational safety practices, or measures taken to safeguard machinery are excluded. 3.4.3.1. Integration of PCE into the Safety Concept As a rule, safety requirements under DIN V 19250 and the measures required are more stringent the greater the risk that must be covered. The risk must be reduced at least to the acceptable limit (DIN VDE 31000 [170]; see Section 1.3) by non-PCE or PCE safety measures. Non-PCE safety measures and equipment may include elimination of ignition sources, isolation to control explosions, pressure-proof design, safety valves, interception spaces, and retention systems. PCE safety measures commonly reduce only a portion of the risk resulting from a unit. The risk on which the following discussion is based is just the portion to be covered by PCE safety measures. The remainder can be dealt with by non-PCE safety measures. Safety tasks allotted to PCE vary widely in character and significance. Similarly diverse are the safety requirements on PCE equipment and practice (technical; nontechnical, e.g., organizational). The use of PCE equipment for plant safety, the tasks assigned, and their performance are among the matters to be decided in a general

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safety review. The following are determined in the safety conference: – The safety objectives (personnel, environment, property) – The PCE equipment to be used for plant safety – The classification of PCE equipment by task: Operating systems Monitoring systems Safety systems Damage minimizing systems – The required technical and organizational measures (e.g., cycle of functional testing) The safety review is the basis for the taskdriven planning, construction, and operation of PCE safety systems, at a reasonable expense and with clearly defined functional scope. The selection of measures for maximum simplicity and direct impact generally leads to a safe and economic solution. PCE safety measures find use when other approaches are not applicable, inadequate, or uneconomic, given a comparable level of risk reduction. The economic comparison takes in not only one-time investment costs, but also recurring maintenance expenses. The requirements for PCE safety and monitoring systems are derived from this definition. Inadvertent triggering of the PCE safety and monitoring systems during any phase of the process must not lead to an unacceptable fault condition of the plant. 3.4.3.2. Classification of PCE Systems In chemical process plants, PCE systems are classified as operating systems, monitoring systems, and safety systems (Fig. 50), in relation to the ranges of a process variable: – Specified operation, subdivided into Normal operating range Admissible error range – Nonspecified operation – nonadmissible error range Region 1 of Figure 50 corresponds to the situation in which factors inherent in the process keep the process variable from reaching the nonadmissible error range. A monitoring system is sufficient. The process variable is brought into

the normal operating range by automatic or (after a status signal) manual action. Region 2 represents the case where the process variable exceeds the limit for the nonpermissible range. Because another protective device is present (safety valve, rupture disk, rapid-opening or rapid- closing valve), a PCE device connected ahead of it to signal or limit the increase in the process variable is classified under monitoring system. In region 3, the PCE device keeps the process variable from entering the nonpermissible range, and is therefore classed as a safety system. PCE Operating System. PCE operating systems are used in specified operation of the plant in its normal operating range. These devices implement the automation functions required for production: measurement and control of all variables relevant to operation, including related functions such as logging and report generation. High-level control algorithms, complex control sequences, automated recipe processing, and optimization strategies are increasingly employed. Many binary, digital, and analog signals have to be processed if all these tasks are to be performed. Because the functions of PCE operating systems are called on continuously or frequently in operation, these devices are subject to plausibility checking by plant personnel, so that failures and malfunctions can be detected immediately. PCE Monitoring System. When the plant is in specified operation, PCE monitoring systems responds to conditions in which one or more process variables are outside the normal operating range, but there is no safety reason to discontinue operation; i.e., these monitoring devices respond at the boundary between normal operating and admissible error ranges of process variables. Acceptable fault conditions of the plant are reported, in order to evoke heightened awareness or direct action by operating personnel; the monitoring system may even initiate action itself to bring process variables back into the normal operating range. Also included are PCE devices connected ahead of PCE or non-PCE safety systems in order, if possible, to prevent them from responding.

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Figure 50. Functions of PCE systems

PCE Safety Systems. In contrast to the functions of PCE operating and monitoring systems, PCE safety devices have the function of preventing nonadmissible error state of the plant. A PCE safety system is necessary if its absence would allow the plant to reach states that could lead directly to personal injury, major environmental damage, or major equipment damage. The task of a PCE safety system is usually to monitor a process safety variable and take one of the following actions if the variable goes outside the admissible error range: 1) Initiate a control process 2) Notify the operating personnel (who are always present) so that the proper action can be initiated ahead of time The functions of PCE safety systems always take priority over those of PCE operating and monitoring systems, and must be executed at a level near the process level with minimum complexity. The functions of PCE safety systems, in contrast to those of PCE operating equipment, are called on extremely seldom, because the probability of occurrence of the undesired event is low, and because PCE operating, monitoring, and safety systems are often in a staggered configuration (Fig. 50).

Because they are called on so infrequently, it may be desirable to allow sharing of PCE safety components, such as actuators, by PCE operating systems, to enhance availability and facilitate plausibility checking. Such common components must meet the requirements for PCE safety equipment. PCE safety systems employed for prevention of major equipment damage are designed solely on economic grounds and are not discussed here. PCE Damage Minimizing Systems. PCE damage minimizing systems come into action: when the plant is in nonspecified operation; when an undesired event occurs. They limit the impacts on persons and the environment, limiting the extent of harm under these extremely rare circumstances. If PCE equipment is used to detect the undesired event, what is monitored is not a process variable (e.g., pressure, temperature), but some other quantity such as the concentration of gases released in the atmosphere. Effectors put into action do not influence the process, but have to do with the threatened region outside the vessels, apparatus, and piping (e.g., initiation of a water curtain to prevent the spread of ammonia). PCE damage control devices are often combined with non-PCE damage minimizing devices and organizational measures.

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3.4.3.3. Requirements for PCE Equipment for Process Plant Safety and Design Principles PCE Monitoring Systems. No special requirements apply to PCE monitoring devices; they are treated like operating equipment. PCE Safety Systems. If the interdisciplinary safety review establishes the need to employ a PCE safety system, the following procedure is employed [168]: 1) Qualitative estimation of risk 2) Setting requirements for PCE safety system 3) Definition of technical and organizational measures In particular, the following steps are identified and documented: – – – –

Task statement, safety problem Function of PCE safety system Technical design (principle) Nature and frequency of scheduled function testing – Other organizational measures (e.g., scheduled maintenance; see Section 4.4) Two points must be considered in the selection of PCE safety devices: the minimum risk reduction aimed for; and the safety-related availability of PCE safety systems. Fundamental Requirement. PCE safety systems must be designed and operated in such a way that if one passive fault occurs, the protective function is still performed. In the design of PCE safety systems, the safety-relevant availability must therefore be chosen such that the risk is reduced to a residual level lower than the limiting risk, even if one passive fault occurs. The safety-relevant availability of PCE safety systems depends on: – The failure rate due to passive faults – The mean time to detect and remedy passive faults – The degree of redundancy of the PCE safety system The safety-relevant availability can thus be enhanced by the following practices, or a combination of them:

– Redundancy: the presence of more operable technical means than are required to fulfill the intended basic functions (e.g., special safety or reliability requirements) – Homogeneous redundancy: redundant design of a device or parts thereof such that the redundant channels are identically structured, and operate by identical physical processes – Heterogeneous redundancy or diversity: redundant design of a device or parts thereof such that the redundant channels operate by different physical processes or are differently structured – Fail-safe quality: ability of a safety device to hold a process safety variable in a safe state, or to modify the variable directly into another safe state, on the occurrence of certain faults in the safety device – Self-monitoring: a protective device is selfmonitoring if, aside from certain faults, it is so constructed that all other faults are detected by the self-monitoring, and thus the safe state is achieved To meet the requirements on PCE safety systems after risk estimation, distinction is made between high and low risk application [166]. Low Risk Application Requirement: One passive fault must be detected and remedied within a time interval in which no violation of specified operation is anticipated. Response: Single- channel PCE safety device with: 1) Short fault-detection time (e.g., high frequency of function testing, continuous plausibility checking); or 2) Low probability of passive faults of the PCE safety device. A reduced risk may be present if non-PCE safety measures of a technical or organizational nature are employed. High Risk Application Requirement: One passive fault must not impair the ability of the PCE safety device to carry out the safety function. Independently of the process behavior, such a fault must be detected and remedied within a time interval in which the simultaneous occurrence of a second, independent fault is not anticipated.

Plant and Process Safety Response: Redundancy of PCE safety device. In general, one-out of-two or (for simultaneously high production availability) two-out ofthree design combined with regular functional checking is sufficient. Diversity (e.g., in acquisition of measurements) does not automatically enhance safety over homogeneous redundancy. It is a supplementary measure for preventing possible systematic faults. Fail-safe or self-monitoring systems are equivalent to redundant devices. Within a PCE safety system, combinations of these measures are possible depending on the particular availability requirements; e.g., acquisition of measured values may be redundant, control fail-safe, and actuator single- channel. There are several ways to estimate risk. Quantitative evaluation is not possible for the chemical production plants under consideration here (see also Sections 1.3 and 3.3), because the wide variety of processes and the comparatively short lives of plants mean that adequate statistical material cannot be obtained. This holds equally for the safety relevant availability of PCE safety devices. Accordingly, the qualitative scheme of Table 13 is used in the design of PCE safety systems. It involves measurements of the risk to be dealt with as well as the safety-relevant availability of the PCE safety systems (the availability of the single- channel device when there is redundancy). Table 13. Grading of PCE safety systems by risk level and safetyrelated availability of PCE safety system (availability of singlechannel device when there is redundancy) Safety availability

Higher Lower

Risk to be covered Lower

Higher

I II

II II

Damage Minimizing Systems. Because the undesired event is expected to occur extremely seldom, damage minimizing systems are commonly single- channel units, and must be tested for function in regular intervals. Principles for Design and Construction of PCE Safety Systems. A number of important principles must be considered in the design and construction of PCE safety systems:

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– Proven, reliable hardware and installation methods must be employed. – The PCE safety device must be simple in construction. Fault effects (e.g., secondary or sequential faults in the PCE safety device) should, if possible, be limited by suitable barriers to fault propagation: high-impedance decoupling, short- circuit strength, galvanic isolation, etc. – Harmful effects due to environment and products, e.g., vibration, impact, static strain forces, thermal action, corrosion, contamination, wear, and electromagnetic effects (including those resulting from lightning, ripple content in power supply, grid malfunctions, grid noise, etc.), must be taken into account. – Fail-safe properties of equipment must be utilized (actuator with spring return to safe position, closed circuit to reset, etc.). – When operating and monitoring systems are shared with PCE safety systems: the safety function must take priority over other functions; and the shared elements must be rated as for the safety device. – The measurement of process safety variables, processing operations, and the implementation of the safety function must be done with accuracy and speed suited to the safety problem. – The measurement ranges of process safety variables must be chosen so as to insure adequate resolution. Limiting values must be far enough from range end points so that resolution is still guaranteed if measurement errors are within tolerance. – The correct setting of limiting values must be protected against inadvertent change. – As a rule, automatic reactivation after initiation of the safety function should be disabled. – Whenever possible, process safety variables should be selected such that they can be measured directly, simply, and by a proven method. Indirect derivation of process safety variables by combining measurement signals should be employed only when direct measurement is impossible or unreliable. – It is desirable to record process safety variables. – Analog process safety variables should be displayed, together with the limiting values, in the monitoring (control) room, or at local

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control panels. In this way, operating personnel can do plausibility checking, so that fault detection times are kept short and the limiting value setting can be checked easily. – The design of PCE safety systems must also take account of maintenance and start-up needs. Ease of inspection and accessibility of all components of the PCE safety system are important, even in the planning phase. – Manual overrides can be provided to allow inspection or repair of PCE safety systems during plant operation. – In isolated cases, redundant PCE safety systems should be examined to determine whether fire hazard or the possibility of mechanical damage necessitates: a split construction; or a protected and/or separate power supply and spatial separation of cable runs for both channels. Use of Programmable Electronic Systems (PES). Most PCE safety systems are hardwired. The main arguments for this are: 1) The setup must be simple and straightforward. 2) The PCE safety device is seldom, or never, modified while in service. 3) Only a few of the PCE devices in a process plant are safety devices. In special cases (more complex protective devices), if the use of PES is more economic, the following principles apply: 1) Certified systems can be used for both singlechannel and multichannel safety devices in the applications, and under the conditions, set forth in the test certificate. 2) Noncertified systems may be used for safety measures only as part of a multichannel safety device, and then for at most one channel. The other channels must be either hardwired (as required for PCE safety devices) or constructed with certified systems. The following minimum requirements apply to the noncertified system: – The system must be proven in former applications – Safety and nonsafety functions must be implemented separately whenever possible (in software and, if applicable, in hardware) to avoid influencing the safety section.

– Along with the standard system monitoring diagnostics (e.g., “watchdog” circuits), which return outputs to the safe state in malfunctions or failures, further system-specific requirements (fan monitoring, climate control, etc., as instructed by the manufacturer) are to be implemented. – Shutdowns may be implemented only via binary outputs. Resetting analog outputs to 0/4 mA is not acceptable as the sole shutdown action. – Reports of a safety function response must be distinct from reports of operating and monitoring equipment (e.g., different colors). – Application software must be created only by trained personnel, written in an easyto-follow form (structured, modular software), and easy to test. – The safety portion of the application program must be subjected to initial examination by the persons involved, and by a specialist, along with the planning documentation. – When major changes are made in software, even in the non-safety-relevant portion, a functional test of the safety-relevant functions should be performed. – Scheduled functional testing should also include the current software status (application software and firmware), including software documentation. Labeling. All important components of the PCE safety system must be labeled as such in the documentation, locally in the instrument and the control room. Testing Before Commissioning. Before the PCE safety system is first placed in service, a test must be performed to determine whether its design and function comply with the provisions of the safety review. At this time, the documentation of the PCE safety system and the test instructions for periodic testing must be available. Testing must be done in such a way that proper functioning is demonstrated in the interplay of all components. The first functional test must be documented in writing.

Plant and Process Safety 3.4.3.4. Operation of PCE Safety System Organizational Measures. Organizational measures are required for the operation of the PCE safety systems. They fall into four groups: Continuous monitoring (surveillance) Inspection (functional testing) Maintenance Repair Continuous Monitoring. Malfunctions of PCE safety systems must be detected by regularly observing process variables, and checking them for plausibility. Such observations are done by qualified personnel. Outwardly noticeable defects or damage to the PCE safety equipment or their installations are identified by regular visual inspections, and must be immediately repaired. Functional Testing. Functional tests are needed in order to reveal passive failure. Test instructions must be prepared, summarizing the nature and extent of periodic testing. These instructions must contain information on the nominal state and nominal performance of the safety device, along with a description of the properties and functions to be tested. This includes, in particular, statements of limiting values and measurement ranges, and other specified features to be inspected (such as actuating times for valves, time lags for trigger signals, and other properties important for the performance of the safety task). The testing procedure must be described in test instructions (e.g., checklists) that the checking personnel can understand. The limiting values must be documented in writing by the operations manager. The test cycle is set in the general safety review. Differences in availability may dictate that some parts of a protective device be tested more often than others. If no comparable experience exists, an appropriately short testing interval should be set at first. If the tests reveal sufficient safety-related availability, the test interval can be lengthened as the time in service increases. By analogy with pertinent technical guidelines, inspection of the entire PCE safety system (from sensor to actuator) must take place at least once a year. The operations manager is responsible for seeing that functional tests are performed.

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It is desirable that testing be done under conditions corresponding to the demand case and with the least possible modification of the PCE safety system. Whenever modifications are necessary to perform the check, special care should be taken to restore the PCE safety system to its proper state. If manual overrides are installed in multichannel PCE safety systems, only one channel at a time may be bridged. Care must be taken that neither safety nor availability is substantially impaired during the test. The method used should start with the assumption that faults may be present in the safety device; thus suitable (e.g., organizational) measures should be taken to insure that the plant remains in specified operation. Further details on testing, especially testing methods for PCE safety devices, are explained in VDI/VDE 2180, Sheet 4. In addition, tests should be performed after long shutdowns and repairs to the device. Maintenance. When service conditions are severe, or in the case of certain measurement techniques (e.g., process analytical instruments), scheduled maintenance may be necessary. Work schedules must be prepared summarizing the nature and extent of periodic maintenance, and the required permits from plant management. Maintenance operations are performed by qualified personnel in accordance with these work schedules, and the work must be documented. Repair. Repair of PCE safety devices must be performed without delay by qualified personnel whenever defects are found in the devices, and there is no alternative that will maintain the level of safety. Documentation. Testing, maintenance, and repair of PCE safety and damange control devices must be documented. In particular, documentation of functional tests must include at least the following information: – Identification of test objective – Results of test, with detailed information on faults corrected – Date of test – Signature of tester – Signature of operations manager The signatures of the tester (inspector) and operations manager confirm the release and ac-

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ceptance of the functional PCE safety or damage control device. The test report must be retained for at least five years so that performance of specified tests can be proved. Fault Analysis. To improve the reliability of protective devices, any faults discovered must be carefully analyzed; longitudinal documentation of test results reveals weak points. If the same causes are seen to produce faults again, this indicates a weak point and calls for improvement or testing at shorter intervals. Decommissioning, Restart, and Change of Limiting Values. Decommissioning and ShortTime Overrides. If a PCE safety device must be temporarily taken out of service or bypassed, e.g., during plant start-up, this must be documented in writing. Other technical or organizational measures must be put into effect to insure safety while the device is out of service or bypassed. The decommissioning or override status must be clearly identifiable. Appropriately labeled technical devices, e.g., key switches, can be provided when a safety device has to be bypassed repeatedly. A clear signal must be given to indicate bypassing; an automatic interlocking involving the bypassed PCE safety device must be released if necessary. There are three options for restart, and these must likewise be documented: 1) Restart without functional testing (e.g., after overriding for calibration) 2) Restart with partial or special testing (e.g., after replacement of a unit, cable, or data line, or after correction of a malfunction). A complete test (as under 3) should be performed as soon as possible. 3) Restart with full functional testing, as provided in the testing specification (e.g., after prolonged interruption of service) Restart after Initiation of Safety Function. Automatic restart after initiation of the safety function is to be “disabled,” i.e., restarting must be prevented when the plant is placed in operation again. This applies particularly when the process safety variable has already returned to the normal operating range. Until the plant state has not been checked portions of the plant affected by the initiation of PCE safety devices cannot be returned to service by manual action.

PCE devices used for restoring the PCE safety device to service are considered PCE operating equipment. Setting Limiting Values. Limiting values for PCE safety systems may be changed only on the instructions of the operations manager, who must also determine the extent to which this action affects the results of the safety review. A report must be prepared to document the change in limiting value (time of change, authority, person performing the change), and the correct setting of the new limiting value must be verified. 3.4.3.5. Summary In the technical safety concept, non-PCE and PCE safety measures are employed to reduce the risk arising from a chemical process plant to a value below the limiting risk. The risk to be countered is evaluated qualitatively, and the result is used in “grading” protective measures by scope. Technical measures are implemented as safety systems. The task of a PCE safety system is to reduce the risk at least to the limiting risk, possibly in cooperation with non-PCE safety systems or organizational measures. From the risk assessment and the associated requirements, safety requirements on PCE safety systems are first determined, and from these are derived the technical and nontechnical measures needed to fulfill the safety function. In a second step, graded measures are devised to meet the requirements. The requirements are more stringent the greater the risk to be covered by the PCE safety system.

3.5. Special Safety Equipment 3.5.1. Pressure-Relief Devices 3.5.1.1. Introduction Liquids and gases in chemical plants are frequently held in vessels that can be closed. A variety of mechanisms (Section 3.5.1.4) can cause overpressure to develop in such vessels. Initially, the vessel walls withstand this overpressure, but if it exceeds some limit, the “maximum allowable working pressure,” which depends on vessel

Plant and Process Safety shape, wall thickness, and material properties, the vessel bursts and the fluid contained is released, unless some sort of relief device limits the rising pressure to the maximum allowable working pressure.

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Classical devices that act automatically to prevent excessive pressure in pressurized compartments are the safety valve and the bursting disk (Figs. 51 and 52) [171–176]. Vents on tanks are operating equipment, not actual safety devices. Not until modern plants were built did their function change as artificial ventilating and gas-displacement systems were installed on tanks. 3.5.1.2. Safety Valves

Figure 51. Safety valve a) Valve disk; b) Flow area

Figure 52. Bursting disk a) Disk; b) Holder

A safety valve opens automatically when the set pressure p0 of the valve is exceeded. The valve disk is lifted by an amount h and makes the flow area available for pressure relief. In general, the full lift is attained within a pressure rise of no more than 10 % above the set pressure. The disk reseats when the pressure has dropped below the set pressure. In this way, the amount of vessel contents released is only that needed for instantaneous pressure limitation. The “opening characteristic” of a safety valve is described by the lift of the valve disk as a function of the pressure p in the vessel. This characteristic depends on the interaction of opening and closing forces on the disk. Opening forces are the pressure force on the disk and, after the disk begins to lift, the fluid dynamic forces exerted by the escaping fluid. These fluid dynamic forces are a function of the lift and strongly influenced by the geometry of the disk, which is designed to achieve a suitable opening characteristic. Closing forces are a constant weight or a lift-depending spring force and the back pressure on the valve disk, which is the sum of an external “superimposed” pressure and the pressure generated by the escaping fluid in the discharge line after the valve opens. Figure 53 illustrates the interaction of opening and closing forces. Where the opening forces, as a function of disk lift at constant pressure, increase more slowly than the spring force, the valve can open only when the pressure rises further. In this range (A – B in Fig. 53), it opens continuously. If the opening forces increase more rapidly than the closing force (to the right of B), the valve opens abruptly along the constant-pressure curve, up to the next equilibrium point D or the full lift limit C. Similarly, as the pressure declines, the valve reacts by closing

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continuously from C (or D) to E and on to F. It then closes abruptly (F – G) because the remaining opening forces at constant pressure and decreasing lift (to the left of F) are smaller than the closing force exerted by the spring. The pressure increase to achieve the full lift between A and C (or D) is the “opening pressure difference.” The pressure drop between A and G is called the “reseating pressure difference,” or “blowdown.”

Figure 53. Lift – force – pressure diagram See text for details

Specially designed safety valves have supplementary loads, which act as a function of pressure to reinforce the closing force until the set pressure is reached, or reinforce the opening force when the set pressure is exceeded. These loads can completely replace weights or springs. In the bellows-type safety valve, a bellows between the disk and housing largely eliminates the effect of back pressure on the discharge side. Safety valves are classified by opening characteristic and loading principle. The full-lift safety valve opens almost abruptly to full lift within a pressure rise of ≤5 % above the set pressure, thereby attaining maximum discharge capacity very quickly. It also closes abruptly. The proportional safety valve, in contrast, opens continuously (though not necessarily linearly) as the pressure increases above the set pressure; it also closes gradually. There are no special requirements on the opening characteristic of the standard safety valve, and its performance falls

between those of the full-lift and proportional types. With weight-loaded and spring-loaded safety valves, the set pressure must be significantly higher than the normal working pressure in the protected compartment so the valve remains tight during normal operation. The working and set pressures can be much closer together in supplementary loaded and in controlled safety valves, because the (controlled) closing and opening forces at the setpoint can change almost discontinuously. Pilot-operated safety valves have to be classed as controlled safety valves because of their construction and functional principles. The full-lift safety valve is preferred when large volumes of gas have to be discharged suddenly. The proportional type adapts better to a fluctuating mass flow, and is well suited to low volume flow rates such as those resulting from thermal expansion, and also to variable flows of mostly incompressible fluids (liquids). Supplementary loaded and controlled safety valves are used chiefly in power plants, where large mass flow rates have to be handled and the generally constant service conditions make it possible to run the plant continuously at the limit of its design pressure. Because it permits good sealing of the interior space against the surroundings, the bellows valve is preferred with fluids that must never be released to the environment. It is suitable where back pressure in the discharge line is high or fluctuating, because it restricts the backpressure effect. The mechanical reliability and discharge capacity of safety valves are commonly determined by type testing. The discharge capacity is characterized by the certified coefficient of discharge αw , which holds only for a range of set pressures established by type testing. Normally αw = 0.6 –0.8 for full-lift valves and gaseous discharge fluids; for proportional valves and liquids αw = 0.1 – 0.35. Against the actual discharge performance, αw includes a safety margin of ca. 10 %. Unauthorized changes made on a safety valve, affect the basis of the type testing and the discharge capacity is no longer certified. In particular, the installation of an incorrect spring can keep the valve from fully opening, even though the set pressure is correct, so that the required discharge capacity is not available. For proper functioning, a safety valve must have the correct set pressure, and must be able

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to discharge the maximum possible mass flow on opening. Spring-loaded full-lift valves tend to go into self-excited oscillation (chattering) when there are abrupt pressure changes close to the set pressure; the valve opens and closes at short intervals, and the discharge capacity is markedly reduced. Chattering of the valve disk damages the seat and possibly the entire valve, and tightness is impaired. To prevent chattering, the pressure drop in the feed line during discharging must not exceed 3 % of the set pressure. Chattering can also occur if the safety valve is too large for the mass flow rate it is to discharge. If chattering cannot be eliminated, the safety valve can be equipped with a vibration damper. A design modification to limit the full lift is also possible in oversized valves. The discharge capacity is also reduced if the back pressure in the discharge line is too high. It should therefore not exceed 15 % of the set pressure, or 50 % for bellows safety valves.

rapidly rising pressures and large volume flow rates. Bursting disks are less sensitive than safety valves to gumming and polymerizing media. They have the disadvantage that they do not reclose, and if a disk fails a large quantity of product can be released. Depending on the design, bursting disks are often sensitive to underpressure in the protected compartment and back pressure on the discharge side. Membranes are employed chiefly for large flow areas and low overpressures. Panels are used when the corrosiveness or other properties of the medium dictate the use of chemically resistant materials, e.g., graphite, which are not appropriate for the fabrication of thin shells.

3.5.1.3. Bursting Disks

1) 2) 3) 4) 5)

A bursting disk is a closure element that fails when its “bursting pressure” is reached, leaving open the flow area of a discharge line. The design is governed by the construction materials. Disks can be classified as follows: Membrane-type disks are loaded by membrane stresses. They fail when the yield point is reached. Brittle plates are loaded by bending stresses. They fracture when the breaking stress is reached. The yield point and breaking strength are functions of temperature, so the bursting pressure varies with service temperature. Shells of revolution fail by buckling when a stability limit, governed by the modulus of elasticity, is reached (reverse-acting disks). The stability limit is less temperature dependent than the yield point and breaking strength. The bursting of the shell after buckling can be aided by special devices, e.g., a fixed knife in the form of a cross, located behind the bursting disk. The serviceability of bursting disks is determined by bursting-pressure tests on samples. A bursting disk causes only a slight increase in the flow resistance of the discharge line; it opens spontaneously, and frees a larger flow area than does a safety valve of the same nominal diameter. It is thus particularly well suited to

3.5.1.4. Sizing of Safety Valves The protection of a plant section with a pressurerelief device such as a safety valve involves five steps: Definition of scenario Calculation of relief cross section Valve selection Determination of mass flow capacity Impact analysis

The first step in defining the scenario is a systematic safety analysis of the plant section under consideration, e.g., in a HAZOP study. These scenario can be divided into two groups: 1) An energy or material flow is supplied from outside (physical action) 2) The energy or material flow is generated in the system (chemical reaction) A simple systems analysis makes it possible to break the first group down further: 1) The system is heated. Pressure builds up as a result of thermal expansion, rising vapor pressure, or evolution of dissolved substances. The mass flow rate to be relieved depends on the maximum quantity of heat per unit time that can be supplied to the system. 2) A substance flows from a higher-pressure system into the system under consideration. The mass flow rate to be relieved depends on the rate at which mass can be delivered to the system.

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3) A flow of matter is supplied to the system by a pressurizing device, e.g., a pump. The rate at which mass must be discharged depends on the delivery capacity of the pressurizer. The second group includes systems in which an unacceptable rise in pressure would result if a reaction went to completion. The pressure rise can result from the increase in vapor pressure as the temperature increases, and/or from the liberation of gas, e.g., by decomposition of the condensed phase. Possible causes and mechanisms of such reactions are described further in Section 2.3, while Section 2.3.2 lists methods by which the progress of an exothermic reaction can be determined experimentally. As a rule, studies of this kind are essential in the design of a pressure-relief device. Useful findings include the rate of pressure rise and the rate of heat production. This forms the basis for determining the rate of mass discharge, and hence for designing the safety valve, with the help of fundamental hydrodynamic equations [171], [172], [177]. If the required minimum cross section of the valve is known, a valve is selected such that its smallest cross section is larger than the required value. The cross section gives the maximum rate of discharge through the valve, which makes it possible to calculate the pressure drops on the high-pressure and discharge sides of the valve. These are compared with the limits discussed in Section 3.5.1.2, which must not be exceeded. It must also be determined whether twophase discharge can occur. Correlations useful for this purpose have been published by DIERS ( Design Institute of Emergency Relief Systems) [178]. Figure 54 illustrates what happens in a pressure-relief event with two-phase flow [179]. When the pressure in the gas space of a vessel is relieved, single-phase discharge occurs first; the pressure drops rapidly. The liquid cannot immediately react to this change of state with a corresponding drop in temperature, and becomes superheated. To restore thermodynamic equilibrium, intensified formation of vapor bubbles begins (after a boiling delay time). The growing and ascending bubbles both displace the liquid and entrain it, setting the contents of the vessel in upward motion. If the free surface of the liquid reaches the discharge opening, a two-phase flow results.

Figure 54. Pressure variation and behavior of vessel contents when pressure on the vapor space above a singlecomponent mixture is relieved

This process depends on both the properties of the reactor contents (viscosity, surface, tension, foaming tendency) and the reactor geometry, fill level, and discharge capacity of the valve [178]. Thus low surface tension and high viscosity promote two-phase flow, while low fill level and a pressure-relief opening small in comparison with the vessel cross section help suppress two-phase flow.

Figure 55. Retention system with liquid separator and direct condenser a) Reactor; b) Liquid separator with retention volume and cooling jacket; c) Direct- contact condenser with dip tube and cooling jacket

The development of two-phase flow has the consequences that more liquid is removed, more total mass is removed, and the mass flow rate is increased substantially. On the other hand, the gas/vapor mass flow rate that has to be discharged for safe pressure relief is smaller than in single-phase flow. This means that a valve designed for single-phase discharge can be un-

Plant and Process Safety dersized for safe pressure relief under two-phase discharge conditions, and a new test must be performed for this case (the pressure drops on the pressure and discharge sides of the valve must also be remeasured). The occurrence of two-phase flow can result in markedly higher inertial forces and momentum transfer. An iterative procedure is usually followed until all constraints are met. A thorough discussion of models to describe two-phase flow has appeared in [180]. 3.5.2. Blowdown Systems 3.5.2.1. Procedure The substances discharged for pressure relief must be handled in such a way that they do not cause personal injury or environmental damage [181]. In general, it is necessary first to determine [178] whether the discharge flow is singlephase, gas/vapor, or two-phase. In the two-phase case, it is expedient to separate the liquid phase [182] and retain it in order to suppress any further reactions. When separation is accomplished in separate vessels, e.g., impingement separators or cyclones, the reaction is stopped by a large quantity of cold water, containing a reaction inhibitor if necessary. On grounds of space and cost, it is desirable to use one apparatus for both functions, if possible (Fig. 55) [183]. The safe removal of substances in gas/vapor form must also be insured. Three approaches to gas/vapor handling are possible, depending on the total quantity of the substances discharged and their properties [184]: Discharge into the atmosphere Retention by treatment systems Retention in closed recovery systems Substances may be released only if it has been demonstrated for the particular case, e.g., by a propagation calculation [185], that the impact of the released substances on humans and the environment stays within acceptable limits, otherwise, retention in treatment systems is necessary. Industrial treatment systems include thermal cleanup systems, flare systems, scrubbers, and dip-tube and other condensation units.

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When systems of these kinds are employed, the first point to check is that the risk potential is reduced to an adequate degree, since both untreated residues and newly formed degradation products are released into the environment. The second point is whether the risk potential is merely shifted; a scrubber may clean up the exhaust air, but then generate highly contaminated wastewater. Finally, it must be demonstrated that the disposal systems are not harmed by the reaction forces and momentum transfer occurring during discharge. If no satisfactory answers are obtained to these three questions, the substances must be delivered to a closed retention system [183], [186]. In the simplest case, they can be captured in a static vessel, or in systems that can be blown up, but devices of this type may quickly run into practical limits because the volumes of gas to be intercepted are often very large. One way to minimize the volume needed is to dissolve or condense the discharged substances, or to fix them chemically. Often, this can be done by introducing the gases or vapors into a waterfilled vessel [187]. The degree of condensation or absorption can be enhanced, and the necessary volume minimized, by spraying the substances into the vessel through nozzles or distributor pipes [188], [189]. Other variables affecting the quality of condensation or absorption are the temperature of the liquid in the vessel, the fill level, and the pressure in the retention system, which should be as high as possible, consistent with the requirements of the pressure-relief device (Fig. 55). 3.5.2.2. Alternatives The safe design of a pressure-relief system combined with suitable facilities for handling the discharge can be a time- consuming iterative process. When costly retention or treatment units appear necessary, economic and environmental factors dictate that alternative pressure-relief arrangements must also be evaluated. For example, pressure relief can be dispensed with if the vessel is constructed so that neither physical nor chemical processes can generate a pressure higher than its maximum allowable working pressure.

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In addition, no pressure-relief device is needed if appropriate instrumentation and control [190] (Section 3.4.3) can insure that an unacceptable pressure rise cannot occur. These features must reliably control the reaction, or provision must be made to inject an emergency suppressor or emergency coolant to stop the reaction immediately [191]. Which solution is preferable can be decided only when the specific constraints of the application are known. 3.5.3. Flame Arresters and Explosion Barriers [192–201]

Figure 56. Cutaway drawing of a crimped-ribbon-type flame arrester [201]

3.5.3.1. Introduction The initiation and propagation of explosions in (chemical) plants where explosive mixtures are present can lead to significant harm to persons, the environment, and property. One feasible, and often economic, way to control and prevent such explosions in closed systems is to separate parts of the plant with flame arresters. These block the propagation of an explosion, and help limit and control its impacts. This type of isolation can be applied to mixtures of flammable gases/vapors and oxidants as well as explosive dust – oxidant mixtures.

Blocking by Liquids. There are two main types of liquid-type flame arresters, the liquid seal in the form of a siphon inserted in a liquid- conveying pipe, and the hydraulic arrester in which a stream of gas passes through a dip tube and is divided into noncoalescing bubbles. The siphon, by virtue of the configuration, is continuously filled with the conveyed medium (Fig. 57). Water is the most common barrier liquid in the hydraulic flame arrester.

