Diffusion and Heat Exchange in Chemical Kinetics 9781400877195

Table of contents :
FOREWORD
PREFACE TO TRANSLATION
FROM THE AUTHOR
CONTENTS
CHAPTER I: INTRODUCTION
Notions of Chemical Kinetics
Reaction Velocity
Simple and Complex Reactions
Order of the Reaction and Energy of Activation
Autocatalysis and Intermediate Products
Chain Reactions
Stationary and Nonstationary Course of the Reaction
Heterogeneous Reactions
Notions of Diffusion and Heat Transfer Theory
Similarity of Diffusion and Heat Transfer Processes
Heat Conductance and Diffusion in an Immobile Medium
Free and Forced Convectidn
Laminar and Turbulent Flow
Transfer Coefficients
Turbulent Exchange Coefficient
Heat Exchange Coefficient
Diffusion Velocity Constant
Similitude Theory
Effective Film
The External and Internal Problem
Resistance Coefficient and Reynolds' Analogy
Relations between the Criteria
Longitudinal Flow Around a Plate
Convection in a Layer of Particles
Differential Equations of the Heat Conductance and the Diffusion
Analysis of the Differential Equations by the Method of the Theory of Similitude
CHAPTER II: DIFFUSIONAL KINETICS
Method of the Uniformly Accessible Surface
First-Order Reaction
Diffusional and Kinetic Region
Example of Chemical Reactions in the Diffusional Range
Combustion of Carbon
Fractional-Order Reactions
Kinetics of Dissolution
Heterogeneous Termination in Chain Reactions
Reaction at the Wall of a Closed Vessel
Diffusional Kinetics of Complex Reactions
Case of Several Diffusing Substances
Reversible Reactions
Parallel and Consecutive Reactions
Autocatalytic Reactions
Uniformly Accessible Surface
Porous Surface
Diffusion Across Membranes
Diffusion Through Pores
Formation of Solid Films
Microheterogeneous Processes
CHAPTER III: THE STEPHAN FLOW
The Maxwell-Stephan Method
Condensation of Vapors in the Presence of Incondensible Gases
CHAPTER IV: NONISOTHERMAL DIFFUSION
Equations of Heat Conductance and of Diffusion in the Simultaneous Presence of Both Processes
The Enskog-Chapman Laws
The Thermal Diffusion Law in Fick's Form
Approximate Theory of Nonisothermal Diffusion
Approximate Form of the Law of Diffusion Thermoeffect
Differential Equations of Simultaneous Heat Conductance and Diffusion
CHAPTER V: CHEMICAL HYDRODYNAMICS
Convective Diffusion in Liquids and Velocity
Distribution Near a Solid Surface
Diffusional Layer
Laminar Sublayer
Dimensional Analysis
General Formulas
Absence of a Laminar Sublayer
Theory of Landau and Levich
Laminar Sublayer Without Transition Zone
Einplrical Formulas
Velocity Distribution in the Turbulent Flow
Laminar Sublayer With Transition
Comparison of the Different Assumptions on the Velocity Distribution
Diffusion to Suspended Particles and Local Structure of Turbulence
CHAPTER VI: THEORY OF COMBUSTION FROM THE POINT OF VIEW OF THE SIMILITUDE THEORY
Theory of Thermal Ignition
Stationary Theory of Thermal Explosion
Nonstationary Theory of Thermal Explosion
Thermal Propagation of a Flame
Theory of Daniell
Theory of Zeldovich
Diffusional Flame Propagation
CHAPTER VII: TEMPERATURE DISTRIBUTION IN THE REACTION VESSEL AND STATIONARY THEORY OF THE THERMAL EXPLOSION
StationaStationary Theory
CorrectiCorrection for the Loss Through Combustion over theInduction Period
Thermal Explosion in Autocatalytic Reactions
Experimental Verification of the Theory of Thermal Explosion
CHAPTER VIII: FLAME PROPAGATION
Equation and Boundary Conditions
Uniqueness of the Solution
Thermal Propagation of the Flame
Method of the Heat Flux
Diffusional (Chain) Flame Propagation In Second-order Auto-cataly3is
Combustion In a Moving Gas
Turbulent Combustion
CHAPTER IX: THERMAL REGIME OF HETEROGENEOUS EXOTHERMAL REACTIONS
Qualitative Theory of the Phenomena of Ignition and Extinction Regardless of the Kinetics of the Reaction
Mathematical Theory of the Phenomena of Ignition and Extinction for Reactions of the First Order
Stationary Surface Temperature Rise
Correction for Thermal Diffusion and Diffusion Thermoeffect
Correction for the Stephan Flow
Experimental Data
Thermal Regime of a Layer or Channel
Applications
Thermal Regime of Contact Apparatus
Combustion of Coal
Catalytic Oxidation of Isopropyl Alcohol
Catalytic Gas Analyzers
CHAPTER X: PERIODIC PROCESSES IN CHEMICAL KINETICS
Conditions for an Oscillating Course of the Reaction In the Neighborhood of Quasistationary Concentrations
Relaxation and Kinetic Oscillations
The System of Volterra-Lotka
Periodic Processes in the Oxidation of Hydrocarbons
Thermokinetic Oscillations
INDEX

Citation preview

DIFFUSION AND HEAT EXCHANGE IN CHEMICAL KINETICS

DIFFUSION AND HEAT EXCHANGE IN CHEMICAL KINETICS By D. A. FRANK-KAMENETSKII Translated by N. Thon

1955 Princeton University Press Princeton, New Jersey

Copyright, 1955, "by Princeton University Press London: Geoffrey Cumberlege, Oxford University Press L. C. Card 55-6250

This book, originally published in Russian in 19^7, was issued through the Publishing House of the U.S.S.R. for the Institute of Chemical Physics of the U.S.S.R. Acadeny of Sciences; Ν. N. Semenov, Responsible Editor. This translation was prepared under contract with the National Science Foundation.

Officers, agents,

and employees of the United States Government, acting within the scope of their official capacities, are granted an irrevocable, royalty-free, nonexclusive right and license to reproduce, use, and publish or have re­ produced, used and published, in the original or other language, for any governmental purposes, all or any portion of the translation.

Printed in the United States of America

FOREWORD In recent years, a new original trend has dev­ eloped In chemical kinetics, aiming at a complex study of the chemical process In combination with the physical processes of transfer of heat and matter. Unexpectedly, It appeared possible to synthesize such apparently distant branches of science as chemical kinetics, on the one hand, and the theory of heat transfer, diffusion, hydro­ dynamics, on the other. Processes which earlier, In the classic chemical kinetics, were viewed as perturbations of the course of a chemical reaction, took on a particular interest in combination with the chemical process. This combination of kinetics with the theory of diffusion, heat transfer, and hydrodynamics yielded a series of new methods of study of rates of reactions, and laid the scientific foundations for the theory of such technically important processes as combustion and disso­ lution, the basic process of chemical technology. Investigators of many countries have taken part in the development of this trend In science. Predvoditelevj Knorre, Vulis (in the theory of combustion of coal), Zharovnikov (in the theory of processes and apparatus of chemical technology), Temkin, Boreskov (in the theory of catalysis), in the U-S.S.R.; Hottel, Mayers, Bxirke and Schumann, Sherwood, Chilton and Colburn, Lewis and von Elbe, Damkohler, Fischbeck, abroad. At the Institute of Chemical Physics of the Academy of Sciences of the U.S.S.R. which devotes itself especially to problems of kinetics of chemical reactions and the theory of combustion, work has been in progress particularly in macroscopic kinetics. It suffices to quote the well known work of Elovich, Todes, Zeldovich

and his school. The author of this book has come from this school. In this book, the attentive reader will find for the first time, not only In the Soviet but In the world of scientific literature, a detailed systematic exposition of all chapters of macroscopic kinetics from one single, original point of view. The author has successfully developed new fruitful methods, Isolated important limiting ranges, introduced a number of new physical concepts, and obtained valuable scientific re­ sults. One can say that this work signals the beginning of the transformation of macroscopic kinetics into an autonomous branch of science. The author has successfully shown that these problems have not only a special and applied, but also a general scientific and theoretical interest. This book represents an original monograph deal­ ing with problems In which the author has taken an active part. Without pretending to absolute completeness and objectivity in the choice of material, it features a unique approach and an original and logically followed point of view· Not all of the author's opinions on the problem dealt with in this book have been generally accept­ ed. In particular, the ideas about the properties of the laminar sublayer, developed in Chapter V, are contro­ versial. But, at any rate, the very formulation of the problems will stimulate further development of science. Soviet science has already achieved an honorable place In the field of chemical kinetics in general and in macroscopic kinetics in particular. There is every reason to expect that this book will stimulate further development in this field and will prove helpful to many chemists, physicists, and engineers who work on applied problems. Academician N. N- Semenov

PREFACE TO TRANSLATION

Theories and applications of interlocking chemi­ cal and physical rate processes have attracted the atten­ tion of chemists and chemical engineers throughout the world. It has become recognized that combustion processes and the rapid chemical reactions of industry, for example, are complex and that analytical treatment requires know­ ledge of chemistry, fluid dynamics and heat transfer. An extensive literature has been developed in this pioneering area of applied science in the United States, in Europe, and in Russia. This volume by Prahk-Kamenetskii has been translated to bring conveniently to hand, phases of Russian work to 19^7· PrankTKamenetskii is a disciple of the Ν. N. Semenov school of physical chemistry which has worked on combustion, explosives, heat exchange, diffusion in heterogeneous systems and chain reactions. He has pub­ lished on reaction ignition, quenching and on periodic processes in chemical kinetics as well. The mathematical treatment of these and related topics for flames, com­ bustion of 3olids and chemical reactions, together with experimental comparisons, comprise the subject of this book. References to work in other countries are given, but references to Russian studies predominate and consti­ tute a selected bibliography. In a few instances, Russian references are supplemented in the translation by editorial footnote references to work elsewhere. The translation was ably performed by the late Nathaniel Thon, a man who has labored extensively in the task of making the results of Russian chemical work avail­ able to English speaking scientists. Editing was restrict­ ed to verifying technical consistency in translation and consistency with usage of expression in the field. R. H. Wilhelm Princeton University

FROM THE AUTHOR "I not only saw from other authors, but am convinced by ny own art, that chemical experi­ ments combined with physical, show peculiar effects." Μ. V. Lomonosov (Project of foundation of a chemical laboratory at the Imperial Academy of Sciences, March,17^5·) The theme of this book is the role of physical factors in the course of chemical reactions. Lomonosov, two hundred years ago, foresaw how fruitful the combination of chemistry and physics can be. To this statement, the development of science up to the present day provides ever new confirmation. Many have con­ tributed to the ideas to which this book is devoted. I express my deep gratitude to my teachers:

Ya.

B. Zeldovlch, L. D. Landau, Ν. N. Semenov. I cordially thank my closest collaborators: Erna Blyumberg, N- Ya. Buben, Ts. M. Klibanov, I. E. Salnikov, Elena Fridman. For their interest and valuable advice and comments, I am indeb­ ted to G· N. Abramovich, A. A. Andronov, A. P. Vanichev, V. V. Voevodskii, L. A. Vulis, V. I. Goldanskii, Μ. V. Keddysh, G. F. Knorre, E. M. Minskii, M. B. Nieman, V. V. Pomer­ antsev.

CONTENTS Page FOREWORD

v

PREFACE TO TRANSLATION

vii

FROM THE AUTHOR

viii CHAPTER I:

INTRODUCTION

Notions of Chemical Kinetics Reaction Velocity Simple and Complex Reactions Order of the Reaction and Energy of Activation Autocatalysis and Intermediate Products Chain Reactions Stationary and Nonstationary Course of the Reaction Heterogeneous Reactions Notions of Diffusion and Heat Transfer Theory Similarity of Diffusion and Heat Transfer Processes Heat Conductance and Diffusion in an Immobile Medium Free and Forced Convectidn Laminar and Turbulent Flow Transfer Coefficients Turbulent Exchange Coefficient Heat Exchange Coefficient Diffusion Velocity Constant Similitude Theory Effective Film The External and Internal Problem Resistance Coefficient and Reynolds' Analogy Relations between the Criteria Longitudinal Flow Around a Plate Convection in a Layer of Particles Differential Equations of the Heat Conductance and the Diffusion Analysis of the Differential Equations by the Method of the Theory of Similitude

3 3 h it5 6 7 io 1i 11 11 12 13 1it15 16 17 18 23 2525 29 36 37 39 Uo

CHAPTER II: DIFFUSIONAL KINETICS Method of the Uniformly Accessible Surface First-Order Reaction Diffusional and Kinetic Region Example of Chemical Reactions in the Diffusional Range

ix

it-7 50 52

CONTENTS Page Combustion of Carbon Fractional-Order Reactions Kinetics of Dissolution Heterogeneous Termination in Chain Reactions Reaction at the Wall of a Closed Vessel Diffusional Kinetics of Complex Reactions Case of Several Diffusing Substances Reversible Reactions Parallel and Consecutive Reactions Autocatalytic Reactions Uniformly Accessible Surface Porous Surface Diffusion Across Membranes Diffusion Through Pores Formation of Solid Films Microheterogeneous Processes CHAPTER III:

5k 60 65 70 72 81 81 87 90 9b 96 96 111 115 115 117

THE STEPHAN FLOW

The Maxwell-Stephan Method Condensation of Vapors in the Presence of Incondensible Gases

138 1

CHAPTER IV: NONISOTHERMAL DIFFUSION Equations of Heat Conductance and of Diffusion in the Simultaneous Presence of Both Processes The Enskog-Chapman Laws The Thermal Diffusion Law in Fick's Form Approximate Theory of Nonisothermal Diffusion Approximate Form of the Law of Diffusion Thermoeffect Differential Equations of Simultaneous Heat Conductance and Diffusion CHAPTER V:

A6 1^7 150 152 1 151*

CHEMICAL HYDRODYNAMICS

Convective Diffusion in Liquids and Velocity Distribution Near a Solid Surface Diffusional Layer Laminar Sublayer Dimensional Analysis General Formulas Absence of a Laminar Sublayer Theory of Landau and Levich Laminar Sublayer Without Transition Zone Einplrical Formulas Velocity Distribution in the Turbulent Flow Laminar Sublayer With Transition

x

158 160 161 163 166 170 172 175 178 181 184

CONTENTS

Comparison of the Different Assumptions on the Velocity Distribution Diffusion to Suspended Particles and Local Structure of Turbulence

192 196

CHAPTER VI: THEORY OF COMBUSTION FROM THE POINT OF VIEW OF THE SIMILITUDE THEORY Theory of Thermal Ignition Stationary Theory of Thermal Explosion Nonstationary Theory of Thermal Explosion Thermal Propagation of a Flame Theory of Daniell Theory of Zeldovich Diffusional Flame Propagation CHAPTER VII:

203 205 213 218 223 226 233

TEMPERATURE DISTRIBUTION IN THE REACTION VESSEL AND STATIONARY THEORY OF THE THERMAL EXPLOSION

Stationary Theory Correction for the Loss Through Combustion over the Induction Period Thermal Explosion in Autocatalytic Reactions Experimental Verification of the Theory of Thermal Explosion CHAPTER VIII:

255 256 257

FLAME PROPAGATION

Equation and Boundary Conditions Uniqueness of the Solution Thermal Propagation of the Flame Method of the Heat Flux Diffusional (Chain) Flame Propagation In Second-order Auto-cataly3is Combustion In a Moving Gas Turbulent Combustion CHAPTER IX:

236

267 269 272 276 278 281 282

THERMAL REGIME OF HETEROGENEOUS EXOTHERMAL REACTIONS

Qualitative Theory of the Phenomena of Ignition and Extinction Regardless of the Kinetics of the Reaction Mathematical Theory of the Phenomena of Ignition and Extinction for Reactions of the First Order Stationary Surface Temperature Rise Correction for Thermal Diffusion and Diffusion ThjBrmoeffect

288 292 30k

310

CONTENTS Page Correction for the Stephan Plow Experimental Data Thermal Regime of a Layer or Channel Applications Thermal Regime of Contact Apparatus Combustion of Coal Catalytic Oxidation of Isopropyl Alcohol Catalytic Gas Analyzers CHAPTER X:

325 330 338

339 3U2 3^5 3^8

PERIODIC PROCESSES IN CHEMICAL KINETICS

Conditions for an Oscillating Course of the Reaction In the Neighborhood of Quasistationary Concentrations Relaxation and Kinetic Oscillations The System of Volterra-Lotka Periodic Processes in the Oxidation of Hydrocarbons Thermokinetic Oscillations INDEX

316

350 353 35U 356 358 362

CHAPTER I:

INTRODUCTION

Classic chemical kinetics studies the course of a. chemical reaction under idealized conditions, namely at a temperature constant in time and space, and at concentrations constant in space. The task of macroscopic kinetics Is to study the chemical reaction under the real conditions of its occurrence In nature or in industry, i.e., with the physical processes attendant on the chemical reaction included.

The most impor­

tant of these physical processes are the diffusion of the reactants and products and the evolution and propagation of heat.

Both these processes are strongly influenced by the

hydrodynamic conditions, i.e., the kind of motion of the gas or liquid which determines the convectlve transfer of heat and of matter. Thus, the specific scope of macroscopic kinetics is the study of the role of diffusion, heat transfer, and convectipn (I.e., the motion of the gas or liquid) in the course of chemical reactions. Investigation of the complex processes which, along with the chemical conversion, are governed also by transfer of heat and of matter, has a three-fold purpose. In the first place, it yields the laws of the course of the chemical process under conditions encountered

2

I. INTRODUCTION

in practice. The practically Important theories of com3 and technical catalytic probustion 1'2,of dissolution^ cesses ' are built on the foundations of macroscopic kinetics. The second purpose of macroscopic kinetics is to provide ways to uncover the true mechanism of the chemi­ cal reaction where direct investigation is hampered by the complicating factors of diffusion and heat evolution. In this case, study of the thermal and diffusional pro­ cesses permits elimination of these factors and isolation of the pure chemical kinetics. Particularly fruitful in this respect have been thermal methods of investigation of the kinetics of strongly exothermal reactions: studies of the kinetics of combustion reactions by the ignition limits^ and by flame propagation velocities^, and of the kinetics of heterogeneous exothermal reactions by the O critical conditions of inflammation and extinction . The third fruit of the study of macroscopic kinetics of chemical reactions is disclosure of the laws of the very processes of transfer of matter and of heat, particularly of the processes of convective diffusion which are most appropriately investigated in connection with chemical processes^'9. Convective transfer of matter is usually linked with turbulent motion of the gas of liquid· Therefore, studies of such processes as dissolution of metals or salts in a turbulently moving liquid can supply valuable information on the very properties of turbulent motion and are thus of considerable interest from the point of view of hydrodynamics. We shall designate that branch of macroscopic kinetics which seeks to elucidate hydrodynamic problems through investigation of chemical processes, as chemical hydrodynamics. The main subdivisions of macroscopic kinetics are: 1. Diffusional kinetics, i.e., study of

I. INTRODUCTION

3

the role of diffusion In the course of heterogeneous chemical reactions In cases where thermal factors can be disregarded. 2. Combustion theory, I.e., study of the role of heat transfer In the course of homogeneous exothermal reactions. 3• Thermal theory of heterogeneous exothermal reactions, bearing on the most complex case where both diffusion arid heat transfer are essential factors. We shall first discuss the fundamentals of these two branches. NOTIONS OF CHEMICAL KINETICS Reaction Velocity Chemical reactions are classified Into homogeneous and heterogeneous processes, depending on whether they take place in a single phase or at an Interface. The rate of a homogeneous chemical reaction Is the amount of matter reacting In unit volume per unit time. This magnitude represents the change of the concen­ tration of the substance per unit time, I.e., the deriva­ tive of the concentration with respect to time. The rate of a heterogeneous reaction Is the amount of substance reacting on unit surface area per unit time; It Is not directly linked with the change of concentration. In both cases, the rate of reaction is a function of the temperature and of the concentrations of the reactants.

k

I.

HOTRODUCTION

Simple and Complex Reactions All reactions, whether homogeneous or hetero­ geneous, can be classified Into simple or complex. We apply the term simple to reactions In which the rate depends only on the concentrations of the original reactants and does not depend on the concentrations of the reaction products. By the term complex, we shall designate reactions in which the rate depends on the concentrations of both the reactants and the final products, or, much more fre­ quently, of the intermediate products. As the concentration of the latter is often difficult to determine, and is subject to considerable variation with time, that dependence will simulate an apparent dependence of the rate of the reaction on time. Order of the Reaction and Energy of Activation The dependence of the reaction rate on the con­ centrations of the participating reactants is usually described by a power law of the form n

ν

A = kc/ ·

nR Cbb . . .

(I,

ι )

where ν is the rate of the reaction, and C^, Cg .•. are the concentrations of the reactants A, B ... taking part in the reaction. The coefficient k which depends only on the temperature is termed the rate constant, and the exponent n, the order of the reaction. There is a distinction be­ tween the order of the reaction with respect to an indi­ vidual reactant (e.g., nA is the order with respect to reactant A) and the overall order, equal to the stun of the Individual orders. In gaseous reactions, the overall order is also referred to as the order with respect to total pressure.

I. INTRODUCTION

5

The temperature dependence of the rate constant is given by Arrhenlus1 law k = ze"E/RT

(I, 2)

Here T is the absolute temperature, R, the gas constant, and E and ζ are constants characteristic of the given chemical reaction. The magnitude E is termed the acti­ vation energy and, z, the pre-exponential factor. The activation energy represents the amount of energy which is necessary for the molecule to possess in order to react. In the case of a simple reaction, the magnitudes C in formula (I, ι ) represent the concentrations of the original reactants only. In the case of complex reactions, the formula includes also the concentrations of the re­ action products, sometimes the final ones, but much more frequently the intermediates. Autocatalysls and Intermediate Products If the rate of a reaction increases with in­ creasing concentration of a product formed in that re­ action, it is termed autocatalytic. There can be autocatalysis by a final or by an intermediate product. The species which causes the increase of the rate is called the active species. The majority of real chemical processes are com­ plex and proceed over active intermediates. Reactions be­ tween two stable molecules require a large activation energy and, therefore, such reactions are slow. The actually observed reactions usually proceed by a rounda­ bout path which circumvents this high energy barrier 1 0 ' 1 1 > In the case of homogeneous reactions, the first stage usually consists in the formation, from the original reactants, of some kind of intermediates. Their nature is not well enough elucidated at this stage, but it is safe to assume that it can be different in different

12

6

I.

