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Coulson and Richardson's Chemical Engineering, Fourth Edition: Volume 3A: Chemical and Biochemical Reactors and Reaction EngineeringR. Ravi, R. Vinu, S. N. Gummadi
Coulson and Richardson's Chemical Engineering: Volume 3A: Chemical and Biochemical Reactors and Reaction Engineering, Fourth Edition, covers reactor design, flow modelling, gas-liquid and gas-solid reactions and reactors.
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Coulson & Richardson's Chemical Engineering Chemical Engineering, Volume 1A: Fluid Flow, Seventh edition Raj Chhabra and V. Shankar Chemical Engineering, Volume 1B: Heat and Mass Transfer, Seventh edition Raj Chhabra and V. Shankar Chemical Engineering, Volume 2A: Particle Technology Particulate processes, Fifth edition Raj Chhabra and Basa M. Gurappa Chemical Engineering, Volume 2B: Separation Process, Fifth edition A. K. Ray Chemical Engineering, Volume 3A, Fourth edition Chemical & Biochemical Reactors R. Ravi, R. Vinu and S. N. Gummadi Chemical Engineering, Volume 3B: Process Control, Fourth edition Sohrab Rohani Chemical Engineering Solutions to the Problems in Volume 1 J. R. Backhurst, J. H. Harker and J. F. Richardson Chemical Engineering Solutions to the Problems in Volumes 2 and 3 J. R. Backhurst, J. H. Harker and J. F. Richardson Chemical Engineering, Volume 6, Third edition Chemical Engineering Design R. K. Sinnott Coulson and Richardson's Chemical Engineering Volume 3A: Chemical and Biochemical Reactors and Reaction Engineering Fourth Edition Edited by R. Ravi Indian Institute of Technology Madras, Chennai, India R. Vinu Indian Institute of Technology Madras, Chennai, India S.N. Gummadi Indian Institute of Technology Madras, Chennai, India Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2017 Elsevier Ltd. 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: 978-0-08-101096-9 For information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals Publisher: John Fedor Acquisition Editor: Anita A. Koch Editorial Project Manager: Ashlie M. Jackman Production Project Manager: Mohanapriyan Rajendran Cover Designer: Victoria Pearson Typeset by TNQ Books and Journals List of Contributors Sathyanarayana N. Gummadi Indian Institute of Technology Madras, Chennai, India Ramamurthy Ravi Indian Institute of Technology Madras, Chennai, India Ravikrishnan Vinu Indian Institute of Technology Madras, Chennai, India xi About Prof. Coulson John Coulson, who died on January 6, 1990 at the age of 79, came from a family with close involvement with education. Both he and his twin brother Charles (renowned physicist and mathematician), who predeceased him, became professors. John did his undergraduate studies at Cambridge and then moved to Imperial College where he took the postgraduate course in chemical engineeringdthe normal way to qualify at that timedand then carried out research on the flow of fluids through packed beds. He then became an Assistant Lecturer at Imperial College and, after wartime service in the Royal Ordnance Factories, returned as Lecturer and was subsequently promoted to a Readership. At Imperial College, he initially had to run the final year of the undergraduate course almost single-handed, a very demanding assignment. During this period, he collaborated with Sir Frederick (Ned) Warner to write a model design exercise for the I. Chem. E. Home Paper on "The Manufacture of Nitrotoluene." He published research papers on heat transfer and evaporation, on distillation, and on liquid extraction, and coauthored this textbook of Chemical Engineering. He did valiant work for the Institution of Chemical Engineers which awarded him its Davis medal in 1973, and was also a member of the Advisory Board for what was then a new Pergamon journal, Chemical Engineering Science. In 1954, he was appointed to the newly established Chair at Newcastle-upon-Tyne, where Chemical Engineering became a separate Department and independent of Mechanical Engineering of which it was formerly part, and remained there until his retirement in 1975. He took a period of secondment to Heriot Watt University where, following the splitting of the joint Department of Chemical Engineering with Edinburgh, he acted as adviser and de facto Head of Department. The Scottish university awarded him an Honorary DSc in 1973. John's first wife Dora sadly died in 1961; they had two sons, Anthony and Simon. He remarried in 1965 and is survived by Christine. JFR xiii About Prof. Richardson Professor John Francis Richardson, Jack to all who knew him, was born at Palmers Green, North London, on July 29, 1920 and attended the Dame Alice Owens School in Islington. Subsequently, after studying Chemical Engineering at Imperial College, he embarked on research into the suppression of burning liquids and of fires. This early work contributed much to our understanding of the extinguishing properties of foams, carbon dioxide, and halogenated hydrocarbons, and he spent much time during the war years on large-scale fire control experiments in Manchester and at the Llandarcy Refinery in South Wales. At the end of the war, Jack returned to Imperial College as a lecturer where he focused on research in the broad area of multiphase fluid mechanics, especially sedimentation and fluidization, two-phase flow of a gas and a liquid in pipes. This laid the foundation for the design of industrial processes like catalytic crackers and led to a long lasting collaboration with the Nuclear Research Laboratories at Harwell. This work also led to the publication of the famous paper, now common knowledge, the so-called Richardson-Zaki equation which was selected as the Week's citation classic (Current Contents, February 12, 1979)! After a brief spell with Boake Roberts in East London, where he worked on the development of novel processes for flavors and fragrances, he was appointed as Professor of Chemical Engineering at the then University College of Swansea (now University of Swansea), in 1960. He remained there until his retirement in 1987 and thereafter continued as an Emeritus Professor until his death on January 4, 2011. Throughout his career, his major thrust was on the wellbeing of the discipline of Chemical Engineering. In the early years of his teaching duties at Imperial College, he and his colleague John Coulson recognized the lack of satisfactory textbooks in the field of Chemical Engineering. They set about rectifying the situation and this is how the now well-known Coulson-Richardson series of books on Chemical Engineering was born. The fact that this series of books (six volumes) is as relevant today as it was at the time of their first appearance is a testimony to the foresight of John Coulson and Jack Richardson. Throughout his entire career spanning almost 40 years, Jack contributed significantly to all facets of professional life, teaching, research in multiphase fluid mechanics and service to xv About Prof. Richardson the Institution of Chemical Engineers (IChem E, UK). His professional work and long standing public service was well recognized. Jack was the president of IChem E during the period 1975e76 and was named a Fellow of the Royal Academy of Engineering in 1978. He was also awarded OBE in 1981. In his spare time, Jack and his wife Joan were keen dancers, being the founder members of the Society of International Folk Dancing and they also shared a love for hill walking. RPC xvi Preface Volume 3 has now been split into two separate volumes 3A and 3B with 3A containing chapters on chemical and biochemical reaction engineering and 3B consisting of process dynamics and control. In volume 3A, there are now a total of six chapters with two on biochemical reaction engineering (Chapters 5 and 6) compared to a single chapter in the third edition. Overall, the basic structure of the third edition has been retained. At the outset, we express our gratitude to the contributors of the third editiondProfessors J. C. Lee, W. J. Thomas, R. Lovitt, and M. Jonesdfor providing an enduring framework for us to build on. We highlight here the few changes we have made in the presentation of some of the material. These changes have been motivated by our own personal experience in teaching reaction engineering and also by the evolution of the presentation of the subject in various textbooks over the past two decades. Each chapter in the present edition starts with a set of Learning Outcomes, which we hope would enable the reader to assess the benefits that would accrue from reading the chapter. In Chapter 1, greater emphasis is placed on deriving the material and energy balance equations from fundamental continuum principles. This, we hope, would enable the reader to more clearly see the assumptions inherent in the design equations of the ideal reactors and also be better equipped to handle more complicated reactors, if the need arises. Further, the section on obtaining kinetics from batch reactor data has been expanded. In Chapter 2, on nonideal reactors, introduction of the two principal distribution functions is decoupled from the experimental methods used to determine them. The convolution theorem forms the centerpiece of the developments leading to the determination of the distribution functions from step and impulse responses. A detailed exposition of the zero parameter models is undertaken to illustrate the interplay of "macromixing," "micromixing," and kinetics in determining conversions in nonideal reactors. Chapter 3 now includes more information related to the derivation of kinetic rate expressions for heterogeneous catalytic reactions where the importance of elementary steps is highlighted. The mechanism of catalyst poisoning is discussed. The section on gasesolid noncatalytic reactions is expanded to include the effect of different controlling resistances on the overall conversion of the solid. In Chapter 4, a rigorous framework is provided for xiii Preface the derivation of rate equations for gaseliquid reactions so that the reader is able to appreciate the importance of various nondimensional numbers. Chapters 5 and 6 deal with biochemical engineering. It is important for the reader to first understand biomolecules (carbohydrates, proteins, nucleic acids, fats, and lipids), and hence this is covered before introducing the principles of microbiology. Chapter 5 thus provides the background to the process engineering aspects of biochemical reactors which are covered in Chapter 6. A section on specificity of enzymes, which discusses the various theories such as lock and key mechanism, induced fit model, and transition state stabilization, is included. The revised section on microbial stoichiometry now contains cell composition, yields, growth stoichiometry, elemental balance, degree of reduction, electron balance, and product stoichiometry. The detailed derivations of the equations governing enzyme kinetics are now covered in the main text of Chapter 6. R. Ravi and R. Vinu Department of Chemical Engineering S.N. Gummadi Department of Biotechnology Indian Institute of Technology, Madras India xiv Introduction Welcome to the next generation of Coulson-Richardson series of books on Chemical Engineering. I would like to convey to you all my feelings about this project which have evolved over the past 30 years, and are based on numerous conversations with Jack Richardson himself (1981 onwards until his death in 2011) and with some of the other contributors to previous editions including Tony Wardle, Ray Sinnott, Bill Wilkinson, and John Smith. So what follows here is the essence of these interactions combined with what the independent (solicited and unsolicited) reviewers had to say about this series of books on several occasions. The Coulson-Richardson series of books has served the academia, students, and working professionals extremely well since their first publication more than 50 years ago. This is a testimony to their robustness and to some extent, their timelessness. I have often heard much praise, from different parts of the world, for these volumes both for their informal and user-friendly yet authoritative style and for their extensive coverage. Therefore, there is a strong case for continuing with its present style and pedagogical approach. On the other hand, advances in our discipline in terms of new applications (energy, bio, microfluidics, nanoscale engineering, smart materials, new control strategies, and reactor configurations, for