3.5.3.2. Flame Arresters for Mixtures of Vapors (Gases) and Air In normal plant operation, flame arresters permit the free passage of explosive mixtures. If ignition occurs, however, they block the passage of the flame, preventing the ignition from being transmitted to the protected side. The extinguishing action in present-day flame arresters, regardless of design, is based on one or more of three mechanisms: Flame Quenching in Narrow Channels. Through intimate contact with the cold walls of a filter element comprising many narrow channels, heat and free radicals are withdrawn from the combustion process, and the flame is extinguished. This mechanism is employed in dry flame screens (e.g., the crimped-ribbon arrester shown in Fig. 56).

Figure 57. Wet-type flame arrester (siphon arrester) [201]

Dynamic Blocking. The geometry of the flame arrester includes a narrow clearance, in which the flow velocity is always higher than the

Plant and Process Safety turbulent flame velocity (combustion velocity) in the explosive mixture. Upstream propagation of the reaction is thus impossible. Use of Flame Arresters in Practice. The selection of an arrester is governed by a variety of constraints. Only exact knowledge of the system to be protected can provide the basis for choosing suitable and economical units. First, it must be established what parts of the plant are to be safeguarded against what effects, i.e., What are the potential ignition sources? Is ignition due to external or internal sources? This is crucial in fixing the location, installation direction, and layout for the arrester. Two applications must be considered for the layout: deflagration and detonation (i.e., coupled combustion and shock waves with rapid pressure rise); the type of protection chosen will depend on which installation is used. Under some circumstances, a run of a few meters of pipe is enough to allow detonation, so that detonation-type arresters are commonly selected for installation in piping. Deflagration-type arresters can be employed where run-up distances are very short, e.g., where tank vents to the atmosphere are to be protected. In a second step, the construction of the flame arrester is considered. The properties of the medium being transported are important: contamination with solids, solidification temperatures (heatability, frost protection), and condensation or solidification of impurities from the stream. Quenching-type flame arresters in some services may become contaminated and suffer corrosion; these devices need regular inspection and cleaning. It makes sense to install several such flame arresters in parallel so that one at a time can be bypassed. If even a few of the passages in an arrester become enlarged, or if the flame filter is incorrectly inserted in the holder, the unit may fail. In the case of pipework not continuously filled with liquid, the wet types of arrester are desirable on grounds of maintenance and availability; these can be obtained in a variety of designs.

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Hydraulic arresters are often used to protect contaminated exhaust air streams. Safe operation depends on correct sizing and design, as well as the use of suitable instrumentation for continuous monitoring of important parameters. An automatic alarm should be provided, together with automatic initiation of emergency response when limiting values are violated. Safety-relevant parameters of this type of arrester include a sufficiently high liquid level over the end of the dip tube, limitation of maximum air flow rate, monitoring of liquid temperature (frost protection), and monitoring of temperature in the gas space of the arrester. As a rule, buildup of contaminants and condensing species from the air stream necessitates continuous cycling of liquid in and out of the arrester. To prevent the formation of a continuous explosion channel by the bursting of explosive gas bubbles in the ascending cloud of bubbles – which would allow an ignition to get through the arrester – it is important to have a hydraulic arrester designed for maximum air flow rate and minimum dip depth. For the correct design of the flame arrester, it is essential to know the maximum experimentally safe gap for the substance being handled. This parameter serves as the basis for classifying substances into Explosion Groups II A, II B, and II C. In Germany, for cases covered by the Flammable Liquids Regulation (VbF), flame arresters must be type-tested (type-approved) for the appropriate explosion group, and for one of the two layouts (deflagration or detonation). Type approval tests are done with appropriate mixtures of fuel and air. Any change in oxidant (e.g., oxygen or chlorine instead of air) and any increase in oxygen partial pressure can lead to a change in performance, which can be assessed only after further experimental tests. Where the VbF does not apply, Germany does not yet have a type-testing requirement. The quenching of flame in an arrester presupposes adequate removal of heat from the reaction or flame zone. In some arrester designs, handling flowing explosive mixtures, heat removal may be hindered so that a flame can stabilize on the arrester. The filter element and piping may be heated to the ignition point of the flowing mixture, and the flame can be propagated through the arrester. As a rule, the only way to insure adequate heat removal is by using end-of-line ar-

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resters. Experimental demonstration of the “endurance burning safety” (ability to prevent flame transmission for an unlimited time) is required. Any flame arrester that does not have this property must be expected to fail after stabilizing the flame for a certain time unless countermeasures are taken. Therefore continuous temperature monitoring (Fig. 58) is required, along with initiation of emergency functions (e.g., cutting off the gas stream).

of explosive atmospheres (classified in zones) and potential ignition sources. Even when flame arresters are used, the zone and the ignition probability must still be analyzed. As a result of this interplay of occurrence probabilities redundancy of flame arresters is necessary. The important factor is the number of independent protective measures (Table 14). Table 14. Number of measures to prevent flame transmission Ignition sources anticipated in plant

Number of precautionary measures Zone 0 Zone 1 Zone 2

Normal operation (e.g., burner flame) 3 Failures that are likely to occur 2 Failures that are not likely to occur 1

2 1 0

1 0 0

German regulations mandate the use of flame arresters to protect non-pressure-proof plant sections in storage facilities and filling points for flammable liquids. Further details can be found in the Flammable Liquids Regulation [196] and the Engineering Code on Flammable Liquids (TRbF). 3.5.3.3. Flame Arresters for Dust – Air Mixtures

Figure 58. Flame arrester with temperature monitoring and steam injection [201] a) Electronics; b) Solenoid valve

The use of a flame arrester leads to quenching of the flame at the arrester. However, the pressure generated in the explosion must be safely accomodated through design features (pressureproof design) in conjunction with adequate provisions for pressure relief. Safety Concepts. The use of flame arresters to separate plant sections that are not proof against explosion pressure shocks or explosion pressures is just one of a range of concepts applicable to explosion protection. As is usual in explosion protection, all protective actions are based on risk analyses, which call for countermeasures with appropriate availability, depending on the probability of occurrence

The flame arresters that have been discussed are not designed for use in plant areas and piping where there is a danger of dust explosion. Other approaches must therefore be employed in such cases. Among the means used for isolation are star-wheel feeders, active barriers with extinguishing media, rapid- closing valves (explosion valves) and doors, and explosion vents. These methods are described more fully in VDI Guideline 2263 [200]. The use of star-wheel feeders requires prior testing to determine the ability to prevent flame propagation, as well as to withstand the explosion pressure. In order that no smoldering product is transported into downstream parts of the plant, the star wheel must be automatically cut off (e.g., with a pressure monitor) in the event of an explosion. When active barriers and rapid- closing doors are used, injection of extinguishing agent into the flame front or closing of the doors is initiated in a few milliseconds by a flame detector. Specialists must perform tests to determine

Plant and Process Safety the effectiveness (speed from detection to response) in the real plant geometry, and the ability to withstand pressure. Explosion valves, acting without auxiliary sources of energy, close at the higher flow velocities occurring during explosive pressure relief, automatically blocking off pipe cross sections, and preventing the explosion from propagating. When explosion vents are used, the formation of long flames and the release of large amounts of product, possibly of environmental relevance, must be anticipated.

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go to practices that safeguard the employee or mitigate risks, regardless of employee behavior. Despite many technical measures, ranging up to “enclosed” plants, contact of employees with hazardous substances cannot be completely prevented. Normal operation entails filling, transferring, and emptying media, taking samples, and cleaning and inspecting vessels. In abnormal operation, for example, substances can get into work areas by leakage and when pressurerelief devices operate. Selected practices for the safe handling of chemicals are discussed in what follows. These should prevent the release of hazardous substances and threats to employees and the environment.

4.1. Safe Handling of Chemicals 4.1.1. Introduction

4.1.2. Normal Operation

Substances and formulations in the form of raw materials, intermediates, and finished products are employed in many branches of industry. Some substances and formulations have dangerous properties (see also Sections 2.1 and 2.2); these are referred to as hazardous substances. When hazardous substances are present, safety precautions must be taken so that concentrations stay below the limiting values in the workplace air, so that employees do not come into direct contact with hazardous substances, and so that fires and explosions do not occur. Measures must be taken to prevent upsets in operation. If upsets come about anyway, the threat to employees must be limited. Systematic safety analyses should be performed to determine the measures required [202]. Hazard sources and the conditions for their activation are systematically and comprehensively examined so that the hazard potentials can be ascertained. The knowledge gained makes it possible to identify the necessary measures case by case. Points to be considered in establishing safety practices include provisions of legislation, engineering codes, industry and factory standards, and safety information relating to specific processes. Technical measures, including the selection of processes with the lowest possible hazard potential, should be favored over administrative measures whenever possible. Preference should

Transferring, Filling, and Emptying. In the chemical industry, the handling of liquids is part of daily routine. Liquids must be drained from tanks, drums, and reactors, or masterbatch tanks and reactors must be filled with liquids. It is often necessary to meter exact quantities in specified times [203]. Preventing Hazardous Concentrations of Substances in the Gas Phase. Some ways to prevent hazardous concentrations are: 1) Transferring liquids in systems that are enclosed or can be equalized (e.g., fixed piping from storage tank to plant); to be employed especially when transferring carcinogenic, highly toxic, and toxic liquids 2) Using gas-displacement devices for pumping 3) Extracting vapors at the point of escape 4) Providing adequate ventilation at the workplace Avoiding Escape of Liquids. While being transferred, liquids can escape by splashing, dripping, spilling, or overflowing. Some ways to prevent the escape of liquids are: 1) Restricting transfer operations 2) Selecting and properly installing suitable filling devices 3) Using appropriate designs for piping disconnections

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4) Employing process-specific practices as well as instrumentation and control, examples being: Emptying into calibrated vessels Emptying through volumetric meters Running drain lines to installed scales Avoiding Electrostatic Charging. Liquids can become electrostatically charged while flowing next to walls or when being sprayed, filtered, or stirred. Charges transported along with the liquid can charge vessels to the extent that static electricity is discharged at the liquid surface; this can result in the ignition of flammable gas – air or vapor – air mixtures. The amount of charging depends strongly on the conductivity, the flow velocity, and the quantity of liquid transported. A high level of charging must always be expected when interfaces are present in nonconductive liquids, e.g., when immiscible liquids are transported, or when liquid mists or vapors are generated. Liquids with conductivities > 104 pS/m do not acquire a charge while flowing. Avoiding Mix-up of Liquids and of Containers. Dangerous reactions may occur if substances are mistaken for others. Mix-up can be avoided by: 1) Design practices; connections for vessels containing different liquids can differ in diameter from one liquid to another or be made with incompatible flange constructions (grooves, lugs) 2) Unambiguous labeling; vessels, piping, and connections for individual liquids can be labeled, e.g., with substance names, colors, or numbers 3) Identity checking before liquids are drained Avoiding, Restricting, or Assisting Manual Transport Operations. Ways of avoiding transport operations include: 1) Fixed piping between storage tanks and consumer (reactor, tap); the preconditions for this is that pipe runs be as short as possible, i.e., all parts of the plant are arranged compactly 2) Mechanization of operations Ways of restricting transport operations include:

1) Purchase of liquids in larger vessels, e.g., 1000-L containers instead of 200-L drums 2) Suitable placement of vessels and other measures to minimize the need for transfers If manual transfers are unavoidable, plantdesigned equipment can make them easier and thus less dangerous. 4.1.3. Sampling Samples are required for the following reasons: 1) Determining identity and quality of feedstocks 2) Monitoring and controlling chemical reactions and other processes in the plant 3) Assessing quality of intermediate and end products Employees conducting sampling operation may be endangered by hazardous substances and the sampling conditions [204]. Safe Design of Sampling Devices. The following points must be considered in the design of sampling stations: 1) The sample should be withdrawn at a point in the plant where the pressure and temperature are as low as possible. 2) The cross-sectional area of the sampling device must be kept as small as possible. 3) Sampling must be designed such that large quantities of hazardous substances cannot escape from a plant or part of a plant, e.g., because of a malfunctioning or damaged sampling device. 4) The inevitable pre-sample liquid must be returned to the closed system, as far as possible. Two points merit special attention: the sampling point should be designed such that there is no pre-sample flow, or the unavoidable pre-sample flow is safely returned to the processing system. When the sample is transferred into the sample container, splashing, vaporization, dripping, overflowing, and escape of hot liquids must be prevented. It may be necessary to take steps to prevent static charging, such as:

Next Page Plant and Process Safety If possible, avoid sampling from stirred vessels through the manhole (especially if the vessel contains flammable substances). Otherwise, turn off stirrer, sprayers, and other charge-producing components before sampling and wait for a “calming” time. Do not allow to spray flammable, chargeable liquids into open sample containers, especially if partial vaporization of the liquid is anticipated. If possible, conduct sampling of flammable, chargable liquids in a closed system only, e.g., in a nitrogen-purged sample vessel. Limit the outflow velocity of the liquid; ruleof-thumb ca. 1 m/s.

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2) Blow dip tube (a) empty through fitting (d) (e.g., with nitrogen) 3) Close fitting (d) 4) Apply vacuum to fitting (e) in order to draw product into container (f) 5) Close ball valve (b) 6) Close fitting (e) 7) Apply nitrogen pressure through fitting (c) to transfer sample into suitable vessel 8) Close fitting (c) Nonreturn valves in the vacuum line prevent the working fluids in the vacuum generator from getting into the product vessel. 4.1.4. Cleaning of Vessels Cleaning methods should be designed so that worker health is not endangered by residues, contaminants, or cleaning agents [205]. Before cleaning is begun, the status of the vessels to be cleaned must be determined. It must be known what substances, with what hazardous properties, are or have been present (solid deposits, liquids, vapors, gases). Hazards can arise from: 1) Reaction of the cleaning liquid with residues and contaminants 2) Flammable cleaning agents 3) Static charging when liquids are sprayed under high pressure 4) Materials unstable with respect to the substances, pressures, temperatures, and mechanical stresses present

Figure 59. Sampling with vacuum a) Dip tube; b) Stopcock; c), d), e) Fittings; f ) Container

Sampling with Vacuum. Figure 59 shows a setup for taking samples with no pre-sample flow. The procedure is shown by the numbers in the diagram: 1) Open ball valve (b)

The proper precautions must be ascertained. Cleaning must be performed by trained personnel under expert supervision, in accord with written information. The cleaning method must be described in an instruction manual. Whenever possible, closed cleaning systems should be used. 4.1.5. Safety Practices when Working with Hazardous Substances Under Abnormal Operation Safety practices must prevent employees being exposed to excessive concentrations of hazardous substances, even in abnormal or upset operation. Pressure vessels and pressure piping

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must be designed, constructed, and operated in such a way that personnel and third parties are not endangered. The action of pressure-relief devices must not threaten employees or third parties. The quantity and nature of the substances present may make it necessary to conduct releases to the atmosphere or to provide supplementary features, e.g., scrubbers, flare systems, or blowdown tanks. Leaks represent a further source of danger to personnel and the environment. Seal systems installed on shafts of pumps, drives, mixers, stirrers, etc. must be selected in accord with the media present and the regulatory requirements [206]. Along with proper design, the effectiveness of the seal systems must be tested regularly, and any leaks must be detected. When media are under pressure, splash deflectors must be used. Technical protective practices must also be coordinated with administrative and personnel practices, especially for the case of abnormal operation.

4.2. Safety in Batch and Continuous Processes 4.2.1. Safety in Batch Processes [207–210] 4.2.1.1. Introduction Safe production imposes stringent requirements on the owners and operators of chemical plants, who must observe and implement a body of legislation, rules, and regulations. What is more, the manufacturer of chemical products must make compromises between productivity, quality, safety, and environmental protection, but safety must take priority. The principles of safety and environmental protection are established by management in accordance with the pertinent laws. Every chemical process involves a special combination of chemicals, equipment, and process conditions. Safe operation makes it essential to study each process separately. The written set of instructions, the operating manual, occupies a central place in the safety effort.

4.2.1.2. The Operating Manual Principles. The operating manual is a comprehensive document that fully describes a particular chemical or physical process used in making a product. It must guarantee that the process can be carried on in a safe, environmentally benign, economical, quality-oriented way. A fundamental requirement is that the manual must describe in full scope and depth the steps that must be performed in the process. It must be unambiguous in that the employee at the workplace knows at all times precisely what actions are to be taken, in what order (e.g., numbering of steps). In extreme cases, every detail of a manipulation may have to be described. While the manual must be exact and complete, it must also be easily understood. It may be expedient to write separate SOPs (standard operating procedures) when complicated and extended actions must be repeated for each batch (e.g., preparation of a hydrogenator). Aids. The operating manual is based on a wide range of documents and other information sources (Fig. 60). For reasons of bulk, it cannot include all this information down to the last detail; a vital task of the author is to set priorities and to pick out specific points about a process and set them apart from what is obvious. Operators of multipurpose plants face a special challenge, for they must diagnose continuously varying conditions in the same apparatus and decide what process-specific information and actions are appropriate for each case. Electronic data processing systems are nearly universal today, and the use of standard forms makes it possible to prepare the operating manual in a consistent way and present it in an easily understood fashion. Modifications and additions can be done quickly (e.g., if production is to be shifted to other plants). The author of the manual must harmonize process data with plant-specific features while observing all guidelines on safety, environmental protection, occupational health, etc. Many documents and other information sources can be used as aids in describing the process, but it is not enough merely to have these to hand. They must be correctly interpreted and applied. A plant manager often finds it difficult to read documents written by specialists in other disci-

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Figure 60. Operating manual: documents and information sources

plines (e.g., some chemists find technical plant flowsheets incomprehensible). Close cooperation with these specialists is needed to insure safe operation. The following information must be checked for completeness: – Correctly acquired and interpreted material and process safety data – Knowledge about the plant (cooling capacities, construction materials, plant status, etc.) – Problems of high-technology facilities (automation, measurement and control, process control engineering systems) – Limitations of technical safeguards (safety equipment) – Utilization of infrastructure (energy supply, alarm systems, emergency systems, gas emission, wastewater, etc.) – Correct and complete risk analysis (process and plant) and transfer of results into practice For some plants or chemical processes, further points may have to be added to this list. A good operating manual is never the product of one individual; it can be created only based on effective collaboration among specialists. As soon as the definitive operating manual is in being, the involved departments involved should

apply their special knowledge to prepare opinions on the process. It is desirable to set up a fixed routing (toxicology/occupational health, safety, environment, registration, etc.) and have each activity signed off on the manual. Contents and Organization. The organization of the operating manual is dictated by company policy, and there is no universal rule. The following three-part breakdown has been tested in practice: 1) Information section: intended chiefly for management; a quick overview of the process, presenting all safety-relevant aspects 2) Operations section: focused on the operating procedures; containing all information needed for the practical conduct of the process in the plant 3) Appendixes: data, details, and special information on the process; can be used to gain a deeper understanding of the process A possible table of contents reflecting this overall breakdown is shown below. Overall Organization of Operating Manual Information Section

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A. Changes from Precedent Operating Procedures B. Chemical Scheme of Process C. Materials (Chemicals, Yield) D. Environmental and Safety Aspects D.1. Safety Data D.2. Summary of Risk Analysis D.3. Waste Disposal D.4. Environmental Impacts (Ecogram) E. Plant and Equipment E.1. Flowsheet (Streams in Apparatus) E.2. Apparatus List F. Material Data G. Safety and Occupational Health G.1. General Provisions G.2. Process-Specific Provisions H. Response to Accidents, Fire Alarms, Power Outages Operations Section I. Process Performance I.1. Step-by-Step Description I.2. Process Instructions I.3. Process Deviations, Remarks J. Intermediate and End Product Inspection K. Packaging and Storage L. Equipment Cleaning on Product Changeover Appendixes M. Safety Tests, Wastewater Analyses, Operating Instructions for Individual Apparatus, Distribution List Safety-relevant items are discussed as follows. D.2, Summary of Risk Analysis gives a quick overview of the risks involved in the chemical process and the apparatus used. Known major risks are described along with key safety practices for the process. These include administrative practices, special technical safeguards, and important limit values (temperature, pressure, etc.). G.1, Safety and Occupational Health: General Provisions lists general directives and rules to be followed in the chemical process. These may relate to reporting of deviations from the operating specifications, grounding, inert gas blanketing, etc. The instructions and rules must be formulated so that they can be observed in practice. G.2, Safety and Occupational Health: Process-Specific Provisions gives detailed in-

formation and practices derived from the risk analysis. It may include special characteristics of equipment, special protective measures, handling of hazardous chemicals, increased risks, temperature limits, etc. This section is crucial for the operating personnel, telling how this process differs from others. It is especially important in multipurpose plants where the hazard potential varies continuously. H, Response to Accidents, Fire Alarms, Power Outages is one of the hardest sections to write because problems often occur where they are least expected. With the help of the risk analysis, the steps in the process are analyzed, device by device; the analysis suggests what problems may occur and need to be considered in the planning of emergency response practices. Critical quantities are temperatures, times, and mixing; these conditions have to be determined separately for a variety of potential events. Practices are defined for each step in the process so that any kind of deviation can be responded to and the process returned to a safe condition. I, Process Performance is the heart of the operating manual. The sequence of operations is detailed and described in chronological order. Sequential numbers are assigned to the process steps, which must be formulated in a clear and unambiguous way. The process conditions to be maintained and the quantities of chemicals to be used are written down as nominal values; the person controlling the process must acknowledge these in the form of actual values. When key reactants are present, or when process conditions are especially safety-relevant, it is desirable to confirm maintenance of the nominal values by having a second person (possibly a supervisor) take readings and sign. In multipurpose plants, the base settings of the apparatus take on central significance for safety. The frequent changes of conditions in a given apparatus mean that adjustments must be made as a function of process conditions (temperature limits, stack gas downstream of scrubber, or stack gas cleanup unit, etc.), and some apparatus may have to be modified (cartridge filters, charging hoppers, etc.). Before a piece of equipment is placed in service, these base values must be set and checked. The operations to be performed may be identical for a number of apparatus items (e.g., charging of a centrifuge). In such cases, reference may

Plant and Process Safety be made to separate operating instructions or unit operation instructions (SOPs). When this is done, all that has to be set down step-by-step are the parameters specific to the process (e.g., speed of a centrifuge). I.3, Process Deviations, Remarks is a form, to be completed on site for each batch. It helps in documenting irregularities in the process. The plant manager thus has a powerful tool for early detection of safety-relevant excursions; a shift log serves the same purpose. L, Equipment Cleaning on Product Changeover is very important. From the safety standpoint, it is essential to describe the individual process steps chronologically and in detail, precisely as in I, Process Performance. It is a mistake to think that cleaning with water is completely harmless; if, for example, warm water is run into a reactor thought to be empty, solvent residues in the vapor phase may reach an explosive concentration. The mechanical removal of tenacious crusts with various tools can be critical if solvent residues are still present. It is therefore important that all safety principles are observed, even when cleaning the equipment. Appendixes give standard operating procedures (SOPs), checklists, piping and electrical diagrams, etc. The exact distribution list is also important, because such a list is the only way to make certain that changes are made in all existing copies of the operating manual. Introduction of a New Operating Manual. Careful, comprehensive documentation is just one important condition for safety in chemical production. Equally important is that employees properly understand the instructions. Often, the use of technical language and foreign words in the operating manual places it at too high a linguistic level for the workers. It is also remarkable to see the wide range of interpretations different people can put on imprecisely formulated directions! Another problem arises when some employees are not fluent in the language of the instructions, and thus fail wholly or in part to understand an exact description. Any new operating manual must be presented and explained, one step at a time, by oral instruction. Actions that are difficult to describe in speech or writing must be demonstrated in the plant. The objective of such training is not simply that the staff

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understand the instructions; as comprehension grows, so do acceptance and safety awareness. The instruction program must end in a test to ascertain whether the operating manual has been understood. This can take the form of a game or competition, or can be administered as a moderated discussion. Another proven method is to have the employees make a display, illustrating and explaining the steps in the process. 4.2.1.3. The Human Aspect of Safety Responsibility. The chemical process employee works in an environment with a substantial hazard potential. Every day brings the need to work with harmful substances, dangerous reactions, and large quantities of chemicals. Each employee thus bears a high degree of responsibility for co-workers, the environment, and the plant. Even small mistakes can lead to grave accidents. In multipurpose plants, the situation is even more critical: Processes differing in their demands on work methods and safety are performed in succession. Close concentration and an ability to adapt quickly to new conditions are required. One of the most urgent tasks for management is to provide each employee with the conditions for safe work and the resources for performing the job. Occupational safety becomes a task for management, to insure that workers are safety-motivated, and to aid them in carrying out their demanding tasks with discipline and selfconfidence – and without accidents. Training. The training of plant operating personnel is a continuing task for plant managers. Thorough training greatly enhances onthe-job safety by helping workers to deal better with the pressure of responsibility, and to react calmly and correctly in stress situations. Training must be related to practice in the plant; an ideal example is discussion of the operating manual before production is started. Special attention must be focused on the training of new employees, who possess good technical knowledge, but not the specific skills for their jobs. During their first months, these employees must be given technical orientation and support so they can perform in accordance with safety rules.

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A particularly important point in training is how to respond in abnormal situations. In the context of the operating manual, responses must be explained in the clearest possible way. Practical and theoretical emergency training (e.g., with simulator programs) help establish a routine for dealing with abnormal conditions. It is important to instill in the worker the self- confidence that will lead to calm, considered actions in a real emergency. This training is documented and evaluated. Feedback, including worker self-assessment, helps reveal gaps so that the appropriate steps can be taken. Motivation. People are not inherently motivated to work safely; external motivation must be provided. Repeated use of convenient, but unsafe, work methods, not immediately leading to an accident, gives rise to bad habits. Administrative measures (e.g., checkups) and technical precautions can evoke safe working methods, and make unsafe behavior difficult or impossible. Correct behavior should be associated with all possible advantages: recognition, easier job performance, higher qualifications, etc. Cooperation. In the course of time, process workers accumulate much practical experience. It is essential for plant management to make use of this knowledge by integrating the workers in the safety process. It is an important distinction whether the wearing of protective clothing and respirators is ordered or self-prescribed by a group within the plant working on a safety problem. Safety awareness, and above all the acceptance of key safety practices, can be markedly enhanced in this way. 4.2.1.4. Internal Organization and Policies The processes that follow one another in a multipurpose facility call for intensified administrative policies. Without clear direction and clear structures, an operation where products are changed over, say, weekly slides into chaos, and an accident becomes inevitable. In what follows, some points are listed that should help avoid such problems. Weekly Program. The weekly program provides a quick survey of planning for the next

week. It yields information on personnel allocations, production program, maintenance activities, and special events. The weekly program is presented orally to the operating personnel and is posted in the plant. Every worker can then make the optimal preparations for coming assignments. Surprises and improvisation are largely eliminated. Weekly Conference. All employees take part in a brief conference every week. This is an occasion for comments on the preceding week and the coming one, and is also very important for exchanging safety information. Absentees must be identified and informed later. Shift Change. A formalized shift change must be instituted, so that the chemical process can be carried on without pause. The procedure should include explicit allocation of tasks, provided in writing if necessary (e.g., in a shift log). Responsibility. Clear structures in work allocation is essential. The chain of command (hierarchical levels) must be preserved. Unless a situation arises calling for prompt intervention, the plant manager should issue instructions to process personnel only through supervisors and team leaders. This avoids, the confusion generated by instructions and counterinstructions, and by indistinct areas of responsibility. Monitoring of Administrative Practices. Procedures on paper are not enough; they must be carried out. Spot checks should be done to find out whether selected points in the operating manual are being complied with. Above all, misunderstandings and gradual safety-relevant changes in work procedures must be detected promptly. Information Wall. An information or notice wall can be used to promote safety awareness, but this can fulfill its purpose only if it is actually read, so it must be prominently located. Attractive presentation (generous use of color, graphics, etc.) makes it more likely that employees take an interest in the information, but no information wall can replace the oral communication of important instructions.

Plant and Process Safety 4.2.1.5. Safety in Production Practice Before the Start of Production. The saying “as you make your bed, so you must lie on it” takes on great significance when preparations are being made for a new process. The decisions made at this stage must lead to safe, accident-free operation later. Serious preparation is particularly important in multipurpose plants where chemical processes change frequently, so that the plant must continually be modified to fit changed process conditions and operations. Plant personnel must also adapt. There must be a distinct “breathing space” between finished and following production. Overlap allows the possibility of confusion, and increases risk. Central importance attaches to the production conference with all employees concerned. The operating manual is reviewed step by step and supplementary instructions are given by management. This production conference is an intensified form of the conference discussed under the heading Introduction of a New Operating Manual at the end of Section 4.2.1.2. Its timing has much to do with its success; it should be held immediately before the start of production, so that employees can concentrate on the new process without distraction, and get themselves mentally set for it. The preparations for a production changeover are extensive, but identical in their main features. It is worth standardizing these procedures; ideally with a checklist. Checklist for Production Preparation 1) Information for leadership – Scope and timing of production – Study of process documentation – Leadership conference 2) Operating manual – Check operating manual (validity, changes) – Make available report forms – Post necessary documents (e.g., chemical data sheets) in plant 3) Feedstocks – Verify availability of reactants required – Verify quality of reactants 4) Training – Conduct process-specific training – Discuss process with all employees concerned

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5) Plant/facilities – Check maintenance completed – Carry out functional tests – Install additional safety features – Make piping hookups according to diagrams – Perform leak tests – Check cleanness of equipment 6) Trial run – Carry out and document trial run with solvent and prepare report 7) Analysis – Prepare analytical sampling 8) Administrative – Order necessary placards and labels – Prepare folders and forms for process documentation 9) Wastes – Check disposal system 10) Storage – Prepare packages/containers – Set aside storage capacity 11) Safety – Verify completeness of safety documentation – Consider safety and occupational health checks During Production. In a correctly prepared start-up, the management concentrates on inspections, monitoring work procedures, and verifying process safety. This task includes: Surveillance of Action on the Shop Floor. It is obvious that a safety problem exists when the author of an operating manual is convinced that the instructions are being strictly followed while, for whatever reason, something entirely different is going on. The comment, “That’s what it says in the book, but we’ve always done it differently,” must not be tolerated. Discrepancies should be identified and immediately corrected. Detection of Irregularities. A permanent task for both leadership and operating personnel is to identify process irregularities promptly; an important aid is a form for reporting process excursions. Prompt response to irregularities is a key element in accident prevention. Assessment of Process Modifications. If deviations from the process instructions are necessary (e.g., because of defective apparatus), the plant manager must consider carefully the safety

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consequences of such a change. If a distillation is carried out in a different piece of equipment under the same conditions, but the reactor jacket offers worse heat transfer, or the stirrer is less effective, the operation may last longer than specified. This type of change can lead to a dangerous condition. Feedback. To obtain feedback on the practicability of an operating manual, it is essential to make contact with the workers at the workplace. An amazing amount of information and suggestions can be conveyed in a short, relaxed conversation. Working Conditions. Noise, heat, uncomfortable posture, and other conditions can strongly affect human performance; working conditions should continually be examined and improved. At Termination of Production. When a process is terminated, extensive process documentation is generated: production reports, analysis reports and certificates, printouts, etc., as well as a report summarizing all vital aspects of the process. This documentation yields information on potentially safety-relevant deviations in process chemistry and technology. Process Documentation 1) Title/production stage Product, reactant, duration 2) Operating manual Applicable operating manual, deviations from valid procedure 3) Production Technical and chemical feasibility 4) Yields Comparison with earlier productions 5) Analysis/quality Remarks on product and reactants 6) Modifications/remodelling Summary comment 7) Economic aspects Calculation 8) Documentation Identification of changes, defects, and missing or outdated documents (e.g., safety tests); all changes made in flowsheets and other schemes in the course of repairs 9) Conference notes on termination of production

Proposals for improving safety and process conditions Good process documentation guarantees continuity of production with regard to work procedures and safety. Reading these documents when the same process is resumed is an absolute must. Finally, the operating manual must be revised on the basis of the process documentation and, if necessary, rewritten. This insures that the manual reflects the reality of the plant and is ready to use for a new production at any time. 4.2.2. Safety in Continuous Processes 4.2.2.1. Introduction A continuously operating production plant usually makes a single product. The steps involved can include both chemical reactions and physical separation operations. The quantity of material involved in both cases can be extraordinarily large, and special care is required, particularly when the substances have dangerous properties (flammable, explosive, toxic, or environmentally harmful). Continuous production plants are highly automated, and are seldom run without process control systems. Apart from start-up and shut-down modes, they operate according to fixed, specified process parameters. Intervention by operating personnel is necessary only during indicated or infrequent deviations from normal operating conditions. Because the process does not change, the danger of neglecting to readjust safety-relevant process variables on a change of product is not crucial. Typical continuous processes are: – – – –

Refinery processes Cracking processes for olefin production Polymerizations Processes for making organic and inorganic bulk products

4.2.2.2. Analogies with Batch Processes While the objectives, procedures, and plant sizes in the batch case are much different from those in the continuous case, quite a few of the basic statements in Section 4.2.1 also apply to continuous processes.

