conservation equations and modeling of chemical and biochemical processes

While the early part of thebook is restricted to homogeneous systems, a later chapter introduces anovel systems approach and presents, in an easy-to-understand manner,the modeling of het

CONSERVATION EQUATIONS ffND MODELING OF CHEMICfiL fiND BIOCHEMICAL

PROCESSES

Said S E M Elnashaie Parag Garhyan

Auburn University Auburn, Alabama, U.S.A.

M A R C E L

MARCEL DEKKER, INC NEW YORK • BASEL

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

ISBN: 0-8247-0957-8

This book is printed on acid-free paper

Headquarters

Marcel Dekker, Inc

270 Madison Avenue, New York, NY 10016

Copyright gC 2003 by Marcel Dekker, Inc All Rights Reserved

Neither this book nor any part may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopying, microfilming, andrecording, or by any information storage and retrieval system, without permission

in writing from the publisher

Current printing (last digit):

10 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA

2 Ethylene: Keystone to the Petrochemical Industry, Ludwig Kniel, Olaf

Winter, and Kari Stork

3 The Chemistry and Technology of Petroleum, James G Speight

4 The Desulfunzation of Heavy Oils and Residua, James G Speight

5 Catalysis of Organic Reactions, edited by William R Moser

6 Acetylene-Based Chemicals from Coal and Other Natural Resources,

Robert J Tedeschi

7 Chemically Resistant Masonry, Walter Lee Sheppard, Jr.

8 Compressors and Expanders: Selection and Application for the Process

Industry, Heinz P Bloch, Joseph A Cameron, Frank M Danowski, Jr,

Ralph James, Jr., Judson S Sweanngen, and Marilyn E Weightman

9 Metering Pumps Selection and Application, James P Poynton

10 Hydrocarbons from Methanol, Clarence D Chang

11 Form Flotation: Theory and Applications, Ann N Clarke and David J.

Wilson

12 The Chemistry and Technology of Coal, James G Speight

13 Pneumatic and Hydraulic Conveying of Solids, O A Williams

14 Catalyst Manufacture: Laboratory and Commercial Preparations, Alvin B.

Stiles

15 Charactenzation of Heterogeneous Catalysts, edited by Francis

Delannay

16 BASIC Programs for Chemical Engineering Design, James H Weber

17 Catalyst Poisoning, L Louis Hegedus and Robert W McCabe

18 Catalysis of Organic Reactions, edited by John R Kosak

19 Adsorption Technology A Step-by-Step Approach to Process Evaluation

and Application, edited by Frank L Slejko

20 Deactivation and Poisoning of Catalysts, edited by Jacques Oudar and

Henry Wise

21 Catalysis and Surface Science: Developments in Chemicals from

Meth-anol, Hydrotreating of Hydrocarbons, Catalyst Preparation, Monomers and Polymers, Photocatalysis and Photovoltaics, edited by Heinz Heinemann

and Gabor A Somorjai

22 Catalysis of Organic Reactions, edited by Robert L Augustine

23 for the T H Tsai, J.