INTRODUCTION

Instances. Doubtlessly, in many cases, free radicals or free atoms can play the role of such active intermediates. In other cases, it can be complex and relatively stable molecules which, however, for various reasons possess a high degree of reactivity, as organic peroxides. The active intermediates react with the original reactants to form the final products. These steps require relatively low activation energies (particularly when the active species are free radicals or atoms) and are there­ fore very fast. In contrast, the primary formation of the active intermediates from the stable original molecules does require a high activation energy and cannot therefore be fast. In order for the reaction over active inter­ mediates to proceed rapidly, it is necessary that these intermediates be regenerated in the course of the re­ action; in other words, it is necessary that the reaction between the intermediate and the original reactant yield not only the stable final product but also new active intermediates. Chain Reactions Reactions in which active intermediates are re­ generated are termed chain reactions. It has been estab­ lished by Semenov10, Hinshelwood, and others, that the majority of the actual homogeneous complex reactions are chain processes. There are two characteristic ways in which a chain reaction can progress; it can be either stationary or non-stationary. Let χ designate the concentration of the active intermediate. Its change with time is governed by the kinetic equation ^ = nQ

+

fx - gx

(I, 3)

I. INTRODUCTION

7

where nQ, f, g are kinetic constants usually termed as follows: nQ Is the rate of generation of chains, f is the rate of chain branching, and g is the rate of chain termination. By chain generation, one means the initial process of formation of the active Intermediate from the original reactant; chain branching is the process wherein a molecule of the active intermediate, reacting with an original re­ actant, gives rise to two or several molecules of an active intermediate: chain termination is the process in which the active intermediate is irr'evocably annihilated. In addition to these processes, included In equation (I, 3), the chain can be carried on through a process in which a molecule of the intermediate, reacting with an original reactant, forms the final product and one new active-intermediate molecule. While this process does regenerate the chain, its rate does not figure in equa­ tion (I, 3) as it does not result in any change of the amount of the active intermediate; it merely regenerates the exact amount that has been expended. But the rate of chain continuation, multiplied by the concentration of the active species, determines the rate of the overall con­ version of the original reactants into the final products. Stationary and Nonstatlonary Course of the Keaction Solutions of the equation (I, 3) vary depending on the relative magnitudes of f and g. if g > f, the course of the reaction is stationary, the concentration χ of the active intermediate will, with time, tend to the stationary value χ•

>

Once this stationary value has been attained, the concen­ tration of the active Intermediate will remain constant and the reaction will proceed at the constant rate

8

I.

INTRODUCTION ν = kX

where

k

(I, 5)

Is the rate constant for the chain continuation. Actually,

nQ, g,

and

f

depend on the concen­

trations of the original species and therefore change slow­ ly over the course of the reaction. magnitude

At the same time, the

also changes according to equation (I, 1O-

X

It is therefore more correct to refer to it not as a stationary but as a quasi-stationary concentration of the active species.

The Initial change of

reaches its quasi-stationary value

X,

x,

until it

takes place with­

in a very short time Interval during which the concentra­ tions of the original substances have not had time to change significantly· An entirely different situation will arise if f > g, ary.

the course of the reaction will become nonstatlon-

The solution of equation (I, 3) has the form

x

where

φ » f - g,

sponding to

t

and

=

^ept t

(I,

- 1 )

is the time.

much greater than

,

6 )

At stages corre­ the concentration

of the active species and the rate of the reaction will increase with time proportionally to an exponential law.

,

according to

The initial period during which the

concentration of the active species and the rate of the reaction are comparatively small is called the induction period.

Its duration is of the order of If the reaction Is conducted In such a way that

g

and

f

vary, a sharp transition from stationary to

nonstationary will occur at

g = f.

This penomenon is

referred to as chain ignition. Thus far, we have considered the simplest case where only one active species Is Involved in the reaction and where the kinetic equations are linear.

In reality,

I.

INTRODUCTION

9

the equations can be of more complex form; however, in principle, the picture remains the same. Let the reaction involve) several intermediate active species at the concentrations x^· For the change of any one of them with time, we can write

= P1Cx, a)

where

a

is the Initial concentration, and

(I, 7)

F

some

function. Equating the right-hand members to zero, we get the system of algebraic equations. F1Cx, A) - 0; F2(x, A) = Oj ... F±(x, a) = 0 (I, 8) Solution of this system should give the quasi-stationary concentrations X of the intermediate species as a func­ tion of the initial concentrations a X - f(a)

(I, 9)

If the system of equations (I, 8) has finite real positive roots, and the conditions derived In our paper11 are ful­ filled, the reaction will take place under stationary con­ ditions. After the lapse of a short Initial period, the concentrations of all intermediate species will be very close to the quasi-stationary values X and from then on will vary only with the variation of the initial concen­ trations. If there are not finite real positive solutions to the system (I, 8), the course of the reaction will be nonstationary, i.e., the concentrations of the active intermediates and the rate of the reaction will increase with time. At constant initial concentrations, this growth will be unlimited, and come to a halt only as a result of exhaustion of reactants. Mathematically, the transition from the station­ ary to the nonstationary course of the reaction corresponds

I.

INTRODUCTION

to a point where the system of equations (I, 8) ceases to have finite real positive roots. That Is the condition of chain Ignition In Its most general form· Heterogeneous Reactions In heterogeneous reactions, the role of active Intermediates Is usually taken over by molecules bound to the surface by chemical forces, or, following the usual terminology, chemlsorbed to the surface. Therefore, In the chemical mechanism of a heterogeneous reaction, some adsorption stage usually plays a decisive role. The kinetics of actual heterogeneous reactions Is usually strongly complicated by lnhomogenelty of the surface. A real solid surface Is never uniform In the kinetic and energetic sense. Different portions of the surface are characterized by different values of the heat of adsorption and the activation energy. Adsorption on a uniform surface is governed by Langmuir's adsorption Isotherm which expresses the de­ pendence of the amount g of adsorbed substance on its partial pressure ρ

in the gas phase by the formula

® where

gQ

and

b

=

®o ρ + b

are constants.

In the so-called Langmulr kinetics on a uniform surface, the dependence of the rate of reaction on the concentrations of the reactants In the gas phase should be of an analogous form. On a nonuniform surface, the dependence of the adsorbed amount of a substance on its partial pressure In the gas phase is usually described by the Freundllch Isotherm 1 g = CPn

I. INTRODUCTION where C is a constant, and

η > ι.

With this type of adsorption isotherm, the re­ action kinetics usually follows a fractional order. Theoretical analysis of the kinetics of complex heterogeneous reactions on a nonuniform surface is beyond the scope of this book. References to the pertinent lit­ erature can be found in our review1^. For a theoretical interpretation of the Preundlich isotherm, see the work of Zeldovich NOTIONS OP DIFFUSION AND HEAT TRANSFER THEORY Similarity of Diffusion and Heat Transfer Processes Phenomena of transfer of heat and of matter are similar. Heat transfer by molecular heat conduction has its analog in molecular diffusion, and heat transfer by convection, in convective diffusion. Only heat transfer by radiation has no analog in transfer of matter. Thanks to the similarity of the processes of ,transfer of heat and of matter, it is not necessary to consider each class separately. All theoretical and ex­ perimental results obtained in the study of heat trans­ can be directly applied to diffusion phenomena, fer1^' and vice versa. Heat Conductance and Diffusion in an Immobile Mediim The fundamental law of transfer of heat in an Immobile medium (molecular heat conduction) is Fourier's law according to which the heat flow q, i.e., the quantity of heat transferred across unit surface area per unit time, is proportional to the temperature gradient q. = - λ H

(I, 10)

I. where

χ

INTRODUCTION

is the coordinate perpendicular to the surface

across which the heat is transferred, and λ is the heat conductivity, a physical constant of the substance in which the heat is propagated. The minus sign Indicates that the heat is transferred In the direction along which the temperature decreases, i.e., in the direction of the negative temperature gradient. The corresponding law for diffusion is Fick1s law according to which the diffusion flow Is proportional to the concentration gradient q. = - D

(I, 11 )

where C is the concentration of the diffusing substance, and q is the diffusion flow, I.e., the quantity of substance transferred across unit surface area per unit time. We use the same symbol q for both the flow of heat and of matter, because of the similarity of heat and diffusion flow; this can hardly give rise to any misunder­ standings. The proportionality coefficient D is termed the diffusion constant. Heat conductance and diffusion in an immobile medium can be observed in a pure form only In solids. In liquids and in gases they are Tinavoidably accompanied by a motion of the liquid or the gas in free or forced convention. Free and Forced Convection In contrast to molecular heat conduction and diffusion, where the transfer of heat or of matter is brought about by the motion of individual molecules, the transfer of heat or of matter due to a motion of the gas or the liqviid as a whole is termed convection. If that motion is caused by the same difference of temperatures or concentrations which determines the transfer of heat or of matter, the convection is referred to as free or

I. natural.

13

INTRODUCTION

If the motion is brought about by external forces,

the process is termed forced convection. In the presence of convection, the laws of Fotirier and of Pick must be supplemented by additional terms expressing the transfer of heat or of matter by the mass flow. With the latter designated by

v,

Pick's law

takes on the form d = - o g - + VxC

(I, 1 la)

and Fourier's law becomes q = - λ g|· + βρνχΤ

(I, ioa)

where νχ is the component of ν in the direction of the χ coordinate, c is the heat capacity, and ρ the density. Lmnlnar and Turbulent Flow The character of the convective transfer of heat or of matter depends on the type of motion of the gas or liquid. Depending on hydrodynamic conditions, this motion can be either laminar or turbulent. A laminar flow is an ordered stationary motion wherein the velocity at each point is constant in time, and the velocities at neighbor­ ing points sire parallel to each other. A turbulent flow is a disordered nonstationary motion wherein the velocity at any point changes continually, with time In a random fashion. In a laminar flow, the mechanism of the transfer of heat or of matter is essentially the same as in an immobile medium. As before, the transfer takes place through molecular heat conductance or diffusion, and only the external conditions vary as a result of the mass flow.

I.

INTRODUCTION

Not so In a turbulent flow where the transfer takes place through turbulent pulsations, I.e., disordered motions of small volumes of gas or liquid. Transfer Coefficients In addition to the similitude, there is one more deep analogy between the mechanism of the processes of diffusion and heat transfer, and the process of transfer of momentum which underlies the resistance of a gas or liqiold to motion. In the absence of turbulence, the in­ tensity of all these three processes is characterized by the coefficients of molecular transfer. These co­ efficients will be defined in the following way. For the transfer of heat, we shall introduce the so-called thermometric conductivity coefficient a, related to the usual heat conductivity coefficient

λ

by

where c is the heat capacity, and ρ the density. In other words, the temperature conductivity coefficient is the heat conductivity divided by the heat capacity of unit volume. For the transfer of matter, we shall use the ordinary diffusion coefficient ( I ,

D

defined by formula

1 1 ) ·

For the transfer of momentum, we Introduce the kinematic viscosity ό, related to the usual viscosity μ by « - £

(I, 13)

All three transfer coefficients, a, D, and υ have the same dimension, cm2/sec. In gases, where the mechanism of transfer of all three magnitudes, heat, matter, and

I. INTRODUCTION

15

momentum is the same and Is determined by the thermal motion of the molecules, all three coefficients, a, D, and x> are numerically of the same order of magnitude. In liquids, the kinematic viscosity can be much greater than the temperature conductivity, and, particularly, the diffusion coefficient. According to the kinetic gas theory, the trans­ fer coefficients for an ideal gas are of the order of the product of the mean free path Λ and the root mean square velocity U of the molecules a » D =» « ® Λϋ

(I, 14)

Turbulent Exchange Coefficient In the case of turbulent motion in a gas or liquid, the role of all three transfer coefficients is taken over by the so-called turbulent exchange coefficient A. The disordered turbulent motion of a ga3 or liquid is similar to the disordered thermal motion of molecules, ex­ cept that it involves not individual molecules, but small volimes of gas or liquid which keep their Individuality for some time· Such small volumes will be referred to as gaseous or liquid particles. An illustration is Figure 1 which represents the pulsations of the velocities of wind. The longitudinal dimensions which characterize the turbu­ lent motion are referred to as the scale of the turbulence In hydrodynamics, two different methods are used to describe the motion of a gas or liquid. The first, proposed by Lagrange, follows the motion of a given indi­ vidual liquid particle; the second, due to Euler, con­ siders the distribution of velocities in space at a given moment. Correspondingly, the two basic scales of turbu­ lence are termed, respectively, Lagrangian or Eulerian· The Lagrangian scale of turbulence is the path along which the particle keeps its Individuality. The Eulerian scale of turbulence is the mean dimension of such an individual

16

I.

INTRODUCTION

particle. One can define a magnitude which, for turbulence, plays the role of a mean free path. That magnitude Is the mixing length 1. It is related with the tixpbulence scales, and the form of that relation can depend on the character of the motion. In the simplest case of isotropic turbulence, when the pulsations of the velocity are the same in all directions, the mixing path coincides with the Lagrangian turbulence scale. The analog in turbulence of the root mean square velocity of molecular motion is the mean

2-

pulsation velocity PIODRE 1 . PULSATIONS OP WIND VELOCITIES Th© abscissa represents the horizontal and the ordinate the vertical components of the velocities, as measured at the airport of Akron, Ohio (USA). Th© figure is taken from "Problems of turbulence , edited by Velikanov and Shveikovskii (ΟΗΓΙ, Moscow 1936)·

u.

In analogy with the gas kinetic theory, the coefficient of turbulent exchange is repre­ sented as the product

A = Iu

(I, 15)

Heat Exchange Coefficient Processes of transfer of heat and matter in convective motion are not always susceptible to analytic cal­ culation, particularly when the motion is turbulent. There­ fore, such problems are usually approached with the aid of empirical coefficients. Por heat transfer, the ratio of the heat flow q. and the temperature difference ΔΤ is usually termed the heat exchange coefficient or,

I.

INTRODUCTION q = αΔΤ

17

(I, 16)

This relation Is known as Newton's law of heat exchange. Actually, It Is not the expression of a law of nature but merely a definition of the heat exchange coefficient. Formula (I, 16) does not, of course, solve the problem of calculation of the heat transfer process, but reduces It to the determination of the heat exchange co­ efficient. This can be done either from experimental data and empirical formulas derived therefrom, or, with the aid of the theory of similitude as will be described below. Nonetheless, the use of the heat exchange co­ efficient has proved a very convenient method of calcula­ tion and has taken root In practice. Diffusion Velocity Constant In analogy with the heat exchange coefficient, we shall introduce, for the transfer of matter in the presence of convection, a magnitude to be termed the diffusion velocity constant, β, defined as the ratio of the diffusion flow q and the concentration difference AC, q = βΔΟ This relation is to be viewed merely as a diffusion velocity constant β. Inasmuch flow is expressed In moles/cm χ sec. and moles/cm^, the dimension of the diffusion is cm/sec, i.e., that of a velocity.

(I, 17) definition of the as the diffusion the concentration in velocity constant

Dimensionally, β does not correspond to a, but to where c is the heat capacity and p, the density. That is because, in diffusion, there is no mag­ nitude analogous to the heat capacity. Concentration is defined, simply, as the amount of matter per unit volume, whereas temperature is not simply equal to the amount of

18

I. INTRODUCTION

heat contained in unit volume but to its quotient by the heat capacity of unit volume·

cp,

Similitude Theory It is seldom possible to find the value of the heat exchange coefficient a or of the diffusion velocity constant β by analytical calculation. In the majority of cases, it is necessary to make use of experimental data. Methods of generalization of the experimental data thus become very important. In this way, experimental results on heat transfer can be utilized for the calculation of diffusion processes and vice versa. Results of experi­ ments made on a small model can be utilized for the cal­ culation of processes in a large-scale setup, and results of experiments made with one substance can be utilized for the calculation of processes with another substance participating. Ways of such generalizations of experimental data are supplied by the similitude theory. This theory rests on the postulate that the real laws of nature cannot depend on the choice of the system of units of measurement. Therefore, any real law can be represented in the form of a relation between dimensionless magnitudes, the so-called similitude criteria. Once this relation has been establish­ ed, for certain geometric and physical conditions, by means of an evaluation of experimental data, it can further serve for the calculation of any process taking place under the same geometric and physical conditions but with different dimensions, velocities, and physical properties of the substances. Moreover, the relation between dimensionless magnitudes, obtained through an analysis of heat transfer phenomena, can be directly utilized for the calculation of diffusion processes. We are interested in the determination of the heat exchange coefficient a and of the diffusion velocity constant β. The similitude theory shows that it is

I. INTRODUCTION

19

possible to construct two different dimensionless para­ meters with these magnitudes. One is the so-called * Nusselt criterion, the other the Margoulis criterion. The Nusselt criterion for heat transfer is de­ fined by Nu =

(I, 18)

where a is the heat exchange coefficient, d, a linear dimension, and λ the heat conductivity of the medium in which the heat transfer takes place. For the diffusion process, Nusselt1s criterion is of the form Nu = ^ where α is the diffusion velocity constant, dimension, and D the diffusion coefficient.

(I, 19) d,

a linear

The Nusselt criterion has proved to be most con­ venient for the calculation of transfer processes in an immobile medium or in a laminar flow. In the instance of purely molecular transfer, Nusselt1s criterion is a constant magnitude, depending only on the geometric shape of the body. In the presence of significant turbulence, another dimensionless magnitude, called the Margoulis ι7 criterion , has proved to be more convenient. This mag­ nitude characterizes the ratio of the velocity of trans­ verse transfer of heat or matter and the linear velocity V of the flow. For the heat transfer process, Margoiilis criterion is expressed by M = cW

(I' 20)

For diffusion, Margoulis criterion is defined simply as the ratio of the diffusion velocity constant β and the linear velocity of the flow V, also called Stanton Group.

20

I.

INTRODUCTION M| =

(I, 21 )

In the limiting case of highly developed turbulence, Margoulis criterion ought to tend to a constant value. This limiting case Is referred to as the region of pure turbulence. It cannot be actually reached but only be approached asymptotically. The conclusion that Margoulls criterion should tend to constancy at the limit of pure turbulence follows from the consideration that under these conditions the process should become Independent of molecular constants and be governed only by magnitudes characteristic for turbulent transfer. The similitude theory leads to the conclusion that for a given geometric shape the Nusselt and the Margoulis criteria should be functions of other dimensionless magnitudes expressing the physical properties of the medium and the nature of the motion of the gas or liqiild. The physical properties of the medium In which the transfer of heat or of matter takes place are charac­ terized by the value of a dimensionless magnitude termed the Prandtl criterion and are defined as the ratio of two coefficients of molecular transfer. For the heat transfer process, Prandtl's cri­ terion is expressed by Pr = I

(I, 22)

and for diffusion, by Pr = § where

x>

(I, 23)

is the kinematic viscosity,

conductivity coefficient, and

D

a,

the temperature

the diffusion coefficient.

From what has been said above about the order of magnitude of the coefficients of molecular transfer, the

I.

INTRODUCTION

21

values of Prandtl1s criterion for gases are close to unity. Por liquids, they are much greater than unity. For viscous liquids, the thermal Prandtl criterion can attain several hundreds. The diffusional Prandtl cri­ terion is even greater and attains the order of a thou­ sand for ordinary aqueous solutions. The character of the motion of a gas or liquid in forced convection is determined by the value of Reynolds' criterion Re = ^

(I, 24)

where V is the linear velocity of the flow, d, a linear dimension, and υ the kinematic viscosity. Reynolds' criterion is a magnitude of purely hydrodynamic nature. It includes no magnitudes characteristic of the process of transfer of heat or matter itself. Therefore, there is no difference in the definition of the Reynolds' criterion for the heat transfer or the diffusion process. The value of Reynolds' criterion determines the nature of the motion of the gas or liquid in the flow. At low values of the Reynolds number, the motion is laminar and at high values, turbulent. It is shown in the similitude theory that in pro­ cesses of forced convection the Nusselt or the Margulis criterion, for given geometric and physical conditions, is a definite function of the criteria of Reynolds and Prandtl: Nu = f(Re, Pr) M = φ(Re, Pr)

(I, 25) (I, 26)

The form of that dependence is found by analysis of experimental data of heat transfer or diffusion. Once that dependence has been established, it can be used to calculate the coefficient of heat transfer or the diffusion velocity constant for any processes taking place under

22

I.

INTRODUCTION

similar geometric and. physical conditions.

The only

difference between the calculation of heat transfer and of diffusion Is that In the first Instance one has to In­ troduce In formulas (I, 2?) and (I, 26) the value of the thermal, and In the second Instance, the value of the dlffuslonal Prandtl criterion. Having calculated the value of the Nusselt or the value of the Margoulls criterion from (I, 25) or (I, 26), we can easily determine the coefficient of heat exchange a or the diffusion velocity constant β with the aid of formulas (I, 18) and (I, 19), or (I, 20) and (I, 21 ) which for this purpose are of the form a =

= McpV

The form of the relations (I, 25) or (I, 2 6 ) is different for laminar and for turbulent flow. In laminar flow, the Nusselt criterion tends to become constant, whereas in turbulent flow the same applies to Margoulis criterion. Consequently, (I, 25) is more suitable for laminar flow and (I, 26), for turbulent flow. In all transfer processes, one can observe two limiting regions. Iiihen the Reynolds criterion tends to zero, one observes purely molecular transfer, realized in an immobile medium or in a laminar flow· In this case the laws of transfer of heat or matter have the form Nu = const The contrary limiting case corresponds to the Reynolds' criterion tending to Infinity, when the laws of heat transfer and diffusion tend to the limiting form charac­ teristic of the turbulent region M = const

I.

INTRODUCTION

23

The latter region cannot be actually realized.

With the

Reynolds criterion tending to Infinity, the heat' transfer and diffusion laws tend to their limiting turbulent form only logarithmically. Por processes of free convection, the criterion governing the character of the motion is, instead of Reynolds' criterion, another dimensionless magnitude call­ ed the Grashof criterion and defined by 7δΤ

Gr

where

g

Is the gravity acceleration,

(I, 27)

d,

a linear

dimension, x> the kinematic viscosity, 7 the volume ex­ pansion coefficient of the medium, and ΔΤ the tempera­ ture difference giving rise to the convection. For gases 7 = ijr, where T is the absolute temperature, and the Grashof criterion takes the form Gr

gd O2

. ΔΤ

(I, 27a)

T

For free convection, the Nusselt or the Margoulls criterion is a definite function of the Grashof and Prandtl criteria. Effective Film In describing processes for transfer of heat or matter between a stream of gas or liquid and a solid sur­ face, it is often convenient to introduce the conventional concept of the effective film. Let us assume that at a distance from the surface the temperature and the concen­ tration are constant (this assumption is not all too far from reality In a turbulent flow) and that changes of these magnitudes occur only In a layer of thickness δ immediately adjacent to the surface. This fictitious layer is what is called the effective film. Its thickness

2k

I.

INTRODUCTION

δ is so chosen that the actual Intensity of the transfer Is obtained on the assumption that Its mechanism Is purely molecular within the film. In this way we have a

=

β β

D

~ B

These relations should be considered as definitions of

δ.

Comparing (I, 18) and (I, 19)> we find δ =^

(I, 28)

In this way, the thickness δ of the effective film appears as an auxiliary magnitude replacing Nusselt1s cri­ terion. In diffusion processes, the effective film Is often termed the diffusion layer. The External and Internal Problem The Influence of Reynolds' criterion on the char­ acter of the motion of the liquid differs depending on the geometric conditions. In hydrodynamics, one distinguishes two types of problems depending upon the character of the motion of a liquid - the so-called external problem and the internal problem· The external problem is that of a flow around an isolated body, the overall dimensions of which can be considered infinite. The linear dimension figuring in the similitude criteria is in that case the dimension of the body placed in the flow. The internal problem deals with a flow within a tube or channel. In this case, the linear dimension d is the diameter of the tube. In the external problem, the transition from laminar to turbulent takes place without discontinuity.