Plant and Process Safety For example, exact knowledge of the process steps is important for process safety, in particular the acceptable deviations from the specified process parameters. The safety parameters and their acceptable ranges, established before the process plant is designed, must be set forth in an operating manual, and embodied in safe procedures for use by the operating personnel. In the context of risk or safety analysis (Section 3.3), the conditions for the occurrence of conceivable upsets must be identified, the consequences must be analyzed in terms of occurrence probability and severity, and appropriate safety practices must be instituted and covered in detail by bulletins and training of operating personnel. These studies must include not only the failure of technical devices (compressors, pumps, mills, and centrifuges) along with malfunctions of instrumentation and control devices, but also errors made by operating personnel, and partial and total power outages. There must be a plant emergency plan, prescribing how workers should respond to a variety of abnormal conditions and accidents. A higherorder emergency and hazard- control plan dealing with major accidents, i.e., those that can pose a serious danger to employees, the immediate neighborhood of the plant, or the environment, must establish internal report routing and hazard- control measures. This plan must also designate those accidents in which external assistance, e.g., the public fire brigade or disaster response teams, must be requested, and what public institutions, e.g., the police and regulatory authorities, are to be informed, immediately or later. Before production is started, operating personnel must receive training and orientation based on: Process documentation: Process flowsheet Chemical reactions Physical steps in the process Parameter values to be maintained Safety rules Functioning of the process monitoring system Safety information: Working with dangerous substances Use of personal protective equipment Periodic activities in the plant

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Operating and safety instructions: Proper operation of apparatus and machinery Repair procedures Response to abnormal events This information must be presented to all employees before they take up work in a plant, and at regular intervals thereafter. It is essential to test whether the knowledge presented has been understood and whether it can be promptly and correctly put into practice in stress situations. When the plant is being shut down on purpose, it is recommended that the shut-down process be combined with a simulated emergency. Another vital activity in continuous plants is the keeping of a shift log. This helps inform later shifts about all important events in the plant. It is documentary in character, and must contain the following items: – – – – – –

Bulletins from management Process upsets Deviations from normal operation Defects found in the plant Repairs needed and those completed Availability of standby equipment and machinery – Warnings from supplier and customer plants

4.2.2.3. Special Features of Continuous Processes Usually, large quantities of materials are involved in continuous processes. Large quantities, especially of flammable gases or toxic liquids or solids, represent high hazard potentials that must be taken into consideration when starting up the process for the first time or restarting after a temporary halt, during operation, and in the shut-down phases. A continuous process has the advantage of inherent safety. While reactors have large dimensions and high throughputs, the quantities of products directly involved in the reaction and the holdup in the reactor are small; the hazard potential due to the reaction is minimal, especially for strongly exothermic reactions. Continuous plants must be carefully inspected before they are first brought on stream:

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1) All installation work, including subordinate work such as steam-trace lines and thermal insulation, has been completed 2) Scaffolding has been removed from the plant 3) Craft workers and assistants not actually needed in starting up the process must have left the plant 4) The entire system (logically subdivided into functional units) has been pressurized with gases (air or nitrogen) or liquids (water or harmless products) and checked for leaks 5) Auxiliary loops, coolers and chillers, special product or scrubber circuits, cleaning loops, and the wastewater and waste air cleanup units have been put through successful test runs 6) All instrumentation and control equipment, particularly their safety-relevant and failsafe functions have been subjected to functional testing by qualified specialists. These functional checks must include not only the operation of alarms and safety interlocks on partial or total power outage, but also the emergency shut-down systems for plant sections or the entire plant Almost all large continuous plants are now run with computer-assisted process control systems. Operating personnel must be thoroughly trained in the use of these systems so that they can take the proper risk-reducing actions in case of trouble, even in stress situations. Safetyrelated circuits are commonly hard-wired in parallel with the process control system. Switch points (temperatures, pressures, quantities, flow rates) are then fixed; they may be changed only after plant management and/or a safety expert has performed a safety analysis and issued a written directive. Where safety circuits are integrated into the process control system, technical or administrative policies must insure that safetyrelevant switch points cannot be altered by operating personnel on their own authority. When starting up a continuous chemical process, special attention should be focused on “priming” the reaction. The pressure, temperature, quantities, and concentrations of reactants or catalysts must meet the criteria specified for the start-up phase. To avoid dangerous conditions due to delayed reactions and the associated buildup of unreacted feeds, the reaction start should be precisely monitored for rates of

change (e.g., temperature or pressure rise) or with real-time analytical methods. If irregularities are seen, the reaction start should be interrupted, the fault identified and remedied, and the plant readied for another attempt. Many process control systems have sequence controls programmed to halt start-up automatically if an upset occurs. During on-stream operation, the plant personnel continuously observe the process parameters, comparing them with the acceptable nominal ranges. Analyses of intermediate and end products not only help keep the commerical product within specification, but also provide information for use in maintaining the process. In particular, buildup of undesired byproducts due to recycles within a process or in closed- cycle scrubber units must elicit a high level of operator attention, and lead to countermeasures if appropriate. Important process parameters that are recorded or stored by the process control system must be retained for up to ten years (depending on legislative provisions). These can be used to demonstrate that the process was run properly, or to prove compliance with emission limits and other air and water pollution control standards during regulatory audits; they are also indispensable in post-accident reconstruction. As a rule, a continuous plant is shut down once a year for cleaning, inspection, and major repairs. After the process is stopped and before these operations are started, the plant must be carefully drained, purged, and cleaned according to the operating manual. For especially hazardous work (entry into vessels and tight spaces, or welding in plants that process flammable or explosive substances) plant management must establish safety practices that must be specified in detail in written work permisssions given to the employees performing the jobs. Exact instructions for the performance of repairs must be given (see Section 4.4). When several craft teams are working at the same time, a coordinator must be designated to prevent them from endangering each other. The coordinator plans the work in time and space and is authorized to issue instructions to the craft teams. Large continuous plants where explosive, flammable, acutely toxic, or environmentally harmful substances are handled in such quan-

Plant and Process Safety tities as to create a high latent hazard potential present a special concern. Such a plant should have a rapid-response service, organized at each key level of the hierarchy, which can provide expert advice to emergency workers, to minimize the consequences of an accident.

4.3. Technical Inspection [211–217] Introduction. In what follows, the term “technical inspection” denotes the inspection of technical devices by independent third parties; its purpose is to find out whether the equipment meets requirements set forth in legislation, regulations, or standards. In this way, unacceptable risks to employees working with the equipment – and to third parties and the environment – are prevented. Plant hardware subject to technical inspection includes pressurized systems (vessels, piping); storage facilities for flammable substances or water pollutants; elevating, hoisting, and conveying equipment for the transportation of persons or freight; electrical equipment; nuclear facilities; means of transport such as automobiles and trucks, railroads, ships, and aircraft; power plants; petrochemical plants; and chemical plants. Experts. As a rule, inspection is performed by an expert recognized by the responsible authority. Conditions for recognition include not only personal integrity, but also appropriate specialist qualifications (through education and continuing education), and it must be determined that the expert is not subject to instruction or restraint having to do with the inspection activity, i.e., independent. The specialist qualification is generally a natural science or engineering degree. In many cases, a specified period of employment in the field is also required. The expert may also be required to take part in an exchange of experience, which has two purposes: to assure consistency of technical inspection, and to identify crucial points of concern for a certain type of technical equipment. The results of such an exchange can also provide a basis for technical progress, and serve as input for standards development.

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The role of the expert in technical inspection has long been established in Germany. The high point of the independent expert tradition was the field of law dealing with facilities “subject to surveillance.” The experts recognized in this area are usually comprised in Technical In¨ spection Boards (TUVs), which have in turn combined (along with some industrial firms that support their own inspection services, staffed by recognized experts) in the Association of Technical Inspection Boards. Official recognition as an expert used to be limited to an individual on the German system. At European level, organizations will be entrusted with the performance of safety tests in the context of equipment manufacturing. Such organizations, after gaining accreditation at national level, can apply to the European Commission for “notification”. These inspection organizations or “notified bodies” are to be engaged by manufacturers, especially for certain conformity tests pursuant to EU directives. The requirements on notified bodies are set forth in the Directives and relate chiefly to independence, personnel and hardware resources, and employee training. While expert inspection in Germany has been handled on a regional basis, notification as a testing service is valid throughout Europe, i.e., a laboratory accredited by a European Union member country and notified by the Commission can also do the pertinent conformity tests in the other member countries. German Federal Water Quality and Pollution Control Act (Wasserhaushaltsgesetz) is following a similar path. Where tests by experts used to be prescribed, jurisdiction is shifting to inspection organizations, whose recognition in one Federal State is valid in all states. Legal Background. The purpose of technical inspection is to make certain that the equipment inspected satisfies the quality and operational requirements set forth in regulations and standards. Regulations are based on legislation that designates goods for protection and identifies technical devices to be regulated. The legislature thus authorizes regulatory bodies to set requirements, e.g., quality, operation, and inspection. Administrative actions such as permits and approvals are also authorized in this way.

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A regulation commonly defines the scope of application, defines exceptions, states what technical devices are covered, establishes the scope and intervals of inspection, designates the persons or institutions that perform the inspections, and states what agencies have surveillance authority. The requirements set forth in such regulations are then implemented in instructions to the agencies or in engineering codes developed by private or public standardization bodies. The harmonized European standards promise to be of great importance in this respect. These standards implement the basic safety requirements stated in European directives. Although their application is voluntary, they allow the presumption that a technical device complying with the harmonized European standards also fulfills the essential safety requirements of the pertinent directives. Expert Inspection. Inspection and testing by the expert usually takes place in two phases. In the first, the existence and completeness of all necessary documents for the technical device are verified (official approvals, type approvals, type test certificates, certificates covering the inspection and use of materials, technical drawings and plans, operational descriptions, and reports from individual tests already performed). The second phase comprises the actual testing of physical facts. The process of expert inspection can also be broken down in time. The first point examined is whether the design being inspected is suitable for the intended service. In the second phase, it is determined whether the technical device as fabricated complies with the submitted and examined design. The next step is to verify that the finished technical device is operated in such a way that no stresses are imposed on it other than those considered in the first phase. An integrated system analysis may be called for here, especially if interaction with other technical devices or within the system as a whole must be taken into account. Finally, after a set inspection interval has passed, so- called periodic tests are conducted. These verify whether changes in the technical device due to its service, e.g., those resulting from wear, suggest that the stresses experienced in the next inspection cycle may no longer be tolerated. Another question at this stage is whether

there have been safety-relevant changes within the system in which the component is integrated. In addition to inspections before and during service, technical devices may also have to be inspected after they are taken out of service, if significant danger can result from improper operation. When technical devices or their service conditions are modified in such a way that there may be safety-relevant effects, a new expert inspection generally has to be performed. Finally, the responsible agency can instruct the expert to assess departures from the regulations or to investigate accidents. The methods used in technical inspection range from simple visual examination to complex, specialized testing methods such as nondestructive ultrasonic tests or electron microscopy in connection with accident investigations. Such tests are generally performed, and the results interpreted, by specialists acting in support of the expert, who evaluates the results and cites them in the final report. Integrated Plant Inspection. The interdisciplinary efforts of specialists in assessing safety problems is particularly important in the inspection of facilities subject to approval under the German Federal Pollution Control Act (Bundesimmissionsschutzgesetz). “Whole-plant inspection” means a comprehensive system inspection and monitoring of a plant with regard to safety functions. This procedure takes in material properties, process conditions, interaction between apparatus and other plant components, interaction between human beings and hardware, administrative policies, and the effect of external safetyrelevant factors. Inspections in such a program are concerned with whether the operation complies with legislative requirements (pollution control, occupational safety and health, water quality, soil conservation), pertinent regulations, and recommendations. These inspection and surveillance tasks concern all those concerned. The three pillars of whole-plant inspection are thus: 1) Supervision of the plant as part of the operator’s responsibility 2) Inspection and surveillance by public agencies

Plant and Process Safety 3) Safety inspections by experts Such a procedure can also serve as a model for the inspection of plants, as proposed in a draft European Council Directive for avoiding the dangerous consequences of serious accidents with hazardous substances.

4.4. Maintenance A plant operating in a trouble-free manner is in its safest condition. However, because technical components are subject to mechanical and process-related wear, upsets can occur. As a result, maintenance is necessary to keep the plant in a safe condition or restore it to such a condition. On the other hand, maintenance can create new dangers in the plant. This maintenance-related risk can be minimized only by good safety practices in the planning and execution of maintenance work. In the past, mistakes in the preparation or execution of maintenance were a significant cause of accidents in the chemical industry. The gas explosion at Flixborough in 1974 was traced back to a makeshift repair procedure [218], and many smaller and less spectacular events have resulted from maintenance errors [219], [220]. In a representative time interval, 30 % of all events occurred during maintenance [220]. Therefore, the value of safe maintenance in the chemical industry cannot be overstated. This is not surprising if the cost of maintenance is analyzed. It amounts to 3 – 5 % of the plant replacement value per year; in large companies, this can add up to more than DM 109 a year [221], [222]. These costs arise from a large number of small operations, and are made up mainly of personnel costs (including those incurred by vendors). A high level of administrative and logistical expenditure is called for. Maintenance actions, especially repair work, are only partially recurrent; many repair jobs must be planned and carried out individually. These points imply the following three fundamental goals: 1) Plants should be designed and constructed in such a way that they are easy to maintain, i.e., easy to repair (good cleaning and draining capabilities, proper access, etc.); require little maintenance; and are made up of

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the most reliable and long-lived components possible 2) The proportion of unforeseen repairs caused by damage should be reduced in favor of scheduled (including preventive) operations 3) Safe procedures must be adopted and enforced for the performance of maintenance work, to reduce the risk arising from maintenance

4.4.1. Actions During Plant Design and Construction The task ensuring safety during maintenance work is to create the technical and administrative conditions for safe performance of the work. This requires a systematic procedure (Fig. 61). A good maintenance strategy begins with the design of a low-maintenance, easy-to-maintain plant. Every intervention in the plant can lead to danger during maintenance work and on restarting. The achievement of long life and high reliability at this stage is also economically important. Systematic safety analysis (e.g., HAZOP) can identify the consequences of malfunctions and their possible causes at the design stage. Two types of event must be distinguished: – Events that follow immediately from the technical failure and the resulting deviation of the process parameters from the design conditions – Events that can occur during a maintenance operation The objective in event analysis is to minimize hazards: – To achieve the highest possible reliability in terms of failures, and to limit the consequences of failures – To minimize the repair risk, i.e., to reduce the number of maintenance actions and to perform them as safe as possible Selection Criteria for Apparatus and Machinery. High reliability is obtained by designing and selecting components on the basis of their requirements. Key criteria are: Construction materials

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Figure 61. Maintenance strategy: central significance in a chemical plant

Functional principle Robustness Secondary points are hardware safeguards against improper operating conditions, e.g., flow monitors on pumps, which help to prevent damage. Because a component may fail anyway, the consequence of such a failure must be analyzed. Even if the probability of occurrence is low, an event (failure mode) with serious consequences can influence the above criteria and make it necessary to seek a different solution. Once these criteria are set, quality assurance systems must be instituted to monitor and guarantee compliance with the standards during the subsequent fabrication and installation of apparatus and machinery. Design Criteria. Because maintenance and repair are inevitable, danger arising from them can be most effectively minimized by good installation and design practices. The criteria at

this stage include clear layout (accessibility), easy and safe disassembly, and draining and pressure-relief capabilities. In a chemical plant, special importance attaches to draining. Some maintenance jobs can be made safer by the use of assembly aids, local spot-vent systems, etc. Aids for Monitoring Plants and Performing Safe Maintenance. The documents needed for reliable operation and safe maintenance should be created at the design stage. These documents can usually be assembled from documents required for plant design and construction, e.g., manufacturers’ data sheets, internal experience reports, and piping and instrumentation diagrams. Periodic inspections are supported by maintenance and inspection schedules, which must be present when the plant is commissioned. Test methods involving no opening of equipment or piping in the plant (e.g., ultrasonic inspection) are preferred. Where safety-relevant process control equipment is not redundant, self-

Plant and Process Safety monitoring is recommended. Inspections, including functional tests and plausibility checks, help todiscover concealed defects and to identify needed repairs. In some situations, automatic monitoring devices such as vibration monitors can also be used. Anticipated, i.e., recurrent repairs must be thought out ahead of time. Detailed job instructions, supported by extracts from technical drawings (job cards, Figs. 62, 63), cut the number of mistakes during a repair or preparation for one. Other documents needed later for the proper execution of maintenance tasks include: piping and instrumentation diagrams with piping and seal specifications; piping and fitting lists; apparatus specifications; measurement station data sheets (to ensure compliance with standards); and functional diagrams of the process control system with lists of setpoints, limits, and alarm settings (unambiguous transfer of process parameters to process control system). 4.4.2. Actions During Plant Operation Figure 64 illustrates the maintenance strategy for this phase. The maintenance task is to keep the plant in its nominal condition so that requirements on product quality, safety, and availability are always met. According to DIN 31 051, maintenance is defined in the following way: 1) Preventive maintenance: monitoring and keeping plant in nominal state 2) Inspection: determination and assessment of actual state 3) Repair: restoration of nominal state There is a fundamental distinction between scheduled maintenance and breakdown maintenance. An attempt is made to minimize the proportion of breakdown maintenance caused by unforeseen failures. Scheduled maintenance is particularly concerned with components having safety and quality relevance; it includes upkeep and inspection as well as measures to obviate repairs or cut their cost [223], [224]. Preventive maintenance and inspection are usually done according to predetermined schedules and are thus systematic (timing and scope set in advance). Maintenance and inspection schedules are used for this purpose. Inspections are scheduled on the basis of the anticipated wear

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of the components involved, known either from manufacturers’ data or from operating experience. Preventive maintenance and inspection are done both while the plant is on stream, nominal/actual comparisons being made continuously and documented during regular rounds, and during scheduled shutdowns. Maintenance and inspection schedules are a reliable and efficient way to organize the latter. Unforeseen failures of plant equipment occur despite scheduled maintenance, and these must be covered by administrative policies (usually operating procedures) providing for later repairs. This “breakdown” maintenance is also supported by repair instructions and aids rendering the work more reliable and faster. 4.4.2.1. Maintenance Activity The procedures for scheduled and breakdown maintenance are similar (Fig. 64). The key step in minimizing risk of breakdown is early detection of a deviation from nominal conditions, which can be apparent in: – Obvious defects: drips, noise, etc. – Concealed defects, e.g., gradual slippage of process parameters Regular rounds by shift personnel under a fixed plan can aid detection of the problem and thus contribute to minimizing risks. The maintenance and inspection schedule for every technical component contains information on: – Test criteria applied in maintenance and inspection – Methods practiced – Timing The action ends with a finding, documented in a report signed by the inspector. If problems are identified, but cannot be remedied immediately, they are recorded in the plant shift log. The description should be oriented to the concrete observation, and should not contain speculation about the cause. The proper remedy cannot be established until the malfunction or the damage has been jointly assessed by plant management, the shift supervisor, the shop supervisor, and in some cases a specialist. A meeting must be held to determine how and with

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Figure 62. A) Example of a job card

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Figure 63. B) Example of a job card

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Figure 64. Maintenance activity chart

what precautions the repair work is performed (Section 4.2.2.2). A further aid to proper reporting of technical problems by operating personnel to maintenance personnel is the technical shift log, which simultaneously serves as a tool for evaluating damage; the shift personnel enter the following information: – Trouble during the shift – Exact location (i.e., identification of component) – Nature of trouble or malfunction observed

– Time of discovery The log should also show what maintenance action was taken; the person responsible must sign it. The purpose of the technical shift log is to state clearly what was done, and who was responsible, to prevent ill- considered and uncoordinated repair efforts. With the help of the technical shift log – including the signature of the responsible person – it is always possible to verify that necessary action has been done properly.

Plant and Process Safety The technical shift log performs two other functions: – Identification of temporary and makeshift repairs – Evaluation in terms of recurrent problems (analysis of weak points) – Computer support is recommended so that: – Temporary repairs are not left in place permanently – Statistical analysis can be done and appropriate countermeasures instituted to extend service life and reduce risk by cutting the number of repair projects [225] Safety procedures must be followed whether the maintenance job is: – Anticipated when preventive maintenance, inspection, and repairs are to be carried out during a scheduled shut-down (scheduled maintenance); or – Unplanned, i.e., caused by a failure (breakdown maintenance) In either case, the procedures must be designed to cope with the risks occurring at every stage. If a problem arises, e.g., failure of a safety device, it may directly affect plant safety. This aspect is not pursued here; the proper response is always set out in the context of the safety analysis. As a rule, the plant or sections of it must be put in a safe condition by a shut-down procedure when a problem occurs; this must not be confused with the “emergency off” or “panic” button for a section of the plant. While planned or “checklist” shutdown of the plant is a frequent procedure, the failure of a technical component required for shutdown may lead to the isolation of specific apparatus or sections of the plant, or to overriding the safety interlocks. Examples include bypassing a leaking agitated vessel and releasing an interlocked fitting that normally cannot be opened except at a certain point in the reaction. Shutdown is not always necessary. In many instances, countermeasures can be taken that do not lead directly to a solution of the problem. There are three possibilities: 1) Postponing the proper repair (e.g., temporary plugging of leaks)

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2) Using planned fallbacks (e.g., putting standby systems on stream) 3) Taking alternative measures (e.g., blowing a vessel down instead of pumping it dry) Alternative actions are temporary modifications; such makeshifts may be more desirable than immediate shutdown of the plant, e.g., in a large continuous process where operation can be maintained until the next planned shutdown. Any modification must be checked first by a systematic safety analysis. 4.4.2.2. Performance of Maintenance Scheduled and breakdown maintenance differ, especially in the documentation existing at the start of production, which describes the specific work to be done and takes the form of instructions: – Scheduled maintenance is covered by preventive maintenance and inspection schedules describing the nature and scope of the work. Repair instructions (e.g., job cards) are generally used. Plans for shutdown maintenance are of this type, especially in continuous process plants. – For breakdown maintenance, the problem and the corrective action taken are recorded in the technical shift log. Special repair instructions are not always available, and the documentation is then limited to the written plant documentation, including updates. There are essentially two further possibilities: 1) Maintenance work while the plant is on stream 2) Maintenance work during a shutdown In what follows, only maintenance during shutdown is discussed. There is no fundamental difference between total and partial shutdown. The only routine actions that should be taken in an operating plant are those that raise no safety concerns. Maintenance can be broken down into three phases: Preparation Execution Post- completion measures

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When danger can arise during the performance of this work, all workers involved must follow special safety practices, which are written up in a “work permit”. The purpose of this is twofold: to make certain that the proper safety procedures are followed; and to ensure that responsibility for the components of the work are clearly defined by signature. Work Permit

No.

Equipment:

Polymerization reactor 1 (R 3.02)

Description of work (incl. to whom assigned) Reattachment of submerged product feed pipe by fitters 1. Isolation of plant section/component: Depressurize all product lines connected; open pipe unions and close off with blank flanges; disconnect N2 lines and close off with blank flanges. Disconnect external steam tracing. Disconnect power to agitator motor. Release interlock on bottom drain valve. Completely empty vessel through R 2.03, and leave valve open. 2. Actions on isolated plant section/component: Using high-pressure cleaning unit, clean vessel with water through manhole. Close bottom drain valve and fill with water. Drain both cleaning waters through R 2.04 for analysis. Have specialist team perform radiometric measurement of level from outside pipes. Suspend ventilation device in vessel. Close bottom drain valve. Date: Time: Signature: 3. Action before beginning work: Instruct fitters that no welding or driving is permitted. Instructions given to: Signature: 4. Occupational safety measures: Introduce conductors safely into tank; post lookout at manhole; ventilation unit; protective clothing for fitters in vessel; only ordinary work clothing (note: hardhat, protective goggles, safety shoes, . . .). 5. Actions after completion of work: 6. Management approval (by plant manager) for performance of work: From 1 March 1993, 10 : 30 am, to 1 March 1993, 4 : 30 pm Date: Time: Signature: 7. Signature of shop supervisor responsible for compliance with items 4 and 5 Date: Time: Signature: 8. Approval to proceed on site by engineering supervisor of plant Date: Time: Signature: 9. After completion of work: approval by plant supervisor 10. All- clear and technical acceptance All clear: Date: Time: Signature of shop supervisor: Technical acceptance: Date: Time

Signature:

11. Waiver of stated procedures (see 1 and 2) 12. Acceptance before restart

Preparatory steps include isolating single components or whole sections of the plant. This requires proper shutdown of the section concerned to a safe condition. Where redundant components exist for the malfunctioning components, plant management and/or the set operating instructions may allow these to be placed on stream. Other temporary actions such as: – Blowing down instead of pumping dry – Changing the preset process control switching or alarm values – Interfering with interlocks are permitted only after a special safety check, unless they are provided for in special operating procedures. After shutdown, the critical danger points are release of substances by opening up the system to drain it (danger of poisoning or fire/explosion), and the possible introduction of undesired substances during cleaning. Ideally, a plant should have emptying devices and holding tanks for the contents of the equipment, so the plant does not have to be opened for drainage. This concept is desirable for substances with high hazard potentials, but the cost may be substantial. Regardless of whether fixed holding tanks are installed, emptying by gravity should be possible. Normally it is necessary to drain only certain portions of the plant. Such areas can be safely separated from other, product- containing sections by closing some fittings and inserting blanks. Residual quantities of product remain in the drained parts of the plant, however, especially in fittings, filters, and pumps. The individual draining features of these must be used to free them of product. Drainage is frequently aided by nitrogen pressure. Subsequent cleaning is not required with highly volatile substances. The vented air from such an operation must be considered; a vent system may be necessary, along with downstream treatment of the gases. When cleaning is required, the agents used must be compatible with the product and, physiologically, as innocuous as possible. If inert media cannot be used, the following become particularly important: – Medium adequately diluted in the shortest possible time

Next Page Plant and Process Safety – Agent reactive only with the small quantity of product found in the drained segment of the plant – Cleaning liquid safely separated from sections of the plant containing larger amounts of product Failure to observe these conditions has frequently led to events, often because, in a nonroutine cleaning operation, the flow direction, pressure, and valve configuration were different from the normal conditions, and so were interpreted wrongly. All connections to other sections of the plant must therefore be examined, including those not ordinarily filled with product (e.g., off-gas piping). When maintenance work calls for disconnection or disassembly of a plant section, personal injury hazards can arise if the repair site is not rendered safe, e.g., if the electric power to drives is not disconnected or product-filled pipes are not safely closed with a blank flange after disassembly of a subsystem. The latter point is crucial when equipment such as pumps and fittings are disassembled, because the safety measures are carried out right at the disassembly site; in larger maintenance operations, entire piping systems are generally made product-free. Engineering codes, accident prevention regulations, and special safety instructions aid in the proper execution of these operations. To avoid misunderstandings at the site, particularly in a complex installation, the defective item must be identified, and its isolation must be indicated visually. The area can be further marked with a tag stating which craft has jurisdiction and what kind of work is being done. The key requirement for safe repairs is upto-date plant documentation. On the basis of the technical specifications (complete and unambiguous description of performance data, media, construction materials, and design data), it must be ensured that parts and materials used (e.g., greases) are such that no danger arises when the affected part of the plant is on stream. Hazards in this sense are leaks due to improper materials (e.g., gaskets) or incorrect temperature/pressure design; unwanted reactions due to the catalytic action of construction materials or other media present; and parameters outside safe ranges owing due to the installation of unsuitable equipment (e.g., a pump with

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too large an impeller generating an excessive pressure). Similar hazards must be considered when replacing or repairing process control system components. A typical mistake involving an instrument is the use of the wrong range. Just as a repair or inspection site must be labeled to prevent confusion, replacement parts or newly delivered repaired components must be labeled so they can be recognized as equipment meeting the specifications. Often there is no outward difference between two parts, e.g., when only the materials inside the device differ. These practices are intended to ensure that the on-site activity can be finished in the shortest possible time, without interruption or risk. Safety and economy are not in conflict. Virtually every maintenance activity comprises many single activities performed by different crafts. Because economics does not permit craft specialists to be employed especially for all such tasks, workers unfamiliar with the special hazards of the production plant have to be brought in; the requirement of up-to-date plant documentation must be supplemented by explicit instructions to the craftspeople at the work site, and a clear task description (see the job card in Fig. 62). It is necessary to establish both the plant documentation needed for maintenance, and the scheduling of various maintenance activities in a plant area. Only by matching and skillfully coordinating maintenance activities to the rhythm of the process they can be prevented from threatening each other. An aid in this respect is a working plan, which describes the sequence of tasks and their duration from their starting times (e.g., a bar chart). In an area where welding, driving, and caulking work is done, for example, explosive mixtures are not permitted. Production not connected to the maintenance action may have to be interrupted. The same holds when flammable or explosive substances are present in the course of maintenance, e.g., when a protective coating is applied to a surface. Some kinds of work also require a second person to be present, even if the job requires only one craft worker. This is the case when a hazard such as the escape of toxic or explosive mixtures cannot be positively ruled out, or when work has to be done inside vessels. Maintenance jobs can extend over more than a day, so interruptions cannot always be avoided.

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The area of the plant where repairs are being done should be marked off with signs throughout the job, especially to prevent an inadvertent return to service before the work is complete. Approval to restart can be given only by persons whose functions include this task, e.g., the responsible plant engineer. Only a single person should be made responsible for maintenance. This person is the contact point throughout maintenance, otherwise there is a danger that someone who does not have an overview of the work may make a wrong decision. To avoid the danger of premature startup, interruptions of the work must be recorded, e.g., in an attendance log, and completion must be documented in the technical shift log. For work that requires a permit, the permit includes a space for approval after completion of the job. A maintenance action cannot be considered finished until all the unit activities have been checked and accepted by the coordinator of the overall action. This includes both inspection for correct execution of a task, and functional testing. Proper execution means both the use of the components provided (correct temperature and pressure ratings, material, etc.) and their functionally correct installation. Functional testing is then done to make certain that all technical components work together in the way necessary for the process. The simplest example is leak testing of a repaired section of the plant; a very complicated one is the comprehensive testing of a process control system. Where preparatory action has been taken, e.g., isolation of sections of the plant, these must be reversed in the same way as temporary modifications. Approval can then be given for production. In some cases, the plant is restarted a step at a time, extra functional tests with reduced throughput or enlarged crew being done first. Such precautions are needed especially when extensive maintenance work has been carried out. 4.4.3. Continuing Plant Development (Analysis of Weak Points) The elimination of problems by repair is a necessity, but it also offers the plant operators a chance to analyze recurring problems, and to use the results in improving the system. Analysis of maintenance actions for continuing development pre-

supposes that the technology and the operating conditions are adequately documented (this includes unambiguous nomenclature in the plant). Entries in the technical shift log aid in evaluating the frequency of problems. If plant documentation is computerized, the cost of such a “weak point analysis” can be substantially reduced. The goal is to use technically feasible and economically acceptable means to optimize sections of the plant, i.e., to minimize the frequency and/or severity of trouble. Before such an analysis, the possibility of human error causing the problem must be eliminated. This analysis of weak points must not be limited to the technology. Often, the technical defect can be eliminated more simply by changing the process conditions. The weak point found may not be the cause, but rather the effect, of defects in the process, e.g., process conditions or product specifications. In such cases, the analysis of weak points becomes a process analysis (Section 4.5). The causes of problems may well be hard to identify. A final statement of causes often has to be based on the vigilance of the operating personnel on the spot, as well as the years they have spent learning the details. The plant manager must therefore make a continuing effort to motivate the personnel to be observant.

4.5. Modification of Plants 4.5.1. Reasons for Modifications While the objective of maintenance is to keep the plant in its nominal condition, modifications to the process and plant involve changing the documented nominal state of either, or both. A variety of factors can lead to process and plant modifications: – Increases in capacity – Change in process to lower manufacturing costs and/or improve product quality, or to modify the product – Measures to eradicate weak points – Upgrading to the state of the art (technology or safety) Any change in the nominal state means that the corresponding parts of the process and plant documentation must be revised or rewritten.

