Simultaneous Mass Transfer and Chemical Reactions in Engineering Science: Solution Methods and Chemical Engineering Applications 0128191929, 9780128191927

Table of contents :
Cover
SIMULTANEOUS MASS TRANSFER AND CHEMICAL REACTIONS IN ENGINEERING SCIENCE: Solution Methods and Chemical Engineering Applications
Copyright
Dedication
PREFACE
1 . Introduction to simultaneous mass transfer and chemical reactions in engineering science
1.1 Gas–Liquid Reactions
1.1.1 Simultaneous biomolecular reactions and mass transfer
1.1.1.1 The biomedical environment
1.1.1.2 The industrial chemistry and chemical engineering environment
1.1.1.2.1 Conclusions
1.1.1.2.2 Summary
1.2 The modeling of gas–liquid reactions
1.2.1 Film theory of mass transfer
1.2.2 Surface renewal theory of mass transfer
1.2.3 Absorption into a quiescent liquid[∗1]
1.2.3.1 Absorption accompanied by chemical reactions[∗1]
1.2.3.2 Irreversible reactions[∗1]
1.2.3.2.1 First-order reactions
1.2.3.2.2 Instantaneous reactions
1.2.3.2.2.1 Remarks
1.2.3.2.2.1.1 Worked example 1.1
1.2.3.2.3 Simultaneous absorption of two reacting gases [∗3]
1.2.4 Absorption into agitated liquids [∗4]
1.2.4.1 Further references
1.2.5 The mathematical theory of simultaneous mass transfer and chemical reactions[∗1]
1.2.5.1 Physical absorption
1.2.6 Chemical absorption
1.2.6.1 Preliminary remarks on simultaneous mass transfer (absorption) with chemical reactions
1.2.6.2 Some solutions to the mathematical models of the theory of simultaneous mass transfer and chemical reactions
1.2.6.3 Approximate closed form solutions∗∗
1.2.7 Numerical solutions
1.3 Diffusive models of environmental transport
2 . Data analysis using R programming
Preamble
2.1 Chemical engineering data analysis
Introduction [2]
Chemical engineering data coding
Coding
Data capture
Data editing
Imputations
Data quality
Data quality assurance
Data quality control
Quality management
Review questions for Section 2.1
2.2 Beginning R
R and statistics or biostatistics
2.2.1 A first lesson using R
Additional references
3 . Theory of simultaneous mass transfer and chemical reactions, with numerical solutions
3.0 The concept of diffusion
3.0.1 Fick's laws of diffusion
3.0.1.1 Fick's first law of diffusion (steady state law)
3.0.1.2 Fick's second law of diffusion
3.1 Simultaneous biomolecular reactions and mass transfer
3.2 The concept of the mass transfer coefficient
3.3 Theoretical models of mass transfer
3.3.1 Nernst One-Film theory model and the Lewis-Whitman Two-Film model
3.3.2 Higbie penetration theory model
3.3.3 Danckwerts surface renewal theory model
3.3.4 Boundary layer theory model
3.3.5 Mass transfer under laminar flow conditions
3.3.6 Mass transfer past solids under turbulent flow
3.3.7 Some interesting special conditions of mass transfer
3.3.8 Applications to chemical engineering design
3.3.8.1 Designing a packed column for the absorption of gaseous CO2 by a liquid solution of NaOH, using the mathematical model of s ...
3.3.8.2 Calculation of packed height requirement for reducing the chlorine concentration in a chlorine–air mixture
3.4 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions
3.4.1 Numerical solutions of a model in terms of simultaneous semi-linear partial differential equations
3.4.2 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations
3.4.3 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations
3.5 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions
3.5.1 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations
3.6 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations
4 . Numerical worked examples using R for simultaneous mass transfer and chemical reactions
Worked Examples
Example A: Solving Reactive Transport Equations Using R
Examples B: Modeling Framework for Cellular Communities in their Environments
References
B
C
D
F
H
L
M
O
P
Index
Back Cover

Citation preview

SIMULTANEOUS MASS TRANSFER AND CHEMICAL REACTIONS IN ENGINEERING SCIENCE Solution Methods and Chemical Engineering Applications BERTRAM K.C. CHAN PhD (The University of Sydney, Australia) PE (California, USA) IEEE (Life Member)

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright Ó 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-819192-7 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Susan Dennis Acquisition Editor: Anita Koch Editorial Project Manager: Redding Morse Production Project Manager: Vignesh Tamil Cover Designer: Matthew Limbert Typeset by TNQ Technologies

This book is dedicated to: the glory of God, my better half Marie Nashed Yacoub Chan, and the fond memories of:

my physical science teacher Brother Vincent, B.Sc., at the De La Salle College, Cronulla, Sydney, New South Wales, Australia, as well as my professors in Chemical Engineering in Australia, including: at the University of New South Wales: Professor Geoffrey Harold Roper and Visiting Professor Thomas Hamilton Chilton, from the University of Delaware and at the University of Sydney: Professor Thomas Girvan Hunter and Professor Rudolf George Herman Prince

PREFACE Among the unit operations in biomolecular, chemical, and process engineering, the one item that has uniquely, fundamentally, and historically, belonged to the discipline of chemical engineering is concerned with operations involving mass transfer. The principal mass transfer operations, in alphabetical order, include[*]: • Adsorption and fixed-bed separations; • Crystallization; • Distillation; • Drying of solids; • Equilibrium-stage operations; • Gas absorption; • Humidification operations; • Leaching and extraction; • Membrane separation processes. More often than not, such operations involve a combination of both mass transfer and chemical reactions. Hence, analysis and design of such unit operations should start with a basic understanding of simultaneous mass transfer and chemical reactions. After about a century of the academic discipline of chemical engineering, to bring into focus the theory and practice of the study and research into chemical engineering, it is useful to take an in-depth look at this important subject, supported by all available modern computational tools to bring to bear a cogent review of the discipline. Theoretical approaches to simultaneous mass transfer and chemical reactions often begin with mass balances of each of the interacting species at the differential levels, often resulting in a set of simultaneous partial differential equations: usually one equation for each participating species. In this work, theoretical and mathematical analyses of these systems of differential equations are undertaken, resulting in existence and uniqueness theorems for such systems. Upon assuming a given set of initial and/or boundary conditions, with respect to each system of partial differential

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Preface

equations in both space and time, the system solution is then reduced to solving the resultant set of partial differential equations, with radical but realistic assumptions of a set of initial and boundary conditions with respect to space and time. Solution methodologies, starting with numerical analyses, and using various computer methodologies, including: (1) Programming, in FORTRAN-IV, for digital computation, (2) Curve fitting, with resultant graphical outputs, using the open-sourced program R, (3) Computation and curve fitting graphical outputs using the programs written in some high-level language, such as BASIC, etc. are used to obtain useful parametric curves for equipment design. [*](i) McCabe, W. L., and Smith, J. (1956). “Unit Operations of Chemical Engineering”, 1st Edition, The McGraw-Hill Companies, Inc., Series in Chemical Engineering, McGraw-Hill Book Company, New York. (ii) McCabe, W. L., Smith, J. C., and Harriott, P. (2005). “Unit Operations of Chemical Engineering”, 7th Edition, The McGrawHill Companies, Inc., Series in Chemical Engineering, McGrawHill Book Company, New York. “Chemical Engineering Science” A semiofficial recorded documentation of Chemical Engineering Science may be represented by “Perry’s Chemical Engineers’ Handbook,” published by McGraw-Hill Book Company, now on its way to the Nineth Edition, spanning over about a century of the illustrious history of the profession. This latest version, under the general editorship of Don W. Green and Marylee Z. Southard, is scheduled to be published in 2019!

Preface

“Perry’s Chemical Engineers’ Handbook,” 9th Edition

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“Perry’s Chemical Engineers’ Handbook”, 1st through 8th Editions This monogram focuses on one single aspect of the vast literature of chemical engineering science, known as “simultaneous mass transfer and chemical reactions,” which is discussed as a critical theoretical analysis of the subject. It is to be hoped that it will enhance an important aspect of this subject. Special Note: Some introductory materials are taken from well-known engineering literatures, which are thoroughly referenced as they occur in the text.

1 Introduction to simultaneous mass transfer and chemical reactions in engineering science Chapter outline 1.1 GaseLiquid Reactions 2 1.1.1 Simultaneous biomolecular reactions and mass transfer

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1.1.1.1 The biomedical environment 2 1.1.1.2 The industrial chemistry and chemical engineering environment

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1.2 The modeling of gaseliquid reactions 10 1.2.1 Film theory of mass transfer 10 1.2.2 Surface renewal theory of mass transfer 12 1.2.3 Absorption into a quiescent liquid[*1] 15 1.2.3.1 Absorption accompanied by chemical reactions[*1] 1.2.3.2 Irreversible reactions[*1] 17

1.2.4 Absorption into agitated liquids [*4] 1.2.4.1 Further references

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1.2.5 The mathematical theory of simultaneous mass transfer and chemical reactions[*1] 26 1.2.5.1 Physical absorption

1.2.6 Chemical absorption

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1.2.6.1 Preliminary remarks on simultaneous mass transfer (absorption) with chemical reactions 27 1.2.6.2 Some solutions to the mathematical models of the theory of simultaneous mass transfer and chemical reactions 28 1.2.6.3 Approximate closed form solutions** 29

1.2.7 Numerical solutions 37 1.3 Diffusive models of environmental transport 37 In many biochemical, biomedical, and chemical processes, in both physiological systems and in the chemical industry, including environmental sciences, mass transfer accompanied

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science. https://doi.org/10.1016/B978-0-12-819192-7.00001-1 Copyright © 2020 Elsevier Inc. All rights reserved.

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with reversible, complex biochemical or chemical reactions in gaseliquid systems are frequently found. From the viewpoint of biochemical and/or chemical purposes in design, the absorption rates of the participating reactants should be accurately estimated. The associated mass transfer rates may significantly affect the process, such as the yield and selectivity. Much work had been done in describing these processes analytically. These approaches will be used later in this work. For example, the absorption of a gas is followed by a firstorder reversible reaction. Thus, for all mass transfer models, including the penetration and surface-renewal models, analytical solutions have been obtained. For other models, limited work had been done, except for special cases. In this work, both analytical and numerical solutions are presented in some detail.

1.1 GaseLiquid Reactions Reference: [*1] Danckwerts, P. V. (1970). Gas Liquid Reactions, McGraw-Hill, NY; van Elk, E. P. (2001). Gas-Liquid ReactionsdInfluence of Liquid Bulk and Mass Transfer on Process Performance. http:// www.vanelk.nl/edwin/cv/thesis.pdf. It is well-known that many biochemical and chemical processes involve mass transfer of one or more species from the gas phase into the liquid phase. In the liquid phase the species from the gas phase is converted by one or more (possibly irreversible) biochemical or chemical reactions with certain species present in the liquid phase. Typical of such examples are the following.

1.1.1 Simultaneous biomolecular reactions and mass transfer 1.1.1.1 The biomedical environment In epidemiologic investigations, occurrences of simultaneous biomolecular reactions and mass transfer are common in many biomedical environments. Some typical examples are: (1) Intestinal drug absorption involving bio-transporters and metabolic reactions with enzymes[*2]: the absorption of drugs via the oral route is a subject of on-going and serious investigations in the pharmaceutical industry since good bioavailability implies that the drug is able to reach the systemic circulation via the oral path. Oral absorption depends on both

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

the drug properties and the physiology of the gastrointestinal tract, or patient properties, including drug dissolution, drug interaction with the aqueous environment and membrane, permeation across membrane, and irreversible removal by organs such as the liver, intestines, and the lung. (2) Oxygen transport via metal complexes[*2]: On average, an adult at rest consumes 250 ml of pure oxygen per minute to provide energy for all the tissues and organs of the body, even when the body is at rest. During strenuous activities, such as exercising, the oxygen needs increase dramatically. The oxygen is transported in the blood from the lungs to the tissues where it is consumed. However, only about 1.5% of the oxygen transported in the blood is dissolved directly in the blood plasma. Transporting the large amount of oxygen required by the body, and allowing it to leave the blood when it reaches the tissues that demand the most oxygen, require a more sophisticated mechanism than simply dissolving the gas in the blood. To meet this challenge, the body is equipped with a finely tuned transport system that centers on the metal complex heme. The metal ions bind and then release ligands in some processes, and oxidize and reduce in other processes, making them ideal for use in biological systems. The most common metal used in the body is iron, which plays a central role in almost all living cells. For example, iron complexes are used in the transport of oxygen in the blood and tissues. Metaleion complexes consist of a metal ion that is bonded via "coordinate-covalent bonds” to a small number of anions or neutral molecules called ligands. For example the ammonia (NH3) ligand is a mono-dentate ligand; i.e., each mono-dentate ligand in a metaleion complex possesses a single electron-pair-donor atom and occupies only one site in the coordination sphere of a metal ion. Some ligands have two or more electron-pair-donor atoms that can simultaneously coordinate to a metal ion and occupy two or more coordination sites; these ligands are called polydentate ligands. They are also known as chelating (Greek for "claw") agents, because they appear to grasp the metal ion between two or more electron-pair-donor atoms. The coordination number of a metal refers to the total number of occupied coordination sites around the central metal ion (i.e., the total number of metaleligand bonds in the complex). This process is another important example of biomolecular reaction and transport. (3) Carotenoid transport in the lipid transporters SR-BI, NPC1L1, and ABCA1: The intestinal absorption of carotenoids in vivo involves several crucial steps:

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1) release from the food matrix in the lumen, 2) solubilization into mixed micelles, 3) uptake by intestinal mucosal cells, 4) incorporation into chylomicrons, and 5) secretion into the lymph. Research has shown that: A) EZ is an inhibitor of the intestinal absorption of carotenoids, an effect that decreased with increasing polarity of the carotenoid molecule; B) SR-BI is involved in intestinal carotenoid transport; and C) EZ acts not only by interacting physically with cholesterol transporters as previously suggested, but also by downregulating the gene expression of three proteins involved in cholesterol transport in the enterocyte, the transporters SR-BI, NPC1L1, and ABCA1. The intestinal transport of carotenoid is thus a facilitated process resembling that of cholesterol; therefore, carotenoid transport in intestinal cells may also involve more than one transporter. Hence, the study of biomolecular reaction and transport is an area of importance in biomedical processes and their occurrences in epidemiologic investigations. In this section, one applies the facilities available in the R environment to solve problems that have arisen from these processes. This study is being approached from two directions: n Using the R environment as a support to numerical analytical schemes that may be developed to solve this class of problems. n Applying the R functions in the CRAN package ReacTran.[*1] [*2] Chan, B. K. C., Biostatistics for Epidemiology and Public Health Using R”, Springer Publishing Company, New York, NY, 2016. Supplemental Chapter: “Research-Level Applications of R’, available at: www.springerpub.com/chan-biostatistics.

1.1.1.2 The industrial chemistry and chemical engineering environment Typical examples of industrial chemical and chemical engineering processes in which this phenomenon occurs include chlorination, gas purification, hydrogenation, and oxidation processes. To undertake the process and equipment design of new reactors and the optimization of existing reactors, applicable theoretical models for reactors are helpful and most likely needed. In general, models of liquidegas contactors consist of two main parts: the micro model and the macro model:

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

• The micro model describes the interphase mass transfer between the gas phase and the liquid phase, and the macro model describes the mixing behavior in both phases. Both parts of the overall model may be solved sequentially, but solving micro and macro models simultaneously is preferred because of optimization of computational time. Gaseliquid mass transfer modeling has been well studied. The Whitman Stagnant Film Model was first described in 1923 by W.G. Whitman, and it was concluded that some phenomena of gaseliquid mass transfer may be regarded as nearly incompletely explained. Moreover, the Higbie Penetration Model has been used as a basis for the development of some new reactor models. The influence of the bulk liquid on the mass transfer process has been studied in some detail. More attention has been paid to the dynamical behavior and stability of gaseliquid reactors and the influence of mass transfer limitations on the dynamics. Also, some important differences between the results of the Higbie Penetration Model and the Whitman Stagnant Film Model have been found. Analytical solution of micro models for mass transfer (accompanied by chemical reactions) is restricted to asymptotic cases in which many simplifying assumptions had to be made (e.g., reaction kinetics are simple and the rate of the reaction is either very fast or very slow compared to the mass transfer). For all other situations numerical-computational techniques are required for solving the coupled mass balances of the micro model. In general, it seems that mostly numerical solution techniques have been applied. Wherever possible, analytical solutions of asymptotic cases have been used to check the validity of the numerical solution method. For example, by modifying one of the boundary conditions of the Higbie Penetration Model it had been found that the mass transfer may be affected by the presence of the bulk liquid. For example, in a packed column, the liquid flows down the column as a thin layer over the packings. It has been examined whether or not the Penetration Model may be applied for these configurations. Both physical absorption and absorption accompanied by first- and second-order chemical reactions have been investigated. From model calculations, it is concluded that the original Penetration Theory, by assuming the presence of a well-mixed liquid bulk, may be applied also to systems where no liquid bulk is present, provided that the liquid layer is sufficiently thick!

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• For packed columns this means, in terms of the Sherwood number, NSh ¼ 4, for both physical absorption and absorption accompanied by a first-order reaction. • In case of a second-order 1,1-reaction, a second criterion, [NSh  4O(Db/Da)], has to be fulfilled. • For very thin liquid layers [NSh < 4, or NSh < 4O(Db/Da)], the original penetration model may give erroneous results, depending on the exact physical and chemical parameters, and a modified model is required. Analytical solution of models for gaseliquid reactors is restricted to a few asymptotic cases, while most numerical models make use of the physically less realistic stagnant film modeldthis is relatively simplistic and easy to apply using the “Hinterland Model.” The Hinterland Model assumes the reaction phase to consist of ONLY a stagnant film and a well-mixed bulk. Inflow and outflow of species to and from the reactor proceeds via the non-reaction phase or via the bulk of the reaction phase, but never via the stagnant film. [“Hinterland” is a German word meaning “the land behind” (a port, a city, .) in geographic usages!] By modifying one of the boundary conditions of the Higbie penetration model it illustrated how the mass transfer may be affected by the presence of the liquid bulk. Thus, for example, in a packed column the liquid flows as a thin layer over the structured or dumped packing. It has been examined whether or not the penetration model can be applied for these situations. Both physical absorption and absorption accompanied by first- and second-order chemical reaction have been investigated. From model calculations it is concluded that the original penetration theory, which assumes the presence of a wellmixed liquid bulk, may be applied also to systems where no liquid bulk is present, provided that the liquid layer has sufficient thickness. For packed columns, this means, in terms of Sherwood number, Sh > 4 for both physical absorption and absorption accompanied by a first-order reaction. In case of a second-order 1,1-reaction a second criterion Sh  4 O(Db/Da) has to be fulfilled. For very thin liquid layers, Sh < 4 or Sh < O(Db/Da), the original penetration model may give erroneous results, depending on the exact physical and chemical parameters, and the modified model is required. Most numerical models of gaseliquid reactors make use of the physically less realistic stagnant film model because implementation of the stagnant film model is relatively easy using the Hinterland concept. The combination of a stagnant film model and

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Hinterland concept may successfully predict many phenomena of gaseliquid reactors. The Higbie penetration model is however preferred as a micro model because it is physically more realistic. Direct implementation of the Hinterland concept is not possible with the Higbie penetration model. Nevertheless, numerical techniques have been applied to develop a new model that implements the Higbie penetration model for the phenomenon mass transfer accompanied by chemical reaction in well-mixed two-phase reactors: assuming a stagnant film. A model was developed that simulates the dynamic behavior of gaseliquid tank reactors by simultaneously solving the Higbie penetration model for the phenomenon of mass transfer accompanied by chemical reaction and the dynamic gas and liquid phase component balances. The model makes it possible to implement an alternative for the well-known Hinterland concept, which is usually used together with the stagnant film model. In contrast to many other numerical and analytical models the present model can be used for a wide range of conditions, the entire range of Hatta numbers, (semi-)batch reactors, multiple complex reactions and equilibrium reactions, components with different diffusion coefficients, and also for systems with more than one gas phase component. By comparing the model results with analytical asymptotic solutions it was concluded that the model predicts the dynamic behavior of the reactor satisfactorily. It had been shown that under some circumstances substantial differences exist between the exact numerical and existing approximate results. It is also known that, for some special cases, differences can exist between the results obtained using the stagnant film model with Hinterland concept and implementation of the Higbie penetration model. [*3] van Elk, E.P., Borman, P.C., Kuipers, J.A.M., Versteeg, G.F. Modeling of gaseliquid reactorsdimplementation of the penetration model in dynamic modeling of gaseliquid processes with the presence of a liquid bulk Received 14 April 1999; received in revised form 8 November 1999; accepted 29 November 1999 Analytical solution of models for gaseliquid reactors is restricted to a few asymptotic cases, while most numerical models make use of the physically less realistic stagnant film model. A model was developed that simulates the dynamic behavior of gaseliquid tank reactors by simultaneously solving the Higbie penetration model for the phenomenon of mass transfer accompanied by chemical reaction and the dynamic gas and liquid phase component balances. The model makes it possible

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to implement an alternative for the well-known Hinterland concept, which is usually used together with the stagnant film model. In contrast to many other numerical and analytical models, the present model may be used for a wide range of conditions, the entire range of Hatta numbers, (semi-)batch reactors, multiple complex reactions and equilibrium reactions, components with different diffusion coefficients, and also for systems with more than one gas phase component. By comparing the model results with analytical asymptotic solutions it was concluded that the model predicts the dynamic behavior of the reactor satisfactorily. It has been shown that under some circumstances substantial differences exist between the exact numerical and existing approximate results. It is also known that for some special cases, differences can exist between the results obtained using the stagnant film model with Hinterland concept and the implementation of the Higbie penetration model. 1.1.1.2.1 Conclusions 1. The penetration model is preferred for the phenomenon of mass transfer accompanied by chemical reaction in wellmixed two-phase reactors. 2. By comparing the model results with analytical asymptotic solutions it is concluded that the model predicts the reactor satisfactorily. It is shown that for many asymptotic cases the results of this new model coincide with the results of the stagnant film model with Hinterland concept. 3. For some special conditions, differences may exist between the results obtained using the stagnant film model with Hinterland concept and the implementation of the Higbie penetration model. 4. An important result is that for 1,1-reactions the saturation of the liquid phase with gas phase species does not approach zero with increasing reaction rate (increasing Hatta number), contrary to what is predicted by the film model with Hinterland concept. Another important deviation may be found at the specific conditions of a so-called instantaneous reaction in combination with the absence of chemical enhancement of mass transfer. 5. Application of the penetration model does not provide any numerical difficulties, while application of the stagnant film model would lead to a discontinuity in the concentration gradient.

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

6. Another disadvantage of the Hinterland concept is that it can strictly only be applied to isothermal systems, whereas in the systems investigated in this thesis the reaction enthalpy is an important parameter that may significantly influence the phenomena of gaseliquid mass transfer. A rigorous model may be developed that simulates the dynamic behavior of stirred non-isothermal gaseliquid reactors by simultaneously solving the Higbie penetration model for the phenomenon mass transfer accompanied by chemical reaction and the dynamic gas and liquid phase component and heat balances. This is achieved by coupling the ordinary differential equations of the macro model mass and heat balances to the partial differential equations of the penetration model. This model is not yet published! Using the newly developed rigorous reactor model it is shown that dynamic instability (limit cycles) can occur in gaseliquid reactors. The influence of mass transfer limitations on these limit cycles has been studied and is has been found that mass transfer limitations make the process more stable. 1.1.1.2.2 Summary Although the rigorous model is believed to be a very accurate model, it has the disadvantage that owing to the complex numerical methods applied it is a rather time-consuming model. On behalf of a more efficient prediction of the possible occurrence of limit cycles, the reactor model was simplified. The simplified model is suited for the prediction of limit cycles using a stability analysis. A stability analysis is a very efficient method to predict the dynamic behavior and stability of a system of ordinary differential equations by linearization of the governing non-linear ODE’s in the neighborhood of the steady state and analyzing the eigenvalues. This method is very powerful for attaining design rules for stable operation of stirred gaseliquid reactors. The influence of mass transfer limitations on the limit cycles is predicted very well using the simplified model, though small discrepancies are found with the more accurate rigorous model. The developed reactor models have been used to model the dynamics of a new, to be developed, industrial hydroformylation reactor. At a certain design of the reactor, the model predicts serious and undesired limit cycles. These conditions have to be avoided by an appropriate reactor design. Hydroformylation reactions are often characterized by a negative reaction order in carbon monoxide. Model calculations showed that this may lead to interesting phenomena: at certain process

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conditions, an improvement of the mass transfer (higher kla, for example, owing to improved mixing) may give rise to a less stable reactor, without increasing the conversion. This unusual phenomenon is explained by the negative reaction order of carbon monoxide. Apparently, the increasing hydrogen and carbon monoxide concentrations cancel each other out and the overall reaction rate remains unchanged. The increasing hydrogen and carbon monoxide concentrations do however make the process more sensitive for the occurrence of limit cycles. Finally, a start has been made with studying the influence of macro-mixing on the dynamical behavior of gaseliquid reactors. For this purpose a cascade of two reactors in series is compared to a single reactor. Initial results indicate that a cascade of reactors in series provides a dynamically more stable design. The total required cooling surface to prevent the occurrence of temperatureeconcentration limit cycles decreases significantly with an increasing number of reactors in series. The first reactor in the cascade is the one with the highest risk of dynamic instability.

1.2 The modeling of gaseliquid reactions This process has evolved through a number of theoretical processes, including the following.

1.2.1 Film theory of mass transfer In typical industrial absorption processes, one should consider the absorption of gases into liquids which are agitated such that the dissolved gas is transported from the surface to the interior by convective motions. The agitation may occur in various ways, including: (i) The gas, or vapor, may be blown through the liquid as a stream of bubblesdas, for example, on a perforated plate or in a sparged vessel. (ii) The liquid may be running in a layer over an incline or vertical surface, and the flow may be turbulent (as, for example, in a wetted-wall cylindrical column operating at a sufficiently high Reynolds number), or ripples may develop and enhance the absorption rate by convective motion. Discontinuities on the surface may cause periodic mixing of the liquid in the course of its flow, or strings of discs or of spheres. (iii) The liquid may be advantageously agitated by a mechanical stirrer, which may also entrain bubbles of gases into the liquid.

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

(iv) The liquid may be sprayed through the gas as jets or drops. First, consider a steady-state situation in which the composition of the liquid and gas, averaged over a specified region and also with respect to any temporal fluctuations, are statistically constant. For example, one may consider an agitated vessel through which liquid and gas flow steadily, both being so thoroughly mixed that their time-average compositions are the same at all points; or one may consider a short vertical section of a packed column (or sphere or disc or wetted-wall column) operating at steady state, such that the average compositions of the liquid and gas in the element remain constant with time. Clearly the situation is a complicated one: the concentrations of the various species are not uniform or constant when measured over short length and time scales. Diffusion, convection, and reaction proceed simultaneously. The natures of the convective movements of liquid and gas are difficult to define: any attempt to describe them completely would encounter considerable complications. Thus, to obtain useful predictions about the behavior of such systems for practical purposes, it is necessary to use simplified models which simulate the situation sufficiently well, without introducing a large number of unknown parameters. This approach may take a number of simplifying steps, as follows: (A) Physical absorption[*1]. Consider first physical absorption, in which the gas dissolves in the liquid without any reaction; it is found experimentally that the rate of absorption of the gas is given by 
Ra ¼ kL a A  A0 (1.I) in which A* is the concentration of dissolved gas at the interface between gas and liquid, assuming this partial pressure to be uniform throughout the element of space under discussion. The area of interface between the gas and liquid, per unit volume of the system, is a and kL is the “physical mass-transfer coefficient.” R is the average rate of transfer of gas per unit area; the actual rate of transfer may vary from point to point, and from time to time. A0 is the average concentration of dissolved gas in the bulk of the liquid. It is usually not possible to determine kL and a separately, by measurements of physical absorption. For example, in a packed column, the fraction of the surface of packings which is effectively wetted is unknown, and in a system containing bubbles, the

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

interfacial area is not generally known! Thus the quantity directly measurable by physical absorption measurements is the combined quantity kLa. Hence, the validity of Eq. (1.I) has been established in numerous experimental studies, and an expression of this form would be predicted from first principles, provided that certain conditions are met. The chief of these are that the temperature and diffusivity at the surface (where the concentration is A*) should be very different from those in the bulk of the liquid; and that no chemical reaction occurs, so that all molecules of dissolved gas are in the same condition. It is sometimes difficult to decide whether a solute reacts chemically with a liquid or merely interacts with it physically. For the present purpose, “physical” solution means that its molecules are indistinguishable.