I.

INTRODUCTION

25

With gradual change of the Reynolds criterion, all magni­ tudes characteristic of the flow, in particular the Nusselt and Margoulis criteria, change continuously. In the internal problem, the transition from laminar to turbulent occurs discontinuously at a certain critical value of the Reynolds criterion, which, for a straight circular tube, lies between 2100 and 2300. This phenomenon is referred to as the hydrodynamic critical condition. At this condition, all magnitudes character­ istic of the flow, in particular the Nusselt and Margoulis criteria, change discontinuously. In the external problem, the purely laminar and the purely turbulent as well are only limiting cases for very small and very large values of the Reynolds criterion. A characteristic and significant property of the internal problem is the range of purely laminar motion of the liquid, the so-called Poiseuille flow, in which turbu­ lence is entirely absent. This is the only possible flow at values of the Reynolds criterion below the critical. In this flow, the velocity distribution over the cross-section of the tube satisfies the parabolic law of Poiseiiille

Such a distribution is called the laminar velocity profile. In a turbulent flow the velocity profile is considerably steeper. Over the major part of the cross-section, the velocity is almost constant; there is a sharp fall of the velocity only in close vicinity to the wall, in the socalled boundary layer. Resistance Coefficient and Reynolds' Analogy In every contact between flowing gas or liquid and a solid body, the flow exerts on the body a definite

26

I.

INTRODUCTION

force and meets a corresponding resistance. The force acting on the unit surface area of the body is termed the tangential stress and is designated by τ. The ratio of the resistance force and the velocity push of the oncoming flow V2 P — is termed the resistance coefficient. In an external flow around a body", the resist­ ance coefficient usually means the ratio of the force act­ ing on unit surface area of the middle section, (i.e., the greatest section of the body perpendicular to the on­ coming flow) and the velocity push F

C =

JJ2 u

2

where F is the total force acting on the body and the surface area of the middle section.

S,

In the internal problem, the resistance co­ efficient means the ratio of the drop of pressure over a definite length and the kinetic energy in the fluid.

efficient

Two different definitions of the resistance coare ueed in the literatiire for that case.

In the German literature, the resistance co­ efficient is defined as the ratio of the drop of pressure over a length equal to the diameter of the tube, and the kinetic energy ^

· Ψ ·

4T = · f pI C 2

where

Δρ

(I' 29)

2

is the drop of pressure over the length

L-

In the American literature, the resistance co­ efficient is used in the sense of the ratio of the *

-"

commonly called friction factor.

I.

INTRODUCTION

27

tangential stress acting on unit surface area of the tube wall, and the kinetic energy f = -ψ

where

tq

(I, 30)

Is the tangential stress at the wall. As the pressure acts on an area equal to the

cross-section of the tube and the tangential stress on an area equal to the product of the perimeter and the length, there is between the drop of pressure and the tangential stress the identical relation ApS = I0IL

(I, 3 1 )

where S is the surface area of the cross-section of the tube, and Π its perimeter. Taking the value of tq from (I, 31 ) and substituting it in (I, 30), one has instead of (I, 30)

r - -% • I

(i,

32)

In this way, the resistance coefficient is defined by the drop of pressure over the length S = £ π

(I, 33)

equal to the ratio of the surface area of the cross-section of the tube and its perimeter. This length is called the hydraulic radius. For a circular tube, S = 2y-; π = Kd 1)· whence

28

I. INTRODUCTION * · • $

(I, 3*0

Thus, for a circular tube, (I, 35) " £

The resistance force is composed of two parts, the fric­ tion resistance and the form resistance. The latter (or, as it is also called, the pressure resistance) is linked with phenomena of flow separation and formation of a zone of reverse circulation behind the body. The character of these phenomena is illustrated by Figure 2, representing the pattern of the motion in a transverse flow around a cylinder (after data of Lohrisch). A change of the conditions of the rupture of streams in connection with the formation of turbulence In the boundary layer can, under certain circumstances, result in a sharp change of the resistance coefficient in an external flow. This is clearly seen from the graph (Pigiire 3) giving the de­ pendence of the resistance coefficient on Reynolds' cri­ terion in a flow around a FIGURE 2. Pattern of the motion In sphere. There Is a sharp drop transverse flow around a cylinder (after Lohrlsch). of the resistance coefficient at a Reynolds number of about 100,000. All these phenomena are essential only in the external problem, In particular for transverse flow around bodies. For a straight tube with smooth walls, only the friction resistance plays a role-

I.

INTRODUCTION

29

In the mechanism of Internal friction consisting in transfer of momentum, the similitude of diffusion and heat transfer can be extended to frlctlonal resistance. This similitude between heat exchange (and, consequently, also diffusion) and frlctlonal resistance was first es­ tablished by Reynolds and is called the Reynolds analogy. If the resistance is due only to friction, the resistance coefficient corresponds to the Margoulls criterion, and the two magnitudes are related by the simple expression M = i = |

(I, 36)

C 0.5 0.4 0.3

0.2

5.0

5.1

5.2

5.3

5.4

5.5

5.6 Ig Re

FIGURE 3. Dependence of the resistance coefficient on Reynolds' criterion In a flow around a sphere. Ordinate: Abscissa:

resistance coefficient log Re

Relations Between the Criteria The form of the relations between the Nusselt or Margoulia criteria and the Reynolds and Prandtl criteria for systems of a definite geometric shape Is determined by an analysis and a generalization of experimental data, or, In simpler cases where there is no influence of turbulence, by analytical calculation· In the Internal problem, we have two entirely different relations for laminar and for turbulent flow conditions. In a flow, the Nusselt criterion depends very little on the Reynolds criterion; in turbulent flow,

30

I. INTRODUCTION

it is proportional to a power of the Reynolds close to unity.

number

In the most important and practically most fre­ quent case of turbulent motion in the internal problem, the dependence of Nusselt1s criterion on Reynolds' cri­ terion can be expressed only by empirical formulas obtain­ ed through evaluation and generalization of experimental data. A number of such formulae, differing little from each other, have been proposed by different authors. The most popular is Kraussold's formula* Nu =

0 . 0 2 l j -Re0,8Pr0·35

^1'

37^

In laminar motion in the internal problem, which law of transfer of heat or of matter to follow depends strongly on the length of the tube. The velocity profile is shaped in the initial portion of the tube, then temperature or concentration profiles are established. Only at a sufficient distance from the origin of the tube is the flow fully stabilized, with the distribution of velocities, temperatures, and concentrations over the cross-section remaining unchanged along the tube. At this stage, the velocity profile obeys the well known parabolic Poiseuille law. In a completely stabilized flow, i.e., at a con­ siderable distance from the origin of a sufficiently long tube, Husselt's criterion for a laminar flow tends to the constant value Nu0 = 3-659

(I, 38)

arrived at by Nusselt by way of analytical calculation. For a flow not stabilized in the hydrodynainic * Known by a variety of names. See Heat Transmission, W. H. McAdams, McGraw-Hill Book Co., New York, 1 9^2.

I.

INTRODUCTION

31

sense, the problem becomes indefinite, with the process depending on the shape of the velocity profile, determined by the conditions at the entrance of the tube. For a laminar flow stabilized in the hydrodynamic, but not stabilized in the thermal or diffusional sense (i.e., with a well-formed velocity profile, but not yet formed temperature or concentration profile), the analy­ tical solution was found by LevSque· According to LevSque1s law1®, the relation between the similitude cri­ teria is of the form N u = 1.615

^Re-Pr^

(I, 39)

where d is the diameter of the tube and L is its length. This formula gives the mean value of Nusselt's criterion over the whole length of the tube. An analogous formula gives the local value of Nu at a point distant by ζ from the origin of the tube, but with a coefficient of 1.077 in place of 1.6 7 5· The product Re-Pr in Levique1s formula is an­ other dlmensionless magnitude (similitude criterion) some­ times called the P^clet criterion: Pe = Re-Pr = ^ d

(I, ^o)

Thus, for a laminar flow, the Nusselt criterion is pro­ portional to the cube root of Reynolds' criterion at a short tube length, and independent thereof at greater tube lengths. The law of Leveque (I, 39) and the law of sta­ bilized flow are limiting laws, valid for very large and for very small values of the magnitude Pe · With a -Li practically acceptable accuracy, one can say that Leveque1s law applies at values of this latter quantity Pe ^ > 50, IJ and the law (I, 38) at values smaller than unity. In be­ tween, there is an intermediate range where only empirical

I.

32

INTRODUCTION

formulas are usable or else a Nusselt solution In the form of an Infinite series - very inconvenient for practical computations. The dependence of the mean and the local values of Nusselt1s criterion on

Pe j-

and

Pe ^ is represent­

ed graphically in Figures 4 and 5, taken from the work of 19 Ya. M. Rubinshtein . The sloping straight line was traced with the aid of Ley^que1s formula, the curve, by Nusselt's solution; the horizontal line shows the limit­ ing value of Nusselt1s criterion, with the corresponding experimental data of Rubinshteiri· log Nu 1.5 1.4 1.3 1.2 1.1

O O0 O O

O

O

1.0 0.9

O SP

0.8 0.7

Lfev §qu< Nu

0.6

ri

0.5 FIOTHE 4.

eS

.61 i(P 1Ju

= 3.659

Formula i/ of Y Nusselt

l09(Pei)

MEAM VALUE OP NOSSELT'S CRITERION IN A LAMINAR FLOW,

FOLLOWINa RUBINSHTEIN Th© ordinate is log Nu and the abscissa log Pe ^ -

The inclined line

represents the theoretical solution of Nusselt. Its upper rectilinear part coincides with L^vSque's formula (I, 30); its applicability is shown by the dotted sloping straight line. The horizontal dotted line shows the limiting value of Nusselt1 s criterion for an infinitely long tube (I, 38). The circles represent the experimental data of Rublnshtein-

If in the Internal problem, the dependence between the criteria can be expressed by general formulas, in the external problem it depends strongly on the shape of the body and the characteristics of the flow (e.g., the degree of its turbulization). Ordinarily, Nusselt's criterion appears to be proportional to Reynolds' criterion to the power 0.5, this relation holding in a wide range of vari­ ation of the Reynolds criterion. The sharp transition

I. INTRODUCTION

33

from laminar to turbulent flow is absent in the external problem.

FIGURE 5. LOCAL VALUE OF NOSSELT'S CRITERION IN A LAMINAR FLOWFOLLCWING RTJBINSHTEDP 9 (The designations are the same as in Figure ^.)

Thus, for the external problem, one can adopt

where the exponent m varies between 0.4 and 0.67, and n between 0.3 and 0.4. Both these magnitudes and the coefficients k, depend on the geometric shape of the body and the degree of turbulization of the oncoming flow. The most frequently occurring values of the exponents in formula (I, in ) are

In the simplest case of the external problem — longitudinal flow around a plate — these values of the exponents were obtained analytically by Pohlhausan20.

3b

I.

FIGURE 6.

INTRODUCTION

DEPENDENCE OF NUSSELT'S CRITERION ON REYNOLDS' CRITERION

IH A FLOW AROUND A SPHERE, ACCORDING TO VYRUBOV AND SOKOLSKII Ths Ordinate is Nu and the abscissa Re. The full line corresponds to the Sokolskii formula (I, ii5), the dotted line to the Vyrubov foimula (I, lis). The graph Is plotted In a logarithmic scale.

For another simple case, gas flow around a sphere (I.e., at a value of Prandtl's criterion close to unity), experimental data obtained by Vyrubov21 lead, for larger values of Re, to the formula

This formula holds at Re > 200. At small Re, Nusselt's criterion for a sphere tends to the limiting constant value

which is very easily obtained analytically. In the intermediate range at Re < 200, one has the formula of Sokolskii22

rhe complete form of the dependence of Nusselt's criterion Dn Reynolds1 criterion in the case of a gas flow around a sphere, according to the experiments of Sokolskii, is

I. INTRODUCTION

35

represented in Figure 6. The dotted straight line corresponds to the formula of Vyrubov (I, ^3). For thin fibers, the experiment gave, at small Re, the limiting constant value of Nusselt's criterion.

even though the analytical solution of the thermal or diffusional problem for a cylinder did not yield such a limit. In the foregoing, the relations between the criteria were given in the form (I, 25), i.e., the expressions of Nusselt's criterion were given as is usually done in the literature. If Nusselt's criterion is known, it is very easy to compute the criterion of Margoulis, with the aid of the obvious relation

In this way one can pass from a relation in the form (I, 25) to the form (I, 26). The dependence of the resistance coefficient on Reynolds' criterion is approximately the same as the dependence of Margoulis criterion, as can also be inferred from Reynolds' analogy [formula (I, 36)]. For turbulent motion in the internal problem, Blasius1 formula

is very popular. According to it, the resistance coefficient is inversely proportional to Reynolds' criterion to the power 0.25, whereas according to formulas (I, 37) and (I, ^7), Margoulis criterion is Inversely proportional to the Reynolds criterion to the power 0.2. The dependence

36

I.

INTRODUCTION

of the resistance coefficient for tubes of different roughness is represented graphically in Pigxare 7· 1.0

0.6

0.4 02

o.io 0.06

0.04 0.02

0.010 0.006

0.010 0.006

0.004

0.004

0.002 0.001

IO3

IO2 FIGURE 7-

K IO4

0.002 0.001

IO5

IO5

IO4

Re

DEPENDSiCE OP THE RESISTANCE COEFFICIENT ON REYNOLDS' CRITERION IN THE PLOW IN A TUBE

The ordinate is the resistance coefficient; the abscissa is the Reynolds cri­ terion. The graph is plotted in a logarithmic scale· The vertical arrow corresponds to the critical value of Reynolds1 criterion; it separates the laminar region to the left from the turbulent region to the right. In the laminar region, the full line corresponds to Poiseuille1 s law; the dotted line prolongs it into the region where laminar motion is still possible, but is unstable· In the turbulent region, the full lines correspond to turbulent motion in smooth (lower curve) and in rough tubes (upper curve). The graph was talien from the book of McAdams, Heat transfer.

Longitudinal Plow Around a Plate The simplest case of the external problem is the longitudinal infinite flow around a plate. In this case there is a distinction to be made between local and mean values of all the significant magnitudes, the Reynolds, Nusselt, and Margoulls criteria, and the resistance coefficient. By local value of a magnitude, we mean its val­ ue at a distance χ from the origin of the plate; this χ is taken as the determining dimension in the computa­ tion of the local values. The local value of Reynolds' criterion is ex­

Vx

pressed by — it increases proportionally to x. The J local value of Nusselt's criterion is proportional to the square root of Reynolds1 criterion, i.e., inversely

I.

INTRODUCTION

37

proportional to the square root of the distance from the origin of the plate. The local values of Margoulis cri­ terion and of the resistance coefficient are inversely proportional to the square root of Reynolds' criterion, i.e., inversely proportional to the square root of the distance from the origin of the plate. The same applies also to the local values of the heat exchange coefficient and of the diffusion velocity constant. These rules are valid for the initial portion in the flow around any body and in particular also for the walls of a tube in its initial portion. Only at a STifficient length of the tube, when the flow has become stabilized in the hydrodynamic sense, are these rules superseded by the entirely different relations which govern the internal flow. Convection in a Layer of Particles The flowing motion through mass consisting of particles of matter (e.g., a heap of coal) represents another case, Intermediate between the external and the internal problem· Such a heap can be considered^ with equal justification, either as an assembly of bodies bathed by an external flow, or as an assembly of channels between these bodies. Clearly, the laws of transfer of heat and of matter in such a layer must be intermediate between those applying to the external and those apply­ ing to the internal problem. The relation between the similitude criteria can, as always, be represented empirically by a formula of the forai (I, 1J-I ), with the value of the exponent m lying between the value of 0.5, characteristic of the external problem, and the value 0.8, characteristic of the internal problem. In the latest and most thorough work of Bernshtein, carried out in the laboratory of

I. INTRODUCTION

38 pp*

Knorre , the following relation was obtained for an air stream flowing through a layer of regular spherical particles Nu = A · Rem

(I, ^9)

with m = 0.6. Here, the velocity Involved In Reynolds' criterion Is calculated for the whole cross-section of the layer (not for the free cross-section). For the dimension d involved in the criteria of Nusselt and Reynolds, one takes the diameter of a sphere. The magnitude A in formula (I, 49) depends on the separation of the layer. By separation, one means the ratio of the volume of all intergranular spaces and the total volume of the mass (without allowing for the inner particle porosity). The dependence of the co­ efficient A in formula A (I, ^9) on the separation 2.0 of the layer, obtained by Bernshtein, is represented in Figure 8. This graph shows that, as a function of the separation, the co­ efficient A passes through a maximum. The reason is that with increasing sep­ aration, the free surface of the particles increases, 0 50 100% but, for a given value of FIGURE 8. DEPENDENCE OF OHE COEFFICIENT A OF BERNSHTEIN S the velocity, the true FORMULA (I, 1»9) ON THE POROSITY OF THE Rren velocity calculated for the Ordinates: coefficient A Abscissas: porosity of the bed whole cross-section of the in percent. layer decreases. 1

l

* See also: Colburn, A., Industrial Engineering Chemistry, 23 910 (1931 ) ·

I.

INTRODUCTION

39

Differential Equations of the Heat Conductance and the Diffusion The foregoing elementary exposition of the main results of the theory of heat transfer in 1TTimobile and In moving media are sufficient for the solution of many prob­ lems of macroscopic kinetics.

However, in many cases we

shall have to use differential equations describing the transfer of heat or of matter. reviewed In the following.

These equations will be

The equation of diffusion In an immobile medium is of the form = div D grad C + q

(I, 50)

where C is the concentration, D the diffusion coef­ ficient, and q' the density of the sources of substance, i.e., the quantity of substances formed as a result of chemical reactions in unit volume per unit time. The symbols div and grad refer to vector-analysis opera­ tions termed, respectively, divergence and gradient.

q' = 0,

If the diffusion coefficient equation (I, 50) becomes

D

is constant and

(I, 50a) where Δ = div grad is the Laplace operator which, in a rectangular coordinate system is equal to the sum of the second derivatives with respect to all three coordinates. The equation of heat conductance in an immobile layer is of the form' cp where

T

= div λ grad T + q

is the temperature,

the density,

λ

c

(I, 51 )

the heat capacity,

the heat conductivity, and

q1

the

ρ

I.

INTRODUCTION

the density of the sources of heat, i.e., the quantity of heat evolved as a result of chemical reactions in unit volrne per unit time. If the heat conductivity

λ

can he considered

constant, equation (I, 51) becomes

3T

=

^

aj

t

'

+

For stationary processes, the terms

(I

and

Am

51a)

are

equal to zero. In the presence of convection, the equations (I, 50) and (I, 51 ) must be supplemented by the terms ν grad C and ν grad T (where ν is the flow velocity) and be solved simultaneously with the hydrodynamic equations. Analysis of the Differential Equations by the Method of the Theory of aimilltuae The theory of similitude enables one, without resorting to integration, to find the general forms of the solution save for the unknown function, i.e., to es­ tablish what dimensionless parameters should figure in the solution sought. To this end, it is necessary to ex­ press the equations in dimensionless variables by intro­ ducing as yardsticks for all the variables, magnitudes figuring in the conditions of the problem. In this way, all constant coefficients become magnitudes of the same dimension. By dividing the whole equation by one of these coefficients, we obtain a dimensionless equation in which dimensionless parameters play the role of constant co­ efficients. Its solution can contain, besides dimension­ less variables, only such dimensionless parameters; all laws describing the given physical process can be repre­ sented by relations between these dimensionless parameters. We will apply this method to the above differ­ ential equations.

We introduce a natural length yardstick

I. INTRODUCTION

41

d for which we take a dimension characteristic of the system, and we replace the usual coordinates x by the dimensionless coordinates

For the temperature and the concentration, we introduce as natural yardsticks the characteristic temperature difference AT and the characteristic concentration difference AC; the dimensionless temperature will then be defined as

and the dimensionless concentration as

where T Q and C Q are magnitudes arbitrarily taken as the zero of temperature and of concentration, and figuring In the conditions of the problem. Finally, as a natural yardstick of time, we introduce the characteristic time t and we define the dimensionless time as

After this transformation, the equations of heat conductance and of diffusion in an Immobile medium (I, 50a) and (I, 51a) take the forai

I.

b-2

INTRODUCTION

Each of these equations contains one dlmensionless para­ meter, a.T —5· d

, and

DT —j· d

These parameters are sometimes called the thermal and the dlffuslonal criteria of homochronlclty. If we formulate the conditions of the problem in such a way that they contain a magnitude of the dimension of time, which we shall take as the characteristic time, then, inasmuch as the equation contains only one parameter, its solution should give the dependence of the dimensionless temperature or concentration on dlmensionless co­ ordinates and a dlmensionless time, with only that one dimensionless parameter:

θ = f [ ξ, t1, p )

(Γ, 52)

ζ = f ( ξ, t', p- ]

(I, 53)

The laws governing the process as a whole should be ex­ pressible by relations between dimensionless magnitudes. In this instanoe, as there is only one dlmensionless para­ meter, such a relation is reduced to a constant value for the parameter aT

= const

d2 for heat transfer and = const d for diffusion. As a very useful practical conclusion, it follows

I.

INTRODUCTION

h3

that the characteristic time of the heat transfer or of diffusion in an Immobile medium is proportional to the square of the linear dimension of the body. If no magnitudes of the dimension of time are involved in the conditions of the problem, one will have to take for the natural yardstick of time the only magni­ tude of the dimension of time that can be constructed from other magnitudes which are Included in the conditions of the problem, d2

for heat transfer, and t

Ί5

for diffusion The dimensionless time will be expressed as t' =

or

t' =

d

dd

and the solutions of the equations will be of the fom θ = f ^l, ||)

(i, 5¾)

ζ = f ( I,

(I, 55)

]

These expressions can be obtained from (I, 5 2 ) and (I, 53) by replacing the yardstick of time τ by the variable time t. In an entirely analogous way, we can transform the laws of Fourier and Pick. Introducing the dimensionless temperature

I.

INTRODUCTION

the dlmensionless concentration

and the dlmensionless coordinate

we transform (I, 10) into

and (I, 11) into

The left-hand members of these expressions are the familiar Nusselt criterion. If the laws of Fourier and Fick are supplemented by convective members, their conversion to dlmensionless variables will bring forth the Peclet criterion, and, in the same way, Reynolds' criterion will appear in hydrodynamic equations. Combination of these criteria will yield all the remaining similitude criteria discussed above. Literature 1. 2. 3. it-.

ZELDOVICH. Teoriya goreniya i detonatsii gazov (Theory of combustion and detonation of gases) Moscow (1944). FRANK-KAMENETSKII, Gorenie uglya (Combustion of coal). Uspekhi Khim. 7., 1277 (1938). BUBEN and FRANK-KAMENETSKII, Zhur. Flz- Khim. 20, 225 (1946). DAMKOHLER, in Eucken-Jakob's Chemie Ingenieur, III, T. 1, 448 (1937).