instance) are occurring so rapidly as well as in such a significant manner that it will be naive, even detrimental, to ignore them. Therefore, while we have tried to retain the basic structure of this series, the contents have been thoroughly revised. Wherever, the need was felt, the material has been updated, revised, and expanded as deemed appropriate. Therefore the reader, whether a student or a researcher or a working professional should feel confident that what is in the book is the most up-to-date, accurate, and reliable piece of information on the topic he/she is interested in. Evidently, this is a massive undertaking that cannot be managed by a single individual. Therefore, we now have a team of volume editors responsible for each volume having the individual chapters written by experts in some cases. I am most grateful to all of them for having joined us in the endeavor. Further, based on extensive deliberations and feedback from a large number of individuals, some structural changes were deemed appropriate, as detailed here. xix Introduction Due to their size, each volume has been split into two sub-volumes as follows: Volume Volume Volume Volume Volume Volume 1A: Fluid Flow 1B: Heat and Mass Transfer 2A: Particulate Technology and Processing 2B: Separation Processes 3A: Chemical Reactors 3B: Process Control Undoubtedly, the success of a project with such a vast scope and magnitude hinges on the cooperation and assistance of many individuals. In this regard, we have been extremely fortunate in working with some outstanding individuals at Butterworth-Heinemann, a few of whom deserve to be singled out: Jonathan Simpson, Fiona Geraghty, Maria Convey, and Ashlie Jackman who have taken personal interest in this project and have come to help us whenever needed, going much beyond the call of duty. Finally, this series has had a glorious past but I sincerely hope that its future will be even brighter by presenting the best possible books to the global Chemical Engineering community for the next 50 years, if not for longer. I sincerely hope that the new edition of this series will meet (if not exceed) your expectations! Lastly, a request to the readers, please continue to do the good work by letting me know if, no not if, when you spot a mistake so that these can be corrected at the first opportunity. Raj Chhabra Editor-in-Chief Kanpur, July 2017 xx CHAPTER 1 Reactor DesigndGeneral Principles Ramamurthy Ravi Indian Institute of Technology Madras, Chennai, India Learning Outcomes 1. The basic objectives of reactor design, classification of reactors, and guidelines for choice of process conditions (Sections 1.1e1.3). 2. The principles of material and energy balances, steady-state and unsteady-state cases, as well as integral and differential balance equations (Section 1.4). 3. The most common forms of rate equations (kinetics) and how to obtain them from a given mechanism (Section 1.5). 4. Derivation of design equations for a batch reactor and application of them to isothermal and nonisothermal operations; various methods of obtaining kinetics from batch reactor data for liquid-phase and gas-phase reactions (Section 1.6). 5. The assumptions inherent in the plug flow reactor (PFR) and how they lead to the design equations; application to gas-phase and liquid-phase reactions; energy balance considerations for nonisothermal operation (Section 1.7). 6. Setup and application of design equations for single and multiple (in series) continuous stirred tank reactors (CSTRs); interpretation of graphical constructions; nonisothermal operations (Section 1.8). 7. Formulation and application of performance criteria to compare batch/PFR and CSTR for single reactions (Section 1.9). 8. Types of multiple reactions; performance criteria and reactor design for multiple reactions (Section 1.10). 1.1 Basic Objectives in Design of a Reactor In chemical engineering, physical operations such as fluid flow, heat transfer, mass transfer, and separation processes play a very large part; these have been discussed in Volumes 1 and 2. In any manufacturing process where there is a chemical change taking place, however, the chemical reactor is at the heart of the plant. Coulson and Richardson's Chemical Engineering. http://dx.doi.org/10.1016/B978-0-08-101096-9.00001-7 Copyright © 2017 Elsevier Ltd. All rights reserved. 1 2 Chapter 1 In size and appearance it may often seem to be one of the least impressive items of equipment, but its demands and performance are usually the most important factors in the design of the whole plant. When a new chemical process is being developed, at least some indication of the performance of the reactor is needed before any economic assessment of the project as a whole can be made. As the project develops and its economic viability becomes established, further work is carried out on the various chemical engineering operations involved. Thus, when the stage of actually designing the reactor in detail has been reached, the project as a whole will already have acquired a fairly definite form. Among the major decisions, which will have been taken, is the rate of production of the desired product. This will have been determined from a market forecast of the demand for the product in relation to its estimated selling price. The reactants to be used to make the product and their chemical purity will have been established. The basic chemistry of the process will almost certainly have been investigated, and information about the composition of the products from the reaction, including any by-products, should be available. On the other hand, a reactor may have to be designed as part of a modification to an existing process. Because the new reactor has then to tie in with existing units, its duties can be even more clearly specified than when the whole process is new. Naturally, in practice, detailed knowledge about the performance of the existing reactor would be incorporated in the design of the new one. As a general statement of the basic objectives in designing a reactor, we can say therefore that the aim is to produce a specified product at a given rate from known reactants. In proceeding further, however, a number of important decisions must be made and there may be scope for considerable ingenuity to achieve the best result. At the outset the two most important questions to be settled include the following: 1. The type of reactor to be used and its method of operation. Will the reaction be carried out as a batch process, a continuous flow process, or possibly a hybrid of the two? Will the reactor operate isothermally, adiabatically, or in some intermediate manner? 2. The physical condition of the reactants at the inlet to the reactor. Thus, the basic processing conditions in terms of pressure, temperature, and compositions of the reactants on entry to the reactor have to be decided, if not already specified as part of the original process design. Subsequently, the aim is to reach logical conclusions concerning the following principal features of the reactor: 1. The overall size of the reactor, its general configuration, and, more important, dimensions of any internal structures. Reactor DesigndGeneral Principles 3 2. The exact composition and physical condition of the products emerging from the reactor. The composition of the products must of course lie within any limits set in the original specification of the process. 3. The temperatures prevailing within the reactor and any provisions, which must be made for heat transfer. 4. The operating pressure within the reactor and any pressure drop associated with the flow of the reaction mixture. 1.1.1 By-products and Their Economic Importance Before taking up the design of reactors in detail, let us first consider the very important question of whether any by-products are formed in the reaction. Obviously, consumption of reactants to give unwanted, and perhaps unsaleable, by-products is wasteful and will directly affect the operating costs of the process. Apart from this, however, the nature of any by-products formed and their amounts must be known so that plant for separating and purifying the products from the reaction may be correctly designed. The appearance of unforeseen by-products on start-up of a full-scale plant can be utterly disastrous. Economically, although the cost of the reactor may sometimes not appear to be great compared with that of the associated separation equipment such as distillation columns, etc., it is the composition of the mixture of products issuing from the reactor that determines the capital and operating costs of the separation processes. For example, in producing ethylene1 together with several other valuable hydrocarbons such as butadiene from the thermal cracking of naphtha, the design of the whole complex plant is determined by the composition of the mixture formed in a tubular reactor in which the conditions are very carefully controlled. As we shall see later, the design of a reactor itself can affect the amount of by-products formed and therefore the size of the separation equipment required. The design of a reactor and its mode of operation can thus have profound repercussions on the remainder of the plant. 1.1.2 Preliminary Appraisal of a Reactor Project In the following pages we shall see that reactor design involves all the basic principles of chemical engineering with the addition of chemical kinetics. Mass transfer, heat transfer, and fluid flow are all concerned and complications arise when, as so often is the case, interaction occurs between these transfer processes and the reaction itself. In designing a reactor it is essential to weigh up all the various factors involved and, by an exercise of judgment, to place them in their proper order of importance. Often the basic design of the reactor is determined by what is seen to be the most troublesome step. It may be the chemical kinetics; it may be mass transfer between phases; it may be heat transfer; or it may even be the need to ensure safe operation. For example, in oxidizing naphthalene or 4 Chapter 1 o-xylene to phthalic anhydride with air, the reactor must be designed so that ignitions, which are not infrequent, may be rendered harmless. The theory of reactor design is being extended rapidly and more precise methods for detailed design and optimization are being evolved. However, if the final design is to be successful, the major decisions taken at the outset must be correct. Initially, a careful appraisal of the basic role and functioning of the reactor is required and at this stage the application of a little chemical engineering common sense may be invaluable. 1.2 Classification of Reactors and Choice of Reactor Type 1.2.1 Homogeneous and Heterogeneous Reactors Chemical reactors may be divided into two main categories: homogeneous and heterogeneous. In homogeneous reactors, only one phase, usually a gas or liquid, is present. If more than one reactant are involved, provision must of course be made for mixing them together to form a homogenous whole. Often, mixing the reactants is the way of starting off the reaction, although sometimes the reactants are mixed and then brought to the required temperature. In heterogeneous reactors, two, or possibly three, phases are present; common examples being gaseliquid, gasesolid, liquidesolid, and liquideliquid systems. In cases where one of the phases is a solid, it is quite often present as a catalyst; gasesolid catalytic reactors particularly form an important class of heterogeneous chemical reaction systems. It is worth noting that, in a heterogeneous reactor, the chemical reaction itself may be truly heterogeneous, but this is not necessarily so. In a gasesolid catalytic reactor, the reaction takes place on the surface of the solid and is thus heterogeneous. However, bubbling a gas through a liquid may serve just to dissolve the gas in the liquid where it then reacts homogeneously; the reaction is thus homogeneous but the reactor is heterogeneous in that it is required to effect contact between two phasesdgas and liquid. Generally, heterogeneous reactors exhibit a greater variety of configuration and contacting pattern than homogeneous reactors. Initially, therefore, we shall be concerned mainly with the simpler homogeneous reactors, although parts of the treatment that follows can be extended to heterogeneous reactors with little modification. 