1.2.2 Surface renewal theory of mass transfer Models evolved under this theory take as their basis the replacement at time intervals of liquid at the surface by liquid from the interior which has the local mean bulk composition. While the liquid element is at the surface and is exposed to the gas, it absorbs gas as though it were infinitely deep and quiescent: the rate of absorption, R, is then a function of the exposure time of the liquid element and will be described by a suitable expression such as those to be described by the reaction kinetics of the system. In general, the rate of absorption is fast or infinite initially, decreasing with time. The replacement of liquid at the surface by fresh liquid of the bulk composition may be due to the turbulent motion of the body of the liquid. Moreover, when liquid runs over the surface of a packing, it may be in a state of undisturbed laminar flow at the top of each piece of packing, except at the discontinuities between pieces of packing, where it may mix thoroughly: at the top of each piece of packing a fresh surface would then be developed and moved discontinuity, when it would then be replaced again by fresh liquid. With this scenario, the surface-renewal models propose that the surface of an agitated liquid, or a liquid flowing over a packing, is a collection of elements which have been exposed to the gas for different durations of time, and which may well be, in general, absorbing at different specific rates. Thus, different versions of the model will lead to different specific rates. Moreover, different versions of the model will lead to different distributions of surface ages about the mean value.

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

The form of the surface-renewal model first proposed by Higbie, in 1935, assumed that every element of the surface is exposed to the gas for the same duration of time, q, before being replaced by liquid of the bulk composition. During this time, the element of liquid absorbs the same amount Q of gas per unit area as though it were infinitely deep and stagnant. The average rate of absorption is therefore Q/q, and this is also the rate of absorption R per unit area averaged over the interface in a representative region of a steady-state absorption system in which the bulk composition is statistically uniformde.g., in a small, but representative, volume element of a packed column. The exposure-time q may be determined by the hydrodynamic properties of the system, and is the only parameter required to account for their effect on the transfer coefficient kL. The relation between q and kL is derived belowdin physical absorption. Under such circumstances, the variation in time and space of the concentration a of dissolved gas in the liquid in the absence of a reaction is governed by the diffusion equation: DA v2 a=vx2 ¼ va=vt

(1.II)

And the rate of transfer of dissolved gas, initial concentration of the passage of gas across the interface, then the concentration of the surface might vary with time. For the present, it is assumed that the diffusion of dissolved gas into the latter. This assumption generally holds when the solubility of the gas is not very great, so that A* represents a mole fraction much less than unity. It would not be true, for example, if ammonia at atmospheric pressure were diffusing into pure water (in which there will be a substantial temperature rise). Under these conditions, the variation in time and space of the concentration, a, of dissolved gas in the liquid in the absence of a reaction is governed by the diffusion from bubbles or absorption by wetted-wall columns, the mass transfer surface is formed instantaneously and transient diffusion of the material takes place. Assuming that a bubble is rising in a pool of liquid (where the liquid elements are swept on its surface) and remains in contact with it during their motion and finally detached at the bottom. The basic assumptions of the penetration theory are: • Unsteady state mass transfer occurring to a liquid element as long as it is in contact with the gas bubbles • Equilibrium existing at the gaseliquid interface • Each liquid element staying in contact with the gas for the same period of time. (The liquid elements are moving at the

13

14

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Figure 1.1 Schematic of the penetration model.

same rate, and there is not a velocity gradient within the liquid.) Under these assumptions, the convective terms in the diffusion may be neglected and the unsteady-state mass transfer of gas (penetration) into the liquid element may be written from Fick’s second law for unsteady-state diffusion as (Fig. 1.1)
 vc=vt ¼ DAB v2 c = vz2 (1.1) and the boundary conditions are: t ¼ 0; z > 0:

c ¼ cAb

t > 0; z ¼ 0:

c ¼ cAi

and

where cAb / The concentration of solute A at an infinite distance from the surface (viz., the bulk concentration) cAi/ The interfacial concentration of solute A at the surface. On solving the above partial differential equation, one obtains: ðcAi  cÞ=ðcAi  cAb Þ ¼ erf½z = f2OðDAB tÞg

(1.2)

If the process of mass transfer is a unidirectional diffusion and the surface concentration is very low: i.e., cAb z 0; then the mass flux of solute A, given by NA (kgm2 s1) may be estimated by the following equation: NA ¼ ½f rDAB =ð1  cAb Þgðvc=vzÞz¼0 z  rðvc=vzÞz¼0

(1.3)

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

From the above two expressions, the rate of mass transfer at time t is given by the following equation: NA ðtÞ ¼ ½OðDAB =ptÞðcAi  cAb Þ

(1.4)

And the mass transfer coefficient is given by kL ðtÞ ¼ OðDAB =ptÞ

(1.5)

Moreover, the average mass transfer coefficient during a time interval tc(t) may be obtained by integrating Eq. (1.2) as Z tc kL;av ¼ ð1=tc Þ kðtÞdt ¼ 2OðDAB =ptc Þ (1.6) 0

Thus, from the above equation, the mass transfer coefficient is proportional to the square root of the diffusivity. This was first proposed by R. Higbie in 1935 and the theory is called the Higbie Penetration Theory.

1.2.3 Absorption into a quiescent liquid[*1] First, consider the case in which no chemical reaction occurs between the dissolved gas and the liquid. The surface of the liquid first contacts the gas at time t ¼ 0, and it may be assumed that, from that instance onward, the concentration in the plane of the surface is uniformly equal to A*dthis concentration corresponds to the solubility of the gas at the prevailing partial pressure above the surface of the liquiddand is assumed to be constant. If this gas were mixed with another gas of different solubility, or if there were a resistance to the passage of gas across the interface, then the concentration at the surface may vary with time. Further, it is assumed that the diffusion of dissolved gas into the liquid does not appreciably affect the temperature, or other physical properties of the latter. This is likely to be true only when the solubility of the gas is not very great, so that A* represents a mole fraction much less than unity. Under these special circumstances, the variation in time and space of the concentration a of dissolved gas in the liquid in the absence of reaction is governed by the diffusion equation: DA v2 a=vx2 ¼ va=vt

(1.7)

and the rate of transfer of dissolved gas across unit area, Rx of any plane parallel to the surface is Rx ¼  DA va=vx

(1.8)

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Here, x is the distance measured from the surface, where x ¼ 0, and DA is the diffusivity or diffusion coefficient of the dissolved gas. Hence, the rate of absorption of gas at any time is R ¼  DA ðva=vxÞx¼0

(1.9)

The term (va/vx)x¼0 is the concentration gradient at the surface and is a function of time. Let the initial concentration of A be uniformly equal to A0, and its concentration remote from the surface remains A0. Then the solution of Eq. (1.7) with boundary conditions: 9 a ¼ A ; x ¼ 0; t > 0 > = a ¼ A0 ; x > 0; t ¼ 0 (1.10) > ; 0 a ¼ A ; x ¼ N; t > 0 
is aeA0 ¼ A A0 erfcfx =2OðDA tÞg
 ¼ A  A0 ½1  erffx = 2OðDA tÞg

(1.11)

Giving the distribution of concentration in the case where the initial concentration is A0, and the function erfcfx = 2OðDA tÞg ¼ ½1  erffx = 2OðDA tÞg

(1.12)

is the error function of x/2O(DAt), and is defined by Z x=2OðDA tÞ
 erffx=2OðDA tÞg ¼ ð2 =OpÞ exp z 2 dz

(1.13)

0

Values of the error function may be found in standard mathematical tables, etc. From Eqs. (1.9) and (1.11), it follows that
 (1.14) R ¼ A  A0 OðDA =ptÞ Thus, the rate of absorption is infinite when the gas and liquid are first in contact, decreasing with time, and the amount Q absorbed by a unit area of surface, in time t, is given by Z t Rdt Q ¼ 0 (1.15) 
 0 ¼ 2 A  A OðDA t=pÞ

1.2.3.1 Absorption accompanied by chemical reactions[*1] If the dissolved gas reacts with the liquid, or with a substance dissolved in the liquid, then Eq. (1.7) should be replaced by

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

DA v2 a=vx2 ¼ va=vt þ rðx; tÞ

(1.16)

in which r(x, t) is the rate per unit volume of liquid at which the reaction is using up the solute gas at time t and at a distance x below the surface. This rate will depend on the local concentration of the gas, and of any other solute with which it reacts. For some cases, numerical and/or analytical solutions of the diffusion-reaction equations are available. It is assumed throughout that the temperature, and the values such as physico-chemical quantities, diffusivities, reaction-rate constants, and solubilities remain constant and uniform. Moreover, enhancement factors E may be computed ,which is the ratios of the amount which would be absorbed if there were no reaction, viz., 2(A* e A0) O(DAt/p).

1.2.3.2 Irreversible reactions[*1] 1.2.3.2.1 First-order reactions Here r ¼ k1 a

(1.17)

in which k1 is the first-order rate constant for the reaction. The rate of reaction of dissolved gas at any point is directly proportional to its concentration. Under these circumstances, the solution to Eq. (1.16), with boundary conditions (10), and with A0 ¼ 0, is [*] [*] Danckwerts, P. V. (1950). Transactions of the Faraday Society, 46:300 a=A ¼ 1 2 expf xOðk1 =DA Þgerfc ½x=f2OðDA tÞgeOðk1 tÞ =

þ1 2 expfxOðk1 =DA Þgerfc½x=f2OðDA tÞg þ Oðk1 tÞ (1.18) =

so that R ¼ AOðDA k1 Þ½erfOðk1tÞ þ expðk1 tÞ =Oðpk1 tÞ

(1.19)

and  
Q ¼ AOðDA =k1 Þ k1 t þ 1 2 erfOðk1 tÞ þ Oðk1 t=pÞexpðk1 t (1.20) =

Thus, when k1t is large, the distributions of concentration and absorption rate tend to limiting values and no longer change with time: for k1t >> 1. a = A ¼ expf  xOðk1 = DA Þ

(1.21)

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18

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

R ¼ A OðDA k1 Þ

(1.22)

Q ¼ A OðDA k1 Þft þ ð1 = 2k1 Þg

(1.23)

and The error in Eq. (1.23) is less than 3% when k1t > 2. When k1t is large, Q ¼ tA OðDA k1 Þ

(1.24)

to within 5% when k1t > 10. For short times of exposures, for k1t 2bOt

(1.29)

b=B0 ¼ ferf½x = 2OðDB teerfðb = ODB Þg=erfcðb = ODB Þ; x > 2bOt (1.30) b=B0 ¼ 0; 0 < x < 2bOt

(1.31)

R ¼ fA = erfðb =ODA ÞgOðDA = pÞ ¼ Et A  OðDA = pÞ

(1.32)

Q ¼ f2A = erfðb =ODA ÞgOðDA t = pÞ ¼ 2Et A  OðDA t = pÞ

(1.33)

where Ei ¼ 1=erfðb = ODA Þ

(1.34)

and b is defined by:

0    exp b2 = DB erfcðb
2 =ODB Þ ¼ B = zA OðDB = DA Þ exp b = DA erf ðb =ODA Þ

(1.35)

0

Thus, the factor Ei is a function of DB/DA and B /zA*. Here a, b are the local concentrations of dissolved gas and reactant, respectively, and DA, DB are their diffusivities. The reaction plane is at a depth of 2bOt beneath the surface. The quantity Ei is the factor by which the reaction increases the amount absorbed in a given time, as compared to absorption without reaction. The concentration p of the product (assuming y moles to be formed from each mole of reacting gas) is:  

 p ODp Þexp  b2 = DA = 
¼ yA OðDA = DP Þ
erfcðb ; 0 < x < 2bOt (1.36) = erfcðb = ODA Þexp  b2 = DP 

  2 p¼ = 2OD p Þexp  b = DA 
yA OðDA = DP Þ
erfcðx 2 ; x > 2bOt (1.37) = erfcðb =ODA Þexp  b = DP where DP is the diffusivity of the product. When the diffusivities are equal: p ¼ yB0 =z; 0 < x < 2bOt

(1.38)

If, in addition, one mole of reactant reacts with one mole of gas to produce one mole of product, then y ¼ z ¼ 1, and the

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

concentration of product at the surface is the same as that of the reactant in the bulk. When Ei is much greater than unity, then
 Ei ¼ OðDA = DB Þ þ B0 = zA OðDB = DA Þ (1.39) the error being of order 1/2Ei. When DA ¼ DB, Ei ¼ 1 þ B0 =zA

(1.40)

for all values of the variables. Under these conditions the rate of absorption is equal to that of physical absorption of a gas of solubility (A* þ B0/z). 1.2.3.2.2.1 Remarks In many cases, the reactions of the dissolved gases are so fast that they may be considered as instantaneous under all circumstances, for example: (1) H2 S þ OH /HS þ H2 O  (2) H2 S þ RNH2 /RNHþ 3 þ HS  þ (3) NH3 þ HCl/ NH4 þ Cl Other reactions may have finite rates which may control the rate of absorption under certain circumstances, for example: (4) (5) (6) (7)

þ CO2 þ H2 O/ HCO 3 þH   CO2 þ OH /HCO3 CO2 þ 2RNH2 /RNHCOO þ RNHþ 3 4Cuþ þ 4Hþ þ O2 / 4 Cuþþ þ 2H2 O

In all these reactions, in which the reaction rates may be controlled by diffusion under some circumstances, and then the chemical steps in the reaction may be treated as though they were instantaneous. In general, irreversible reactions may be treated as instantaneous if the following condition is fulfilled:
 ðQ0 = 2A ÞOðp = DA tÞ >> 1 þ B0 = zA (1.41) in which Q0 is the amount of gas which would be absorbed in time t if there were no depletion of reactant B in the neighborhood of the surface; if the condition is fulfilled, then the reactant B is entirely depleted in the neighborhood of the surface and the rate of reaction is only controlled by diffusion. 1.2.3.2.2.1.1 Worked example 1.1 H2S at 1 atmosphere is absorbed into quiescent water and into a 0.1 M solution of monoethanolamine (MEA) at 25o C. The reaction H2 S þ RNH2 /HS þ RNHþ 3

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

may be taken to be instantaneous and irreversible. Calculate the amount of H2S absorbed per cm2 of surface in 0.1 s in water. The diffusivity of H2S in water is 1.48  105 cm2/s. The solubility of H2S is 0.10 g-mole/1 atm. Solution: In water:
 Q ¼ 2A OðDA t=pÞ ¼ 2X0:1X10 3 XO 1:48x10 5 X0:1=p ¼ 1:373X10 7 g  mole=cm2 1.2.3.2.3 Simultaneous absorption of two reacting gases [*3] Consider the case of a mixture of two gases being absorbed simultaneously, in which each undergoes a second-order reaction with the same dissolved reactant. As a typical example, this scenario may arise when a mixture of H2S and CO2 is absorbed by a solution of alkali amine. Here, the two solute gases are A and B, their concentrations at the surface, A* and B*, will in general be different, as will their diffusivities DA and DB, and their rate constants kA and kB, governing their rates of reaction with the reactant B. The reactions are both supposed to be second-order, so that the local rates of reaction are kAab and kBab respectively. Stoichiometrically: A þ zB/P A þ zB/P The cases of a second-order reaction between A and B and an mth-, nth-order reaction between A and B will be considered in Chapter 3. In the case of a second-order reaction, the necessary conditions will be discussed in Section 1.2.3.2.X concentration where y moles of product arise from the reaction of one mole of dissolved gas. For large values of k1t, it approaches 2O(k1t/p). [*3] Danckwerts, P. V. (1967). Chemical Engineering Science 22: 472.

1.2.4 Absorption into agitated liquids [*4] In most commercial absorption mass transfer equipment, such as plate or packed towers and columns, in almost all situations, one may and should consider the absorption of gases into liquids which are agitated vigorously so that the dissolved gases are transported from the surface to the interior by convective motion. Such agitations may occur in various ways, such as:

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

(1) The liquid may be running in a layer over a vertical surface or an inclined plane, and the flow may be laminar or turbulent, or ripples may develop and enhance the absorption rate by convective motion. Discontinuities on the surface may cause periodic mixing of the liquid layer along its flow (as in a packed column). (2) The gas may be blown through the liquid as streams of bubbles (as on a bubble plate). (3) The liquid may be agitated by a mechanical stirrer, which may also entrain bubbles of gas into the liquid. (4) The liquid may be sprayed. [*4] Roper, G. H., Hatch, Jr., T. F., and Pigford, R. L. (1962). Theory of absorption and reaction of two gases in a liquid, Industrial Engineering Chemistry Fundamentals 1 (2) 144e152. The theory of diffusion and simultaneous chemical combination of several gases dissolving in a liquid and reacting near the interfaces has been discussed mathematically. The resultant set of nonlinear partial differential equations representing mass balances and corresponding to the penetration theory of mass transfer may be solved numerically. The chemical reaction rate expressions are second (or higher) order, corresponding to a bimolecular (or multi-molecular) mechanism. The results indicate that chemical reaction between two (or multiple) dissolved substances increases the rate of absorption of each by an amount (or amounts) that depend(s) on the ratio of solubilities, the reaction rate constant(s), and the diffusion coefficients. Mathematical solutions are presented for the simultaneous absorption of two gases in a liquid which contains a third component capable of reacting chemically with both of the dissolved gases. Based upon theoretical and experimental studies, one may express the rate of absorption of a gas in a liquid with which the gas reacts chemically in terms of diffusion and reaction kinetic mechanisms. The results are usually expressed by showing the ratio of absorption of the gas to the rate that would occur without the reaction, but with all other conditions held constant. This reaction rate enhancement ratio depends on: • The reaction rate constant • The time of exposure of the liquid surface to the gas • The diffusion coefficients in the liquid, and • Certain stoichiometric ratios. The theoretical computation of such phenomena may be based on the assumption that diffusion and reaction take place in a laminar flow near the interface, the reaction rate per unit volume being so fast that only a small space in the liquid phase is

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

sufficient to keep up with the diffusion of reactants to the reaction zone. With these simplifying assumptions, the resultant mathematical problem is that of solving the unsteady-state diffusion equation with a reaction term included. Thus, there will be as many different cases as there are orders of reaction and ways of bringing various numbers of reactants together. The simplest problems are those in which linear partial differential equations are obtained, corresponding to first-order reactions of a single diffusing species after it dissolves in the liquid at the interface. Results of computations for both reversible and irreversible first-order reactions are available. Theoretical computations have been undertaken involving two dissolved substances: A and B, when A is initially in the gas, and B is a non-volatile reagent initially present in the liquid. This situation is more complex than the linear problem, however, exact solutions have been obtained for fast reactions which are limited by diffusion of A and B to a reaction boundary that moves into the liquid as time elapses. When the reaction occurs at a finite rate, the reaction zone is diffuse, and no exact solutions have been obtained. No theoretical calculations have been available for another physically interesting situation in which substances A and B are both gases that are brought into contact with a laminar layer of liquid in which both are soluble and within which they react with each other by a second-order chemical mechanism. A typical practical example is the simultaneous absorption of CO2 and NH3 into water. These gases do not react in the vapor phase, but in the dissolved state they react with each other to form ammonium carbamate. The speed of the homogeneous reaction has been measured. As an additional example, gases containing both CO2 and H2S are sometimes washed with an alkaline liquid in which both gases dissolve and react, the two species competing for the same alkaline reagent. However, it should not escape one’s notice that these two examples are of fundamentally different types: • In the former, a third substance is not involved, unless it is the solvent, which is present in a large excess and is not depleted by the reaction, or a homogeneous catalyst that is not consumed • In the first example, the chemical equation is of the type: A þ B / • In the second example, a third substance is required for each dissolving gas to react, and its depletion from the solution near the interface limits the rate of absorption of both A and

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

B. Then the chemical steps are typified by amounts that depend on the ratio(s) of solubilities, the various reaction rate constants, and the applicable diffusion coefficients. Mathematical solutions are presented for the simultaneous absorption of two gases in a liquid medium which contains a third component capable of reacting chemically with both of the dissolved gases. Based on the chemical steps beginning with • AþC/ and, simultaneously: • BþC/ A mathematical model for the simultaneous diffusion and reaction of two dissolved gases results in the following model differential equations: A material balance on a differential element having dimensions dx  dz in the direction perpendicular to the interface and the direction of flow perpendicular to the interface and the direction of flow parallel to the interface respectively, leads to a pair of non-linear partial differential equations: DA v2 A=vx2 ¼ uvA=vz þ kAB

(1.42A)

DB v2 B=vx2 ¼ uvB=vz þ nkAB

(1.42B)

where the last terms on the right-hand side of the two equations represent the rate of the homogeneous chemical reaction between substances A and B in the solution per unit volume of solution. The reaction is assumed to have the stoichiometry corresponding to the chemical equation: A þ nB/

(1.43)

and to be irreversible. Kinetically it is assumed to be second order with respect to the two concentrations.

1.2.4.1 Further references Emmert, R. E., Pigford, R. L. Section 14 Gas Absorption and Solvent Extraction, Section 14 in Perry, R. H., Chilton, C. H., Kirkpatrick, S. D., (Eds) (1963). Perry’s Chemical Engineers’ Handbook, fourth ed., McGraw-Hill Book Company, New York, NY. Roper, Hatch, and Pigford have obtained a theoretical solution, using graphic presentations, for the case where two absorbing gases react with one another. Hatch and Pigford [Ind. Eng. Chem., Fund. Quarterly:1:209 (1962)] described an experiment in which CO2 and NH3

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

simultaneously dissolved in water and reacted with one another. Their results thus confirmed the foregoing theory for the pseudo first-order case. An example of a first-order reaction: Consider a first-order irreversible reaction of a dissolved gas. It may be used to show how the effect of a chemical reaction may be calculated on the basis of three models. In this case, the local rate of reaction per unit volume is r ¼ k1 a

(1.44)

k1 being the first-order reaction-rate constant, and a is the local content of dissolved gas. The film model: The equation for the film model now takes the form  a ¼ A ; x ¼ 0 DA d 2 a=dx2 ¼ k1 a (1.45) a ¼ A0 ; x ¼ d where d ¼ DA/kL, kL is the transfer coefficient for physical absorption. Thus  a ¼ ð1 =sinhOMÞ A0 sinhxOðk1 = DA Þ þ A sinh½ðDA = kL Þex Oðk1 = DA Þ (1.46) where M ¼ DAk1/k2L, and the absorption rate is
  R ¼  DA ðda=dxÞx¼0 ¼ kL A  A0 = coshOM fOM = tanhOMg (1.47) Remarks: (1) R is not directly proportional to (A* e A0), as in physical absorption. This is generally the case when chemical reactions occur, and the value of A0 is nonzero. Moreover, it is not generally appropriate to define a transfer coefficient as R/ (A* e A0), except in the case of physical absorption only. (2) The result expressed in Eq. (1.10) was first obtained by Hatta, S. (1932): Technol. Repts. Tohoku Imp. University, 10:119. (3) When OM >> 1, A0 tends to zero, and Eq. (1.10) becomes R ¼ A OðDA k1 Þ

(1.48)

E ¼ OðDA k1 Þ=kL ¼ OM

(1.49)

giving

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Here, E is the factor by which the chemical reaction increases the rate of absorption compared to merely physical absorption (with A0 ¼ 0 in both cases). When OM >> 1, the dissolved gases all react in the film and none diffuses in the unreacted state into the bulk of the liquid. Thus the film thickness, or the value of kL, is irrelevant and does not appear in the expression for R. For the Higbie model, one finds the expression for transient absorption, accompanied by irreversible first-order reaction.

1.2.5 The mathematical theory of simultaneous mass transfer and chemical reactions[*1] Consider the absorption of gases into liquids which are in a state of agitation such that the dissolved gases are transported from the surface to the interior by agitated convective motion. The agitation may occur in one or more of several ways, including: (1) The liquid may be blown through the liquid as a moving stream of bubbles, as, for example, on a bubble plate or in a sparged vessel. (2) The liquid may be running in a layer over a vertical or an inclined surface, and the liquid flow may be turbulent (as, for example, in a wetted-wall column operating at a sufficiently high Reynolds number), or ripples may develop and, thus, enhance the absorption process by convective motion. Discontinuities in the surface may result in periodic mixing of the liquid layer in the course of its flowdas, for example, in a packed column, strings of discs, or strings of spheres. (3) The liquid may be sprayed, as drops or jets, through the gas, as in a spray column of chamber. (4) The liquid may be agitated by a stirrer which may also entrain bubbles of gas into the liquid. (5) The liquid and gases may be interfaced in any physical contacting configurations and equipment.

1.2.5.1 Physical absorption In physical absorption, the gas dissolves in the liquid without reacting. The rate of absorption, found by experiment, is given by 
Ra ¼ kL a A  A0 (1.50) where: R is the average rate of transfer of gas per unit area (a is the area of interface between the gas and the liquid, per unit volume of the system), kL is the physical mass transfer coefficient, A* is the

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

concentration of the dissolved gas corresponding to equilibrium with the partial pressure of the gas at the interface between gas and liquid, and A0 is the average concentration of dissolved gas in the bulk liquid. In general, it is not possible to determine kL and a separately by measurement of physical adsorption. For example, in a packed column, the fraction of the surface of packings which is effectively wetted is unknown; and in a system containing bubbles, the interfacial area is not known in general. Hence the quantity that may be directly measurable by physical absorption measurements is the combined quantity kLa. However, Eq. (1.50) has been established in many experimental investigations so that this equation would be predicted from basic principles whenever certain conditions are metdthe main conditions are that the diffusivity and temperature at the surface (where the concentration is A*) should not be very different from those in the bulk of the liquid, and that no chemical reaction occursdso that all molecules of the dissolved gas are in the same conditions.

1.2.6 Chemical absorption 1.2.6.1 Preliminary remarks on simultaneous mass transfer (absorption) with chemical reactions It is difficult to ascertain whether a solute, in fact, chemically reacts with a liquid or just interacts with it physically! For the present purpose, a “physical” solution means that the dissolved molecules are indistinguishable, viz., not distinguishable as “unreacted” and “reacted” portions! It has not escaped one’s attention that the effects of the hydrodynamics of agitated liquids may well have an important bearing on the issues at hand. Nevertheless, in the present context, it seems indeed useful to use some simplified models of the physical absorption process to describe and analyze the challenge at hand! This approach will therefore proceed with a survey of some well-established models, and proceed to describe the effect of chemical reactions on the mass transfer process! In particular, the mass transfer models based upon: (1) The Nernst One-Film Theory Model, (2) The Whitman Two-Film Theory Model, (3) The Danckwerts Surface Renewal Theory Model, (4) The Higbie Penetration Theory Model, and (5) The Boundary Layer Theory Model.

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

A full presentation and discussion of these various approaches are given in Chapter 3.

1.2.6.2 Some solutions to the mathematical models of the theory of simultaneous mass transfer and chemical reactions Some typical cases are described below. (i) Simultaneous absorption of two reacting gases[*1] Consider two gases being absorbed simultaneously, with each undergoing a second-order reaction with the same dissolved reactant. In practice, a typical example is the case of a mixture of H2S and CO2 being absorbed by a solution of amine or alkali. Let A1 and A2 be the two solute gases, and their respective concentrations at the liquid surface be A1* and A2*, respectively, and their diffusivities be DA1 and DA2, respectively, the rate constants k1 and k2 be governing the rates of their reactions with reactant B. The reactions are both assumed to be of second order, so that the local rates of reaction are, respectively, k1ab and k2ab. Now, stoichiometrically, one may write: A1 þ z1 B/P1

(1.51)

A2 þ z2 B/P2

(1.52)

Danckwerts reported that: (a) As O(k2Bt) approached N, the system approaches that for infinitely fast reaction of both solute gases, and an analytical solution is possible. (b) In general, the effect of the presence of A1 is to reduce the rate of absorption of A2, and vice versa. (ii) Irreversible reaction of general order Here, one may assume that the reaction rate is given by: m

n

r ¼ kmn ðaÞ ðbÞ

(1.53)

Reactions of such type include: (a) The zero-order case: m ¼ 0, n ¼ 0, so that (m þ n) ¼ 0 þ 0¼0 (b) The first-order case: m ¼ 1, n ¼ 0, so that (m þ n) ¼ 1 þ 0¼1 (c) The second-order case: m ¼ 1, n ¼ 1, so that (m þ n) ¼ 1þ1¼2 Defining the dimensionless quantity M by m1
0 n M ¼ ½p = f2ðm þ 1Þgkmn ðA  Þ B t (1.54)

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Then: (1) If OM >> (B0/zA*)O(DB/DA), then the rate of absorption is governed by diffusion alone, viz., the reaction may be considered as “instantaneous,” and R ¼  DA ðva=vxÞx¼0

(1.55)

applies. The term (va/vx)x¼0 is the concentration gradient at the surface, and is a function of time. (2) If OM > 1 þ B0 = 2A1 þ A2 (1.56) where: A1* and A2* are the surface concentrations of H2S and CO2, and the factor 2 is the stoichiometric factor for the reaction of CO2 with NH3 and the enhancement factors are:   Ei1 z 1 þ DB B0 = ðzA DA þ zA DA gOðD1A = DB Þ (1.57)   Ei2 z 1 þ DB B0 = ðzA  DA þ zA DA gOðD2A = DB Þ (1.58) These expressions show that the presence of A2 diminishes Ei, and hence the rate of absorption of A1, and vice versa.