Plant and Process Safety In this respect, modification differs in scope, but not in basic execution, from a new design project. The reason it is so important to treat a modification like a new design is that there is no way to know ahead of time how widely the modification will ramify itself. Even minor changes, e.g., replacing one apparatus by another, or raising the temperature in a unit operation, can cause a violation of the design or process control parameters. The systematic methods already described for evaluating new plants must be applied here too, because they cover the entire field of possible hazards. First, all areas of concern, e.g., explosion hazards, are studied to determine whether the proposed change affects them at all. Because a change often relates to only a few danger spots, the number of areas of concern quickly becomes small, as does the number of aspects to be considered; detailed analysis is needed for only a few aspects in most cases. Knowledge of the existing safety concept is essential for assessment of all new practices. Any change represents a “compatible” expansion of the whole concept. 4.5.2. Procedure for Modification A suitable procedure must be established if modifications are to be carried out correctly. As soon as a major change becomes necessary in the plant, e.g., because a new unit operation is being introduced, the project is implemented in the same way as any other design project, right up to the construction of a new plant (Section 3.2). In the case of minor changes, the amount of investigative effort required is generally small, so a simplified procedure is useful in such cases. The principle remains the same, but the effort invested in safety analysis and documentation updating is matched to the scope of the planned change. A useful aid is the change sheet, which has two functions: to describe the objective and the implementation of the planned change; and to insure that necessary safety questions have been asked and properly answered. The change sheet also documents the work and its approval, and is used in identifying the parts of the plant documentation affected by the change (Section 4.4.1). A signature is required for each approval and each inspection of each individual step in the process. The work permit already mentioned

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complements this task by listing actions to be taken during its execution. The items contained in the change sheet are: 1) Identification of what is to be changed (including nature of change and other ordering criteria; suitable tools for this purpose are job cards, piping and instrumentation diagrams, and operating manuals) 2) Purpose of the change 3) Description of the change 4) Identification of essential danger spots with the help of a checklist 5) Discussion with specialist departments on the danger spots relevant to each 6) Inquiry into necessary updating of safety analysis 7) List of plant and process documents affected, with possibility of monitoring progress of the change; testing steps after completion (e.g., acceptance test, test run) 8) Questions as to possible effects on product quality Important consultations with the specialist departments and other safety information must be recorded, and the safety analysis must be attached. The following fictitious examples show how small changes can have wide-ranging consequences. 1) An additive used in a thermally sensitive mixture is purchased from another vendor; the composition appears to be only slightly different, but it lowers the safe temperature limit. The mixture is to be subjected to distillation at high temperature. The result is thermal decomposition of the bottoms, faster heat evolution, and a rise in pressure in the reboiler. Corrective action: Perform safety study in the lab with the new feedstocks; lower the temperature set point. 1) A pump delivering acid from a storage tank must fill a vessel more rapidly. On the basis of the characteristic curve, the old pump was selected so that the vent line of the vessel cannot be completely flooded. As a result, releases of substances are prevented without a level-limiting feature having to be installed. The new pump, with a larger impeller, attains a higher pressure. It is now possible to flood

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the vent completely so that acid is released. The danger is not noticed as long as overfilling does not occur. The inherent safety formerly present no longer exists. Corrective action: Select pump according to proper safety documentation; see change sheet. 4.5.3. Summary of Plant Maintenance and Process Modification The systematic procedure described, with the associated aids, contributes to plant safety from design to operation and process upgrades. By itself, however, it is not sufficient to guarantee plant safety. While it cannot replace the knowledge of experienced workers, it offers a guide and support for daily work. The reasons for the increasing volume of technical documentation and administrative support are: – Plants themselves have become far more complex (process control engineering and other advances) – Maintenance work is being done by a larger number of specialized teams – There is increased public interest in readily understandable safety practices

5. Hazard Control 5.1. Means of Limiting Accident Impacts The responsible care of every plant-operating company to take active measures to protect employees, third parties, and the environment against potential harm means that the operator must take responsibility for technical and administrative precautions to limit the impacts of minor and major accidents in the facility. Legal and regulatory provisions independently require limitation of the consequences of accidents in plants; e.g., the “Seveso Directive” [226], [227] and the implementing regulations in EC member countries. Action to limit accident impacts is also mandated by other regulations, e.g., the German Pressure Regulation [228]. The release of substances from pressure-relief features of a plant

(safety valves, rupture disks, emergency blowdown devices) must present “no hazard” to employees, third parties, and the environment [228], [229]. Whether such a release is permissible must be assessed in view of the properties of the substances, the location of the plant, the propagation behavior of the substances, the duration of the release, and the occurrence of hazardous concentrations. If a release is not possible without hazard, retention or disposal equipment must be provided to limit the consequences. German water law also requires that plants be designed, built, and operated in such a way that water pollutants cannot escape. Substances that escape anyway must be retained quickly and reliably so that contamination of receiving waters does not become a concern [230], [231]. The basic idea of a two-barrier system is contained in these provisions. The enclosing shell of a storage tank, e.g., for fuel oil, or the pressureresistant construction of a reactor in the chemical industry, is the first barrier; it must safely withstand the mechanical, thermal, and chemical stresses imposed on it in normal operation, and in the event of accidents. If a release occurs in spite of the preventive measures, a second barrier must limit the consequences of the release in such a way that employees and third parties, as well as the neighborhood of the plant and the environment, cannot be threatened, or that such threat is held to an acceptable level. The chemical industry uses a variety of means to limit the impacts of accidents. As a rule, these also have functions relating to occupational safety, pollution control, and water and soil conservation. In addition to retaining the product in liquid, solid, or gaseous form in case of an upset in the plant, they also provide for retention of contaminated wastewater or cooling water, and the retention of water contaminated by fire fighting. The necessity of retention systems and the way in which they are implemented depends chiefly on the properties of the substances being handled, or those produced in the plant during an accident, and on the actions of such substances on persons and the environment – not just in terms of the immediate effect of exposure, but also the effects produced by a reaction (fire, explosion) of such substances. The hazard

Plant and Process Safety potential of a substance is assessed on the basis of its physical, chemical, and biological properties. The water-polluting effect of a substance is generally evaluated from its assignment to a pollutant class [232], [233], based on the acute toxicity of the substance to mammals, bacteria, and fish (sometimes to algae and Daphnia), its degradability, long-term effects, and distribution. The principal ways to limit the impacts of potential accidents are: – Retention systems, catch wells – Quick- closing valves, emergency compartmentalization systems – Emergency drain and collection systems – Blowoff and disposal systems – Spray- curtain systems – Partial and complete containment – Fire protection – Retention of water contaminated by fire fighting The choice of the means to limit accident consequences must be based on the operating conditions (e.g., pressure, temperature, substance properties), the type of plant, its location, and the situation in its vicinity. 5.1.1. Retention Systems, Catch Wells The best-known retention system for liquids is the catch pan or well. In Germany, the Engineering Code for Flammable Liquids (TRbF) prescribes this way of retaining any liquid that leaks from tanks located in the well [234], [235]. Catch wells not only prevent harm to soils and receiving waters by flammable liquids, but also limit the spread of fires and emissions. If the liquid surface is completely covered (e.g., with a low-expansion fire-fighting foam), fires and emissions can be almost entirely suppressed. In addition to product retention, a catch pan or well can also hold back contaminated wastewater and water contaminated by fire fighting. Special requirements apply to the design and selection of material of chemically resistant, leakproof retention systems such as catch wells, pans, and baseplates. These standards are chiefly dictated by the stresses occurring, for example: – Mechanical stress due to loads, impacts, friction, building subsidence, internal stresses, and contraction

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– Thermal stresses due to ambient temperature cycles (day – night, summer – winter) or hot media – Chemical stresses dependent on water pollutant class, intermittent attack, penetration, or aggressive properties of retained substances relative to construction materials The resistance to chemical attack needed in a retention system, and hence the material chosen, generally depend on the water pollutant classification and chemical nature of the substances present. Chemical stresses are classed by severity: 1) Low load: no water pollutants present in normal operation; retention of water contaminated by fire fighting 2) Moderate load: leaks with action limited in time (nonrecurring or recurring but brief) 3) High load: intermittent contact with waterpolluting liquids; regular contact with heavily laden cleaning water The duration of exposure must be determined for each component. If the loads are low or moderate, the design can be based on a fairly lowprobability event, occurring once and of limited duration. Infrastructural administrative or technical measures, e.g., provision for prompt detection and remedy of leaks, may also be taken [236–239]. If required by the load assessment, retention systems can be rendered tight and chemically resistant by special coatings (resins, paints, etc.) or plastic or metal foil liners. A large industrial plant can set up a further barrier for retention of chemicals and contaminated fire-fighting water by connecting production equipment or storage tanks to biosewers leading to the plant’s wastewater treatment system. A special type of retention system involves double-wall construction of vessels, pipes, etc. The retention space is attached to the containing barrier (in the case of internally jacketed vessels, placed inside it) in such a way that the space between the two walls catches any leaks and can be fitted with an alarm [239–242]. Double-wall design is not always safer than single-wall construction. The reasons have to do with fabrication cost and additional stresses

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(e.g., those due to constrained expansion at penetrations and supports) as well as stability against denting and buckling. Inspection and maintenance of a double-wall component is difficult or impossible. 5.1.2. Rapid-Closing Valves, Emergency Compartmentalization Systems In plants containing large amounts of products where the release of a substance would represent a high hazard potential, it may be useful to subdivide the plant with rapid- closing valves and fittings. These can be actuated after the detection of a release, e.g., by a gas warning system [243], [244], and after the scope of the release has been evaluated. They can be activated manually or automatically. The effect is to isolate the area of the leak from the rest of the facility. The quantity released at a leak can thus be limited to a value estimated in advance. The use of a rapid- closing valve system presupposes detailed advance planning and design of the plant sections, knowledge of the amounts of products contained, the way in which a leak is detected, etc. The analysis may also include the emergency shut-down of the entire plant if dangerous conditions in the plant can be created by blocking individual sections [245], [246]. 5.1.3. Emergency Drain and Collection Systems The product in a plant or in an isolated portion of the plant may be drained or released into a retention system (e.g., a catch tank or blowdown tank permanently installed in the plant) if a leak or other hazardous condition arises (e.g., fire). The retention system may be fitted with a separator (e.g., a cyclone) if the properties of the substance and the system pressure demand separation of the phases of the substance. It is also useful to configure the system in such a way that the plant drains by gravity. A pressure-relief system may also be needed to reduce the pressure of the gas phase [247], [248]. 5.1.4. Blowoff and Disposal Systems In plants handling gases, flares, scrubbers, absorbers, and condensers are useful for pressure

relief and gas disposal. Systems of this type function not only as emergency pressure-relief systems, but are also used for the disposal of gases and vapors arising during normal operation, or when the plant is up or shut down, e.g., where large amounts of low-quality gas are produced for a short time. If these gases or vapors are flammable, the plant design must also meet the safety standards for explosion protection; other safeguards against spontaneous ignition and fire may also be necessary, e.g., with absorption on activated- carbon filters [249], [250]. 5.1.5. Spray-Curtain (Drench) Systems Spray curtains are often employed in plants with critical gases or vapors, where a release at an unpredictable location cannot be ruled out and may lead to a hazard to employees, third parties, or the environment. The plant section in question, or the whole plant, is surrounded by a pipe system fitted with closely spaced spray nozzles. If a pollutant escapes, a high-pressure mixture of water and steam issues from the nozzles, surrounding the plant in a dense water mist that can condense or capture the released substances. Neutralizing agents can also be added to the water; e.g., ammonia for phosgene plants. Due to the high-pressure and thermodynamic effects the effectiveness of these systems can be maintained, even in windy weather. Mobile equipment such as articulated booms and water cannons operated by the plant fire fighting service can be used to create spray curtains in smaller plants, or as a supplementary measure in larger ones [251]. 5.1.6. Partial and Complete Containment A technique well known from the nuclear industry is to place the entire plant inside a pressuretight containment, enclosed on all sides, but with personnel entry points. This approach has also been discussed for chemical plants and portions of plants where very dangerous substances are handled. This approach may be correct from the environmental protection standpoint, but in practice there are limits, since some chemical plants

Plant and Process Safety cannot be remotely controlled and cannot be run without operating personnel. For reasons of occupational safety and health, a chemical plant containment cannot be hermetically sealed from the outside world while the plant is in normal operation. Entry for operation and maintenance must be possible, and continuous ventilation is required, both for waste heat removal and on occupational health grounds. A release would have to be detected very promptly in order for all openings to be shut and the stackgas cleanup system turned on so that retention could be achieved. A further problem arises with flammable substances. Even a small cloud of an explosive mixture released because of a malfunction is diluted much more slowly in the containment than in an open plant, so it remains flammable over a long time. Containment also exacerbates the effect of an explosion; because there are no pressurerelief openings, the pressure wave and heat of an explosion affect humans and structures many times more than in the open case. A large fire in a containment plant inevitably causes a huge amount of heat to accumulate in the building. Thermal stresses on the structure and plant equipment result, possibly causing their complete failure. Mobile fire fighting is far more difficult in such a facility. It is possible that an initially minor event in a containment plant may lead to substantial personal injury and a dangerous release of substances. A variety of plant subdivisions, where special process or design conditions bring about a severe hazard potential for the environment, are already being built with double walls or enclosed in boxes or compartments. Aside from this form of enclosure and double-walled construction, whole-plant containment cannot be recommended for general use. Complete containment is an option only in rare cases, e.g., where there are no flammable or explosive substances; even in such cases it is necessary to strike a careful balance between advantages and disadvantages (pollution control, industrial safety, fire and explosion protection, plant reliability, and safe maintenance practices) [251– 253].

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5.1.7. Fire Protection; Retention of Water Contaminated by Fire Fighting Effects on employees, third parties, and the environment must also be expected when fire occurs in the plant. These can result from the fire itself (heat, combustion gases, etc.), or from the extinguishing agents (e.g., the Sandoz accident at Basel). Fire protection legislation requires that the fire fighting service seek to extinguish a fire virtually always and under all circumstances. From the environmental protection standpoint, it may be preferable not to extinguish a fire that has already passed a certain size, but to let it burn at a high temperature while keeping it from spreading to the surroundings; the amount of environmentally harmful combustion gases generated is smaller, and no contaminated extinguishing agent is produced. There must, however, always be fire protection measures to prevent the start and growth of a fire, or to safeguard the surroundings from the effects of fire (Section 3.4.1). Fixed facilities for the retention of water contaminated by fire fighting must be provided, depending on the substances involved, their water pollution classifications, the construction of the first barrier, and other factors [254]. Mobile retention systems can also be used as a barrier. A number of options, e.g., mobile folding vessels and rapid-set-up tanks, are now on the market. Even simple devices such as magnetic plates and compressed-air bags to close off sewer intakes, have their place in contaminated fire-fighting water retention.

5.2. Hazard Control Plans at Plant Level and Beyond 5.2.1. Hazard Control Plans In what follows, plans established by a plant or works to control hazards inside the battery limits are referred to as plant alarm and hazard control plans. In Europe, plants subject to the “Seveso Directive” (implemented in Germany as the Industrial Emergencies Regulation) are required to create an alarm and hazard control plan [255], [256]. An alarm and hazard control plan describes administrative and technical actions to minimize the impacts of accidents; relates to various levels

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of organization (e.g., constituent plants and the larger facility to which they belong) and is prepared with an eye to the immediate surroundings of each unit. The actions prescribed include: Management Personnel Technical resources Communications Documentation Procedures and jurisdictions In facilities containing a large number of varied units, alarm and hazard control plans are prepared at “plant” (smaller unit) and “works” (larger unit) levels [257].

to emergency action for management, and provides a pattern for the allocation of technical resources. 5.2.1.2. Works Hazard Control Plan The alarm and hazard control plan of the works relates to a higher organizational level than the plant, and documents the emergency response organization adopted by the works when an event occurs. It defines, for all personnel involved: The essential tasks to be performed The way in which efforts are coordinated It also establishes duties relating to information and prescribes the forms of cooperation with:

5.2.1.1. Plant Hazard Control Plan The plant alarm and hazard control plan sets forth the plant safety organization. It contains information necessary to plan and specify the emergency response. The coordinated administrative measures taken in response to an accident are such as to afford the highest possible level of protection to humans and facilities. The hazard control plan identifies all potential threats within the plant (e.g., those due to fire, explosion, or other accident) and external to the plant (e.g., trouble in nearby plants, flooding), lists safety equipment in the plant, and prescribes the desired actions of plant personnel in a dangerous situation. The plant hazard control plan is aimed primarily at the people in charge (e.g., plant managers and engineers, laboratory and workshop managers), who must ensure that their personnel are properly instructed as to how to act in a particular hazard situation. The plan is a document for use in plant personnel instruction and drills, as provided in the Industrial Emergencies Regulation. The plant hazard control plan serves as a guide for people in charge, giving instructions for actions in a particular hazard situation. It lays down guidelines both for measure within the plant and for measures by technical task forces and service groups (e.g., fire safety, occupational safety, plant security, maintenance). The plan is also an informational aid and guide

External organizations Government agencies

5.2.1.3. Required Contents of the Plan Many suggestions have appeared concerning what alarm and hazard control plans should contain [258], [259]. In the German state of North Rhine – Westphalia, a working group, including independent experts as well as agency and industry representatives, has developed a model plant alarm and hazard control plan [257], which can serve as a pattern for plant or works plans. Major headings in the model plan are: Introduction. Mailing address, scope, revision (update) status General Information on the Facility and its Surroundings. Orientation for emergency workers on: 1) 2) 3) 4)

Access routes and staging areas Information on the work force Danger sources, major hazards Safety equipment and plans in place

Alarm Plan. The routing of alarms in the event of an accident is summarized, as is mandatory internal and external reporting. Hazard Control by Internal Activities. This lists departments in the works (works manager, plant, works fire service, central safety office, industrial medicine, safety engineer, etc.) and defines their jurisdictions and assignments.

Plant and Process Safety Warnings. Options for informing persons in the works or plant about possible hazards are discussed, along with ways of delivering warnings and other information to those living nearby. Instructions for Particular Events. In view of local circumstances, precautions and responses are described for a variety of events (flood, smog, power outage, product release, need for outside assistance, etc.). Notification of Government Agencies and the Public. This describes the internal location, where all information required to be reported externally can be retrieved. Resources and Technical Specialists. A table lists all current technical documents, devices, equipment, and technical specialists, and records their availability and contact information. Telephone List. Telephone numbers that may be vital for government agencies (for inquiries within the works) and for responsible works management personnel (e.g., for reports to agencies) must be listed here. Definitions, Regulations, Conventions. This lists concepts and procedures defined in regulatory documents; key passages may be quoted verbatim. Appendixes. These include maps, keyword index, and other items. 5.2.2. Off-Battery Hazard Control Plans An example of an off-battery hazard control plan is the special accident response plan, created jointly by the disaster control agency and the plant operator. Such a plan relates to a plant or works in which, because of the production process, storage, and other features, hazards to nearby inhabitants may occur and disasters cannot be ruled out. The special accident response plan repeats portions of the works alarm and hazard control plan that are vital for response by the agency; these include: 1) Description of the facility 2) Routing of reports and alarms 3) Immediate response to accident/disaster Identification of threat Identification of threatened area Measurement of pollutant levels Warning/notification of populace

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4) Subsequent actions The disaster response plan of a municipality covers all events that may figure in a disaster. It is not restricted to any one plant or works.

5.3. Public Awareness and Responsible Care Responsible Care. The chemical industry’s worldwide responsible care program obliges companies to strive for steady improvements in health, safety, and environmental protection. It mandates that corporations seek a stronger dialog with the public, to enhance public knowledge of product and manufacturing safety. The centerpiece of an active environmental philosophy, as the Association of the German Chemical Industry made binding on its member companies in 1992, consists of ten Guiding Principles and six Management Practice Codes intended to implement them. CEFIC (the European Chemical Industry Council) has published similar guiding principles for member societies [260]. Member companies are bound by the following guiding principles: 1) To acknowledge the public interest in chemical products and the activities of manufacturers, and to respect this interest 2) To develop and manufacture only those products that can be safely produced, transported, used, and disposed of 3) In the planning of new products and production processes and the improvement of existing products and processes, to assign a high priority to health, safety, and environmental aspects 4) In any chemical-related health and environmental activity, to provide full information to government agencies, employees, customers, and the public, and to recommend suitable protective practices 5) To advise customers on the safe use and transportation of chemical products, as well as their safe disposal 6) To operate production facilities so as to safeguard the environment as well as the health and safety of employees and the public

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7) To conduct research to increase knowledge about the possible health, safety, and environmental impacts of products, production processes, and waste products 8) To cooperate with other companies in solving problems occasioned in the past by the handling and disposal of hazardous substances 9) To cooperate with the government and government agencies in devising responsible legislation, procedures, and standards that enhance the safety and protection of the public, workers, and the environment 10) To promote the principles and practices of the responsible care program by sharing experience with all the other companies manufacturing, using, transporting, and disposing of chemical products, offering all kinds of support to them The six Management Practice Codes deal with: Information and emergency response Process and plant safety Environmental protection Transportation safety Occupational health and safety Product responsibility CEFIC is promoting and developing the responsible care concept throughout Europe. Public Awareness. Article 13 of the Seveso Directive for Europe in general, and § 11a of the Industrial Emergencies Regulation in Germany, mandates that the operators of accident-relevant plants provide information on safety practices and proper accident response to persons who may be affected by an accident as well as to the public at large. This information must be made available in suitable form, and without having to be solicited [255]. The objective is to create a bulletin, understandable to the general public, containing the following information: – Name of plant operator and location of plant – Name and title of the person giving information – Brief description of the nature and purpose of the plant

– Identification of substances or preparations that can cause an accident, along with their essential hazardous attributes – Nature of dangers in an accident, including possible impacts on humans and the environment – Nature of warning given to affected persons and follow-up – Correct behavior and actions of affected persons in the event of an accident – Pertinent safety practices – Internal and external hazard control plans – Coordination of plans between plant operator, municipality, and agencies responsible for hazard control In areas of high industrial concentration, a joint bulletin may be issued by the operators of plants where emergencies may occur.

6. References General References 1. “Das Sicherheitskonzept f¨ur die chemische Technik,” Dechema Monogr. 88 (1980) no. 1818 – 1835. BASF (eds.): Sicherheit in der Chemie, Verlag Wissenschaft und Politik, Berend von Nottbeck, K¨oln 1979. Bayer AG (eds.): Sichere Chemietechnik, Leverkusen 1982. F. P. Lees: Loss Prevention in the Process Industries, vols. 1 and 2, Butterworths, London 1980. G. L. Wells: Safety in Process Plant Design, J. Wiley & Sons, New York – Toronto 1980. S. Lange: Ermittlung und Bewertung industrieller Risiken, Springer Verlag, Berlin – Heidelberg 1984. “Fortschritte der Sicherheitstechnik I,” Dechema Monogr. 107 (1987). D. A. Crowl, J. F. Louvar: Chemical Process Safety: Fundamentals with Applications, Prentice Hall Inc., New Jersey 1990. J. Whiston: Safety in Chemical Production, Blackwell Sci. Publ., Oxford 1991. H. Pohle: Chemische Industrie – Umweltschutz, Arbeitsschutz, Anlagensicherheit, VCH, Weinheim 1991. G. Strohrmann: Anlagensicherung mit Mitteln der MSR-Technik, 2nd ed., R. Oldenburg Verlag, M¨unchen – Wien 1983. DECHEMA (ed.): “Anlagensicherung mit Mitteln der MSR-Technik,” Praxis der Sicherheitstechnik, vol. 1, Frankfurt/Main 1988. NAMUR (eds.): NAMUR-Empfehlung: Anlagensicherung mit Mitteln der Prozeßleittechnik, Erstausgabe

Plant and Process Safety 30. 10. 92, Vertrieb: NAMUR-Gesch¨aftsstelle. Berufsgenossenschaft der chemischen Industrie (BG Chemie) (eds.): Ratgeber Anlagensicherheit; Grundlagen und Anwendungshilfen zur Anlagensicherheit, Verlag Kluge, Berlin 1992. Specific References 2. Berufsgenossenschaft der chemischen Industrie (BG Chemie), Heidelberg, annual reports 1983 – 1992. 3. Her Majesty’s Stationery Office (HMSO), The Flixborough Disaster – Report of the Court of Inquiry, London 1975. 4. V. C. Marschall: Major Chemical Hazards (Ellis Horwood series in chemical engineering), Ellis Horwood, Chichester 1987. 5. H. Steen et al.: Das Bhopal-Ungl¨uck 1984, Umweltbundesamt (eds.), Berlin 1987. 6. Bundesminister f¨ur Umwelt, Naturschutz und Reaktorsicherheit (eds.): “Rhein-Bericht, Bericht der Bundesregierung u¨ ber die Verunreinigung des Rheins durch die Brandkatastrophe bei der Sandoz AG/Basel,” Bonn, Feb. 12, 1987. 7. DIN/VDE 31 100,part 2 (Deutsche Norm), Begriffe der Sicherheitstechnik, Beuth Verlag, Berlin 1987. 8. H. Schacke et al.: “Redundanz im Staubexplosionsschutz? – Konzept komplement¨arer Schutzmaßnahmen,” Staub Reinhalt. Luft 53 (1993) 453 – 459. 9. Statistisches Bundesamt (eds.): Statistisches Jahrbuch 1993 f¨ur die Bundesrepublik Deutschland, Wiesbaden 1993. 10. T. A. J¨ager: “Zur Sicherheitsproblematik technologischer Entwicklungen,” Qualit¨at Zuverl¨assigkeit 19 (1974) no. 1, 2 – 9. 11. Polizeiliche Kriminalstatistik des Bundeskriminalamtes, Wiesbaden (f¨ur die Jahre 1985 bis 1991). 12. Amtliche Mitteilungen der Bundesanstalt f¨ur Arbeitsschutz, no. 2, Dortmund, April 1993. 13. V. Pilz: “Grundlagen f¨ur die Vorhersage der Auswirkungen von St¨orf¨allen,” VFDB 2. (1981) no. 3, 116 – 125. 14. H. Giesbrecht et al.: “Analyse der potentiellen Explosionswirkung von kurzzeitig in die Atmosph¨are freigesetzten Brenngasmengen,” Chem. Ing. Tech. 52 (1980) no. 2, 114 – 122 (part 1); 53 (1981) no. 1, 1 – 10 (part 2). 15. W. C. Brasie, D. W. Simpson: “Guidelines for Estimating Damage Explosion,” Reprint 21 A,

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32. K. Nabert, G. Sch¨on: Sicherheitstechnische Kennzahlen brennbarer Gase und D¨ampfe, 2nd rev. ed., Deutscher Eichverlag, Braunschweig 1980. 33. VDI-Kommission Reinhaltung der Luft, VDI 2263, Blatt 1, Untersuchungsmethoden zur Ermittlung von sicherheitstechnischen Kenngr¨oßen von St¨auben, D¨usseldorf 1990. 34. Richtlinie 84/449/EWG, Pr¨ufmethoden f¨ur physikalisch chemische Eigenschaften, Anhang A, ver¨offentlicht im Amtsblatt der Europ¨aischen Gemeinschaft L 251/89. 35. J. Troitzsch: International Plastics Flammability Handbook, 2nd ed., Hanser Verlag, M¨unchen 1990. 36. W. Jost: Explosions- und Verbrennungsvorg¨ange in Gasen, Springer Verlag, Berlin 1939. 37. B. Lewis, G. v. Elbe: Combustion, Flames and Explosions of Gases, 2nd ed., Academic Press, New York 1961. 38. H. H. Freytag: Handbuch der Raumexplosionen, Verlag Chemie, Weinheim 1965. 39. D. A. Franck-Kamenetzkii: Stoff- und W¨arme¨ubergang in der chemischen Kinetik, Springer Verlag, Berlin 1959. 40. W. Bartknecht: Explosionsschutz – Grundlagen und Anwendung, Springer Verlag, Berlin 1993. 41. F. Funk: “Berechnung sicherheitstechnischer Kennzahlen,” Chem. Tech. (Leipzig) 29 (1977) no. 9, 494 –497. 42. K. Nabert, G. Sch¨on: Sicherheitstechnische Kennzahlen brennbarer Gase und D¨ampfe, Deutscher Eichverlag, Braunschweig 1980/89 mit Nachtr¨agen. 43. H. F. Coward, G. W. Jones: “Limits of Inflammability of Gases and Vapors,” Bull. U.S. Bur. Mines 279 (1952) no. 503. 44. DIN 51 649,part 1, Bestimmung der Explosionsgrenzen von Gasen und Gasgemischen in Luft, Beuth Verlag, Berlin 1986. pr EN 1839, Determination of Explosion Limits of Gases, Vapours and their Mixtures. 45. H. Le Chatelier, Ann. Mines. (Ser. 8) 19 (1891) 388 –395. 46. W. Berthold, U. L¨offler: Lexikon Sicherheitstechnischer Begriffe in der Chemie, Verlag Chemie, Weinheim 1981. 47. D. Conrad: “Inertisierung explosionsf¨ahiger Gassysteme als Maßnahme des prim¨aren Explosionsschutzes,” 2. Int. Kolloquium der IVSS, Frankfurt 1973.

48. Hauptverband der gewerblichen Berufsgenossenschaften, Explosionsschutz-Richtlinien (EX-RL), C. Heymanns Verlag, K¨oln 1990. 49. M. M¨uller, H. Schacke, C.-D. Walther: Explosion Characteristics of Carbon Monoxide at High Temperatures, Loss Prevention and Safety Promotion in the Process Industries, Proc. Int. Symp. 6th, Oslo, June 1989, vol IV, p. 104/104-12, European Federation of Chemical Engineering, 1989. 50. D. Conrad, R. Kaulbars: “Druckabh¨angigkeit der Explosionsgrenzen von Wasserstoff,” Chem. Ing. Tech. 67 (1995) no. 2, 185 – 188. 51. H.-J. Heinrich: “Bemessung von Druckentlastungs¨offnungen zum Schutz explosionsgef¨ahrdeter Anlagen in der chemischen Industrie,” Chem. Ing. Tech. 38 (1966) 1125 – 1133. 52. H.-J. Heinrich: “Zur Bemessung von Druckentlastungs¨offnungen bei Gas- und Staubexplosionen,” Wissenschaftliche Berichte aus der Bundesanstalt f¨ur Materialpr¨ufung (BAM), Berlin 1973, pp. 277 –280. 53. S. Crescitelli, G. Russo, V. Tufano, J. Occup. Accid. 2 (1979) 125 – 133. 54. W. Jost: Probleme der Flammenfortpflanzung, report no. 226, Deutsche Versuchsanstalt f¨ur Luft- und Raumfahrt e.V., Oberpfaffenhofen 1962, pp. 3 – 24. 55. J. Nagy et al.: “Explosion Development in Closed Vessels,” Rep. Invest. U.S. Bur. Mines 7507 (1971) 1 – 50. 56. D. Conrad, S. Dietlen: “Untersuchungen zur Zerfallsf¨ahigkeit von Distickstoffoxid,” BAM-Forschungsbericht, no. 89, Verlag f¨ur neue Wissenschaft, Bremerhafen 1983. 57. H. R. Christen: Thermodynamik und Kinetik chemischer Reaktionen, Diesterweg u. Salle, Frankfurt 1974. 58. D. Conrad: “Vermeidung von Gefahren beim Umgang mit zerfallf¨ahigen Gasen,” 2. Sicherheitstechnische Vortragsveranstaltung u¨ ber Fragen des Explosionsschutzes, PTB Braunschweig, March 1983. 59. D. Conrad, R. Kaulbars: “Untersuchungen zur ¨ chemischen Instabilit¨at von Athylen,” Chem. Ing. Tech. 47 (1975) 265. 60. D. Oberhagemann: “Z¨undtemperaturen von Ein- und Mehrkomponentensystemen,” VDI-Fortschrittber. , Reihe 3, no. 185, VDI Verlag, D¨usseldorf 1989. 61. P. Field: Dust Explosions, Elsevier Sci. Publ., Amsterdam 1982.

Plant and Process Safety 62. J. Cross, D. Farner: Dust Explosions, Plenum Press, New York 1982. 63. W. E. Baker et al.: Explosion Hazards and Evaluation, Elsevier Sci. Publ., Amsterdam 1983. 64. M. Glor: Electrostatic Hazards in Powder Handling, Research Studies Press, Letchworth, 1983. 65. R. K. Eckhoff: Dust Explosions in the Process Industries, Butterworth-Heinemann, Oxford 1991. 66. W. Bartknecht: Explosionsschutz, Grundlagen und Anwendung, Springer Verlag, Berlin 1993. 67. VDI-Guideline 2263, part 1, Test Methods for the Determination of the Safety Characteristics of Dusts, Beuth Verlag, Berlin 1990. 68. W. Berthold: “Bestimmung der Mindestz¨undenergie von Staub/Luft-Gemischen,” VDI Fortschrittber. 134, Reihe 3, VDI-Verlag, D¨usseldorf 1987. 69. ISO 6184/1- 1985 (E) Explosion Protection Systems, part 1, Determination of Explosion Indices of Combustible Dusts in Air. 70. VDI-Guideline 3673, part 1, Pressure Release of Dust Explosions, Beuth Verlag, Berlin 1979 (Draft 1992). 71. ISO 6184/4- 1985 (E) Explosion Protection Systems, part 4, Determination of Explosion Suppression Systems. 72. J. Zehr: “Anleitungen zu den Berechnungen u¨ ber die Z¨undgrenzwerte und die maximalen Explosionsdr¨ucke,” VDI-Ber. 19 (1975) 62. 73. J. Sch¨onewald: “Vereinfachte Methode zur Berechnung der unteren Z¨undgrenze von Staub-Luft-Gemischen,” Staub Reinhalt. Luft 31 (1971) no. 9, 376. 74. DIN 55 990,part 6, Pulverlacke, Berechnung der unteren Z¨undgrenze. 75. L. Bretherick: Handbook of Reactive Chemical Hazards, 4th ed., Butterworths, London 1990. 76. R. King: Safety in the Process Industries, Butterworths-Heinemann, London 1990. 77. T. Grewer, O. Klais: Exotherme Zersetzung – Untersuchung der charakteristischen Stoffeigenschaften, VDI-Verlag, D¨usseldorf 1988. 78. W. H. Seaton, E. Freedman, D. N. Treweek: “CHETAH: The ASTM Chemical Thermodynamic and Energy Release Potential Evaluation Program,” ASTM Data Ser. DS 51 (1974). 79. M. A. Cook: The Science of High Explosives, Robert E. Krieger Publ. Corp., Huntington, New York, 1971.

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252. V. Pilz: internal report, Bayer AG, Leverkusen, Mai 1992. ¨ 253. DOW Offentlichkeitsarbeit: Die Herstellung von Polymeric-MDI . . . ein sicherer chemischer Prozeß. 254. Richtlinie zur Bemessung von L¨oschwasser-R¨uckhalteanlagen beim Lagern wassergef¨ahrdender Stoffe (L¨oR¨uRL), Ministerialblatt f¨ur das Land Nordrhein-Westfalen, no. 71 vom 20. 11. 1992, pp. 1719 ff 255. “Seveso-Richtlinie” 82/501/EWG, 1982, u¨ berarbeiteter Entwurf von 1992 in H. J. Uth: St¨orfall-Verordnung, Kommentar, 2nd ed., Bundesanzeiger Verlagsgesellschaft, K¨oln 1994. 256. Bekanntmachung der Neufassung der 12. Verordnung zur Durchf¨uhrung des Bundes-Immissionsschutzgesetzes (St¨orfall-Verordnung), Sep. 20, 1991, BGBl. I, p. 1891. 257. NRW Arbeitskreis “Alarm- und Gefahrenabwehrplan”: Betrieblicher Alarmund Gefahrenabwehrplan, Stand 23. 08. 1993, D¨usseldorf. 258. Leitlinie zur Erstellung betrieblicher Alarmund Gefahrenabwehrpl¨ane, VCI, Frankfurt/Main, Mar. 1988. 259. W. Steuer: Fachveranstaltung “Der chemische Unfall,” Haus der Technik, Essen 1991. 260. Responsible Care, A Chemical Industry Commitment to Improve Performance in Health, Safety and the Environment, CEFIC Schrift 5/1993. 261. DFG MAK- und BAT-Werte-Liste, 2001, Wiley-VCH Weinheim

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Environmental Management in the Chemical Industry Louis Jourdan, European Chemical Industry Council (CEFIC), Brussels, Belgium Filip Jonckheere, European Chemical Industry Council (CEFIC), Brussels, Belgium Ulrich Geffarth, European Chemical Industry Council (CEFIC), Brussels, Belgium Jaques Busson, European Chemical Industry Council (CEFIC), Brussels, Belgium

1. 1.1. 1.2.

1.3. 2. 2.1. 2.2. 2.3. 2.4. 2.4.1. 2.4.2. 2.4.3. 2.4.4. 2.5.

2.5.1. 2.5.2. 2.5.3. 2.6. 2.6.1. 2.6.2. 2.6.3. 2.6.4. 2.6.5. 2.7. 3. 3.1. 3.2. 3.3. 3.3.1. 3.3.2. 3.3.3. 3.3.4.

Introduction . . . . . . . . . . . . . . . Historical Development . . . . . . . . Factors Influencing Companies to Improve Their Environmental Performance . . . . . . . . . . . . . . . Scope of This Contribution . . . . . . Environmental Management in Chemical Companies . . . . . . . . . . Preamble . . . . . . . . . . . . . . . . . Environmental Management Systems . . . . . . . . . . . . . . . . . . Environmental Policy . . . . . . . . . Organization and Structure . . . . . Typical Organization . . . . . . . . . . . Staffing of HSE Functions . . . . . . . Employee Motivation and Training . . Research and Development . . . . . . . Standards, Regulations, Monitoring, and Environmental Programs . . . . . . . . . . . . . . . . . Register of Standards . . . . . . . . . . Measuring Environmental Performance . . . . . . . . . . . . . . . . Environmental Programs . . . . . . . . Environmental Audits . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . The Audit Plan . . . . . . . . . . . . . . Environmental Auditors . . . . . . . . . Audit Methodology and Process . . . Audit Report and Action Program . . Environmental Management Reviews . . . . . . . . . . . . . . . . . . Environmental Communication . . Environmental Communication: A New Aspect of Management . . . Audiences and Vehicles . . . . . . . . The Environmental Report . . . . . . Audiences . . . . . . . . . . . . . . . . . Corporate or Site Reports . . . . . . . . Contents of the Report . . . . . . . . . . Links to Environmental Management Systems . . . . . . . . . . . . . . . . . . .

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3.3.5. 3.4. 3.5. 4. 4.1. 4.2. 4.3. 4.4.

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5. 5.1. 5.2. 5.3. 5.4. 5.4.1. 5.4.2. 5.4.3. 5.4.4. 5.4.5. 6.

6.1. 6.1.1. 6.1.2. 6.1.3. 6.2. 6.2.1. 6.2.2. 7. 7.1. 7.2. 8.

Trends . . . . . . . . . . . . . . . . . . . . Stakeholder’s Reactions . . . . . . . . Who Communicates? . . . . . . . . . The Impact of Legislation on Environmental Management . . . . Operating Permits . . . . . . . . . . . Process Safety Legislation . . . . . . The EC Eco-Management and Audit Regulation . . . . . . . . . . . . . . . . . Mandatory Environmental Communication . . . . . . . . . . . . . Voluntary Action by Industry . . . . Internal Statements . . . . . . . . . . . Voluntary Agreements . . . . . . . . . Public Commitments . . . . . . . . . . Responsible Care . . . . . . . . . . . . Responsible Care; What Is It? . . . . . The Guiding Principles . . . . . . . . . National Programs . . . . . . . . . . . . The Scope of Responsible Care; Product Stewardship . . . . . . . . . . . The Challenges of Responsible Care . Standardization and Certification of Environmental Management Systems . . . . . . . . . . . . . . . . . . Standards for Environmental Management Systems . . . . . . . . . Generic or Specific Standards . . . . . EMS Standards . . . . . . . . . . . . . . The ISO 9000 and 14 000 Standards . Certification of Environmental Management Standards . . . . . . . . Certification of an EMS . . . . . . . . . Certification Against the EC Regulation . . . . . . . . . . . . . . . . . International Aspects of Environmental Management . . . . Environmental Requirements in Different Countries . . . . . . . . . . . Technological Cooperation . . . . . . References . . . . . . . . . . . . . . . . .