1.2.2 Batch Reactors and Continuous Reactors Another kind of classification which cuts across the homogeneouseheterogeneous division is the mode of operationdbatchwise or continuous. Batchwise operation, shown in Fig. 1.1A, is familiar to anybody who has carried out small-scale preparative reactions in the laboratory. There are many situations, however, especially in large-scale operation, where considerable advantages accrue by carrying out a chemical reaction continuously in a flow reactor. Reactor DesigndGeneral Principles 5 (A) Reactants charged at beginning of reaction A (B) B A Reactants Products B Products removed at and of reaction (C) (D) Reactants A B Reactants A B Products Products Figure 1.1 Basic types of chemical reactors. (A) Batch reactor. (B) Tubular flow reactor. (C) Continuous stirred tank reactor (CSTR) or "backmix reactor." (D) CSTRs in series as frequently used. Fig. 1.1 illustrates the two basic types of flow reactor which may be employed. In the tubular flow reactor (B) the aim is to pass the reactants along a tube so that there is as little intermingling as possible between the reactants entering the tube and the products leaving at the far end. In the continuous stirred tank reactor (CSTR) (C) an agitator is deliberately introduced to disperse the reactants thoroughly into the reaction mixture immediately after they enter the tank. The product stream is drawn off continuously and, in the ideal state of perfect mixing, will have the same composition as the contents of the tank. In some ways, using a CSTR, or backmix reactor as it is sometimes called, seems a curious method of conducting a reaction because as soon as the reactants enter the tank they are mixed and a portion leaves in the product stream flowing out. To reduce this effect, it is often advantageous to employ a number of stirred tanks connected in series as shown in Fig. 1.1D. The stirred tank reactor is by its nature well suited to liquid-phase reactions. The tubular reactor, although sometimes used for liquid-phase reactions, is the natural choice for gas-phase reactions, even on a small scale. Usually the temperature or catalyst is chosen so that the rate of reaction is high, in which case a comparatively small tubular reactor is sufficient to handle a high volumetric flow rate of gas. A few gas-phase reactions, examples being partial combustion and certain chlorinations, are carried out in reactors, 6 Chapter 1 which resemble the stirred tank reactor; rapid mixing is usually brought about by arranging for the gases to enter with a vigorous swirling motion instead of by mechanical means. 1.2.3 Variations in Contacting PatterndSemibatch Operation Another question which should be asked in assessing the most suitable type of reactor is whether there is any advantage to be gained by varying the contacting pattern. Fig. 1.2A illustrates the semibatch mode of operation. The reaction vessel here is essentially a batch reactor, and at the start of a batch it is charged with one of the reactants A. However, the second reactant B is not all added at once, but continuously over the period of the reaction. This is the natural and obvious way to carry out many reactions. For example, if a liquid has to be treated with a gas, perhaps in a chlorination or hydrogenation reaction, the gas is normally far too voluminous to be charged all at once to the reactor; instead it is fed continuously at the rate at which it is used up in the reaction. Another case is where the reaction is too violent if both reactants are mixed suddenly together. Organic nitration, for example, can be conveniently controlled by regulating the rate of addition of the nitrating acid. The maximum rate of addition of the second reactant in such a case will be determined by the rate of heat transfer. A characteristic of semibatch operation is that the concentration CB of the reactant added slowly, B in Fig. 1.2, is low throughout the course of the reaction. This may be an advantage if more than one reaction is possible, and if the desired reaction is favored by a low value of CB. Thus, the semibatch method may be chosen for a further reason, that of improving the yield of the desired product, as shown in Section 1.10.4. (B) (A) First reactant dumped in A B Reactants A B Second reactant added continuously Products (C) B Reactants A Products discharged at end Products Figure 1.2 Examples of possible variations in reactant contacting pattern. (A) Semibatch operation. (B) Tubular reactor with divided feed. (C) Stirred tank reactors with divided feed (in each case the concentration of B, CB, is low throughout). Reactor DesigndGeneral Principles 7 Summarizing, a semibatch reactor may be chosen: 1. to react a gas with a liquid, 2. to control a highly exothermic reaction, and 3. to improve product yield in suitable circumstances. In semibatch operation, when the initial charge of A has been consumed, the flow of B is interrupted, the products are discharged, and the cycle will begin again with a fresh charge of A. If required, however, the advantages of semibatch operation may be retained, but the reactor system is designed for continuous flow of both reactants. In the tubular flow version (Fig. 1.2B) and the stirred tank version (Fig. 1.2C), the feed of B is divided between several points. These are known as cross-flow reactors. In both cases CB is low throughout. 1.2.4 Influence of Heat of Reaction on Reactor Type Associated with every chemical change there is a heat of reaction (or more precisely, an enthalpy change due to reaction), and only in a few cases it is so small that it can be neglected. The magnitude of the heat of reaction often has a major influence on the design of a reactor. With a strongly exothermic reaction, for example, a substantial rise in temperature of the reaction mixture will take place unless provision is made for heat to be transferred as the reaction proceeds. It is important to try to appreciate clearly the relation between the enthalpy of reaction, the heat transferred, and the temperature change of the reaction mixture; quantitatively this is expressed by an energy balance (Section 1.4). If the temperature of the reaction mixture is to remain constant (isothermal operation), the heat equivalent to the heat of reaction at the operating temperature must be transferred to or from the reactor. If no heat is transferred (adiabatic operation), the temperature of the reaction mixture will rise or fall as the reaction proceeds. In practice, it may be most convenient to adopt a policy intermediate between these two extremes; in the case of a strongly exothermic reaction, some heat transfer from the reactor may be necessary to keep the reaction under control, but a moderate temperature rise may be quite acceptable, especially if strictly isothermal operation would involve an elaborate and costly control scheme. In setting out to design a reactor, therefore, two very important questions to ask include the following: 1. What is the heat of reaction? 2. What is the acceptable range over which the temperature of the reaction mixture may be permitted to vary? 8 Chapter 1 The answers to these questions may well dominate the whole design. Usually, the temperature range can only be roughly specified; often the lower temperature limit is determined by the slowing down of the reaction and the upper temperature limit by the onset of undesirable side reactions. 18.104.22.168 Adiabatic Reactors If it is feasible, adiabatic operation is to be preferred for simplicity of design. Fig. 1.3 shows the reactor section of a plant for the catalytic reforming of petroleum naphtha; this is an important process for improving the octane number of gasoline. The reforming reactions are mostly endothermic so that in adiabatic operation the temperature would fall during the course of the reaction. If the reactor were made as one single unit, this temperature fall would be too large, i.e., either the temperature at the inlet would be too high and undesired reactions would occur, or the reaction would be incomplete because the temperature near the outlet would be too low. The problem is conveniently solved by dividing the reactor into three sections. Heat is supplied externally between the sections, and the intermediate temperatures are raised so that each section of the reactor will operate adiabatically. Dividing the reactor into sections also has the advantage that the intermediate temperature can be adjusted independently of the inlet temperature; thus an optimum temperature distribution can be achieved. In this example, we can see that the furnaces where heat is transferred and the catalytic reactors are quite separate units, each designed specifically for one function. This separation of function generally provides ease of control, flexibility of operation, and often leads to a good overall engineering design. Reactors 1 2 3 Reactants Reactor charge furnace Products Intermediate furnace Figure 1.3 Reactor system of a petroleum naphtha catalytic reforming plant. (The reactor is divided into three units each of which operates adiabatically, the heat required being supplied at intermediate stages via an external furnace). Reactor DesigndGeneral Principles 9 (A) (B) (C) Figure 1.4 Batch reactors showing different methods of heating or cooling. (A) Jacket. (B) Internal coils. (C) External heat exchangers. 188.8.131.52 Reactors With Heat Transfer If the reactor does not operate adiabatically, then its design must include provision for heat transfer. Fig. 1.4 shows some of the ways in which the contents of a batch reactor may be heated or cooled. In A and B, the jacket and the coils form part of the reactor itself, whereas in C, an external heat exchanger is used with a recirculating pump. If one of the constituents of the reaction mixture, possibly a solvent, is volatile at the operating temperature, the external heat exchanger may be a reflux condenser, just as in the laboratory. Fig. 1.5 shows ways of designing tubular reactors to include heat transfer. If the amount of heat to be transferred is large, then the ratio of heat transfer surface to reactor volume will be large, and the reactor will look very much like a heat exchanger as in Fig. 1.5B. If the reaction has to be carried out at a high temperature and is strongly endothermic (for example, the production of ethylene by the thermal cracking of naphtha or (B) (A) (C) Reactants Convection section Radiant section Products Burners Figure 1.5 Methods of heat transfer to tubular reactors. (A) Jacketed pipe. (B) Multitube reactor (tubes in parallel). (C) Pipe furnace (pipes mainly in series although some pipe runs may be in parallel). 10 Chapter 1 ethanedsee also Section 1.7.1, Example 1.5), the reactor will be directly fired by the combustion of oil or gas and will look like a pipe furnace (Fig. 1.5C). 184.108.40.206 Autothermal Reactor Operation If a reaction requires a relatively high temperature before it will proceed at a reasonable rate, the products of the reaction will leave the reactor at a high temperature and, in the interests of economy, heat will normally be recovered from them. Since heat must be supplied to the reactants to raise them to the reaction temperature, a common arrangement is to use the hot products to heat the incoming feed as shown in Fig. 1.6A. If the reaction is sufficiently exothermic, enough heat will be produced in the reaction to overcome any losses in the system and to provide the necessary temperature difference in the heat exchanger. The term autothermal is used to describe such a system which is completely self-supporting in its thermal energy requirements. The essential feature of an autothermal reactor system is the feedback of reaction heat to raise the temperature and hence the reaction rate of the incoming reactant stream. Fig. 1.6 shows a number of ways in which this can occur. With a tubular reactor the feedback may be achieved by external heat exchange, as in the reactor shown in Fig. 1.6A, or by internal heat exchange as in Fig. 1.6B. Both of these are catalytic reactors; their thermal characteristics are discussed in more detail in Chapter 3, Section 3.6.2. Being catalytic the reaction can only take place in that part of the reactor which holds the catalyst, so the temperature profile has the form indicated alongside the reactor. Fig. 1.6C shows a CSTR in which the entering cold feed immediately mixes with a large volume of hot products and rapid reaction occurs. The combustion chamber of a liquid-fueled rocket motor is a reactor of this type, the products being hot gases which are ejected at high speed. Fig. 1.6D shows another type of combustion process in which a laminar flame of conical shape is stabilized at the orifice of a simple gas burner. In this case the feedback of combustion heat occurs by transfer upstream in a direction opposite to the flow of the cold reaction mixture. Another feature of the autothermal system is that, although ultimately it is self-supporting, an external source of heat is required to start it up. The reaction has to be ignited by raising some of the reactants to a temperature sufficiently high for the reaction to commence. Moreover, a stable operating state may be obtainable only over a limited range of operating conditions. This question of stability is discussed further in connection with autothermal operation of a CSTR (Section 1.8.4). 