I. INTRODUCTION

45

5-

FRANK-KAMENETSKII, KKENTSEL and ZBEREV, Khlm. Promyshlennost' No. i, 31 (1946).

6.

PRANK-KAMENETSKII, Zhur. Flz. Khlm- J_3, 738 (l 939) -

7.

ZELDOVICH and SEMENOV, ZHDR. eksptl• teoret. Flz. 10, 1116 (1940).

8.

KLIBMOVA and FRANK-ICAMENETSKII, Acta Phyaicochltn. !§., 387 (19^3)-

9-

LEVICH, Zhur. Flz. Khlm- 18, 335 (19^4).

10. 11.

SEMENOV, Tsepnye reaktsli (Chain reactions). (1934). FRANK-KAMENETSKII, Zhur. Flz- Khlm. 14, 695 (l94o).

12.

FRANK-KAMENET SKI I, Klnetlka slozhnykh reaktsli (Kinetics of complex reactions) Uspekhi Khim. K>, 373 (1941 ).

13-

FRANK-KAMENETSKII, Uspekhi Khlm. H>, 544 (1941 ).

14.

ZELDOVICH, Acta Physlchochlm. J_, 449 (1934).

15-

GUKHMAN, Fizicheskie osnovy teploperedachi (Physical foundation of heat transfer) (1934).

16.

KIRPICHEV, MIKHEEV, and EIGENSON, Teploperedacha (Heat Transfer) (19^0).

17.

MARGOULIS, ¥., Chaleur et Industrie, 134, 135, 269, 352, (1931).

18.

LEVEQUE, Ann. des Mines, (12), J_3, 201, 305, 381, (1928).

19-

RUBINSHTEIN, Metod analogli s diffuziei; prlmenenle ego dlya issledovaniya teploperedachi v nachalnom uchastke truby (Method of analogy with diffusion and its application to the Investigation of heat transfer in the initial portion of a tube). Article in the volume "issledovanle protsessov regulirovaniya, teploperedachi i obratnogo okhlazhdeniya" (Investigation of processes of regulation, heat transfer and reverse cooling) (1938).

20.

POHLHAUSEN,

1, 115 (1921 ).

21. VYRUBOV, 22.

"Issledovanle protsessov goreniya naturalnogo topliva" (Investigation of processes of combustion of natural fuel) edited by Knorre (In press).

CHAPTER II: DIFFUSIONAL KINETICS

In the real course of a heterogeneous reaction in nature or in industry, the observed rate of reaction is determined partly by its true chemical kinetics at the surface and partly by the rate of transport of reactants to that surface through molecula.r or convective (or turbu­ lent) diffusion. The study of the course of chemical reac­ tions under such conditions is the object of diffusiona.1 kinetics. In principle, the problem can be approached by three methods, all three of which a.re used. The first con­ sists in an exact analytical solution of the differential equation of the diffusion under boundary conditions which are determined by the kinetics of the reaction at the sur­ face. In mathematical language these will be combined bound­ ary conditions of the type -D gra,d C = kCn where the kinetics of the reaction is expressed by a. formula of the type (I, 1). This first approach is used only in certain specific problems in the series of papers of Paneth and Herzfeld1, Damkohler2, Predvoditelev and Tsukhanova , Levich , and k6

II. DIFFUSIONAL KINETICS Seraenov51. These authors confined themselves to first-order reactions. Each individual case, characterized by definite geometric and physical conditions, becomes an independent complex problem of its own. There can be no question of any general results applicable to any geometric configuration or any kind of motion of the gas. The second method involves complete sacrifice of an analytical solution and application of the theory of si­ militude not only to transfer processes but to the chemical process itself. This method was sketched in a series of paQ pers by Vulis and was recently developed systematically by 56 D'yakonov . In our own work extensive use was made of a third method-' which we call the "quasi-stationary method" or the "method of uniformly accessible surface." This method of approximation not only simplifies the calculation consider­ ably but, which is even more important, permits isolation of the physically essential limiting cases. METHOD OF THE UNIFORMLY ACCESSIBLE SURFACE This approximate but generally applicable method of solution of diffusional kinetics consists in the follow­ ing. It is assumed that the conditions of diffusional trans­ ference can be considered approximately independent of the course of the reaction at the surface. This method is ap­ plicable to cases where all portions of the surface can be considered equally accessible with respect to diffusion. Such surfaces will be referred to as uniformly accessible. The diffusion process can be described either by integration of the diffusion equation with a simple ( C = O at the surface) rather than a combined boundary condition or by means of the experimental data of diffusion or heat ex­ change. This permits broad utilization of the methods and the results of the theory of similitude. Let the concentration of the reactant in space be

II. DIFTPUSIONAL KINETICS C and its concentration at the surface, where the reaction takes place, be designated by C1. In the external prob­ lem, C will mean the concentration at infinite distance from the surface and in the internal problem, the mean volume concentration. The rate of reaction at the surface depends on

C1

and is given by the true chemical reaction kinetics at the siirface. Let the rate be expressed as a function of the concentration of the reactant at the surface f(C'). In the stationary or quasi-stationary state, the rate of re­ action should be equal to the amount of reactant supplied to the surface by molecular or turbulent diffusion, which according to formula (I, 17) can be expressed q = β«3 - C«) where

β

(II, 1 )

is the diffusion velocity constant.

Equating the amount of substance consumed by re­ action at the surface and the amount supplied by diffusion, we get the algebraic equation β(0 - C') = f(C')

-

(II, 2)

This reaction can be solved for C1 for any con­ crete function f(C'). Substitution of the thus obtained C1 in (II, 1) or in f(C') gives the desired value of the overall reaction rate. The diffusion velocity constant

β

can be found

either analytically or from the experimental data of diffusion or heat transfer with the aid of the theory of similitude according to which β =

= MV

(II, 3)

II. DIFFUSIONAL KINETICS

1+9

First-Order Reaction Consider'the simplest case^'6'7 where the reaction at the surface is of the first order

Equation (II, 2) then becomes

and its solution is

The quasi-stationary reaction rate is

In this instance the macroscopic rate of reaction also follows the first order in reactant concentration. It can be written

where

This relation can be written in the instructive form

which means that in this simplest case there is additivity of the reciprocals of the reaction rate and of the diffusion constants, i.e., of the diffusional and kinetic "resistances"6'7.

50

II. DIFtPDSIONAL KINETICS Dlffuslonal and Kinetic Region

Formula (II, 7) takes on a simple form in the two limiting ranges where one of the two magnitudes k and β is much greater than the other. If k » β, one has k* = β and, according to (II, k), Ct =

k

C« C

in which case the rate of the overall process is entirely determined by the rate of diffusion, if k « β, equa­ tion (II, 7) becomes k* ** k and (II, if) becomes C1 « C, i.e., the overall rate is entirely determined by the true chemical kinetics of the reaction at the surface and does not depend on the conditions of the diffusion. The same limiting ranges will exist also in cases of more complex types of the true kinetics at the surface. If the diffusion resistance is much smaller than the chemical resistance, i.e., βΟ » f(C) the macroscopic rate of reaction will coincide with the true reaction rate at the surface, i.e., will be f(C), and the concentra­ tion of the reactant at the surface will be identical with its concentration in space, C1 1 this depth is essentially Infinite, and the effective depth of penetration of the reaction corresponds only to the distance at which there is a significant fall of the concentration. As we have pointed out in the Introduction, the case of a fractional-order reaction is in no way unreal. For heterogeneous reactions the order Is very often less than unity. In the particular case tion) the expression for lifted gives

η= 1

(first-order reac­

becomes indefinite, and when

102 -

II • DIFFUSIONAL KINETICS

For any n

one gets for the proportionality factor

(according to the boundary condition, at For the macroscopic rate of reaction we get, finally

where

n

is the order of the reaction.

We now can formulate accurately the conditions under which it is permissible to use the assumption of infinite thickness of the layer of the material. By formula (II, 65) the natural yardstick of length (obtained from the equation itself) Is in this case the magnitude

if the thickness of the porous material is great compared with this magnitude, the assumption is permissible. Thus, the greater is the rate of reaction, the smaller the thickness of the layer at which the assumption is usable. Conversely, at a finite thickness of the layer, decrease of the rate of reaction leads to a region where the thickness of the layer becomes very much smaller than

here, the whole inner surface of the material will behave

II.

DrF7PUSIOKAL KINETICS

103

entirely as a free surface, I.e., we shall find ourselves in a pure kinetic region. For the sake of lnstructlveness we can Introduce the notion of effective depth of penetration of the reaction Into the layer of the porous material L. Except for a dimensionless factor of the order of unity, one can define the depth of penetration of the reac­ tion as

L

In particular, for a first-order reaction

L ~ yp

(II, 70a)

We see that the greater the rate of the reaction the smaller the depth of Its penetration Into the layer of the material. If the total thickness of the layer H Is great compared with the depth of penetration L3 one can make use of the calculation just considered, which rests on the as­ sumption of an infinite thickness of the layer. If L be­ comes of the order of H, the macroscopic rate of reaction will depend on the ratio g. Finally, if the rate of reac­ tion becomes sufficiently fast for the depth of penetration L to fall to values comparable with the diameter of Indi­ vidual pores h, the whole approach of Zeldovich becomes unsuitable. It will neither be possible to introduce an effective diffusion coefficient nor an effective rate con­ stant per unit volume, as the reaction will take place only at the outer surface of the lump. Further Increase of the rate of reaction will no longer result in a decrease of the depth of its penetration into the interior of the liamp, as this depth is of the order of the diameter of individual pores, or smaller. Consequently, at

II.

DIFFUSIONAL ICTNEriCS L < h

one can disregard completely the effect of the porosity and consider that the reaction takes place only at the outer surface of the lump. If the assumption of Infinite thickness of the layer is applicable, the overall rate of the reaction is described by formula (II, 69).

The temperature dependence

of the rate of reaction should correspond to half the acti­ vation energy.

With the aid of the notion of effective

depth of penetration into the layer of material, we can give a very instructive interpretation to formula (II, 69).

With

increasing true rate of the reaction, the depth of penetra­ tion decreases; according to (II, 70), it is inversely pro­ portional to the square root of the rate of reaction.

On

the other hand, the macroscopic rate of the reaction is pro­ portional to the product of the true rate of the reaction and to the depth of penetration.

Consequently, the macro­

scopic rate of the reaction Increases proportionally to the square root of the true rate of reaction. In (II, 69) and the preceding formulas, we con­ sidered the concentration

C

of the reactant directly at

the free surface of the layer of porous material as given, and we took into account only the diffusion within that layer.

If the reaction is fast, diffusion In the gaseous

(or liquid) phase to the surface of the layer can also be­ come significant.

It can be easily taken into account by

methods already known to us.

The diffusion flow from the

volume to the surface is q = where C,

CQ

(C0 - C)

is the concentration of reactant in the volume;

at the surface of the layer; and

D

is the true diffu­

sion coefficient of the reactant in the volume. Equating the diffusion flow to the right-hand

II. DIFFUSIONAL KINETICS

1+9

member of the expression (II, 69), we get the algebraic equation

for the determination of the quasistationary concentration C of the reactant at the surface. In the particular case when the true kinetics of the reaction is of the first order, equation (II, 71) takes the form

hence

and

In this problem, there can be four limiting regions: 1. At the rate of the overall process is determined by diffusion in the volume

The concentration of the reactant even at the surface of the layer, and even more in the interior of the pores, is considerably smaller than in the volume

Following a proposal of Vulis^, this region is

106

II. DIFFUSIONAL KINETICS termed the "outer diffusional region". 2. At »VD'k' and H » L » 1 (where L Is the depth of penetration of the reaction into the interior of the lump, defined by formula (II, 70); H is the thickness of the lump or of the layer of porous material; and h is the mean diameter of single pores). The determining stage is diffusion in pores. The concentration of the reactant at the free surface of the layer is very close to its concentration in the volume, C =3 CQ, whereas the concentration in the pores falls prac­ tically to zero. The rate of the reaction in this region is expressed by formula (II, 69)· This region could be termed the region of diffusion in pores or, following a very pertinent proposal of Vulis, the "inner diffusional region". D» VD'k' 3. At and L» H, the macro­ scopic kinetics coincides with the true kinetics at the surface. In this "inner kinetic region", the concentration of the reactant in the pores throughout the thickness of the layer coincides with the concentration in the volume. In the inner kinetic region, the whole inner surface of the porous material is in operation.

Finally, at L < h and ^D » k (where k is the true rate constant of the reaction at the surface) we observe the "outer kinetic region" where the macroscopic kinetics also coincides with the true kinetics at the surface, but the reaction takes place only at the outer surface of the lumps of the porous material. Evidently, in the inner kinetic region the macro­ scopic rate of reaction is proportional to the volume of the porous material, and in the outer kinetic region it is pro­ portional to its surface area.

II. DIFFUSIONAL KINETICS

107

In the region Intermediate between the outer and the inner diffusional regions, the concentration field and the rate of the overall process can be obtained by solving equation (II, 71). For the simplest case of a monomolecular reaction this was done above. The corresponding formulas for the region intermediate between the inner diffusional and the inner kinetic regions can be obtained by solving equation (II, 58) -under the changed boundary conditions,

to =

0

at

x = L

With these boundary conditions, the equation can be inte­ grated by elementary functions only for the simplest case of a monomolecular reaction, but even in this simplest case the final formulas are quite Involved and will not be given here. It still remains to clarify the significance of the effective diffusion coefficient D1 and the effective rate of reaction f'(ξ) or the effective rate constant k'. These magnitudes have a very simple meaning when one con­ siders not a reaction in a porous material but a homogeneous reaction in which the reactant is supplied by diffusion from another phase; for example, the absorption of a gas by a liquid accompanied by a chemical reaction in the liquid. If convection in the liquid is taken to be absent, this process will be described by formula (II, 71), with D1 designating simply the diffusion coefficient of the reactant in the liq­ uid, and k' the rate of the homogeneous chemical reaction in the liquid. In the case of a porous or a powdery material, the meaning of these magnitudes will be somewhat different. It is related to the structure of the porous surface. Let us take the simplest model of a porous material and consider the pores as capillaries running from the free surface, with­ out breaks or intersections throughout the whole thickness of the layer. As characteristics of the pores, we Introduce

108

II.

DIFFUSIONAL KINETICS

the mean pore diameter h, the number N of pores per unit surface area, and the "labyrinth coefficient" X which is defined as the mean distance along the pores corresponding to unit lengths In the direction perpendicular to the surface dx 1 is the distance measured along the direction of the pores, and χ is the distance measured perpendicularly to the sur­ face. The surface area of the pores per unit volume of the layer is equal X Null ·

Consequently, the effective rate of reaction f *(ζ ) per unit volume is related with the true rate of reaction f(ζ) per unit reacting surface area by the expressions f'(S) =XNrthf(S)

(II, 72)

k' = XNrthk

(II, 72a)

where the rate constant k

is defined as in (II, 2a).

Fick's law of diffusion in the pores will be written

C where ω is the volume of the liquid phase in which the reaction takes place. Hence we obtain the equation for the determination of the quasistationary concentration of the reactant in the volume of the liquid phase S S(C0 - C) = k'tuC whence the concentration

C

(II, 80)

will "be expressed as

Fs C f S + k'tu 0

c = J5-S

(II, 81)

and the rate of the reaction per unit volume of the liquid phase as

k •£-*·-

'

1

s

^ S + k' ο) and away from the surface for reactions involving an increase of the volume (ςό^ < ο). Let us now examine the effect of the Stephan flow on the rate of the reaction. In an accurate calculation, the geometric shape of the surface may prove essential. Usually, however, the Stephan flow is localized in a thin

III. THE STEPHAN FLOW

133

layer adjacent to the surface ("boundary layer"); therefore, the surface curvature can be disregarded for purposes of an approximate calculation and the surface can be considered plane. This approximation is termed the "boundary layer" method. In that approximate solution, the total flow of matter is considered constant, independent of y; the mass flow velocity v should also be independent of y. On the other hand, the partial pressures p of the different species (and their concentrations as well) change with the distance y from the surface. The diffusion coefficients D^ In a polycomponent gas mixture are functions of the concentrations and may, therefore, also be dependent on y. Let formula (III, 9) be written in the form

1 The left-hand member of this equation does not depend on y; therefore, the right-hand member should not be dependent on y. Consequently, the dependence of the partial pressures on y should be such that

be constant. It can now be seen that the total flow of the species q1 is proportional to the velocity v of the mass flow

134

III. THE STEPHAN FLOW

To Interpret the physical meaning of the magnitude F, we introduce the mean diffusion coefficient 15, defined by

and the symbol

7

defined by

The magnitude 7 is a dimensionless number of the order of unity. In the case of equality of all diffusion coefficients,

represents the volume decrease in the reaction per unit volume of the species A1 which limits the process. For 7 > 0 the Stephan flow is directed to the surface, for 7 < 0, away from it, and for 7 = 0 it is zero. The mean diffusion coefficient D can change with the distance y from the surface. But, substituting the above notations in the expression for F, we find

from which it follows that 7 (and P as well) does not depend on y. Substituting (III, 15) in (III, 12), we find for the total velocity of diffusion

All magnitudes in this expression are independent of y.

III.

THE STEPHM FLOW

Expressing, I n ( I I I ,

3)>

135 by ( I I I ,

16), we

find

The d i f f u s i o n c o e f f i c i e n t D 1 i s a f u n c t i o n o f t h e c o n c e n t r a t i o n s o f t h e components, s p e c i f i c a l l y o f p 1 , and cannot t h e r e f o r e be c o n s i d e r e d c o n s t a n t . However, i n s i m p l e r cases t h i s c o n c e n t r a t i o n dependence o f t h e d i f f u s i o n c o e f f i c i e n t i s weak enough t o be d i s r e g a r d e d i n a f i r s t approximat i o n . I n t e g r a t i o n of equation ( I I I , assumption o f a c o n s t a n t D 1 g i v e s in (y - p j where

C

17) on t h e

= Tjj- y + C

(III,

18)

i s an i n t e g r a t i o n c o n s t a n t .

Inasmuch as we have used t h e boundary l a y e r method, t h e i n t e g r a t i o n s h o u l d extend o n l y over t h e bounda r y l a y e r , t h e t h i c k n e s s o f w h i c h can be t a k e n equal t o the e f f e c t i v e f i l m thickness 6 = ^ ( I , 28). Let substance A 1 we have

p,j 0 r e p r e s e n t t h e p a r t i a l p r e s s u r e o f t h e I n space, and p 1 ' at the surface. Then

1 ) a t y = 0, p 1 = p., 0 2) a t y = 5, p 1 = p 1 ' Prom t h e f i r s t c o n d i t i o n , we f i n d t h e i n t e g r a t i o n c o n s t a n t C: c =

and t h e n f o m u l a ( I I I ,

m ( f - p , ° )

18) t a k e s t h e f o r m

136

III.

THE STEPHA1T FLOW

or where

i s t h e mole f r a c t i o n • Prom t h e second c o n d i t i o n we can f i n d t h e v a l u e

o f t h e mass f l o w v e l o c i t y

v

)

W i t h t h e a i d o f f o r m u l a ( I I I , 16), we can pass t o t h e r a t e of the reaction, i . e . , t o the t o t a l flow of the l i m i t i n g species q1:

S u b s t i t u t i n g f o r t h e t h i c k n e s s o f t h e reduced f i l m

6

v a l u e f r o m ( I , 2 8 ) , we g e t

or

I f t h e r e a c t i o n m i x t u r e i s s t r o n g l y d i l u t e d by an i n e r t gas o r an excess o f one o f t h e r e a c t a n t s , 30 t h a t b o t h

its

III. and

THE STEPHAN FLOW

137

a r e s m a l l as compared w i t h u n i t y ,

e f f e c t o f t h e Stephan f l o w on t h e r a t e o f t h e w i l l vanish. (III,

In fact,

the

reaction

expansion o f t h e l o g a r i t h m

in

23) i n t o a s e r i e s g i v e s

and, w i t h

p = RTC,

w h i c h i s i d e n t i c a l w i t h t h e e x p r e s s i o n used w i t h o u t Stephan f l o w t a k e n i n t o

the

account.

However, I n t h e presence o f a h i g h

concentration

of the species, the d i f f u s i o n of which l i m i t s the process, t h e Stephan f l o w can a l t e r t h e r e a c t i o n r a t e c o n s i d e r a b l y I n an i r r e v e r s i b l e r e a c t i o n i n t h e d i f f u s i o n region,

pj = 0

and t h e r a t e o f t h e r e a c t i o n , w i t h t h e

Stephan f l o w t a k e n i n t o a c c o u n t , w i l l be expressed by

P a r t i c u l a r l y s i g n i f i c a n t i s the e f f e c t of the Stephan f l o w on processes o f c o n d e n s a t i o n o f v a p o r s .

In

t h i s case, t h e r e i s o n l y one r e a c t i n g substance and t h e r e a r e no gaseous r e a c t i o n p r o d u c t s ; c o n s e q u e n t l y

=

•

The gaseous m i x t u r e c o n s i s t s o f two s p e c i e s , t h e condensing A d i f f e r e n t c o n s i d e r a t i o n o f t h e Stephan f l o w was g i v e n b y DamkohlerS who i n t r o d u c e s t h e w h o l l y u n s u b s t a n t i a t e d assumption

H i s r e s u l t s t h e r e f o r e cannot be a c c e p t e d .

P.-K.

III.

138

THE STEPHAN JLOW

vapor and the Inert gas; but, In the case of a binary mixture, D1 = Dq = D, and., consequently, In this case, 7=1. The partial pressure of the vapor at the surface is equal to the saturation pressure at the temperature of the surface

P' = Psat i

In this case, we obtain from (III, 21 ) the well known formula of Stephan

*

=

D P δ W

P

111

Psat

ρ _ po

fTTT (I11

'

26)

for the rate of condensation of vapors In the presence of an Inert gas. to zero,

If the amount of Inert gas in the mixture tends p° will tend to P, and the rate of condensa­

tion expressed by Stephan1s formula will tend to infinity. Actually this means that at low concentrations of the Inert gas the rate of condensation is no longer governed by the diffusion of the condensing vapor to the siarface but by other stages of the process. Practically, that stage is usually the removal of the heat evolved In the condensation; that factor is examined in detail In Nusselt1S theory of film condensation. In reactions involving formation of gaseous products, the velocity of the Stephan flow can never be­ come Infinite, and the change of the rate of the process cannot be very great due to the Stephan flow. The Maxwell-Stephan Method A very elegant approach to diffusion processes in gases, differing somewhat In form from the preceding, was proposed by Maxwell and extensively applied by Stephan1.

III. THE STEPHAN PLOW

139

So far, we have been treating separately the molecular flow D grad C and the mass flow vO. In the molecular-kinetic analysis of diffusion phenomena In gases, it is difficult to adhere to such a separation since diffusion is unavoidably accompanied by mass flow. The Maxwell-Stephan method deals exclusively with the total flow of matter q, without separating it into molecular and mass flow. In the absence of diffusion, the amount of sub­ stance transferred by the mass flow Is related with the linear flow velocity xt by q=^nC The index η refers to the vector component perpendicular to the surface. In the presence of diffusion one can introduce a mean linear velocity tt of motion of the molecules, re­ lated witk the total flow of substance, q, in the same manner as the linear velocity α of the mass flow is re­ lated with the amount of substance q transferred by that flow, «η = J

(III, 27)

This mean velocity of molecular motion is the basic magni­ tude In the Maxwell-Stephan method· It must not be con­ fused with the root mean square velocity Va1 of the kinetic gas theory. The root mean square velocity is a scalar, whereas u is a vector. At equilibrium, i.e., in the absence of diffusion and of mass flow, a = o, but

1 if0

III.