1.2.6.3 Approximate closed form solutions** In the mathematical analysis of simultaneous gas absorption and chemical reaction, a number of closed-form solutions for special cases have been reported.** ** Sherwood, T. K. and Pigford, R. L. (1952). Absorption and Extraction, McGraw-Hill Series in Chemical Engineering, Kirkpatrick, S. D. Consulting Editor, pp. 317e390: Chapter IX Simultaneous Absorption and Chemical Reaction, McGraw-Hill, New York, NY. Example 1. Theory of the stagnant film of finite thicknessdslow first-order reaction [SP] In general, for a particular problem, the approach of using numerical analysis to obtain a useful solution, is recommended. Much has been written on this methodology. A reference to the latest standard collection is readily available in the most recent edition of “Perry’s Chemical Engineers’ Handbook,” which should be a fruitful starting point.

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

If the reaction is not very fast compared with the diffusion rate, then the zone of reaction will not be restricted to a thin region, but will be spread throughout the liquid film. Consider the following derivation for the absorption and simultaneous slow irreversible reaction of a solute. A, with a large excess of the solvent. In this system, the solute diffuses through the gas film under the influence of the driving force (p e pi). Upon entry into the liquid phase, it begins to diffuse toward the main body of the liquid and immediately starts to react with the dissolved substance B. Since the diffusing current of solute is being depleted as it diffuses into the liquid, the concentration gradient will be concave upward. The total rate of diffusion of reacted and unreacted forms of A is constant, since there is no accumulation of A in the film. If the rate equation for the reaction is known, then the conditions of the process may be expresses mathematically. For example, assume that the reaction is of the first order, and that the rate of elimination of A is proportional to the concentration of A at any point in time, viz., dA=dq ¼ kc cV

(1.59)

where kc is the specific reaction-rate constant, c is the variable concentration of A, and V is the liquid volume considered. Next, consider a differential element, of unit cross-sectional area, and of thickness xL . The diffusion rate into this element will be NA ¼  DA dc=dx

(1.60)

at x ¼ x. The rate of diffusion out of this element, at x ¼ x þ dx, will be 
  NA ¼  DA dc = dx þ d 2 c = dx2 dx (1.61) The disappearance of A within the element, owing to reaction with B, is dA=dq ¼  kc cdx

(1.62)

since the volume of the element is dx. The difference in diffusion rates in and out of the element must be equal to the rate of elimination of A by reaction with B, hence DA d 2 c=dx2 ¼ kc c

(1.63)

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Now, assuming that kc remains constant throughout the film, the solution of Eq. (1.63) may be expressed as c ¼ C1 expða0 xÞ þ C2 expða0 xÞ

(1.64)

where a0 ¼ O(kc/DA). Substituting the limits: c ¼ ci at x ¼ 0, and c ¼ cL at x ¼ xLa the constants C1 and C2 are obtained, and the solution takes the form: c ¼ ½cL sinhða0 xÞ þ ci sinhfa0 ðxL exÞg=sinhða0 xL Þ

(1.65)

and for the case of cL ¼ 0, Eq. (1.65) reduces to c ¼ ci sinhfa0 ðxL exÞg=sinhða0 xL Þ

(1.66)

Remark I: (1) Eq. (1.66) is mathematically analogous to the description of heat conduction along a fin that is maintained at a constant temperature at the base, with heat dissipation along the fin in proportion to the temperature profile maintained on the fin. (2) By differentiating Eq. (1.65), the slope of the concentration curve is obtained as follows: dc=dx ¼ ½a0 cL coshða0 xÞ þ a0 ci coshfa0 ðxL  xÞg=sinhða0 xL Þ (1.67) (3) The rate of diffusion into the liquid may be obtained by multiplying the slope at x ¼ 0 by the diffusivity DA, viz., NA ¼ DA ðdc=dxÞx¼0 ¼ DA a0 ½ci coshða0 xL Þ  cL g=sinhða0 xL Þ

(1.68)

(4) Similarly, the rate of diffusion of A into the main body of the liquid may be obtained by substituting x ¼ xL: NA ’ ¼ DA ðdc=dxÞx ¼ xL ¼ DA a0 fci  cL coshða0 xL  xÞg=sinhða0 xL Þ

(1.69)

(5) With respect to the solute A entering the liquid phase, the fraction F arriving at the main body of the liquid without reacting is: F ¼ NA’/NA., and upon substituting from Eqs. (1.68) and (1.69) ¼ ½fci  cL coshða0 xL  xÞg=fci coshða0 xL Þ  cL gjx¼ 0 ¼ fcL coshða0 xL Þ  ci g=fcL  ci coshða0 xL Þg

(1.70)

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Example 2. Theory of the unsteady-state absorption in a stagnant liquiddslow first-order reaction [SP] At the initial exposure of a liquid to a gas during which the solute may accumulate in the liquid, an additional term should be added to Eq. (1.63)

DA d 2 c=dx2 ¼ kc c Remark II: (1) Consider an elemental section make-up of the liquid phase near the interface, having a vertical side of unit cross-section parallel near the interface, and a thickness dx. The rates of diffusion in and out of this liquid section are described by the following two equations representing mass balances: NA ¼  DA ðdc = dxÞ 
  NA ¼  DA ðdc = dxÞ þ d 2 c = dx2 dx

(1.71) (1.72)

while the net rate of disappearance per unit volume owing to reaction is given by: kc cA ekc’ cB

(1.73)

where kc and kc’ are the rate constants for the forward and reverse reactions: A4B

(1.74)

(2) Now, at equilibrium, the opposing rates are equal: Kc ’ ¼ kc =K

(1.75)

where K is the chemical equilibrium constant. (3) The rate of depletion of A from the elemental section is (vcA/ vq) dx in the unsteady state. Equating the rate of input of A by diffusion to the sum of the rates of output by diffusion, disappearance by reaction, and depletion, the result is
 DA v2 cA = vx2 ¼ ke cA eðke = K ÞcB þ vcA =vq (1.76) Similarly, a mass balance for B results in
 DA v2 cA = vx2 ¼ ðke = K ÞcB  kc cA þ vcA =vq

(1.77)

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

(4) Should the concentration of B be eliminated and the diffusivities are assumed equal, the result is a fourth-order linear partial differential equation:

  D v4 cA =vx4 e2v3 cA =vqvx2 eke f1 þ ð1=K Þg v2 cA =vx2 (1.78) þð1=DÞv2 cA =vq2 þ ðke =DÞf1 þ ð1=K ÞgðvcA =vqÞ (5) It is assumed that: (a) the product of the reaction is non-volatile, (b) the interfacial concentration of A is equal to the value in equilibrium with the gas, where the partial pressure of A is constant, (c) the initial concentrations of A and B in the liquid before exposure to the gas are assumed to be uniformly equal to their equilibrium values, and (d) The body of liquid is considered to be infinite effectively. Hence, correspondingly, the following boundary conditions are applicable: (A) At x ¼ 0 : cA ¼ ci ¼ pA =H (B) At x ¼ 0 : vcB =vx ¼ 0 (C) At q ¼ 0: cB ¼ KcA ¼ {K/(1 þ K)}c0, a constant (D) At x ¼ N: cB ¼ KcA ¼ {K/(1 þ K)}c0, a constant where c0 represents the sum of the initial values of cA and cB, which are assumed to be in equilibrium with each other. (6) Since Eq. (1.78) is linear (with respect to the derivative terms) with simple boundary conditions, the Laplace transform method may be used to obtain its solutions, with the following results: fðK þ 1Þcav ec0 g=fðK þ 1Þci ec0 g  
¼ ð3 = OpÞ O DL q = BF2 ff ðK ; kc qÞ = ðK þ 1Þg

(1.79)

where  1=2 f ðK ; kc qÞ ¼ K þ 1 þ pK 5 =f4ðK  1Þðkc qÞg exp½kc q = fK ðK e1Þg    1=2 1=2  erf Kkc q=ðK e1g eerf kc q=fK ðK e1Þg  3  1=2 pK ðK þ 1Þ =4kc q erf½fðK þ 1Þkc qg=K 1=2

(1.80)

where (Kþ1)cav is the average total concentration of solute in the liquid in both the reacted and the unreacted forms and cav represents the average concentration of A after absorption if A and B were in chemical equilibrium at every point.

33

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

(7) Similarly, (K þ 1)ci represents the total concentration of unreacted and reacted solute in the liquid if it were allowed to come to chemical and physical equilibrium with the gas. (8) Using the following functional notations: Z q x expðxÞ ¼ e and erfðxÞ ¼ ð2 = OpÞ ep2 dp (1.81) 0

the latter being the error function. (9) (i)

For large values of kcq, the approximate equivalent of Eq. (1.79) is h
 1=2 1=2 f ðK ; ke qÞ ¼ ð1 þ K Þ 1e p1=2 = 2 K fK =ðK þ 1Þg ðkc qÞ i   3 1 þ K = f2ðK þ 1Þg ðkc qÞ (1.82)

(ii) For small values of kcq approximately, Eq. (1.82) simplifies to hn 3=2 2 f ðK ; ke qÞ ¼ 1 þ K 1=2 ðK þ 1Þ ðK e1Þ o i (1.83) 2  K 4 þ 3K 2  2K = 3ðK  1Þ ðkc qÞ (iii) For the irreversible reaction, viz., K ¼ N f ðN; ke qÞ ¼ 1 2 expðke qÞ n o 1=2 1=2 þOðp = 4Þ ð1 þ 2ke qÞ = ðke qÞ erf ðkc qÞ =

34

(1.84)

When these equations are applied to absorption from a gas in a falling liquid film, the time of exposure is that of the liquid surface, which may be calculated from the surface velocity: q ¼ Z=us ¼ ð2 = 3ÞZ=uav

(1.85)

where Z is the height of the surface. By means of a mass balance on the liquid layer of thickness Bn, the liquid-film absorption coefficient kL is obtained as follows: uav Bn ½ðK þ 1Þcav ec0  ¼ kL Z½ðK þ 1Þci ec0 

(1.86)

kL Z=uav Bn ¼ ½ðK þ 1Þcav  c0 =½ðK þ 1Þci  c0  1=2
¼ ð3 = OpÞ DL q=Bn2 ff ðK ; kc qÞ = ðK þ 1Þg

(1.87A)

and

As before, in these equations:

(1.87B)

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

• c0 represents the concentration of total dissolved solute (both A and B forms) before exposure to the liquid surface, and • (K þ 1)ci represents the concentration of total dissolved solute, both A and B forms, in a liquid in phase equilibrium and reaction equilibrium with the gasdthis latter quantity is not the actual solute concentration at the interface, where chemical equilibrium may not exist. It is the concentration which would be found in the usual static equilibrium experiment. The absorption coefficient kL is defined in Eq. (1.86) on the basis of total dissolved-solute concentrations as a driving force. Comparing Eq. (1.87) with the following equation, Eq. (1.88), developed for pure physical adsorption: kL ¼ Oð6 = pÞOðDL uav = ZÞ
 viz.; kL Z=uav Br ¼ ð3 = OpÞO DL q = Br 2

(1.88) (1.89)

shows that f(K, keq)/(K þ 1) is a multiplying factor which leads to the effect of the chemical reaction on the coefficient for no reaction. • Inspection of Eq. (1.83) shows that f(K. keq) becomes equal to unity for the condition of a very unfavorable equilibrium (K ¼ 0). Under this condition, the correction factor to the absorption coefficient becomes equal to one, as expected. • Also, f(K, keq) becomes equal to unity when the equilibrium favors the production of B(K > 0) but the reaction is very slow (keq w 0). • Eq. (1.87) shows that under these conditions, the absorption coefficient is lower than that for purely physical absorptiond owing to the fact that the solute dissolves at the interface in A form and does not react immediately: the interfacial concentration is only pi/H, therefore, rather than (K þ 1) (pi/H), which is used to define kL. The driving force causing diffusion is smaller, and the rate of absorption is also smaller, owing to the reaction. • When the reaction is fast: ke / N, Eq. (1.82) shows that f(K, keq) approaches (K þ 1) as a limiting value: the absorption coefficient then has its normal value, irrespective of K. Example 3. Theory of the unsteady-state absorption in a stagnant liquiddrapid second-order, irreversible reaction For a chemical reaction occurring in the liquid, after the absorption of gaseous substance A, it may be stated as

A þ B/AB

(1.90)

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

If the rate of absorption of A is sufficiently rapid, the rate may be limited by the rate at which B can diffuse from the liquid toward the gaseliquid interfacedin order to react with A. Thus, the faster B is able to diffuse, the shorter the distance A will need to diffuse into the liquid to meet A. Thus, in contrast with the scenario for the first-order reaction: A/C

(1.91)

for which the absorption rate is independent of the concentration of B, in contrast to the second-order case, wherein the absorption rate of A depends on the concentration of B. When the reaction rate is extremely fast compared with the rate of diffusion of the reactants, the theory may be developed as shown below. In the absorption scenario, the following liquid layer has a uniform concentration of B when it is exposed to a gas containing A. As A is absorbed, molecules of B which are near the interface are consumed in the reaction and are replenished by the diffusion of additional molecules of B from the main body of the liquid. As the reaction is assumed to take place instantaneously when A and B are brought together, regardless of the concentrations in the reaction zone, the point where A and B molecules meet each other will recede farther away from the interface. Hence, the absorption rate of A diminishes because the reacting molecules have to diffuse farther into the liquid to react! Thus, there is little or no opportunity for A to react until it arrives at the reaction zone at a distance x0 from the interface. Hence, within the region 0 < x < x0 , the diffusion of A is represented by the second-order partial differential equation: DA v2 cA =vx2 ¼ vcA =vq

(1.92)

Similarly, component B does not react with A until B reaches the point x ¼ x0 ; and for x > x0 , the diffusion of B is governed by a similar second-order partial differential equation: DB v2 cB =vx2 ¼ vcB =vq

(1.93)

Moreover, for short times of exposure of the liquid surface, concentrations of A and B vary within a region near the interface, so that under these conditions, it may be assumed, without affecting the results significantly, that the liquid layer is infinitely deep. Thus Eq. (1.93) applies within the region x0 < x < N.

Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

Sherwood and Pigford [***] showed, geometrically from the profiles of the concentration curves for A þ B at times q and q þ dq, *** Sherwood, T. K. and Pigford, R. L.(1952). Absorption and Extraction, 334, McGraw-Hill Book Company, New York, NY. that solutions of Eqs. (1.92) and (1.93), will be suitable if they can be made to satisfy the necessary boundary conditions, are: cA ¼ A1 þ B1 erf½x = f2OðDA qÞg

(1.94)

cB ¼ A2 þ B2 erf½x = f2OðDB qÞg

(1.95)

where the functional notation of Eq. (1.81) is used.

1.2.7 Numerical solutions In practice, when seeking numerical solutions for similar system sets of partial differential equations, it is much more straightforward to use numerical techniques such as techniques available in numerical analysis and/or solving in the R domain. Fully worked-out examples are provided in Chapters 3 and 4.

1.3 Diffusive models of environmental transport If one considers only the diffusive element, leaving out the reactive element, in a similar problem, a comprehensive collection of formulations have been systematically worked out by Choy and Reible [CR], including: * Diffusive models of environmental transport * Equilibria within environmental phases * Diffusion in semi-infinite systems * Diffusion in finite layers * Diffusion in two-layer composite systems * Diffusion in three-layer composite systems * Advective diffusive models * Diffusion with volatile liquid evaporation * Diffusion with time-dependent partition coefficients * Diffusion with constant flux liquid evaporation * Some typical numerical examples of diffusive models of

environmental transport.

[CR] Choy, B. and Reible, D. D. (1999). Diffusion Models of Environmental Transport, CRC Press, New York, NY.

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Chapter 1 Introduction to simultaneous mass transfer and chemical reactions

It certainly should not escape one’s attention that, should there be reactive processes involved in any one of these cases, suitable reactive terms may be added to any of these models to provide the concomitant model description to reflect the simultaneous mass transfer and chemical reactions processes.

2 Data analysis using programming

R

Chapter outline Preamble 39 2.1 Chemical engineering data analysis 41 Introduction [2] 41 Chemical engineering data coding 41 Coding 42 Data capture 42 Data editing 43 Imputations 43 Data quality 44 Data quality assurance 44 Data quality control 44 Quality management 44 Review questions for 45 2.2 Beginning R 45 R and statistics or biostatistics 45 2.2.1 A first lesson using R 49

Additional references 58

Preamble This chapter presents the methodology of R programming in a practical setting. A large number of simple, yet instructive, worked examples are used. A progressive, self-taught, approach is outlined.

In an Internet online advertisement for a job vacancy for a chemical statistician or a biostatistician, the job description was presented as follows: Job Summary Chemical Biostatistician I Salary: Negotiable

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science. https://doi.org/10.1016/B978-0-12-819192-7.00002-3 Copyright © 2020 Elsevier Inc. All rights reserved.

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Chapter 2 Data analysis using R programming

Employer: ABC Biostatistics Location: City XX, State YY Type: Entry level Category: Statistics/biostatistics Data analysis/processing, biostatistical Education: A relevant Bachelor’s degree, and a Master’s degree preferred ABC Biostatistics is a leader in analyzing statistical data. ABC partners with others to offer statistical expertise supported by web-based data management systems, and provides timely implementation in data analyses. Job Summary Position Summary: An opportunity is available for a biostatistician to join a growing group focused on analysis and related translational research. ABC, located in downtown City XX, is responsible for the design, management, and analysis of a variety of industrial projects, as well as the analysis of associated market data. The successful candidate will work with fellow statistics and biostatistics staff and financial investigators to undertake investment studies. Responsibilities: Study investment situations and associated ancillary studies in collaboration with fellow staff. Prepare reports of study results; interpret results in collaboration with other financial departments, assist in the preparation of manuscripts, provide chemical biostatistical consultations with collaborating staff. Perform other job-related duties as needed. Additional Requirements Additional Required Qualifications: A Master’s degree in biostatistics, or related field. A working knowledge of applications in statistics and/or biostatistics. Skill in statistical computing using the R software. Other Qualifications: Some biostatistical consulting experience. S-Plus or R programming experience, such as analysis of medical data. Experience in communicatingdorally and in writing. Interpersonal working skills. Other languages also preferred. To learn more about ABC, visit www.ABC.com. Be cognizant of the requirements of an acceptable level of proficiency in data analysis with R.programming! A skill set that would include R programming would be helpful.

Chapter 2 Data analysis using R programming

2.1 Chemical engineering data analysis Data consist of facts or figures from which conclusions may be drawn. When recorded data are classified and organized, they become information. The steps involved in turning data into information are known as data processing. This section describes data processing and how computers undertake these steps effectively and efficiently. These processing activities may use R programming.

Introduction [2] Coding Automated coding systems The simplified flowchart below shows how raw data are transformed into information: Data / Collection/ Chemical engineering data processing Data processing take place when all of the relevant data have been collected. They are processed to produce informationdthe output. Data processing includes the following steps: 1. Data coding 2. Data capture 3. Editing 4. Imputation 5. Quality control 6. Results

Chemical engineering data coding First, raw data must be coded. Survey responses must be labeled, usually with simple, numerical codes, by the interviewer in the field or by an office worker. This step is important because it renders data entry and data processing easier. In surveys, there are two types of questionsdclosed and open. These responses affect the type of coding required. A closed question has only a fixed number of predetermined survey responses permitted. These responses will have already been coded. The following example, on sporting activities, is an example of a closed question: In what way is sport important in providing the following benefits? 1. Very important 2. Slightly important 3. Not important.

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Chapter 2 Data analysis using R programming

An open question admits any response, rendering subsequent coding more difficult. In coding an open question, the processor should first sample a number of responses, and then choose a code structure that includes all possible answers. An example of an open question is: What water sports do you participate in? Specify (about 35 characters) _____________ In a given census, the code of each question is premarked on the questionnaire. To process the questionnaire, these codes are entered directly into the database and prepared for data capturing. The following is an example of premarked coding: What language does this person normally speak at home? 1. Spanish 2. English 3. Other (please specify) ____________

Coding Most computer programs in use will automate routine jobs. The advantages of an automated coding system include fast processing, which increasingly becomes faster, more consistent, and more economical. The next step is inputting the coded biodata into a computer database: this is known as data capture.

Data capture In this process, data are transferred from a paper copy, such as questionnaires and survey responses, to electronic files, then inserted into a computer: the questionnaires should be prepared for chemical engineering data capture. Here, the questionnaire is scrutinized to ensure that all of the required data have been reported, and are decipherable. The methods used for capturing data are: (1) Tally charts: to record data such as the number of occurrences of a particular event and to develop frequency distribution tables. (2) Batch keying: which uses a computer keyboard to type in the data. (3) Interactive capture: also known as intelligent keying. Usually, captured data are edited before they are inputted. This method combines data editing and data capture in one function.

Chapter 2 Data analysis using R programming

(4) Optical character readers (barcode scanners) recognize alpha or numeric characters. These readers can scan lines and translate them into the program. (5) Magnetic and other recordings allow for both reading and writing capabilities: used when data security is important. A computer keyboard is one of the best known input (or data entry) devices in current use. Some modern examples of data input devices are: optical mark reader barcode reader scanner used in desktop publishing light pen trackball mouse, etc. Once the data have been entered into a computer database, one must ensure that all of the responses are accurate. This procedure is known as data editing.

Data editing Data should be edited. This ensures that the information provided is accurate, complete, and consistent. Two levels of data-editing are: micro- and macro-editing. Microediting corrects the data at the record level: to determine the consistency of the data and correct the individual data records. Macroediting also detects errors in data, but does this through the analysis of aggregate data (totals), determining the compatibility of data.

Imputations Imputation may overcome the problems of missing, invalid, or incomplete responses identified during the editing process, including editing errors that may have been found. Here, the data are searched for errors since respondents are not the only ones capable of making mistakes; errors may also occur during editing as well as coding. Other imputation methods available are: * Hot deck: using other records as “donors” to answer the

question that needs imputation.

* Substitution: relying on comparable data, imputed data may

be extracted from the respondent’s record, or the imputed data may be taken from the respondent’s alternative source file.

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Chapter 2 Data analysis using R programming

* Estimator: using information from other questions or from

other answers, and through mathematical operations, deriving plausible values for the incorrect or missing field. * Cold deck: making use of a fixed set of values, covering all the data items. These values may be constructed from the historical data or subject-matter expertise. The donor may be found by the nearest neighbor imputation. In this case, some criteria may be developed to ascertain which responding unit “most resembles” the unit with the missing value in accordance with some predetermined characteristics. The closest unit to the missing value is then used as the donor. Imputation methods may be performed automatically.

Data quality • Quality assurance • Quality control • Quality management in statistical agencies Data quality is an essential element at all levels of processing: to ensure the quality of a product or service in survey development activities, both quality assurance and quality control methods are used.

Data quality assurance Data quality assurance covers planned activities necessary in providing confidence that a product will satisfy its purpose and the users’ needs.

Data quality control Data quality control is a regulatory procedure through which one measures quality, with some predetermined standards. It responds to observed problems, and uses on-going measurements to make decisions on the product, processes, or products.

Quality management The quality of the data should be defined in the context of being “fit for use,” which will depend on the intended function of the data and the fundamental quality characteristics. The elements of quality include: * Accessibility * Accuracy

Chapter 2 Data analysis using R programming

* * * *

Coherence Interpretability Relevance Timeliness

Specialized programs may be developed to edit, clean, impute, and process the final tabular output.

Review questions for Section 2.1 1. In the job description for an entry-level statistician, from the viewpoint of a prospective applicant, what basic statistical computing languages are important to meet the requirement? And why? 2. For a typical MBA (Master of Business Administration) program in Business and Finance, should the core curriculum include the development of R? Why? 3. (a) Compare the concepts of information and data. (b) How does data processing convert data to information? 4. How are statistics and computing applied to data processing? 5. (a) Contrast the concepts of quality assurance and quality control in the world of data processing. (b) In what way does statistics occur in data processing?

2.2 Beginning R R is an open-sourced, free, integrated software environment for data analysis, computation, and graphical display. The R environment consists of: * data handling and storage facility * computations on arrays and matrices * tools for data analysis * graphical capabilities for analysis and display and * an efficient, and continuing developing programming lan-

guage which consists of loops, conditionals, user-defined functions, and input and many output capabilities.

The term “environment” is used to show that it is indeed a planned and coherent system [3,4].

R and statistics or biostatistics R was originally developed by Robert Gentleman and Ross Ihaka of the Statistics Department of the University of Auckland, in New Zealand, in 1997. Since then there has been a R-development core group (of about 20 people with write-access to the R source code).

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Chapter 2 Data analysis using R programming

The introduction to the R environment was not primarily directed toward statistics. Since its development in the 1990s, it appeared to have been “hijacked” by many working in the areas of classical and modern statistics, including applications in financial engineering, econometrics, and biostatistics with respect to epidemiology! These applications provides the raison d’être for authoring this book. For this introductory discussion, the version of R is R-3.3.2, officially released on October 31, 2016. The R package is the Comprehensive R Archive Network, CRAN, at http://cran.r-project.org/. R packages may also be found in numerous publications, e.g., the Journal of Statistical Software, 45th volume, at http://www. jstatsoft.org/v45. To get started, the R-3.3.2 version is being used here. Recall that in Section 2.1, the R environment was obtained as follows: Download the open-source program R from the Internet and take a look at the R computing environment. Remark: Access the Internet at the website of CRAN (the Comprehensive R Archive Network: http://cran.r-project.org/

To install R: R-3.3.2-win32.exe http://www.r-project.org/ [> download R [> Select: USA http://cran.cnr.Berkeley.edu University of California, Berkeley, CA [> http://cran.cnr.berkeley.edu/ [> Windows (95 and later) [> R-3.3.2-win32.exe After downloading: [> Double-click on: R-3.3.2-win32.exe (on the DeskTop) to unzip and install R [> The icon (Script R 3.3.2) will appear on the computer “desktop” as illustrated in Fig. 2.1. In this book, the following special coloring scheme legend is used for all statements during the computational activities in the Renvironmentdto clarify the various inputs and outputs of the process: 1. 2. 3. 4.

Texts in this book (Times New Roman font) Line Input in R code (Verdana font) Line output in R code (Verdana font) Line Comment Statements in R code

Roman font)

(Italicized Times New

Chapter 2 Data analysis using R programming

Figure 2.1 The R icon on the computer DeskTop (the icon for R 3.3.2 is exactly the same as that for R 2.9.1.)

Note: The # sign is the Comment Character: the text in the line following this sign is considered as a comment, that is, no computational action will be taken. The computational activities will proceed as though the comment statements are ignored. These comment statements will remind the programmers and users by providing some clarification of the purposes involved in the remainder of the R environment. # is known as the Number Sign, or the “pound” key, the hash key, and, less commonly, as the octothorp, octothorpe, octathorp, octotherp, octathorpe, and octatherp! To use R under Windows: Double-click on the R 3.3.2 icon ..... Upon selecting and clicking on R, the R-window opens, with the following declaration: R version 3.3.2 (2016-10-31) Copyright (C) 2016 The R Foundation for Statistical Computing. ISBN 3-900051-07-0. R is a free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. (1) Type “license()” or “licence()” for distribution details. R is a collaborative project with many contributors. (2) Type “contributors()” for more information and “citation* Coherence ()” on how to cite R or R packages in publications. (3) Type “demo()” for some demos, “help()” for online help, or “help.start()” for an HTML browser interface to help. (4) Type ’q()’ to quit R.

47

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Chapter 2 Data analysis using R programming

># >#

This is the R computing environment. Computations may begin now!

> >#

First, use R as a calculator, and try a simple arithmetic say: 1 þ 1

> # operation, >1+1 >[1] 2

# This is the output! # WOW! It’s really working! ># The [1] in front of the output result is part of R’s way of printing ># numbers and vectors. Although it is not so useful here, it does ># become so when the output result is a longer vector >

From this point on, this book is best read with the R environment at hand. It will be a very effective learning experience if one practices each R command as one goes along the textual materials! Statistical Data Analysis Manuals An Introduction to R

The R Language Definition

Writing R Extensions

R Installation and Administration

R Data Import/Export

R Internals

Reference Search Engine & Keywords

Packages

Miscellaneous Material About R

Authors

Resources

License

Frequently Asked Questions

Thanks

NEWS

User Manuals

Technical papers

Material specific to the Windows port CHANGES

Windows FAQ

Figure 2.2 Output of the R Command.