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1. Introduction 1.1. Historical Development A few decades ago, the term “environmental management,” not to mention “environmental management system,” would have raised eyebrows. Despite the fact that some advanced companies were already behaving quite properly as far as environmental protection was concerned – at least relative to the prevailing standards – the term had not yet been coined, and companies did not have the type of environmental organization considered necessary today. Nevertheless, environmental considerations had been taken into account, consciously or unconsciously, in the management of chemical industry for a long time: one might quote, for example, a French imperial decree from 1810 controlling “dangerous, noxious and infectious establishments” which could be considered as a precursor to the EC “Seveso” Directive. Closer to the present, the Solvay process for the manufacture of soda ash, which in the second half of the nineteenth century replaced the Leblanc process, brought substantial environmental improvements. In fact, safety considerations were integrated into industrial management long before the need to protect the environment was recognized. Following dramatic accidents and severe health damage that were common in the early stages of the industrial revolution, governments, supported by a few enlightened industrialists, established the first of many standards for process, product, and human safety. The efforts of certain politicians to protect workers against the effects of white lead, and of Chancellor Bismarck to lay the foundations for accident prevention and social security, are well known examples. Some major companies, such as E.I. du Pont de Nemours, have built their reputations on the basis of very strict safety standards. But it was not before the 1960s or 1970s that the impact of chemical products and chemical industry on the environment were widely publicized – first in 1962 with Rachel Carson’s “Silent Spring,” then with the UN Conference in Stockholm in 1972. Since that time, local and national governments as well as intergovernmental organizations, in particular the Organization for Economic Cooperation and Development (OECD)

and various United Nations Agencies, have established a number of legislative actions, conventions, or recommendations that have necessarily been integrated into the management of chemical companies. The concept of “Sustainable Development” was introduced in the 1987 report “Our Common Future” from the UN Commission for Environment and Development, led by G. H. Brundtland [1]. The report recognized the threats imposed on the environment by both increasing population and ongoing industrial development, but it also identified challenges that must be met if such development is to be sustained without compromising the needs of future generations. Industry has an indispensable role to play if these objectives are to be achieved, in particular with respect to energy consumption, food production, and chemicals. The Brundtland report paved the way to organization of the UN Conference on Environment and Development (UNCED), held in Rio de Janeiro in June 1992 [2]. Whatever can be said about the meager immediate results of the Conference, its products, in particular the “Agenda 21”, will have to be taken into consideration by industrial managers. Two chapters of Agenda 21 are of direct relevance to chemical industry, and a United Nations Commission for Sustainable Development (UNCSD) has been set up to implement the Conference’s decisions. The integration of environmental considerations into industrial management is influenced most obviously by the level of development of the country in question: even within the European Community, after more than twenty years of legislative actions to harmonize (“approximate”) national legislation, some differences still exist between the behavior of certain companies in different Member States. When one looks at developing countries, the differences are often even greater. Political regimes also have an influence on environmental awareness and requirements: the state of the environment left by the centrally planned economy regimes of the former communist countries clearly shows that free-market economy, democracy, and a high state of development all contribute to superior environmental protection in industrialized or industrializing countries.

Environmental Management in the Chemical Industry

1.2. Factors Influencing Companies to Improve Their Environmental Performance While companies undoubtedly must satisfy first their “bottom line” (profitability), as well as a second crucial one – socio-economic responsibility – they now must satisfy a third one as well, namely environmental responsibility. Several factors are pushing industrial managers to meet this third challenge and to improve the environmental performance of their companies. Some of these are listed below, in an order that might vary depending on a particular manager’s personal commitment, the company’s culture, the local legal and economic conditions, etc. Legal Requirements and Regulations. In all industrialized countries, environmental regulations are now being established at an ever increasing pace (Fig. 1), and they are setting ever tighter standards. Although enforcement requirements may differ between countries, no reputable company can afford not to comply with at least national and local regulations.

Figure 1. Evolution of environmental legislation in the European Community (EC)

Environmental Liability. Fears of prosecution triggered by civil or even criminal liability for environmental damage are a strong incentive toward excellence in environmental performance. Some well-known “environmental scandals” have been settled, either between the parties involved or by the courts, at levels that could endanger the financial survival of a company. A matter of especially great concern is “past pollution”; i.e., the ecological damage created by

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an industrial operation that was complying fully with the legal requirements of some past period. Even if not yet common, criminal liability for environmental damage may soon become a reality, as is already the case for safety failures. Several pieces of legislation (Council of Europe, European Community) based on the concept of “strict” liability will make the matter even more important (see Chap. 4 for more detailed information). Image. As the public, under the influence of the media and many environmental nongovernmental organizations (NGOs), becomes increasingly aware of the need to protect the environment, companies will need to build public images as “good environmental citizens” if they hope to maintain their licenses to operate. There have already been cases of companies boycotted by consumer organizations in response to some environmental affair. “Ecolabels” or “Green Awards” have become strong elements in marketing strategies, and environmental performance criteria are becoming subject to more and more consideration in relations between suppliers and consumers. Employees. Employees and unions have traditionally been particularly concerned by labor relations, and especially wages, working conditions, and safety, but they are becoming increasingly interested as well in the environmental performance and reputation of the companies with which they are associated. Some companies have established “Environment Committees” at the shopfloor level paralleling the longestablished Safety Committees. There are also clear indications that a good environmental image helps a company recruit high-quality personnel, especially at the management and executive level. Finance. It is now recognized by the financial community – shareholders, bankers, insurers – that good environmental performance is an asset, even if in the short term it entails some capital expenditures and increased operating costs. There are more and more cases of corporate annual general meetings of shareholders where questions on environmental performance are debated. Moreover, insurance companies now adjust their premiums according to the environ-

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mental reputations of their clients. Some organizations have proposed to their customers – mostly in the financial community – an “ecorating” of major industrial corporations, similar to financial ratings. Professional Associations. Industrial associations at the national or international level have no real power over their member companies, but they are nevertheless an effective platform for establishing common principles and guidelines that set a high-level yardstick of environmental performance for all their members. Ethical Considerations. Some industry leaders are already well-known for their personal commitments to environmental protection, and they manage their companies accordingly. Many other company executives are beginning to transfer to their management activities their personal ethics with respect to the environment.

1.3. Scope of This Contribution When speaking about environmental management it has become current practice to discuss health-protection and safety-management aspects together with purely ecological matters. This contribution devoted to “environmental management” will therefore include various aspects of health, safety, and environmental (HSE) management. It must be recognized, however, that in some companies these elements are not integrated, and that health protection (H), safety (S), or health protection and safety (H + S), are managed separately from environmental protection (E). Conversely, some companies include quality under the same management structure. Because of their specificity, safety and environmental aspects of transport and distribution are integrated into an HSE management structure by only a few companies.

2. Environmental Management in Chemical Companies 2.1. Preamble As noted in the previous chapter, the concept of environmental management is relatively new;

it has developed without established standards, in a “learning-by-doing” process. It is therefore no wonder that the environmental management structures adopted by individual companies differ significantly according to company size, geographical location, and coverage; the legislative framework of the country(ies) in which they operate; their culture; etc. What is described below is the typical structure, organization, and content of an environmental management system adopted by a typical large or multinational company with headquarters in one of the industrialized countries. Companies of smaller size or operating under different conditions should adapt it accordingly.

2.2. Environmental Management Systems Basically, the functioning of a company relies on two systems: 1) An operational system, which achieves the basic purposes of the company (e.g., the manufacture of a certain range of products under certain financial and social conditions) 2) A management system, which acts on the operational system to help it function and improve Management systems applied to the various aspects of management are an important key to the long-term success of any company. Fundamentally, such a system necessitates defining a policy, establishing standards and objectives, planning for achievement, and allocating whatever resources are required, as well as documenting, measuring, auditing, and reviewing for improvement. An environmental management system (EMS) is built along this “loop,” as represented by Figure 2 (see next page). Its main elements are described below.

2.3. Environmental Policy Although it must be recognized that establishing and publicizing a policy for HSE matters is not, or at least not yet, a part of the culture of some companies, a clearly stated environmental policy is undoubtedly the foundation of any environmental management system.

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lar the company’s employees, shareholders, and major customers.

2.4. Organization and Structure To implement a company’s environmental policy the necessary human resources must be allocated and organized in such a way that they work with the greatest possible effectiveness, in particular in their relationship with the operating system. Organization implies a structure that identifies areas of responsibility and authority for key personnel.

Figure 2. The environmental management system (EMS) “loop”

An environmental policy can be defined as a public statement of the intentions, principles of action, and objectives that govern the management of a company with respect to the environment. Although the details and extent of such a policy may vary from company to company, the policy itself is generally expressed in a concise statement. Typical elements include: 1) A commitment to conduct the company’s activities in a manner that safeguards the environment, the health, and the safety of its personnel, local communities, consumers, and the general public 2) Compliance with environmental legislation, or even an intention to go beyond legislative compliance when appropriate 3) Communication of environmental information to concerned stakeholders within society 4) Information for customers on the nature and safe use of the company’s products 5) Environmental education and training of employees and contractors 6) Support for environmental research 7) Access to environmentally sound technologies A corporate environmental policy is a document approved and endorsed by the highest level of management; the President, Chief Executive Officer (CEO), or Executive Committee. The policy is also made available to the public and communicated to all interested parties, in particu-

2.4.1. Typical Organization Again, the personnel structure of an environmental management system may differ quite considerably from company to company, but a typical organization is that described by Figure 3 (note, however, that in some companies the functions H + S and E are separated). The basic principles are as follows:

Figure 3. Structure of a typical health, safety, and environmental (HSE) organization

The Board of Directors is responsible for defining the company’s environmental policy. Some companies have established an “HSE Advisory Committee” to assist the Board in this function; members of the Advisory Committee are members of the Board or persons external to the company selected for their interest, competence, or representativity in HSE matters. The Managing Director (President, CEO) is responsible for the adoption and implementation

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of the company’s environmental policy. His personal involvement should be made clearly visible across the company (in the famous words of DuPont’s E. Woolard: “as Chief Officer of this company, I am also its Chief Environmentalist” [3]). A senior manager (Vice-President, Director) reporting directly to the President, heads the company’s central HSE department. This centralized function is not directly responsible for policy implementation, but ensures the proper working of the various elements of the EMS. It includes specialized activities such as toxicology/ecotoxicology, occupational health, emissions, waste management, environmental audits, product safety, regulatory affairs, etc. Most of the members of the central HSE departments act in an advisory capacity to support the “line” management. Business units (product divisions, manufacturing companies, geographical areas) are responsible for policy implementation. Here again, HSE directors or coordinators at the level of the division or site function in an advisory capacity. 2.4.2. Staffing of HSE Functions The number of employees holding a full-time HSE position varies widely, depending on such factors as process and product hazards, local regulations, etc. An approximate range is from 2 to 25 HSE staff members per 1000 employees, with an average around ten. Allocation of professionals between a central corporate HSE department and the HSE departments in various divisions and sites is again a matter to be decided by the company: Some have large centralized departments, with up to 150 experts competent in all aspects of the corporation; others prefer to decentralize to the greatest extent possible at the level of divisions or sites, maintaining only a light core group with mainly coordinating functions. Whether the HSE functions should be staffed by generalists or by professionals educated in HSE matters is much debated, especially since the educational system now provides explicit technical or university degrees in HSE disciplines. Equally important are concerns about the possibility of career development for specialists

in HSE. Many companies regard an HSE position as a necessary step toward general management; others ensure that HSE specialists have career development opportunities similar to those offered to generalists. 2.4.3. Employee Motivation and Training As noted previously, employees in business units and operational sites are responsible for the implementation of a company environmental policy. This requires that staff at all levels be clearly aware of their accountability, as well as properly trained and motivated. Most companies hold employees accountable through evaluation of their individual environmental performance, or at least safety performance, which is more easily measurable. Plant and site managers are often required to set HSE objectives for their specific areas of responsibility, with possible monetary bonuses linked to achieving the goals. Several companies have prize programs with monetary awards or other forms of recognition for environmental improvements. Training in HSE matters is a key element in the effectiveness of an EMS. Training is especially necessary when a generalist is appointed to an HSE position. Several companies include HSE elements in their training programs for managers at various levels. Training of all employees in HSE matters is more and more considered to be a major priority, even if for the moment only a few companies have implemented such comprehensive programs. The case of line middle-managers deserves special consideration, because these persons sometimes perceive environmental demands as coming from people outside the plant with little knowledge about what is happening in practice. It is therefore important that senior management demonstrate its commitment to environmental quality, for example, by facilitating a two-way communication channel on HSE matters, allocating the necessary human and financial resources, and properly rewarding positive achievements. Also, programs that include a strong motivation element, such as Responsible Care (see Section 5.4), are helpful in convincing line managers, as well as staff people like sales managers, of their environmental duties.

Environmental Management in the Chemical Industry Another difficult case is that of contractors working on behalf of a company for construction or maintenance work, waste disposal, transport, etc. It is normally part of a company’s policy to insist that contractors work under the same HSE standards as the company’s own personnel, but accountability, communication lines, training possibilities, etc. are obviously different. The problem requires special attention on the part of all those involved. 2.4.4. Research and Development Most major chemical companies have R & D programs devoted exclusively to HSE matters and representing some 10 – 25 % of all their R & D expenditures. Environmental R & D topics are linked most obviously to a company’s products and processes, and to its HSE priorities and programs.

2.5. Standards, Regulations, Monitoring, and Environmental Programs A fundamental element of an EMS is a comprehensive register of applicable legislation and company standards. It is against these governmental or company standards that environmental performance will be measured, permitting the necessary programs for correction or improvement to be developed. 2.5.1. Register of Standards The number and complexity of environmental statutes at the local, national, and international level is constantly increasing. It is the role of a central business unit or individual site HSE functions to determine those that are applicable in the various sectors of the company. It may also happen that, because of gaps in the applicable legislation or difficulties in adapting it to the specialties of a certain activity, a given company is forced to establish its own standards, which would of course go beyond the legislated criteria. In general such standards are established at the highest level of the company, and are applicable to all its sectors.

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To ensure proper implementation by line personnel, legislation and company standards may require more detailed guidelines or guidance notes. Such guidelines, generally drafted by professionals in the central HSE department on the basis of their own experience and outside information, provide site managers with key principles that are to be incorporated into local procedures. Maintaining the currency of all documentation of applicable laws and standards, guidelines, and local procedures, and communicating modifications and additions to the affected personnel, is a difficult task; a written procedure with clear allocation of responsibilities may be necessary. 2.5.2. Measuring Environmental Performance “You can’t manage what you can’t measure.” Therefore, to measure the effectiveness of an environmental management system, real data are required on the precise effects of the company’s activities on the environment and individual people. It is thus necessary to conduct a planned, regular sampling and monitoring program at the various company sites as well as outside the company (e.g., regarding the effects of certain products). It is generally accepted that the parameters to be monitored should include: Point sources and fugitive emission to the atmosphere Point sources and fugitive discharges to water or sewers Solid and other wastes, particularly hazardous wastes Contamination of land Use of water, fuels, and energy Discharge of thermal energy, noise, odor, and dust Effects on specific parts of the environment and ecosystems On-site accidents and near-misses Personnel injuries Transport accidents Complaints from community residents Complaints from customers related to HSE matters

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All these data should be obtained by prescribed or accepted (standardized) measurement techniques and collected in registers established at various levels in the company. Some companies collect only absolute figures, while others combine certain data with appropriate weighting factors into an index or several indices. Such indices may be useful management tools for fixing objectives and demonstrating improvements. Results of all measurements are communicated to the various interested parties, from plant operators to senior management, at the appropriate frequency and in the relevant form. 2.5.3. Environmental Programs On the basis of global, long-term targets, set at the highest achievable level (e.g., 50 % reduction in waste generation within the next 10 years), together with data from the register of environmental effects, individual environmental programs are established for the various sites and units of a company. Programs usually cover a one-year period within a multi-year global program. Proposals for individual programs are checked for feasibility, relevance, and priority, usually by the central HSE department. A proposal for a corporate environmental program is then submitted to senior management, which decides on the proposals to be accepted and allocates the necessary human and financial resources. The corresponding investments are included in the global investment program of the company. It is not easy to determine the amount of capital investment currently devoted to HSE by chemical companies, in part because there is no established methodology for computing these expenses. Furthermore, while it may be reasonably easy to determine the cost of end-of-pipe equipment, how is one to isolate the environmental component in the cost of a “cleaner,” integrated process? Estimates of capital expenditure related to HSE average 10 – 20 % of total capital investment for most companies. Operating expenses for HSE are in a similar range, and these are expected to increase, at least as long as processes intrinsically safe and “clean” are not generalized.

2.6. Environmental Audits All companies have long been accustomed to having their financial management systems audited by accredited experts. A few decades ago the auditing of safety practices became a current practice for many companies. Only a few are so far familiar with the concept of “environmental auditing,” but a trend in this direction is spreading rapidly across industry. 2.6.1. Definition The International Chamber of Commerce (ICC) has proposed the following definition for an environmental audit [4]: “A management tool comprising a systematic, documented, periodic, and objective evaluation of how well environmental organization, management and equipment are performing with the aim of helping to safeguard the environment by: 1) Facilitating management control of environmental practices 2) Assessing compliance with company policies, which would include meeting regulatory requirements” Such a definition might appear somewhat narrow when one considers that many corporate policies tend to cover unregulated risks. Indeed, the United States Environmental Protection Agency (EPA) has included such unregulated risks in its definition of environmental auditing. Environmental audits are therefore a necessary component of a company’s environmental management system, permitting a determination of whether or not the system is implemented effectively and is suitable for fulfilling the company’s environmental policy. Given the many activities of a company that have a bearing on the environment, environmental audits may serve various purposes: Compliance audits verify that an industrial activity meets its legal obligations (permits, emission limits, etc.). They do not differ in principle from inspections by government inspectors.

Environmental Management in the Chemical Industry Health and safety audits are still carried out separately in many companies, even though the tendency now is to integrate them into global environmental audits. Site audits are the most common type of environmental audits. They are designed to check individual site activities against those elements of a company’s environmental policy and management system applicable to the site in question. For large sites, audits may be limited to a single plant or a series of units. Specific issues may also be the subject of an environmental audit at the corporate or site level, such as waste management, transport and distribution activities, etc. 2.6.2. The AuditPlan To ensure proper verification of a global environmental management system, auditing of the various facets of a company’s activities should be carefully planned. The central HSE department is generally responsible for establishing an appropriate audit plan, which should deal with the following points: 1) Setting priorities 2) Establishing specific areas and activities to be audited and the frequency for auditing each particular site as a function of the nature and risk of its activities as well as conclusions reached in the previous audit; experience shows that a frequency of one to five years covers most situations encountered in the chemical industry 3) Objectives of the various audits 4) Personnel requirements 5) A protocol for conducting the audits 6) Procedures for reporting audit findings

2.6.3. Environmental Auditors It is generally accepted, even in legislation (see Section 4.4), that environmental audits should be performed either by personnel from within

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the company (but wherever possible independent of the unit being audited) or people external to the company. A growing number of organizations and consulting companies are offering their expertise for environmental audits, which may be useful when internal expertise or capacity is lacking. Because of the rapid development of this market, great care should be exercised in the selection of an external consultant. Simultaneously with the publication of the ISO 14 001 Standard for Environmental Management Systems, ISO issued in 1996 a series of guidelines for environmental auditing: – ISO 14 010: General principles on environmental auditing – ISO 14 011: Audit procedures – Auditing of environmental management systems – ISO 14 012: Qualification criteria for environmental auditors A large number of environmental auditors have already been accredited to carry out third party ISO 14 001 audits according to the abovementioned auditing requirements. In its efforts to make the ISO 9000 and ISO 14 000 series of standards more compatible, ISO decided in 1998 to develop one common auditing standard for quality and environmental management systems, replacing ISO 10 011 and ISO 14 010/14 011/14 012. Companies may want to carry out environmental audits using their own teams of auditors, supplied either by the central HSE department or drawn from various line activities within the company. It is highly recommended that the auditing requirements expressed in ISO 14 010/14 011/14 012 should be taken into consideration. A typical team includes three to four persons under the recognized authority of a leader. At least two members of the team should be experts in environmental matters, and one or more should be familiar with the particular type of activity to be audited. All members should have the necessary skills to obtain, from observation and interviews, clear, unbiased information on the operation in question without being regarded as “policemen” by the local employees. To facilitate contacts and technical arrangements an employee of the unit subject to audit may be nominated to act as a liaison with the auditing team.

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The time required for an environmental audit varies considerably according to the size of the site, the nature of the operation, the cooperation of local personnel, etc. It may range from a few days to several weeks. 2.6.4. AuditMethodology and Process The detailed actions required to carry out an environmental audit are highly dependent on the type and nature of the activities to be audited, but the following features are common to most audits: 1) Identification of elements of the management system, including instruction manuals, written procedures, communication, and reporting schemes 2) Evaluation of the control system 3) Assessment of strengths and weaknesses of the management system, especially deviations from written procedures and failures in the reporting lines 4) Examination of records of data on controlled or accidental emissions, energy use, complaints, etc. 5) Evaluation of training and motivation of personnel Auditors collect the necessary information through direct observation and interviews of personnel at the plant, drawn from all levels. Crosschecking of information is a common practice, but care should be exercised to avoid friction with and between employees. Contacts may be facilitated by starting the audit activities with a brief meeting with plant personnel, where the audit objectives, plan, and methods are explained. The full cooperation of local management is in any case required. Immediately at the conclusion of an audit the leader of the auditing team should present the major findings to the local (plant or site) management. Already at this stage a number of immediate corrections can often be achieved. 2.6.5. AuditReport and Action Program The final report is drafted by the team leader, and usually checked by the site manager before being sent to senior management (President and/or Board).

A typical report includes three parts: 1) An executive summary of major findings 2) A section on detailed observations, the part most useful to local management for improving the operating procedures 3) A proposal for corrective actions, including capital investment, major equipment modification, or changes to managerial procedures when appropriate The corrective action program is discussed at senior management level. When approved, in conjunction with an appropriate time scale, it is integrated into the global environmental program of the company.

2.7. Environmental Management Reviews The suitability and effectiveness of an environmental management system must be assessed at regular intervals. This is accomplished at regular management meetings involving senior managers of the company together with the head of the central HSE department. Such meetings are intended to review the latest data on environmental performance, trends, deviations from predetermined objectives and targets, how effectively the environmental program has been implemented, etc. When necessary, a review meeting may lead to decisions to modify or adapt the environmental objectives, the program, or even the policy if this seems justified. With this last element the environmental management system “loop” is closed.

3. Environmental Communication 3.1. Environmental Communication: A New Aspect of Management Traditionally, industry and particularly the chemical industry have not been models of good communication with society; indeed, secretiveness and confidentiality were deeply entrenched in the cultures of many reputable companies.

Environmental Management in the Chemical Industry Starting with the disclosure of financial information, however, enterprises have increasingly adopted a more open attitude. A similar evolution has occurred in the environmental domain: a few decades ago, no company would have considered going beyond minimal “instructions for use” on the labels of their products. Then came recognition of the “dutyto-inform,” mostly in the form of information provided to the local community in the vicinity of an industrial site regarding actions to be taken in case of an accidental emergency at the plant. Finally, the public “right-to-know” was recognized by industry, first in North America, then in Europe, and ultimately even in Japan, where public-opinion pressures are less severe. This is not to say that there is now full transparency on HSE matters between industry and society. It is worth noting in this context that environmental organizations talk in terms of “corporate disclosure,” whereas industry prefers to talk in terms of “corporate communication and reporting.” Some companies still believe that the public is interested only in being told about risks to which they are exposed, not absolute figures regarding emissions of dangerous substances. Industrial property rights are often used as an argument to limit disclosure. Nevertheless, environmental communication is now featuring in the environmental policy of a growing number of companies, and it is developing rapidly across the world chemical industry. A number of factors explain this trend: 1) The disclosure of environmental information has become compulsory under some existing national legislation (see Section 4.4). In the future more legal requirements are expected in many countries. 2) Within the broad concept of “sustainable development” environmental reporting has been identified as a major challenge for industry. As mentioned in the Agenda 21 document [5], “companies are encouraged to report annually on their environmental records, as well as on their use of energy and natural resources.” 3) The credibility of industry, in particular of the chemical industry, is poor in the eyes of the public. Providing data on environmental performance (showing not only positive

4)

5)

6) 7)

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achievements but also difficulties and failures) can help restore industry’s image and build credibility. Customers have begun to integrate environmental considerations, including the environmental performance of suppliers, into their purchasing decisions. Shareholders, financial institutions, and the insurance branch require information to help them assess the real present and future financial risks to which companies they are dealing with are exposed. Employees and local communities need information about their working and living environments. Environmental NGOs are likely to campaign more aggressively against corporations that give the impression they have something to hide about their environmental performance.

In fact, more and more companies consider environmental reporting to be a definite competitive advantage and a major factor in their “license to operate”.

3.2. Audiences and Vehicles Many different stakeholders in society have an interest in environmental information supplied by industry. The key ones include: Local and national governmental agencies Local communities The general public Media (general and specialized) Employees Shareholders Banks, financial institutions, and insurers Domestic and trade customers Schools, colleges, universities, and business schools Environmental NGOs Satisfying the needs of these various audiences may require specific vehicles for environmental communication. Some examples of appropriate communication forms are: Labels. The chemical industry has for several decades been using regulatory or standardized labels, in particular for hazardous products. Specification sheets, such as the standardized

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“Material Safety Data Sheets” (MSDS), provide supplementary information on the hazards and risks of chemicals for various user categories. Product advertising increasingly includes some environmental information. To limit the use of ill-founded or misleading statements, certain industrial organizations have drafted special guidelines on the matter [6]. The eco-labeling schemes operative in the EC and other countries represent a form of regulated environmental advertisement. Media, including the general press and TV, are becoming increasingly interested in the environmental performance of enterprises. However, they are attracted more by bad news than good news, and the publicity afforded by the media to unfortunate accidents or incidents may seriously endanger a company’s reputation. Company newsletters and magazines offer a privileged form of support to environmental information directed at employees or shareholders. “Open house” days and plant visits provide the best opportunity to convey environmental information about a site to local communities (including local media and environmental organizations), to employees, and to schools and other educational bodies. “Hot lines”, toll-free telephone numbers, and information centers are being made available by some companies to their customers and the public.

3.3.1. Audiences The annual environmental report is certainly the communication vehicle best suited to the needs of a number of audiences, in particular the employees, local communities, shareholders and financial institutions, environmental organizations, etc. This underscores a major difficulty in the concept of the annual environmental report: finding the right style and the right balance between technical information, financial data, and easyto-understand words, graphs, and figures. From the experience gained so far it is apparent that some reports would benefit from an improvement in style and presentation. 3.3.2. Corporate or Site Reports Perhaps for fear of local adverse reactions, some corporations publish only a report covering the company as a whole, with consolidated data. However, if the report is to meet the needs of local communities and local audiences it is obvious that data pertaining to the site in question must also be published. This fact is accepted by a growing number of companies. Some publish a single report with a section for the entire corporation together with various sections dealing with individual sites; others publish a corporate report supplemented by free-standing individual reports for their (main) sites.

3.3. The Environmental Report The concept of an environmental report, especially an annual environmental report, is becoming more and more popular among corporations in North America and Europe, and also Japan. Many companies consider the annual environmental report as a key element in their environmental communication strategy. Because this concept is still in its infancy (the first full free-standing environmental reports were published in the late 1980s or early 1990s), there is as yet no standard or commonly accepted format for such reports. What follows is therefore simply a review of current trends, which may evolve significantly in the coming years.

3.3.3. Contents of the Report Some reports are still mainly textual, providing only a few numeric figures for principal indicators, generally consolidated at the corporate level. Certainly reports should include a significant textual part, presenting, for example, the company’s environmental policy, its environmental program and organization, the major achievements, objectives, and targets. However, a growing need is perceived to publish numerical data for both corporate and site reports as a response to the now recognized “right-to-know” of the public. Indeed, publishing such numerical data within a report offers a definite advantage

Environmental Management in the Chemical Industry over provision of the “dry” figures required under certain legislation: the figures are put in the perspective of a real business (size of the site, products manufactured, etc.), thus avoiding – at least partially – easy misinterpretation by certain groups. It is mentioned in Section 2.5.2 that some companies establish “environmental indices” from measured data on emission. Certainly such indices are useful for demonstrating the trends in environmental performance, but using them as a substitute for the publication of absolute figures can generate harsh criticism from some concerned stakeholders. Another issue of concern is whether reports of all companies (at least in the same industrial sector) should include data covering the same parameters for a company or a site. This is certainly reasonable in terms of environmental relevance. Furthermore, one cannot deny the right on the part of many stakeholders (including regulatory authorities) to undertake certain comparisons between companies and sites, and this consideration would also militate in favor of a certain comparability of data. At least a “core list” of common parameters should be included in all reports. In order to improve transparency and consistency of the reports the European Chemical Industry Council (CEFIC) issued in 1998 the Responsible Care – Health, Safety and Environmental guidelines. It lists 16 core parameters and their definitions according to which all chemical companies will report. The collation of comparable data will allow monitoring continual improvements.

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3.3.5. Trends There are many indications that corporate environmental reports, published at regular intervals, will eventually become obligatory for all major companies, in particular in the chemical industry. The quality and quantity of information disclosed in the reports is likely to evolve. A research study [7] has identified five stages in the way environmental reports develop to meet the stakeholders’ information needs: 1) Green “glossies,” newsletters, videos; short statements in annual reports 2) One-off environmental reports, often linked to a first formal policy statement 3) Annual reporting, linked to an environmental management system but providing more text than figures 4) Provision of full environmental performance data on an annual basis, available on diskette or on-line, with an environmental report cited in the annual report 5) Sustainable-development reporting linking environmental, economic, and social aspects of corporate performance and supported by indicators of sustainability Presently, most examples of environmental reports meet the definition of stage 3; only a few have reached stage 4. To attain stage 5 companies will need to develop and adopt methodologies for full environmental cost pricing and lifecycle assessment of their activities and products.

3.4. Stakeholder’s Reactions 3.3.4. Links to Environmental Management Systems Environmental communication is not stricto sensu an element of the EMS. However, it is becoming more and more accepted that the results of environmental audits – at least those not containing confidential information, the disclosure of which could endanger competitiveness – must be made public. The EC Regulation on EcoAudits (see Section 4.3) requires the publication of a “public statement” after every audit. Site environmental reports would be a perfectly appropriate support vehicle for these statements.

The argument most commonly used by those reluctant to embark upon environmental communication is a fear of adverse reactions from various groups within society. Experience shows, however, that with the necessary skills and precautions, the release of environmental information and in particular of annual environmental reports generates only occasional negative reactions, and that the benefits definitely outweigh the difficulties. For example, local communities and employees always welcome environmental information, especially that related to specific sites. Shareholders and financial analysts have begun to

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take an interest in environmental reports, but are likely to demand deeper assessments. The media have reacted positively to the release of environmental reports, but may be critical of reports that are too “textual.” Environmental organizations are for the moment granting the benefit of doubt to corporations strongly involved in environmental reporting. Certainly the credibility of information released by companies, in particular by those with a poor public image, may become an issue, and it will take some time for companies to demonstrate that the information they publish is candid and accurate. This has prompted some companies to request validation of their information, in particular of their environmental reports, by independent third parties, such as accredited inspection organizations. For the moment, few reports have received such external validation, but several companies envisage having future reports validated by reputable, undisputed thirdparty organizations.

munications. There have been too many examples of accidents or isolated incidents that have ruined several years of effort by reputable companies to build trust with the public, simply because communications have been poorly managed during the event. Many companies now include communication as an element in their training programs, in particular in the training of their HSE specialists.

3.5. Who Communicates?

4.1. Operating Permits

Many companies still have a policy of relegating the responsibility for environmental communication to their professional communicators, backed as far as technical information is concerned by staff people in the HSE department. Professional communicators will always be needed to coordinate a company’s information policy and define its rules and practices, but there is a recent trend toward inviting line and staff managers to participate in environmental communication. For example, open house days obviously involve many employees, who must be prepared to answer questions about a company’s environmental performance. Site managers or superintendents may also be interviewed by journalists or media reporters, and employees are the company’s best ambassadors in the local community. Involving various levels of HSE managers in communication becomes all the more important when a communication crisis bursts following an accident in a plant, for example, or at any event attracting the attention of the media. All companies and sites should establish crisis management plans, duly tested and identifying all the major players, especially with respect to com-

Any industrial operation is subject to the granting of a permit by the competent authority. The dossier submitted with an application should include all information required to evaluate the environmental impact of the planned operation. It is then up to the authority to establish conditions that may be attached to the permit. These conditions might in some cases mandate a revision of certain processes or procedures as they were initially envisaged by the applicant. The permit process therefore implies a dialogue between the authority and the applicant, and it also implies the release of information to members of the public who might be affected by the planned operation. Even if the power of decision remains with the authority, the result of such consultation might be quite important for some “sensitive” operations, such as waste disposal sites. The whole permit process can extend over several years, which underscores the extent to which environmental questions can impact on industrial management decisions. Certain conditions attached to such a permit are of a managerial nature (e.g., emergency plans, reporting lines, etc.). Most of them, however, establish emission limit values (ELVs) for

4. The Impact of Legislation on Environmental Management Complying with regulatory standards is clearly an important element of any environmental management system. The purpose of this section is not to describe all legislative texts applicable to industrial activities, but rather to identify major trends in environmental legislation influencing the management of chemical companies.