W Lane, and C S Lin

24 Temperature-Programmed Reduction for Solid Materials

Character-ization, Alan Jones and Brian McNichol

25 Catalytic Cracking: Catalysts, Chemistry, and Kinetics, Bohdan W.

Wojciechowski and Avelino Corma

26 Chemical Reaction and Reactor Engineering, edited by J J Carberry

and A Varma

27 Filtration: Principles and Practices, Second Edition, edited by Michael J.

Matteson and Clyde Orr

28 Corrosion Mechanisms, edited by Florian Mansfeld

29 Catalysis and Surface Properties of Liquid Metals and Alloys, Yoshisada

Ogino

30 Catalyst Deactivation, edited by Eugene E Petersen and Alexis T Bell

31 Hydrogen Effects in Catalysis: Fundamentals and Practical Applications, edited by Zoltan Paal and P G Menon

32 Flow Management for Engineers and Scientists, Nicholas P

Chere-misinoff and Paul N ChereChere-misinoff

33 Catalysis of Organic Reactions, edited by Paul N Rylander, Harold

Greenfield, and Robert L Augustine

34 Powder and Bulk Solids Handling Processes: Instrumentation and

Control, Koichi linoya, Hiroaki Masuda, and Kinnosuke Watanabe

35 Reverse Osmosis Technology: Applications for High-Purity-Water

Production, edited by Bipin S Parekh

36 Shape Selective Catalysis in Industrial Applications, N Y Chen, William

E Garwood, and Frank G Dwyer

37 Alpha Olefms Applications Handbook, edited by George R Lappin and

Joseph L Sauer

38 Process Modeling and Control in Chemical Industries, edited by Kaddour

Najim

39 Clathrate Hydrates of Natural Gases, E Dendy Sloan, Jr.

40 Catalysis of Organic Reactions, edited by Dale W Blackburn

41 Fuel Science and Technology Handbook, edited by James G Speight

42 Octane-Enhancing Zeolitic FCC Catalysts, Julius Scherzer

43 Oxygen in Catalysis, Adam Bielanski and Jerzy Haber

44 The Chemistry and Technology of Petroleum: Second Edition, Revised

and Expanded, James G Speight

45 Industnal Drying Equipment: Selection and Application, C M van't Land

46 Novel Production Methods for Ethylene, Light Hydrocarbons, and

Aro-matics, edited by Lyle F Albnght, Billy L Crynes, and Siegfried Nowak

47 Catalysis of Organic Reactions, edited by William E Pascoe

48 Synthetic Lubncants and High-Performance Functional Fluids, edited by

Ronald L Shubkin

49 Acetic Acid and Its Derivatives, edited by Victor H Agreda and Joseph R.

Zoeller

50 Properties and Applications of Perovskite-Type Oxides, edited by L G.

Tejuca and J L G Fierro

51 d of edited by E Robert Becker andCarmo J Pereira

52 Models for Thermodynamic and Phase Equilibria Calculations, edited by

Stanley I Sandier

53 Catalysis of Organic Reactions, edited by John R Kosak and Thomas A.

Johnson

54 Composition and Analysis of Heavy Petroleum Fractions, Klaus H Altgelt

and Mieczyslaw M Boduszynski

55 NMR Techniques in Catalysis, edited by Alexis T Bell and Alexander

Pines

56 Upgrading Petroleum Residues and Heavy Oils, Murray R Gray

57 Methanol Production and Use, edited by Wu-Hsun Cheng and Harold H.

Kung

58 Catalytic Hydroprocessing of Petroleum and Distillates, edited by Michael

C Oballah and Stuart S Shin

59 The Chemistry and Technology of Coal: Second Edition, Revised and

Expanded, James G Speight

60 Lubricant Base Oil and Wax Processing, Avilino Sequeira, Jr.

61 Catalytic Naphtha Reforming: Science and Technology, edited by

George J Antos, Abdullah M Aitani, and Jose M Parera

62 Catalysis of Organic Reactions, edited by Mike G Scares and Michael L.

Prunier

63 Catalyst Manufacture, Alvin B Stiles and Theodore A Koch

64 Handbook of Grignard Reagents, edited by Gary S Silverman and Philip

E Rakita

65 Shape Selective Catalysis in Industrial Applications: Second Edition,

Revised and Expanded, N Y Chen, William E Garwood, and Francis

G Dwyer

66 Hydrocracking Science and Technology, Julius Scherzer and A J.

Gruia

67 Hydrotreating Technology for Pollution Control: Catalysts, Catalysis,

and Processes, edited by Mario L Occelli and Russell Chianelli

68 Catalysis of Organic Reactions, edited by Russell E Malz, Jr.

69 Synthesis of Porous Materials: Zeolites, Clays, and Nanostructures,

edited by Mario L Occelli and Henri Kessler

70 Methane and Its Denvatives, Sunggyu Lee

71 Structured Catalysts and Reactors, edited by Andrzei Cybulski and

Jacob Moulijn

72 Industnal Gases in Petrochemical Processing, Harold Gunardson

73 Clathrate Hydrates of Natural Gases: Second Edition, Revised and

Expanded, E Dendy Sloan, Jr.

74 Fluid Cracking Catalysts, edited by Mario L Occelli and Paul O'Connor

75 Catalysis of Organic Reactions, edited by Frank E Herkes

76 The Chemistry and Technology of Petroleum, Third Edition, Revised

and Expanded, James G Speight

77 Synthetic Lubricants and High-Performance Functional Fluids, Second

Edition- Revised and Expanded, Leslie R Rudnick and Ronald L.

Shubkin

78 The of and Second Edition,

79 Reaction Kinetics and Reactor Design: Second Edition, Revised and

John B Butt

80 Regulatory Chemicals Handbook, Jennifer M Spero, Bella Devito, and

Louis Theodore

81 Applied Parameter Estimation for Chemical Engineers, Peter Englezos

and Nicolas Kalogerakis

82 Catalysis of Organic Reactions, edited by Michael E Ford

83 The Chemical Process Industries Infrastructure: Function and

Eco-nomics, James R Couper, O Thomas Beasley, and W Roy Penney

84 Transport Phenomena Fundamentals, Joel L Plawsky

85 Petroleum Refining Processes, James G Speight and Baki Ozum

86 Health, Safety, and Accident Management in the Chemical Process

Industries, Ann Marie Flynn and Louis Theodore

87 Plantwide Dynamic Simulators in Chemical Processing and Control,

William L Luyben

88 Chemicial Reactor Design, Peter Harriott

89 Catalysis of Organic Reactions, edited by Dennis Morrell

90 Lubricant Additives: Chemistry and Applications, edited by Leslie R.

Rudnick

91 Handbook of Fluidization and Fluid-Particle Systems, edited by

Wen-Ching Yang

92 Conservation Equations and Modeling of Chemical and Biochemical

Processes, Said S E H Elnashaie and Parag Garhyan

93 Batch Fermentation: Modeling, Monitoring, and Control, Ah Cmar,

Satish J Parulekar, Cenk Undey, and Gulnur Birol

94 Industrial Solvents Handbook, Second Edition, Nicholas P

Chere-misinoff

ADDITIONAL VOLUMES IN PREPARATION

Chemical Process Engineering: Design and Economics, Harry Silla

Process Engineering Economics, James R Couper

Petroleum and Gas Field Processing, H K Abdel-Aal, Mohamed

Aggour, and M.A Fahim

Thermodynamic Cycles: Computer-Aided Design and Optimization,

Chih Wu

Re-Engineering the Chemical Processing Plant: Process tion, Andrzej Stankiewicz and Jacob A Moulijn

We would like readers—instructors and students—to read this preface fully before using this book This preface is classified into three parts:

care-1 Background and Basic Ideas explains the fundamentals of using

a system approach as a more advanced approach to teachingchemical engineering It also discusses very briefly how thisapproach allows compacting the contents of many chemical engi-neering subjects and relates them with one another in a systema-tic and easy-to-learn manner More details on this aspect of thebook are given inChapter 1