Chapter 2 Data analysis using R programming

2.2.1 A first lesson using R This section introduces several important and practical features of the R Environment (Fig. 2.2). LogIn and start an R session in the Windows system of the computer: >

# This is the R environment. Outputting the page shown in Fig. 2.1. > # Statistical Data Analysis Manuals [31] >

> help.start()#

starting httpd help server ... done If nothing happens, one should open At “http://127.0.0.1:28103/doc/html/index.html”oneself. this point, explore the HTML interface to online help right from the desktop, using the mouse pointer to note the various features of this facility available within the R environment. Then, returning to the R environment: >help.start() Carefully read through each of the sections under “Manuals”dto obtain an introduction to the basic language of the R environment. Then look through the items under “Reference” to reach beyond the elementary level, including access to the available “ R Packages”dallR functions and datasets are stored in packages. For example, if one selects the Packages Reference, the following R Package Index window will open up, showing Fig. 2.3, listing a collection of R program packages under the R library: C:\Program Files\R\R-2.14.1\library.

One may now access each of these R program packages, and use them for further applications as needed. Returning to the R environment: > > x

# Generating a pseudo-random 100-vector x

> y # Generating > plot (x, y) > >

another pseudo-random 100-vector y

# Plotting x versus y in the plane, resulting in a graphic # window: Fig. 2.4.

Remark: For reference, Appendix contains the CRAN documentation of the R function plot(), available for graphic outputting, which may be found by the R code segment: > ?plot CRAN has

documentations for many R functions and packages.

49

50

Chapter 2 Data analysis using R programming

Packages in C:\Program Files\R\R-2.14.1\library base

The R Base Package

boot

Bootstrap Functions (originally by Angelo Canty for S)

class

Functions for Classification

cluster

Cluster Analysis Extended Rousseeuw et al.

codetools

Code Analysis Tools for R

compiler

The R Compiler Package

datasets

The R Datasets Package

foreign

Read Data Stored by Minitab, S, SAS, SPSS, Stata, Systat, dBase, ...

graphics

The R Graphics Package

grDevices

The R Graphics Devices and Support for Colours and Fonts

grid

The Grid Graphics Package

KernSmooth

Functions for kernel smoothing for Wand & Jones (1995)

lattice

Lattice Graphics

MASS

Support Functions and Datasets for Venables and Ripley's MASS

Matrix

Sparse and Dense Matrix Classes and Methods

methods

Formal Methods and Classes

mgcv

GAMs with GCV/AIC/REML smoothness estimation and GAMMs by PQL

nlme

Linear and Nonlinear Mixed Effects Models

nnet

Feed-forward Neural Networks and Multinomial LogLinear Models

parallel

Support for Parallel computation in R

rpart

Recursive Partitioning

spatial

Functions for Kriging and Point Pattern Analysis

splines

Regression Spline Functions and Classes

stats

The R Stats Package

stats4

Statistical Functions using S4 Classes

survival

Survival analysis, including penalised likelihood.

tcltk

Tcl/Tk Interface

tools

Tools for Package Development

utils

The R Utils Package

Figure 2.3 Package index.

Chapter 2 Data analysis using R programming

Figure 2.4 Graphical output for plot (x, y).

Again, return to the R workspace, and enter: > >

>ls()# (This is a lower-case “L” followed by “s”, that is, the “list” >

# command.) # (NOT 1 ¼ “ONE” followed by “s”) > # This command will list all the Robjects now in the > # R workspace: > # Outputting:

>

[1] "E" "n" "s" "x" "y" "z"

Again, return to the R workspace, and enter: > >rm (x, y)# Removing >x # Calling for x

all x and all y from the R workspace

Error: object “x” not found >

# Of course, the xs have just been removed! # Calling for y Error: object “y” not found # Because the ys have also been # removed! >y

>

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Chapter 2 Data analysis using R programming

# Let x ¼ (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) # Outputting x (just checking!)

> x x

[1] 1 2 3 4 5 6 7 8 9 10 > >w # standard deviations >dummy # Making a data frame of two columns, x and y, for inspection >dummy

# Outputting the data frame dummy

1 2 3 4 5 6 7 8 9 10

x

y

1 2 3 4 5 6 7 8 9 10

1.311612 4.392003 3.669256 3.345255 7.371759 -0.190287 10.835873 4.936543 7.901261 10.712029

> >fm # Doing a simple linear regression >summary(fm)# Fitting a simple linear

regression of y on x, # then inspect the analysis, and outputting:

> Call:

lm(formula = y w x, data = dummy) Residuals:

Min -6.0140

1Q -0.8133

Median -0.0385

3Q 1.7291

Max 4.2218

Coefficients:

(Intercept) x

Estimate

Std. Error

t value

Pr(>jtj)

1.0814 0.7904

2.0604 0.3321

0.525 2.380

0.6139 0.0445 *

Chapter 2 Data analysis using R programming

--Significant codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 3.016 on 8 degrees of freedom Multiple R-squared: 0.4146, adjusted R-squared: 0.3414 F-statistic: 5.665 on 1 and 8 DF, p-value: 0.04453 >fm1 summary(fm1)#

Knowing the standard deviation, then doing a # weighted regression and outputting:

> Call:

lm(formula = y w x, data = dummy, weights = 1/w^2) Residuals:

Min -2.69867

1Q -0.46190

Median -0.00072

3Q 0.90031

Max 1.83202

Coefficients:

(Intercept) x

Estimate

Std. Error

t value

Pr(>jtj)

1.2130 0.7668

1.6294 0.3043

0.744 2.520

0.4779 0.0358 *

--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 1.356 on 8 degrees of freedom Multiple R-squared: 0.4424, adjusted R-squared: 0.3728 F-statistic: 6.348 on 1 and 8 DF, p-value: 0.03583 >attach(dummy)# >

Making the columns in the data # frame as variables

The following object(s) are masked _by_ ’.GlobalEnv’: x >lrf # regression functionlrf >plot (x, y)#

Making a standard point plot, outputting:

Fig. 2.5. Remark: For reference, Appendix contains the CRAN documentation of the R function plot(), available for graphic outputting, which may be found by the R code segment:

53

54

Chapter 2 Data analysis using R programming

>?plot

Figure 2.5 Graphical output for plot (x, y).

># CRAN has documentations for many Rfunctions and packages. Again, return to the R workspace, and enter: >

>ls()# (This is a lower-case “L” followed by “s”, that is, the “list” >

# command.) # (NOT 1 ¼ “ONE” followed by “s”) > # This command will list all the R objects now in the > # R workspace: > # Outputting: >

[1] "E" "n" "s" "x" "y" "z"

Again, returning to theR workspace, and enter: >

# Removing all x and all y from the R workspace # Calling for x

>rm (x, y) >x

Error: object “x” not found >

# Of course, the xs have just been removed!

> >y

# Calling for y

Error: object “y” not found > # Because the y s have been > > x

# Outputting x (just checking!)

[1] 1 2 3 4 5 6 7 8 9 10 > w

>dummy #

Making a data frame of two columns, x, and y, for inspection > dummy # Outputting the data frame dummy

1 2 3 4 5 6 7 8 9 10

x

y

1 2 3 4 5 6 7 8 9 10

1.311612 4.392003 3.669256 3.345255 7.371759 -0.190287 10.835873 4.936543 7.901261 10.712029

> >fm # Doing a simple linear regression >summary(fm) >

# Fitting a simple linear regression of y on x, # then inspect the analysis, and outputting:

Call: lm(formula = y w x, data = dummy) Residuals:

Min -6.0140

1Q -0.8133

Median -0.0385

3Q 1.7291

Max 4.2218

Coefficients:

(Intercept) x

Estimate

Std. Error

t value

Pr(>jtj)

1.0814 0.7904

2.0604 0.3321

0.525 2.380

0.6139 0.0445 *

55

56

Chapter 2 Data analysis using R programming

--Significant codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 3.016 on 8 degrees of freedom Multiple R-squared: 0.4146, adjusted R-squared: 0.3414 F-statistic: 5.665 on 1 and 8 DF, p-value: 0.04453 >fm1 summary(fm1)#

Knowing the standard deviation, # then doing a weighted # regression and outputting:

> > Call:

lm(formula = y w x, data = dummy, weights = 1/w^2) Residuals:

Min -2.69867

1Q -0.46190

Median -0.00072

3Q 0.90031

Max 1.83202

Coefficients:

(Intercept) x

Estimate

Std. Error

t value

Pr(>jtj)

1.2130 0.7668

1.6294 0.3043

0.744 2.520

0.4779 0.0358 *

--Significant codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 1.356 on 8 degrees of freedom Multiple R-squared: 0.4424, adjusted R-squared: 0.3728 F-statistic: 6.348 on 1 and 8 DF, p-value: 0.03583 >attach(dummy)# >

Making the columns in the dataframe as # variables

>lrf lrf >plot (x, y)#

Making a standard point plot, # outputting: Fig. 2.6. >abline(0, 1, lty=3)# adding in the true regression line: > # (intercept ¼ 0, slope ¼ 1), > # outputting: Fig. 2.7. >abline(coef(fm))# adding in the unweighted regression line: > # outputting Fig. 2.8 >

Chapter 2 Data analysis using R programming

Figure 2.6 Adding in the local regression line.

Figure 2.7 Adding in the true regression line (intercept ¼ 0, slope ¼ 1).

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Chapter 2 Data analysis using R programming

Figure 2.8 Adding in the unweighted regression line.

Additional references [1] W.N. Venables, D.M. Smith, The R Development Core Team, An Introduction to R, rev. ed., Network Theory Limited, Bristol, UK, 2005. [2] R.D. Peng, F. Dominici, Statistical Methods for Environmental Epidemiology with R: A Case Study in Air Pollution and Health (Springer Use R! Series), Springer, New York, NY, 2008. [3] H.E. Wichmann, W. Mueller, P. Allhoff, M. Beckmann, N. Bocter, M.J. Csicsaky, G. Schoeneberg, Health effects during a smog episode in West Germany in 1985, Environmental Health Perspectives 79 (1989) 89e99. [4] A. Bradley, Wellcome Trust Sanger Institute Annual Review, 2009. Retrieved from: http://www.sanger.ac.uk/about/how/assets/intro-review. pdl. [5] A.S. Foulkes, Applied Statistical Genetics with RdFor Population-based Association Studies (Springer Series in Use R!), Springer, New York, NY, 2009. [6] E. Kilbourne, The molecular epidemiology of influenza, The Journal of Infectious Diseases 127 (4) (1973) 478e487. [7] P.A. Schulte, F.P. Perera, Molecular Epidemiology: Principles and Practice, Academic Press, New York, NY, 1993, p. 588. [8] Wikipedia. Spline (mathematics). Retrieved from: http://en.wikipedia. org/wiki/Spline_(mathematics). [9] M.H. Schultz, Spline Analysis (Prentice-Hall Series in Automatic Computation), Prentice-Hall, Englewood Cliffs, NJ, 1973.

Chapter 2 Data analysis using R programming

[10] D.B. Rubin, Interference and missing data, Biometrika 63 (1976) 581e592. [11] D.B. Rubin, Multiple Imputation for Nonresponse in Surveys (Wiley Classics Library Series), Wiley, Hoboken, NJ, 1987. [12] R.J.A. Little, D.B. Rubin, Statistical Analysis with Missing Data (Wiley Applied Probability and Statistics Series), Wiley, Hoboken, NJ, 1987. [13] R.J.A. Little, D.B. Rubin, Statistical Analysis with Missing Data, Wiley Probability and Statistics Series, second ed., Wiley-Interscience, Hoboken, NJ, 2002. [14] J.L. Schafer, Analysis of Incomplete Multivariate Data (Chapman & Hall Monographs Series on Statistics and Applied Probability), Chapman & Hall/CRC Press, Boca Raton, FL, 1997. [15] G. Molenberghs, M.G. Kenward, Missing Data in Clinical Studies (Wiley Series in Statistics in Practice), Wiley, West Sussex, UK, 2007. [16] M.J. Daniels, J.W. Hogan, Monographs on Statistics and Applied Probability, Chapman & Hall/CRC, Boca Raton, FL, 2008. [17] S. Van Buuren, K. Groothuis-Oudshoorn, Mice: multivariate imputation by chained equations in R, Journal of Statistical Software 45 (3) (2011) 1e67. Retrieved from: http://www.jstatsoft.org/v45/i03/. [18] S. Van Buuren, Flexible Imputation of Missing Data, Chapman & Hall/ CRC Press, Boca Raton, FL, 2012. [19] S. Van Buuren, J.P.L. Brand, C.G.M. Groothuis-Oudshoorn, D.B. Rubin, Fully conditional specification in multivariate imputation, Journal of Statistical Computation and Simulation 76 (12) (2006) 1049e1064. Retrieved from: http://www.stefvanbuuren.nl/publications/ FCSinmultivariateimputation-JSCS2006.pdf. [20] S. Van Buuren, Multiple imputation of discrete and continuous data by fully conditional specification, Statistical Methods in Medical Research 16 (3) (2007) 219e242. Retrieved from: http://www.stefvanbuuren.nl/ publications/MIbyFCS-SMMR2007.pdf. [21] S. Van Buuren, H.C. Boshuizen, D.L. Knook, Multiple imputation of missing blood pressure covariates in survival analysis, Statistics in Medicine 18 (1999) 681e694. Retrieved from: http://www.stefvanbuuren. nl/publications/Multipleimputation-StatMed1999.pdf. [22] J.P.L. Brand, Development, Implementation and Evaluation of Multiple Imputation Strategies for the Statistical Analysis of Incomplete Data Sets (Dissertation), Erasmus University, Rotterdam, 1999. [23] A. Krul, H.A.M. Daanen, H. Choi, Self-reported and measured weight, height and body mass index (BMI) in Italy, The Netherlands and North America, European Journal of Public Health 21 (4) (2010) 414e419. [24] H.M. Van Keulen, A.M.J. Chorus, M.W. Verheijden, Monitor Convenant Gezond Gewicht Nulmeting (determinanten van) beweeg- en eetgedrag van kinderen (4-11 jaar), jongeren (12- 17jaar) en volwassenen (18þ jaar), TNO/LS 2011.016, TNO, Leiden, 2011. Retrieved from: http://www. gezondeschool.info/object_binary/o11783_2011-016-monitor-convenantgezond-gewicht-def.pdf. [25] M. Van der Klauw, H.M. Van Keulen, M.W. Verheijden, Monitor Convenant Gezond Gewicht Beweeg- en eetgedrag van kinderen (4-11 jaar), jongeren (12-17 jaar) en volwassenen (18þ jaar) in 2010 en 2011, TNO/LS 2011.055, TNO, Leiden, 2011 (in Dutch). Retrieved from: http:// www.convenantgezondgewicht.nl/download/131/2011_055_monitor_ convenant_gezond_gewicht.pdf. [26] B. Pfaff, VAR, SVAR and SVEC models: implementation within R package vars, Journal of Statistical Software 27 (4) (2008). Retrieved from: http:// www.jstatsoft.org/v27/i04/. [27] F. Hahne, W. Huber, R. Gentleman, S. Falcon, Bioconductor Case Studies (Use R! Series), Springer, New York, NY, 2008.

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[28] K.S. Pang, Modeling of intestinal drug absorption: role of transporters and metabolic enzymes (for the Gillette Review Series), Drug Metabolism and Disposition 31 (12) (2003) 1507e1519. [29] Casidy, R., & Frey, R. Hemoglobin and the Heme Group: Metal Complexes in the Blood for Oxygen Transport. Inorganic Synthesis Experiment, Department of Chemistry, Washington University, St. Louis, MO. [30] A. During, H.D. Dawson, E.H. Harrison, Biochemical and molecular actions of nutrients: carotenoid transport is decreased and expression of the lipid transporters SR-BI, NPC1L1, and ABCA1 is downregulated in Caco-2 cells treated with ezetimibe, in: Presented in Abstract Form at Experimental Biology ’05, San Diego, April 2005. [31] K. Soetaert, T. Petzoldt, R.W. Setzer, Solving differential equations in R: package deSolve, Journal of Statistical Software 33 (9) (2010) 1e25. Retrieved from: http://www.jstatsoft.org/v33/i09. [32] G.H. Roper, T.F. Hatch Jr., R.L. Pigford, Theory of absorption and reaction of two gases in a liquid, I & E C Fundamentals 1 (1962) 144e152. [33] T.F. Hatch, R.L. Pigford, Simultaneous absorption of carbon dioxide and ammonia in water, I & E C Fundamentals 1 (1952) 209e214. [34] G.E. Forsythe, W.R. Wasow, Finite-Difference Methods for Partial Differential Equations, Wiley, New York, NY, 1960. [35] T.F. Yin, M. Lawrence, D. Dianne Cook, Notes on the Bioconductor Package “biovizBase”, 2012. Retrieved from CRAN site). [36] K. Soetaert, P.M.J. Herman, A Practical Guide to Ecological Modelling: Using R as a Simulation Platform, Springer, New York, NY, 2009. [37] G.E. Fraser, Diet, Life Expectancy, and Chronic Disease: Studies of Seventh-Day Adventists and Other Vegetarians, Oxford University Press, New York, NY, 2003. [38] Living longer and better, SCOPE 47 (2) (Fall 2011) 20e23. Retrieved from: www.LLU.edu/escope. , K. Jaceldo-Siegl, G.E. Fraser, Vegetarian dietary [39] N.S. Rizzo, J. Sabate patterns are associated with a lower risk of metabolic syndrome, Diabetes Care 34 (5) (May 2011) 1225e1227. Retrieved from: http://www. ncbi.nlm.nih.gov/pmc/articles/PMC3114510/figure/F1. [40] S.N. Wood, Generalized Additive Models: An Introduction with R, Chapman & Hall/CRC, Boca Raton, FL, 2005. [41] S. Dudoit, R. Gentleman, Bioconductor Basics Tutorial, 2002. Retrieved from: http://www.bioconductor.org/help/course-materials/2002/ Summer02Course/Labs/basics.pdf. [42] L.S. Freedman, V. Fainberg, V. Kipnis, D. Midthune, R.J. Carroll, A new method for dealing with measurement error in explanatory variables of regression models, Biometrics 60 (2004) 172e181. [43] S.R. Cole, H. Chu, S. Greenland, Multiple-imputation for measurementerror correction, International Journal of Epidemiology 35 (2006) 1074e1081. [44] N. Matloff, R for Programmers, 2008. Retrieved from: http://heather.cs. ucdavis.edu/wmatloff/R/RProg.pdf. [45] N. Matloff, Getting Started with the R Data Analysis Package, 2012. Retrieved from: http://heather.cs.ucdavis.edu/wmatloff/r.html. [46] N. Matloff, The Art of R Programming: A Tour of Statistical Software Design, No Starch Press, San Francisco, CA, 2011.

3 Theory of simultaneous mass transfer and chemical reactions, with numerical solutions Chapter outline 3.0 The concept of diffusion 67 3.0.1 Ficks laws of diffusion 68 3.0.1.1 Ficks first law of diffusion (steady state law) 3.0.1.2 Ficks second law of diffusion 70

68

3.1 Simultaneous biomolecular reactions and mass transfer 73 3.2 The concept of the mass transfer coefficient 74 3.3 Theoretical models of mass transfer 74 3.3.1 Nernst One-Film theory model and the Lewis-Whitman Two-Film model 74 3.3.2 Higbie penetration theory model 78 3.3.3 Danckwerts surface renewal theory model 81 3.3.4 Boundary layer theory model 82 3.3.5 Mass transfer under laminar flow conditions 83 3.3.6 Mass transfer past solids under turbulent flow 84 3.3.7 Some interesting special conditions of mass transfer 85 3.3.8 Applications to chemical engineering design 86 3.3.8.1 Designing a packed column for the absorption of gaseous CO2 by a liquid solution of NaOH, using the mathematical model of simultaneous gas absorption with chemical reactions 86 3.3.8.2 Calculation of packed height requirement for reducing the chlorine concentration in a chlorineeair mixture 94

3.4 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions 97 3.4.1 Numerical solutions of a model in terms of simultaneous semilinear partial differential equations 97 3.4.2 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations 97

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science. https://doi.org/10.1016/B978-0-12-819192-7.00003-5 Copyright © 2020 Elsevier Inc. All rights reserved.

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

3.4.3 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations 97 3.5 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions 98 3.5.1 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations 132 3.6 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations 139 A Classical Experimental Study of Simultaneous Absorption of Carbon Dioxide and Ammonia in Water Hatch and Pigford [**1] studied simultaneous gas absorption accompanied by chemical reaction, using the system CO2eNH3ewater. These gases were combined and brought into contact with a laminar water jet. After reaction by a bimolecular reaction, the absorption rates were assessed by analysis of the liquid phase departing the jet absorption equipment. It was found that the rate of absorption of CO2 was significantly affected by reaction with an excess of NH3 in the liquid. (This result is in agreement with predictions based upon the penetration theory for gas absorption of two gases followed by a second order, and irreversible reaction.) [**1] Hatch, Jr., T. F. and Pigford, R. L. (1962). “Simultaneous Absorption Carbon Dioxide and Ammonia in Water”, Ind. Eng. Chem. Fundamen. 1(3), 209e214, August. The very high solubility of NH3 in water results in a high gasside resistance to its absorption, making it difficult to assess the effect of the rate of reaction on its rate of absorption owing to the difficulty in determining the ammonia concentration at the gaseliquid interface, so that the confirmation of the diffusionreaction theory was limited to the observations on CO2 absorption, its validity basing upon the accuracy of the NH3 gas-phase resistance measurements. This classical experimental study illustrates the industrially important cases in which two active gases are simultaneously dissolved in a liquid, diffusing and reacting rapidly near the interface immediately after a reaction occurs. Two important practical examples of this situation are: (i) In the production of synthesis gas: the simultaneous removal of CO2 and H2S from a gas mixture by washing with an alkaline liquid. (ii) In the Solvay process: the simultaneous washing, with water, of a gas mixture containing both NH3 and CO2.

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

In these situations, the rate of absorption of each component depends on the rate of the other component, because the concentration distribution near the interface of one component is affected by its reaction with the other component. The experimental study seemed to show that the results closely followed the penetration theory when the effect of reaction is allowed for with the aid of known reaction mechanisms. The chemical system in this study involves NH3 and CO2 as the gaseous components, and distilled water as the solvent. The chemistry of this system consists of the gases reacting in solution via the following steps: CO2 þ NH3 /NH2 COOH

(R1)

 NH2 COOH þ NH3 /NHþ 4 þ NH2 COO

(R2)

NH2 COO þ H2 O/NH4 CO 3

(R3)

 þ NH4 CO 3 / NH4 þ CO3

(R4)

For this chemical system: Reaction (R1) shows the formation of carbamic acid NH2COOH. Reaction (R2) shows the carbamic acid picking up another NH3 to form ammonium carbamate, which partially ionizes into the  ammonium ion NHþ 4 and the carbamate ion NH2COO .  Reaction (R3) shows the carbamate ions NH2COO slowly react with water to form the ammonium bicarbonate ions NH4CO 3. Reaction (R4) shows the ammonium bicarbonate ions þ NH4CO 3 may then dissociate into ammonium ions NH4 and  carbonate ions CO3 . In this reaction system, ionic equilibria are present: Reaction (R1) is the rate-determining step in the absorption process because reaction (R2) is very fast and irreversible. Taken together, these two reactions consist of the consumption of two molecules of NH3 per molecule of CO2 via the following net reaction:  CO2 þ 2NH3 /NHþ 4 þ NH2 COO

(R5)

which is a second-order reaction rate law since the rate is proportional to the product of the concentrations of CO2 and NH3. Reaction (R3) is slower than the others, and occurs outside the experimental apparatus for the absorption process.

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Physical absorption For purely physical absorption into a laminar liquid phase, the well-known result from the penetration theory is: kL0 ¼ 2OðDA = ptÞ

(3.1)

And, for the experimental jet absorber, in which t is the time of flight of a liquid particle in the jet, it is approximately given by: t ¼ hpd 2 =4qL

(3.2)

where t ¼ elapsed time of exposure of liquid to gas h ¼ jet length d ¼ jet diameter and qL ¼ volumetric flow rate of liquid stream through the nozzle for a jet of uniform diameter, and having a rod-like velocity distribution. For such as ideal jet, the total rate of absorption for the whole surface is: f0A ¼ 4OðDA qL hÞðAi  A1 Þ

(3.3)

where f0A ¼ absorption rate for the whole jet DA ¼ diffusivity of dissolved gas A qL ¼ volumetric flow rate of liquid stream through the nozzle h ¼ jet length Ai ¼ interfacial concentration of A (CO2) in the liquid phase A1 ¼ initial concentration of A (CO2) in the liquid phase Results: The data for both NH3-N2 and NH3-He, in which the NH3 diffusion coefficient was about four times greater, had been satisfactorily correlated by a log-log plot, giving a straight line for an empirical equation:
  0:28 kGB phdPM = qv ðP = RT Þ ¼ 0:76ðDvB h=qv Þ (3.4) where k*GB ¼ average mass transfer coefficient for B d ¼ jet diameter PM ¼ arithmetic-average partial pressure of the inert gas qv ¼ volumetric flow rate of gas mixture into the apparatus P ¼ total pressure R ¼ DB/DA T ¼ temperature DvB ¼ diffusivity of B in the gas phase h ¼ jet length

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Eq. (3.4), the empirical correlation, showed that the Stanton number for gas phase diffusion was slightly influenced by flow, diffusion, or jet length. (The Stanton number for diffusion in the gas phase, St ¼ DvB/ rucp where: r ¼ density of the fluid, u ¼ speed of the fluid, and cp ¼ specific heat of the fluid.) Overall, it was found that the absorption rate for CO2 seems to correspond closely to the penetration theory when the effect of the reaction is assumed by using known reaction mechanisms. 1. Biomolecular reaction occurrences of simultaneous biomolecular reactions and mass transfer are common in many biomedical environments: Some typical examples are [***]: [***] Chan, B. K. C. (2016). “Biostatistics for Epidemiology and Public Health Using R ”, Springer Publishing Company, New York, NY, Available at: www.springerpub.com/chan-biostatistics: “Supplemental Chapter: Research-Level Applications of R .” Research has shown that: A) EZ is an inhibitor of the intestinal absorption of carotenoids, an effect that decreased with increasing polarity of the carotenoid molecule; B) SR-BI is involved in intestinal carotenoid transport; and C) EZ acts not only by interacting physically with cholesterol transporters as previously suggested, but also by downregulating. The gene expression of three proteins is involved in cholesterol transport in the enterocyte, the transporters SR-BI, NPC1L1, and ABCA1. The intestinal transport of carotenoid is thus a facilitated process resembling that of cholesterol; therefore, carotenoid transport in intestinal cells may also involve more than one transporter. Hence, the study of biomolecular reaction and transport is an area of importance in biomedical processes and their occurrences in epidemiologic investigations. In this section, one applies the facilities available in the R environment to solve problems that arise from these processes. This study is being approached from two directions: • Using the R environment as a support to numerical analytical schemes that may be developed to solve this class of problems. • Applying the R functions in the CRAN package ReacTran.[24]

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

2. Some examples in chemical engineering sciences [*]: Simultaneous chemical reactions and mass transfer A large number of highly significant industrial systems consist of simultaneous mass transfer operations and chemical reactions, some of these are: (1) Absorption of carbon dioxide gas into a dilute aqueous solution of sodium hydroxide in a packed column, to remove the CO2 gas from a vapor stream. (2) Absorption of CS2 in aqueous amine solutions for the manufacture of dithiocarbamates. (3) Absorption of O2 in aqueous solutions of CuCl for the conversion to CuCl2 and copper oxychloride. (4) Addition chlorination: Reaction between Cl2 and C2H4 in a C2H4Cl2 medium. (5) Substitution chlorination: Chlorination of many organic compounds such as benzene, toluene, phenols, etc. [*] Danckwerts, P. V. (1970). “Gas-Liquid Reactions”, McGrawHill Chemical Engineering Series, McGraw-Hill Book Company, New York, NY. 3. Some models in the diffusional operations of environmental transport unaccompanied by chemical reactions [**] [**] Choy, B. and Reible, D. D. (1999). “Diffusion Models of Environmental Transport”, CRC Press, New York. In a theoretical model analysis of environmental contamination of liquid media, Choy and Reible [**] provided a similar analysis of the transport of contaminants in a medium. However, in that study, no chemical interactions were considered, and only the movements of the contaminants were modeled. Two principal cases were analyzed: (1) Diffusion models of environmental transport The concentration of the contaminant A, cA, may be found from the solutions of the following equation: vcA =dt ¼ ðDA = Rf Þv2 cA =vx2

x˛½0; NÞ

in which DA ¼ the diffusivity, or dispersion coefficient, of species A Rf ¼ the retardation factor of the solution medium. (2) Advection-diffusion models of environmental transport The concentration of the contaminant A, cA, may be found from the solutions of the following equation: vcA =dt ¼ ðDA = Rf Þv2 cA =x2  ðv = Rf ÞvcA =vx in which

x˛½0; NÞ

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

DA ¼ the diffusivity, or dispersion coefficient, of species A Rf ¼ the retardation factor of the solution medium and n ¼ the Darcy velocity of the mobile phase. For simple model geometries, analytical expressions were obtained for these cases. Remarks: (1) Should chemical reactions take place among the interactive species, and with the liquid medium itself, more complicated reactions will occur. (2) Nevertheless, for such case, solutions may be obtained in the manner discussed in the rest of this book, to provide the appropriate and applicable analytical solutions. This solution may then be used to guide a full understanding of the systems considered.