1.3 Choice of Process Conditions The choice of temperature, pressure, reactant feed rates, and compositions at the inlet to the reactor is closely bound up with the basic design of the process as a whole. In arriving Reactor DesigndGeneral Principles 11 T (A) Position in reactor Reactor Heat exchanger Products Position in heat exchanger Inlet reactants Reactants Outlet products T Position in reactor (B) Reactants Inlet reactants Products Outlet products Cold reactants Products (C) T Warm products Reactants Position (D) Conical flame front z Hot products T Cold fuel gas Burner Cold fuel gas z Position through flame front Figure 1.6 Autothermal reactor operation. (A) Tubular reactor with external heat exchange. (B) Tubular reactor with internal heat exchange. (C) CSTR. (D) Gas burner in a combustion process. 12 Chapter 1 at specifications for these quantities, the engineer is guided by knowledge available on the fundamental physical chemistry of the reaction. Usually he or she will also have results of laboratory experiments giving the fraction of the reactants converted and the products formed under various conditions. Sometimes he or she may have the benefit of highly detailed information on the performance of the process from a pilot plant, or even a largescale plant. Although such direct experience of reactor conditions may be invaluable in particular cases, we shall here be concerned primarily with design methods based upon fundamental physicochemical principles. 1.3.1 Chemical Equilibria and Chemical Kinetics The two basic principles involved in choosing conditions for carrying out a reaction are thermodynamics, under the heading of chemical equilibrium, and chemical kinetics. Strictly speaking, every chemical reaction is reversible and, no matter how fast a reaction takes place, it cannot proceed beyond the point of chemical equilibrium in the reaction mixture at the particular temperature and pressure concerned. Thus, under any prescribed conditions, the principle of chemical equilibrium, through the equilibrium constant, determines how far the reaction can possibly proceed given sufficient time for equilibrium to be reached. On the other hand, the principle of chemical kinetics determines at what rate the reaction will proceed towards this maximum extent. If the equilibrium constant is very large, then for all practical purposes the reaction may be said to be irreversible. However, even when a reaction is claimed to be irreversible an engineer would be very unwise not to calculate the equilibrium constant and check the position of equilibrium, especially if high conversions are required. In deciding process conditions, the two principles of thermodynamic equilibrium and kinetics need to be considered together; indeed, any complete rate equation for a reversible reaction will include the equilibrium constant or its equivalent (see Section 1.5.4), but complete rate equations are not always available to the engineer. The first question to ask is in what temperature range will the chemical reaction take place at a reasonable rate (in the presence, of course, of any catalyst, which may have been developed for the reaction)? The next step is to calculate values of the equilibrium constant in this temperature range using the principles of chemical thermodynamics. (Such methods are beyond the scope of this chapter and any reader unfamiliar with this subject should consult a standard textbook.2) The activity-based equilibrium constant Ka of a reaction depends only on the temperature as indicated by the relation: d ln Ka Dhrxn ¼ s2 (1.1) dT RT where Dhrxn s is the standard enthalpy change of reaction or more commonly known as the standard heat of reaction. If the reaction mixture is an ideal gas mixture, then Ka may be Reactor DesigndGeneral Principles 13 replaced by Kp, the partial pressureebased equilibrium constant, in Eq. (1.1). The equilibrium constant is then used to determine the limit to which the reaction can proceed under the conditions of temperature, pressure, and reactant compositions which appear to be most suitable. 1.3.2 Calculation of Equilibrium Conversion Whereas the equilibrium constant itself depends on the temperature only, the conversion at equilibrium depends on the composition of the original reaction mixture and, in general, on the pressure. If the equilibrium constant is very high, the reaction may be treated as being irreversible. If the equilibrium constant is low, however, it may be possible to obtain acceptable conversions only by using high or low pressures. Two important examples are the following reactions: C2H4 þ H2O # C2H5OH N2 þ 3H2 # 2NH3 both of which involve a decrease in the number of moles as the reaction proceeds, and therefore high pressures are used to obtain satisfactory equilibrium conversions. Thus, in those cases in which reversibility of the reaction imposes a serious limitation, the equilibrium conversion must be calculated in order that the most advantageous conditions to be employed in the reactors may be chosen; this may be seen in detail in the following example of the styrene process. A study of the design of this process is also very instructive in showing how the basic features of the reaction, namely equilibrium, kinetics, and suppression of by-products, have all been satisfied in quite a clever way by using steam as a diluent. Example 1.1 A Process for the Manufacture of Styrene by the Dehydrogenation of Ethylbenzene Let us suppose that we are setting out from first principles to investigate the dehydrogenation of ethylbenzene, which is a well-established process for manufacturing styrene: C6H5 $ CH2 $ CH3 ¼ C6H5 $ CH:CH2 þ H2 There is a catalyst available, which will give a suitable rate of reaction at 560 C. At this temperature the pressure-based equilibrium constant for the reaction above is PSt PH ¼ Kp ¼ 100 mbar ¼ 104 N m2 (A) PEt where PEt, PSt, and PH are the partial pressures of ethylbenzene, styrene, and hydrogen, respectively. 14 Chapter 1 Part (i) Feed pure ethylbenzene: If a feed of pure ethylbenzene is used at 1 bar pressure, determine the fractional conversion at equilibrium. Solution This calculation requires not only the use of the equilibrium constant, but also a material balance over the reactor. To avoid confusion, it is as well to set out this material balance quite clearly even in this comparatively simple case. First it is necessary to choose a basis; let this be 1 mol of ethylbenzene fed into the reactor: a fraction ae of this will be converted at equilibrium. Then, from the above stoichiometric equation, ae mole styrene and ae mole hydrogen are formed, and (1 ae) mole ethylbenzene remains unconverted. Let the total pressure at the outlet of the reactor be P which we shall later set equal to 1 bar. C6H5 · C2H5 C6H5 · C2H5 → REACTOR → C6H5 · C2H3 Temperature 560°C = 833 K H2 Pressure P (1 bar = 1.0 × 105 N/m2) In C6H5 $ C2H5 Mole 1 a Mole 1 ae C6H5 $ C2H3 e H2 e Total Out b Mole fraction 1ae 1þae c Partial pressure 1ae 1þae P ae ae 1þae ae 1þae P ae ae 1þae ae 1þae P 1 þ ae Since for 1 mol of ethylbenzene entering, the total number of moles increases to 1 þ ae, the mole fractions of the various species in the reaction mixture at the reactor outlet are shown in column b above. At a total pressure P, the partial pressures are given in column c. If the reaction mixture is at chemical equilibrium, these partial pressures must satisfy Eq. (A) above: ae ae P P PSt PH ð1 þ ae Þ ð1 þ ae Þ a2e ¼ Kp ¼ ¼ P ð1 ae Þ PEt 1 a2e P ð1 þ ae Þ i.e., a2e P ¼ 1:0 104 N m2 2 1 ae Thus, when P ¼ 1 bar, ae ¼ 0.30; i.e., the maximum possible conversion using pure ethylbenzene at 1 bar is only 30%; this is not very satisfactory (although it is possible in some processes to operate at low conversions by separating and recycling reactants). Ways of improving this figure are now sought. (B) Reactor DesigndGeneral Principles 15 Note that Eq. (B) shows that as P decreases, ae increases; this is the quantitative expression of Le Chatelier's principle that, because the total number of moles increases in the reaction, the decomposition of ethylbenzene is favored by a reduction in pressure. There are, however, disadvantages in operating such a process at subatmospheric pressures. One disadvantage is that any ingress of air through leaks might result in ignition. A better solution in this instance is to reduce the partial pressure by diluting the ethylbenzene with an inert gas, while maintaining the total pressure slightly in excess of atmospheric. The inert gas most suitable for this process is steam; one reason for this is that it can be condensed easily in contrast to a gas such as nitrogen, which would introduce greater problems in separation. Part (ii) Feed ethylbenzene with steam: If the feed to the process consists of ethylbenzene diluted with steam in the ratio 15 mol steam:1 mol ethylbenzene, determine the new fractional conversion at equilibrium a0 e . Solution Again we set out the material balance in full, the basis being 1 mole ethylbenzene into the reactor. C6H5 · C2H5 H2O C6H5 · C2H5 → REACTOR → C6H5 · C2H3 Temperature 560°C = 833 K H2 Pressure P (1 bar = 1.0 × 105 N/m2) H2O In C6H5 $ C2H5 Mole 1 C6H5 $ C2H3 e H2 e H2 O 15 Out b Mole fraction a Mole 1 a0 e 0 a0 e 16þa0 e ae 0 a0 e 15 Total 1a0 e 16þa0 e 16 þ a0 e ae 16þa0 e 15 16þa0 e c Partial pressure 1a0 e 16þa0 'e a0 e 16þa0 e P P 0 ae 16þa0 e P i.e., a0 e a0 e P P PSt PH ð16 þ a0 e Þ ð16 þ a0 e Þ Kp ¼ ¼ PEt ð1 a 0 e Þ P ð16 þ a0 e Þ a0 e P ð16 þ a0 e Þð1 a0 e Þ 2 ¼ a0 e P ¼ 1:0 104 0 ð16 þ a e Þð1 a0 e Þ 2 (C) 16 Chapter 1 Thus when P ¼ 1 bar, a0 e ¼ 0:70, i.e., the maximum possible conversion has now been raised to 70%. Inspection of Eq. (C) shows that the equilibrium conversion increases as the ratio of steam to ethylbenzene increases. However, as more steam is used, its cost increases and offsets the value of the increase in ethylbenzene conversion. The optimum steam:ethylbenzene ratio is thus determined by an economic balance. Part (iii) Final choice of reaction conditions in the styrene process. Solution The use of steam has a number of other advantages in the styrene process. The most important of these is that it acts as a source of internal heat supply so that the reactor can be operated adiabatically. The dehydrogenation reaction is strongly endothermic, the standard heat of reaction at 560 C being Dhrxn s ¼ 125,000 kJ/kmol. It is instructive to look closely at the conditions that were originally worked out for this process (Fig. 1.7). Most of the steam, 90% of the total used, is heated separately from the ethylbenzene stream, and to a higher temperature (710 C) than is required at the inlet to the reactor. The ethylbenzene is heated in the heat exchangers to only 520 C and is then rapidly mixed with the hotter steam to give a temperature of 630 C at the inlet to the catalyst bed. If the ethylbenzene were heated to 630 C more slowly by normal heat exchange decomposition and coking of the heat transfer surfaces would tend to occur. Moreover, the tubes of this heat exchanger would have to be made of a more expensive alloy to resist the more severe working conditions. To help avoid coking, 10% of the steam used is passed through the heat exchanger with the ethylbenzene. The presence of a large proportion of steam in the reactor also prevents coke deposition on the catalyst. By examining the equilibrium constant of reactions involving carbon such as C6H5 $ CH2 $ CH3 # 8C þ 5H2 C þ H2O # CO þ H2 it may be shown that coke formation is not possible at high steam:ethylbenzene ratios. The styrene process operates with a fractional conversion of ethylbenzene per pass of 0.40 compared with the equilibrium conversion of 0.70. This actual conversion of 0.40 is determined by the rate of the reaction over the catalyst at the temperature prevailing in the reactor. (Adiabatic operation means that the temperature falls with increasing conversion and the reaction tends to be quenched at the outlet.) The unreacted ethylbenzene is separated and recycled to the reactor. The overall yield in the process, i.e., moles of ethylbenzene transformed into styrene per mole of ethylbenzene supplied, is 0.90, the remaining 0.10 being consumed in unwanted side reactions. Notice that the conversion per pass could be increased by increasing the temperature at the inlet to the catalyst bed beyond 630 C, but the undesirable side reactions would increase and the overall yield of the process would fall. The figure of 630 C for the inlet temperature is thus determined by an economic balance between the cost of separating unreacted ethylbenzene (which is high if the inlet temperature and conversion per pass are low) and the cost of ethylbenzene consumed in wasteful side reactions (which is high if the inlet temperature is high). Reactor DesigndGeneral Principles 17 Vaporiser Ethylbenzene Feed/product heat exchangers 160°C 10% 520°C 710°C Steam 90% 630°C Superheater 560°C Styrene-hydrogenethylbenzene product Reactor Figure 1.7 A process for styrene from ethylbenzene using 15 mol steam: 1 mol ethylbenzene, operating pressure 1 bar, conversion per pass 0.40, overall relative yield 0.90. 