2 Review of Chapters and Appendices briefly describes the contents

of each chapter and the educational philosophy behind choosingthese materials

3 Relation of the Book Contents to Existing Chemical EngineeringCourses shows how this book can be used to cover a number ofcourses in an integrated manner that unfortunately is missing inmany curricula today The relation of the contents of the book

to existing courses is discussed Although our frame of reference

is the curricula of the Chemical Engineering Department atAuburn University, the discussion can be applied to many curri-cula worldwide

1 BACKGROUND AND BASIC IDEAS

We have adopted a novel approach in the preparation of this rather tionary undergraduate-level chemical engineering textbook It is based onthe use of system theory in developing mathematical models (rigorous designequations) for different chemical and biochemical systems After a briefintroduction to system theory and its applications, the book uses the gen-eralized modular conservation equations (material and energy balances) asthe starting point

revolu-This book takes as its basis the vision of chemical engineering formed, as expressed in the Amundson report of 1989, in which areas new tothe traditional subject matter of the discipline are explored These new areasinclude biotechnology and biomedicine, electronic materials and polymers,the environment, and computer-aided process engineering, and encompasswhat has been labeled the BIN—Bio, Info, Nano—revolution The bookaddresses these issues in a novel and imaginative way and at a level thatmakes it suitable for undergraduate courses in chemical engineering.This book addresses one of the most important subjects in chemicalengineering—modeling and conservation equations These constitute thebasis of any successful understanding, analysis, design, operation, and opti-mization of chemical and biochemical processes The novel system approachused incorporates a unified and systematic way of addressing the subject,thus streamlining this difficult subject into easy-to-follow enjoyable reading

trans-By adopting a system approach, the book deals with a wide range

of subjects normally covered in a number of separate courses—mass andenergy balances, transport phenomena, chemical reaction engineering,mathematical modeling, and process control Students are thus enabled toaddress problems concerning physical systems, chemical reactors, and bio-chemical processes (in which microbial growth and enzymes play key roles)

We strongly believe that this volume strikes the right balance betweenfundamentals and applications and fills a gap in the literature in a uniqueway It efficiently transmits the information to the reader in a systematic andcompact manner The modular mass/energy balance equations are formu-lated, used, and then transformed into the design equations for a variety ofsystems in a simple and systematic manner

In a readily understandable way, this book relates a wide spectrum ofsubjects starting with material and energy balances and ending with processdynamics and control, with all the stages between The unique systemapproach shows that moving from generalized material and energy balanceequations to generalized design equations is quite simple for both lumpedand distributed systems The same has been applied to homogeneous andheterogeneous systems and to reacting and nonreacting systems as well as to

steady- and unsteady-state systems This leads the reader gracefully andwith great ease from lumped to distributed systems, from homogeneous toheterogeneous systems, from reacting to nonreacting systems, and fromsteady-state to unsteady-state systems.

Although steady-state systems are treated, we have provided enoughcoverage of transient phenomena and unsteady-state modeling for students

to appreciate the importance of dynamic systems While the early part of thebook is restricted to homogeneous systems, a later chapter introduces anovel systems approach and presents, in an easy-to-understand manner,the modeling of heterogeneous systems for both steady-state andunsteady-state conditions, together with a number of practical examples.Chemical and biochemical units with multiple-input multiple-output(MIMO) and with multiple reactions (MRs) for all of the above-mentionedsystems are also covered Nonreacting systems and single-input single-out-put (SISO) systems are treated as special cases of the more general MIMO,

MR cases The systems approach helps to establish a solid platform onwhich to formulate and use these generalized models and their special cases

As the book covers both steady- and unsteady-state situations, itlogically includes a chapter on process dynamics and control that is anexcellent introduction to a more advanced treatment of this topic, withspecial emphasis on the industrially more relevant digital control systemsdesign

Given that all chemical/biochemical engineering processes and systemsare highly nonlinear by nature, the book discusses this nonlinear behavior insome detail All the necessary analytical and numerical tools required areincluded Matrix techniques are also covered for large-dimensional systemsthat are common in chemical/biochemical engineering The book alsocovers, in a manner that is clear and easy to understand for undergraduatechemical engineers, advanced topics such as multiplicity, bifurcation, andchaos to further broaden the student’s perspective It is increasingly impor-tant for undergraduate students to think outside the conventional realm ofchemical engineering, and we have shown that these phenomena are relevant

to many important chemical/biochemical industrial systems It is also shownthat these phenomena cannot be neglected while designing these systems ortheir control loops In the past these subjects—multiplicity, bifurcation, andchaos—have tended to be relegated to advanced research treatises We treatthem here in a manner that undergraduate students can understand andappreciate

In our fast-changing world the chemical/biochemical industry is alsorapidly changing Today’s chemical/biochemical engineering graduatesshould be exposed to training in creativity as applied to these systems.Therefore a chapter on novel configurations and modes of operations for

two important processes is presented in the form of detailed exercises Thisimportant chapter requires a special effort from the instructor to make it theexercise on creativity that it is meant to be.