3.0 The concept of diffusion References: [1] Two-film theory: 1. Whitman, W. G. (1923). Chem. & Met. Eng., 29:147. 2. Lewis, W. K., and Whitman, W. G. (1924). Ind. Eng. Chem., 16: 215. [2] Penetration theory: 1. Danckwerts, P. V. (1950). Trans. Faraday Soc., 46:300. 2. Danckwerts, P. V. (1951). Trans. Faraday Soc., 47:300. 3. Danckwerts, P. V. (1970). “Gas-Liquid Reactions”, McGrawHill, New York. 4. Higbie, R. (1935). J. Am. Inst. Chem. Engrs., 31:365. [3] Surface renewal theory: Danckwerts, P. V. (1951). Ind. Eng. Chem., 43:1460. [4] ceeserver.cee.cornell.edu. [5] Majumder, S.K., and Das, C. (2012). “National Programme on Technology Enhanced Learning (NPTEL) e Mass Transfer Operation I” Dept. of Chem. Eng., IIT. Guwahati, India http://nptel.ac.in/courses/103103035/1. [6] Green, D. W., and Perry, R. H., Editors (2008). “Perry’s Chemical Engineers’ Handbook”, 1/e to 8/e, Section 14: Absorption and Gas-Liquid System Design. McGraw-Hill Professional, New York, 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934, ftp://ftp.feq.ufu.br/Claudio/Perry/DOCS/Chap14.pdf [7] Green, D. W., and Southard, M. Z., Editors (August 17, 2018). “Perry’s Chemical Engineers’ Handbook”, 9/e, McGraw-Hill Professional, New York, 2018

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Publishers’ remarks: Now in its 85th Anniversary Edition, this industry-standard resource has equipped generations of engineers and chemists with vital information, data, and insights. Thoroughly revised to reflect the latest technological advances and processes, Perry’s Chemical Engineers’ Handbook, Ninth Edition provides unsurpassed coverage of every aspect of chemical engineering. You will get comprehensive details on chemical processes, reactor modeling, biological processes, biochemical and membrane separation, process and chemical plant safety, and much more.

This fully updated edition covers: Unit Conversion Factors and Symbols • Physical and Chemical Data including Prediction and Correlation of Physical Properties • Mathematics including Differential and Integral Calculus, Statistics, Optimization • Thermodynamics • Heat and Mass Transfer • Fluid and Particle Dynamics *Reaction Kinetics • Process Control and Instrumentation• Process Economics • Transport and Storage of Fluids • Heat Transfer Operations and Equipment • Psychrometry, Evaporative Cooling, and Solids Drying • Distillation • Gas Absorption and Gas-Liquid System Design • Liquid-Liquid Extraction Operations and Equipment • Adsorption and Ion Exchange • GasSolid Operations and Equipment • Liquid-Solid Operations and Equipment • Solid-Solid Operations and Equipment •Chemical Reactors • Bio-based Reactions and Processing • Waste Management including Air, Wastewater and Solid Waste Management• Process Safety including Inherently Safer Design • Energy Resources, Conversion and Utilization* Materials of Construction.

3.0.1 Ficks laws of diffusion 3.0.1.1 Ficks first law of diffusion (steady state law) JA fðdCA = dZ Þ

(3.1A)

JA ¼  DAB ðdCA = dZÞ

(3.1B)

or

in which JA ¼ the molar flux of component A in the Z direction CA ¼ the concentration of A, Z ¼ the distance of diffusion, and the constant of proportionality, DAB, is the diffusion coefficient, or the diffusivity, of the

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.1 A demonstration of the process of diffusion (Wikipedia).

molecule A in B. This is valid only at steady state condition of diffusion. Eq. (3.1B) is called Fick’s first law of diffusion. Illustrating Fick’s laws of diffusion For the technique of measuring cardiac output, see Fick’s principle. Fick’s laws of diffusion describe diffusion and were derived by Fick in 1855 (Fig. 3.1). They can be used to solve for the diffusion coefficient, D. Some remarks on Fick’s first law of diffusion Fick’s first law relates the diffusive flux to the concentration under the assumption of steady state J ¼  D d4=dx

(3.2)

where • J is the “diffusion flux” • D is the diffusion coefficient or diffusivity • 4 (for ideal mixtures) is the concentration, of which the dimension is amount of material per unit volume. It may be expressed in units of mol/m3 • x is the position coordinate, the dimension of which is length.

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In two or more dimensions one may use V, the del or gradient operator, which generalizes the first derivative, obtaining J ¼  DV4

(3.3)

where J denotes the diffusion flux vector. Then Fick’s first law (one-dimensional case) may be expressed as: Ji ¼  ðDci = RT Þðvmi = vxÞ

(3.4)

where the index i denotes the ith species. c is the concentration (mol/m3), R is the Universal Gas Constant [J/(K$mol)], T is the absolute temperature (K) and m is the chemical potential (J/mol). If the primary variable is mass fraction (yi, given, for example, in kg/kg), then the equation changes to: Ji ¼  rDVyi

(3.5)

where r is the fluid density (for example, in kg/m3).

3.0.1.2 Ficks second law of diffusion Fick’s second law predicts how diffusion causes the concentration to change with time. v4=vt ¼ Dv2 4=vx2

(3.6)

where 4 is the concentration in dimensions of [(amount of substance) length3] 4 ¼ 4(x,t) is a function that depends on location x and time t t is time (s) D is the diffusion coefficient in dimensions of (length2 time1), Example: m2/s, x is the position (length). In two or more dimensions, one should use the Laplacian D ¼ V2, which generalizes the second derivative: v4=vt ¼ DD4

(3.7)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Derivation of Fick’s laws of diffusion Reference: Berg, H. C. (1977). “Random Walks in Biology”, Princeton. Remarks: Additional Remarks on Fick’s Laws of Diffusion: Fick’s First Law of Diffusion Consider particles performing a random walk in one dimension with length scale Dx and timescale Dt. Let N(x,t) be the number of particles at position x at time t. At a given time step, half of the particles would move left and half would move right. Since half of the particles at point x move right, and half of the particles at point x þ Dx move left, the net movement to the right is: ½ 1=2fN ðx þ Dx; tÞ  Nðx; tÞg The flux, J, is this net movement of particles across some area element of area a, normal to the random walk during a time interval Dt. Hence one may write: J ¼ ½ 1=2fN ðx þ Dx; tÞ  N ðx; tÞg = ðaDtÞ ¼  1=2½fN ðx þ Dx; tÞ = ðaDtÞg  fN ðx; tÞ = ðaDtÞg

(3.8)

Multiplying the top and bottom of the right-hand side by (Dx)2 and rewriting, one obtains: ohn n o n oi N ðx þ Dx; tÞ = aðDxÞ2  N ðx; tÞ = aðDxÞ2 J ¼ ðDxÞ2 = 2Dt (3.9) Defining concentration as the number of particles per unit volume, viz. 4ðx; tÞ ¼ N ðx; tÞ=aDx

(3.10)

2

Moreover, (Dx) /2Dt is the definition of the diffusion constant in one dimension, D. Thus, the expression simplifies to J ¼ D½f4ðx þ Dx; tÞ = Dxg  f4ðx; tÞ = Dxg

(3.11)

In the limit when Dx approaches an infinitesimal limit, the right-hand side becomes a space derivative: J ¼  Dðv4 = vxÞ

(3.12)

Fick’s second law of diffusion Fick’s second law of diffusion may be obtained from Fick’s first law of diffusion together with the property of mass conservation in the absence of any chemical reactions: v4=vt þ vJ=vx ¼ 00v4=vt  v=vxðDv4 = vxÞ ¼ 0   v=vxðDv4 = vxÞ ¼ Dðv = vxÞðv = vxÞ4 ¼ D v2 4 = vx2

(3.13) (3.14)

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

and thus receive the form of Fick’s equations as stated above. For the case of diffusion in two or more dimensions, Fick’s second law becomes v4=vt ¼ DV2 4

(3.15)

If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick’s second law yields v4=vt ¼ V$ðDV4Þ

(3.16)

In two or more dimensions one obtains V2 4 ¼ 0

(3.17)

which is the Laplace equation, the solutions to which are known as harmonic functions. A derivation of Fick’s second law of diffusion Fick’s second law of diffusion is a special case of the convection-diffusion equation in which there is no advective flux and no net volumetric source, viz., in the absence of any chemical reactions. It may be derived from Fick’s first law of diffusion and the mass conservation continuity equation: v4=vt þ vJ=vx ¼ 00v4=vt  ðv = vxÞðDv4 = vxÞ ¼ 0

(3.18)

Assuming the diffusion coefficient D to be constant, one may exchange the orders of the differentiation and multiply by the constant   ðv = vxÞðDv4 = vxÞ ¼ Dðv = vxÞðv = vxÞ4 ¼ D v2 4 = vx2 (3.19) and thus result in the form of the equation for Fick’s second law of diffusion: v4=vt ¼ DV2 4

(3.20)

(This is analogous to the heat conduction equation.) If the flux were the result of both diffusive and advective fluxes, then the convection-diffusion equation is the result. Example solution in one dimension: diffusion length A simple case of diffusion with time t in one dimension (taken as the x-axis) from a boundary located at position: x ¼ 0, where the concentration is maintained at the value n0, is nðx; tÞ ¼ n0 erfcfx = 2OðDtÞg

(3.21)

where erfc{ } is the complementary error function. This models the case when corrosive gases diffuse through the oxidative layer

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

toward the metal surface, if it is assumed that concentration of gases in the environment is constant and the diffusion space (i.e., corrosion product layer) is semi-infinitedstarting at 0 at the surface and spreading infinitely deep in the material. If the diffusion space is infinite: nðx; 0Þ ¼ 0; x > 0

(3.22)

nðx; 0Þ ¼ n0 ; x  0

(3.23)

and

then the solution is amended only with coefficient 1⁄2 in front of n0 (this is quite obvious, as the diffusion now occurs in both positive and negative directions). This case is valid when some solution with concentration n0 is put in contact with a layer of pure solvent. The length 2O(Dt) is called the diffusion length, and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t. As an approximation of the error function, the first two terms of the Taylor series may be used: nðx; tÞ ¼ n0 ½1  2fx = 2OðDtpÞg

(3.24)

If D is time-dependent the diffusion length becomes Rt 2Of 0 Dðt 0 Þdt 0 g This idea is useful for estimating a diffusion length over a heating and cooling cycle, where D varies with temperature.

3.1 Simultaneous biomolecular reactions and mass transfer In epidemiologic investigations, occurrences of simultaneous biomolecular reactions and mass transfer are common in many biomedical environments. Some examples are: (1) Intestinal drug absorption involving bio-transporters and metabolic reactions with enzymes[21] Oxygen Transport via Metal Complexes.[22] (2) Carotenoid Transport in the Lipid Transporters SR-BI, NPC1L1, and ABCA1.[23]

73

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

3.2 The concept of the mass transfer coefficient Movement of the bulk fluid particles in the turbulent condition is, as in any turbulent fluid dynamics environment, highly complex.

3.3 Theoretical models of mass transfer 3.3.1 Nernst One-Film theory model and the Lewis-Whitman Two-Film model References: (1) ceeserver.cee.cornell.edu (2) Majumder, S.K., and Das, C., Dept. of Chemical Engineering, Indian Institute of Technology, Guwahati, Guwahati-781039, Assam, India “National Programme on Technology Enhanced Learning (NPTEL) e Phase II, Course Name: Mass Transfer Operation- I00 http://nptel.ac.in/courses/103103035/1 In chemical engineering analysis of mass transfer, the film theory is based on the following assumption: Mass transfer occurs by molecular diffusion through a fluid layer at the phase boundary. Gas transfer rates If either phase concentration cannot be predicted by Henry’s law then there will be a transfer of mass across the interface until equilibrium is reached. One common conceptualization method is the film theory, or the one-film theory or two-film theory: The Nernst one-film model A simple model of the gaseliquid transfer process is attributed to Nernst (1904): J ¼ DvC=vX

(3.25)

2

(typical units: mg/cm $s), where D is the diffusivity or the diffusion coefficient. If the thickness of the stagnant film is given by dn, then the gradient may be approximated by: vC=vX z ðCb  Ci Þ=dn

(3.26)

in which Cb and Ci are concentrations in the bulk and at the interface, respectively. At steady-state, if there are no reactions in the stagnant film, then there will be no accumulation in the film

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

(assuming that D remains constant). Hence the gradient must be linear and the approximation is appropriate, and: J ¼  DðCb  Ci Þ=dn

(3.27)

Mass transfer coefficients To proceed with calculations of the mass transfer coefficients, define a mass transfer coefficient for either the liquid or gas phase as kl or kg (with dimensions ¼ L/T ¼ LT1): kl;g ¼ D=dn

(3.28)

Thus, at this stage of the analysis: Ci ¼ Cg =Hc

(3.29)

(if the film side is liquid and the opposite side is the gas phase). The Lewis-Whitman two-film model In cases with gaseliquid transfer, one may have transfer considerations from both sides of the interface. In such cases, one may use the Lewis-Whitman (1923) two-film model as described here: Calculation of Ci is done by assuming that equilibrium (viz., Henry’s law) is attained instantly at the interface (i.e., using Henry’s law based on the bulk concentration of the other bulk phase). This assumes that the other phase does not have a “film.” For the moment: Ci ¼ Cg =Hc

(3.30)

(if the film side is liquid and the opposite side is the gas phase). The two-film model In many cases with gaseliquid transfer one has transfer considerations from both sides of the interface. Therefore, one may use the Lewis-Whitman two-film model as described below (Fig. 3.2). The same assumptions will be applied to the two-film model as apply in the single Nernst film model. The problem is that one may now have difficulty in finding the interface concentrations, Cgi or Cli. However, one may assume that equilibrium will be attained at the interface (since, generally, gas solubilization reactions occur rather fast), so that: Cli ¼ Cgi =Hc

(3.31)

A steady-state flux balance (assumed for thin films) through each film may now be performed. The fluxes are then given by: J ¼ kl ðCl  Cli Þ

(3.32A)

75

76

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.2 The Lewis-Whitman two-film model for mass transfer with chemical reactions (Wikipedia).

and J ¼ kg ðCgi  Cg Þ

(3.32B)

assuming that the flux is in the X direction, viz., from bulk gas phase to bulk liquid phase. Now, define:   J ¼ Kl Cl  Cl (3.33A)   (3.33B) J ¼ Kg Cg  Cg where. Kl ¼ overall mass transfer coefficient based on liquid-phase concentration. Kg ¼ overall mass transfer coefficient based on gas-phase concentration. Note: Both Kl and Kg have dimensions L/T, or LT1. Cl* ¼ liquid phase concentration that would be in equilibrium with the bulk gas concentration. ¼ Cg/Hc (typical dimensions are moles/m3). Cg* ¼ gas phase concentration that would be in equilibrium with the bulk liquid concentration. ¼ HcCl (typical dimensions are moles/m3). Expanding the liquid-phase overall flux equation to include the interface liquid concentration:

Chapter 3 Theory of simultaneous mass transfer and chemical reactions


 J ¼ Kl $ ½Cl  Cli  þ Cli  Cl

(3.34)

Then substituting Cli ¼ Cgi =Hc

(3.35A)

Cl ¼ Cg =Hc

(3.35B)

and

one obtains: J ¼ Kl fðCl  Cli Þ þ ðCgi  Cg Þ = Hc g

(3.36)

In the steady-state, fluxes through all films must be equal. Let all these fluxes be equal to J. On an individual film basis: ðCl  Cli Þ ¼ J=kl

(3.37A)

ðCgi  Cg Þ ¼ J=kg

(3.37B)

and

Since all the Js are equal: J ¼ Kl fðJ = k1 Þ þ ðJ = Hc $ kg Þg

(3.38)

which may can be rearranged to give: ð1 = Kl Þ ¼ ð1 = kl Þ þ ð1 = Hc $ kg Þ

(3.39A)

A similar analysis starting with the overall flux equation based on gas phase concentration will give: ð1 = kg Þ ¼ ðHc = kl Þ þ ð1 = kg Þ

(3.39B)

These last two equations [(3.39A) and (3.39B)] may be viewed as “resistance” expressions, where 1/kg, or 1/kl, represent total resistance to mass transfer based on gas or liquid phase concentration, respectively. The total resistance to mass transfer consists of three resistances in series: (1) Liquid film, (2) Interface, and (3) Gas film. In this model analysis, it was assumed that there was instant equilibrium at the interface so there is no transfer limitation here.

77

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Single film control It is possible that one of the films exhibits relatively high resistance and therefore dominates the overall resistance to transfer. This, of course, depends on the relative magnitudes of kl, kg, and Hc. Thus, the solubility of the gas and the hydrodynamic conditions which establish the film thickness or renewal rate (in either phase) determine if a film controls. In general, highly soluble gases (low Hc) have transfer rates controlled by gas film (or renewal rate) and vice versa. For example, (i) oxygen (slightly soluble) transfer is usually controlled by a liquid film (ii) ammonia (highly soluble) transfer is usually controlled by a gas phase film. Applications Transfer of gas across a gaseliquid interface may be accomplished by bubbles or by creating large surfaces (interfaces).

3.3.2 Higbie penetration theory model Many of the industrial processes of mass transfer with chemical reactions (such as gas absorption) are unsteady state processes where the gaseliquid contact time is too short to achieve a steady-state condition. For example, in the absorption of gases from bubbles or absorption by wetted-wall columns, the mass transfer surface is formed instantaneously and transient diffusion of the material takes place (see Fig. 3.3). Assume that a bubble is rising in a pool of liquid (as in a bubble column) where the liquid elements are swept on its surface, remain in contact with it during their motion and finally get detached at the bottom: Inside a Gas Absorption Column The assumptions of the penetration theory of mass transfer are: • Unsteady state mass transfer occurring in a liquid element as long as it is in contact with the gas bubbles • Equilibrium existing at the gaseliquid interface • Each liquid element staying in contact with the gas for the same period of time (the liquid elements are moving at the same rate and there is no velocity gradient within the liquid). Under these assumptions,   vc=vt ¼ DAB v2 c = vz 2 (3.40)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

79

Figure 3.3 The penetration model for mass transfer with chemical reactions.

subject to the following boundary conditions: t ¼ 0, z > 0: c ¼ cAb, and t > 0, z ¼ 0: c ¼ cAi where. cAb ¼ The concentration of solute A at infinite distance from the surface (viz., the bulk concentration) cAi ¼ The interfacial concentration of solute A at the surface On solving the above partial differential equation, the result is: ðcAi  cÞ=ðcAi  cAb Þ ¼ erffz = 2OðDAB tÞg

(3.41)

in which erf{} is the error function. • Some remarks on the error function erf(x) In error function (also known as the Gaussian error function) is a nonelementary function of sigmoid shape that occurs in the theory of probability, statistics, and partial differential equations when describing diffusion. It is defined as follows:[1][2] • In mathematical statistics, for nonnegative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and variance 1/2, erf(x) describes the probability of Y falling in the range [x, x].

80

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.4 A plot of the error function erf(x) versus x (Wikipedia).

The function / e1 and / þ1, asymptotically, for x / eN and þN, respectively, and passes through the point (0, 0) (Fig. 3.4). Remarks: (1) The name “error function,” and the abbreviation for the error function (and the error function complement) were developed from “the theory of probability, and notably the theory of errors,” for the “law of facility of errors,” viz, the normal distributiondwhose density is given by   1=2 f ðxÞ ¼ ðc=pÞ exp cx2 (3.42) The chance of an error lying between p and q is Z q   1=2 ðc=pÞ exp cx2 dx ¼ 1=2½erfðqOcÞ  erfðpOcÞ

(3.43)

p

(2) Complementary error function The complementary error function, denoted as erfc(x), is defined as erfcðxÞ ¼ 1  erfðxÞ Z N 2 2 ¼ pffiffiffi et dt p x ¼ ex erfcxðxÞ; 2

(3.44)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Now, if mass transfer is a unidirectional diffusion and the surface concentration is very low, viz., CAb z 0

(3.45)

then the mass flux of solute A, viz., NA (kgm2 s1) may be calculated by the following equation: NA ¼ f rDAB = ð1  cAb Þgðvc=vzÞz¼0 z  rðvc=vzÞz¼0

(3.46)

• The mass transfer coefficient kL: Thus, the rate of mass transfer at time t is given by the following equation: NA ðtÞ ¼ fOðDAB = ptÞgðcAi  cAb Þ

(3.47)

And the instantaneous mass transfer coefficient kL is given by kL ðtÞ ¼ OðDAB = ptÞ

(3.48)

since the instantaneous mass transfer coefficient kL is defined by the following equation: NA ðtÞ ¼ kL ðtÞðcAi  cAb Þ

(3.49)

The time-averaged mass transfer coefficient during a time interval tc is obtainable by integrating Eq. (3.49) over one time period from t ¼ 0 to t ¼ tc, as follows: Z tc (3.50) kðtÞdt ¼ 2OðDAB = ptcÞ kL;av ¼ ð1 = tcÞ 0

Thus, from the above equation, the time-averaged mass transfer coefficient is proportional to the square root of the diffusivity. This was first proposed by Higbie in 1935 and the theory is also known as the Higbie penetration theory.

3.3.3 Danckwerts surface renewal theory model The surface renewal theory model for mass transfer is an unsteady-state theory where molecules of solute are in random motion. It assumes that these solute molecules form a cluster which goes to the interphase where they remain for some unspecified time which basically depends on experiments. Some of the solute then pass through this interphase while the rest mixes back to the bulk. Here the specific term is uncertain time of contact.

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Chapter 3 Theory of simultaneous mass transfer and chemical reactions

The surface-renewal theory has been developed so that the deficiencies of the first two theories can be rectified through incorporation of some statistical components into the description of interphasal mass or heat transport. For mass transfer in the liquid phase, Danckwerts (1951) modified the Higbie penetration theory as follows: assume that a portion of the mass transfer surface is replaced with a new surface by the motion of eddies near the surface and applied the following assumptions: (1) The liquid elements at the interface are randomly swapped by fresh elements from bulk liquid (2) Each of the liquid elements at the surface has the same probability of being substituted by fresh elements (3) Unsteady-state mass transfer occurs at the element during its sojourn at the interface. Hence, the average molar flux, NA,av, is NA;av ¼ ðCAi  CAb ÞOðsDAB Þ

(3.51)

Comparing Eq. (3.51) with Eq. (3.49) one obtains Eq. (3.52): kL;av ¼ OðsDAB Þ

(3.52)

where s is fraction of the surface renewed in unit time, i.e., the rate of surface renewal (s1).

3.3.4 Boundary layer theory model Boundary layer theory does take into account the hydrodynamics or the flow field that characterizes a system. Fluidefluid interfaces Fluidefluid interfaces are typically not fixed and are strongly affected by the flow leading to heterogeneous systems that make it difficult to develop a general theory to support the mass transfer correlations. Fluidesolid interfaces Solids typically have fixed and well-defined surfaces, allowing one to develop theoretical models for the empirical mass transfer correlations In contrast to fluidefluid interfaces the models are far more rigorous but, unfortunately, computationally intensivedas shown in Section 3.4. Example: Prandtl’s experimental mass transfer from a flat plate* As an example, consider the classical case of mass transfer from a “flat plate” configuration: a sharp-edged, flat plate that

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

83

Figure 3.5 Prandtl: Flat plate experimentdthe concept of boundary layers. ** Schlichting, H., and Gersten, K. (1999).

is sparingly dissolvable, is immersed in a rapidly flowing solvent (Fig. 3.5). * Bird, R. B., Steward, W. E., and Lightfoot, E. N. (2011). “Transport Phenomena”, 2nd ed., Wiley. ** Schlichting, H., and Gersten, K. (1999). “Boundary Layer Theory”, 8th ed., Springer. L: plate length; UN: bulk fluid velocity From experimental data, using dimensionless parameters, the following mass transfer correlation was established (Figs. 3.6 and 3.7): NSh¼ 0:646ðNRe Þ

1=2

ðND Þ

1=3

(3.53)

where NSh ¼ the Sherwood Number ¼ kL/D NRe ¼ the Reynolds Number ¼LUN/n ND ¼ the diffusivity number ¼ n/D L ¼ the plate length UN ¼ the bulk fluid velocity D ¼ the diffusivity of the material from the plate into the bulk fluid n ¼ the diffusivity/viscosity of the fluid

3.3.5 Mass transfer under laminar flow conditions Mass transfer coefficient does not play a large role in laminar flow conditions as molecular diffusion exists there. In the laminar flow

84

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.6 The flat plate experiment for boundary layer mass transfer for a flat plate (1 of 2). ** Schlichting, H., and Gersten, K. (1999).

Figure 3.7 The flat plate experiment for boundary layer mass transfer for a flat plate (2 of 2). ** Schlichting, H., and Gersten, K. (1999).

regime, the average liquid phase mass transfer coefficient, kL,av is correlated with Sherwood number (NSh) and DAB as follows: kL;av ¼ 3:41ðDAB = dÞ

(3.54)

NSh ¼ kL;av d=DAB ¼ 3:41

(3.55)

3.3.6 Mass transfer past solids under turbulent flow Mass transfer under flow past solids is important in practical applications. Several theories have attempted to clarify the behavior of the corresponding mass transfer coefficients. All the theories have some assumptions and some drawbacks. Hence, they are revised frequently. In turbulent flow medium,

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

small fluids in element of different sizes, called eddies, move randomly. These eddies form and interact among others and disappear in the flow path. The total molar flux of a solute “A” owing to molecular diffusion and eddy diffusion, JA is as follows: JA ¼  ðDAB þ DE ÞdCA =dZ

(3.56)

where DE is the eddy diffusivity. (Eddy diffusivity depends on the intensity of local turbulence.)

3.3.7 Some interesting special conditions of mass transfer A. Equimolar counter-diffusion of A and B (NA ¼ e NB) Now, for gas phase diffusion, it is known that NA ¼ kG' ðpA1  pA2 Þ ¼ ky' ðyA1  yA2 Þ ¼ kc' ðCA1  CA2 Þ

(3.57)

/Gas Phase NA ¼ kx' ðxA1  xA2 Þ ¼ kL' ðCA1  CA2 Þ

(3.58)

/Liquid Phase For gas phase diffusion, it is known that NA ¼ DAB ðpA1  pA2 Þ=RT d

(3.59)

Equating Eq. (3.57) and Eq. (3.59), the result is kG' ¼ DAB =RT d

(3.60)

Again, NA ¼ DAB ðpA1  pA2 Þ=RT d ¼ DAB PðyA1  yA2 Þ=RT d

(3.61)

Equating Eq. (3.57) and Eq. (3.61), kG' ¼ DAB =RT dy

(3.62)

Also, NA ¼ DAB ðpA1  pA2 Þ=RT d ¼ DAB ðCA1  CA2 Þ=d

(3.63)

Equating Eq. (3.57) and Eq. (3.63), kc' ¼ DAB =d

(3.64)

B. For liquid phase diffusion, the flux may be written as:

85

86

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

NA ¼ ðDAB = dxBM Þðr=MÞav ðxA1  xA2 Þ

(3.65)

xBM ¼ ðxB2  xB1 Þ=lnðxB2 = xB1 Þ

(3.66)

in which,

Equating Eq. (3.58) and Eq. (3.65), one obtains kx' ¼ ðDAB = dxBM Þðr=MÞav

(3.67)

NA ¼ ðDAB = dÞðr=MÞav ðxA1  xA2 Þ ¼ ðDAB = dÞðCA1  CA2 Þ

(3.68)

NA ¼ ðDAB = d xBM Þðr = MÞavðxA1  xA2 Þ ¼ ðDAB = dxBM ÞðCA1  CA2 Þ

(3.69)

Also

Also,

Equating Eq. (3.57) and Eq. (3.69) kL ¼ DAB =dxBM

(3.70)

C. Conversion formulas for mass transfer coefficients in different forms: kc ¼ RTkG

(3.71A)

ky ¼ PkG

(3.71B)

kx ¼ ðr=MÞav kL

(3.71C)

3.3.8 Applications to chemical engineering design 3.3.8.1 Designing a packed column for the absorption of gaseous CO2 by a liquid solution of NaOH, using the mathematical model of simultaneous gas absorption with chemical reactions Special reference: Olutoye, M. A., and Mohammed, A. (2006). “Modeling of a Gas-Absorption Packed Column for Carbon Dioxide-Sodium Hydroxide System,” AU J. T. 10(2): 132e140, October. https://www. researchgate.net/publication/254071709 Modeling of a Gas-Absorption Packed. (PDF Download Available). Available from: https://www.researchgate.net/publication/ 254071709_Modelling_of_a_Gas-Absorption_Packed_Column_ for_Carbon_Dioxide-Sodium_Hydroxide_System (accessed Apr 23, 2018).