1.3.3 Ultimate Choice of Reactor Conditions The use of steam in the styrene process mentioned above is an example of how an engineer can exercise a degree of ingenuity in reactor design. The advantages conferred by the steam may be summarized as follows: 1. it lowers the partial pressure of the ethylbenzene without the need to operate at subatmospheric pressures; 2. it provides an internal heat source for the endothermic heat of reaction, making adiabatic operation possible; and 3. it prevents coke formation on the catalyst and coking problems in the ethylbenzene heaters. As the styrene process shows, it is not generally feasible to operate a reactor with a conversion per pass equal to the equilibrium conversion. The rate of a chemical reaction decreases as equilibrium is approached, so that the equilibrium conversion can only be attained if either the reactor is very large or the reaction is unusually fast. The size of reactor required to give any particular conversion, which of course cannot exceed the maximum conversion predicted from the equilibrium constant, is calculated from the 18 Chapter 1 kinetics of the reaction. For this purpose we need quantitative data on the rate of reaction, and the rate equations which describe the kinetics are considered in Section 1.5. If there are two or more reactants involved in the reaction, both can be converted completely in a single pass only if they are fed to the reactor in the stoichiometric proportion. In many cases, the stoichiometric ratio of reactants may be the best, but in some instances, where one reactant (especially water or air) is very much cheaper than the other, it may be economically advantageous to use it in excess. For a given size of reactor, the object is to increase the conversion of the more costly reactant, possibly at the expense of a substantial decrease in the fraction of the cheaper reactant converted. Examination of the kinetics of the reaction is required to determine whether this can be achieved and calculate quantitatively the effects of varying the reactant ratio. Another and perhaps more common reason for departing from the stoichiometric proportions of reactants is to minimize the amount of by-products formed. This question is discussed further in Section 1.10.4. Ultimately, the final choice of the temperature, pressure, reactant ratio, and conversion at which the reactor will operate depends on an assessment of the overall economics of the process. This will take into account the cost of the reactants, the cost of separating the products, and the cost associated with any recycle streams. It should include all the various operating costs and capital costs of reactor and plant. In the course of making this economic assessment, a whole series of calculations of operating conditions, final conversion, and reactor size may be performed with the aid of a computer, provided that the data are available. Each of these sets of conditions may be technically feasible, but the one chosen will be that which gives the maximum profitability for the project as a whole. 1.4 Material and Energy Balances The ability to write down material and energy balance equations for a particular reactor configuration is the first step toward reactor analysis and design. It is important to be familiar with both an overall balance for the reactor as a whole and also a local balance valid at each point in the reactor. The latter is required in cases such as the plug flow reactor (PFR) where concentrations and/or temperature vary with position within the reactor. The treatment of material and energy balances adopted here is different from those in the standard reaction engineering textbooks and is more akin to what is found in books on transport phenomena.3 Reaction engineering textbooks can afford a simplified treatment because the systems they primarily deal with are ideal reactors. There is an attempt here to show that the fundamental forms of the equations found in the latter3 provide a secure starting point on two counts: (1) the simplifications that lead to the final equations are made transparent and (2) a pointer to handling more complicated reactor systems is Reactor DesigndGeneral Principles 19 provided. But there is a price to be paid. The reader has to become familiar with a few simple vector operations such as the scalar or dot product, gradient, and divergence. These are explained briefly wherever they occur in this chapter. For a more detailed treatment, the reader may consult Bird et al.3 We first consider the simpler case of material balances and then undertake a derivation of the energy balance equations. 1.4.1 Material Balance and the Concept of Rate of Generation of a Species We start with the material balance for a fixed control volume, Vc, through which a single component fluid flows. The mass balance can be written as Rate of accumulation ¼ input rate output rate. (1.2) A very general representation of the above equation can be given within the framework of continuum mechanics: Z Z dm d ¼ r dV ¼ rv$n dA (1.3) dt dt Vc Ac R where m ¼ Vc rdV represents the mass of fluid inside the control volume at any time, t, with r ¼ r(x,t) denoting the fluid density at a location indicated by the position vector x at an instant of time t. The quantity v is the velocity vector, whereas n is the outward unit normal vector. The area integral in the equation has the following interpretation: the quantity rv is a vector and has the unit of mass/area/time. Thus it is denoted as the mass flux vector. Of course, mass itself is a scalar quantity and the term rv$n represents the flux of mass out of the control volume at a particular location on the surface enclosing the control volume where the fluid density is r, the velocity is v, and the unit outward normal is n. The quantity rv$n is the scalar dot product of the vector rv with the vector n and is given by rv$n ¼ rvx nx þ rvy ny þ rvz nz in the familiar rectilinear Cartesian coordinate system, where (vx, vy, vzR) and (nx, ny, nz) are the components of the vector v and n, respectively. Hence the integral R Ac rv$n dA represents the net efflux of mass from the control volume whereas Ac rv$n dA represents the net influx (ineout) of mass into the control volume with Ac representing the area enclosing the control volume. Thus Eq. (1.3) may be written in a more familiar form as where mr,in and mr,out control volume. dm ¼ mr;in mr;out dt are, respectively, the rate at which mass flows in and out of the 20 Chapter 1 An extension of the above equations to mixtures (having n components, say) would involve two factors: (1) a mass balance for each of the n components, (2) generation (or disappearance) of mass of a component due to chemical reactions. Thus Eq. (1.2) is to be modified as ðRate of accumulationÞi ¼ ðinput rateÞi ðoutput rateÞi þ ðgeneration rateÞi ; i ¼ 1; .; n. Correspondingly, Eq. (1.3) may be generalized as Z Z Z dmi d r dV ¼ ri vi $n dA þ Ri dV; i ¼ 1; .; n ¼ dt Vc i dt Ac Vc (1.4) where ri is the mass density of species i and vi is the velocity of species i. The quantity Ri represents the net rate of generation per unit volume of species i due to chemical reactions. It is expressed as mass/volume/time for a homogeneous reaction, which is our R focus in this chapter. Thus the integral Vc Ri dV is the net rate of generation of species i due to chemical reactions in the control volume. If the quantity Ri is negative, then there is a net disappearance of species i due to chemical reactions. This means that the rate at which species i is being produced by reactions is less than the rate at which it is being consumed. The other two terms in Eq. (1.4) have an interpretation similarRto that of Eq. (1.3), except now that they refer to a particular component i. Thus, Ac ri vi $n dA is R the net influx of species i into the control volume and Vc ri dV ¼ mi is the mass of species i in the control volume at time t. Summing all the equations in (1.4), we obtain d n P i¼1 dt mi Z ¼ n X Ac i¼1 ! ri vi $n dA þ Z n X Vc i¼1 ! Ri dV. (1.5) P We denote m ¼ nj¼1 mj as the total mass of the mixture in the control volume, r ¼ P . Pn n r as the total local density of the mixture, and v ¼ r v r as the mass i j¼1 j i¼1 i average velocity of the mixture. Further, we note that n X i¼1 Ri ¼ 0. (1.6) This expresses the fact while one species may be converted to another through chemical reactions, there can be no net creation or destruction of mass due to reactions. Thus Eq. (1.5) may be written as Z dm (1.7) ¼ rv$n dA dt Ac Reactor DesigndGeneral Principles 21 which looks identical to Eq. (1.3), but the quantities in Eq. (1.7) refer to mixture quantities. Dividing each of the equations in Eq. (1.4) by the molecular weight of the corresponding species i, we obtain the molar analog of Eq. (1.4): Z Z Z dni d ¼ Ci dV ¼ Ci vi $n dA þ ri dV; i ¼ 1; .; n dt Vc dt Ac Vc (1.8) where ni is the number of moles of component i within the control volume at time t, Ci is the molar concentration of species i, and rRi is the net molar rate of generation of species i expressed as moles/volume/time. Since Ac Ci vi $n dA represents the net molar rate of species i entering the reactor, Eq. (1.8) may be written as Z dni (1.9) ¼ Fi;in Fi;out þ ri dV; i ¼ 1; .; n dt Vc where Fi,in and Fi,out are the inlet and outlet molar rates, respectively, of species i. In the most common case, there is one inlet and one outlet to a reactor. The quantity ri occupies a central place in reaction engineering and its determination and representation will be discussed in detail in Section 1.5. Often rA,, which represents the rate of disappearance of a component A, is used to represent the rate of a reaction in which A is a reactant. Eqs. (1.4), (1.8), and (1.9) are sufficient for the analysis of the well-mixed batch and CSTRs. For a batch reactor, the flow terms in Eq. (1.9) are zero and hence the rate of accumulation of species i equals its rate of generation by reaction. For a CSTR, if operating in a steady state, the rate of accumulation term (left-hand side of Eq. 1.9) is zero, and hence the rate of generation of species i by reaction is just balanced by the difference between outflow and inflow. For unsteady state operation of a flow reactor, it is important to appreciate the distinction between the reaction term and the accumulation term, which are equal for a batch reactor. Transient operation of a flow reactor occurs during start-up and in response to disturbances in the operating conditions. The nature of transients induced by disturbances and the differences between the reaction and accumulation terms can best be visualized for the case of a CSTR (Fig. 1.8). In Fig. 1.8A, the reactor is operating in a steady state. In Fig. 1.8B, it is subject to an increase in the input of reactant owing to a disturbance in the feed composition. This results in a rise in the concentration of