2 REVIEW OF CHAPTERS AND APPENDICES

This book presents a unified approach to the analysis of a wide range ofchemical and biochemical systems It begins with a summary of the funda-mental principles governing thermodynamics and material and energy bal-ances and proceeds to consider the mathematical modeling of a range ofsystems from homogeneous steady state to heterogeneous unsteady state Anovel feature is the inclusion of the concepts surrounding chaotic systems atundergraduate level—an area of growing importance but one sadlyneglected in most texts of this kind The last chapter deals with two indus-trial processes—reforming and fermentation—in which the foregoing prin-ciples are applied and illustrated for novel configurations and modes ofoperation The useful appendices deal with many of the mathematical tech-niques such as matrix algebra, numerical methods, and the Laplace trans-form that are utilized in the book

Chapter 1: System Theory and Chemical/Biochemical

Engineering SystemsThis chapter, one of the most important, introduces the main components ofthe philosophy governing the entire book It covers in a simple manner themain ideas regarding system theory and its application to chemical andbiochemical systems These systems are classified according to the principles

of system theory, and this more novel classification is related to the moreclassical classifications This chapter also covers the main differencesbetween material and energy balances (inventory) and design equations,the concepts of rate processes together with their relation to state variables,and the general modeling of processes The thermodynamic limitation ofrate processes in relation to modeling and simulation is examined A briefdiscussion of the new approach adopted in this book in connection withrecent advances in the profession based on the Amundson report is alsopresented

Chapter 2: Material and Energy Balances

This chapter addresses materials and energy balances for reacting (single aswell as multiple reactions) and nonreacting systems in a compact way Italso covers SISO as well as MIMO systems A generalized material andenergy balance equation for a MIMO system with MRs is rigorously devel-

oped All other cases can be easily considered as special cases of this generalcase A large number of solved illustrative examples are provided, andunsolved problems are given as exercises at the end of the chapter.

The modular system approach used in this chapter ultimately requires thereader to know only two generalized equations (material and energy bal-ances), with all other cases being special cases This approach makes thesubject easy to comprehend and utilize in a short time, and will also proveextremely useful in preparing the reader to modify these equations intodesign equations (mathematical models)

Chapter 3: Mathematical Modeling (I): Homogeneous

Lumped SystemsThis chapter covers in an easy and straightforward manner the transforma-tion of the material and energy balance equations to design equations(mathematical models) It explores closed, isolated, and open lumped homo-geneous systems Steady-state as well as unsteady-state models are devel-oped and solved for both isothermal and nonisothermal systems Bothchemical and biochemical systems are addressed Again, generalized designequations are developed with all other cases treated as special cases of thegeneral one This approach helps to achieve a high degree of efficiencyregarding rational transformation of knowledge in a concise and clear man-ner We concentrate our efforts on reacting systems for two reasons: the first

is that for homogeneous systems the nonreacting systems are rather trivial,and the other is that the nonreacting system can be considered a special case

of reacting systems when the rates of reactions are set equal to zero A goodnumber of solved and unsolved problems are given in this chapter

Chapter 4: Mathematical Modeling (II): Homogeneous

Distributed Systems and Unsteady-State Behavior

This chapter covers the transformation of the material and energy balanceequations to design equations (mathematical models) for distributed sys-tems Steady-state as well an unsteady-state models are developed andsolved for both isothermal and nonisothermal systems Again, generalizeddesign equations are developed with all other cases treated as special cases ofthe general one, and this approach facilitates efficient transformation ofknowledge We concentrate on reacting systems for the same reasons pre-viously discussed Chapter 4 gives detailed coverage of the mathematicalmodeling and analytical as well as numerical solution of the axial dispersion

model for tubular reactors as an illustrative example for diffusion/reactionhomogeneous systems The same example is extended to provide thesolution of the two-point boundary value differential equations and itsassociated numerical instability problems for nonlinear systems Severalunsolved problems are provided at the end of this chapter.

Together, Chapters 3 and 4 provide systematic, easy-to-understandcoverage of all types of homogeneous models, both lumped/distributedand isothermal/nonisothermal systems Both chapters can also be used asthe necessary materials for a thorough course on chemical reaction engineer-ing based on a well-organized approach utilizing system theory

Chapter 5: Process Dynamics and Control

In the last 20 years, digital control has completely replaced analog control inindustry and even in experimental setups It is our strong belief that theclassic complete course on analog control is no longer necessary Controlcourses should be directed mainly toward digital control systems, which arebeyond the scope of this book It is useful, however, for readers to have abasic background in analog control (only one well-chosen chapter, notnecessarily an entire course) to prepare them for a next course on digitalcontrol Chapter 5 aims to do this by introducing the basic principles ofprocess dynamics and classical control, including the various forms of pro-cess dynamic models formulation, basic process control concepts, the use ofLaplace transformation and its utilization, the transfer function concepts,ideal forcing functions, block diagram algebra, components of the controlloop, and a limited number of simple techniques for choosing the controlconstants for PID controllers All these important concepts are supplemen-ted with useful solved examples and unsolved problems

Chapter 6: Heterogeneous Systems

Most chemical and biochemical systems are heterogeneous (formed ofmore than one phase) The modular system approach we adopt in thisbook makes the development of material and energy balances, as well asdesign equations for heterogeneous systems quite straightforward.Heterogeneous systems are treated as just a number of homogeneous sys-tems (each representing one phase), and these systems are connected toeach other through material and energy exchange This approach proves

to be not only rigorous and general but also easy to comprehend andapply to any heterogeneous system utilizing all the knowledge and experi-ence gained by the reader through the previous chapters on homogeneoussystems Chapter 6 introduces these concepts and develops generalizedmaterial and energy balance equations as well as design equations for all

types of systems—isothermal/nonisothermal, lumped/distributed, andsteady-/unsteady-state A number of chemical and biochemical examples

of varying degrees of complexity and unsolved problems are presented forbetter understanding of the concepts