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Abstract This paper presents a research work on the modeling of a gas absorption packed column with the aim of formulating a mathematical model and simulation of the model using a computer software to obtain the rate of absorption and the amount of absorbed carbon dioxide (CO2) into dilute sodium hydroxide (NaOH). Arm field gas absorption column titration techniques were used for the analysis. The total concentration of carbonate and hence the amount of carbon dioxide absorbed was determined using gravimetric methods. The comparison between the experimentally obtained and simulated results shows that the formulated model is a good representation of the system. (A program was written in Basic for simulation of any gas absorption packed column, the model and computer program could therefore be used in determining the condition in a counter-current continuous column at any given time, for any chemically reacting absorption system.) Description and operation of the gas absorption column A gas absorption experiment is usually carried out in a vertical counter-current packed column, about 75 mm in diameter, in which there are two lengths of branching. The liquid solvent is fed at the top of the column and is distributed over the surface of the packing either by nozzle or distribution plates. Pressure tappings are provided at the base, center, and top of the column to determine pressure drops across the column. Sampling points are also provided for the gas at these same three points. The liquid outlet stream and feed solution are also equipped with a sampling point. Suitable manometer measurement is included. Water/solvent is taken from a sump tank, and pumped to the column via a calibrated flow meter. Gas is taken from a pressure cylinder through a calibrated flow meter, and mixed with air, supplied and monitored from a small compressor in a predetermined (but variable) mixed ration. The mixture is fed to the base of the tower, in which a liquid seal is provided. The effluent gas leaves the top of the column and is intended to be exhausted to the atmosphere outside the laboratory building. The apparatus is designed to absorb a CO2eair mixture into an aqueous solution flowing down the column. Gas analysis is provided for this system (see Fig. 3.8). Derivation of a mathematical model for an arm field gas absorption column: overall material balance The material balance equation for the main component contains the following terms: Gt ¼ Gd þ Gr þ Ga

(3.73)

87

88

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.8 Gas absorption packed column.

where: Gt is the amount of the component fed to the elementary volume per unit time; Gd is the amount of the component out from the elementary volume per unit time; Gr is the rate of consumption of the component in the chemical reaction in the elementary volume; and Ga is the rate of accumulation of the component in the elementary volume. This may be written for the whole reactor if there is uniform distribution of materials over the reactor volume. Differential material balance The material balance of “A” (CO) and “B” (NaOH) is obtained by noting that of the two reactants, A alone is present in the main body of gas and B alone is present in the main liquid. Secondly, for each mole of “A” reacted, “b” moles of “B” are consumed. In this case, for each mole of CO2 reacted, 2 moles of NaOH are consumed: CO2 þ 2NaOH/Na2 CO3 þ H2 O

(3.74A)

given by A þ bB / Products, b ¼ 2. Referring to Fig. 3.9, AðgasÞ þ bBðliquidÞ ðfastÞ/Product

(3.74B)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Figure 3.9 Material balances.

However, a material balance about the end of the packed column gives composition at any point in the column: Now: ½A’lost by gas ¼ ð1 = bÞ½B’lost by liquid

(3.75)

or GdYA ¼ ðL = bÞdXB ¼  Ld½CB = CU  ¼ d½G’PA = p ¼ ð1 = bÞd½L’CB = CT 

(3.76)

89

90

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Compositions at any point in the tower are found in terms of the end conditions by integrating the material balance equation. Thus G½YA  YAI  ¼  L½ðXB  XBI Þ = b ¼ G½ðPA = PU Þ  ðPAI = PUI Þ ¼  ðL = bÞ½ðCB = CU Þ  ðCBI = CUI Þ ¼ ðG”PA = pÞ  ðG”PAI = pÞ ¼ ðI = bÞ  ½ðL’CB = CT Þ  ðL’CBI = CTI Þ

(3.77)

The above expressions give the concentrations of reactants A and B in the phases throughout the tower, PU z pCU z CT

(3.78)

Eqs. (3.77) and (3.78) for differential material balance become GDPA =bCT ¼  LdCB ;

(3.79)

and for point conditions. GðPA  PAI Þ ¼  ðL = bCT ÞðCB  CBI Þ

(3.80)

Height of tower equation The height of the tower (column) is described by combining rate equation and material balance for a differential element of tower and is given by: h ¼ gA ¼ ð1 = KAg Þ þ HA=OðALKCB Þ

(3.81)

rearranging and integrating, in terms of A and B. For a specific system: dQ=dt ¼ DvC=vX .

(3.82)

This holds for the diffusion of unreacted solute through the medium, and also that the mass of solvent per unit volume of the medium is virtually constant throughout. Let C* be the saturated concentration of the solute reigning at the surface (X ¼ 0); C ¼ concentration at a distance X below the surface; t ¼ time; and k ¼ velocity constant of the reaction between solute and medium. The initial concentration of solute in the medium is uniformly equal to zero. Hence, the rate at which the solute crosses a unit area of any plane of c constant X in the direction of increasing X is ðdQ = dtÞ ¼ DvC=vX

(3.83)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Consider the element of volume of unit cross-sectional area between planes X and X þ dX. The following changes in its content of solute occur in timidity: DiffusionðinÞ ¼  ðDvC = vXÞdt;
    DiffusionðoutÞ ¼  Ddt ðvC = vX Þ þ v2 C = vX 2 dX ; Reacting ¼  kcdt$dx;
 Net Increase ¼ Dv2 C = vX 2  kC dt$dx

(3.84) (3.85) (3.86) (3.87)

If the increase in concentration is dc, then the net increase above may be equated to dc/dx, giving dc=dt ¼ Dv2 C=vx2  KC.

(3.88)

With the boundary conditions: C ¼ C  ; x ¼ 0; t > 0; C ¼ 0; x > 0; t ¼ 0;

(3.89)

C ¼ 0; x ¼ N; t > 0 the solution to Eq. (3.89) is C=C  ¼ 1=2exp½ X OðK =DÞ

erfc

;

(3.90)

where erfc ¼ 1  erfðzÞ; and

Z z ¼ 1  ð2 = OpÞ

z

  exp y 2 dy

(3.91)

(3.92)

0

Differentiating Eq. (3.91), vc=vxðvc=vxÞx¼0 ¼  C  OðD = kÞ½erfOðKtÞ þ fexpðktÞ = OðpktÞg (3.93) The rate of absorption dQ/dt per unit area of surface becomes dQ=dt ¼  Dðvc=vxÞx¼0 ¼ C  OðD = kÞ½erfOðKtÞ þ fexpðktÞ = OðpktÞg

(3.94) (3.95)

The quantity Q absorbed in time ‘t’ is Q ¼ C  OfðD = K Þ½ðKt þ 1=2ÞexfOðktÞ þ Ofðkt = pÞekt Þgg

(3.96)

91

92

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

when Kt is sufficiently large, erf ¼ O(Kt) z 1, and ðC = C  Þ z expf xOðK = DÞg

(3.97)

ðvQ = vxÞdQ z C  OðDK Þ

(3.98)

Q ¼ C  OðD = K ÞðKtþ 1=2Þ.

(3.99)

Then

and

Determination of the absorption acceleration factor in the case of a chemical reaction “ε” If aA(g) þ bB(l) / Product, the true rate equation is urA ¼ KCA CB

(3.100)

which could also be written as urA ¼ K 1 CA

(3.101)

K 1 ¼ KCBn

(3.102)

where

As a result, the individual mass transfer coefficient from the gas to the liquid phase is increased, WA ¼ DgradCA which can be expressed as      WA ¼ bi CAb  CA1 cos h M 1=2 M 1=2 tan h M 1=2

(3.103)

(3.104)

where bi ¼ D=d ¼ ε

(3.105)

which describes absorption accompanied by a chemical reaction and differs from the equation WA ¼ bi ðCA  CAL Þ

(3.106)

which describes physical absorption. When a fast chemical reaction takes place in the film and CAL ¼ 0, Eq. (3.106) takes a simpler form:  (3.107) uA ¼ bCAB M 1=2 = tan h M 1=2

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

The absorption accelerator factor ε may be found by equating Eqs. (3.107) and (3.103) at CAL ¼ 0. Here, ε is the factor in the equation describing absorption accompanied by a chemical reaction. For the physical absorption equation: at CAL ¼ 0,   ε ¼ ðbCAb = b1 CAb ÞM 1=2 tan h M 1=2 (3.108)   ¼ M 1=2 tan h M 1=2 or at M1/2 >> 1 (because then tan h M1/2 tends to unity), ε¼ ¼ ¼

M 1=2  1=2 DK =B2 L 1=2 ðKDÞ = ðb2 LÞ

(3.109)

Thus, it is seen that an increase in the reaction rate constant K leads to greater absorption acceleration factors ε. The predictive model equation for the quantity of absorbed CO2 in a given time (t) by diffusion accompanied by chemical reaction is given by Q ¼ C  OðD = K Þ½ðKt þ 1=2ÞerfOðKtÞ þ OfKt = pÞexpðktÞg Q ¼ C  OðD = K ÞðKt þ 1=2Þ

(3.110) (3.111)

From these expressions, one would expect the Qv,t plot to be linear with a slope of C*O(KD) until the depletion of the OHe near the surface, which disturbs the first-order character of the reaction and makes it impossible to deduce the velocity reaction constant from the observation. Conclusions Mathematical modeling of a gas absorption packed column carbon dioxide sodium hydroxide system had been performed, and a model equation was developed and tested. The result obtained shows that it can be used to predict the quantity of CO2 absorbed in a given time. And the Qv,t plot cannot be linear as expected due to the depletion of OH near the surface which disturbed the first-order characteristics of the reaction and makes it impossible to deduce the reaction velocity constant.

93

94

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

3.3.8.2 Calculation of packed height requirement for reducing the chlorine concentration in a chlorineeair mixture References: James, R., Fair, Ph.D., P.E., Professor of Chemical Engineering, University of Texas; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society, American Society for Engineering Education, National Society of Professional Engineers (Section Editor, Absorption, Gas-Liquid Contacting) D. E. Steinmeyer, M.A., M.S., P.E., Distinguished Fellow, Monsanto Company; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society (Liquid-in-Gas Dispersions) W. R. Penney, Ph.D., P.E., Professor of Chemical Engineering, University of Arkansas; Member, American Institute of Chemical Engineers (Gas-in-Liquid Dispersions) B. B. Crocker, S.M., P.E., Consulting Chemical Engineer; Fellow, American Institute of Chemical Engineers; Member, Air Pollution Control Association (Phase Separation) Section 14 Gas Absorption and GaseLiquid System Design: ftp://ftp.feq.ufu.br/Claudio/Perry/DOCS/Chap14.pdf Packed Tower Design Methods for calculating the height of the active section of countercurrent differential contactors such as packed towers, spray towers, and falling-film absorbers are based on rate expressions representing mass transfer at a point on the gaseliquid interface and on material balances representing the changes in bulk composition in the two phases that flow past each other. The rate expressions are based on the interphase mass-transfer principles. Combination of such expressions leads to an integral expression for the number of transfer units (NTU) or to equations related closely to the number of theoretical plates. The following discussions describe a direct method for using such equations, firstly in a general case, and then for cases in which simplifying assumptions are valid. Use of mass transfer-rate expression (Fig. 3.10) shows a section of a packed absorption tower together with the nomenclature that will be used in developing the equations which follow. In a differential section dh, one may equate the rate at which solute is lost from the gas phase to the rate at which it is transferred through the gas phase to the interface as follows: dðGM yÞ ¼ GM dy  y dGM ¼ NA a dh

(3.112)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

GM2 LM2 Y2

x2

↑

↓

_______ _______ y

Packed Tower

x

GM LM

dh

↑

↓

y1 x1 GM1

LM1

Figure 3.10 Mass balances in a packed tower stripper or absorber.

When only one component is transferred, dGM ¼  NA a dh

(3.113)

Substitution of this relation into Eq. (3.112) and rearranging yields dh ¼  GM dy=½NA að1  yÞ

(3.114)

For this derivation, one uses the gas-phase rate expression NA ¼ kGðy  yi Þ and integrates over the tower to obtain Z y1 ½GM = fkG að1  yÞðy  yiÞgdy hT ¼ y2

(3.115)

(3.116)

95

96

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Multiplying and dividing by yBM places Eq. (3.116) in the HGNG format: Z y1 ½GM = kG ayBM fyBM = ð1  yÞðy  yi Þgdy (3.117A) hT ¼ y2

¼ HG;av

Z

y1

fyBM = ð1  yÞðy  yi Þgdy

(3.117B)

y2

¼ HG;av NG

(3.117C)

The general expression given by Eq. (3.117) is more complex than generally is required, but it should be used when the mass transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (3.117) depend on the local liquid composition xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as one moves along the packed column. The following worked example illustrates the use of Eq. (3.117) for scrubbing chlorine from air with aqueous caustic solution. For this case one may make the simplifying assumption that yi, the interfacial partial pressure of chlorine over the aqueous caustic solution, is zero owing to the rapid and complete reaction of the chlorine after it dissolves. Note that the feed gas is not dilute. Example 2: Packed Height Requirement Calculations: The Problem One may compute the height of packing needed to reduce the chlorine concentration of 0.537 kg/(s$m2), or 396 lb/(h$ft2), of a chlorineeair mixture containing 0.503 mole-fraction chlorine to 0.0403 mole fraction. On the basis of test data described by Sherwood and Pigford (Absorption and Extraction, McGraw-Hill, 1952, p. 121) the value of kG ayBM at a gas velocity equal to that at the bottom of the packing is equal to 0.1175 kmol/(s$m3), or 26.4 lb$mol/(h$ft3). The equilibrium back pressure yi may be assumed to be negligible. The Solution By assuming that the mass-transfer coefficient varies as the 0.8 power of the local gas mass velocity, one may derive the following relation: KGa ¼ kG ayBM 0:8 ¼ 0:1175½f71y þ 29ð1  yÞg=f71y1 þ 29ð1  y1 Þgfð1  y1 Þ=ð1  yÞg (3.118)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

where 71 and 29 are the molecular weights of chlorine and air, respectively. Noting that the inert-gas (air) flow rate is given by

or

0 ¼ GM ð1  yÞ GM   ¼ 5:34  103 kmol= s $ m2

(3.119B)

  ¼ 3:94lb $ mol= h $ ft2

(3.119C)

(3.119A)

Upon introducing these expressions into the integral gives Z 0:503 n io h hT ¼ 1:82 ½ð1  yÞ=ð29 þ 42yÞ0:8 = ð1  yÞ2 lnf1 = ð1  yÞg dy 0:0403

(3.120) This definite integral may be evaluated numerically by the use of Simpson’s rule to obtain hT ¼ 0:305 m

(3.121)

(or, approximately 1 ft). For other conditions, such as the case in which the back pressure cannot be neglected, other design methods are available, such as those listed in Perry,* and they should be consulted. *ftp://ftp.feq.ufu.br/Claudio/Perry/DOCS/Chap14.pdf.

3.4 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions 3.4.1 Numerical solutions of a model in terms of simultaneous semi-linear partial differential equations 3.4.2 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations 3.4.3 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations (To facilitate referencing, some of the Reaction-Transport modeling equations in this section is pre-designated by RT-.)

97

98

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

In formulating a mathematical model representing a typical bimolecular reaction and transport such as the examples cited at the beginning of this chapter, consider the two principal reactants A and B being brought into contact with a medium for reaction. There are physically interesting situations in which substances A and B are both gases that are brought into contact with a liquid in which both are soluble and within which they react with each other. Such theoretical and experimental results have been reported by Roper et al.[32] This analysis presents useful and efficient numerical methods for solving the systems equations arising from such situations in which A and B are brought into contact with a laminar layer of liquid in which they react by a general (n1 þ n2)-order biochemical or chemical mechanism. Some aspects of the nature of the mathematics of these equations will be discussed as their solutions are sought. Some numerical results are obtained and are presented. The R environment will be used to present the theoretical results graphically.

3.5 Theory of simultaneous bimolecular reactions and mass transfer in two dimensions A mathematical model for simultaneous biochemical, or chemical, reaction and mass transport of two species in a medium NB: All equations in Sections 3.4 and 3.5 are designated in the form of (RT-n.), where n ¼ 1, 2, 3, ., where RT denotes Reaction and Transport. The reaction is assumed to have the stoichiometry corresponding to the irreversible biochemical or chemical equation A þ yB/Products Let y and z be two mutually perpendicular axes in space. Consider a material balance on a differential element having dimensions dy  dz in the direction perpendicular to the interface and the direction of flow parallel to the interface. Let A ¼ A(y, z), and B ¼ B(y, z) be the concentrations of biochemical, or chemical, species A and B at the point (y, z). Let the absorbing medium flow with velocity u in the direction of positive z-axis, with diffusion occurring in the direction of positive y-axis. Consider the mass balance of A and B over a volume element dV ¼ (dy) (dz) (1), with A and B reacting in the medium phase after

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

absorption, according to the irreversible biochemical mechanism indicated, viz., A þ yB / Products. Now for species A: Net rate of diffusion ¼ net rate of accumulation þ net rate of reaction viz., [(rate of diffusion into dV) e (rate of diffusion out of dV)] ¼ [(rate of flow into dV) e (rate of flow out of dV)] þ [rate of biochemical, or chemical, reaction] Hence,


    ½ ðdz $ 1ÞDA ðvA = vyÞ   ðdz $ 1ÞDA ðvA = vyÞ þ v2 A = vy 2 dy ¼ ½ðdy $ 1ÞuAg  fðdy $ 1ÞuðA þ ðvA = vzÞdzÞg þ kAn1 Bn2 ðdy $ dz $ 1Þ which results in Eq. (RT-1A). For species B, a similar mass balance results in Eq. (RT-1B). DA v2 A=vy2 ¼ uvA=vz þ kAn1 Bn2

(RT-1A)

DB v2 B=vy2 ¼ uvB=vz þ ykAn1 Bn2

(RT-1B)

with the following boundary conditions on A(y, z) and B(y, z):  Að0; zÞ ¼ Ai ; Bð0; zÞ ¼ Bi (RT-1C) Aðy; 0Þ ¼ 0; Bðy; 0Þ ¼ 0 Now, let A(y, z) and B(y, z) be bounded as y and z tend to infinity. The three terms in each of the above pairs of equations, (RT-3.1A) and (RT-3.1B), represent the diffusion, convection, and reaction terms, respectively.

99

100

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Transformation of the mathematical model into condensed and/or dimensionless forms For the simple case of reaction for a second-order biochemical, or chemical, mechanism: Let n1 ¼ 1 ¼ n2 . Let ) a ¼ A=Ai ; b ¼ B=Bi (RT-2) 1=2 x ¼ ðkBi =DAÞ y; t ¼ ðkBi =uÞz then Eqs. (RT-1A) and (RT-1B) become v2 a=vx2 ¼ va=vt þ ab

(RT-3A)

Rv2 b=vx2 ¼ vb=vt þ mab

(RT-3B)

with the boundary conditions on a(x, t) and b(x, t) given by: 9 að0; tÞ ¼ 1 ¼ bð0; tÞ > = aðx; 0Þ ¼ 0 ¼ bðx; 0Þ (RT-3C) > ; aðx; zÞand bðx; tÞare bounded as x; t/N and the following dimensionless parameters: R ¼ DB =DA ;

m ¼ yAi =Bi

(RT-4)

In general, the model system Eqs. (RT-1A)e(RT-1C) may be transformed into a condensed form by the following substitution of new variables. Let the variables A, B, y, and z be represented by the following condensed variables p

a ¼ A=Ai

p

b ¼ B=Bi  1=2 x ¼ kAri Bis =DA y 1=2  z t ¼ kAri Bis =u respectively, where p, q, r, and s are exponents to be determined.

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Substituting these new condensed variables into Eqs. (RT-1A) and (RT-1B), the result is:       pþr pþr pn1 qn2 kAi Bis v2 a=vx2 ¼ kAi Bis va=vt þ kAi Bi an1 bn2     qþs qþs kAri Bi Rv2 b=vx2 ¼ kAri Bi vb=vt   pn11 qn2þ1 þ kAi Bi man1 bn2 where a ¼ A=Ai ; b ¼ B=Bi 1=2  x ¼ kAn11 Bin2 =DA y; i

   t ¼ kAn11 Bin2 =u z i

(RT-5)

resulting in v2 a=vx2 ¼ va=vt þ an1 bn2

(RT-6A)

Rv2 b=vx2 ¼ vb=vt þ man1 bn2

(RT-6B)

together with boundary conditions given by Eq. (RT-3C) and dimensionless parameters given by Eq. (RT-4) as before. In fact, the condensed equations, Eqs. (RT-6A) and (RT-6B), are obtained if the respective exponents of Ai and Bi are equated, viz., p þ r ¼ pn1 ;

s ¼ qn2 ;

r ¼ pn1  1;

q þ s ¼ qn2 þ 1

Choosing p ¼ 1 ¼ q, one obtains: r ¼ n1 e 1, and s ¼ n2. This choice of p and q makes the condensed variables a and b dimensionless, and the boundary condition Eq. (RT-1C) is reduced to Eq. (RT-3C). Hence the new condensed variables are defined by Eq. (RT-2). Solutions to the system equations For somewhat different boundary conditions, Roper et al.[32] obtained some approximate solutions using the method of moments in which concentration profiles for A and B were assumed to be similar cubic polynomials. The approximate solutions were forced to satisfy the boundary conditions exactly but to satisfy the differential equations only on the average. Empirical factors were used to adjust the solutions to known asymptotic solutions, with a reported error rate of about 3%. Since the boundary conditions appear to be incorrect, as shown at the end of this section, it seems that any success of their work may be fortuitous. However, in this work, a versatile scheme will be developed to solve the system of equations. The remainder of this section will discuss this scheme.

101

102

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Numerical schemes Several finite difference discretization schemes are used to solve the equation system (RT-3.1A)e(RT-3.1C): (a) Fully-explicit scheme (b) Semi-explicit scheme (c) Semi-implicit schemes, for which the following variations are possible for solving the corresponding system of linear equations: (i) Calculating the inverse (ii) For triangular systems, solution by elimination (iii) Choleski’s method, with and without iterations (d) Fully-implicit scheme (e) Crank-Nicholson method (a) Fully-explicit scheme Now, Eqs. (RT-3A) and (RT-3B) may be rewritten as va=vt ¼ v2 a=vx2  ab

(RT-7A)

vb=vt ¼ Rv2 b=vx2  mab

(RT-7B)

Let a w A, and b w B, also Aji ¼ a(xi, tj) and Bji ¼ b(xi,tj). Then the boundary conditions, Eq. (3C) become j

j

að0; tÞ ¼ A1 ¼ 1 ¼ B1 ¼ bð0; tÞ aðx; 0Þ ¼ A1i ¼ 0 ¼ Bi1 ¼ bðx; 0Þ

9 = ;

(RT-7C)

aðx; tÞand bðx; tÞare bounded as x; t/N The corresponding finite difference equations, obtainable by discretizing Eqs. (RT-7A) and (RT-7B), are, respectively:     jþ1 j j j j 2 j j (RT-8A) Ai  Ai =Dt ¼ Ai1  2Ai þ Aiþ1 =ðDxÞ  Ai Bi     j j 2 j j  2Bij þ Biþ1 Bijþ1  Bij =Dt ¼ R Bi1 =ðDxÞ  mAi Bi

(RT-8B)

with the restriction that Dt/(Dx)2  ½. Eqs. (RT-8A) and (RT-8B) may now be solved explicitly for Ajþ1 and Bjþ1 to yield: i i h h i i  jþ1 2 j j 2 j j Ai ¼ 1  2Dt = ðDxÞ  DtBi Ai þ Dt = ðDxÞ Aiþ1 þ Ai1

jþ1

Bi

(RT-9A) h h i i  j j j j ¼ 1  2RDt = ðDxÞ2  mDt Ai Bi þ RDt = ðDxÞ2 Biþ1 þ Bi1 (RT-9B)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

which may be used for direct computation of Ajþ1 and Bjþ1, given i i j j An and Bn, for n ¼ ie1, i, iþ1. This algorithm is programmed for digital computer calculation, and convergent results are obtained. (b) Semi-explicit scheme In this scheme, the finite difference equations, discretizing Eqs. (RT-7A) and (RT-7B), are:     jþ1 j j j j jþ1 j Ai  Ai =Dt ¼ Ai1  2Ai þ Aiþ1 =ðDxÞ2  Ai Bi (RT-10A) 

jþ1

Bi

   j j j j 2 jþ1 jþ1  Bi =Dt ¼ R Bi1  2Bi þ Biþ1 =ðDxÞ  mAi Bi (RT-10B) jþ1

jþ1

Solving for Ai from Eq. (RT-10A), then Bi from Eq. (RT-10B), one obtains hn o  i h i jþ1 j j j j Ai ¼ Dt = ðDxÞ2 Ai1  2Aj þ Aiþ1 þ Ai = 1 þ DtBi

jþ1

Bi

¼

hn

RDt = ðDxÞ

2

o

j

j

j

Bi1  2Bi þ Biþ1



(RT-11A) i h i j j þ Bi = 1 þ mDt Ai (RT-11B)

For this procedure, it is seen that Ai must first be evaluated, and then Bjþ1 is obtained. The rest of the computation may proi ceed as in the fully explicit case. This algorithm is programmed for digital computer calculation, and convergent results are obtained. (c) Semi-implicit scheme In this scheme, the finite difference equations, discretizing Eqs. (RT-7A) and (RT-7B), are:     jþ1 j jþ1 jþ1 jþ1 jþ1 j Ai  Ai =Dt ¼ Ai1  2Ai þ Aiþ1 =ðDxÞ2  Ai Bi (RT-12A) jþ1

    jþ1 j jþ1 jþ1 jþ1 jþ1 jþ1 Bi  Bi =Dt ¼ R Bi1  2Bi þ Bi1 =ðDxÞ2  mAi Bi (RT-12B) Eqs. (RT-12A) and (RT-12B) may be rewritten in matrix form: C Ajþ1 ¼ Aj

(RT-13A)

D Bjþ1 ¼ Bj

(RT-13B)

103

104

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

where C is a tridiagonal matrix given by

C = [ I – {Δt/(Δx)2} A + Δt B j ] = [c1 u1 0 0 0 0 0 … 0 0 ] [u1 c2 u2 0 0 0 0 … 0 0 ] [ 0 u2 c3 u3 0 0 0 … 0 0 ] [ 0 0 u3 c4 u4 0 0 0 …0 0 ] [ ................................................... ] [0 0 0 - - - - -- 0 ] [ 0 0 0 - - - - 0 cn-1 un-] [ 0 0 0 - - - 0 un-1 cn] . . j with ci ¼ 1  2Dt ðDxÞ2 þ Dt Bi ; and ui ¼ Dt ðDxÞ2 . C is a tridiagonal matrix given by = [-2 +1 0 0 0 0 0 0 0] [+1 -2 +1 0 0 0 0 0 0] [ 0 +1 -2 +1 0 0 0 0 0] [ 0 0 +1 -2 +1 0 0 0 0] [............................................. ] [ 0 0 0 - - +1 -2 +1 ] [ 0 0 0 - - - +1 -2 ]

C

B j is a diagonal matrix given by B

j

=

[B1j 0 0 0 0 0 0 0 0 ] [ 0 B2j 0 0 0 0 0 0 0 ] [ 0 0 B3j 0 0 0 0 0 0 ] [ 0 0 0 B4j 0 0 0 0 0 ] [ ............................................... ] [ 0 0 0 0 0 0 0 0 B nj ]

D is a tri-diagonal matrix given by. D

= [ I - R{Δt/(Δx)2} A + m Δt Aj

]

= [d1 v1 0 0 0 0 0 … 0 0] [v1 d2 v2 0 0 0 0 … 0 0 ] [ 0 v2 d3 v3 0 0 0 … 0 0] [ 0 0 v3 d4 v4 0 0 0 …0 0] [.....................................................] [ 0 0 0 - - - - - - - - 0] [ 0 0 0 - - - -0 dn-1 vn-1] [ 0 0 0 - - - - 0 vn-1 dn]

with . . jþ1 di ¼ 1  R 2Dt ðDxÞ2 þ mDt Ai ; and vi ¼ RDt ðDxÞ2 .