the reactant within the reaction vessel corresponding to the accumulation term which is quite distinct from, and additional to, the reactant removal by the reaction term. Fig. 1.8C shows another kind of transient which will cause compositions in the reactor to change, namely a change in the volume of reaction mixture contained in the reactor. Other variables which must be controlled, apart from feed composition and flow rates in and out, are temperature and, for gas reactions particularly, pressure. Variations of any of these quantities with time will cause a change in the composition levels of the reactant in the reactor, and these will 22 Chapter 1 (A) (B) (C) Feed composition changed Constant feed Feed flow rate increased Level constant Constant level Level rises Reactor compositions change Constant reactor compositions Reactor compositions change Figure 1.8 Continuous stirred tank reactor showing steady-state operation (A), and two modes of unsteady state operation (B) and (C). appear in the accumulation term in the material balance. Of course, since the reaction rate is also affected by the composition levels in the reactor, the reaction term in Eq. (1.9) will also be affected. For the PFR, where there is a continuous variation of composition along the length of the reactor, a local continuity equation is required. Towards this end, we use the divergence theorem3 given by Z Z ri vi $n dA ¼ V$ri vi dV (1.10) Ac Vc where "V$" is the divergence operator. [In rectilinear Cartesian coordinate system, the divergence of vector a, with components ax, ay, and az, is given by V$a ¼ vax vay vaz þ þ vx vy vz whereas in the cylindrical coordinate system, defined by the coordinates (rc, f, z), V$a ¼ 1 v 1 vaf vaz þ ; ðrc arc Þ þ rc vrc rc vf vz where arc ; af ; az are the components of the vector a in the cylindrical coordinates.] Using Eq. (1.10) in Eq. (1.4), and bringing all the terms into a single volume integral, we obtain Z vri (1.11) þ V$ri vi Ri dV ¼ 0; i ¼ 1; .; n. Vc vt It must be noted that the time derivative operator can be taken inside the integral in Eq. (1.4) because the control volume is fixed in space. The vanishing of an integral does not Reactor DesigndGeneral Principles 23 necessarily mean vanishing of the integrand. But in this case, using standard limiting arguments, we may conclude that vri þ V$ri vi ¼ Ri ; i ¼ 1; .; n (1.12) vt While Eqs. 1.3, 1.4, 1.8, and 1.9 are global balances valid for the system as a whole, Eq. (1.12) is a local balance equation, valid at every point within the system. Summing Eq. (1.12), we obtain the overall mixture continuity equation: vr þ V$rv ¼ 0; vt where Eq. (1.6) has been used. The molar analog of Eq. (1.12) is vCi þ V$Ci vi ¼ ri ; i ¼ 1; .; n. vt (1.13) (1.14) 1.4.2 Energy Balance The earliest instance of an energy balance came out of classical thermodynamics through the first law of thermodynamics for a closed system containing a homogeneous mass of a single component fluid. It may be expressed as dEI ¼ Qr þ Wr dt where EI is the internal energy of the mass of fluid in the system, Qr is the rate at which heat is added to the system, and Wr is the rate at which work is done on the system. To extend this equation to open systems would require taking into account the kinetic energy as well as the flow of energy in and out of the system. A general balance for a fixed control volume may be expressed as dðEI þ EK Þ ¼ net influx ðEI þ EK Þ þ Qr þ Wr (1.15) dt where EK refers to the kinetic energy. The heating rate can be divided into two parts, one arising out of transfer of heat by conduction across the surface enclosing the control volume and another by radiation. In most chemical engineering applications, the effect of radiation can be neglected. The rate of work done can also be classified into two types, one termed as "flow work" and another referred to as "shaft work." The former is the work done on the fluid to cause its flow through the control volume, whereas the latter arises out of work done by stirrer, mixer, etc. The flow work can be further divided into two parts: work due to body forces and work due to contact force or stress force. The work done by 24 Chapter 1 the contact force consists in turn of work due to pressure force and that due to shear force. If the body forces are conservative, that is, if they are derivable from a time-independent potential, then the potential energy term can be added to the internal and kinetic energy in Eq. 1.15.3 Similarly the work due to the pressure force can be combined with the influx of internal energy to yield the influx of enthalpy. In summary, with these modifications, Eq. (1.15) takes the form dðEI þ EK þ EP Þ ¼ net influx ðH þ EK þ EP Þ þ Qr þ Wr;shear þ Wr;shaft dt where H refers to the enthalpy and EP to the potential energy. (1.16) We first deal with simpler forms of the above equation because in reaction engineering, usually, kinetic energy and potential energy are neglected when compared to the internal energy or enthalpy. The shear stress work term is also neglected. Further, following the continuum hypothesis, we may express the total internal energy of the fluid within the control volume as Z EI ¼ Ce εdV Vc where C is the molar concentration and e ε is the specific internal energy per mole. Further, the net influx of enthalpy may be written as Z e Net influx of enthalpy into control volume ¼ C hv$ndA Ac where he is the specific enthalpy per mole. With the above approximations and representations, Eq. (1.16) becomes Z Z d e Ce εdV ¼ Chv$n dA þ Qr þ Wr;shaft dt Vc Ac (1.17) All of the above equations apply directly to a fluid with only one component. For a mixture, which is what we have to deal with in reaction engineering, it appears the convention to adopt the same form as the above equations except that the quantities in the equation are to be regarded as corresponding to the mixture. The simplest representation of mixture properties is in terms of the pure component properties and is obtained by neglecting the so-called change in properties associated with mixing. Thus, to get the mixture equations, we apply the following rules: e ε/ n X i¼1 e xie εi ; Chv/ n X i¼1 Ci hei vi Reactor DesigndGeneral Principles 25 where xi is the mole fraction of species i in the mixture, eεi and hei are the pure component-specific internal energy and enthalpy of species i at the temperature and pressure of the mixture. Thus Eq. (1.17) may be written as ! Z X Z n n X d Cie Ci hei vi $ndA þ Qr þ Wr;shaft . εi dV ¼ dt Vc i¼1 Ac i¼1 (1.18) At steady state, we obtain Z X n Ac i¼1 Ci hei vi $ndA þ Qr þ Wr;shaft ¼ 0: (1.19) In the most common case of one inlet and one outlet and when hei is independent of position on the control surface at the inlet and outlet, we may write Eq. (1.19) as n X i¼1 Fi;in hei;in n X i¼1 Fi;out hei;out þ Qr þ Wr;shaft ¼ 0 (1.20) R We note that the magnitude of Ci vi $n dA evaluated at the inlet will equal Fi,in, whereas that evaluated at the outlet will yield Fi,out. Eq. (1.20) is a convenient starting point for analyzing the nonisothermal operation of a CSTR under steady-state conditions (Section 1.8.4). The developments leading from Eqs. (1.15) to (1.20) serve to bring out the generalizations and simplifications inherent in Eq. (1.20). To develop further the unsteady state equation, we express the accumulation term too in terms of enthalpy. Noting that hei ¼ eεi þ Pe vi (1.21) where vei is the molar specific volume of pure component i at the temperature (T) and pressure (P) of the mixture, we obtain n X i¼1 Cieεi ¼ n X i¼1 Ci hei P n X i¼1 Ci vei However, n X i¼1 Ci vei ¼ C n X i¼1 xi vei ¼ Ce v¼1 where ve is the molar volume of the mixture (volume/mole) and is the inverse of C, the P molar concentration of the mixture (mole/volume). In equating ni¼1 xi vei to ve in (1.22) 26 Chapter 1 Eq. (1.22), we follow the spirit of our earlier stated assumption that change in properties on mixing is neglected. Hence from Eqs. (1.21) and (1.22), n X i¼1 Cieεi ¼ n X i¼1 Ci hei P (1.23) Therefore, v vt n X i¼1 ! Cie εi v ¼ vt n X i¼1 ! Ci hei n vP X vhei e vCi Ci ¼ þ hi vt vt vt i¼1 ! vP vt (1.24) Further, hei is a function of P and T so that ! ! vhei vhei vP vhei vT þ . (1.25) ¼ vt vt vP vt vT T P The derivative vhei vP may be obtained from an equation of state for the component i T and is usually neglected for liquid-phase reactions. The derivative vhei vT is the P (molar) specific heat at constant pressure denoted by cePi. Taking into account these definitions and using Eqs. (1.24) and (1.25) in Eq. (1.18), we obtain ! ! ! ! Z Z Z n n n X X vP X vhei vT vC i dV þ dV hei Ci 1 dV þ Ci cePi vt vt vP vt Vc Vc i¼1 Vc i¼1 i¼1 T Z X n ¼ Ci hei vi $n dA þ Qr þ Wr;shaft . Ac i¼1 (1.26) Eq. (1.26) is adequate for analyzing the nonisothermal operation of a batch reactor (Sections 1.6.4 and 1.6.5). As we have remarked above, Eq. (1.20) is a good starting point for analyzing a CSTR operating at steady state. For the PFR, we need a local balance. A simplified set of equations useful for applying directly to a CSTR and a PFR operating at steady state is presented in the Appendix at the end of this chapter. 1.5 Chemical Kinetics and Rate Equations When a homogeneous mixture of reactants is passed into a reactor, either batch or tubular, the concentrations of the reactants fall as the reaction proceeds. Experimentally it has been found that, in general, the rate of the reaction decreases as the concentrations of the reactants decrease. To calculate the size of the reactor required to manufacture a particular product at a desired overall rate of production, the design engineer therefore needs to Reactor DesigndGeneral Principles 27 know how the rate of reaction at any time or at any point in the reactor depends on the concentrations of the reactants. Since the reaction rate varies also with temperature, generally increasing rapidly with increasing temperature, a rate equation, expressing the rate of formation of a species i (the quantity ri in Eqs. 1.8, 1.9, and 1.14) as a function of concentrations and temperature, is required to design a reactor. 1.5.1 Definition of Order of Reaction and Rate Constant For concreteness, let us consider a homogeneous irreversible reaction: aA þ bB / cC þ dD ð1:27Þ where A and B are reactants, C and D are products. The above equation represents the fact that a moles of A and b moles of B react to yield c moles of C and d moles of D. A fundamental relation between the rates of generation of the species involved in the above reaction is given by rA rB rC rD ¼ ¼ ¼ nA nB nC nD (1.28) where vA, vB, vc, and vD are the corresponding stoichiometric coefficients in the reaction (1.27). In the reaction as written, nA ¼ a; nB ¼ b; nC ¼ c; nD ¼ d. That is, as per convention, the stoichiometric coefficients of the components that appear on the reactant side are assigned a negative sign. It must be noted that Eq. (1.27) is not a unique representation of the stoichiometry of the reaction. For instance, it can also be represented by aaA þ abB / acC þ adD in which case nA ¼ aa; nB ¼ ab; nC ¼ ac; nD ¼ ad. However, as may be seen from Eq. (1.28), the relationship between the rates of generation of the various species remains unaltered. Eq. (1.28) may be more generally represented by ri rj ¼ ni nj (1.29) for any two species i and j involved in a single reaction. Eq. (1.29) suggests a definition of a reaction rate, r, which is independent of the species, namely, ri rh ; ni (1.30) 28 Chapter 1 where i is any species involved in the reaction. Following convention in chemical engineering, we prefer to deal with a particular rate ri than r. At constant temperature, the rate of generation of a species A, rA, or equivalently the rate of disappearance of A, rA,, in a reaction such as 1.27, is a function of the concentrations of the reactants. Experimentally, it has been found that often (but not always) the function has the mathematical form rA ¼ kCAp CBq (1.31) CA is the molar concentration of A and CB the molar concentration of B. The exponents p and q in this expression are quite often (but not necessarily) whole numbers. When the functional relationship has the form of Eq. (1.31), the reaction is said to be of order p with respect to reactant A, q with respect to B. The overall order of the reaction is (p þ q). The coefficient k in Eq. (1.31) is referred to as the rate constant of the reaction. Its dimensions depend on the exponents p and q (i.e., on the order of the reaction); the