Chapter 7: Practical Relevance of Bifurcation, Instability,

and Chaos in Chemical and Biochemical Systems

This chapter covers the basic principles of multiplicity, bifurcation, andchaotic behavior The industrial and practical relevance of thesephenomena is also explained, with reference to a number of importantindustrial processes Chapter 7 covers the main sources of these phenom-ena for both isothermal and nonisothermal systems in a rather pragmaticmanner and with a minimum of mathematics One of the authors haspublished a more detailed book on the subject (S S E H Elnashaieand S S Elshishini, Dynamic Modelling, Bifurcation and ChaoticBehavior of Gas-Solid Catalytic Reactors, Gordon & Breach, London,1996); interested readers should consult this reference and the other refer-ences given at the end of Chapter 7 to further broaden their understanding

by offering two examples of novel processes: one for the efficient production

of the ultraclean fuel hydrogen and the other for the production of the cleanfuel ethanol through the biochemical path of utilizing lingo-cellulosicwastes Readers can expect to use the tools provided earlier in this book

in order to develop these novel processes and modes of operation withoutthe need of the expensive pilot plant stage

Appendices

Although it is difficult to make a book completely comprehensive, we tried

to make this one as self-contained as possible The six appendices cover anumber of the critical mathematical tools used in the book Also included is

a short survey of essential available software packages and programmingenvironments These appendices include analytical as well as numerical toolsfor the handling and solution of the different types of design equations,

including linear and nonlinear algebraic and ordinary differential and partialdifferential equations.

3 RELATION OF THE BOOK CONTENTS TO

EXISTING CHEMICAL ENGINEERING

COURSES

(CHEN 2100, Principles of Chemical Engineering, which covers the cation of multicomponent material and energy balances to chemical pro-cesses involving phase changes and chemical reactions)

Engineering Analysis (which covers mathematical modeling and analytical,numerical, and statistical analysis of chemical processes) Statistical processcontrol (SPC) is not, of course, covered in this book and the course readingshould be supplemented by another book on SPC (e.g., Amitava Mitra,Fundamentals of Quality Control and Improvement, Prentice Hall, NewYork, 1998)

Chapters 4, 5, 6, and 8 are suitable for a senior class on modeling ofdistributed systems and process dynamics and control (CHEN 4160, ProcessDynamics and Control, which covers steady-state and dynamic modeling ofhomogeneous and heterogeneous distributed chemical processes, feedbacksystems, and analog controller tuning and design) prior to the course ondigital control (CHEN 6170, Digital Process Control)

Chapters 3 and 4 and the first part of Chapter 8 can be used for anundergraduate course on chemical reaction engineering (CHEN 3700,Chemical Reaction Engineering, which covers design of chemical reactorsfor isothermal and nonisothermal homogeneous reaction systems)

University) I also thank Professor A A Adesina (University of New SouthWales, Australia) and Professor N Elkadah (University of Alabama,Tuscaloosa).

Last but not least, I express my love and appreciation for the extensivesupport and love I receive from my wife, Professor Shadia Elshishini (CairoUniversity, Egypt), my daughter Gihan, and my son Hisham

Said Elnashaie

I would like to express my sincere thanks to Dr Said Elnashaie for giving

me the opportunity to work with him as his graduate student and lateroffering me the chance to be the coauthor of this book I express my gra-titude to my grandfather, Shri H P Gaddhyan, an entrepreneur fromChirkunda (a small township in India) for always being an inspiration to

me Without the motivation, encouragement, and support of my parentsSmt Savita and Shri Om Prakash Gaddhyan, I would have not been able

to complete this book A special note of thanks goes to my brother Anuragand his wife Jaishree Finally, I express my love and thanks to my wifeSangeeta for her delicious food, endurance, and help; her smile alwayscheered me up and provided the impetus to continue when I was busyworking on the manuscript

Parag Garhyan

of the State Variables

1.4.3 Energy Equation (Conservation of Energy, FirstLaw of Thermodynamics for an Open System)

System Theory and Mathematical Modeling

Undergraduate Chemical Engineering Education

Approach Used in This Book

Engineering Education

More Efficient Undergraduate Chemical EngineeringEducation

1.12.1 Different Types of Systems and Their MainCharacteristics

1.12.2 What Are Models and What Is the DifferenceBetween Models and Design Equations?

1.12.3 Summary of Numerical and Analytical SolutionTechniques for Different Types of ModelReferences

Problem

Selectivity

2.2.3 Relations Among Rate of Reaction, Conversion,and Yield

for the Reactants

Single Reaction

Dependence and Linear Independence of MultipleReactions)

(Multiple-Input, Multiple-Output, and MultipleReactions)

Heat of Reaction

Reactions

Special Case of a Single Reaction

(for Multiple Reactions and Multiple-Input andMultiple-Output System Reactor with MultipleReactions)

ReferenceProblems

3.1.3 What Are Mathematical Models and Why Do

Rigorous Heterogeneous Models

Simulators and Putting Mathematical Models intoUser-Friendly Software Packages

Design Equations (Steady-State Mathematical Models)

ReferencesProblems

Systems and Unsteady-State Behavior

4.1.2 Nonisothermal Distributed Systems

Heterogeneous Systems

Axial Dispersion Model

Differential Equations and NumericalInstability Problems

Problems

5.6.3 The Transform of Derivatives

of Differential Equations

and Inverse Transformations

Types of Classical Control

5.12.1 Typical Feedback Control Loop and theTransfer Functions

5.12.2 Algebraic Manipulation of the Loop TransferFunctions

5.12.3 Block Diagram and Transfer Functions

5.13.1 Choosing the Controller Settings5.13.2 Criteria for Choosing the Controller Settingsfrom the Time Response

5.13.3 Cohen and Coon Process Reaction CurveMethod

Solved ExamplesProblems

6.2 Design Equations (Steady-State Models) for Isothermal,Heterogeneous Lumped Systems

Distributed Heterogeneous Systems

Steady-State, and Equilibrium Stages System)

(Porous Catalyst Pellet)