Environmental Management in the Chemical Industry a list of relevant substances or indicators (waste, energy, noise, etc.) that the planned operation is expected to meet at all times. In theory, these ELVs should result from a risk assessment process, starting with a definition of environmental quality objectives (EQOs) for the area concerned. Defining an EQO is a political decision by the authority: the fact that the area in question is of a residential nature, for example, or that rivers should continue to sustain aquatic life, and so forth. EQOs lead to the determination of environmental quality standards (EQSs) for every environmental compartment, such as the maximum concentration of SO2 permitted in the atmosphere or heavy-metal limits in water, the maximum level of noise, etc., which in turn make it possible to achieve the EQOs. EQOs are fixed at the local level, or more often internationally, on the basis of scientific assessments. Finally, a model is used to calculate various ELVs of the planned operation to ensure that the EQOs are met, taking into account the existing levels of various parameters in the area concerned. This process, although complicated and possibly leading to discrepancies between the conditions of similar operations in different areas, is the only one that can assure that the environmental risk of an industrial activity is limited to an acceptable level while at the same time being optimally cost-effective. Nevertheless, another process is becoming increasingly popular for establishing ELVs: the use of “best available technique” (BAT). A BAT may be defined as that set of technical and managerial processes and procedures available at a given time which provides for the most costeffective manufacture of a certain product with minimal impact on the environment. The use of BAT is attractive to the authorities because it simplifies the process of establishing ELVs. It may also attract industrialists, because it ensures a “level playing field” for all companies in a particular business. However, it does raise certain questions, such as how long a BAT remains a BAT before becoming outdated by a new development, the possibility that some developers might enjoy a monopolistic situation, or the best way of ensuring that EQOs are met in heavily industrialized areas. Whatever the process chosen for determining the ELVs, it is necessary that an integrated approach be adopted so that the impact on all

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compartments of the environment is minimized: taking care that atmospheric pollution, for example, is not transferred to the aquatic environment. A series of modern legislative texts follows this trend, including the EC Directive on Integrated Pollution Prevention and Control (IPPC).

4.2. Process Safety Legislation Safety, because of its probabilistic aspects, cannot be regulated simply by standards. Assessing the accidental risk of an activity with respect to both equipment and management practices is therefore the key to safe operation (→ Plant and Process Safety). In principle, regulation is impossible because, at least in the chemical industry, every case is a special one. However, legislation can require an operator to carry out a risk assessment and present the results to the authorities. Here again, a dialogue between the competent authority and the applicant will determine the level of acceptable risk and various conditions the activity should satisfy to meet this acceptable risk. EC Directive 96/82/EC, the “Seveso” Directive, now being revised, is typical of legislation that fixes not a standard, but a management practice (i.e., the risk assessment procedure) in order to achieve its objective.

4.3. The EC Eco-Management and AuditRegulation In 1993, the EC Council adopted a Regulation (EEC/1836/93) introducing a “Voluntary Community Scheme of Eco-Management and Audit (EMAS).” The initial intent was to facilitate development of environmental auditing practices within industry, but it was soon recognized that environmental audits could not be isolated from environmental management systems, and the title and content of the regulation were modified accordingly. An enterprise deciding to take part in the scheme is committed to take the following actions: 1) Carry out an initial review of its environmental performance

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2) Establish an EMS 3) Conduct periodic environmental audits of its various sites 4) Produce a “public statement” after each audit 5) Have the public statement “verified” by an independent and accredited third party When the competent national authorities have satisfied themselves that a particular enterprise has complied with the requirements of the regulation they then grant what is known as a “statement,” which the company is free to use at its discretion. The regulation includes annexes specifying the elements of an EMS, the content and methodology of environmental audits, the role of verifiers, and the requirements for accreditation. At the end of 1998, some 2 000 industrial sites won EMAS registration. Around two thirds of these registrations are in Germany, whereas the take-up elsewhere has been patchy. Many companies went to the ISO 14 001 certificate instead and were not willing to duplicate part of the efforts in order to get an EMAS registration, which is limited to Europe and does not offer tangible additional benefits. The Regulation is currently revised and is likely to show the following features: – EMAS will no longer be limited to “sites” but will be open to “organizations”, which is in line with ISO 14 001 – The environmental management requirements of ISO 14 001 will be taken over as this will facilitate the step towards EMAS – EMAS will retain a number of specific aspects which, in addition to the (extended) environmental statement, will differentiate it form ISO 14 001 Industry welcomes the alignment with ISO 14 001 but remains concerned that in the absence of any regulatory relief for EMAS registered companies many companies will stay with ISO 14 001

4.4. Mandatory Environmental Communication Operating permits require an operator to monitor emissions and all other parameters subject to ELVs, and to report the measured results to

the competent authority. In many countries, including the EC Member States, legislation of the “Freedom of Information” type make any information (especially environmental information) collected by governmental authorities available to the public, either upon request or directly at the initiative of the authorities themselves. In addition, many statutes related to accidental risk, such as the EC Directive on Major Hazards (the “Seveso” Directive) require active informing of any population likely to be affected by an accident from a chemical site [8]. Through this mechanism a form of mandatory environmental communication already exists in many countries. Some countries have gone further in requiring industrial activities to measure the emissions of a specific list of substances and then report them to a “register” freely available to the public, sometimes on-line or on diskettes. In the United States, for example, the 1986 Emergency Planning and Community Right-to-Know Act (EPCRA) introduced under Section 313 of Title III the concept of the Toxics Release Inventory (TRI) [9]. Under this legislation, companies employing more than 10 full-time employees must provide the EPA with annual emission data for some 300 toxic chemicals manufactured, processed, or used in excess of certain threshold amounts. Data on emissions released and waste generated by individual sites are made available to the public through printed reports and electronic databases. Certainly the introduction of the TRI has met the expectations of many stakeholders; it has also been a powerful incentive to industry in the United States to monitor and reduce its emissions and waste generation. It should be recognized, however, that making raw data available out of context opens the door to considerable misinterpretation, for example, identifying the largest companies or sites as “major polluters.” Certain other countries, in Europe and elsewhere, also maintain registers of the TRI type, albeit ones that are less comprehensive and less readily available. The EC has been considering introduction of a “Polluting Emission Register,” possibly modeled along the lines of the TRI.

Environmental Management in the Chemical Industry

5. Voluntary Action by Industry Environmental legislation, however comprehensive and detailed it may be in many industrialized countries, cannot cover all the cases encountered in day-to-day industrial practice. Moreover, legislation is rigid, and it cannot be adapted to follow the pace of technological and managerial development. It is also not the most cost-effective approach, because it requires that resources be committed to inspection and enforcement. Finally, the pressure of legislation is not perceived as a real motivation on the part of the workforce. This has induced industry, led by a few large, advanced companies, to develop voluntary environmental activities and programs. These steps have been further encouraged by industrial associations and supported by certain enlightened governments. Some of the resulting acts can be seen as full substitutes for legislation. More often they complement existing legislation in facilitating its implementation and bridging gaps that legislation cannot cover. Voluntary corporate action can take several forms. These are described briefly below, before focusing on the worldwide chemical industry initiative known as Responsible Care.

5.1. Internal Statements Several elements of the environmental management systems discussed in Chapter 3 constitute voluntary actions, provided they are not made mandatory, for example, through an extension of the EC Eco-Management and Audit Regulation (see Section 4.3). These include in particular: 1) The HSE policy (see Section 2.3); this policy, endorsed at the highest level of corporate management, states the principles and objectives that govern management as far as HSE matters are concerned 2) The environmental program (see Section 2.5.2), which defines actions to be taken in a certain area to reach an objective within the framework of a set timetable Both overall environmental policy and specific environmental programs are internal management tools, even if many companies publicize them (or at least a few of their environmental

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programs, avoiding the disclosure of competitively sensitive information) in their annual environmental reports. Experience shows that the drafting of a set of environmental programs contributes greatly to the development of environmental awareness within the workforce.

5.2. Voluntary Agreements At the other extreme of the spectrum are “voluntary agreements” signed between a company and an authority for the purpose of achieving a target, with the choice of means left to the company. The term “voluntary agreement” may not appear strictly appropriate: in one sense, entering into an agreement is always a voluntary action; on the other hand, the alternative to not signing such an agreement is, in most cases, a law. Other terms are sometimes used as well, in particular “covenant.” The components of a voluntary agreement are: 1) The aim; e.g., an emissions reduction target, a labeling scheme, the phaseout of a certain product, etc. 2) The parties to the agreement: a) On the governmental side, a local authority (e.g., for reduction of discharges to water from a site), a national government (for a global plan involving an industrial sector), or the EC Commission (although monitoring the implementation may be left to national authorities). b) For industry, the most straightforward case is when the agreement is signed by a company, but when the agreement covers an entire industrial sector at the national or regional level, the responsible party is usually the trade association concerned. This immediately raises the problem of the extent of the authority an association may be able to wield over its members. 3) The penalties in case of failure to achieve the aims of the agreement. Some agreements include financial penalties; this may lead to serious difficulties if the agreement has been signed by a trade association. More often the penalty is simply that a statute is subsequently enacted, which is more burdensome and less cost-effective for both parties.

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4) A monitoring system and an administration to facilitate implementation of the agreement. Experience has shown that this can be done through light, cost-effective organizations. The following examples are representative of voluntary agreements within the chemical sector: 1) The “33/50 Program” signed between the U. S. Environmental Protection Agency and a number of chemical companies for the voluntary reduction of their emissions of 17 of the most polluting substances by 33 % in 1992 and by 50 % in 1995 relative to 1988 levels 2) EC Commission Recommendation 90/437/ EEC of June 1990 formalizing a voluntary program of reduction in the use of chlorofluorocarbons by the foam plastics industry, represented by its European associations 3) EC Commission Recommendation 89/542/ EEC introducing a voluntary system of detergent labeling under the control of the European industry federation of detergent manufacturers 4) The various “covenants” signed between the Dutch government and Dutch industry associations for implementing the National Environment Policy Plan (NEPP) of 1989, which established a long-term policy and a program of reduction of emissions, negotiated within the various sectors

5.3. Public Commitments Half-way between internal statements and binding agreements are the various forms of public commitments by which a company, through its President or CEO, commits itself to adopting an environmental behavior expressed in a set of principles. There are indeed certain similarities between this form of voluntary action and the voluntary agreements discussed above: 1) The principles define the aims of the commitment. In general they are drafted in the form of a set of 10 – 15 short guidelines. For effective, practical implementation these must be supplemented by Codes of Practice or Guidance Notes.

2) The principles are drawn up by an organization such as the national or international trade association of the industrial sector concerned, and then endorsed publicly by the senior management of individual member companies. 3) The other party is the public, in general represented by environmental non-governmental organizations (NGOs) that check company environmental performances against the content of the corresponding guiding principles. Companies may suffer a severe penalty when failure to implement the principles is publicized by a credible NGO. The best known example of such a commitment is the “Business Charter for Sustainable Development” launched by the International Chamber of Commerce (ICC) in April 1991, and now endorsed by over 1100 major corporations around the world [10]. The Charter is a set of 16 principles covering various aspects of the integration of environmental considerations into management practice, such as: environmental policy, continuous improvement of environmental performance, employee education, research, technology transfer, impact assessment, product and services stewardship, sound design and operation, contractors and suppliers, emergency preparedness, dialogue with stakeholders, and reporting. The ICC Business Charter for Sustainable Development is recognized as a major driving force for continuous improvement in the environmental performance of enterprises through voluntary public commitment. Similar sets of guiding principles have been established by a number of national and international organizations. Some are less comprehensive than the ICC Business Charter, especially with respect to the requirements on research, product stewardship, and reporting; others stress specific aspects, such as emergency preparedness or overseas operations. Two of these Codes are perhaps worthy of special mention. One is the set of CERES principles (formerly the “Valdez” principles), drawn up in 1989 by the Coalition for Environmentally Responsible Economies (CERES) after the Exxon Valdez oil spill [11]. Signatory companies, based mostly in

Environmental Management in the Chemical Industry the United States, are particularly committed to environmental reporting (see Chap. 3) and compensation for environmental damage. A second is the Global Environment Charter, launched in 1991 by Keidanren, the Japanese Federation of Economic Organizations [12]. The Keidanren Charter emphasizes the role of corporations in technology transfer, including transfer to developing countries, but does not require public reporting.

5.4. Responsible Care Several chemical companies are signatories to the ICC Business Charter or other sets of principles like those mentioned above. But most also take part in “Responsible Care,” the chemical industry’s own initiative that originated in 1984 in Canada and is now spreading throughout the world [13]. 5.4.1. Responsible Care; What Is It? Responsible Care relies on two fundamental features: 1) A commitment by company leaders to seek continuous improvement in the environmental performance of a company’s activities, including employee involvement and relations with customers and the society’s stakeholders. This commitment is expressed by the signature of the company President or CEO on a set of Guiding Principles. 2) A program established and operated by national chemical industry associations to assist their member companies in implementation of the Guiding Principles and verification of the true achievements resulting from their commitments.

5.4.2. The Guiding Principles There is not a single set of Responsible Care Guiding Principles: these are instead established on a national basis by the corresponding chemical industry associations with consideration directed to local circumstances. However, the Principles everywhere are remarkably similar in content and format; indeed, they are all modeled

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along the lines of other well-known principles, such as the ICC Business Charter for Sustainable Development or the CEFIC Guidelines for Protection of the Environment. 5.4.3. National Programs Similarly, the details involved in facilitating and monitoring specific programs are left to national chemical industry associations, who must adapt them to local considerations such as the national legislative framework, the state of industrial development, the companies’ cultures, etc. However, to ensure consistency in the initiative in the various countries, all national programs are expected to include seven fundamental features: 1) A formal commitment on behalf of each company to a set of guiding principles (see above. In the majority of cases the commitment is signed by the company CEO, but in some countries it is fulfilled by endorsement by the Board of the national association). 2) A series of codes, guidance notes, and checklists to assist companies in implementing the commitment. In some countries (e.g., the United States) a set of detailed codes has been drafted to cover the following areas: control of emissions, process safety, occupational health and safety, transport and distribution, community awareness and emergency response, and product stewardship. Elsewhere, especially in Europe, relevant associations had already developed a number of guidance documents on environmental management long before the introduction of Responsible Care. Producing and updating such codes and guidance documents is a continuing process. 3) The progressive development of indicators against which improvements in performance can be measured. In general, these indicators include sets of emission data for a number of selected substances and data on waste generation and recycling, as well as data on energy consumption and accidents. 4) An ongoing process of communication on HSE matters with interested parties outside the industry. The issue of environmental communication, discussed in Chapter 3,

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is therefore an integral part of Responsible Care. Obviously the recommendations provided by the association to its member companies regarding the communications process influence the selection of indicators of performance. 5) Provision of fora in which companies can share views and exchange experiences on implementation of their commitments. Some associations provide extensive service to their member companies in the organization of seminars, workshops, local cells, etc. This is particularly valuable for increasing the motivation of people who in general do not respond easily to environmental concerns, such as sales managers or line middle-managers. 6) Adoption of a title and logo that clearly identify a national program as being consistent with and part of the concept of Responsible Care. The standard logo (Fig. 4) has been registered in many countries, and is used throughout the world by the chemical industry. Many associations, even in non-English speaking countries, have retained the registered term “Responsible Care;” others have adopted a different expression that communicates in translation the spirit of the term Responsible Care.

tions have made or will make company commitments to Responsible Care a condition of membership. This implies that if a company does not fulfill its commitment, the association has the right to expel it. Before taking this extreme action, however, strong peer pressure will be exerted by other members on the noncomplying company. 5.4.4. The Scope of Responsible Care; Product Stewardship Responsible Care is sometimes understood as covering nothing more than the manufacturing activities of a company. Fundamentally, however, the Guiding Principles of Responsible Care apply to all aspects of industrial operation, from basic research to product development, manufacture, distribution, and use. They are thus more nearly consistent with a “cradle-to-grave” approach. In this respect, product stewardship is a vital component of Responsible Care. Product stewardship may be defined as “the responsible and ethical management of the health, safety, and environmental aspects of a product throughout its life cycle, from its design to manufacture and distribution, including resource consumption, transport, and final disposal options.” Responsible Care therefore aims at developing an “environmental culture” among all employees of a company, whatever their position and function. Researchers are motivated to assess the HSE effects of a new product at an early stage of development, and to find substitutes if the risk appears unacceptable. Engineers develop processes with a concern for minimal consumption of resources and the lowest possible impact on the environment. Production people operate their plants safely, and with full respect for environmental standards. People in charge of transport and distribution seek the safest means of transport. Sales managers advise their customers on the safe use of products, including problems raised after normal use to avoid creating waste problems.

Figure 4. The Responsible Care logo

7) Consideration of the best ways to encourage all member companies to commit to and participate in Responsible Care. Many associa-

5.4.5. The Challenges of Responsible Care The successful development of the Responsible Care initiative by chemical industry in many

Environmental Management in the Chemical Industry countries throughout the world entails two major challenges: 1) Finding ways to maintain the integrity of the initiative as national associations adapt their programs to local conditions and cultures – which may vary significantly among countries like the United States, Japan, and the nations of South America, Europe, or Africa. In addition, Responsible Care is a chemical industry initiative, but other industrial sectors more or less closely related to the chemical industry would benefit from similar programs adapted to their specific needs. The International Council of Chemical Associations (ICCA) has established a procedure for checking the consistency of programs developed by national associations against the seven fundamental features, both at the time of introduction of the program and periodically thereafter. The ICCA has also established rules for “partnership” with other industrial sectors that deal with chemicals (transport and distribution companies, waste disposal contractors, etc.) and are also interested in Responsible Care. 2) Finding ways to make Responsible Care acceptable and credible to the public, regulatory authorities, and environmental organizations. Responsible Care is not simply a publicrelations exercise: it is based strictly on improved performance – not on words, but on actions. Credibility therefore will be gained by demonstrable achievements, which will need to be communicated effectively. Many national programs include “public advisory panels” consisting of representatives of the various stakeholders; others have developed alternative channels of communication, but all companies are encouraged to adopt an open attitude toward information. Honest dialogue between industry and the public and between plants and local communities is a crucial element in bringing about the industrywide cultural change required by Responsible Care, and in building public confidence and trust, a key to enabling the chemical industry to retain its “license to operate.”

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6. Standardization and Certification of Environmental Management Systems As discussed in Chapter 2, companies first establish policies for health, safety, and environmental matters, and then set up environmental management systems to ensure that objectives defined in the policies are effectively achieved. An EMS cannot really be defined by law, even if the EC Eco-Management and Audit Regulation (see Section 4.3) does describe the basic elements of an EMS and encourages companies to establish such EMSs. Conversely, leaving the companies without adequate guidance on how to establish an EMS may result in serious discrepancies in the structure and organization of EMSs as adopted by individual firms, with the accompanying risk of some stakeholders coming to question the credibility of the EMSs of certain companies. This dilemma has led to the concept of standardization of environmental management systems. In addition to helping companies achieve their objectives, appropriate standards can, through certification, offer assurance to others that the stated objectives will be achieved.

6.1. Standards for Environmental Management Systems 6.1.1. Generic or Specific Standards Although there is a long history associated with the drafting of standards for product specifications, testing methods, etc., the concept of standards for management systems is quite new: it indeed first appeared in the mid 1980s with the issuance of the series of standards for quality management systems ISO 9000 (and its national or regional equivalents, EN 29 000, BS 5750). Whereas the ISO 9000 standards were restricted in their initial scope to that which was necessary to achieve product quality, it was soon recognized that the same requirements could also have a generic interpretation, and that, for example, the ISO 9001 and ISO 9004 standards could be adapted to cover environmental management systems. Nevertheless, a few standardization bodies undertook the drafting of standards specific to

1352

Environmental Management in the Chemical Industry

EMS, such as BS 7750 (see below), issued in 1992. A few similar national standards have since been issued elsewhere, and the International Organization for Standardization (ISO) is taking the same route. A certain number of standardization bodies share with some industrialists the view that there should be further development of the concept of a generic management-system standard. The standard would include appendices covering specific requirements for product quality, environmental management, etc. The vision is that this structure would eventually replace the emerging sets of separate management-systems standards, but it probably will not come to pass for several years. 6.1.2. EMS Standards In March 1992 the British Standards Institute (BSI) issued the first standard specifically designed for environmental management systems, under the reference BS 7750. In April 1993 the Association Franc¸aise de Normalisation (AFNOR) published an experimental standard (X30-200) which, although somewhat different in its wording, follows the same structure as the British Standard. Under the impulse of S. Schmidheiny from the Business Council for Sustainable Development, the International Organization for Standardization (ISO) established in 1991 a Strategic Advisory Group on the Environment (SAGE). The ISO-SAGE in turn created six subgroups, one of them in charge of making recommendations on a standard for EMS. In June 1993 the work of ISO-SAGE was taken over by a new Technical Committee (TC): ISO/TC 207, “Environmental Management.” Six subcommittees (SC) were set up – SC 1 – Environmental management systems (ISO 14 00x series of standards) – SC 2 – Environmental auditing (ISO 14 01x series of standards) – SC 3 – Environmental labeling (ISO 14 02x series of standards) – SC 4 – Environmental performance evaluation (ISO 14 03x series of standards) – SC 5 – Life cycle assessment (ISO 14 04x series of standards)

– SC 6 – Terms and definitions ISO 14 05x series of standards) The first standards were issued in 1996. Amongst them ISO 14 001 (Environmental management systems – Specification with guidance for use) which has since substituted all national EMS standards. 6.1.3. The ISO 9000 and 14 000 Standards There has been a growing perception that although separate standards may be required for quality and environmental aspects for management systems, these standards should be made as compatible as possible. Compatibility is understood to mean that the common elements (e.g., training, auditing, document control, etc.) of the standards can be implemented without unnecessary duplication or the imposition of conflicting requirements. In 1998, ISO therefore decided that ISO 14 001/14 004 should be revised in advance of its envisaged revision schedule, in an effort to make it as compatible as possible with ISO9001/9004 which are currently subject to fundamental revision. Both series of standards would then be simultaneously issued in 2000.

6.2. Certification of Environmental Management Standards Standards for management systems, in addition to their “educational” aspects, also allow for “certification” (or “registration”) of the activity covered by the standard by an independent, accredited third party, thus giving confidence to the company and outside stakeholders that the objectives of the management system will be achieved. 6.2.1. Certification of an EMS Certification is a process in which a certification body (often a commercial firm) assesses a particular system against the relevant standard and then issues a certificate of compliance. This is followed up with a series of regular surveillance visits (usually once or twice a year). Certification bodies may themselves be accredited for this

Environmental Management in the Chemical Industry function by a governmental or similar agency, lending additional credibility to their work and resulting in an “accredited certification.” ISO has developed standards for the auditing of an EMS and has furthermore decided to develop a common auditing standard for quality and environmental management systems (see Section 2.6.3). The latter will lead to significant savings for the industry as one single accredited certification body will be able to certify both management systems. It can be expected that the existence of a certificate awarded by an international or national accredited body will become increasingly important to regulators and stakeholders as a form of assurance regarding the professionalism of the company and its EMS. This will assist many companies in meeting the key objective of being able to demonstrate and promote their management of environmental issues in a tangible and credible way. Lack of certification will in time increasingly expose a company to multiple audits by customers seeking to satisfy themselves with regard to the company’s environmental probity. 6.2.2. Certification Against the EC Regulation One article in the EC Eco-Management and Audit Regulation stipulates that “Companies implementing national, European or international standards for environmental management systems and certified, according to appropriate certification procedures, as complying with those standards, shall be considered as meeting the corresponding requirements of this Regulation . . .” (certification, however, will not exempt the “environmental statement” from being validated by a certified verifier). The introduction of this article was of crucial importance as a way of linking the Regulation and voluntary participation on the part of companies to certification against the standards for EMS; because the public statement is an integral part of the scheme, it will quite normally be “verified” by the certification body as a standard part of the certification process. The revised EMAS Regulation will now formally take on board the environmental manage-

1353

ment system requirements of ISO 14 001 (see Section 4.3).

7. International Aspects of Environmental Management Most chemical companies, at least above a certain size, operate in several countries. In some cases all the countries may be at a similar state of development, but both industrialized and developing countries may also be involved. Especially in the latter case, special consideration must be given to environmental management to avoid the problems that too often have tarnished the images of “multinational” or “transnational” companies.

7.1. Environmental Requirements in Different Countries Many reasons, mostly of a commercial nature, may prompt a company to establish an operation in another country, one of which might be a less stringent environmental legislative atmosphere in the host country. There have been a few cases in which “relocation” of a company’s activity from its home country to another was perfectly justified because of the sudden introduction of exceptionally stringent environmental legislation not considered sensible or appropriate by most countries, even the most developed ones. Unfortunately, however, there have been more cases in which companies have established operations in countries with little environmental legislation simply to benefit from standards less demanding and costly than those reasonable standards in effect in their home country. Such “double-standard” practices, including in the environmental domain, have contributed significantly to the bad reputation associated with “multinationals” by some environmental organizations and the governments of some developing countries. These practices are no longer considered to be ethical by reputable corporations that, in their foreign operations, comply not only with local legislation but also with their own company standards, which are generally equivalent to or exceed the requirements of their home country.

1354

Environmental Management in the Chemical Industry

This is a basic principle expressed by many companies in their environmental policies. It must be recognized, however, that companies of foreign parentage following this course of action put themselves in a position of competitive disadvantage vis-`a-vis local companies satisfied with complying, at lower cost, only with national environmental requirements. Foreign companies can sometimes overcome this difficulty by “educating” both national companies (e.g., through the intermediacy of national chemical industry associations) and local governments in modern aspects of environmental protection. The role of transnational corporations in raising the environmental awareness and behavior of developing countries is now clearly recognized [1, Chap. 3.III].

7.2. Technological Cooperation Technological cooperation between countries at different levels of development has been identified as a key component of sustainable development. Technological cooperation between companies in industrialized and developing countries can take several forms: Technical Assistance. A company or an organization sends experts to a company requesting expertise and assistance, perhaps for an audit of environmental performance, a troubleshooting mission, an education or training session, etc. The mission might be accomplished on a commercial basis, but more often it is carried out under the aegis of a voluntary program operated by an industrial association, a government, or an international organization. Transfer of Technology. A company makes available, within the terms of a commercial contract, a process or a technique to which it holds title, generally as the result of a research and development program on its own premises. Such a transfer of technology should obviously include technological cooperation before, during, and after actual start-up of the operation. Foreign Investment. A company may elect to set up its own operation in a foreign country, either in the form of a fully owned subsidiary

or as a partnership with some other company at an appropriate level of equity. The industrial operation might be built from a green site, or the foreign investor might acquire an existing facility. In general, any such foreign investment implies a transfer of technology from the home company to the foreign one. Several industrial associations have produced guidelines for an environmentally sound transfer of technology [15].

8. References 1. The World Commission on Environment and Development: Our Common Future, Oxford University Press, Oxford 1987. 2. S. P. Johnson: The Earth Summit: The United Nations Conference on Environment and Development (UNCED), Graham and Trotman Ltd., London 1993. 3. S. Schmidheiny (Business Council for Sustainable Development): Changing Course, MIT Press, Cambridge 1992. 4. International Chamber of Commerce: Environmental Auditing, Publication 486, ICC Publishing, Paris 1989. 5. S. P. Johnson: The Earth Summit, Graham and Trotman, London 1993, Agenda 21. 6. Confederation of Finnish Industries: Guidelines for Environmental Arguments in Marketing, Helsinki 1990. 7. Deloitte Touche Tohmatsu International, International Institute for Sustainable Development, Sustainability: Coming Clean, Corporate Environment Reportin g, Deloitte Touche Tohmatsu, London 1993. 8. M. S. Baram, D. G. Partan: Corporate Disclosure of Environmental Risks, US and European Law, Butterworth Legal Publ., Salem 1992. 9. U.S. Environmental Protection Agency: Emergency Planning and Community Right-to-Know Fact Sheet, EPA, Washington D.C., 1988. 10. International Chamber of Commerce: Business Charter for Sustainable Development, Publication 210/356A, ICC Publishing, Paris 1991. J. O. Willums, U. Goluke: From Ideas to Action, Publication 504, ICC Publishing, Paris 1992. 11. The Coalition for Responsible Economies: The CERES Principles, CERES, Boston 1989.

Environmental Management in the Chemical Industry 12. Keidanren: The Global Environmental Charter, Keidanren, Tokyo 1991. 13. European Chemical Industry Council: Responsible Care, CEFIC, Brussels 1993. 14. Chemical Industries Association: Responsible Care Management Systems, Guidelines for

1355

Certification to ISO 9001, CIA, London 1992. 15. CEFIC Guidelines on Transfer of Technology (Safety, Health and Environment Aspects), CEFIC, Brussels 1991.

Volumes 1--2

Author Index

1359

Author Index Alt, Christian, M¨unchen, Federal Republic of Germany Solid – Liquid Separation, Introduction, 923 Bartels, Klaus, Berufsgenossenschaft der chemischen Industrie, Heidelberg, Federal Republic of Germany Plant and Process Safety, 1205 Bender, Herbert, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Plant and Process Safety, 1205 Berthold, Werner, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Plant and Process Safety, 1205 Biegler, Lorenz T., Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States Mathematics in Chemical Engineering, 3 Bockhorn, Henning, Technische Hochschule, Darmstadt, Federal Republic of Germany Mathematical Modeling, 165 Boger, David V., Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia Fluid Mechanics, 371 Borho, Klaus, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Solids Technology, Introduction, 941 Busson, Jaques, European Chemical Industry Council (CEFIC), Brussels, Belgium Environmental Management in the Chemical Industry, 1331 Cohen, Henry E., Imperial College of Science and Technology, Royal School of Mines, London, United Kingdom Solid – Solid Separation, Introduction, 931 Conrad, Dietrich, Berlin, Federal Republic of Germany Plant and Process Safety, 1205 Drees, Stefan, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Eberz, Albert, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 F¨orster, Hans, Physikalisch-Technische Bundesanstalt, Braunschweig, Federal Republic of Germany Plant and Process Safety, 1205 Fatemi, Ali, The University of Toledo, Toledo, Ohio 43606, United States Mechanical Properties and Testing of Metallic Materials, 761 Faulhaber, Friedrich Richard, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Solids Technology, Introduction, 941 Finlayson, Bruce A., Department of Chemical Engineering, University of Washington, Seattle, Washington 98195, United States Mathematics in Chemical Engineering, 3 Fischer, Kai, Fachbereich Technische Chemie, Universit¨at Oldenburg, Oldenburg, Federal Republic of Germany Estimation of Physical Properties, 537 Flumerfelt, Raymond W.See also Mathematics in Chemical Engineering and Fluid Mechanics , Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843, United States Transport Phenomena, 271 Gatica, Jorge E., Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Model Reactors and Their Design Equations, 487 Geffarth, Ulrich, European Chemical Industry Council (CEFIC), Brussels, Belgium Environmental Management in the Chemical Industry, 1331 Geiger, Adrian, Sandoz, Basel, Switzerland Plant and Process Safety, 1205 Glor, Martin, Ciba-Geigy AG, Basel, Switzerland Plant and Process Safety, 1205 Glover, Charles J., Department of Chemical Engineering, Texas A & M University, College Station, Texas 77843, United States Transport Phenomena, 271 Gr¨afen, Hubert, Bayer AG, Leverkusen, Federal Republic of Germany Construction Materials in Chemical Industry, 605; Corrosion, 653 Grant, Colin D., University of Strathclyde, Glasgow, United Kingdom Energy Management in Chemical Industry, 1187 Grenner, Dieter, Bayer AG, Dormagen, Federal Republic of Germany Plant and Process Safety, 1205 Grossmann, Ignacio E., Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States Mathematics in Chemical Engineering, 3 Hagen, Hans, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Harbordt, J¨urgen, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Henkel, Klaus-Dieter, Buna AG, Schkopau, Federal Republic of Germany Reactor Types and Their Industrial Applications, 953 Hergeth, Wolf-Dieter, Wacker Polymer Systems, Burghausen, Federal Republic of Germany On-Line Monitoring of Chemical Reactions, 821 Hesse, G¨unter, Bayer AG, Brunsb¨uttel, Federal Republic of Germany Plant and Process Safety, 1205 Hlavacek, Vladimir, Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Model Reactors and Their Design Equations, 487 Horn, Elmar-Manfred, Bayer AG, Leverkusen, Federal Republic of Germany Corrosion, 653 Jonckheere, Filip, European Chemical Industry Council (CEFIC), Brussels, Belgium Environmental Management in the Chemical Industry, 1331 Jourdan, Louis, European Chemical Industry Council (CEFIC), Brussels, Belgium Environmental Management in the Chemical Industry, 1331 King, C. Judson, University of California, Berkeley CA 94 720, United States Separation Processes, Introduction, 915 Krahe, Martin, Bioengineering AG, 8636 Wald, Switzerland Biochemical Engineering, 1117 Linnhoff, Bodo, University of Manchester Institute of Science and Technology, Manchester, United Kingdom Pinch Technology, 1075 M¨uller, Edmund, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Plant and Process Safety, 1205 M¨uller, Michael, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Mix, Karl-Heinz, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Mosberger, Erich, Lurgi AG, Frankfurt, Federal Republic of Germany Chemical Plant Design and Construction, 987 Noha, Klaus, Hoechst Aktiengesellschaft, Frankfurt, Federal Republic of Germany Plant and Process Safety, 1205

1360

Volumes 1--2

Author Index

Onken, Ulfert, Fachbereich Chemietechnik, Universit¨at Dortmund, Dortmund, Federal Republic of Germany Estimation of Physical Properties, 537 Ono, Kanji, Department of Materials Science and Engineering, University of California, Los Angeles, California 90024, United States Nondestructive Testing, 785 Orth, Andreas, University of Applied Sciences, Frankfurt am Main, Germany; Umesoft GmbH, Eschborn, Germany Design of Experiments, 423 Palluzi, Richard P., Exxon Research and Engineering Company, Annandale, NJ 08801, United States Pilot Plants, 1083 Paschedag, Anja R., Technische Universit¨at Berlin, Germany Computational Fluid Dynamics, 463 Pilz, Volker, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Polke, Reinhard, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Solids Technology, Introduction, 941 Puszynski, Jan A., Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Model Reactors and Their Design Equations, 487 Rarey, J¨urgen, Fachbereich Technische Chemie, Universit¨at Oldenburg, Oldenburg, Federal Republic of Germany Estimation of Physical Properties, 537 Sahdev, Vimal, Linnhoff March Ltd., Manchester, United Kingdom Pinch Technology, 1075 Schacke, Helmut, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Schindler, Helmut, Bayer AG, Leverkusen, Federal Republic of Germany Corrosion, 653 Schlecker, Hartmut, Bayer AG, Leverkusen, Federal Republic of Germany Corrosion, 653 Schneemann, Klaus, H¨uls AG, Marl, Federal Republic of Germany Abrasion and Erosion, 735 Schr¨ors, Bernd, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Schulz, Nikolaus, Bayer AG, Dormagen, Federal Republic of Germany Plant and Process Safety, 1205 Soravia, Sergio, Process Technology, Degussa AG, Hanau, Germany Design of Experiments, 423 Steinbach, J¨org, Schering AG, Berlin, Federal Republic of Germany Plant and Process Safety, 1205 Stoessel, Francis, Ciba-Geigy AG, Basel, Switzerland Plant and Process Safety, 1205 Thoma, Peter, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Solids Technology, Introduction, 941 Viard, Richard, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Viljoen, Hendrik J., Laboratory for Ceramic and Reaction Engineering, Department of Chemical Engineering, University of Buffalo, Buffalo, NY 14260, United States Model Reactors and Their Design Equations, 487 Vogel, Herbert, BASF Aktiengesellschaft, Ludwigshafen, Federal Republic of Germany Process Development, 873 Walper, Matthias, Bayer AG, Brunsb¨uttel, Federal Republic of Germany Plant and Process Safety, 1205 Walther, Claus-Diether, Bayer AG, Leverkusen, Federal Republic of Germany Plant and Process Safety, 1205 Wandrey, Peter-Andreas, Bundesanstalt f¨ur Materialforschung und –pr¨ufung, Berlin, Federal Republic of Germany Plant and Process Safety, 1205 Weidlich, Stephan, Hoechst Aktiengesellschaft, Frankfurt, Federal Republic of Germany Plant and Process Safety, 1205 Westerberg, Arthur W., Carnegie Mellon University, Pittsburgh, Pennsylvania 15231, United States Mathematics in Chemical Engineering, 3 Widmer, Ulrich, Sandoz, Basel, Switzerland Plant and Process Safety, 1205 Yeow, Y. Leong, Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia Fluid Mechanics, 371 Zimmermann, J¨urgen, Bayer AG, Dormagen, Federal Republic of Germany Plant and Process Safety, 1205 Zlokarnik, Marko, Graz, Austria Scale-Up in Chemical Engineering, 1093 Zlokarnik, Marko, Bayer AG, Leverkusen, Federal Republic of Germany Mixing, Introduction, 939