THE STEPHAN FLOW

i s by no means equal t o z e r o . I n t h e f o r e g o i n g , we made use o f t h e e x p r e s s i o n f o r t h e t o t a l f l o w o f substance (III,

28)

We s h a l l w r i t e such e x p r e s s i o n s f o r two d i f f e r e n t components o f t h e m i x t u r e w h i c h we s h a l l d e s i g n a t e by t h e s u b s c r i p t s 1 and 2.

With the a i d of formula ( I I I ,

2 7 ) , we f i n d f o r t h e mean

v e l o c i t i e s of molecular motion

S u b t r a c t i n g t h e second f r o m t h e f i r s t e q u a t i o n , we have (III,

29)

Thus, t h e d i f f e r e n c e o f t h e mean v e l o c i t i e s o f m o l e c u l a r m o t i o n f o r two components o f t h e m i x t u r e i s independent o f t h e presence o f a mass f l o w and i s determined s o l e l y by the conditions of the d i f f u s i o n . As l o n g as we expressed i t was necessary t o d e s i g n a t e t h e i n d i c a t i n g t h e v e l o c i t y component t o t h e s u r f a c e . The same a p p l i e d

t h e f l o w by t h e v e l o c i t y , l a t t e r by t h e s u b s c r i p t along the perpendicular t o t h e g r a d i e n t . Now

III.

THE STEPHAN FLOW 137

t h a t we have equated two e x p r e s s i o n s f o r t h e f l o w and have expressed t h e v e l o c i t y b y t h e g r a d i e n t , t h e r e s u l t has b e come independent o f t h e e n t i r e l y a r b i t r a r y d i r e c t i o n o f t h a t p e r p e n d i c u l a r , and t h e s u b s c r i p t

n

has become

superfluous. I n t h e case o f a b i n a r y m i x t u r e , and, because o f t h e constancy o f t h e t o t a l p r e s s u r e

Formula ( I I I ,

29) t h e n g i v e s

or This i s t h e Maxwell-Stephan f o r m u l a . (III,

We s h a l l r e f e r

to

30) as t h e d i f f u s i o n l a w i n t h e M a x w e l l - S t e p h a n

f o r m , and t o ( I I I ,

28) as t h e d i f f u s i o n l a w i n t h e P i c k

form. M a x w e l l o b t a i n e d t h e f o r m u l a ( I I I , 30) d i r e c t l y f r o m t h e k i n e t i c gas t h e o r y w h i c h , o f c o u r s e , can y i e l d t h e u s u a l d i f f u s i o n f o r m u l a s b y r e v e r s i n g t h e above derivation. Formula ( I I I ,

30) enables one t o o b t a i n i n a

v e r y easy and s i m p l e manner S t e p h a n ' s f o r m u l a f o r t h e r a t e o f c o n d e n s a t i o n i n t h e p r e s e n c e o f an i n e r t g a s .

Let

11+2

III.

THE STEPHM FLOW

C o n s e q u e n t l y , I n t h e case under

According t o ( I I I ,

consideration

2 7 ) we can I m m e d i a t e l y g e t t h e t o t a l

o f condensing v a p o r , i . e . ,

the r a t e of

flow

condensation 1

I n many t e x t b o o k s and t a b l e s ( p a r t i c u l a r l y i n t h e American literature),

t h i s f o r m u l a i s g i v e n as t h e g e n e r a l e x -

p r e s s i o n f o r t h e r a t e o f d i f f u s i o n i n s t e a d o f as F i c k ' s law.

Actually formula ( I I I ,

33) a p p l i e s o n l y t o t h e

case o f a b i n a r y m i x t u r e , one component o f w h i c h i s

special neither

produced n o r consumed anywhere. I n c o n d e n s a t i o n on a p l a n e

surface

Passing from concentrations t o p a r t i a l we g e t f r o m ( I I I ,

3 3 ) , because

pressures,

p = RTC,

t)

where

q1

where

B

and

P

do n o t depend on

i s the i n t e g r a t i o n Let the

y

y.

I n t e g r a t i o n gives

constant.

a x i s be d i s p o s e d i n t h e same way as

i n t h e f o r e g o i n g c h a p t e r ; we t h e n f i n d t h e boundary conditions

III.

THE STEPHAN FLOW

at

y = 0,

p2 = P -

at

y = 6,

p2 = P - p a a t

From t h e f i r s t

p^

c o n d i t i o n we d e t e r m i n e B = ln(P -

137

B

p,0)

and f r o m t h e second c o n d i t i o n we f i n d

q1

w h i c h i s i d e n t i c a l w i t h Stephan 1 s f o r m u l a ( I I I ,

26).

I n many t e c h n i c a l a p p l i c a t i o n s one f i n d s d e n s a t i o n o f a vapor s t r o n g l y d i l u t e d w i t h gases.

con-

incondensible

Examples a r e t h e c o n d e n s a t i o n r e c o v e r y o f

volatile

s o l v e n t s , t h e c o n d e n s a t i o n o f spent steam i n t h e condensers o f steam e n g i n e s , t h e c o n d e n s a t i o n o f w a t e r v a p o r p r i o r absorption o f n i t r o g e n oxides t o o b t a i n concentrated

to

nitric

a c i d , t h e c o n d e n s a t i o n o f ammonia f r o m t h e n i t r o g e n hydrogen m i x t u r e a f t e r s y n t h e s i s , t h e condensation o f

sul-

f u r i c a c i d vapors i n t h e c o n c e n t r a t i o n and m a n u f a c t u r e o f oleum. The most i n t e r e s t i n g and f r e q u e n t case i s where the task consists i n the f u l l e s t p o s s i b l e condensation o f the vapor, c a l l i n g f o r a considerably lower

concentration

i n t h e o u t g o i n g as compared w i t h t h e i n f l o w i n g g a s .

The

method o f such a c a l c u l a t i o n was p r o p o s e d b y us^ and was k* made more p r e c i s e b y A m e l i i i There i s a deep f u n d a m e n t a l d i f f e r e n c e between t h e c o n d e n s a t i o n o f p u r e vapors and t h e c o n d e n s a t i o n o f See a l s o : C o l b u r n , A. P . , and Hougen, Chem., 26^ 1178 ( 1 9 3 ^ ) -

144

III.

THE STEPHAN FLOW

pure vapors In the presence of a large excess of lncondensIble gases. In the first instance, the rate of the process is determined by the removal of the evolved latent heat of condensation; in the second, by the velocity of diffusion of the vapor to the surface at which the con­ densation takes place, across a layer of lncondensible gas which forms at that surface. In the first Instance, it Is reasonable to calculate by the methods of the theory of heat transfer and to use the heat exchange coefficient as the basic magni­ tude. This procedure loses its meaning in the second instance, although such uncritical extension of a method suitable only for pure vapors, to vapors diluted with ln­ condensible gases, is not uncommon in technical calcula­ tions. Actually, in the presence of lncondensible gas, the rate of condensation is determined by the diffusion of the vapor to the surface, and the Stephan flow has to be taken into account; its role is more significant the lower the content of lncondensible ga,s in the mixture. At each point, the velocity of condensation is given by Stephan's fomula (III, 26). To find the final results of the process, it is necessary to average that formula over the whole surface at which the condensation takes place. Let us consider the condensation process in a tube of length L and diameter d. Let P denote the total pressure, pQ the partial pressure of the con­ densing vapor at the entrance of the tube, and pf at the exit, and let Psa^ be the saturation pressure at the temperature of the wall, assumed to be uniform over its whole length; Integration of Stephen's formula over the length of the tube gives the result

(III, 36)

III.

THE STEPHAN PLOW

145

where the magnitude Z depends on the dimensions of the tube and the conditions of the diffusion. The latter are best described by Margoulis criterion M with the aid of which Z is expressed simply by Z = -^M

(III, 37)

In the practically most important case of turbulent flow, Margoulis criterion depends only little on the properties of the gas and the hydrodynamlc flow. Consequently, in the case of turbulent flow, attainment of a stated degree of condensation requires a definite ratio between length and diameter. Simultaneously, with the condensation of vapor at the surface, the gas mixture is cooled down as a result of heat exchange. If this cooling Is too rapid, the vapor can become supersaturated and volume condensation will set in. Voliane condensation is usually undesirable since it leads to formation of fine droplets of liquid, which are carried away by the gases in the form of mist- It is much more difficult to collect such a mist than to condense the vapor. Particiilarly harmful is the formation of misty sulfuric acid In the processes of concentration of sulfuric acid and manufacture of oleum. To combat it, it is nec­ essary to calculate correctly the thermal conditions of the process. The main requirement is that the cooling of the gas be not too fast in comparison with the diffusion process. The paradoxical conclusion is that all too in­ tense cooling can defeat the results of the condensation process. Literature

Ί,

5 5 0 ( 1 8 8 2 ) ; 4l_, 725 ( 18 9 0 ) -

1.

STEPHAN, Ann, der Physik, 1

2.

DAMKQHiiKR, Der Chemie-Ingenieur III, Th. 1, 448 ff.(l937)·

3·

FRANK-KAMENETSKII, Zhur. Tekh- Piz. 12, 327 (1942).

4.

AMELIN, Zhur. Tekh. Fiz. J_5.> 2 8 7 (19^5)·

5·

BUBEN, Sbornik rabot po fizicheskol Khlmii, (Supple­ ment to Zhur. Fiz· Khlm. 1946). p. 148, 154 (1947)·

CHAPTER IV: NONISOTHERMAL DIFFUSION

Equations of Heat Conductance and of Diffusion in the Simultaneous Presence of Both Processes Up to this point we have dealt with heat con­ ductance in a medium in which the concentration is constant and the same at all points, and we have dealt with iso­ thermal diffusion· In the first instance, only heat trans­ fer processes occur In the system, to the exclusion of diffusion; in the second, only diffusion took place, to the exclusion of heat transferIn practice, very frequently both a. concentration and a temperature gradient are present in the same system, and heat transfer and diffusion occur simultaneously. The process then becomes more complex, and entirely new phe­ nomena, known as thermal diffusion and diffusion thermoeffect, arise. This is essentially due to the fact that the heat flow depends not only on the temperature gradient but also on the concentration gradient; and the diffusion flow de­ pends not only on the concentration, but also on the temperature gradient. This necessitates introduction, in the expression for the heat flow, in addition to the - λ grad T term, H6

IV.

NONISOTHERMAL DIFFUSION

of still another term proportional to the concentration gradient; and in the expression for the diffusion flow, In addition to the - D grad C term, of another term pro­ portional to the temperature gradient. The phenomenon of thermal diffusion has recent­ ly become important, as it has become the basis of a very effective method of separation of isotopes proposed by Clusius. The Enskog-Chapman Laws The laws of thermal diffusion and diffusion thermoeffect have been derived from the kinetic gas theory by Enskog and Chapman. A detailed exposition of that theory will be found in the book of Chapman and Cowling1. Its final result is the following expression for the law of diffusion in the Maxwell-Stephan form:

S1 - e2 = -

D^jsgrad P1 +

TJ

(IV, 1 )

and the corresponding expression for the heat flow q = - λ(grad T)n + ϊ>η + JPk^Ca1 - «2)

(IV, 2)

Here, kip is the so-called thermal diffusion ratio, a dlmensionless magnitude characteristic of the given gas pair as its physical constant (sometimes, the magnitude Hp = krpD, termed the thermal diffusion coefficient, is Introduced instead; kp is thus defined as the ratio of the thermal diffusion and the diffusion coefficient); T = cpT Is the mean heat content of the mixture; J Is the thermal equivalent of the work, converting the second term of the right-hand member of (IV, 2), which has the dimension of mechanical work, Into heat units. Foniiula (IV, 1 ) and all expressions for the

148

IV.

NONISOTHERMAL DIFFUSION

diffusional flow formulated In the Maxwell-Stephan form are valid only for a binary mixture. Foiraulas (IV, 1) and (IV, 2 ) show that thermal diffusion and diffusion thermoeffect phenomena are closely related and that their intensity is determined by the val­ ue of the same physical constant Ict, there is no special coefficient of diffusion thermoeffect. This is but one particular instance of a general law of nature, the socalled principle of symmetry of kinetic coefficients, or the Onsager principle. The numerical value of the thermal diffusion ratio kip depends on the individual properties of the given gas pair. The kinetic gas theory shows that this magnitude is extremely sensitive to the particular mech­ anism of collision between molecules, specifically, to the law of the repulsive forces acting between the molecules at close ranges in the act of collision. It is not enough to specify that the gas is con­ sidered ideal, i.e., that interaction forces are negligible at mean intermolecular distances. Even with the magnitude kp can have widely different pending on the law of the interaction forces tween the molecules at distances much closer distance, such as are involved in the act of

an ideal gas, values de­ arising be­ than the mean collision.

If the molecules are viewed as material points (force centers) which at close range repel each other with a force inversely proportional to a nth power of the distance, one has for η = 5, ICT = O, i.e., thermal diffusion is totally absent. Maxwell In his classic studies of kinetic gas theory considered particularly the case η = 5, mathematically the simplest. Consequently, there was no room in his theory for thermal diffusion or diffusion thermoeffect. These phenomena for ideal gases were predicted in 1912 - 1915 by Enskog and Chapman who developed this theory following Maxwell, but for any

IV. NONISOTHERMAL DIFFUSION

149

values of n. For all ordinary gases η > 5· If, in formula (II, 1), the subscript 1 refers to the heavier gas (or, in the case of gases of equal molecular weight, to the gas with larger molecules), kT will be a positive magnitude. With the subscripts reversed, krp will be negative. This means that heavier (or, at equal molecular weight, larger) molecules tend as a result of thermal diffusion to con­ centrate in the colder parts of the system. For η < 5, the signs are reversed. This case appears to be realized only in strongly ionized gases. The numerical value of kip depends very strongly on the composition of the mix­ ture. At a low concentration of one component of the mixture, kT is proportional thereto, Ior = bx

(IV, 3)

where χ is the mole fraction of the substance present at a low concentration. The proportionality coefficient b does not for any known gas pair exceed 0.2 - 0.3. With increasing concentration, the growth of kip with the concentration becomes slower, k^ passes through a maximum, and then falls to zero (with decreasing concentration of the second component). Figure 17, taken from the book of Chapman and Cowling, represents the dependence of the thermal diffusion ratio kqi on the composition for a hydrogen-nitrogen mixture. The experimentally observed maximum values of kp did not exceed 0.1. The value of krj, is greater, the greater the difference between the molecular weights of the two gases. Therefore, thermal diffusion is very significant in mix­ tures of hydrogen with other gases. It is far less impor­ tant in pairs of gases with close molecular weights.

150

IV.

N0NIS0THERMAL DIFFUSION

I n o r d e r t o w r i t e down t h e e q u a t i o n s o f t h e r m a l d i f f u s i o n i n F i c k ' s form, i . e . ,

t o express t h e f l o w o f

substance i n an e x p l i c i t f o i r o , i t i s necessary t o r e v e r s e the t r a n s f o r m a t i o n through which, i n a preceding chapter [formulas ( I I I ,

29) - ( I I I ,

3 0 ) ] , we have passed f r o m t h e

law o f d i f f u s i o n i n F i c k ' s f o r m t o t h e law i n t h e M a x w e l l Stephan f o r m , as o b t a i n e d f r o m t h e k i n e t i c t h e o r y ;

for

p r a c t i c a l a p p l i c a t i o n , i t must be t r a n s f o r m e d i n t o t h e Fick form. The t o t a l f l o w o f substance i s equal t o t h e sum o f t h e f l o w s o f t h e two components o f t h e m i x t u r e

I n o t h e r words, t h e l i n e a r v e l o c i t y o f t h e mass f l o w r e p r e s e n t s t h e w e i g h t e d mean o f t h e mean v e l o c i t i e s o f m o t i o n o f t h e molecules

I n o r d e r t o pass f r o m c o n c e n t r a t i o n s t o p a r t i a l

IV. pressures,

it

NONISOTHERMAL DIFFUSION

151

i s necessary t o m u l t i p l y t h e numerator and

t h e denominator o f t h e r i g h t - h a n d member o f (XV, 5) by KT;

t h i s gives

where

P = p1 + p 2

i s the t o t a l pressure-

o f ( I V , 6) we can express

by

and

S u b s t i t u t i n g t h i s expression f o r assembling a l l terms c o n t a i n i n g

With the a i d

and i n the

left-hand

member, we g e t

Because

t h i s e x p r e s s i o n can be r e w r i t t e n as

We now can express t h e t o t a l f l o w o f t h e substance d e s i g n a t e d by t h e s u b s c r i p t 1 as

IV.

NONISOTHERMAL DIFFUSION 152

T h i s i s t h e law o f d i f f u s i o n ,

including thermal d i f f u s i o n ,

i n t h e F i c k formFrom p a r t i a l p r e s s u r e s one can pass t o c o n c e n t r a tions. Because p = RTC, and P = RTSC, where EC i s t h e sum o f c o n c e n t r a t i o n s o f a l l t h e s u b s t a n c e s , ( I V , 9) can be w r i t t e n i n t h e f o r m

since

The o r i g i n a l Enskog-Chapman f o r m u l a i s

applicable,

as has been p o i n t e d o u t , o n l y t o b i n a r y m i x t u r e s . formulas (IV,

9) and ( I V ,

But

10) can be extended a l s o t o m u l t i -

component m i x t u r e s ; o n l y , I n t h a t case, t h e v a l u e s o f and

kj

D

w i l l be d i f f e r e n t f o r d i f f e r e n t components and

s t r o n g l y dependent on t h e c o m p o s i t i o n o f t h e m i x t u r e . Approximate Theory o f N o n l s o t h e r a a l

Diffusion

F o r p a i r s o f gases w i t h c l o s e m o l e c u l a r w e i g h t s the numerical value of

k,p

i s small.

Consequently,

in

most p r a c t i c a l l y i m p o r t a n t cases ( w i t h t h e e x c e p t i o n o f m i x t u r e s c o n t a i n i n g hydrogen) t h e term w i t h

kp

in

( I V , 9) can be d i s r e g a r d e d . The law o f n o n i s o t h e r m a l d i f f u s i o n t h e n t a k e s the form

Thus, i f

t h e o r d i n a r y d i f f u s i o n e q u a t i o n s can

be used, w i t h o u t a l l o w a n c e f o r t h e r m a l d i f f u s i o n , a l s o t h e case o f n o n i s o t h e r m a l d i f f u s i o n ;

only, instead of

in

IV. NONISOTHERMAL DIFFUSION concentrations,

153

one must i n t r o d u c e p a r t i a l p r e s s u r e s ,

one must r e p l a c e

D

and

and

T h i s i s t h e most w i d e s p r e a d method o f

calcula-

t i o n o f processes o f n o n i s o t h e r m a l d i f f u s i o n , and t h e one most g e n e r a l l y used by many a u t h o r s .

It

s h o u l d , however,

be b o r n e i n mind t h a t t h i s method i s l e g i t i m a t e o n l y ICiji

if

i s v e r y s m a l l , i . e . , when t h e m o l e c u l a r w e i g h t s o f

components d i f f e r

little.

One can see f r o m f o r m u l a ( I V , l a r g e r values of

all

l&p,

10) t h a t ,

with

t h e e r r o r committed t h r o u g h d i s -

r e g a r d i n g t h e t h e r m a l d i f f u s i o n t e r m i s o f t h e same o r d e r o f magnitude as t h e e r r o r o f u s i n g c o n c e n t r a t i o n s of p a r t i a l pressures.

instead

N u m e r i c a l l y , however, t h e f o r m e r

e r r o r i s a c t u a l l y s m a l l e r , as t h e n u m e r i c a l v a l u e o f i s never g r e a t e r than

krp

o.i.

As was p o i n t e d o u t , i f

formula (IV,

10) t h e n becomes

I n the c a l c u l a t i o n of the d i f f u s i o n of the e r gas,

k,p

and, c o n s e q u e n t l y ,

the absolute value of

b

b,

are negative.

i s n e v e r i n excess o f

lightBut

0.2 -

0.3;

c o n s e q u e n t l y , i f we d i s r e g a r d t h e second t e r m i n p a r e n t h e s e s i n (IV,

1 0 ) , we always commit a g r e a t e r e r r o r t h a n i f we

disregard the thermal d i f f u s i o n term i n (IV,

9).

I n other

words, use o f t h e g r a d i e n t o f p a r t i a l p r e s s u r e i n noni s o t h e r m a l d i f f u s i o n i s always more a c c u r a t e t h a n t h e use of the concentration gradient.

15^

IV.

NONISOTHERMAL DIFFUSION

With small

,

we can always use as a f i r s t

a p p r o x i m a t i o n t h e simple e x p r e s s i o n o f F o u r i e r ' s law f o r t h e h e a t f l o w , inasmuch as t h e a d d i t i o n a l t e r m w h i c h stands i n f o r m u l a ( I V , 2) i s p r o p o r t i o n a l t o

kj.

As a second a p p r o x i m a t i o n , we can s u b s t i t u t e i n ( I V , 2) t h e e x p r e s s i o n ( I V , regard the term containing

1) f o r

and d i s -

the expression f o r the

h e a t f l o w t h e n takes t h e f o r m

This expression i s s u i t a b l e only f o r a binary mixture.

In

t h e case o f a multicomponent m i x t u r e , i n w h i c h o n l y one gas d i f f u s e s , one can t r e a t t h e m i x t u r e o f a l l t h e o t h e r gas as an i n e r t gas;

w i l l t h e n mean t h e p a r t i a l

p r e s s u r e o f t h e d i f f u s i n g gas, and

p2

w i l l be

I n t h e simultaneous d i f f u s i o n o f s e v e r a l gases, t h e laws o f t h e t h e r m a l d i f f u s i o n and o f d i f f u s i o n thermoe f f e c t become more complex- Equations o f t h e t y p e o f (IV,

1) and ( I V , 2) f o r t h i s case were r e c e n t l y d e r i v e d by

2

Hellund .

Because o f t h e unusual c o m p l e x i t y o f these e x -

pressions, t h e i r p r a c t i c a l u t i l i z a t i o n i s

difficult.

Using t h e common method o f d e r i v a t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s o f h e a t conductance and d i f f u s i o n , b u t r e p l a c i n g t h e laws o f F o u r i e r and o f F i c k by t h e f o r m u l a s ( I V , 13) and , we o b t a i n t h e d i f f e r e n t i a l equations o f h e a t conductance and d i f f u s i o n I n an immobile medium i n t h e f o l l o w i n g f o r m :

IV.