Fluidized-Bed Reactors

Application to Fluidized-Bed Catalytic Reactors

Catalytic Reactor

for the Alcoholic Fermentation Process

6.10.1 Background on the Problems Associated withthe Heterogeneous Modeling of AlcoholicFermentation Processes

6.10.2 Development of the Model6.10.3 Solution Algorithm6.10.4 Comparison Between the Model andExperimental/Industrial DataReferences

Problems

Chemical and Biochemical Systems

7.1.1 Isothermal Multiplicity (or ConcentrationMultiplicity)

of the Ultraclean Fuel Hydrogen from Hydrocarbonsand Waste Materials

Ultraclean/Efficient Reforming ProcessConfiguration

Process for the Production of the UltracleanFuel Hydrogen

References

Functions

System Theory and Chemical/

Biochemical Engineering Systems

1.1 SYSTEM THEORY

1.1.1 What Is a System?

The word system derives from the Greek word ‘‘systema’’ and means anassemblage of objects united by some form of regular interaction or inter-dependence A simpler, more pragmatic description regarding systemsincludes the following:

The system is a whole composed of parts (elements)

The concept of a system, subsystem, and element is relative anddepends on the degree of analysis; for example, we can take theentire human body as a system, and the heart, the arms, the liver,and so forth as the elements Alternatively, we can consider theseelements as subsystems and analyze them with respect to smallerelements (or subsystems) and so on

The parts of the system can be parts in the physical sense of theword or they can be processes In the physical sense, the parts ofthe body or of a chair form a system On the other hand, forchemical equipment performing a certain function, we considerthe various processes taking place inside the system as the elementswhich are almost always interacting with each other to give thefunction of the system A simple chemical engineering example is a

chemical reactor in which processes like mixing, chemical reaction,heat evolution, heat transfer, and so forth take place to give thefunction of this reactor, which is the changing of some reactants tosome products.

The properties of the system are not the sum of the properties ofits components (elements), although it is, of course, affected bythe properties of its components The properties of the systemare the result of the nonlinear interaction among its components(elements) For example, humans have consciousness which is not

a property of any of its components (elements) alone Also, masstransfer with chemical reaction has certain properties which arenot properties of the chemical reaction or the mass transferalone (e.g., multiplicity of steady states, as will be shown later inthis book)

This is a very elementary presentation of system theory We will revisit thesubject in more detail later

1.1.2 Boundaries of a System

The system has boundaries distinguishing it from the surrounding ment Here, we will develop the concept of environment The relationbetween the system and its environment gives one of the most importantclassifications of a system:

environ-1 An Isolated System does not exchange matter or energy with thesurroundings Thermodynamically it tends to the state of thermo-dynamic equilibrium (maximum entropy) An example is a batchadiabatic reactor

2 A Closed System does not exchange matter with the surroundingsbut exchanges energy Thermodynamically it tends to the state ofthermodynamic equilibrium (maximum entropy) An example is

a batch nonadiabatic reactor

3 An Open System does exchange matter and energy with the roundings Thermodynamically, it does not tend to the thermo-dynamic equilibrium, but to the steady state or what should becalled the ‘‘stationary non equilibrium state,’’ characterized byminimum entropy generation An example is a continuous stirredtank reactor

sur-This clearly shows that the phrase we commonly use in chemical ing, ‘‘steady state,’’ is not really very accurate, or at least it is not distinctiveenough A better and more accurate phrase should be ‘‘stationary non-equilibrium state,’’ which is a characteristic of open systems and distin-

engineer-guishes it from the ‘‘stationary equilibrium state,’’ associated with anisolated and closed systems.

1.2 STEADY STATE, UNSTEADY STATE, AND

THERMODYNAMIC EQUILIBRIUM

As briefly stated above, the steady state and unsteady state are conceptsrelated to open systems (almost all continuous chemical engineering pro-cesses are open systems) Steady state is when the state of the system doesnot change with time, but the system is not at thermodynamic equilibrium(i.e., the process inside the system did not stop and the stationary behaviorwith time is due to the balance between the input, output, and processestaking place in the system) The thermodynamic equilibrium is stationarywith time for isolated and closed systems because all processes have stopped,and the nonequilibrium system is changing with time but tending to thethermodynamic equilibrium state We will come back to these conceptswith more details later in this book

1.2.1 The State of the System

We have used the term ‘‘state of the system’’ many times; what is the state of

a system? The state of a system is rigorously defined through the statevariables of the system The state variables of any system are chosen accord-ing to the nature of the system The state of a boiler can be described bytemperature and pressure, a heat exchanger by temperature, a nonisother-mal reactor by the concentration of the different components and tempera-ture, an isothermal absorption tower by the concentration of differentcomponents on different plates, a human body by blood pressure and tem-perature, flow through a pipe by the velocity as a variable varying radiallyand axially, and so on

Thus, state variables are variables that describe the state of the system,and the state of the system is described by the state variables

1.2.2 Input Variables

Input variables are not state variables; they are external to the system, butthey affect the system or, in other words, ‘‘work on the system.’’ For exam-ple, the feed temperature and composition of the feed stream to a distillationtower or a chemical reactor or the feed temperature to a heat exchanger arethe input variables They affect the state of the system, but are not affected

by the state of the system (except when there is a feedback control, and inthis case, we distinguish between control variables and disturbances or inputvariables)

Later, we will discuss the distinction between different types of ables in more detail; for example, design and operating variables and thedifference between variables and parameters.

vari-1.2.3 Initial Conditions

These are only associated with unsteady-state systems An unsteady-statesystem is a system in which its state variables are changing with time.Unsteady-state open systems will change with time, tending toward the

‘‘stationary nonequilibrium state’’, usually called the ‘‘steady state’’ in thechemical engineering literature On the other hand, for closed and isolatedsystems, the unsteady-state behavior tends toward thermodynamic equili-brium