SUBJECT INDEX

Index Terms

Links

A Abrasion description

737

of construction materials

735

Abrasion wear examples for

753

types of

737

Absorption in a falling film with reaction, mathematical treatment AC 6 X Accident Prevention Regulation Acentric factor

345 620 1251 546

Acetamide calculation of surface tension

593

Acetone (2-propanone) (DMK) –cyclohexane, vapor–liquid equilibria

564

–decane, vapor–liquid equilibria

564

diffusion coefficient in ethyl acetate

590

–2,3-dimethylbutane, vapor–liquid equilibria

564

–heptane, vapor–liquid equilibria

564

–hexane, vapor–liquid equilibria

564

–2-methylbutane, vapor–liquid equilibria

564

–pentane, vapor–liquid equilibria

564

Acetonitrile surface tension of aqueous solutions

595

Acetophenone –benzene, surface tension of mixture

594

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Acetylene (ethyne) temperature dependence of the Gibbs energy of formation Acheson furnace

574 975

Acoustic emission activation and detection

803

for nondestructive testing

803

sources for and source location

804

Adams–Bashforth integration methods

63

Adams integration method

67

Adams–Moulton integration method

67

815

Adiabatic reactor heat exchange in

528

temperature profile in

528

Aeration cell

665

Air heat-transfer coefficient

296

thermophysical properties

304

viscosity at different temperatures

381

Airlift reactor in biochemical engineering

1135

Alcohols prediction of vaporization enthalpy

559

Algebraic equations solution of sets of Alitizing

6 708

Alloys nondestructive testing for identification

787

oxidation

698

sulfurization

698

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Alumina ceramics mechanical and physical properties of WC hard metal

647

refractory bricks as construction

647

Aluminum as construction material in chemical industry

631

corrosion of aluminum and aluminum alloys

631

elastic constants

764

hardness and jet wear

748

plane-strain fracture toughness

781

thermal properties

295

wear by flow of pure water and seawater

746

Aluminum alloys as construction material in chemical industry

631

corrosion behavior

632

layer-type corrosion

679

pitting corrosion

672

standardized corrosion test methods

725

stress corrosion cracking

686

tensile properties

767

Aluminum bronze wear by flow of pure water and seawater

635 746

Ambrose estimation of critical data

557

American Society for Nondestructive Testing (ASNT)

786

Ammonia diffusion coefficient in air

589

viscosity data

577

Ammonium nitrate and fuel oil (ANFO) Amperometry

1236 857

Analysis of variance (ANOVA) in design of experiments Andrussow process

446 968

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Aniline diffusion coefficient in water ANSY

591 1066

Antoine equation for prediction of vapor pressure of liquids

556

Approximations piecewise Archimedes number Arc spraying materials for Arrhenius number Arrow matrix ARW value (Arbeitsplatzrichtwerte)

18 1093 713 751 1094 10 1218

Asbestos thermal properties

295

ASOG method for prediction of vapor-liquid equilibria ASPEN PLUS

563 887

Association Franc¸aise de Normalisatio n (AFNOR)

1352

ATERM

1066

ATSB diffusion

340

Attenuation coefficient of radiation intensity in uniform material August equation AUTOCAD Awesta

789 556 1066 620

B BAM oven

1231

Baroncini–Latini method for estimating the thermal conductivities of liquids Batch cultivation

586 1143

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Batch reactor for nonhomogeneous, gas–liquid systems

493

for nonhomogeneous, solid–solid systems

493

homogeneous, isothermal

490

nonisothermal

491

Beek and Singer model

515

Bekaplast

719

494

Belt reactor with mixing head

985

Bend test

769

Benedict–Webb–Rubin equation

549

Benson method for estimation of heat capacity of ideal gases

554

for estimation of thermodynamic reaction data

572

Benzene –acetophenone, surface tension of mixture

594

–diethyl ether, surface tension of mixture

594

diffusion coefficient in air

589

diffusion in carbon tetrachloride

591

–nitrobenzene, surface tension of mixture

594

temperature dependence of the Gibbs energy of formation

574

viscosity

381

Bernoulli distribution

145

Bernoulli equation

378

extended

359

Bessel function, zero-order Best available technique (BAT) BFGS update

75 1345 126

Bhirud method determination of liquid molar volumes of organic compounds Bifurcation theory

552 72

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Bingham fluid

396

tube flow, velocity distribution

402

tube flow, volumetric flow rate

402

Bingham plastics flow in a tube Binomial distribution function

307 145

Biocatalysts, immobilized reactors for biochemical processes Biochemical engineering

979 1117

Bioreactors in biochemical engineering Biot number BISAM

1178 298 1066

Blasius equation

382

Blasius expression

331

Bodenstein number

208

512

Boger fluid

406

409

1094

Bondi–Rowlinson method for the prediction of heat capacities of liquids Bond number Borescope

556 1093 813

Borides melting point, density, and hardness

749

Boundary element method

117

Boundary finite element method

118

Boundary integral equation

117

Boundary layer turbulent

326

Boussinesq approximation

517

Box–Behnken designs

443

Box–Wilson method

526

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Branch of a multiple-valued function Branch and bound method Brandschacht test

36 134 1222

Brass elastic constants

764

hardness and jet wear

748

Brent’s method

13

Bricks lining

711

thermal properties

295

Brinell hardness number

772

Brinell hardness test

772

Brinkman number

294

British Institute for Nondestructive Testing (BINDT)

786

British Standards Institute (BSI)

713

1093

1352

Bromley method for estimating low-pressure viscosity of gases

579

Brown’s method

13

Broyden’s method

13

Bubble column gassing devices for

965 1178

packed

972

with suspended catalyst

969

Bubble flow

972

389

Buildup factor in radiography Bulirsch–Stoer recursion algorithm Bulk viscosity

789 18 380

Burgers viscosity equation

96

Burke–Plummer equation

518

Burner

959

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Bursting disk brittle plates

1283

membrane-type

1283

shells of revolution

1283

Buss loop reactor

965

971

2-Butanol (sec-butanol, sec-butyl alcohol) (SBA) thermal conductivity at 480 K, estimation

583

1-Butanol (butyl alcohol) (NBA) calculation of surface tension

594

surface tension of aqueous solutions

594

2-Butanone –cyclohexane, vapor–liquid equilibria

564

–heptane, vapor–liquid equilibria

564

–hexane, vapor–liquid equilibria

564

–octane, vapor–liquid equilibria

564

surface tension of aqueous solutions

594

Butyric acid surface tension of aqueous solutions

595

C CADAM/AEC

1066

CADEX

1066

CAD method

1054

Cake filtration ranges of application Calorimetry

928 825

adiabatic reaction

828

heat balance

828

heat compensation

828

heat flux

827

isoperibolic reaction

829

isothermal reaction

829

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Calorimetry (Cont.) screening method in process safety

1245

Capillary hydrodynamic fractionation (CHDF)

851

Capillary rheometer

411

Caproic acid surface tension of aqueous solutions

595

Carbides melting point, density, and hardness

749

Carbon dioxide calculation of thermal conductivity at 473.15 K and 30 MPa

585

thermophysical properties

304

viscosity

381

viscosity data

577

Carbon tetrachloride diffusivity in various solvents

589

–nitrobenzene, surface tension of mixture

594

Carman–Kozeny equation

518

Casson equation

396

Cast iron as construction material in chemical industry

628

quality standards

628

thermal properties

295

wear resistance

744

Castor oil

381

Catalysis reactors for gas-phase reactions

965

reactors for solid-catalyzed reactions

965

Catalyst pellet one-phase model

509

two-phase model

509

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Catalysts diffusion and reaction at, mathematical treatment mathematics for reaction and diffusion in

343 78

Cauchy equation of motion

285

Cauchy–Riemann equation

33

Cauchy’s integral

35

Cauchy’s stress principle

284

Cauchy’s theorem

35

Cavitation erosion

739

of metals

745

Cell retention

1146

Central composite designs (CCD)

376

444

Centrifugal pump in biochemical engineering

1182

Ceramic enamel for enameling chemical apparatus

711

712

Ceramics identification by hardness testing

787

resistance to wear

749

CFD

463

Chapman–Enskog equation

576

for diffusivity Chapman–Enskog relation on thermal conductivity

338 288 289

Charge-transfer factor

658

Charpy impact test machine

773

Chebyshev polynomial for spectral method

17 103

CHEMADYN

888

CHEMASIM

887

CHEMCAD

887

Chemical plant design and construction

987

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Chemicals Act

1251

Chemical spot testing material identification by Chemical thermodynamic and hazard evaluation (CHETAH) Chemical vapor deposition (CVD) coatings by

787 1236 751 711

Chen method for the prediction of vaporization enthalpy

560

Chilton–Colburn analogy

349

Chi-square distribution

148

Chloride corrosion

669

Chlorobenzene diffusion in bromobenzene Cholesky decomposition Chromatography see also under individual names

591 8 851 915

Chromium passivation

666

Chueh–Swanson method for the prediction of heat capacities of liquids

554

Chung method for estimating low-pressure viscosity of gases

579

for predicting thermal conductivity of gases at high pressure Churn flow

583 389

Cladding see also Plating

706

with refractory metals by explosion

641

Clarifiers in gravity sedimentation

926

Classification

934

Claus process

968

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Cloe process

971

Coalescence in liquid–gas or liquid–liquid systems

940

Coalition for Environmentally Responsible Economies (CERES)

1348

Coating inorganic nonmetallic on metals

710

of inorganic nonmetallic materials, thermally sprayed

711

organic, of chemical apparatus

714

713

Cobalt alloys oxidation behavior

705

Colburn analogy

331

Colburn j factor

331

Cold composite curve of processes Colebrook equation

348

1075 330

383

Collocation method for ordinary differential equations

82

Combustion gases corrosion by

699

Complex number

30

Complex plane

30

integration in

34

Complex variable

30

analytic functions

33

elementary functions

31

Composite curves of processes

1075

Composite materials fiber-reinforced plastics, determination of fiber content of

788

vessels and piping, nondestructive testing by acoustic emission

803

816

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Compressibility factor of gases

543

547

Compression test of metals

768

Compressive strength

768

Compton scattering

789

Computational Fluid Dynamics

463

Computer-aided engineering (CAE) Concentration overpotential

1052 659

Concrete thermal properties

295

Conductometry

855

Confidence interval

180

Confidence level

148

Construction materials abrasion and erosion of apparatus from classification of flammability and test method for in chemical industry Continuity equation for flows Continuous cultivation Continuous, ideally mixed, stirred-tank reactor

735 1222

1259

605 284 375 1145 498

concentration ratio of different reaction order

499

kinetics

500

Continuous stirred-tank reactor (CSTR) cascade of

505

isothermal heterogeneous system

502

kinetics

500

nonisothermal, mathematical treatment of

502

residence-time distribution

500

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Contract types

1029

lump-sum engineering contract

1029

reimbursable contract with target price

1029

supply contract

1029

turnkey contract

1029

Control system and instrumentation of pilot plants Convective diffusive equation

1088 96

Coordinate system cylindrical

59

spherical

60

Copper as construction material in chemical industry

634

corrosion behavior

634

elastic constants

764

electrolytic corrosion

663

hardness and jet wear

748

thermal properties

295

wear by flow of pure water and seawater

746

Copper alloys as construction material in chemical industry

634

copper–nickel, corrosion behavior

636

corrosion behavior

634

Cu–Ni–Fe, corrosion behavior

636

Cu–Zn, dezincification

678

Cu–Zn, stress corrosion cracking

686

erosion–corrosion by liquid flow

745

standardized corrosion test methods

725

wear by flow of pure water and seawater

746

Corkboard thermal properties

295

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Corrosion

653

corrosion behavior of stainless steel

623

intergranular, of stainless steel

624

pitting, of stainless steel

625

stress corrosion cracking of stainless steel

626

Corrosion cell

657

Corrosion current density determination Corrosion fatigue

661 627

692

Corrosion inhibitors see also Anticorrosives Corrosion protection

720 706

Corrosion rate determination Corrosion testing

661 722

Cost estimate design costs

999

for a chemical plant accuracy of estimation

147

in chemical plants

990

pilot plant

1087

Cost index in plant design

994

Costs calculation of plant

1019

design

999

operating

995

reduction of

907

Couette flow mathematical treatment

303

Coulometry

857

Countercurrent processes

915

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Counterflow separation column mathematical modeling Courant number

235 98

Covariance

147

Cox–Merz rule

407

Crank–Nicolson method

67

Creep test

775

Crevice corrosion

672

Crider and Foss model

515

Critical path method (CPM)

1036

Critical properties estimation from molecular increments

543

544

of organic compounds, estimation

544

546

Crushing

932

Crystallizer continuously operating, size distribution Cubic B-splines

209 90

Cubic equation of state

548

Curvilinear coordinate

59

Cyclohexane –acetone, vapor–liquid equilibria

564

–2-butanone, vapor–liquid equilibria

564

–cyclohexanone, vapor–liquid equilibria

564

Cyclohexanone –cyclohexane, vapor–liquid equilibria

564

Cyclopentanone –octane, vapor–liquid equilibria Cylindrical coordinate system

564 59

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

D Damköhler number

1094

Damköhler number

347

Daniell cell

657

Darcy–Oberbeck–Boussinesq model

517

Darcy’s law

517

modified

518

Darken equation

591

DASSL Deborah number

72 1093

Decane –acetone, vapor–liquid equilibria

564

Decanoic acid surface tension of aqueous solutions

595

Deflagration to detonation transition (DDT)

1237

Del operator

55

Dense-medium separation

935

Densimetry

859

DESIGN II

887

Design Institute of Emergency Relief Systems (DIERS)

281

1284

Design of experiments class of models used in

442

experimental investigations

425

experimental strategies

428

for special purposes

453

optimization methods

451

regression analysis

444

standard

443

visualizing models

444

Design of experiments (DoE)

423

Deterministic mathematical model

172

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Deterministic system probability density functions

205

Detonation spraying

713

Deutsche Gesellschaft für Zerstörungsfreie Prüfung (DGZfP)

786

Development costs

879

Deviatoric stress tensor

380

Dewar test screening method for process safety DFP formula Dialysis cultivation Diaphragm cell process

1244 126 1149 979

Diaphragm pump in biochemical engineering

1183

Dichlorodifluoromethane thermophysical properties Dielectric spectroscopy

304 833

Diethyl ether (DEE) (Ethoxyethane ether) –benzene, surface tension of mixture

594

Diethyl ketone (DEK) surface tension of aqueous solutions

595

Difference equations, linear arising in staged operations, such as distillation ect. Differential–algebraic system

14 69

Differential equation mathematical models, based on

193

Differential equation, ordinary sensitivity of the solution to the value of parameters

74

Diffusion ATSB diffusion

340

ECD diffusion

339

equimolar counterdiffusion

339

of gases, steady-state measurement

339

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Diffusion (Cont.) with heterogeneous reaction e.g., with catalysts

343

with homogeneous chemical reaction

342

Diffusion coating process metal coatings by

708

Diffusion coefficient of gases , calculation

587

of liquids, calculation

589

Diffusion equation Diffusion overpotential

48

98

99

659

Diffusive mass transfer in binary systems

332

Diffusivity of gases and liquids

338

Diisopropyl ether (IPE) from 2-propanol, calculation of equilibrium constant Dilatometry

575 859

Dimensional analysis in scale-up in chemical engineering

1094

Dimensionless equation for transport phenomena

292

2,3-Dimethylbutane –acetone, vapor–liquid equilibria Dirac delta function

564 197

Direct search method for solving nonlinear equations Discounted cash flow (DCF)

186 909

Dispersion state influence on product properties DISPO

941 1066

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Distillation column, mathematics for

69

columns, improvement of control system

1197

columns with vapor recompression

1199

methods of improving energy efficiency

1196

solution of difference equations, arising in split-tower distillation

1198

14 1199

Distillery savings in batch cooking by pinch technology

1080

DIVA

888

Divergence theorem

282

Dobratz method for estimation of ideal gas heat capacities D-optimal designs DOSY

555 443 1066

Dow process

984

Drag coefficient

387

Drop impingement wear

741

examples

757

Dry gas generator

975

745

Drying of solids convective dryers, control and heat recovery

1200

hybrid dryers

1201

with microwaves and radio-frequency radiation

1201

DSS

104

DTPTB

78

Ductility

766

evaluation Duplex steel

769 619

Dust explosion indices of dust–air mixture

1230

ignition sensitivity of dust cloud

1231

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Dust explosibility

1233

Dust roaster

973

DVCPR

82

DVERK

78

Dyadics

52

divergence of

56

divergence theorem

58

Dynamic tear test

774

Dynamic viscosity

407

E Ebonite lining

716

Eddy-current method identification of metals and alloys by

787

Eddy-current nondestructive testing

808

Edmister equation

546

Eicosane temperature dependence of the Gibbs energy of formation

574

Eigenvalue eigenvalue problem, numerical methods for

115

eigenvalue problems solved by initial value techniques

75

of matrices

15

Elastic bending stress

769

Elastic deformation of metals

763

Elastic modulus

763

Elastic rolling

752

Elastomer wear resistance

747

Electrical resistivity for identification of metals and alloys

787

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Electric Power Research Institute (EPRI) NDT Center

786

Electrochemical corrosion protection

721

Electrochemical deposition see also Electroplating; Electrochemical plating

708

Electrochemical element

656

Electrochemical series

655

Electrochemistry corrosion by electrolytic processes

654

electrochemical equilibrium between electrodes and electrolytes

654

Electrolysis inorganic material

979

organic material

979

Electrolytic metal refining

979

Electromagnetic nondestructive testing method

806

Electropolishing in biochemical engineering Electrostatic separation

1158 936

Electrothermal reactor for noncatalytic reactions

973

Elliptic partial differential equation

94

Ellis fluid tube flow, volumetric flow rate

400

Ellis model

398

ELLPACK

107

Elution chromatography

916

Ely-Hanley method for estimating the thermal conductivity of gases at low pressure

586

Emergency Planning and Community Right-to-Know Act (EPCRA)

1346

Emission limit values (ELV)

1344

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Employers’ Liability Insurance Association of Chemical Industries

1211

Enamel for enameling chemical apparatus

710

711

Energy management, in chemical industry

1187

optimization of process energy and costs by pinch technology

1076

Energy transport equation for flow Engineering Code for Flammable Liquids (TRbF) Enthalpy

378 1315 153

difference between two states of gases

553

of formation, estimation

571

of vaporization, calculation

558

reaction enthalpies

569

Entrained-flow reactor

971

Entropy

153

973

Environmental control in chemical plant design and construction Environmental management

1007 1331

audiences and vehicles

1341

audit regulation

1345

communication

1340

environmental report

1342

international aspects

1353

ISO 9000 and 14000 standards

1352

operating permits

1344

organization and structure

1335

process safety legislation

1345

public commitments

1348

responsible care

1349

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Environmental management (Cont.) reviews

1340

stakeholder’s reactions

1343

standards

1337

systems

1334

voluntary agreements

1347

Environmental management systems standardization and certification Environmental quality objective (EQO)

1351 1345

Epoxy resins chemical resistance

644

reinforced with glass fibers, hardness and jet wear

748

Equations linear algebraic linear difference equations

6 14

nonlinear, direct search methods

186

systems of nonlinear, gradient methods for solving

187

Equilibration processes for separation of mixtures

916

Equilibrium of chemical reactions, calculation Equipment Safety Act Ergun relationship

570

574

1251 388

Erosion cavitational

739

description

739

liquid erosion damage

756

of construction materials

735

Erosion–corrosion examples for

742 757

Erosion wear examples for

754

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Error analysis in experiments

150

Esters prediction of vaporization enthalpy

559

Ethane temperature dependence of the Gibbs energy of formation

574

Ethanol (ethyl alcohol) diffusion coefficient in air

589

surface tension of aqueous solutions

595

viscosity behavior

576

Ethoxylation of butanol, calculation of reaction enthalpy

573

Ethyl acetate surface tension of aqueous solutions

595

Ethylene (ethene) calculation of diffusion coefficient in water

589

temperature dependence of the Gibbs energy of formation

574

Ethyl propionate surface tension of aqueous solutions

595

Eucken equation

289

Eucken factor

582

Eucken method for estimating the thermal conductivity of gases at low pressure

582

Euler integration method for ordinary differential equations Euler–Newton continuation Euler number

63 73 1093

Euler’s first law

285

Euler’s second law

285

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Evaporation Coating metal coatings by

708

Evaporation cooler simultaneous mass and heat transfer, dynamic model

232

Evolutionary operation (EVOP) in design of experiments Excess Gibbs energy of mixing

451 561

Experiments error analysis

150

factorial design of, and analysis of variance

150

Expert Commission on Safety in the Swiss Chemical Industry (ESCIS)

1254

Expert system in process development

888

Explosion definition

1235

mechanisms

1236

Explosion barrier

1286

Explosion cladding with refractory metals

706 641

Explosion limit (EL)

1234

Explosion pressure

1225

as a function of time

1234

Explosion prevention

1269

Explosion protection

1269

Explosion range pressure dependence of Explosion risk

1225 1263

Explosion risk assessment keywords for Explosion risk reduction

1270 1270

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Explosives grouping of

1236

Extensional viscosity

410

Extraction solution of difference equations, arising in

14

Extruder

962

Exxon process

962

F Factorial design 22 factorial design 3

432

2 factorial design

435

fractional

436

Factorial design of experiments

150

Factorial designs

430

Failure modes and effects analysis (FMEA)

1241

Falling ball viscometer

387

Falling-film reactor

962

Fanning equation

392

Faraday constant

656

Faraday’s law

656

Fatigue test

778

Fault tree analysis FDRXN Fed-batch cultivation

1256

1257 82 1144

Fedors estimation of critical data

545

Ferralium

255

621

Ferrous alloys visual identification by spark testing

787

Fiberscope

813

Fick’s first law

336

Field-flow fractionation (FFF)

851

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Filtration ranges of application of cake filtration

928

sterile

1127

survey

927

Finite difference method for ordinary differential equations

101

19

Finite volume method

476

88

102

1259

design and construction of chemical plants

1261

equipment

1261

fire-fighting methods

1262

Fischer–Tropsch reactor

971

FISHPAK

107

Fixed-bed catalytic reactor

965

for gas-phase reactions

415

79

Finite element method

Fire protection

1181

965

Fixed-bed processes for separation of mixtures

917

Fixed-bed reactor for catalytic liquid-phase and gas–liquid reactions in biochemical engineering

965 1148

with combustion zone

968

with heating or cooling elements

965

Flame arrester

1286

crimped-ribbon-type

1286

for dust–air mixture

1288

wet-type (siphon arrester)

1286

with temperature monitoring and steam injection

1288

Flame detachment Flame spraying materials for Flammability index test equipment

1223 709

713

751 1221

This page has been reformatted by Knovel to provide easier navigation.

417

Index Terms

Links

Flash point test equipment

1219

Flexicracking

971

Flexure

769

Flocculation

936

Flotation

935

Flow around bodies, heat-transfer coefficients

334

between two concentric cylinders

383

compressible flow in tubes, flow calculations

363

creeping around spheres, mathematical treatment

310

creeping flow past a sphere

385

energy transport equation

378

enhanced by the injection of an immiscible low-viscosity fluid

308

flow rate measurement

394

gas–liquid streams

389

in channels of arbitrary cross section

305

in channels with varying cross sections, lubrication analysis

313

in closed channels, pressure drops and temperature changes

362

in macroscopic systems

391

in packed-bed reactors

517

in pipes

381

in tubes, heat-transfer, and friction coefficients

333

in valves, beds, contractions, and fittings

394

of metals

764

over flat plates, heat-transfer, and drag coefficients

333

Poiseuille flow in a tube

304

rheological constitutive equations

379

steady, one-dimensional, mathematical treatment

302

trough granular beds

387

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Flow (Cont.) two-dimensional, stream function equations

309

two-phase concentric in tubes

308

FLOWPACK

887

Flow phenomenon applications of the law of conservation of mass

229

mathematical models, application of the principle of conservation of momentum

217

mathematical models for

216

reactors with turbulent flow, mathematical model for

210

Flue gas corrosion by Fluidization

699 388

Fluidized bed minimum superficial gas velocity Fluidized-bed reactor

388 959

965

calculation of the effective reaction rate of heterogeneous reactions in for liquid-phase and gas–liquid reactions in biochemical engineering Fluid mechanics numerical methods in vector formulas useful in Fluohm reactor

364 970 1148 371 415 56 975

Fluorescent screen in radiography for intensification of the image Forchheimer model Fourier number Fourier series

792 518 1094 37

Fourier’s law for heat conduction

287

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fourier transform

38

of discrete data

27

Fracture mechanics

780

Fracture strength

766

Fracture toughness

780

testing

781

Frank–Kamenetzki temperature profiles

1239

Franklin method for estimation of thermodynamic reaction data Fredholm equation of the first and second kind

572 110

numerical methods for

113

Fredholm integral equation

109

Free convection on a vertical plate Free heat convection heat-transfer coefficients

323 296 334

Friction factor of flows in pipes

382

Friction factor–Reynolds number relationship for flow in tubes

402

Froth flotation

936

Froude number

293

extended

1093

1093

Fuller method for estimation the diffusion coefficients of gases

589

Furan resins chemical resistance Furnace test

644 1222

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

G Galerkin finite element method Galilei number

19

88

417

1093

Gas diffusion coefficients, calculation

588

estimation of heat capacity of ideal gas

554

estimation of viscosities

576

explosion data for mixtures

1222

gas–liquid equilibria

566

gas mixtures, calculation of heat capacity

554

mixtures, calculation of pseudocritical properties

551

mixtures, virial equation for

551

pressure limit of stability

1227

thermal conductivity at high pressure

583

thermal conductivity, prediction

582

thermophysical properties

304

viscosity of gas mixtures, estimation

578

Gas chromatography (GC)

854

Gas tungsten arc welding (GTAW) in biotechnology

1152

Gauss–Hermite polynomials quadrature formula for

23

Gaussian quadrature points and weights

23

Gauss–Ostrogradski integration principle

221

Gauss–Seidel method GEARB

11 104

Gear’s backward difference formula Geometric mean

67 143

German Federal Water Quality and Pollution Control Act (Wasserhaushaltsgesetz) German Hazardous Materials Regulation (GefStoffV)

1301 1218

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Gibbs energy of formation, estimation

571

of formation, for several hydrocarbons

574

of reaction, calculation

569

Glass as construction material in chemical industry

645

hardness and jet wear

748

in biochemical engineering thermal properties

1159 295

Glass wool thermal properties Global polynomial

295 16

Glycerol (propane-1,2,3-triol) thermophysical properties

304

Glycerol (propane-1,2,3-triol)

381

Godbert–Greenwald furnace

1231

Gomez method for prediction of vapor pressure of liquids

558

Gradient method for solving nonlinear systems of equations

187

Graetz problem for heat transfer

75

Graphite as construction material in chemical industry Grashof number

646 1094

Gravity concentration

935

Green’s function

115

Green’s theorem

58

59

Grewer test apparatus for testing spontaneous flammability of dusts

1221

Grinding see also Milling

932

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Guldager electrolysis method

721

Guldberg rule

544

Gunn–Yamada expression for predicting liquid molar volumes Guthrie’s modular technique

550 992

H Hagen–Poiseuille equation

305

Hagen–Poiseuille flow

218

Hagen–Poiseuille law

861

382

Haggenmacher correction on the prediction of vaporization heats Hamilton–Cayley theorem Hammerstein equation numerical methods for

559 15 110 113

H–and–D curve for exposed X-ray films

791

Hankinson–Thomson method determination of liquid molar volumes of organic compounds

551

Hardmetal mechanical and physical properties of WC hard metal

647

wear resistance

746

Hardness correlation with tensile strength

773

testing

771

testing, material identification by

787

Harrison–Seaton method for estimation of ideal gas heat capacities

555

Hartmann apparatus

1232

Hartmann apparatus, modified

1233

Hastelloy

630

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Hatta number

1094

Hayduk–Minhas estimation of diffusion coefficients in liquids

590

Hazard causes and effects

1208

evaluation of

1257

types and sources of

1207

Hazard and operability study (HAZOP)

1008

1256

Hazard assessment in plant and process safety

1241

Hazard characteristics of exothermic processes

1241

Hazard control in plant and process safety

1314

Hazard control plan off-battery

1319

plants and works

1317

Hazard identification

1263

Hazardous Materials Regulation

1251

Hazardous substances safety practices when working with

1291

symbols and letter codes

1216

Hazard potential from an explosion

1213

from release of a volatile substance

1212

magnitude

1211

mathematical treatment

1212

Heat conduction equation

48

equation, unsteady

108

in solids, mathematical treatment

294

mathematical model for

225

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Heat convection mathematical treatment

316

Heat exchanger design

1193

duty of

1192

overall heat balance

1192

selection

1193

Heat-exchanger network (HEN) Heat-transfer coefficient

1006 296

for hot air at the wall of a pipe with turbulent flow, model for

179

Heat recovery in chemical industry

1192

Heat transfer Graetz problem for

75

in flows, summary of relations for

332

in laminar boundary layers

321

in laminar tube flow

317

mathematical modeling

226

parameter estimation for

27

simultaneous mass and heat transfer, dynamic model

231

turbulent in tubes

329

turbulent on a flat plate

331

with flow, mathematical treatment

316

Heaviside expansion

232

45

Helium diffusion coefficient in air

589

thermophysical properties

304

viscosity data

577

Heptane –acetone, vapor–liquid equilibria

564

–2-butanone, vapor–liquid equilibria

564

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Heptane (Cont.) –2-pentanone, vapor–liquid equilibria

564

–3-pentanone, vapor–liquid equilibria

564

viscosity

381

Heptanoic acid surface tension of aqueous solutions

595

Hermite cubic polynomial

20

Hermite polynomial

18

Herschel–Buckley model

396

Hexane –acetone, vapor–liquid equilibria

564

–2-butanone, vapor–liquid equilibria

564

diffusion coefficient in air

589

–2-pentanone, vapor–liquid equilibria

564

temperature dependence of the Gibbs energy of formation

574

viscosity

381

Heyrovsky´ reaction

662

Hock process

960

H-Oil process

972

Holography for inspection of coatings etc. for lack of bonding Homotopy methods Hooke’s law

813 13 763

Hopf bifurcation analysis

248

Hot composite curve

1075

Houdry flow process

971

Houdry process

968

HTRI

1066

Hydrocyclones

927

Hydrodynamic chromatography (HDC)

850

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Hydrogen corrosion of steel by at elevated temperature by

690

diffusion coefficient in air

589

pressurized, reaction with steel

616

thermophysical properties

304

Hydrogen chloride viscosity data

577

Hydrogen cyanide viscosity data Hydrogen electrode

577 655

Hydrogen overpotential and acid corrosion

662

Hydrogen sulfide viscosity data Hyperbolic partial differential equation

577 94

Hypergeometric distribution function

146

HYSIM

887

I Identity tensor

379

Ignition energy

1229

Ignition sensitivity

1231

Ignition temperature apparatus for determining Impact test

1229 773

Impeller marine-type

499

turbine

499

Incident sequence analysis

1257

Incineration plant

902

Incoloy

625

Inconel

630

627

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Infrared spectroscopy Inoculation

Links 842 1143

Integral equation

109

Integral transform

37

Intergranular corrosion

674

INTERGRAPH/IGDS

1066

Internal energy

154

International Chamber of Commerce (ICC)

1338

International Council of Chemical Associations (ICCA)

1351

International Institute of Welding (IIW)

1348

786

Interpolation two-dimensional, and quadrature

29

Investment dynamic return on

908

return of investment (ROI)

997

static return on

908

elastic constants

764

passivation

666

Iron

Iron ore treatment of, in Minnesota

933

Isobutyl alcohol (IBA) surface tension of aqueous solutions from

594

Isobutyric acid surface tension of aqueous solutions

595

Isopentyl alcohol surface tension of aqueous solutions

594

Isopropylbenzene estimation of thermal conductivity

587

Isovaleric acid surface tension of aqueous solutions

595

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Iterative methods for solution of sets of linear algebraic equations

11

J Jacobian

280

Jacobian matrix

13

Jacobi method

105

Jacobi polynomial

18

Japanese Society for Nondestructive Inspection (JSNDI)

786

Jet-Kote process

709

Jet reactor

983

713

Jet stirred reactor mathematical model for

218

Jet wear examples for

755

of metals

744

of polymers and metals

749

Joback method for estimation of critical data

554

for estimation of enthalpies and free energies of formation

571

for estimation of ideal gas heat capacities

555

for estimation of thermodynamic reaction data

571

for prediction of boiling and melting point of liquids

558

Joback’s increment

544

Joffe’s rule

553

K KAPAZ

1066

Karman momentum integral equation

321

Kay’s rule

553

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Kellogg process

962

Kinematic relation

282

Kinematic viscosity

576

Kinetics modeling of the mean reaction rates

262

of continuous, ideally mixed stirred tank reactors

500

of continuous stirred-tank reactors

500

Kinglor–Metor process

975

Kistiakowsky equation

541

Kneader

975

Knoop hardness number

773

Kobayashi method for the prediction of vaporization enthalpy KOKO Kölbel–Schulze index

560 1066 995

Kolbe–Schmitt process in salicylic acid production Kolmogorov length Kronecker delta

975 218 53

Kuhn–Tucker multiplier

122

Kureha process

959

217

379

L Lack-of-fit test in design of experiments Laguerre polynomial Lamé constant

446 18 796

Laminar flow mathematical model for

217

Lang–Chilton method

992

Langer–Mond process

959

Lang factor method

992

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Langmuir–Hinshelwood theory oxidation rate of NO Laplace equation Laplace number