NONISOTHERMAL DIFFUSION

155

W i t h t h e a i d o f ( I V , 10), t h e l a t t e r e q u a t i o n can be a l s o expressed i n c o n c e n t r a t i o n s :

I f t h e d i f f e r e n c e s o f temperatures and c o n c e n t r a t i o n s i n t h e system are s m a l l , t h e t e m p e r a t u r e and concent r a t i o n dependence o f t h e p h y s i c a l c o n s t a n t s can be d i s regarded. I f t h e temperature dependence o f t h e d i f f u s i o n c o e f f i c i e n t can be d i s r e g a r d e d , i t i s even more l e g i t i m a t e t o consider as t e m p e r a t u r e independent. In f a c t , t h e d i f f u s i o n c o e f f i c i e n t o f a gas b e i n g p r o p o r t i o n a l t o t h e a b s o l u t e temperature t o t h e power magnitudes ent t h a t

1.5

or

2,

the

a r e even l e s s t e m p e r a t u r e dependD

Itself.

On t h a t b a s i s , we can r e p r e s e n t our d i f f e r e n t i a l equations i n the form

At low c o n c e n t r a t i o n s o f t h e d i f f u s i n g species

,

i . e . , f o r m i x t u r e s s t r o n g l y d i l u t e d by an i n e r t gas, can be c o n s i d e r e d c o n s t a n t I n t h a t case, t h e e q u a t i o n s o f h e a t conductance and o f d i f f u s i o n can be r e p r e s e n t e d I n a symmetrical f o r m . To t h a t end, we c o n s i d e r a c o n s t a n t , and, s i n c e f o r any

156

IV.

NONISOTHERMAL DIFFUSION

I n t h e presence o f c o n v e c t i o n , these e q u a t i o n s should be supplemented by t h e u s u a l c o n v e c t i o n terms and

I f t h e temperature o r c o n c e n t r a t i o n d i f f e r e n c e s are l a r g e , i t becomes necessary t o • a l l o w f o r t h e temperat u r e and c o n c e n t r a t i o n dependence o f t h e p h y s i c a l c o n s t a n t s . T h i s i n v o l v e s c o m p l i c a t e d computations. L i t erat lire 1•

CHAPMAN and COWLING, M a t h e m a t i c a l t h e o r y o f nonu n i f o r m gases. Cambridge 19^0.

2.

HELLUND,

CHAPTER V:

CHEMICAL IBfDROICTAMICS

If the transfer of matter is related directly to turbulent motion of the liquid or gas, the investigation of the kinetics of chemical processes in the diffusional region can serve as a means for the study of the hydrodynamic characteristics of the turbulent flow: velocity distribution, pulsation eddy, local structure of the turbulence· This chapter of macroscopic kinetics we call chemical hydrodynamics. Work in this direction is only in its beginnings. What will be discussed in the following, is mainly a pro­ gram of future investigations, with only a few preliminary resialts pertaining to the kinetics of dissolution processes. At the present time, we can point out two main problems of chemical hydrodynamics: ι. Investigation of the velocity distri­ bution near the solid surface (in the so-called laminar sublayer) by the study of the diffusional kinetics of chemical processes at the surface of an immobile solid body. 2.

Investigation of the local structure of turbulence by the kinetics of processes taking place at the surface of suspended particles.

158

V. CHMICAL HYDRODYNAMICS

It might also be of considerable Interest to In­ vestigate transfer processes at the free boundary between a gas and a liquid, or a liquid and a liquid, where the velocity distribution could be different from that at a solid surface. To this end, it would be necessary to make a detailed study of the diffusional kinetics of the solu­ tion of gases in liquids under definite hydrodynamic con­ ditions. However, the experimental material available for that purpose is far from sufficientConvective Diffusion in Liquids and Velocity Distribution Near a Solid Surface If a chemical process takes place In the diffusion­ al region, its rate is determined by that of the transfer of matter to the surface where the reaction occurs. If that process takes place under more or less well defined hydrodynamic conditions, its investigation permits de­ fined hydrodynamic conclusions. Until recently, all our knowledge of processes of convective transfer was derived from studies of heat transfer. Vast experimental material has been accumulated in this field, and is being commonly made use of in hydro­ dynamics and In the theory of turbulence. We set ourselves the task of utilizing for the same purposes, experimental data of transfer of matter, i.e., diffusional kinetics of such simple processes as dlssolutibn1· Study of convec­ tive diffusion phenomena in gases yields little new In comparison with transfer of heat, as both Instances in­ volve values of the Prandtl criterion of the order of unity. In contrast, diffusion in liquids, with its very low dif­ fusion coefficients, easily involves Prandtl numbers of the order of several thousands and more- The diffusion coefficient of substances dissolved in water is of the —5 P order of 1o~ cm /sec, which, with a kinematic viscosity P P of pure water of 10" cm /sec gives a Prandtl criterion of about IO^ even in dilute solutions. In highly

V.

CHEMICAL HmiOIiYliAMICS

159

concentrated solutions, the kinematic viscosity is mark­ edly higher. At the same time, there is a strong decrease of the diffusion coefficient which, in a first approxima­ tion, is inversely proportional to the viscosity; thus, the diffusions! Prandtl criterion increases roughly pro­ portionally to the square of the viscosity. By increasing the viscosity of a ferric chloride solution through add­ ition of calcium chloride, we were able1 to attain a value of the diffusional Prandtl criterion of about 4o,ooo. In this way, study of transfer of matter phenome­ na in a liquid medium, specifically in aqueous solutions, has permitted investigation of the limiting case of convective diffusion at very high values of the Prandtl cri­ terion. This is of great interest not only from the chemi­ cal but also from the hydrodynamic point of view. The best approach to that problem is the utilization of data of kinetics of dissolution1· In heat transfer phenomena such high values of the Prandtl criterion could never be attained. The high­ est values of the Prandtl criterion, observed in the study of heat transfer in viscous oils, did not exceed a few hundred. At that, the accuracy of the results was very poor owing to many basic difficulties, linked with the strong temperature dependence of the viscosity of liquids. Heat transfer phenomena must of necessity be studied under n0nl30thermal conditions, where the values of the vis­ cosity and, consequently, also of the Prandtl criterion, are different at different points; this renders the in­ terpretation of experimental results extremely difficult. In particular, the heat exchange coefficients on heating and on cooling are different, and it is necessary to re­ sort to extrapolation to an infinitely small temperature O difference or to rather artificial and poorly substanti­ ated ad hoc methods of recalculation of experimental data2· Investigation of processes of convective diffusion

16o

V.

CHEMICAL HYDRODYNAMICS

and, In particular, of the kinetics of dissolution, is free from these difficulties. It turns out to be the best method of study of processes of transfer of matter at high values of the Prandtl criterion under strictly isothermal conditions. The higher the value of the Prandtl criterion at which the convection process is studied, the deeper does one penetrate into the structure of the boundary layer in closest vicinity to the solid surface. Diffuslonal Layer At a distance from the surface, the transfer of matter by the mass flow is very intense, and the concen­ tration is completely equalized everywhere· All of the diffusional resistance is concentrated in the immediate vicinity of the surface, where convective transfer is negligible as compared with molecular diffusion. This zone is sometimes called the diffusional boundary layer or simply the diffusion layer. Its thickness can be con­ sidered tantamount with the effective film thickness dis­ cussed in Chapter I. The greater the Prandtl criterion, the smaller is the intensity of molecular transfer, and the smaller must be the coefficient of turbulent exchange for the transfer of matter by mass flow to be negligible in comparison with molecular transfer. This means that the thickness of the diffusion layer decreases with increasing Prandtl criterion. With very high values of the Prandtl criterion, even an insignificant turbulence (or, generally speaking, a normal component of the velocity) will transfer matter with a very much higher intensity than molecular diffusion. Only that zone in which convective (specifically, turbulent) transfer is entirely absent, will play the role of a dif­ fusion zone. By raising the value of the Prandtl cri­ terion, we move the zone in which the diffusion process is studied increasingly close to the solid surface.

V.

CHEMICAL HiDRODYTtAMICS

161

Investigation of the dependence of the transfer of matter process on the Prandtl criterion, will elucidate the vari­ ation of the coefficient of turbulent exchange with the distance from the solid surface. The greater the value of the Prandtl criterion, the closer is the approach to that surface and the deeper the penetration into the boundary layer. The concept of a diffusion layer is purely con­ ventional, corresponding exactly to the concept of effect­ ive film introduced in Chapter I. Physically, the diffusion layer is no different from the rest of the flow. It rep­ resents the zone in which the diffusion of the substance under consideration can be looked upon as purely molecular. It is therefore natural for the diffusion layer to have different thicknesses for different substances diffusing in the same flow.

The smaller the diffusion coefficient,

i.e., the greater Prandtl1s criterion, the thinner is the diffusion layer. In other words, it will be necessary to approach the surface that much closer in order to be able to disregard convective diffusion in comparison with mole­ cular diffusion. The real physical structure of the flow cannot, of coizrse, in any way depend on Prandtl1s criterion which has different values for the diffusion of different sub­ stances in the same flow. The thickness of the diffusion layer stands in no relation to the real physical structure of the flow, and it represents a purely conventional auxiliary magnitude. One could dispense with it, and use only the Kusselt or Margoulis criteria in its place. Laminar Sublayer The concept of a laminar sublayer has an entire­ ly different character. This concept corresponds to a definite representation of the physical structure of the flow. Different assumptions can be made about the nature of the fall of the turbulent exchange coefficient on

162

V. CHEMICAL HYDRODYIiAMICS

approaching the solid surface: 1 . One can assume that the turbulent exchange coefficient vanishes only at the very surface of the solid and that it has a finite value at even the smallest distance therefrom. 2. One can assume that the turbulent exchange coefficient vanishes at a certain finite distance from the solid surface, and that convective diffusion is entirely absent at small­ er distances from the surface. That is the zone referred to as the laminar sublayer. The assumption of the existence of a laminar sub­ layer of finite thickness is a hypothesis requiring ex­ perimental verification. The first of the two abovementioned hypotheses, according to which the turbulent exchange coefficient vanishes only at the very surface of the solid, may be just as legitimate. A choice between the two hypotheses can be made only on the basis of ex­ perimental data on convection at very high values of the Prandtl criterion. If a laminar sublayer does not exist, the thickness of the diffusion layer should decrease con­ tinuously with increasing Prandtl number. If it does ex­ ist, the thickness of the diffusion layer should decrease only as long as it remains greater than the thickness of the laminar sublayer. At the limit, at very high values of the Prandtl criterion, the diffusion layer should co­ incide with the laminar sublayer. At that point, its thickness should cease to decrease with further increasing Prandtl number, and be solely governed by the hydrodynamic characteristic of the flow. In this limiting case, the concept of a diffusion layer, or of a reduced film, will be no longer conventional, but will become identical with the physical laminar sublayer.

V. CHEMICAL HYDRODiIlAMICS

163

Thus, the study of convective diffusion in liquids is linked with the very interesting physical prob­ lems of the nature of the decay of turbulence on approach­ ing the solid surface, and of the law of the change of the turbulent exchange coefficient in its immediate vicinity. Dimensional Analysis Let us consider, following von Karman , the velocity distribution in the vicinity of the solid surface from the point of view of the theory of similitude. Phenome­ na taking place in close vicinity of the surface should be governed by values of magnitudes pertaining to that region, not to the main flow. Such magnitudes are the kinematic viscosity of the liquid υ, its density p, and the tangential stress at the surface tq. Prom these magnitudes one can construct only one magnitude of the dimension of a length, namely, δ»

(ν, 1)

V termed the thickness of the boundary layer. At greater distances from the surface, the root mean square pulsation velocity u tends to a constant value uQ· According to Prandtl, this value is equal to the change of the mean velocity over a length equal to the mixing length

where y is the distance from the surface, 1 the mixing length and ν the mean velocity. In the turbulent flow, the turbulent exchange coefficient plays the role of the kinematic viscosity

\6k

V.

CHEMICAL HYDROEBTNAMICS A = Iu

and the magnitude pA = ρIu plays the role of dynamic viscosity. The tangential stress will be expressed by

Substituting

u

for

1 (V, 2)

If the curvature of the surface is disregarded, the tan­ gential stress τ is independent of the distance y from the surface. The magnitude τ, as defined by (V, 2) can then be put equal to the tangential stress at the surface (V, 3) where uQ is the value of the mean square pulsation ve­ locity at a great distance from, the surface, calculated without allowing for the curvature of the surface. Ex­ pressing u0 with the aid' of (V, 3) by experimentally measurable magnitudes, we get ^

This magnitude is the natural yardstick of velocities for the turbulent flow. It is customary to express the tangential stress through the velocity V of the main flow and the resist­ ance coefficient f, as was done in the Introduction

V.

[fonnula (I, 30)J.

CHEMICAL HYDRODYNAMICS

165

We then find for the pulsation velocity

„ Ο/ξ

vVf

(γ,

and for the thickness of the boundary layer

u

° Vr

(V, 5)

The latter magnitude is the natural yardstick of length for the velocity distribution in the vicinity of the solid surface. In particular, the thickness of the laminar sub­ layer, if it exists, should be proportional to δ'. In the following description of the velocity distribution at the solid surface, we shall make use of the dimensionless variables v* = Σ

and

y* =

In the limiting range of pure turbulence where the resistance coefficient can be considered constant, the thickness of the laminar sublayer at constant velocity of the flow should be proportional to the kinematic vis­ cosity υ. If the laminar sublayer does exist, the diffusion velocity c o n s t a n t , a t t h e l i m i t o f l a r g e v a l u e s o f P r a n d t l S criterion, will tend to the value 1

where

δ*

is the thickness of the laminar sublayer. We introduce the dimensionless thickness of the

V.

CHEMICAL HYDRODYNAMICS

laminar sublayer L, (V, 5)

so defined that In accordance with

6* = L

ο

= L81

(V, 6)

The magnitude L Is a universal constant, the limiting values of the diffusion velocity constant and of Margoulls criterion at high values of Prandtl1s criterion are

β = Μ

O = JL Vi~-

du

LO

Pr

E

(V, 7)

In the limiting range of pure turbulence, the resistance coefficient, and with It the magnitude M Pr, should tend to a constant value. In a first approximation, the dif­ fusion coefficients of dissolved substances in a liquid medium are Inversely proportional to the kinematic vis­ cosity. Consequently, lnthis limiting case, the Margoulis criterion should be Inversely proportional to the square of the kinematic viscosity. The same applies also to the diffusion velocity constant at constant flow velocity. General Formulas Transfer of heat, matter, and momentum can take place either through molecular or through turbulent ex­ change. The Intensity of molecular transfer is character­ ized by the coefficients of diffusion, heat conductivity, and kinematic viscosity. The Intensity of turbulent trans­ fer is characterized by the value of the turbulent exchange coefficient, defined by formula (I, 15); this coefficient will henceforth be designated by A. Its value is the same for the transfer of all the three magnitudes mentioned

V. above.

CHEMICAL HYDRODYNAMICS

2

Thus, as l o n g as we d e a l w i t h t u r b u l e n t

01 transfer,

t h e r e I s always complete s i m i l i t u d e between d i f f u s i o n , h e a t t r a n s f e r , and r e s i s t a n c e .

This similitude Is

f r i n g e d o n l y when m o l e c u l a r t r a n s f e r becomes

In-

significant.

We s h a l l w r i t e t h e e x p r e s s i o n s f o r t h e d i f f u s i o n f l o w , h e a t f l o w , and t h e t a n g e n t i a l s t r e s s ,

the

s e p a r a t i n g t h e terms

c o r r e s p o n d i n g t o m o l e c u l a r and t o t u r b u l e n t

transfer:

These e x p r e s s i o n s f o l l o w d i r e c t l y f r o m t h e laws o f h e a t conductance, d i f f u s i o n , and i n t e r n a l f r i c t i o n ,

if

it

is

k e p t i n mind t h a t i n t u r b u l e n t t r a n s f e r t h e t u r b u l e n t change c o e f f i c i e n t

A

plays the r o l e of d i f f u s i o n ,

c o n d u c t i v i t y , and k i n e m a t i c v i s c o s i t y ,

ex-

heat

and t h a t

I n t e g r a t i o n gives

To s o l v e t h e s e i n t e g r a l s ,

it

i s n e c e s s a r y t o know t h e d e -

pendence o f t h e t u r b u l e n t exchange c o e f f i c i e n t

A

on t h e

168

V.

CHEMICAL HYDRODYNAMICS

distance from the surface

y.

The accurate form of this

dependence for small y is not known at the present time, and we can only make assumptions. With 3ome such relation between A and y assumed, one can substitute it in (V, 11) or (V, 12) and carry out the integration; one will thus find the relation between the heat or diffusion flow and the temperature or concentration difference, i.e., the law of forced convection at any value of Pr. Comparison of the result with experimental data of heat transfer or diffusion will show to what extent the assumptions made correspond to reality. At large

y,

the integrands in (V, 11 ) - (V, 13)

begin to depend on y not only because of the dependence of A on y, but also as a result of purely geometric factors. In all cases, except for a plane surface, the magnitudes q and τ will, on account of these factors, depend on y. Let us designate the values of these magni­ tudes at the surface (at y = 0) by the subscript 0. We can then put

where

q = q δ*, the first term. Formula (V, 23) can now be written as B - Γ»7) ay· & /° ο δ* But, since

δ*

CV, «)

is small compared with the radius of curva­

ture of the surface, f(y) in the first integral can be taken equal to unity, and the integral is simply equal to

V. 5*.

CHEMICAL HYDRODYNAMICS201

Computation o f t h e second i n t e g r a l r e q u i r e s t h e know-

l e d g e o f t h e dependence o f

A

on

y;

we c a n c i r c u m v e n t

t h i s d i f f i c u l t y by e x p r e s s i n g t h e second i n t e g r a l the resistance c o e f f i c i e n t .

through

To t h i s end we t r a n s f o r m

(V, 19) i n a n a l o g y t o (V, 4 5 ) , w h i c h g i v e s

Hence, t h e i n t e g r a l w h i c h i n t e r e s t s us i s

and, s u b s t i t u t i n g i n (V, 4 5 ) , we have eL F R ness and vtd seen haoieeflryu fn ueflto ash oo Pr i lm e irofdofm 'i sntd =h eai e dfa l5r1, afin*a uyM ds.atseuie laofwe irblro fg yOn lgun a has yushave yt lhie tvi[ohsfareon etlrand ocm lfocta rohu tiiyhtriltn ya edoes egrr Pr ci( hand, o to Ieo,nnn=sddo as o 3t sat6 1,n)w depend a tit]ot.ti we t htsheFrom Pr heto ghtu ehe »a tlthidtcit(ckhbe aI1, kni,lcnelkes48) anson tceh scseoso iftrthd ft eh io tnhtfi g hse c ek t- o

V. laminar

CHEMICAL HYDRODYNAMICS

2

01

sublayer. Prom d i m e n s i o n a l c o n s i d e r a t i o n s I t f o l l o w s

that

t h e t h i c k n e s s o f t h e l a m i n a r s u b l a y e r s h o u l d be p r o p o r t i o n a l t o t h e f u l l t h i c k n e s s o f t h e boundary l a y e r , by f o r m u l a (V, 5) •

expressed

D e s i g n a t i n g t h e r a t i o o f t h e s e two

magnitudes by t h e t e r m o f d i m e n s i o n l e s s t h i c k n e s s o f t h e laminar sublayer

L,

and c o n s i d e r i n g i t

a u n i v e r s a l con-

s t a n t , we o b t a i n by s u b s t i t u t i n g t h e v a l u e o f I n (V,

5*

(V, 6)

5)

Formula (V, 50) t h e n t a k e s t h e f o r m

P r a n d t l h i m s e l f p r e f e r s a somewhat d i f f e r e n t f o m o f h i s formula-

He I n t r o d u c e s t h e v a l u e o f t h e v e l o c i t y o f f l o w

a t t h e boundary between t h e l a m i n a r s u b l a y e r and t h e c o r e of the flow, i . e . , v'•

at

y = 5*,

w h i c h he d e s i g n a t e s by

Inasmuch as w i t h i n t h e l i m i t s o f t h e l a m i n a r

t h e v e l o c i t y d i s t r i b u t i o n s h o u l d be l i n e a r ,

sublayer

the v e l o c i t y

g r a d i e n t i n t h e l a m i n a r s u b l a y e r w i l l be expressed by

E S xupbrsetsi st ui nt g i n g Ti n by (V, t h48) e r eand s i s t(V, a n c e50)c owe e f f igcei te n t , we g e t

178

V.

CHEMICAL HYDRODYNAMICS

P r a n d t l ' s f o r m u l a has served as a b a s i s f o r a number o f e m p i r i c a l f o r m u l a s t e n d i n g t o d e s c r i b e t h e e x perimental m a t e r i a l obtained i n the study of heat t r a n s f e r i n viscous l i q u i d s . A d e t a i l e d review of a l l t h i s m a t e r i a l , w i t h a t h o r o u g h a n a l y s i s o f t h e methods o f r e c a l c u l a t i o n o f t h e e x p e r i m e n t a l d a t a , can be f o u n d i n t h e book o f 2 Hofmann • L e t us t h e n d e s i g n a t e t h e r a t i o o f P r a n d t l ' s v e l o c i t y a t t h e boundary o f t h e l a m i n a r s u b l a y e r and t h e mean v e l o c i t y o f t h e f l o w by

P r a n d t l ' s f o r m u l a can t h e n be w r i t t e n i n one o f t h e f o l l o w i n g forms

The m a j o r i t y o f t h e e m p i r i c a l f o r m u l a s f o l l o w s t h e t y p e o f P r a n d t l ' s f o r m u l a , w i t h v a r i o u s a r b i t r a r y assumptions on t h e dependence o f t h e magnitude on t h e p r o p e r t i e s o f t h e f l o w . As a rough a p p r o x i m a t i o n , > can be c o n s i d e r e d o c o n s t a n t . Ten Bosch , f r o m experiments w i t h w a t e r , f o u n d for the value of 0.35.

V.

CHEMICAL HYDRODYNAMICS

179

If the laminar sublayer Is considered as an accurate, and not merely an approximate concept, and its dimensionless thickness L, a constant magnitude, φ should be proportional to the square root of the resist­ ance coefficient. This can be seen, for example, from (V, 51) and (V, 58): φ = L Vf

(V, 61 )

On this basis, Prandtl derives the dependence of φ on the Reynolds criterion from Blasius1 resistance law (I, U8) according to which the resistance coefficient is inversely proportional to the fourth-power root of the Reynolds criterion. It then follows from (V, 61) that φ should be inversely proportional to the eighth-power root of the Reynolds criterion. Prandtl chooses the proportionality factor so as best to fit the experimental data, and, as a result, he obtains for the internal problem (straight-circular tube) the formula 1

φ = 1.7^ Re ^

(V, 62)

Combining (V, 62) with (V, 61) and with Blasius1 law (I, W) according to which 1 yf = 0.395

Re

F

we find for the dimensionless thickness of the laminar sublayer L = 4Λ

(V, 63)

If one attributes a real physical existence to the laminar sublayer, and considers it as a region of complete absence of turbulence, the magnitude φ should be

180

V.