units in which it is to be expressed may be inferred from the defining Eq. (1.31). For example, if a reaction A / Products behaves as a simple first-order reaction, it has a rate equation rA ¼ k1 CA (1.32) If rA is measured in units of kmol/m3/s and the concentration CA in kmol/m3, then k1 has the unit s1. On the other hand, if the reaction above behaved as a second-order reaction with a rate equation rA ¼ k2 CA 2 (1.33) the unit of this rate constant, with rA in kmol/m3 s and CA in kmol/m3, is m3(kmol)1 s1. A possible source of confusion is that in some instances in the chemical literature, the rate equation, for, say, a second-order gas-phase reaction may be written as rA ¼ kp P2A , where PA is the partial pressure of A and may be measured in N/m2, bar, or even in mm Hg. This form of expression results in rather confusing hybrid units for kp and is not to be recommended. If a large excess of one or more of the reactants is used, such that the concentration of that reactant changes hardly at all during the course of the reaction, the effective order of the reaction is reduced. Thus, if in carrying out a reaction which is normally second order with a rate equation rA ¼ k2CACB an excess of B is used, then CB remains approximately constant and almost equal to the initial value CB0. The rate equation may then be written rA ¼ k10 CA , where k10 ¼ k2 CB0 is an effective rate constant and the reaction is now said to be pseudoefirst-order. Reactor DesigndGeneral Principles 29 1.5.2 Influence of Temperature: Activation Energy Experimentally, the influence of temperature on the rate constant of a reaction is well represented by the original equation of Arrhenius: k ¼ A expðE=RTÞ (1.34) where T is the absolute temperature and R the universal gas constant. In this equation, E is termed as the activation energy, and A the frequency factor. There are theoretical reasons to suppose that temperature dependence should be more exactly described by an equation of the form k ¼ A0 T m expðE=RTÞ, with m usually in the range 0e2. However, the influence of the exponential term in Eq. (1.34) is in practice so strong as to mask any variation in A with temperature, and the simple form of the relationship, Eq. (1.34), is therefore quite adequate. The quantity E is called the activation energy because in the molecular theory of chemical kinetics it is associated with an energy barrier that the reactants must surmount to form an activated complex in the transition state. Similarly, A is associated with the frequency with which the activated complex breaks down into products or, in terms of the simple collision theory, it is associated with the frequency of collisions. Values of the activation energy E are in J/kmol in the SI system but are usually quoted in kJ/kmol (or J/mol); using these values, R must then be expressed as kJ/kmol K. For most reactions the activation energy lies in the range 50,000e250,000 kJ/kmol, which implies a very rapid increase in rate constant with temperature. Thus, for a reaction which is occurring at a temperature in the region of 100 C and has an activation energy of 100,000 kJ/kmol, the reaction rate will be doubled for a temperature rise of only 10 C. Thus, the complete rate equation for an irreversible reaction such as 1.27 normally has the form rA ¼ A expðE=RTÞCAp CBq (1.35) Unfortunately, the exponential temperature term exp(eE/RT ) is rather troublesome to handle mathematically, both by analytical methods and numerical techniques. In reactor design, this means that calculations for reactors, which are not operated isothermally tend to become complicated. In a few cases, useful results can be obtained by abandoning the exponential term altogether and substituting a linear variation of reaction rate with temperature, but this approach is quite inadequate unless the temperature range is very small. 1.5.3 Rate Equations and Reaction Mechanism One of the reasons why chemical kinetics is an important branch of physical chemistry is that the rate of a chemical reaction may be a significant guide to its mechanism. The 30 Chapter 1 engineer concerned with reactor design and development is not interested in reaction mechanism per se, but should be aware that an insight into the mechanism of the reaction can provide a valuable clue to the kind of rate equation to be used in a design problem. In the present chapter, it will be possible to make only a few observations on the subject, and for further information the excellent text of Moore and Pearson4 should be consulted. The first point that must be made is that the overall stoichiometry of a reaction is no guide whatsoever to its rate equation or to the mechanism of reaction. A stoichiometric equation is no more than a material balance; thus the reaction KClO3 þ 6FeSO4 þ 3H2SO4 / KCl þ 3Fe2(SO4)3 þ 3H2O is in fact second order in dilute solution with the rate of reaction proportional to the 2þ concentrations of ClOe ions. In the general case the stoichiometric coefficients 3 and Fe vA, vB, vC are not necessarily related to the orders p, q, r for the reaction. However, if it is known from kinetic or other evidence that a reaction, M þ N / Product, is a simple elementary reaction, i.e., if it is known that its mechanism is simply the interaction between a molecule of M and a molecule of N, then the molecular theory of reaction rates predicts that the rate of this elementary step is proportional to the concentration of species M and the concentration of species N, i.e., it is second order overall. The reaction is also said to be bimolecular since two molecules are involved in the actual chemical transformation. More generally, we regard a reaction step as elementary as written if the order corresponding to each of the reactants equals its stoichiometric coefficient. Thus, the reaction between H2 and I2 is known to occur by an elementary bimolecular reaction: H2 þ I2 / 2HI and the rate of the forward reaction corresponds to the equation: r12 ¼ kf CH2 CI2 For many years the hydrogeneiodine reaction was quoted in textbooks as being virtually the only known example of a simple bimolecular reaction. There is now evidence5 that in parallel to the main bimolecular transformation, some additional reactions involving iodine atoms do occur. Whereas in the hydrogeneiodine reaction, atomic iodine plays only a minor part, in the reaction between hydrogen and bromine, bromine and hydrogen atoms are the principal intermediates in the overall transformation. Reactor DesigndGeneral Principles 31 The kinetics of the reaction is quite different from that of the hydrogeneiodine reaction, although the stoichiometric equation: H2 þ Br2 / 2HBr looks similar. The reaction actually has a chain mechanism consisting of the elementary steps: Br2 # 2Br$ chain initiaton and termination ) Br$ þ H2 # HBr þ H$ chain propagation H$ þ Br2 / HBr þ Br$ The rate of the last reaction, for example, is proportional to the concentration of H$ and the concentration of Br2, i.e., it is second order. When the rates of these elementary steps are combined into an overall rate equation, this becomes k0 CH2 CBr2 ¼ CHBr 1 þ k00 CBr2 1=2 rHBr (1.36) where k0 and k00 are constants, which are combinations of the rate constants of the elementary steps. This rate equation has a different form from the usual type given by Eq. (1.31) and cannot therefore be said to have any order because the definition of order applies only to the usual form. A basic principle used to obtain expressions such as 1.36 from a mechanism is the pseudoesteady-state approximation (PSSA). According to this principle, the net rate of formation of any reactive intermediate is taken to be zero. The rationale is that such a species reacts as quickly as it is formed. For the HBr formation mechanism, the principle would apply to the radicals H and Br. Thus, rBr$ ¼ rH$ ¼ 0: (1.37) Further, since each of the reactions in the mechanism is an elementary reaction, rBr$ ¼ 2k1 CBr2 k1 ðCBr$ Þ2 k2 CBr$ CH2 þ k2 CH$ CHBr þ k3 CH$ CBr2 ¼ 0 (1.38) and rH$ ¼ k2 CBr$ CH2 k2 CH$ CHBr k3 CH$ CBr2 ¼ 0 (1.39) The factor 2 in the first term on the right-hand side of the equation indicates that k1 represents the rate constant for the disappearance of Br2 through the forward reaction in the initiation step. That is, rBr2 ¼ k1 CBr2 $ From Eqs. (1.38) and (1.39), one obtains 2k1 CBr2 k1 ðCBr$ Þ2 ¼ 0 (1.40) 32 Chapter 1 This implies that the first step is essentially in equilibrium. This is sometimes invoked as an independent principle: a particular elementary step in a mechanism is in equilibrium and is referred to as the quasi-equilibrium assumption. From Eq. (1.40), we get an expression for CBr$ : CBr$ ¼ 2 k1 CBr k1 2 1=2 (1.41) Substituting in Eq. (1.39), we get an expression for CH$ : CH$ ¼ k2 CH2 k1 CBr 2 ðk3 CBr2 þ k2 CHBr Þ k1 2 1=2 . (1.42) The final rate expression is obtained by noting that rHBr ¼ k3 CH$ CBr2 þ k2 CBr$ CH2 k2 CH$ CHBr . Using Eq. (1.39), we obtain rHBr ¼ 2k3 CH$ CBr2 . (1.43) Substituting for CH, from Eq. (1.42) and rearranging, we get 2k1 k1 2k2 rHBr ¼ 1=2 CH2 ðCBr2 Þ1=2 (1.44) k2 CHBr 1þ k3 CBr2 Comparing with Eq. (1.36), we note that 0 k ¼ 2k2 2k1 k1 1=2 ; k00 ¼ k2 . k3 (1.45) We shall find that the rate equations of gasesolid heterogeneous catalytic reactions (Chapter 3) also do not, in general, have the same form as Eq. (1.31). However, many reactions, although their mechanism may be quite complex, do conform to simple first- or second-order rate equations. This is because the rate of the overall reaction is limited by just one of the elementary reactions that is then said to be rate determining. The kinetics of the overall reaction thus reflects the kinetics of this particular step. An example is the pyrolysis of ethane,5 which is important industrially as a source of ethylene1 (see also Section 1.7.1; Example 1.5). The main overall reaction is C2H6 / C2H4 þ H2 Reactor DesigndGeneral Principles 33 Although there are complications concerning this reaction, under most circumstances it is first order, the kinetics being largely determined by the first step in a chain mechanism: C2H6 / 2CH3$ which is followed by the much faster reactions: CH3 $ þ C2 H6 9 / C2 H5 $ þ CH4 > = C2 H5 $ / C2 H4 þ H$ H$ þ C2 H6 / C2 H5 $ þ H2 > ; chain propagation Eventually the reaction chains are broken by termination reactions. Other free radical reactions also take place to a lesser extent leading to the formation of CH4 and some higher hydrocarbons among the products. 1.5.4 Reversible Reactions For reactions that do not proceed virtually to completion, it is necessary to include the kinetics of the reverse reaction, or the equilibrium constant, in the rate equation. The equilibrium state in a chemical reaction can be considered from two distinct points of view. The first is from the standpoint of classical thermodynamics and leads to relationships between the equilibrium constant and thermodynamic quantities such as standard free energy change and heat of reaction, from which we can very usefully calculate equilibrium conversion (Example 1.1 in Section 1.3.2). The second is a kinetic viewpoint, in which the state of chemical equilibrium is regarded as a dynamic balance between forward and reverse reactions; at equilibrium the rates of the forward reactions and reverse reactions are just equal to each other, making the net rate of transformation zero. Consider a reversible reaction: kf A þ B#M þ N kr which is second order overall in each direction and first order with respect to each species. The hydrolysis of an ester such as ethyl acetate is an example: CH3COOC2H5 þ NaOH # CH3COONa þ C2H5OH The net rate of disappearance of species A is given by rA ¼ kf CA CB kr CM CN (1.46) 34 Chapter 1 At equilibrium, when CA ¼ CAe, etc., rA is zero and we have kf CAe CBe ¼ kr CMe CNe or CMe CNe kf ¼ CAe CBe kr (1.47) But CMeCNe/CAeCBe is the concentration-based equilibrium constant Kc and hence kf /kr ¼ Kc. Often it is convenient to substitute for kr in Eq. (1.46) so that we have a typical example of a rate equation for a reversible reaction: rA ¼ kf CA CB CM CN Kc (1.48) We see from the above example that the forward and reverse rate constants are not completely independent, but are related by the equilibrium constant, which in turn is related to the thermodynamic free energy change, etc. More detailed examination of the kinds of kinetic equations that might be used to describe the forward and reverse reactions shows that, to be consistent with