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

A

jþ1

is the diagonal matrix given by A j+1 = [A1j+1 0 0 0 0 0 0 0 0 ] [0 A2j+1 0 0 0 0 0 0 0 ] [0 0 A3j+1 0 0 0 0 0 0 ] [0 0 0 A4j+1 0 0 0 0 0 ] [ ........................................................] [ 0 0 0 0 0 0 0 0 Anj+1]

The coupled matrix equations, Eqs. (RT-13A) and (RT-13B), may be solved numerically by several procedures, including: (i) Calculating the inverse CA

jþ1

¼ Aj ¼ > Ajþ1 ¼ C1 Aj

DBjþ1 ¼ Bj ¼ > B

jþ1

¼ D1 Bj

This method is generally not used owing to computational inefficiency and large data storage requirements.[26] (ii) For triangular systems, solution by elimination[24]dThis algorithm is programmed for digital computer calculation using one subroutine, and convergent results are obtained. In order to properly include the boundary condition of a(0, 1) ¼ 1 ¼ b(0, t), one may set jþ1

A0

jþ1

¼ 1 ¼ B0

so that, for calculations, Eqs. (RT-13A) and (RT-13B) become: [ [1+{2 Δt/(Δx)2}+ Δt B1j] -Δt/(Δx)2 0 0 [{Δt/(Δx)2}A0j+1+A1j]

0........... ][A1j+1] =

[-Δt/(Δx)2 [1+{2 Δt/(Δx)2}+ Δt B2j] -Δt/(Δx)2 0 ......][A2j+1] = [A2j] [ 0 -Δt/(Δx)2 [1+{2 Δt/(Δx)2}+ Δt B3j] -Δt/(Δx)2 0][A3j+1] = [A3j] [ [

0

0

.......................................................,][........] = ......]

[ [

0

0

-Δt/(Δx)2 [1+{2 Δt/(Δx)2}+ Δt Bnj][Anj+1.] = . Anj]

(RT-14A)

105

106

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

and [ [1+{R 2 Δt/(Δx)2}+mΔt A1j] -Δt/(Δx)2 0 0 0........0][B1j+1] = [{RΔt/(Δx)2}B0j+1 +B1j] [-Δt/(Δx)2 [1+{R 2Δt/(Δx)2}+ mΔt A2j] -Δt/(Δx)2 0..0][B2j+1] = [ B2j] [ 0 -Δt/(Δx)2[1+{R2Δt/(Δx)2}+mΔt A3j] -Δt/(Δx)2 0][B3j+1] = [ B 3j ] [ 0 [

0

[ 0 Bnj]

0 -Δt/(Δx)2 [1+{R 2 Δt/(Δx)2}+ mΔt Anj ][Bnj+1] = [

(RT-14B)

..................................][.......] = …....]

(iii) Computation using Choleski’s methoddSince the matrices C and D, in Eqs. (RT-13A) and (RT-13B), respectively, are each symmetric and positive definite, one can write C ¼ L LT

(RT-15A)

where L is a lower triangular matrix. Now

¼> ¼> ¼>

C Ajþ1 ¼ Aj  T  jþ1 LL A ¼ Aj   L LT Ajþ1 ¼ Aj   L g ¼ Aj ;

where LT Ajþ1 ¼ g and L

LT

=

C

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

=> [D1 0 0 0..............][D1 [ E1 D2 0 0 0.........][ 0 [ 0 E2 D3 0 0 0.....][ 0 [ 0 0........................][ 0 [ 0 0,,.0 0 E n-1 Dn][ 0

E1 0 0................... ] = [c1 u1 0 0 0.............] D2 E2 0 0............. ] [u1 c2 u2 0 0...........] 0 D3 E3 0 0 0..... ] [0 u2 c3 u3 0 0…...] 0.............................] [0 0 0 0...............] 0 0.. …...0 0 0 Dn] [0 0 0 0…..un-1 cn]

=> D1 = √c1

Di = √(ci - Ei-12), i = 2, 3, 4, …, n Ei = ui/Di ,

i = 1, 2, 3, …, n - 1

which forms the algorithm for L and LT, the subroutine for which is denoted by CHOL. Also, Lg = => [D1 [ E1 [0 [0 [0

Aj

0 0 0................0] [g1] = [J1] D2 0 0 0..........0] [g2] [J2] [J3] E2 D3 0 0 0.....0] [g3] 0........................0] [...] [...] 0 0 ....0 En-1 Dn] [gn] [Jn ]

.=> g1 = J1 / D1 => g i = (Ji – E-1gi-1)/Di, i = 2, 3, 4, …, n

which forms the algorithm for g, the subroutine for which is named SOLV. Further LT A j+1 = g => [ D1 E1 0 0 0] [G1] = [g1] [ 0 D2 E2 0 0..........0] [G2] [g2] [g3] [ 0 0 D3 E3 0 0 0..0] [G3] [ 0 0..........................0] [...] [...] [ 0 0 0..........0 0 0 Dn] [Gn] [gn] => Gn = gn / Dn

Gi = (gi – EiGi+1)/Di, i = n - 1,… ,3, 2, 1

which forms the algorithm for Ajþ1, and is computed using the

107

108

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

subroutine SOLV. Similarly, D ¼ MM T and

(RT-15B)

  DBjþ1 ¼ Bj ¼> MM T Bjþ1 ¼ Bj   ¼> M M T Bjþ1 ¼ Bj ¼>

M h ¼ Bj ;

where MTB jþ1 ¼ h, in which M is a lower-triangular matrix given by M

MT

D

=

=> [a1 0 0 0…………..][a1 b1 0 0……….….] = [d1 v1 [ b1 a2 0 0 0……….][ 0 a2 b2 0 0……...] [v1 d2 [ 0 b2 a3 0 0 0….....][ 0 0 a3 b3 0 0 0...] [0 v2 [ 0 0………………...][ 0 0……….… ….….] [0 0 [0 0 [ 0 0….0 0 bn-1 an] [ 0 0 0…..0 0 0 an] => a1 = √d1

ai = √(di – bi-12), bi = vi/ai,,

0 0 0…….....] v2 0 0….......] d3 v3 0 0......] 0 0…….......] 0 0.....vn-1 dn]

i = 2, 3, 4, . . ., n i = 1, 2, 3, . . ., n – 1

which may be computed again by the subroutine CHOL. Again, M

h = Bj

=> [a1 0 0 0…….……0] [h1] = [ b1 a2 0 0 0……....0] [h2] [ 0 b2 a3 0 0 0…...0] [h3] [ 0 0…………………0] [...] [ 0 0 0…./...0 bn-1 an] [hn]

[w1] [w2.] [w3.] [.....] [wn ]

=> h1 = w1 / a1 hi = (wi – bi-1hi-1)/ai, i = 1, 2, 3, …, n

which again may be computed by the forms the subroutine SOLV.

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Finally, MT

B j+1 =

h

=> [a1 b1 0 0……….…0] [ s1] = [h1 ] . [h2 ] [ 0 a2 b2 0 0……...0] [s2 ] [h3 ] [ 0 0 a3 b3 0 0 0...0] [ s3] [ 0 0…….….……….0] […] […] [hn ] [ 0 0 0……0 0 0 an] [ sn ] => sn = hn / an si = (hi – bisi+1)/ai, i = n-1,… ,3, 2, 1

which again may be computed using the subroutine SOLV. Thus, this scheme consists of the following algorithm: CHOL Aj, Bj =======> SOLV CHOL =======> SOLV

iv

CHOL Aj+1 =======> Bj+1 SOLV CHOL Aj+2 =======> Bj+2 , etc. SOLV

This procedure is programmed for digital computer calculations, using these two subroutines, and results are obtained. As in the calculation for (ii), viz., solution by elimination for tridiagonal systems, two vectors Aj and Bj were modified so as to properly include the boundary conditions: a(0, t) ¼ 1 ¼ b(0, t). Fully implicit scheme In this scheme, the finite difference equations, discretizing Eqs. (RT-7A) and (RT-7B), are:     jþ1 j jþ1 jþ1 jþ1 jþ1 jþ1 Ai  Ai =Dt ¼ Ai1  2Ai þ Aiþ1 =ðDxÞ2  Ai Bi (RT-16A)     jþ1 j jþ1 jþ1 jþ1 2 jþ1 jþ1 Bi  Bi =Dt ¼ R Bi1  2Bi þ Biþ1 =ðDxÞ  mAi Bi (RT-16B)

which may be written in matrix form as: i h 2 I  Dt = ðDxÞ A Ajþ1 ¼ Aj  DtB jþ1 Ajþ1 h

where

i 2 I  RDt = ðDxÞ A Bjþ1 ¼ Bj  mDtBjþ1 Bjþ1

(RT-17A) (RT-17B)

109

110

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

A = [-2 +1 0 ] [+1 -2 +1 0 ] [ +1 -2 +1 0 ] [ ………….…………....] [ 0 +1 -2]

A j+ 1 = [A1 j+1 0 ] [ A2 j+1 0 ] [ A3 j+1 0 ] [ ………………. ……...] [ 0 An j+1]

and B j+ 1 = [B1 j+1 0 ] [ B2 j+1 0 ] [ B3 j+1 0 ] [ ……………………... .] [ 0 Bn j+1 ]

As before, to incorporate the boundary conditions: a(0, t) ¼ 1 ¼ b(0, t), the matrices for these schemes become [ [1+{2 Δt/(Δx)2} -Δt/(Δx)2 [{Δt/(Δx)2}1 + A1j - Δt B1j+1A1 j+1] [-Δt/(Δx)2 [ [ [ [

[1+{2 Δt/(Δx)2} -Δt/(Δx)2 A2j - Δt B2j+1A2 j+1]

0

][A2j+1] =

-Δt/(Δx)2 0][A3j+1] =

0

[1+{2 Δt/(Δx)2}] -Δt/(Δx)2 j j+1 j+1 A3 - Δt B3 A3 ]

0

0 …………. ……………………… .][…...] = .………….……….]

0

0 An - Δt Bnj+1An j+1]

[ [ [

][A1j+1] =

0

j

-Δt/(Δx)2

[1+{2 Δt/(Δx)2][Anj+1] =

(RT-18A)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

and [ [1+ R{2 Δt/(Δx)2} -RΔt/(Δx)2 [R{Δt/(Δx)2}1 + B1j - m Δt A1j+1B1j+1 ] [1+R{2 Δt/(Δx)2} [-RΔt/(Δx)2 B2j - m Δt A2j+1B2 j+1]

][B1j+1] =

0

][B2j+1] = [

-RΔt/(Δx)2

0

[ 0 -RΔt/(Δx)2 j j+1 j+1 B3 - m Δt A3 B3 ]

[1+R{2 Δt/(Δx)2}]

-RΔt/(Δx)2

[ 0 0 .……..…………..….] [ 0 0 Bnj – m Δt Anj+1Bnj+1]

…………………………………… ][…...] = [ -RΔt/(Δx)2

0][B3j+1] = [

[1+R{2 Δt/(Δx)2][Bnj+1] = [

(RT-18B) and Aji, and beClearly, the implicit relations between Ajþ1 i jþ1 j tween Bi and Bi, mean that an iterative scheme is to be used. The symmetric tridiagonal matrices on the left-hand side of Eqs. (RT-18A) and (RT-18B) can be readily obtained and factored by the subroutine CHOL. Since it is independent of Aji and Bji, this part of the calculation need be done only once per increment in time or space. Given the vector Aji, the next incremental vector Ajþ1 is obi tained by iteration, using Eq. (RT-18A) and SOLV, to a predetermined accuracy. Similarly Bjþ1 can be obtained from Bji, i using Eq. (RT-18B). This algorithm is programmed for computer calculations, and the results are obtained. (v) Crank-Nicholson semi-implicit scheme In this scheme, the finite difference equations, discretizing Eqs. (RT-7A) and (RT-7B), are:   h  jþ1 2  Aji =Dt ¼ 1 2 Ajþ1 þ Ajþ1 Ajþ1 i i1  2Ai iþ1 = ðDxÞ =

  i   j j j jþ1 jþ1 j j þ Ai1  2Ai þ Aiþ1 = ðDxÞ2  1 2 Ai Bi þ Ai Bi

(RT-19A)

=

  jþ1 j Bi  Bi =Dt h    i jþ1 jþ1 jþ1 j j j ¼ 1 2 R Bi1  2Bi þ Biþ1 = ðDxÞ2 þ Bi1  2Bi þ Biþ1 = ðDxÞ2 =

  jþ1 jþ1 j j  1 2 m Ai B i þ Ai B i =

which, in matrix form, are

(RT-19B)

111

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

C1 Ajþ1 ¼ C 2 Aj

(RT-20A)

D 1 Bjþ1 ¼ D 2 Bj

(RT-20B)

where C1 = [ I – ½{Δt/(Δx)2}A + ½ Δt B j+1] C2 = [ I + ½{Δt/(Δx)2}A + ½ Δt B j ] D1 = [ I – ½{Δt/(Δx)2}A + ½ m Δt A j+1] D2 = [ I – ½{Δt/(Δx)2}A + ½ m Δt A j ]

with A

B

j

= [A1j 0 ] [ A2j 0 ] 0 ] [ A3j [ …………….………...] [ 0 Anj]

j

=[B1 j 0 ] [ B2 j 0 ] [ B3j 0 ] [ ……………….….......] [ 0 Bnj ]

and Aj, Ajþ1, and Bjþ1 are defined as before. Clearly, this required an iterative scheme. Again, to incorporate the boundary conditions: a(0, t) ¼ 1 ¼ b(0, t), one may first rewrite Eqs. (RT-20A) and (RT-20B) as follows: n o 2 1 I  2 ðDt=DxÞ A Ajþ1 =



n o 2 j 1 1 ¼ I  2 ðDt=DxÞ  2DtB Aj  1 2DtBjþ1 Ajþ1 =

=

=

(RT-21A)

n o 2 I  1 2 R ðDt=DxÞ A Bjþ1 =



n o 2 ¼ I  1 2 R ðDt=DxÞ  1 2 mDtAj Bj  1 2 mDtAjþ1 Bjþ1 =

=

=

112

(RT-21B)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Upon inclusion of the boundary conditions, these matrix equations become [ a m 0 0 0 0…0 ][A1j+1] [m a m 0 0 0… 0][A2j+1] [ 0 m a m 0…..0][A3j+1] [ ……………………...][ …..] [ 0 0 0 0….... m. a][Anj+1] = [{Δt/(Δx)2] + [ p1 n 0 0 0 0…0 ][A1j] - ½ Δt [B1j+1 0 [A1j+1]

0

0……0]

p2 n 0 0 0… 0][A2j]

[0

B2j+1 0

p3 n 0 0…..0][A3j]

[0

0

B3j+1 .0….…0]

[0

0

0 …0

[ 0 [A2j+1]

]

[n

[ 0 [A3j+1 ]

]

[0 n

0….…0]

[ ------ ] [ ………………….…..][….] […………………..….…][…...] [

] [ 0 0 0 0….... n pn][Anj]

0

j+1

j+1

Bn ][An ]

(RT-22A) n . o where a ¼ 1 þ Dt ðDxÞ2  n o 2 m ¼  1 2 Dt = ðDxÞ =

n o   2 j pi ¼ 1  Dt = ðDxÞ  1 2 Dt Bi =

 =

n¼

12

n

Dt = ðDxÞ

2

o

113

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

and [ c r 0 0 0 0…0 ][B1j+1] [r c r 0 0 0… 0][B2j+1] [0 r c r 0….....0][B3j+1] [ …………………….][…...] [ 0 0 0 0….... r c][Bnj+1] = [R{Δt/(Δx)2] + [ q1 s 0 0 0 0…...0][B1j] - ½mΔt [A1j+1 0 [

0

]

[

0

[

------

[

0

0

0…….0][B1j+1]

q2 s 0 0 0… 0][B2j] [0 A2j+1 0 0….…0][B2j+1] ] [ 0 s q3 s 0 0…..0][B3j] [0 0 A3j+1 0…. ..0][B3j+1] ] [ ……………………..][….] [……………….....…....][…..] ] [ 0 0 0 0….... s qn][Bnj] [0 0 0 ….… 0 Anj+1][Bnj+1] [s

(RT-22B) n . o where c ¼ 1 þ R Dt ðDxÞ2   n o 2 r ¼  1 2 R Dt = ðDxÞ =

n o   2 j qi ¼ 1  R Dt = ðDxÞ  1 2 mDtAi =

 s¼

12

=

114

 n o 2 R Dt = ðDxÞ

It is easily seen that the implicit relations between Ajþ1 and Aji, i and between Bjþ1 and Bji, require the use of an iterative proi cedure. This procedure for computation is the same as that for the fully implicit scheme, viz., using Choleski’s method. This algorithm is programmed for machine computation, and the results are obtained. Biomedical, biopharmaceutical, and chemical applications An important application of this study is the effects of biochemical, biopharmaceutical, and chemical reactions on the rate of transport into the media, e.g., the enhancement of rates of mass transport with reactions over the rate of physical mass transport. A typical measure of these absorption rates may be expressed in terms of the transport coefficients, defined as the rates of transport of dissolved reactants per unit of interfacial area.

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

For physical mass transport, this mass transfer coefficient is for Reactant A :

kLA ¼  OðkBi DA Þ

(RT-23A)

for Reactant B :

kLB ¼  OðkBi DB Þ

(RT-23B)

The mass transfer coefficient, in general, will be different for A and for Bdeach dependent on the reaction rates between the reacting substances: kLA ðtÞ ¼  ðDA = Ai ÞðvA=vzÞz¼0 ¼ OðkBi DA Þðva=vxÞx¼0 (RT-24A) kLB ðtÞ ¼  ðDB = Bi ÞðvB=vzÞz¼0 ¼ OðkBi DA ÞRðvb=vxÞx¼0 (RT-24B) After Eqs. (RT-7A) and (RT-7B) have been solved, subject to the boundary conditions in Eq. (RT-7C), then the derivatives required in Eqs. (RT-24A) and (RT-24B) may be determined. The mass transport rate enhancement factor FE, may be expressed as the ratio: Mass transfer coefficient for mass transport with biochemical reaction Mass transfer coefficient for physical mass transport only Thus, for Reactant A: Z T FEA ¼ kLA =kLA ðtÞjmean ¼ ð1 = T Þ ðva=vxÞx¼0 dT (RT-25A) 0

where kLA(t)jmean ¼ time-mean mass transfer coefficient for mass transport with biochemical reaction, for Reactant component A, 9 28 3 h xx2 plot (xx2, h) > lines (x, A2) > lines (x, A22) > lines (x, A52) > lines (x, A82) > lines (x, A102) > title ("Progressive Concentration Profiles of Component A in the Liquid Phase Showing Time-dependent Penetration for t = 0, + 0.2T, 0.5T, 0.8T, and 1.0T00 ) # Outputting: Figure 36. # Collating the Data for Component B > g plot (xx2, g) > lines (x, B2) > lines (x, B22) > lines (x, B52) > lines (x, B82) > lines (x, B102) > title ("Progressive Concentration Profiles of Component B in the # Liquid Phase Showing Time-dependent Penetration for # t = 0, 0.2T, 0.5T, 0.8T, and 1.0T00 ) > Outputting: Figure 37. >

Case II: R ¼ 1, m ¼ 1, X ¼ 10, NX ¼ 20, T ¼ 10, NT ¼ 100

# Defining the Displacement Points > x2 A22 B22 A222 B222 A252 B252 A282 B282 A2102 B2102 # Collating the Data for Component A > hA2 xx22 # Plotting the Data for Component A > plot (xx22, hA2) > lines (x2, A22)

131

132

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

> lines (x2, A222) > lines (x2, A252) > lines (x2, A282) > lines (x2, A2102) > title ("Progressive Concentration Profiles of Component A in the # Liquid Phase Showing Time-dependent Penetration for # t = 0, 0.2T, 0.5T, 0.8T, and 1.0T00 ) # Outputting: Figure 38. > # Collating the Data for Component > hB2 # Plotting the Data for Component B > plot (xx22, hB2) > lines (x2, B22) > lines (x2, B222) > lines (x2, B252) > lines (x2, B282) > lines (x2, B2102) > title ("Progressive Concentration Profiles of Component B in the # Liquid Phase Showing Time-dependent Penetration for t = 0, # 0.2T, 0.5T, 0.8T, and 1.0T00 ) # Outputting: Figure 39. >

3.5.1 An existence theorem of the governing simultaneous semi-linear parabolic partial differential equations (In 3.4.1, the system appears to be a well-posed initial and boundary value problem for a system of simultaneous semi-linear parabolic partial differential equations.) Existence theorem For a system defined by Eqs. (RT-27A) and (RT-27B), there exists a stable solution. Proof of the Existence Theorem: First, two preliminary lemmas* are stated as a first step in proving the existence theorem: * For example: Coddington, E. A., and Levinson, N. (1955). “Theory of Ordinary Differential Equations”, McGraw-Hill Book Company, New York.

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Preliminary Lemmas: Existence and uniqueness lemmas for first-order ODEs. The general first-order ODE is dy=dx ¼ F ðx; yÞ;

yðx0 Þ ¼ y0

(*)

One is interested in the following questions: (i) Under what conditions can one be sure that a solution to (*) exists? (ii) Under what conditions can one be sure that there is a unique solution to (*)? Here are the answers. Preliminary Lemma A (Existence). Suppose that F(x, y) is a continuous function defined in some region R ¼ fðx; yÞ: x0  d < x < x0 þ d; y0  ε < y < y0 þ εg containing the point (x0; y0). Then there exists a number d1 (possibly smaller than d) so that a solution y ¼ f(x) to (*) is defined for x0 e d1 < x < x0 þ d1 . Preliminary Lemma 2 (Uniqueness). Suppose that both F(x, y) and (vF/vy) (x, y) are continuous functions defined on a region R as in Theorem 1. Then there exists a number d2 (possibly smaller than d1) so that the solution y ¼ f(x) to (*), whose existence was guaranteed by Preliminary Lemma A (Existence), is the unique solution to (*) for x0 e d2 < x < x0 þ d2. The finite element method may be used for approximating the system solution. The finite element procedure is simply the Rayleigh-Ritz-Galerkin (RRG) approximation applied to the spaces of piecewise polynomial functions. Methodology in numerical analysis using this approach is, probably, better known as spline analysis in engineering sciences. It represents a unified and mathematically vigorous approach to the finite element method for solving continuous or infinite-dimensional problems [**] Schultz, M. H. (1973). “Spline Analysis”, Prentice-Hall, Englewood Cliffs, NJ. [**] For example: Coddington, E. A., and Levinson, N. (1955). “Theory of Ordinary Differential Equations”, McGraw-Hill Book Company, New York. Basically, the Galerkin procedure is used to “discretize” the space variables of the semi-linear system axx ¼ at þ ab

(RT-27A)

Rbxx ¼ bt þ mab

(RT-27B)

where a ¼ a(x, t) m having b ¼ bðx; tÞ

133

134

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

with aðx; tÞ ¼ 0 ¼ bðx; tÞ;

x>0

að0; tÞ ¼ 1 ¼ bðx; tÞ;

t>0

whose equivalent system having homogeneous boundaries is (dropping the suffices): axx ¼ at  ab þ abs þ as b

(RT-28A)

Rbxx ¼ bt  mab þ mabs þ mas b

(RT-28B)

or: at  axx  ab þ abs þ as b ¼ 0

(RT-29A)

bt  Rbxx  mab þ mabs þ mas b

(RT-29B)

with aðx; 0Þ ¼ 0 ¼ bðx; 0Þ;

x>0

bð0; tÞ ¼ 0 ¼ bð0; tÞ;

t>0

Using the notation

Z

N

ðf ; gÞh

fðxÞgðxÞdx 0

and multiplying Eqs. (RT-29A) and (RT-29B) by smooth but arbitrary functions v and w, one obtains the following weak formulations Z N Z N Z N Z N Z N at vdx  axx vdx  abvdx þ bs avdx þ as bvdx 0

0

0

0

0

¼ 0 Z

N

Z

0



Z bt wdx  R N

N

0

Z

N

bxx wdx  m 0

(RT-29A)

Z

N

abwdx þ m

abs wdx þ m

0

as bwdx ¼ 0

0

(RT-29B) for all v, w ˛ CN(0, N) h the set of infinitely differentiable and continuous functions which are nonnegative. Now, let

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Z

N

at vdx ¼ ðat ; vÞ

0

Z 

N

Z axx vdx ¼ 

0

¼  ¼ 

N

Z0 N Z0 N

v2 a=vx2 vdx fvðva=vxÞ=vxgvdx

ðvax =vxÞvdx Z N N ax ðvv=dxÞdx ¼ ½ax v0 þ 0 Z N ax vx dx ¼ 0þ 0

0

¼ ðax ; vx Þ Z N abvdx ¼ ðab; vÞ  0

Z

N

bs avdx ¼ ðbs a; vÞ

0

Z

N

as bvdx ¼ ðas b; vÞ

0

and, similarly:

Z Z

N

N

R Z

bxx wdx ¼ Rðbx ; wx Þ

0 N

m Z

bt wdx ¼ ðbt ; wÞ

0

abwdx ¼ mðab; wÞ

0 N

m

bs awdx ¼ mðbs a; wÞ

0

Z

N

m

as bwdx ¼ mðas b; wÞ

0

Substituting these into Eqs. (RT-3.29A) and (RT-3.29B), the results are: ðat ; vÞ þ ðax; vx Þ  ðab; vÞ þ ðbs a; vÞ þ ðas b; vÞ ¼ 0

(RT-30A)

135

136

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

ðbt ; wÞ þ Rðbx; wx Þ  mðab; wÞ þ mðbss a; wÞ þ mðas b; wÞ ¼ 0 (RT-30B) for all t>0. Now, let Sn be a finite dimensional sum of elements defined in CN(0, N), viz., Sn 3 CN(0, N), that is, Sn is spanned by the basis of {fi(x)}, i ¼ 1, 2, 3, ., n . Defining a semidiscrete Galerkin approximation, let n X

ah ¼ ah ðtÞ ¼

ci ðtÞfi ðxÞ ¼

X

ci fi

(RT-31A)

X di fi

(RT-31B)

i¼1

bh ¼ bh ðtÞ ¼

n X

i

di ðtÞfi ðxÞ ¼

i¼1

i

Thus, if the generalized solution of Eqs. (RT-3.30A) and (RT3.30B) is a and b, then a weak formulation, or semidiscrete Galerkin approximation to a and b for each element i, is given by Eqs. (RT-3.31A) and (RT-3.31B). The coefficients {ci(t)}i¼1,2,3,.,n and {di(t)}i¼1,2,3,.,n are functions of time which are determined as the solution of the linear system of ordinary differential equations:  h   h  at ; v þ ax ; vx  ðah ; bh ; vÞ þ ðbs ah ; vÞ þ ðas bh ; vÞ ¼ 0 (RT-32A)     bht ; w þ R bhx ; wx  mðah ; bh ; wÞ þ mðbs ah ; wÞ þ mðas bh ; wÞ ¼ 0 (RT-32B) where: 1  t  n, for all t > 0, ðah ð0Þ; vÞ ¼ ð0; vÞ; ðbh ð0Þ; wÞ ¼ ð0; wÞ; Now  h  at ; v ¼

Z

N

"

for all v˛Sn ; for all w˛Sn ;

n X

#

fi ðxÞfdCi ðtÞ = dtg fk ðxÞdx 82i ¼ 1 39 = n < Z N X 4 ¼ fi ðxÞfk ðxÞdx5 dCi ðtÞ = dt : 0 ; i¼1

¼

0

n X i¼1

ðfi ðxÞfk ðxÞÞdCi = dt

(RT-33A)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

   axh ; vx ¼

Z

N

"

#

n X

Ci ðtÞfik ðxÞ fk ðxÞdx 2 i¼1 3 Z N n X 4 fix ðxÞfkx ðxÞdx5Ci ðtÞ ¼ 0

0

i¼1

n X ½fi ðxÞfk ðxÞCi ¼

(RT-33B)

i¼1

"

# ! X ci fi di fj ; fk ða b ; vÞ ¼  i" j #" # Z N X X ci f i di fj ; fk dx ¼  0 j 3 2 i X Z N ¼  4 gij fi fj dx5 h h

ij

X

#"

2 X Z ðbs ah ; vÞ ¼  4 i1 j1

N

i2 j2

3 gij fi1 fj1 dx5

(RT-33D)

0

2 X Z ðas bh ; vÞ ¼  4



(RT-33C)

0

N

3 gij fi2 fj2 dx5

(RT-33E)

0

Hence the first equation, for ci, viz., Eq. (RT-33A):    ath ; v þ axh ; vx  ðah ; bh ; vÞ þ ðbs ah ; vÞ þ ðas bh ; vÞ ¼ 0 (RT-33A) becomes

X X Z gij fi fj dx ðfi ðxÞfk ðxÞÞdCi = dt þ ½fi ðxÞfk ðxÞCi  i i;j 3 2 2 3i X Z N X Z N 4 þ 4 g1ij fi1 fj1 dx5 þ g2ij fi2 fj2 dx5 ¼ 0 (RT-34)

X

i1 j 1

0

i2 j2

0

which is a set of linear ordinary differential equations for the ci-s. A similar set of equations for the di-s may be obtained for Eq. (RT-33B). Moreover, the rest of Eq. (RT-33), viz., Eqs. (RT-33C), (RT-33D), and (RT-33E), become

137

138

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Z ða ð0Þ; vÞ ¼ h

N

0

2 Z n X 4 i

N

"

n X i

# ci ð0Þfi ðxÞ fk ðX Þdx 3

fi ðxÞfk ðxÞdx5ci ð0Þ

0

¼

n X ðfi ; fk Þci ð0Þ i¼1

¼ 0 and

ðbh ð0Þ; wÞ ¼

n X ðfi ; fk Þdi ð0Þ ¼ 0 i¼1

Now let the vector

then the system of ordinary differential equations equivalent to Eq. (RT-3.34) may be written as n X

aij dyk ðtÞ=dt þ

X

bij yk ðtÞ þ Gk ðy1 ; y2 ; y3 ; /; yn Þ ¼ 0

i;j ¼ 1

n P with aij yk ðtÞ ¼ 0, for k ¼ 1, 2, 3, .,2n, which, in metric form, i;j ¼ 1 is  PdyðtÞ=dt þ Q yðtÞ ¼ R; t > 0 (RT-35) PyðtÞ ¼ 0g

Since P is symmetric, positive definite, and hence nonsingular, it follows, from a standard theoretical result in the analysis of ordinary differential equations [*], that the equation system

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

(RT-3.35) has a unique solution y*(t) which may be expressed analytically as Z t 
  (RT-36)* y ðtÞ ¼ exp ðs  tÞP1 Q P1 Qds; t > 0 0

Hence, the semidiscrete Galerkin method is well-defined for the semi-linear systems [Eqs. (RT-3.27A) and (RT-3.27B)]. This completes the proof of the existence theorem. [*] For example: Coddington, E. A., and Levinson, N. (1955). “Theory of Ordinary Differential Equations”, McGraw-Hill Book Company, New York.