525 34 1093

Laplace transform

42

43

Laplacian operator

56

282

Laurent series

35

Layer corrosion

679

Lead as construction material in chemical industry

636

hardness and jet wear

748

thermal properties

295

Lead alloys as construction material in chemical industry

636

corrosion behavior

637

Leak testing of an internally pressurized vessel

813

pressurized gas leaks

813

Least-square method

24

Lee–Kesler–Pitzer method for the prediction of vaporization enthalpy

557

Legendre polynomial

17

Leibniz formula

59

Lennard–Jones 12–6 potential

289

542

Letsou–Stiel method for predicting the viscosity of pure liquids Levenberg–Marquardt method Lewis number

579 26 1094

Licensing agreement in process development Likelihood function Limiting oxygen concentration (LOC)

1024 191 1234

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Line-type corrosion

679

Linhoff analysis

899

Lining of chemical apparatus

707

of fusion-cast basalt

711

organic, of chemical apparatus

714

Liquid exclusion chromatography (LEC)

850

713

Liquid–liquid extraction countercurrent extractor Liquid-ring pump

916 963

Liquids diffusion coefficients, calculation

589

estimation of heat capacity of pure liquids

554

estimation of pressure dependence of density

551

estimation of surface tension

592

gas–liquid equilibria

566

liquid–liquid equilibria, calculation

567

mixtures, activity coefficients

561

mixtures, calculation of heat capacity

555

mixtures, calculation of molar volume

549

mixtures, excess Gibbs energy of mixing

561

mixtures, fugacities of components

566

mixtures, vapor–liquid equilibria

566

prediction of boiling and melting point

558

solid–liquid equilibria, calculation

568

surface tension of liquid mixtures

594

thermal conductivity, estimation

585

thermal conductivity of liquid mixtures

581

thermophysical properties

304

viscosities, estimation

578

Loop reactor mathematical modeling

960 242

243

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Loss coefficient

393

Loss modulus of viscoelastic fluids

407

Lower explosion limit (LEL)

1223

LSODAR

72

LSODE

71

104

Lubricants in biochemical engineering

1159

Lucas method for estimating low-pressure viscosity of gases

578

LUPREA

1066

Lurgi Netzplan System

1038

Lurgi Sandcracker

959

Lydersen estimation of critical data

544

M MacCormack method Mach number MacLaurin series

97 1093 35

Magnesium elastic constants

764

Magnetic nondestructive testing method of ferromagnetic materials Magnetic separation

787 935

Magnetic testing identification of alloys by Major Hazards Directive

787 1251

MAK value classification of hazardous substances

1217

Margules method for calculating the molar excess Gibbs energy of mixing Mass-expansion coefficient

561 519

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Mass-transfer coefficient

350

for some specific flow and mass-transfer situations Mass spectrometry (MS)

362 858

Mass transfer convective

338

in laminar and turbulent boundary layers

348

simultaneous mass and heat transfer, dynamic model

231

with chemical reactions, mathematical modeling

237

without chemical reaction, mathematical modelling

230

MASY

232

1066

Materials of construction alloy systems used as

1016

in chemical plants

1014

nature of failure of

1014

Material testing, nondestructive

785

visual and optical methods

812

Mathematical model based on differential equations

193

based on transport equations for probability density functions empirical for a set of experimental data Mathematical modeling

174

175

175 16 165

Mathematics for transport processes in chemical engineering

279 3

Matrix eigenvalues of for solution of sets of linear algebraic equations transformation in scale-up Maximum experimental safe gap (MESG) apparatus for determining

15 6 1097 1229 1230

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Maximum likelihood method

191

McCabe–Thiele diagram

235

Mean deviation

143

Membrane filter in biochemical engineering

1181

Membrane process

979

Membrane reactor

980

Membrane separation

918

Mercury thermophysical properties

304

viscosity

381

Mercury amalgam process

979

Metal carbides resistance to wear

749

Metal coating by immersion

708

Metals elastic constants for some common metallic materials

764

erosion–corrosion by liquid flow

745

factors influencing the mechanical stability

606

forming of oxide films

695

high-temperature corrosion

695

jet wear

744

mechanical properties, definition and testing

761

metal coatings on

706

nondestructive testing for identification

787

protective film formation and passivity

665

sulfurization

698

symptoms of corrosion damage

608

wear and wear resistance

743

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Metals, surface treatment hardening methods

749

hardfacing

749

plating, spraying methods

749

Meter model

397

398

Methane diffusion coefficient in air

589

temperature dependence of the Gibbs energy of formation

574

viscosity data

577

Methanol (methyl alcohol) diffusion coefficient in air

589

surface tension of aqueous solutions

595

Methyl acetate surface tension of aqueous solutions

595

Methylamine (monomethylamine) prediction of vaporization enthalpy

559

2-Methylbutane –acetone, vapor–liquid equilibria

564

Methyl propionate surface tension of aqueous solutions Metra potential method (MPM) Micromixing

595 1036 939

Microorganism growth and bioreaction

1121

growth rates and Michaelis–Menten constant

1122

Microorganisms suitable equipment for specific processes and products of

1150

Microscopy long-distance microscopes

813

Microwave nondestructive testing of metals and other materials by

811

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Middle value

143

Mineral processing

931

Mineral oil forced convection, heat-transfer coefficient

296

light machine, thermophysical properties

304

Minimum ignition energy (MIE)

1232

Minimum ignition temperature (MIT)

1231

Miniplant technique for process development

890

limitations of

894

Misic–Thodos method for estimating the thermal conductivity of gases at low pressure

586

Missenard incremental method for estimation of liquid heat capacities

555

Missenard method for estimating the thermal conductivities of liquids Mixer–settler

588 916

Mixing hydraulic

1132

in biochemical engineering

1130

in heterogeneous systems

940

introduction

939

of homogeneous miscible substances

939

of pastes and granular materials

940

pneumatic

1132

Mixing head with injection mold

962

Model reactor

487

Modulus of resilience

767

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Moisture measurement of materials by microwave technique

812

Molecular interaction estimation

542

Momentum transfer in laminar boundary layers

319

Monod model

1122

Moody chart

383

Morris method for estimating the viscosity of pure liquids

581

MOSY

1066

Motor control centers (MCC)

1051

Moving-bed reactor for catalytic gas-phase reactions

959

965

965

Multibed reactor with cold-gas injection

968

with heat supply

968

with interstage cooling

968

Multichamber tank

962

Multiphase approaches

472

Multiple-hearth furnace

973

Multiple-hearth reactor

975

Multitubular reactor

960

MVS/LUROMAK

1066

965

N National Environment Policy Plan (NEPP)

1348

Navier–Stokes equation

288

dimensionless form

293

Near infrared spectroscopy (NIR)

839

Necking

766

376

381

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Negvekar–Daubert method for estimating the thermal conductivities of liquids Nernst equation

588 656

Neutron rays detection of explosives by Neville’s algorithm Newtonian constitutive equation

788 16 380

395

Newtonian fluid flow of

379

tube flow, volumetric flow rate

400

Newton number Newton–Raphson method

1093 12

523

Newton’s law of cooling

317

Nickel as construction material in chemical industry

629

elastic constants

764

high-nickel materials, sensitivity to sulfur

698

Nickel alloys as construction material in chemical industry

629

corrosion behavior

629

nickel–chromium

630

Nickel–chromium, intergranular damage by H2S and SO2

705

Nickel–chromium, resistance to various gases

705

nickel–molybdenum and nickel–molybdenum–chromium

630

oxidation behavior

705

special type of stress corrosion cracking

687

wear by flow of pure water and seawater

746

Niobium mechanical properties Nirosta

640 620

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Nitrides melting point, density, and hardness

749

Nitrobenzene (oil of mirbane) –benzene, surface tension of mixture

594

–carbon tetrachloride, surface tension of mixture

594

Nitrogen thermophysical properties

304

viscosity data

577

Nondestructive testing

785

Non-Newtonian fluid basic equations for flow

289

flow in tubes, establishing whether laminar or turbulent

403

in scale-up in chemical engineering

1109

Noridur 9.4460

621

Normal probability distribution

144

Notched-bar impact test

773

NRTL method

884

for calculating the molar excess Gibbs energy of mixing Nuclear magnetic resonance (NMR) spectroscopy Nusselt number Nyquist critical frequency

562 862 1094 28

O Octane –2-butanone, vapor–liquid equilibria

564

–cyclopentanone, vapor–liquid equilibria

564

viscosity

381

Octanoic acid surface tension of aqueous solutions

595

Ogata–Tsuchida method for prediction of boiling and melting point of liquids Ohnesorge number

558 1093

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Oil agglomeration

937

Oldroyd-B equation

408

Olive oil

381

One-factor-at-a-time method

429

Optical spectroscopy

837

Optimization

118

dynamic optimization problems

136

linear programming

130

mixed-integer linear programming

133

mixed-integer nonlinear programming

135

successive quadratic programming

127

Ordinary differential equation as initial value problems

61

computer software for solving

71

Ore separation, survey

932

Orifice plate for flow rate measurement

394

Orrick–Erbar method for estimating the viscosity of pure liquids

581

Orthogonal collocation method for ordinary differential equations

82

on finite elements

82

OSBL investment costs

898

Oscillation wear

752

Ostwald–de Waele fluid

1110

Ostwald–de Waele model

397

ougen–Watson–Langmuir–Hinshelwood kinetics

185

Overpotential

658

Oxide ceramics coatings

711

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Oxides melting point, density, and hardness

749

Oxygen thermophysical properties Oxygen electrode

304 655

Oxygen overpotential and oxygen corrosion Oxygen uptake rate

663 1124

P PAAG methods in plant and process safety

1256

Packed-bed reactor

507

bed porosity of

511

energy balance, mathematical treatment of

507

energy transport in

507

mass balance, mathematical treatment of

507

mass transport in

512

mathematical treatment of

68

513

Paints and coatings for chemical apparatus

719

testing for lack of bonding by thermal methods and by holography

812

Parabolic partial differential equation

94

Parseval equation

39

Partial differential equation

94

solution by using transforms

221

48

Particle size analysis

848

Partition coefficient

916

920

Passivity of metals

665

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Pastes mixing

940

Patent Situation in process development

878

PDECOL

104

PDEPACK

104

Péclet number

294

Penetrameter

792

512

1094

Penetrant method for nondestructive testing

805

Penetration theory for mass transfer from a gas to a liquid phase

353

Pentane –acetone, vapor–liquid equilibria

564

viscosity

381

1-Pentanol surface tension of aqueous solutions

594

2-Pentanone –heptane, vapor–liquid equilibria

564

–hexane, vapor–liquid equilibria

564

3-Pentanone –heptane, vapor–liquid equilibria

564

Peristaltic pump in biochemical engineering

1182

Pertinax hardness and jet wear Photochemical reactor

748 980

Physical properties data sources

539

estimation

537

Physical vapor deposition (PVD) coatings by

711

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Pictet rule

540

Pilot plant

894

1083

Pinch technology

1075

1193

Pi number

1093

1094

Pipe in biochemical engineering

1176

in chemical plants

1042

metallic, nondestructive testing

815

relative roughness

383

ultrasonic nondestructive testing

800

Pipeline wear by hydraulic transport of solids

741

Piston pump in biochemical engineering

1182

Pi theorem

1097

Pitot tube

395

Pitting corrosion

668

Pitzer acentric factor

546

Plackett–Burman designs

443

Plane-strain fracture toughness

781

Plant start-up of pilot plant Plant and process safety

1090 1205

assessment of criticality

1248

batch and continous processes

1292

checklists for preparation, production and termination

1297

chemical reactions

1216

definitions

1210

design

1250

explosion data for gas mixtures

1222

flammability ratings

1219

for exothermic reactions

1238

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Plant and process safety (Cont.) human aspect

1295

internal organization and policies

1296

legal aspects

1272

maintenance

1303

measures

1250

methodological aids

1254

methods of investigation

1243

modification of plants

1312

process control

1273

public awareness

1319

requirements for safety

1209

runaway scenario

1247

safe processing

1259

safety analysis methods

1255

sources of hazards

1207

special equipment

1262

technical inspection

1301

ventilation model

1266

Plant design safety in pilot plant

1091

Plant design and construction

987

apparatus and machinery

1043

approvals procedure

1012

cashflow schedule

1041

commissioning

1061

construction materials in

1014

construction team

1058

contract types and provisions

1029

contract writing

1028

cost assessment

999

course of capital investment

996

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Plant design and construction (Cont.) development of a project

988

development of cost indexes

995

development of new product

990

engineering work progress

1038

example of calculation for plant costs

1019

execution of construction

1056

guide drawings

1045

matrix project management

1033

optimization

1004

organization of a project team

1034

piping

1042

principles of cost controlling

1038

process engineering

1041

process simulation

1004

profitability analysis project control Plasma spraying

996 1035 713

of metals

710

Plasma torch

975

Plaster thermal properties

295

Plastic deformation of metals

764

Plastics as construction materials in chemical industry

642

wear by flow of pure water and seawater

746

Plastics, fiber-reinforced determination of fiber content

804

glass fiber-reinforced as construction materials in chemical industry nondestructive testing of pressure vessels and piping

644 816

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Platforming

Links 968

Plating wear resistance plated metals

750

Poiseuille flow of non-Newtonian fluids

306

one-dimensional, mathematical treatment

304

Poisson distribution

146

Poisson ratio

763

determination from ultrasonic wave velocity

305

796

Poisson’s equation

118

Polarization

658

Polyamides (PA) hardness and jet wear

748

Polybutene as construction materials in chemical industry

643

Polyester resins chemical resistance

644

Polyethylene as construction materials in chemical industry

643

energy requirement for high and low pressure polymerization

1202

Polyethylene hardness and jet wear

748

Polyethylene, low-density (LDPE) viscosity at different temperatures

381

Polymerization reactor types for

955

Polymers hardness and jet wear in biochemical engineering

749 1158

linings from, for chemical apparatus

715

solutions as pseudoelastic fluids

397

716

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Polymers (Cont.) viscoelasticity

395

wear resistance

747

Poly(methyl methacrylate) (PMMA) hardness and jet-wear of acrylic glass

748

Polypropylene (PP) as construction materials in chemical industry

643

Poly(tetrafluoroethylene) (PTFE) as construction materials in chemical industry

644

Polyurethanes (PUR) hardness and jet wear

748

wear resistance

748

Poly(vinyl chloride) (PVC) as construction materials in chemical industry

643

hardness and jet wear

748

Poly(vinylidene fluoride) (PVDF) as construction materials in chemical industry

644

Potentiometry

856

Powder-bed reactor

960

Powder coating from plastic powder

719

Powder flame spraying materials for

751

Power law of pseudoplastic fluids

397

Power-law fluid flow in a tube

307

tube flow, volumetric flow rate

401

Poynting factor

561

Prandtl analogy

330

348

Prandtl number

294

1094

Precedence diagram method (PDM)

1036

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Pressure maximum explosion

1225

Pressure limit of stability for unstable gases

1227

Pressure vessel metallic, nondestructive testing

815

Primer for paints PRO/II

711 887

Probability of outcomes, and statistics

142

Probability density function

145

174

examples of calculating

204

205

mathematical models, based on transport equations for

194

single-point, transport equations for

199

PROBAD/FEZEN PROCESS

200

1066 887

1066

Process control engineering (PCE ) damage minimizing systems

1275

monitoring system

1274

operating systems

1274

requirements for plant safety

1273

safety system

1275

system functions

1274

Process design Process development

1001 873

chemical mechanism in

876

cost estimation

896

mathematical approach

883

miniaturization limits for equipment

893

miniaturization limits for instrumentation components

893

physicochemical data in

877

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Process development (Cont.) plant capacity

880

raw materials for

880

simulation programs for

887

testing on a small scale

883

Process energy optimization of process energy and costs by pinch technology Process hazard assessment and safety evaluation (PHASE)

1075 1241

Process modifications examples for cost savings by pinch technology

1077

Program evaluation and review technique (PERT)

1036

Programmable electronic systems (PES)

1278

2-Propanol (IPA) (isopropyl alcohol) surface tension of aqueous solutions

595

1-Propanol (propyl alcohol) (NPA) surface tension of aqueous solutions

595

Propionic acid surface tension of aqueous solutions

595

Propyl acetate surface tension of aqueous solutions

595

Propylamine calculation of the saturated liquid molar volume

551

surface tension of aqueous solutions

595

Propyl formate surface tension of aqueous solutions

595

Propyl propionate surface tension of aqueous solutions

595

Przezdziecki–Sridhar method for estimating the viscosity of pure liquids

581

Pseudoplastic fluid flow of

397

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Pull test

764

Pulseecho technique

817

Q Quadrature formulas for calculation of integrals

21

Quale’s increment

593

Quasi-Newton update

125

R Rabinowitsch–Mooney equation

400

Radial-flow reactor

968

Radiation sorting (solid–solid separation)

934

Radiochemical reactor

984

412

Radiography lead foil and other screens for image intensification

792

nondestructive testing of materials by

788

radiographic images

790

Raman spectroscopy

844

Raoult equation lowering of the freezing point of solution

568

Raoult’s law

561

Raschig process

960

Rate-governed processes for separation of mixtures Rate of strain tensor Rayleigh number

916 379 1094

Rayleigh scattering

789

REACFD

104

REACOL

104

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Reaction, chemical adiabatic temperature increase

1247

heats of reaction for common processes

1239

on-line monitoring of

821

overpressure-inducing exothermic reactions

1240

probability and kinetics of a runaway

1248

runaway potential

1238

Reaction column

962

Reaction enthalpy calculation

569

573

Reaction, equilibrium calculation

569

dependence on temperature and pressure

570

Reaction overpotential

659

Reaction resin fiber-reinforced coatings from

717

reinforced with glass flakes, grouts and spray coatings from

717

Reactor cascade of ideally mixed, mathematical models

241

catalytic, continuous temperature control

965

design equations for model reactors

487

electrochemical processes

975

endothermic gas-phase reactions

956

exothermic gas-phase reactions

956

for bioprocesses

979

gas–liquid reactions

962

gas–liquid reactions over solid catalysts

965

heterogeneous catalytic reactions in, mathematical modeling

249

heterogeneous gas catalysis

965

homogeneous phase, mathematical modeling

239

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Reactor (Cont.) ideally mixed nonisothermal flow reactor, stability analysis

257

ideally mixed stirred-tank reactor, residence time distribution function

205

ideally nonmixed (ideal tubular), residence time distribution function

207

industrial applications of

953

isothermal, mathematical modeling

240

isothermal, sensitivity analysis

248

isothermal, stability analysis

243

jet-stirred fluid-phase with turbulent flow, mathematical model

194

jet stirred, mathematical model for

218

liquid-phase reactions

962

loop reactor, mathematical modeling

242

model for a chemical reactor with axial diffusion

78

noncatalytic gas–solid reactions

971

noncatalytic liquid–solid reactions

973

noncatalytic solid-phase reactions

973

nonisothermal, mathematical modeling

249

nonisothermal, stability analysis

256

optimization of

522

photochemical processes

980

polymerization reactions

962

radiochemical processes

980

residence time distribution, functions for

205

solid-catalyzed reactions

965

stirred tank, mathematics for

70

submerged aerobic processes

980

with external recirculation

962

with finite mixing, residence time distribution function

208

972

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Reactor (Cont.) with fixed bed of inerts

956

with indirect electric heating

975

with internal recirculation

962

with recycle

956

with turbulent flow, mathematical model for

194

Rectification column mathematical modeling

236

Redlich–Kwong equation

548

Redox electrode

656

Reduction arc furnace

975

Reduction resistance furnace

975

Reformer

959

665

Refractory ceramics refractory bricks as construction material in chemical industry

646

Regenerative furnace

959

Regression, linear

176

standardized computer programs

181

used in design of experiments

444

968

Regression measure in design of experiments Regression, nonlinear

445 184

Reichenberg method for estimating low-pressure viscosity of gases Reiner–Rivlin equation REPR

579 290 1066

Residence time distribution function for chemical reactors

205

Resilience

766

Resista Clad process

708

Resistance polarization

659

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Resitex hardness and jet wear

748

Response surface designs

441

Responsible Care logo Reynolds analogy

1350 330

348

349

Reynolds flux mathematical modeling Reynolds number

261 293

1093

Reynolds stress mathematical modeling

261

mathematical models for

211

Reynolds transport theorem

283

Rheniforming

968

Rheometry

861

Rheopectic fluid

398

Richardson extrapolation

219

66

Riedel method for the prediction of vaporization enthalpy Riesz–Fischer theorem

557 39

Rihani–Doraiswamy method for estimation of ideal gas heat capacities Ring-and-disk reactor

555 962

Risk lethal to humans Risk chart RKF45

1211 1211 71

Robbins–Kingrea method for estimating the thermal conductivities of liquids Rockwell hardness test ROHR2 Roll cladding

588 771 1066 706

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Romberg’s method for integrals

24

Root-mean-square

143

Rotary drum

975

Rotary kiln

973

Rotating bending fatigue test

778

975

Roy–Thodos method for prediction of thermal conductivity of gases

583

Rubber hardness and jet wear

748

lining

715

Runge–Kutta–Feldberg method

64

Runge–Kutta–Gill method

64

Runge–Kutta method

64

Rupture strength

766

Rushton impeller

1131

S Sachsse–Bartholomé process

959

Safety in chemical plant design and construction

1007

Safety ratings

1215

Safety valve

1281

controlled

1282

full-lift

1282

proportional

1282

sizing of

1283

standard

1282

supplementary loaded

1282

weight-loaded and spring-loaded

1282

Sampling device safe design of

1290

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Sato method for estimating the thermal conductivity of liquids

586

Sato–Riedel method for estimating the thermal conductivities of liquids SATU

588 888

Scale-up fundamentals of the theory of models

1099

in chemical engineering

1093

procedure at unavailability of model material systems

1103

Scaling

696

Scattering techniques

848

Schmidt number

221

Schroeder’s estimation of molar volumes of liquids

549

Screening

933

Screw-conveyor reactor

975

SDC

703

1094

1066

Secant method

13

Second law of thermodynamics accounting of entropy for a macroscopic transport system

360

Second virial coefficient calculation

548

Sedimentation gravity and centrifugal sedimentation, survey

926

Self-accelerating decomposition temperature (SADT)

1237

Semenov temperature profiles

1239

Sensitivity analysis of a model Separation factor

74

178

916

Separation of homogeneous mixtures introduction

915

Sewage treatment plant

902

Shaft kiln

972

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Index Terms

Links

Shaft reactor, retort

975

Shallow-bed reactor

968

Shear modulus

763

771

Shear rate of flows Shear strain

380 762

Shear strength of a joint, determination

770

Shear stress

382

Sherardizing

708

Sherwood number

762

770

1094

Shooting method for ordinary differential equations

77

Siegert relation

849

Sieve-tray tower

983

Silicon carbide as construction material in chemical industry

647

Silver elastic constants

764

thermal properties

295

Simplex algorithm

132

Simplex method

186

Simpson’s rule for integrals

22

Simulation program for dynamic systems

887

for steady-state processes

888

SIMUSOLV

888

Single complexity factor model

991

high complexity factor

991

low complexity factor

991

medium complexity factor

991

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Index Terms

Links

Size exclusion chromatography (SEC)

850

Size reduction survey

932

Size separation survey

933

Sliding wear

752

Slug flow

389

Small burner test

1222

Snell’s law

797

Soave equation

548

Solid–liquid equilibrium calculation

568

Solid–liquid separation general considerations

923

Solids heat conduction, mathematical treatment

294

mixing of granular materials

940

Solid–solid separation radiation sorting

934

Solid–solid separation introduction

931

Solids technology development of processing strategy

946

introduction

941

product properties

942

working hypothesis

943

Solid technology measurement methods

949

research and development

951

verification of processing strategy

949

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Index Terms

Links

Sparged stirred tank

963

cascade of

971

with suspended catalyst

971

Spark testing

787

SPEEDUP

888

Spherical coordinate system Spinning jet Splines

60 962 20

Spongiosis

677

Spray coating, metallic

708

Spraying methods for coatings on metals Spray reactor

714 962

Stainless steel addition of molybdenum

618

addition of nitrogen

619

as construction material in chemical industry

617

austenitic

618

austenitic, addition of silicon

621

austenitic–ferritic

623

cast types

621

chromium and chromium–nickel steel

618

composition and short names

619

622

copper-containing, special type of stress corrosion cracking

686

corrosion behavior

623

ferritic

618

ferritic–austenitic

618

martensitic

618

mechanical properties

621

pitting corrosion

669

pitting corrosion test

726

621

622

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Index Terms

Links

Stainless steel (Cont.) stress corrosion cracking

684

technical properties

621

thermal properties

295

Standard airlift reactor in biochemical engineering

1164

Standard deviation

143

Standard electrochemical potential

655

Standard operating procedure (SOP)

1295

Standard potential

655

Stanton number

330

Statistical mathematical model

173

1094

Statistical system probability density functions Statistical variable

209 172

Statistics probability of outcomes, use of statical methods Staverman expression

142 563

Steam condensing, heat-transfer coefficient

296

viscosity

381

Steel as construction material

612

corrosion by combustion gases

699

corrosion by synthesis gas, CO, NH3–H2 mixtures

701

corrosion effects by ash deposition

700

corrosion fatigue

692

engineering stress–strain diagram

765

Fe–Cr–Ni phase diagram

618

for low temperatures in chemical industry

616

for use in chemical plants

612

for use in chemical plants, quality standards

612

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Index Terms

Links

Steel (Cont.) hardness and jet wear

748

heat-resistant steels in chemical plants

615

hydrogen corrosion at elevated temperature

690

hydrogen-induced cracking at low temperature

688

intergranular corrosion

674

oxidation behavior

702

plane-strain fracture toughness

781

resistance to CO2 and CO2–CO

703

resistant to pressurized hydrogen

616

standardized corrosion test methods

724

stress corrosion cracking

682

sulfurization

699

vessel material in biochemical engineering

1150

wear by flow of pure water and seawater

746

wear resistance

743

with high-temperature strength in chemical plants

614

Steel alloy tensile properties Steeping press

767 975

Sterilization “empty” sterilization in place (SIP)

1127

“full” sterilization in place (SIP)

1127

continuous water/steam

1127

in autoclave

1126

in biochemical engineering

1125

steam

1126

Stiel polar factor

547

Stiel–Thodos method for estimating the thermal conductivity of gases at low pressure

586

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Index Terms

Links

Stiffness of a system of differential equations of metals

68 763

765

Stirred tank, continuous cascade of

961

Stirred tank reactor in biochemical engineering

1131

1163

Stirred-tank reactor batch

961

continuous

961

mathematics for safety analysis methods applied to

70 1257

semicontinuous

961

with liquid recirculation

975

Stockmayer potential

289

Stokes–Einstein equation

849

for diffusion of liquids Stokes law

543

589 312

Stokes number

1093

Stokes theorem

58

387

Storage modulus of viscoelastic fluids

407

Strain of metals

762

Strain-hardening exponent

768

Strain-induced corrosion

694

Strain-rate sensitivity

768

Strategic Advisory Group on the Environment (SAGE) Strength coefficient

1352 768

Strength property determination

765

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Index Terms

Links

Stress of metals Stress corrosion cracking (SCC)

762 681

test methods

724

Stress–life curve

778

Stress relaxation test

776

Stress–strain diagram of metals

765

true

767

Stress tensor

285

of fluids

379

Stress vector

284

Strouhal number

1093

STRUDL

1066

Student’s t-distribution

144

Styrene by catalytic dehydration of ethylbenzene, mathematical modeling

249

Styrene production, lowering of energy requirements Submerged-jet reactor

1202 983

Sucrose surface tension of aqueous solutions

595

Sulzer loop reactor

956

Sulzer mixer–reactor

960

Surface tension of liquids, estimation

592

Suspended-bed reactor for liquid-phase and gas–liquid reactions

971

Suspension furnace

973

Suspension reactor

969

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Index Terms

Links

Synthesis gas corrosion by Synthol process

701 971

T Tafel equation

658

662

Tantalum coatings by molten salt electrolysis

708

corrosion resistance

639

hydrogen embrittlement

690

mechanical and physical properties

639

mechanical properties

640

Tantalum alloys mechanical properties

640

tantalum–niobium

640

tantalum–tungsten

640

Taylor–Galerkin method

97

Taylor vortex

384

Taylor vortice

1146

Technical Control Board (TÜV)

1301

Tensile strength, ultimate

766

Tensile test

764

Tensor operation

279

Tensors calculus of

281

Texaco coal gasification process

972

Theorem of corresponding states

543

Thermal analysis, differential (DTA) screening method in process safety Thermal carbon black processes

1243 959

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Index Terms

Links

Thermal conductivity of gases from molecular theory

288

289

of gases, prediction

581

582

of liquids, estimation

585

Thermal-expansion coefficient

518

Thermal inspection method

812

Thermodynamics differentials of thermodynamic functions

153

partial derivatives of thermodynamic functions

155

state functions

153

thermodynamic data of chemical reactions, calculations

569

Thermoelectric properties identification of alloys by

788

Thermography nondestructive testing by

812

Thermoplastics lining

716

powder, coatings from

716

Thermosets as construction materials in chemical industry

644

lining, graphite-filled

717

Thickener for sedimentation Thiele modulus

926 78

92

Thinh method for estimation of ideal gas heat capacities Thixotropic fluid

555 398

Thodos method for estimating low-pressure viscosity of gases

582

Thomas method for estimating the viscosity of pure liquids Threshold limit value (TLV)

581 1218

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Index Terms

Links

Ticorex A

639

Tin corrosion behavior

638

Tin alloys tin bronze

635

Titanium as construction material in chemical industry

638

corrosion resistance

639

elastic constants

764

hydrogen embrittlement

690

mechanical and physical properties

639

plane-strain fracture toughness

781

Titanium alloys tensile properties

638 767

Titanium dioxide, C.I. 77891 suspension, shear stress–shear rate for TN process

398 613

Toluene diffusion coefficient in hexane

590

591

diffusion in chlorobenzene

590

591

Tomography for nondestructive testing by radiography

794

Topsoe process

968

Torsional shear stress

770

Toughness

766

Tower reactor

962

Toxics Release Inventory (TRI) Traction

773

1346 284

Transport equation for single-point probability density functions

199

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Index Terms

Links

Transport phenomenon

271

mathematical treatment

271

summary of the basic transport relations

290

summary of the equations for macroscopic systems

356

Trapezoid rule for integrals

22

Trickle-bed reactor

983

Trickle-flow reactor

965

TRK value

1218

Trouton rule

540

Trouton’s constant

541

Trouton viscosity

410

Tsonopoulos equation

548

Tube flow of non-Newtonian fluids

399

Tubular reactor

959

Tubular reformer

965

Turbidimetry in scattering

848

Turbulent diffusion flame combustion of hydrogen in, mathematical modeling

263

combustion of propane, model for

214

Turbulent flow in tubes

326

mathematical treatment

324

mixing length theories

328

nonreactive, free jets, mathematical model for

218

Turbulent flow

382

Two-film theory for mass transfer between two phases

353

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Index Terms

Links

U Ultrasonic microsensor

833

Ultrasonic nondestructive testing

795

Ultrasonic wave

830

compressive and shear waves in materials

796

generation and detection

797

reflection

797

surface (or rayleigh) waves

797

ultrasonic pulses

797

Ultraviolet–visible spectroscopy (UV–VIS)

839

UNIFAC method for prediction of vapor-liquid equilibria

563

Uniquac equation for calculating the molar excess Gibbs energy of mixing

562

United States Environmental Protection Agency (EPA)

1338

Upper explosion limit (UEL)

1223

Urysohn equation

110

Urysohn equation of the second kind numerical methods for Utility targets

113 1076

V Vacuum plasma spraying materials for

751

Vacuum system leak testing

815

Valeric acid surface tension of aqueous solutions

595

Valve in biochemical engineering

1171

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Index Terms

Links

Vanadium oil ash containing V, corrosion by

700

Van der Waals equation of state reduced

548

Van Krevelen–Chermin method for estimation of thermodynamic reaction data

572

Van Velzen method for predicting the viscosity of pure liquids

578

Vaporization enthalpy calculation

558

enthalpy of

540

Variance

143

Variance analysis in design of experiments Vector calculus of

446 51 281

curl of

55

divergence of

55

divergence theorem

58

Vector analysis

51

Vector differential operator

55

Vector differentiation

54

Vectorial search method

186

Vector integration

57

Vector operation

53

Venturi meter

279

395

Verma–Doraiswamy method for estimation of thermodynamic reaction data VERONA

572 1066

Vetere method for the prediction of vaporization enthalpy VEW

559 620

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Index Terms

Links

Vibromixer

1132

Vibrothermography

812

Vickers hardness test

773

Videoscope

813

Vinyl ester resins chemical resistance

644

Virial equation

548

Viscoelastic fluid

379

395

403

404

Viscometer

380

383

Viscosity

576

mechanics

of gases, effect of pressure on

578

of gases from molecular theory

288

of liquid mixtures

579

of pseudoelastic fluids

397

of pure gases, calculation

576

of shear-thickening (dilatant) fluids

397

reduced, of gases, estimation

576

Viscosity ratio number

1093

Volmer reaction

662

Voltaic cell

656

Voltammetry

857

Volterra equation of the second kind

109

numerical methods for

398

111

Volterra integral equation

109

Von Karman analogy

330

Von Karman friction factor relation

330

Von Karman–Nikuradse equation

383

VTPLAN

887

348

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Index Terms

Links

W Wagner method for the prediction of vaporization enthalpy Wagner’s theory of metal oxidation

557 697

Wall heat-transfer coefficient for packed-bed reactors Waste disposal sites

513 902

Waste heat boilers economic analysis

1194

Water boiling, heat-transfer coefficient

296

diffusion coefficient in air

589

forced convection, heat-transfer coefficient

296

surface tension of aqueous solutions of organic compounds

595

thermophysical properties

304

viscosity at different temperatures

381

viscosity data

577

Watson method for the prediction of vaporization enthalpy

560

Watson’s equation for the temperature dependence of vapor pressure

560

Wear hydroabrasive or scouring wear

741

in pipelines by hydraulic transport of solids

741

of construction materials

735

types and mechanisms

735

Wear intensity

737

Weber number

1093

Weissenberg effect Weissenberg number

404 1093

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Index Terms

Links

Welding of stainless steel

623

radiographic investigation of welds

804

Weld Overlay Cladding

707

Wilson method for calculating the molar excess Gibbs energy of mixing

561

Wilson model

884

Wöhler diagram

692

Wood thermal properties

295

X X-ray attenuation in metals by absorption and scattering

788

generation

788

nondestructive testing of materials by

788

X-ray film evaluation of exposured films

790

for radiography

790

X-ray fluorescence analysis (XRF) material identification by

786

Y Yen–Woods method determination of liquid molar volumes of organic compounds

552

Yield point of metals

765

Yield stress of a fluid, measurement

402

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Index Terms

Links

Yoneda method for estimation of ideal gas heat capacities

555

for estimation of thermodynamic reaction data

572

Young’s modulus

763

determination

765

determination from ultrasonic wave velocity

796

Z Zinc as construction material Zinc alloys

637 637

Zirconium corrosion resistance

639

hydrogen embrittlement

690

mechanical and physical properties

639

641

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