CHEMICAL HYDRODJfUAMICS

determined entirely by the velocity distribution In the flow, and, consequently, be a function only of the Reynolds criterion. Several Investigators have attempted to choose the dependence of φ on the Reynolds criterion so that Prandtl's formula will fit the experimental data satisfac­ torily. However, in order to attain a good agreement with the experiment, one must consider φ as a function not only of the Reynolds criterion but also of the Prandtl criterion. Q

Ten Bosch

on the basis of a treatment of experi­

mental results obtained with different substances, has pro­ posed for φ the formula

where

B= 1.4 on heating and

B = 1 ·12

on cooling.

Kupryanov^ believes that best agreement with the experiment­ al data can be obtained with φ considered as a function of the Prandtl criterion only: 1

φ = 0.4¾ Pr

^

ρ Hofmann proposes the formula 1 φ = 1.5 Re ^ Pr

1 τ

It should be borne in mind that experimental data of heat transfer at high values of the Prandtl criterion have been obtained by measurements of the heat transfer in viscous oils. The viscosity of such liquids is very strongly temperature dependent. Therefore, such experiments are always strongly complicated by the effect of the tempera­ ture dependence of the physical constants on the law of convection· This manifests itself Immediately In the ex­ periment by the difference of the heat exchange coefficients on heating and on cooling.

V.

CHEMICAL HYDRODYNAMICS

1 81

Therefore, the experimental material available In the literature does not permit a sufficiently accurate establishment of the law of convection at high values of the Prandtl criterion. More accurate results are obtained by means of studies of turbulent diffusion processes under strictly lsothennal conditions. Practically, such results can be obtained, for example, In the study of dissolution processes taking place In the diffusional range. A further refinement of the above rough calcula­ tions can be attained by abandoning the coarse model of Prandtl, and representing the flow velocity ν as a con­ tinuous function of the distance from the surface y, which amounts to linking the laminar sublayer with the core of the flow. Calculations of this kind will be discussed further below. The fact that φ as a function solely of Re fails to render satisfactorily the experimental results, is proof of the inadequacy of the original model of Prandtl, and the necessity of linking the laminar sublayer and the main stream. However, the data available in the literature do not warrant any conclusions about the physical existence of the laminar sublayer, i.e., provide no clue to whether the turbulent exchange coefficient becomes zero at some finite distance from the surface. To answer this question, there is need for experimental data at very high values of the Prandtl criterion. An extrapolation of empirical formulas obtained at moderate values of this criterion, to very high values, cannot, of course, be considered permissible. Velocity Distribution in the Turbulent Flow Any theory pretending to a complete description of the structure and properties of th'e flow in the vicinity of a solid surface, must be In agreement with the experi­ mental data of velocity distribution. Extensive and very careful measurements of this

V. CHEMICAL HYDRODYNAMICS201 k i n d f o r t u r b u l e n t m o t i o n i n tubes have been done by Nikuradze. by t h e

The r e s u l t s o f h i s measurements a r e expressed

fomula

L e t us compare t h i s v e l o c i t y d i s t r i b u t i o n w i t h t h a t

follow-

i n g f r o m P r a n d t l ' s model. I n t e g r a t i n g e q u a t i o n (V, 1o) n o t t o t h e mean value of the v e l o c i t y

V

but t o y a n a r b i t r a r y value

v(y),

we g e t

U s i n g P r a n d t l ' s model, we s p l i t t h e i n t e g r a n d i n t o two zones, a l a m i n a r s u b l a y e r and t h e core o f t h e f l o w ,

and

we g e t

As a f i r s t a p p r o x i m a t i o n , d i s r e g a r d i n g t h e c u r v a t u r e o f s u r f a c e , we p u t

f ( y ) = 1•

For t h e c o r e o f t h e f l o w , i t

the is

known f r o m hydrodynamics t h a t t h e t r u b u l e n t exchange c o e f f i c i e n t i s p r o p o r t i o n a l t o the distance from the It

i s expressed by f o r m u l a ( I ,

where

1

surface.

15).

i s t h e m i x i n g l e n g t h , and

u

t h e mean p u l s a t i o n

velocity. The m i x i n g l e n g t h tance from the surface

y

1

i s proportional to the d i s -

V.

CHEMICAL HYDRODYNAMICS

2

01

1 = KJ

(V, 67) k

where t h e p r o p o r t i o n a l i t y f a c t o r The mean p u l s a t i o n v e l o c i t y

u

can be c o n s i d e r e d e q u a l t o

uQ

is a universal

constant.

i n the core o f the f l o w

S u b s t i t u t i n g i n (V, 66) and i n t e g r a t i n g , we f i n d

o r , i n t r o d u c i n g the dimensionless thickness of the laminar sublayer

L

w i t h the aid of

F o r a comparison w i t h (V, 6k)

(V,

6)

we w r i t e

(V, 70) I n t h e f o r m

Thus, t h e n u m e r i c a l c o e f f i c i e n t s o f t h e f o r m u l a o f N i k u r a d z e s h o u l d be r e l a t e d w i t h t h e magnitudes I n v o l v e d i n P r a n d t l ' s t h e o r y o f t h e l a m i n a r s u b l a y e r by

The l a t t e r e q u a t i o n g i v e s f o r t h e d i m e n s i o n l e s s of the laminar sublayer the value

thickness

V.

CHEMICAL HYDRODYNAMICS L = 11 .7

(Vj Ik )

The sharp discrepancy between this value and the value (V, 63) is proof of the inadequacy of Prandtl1s model. Laminar Sublayer With Transition The basic shortcoming of Prandtl1s theory is that it fails to rest on a continuous distribution of the normal component of the velocity, but assumes that at the boundary of the laminar sublayer the pulsation velocity changes discontinuously from zero to a finite value. Such a discon­ tinuous variation is physically impossible. To construct a physically satisfactory theory, one must introduce a continuous link between the laminar sublayer and the core of the flow. 1). Karman was the first to propose an improvement on Prandtl's theory, by splitting the flow not into two but into three zones: a laminar sublayer, the core of the flow, and an intermediate linlcing zone. Further similar calculations, involving different arbitrary assumptions on the velocity distribution in the linking zone, were made p 1 1 by Hofmann and by Mattioli . The approach used by these authors complicates Prandtl's theory considerably, but does not eliminate its basic shortcoming - the law of distribu­ tion of pulsation velocities used in these calculations consists of three separate segments, but still is not continuous. We have considered the problem of the calculation of convective diffusion processes on the assumption of a continuous transition from the laminar sublayer to the core of the flow. To this end, we split the flow, following Karman, into three zones: a laminar sublayer of thickness B*, where the turbulent exchange coefficient is strictly zero, the core of the flow where the velocity distribution follows the logarithmic law of Nikuradze (V, 6h), and an intermediate transition zone. On the latter, however, we

V.

CHEMICAL HIDRODyNAMICS

185

impose the condition that it should pass smoothly and con­ tinuously into the laminar sublayer on one side and the core of the flow on the other side. Not only the mean velocity of flow v, but also its derivative with respect to the distance from the surface should have no discontinuity. In the laminar sublayer the velocity distribution should satisfy the law

In the core of the flow, it should tend to the logarithmic law of Nikuradze, and, consequently, Inversely proportional to y.

should become

Let us place the origin of the coordinates at the outer boundary of the laminar sublayer, and let us set ourselves the task of describing the velocity distribution in the intermediate zone and In the core of the flow with the aid of one single analytical expression. The ex­ pression for should at y —> 0 give the value (V, 75), and at larger values of y it should become inversely pro­ portional to y. In order to meet these two conditions, we represent ^ in the fom

§ = T

[ 1

(V, 76)

where f is an unknown function which we shall call the "transition function". The constant y can be termed the width of the transition zone. For the sake of convenience in the following cal­ culations we shall Introduce the auxiliary variable (V, 77)

Por ^ to be continuous, the velocity distri­ bution must satisfy the transition conditions: at the

186

V.

CHEMICAL HYDRODYNAMICS

boundary o f t h e l a m i n a r s u b l a y e r , w i t h finity,

f (T|)

with

tending t o zero,

1 -

.

y

tending to

in-

should vanish; i n t h e core o f the f l o w , f(t])

should tend t o the

Clearly, the simplest t r a n s i t i o n f u n c t i o n

i n g these requirements

limit

satisfy-

is

The e x p r e s s i o n (V, 78) fas? t h e t r a n s i t i o n f u n c t i o n can be considered a very convenient i n t e r p o l a t i o n formula f o r d i s t r i b u t i o n of v e l o c i t i e s i n the v i c i n i t y of the

the

surface.

But f o r t h e most i n t e r e s t i n g case o f l a r g e v a l u e s o f t h e Reynolds c r i t e r i o n ,

the general form o f t h e law o f

c o n v e c t i v e d i f f u s i o n can be o b t a i n e d a c c u r a t e l y , except an unknown f u n c t i o n , o f t h e P r a n d t l c r i t e r i o n , of the f u n c t i o n

f(t]),

for

f o r any f o r m

on t h e s o l e c o n d i t i o n t h a t i t

should

s a t i s f y t h e t r a n s i t i o n c o n d i t i o n s f o r m u l a t e d above. According t o

(V, 10) t h e e x p r e s s i o n f o r t h e b u r -

b u l e n t exchange c o e f f i c i e n t c o r r e s p o n d i n g t o t h e v e l o c i t y d i s t r i b u t i o n (V, 76)

is

I n the core of the f l o w , and,

f(-i))

becomes

1 - t),

with

consequently,

B u t , a c c o r d i n g t o (V, 67) and (V, 6 8 ) , i n t h e ' c o r e o f t h e flow

Combining (V, 80) w i t h (V, 8 1 ) , we conclude t h a t

V. CHmiCAL HYDRODYNAMICS

187

Thus, the width of the linking zone cannot he chosen ar­ bitrarily, but is bound with the fundamental characteristics of the turbulent flow. Taking for the natural yardstick of length, as was done throughout the foregoing discussion, the magnitude Ό

o

u

we find for the dimensionless width of the linking zone the magnitude . Here κ is the proportionality factor be­ tween the mixing length and the distance from the surface, a fundamental universal constant characteristic of the properties of turbulence. Substituting in (V, 23) the expression (V, 79) for the turbulent exchange coefficient, and bearing In mind that the origin of the coordinate η now is not at the surface but at the outer boundary of the laminar sublayer, we get for the thickness of the diffusion layer δ = δ* + 7 1

η0

τ + ΐ(η)(?ίη- 1 ) ' % t

8

I

3)

where η0 is the value of y at the point where the char­ acteristic value of the velocity V is reached (e.g., at the center of the tube). Bearing In mind that the corresponding value of y is equal to half the determining dimension, we get from (V, 82) H0 =

~ ~

(V, 8k)

κ y| Re Thus, the limiting case of high values of the Reynolds criterion corresponds to low values of η0· In order to obtain results for this most

188

V.

CHEMICAL HXUROIKKAMICS

i n t e r e s t i n g l i m i t i n g case, we s h a l l r e p r e s e n t t h e

inte-

g r a n d i n (V, 83) i n t h e f o r m o f a sum o f two terms so t h a t , with

i) 0

t e n d i n g t o z e r o , one o f them w i l l g i v e an i n t e -

g r a l expressed by known f u n c t i o n s

(integral

logarithm),

and t h e o t h e r , an i n t e g r a l t e n d i n g t o a f i n i t e l i m i t . can be a t t a i n e d by i n t r o d u c i n g an a u x i l i a r y

0

2

4

6 8 10 12

14 16 18 20 22 24 26 28 30

FIOURB 18. COMPARISON OF DIFFEEiENT ASSUMPTIONS ON THE VELOCITY DISTRIBUTION IN A TURBULENT FLOW Ordlnates: for the top graph, the effective pulsation velocity; for the bottom graph, velocity of the flow. Abscissas: dimensionless distance from the wall. The scale of velocities is the magnitude O

u

• v/T

the scale of length is the magnitude B' « . ο The curves are drawn with the aid of the formulas proposed by the different authors.

The abscissas are the dimensionless distances from the surface which, according to (V, 5), is defined by

y* = bt = ir y

(v, 96)

V.

CHEMICAL HYDRODYNAMICS

193

The ordlnates represent the effective pulsation velocities and the tangential flow velocities, relative to the magnitude

where V Is the velocity of the main flow at infinite dis­ tances from the surface. The curves correspond to the formulas of Levich, Prandtl, von Karman, and the author. The top graph gives the distribution of the effective pul­ sation velocity, the bottom graph the distribution of the tangential velocity (i.e., the mean velocity of the flow). The scale of velocity is the magnitude

the scale of length is

δ 1.

By effective pulsation velocity

we mean the magnitude which gives the correct value of the turbulent exchange coefficient, If the mixing length is expressed by formula (V, 6 7 ) . Within the limits of the laminar sublayer insofar as it has real physical existence, the effective pulsa­ tion velocity is zero, and the tangential velocity depends on the distance according to (V, 75)· After Integration, and passing to dimensionless coordinates, we get v* = y*

(V, 97)

The full line in the region Immediately adjacent to the surface has been drawn according to this formula. If the laminar sublayer has no real physical existence, and the normal component of the velocity vanishes only at the very surface, then, according to Landau and Levich, the normal component of the velocity in immediate vicinity of the ο surface should Increase proportionally to y · The pro­ portionality coefficient we can indicate only for the

V.

1

CHEMICAL HYDROIHTtAMICS

laminar boundary layer where, according to Levich

u

*

=

v* =

F

y*2

- =

ο

9 8

= / O

(V, 99) 1

u

)

+

KJ*

these formulas being valid only In Immediate vicinity of the surface. Formula (V, 98) is represented in Figure 18. Fomula (V, 99) at small

y*

is practically not

different from the straight line (V, 97); at high

y*,

formular (V, 99) loses its meaning, as the assumptions on which it was derived lose their validity,

ihus, one can­

not use Levich1s method to derive the complete velocity distribution curve.

This method can only give the limit­

ing law of convection for high values of the Prandtl cri­ terion.

Comparison of formulas (V, 6) and (V, 30) shows

that in the Landau and Levich theory, the magnitude 1 n

VFr

plays the role of the dimensionless thickness laminar sublayer.

L

of the

At higher values of the Prandtl cri­

terion, this magnitude should become much smaller than unity. At a distance from the surface, In the core of the flow, all formulas should go over into the formula of Nilcuradze (V, 6k) which rests on reliable direct experi­ mental material; the corresponding formula for the pulsa­ tion velocity is (V, 68). In Prandtl1s model, the pulsation velocity changes abruptly from zero to the value (V, 68) at the outer

V.

CHEMICAL HYDRODYTiAMICS

195

boundary of the laminar sublayer. In order to obtain NUniradze1s formula for the core of the flow, one must assume that this abrupt change occurs at the value y* = 11.7 which has been calculated above (V, 7b)· Naturally, Prandtl's model, lnwhlch the linking zone does not figure at all, requires too high a value of the thickness of the laminar sublayer to give a correct velocity distribution· In von Kannan's model the laaainar sublayer ends at the value y* = 5· From y* = 5 to y* = 30, there is an lntemediate zone in which the velocity distribution is expressed by the formulas u* = 1 -

v* = 5(1 + In

(V, 100)

(V,

101 )

Direct experimental data for convection processes at high­ er values of the Prandtl criterion lead, as has been point­ ed out, to values of the dimensioriless thickness L of the laminar sublayer of about b. We thus come to the conclusion that the existing experimental data can be described with the aid of the concept of the existence of a laminar sublayer of a dimenslonless thickness of about k, in which turbulent transfer is entirely absent. However, there is no room yet for the conclusion that turbulence vanishes altogether at a finite distance from the surface. We can only say that close to the sur­ face, there exists a sublayer of finite thickness, In which convective transfer is practically absent. The problem of the nature of the motion in this sublayer is a matter for further investigation.

196

V. CHEMICAL HYDRODYNAMICS Diffusion to Suspended Particles and. Local Structure of TurbTilence

Up to this point we have discussed processes linked with diffusion from the flow to an Immobile solid surface. A second important problem of chemical hydro­ dynamics is the investigation of processes linked with diffusion to the surface of suspended particles, moving under the action of turbulent motion of a gas or liquid. This problem Is encountered In the study of microheterogeneous processes, the formal diffusional kinetics of which have been discussed in Chapter II. There the diffusion process to the surface of each indi­ vidual suspended particle had been described by the dif­ fusion velocity constant β, without any detailed con­ sideration of the factors determining the value of that magnitude. We now shall set ourselves the task of calcu­ lating the velocity of diffusion to the surface of sus­ pended particles in turbulent motion. This problem has a primary significance for such processes as the evapora­ tion of liquid drops in a turbulent flow or the dissolu­ tion of suspended particles in stirring with an agitator. The latter process is widely used in hydrometallurgy. If -the dimensions of the particles are very small, their motion follows entirely the turbulent pulsa­ tions. The rate of the motion of the gas or liquid rela­ tive to the siarface is in this case zero, and the diffusion process obeys the laws of diffusion in a motionless medium. Assuming the particles in a first approximation to be of spherical shape, the value of the Nusselt cri­ terion can be taken equal to 2, as was done in Chapter II. Such an approach is entirely legitimate in appli­ cation to very small particles, for example, to such important instances of microheterogeneous processes as the dissolution of colloidal sols or the catalytic hydrogenation on suspended catalysts.

v.

CHEMICAL HYDRODmAMICS

197

With the increase of the size of the particles, they cease to follow the turbulent pulsation.

The ve­

locity of the gas or liquid relative to the surface of the particles becomes different from zero, and turbulent trans­ fer begins to play an ever increasing role in the diffusion process. Very large particles are practically not carried along at all by the flow, and are subject to the same laws as a motionless surface placed in a flowHow and at what dimensions will a transition take place from the limiting case of "small" to the limiting case of "large" particles? In principle, the way to solve this problem is indicated by the theory of local structure of the turbu­ lence developed JLn recent years by Eolmogorov and Landau. We shall use this theory in the form in which it is given in the book of Landau and Lifshits^. The mean relative velocity of motion of two par­ ticles of liquid or gas at a distance χ from each other is expressed by

where ρ is the density of the liquid or gas, and e the dissipation energy, i.e., the energy dissipated (converted into heat as a result of viscosity) per unit volume in unit time. The dissipation energy can be expressed by the tangential stress τ or the resistance coefficient f. The total energy dissipated over the whole volume w is equal to the work of the viscosity forces on the friction surface σ, €

2 7 )

we get

- T0)

(IX, 28)

where η is the exponent of the Nusselt criterion in for­ mula (I, 1H). Its value depends on the hydrodynamlc con­ ditions; in most cases, as we have seen In chapter I, it is close to 1/3. In all real cases encountered in practice, the stationary surface temperature rise should be calcu­ lated with the convection taken into account. As can be seen from formula (IX, 28), at D^a the temperature of the surface differs from the theoreti­ cal temperature which can be either higher or lower. If the diffusion coefficients of the rate-determining substance, i.e., of the substance with the lowest value of

IX. THERMAL REGIME

310

is smaller than the temperature conductivity coefficient of the mixture, the surface temperature will be lower than the theoretical temperature· If, on the contrary, D > a, the stationary surface temperature in the upper thermal regime will be higher than the theoretical temperature corresponding to an adiabatic course of the reactionThis last conclusion was confirmed experimentally in study of the catalytic oxidation of hydrogen on platinum in mixtures deficient in hydrogen. Buben1s

Correction for Thermal Diffusion and Diffusion Thermoeffect Formula (IX, 28) is valid if one may disregard, first, the external heat exchange, and secondly, the corrections for thermal diffusion and Stephan flow. By external heat exchange we mean all forms of heat removal from the surface besides the direct heat transfer to the reacting gas, e.g., loss of heat through radiation or cession of heat by the 3olid surface through direct con­ tact to other solid bodies (walls of the vessel, ends of the wire on which the reaction takes place, etc.). The correction for external heat exchange is easy to introduce in each concrete case, by adding the corresponding term in the right-hand member of the equality (IX, 19)· Obviously, this correction depends entirely on the concrete conditions of the experiment, and warrants no general treatment. In contrast, the corrections for thermal diffusion and Stephan flow can be discussed in general formIt is legitimate to disregard thermal diffusion in cases where the molecular weights of all the components of the mixture are close to each other. However, it is entirely inadmissible in such processes as catalytic oxi­ dation of hydrogen. It is legitimate to disregard the Stephan flow in cases where the reacting mixture is strong­ ly diluted by inert gases, but not in the presence of high concentrations of the reactants. The problem of the

IX. THERMAL REGIME

311

calculation of the stationary temperature rise of the surface with the inclusion of thermal diffusion and Stephan flow has been discussed in detail In the work of Buben^. The previous results of Ackerman are erroneous, as this author has considered only the effect of the Stephan flow on diffusion and not Its effect on heat transfer, whereas actually the Stephan flow (as any other convective flow) transfers both matter and heat. Let us consider first the correction for thermal diffusion and diffusion thermoeffect. In the presence of these processes, we must use, for the heat and the diffusion flow, Instead of the usual laws of Pick and Fourier, the expressions (IV, 9) and (IV, 13)In the approximation corresponding to the method of the boundary layer, we shall put

and are the equivalent film thicknesses, where respectively, for the processes of diffusion and of heat transfer; is the partial pressure of the rate-determining gas at a distance from the surface; is the temperature of the surface, and the temperature at a distance from the surface. Formulas (IV, 9) and (IV, 13) give

(IX, 29)

(IX, 30)

312

IX. THERMAL REGIME

The velocity of the mass flow v perpendicular to the surface is at this point taken to be zero, as the correction for the Stephan flow Is to be considered separately. The t h i c k n e s s e s a n d of the equivalent film for diffusion and for heat transfer can be different owing to the different values of the Nusselt criterion. In the stationary state, all the heat evolved by the reaction at the surface, is carried away by heat exchange; consequently, the condition of the stationary thermal state of the surface has the form (IX, 31) where Q is the heat of the reaction. Substituting (IX, 29) and (IX, 30) in (IX, 31) we get

Solving this equation for

we get

(IX, 32)

This formula (IX, 32) is the solution of our problem, as it gives the stationary temperature rise of the surface with the inclusion of thermal diffusion and diffusion thermoeffect• It is easily noted that the temperature rise does not depend on the absolute values of and only on their ratio. Multiplying and dividing the numerator in (IX, 32) b y a n d the denominator by we get

IX. THERMAL REGIME

313

(IX, 33)

According to foraula (I, 28) the thickness of the equivalent film is:

and consequently

where and are the diffusional and the thermal values of the Nusselt criterion.