3.6 A uniqueness theorem of the governing simultaneous semi-linear parabolic partial differential equations From physical considerations, a(x, t) and b(x, t) may be restricted to nonnegative, bounded, and continuous functions. The following uniqueness theorem for a and b will be established. Uniqueness theorem If a, b is a classical solution of Eqs. (RT-3A), (RT-3B), and (RT3C), then this system has at most one solution. Proof of the theoremdThe inhomogeneous boundary conditions in Eqs. (RT-3A) and (RT-3B) may be transformed to homogeneous boundaries by modifying the dependent variables a and b in Eq. (RT-3C). Let as and bs be the steady-state solutions of Eqs. (RT-3A), (RT3B), and (RT-3C): v2 a=vx2 ¼ va=vt þ ab

(RT-3A)

Rv2 b=vx2 ¼ vb=vt þ mab

(RT-3B)

with the boundary conditions on a(x, t) and b(x, t) given by: að0; tÞ ¼ 1 ¼ bð0; tÞ g aðx; 0Þ ¼ 0 ¼ bðx; 0Þ g aðx; zÞand bðx; tÞare bounded as x; t/N

(RT-3C) g

viz., vas/vt ¼ 0 ¼ vbs/vt, and as ¼ as(x), bs ¼ bs(x) hence v2 as =vx2 ¼ as bs

(RT-27A)

Rv2 b=vx2 ¼ mas bs

(RT-27B)

139

140

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Also as ð0Þ ¼ 1 ¼ bs ð0Þ

(RT-27C)

Now, let a1 ¼ as  a b1 ¼ bs  b From physical considerations : as  a; and bs  b;

(RT-28A)

hence a1  0; and b1  0;

(RT-28B)

Writing a ¼ as e a1, and b ¼ bs e b1, and substituting in Eqs. (RT-3A) and (RT-3B), one obtains v2 as =vx2  v2 a1 =vx2 ¼ vas =vt  va1 =vt þ ðas  a1 Þðbs  b1 Þ (RT-29A) Rv2 bs =vx2  Rv2 b1 =vx2 ¼ vbs =vt  vb1 =vt þ mðas  a1 Þðbs  b1 Þ (RT-29B) Similarly, substituting in Eq. (RT-3C), one obtains as ð0Þ  a1 ð0; tÞ ¼ 1 ¼ bs ð0Þ  b1 ð0; tÞ;

t>0

(RT-29C)

as ðxÞ  a1 ðx; 0Þ ¼ 0 ¼ bs ðxÞ  b1 ðx; 0Þ;

x>0

(RT-29D)

Using Eqs. (RT-3A) and (RT-3B), the following transformed system is obtained from simplifying Eqs. (RT-29A) and (RT-29B): v2 a1 =vx2 ¼ va1 =vt  a1 b1 þ ða1 bs þ as b1 Þ

(RT-30A)

Rv2 b1 =vx2 ¼ vb1 =vt  ma1 b1 þ mða1 bs þ as b1 Þ

(RT-30B)

with homogeneous initial and boundary conditions a1 ðx; 0Þ ¼ 0 ¼ b1 ðx; 0Þ;

x>0

(RT-30C)

a1 ð0; tÞ ¼ 0 ¼ b1 ð0; tÞ;

t>0

(RT-30D)

It is seen that homogeneity of the boundaries, Eqs. (RT-29C) and (RT-29D), has been obtained at the expense of inhomogeneity of the system: Eqs. (RT-29A) and (RT-29B), which may be written, using suffix notations for simplicity and convenience (where fxx h v2f/vx2, and ft h vf/vt): a1t  a1xx  a1 b1 þ a1 bs þ as b1 ¼ 0

(RT-31A)

b1t  Rb1xx  ma1 b1 þ ma1 bs  mas b1 ¼ 0

(RT-31B)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Let a0 and b0 be another solution of Eqs. (RT-3A) and (RT-3B), and let a2 ¼ as e a0 b2 ¼ bs  b0 then a2t  a2xx  a2 b2 þ a2 bs þ as b2 ¼ 0

(RT-32A)

b2t  Rb2xx  ma2 b2 þ ma2 bs  mas b2 ¼ 0

(RT-32B)

If A ¼ a1  a2 ;

and

B ¼ b1  b2 ;

then the theorem will be established if it can be shown that A ¼ 0 ¼ B;

for all t.

Now, combining terms from Eqs. (RT-30A) and (RT-30B) and (RT-3.31A) and (RT-3.31B): At  Axx  a1 b1 þ a2 b2 þ a1 bs  a2 bs þ as b1  as b2 ¼ 0 or, At  Axx  ða1 b1 þ a2 b2 Þ þ ða1 bs  a2 bs Þ þ ðas b1  as b2 Þ ¼ 0 (RT-33A) Bt  RBxx  ma1 b1 þ ma2 b2 þ ma1 bs  ma2 bs  mas b1 þ mas b2 ¼ 0 or, Bt  RBxx  mða1 b1  a2 b2 Þ þ mða1 bs  a2 bs Þ  mas ðb1  b2 Þ ¼ 0 (RT-33B) Now, a1 b1 þ a2 b2 ¼ a1 b1 þ a2 b1  a2 b1 þ a2 b2 ¼ ða1  a2 Þb1  a2 ðb1  b2 Þ ¼ Ab1  a2 B

g

a1 bs  a2 bs ¼ ða1  a2 Þbs ¼ Abs

g

as b1  as b2 ¼ as ðb1  b2 Þ ¼ as B

g

(RT-34)

141

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Substituting the relationships expressed in Eq. (RT-3.33) into Eqs. (RT-32A) and (RT-32B), the result is: At  Axx þ ð Ab1  a2 BÞ þ ðAbs Þ þ ðas BÞ ¼ 0 Bt  RBxx  mð Ab1  a2 BÞ þ mðAbs Þ  mðas BÞ ¼ 0 or, At  Axx þ ðbs  b1 ÞA þ ðas  a2 ÞB ¼ 0

(RT-35A)

Bt  RBxx  mðbs  b1 ÞA þ mðas  as2 Þ ¼ 0

(RT-35B)

with homogeneous initial and boundary conditions: Aðx; 0Þ ¼ 0 ¼ Bðx; 0Þ;

x>0

g

(RT-35C)

Að0; tÞ ¼ 0 ¼ Bð0; tÞ;

t >0 g

(RT-35D)

Multiplying Eqs. (3e11A) by A, and integrating with respect to all possible values of x, one obtains: Z N (RT-36A) fAt  Axx þ ðbs  b1 ÞA þ ðas  a2 ÞBgAdx ¼ 0 0

For simplicity, the limits notation of (0, N) are omitted in the remaining analysis, so that Eq. (RT-36A) may be written as Z Z Z Z 2 At Adx  Axx Adx þ ðbs  b1 ÞA dx þ ðas  a2 ÞABdx ¼ 0 (RT-37) Now, (a)

Z

Z

Z

At Adx ¼ ðvA = vtÞAdx ¼ ðAvA = vtÞdx  Z Z  2 1 1 ¼ v 2A = vt dx ¼ 2 ðd = dtÞ A2 dx =

=

142

(3.i)

and writing

1=2

Z jjAjj ¼ ðA; AÞ hence

1=2

¼

Z

AAdx

Z ¼

1=2 2

A dx

 2 A2 dx ¼ A

and combining (3.i) and (3.ii), ʃAt A dx ¼ ½ (d/dt)jjAjj2 (b)

;

(3.ii)

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Z Z  Axx Adx ¼ ½Axx AN þ Ax Ax dx; integrating by parts 0 Z 2 ¼ 0 þ ½Ax  dx; using the conditions of Equation Z 2 ¼ ½Ax  dx (RT e 35C)

0

(b) Using the conditions of Eqs. (RT-34C) and (RT-34D): Z ðbs  b1 ÞA2 dx  0 (3.iii) Hence, the conditions (a), (b), and (c) reduce Eq. (RT-37) to the following inequality: Z   1 2 ðd = dtÞA2 þ ðas  a2 ÞABdx  0 (RT-38A) =

Similarly, from Eq. (RT-35B), multiplying by B, and integrating with respect to all possible values of x, one obtains: Z   1 2 ðd = dtÞB2 þ m ðas  a2 ÞABdx  0 (RT-38B) =

From Eq. (RT-38A):  

1 2 ðd = dtÞA2 

Z 

ðas  a2 ÞABdx

=

or

Z 2 ðd = dtÞjjAjj   2 ðas  a2 ÞAB dx  Z      2 ðas  a2 ÞAB dx   Z Z        2 ðas  a2 Þ dx AB dx;  Z Z 1=2 Z 1=2      2 2    ;     B dx A dx  2 ðas  a2 Þdx    2jjas  a2 jjLN ð0; NÞjjAjj jjBjj  2ajjAjj jjBjj; where a ¼ jjas  a2 jjLN ð0; NÞ      2  2    A j þ   B  j ;   ** Holder’s inequality [1] **Theorem of the arithmetic and geometric means [**]

143

144

Chapter 3 Theory of simultaneous mass transfer and chemical reactions

Hence,

and similarly,

      2    2 2 ðd = dtÞAj  a Aj þ Bj  0

(RT-39)

      2    2 2 ðd = dtÞBj  b Aj þ Bj  0

(RT-40)

where b ¼ ma. Now let l ¼ l(t) ¼ jjAjj2 D jjBjj2, then l0

(RT-41)

And, on adding Eqs. (RT-39) and (RT-40), one obtains: ðd = dtÞl  ða þ bÞl  0 so that l  lð0Þexpða þ bÞt ¼ 0 because l(0) ¼ 0, by the condition of homogeneity of the boundaries. Thus, l0

(RT-42)

The conditions required by Eqs. (RT-41) and (RT-42) imply that lh0

(RT-43)

A ¼ 0 ¼ B; for all t

(RT-44)

which means

This completes the proof of the uniqueness theorem. [**] For example: Hardy, G. H. (2009). “A Course of Pure Mathematics”, 10th Edition, Cambridge University Press, Cambridge, UK (First Edition 1908) Final Remark: These several theorems have been established for a twodimensional system. The same conclusions may likely be extended to a three-dimensional system, as well as to systems of higher dimensions!

4 Numerical worked examples using R for simultaneous mass transfer and chemical reactions Chapter outline Worked Examples 149 Example A: Solving Reactive Transport Equations Using R 149 Examples B: Modeling Framework for Cellular Communities in their Environments 311 Chemical Engineering Science: The Theory of Simultaneous Mass Transfer and Chemical Reactions in Transport PhenomenadAn Introduction Using R

Several numerical examples are chosen to illustrate and highlight the application of the R program for numerical computations in simultaneous mass transfer and chemical reactions in typical transport phenomena in chemical engineering science. Example A: Simultaneous Mass Transfer and Chemical Reactions Using R * 1-D, 2-D, and 3-D Rectangular Geometries * Cylindrical Geometry * Polar Coordinates * 1-D, 2-D, and 3-D Volumetric Advective-Diffusive Transport in an Aquatic System Example B: Diffusive Models of Environmental Transport Using R * Equations for Diffusion in Environmental Transport Example C: Numerical Worked Examples of Advection Using R for: * 1-D, 2-D, and 3-D Rectangular Geometries * Cylindrical Geometry * Polar Coordinates Reference: https://en.wikipedia.org/wiki/Advection Advection[Wikipedia]

In the field of science and engineering, advection is the transport of a substance by bulk motion. The properties of that substance are carried with it. Generally, the majority of the Simultaneous Mass Transfer and Chemical Reactions in Engineering Science. https://doi.org/10.1016/B978-0-12-819192-7.00004-7 Copyright © 2020 Elsevier Inc. All rights reserved.

145

146

Chapter 4 Numerical worked examples

advected substance is a fluid (gas or liquid). The properties that are carried with the advected substance are conserved properties such as energy. An example of advection is the transport of pollutants or silt in a river by bulk water flowing downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as air or water. In general, any substance or conserved, extensive quantity may be advected by a fluid that can hold or contain the quantity or substance. During advection, a fluid transports some conserved quantity or material via bulk motion. The fluid motion may be described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Advection requires currents in the fluid, and so it cannot happen in rigid solids. It does not include transport of substances by molecular diffusion. Advection is not to be equated with the process of convection which is the combination of advective transport and diffusive transport. In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity, or salinity. (Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.) Contents Distinction between advection and convection Meteorology Other quantities Mathematics of advection 4.1. The advection equation 4.2. Solving the equation 4.3. Treatment of the advection operator in the incompressible Navier-Stokes equations 5. Additional information 6. References

1. 2. 3. 4.

Advection Versus Convection The term advection is sometimes considered as a synonym for convection, but technically, convection covers the sum of transport both by diffusion and by advection. Advective transport describes the movement of some quantity via the bulk flow of a fluid (as in a river or pipeline). Meteorology In meteorology and physical oceanography, advection often refers to the horizontal transport of some property of the atmosphere or ocean, such as heat, humidity, or salinity, and convection generally refers to vertical transport (vertical advection).

Chapter 4 Numerical worked examples

Advection is important for the formation of orographic clouds (terrain-forced convection) and the precipitation of water from clouds, as part of the hydrological cycle. Advection also applies if the quantity being advected is represented by a probability density function at each point. The Mathematics of Advection The Advection Equation is the partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. It is derived from the scalar field’s conservation law, together with Gauss’s theorem and taking the infinitesimal limit. A simple visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a “pulse” via advection, as the movement of bulk water itself transports the ink. Note that as it moves downstream, the “pulse” of ink will also spread via diffusion. The sum of these processes is called convection. Note: If the ink is added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source via diffusion alone, which is not advection. The Advection Equation In Cartesian coordinates the advection operator is u , V ¼ ux

v v v þ uy þ uz vx vy vz

(4.1)

where u ¼ (ux, uy, uz) is the velocity field, and V¼ is the del operator. Note that Cartesian coordinates are used here. The advection equation for a conserved quantity described by a scalar field j is expressed mathematically by a continuity equation: vj þ V,ðjuÞ ¼ 0 vt

(4.2)

where V, is the divergence operator, and u is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies the equation V,u ¼ 0

(4.3)

and u is said to be solenoidal. Under such conditions, the above equation may be written as vj þ u,Vj ¼ 0: vt

(4.4)

147

148

Chapter 4 Numerical worked examples

In particular, if the flow is steady, then v/vt ¼ 0, and u , Vj ¼ 0:

(4.5)

which shows that j is constant along a streamline. Hence, vj = vt ¼ 0;

(4.6)

so j does not vary in time. If a vector quantity a (such as a magnetic field) is being advected by the solenoidal velocity field u, the advection equation above becomes: va þ ðu , VÞa ¼ 0: vt

(4.7)

Solving the Advection Equation (Figure 1) The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous “shock” solutions, which are difficult for numerical schemes to solve. Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes vj vj þ ux ¼ 0: vt vx

(4.8)

where j ¼ j(x, t) is the scalar field being advected, and ux is the x component of the vector u ¼ (ux, 0, 0).

1.0 0.8 0.6

phi

0.4 0.2

–2

–1

0 x

1

2

3

4 –2

–1

0

1

2 y

3

0.0 4

Figure 1 A simulation of the advection equation where u ¼ (sin t, cos t) is solenoidal.

Chapter 4 Numerical worked examples

The Advection Operator in the Incompressible Navier-Stokes Equations Numerical simulation can be aided by considering the skew symmetric form for the advection operator: 1 1 u , Vu þ VðuuÞ 2 2

(4.9)

VðuuÞ ¼ ½Vðuux Þ; Vðuuy Þ; Vðuuz Þ

(4.10)

where

and u is the same as above. Since skew symmetry implies only imaginary eigenvalues, this form reduces the “blow up” and “spectral blocking” often experienced in numerical solutions with sharp discontinuities. Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems. ! kuk2 u , Vu ¼ V þ ðV  uÞ  u (4.11) 2 2 1 1 kuk u , Vu þ VðuuÞ ¼ V 2 2 2

!

1 þ ðV  uÞ  u þ uðV , uÞ (4.12) 2

This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is challenging and there is a large scientific literature on this subject!

Worked Examples Example A: Solving Reactive Transport Equations Using R Package ‘ReacTran’

August 16, 2017 Version 1.4.3.1 Title Reactive Transport Modeling in 1D, 2D, and 3D Authors Karline Soetaert , Filip Meysman Maintainer Karline Soetaert

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Depends R (>¼ 2.10), rootSolve, deSolve, shape Imports stats, graphics Description Routines for developing models that describe reaction and advective-diffusive transport in one, two, or three dimensions. Includes transport routines in porous media, in estuaries, and in bodies with variable shape. License GPL (>¼ 3) LazyData yes Repository CRAN

Repository/R-Forge/Project reactran Repository/R-Forge/Revision 100 Repository/R-Forge/DateTimeStamp 2017-08-15 06:57:39 Date/Publication 2017-08-15 22:13:03 UTC Needs Compilation yes R topics documented:

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

ReacTran-package advection.1D fiadeiro g.sphere p.exp setup.compaction.1D setup.grid.1D setup.grid.2D setup.prop.1D setup.prop.2D tran.1D tran.2D tran.3D tran.cylindrical tran.polar tran.volume.1D ReacTran-package Reactive Transport Modeling in 1D, 2D, and 3D Description R-package ReacTran contains routines that enable the development of reactive transport models in aquatic systems (rivers, lakes, oceans), porous media (floc aggregates, sediments, .)

Chapter 4 Numerical worked examples

and even idealized organisms (spherical cells, cylindrical worms, .). The geometry of the model domain is either one-dimensional, two-dimensional, or three-dimensional. The package contains: • Functions to setup a finite-difference grid (1D or 2D) • Functions to attach parameters and properties to this grid (1D or 2D) • Functions to calculate the advective-diffusive transport term over the grid (1D, 2D, or 3D) • Utility functions Details Package: ReacTran Type: Package Version: 1.4.3 Date: 2017-08-14 License: GNU Public License 2 or above Authors: Karline Soetaert (Maintainer) Filip Meysman See Also Functions ode.1D, ode.2D, ode.3D from package deSolve to integrate the reactive-transport model Functions steady.1D, steady.2D, steady.3D from the package to find the steady-state solution of the reactivetransport model rootSolve

tran.1D,tran.2D,tran.3D for a discretization of the general transport equations tran.volume.1D for discretization of the 1-D transport equations using finite volumes tran.cylindrical,tran.spherical for a discretization of 3-D transport equations in cylindrical and spherical coordinates tran.polar, for a discretization of 2-D transport equations in polar coordinates setup.grid.1D,setup.grid.2D

for the creation of grids in 1-D

and in 2-D setup.prop.1D,setup.prop.2D

grids.

for defining properties on these

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Examples ## show examples (see respective help pages for details) ## 1-dimensional transport example(tran.1D) example(tran.volume.1D) ## 2-dimensional transport example(tran.2D) example(tran.polar) ## 3-dimensional transport example(tran.3D) example(tran.cylindrical) example(tran.spherical) ## open the directory with documents browseURL(paste(system.file(package="ReacTran"), sep=""))

"/doc",

## open the directory with fortran codes of the transport functions browseURL(paste(system.file(package="ReacTran"), fortran", sep=""))

"/doc/

## show package vignette with how to use ReacTran and how to solve PDEs ## + source code of the vignettes vignette("ReacTran") vignette("PDE") edit(vignette("ReacTran")) ## a directory transport

with

fortran

implementations

browseURL(paste(system.file(package="ReacTran"), fortran", sep=""))

of

the

"/doc/

## End

Worked Examples Example (A1) trans.1D General One-Dimensional Advective-Diffusive Transport

Description This program estimates the transport term (i.e., the rate of change of a concentration owing to diffusion and advection) in a one-dimensional model of a liquid (volume fraction constant

Chapter 4 Numerical worked examples

and equal to one) or in a porous medium (volume fraction variable and lower than one). The interfaces between grid cells can have a variable crosssectional area, e.g., when modeling spherical or cylindrical geometries (see example). Usage tran.1D(C, C.up = C[1], C.down = C[length(C)], flux.up = NULL, flux.down = NULL, a.bl.up = NULL, a.bl.down = NULL, D = 0, v = 0, AFDW = 1, VF = 1, A = 1, dx, full.check = FALSE, full.output = FALSE)

Arguments C concentration, expressed per unit of phase volume, defined at the C.up concentration at upstream boundary. One value [M/L3] C.down

concentration at downstream boundary. One value

[M/L3] flux.up flux across the upstream boundary, positive ¼ INTO model domain. One value, expressed per unit of total surface [M/L2/T]. If NULL, the boundary is prescribed as a concentration or a convective transfer boundary.

flux across the downstream boundary, positive ¼ domain. One value, expressed per unit of total surface [M/L2/T]. If NULL, the boundary is prescribed as a concentration or a convective transfer boundary.

flux.down OUT of model

a.bl.up convective transfer coefficient across the upstream boundary layer. Flux = a.bl.up*(C.up-C0). One value [L/T] a.bl.down convective transfer coefficient across the downstream boundary layer(L). Flux = a.bl.down*(CL-C.down). One value [L/T] D diffusion coefficient, defined on grid cell interfaces. One value, a vector of length Nþ1 [L2/T], or a 1D property list; the list contains at least the element int (see setup.prop.1D) [L2/T] v advective velocity, defined on the grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). One value, a vector of length Nþ1 [L/T], or a 1D property list; the list contains at least the element int (see setup.prop.1D) [L/T]

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AFDW weight used in the finite difference scheme for advection, defined on grid cell interfaces; backward ¼ 1, centered ¼ 0.5, forward ¼ 0; default is backward. One value, a vector of length Nþ1, or a 1D property list; the list contains at least the element int (see setup.prop.1D) [e] VF Volume fraction defined at the grid cell interfaces. One value, a vector of Length Nþ1, or a 1D property list; the list contains at least the elements int and mid (see setup.prop.1D) [e] A Interface area defined at the grid cell interfaces. One value, a vector of length N+1, or a 1D grid property list; the list contains at least the elements int and mid (see setup.prop.1D) [L2] dx distance between adjacent cell interfaces (thickness of grid cells). One value, a vector of length N, or a 1D grid list containing at least the elements dx and dx.aux (see setup.grid.1D) [L] full.check logical flag enabling a full check of the consistency of the arguments (default ¼ FALSE; TRUE slows down execution by 50 percent) full.output logical flag enabling a full return of the output (default ¼ FALSE; TRUE slows down execution by 20 percent)

Details The boundary conditions may take one of the following values: • • • •

(1) (2) (3) (4)

zero-gradient fixed concentration convective boundary layer fixed flux.

The above order also shows the priority. The default condition is the zero gradient. The fixed concentration condition overrules the zero gradient. The convective boundary layer condition overrules the fixed concentration and zero gradient. The fixed flux overrules all other specifications. To ensure that the boundary conditions are well defined: for instance, it does not make sense to specify an influx in a boundary cell with the advection velocity pointing outward. Transport properties: The diffusion coefficient (D), the advective velocity (v), the volume fraction (VF), the interface surface (A), and the advective finite

Chapter 4 Numerical worked examples

difference weight (AFDW) can either be specified as one value, a vector or a 1D property list as generated by setup.prop.1D. When a vector, this vector must be of length Nþ1, defined at all grid cell interfaces, including the upper and lower boundary. The finite difference grid (dx) is specified either as one value, a vector or a 1D grid list, as generated by setup.grid.1D.

Value dC the rate of change of the concentration C due to transport, defined in the center of each grid cell. The rate of change is expressed per unit of phase volume [M/L3/T] C.up concentration at the upstream interface. One value [M/ L3] only when (full.output = TRUE) C.down concentration at the downstream interface. One value [M/L3] only when (full.output = TRUE) dif.flux diffusive flux across at the interface of each grid cell. A vector of length Nþ1 [M/L2/T] only when (full.output = TRUE) adv.flux advective flux across at the interface of each grid cell. A vector of length Nþ1 [M/L2/T] only when (full.output = TRUE) flux total flux across at the interface of each grid cell. A vector of length Nþ1 [M/L2/T] only when (full.output = TRUE) flux.up flux across the upstream boundary, positive ¼ INTO model domain. One value [M/L2/T] flux.down OUT of model

flux across the downstream boundary, positive ¼ domain. One value [M/L2/T]

Note The advective equation is not checked for mass conservation. Sometimes, this is not an issue, for instance when v represents a sinking velocity of particles or a swimming velocity of organisms. In other cases, however, mass conservation needs to be accounted for. To ensure mass conservation, the advective velocity must obey certain continuity constraints: in essence the product of the volume fraction (VF), interface surface area (A) and advective velocity (v) should be constant. In sediments, one can use setup.compaction.1D to ensure that the advective velocities for the pore water and solid phase meet these constraints.

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In terms of the units of concentrations and fluxes, one may follow the convention in the geosciences. The concentration C, C.up, C.down as well at the rate of change of the concentration dC are always expressed per unit of phase volume (i.e., per unit volume of solid or liquid). Total concentrations (e.g., per unit volume of bulk sediment) may be obtained by multiplication with the appropriate volume fraction. In contrast, fluxes are always expressed per unit of total interface area (so here the volume fraction is accounted for). Authors Filip Meysman , Karline Soetaert References Soetaert and Herman (2009). A practical guide to ecological modelling - using R as a simulation platform. Springer. See Also tran.volume.1D

for a discretization the transport equation using finite volumes. tran.2D, tran.3D advection.1D,

for more sophisticated advection schemes

Examples # =============================================# # EXAMPLE 1: O2 and OC consumption in sediments # # =============================================# # this example uses only the volume fractions # in the reactive transport term #===================# # Model formulation # #===================# # Monod consumption of oxygen (O2) O2.model