the thermodynamic equilibrium constant, the form of the rate equation for the reverse reaction cannot be completely independent of the forward rate equation. A good example is the formation of phosgene: CO þ Cl2 # CO Cl2 3=2 The rate of the forward reaction is given by rf ¼ kf CCO CCl2 . This rate equation indicates that the chlorine concentration must also appear in the reverse rate equation. Let this be p q s ; then at equilibrium, when r ¼ r , we must have rr ¼ kr CCOCl CCl CCO f r 2 2 3=2 p q s kf CCO CCl2 ¼ kr CCOCl C C Cl2 CO 2 eq eq or kf ðq3=2Þ ðs1Þ p ¼ K c ¼ CCOCl C C CO Cl2 2 eq kr (1.49) But we know from the thermodynamic equilibrium constant that Kc ¼ CCOCl2 CCO CCl2 (1.50) eq Therefore it follows that, for consistency with the thermodynamic concept of equilibrium, p ¼ 1, q ¼ ½, and s ¼ 0. The complete rate equation is therefore 3=2 1=2 rCO ¼ kf CCO CCl2 kr CCOCl2 CCl2 Reactor DesigndGeneral Principles 35 or 0 1=2 rCO ¼ kf @CCO CCl2 3=2 CCOCl2 CCl2 Kc 1 A (1.51) A more detailed treatment of the implications of thermodynamic consistency for reaction kinetics may be found in Denbigh.2 1.5.5 Experimental Determination of Kinetic Constants The interpretation of laboratory-scale experiments to determine order and rate constant is another subject which is considered at length in physical chemistry texts.4,5 Essentially, it is a process of fitting a rate equation of the general form given by Eq. (1.31) to a set of numerical data. In a batch reactor, the experiments that are carried out to obtain the kinetic constants may be of two kinds, depending on whether the rate equation is to be used in differential form or in its integrated form. If the differential form is to be used then experiments must be designed so that the rate of disappearance of reactant A, rA, can be measured without its concentration changing appreciably. With batch or tubular reactors, this has the disadvantage in practice that very accurate measurements of CA must be made so that, when differences in concentration DCA are taken to evaluate rA, the difference may be obtained with sufficient accuracy. CSTRs do not suffer from this disadvantage; by operating in the steady state, steady concentrations of the reactants are maintained and the rate of reaction is determined readily. If the rate equation is to be employed in its integrated form, the problem of determining kinetic constants from experimental data from batch or tubular reactors is in many ways equivalent to taking the design equations and working backwards. Experiments using the various reactors to determine kinetic constants are discussed in the sections describing these reactors (Sections 1.6.6, 1.7.4 and 1.8.5). Unfortunately, many of the chemical processes that are important industrially are quite complex. A complete description of the kinetics of a process, including by-product formation as well as the main chemical reaction, may involve several individual reactions, some occurring simultaneously and some proceeding in a consecutive manner. Often the results of laboratory experiments in such cases are ambiguous and, even if complete elucidation of such a complex reaction pattern is possible, it may take several man-years of experimental effort. Whereas ideally the design engineer would like to have a complete set of rate equations for all the reactions involved in a process, in practice the data available to him often fall far short of this. 36 Chapter 1 1.6 Batch Reactors Whereas continuous flow reactors are likely to be most economical for large-scale production, the very real advantages of batch reactors, especially for smaller scale production, should not be overlooked. Small batch reactors generally require less auxiliary equipment, such as pumps, and their control systems are less elaborate and costly than those for continuous reactors, although manpower needs are greater. However, large batch reactors may sometimes be fitted with highly complex control systems. A big advantage of batch reactors in the dyestuff, fine chemical and pharmaceutical industries is their versatility. A corrosion-resistant batch reactor such as an enamel or rubber-lined jacketed vessel (Fig. 1.4A) or a stainless steel vessel with heating and cooling coils (Fig. 1.4B) can be used for a wide variety of similar kinds of reaction. Sometimes only a few batches per year are required to meet the demand for an unusual product. In some processes, such as polymerizations and fermentations, batch reactors are traditionally preferred because the interval between batches provides an opportunity to clean the system thoroughly and ensure that no deleterious intermediates such as foreign bacteria build up and spoil the product. Moreover, it must not be forgotten that a squat tank is the most economical shape for holding a given volume of liquid, and for slow reactions a tubular flow reactor with a diameter sufficiently small to prevent backmixing would be more costly than a simple batch reactor. Although at present we are concerned mainly with homogeneous reactions, we should note that the batch reactor has many advantages for heterogeneous reactions; the agitator can be designed to suspend solids in the liquid and to disperse a second immiscible liquid or a gas. In calculating the volume required for a batch reactor, we shall be specifying the volume of liquid which must be processed. In designing the vessel itself the heights should be increased by about 10% to allow freeboard for waves and disturbances on the surface of the liquid; additional freeboard may have to be provided if foaming is anticipated. 1.6.1 Calculation of Reaction Time: Basic Design Equation Calculation of the time required to reach a particular conversion is the main objective in the design of batch reactors. Knowing the amount of reactant converted, i.e., the amount of the desired product formed per unit volume in this reaction time, the volume of reactor required for a given production rate can be found by simple scale-up as shown in the example on ethyl acetate. The reaction time tr is determined by applying the general material balance Eq. (1.9). In the most general case, when the volume of the reaction mixture is not constant throughout the reaction, it is convenient to make the material balance over the whole volume of the fluid in the reactor Vb. For a batch reactor, the flow terms are zero. Hence, Reactor DesigndGeneral Principles 37 applying Eq. (1.9) for R the reactant A and assuming a uniformly mixed batch reactor (which implies that Vb rA dV ¼ rA Vb ), we obtain dnA ¼ rA Vb (1.52) dt If nA0 moles are charged initially, the number of moles remaining at time t, nA, may be expressed in terms of the fraction of A converted, aA. The fractional conversion of A is defined as aA h1 nA . nA0 (1.53) Thus nA ¼ nA0(1 aA) and hence dnA d daA ¼ ½nA0 ð1 aA Þ ¼ nA0 dt dt dt Substituting into Eq. (1.52), we obtain daA ¼ r A Vb dt Then, integrating over the period of the reaction (tr) to a final conversion aAf, we obtain the basic design equation: Z aAf daA (1.54) tr ¼ nA0 ð rA ÞVb 0 nA0 For many liquid-phase reactions, it is reasonable to neglect any change in volume of the reaction mixture. It can be easily seen from Eq. (1.7) that a sufficient condition for this assumption to be valid is that the density of the reaction mixture is independent of composition. From Eq. (1.7), with the flow term equal to zero, we get dm dðrVb Þ ¼ ¼ 0: dt dt Thus if r remains constant then so does Vb. Eq. (1.54) then becomes Z Z aAf nA0 aAf daA daA ¼ CA0 tr ¼ Vb 0 rA rA 0 (1.55) where CA0 is the initial concentration of A. For a constant-volume batch reactor, it is sometimes more convenient to deal directly with concentration than conversion. For a constant Vb, we may write Eq. (1.52) as dðnA =Vb Þ dCA ¼ ¼ rA dt dt (1.56) 38 Chapter 1 Integrating Eq. (1.56), we obtain the reaction time as Z CAf Z CA0 dCA dCA tr ¼ ¼ CA0 rA CAf rA (1.57) where CAf is the final concentration. Another measure of conversion that is useful in dealing with constant-volume batch reactors is the quantity c defined as chCA0 CA ¼ CA0 aA (1.58) This quantity represents the number of moles of A converted per unit volume of the reactor. Eqs. (1.52)e(1.57) focus on a particular species, usually a reactant in the reaction. The number of moles of any other species (reactant or product) can be determined using Eq. (1.29). Thus for a species i, we have, analogous to Eq. (1.52), dni ¼ ri Vb dt Dividing Eq. (1.59) by Eq. (1.52) and using Eq. (1.29), we obtain ri dni ni ¼ ¼ rA dnA nA (1.59) (1.60) Integrating Eq. (1.60), we obtain ni ni0 ¼ ni ni ðnA nA0 Þ ¼ ðnA0 nA Þ nA nA (1.61) For a constant-volume batch reactor, we get, on dividing Eq. (1.61) by the volume of the reaction mixture, ni ðCA0 CA Þ (1.62) Ci Ci0 ¼ nA The equations that result from the integration of Eq. (1.55) or 1.57 for reactions of various orders are discussed in considerable detail in most texts dealing with the physicochemical aspects of chemical kinetics.4,5 The integrated forms of Eq. (1.55) for a variety of the simple rate equations are shown in Table 1.1 in terms of the quantity c defined by Eq. (1.58). The integrated forms can be easily verified by the reader if desired. One particular point of interest is the expression for the half-life of a reaction t1/2; this is the time required for one half of the reactant in question to disappear. A first-order reaction is unique in that the half-life is independent of the initial concentration of the reactant. This characteristic is sometimes used as a test of whether a reaction really is first order. Also since t1=2 ¼ k11 ln 2, a first-order rate constant can be readily converted into a half-life which one can easily remember as characteristic of the reaction (see Section 1.6.6 for a more detailed discussion on half-life). Reactor DesigndGeneral Principles 39 Table 1.1: Rate equations for constant-volume batch reactors. Reaction Type Rate Equation Integrated Form Irreversible Reactions First order A / products dc dt ¼ k1 ðCA0 cÞ CA0 t ¼ k11 ln ðCA0 cÞ t1=2 ¼ ln2 k1 Second order A þ B / products dc dt ¼ k2 ðCA0 cÞðCB0 cÞ ðCB0 cÞ t ¼ k2 ðCB01CA0 Þ ln CCA0 B0 ðCA0 cÞ CA0 6¼ CB0 2A / products dc dt Order p, one reactant A / products dc dt ¼ k2 ðCA0 cÞ2 t ¼ k12 1 CA0 c C1A0 t1=2 ¼ k2 1CA0 " ¼ kðCA0 cÞ p t¼ # C 1 1 kðp1Þ ðCA0 cÞp1 t1=2 ¼ 1 A0 2p1 1 kðp 1ÞCA0 p1 p1 p 6¼ 1 Reversible Reactions First order Both directions A #kkfr M dc dt ¼ kf ðCA0 cÞ kr ðCM0 þ cÞ t¼ k C k C f A0 r M0 1 ln ðkf þkr Þ kf ðCA0 cÞkr ðCM0 þcÞ or since Kc ¼ kf/kr Kc CA0 CM0 t ¼ kf ð1þK ln K c ðCA0K cc ÞðCM0 þcÞ cÞ If CM0 ¼ 0 1 e ln ðc ccÞ t ¼ ðkf þk rÞ e Second order Both directions A þ B #kkfr M þ N If CA0 ¼ CB0 and CM0 ¼ CN0 ¼ 0 2 dc 2 dt ¼ kf ðCA0 cÞ kr c t¼ pffiffiffiffiffi CA0 þc½1=ðOK C Þ1 KC 2kf CA0 ln CA0 c½1=ðOK C Þþ1 A further point of interest about the equations shown in Table 1.1 is to compare the shapes of graphs of fractional conversion or moles converted versus time for reactions of different orders p. Fig. 1.9 shows a comparison between first- and second-order reactions involving a single reactant only, together with the straight line for a zero-order reaction. The rate constants have been taken so that the curves coincide at 50% conversion. It may be seen that the rate of the second-order reaction is high at first but falls rapidly with increasing time and, compared with first-order reactions, longer reaction times are required for high conversions. The zero-order reaction is the only one where the reaction rate does not decrease with increasing conversion. Many biological systems have apparent reaction orders between 0 and 1 and will have a behavior intermediate between the curves shown. 40 Chapter 1 0 p= Moles converted / unit volume χ CAO p= 1 p=2 CAO/2 0 Time t Figure 1.9 Batch reactions at constant volume: comparison of curves for zero-, first-, and second-order reactions. 1.6.2 Reaction TimedIsothermal Operation If the reactor is to be operated isothermally, the rate of disappearance of A, rA, can be expressed as a function of concentrations only, and the integration in Eq. (1.55) or 1.57 is carried out. We now consider an example with a rather complicated rate equation involving a reversible reaction and show also how the volume of the batch reactor required to meet a particular production requirement is calculated. Example 1.2 Production of Ethyl Acetate in a Batch Reactor Ethyl acetate is to be manufactured by the esterification of acetic acid with et