For these unsteady-state systems, whether open or closed, the systembehavior cannot be defined without knowing the initial conditions, or thevalues of the state variables at the start (i.e., time ¼ 0) When the initialconditions are defined, the behavior of the system is uniquely defined Themultiplicity (nonunique) phenomena that appear in some chemical engineer-ing systems are related to the steady state of open systems and not to theunsteady-state behavior with known initial conditions The trajectorydescribing the change of the state variables with time starting at a specificinitial condition is unique

O ¼ f Ið Þ

where O is the output and I is the input On the other hand, a physical model

is a model based on the understanding of what is happening inside thesystem and the exchange between the system and the surrounding There

is no model which is completely empirical and there is no model which iscompletely physical, but we name the model according to its dominantfeature

1.3.1 Elementary Procedure for Model Building

An elementary procedure for model building is as follows:

1 Define the boundaries of the system (Fig 1.2)

2 Define the type of system: open, closed, or isolated

3 Define the state variables

4 Define the input variables (sometimes called input parameters)

5 Define the design variables (or parameters)

6 Define the nature of the interaction between the system and thesurroundings.

7 Define the processes taking place within the boundaries of thesystem

8 Define the rate of the different processes in terms of thestate variables and rate parameters, and introduce the neces-sary equations of state and equilibrium relations between thedifferent phases Note: Usually, equilibrium relations betweencertain variables are used instead of rates as an approxima-tion when the rate is quite high and the process reachesequilibrium quickly

9 Write mass, heat (energy), and momentum balance equations toobtain the necessary equations (or model equations) relating theinput and output through the state variables and parameters.These equations give the variation of state variables with timeand/or space

1.3.2 Solution of the Model Equations

The model developed should be solved for certain inputs, design parameters,and physicochemical parameters in order to obtain the output and thevariation of state variables within the boundaries of the system To solvethe model equations, we need two main items:

1 Determination of the model parameters (to be determined mentally)

experi-2 A solution algorithm, the complexity of it, and whether it isanalytical or numerical depends on the complexity of the system

Of course, we will discuss the solution of the different types of model in fulldetail later in this book

(labora-1.4 FUNDAMENTAL LAWS GOVERNING THE

PROCESSES IN TERMS OF THE STATE

VARIABLES

Here, we give a very simple presentation of the necessary components fordeveloping model (design) equations for chemical/biochemical processes.Full details and generalization will be given inChapter 3

1.4.1 Continuity Equations for Open Systems

Total continuity equations (mass balances):

Mass flow into system

ð Þ
Mass flow out of systemð Þ

¼ Time rate of change of mass inside the system
Component continuity equations (component mass balances):ðFlow of moles of jth component into the systemÞ

Flow of moles of jth component out the systemð Þ

þ ðRate of formation of moles of jth component bychemical reactionsÞ

¼ ðTime rate of change of moles of jth component insidethe systemÞ

1.4.2 Diffusion of Mass (Transport Law)

Fick’s law gives that the mass transfer (diffusion) is proportional to theconcentration gradient(Fig 1.3):

NA¼
DAdCA

dZwhere, NA is the molar flux in moles=ðunit areaÞðunit timeÞ, DA is thediffusion coefficient, dCA is the concentration driving force, and dZ is thedistance in the direction of diffusion The same equation over a certainthickness (film, interface, etc.) can be approximated as

1.4.3 Energy Equation (Conservation of Energy, First Law of

Thermodynamics for an Open System)

Flow of internal,kinetic and potential

energies into thesystem by convection

A

Flow of internal,kinetic and potentialenergies out of thesystem by convection

or diffusion

0BB

@

1CCA

þ

Heat added

to the system

by conduction,radiation andreaction

A

Work done

by the system

on surroundings,i.e shaft workand PV work

0BB

@

1CCA

¼ fTime rate of change of internal, kinetic and potentialenergies inside the systemg

In most chemical engineering systems that we will study, the above generalform reduces to essentially an enthalpy balance, as will be shown later.Heat Transfer

Fourier’s law describes the flow of heat in terms of temperature gradient asfollows:

q ¼
dT

dZ

The above relation can be approximated in terms of the temperature ence between two points as follows:

differ-q ¼ h
T

where q is the heat flux in units of J/cm2s, is the thermal conductivity, dT

is the temperature driving force, dZ is the distance in the direction of heattransfer, and h ¼=
(analogous to the mass transfer case shown above and

Force ¼ Mass  Acceleration

In this case, the mass is considered to be constant

When the mass varies with time, the equation will have the followinggeneral form:

1

gC

d Mvð iÞ

dt ¼XN j¼1

where _Pyis the time rate of change of momentum caused by the force in the

ydirection Fy (the momentum transfer is in the x direction) and Ax is thearea perpendicular to x, where x is the direction of momentum transfer.Shear Stress

xy¼Fy

Ax

where Fy is the scalar y -component of the force vector and Ax is the areaperpendicular to the x axis We can write

for the British system

¼ 1 for the SI system

For shear stress, we can write

wherexy is the momentum flux

Newton’s Law of Viscosity (or Momentum Transfer)

1.4.5 Equations of State

To write mathematical models, in addition to material and energy balancesand rate of different processes taking place within the boundaries of thesystem, we need equations that tell us how the physical properties, primarilydensity and enthalpy, change with temperature