In the absence of convection

and

In the presence of convection whence (IX, 34) Bearing in mind, besides that where C is the concentration in the gas phase of the substance the

314

IX. THERMAL REGIME

diffusion of which limits the process, we get finally

(IX, 35)

For the magnitudes, and the physical constants In this formula one should take the mean values for the boundary layer. We introduce the theoretical temperature defined by the relation (IX, 23)

Formula (IX, 35), on account o f = the form

cpa, will then take

(IX, 36)

This expression differs from formula (IX, 28) by the two corrective terms in the numerator and the denominator. The first allows for diffusion thermoeffect, the second for thermal diffusion. Bearing in mind that = RTC, where R is expressed in thermal units, and introducing the mole fraction of the diffusing substance in the gaseous mixture

we bring (IX, 36) to the form

IX. THERMAL REGIME

315

(IX, 37)

where R Is expressed In thermal units. Formula (IX, 37) represents the final formula for the computation of the stationary temperature rise at the surface with the inclusion of thermal diffusion and diffusion thermoeffect, in the presence of convection. For mixtures strongly diluted by an inert gas, this formula can be simplified. In this case we can put

and make use of the relation (IV, 3)

where b

is a constant. After that, we get from (IX, 37)

(IX, 38)

For reactions with a large heat effect, only the corrective term in the numerator, i.e., only the correction for thermal diffusion, is essential. The correction for diffusion thermoeffect is negligibly 3mall for such reactions. In this case, if i.e., if the process is limited by the diffusion of the heavier gas, thermal

316

IX. THERMAL REGIME

diffusion will lower the surface temperature, and vice versa. These conclusions of the theory have been con­ firmed experimentally by Buben1study of the thermal regime of the surface in the catalytic oxidation of hy­ drogen on platinum. In the preceding calculation we did not take into account the temperature dependence of the physical constants. That is why the formulas obtained include, in addition to the temperature T1 and TQ which have a definite meaning, also a mean temperature T for the numerical valtie of which one usually takes the arithmetic mean of the temperatures T1 and TQ· This method of calculation, and the use of the arithmetic mean temperature, are adequate only for small temperature differences. Accurate calculation in the case of a large temperature rise calls for the inclusion of the temperature dependence of kip (or b) and of other physi­ cal constants, which results in quite complex expressions. Correction for the Stephan Flow In calculating the effect of the Stephan flow on the stationaiy surface temperature, we can use directly the expression for the flow of material (III, 25). If, as we have been assuming all the time, heat is transferred from the surface only through direct contact with the re­ acting gas, it follows from the law of conservation of energy - X(grad T)n + Σ q^ = 0

(IX, 39)

where w^ is the diffusion flow of the 1th component of the mixture, and I^ its heat content, figured from the common zero point of energy, i.e., Including both physical heat and the chemical energy. The expression (IX, 39) is entirely analagous to the fomula (IX, 31 ) but is more

IX. THERMAL REGIME

317

convenient for the discussion of the Stephan flow, as It spares the calculation of the total heat flow Including t the amount of heat transferred by the Stephan flow. With the aid of the condition of stoichiometry (III, 1 ), we can transform (IX, 39) into (IX, 4o)

where are the stoichiometric coefficients, negative for the original reactants and positive for the final products; the subscript 1 refers to the substance the diffusion of which limits the process, i.e., the substance for which the magnitude

has the smallest value. We

pass now to the individual zero point of energy, thus separating the chemical energy from the physical heat. To that end, we represent the heat content in the form

where is an arbitrary temperature, and is the heat content at that temperature. The heat capacity c is calculated per mole, as Is the heat content. Formula (IX, 4o) then becomes (IX, 4i)

where

is the heat of the reaction at the temperature under constant pressure, per one mole of the substance designated by the subscript 1. Formula (IX, 41) can be written in the form

318

IX. THERMAL REGIME

Combining with (IX, 31) we conclude that the amount of heat transferred by the Stephan flow is expressed by

where is the temperature to which the given heat of reaction refers. Thus, allowance for the amount of heat transferred by the Stephan flow is Identical with an allowance for the temperature dependence of the heat of the reaction. Integration of equation (IX, boundary layer gives

) over the

(IX, k2)

where T 1 is the surface temperature and T Q the temperature in the volume- Considering that according to Kirchhoff's law, the temperature dependence of the heat of reaction, in our notation, is expressed by

we can represent formula (IX, bs ) in the form (IX, 1+3)

where and are the values of the heat of the reaction at the temperatures and This formula

IX. THERMAL REGIME

319

(IX, 43) does not contain the arbitrary temperature T° and, consequently, the choice of that temperature does not alter the result of the calculation. In order to calculate the surface t e m p e r a t u r e i t is necessary to substitute in fonnula (III, 25)

and solve the equation obtained for and substitute f o r i t s we then get

We put in (IX, 42) expression (III, 25);

We introduce the designations

(IX, 1*5)

(IX, 46)

where is the mean heat capacity of the gaseous mixture per one mole, so that

In the notation adopted, (IX, 44) takes the form (IX, 47)

320

IX.

THERMAL REGIME

1 + α(Τ1 - T0) = (1 - rx, )

(IX, W)

Here X1 is the mole fraction, in the gaseous mixture, of the substance the diffusion of which limits the process. In all the formulas of this chapter, heat capacity Is understood as the mean heat capacity in the temperature range between the unknown surface temperature T1 and the temperature TQ In the gas volume. The magnitude

y

i3 defined by the formula

r = •§• Σ J=· υ 1 I

(III, U)

u

Pomula (IX, 48) serves to determine the stationary sur­ face temperature rise with the inclusion of the Stephan flow. It is easily seen that when the Stephan flow is directed away from the surface, it will lower the surface temperature. If, on the contrary, the Stephan flow points to the surface, the temperature can be higher than the theoretical temperature. This conclusion is valid for catalytic reactions involving only gaseous substances. If a solid substance is formed or consumed in the reaction, it is necessary to take Into account the amount of heat remaining in the solid at its formation or evolved from it at its consumption· In this case we have, instead of (IX, Uo) - X(grad T)n +

Σ D1I1

(IX, 49)

where o_ is the stoichiometric coefficient of the solid and Ici its heat content. We consider a quasistationary state when the

321*

IX. THERMAL REGIME

temperature of the solid is equal to the surface temperature and. remains constant. Then

By a calculation analogous to the foregoing, we get instead of (IX, 48) the following equations for the determination of

(IX, 50)

In this case, it is no more necessary to take into account the stoichiometric coefficient of the solid in the calculation of and Let us consider specific concrete cases of application of these formulas. The simplest case is that of a mixture strongly diluted by inert gases or by one of the reacting components. In this case

and, as

is always of the order of unity,

321*

IX. THERMAL REGIME

Developing the right-hand, member of equation (IX, k8) into a Newtonian binomial, we get

(IX, 51 ) whence

Substituting for and their values (III, H), (IX, 45), and (IX, h-6), we get, considering (IX, 23),

If the gaseous mixture is strongly diluted by an Inert gas or an excess of one of the reacting gases, its heat capacity will be determined mainly by the heat capacity of the latter. In this case it is entirely legitimate to disregard the difference between the heat capacities of the original reactants and the reaction products and to consider that the theoretical temperature corresponding to an adiabatic course of the reaction, is defined by formula (IX, 23). Thus, formula (IX, 51) is reduced to

(IX, 52) In a first approximation, at small one can disregard the second term in and then formula (IX, 52) will coincide with (IX, 28). This bears our our previous

321*

IX. THERMAL REGIME

contention that, In dilute mixtures, one can disregard the effect of the Stephen flow. In a second approximation, formula (IX, 52) gives a corrective term which allows for the effect of the Stephan flow on the surface temperature at small If a solid body Is formed or consumed in the reaction, formula (IX, 50) will give at

(IX, 52a)

Thus, In dilute mixtures, formation or consumption of a solid phase affects only the corrective term containing We shall consider another simple case: let the diffusion coefficients of all substances forming the mixture be the same and equal to the temperature conductivity coefficient of the mixture, and let the heat capacities of all reactants and products be equal. In this case, formulas (III, 14), (IX, 45), and (IX, 46) give

We take at once the general case when the -reaction involves not only gaseous substances but also a solid surface at which the reaction takes place [formula (IX, 50)]. The exponent in that formula will be equal to unity and formula (IX, 50) will become

321*

IX. THERMAL REGIME

(IX, 53)

whence

or The magnitude

(IX, 54)

coincides with the definition of according to formula (IX, 23). But, in this case, the temperature thus defined, is not any more equal to the theoretical temperature for an adiabatic course of the reactionAc tually, at high surface temperatures, particularly in heterogeneous combustion processes, there usually arise additional reactions in the volume. In the first place, at high surface temperatures, dissociation of the combustion products becomes significant, as a result of which the final products are formed only after complete cooling. On the other hand, the temperature of the gas in the neighborhood of the surface becomes high enough to give

IX. THERMAL REGIME

325

rise to various homogeneous reactions. All these circum­ stances evidently complicate the problem of the thermal regime of the surface. The foregoing calculations are valid only for the simplest case where homogeneous reac­ tions are absent. This situation can mostly be realized only if the surface temperature is kept sufficiently low through dilution of the reacting mixture by inert gases. But then all the corrections discussed in the foregoing, except the correction for thermodiffusion, become negligible. Thus, the practically most important formulas for the calculation of the thermal regime of the surface are (IX, 28) and, for the case of a large difference in the molecular weights of the gases, (IX, 38). Experimental Data The thermal regime of exothermal heterogeneous reactions was studied, in the instance of the catalytic oxidation of hydrogen and other fuels on platinum wires, 7 3 by Davies , and in greater detail by Bubenj. Figures 25 - 28 show some of the experimental data of Buben· In Pigiares 25 and 26, the ordinates represent the temperature of the wire as a function of the intensity of the heating current. Figure 25 refers to a mixture of hydrogen and air, and Figure 26 to a mixture of ammonia and air. The lower branch of each curve refers to the lower thermal regime. Here, the temperature rise of the surface does not depend on the fuel content in the mixture. This temperature rise is due entirely to the heating of the wire by the current. If the heating current intensity is increased and thus the wire temperature raised, a discontinuous upward jump of the temperature (marked by arrows on the figures) will occur when a definite tempera­ ture is attained (about 100°C for hydrogen, and about 200°C for ammonia). The process passes onto the upper branch of

326

IX-

THERMAL REGIME

FIGURE 25. SURFACE TEMPERATURE AS A FUNCTION OF THEINTENSTTY OF THE CURRENT INT THECATALYTIC HEATING' OXIDATION OF HYDROGEN ON PLATINUM (BUEEN) Ordinates are surface temperatures in degrees centigrade. Abscissas are intensities of the heating current in amperes. The circles represent the results of experiments with a mixture of hydrogen and air, containing 2.75)6 hydrogen; the crosses refer to a mixture with 1-316 hydrogen. The arrow points to the beginning of ignition.

FIGURE 26. SURFACE TEMPERATURE AS A MICTION OF THE INTENSITY" OF THE HEATING CURRENT IN THE CATALYTIC OXIDATION OF AMMONIA ON PLATINUM (BUBEN) (Ordinates and abscissas, as In Figure 25) The circles refer to results of experiments with a mixtur S o f a ™ o n l a a™ 1 al1, containing 5-2 f J™onia; the crosses refer to a mlJcture wlt h 3-1* ammonia,

IX. THEEMAL REGIME

327

the curve, corresponding to the upper temperature regime· In this range, the temperature rise at the surface de­ pends on the fuel content In the mixture; the figures show two curves, each referring to a different fuel content. If, after Ignition of the surface, the heating current In­ tensity Is made to decrease, the upper temperature regime will persist at a substantially lower current Intensity than was needed for the Ignition. At a low fuel content In the mixture, decrease of the heating current Intensity will finally cause extinction of the surface, I.e., sharp passage onto the lower branch of the curve. Extinction takes place at a considerably higher temperature of the surface than the Ignition. For mixtures of hydrogen with air, that temperature is about 3000 and for mixtures of ammonia with air, about 350°C. At high fuel contents In the mixture, feven com­ plete shutting off of the current will not result in an extinction of the surface. In such cases, it Is possible to measure directly the stationary temperature rise of the surface as a result of the reaction in the absence of a heating current. In poor mixtures, where elimination of the heating current results in extinction of the surface, the stationary temperature rise of the surface can be de­ termined from the difference of temperatures on the upper and the lower branches of the curve. In Pig\ares 27 and 28, the stationary temperature rise of the surface is represented as a function of the fuel content of the mixture. Figure 27 refers to mixtures deficient In hydrogen, In which the process is limited by the diffusion of hydrogen. Here, the abscissa represents the percent content of hydrogen in the mixture. In each figure, the curve a represents the stationaiy temperature rise, calculated with inclusion of the effect of the Stephan flow (which, on account of the low content of the diffusing gas In the mixture, Is slight) but without cor­ rections for thermal diffusion and radiation. The curve

328

IX. THERMAL REGIME

FIGURE 27- STATIONARY TEMPERATURE RISE OF THE SURFACE IK MIXTURES DEFICIENT IK HYDROOEK (BUBEN) Ordinate represents the surface temperature rise TJ - T q . Abscissa gives the percent E^. Curve a was drawn with Inclusion of the Stephan flow only; curve b, with inclusion of thermal diffusion; curve SL, with inclusion of loss of heat by radiation- The different kinds of points correspond to different velocities of the gas flow.

b represents the stationary temperature rise of the surface, calculated with the thermal diffusion Included. In mixtures where the process is limited by diffusion of the lighter gas, thermal diffusion enhances the temperature rise of the surface, as in this case the thermal diffusion flow is in the same direction as the ordinary diffusion flow, and therefore the total velocity of diffusion is

IX.

THEIEiMAL RSGrIME

329

T1-T,

700

600

500

400

300

200

100

FIGURE 28. STATIOHAHT TEMPERATURE RISE OP THE STEFAOE IN MIXTURES WITH EXCESS HYDROGEN (BUBEN) Ordlnates represent the surface temperature riae T1 - T0. Abscissas give the percent content of air in the mixture. Other designations are the same as in Figure 27.

greater than without thermal diffusion. In contrast, in mixtures where the process is limited by diffusion of the heavier gas (in this case, in mixtures with excess hydro­ gen), the direction of the thermal diffusion flow is opposed to the ordinary diffusion flow and therefore causes a lowering of the surface temperature. Curve c in Figure 27 was drawn with inclusion of loss of heat by radiation. In mixtures with excess hy­ drogen, this correction can be disregarded, the tempera­ tures involved being low enough for the intensity of emission to be negligible. The experimentally measured surface temperatures are represented by points. The different kinds of points refer to different velocities of the gas flow. The stationary surface temperature rise is,

330

IX.

THEEiMAL REGIME

evidently, practically independent of the gas flow velocity, and is in fair agreement with the calculation. The influence of thermal diffusion on the sur­ face temperature rise, in mixtures of hydrogen and air, turns out to be very strong; this is due to the great difference of the molecular weights of the components of the mixture. In other mixtures of combustible gases, this influence will be very much weaker. Earlier, experimental data of the surface temper­ ature rise of a platinum wire were obtained by Davies^ who also came to the qualitative conclusion that the rate of the process is limited by diffusion. Thermal Regime of a Layer or Channel We have considered the thermal regime of a uni­ formly accessible surface, all points of which are under the same conditions of diffusion and heat exchange. Correspondingly, the reactant concentration and the temper­ ature in the volume were considered to be the same every­ where. Of great practical Importance is the case where the flow of reactants passes through a layer or channel of a length sufficient for the concentration and temperature to change considerably. The picture of the phenomena is not changed in principle; the same stationary thermal regimes and critical conditions are preserved. However, a number of complications arise on closer consideration, linked with the longitudinal transfer of heat and with the possibility of propagation of the reaction zone along the layer or the channel. A complete theory which would take into account the longitudinal heat conduction IXL a general form, does not exist at the present time· What does exist In the lit­ erature is a number of approximate calculations for sundry ι7 simple special cases. Thus, the work of Mayers 1 discusses

IX. THERMAL REGIME

331

the thermal regime of a layer of burning coal, and the work of Todes and Margolis1^, the thermal regime of a catalyst bed for two simple limiting cases. In the first Instance, that of a "long layer" of catalyst, the authors simply disregard the longitudinal heat transfer and take Into account only the change of concentration and tempera­ ture along the layer resulting from the progress of the reaction; in the second instance, that of a "short layer", they consider the longitudinal heat transfer Intense enough for the temperature to be assumed constant over the whole length of the layer. We are not going to enter Into detailed quanti­ tative calculations and shall limit ourselves to funda­ mental qualitative considerations. We shall remark first of all that, in this problem, there can be two forms of critical conditions of ignition and of extinction. We shall designate by the tern of critical conditions of the first kind, the condition of ignition and extinction dis­ cussed in the foregoing and linked with removal of heat as a result of heat transfer. If the longitudinal heat conductivity is large enough to result in an equalization of the temperature along the layer, critical conditions of the second kind, linked with removal of heat by the prod­ ucts of the reaction, will become possible. Critical con­ ditions of the second kind can arise also in a homogeneous exothermal reaction in a flow, under conditions of com­ plete mixing of the ingoing and the reacting mixtures. The idea of the existence of critical conditions of the S and Zysin, second kind was first suggested by Zeldovich 1 ^ particularly with respect to homogeneous reactions. The physical meaning of critical conditions of the second kind can be illustrated in the following way. The cause of the appearance of the critical condition of extinction is the fact that the rate of the reaction can­ not increase indefinitely with the temperature. In the case of critical conditions of the first kind, this increase

332

IX.

THERMftL REGIME

Is halted by the passage of the reaction into the diffusional region. Under the critical conditions of the second kind, an analogous result is attained for an entirely dif­ ferent reason· The total amount of reacting gas within a layer or a channel is limited. With increasing temperature, evolution of heat increases with increasing rate of reac­ tion only up to the point where the whole stock of react­ ing gas is fully spent within the time of its stay in the reaction zone. Further increase of the rate of reaction can lead to no further increase of the overall heat evolu­ tion, unless the velocity of the gas flow is increased. This can result in critical conditions of ignition and ex­ tinction formally entirely analagous to, but basically different from the critical conditions of the first kind. By their nature, these critical conditions are akin to those formulated in the work of Zeldovich 1 5 for homogeneous reactions. Critical conditions of the first kind are inti­ mately linked with the passage of the reaction from the kinetic into the diffusional region and vice versa. In contrast, critical conditions of the second kind have nothing to do with the diffusional region and can be ob­ served within the kinetic region. The stationary surface temperature jumps at ignition, but the reaction remains within the kinetic region. The possibility of critical conditions within the kinetic region, in a reaction in a layer of catalyst, was pointed out in the work of Todes and Margoulis1^ in the instance of the catalytic oxidation of isooctane in a flow. We had, even earlier, pointed out that possibili­ ty in the instance of combustion of a carbon channel. We shall now consider our mathematical theory of critical phenomena of the second kind for a very much idealized scheme, mainly for the purpose of sketching a qualitative picture.

IX.

THERMAL REGIME

333

We shall consider the temperature of the react­ ing surface to be the same along the whole length of the layer or channel.

We shall designate that temperature by

T and seek to determine it from the equation of the heat balance QVS(CQ - C) = ΑΣ(Τ - TQ)

(IX, 55)

where Q is the heat of the reaction, V the linear velocity of the gas flow, S the free cross-section area of the layer or channel, C0 and C the concentrations of the reacting gas at the entrance and exit of the channel,

a

the reacting surface area,

a

the heat trans­

fer coefficient with the surrounding medium, and temperature of that medium.

Tq

the

We shall consider the simplest case where the critical conditions lie within the kinetic region and the kinetics of the reaction is of the first order. The law of the change of the concentration of the reacting gas along the layer or channel will then be of the form (IX, 56) where k is the rate constant, depending on the tempera­ ture according to Arrhenius law

Combining (IX, 55) and (IX, 5 6 ) we get for the determination of the stationary temperature equation

T

the

(IX, 57)

In the following, taking into account that

E » RT,

we

IX.

334

THERMAL RIDIME

shall avail ourselves o~ the development o~ the exponential which we have used previously in the theory o~ thermal ignition (IX, 58)

k

where

is the dimensionless temperature. With the aid o~ the development (IX, 58) we bring (IX, 57) to the ~orm

ze

E - mr=o

e9

VS

or - In( 1

- '09)

(IX, 59)

where

are dimensionless parameters. Solving equation (IX, 59) with respect to 9, we can ~ind the stationary temperature rise as a ~unction o~ two dimensionless parameters, s and u. In order to elucidate the general properties

o~

IX.

THERMAL REGIME

335

the solutions, we write (IX, 59) In the form ζ = - ln(I - ΌΘ )e~ We plot

θ

as abscissas, and

θ

= f(e)

f(e)

(IX, 60)

as ordinates.

The

point of Intersection of the curve with the straight line running parallel to the axis of abscissas at the ordinate ζ

gives the desired stationary temperature. If

f(θ )

has neither maxima nor minima, the

stationary solution will exist always and will be unique. If f(θ ) does have maxima or minima, several stationary states are possible, and the values of the parameters at which a maximum or a minimum occur will give the critical conditions of transition from one regime to another. To find such critical condition it is necessary, therefore, to solve the equation

~3e with respect to

θ

= ο

(IX, 61 )

and to substitute the value of

the equation (IX, 59)· the parameters ζ and

θ

In

Ve then get a relation between u, corresponding to the critical

condition. Eqmtion (IX, 61 ) is reduced to

- (1 - υθ) ln(l - οθ ) = ο or - χ In χ = υ

(IX, 62)

χ = 1 - υθ

(IX, 63)

where

Equation (IX, 62) is easily solved with the aid of Figure 29 where the abscissas are χ and the ordinates χ In χ.

336

IX.

THERMAL REGIME

ν = xlnx

0.4, 0.3 0.2

0

1.0

X

FIGURE 29· SOLUTION OP EQUATION (IX, 62) Ordlnates are « = - χ In χ. Abscissas are x.

It is easily seen that χ In χ has a minimum at χ = 1 , and that the minimum value of χ In χ is equal to - 1 . Consequently, equation (IX, 62) has no solution at all at υ > -1-, and two different values of χ correspond to every ο at ο < ·!•· This means that f(θ) is a monotonous 1

function at υ > —, and has two extreme values, one maxi1 mun and one minimum, at ο < —. In the first case equaν V tion (IX, 59) has always only one solution; in the second case, three solutions are possible. As was shown above in a formally analogous case, only the two extreme solu­ tions out of the three possible correspond to stable states. Thus, there are no critical phenomena at ο > 1; the sur­ face temperature changes continuously with the change of the external conditions. In contrast, at ο < there arise critical conditions of ignition and extinction. To find these conditions, we proceed as follows. Prescribing some value of υ, we find the two correspond­ ing values of χ from the curve in Figure 29, and, accord­ ing to (IX, 63), two values of θ, the smaller of which corresponds to the critical condition of ignition and the larger of which to the condition of extinction. Substi­ tuting these values of θ in equation (IX, 59), we find two values of ζ at which, at the given u, ignition and extinction will occur.

IX.

THERMAL REGIME

337

For practical computations it is convenient to introduce instead of

ζ

the new parameter

δ = I=

B S c ze RT* α °

*

Q

w

while the parameters ζ and ο are both dependent on the velocity V of the gas flow, δ depends only on the temperature, pressure, and composition of the gas, and on the conditions of the heat exchange. This parameter is entirely analogous to that introduced, with the same symbol, in the theory of thermal ignition in homogeneous reactions (see chapter VI). The critical values of δ, calculated in this way for different values of v>, are represented in Figure 30. The upper branch of the curve corresponds to the critical condition of ignition, the lower to the condition of extinction. At x> tending to zero, the critical value of δ tends to 1 . This result has already been obtained In the foregoing.

\g£