Liquid density ¼L¼ f1ðP; T; xiÞ

Vapor density ¼V¼ f2ðP; T; yiÞ

Liquid enthalpy ¼ h ¼ f3ðP; T; xiÞ

Vapor enthalpy ¼ H ¼ f4ðP; T; yiÞ

For simplicity, h is related to CpT and H is related to CpT þV

If Cp is taken as function of temperature and we consider that thereference condition is To (at which h ¼ 0), then

where A3, A4, and A5 are defined in terms of T0, A1 and A2

For Mixture of Components (and Negligible Heat of Mixing)

The enthalpy of the liquid mixture can be expressed as

h ¼

PJ

j¼1xjhjMj

PJ j¼1xjMj

where xj is the mole fraction of the jth component, Mj is the molecularweight of the jth component (g/gmol), and hj is the enthalpy of the purecomponent j (J/gmol) The denominator depicts clearly the average mole-cular weight of the mixture

Density

Liquid densities are usually assumed constant (unless large changes incomposition and temperature occur)

Vapor densities can be obtained from the relation

which gives

 ¼MP

RTwhere P is the absolute pressure, V is the volume, n is the number of moles,

Ris the universal gas constant, T is the absolute temperature, and M is themolecular weight

Notes

1 The reader must be cautious about the use of consistent units for

Rand other variables

2 For a high-pressure and/or high-temperature system, the pressibility factor (z factor) should be introduced, which isobtained from the knowledge of the critical temperature andcritical pressure of the system (the reader is advised to refer to

com-a thermodyncom-amics book; for excom-ample, Ref 1)

3 For an open (flow) system, Eq (1.1) becomes

Pq ¼ nnRTwhere q is the volumetric flow rate (L/min) andnn is the molar flowrate (mol/min)

k ¼ k0eð
E=RTÞ, P is the pressure (usually used for gas-phase reactions), and

Crepresents the concentrations

In many cases, we can write the rate of reaction as the product of threefunctions, each a function in one of the variables (temperature, concentra-tions, or pressure.)

r ¼ f1ð Þ fT 2ð Þ fC 3ð ÞP

Gas–solid catalytic reactions will have the same form, but will refer to theweight of catalyst rather than the volume:

r ¼ f T; C; Pð Þ

where r is the rate of reaction (gmol/g catalyst s) Note the difference in theunits of the rate of reaction for this gas–solid catalytic reaction comparedwith the homogeneous reaction

1.4.7 Thermodynamic Equilibrium

Chemical Equilibrium (for Reversible Reactions)

Chemical equilibrium occurs in a reacting system when

XJ

j¼1jj¼ 0

where j is the stoichiometric coefficient of jth component with the signconvention that reactants have negative sign and products have positivesigns (this will be discussed in full detail later) andjis the chemical poten-tial of the jth component

Also, we have the following important relation:

j¼ 0

j þ RT ln Pjwhere0

j is the standard chemical potential (or Gibbs free energy per mole)

of the jth component, Pj is the partial pressure of the jth component, R isthe universal gas constant, and T is the absolute temperature

For the reaction

aA ,k1

k2 BBwith the forward rate of reaction constant being k1and the backward being

ln PB
ln PA¼ A0

A
B0

B

RT

Function of temperature only

where, Kp is the equilibrium constant; we can write

ln Kp¼ f Tð Þ:

Finally, we can write

Kp¼ ef T ð Þ¼ Kp0e

H=RTð Þ

This equilibrium constant of a reversible reaction is very important in order

to determine the limit of conversion of reactants under any given design andoperating conditions

For Vapor–Liquid Systems

This phase equilibrium leads to the satisfaction of our need for a ship which permits us to calculate the vapor composition if we know theliquid composition or vice versa when the two phases are at equilibrium Wewill give an example regarding the bubble-point calculation

relation-Bubble-Point Calculation(Fig 1.5)

Given the pressure P of the system and the liquid composition xj, we culate the temperature of the system T and the vapor composition yj.This calculation usually involves trial and error (for some other cases, thesituation will be that we know xj and T and want to find P and yj, or weknow P and yj and want to find xj and T)

cal-For Ideal Vapor-Phase Behavior (Needs Correction at High

Pressures)

Dalton’s Law

Dalton’s law applies to the ideal vapor-phase behavior and states that thepartial pressure of component j in the vapor phase is the mole fraction of thecomponent multiplied by the total pressure:

pj¼ Pyj

where pj is the partial pressure of component j in the vapor phase, P is thetotal pressure, and yjis the mole fraction of component j in the vapor phase.Rault’s Law

Rault’s law states that the total pressure is the summation of the vaporpressure of each component ðp0jÞ multiplied by the mole fraction of com-ponent j in the liquid phase:

Now, because the relations

PxjThe vapor pressure p0j is a function of temperature only, as shown by therelation

ln p0j ¼Aj

T þ BjTherefore, the vapor–liquid equilibrium computation can be performedaccording to the above relation

The Utilization of Relative Volatility

The relative volatility of component i to component j is defined as

ij¼yi=xi

yj=xj

¼Volatility of iVolatility of jFor binary systems, we have

be needed; it is not covered in this book The reader is advised toconsult a multiphase thermodynamics book for that purpose.

1.5 DIFFERENT CLASSIFICATIONS OF PHYSICAL

MODELS

Before we move to a more intellectual discussion of the history ofchemical engineering and the role and position of system theory andmathematical modeling, we present the different basis for classification ofmathematical models

I Classification according to variation or constancy of the statevariables with time

Steady-state models: described by algebraic equations orODEs (ordinary differential equations) or PDEs (partialdifferential equations)

Unsteady-state models: described by ODEs or PDEs

II Classification according to the spatial variation of the statevariables

Lumped models (usually called lumped parameter models,which is wrong terminology because it is the state variablesthat are lumped together not the parameters): described byalgebraic equations for the steady state and ODEs for theunsteady state