Luyben, W. L. - Process Modeling, Simulation and Control for Chemical Engineers (1989), Notas de estudo de Engenharia Química

Baixe Luyben, W. L. - Process Modeling, Simulation and Control for Chemical Engineers (1989) e outras Notas de estudo em PDF para Engenharia Química, somente na Docsity! WIlllAM 1. LUYBENILLI . I PROCESS MODELING, l SIMULATION AND\ CONTROL iOR 5 FlitI m CHEMICAL ENGINEERS m SECOND EDITION- Ia I 1 L McGraw-Hill Chemical Engineering Series ’ Editorial Advisory Board James J. Carberry, Profissor of Chemical Engineering, University of Notre Dame James R. Fair, Professor of Chemical Engineering, University of Texas, Austin WilUum P. Schowalter, Professor of Chemical Engineering, Princeton University Matthew Tirrell, Professor of Chemical Engineering, University of Minnesota James Wei, Professor of Chemical Engineering, Massachusetts Institute of Technology Max S. Petem, Emeritus, Professor of Chentical Engineering, University of Colorado Building the Literature of a Profession Fifteen prominent chemical engineers first met in New York more than 60 years ago to plan a continuing literature for their rapidly growing profession. From industry came such pioneer practitioners as Leo H. Baekeland, Arthur D. Little, Charles L. Reese, John V. N. Dorr, M. C. Whitaker, and R. S. McBride. From the universities came such eminent educators as William H. Walker, Alfred H. White, D. D. Jackson, J. H. James, Warren K. Lewis, and Harry A. Curtis. H. C. Parmelee, then editor of Chemical and Metallurgical Engineering, served as chair- man and was joined subsequently by S. D. Kirkpatrick as consulting editor. After several meetings, this committee submitted its report to the McGraw- Hill Book Company in September 1925. In the report were detailed specifications for a correlated series of more than a dozen texts and reference books which have since become the McGraw-Hill Series in Chemical Engineering and which became the cornerstone of the chemical engineering curriculum. From this beginning there has evolved a series of texts surpassing by far the scope and longevity envisioned by the founding Editorial Board. The McGraw- Hill Series in Chemical Engineering stands as a unique historical record of the development of chemical engineering education and practice. In the series one finds the milestones of the subject’s evolution: industrial chemistry, stoichio- metry, unit operations and processes, thermodynamics, kinetics, and transfer operations. Chemical engineering is a dynamic profession, and its literature continues to evolve. McGraw-Hill and its consulting editors remain committed to a pub- lishing policy that will serve, and indeed lead, the needs of the chemical engineer- ing profession during the years to come. PROCESS MODELING, SIMULATION, AND CONTROL FOR CHEMICAL ENGINEERS Second Edition William L. Luyben Process Modeling and Control Center Department of Chemical Engineering Lehigh University McGraw-Hill Publisbing Company New York St. Louis San Francisco Auckland Bogota Caracas Hamburg Lisbon London Madrid Mexico Milan Montreal New Delhi Oklahoma City Paris San Juan SHo Paul0 Singapore Sydney Tokyo Toronto PROCESS MODELING, SIMULATION, AND CONTROL FOR CHEMICAL ENGINEERS INTERNATIONAL EDITION 1996 Exclusive rights by McGraw-Hill Book Co.- Singapore for manufacture and export. This book cannot be m-exported from the country to which it is consigned by McGraw-Hill. 567690BJEPMP9432 Copyright e 1999, 1973 by McGraw-Hill, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, u without the prior written permission of the publisher. This book was set in Times Roman. The editors were Lyn Beamesderfer and John M. Morris.% The production supervisor was Friederich W. Schulte. The cover was designed by John Hite. Project supervision was done by Harley Editorial Services. Ubrury of Congress Cataloging-in-Publlcatlon Data William L. Luyben.-2nd ed. p. cm. Bibliography: p. Includes index. ISBN 6-67-639159-9 1. Chemical process-Math data processing., 3. Chemica TP155.7.L66 1 9 6 9 , 669.2’61-dc19 When ordering this title use ISBN 1 process No.DEADQUiSICION ABOUT THE AUTHOR William L. Luyben received his B.S. in Chemical Engineering from the Penn- sylvania State University where he was the valedictorian of the Class of 1955. He worked for Exxon for five years at the Bayway Refinery and at the Abadan Refinery (Iran) in plant. technical service and design of petroleum processing units. After earning a Ph.D. in 1963 at the University of Delaware, Dr. Luyben worked for the Engineering Department of DuPont in process dynamics and control of chemical plants. In 1967 he joined Lehigh University where he is now Professor of Chemical Engineering and Co-Director of the Process Modeling and Control Center. Professor Luyben has published over 100 technical papers and has authored or coauthored four books. Professor Luyben has directed the theses of over 30 graduate students. He is an active consultant for industry in the area of process control and has an international reputation in the field of distillation column control. He was the recipient of the Beckman Education Award in 1975 and the Instrumqntation Technology Award in 1969 from the Instrument Society of America. f r.-., .,y<i ‘., Overall, he has devoted ove$? 3$,years to, his profession as a teacher, researcher, author, and practicing en&eer: :.: ’ ! 1. I’ vii CONTENTS 1 1.1 1 .2 1 .3 1 .4 1 .5 1 .6 Part I Preface Introduction 1 Examples of the Role of Process Dynamics and Control Historical Background Perspective Motivation for Studying Process Control General Concepts Laws and Languages of Process Control 1.6 .1 Process Control Laws 1.6.2 Languages of Process Control 1 6 7 8 8 11 11 1 2 Mathematical Models of Chemical Engineering Systems Xxi 2 Fundamentals 2 .1 Intreduction 2.1.1 Uses of Mathematical Models 2.1.2 Scope of Coverage 2.1.3 Principles of Formulation 2.2 Fundamental Laws 2.2.1 Continuity Equations 2.2.2 Energy Equation 2.2.3 Equations of Motion 2.2.4 Transport Equations 2.2.5 Equations of State 2.2.6 Equilibrium 2.2.7 Chemical Kinetics Problems 1 5 1 5 1 5 1 6 1 6 1 7 1 7 2 3 2 7 3 1 3 2 3 3 3 6 3 8 xi I 4 X i i C O N T E N T S 3 3 .1 3 .2 3 .3 3.4 3 .5 3.6 3.7 3.8 3 .9 3.10 3.11 3.12 3.13 3.14 Part II Examples of Mathematical Models of Chemical Engineering Systems Introduction Series of Isothermal, Constant-Holdup CSTRs CSTRs With Variable Holdups Two Heated Tanks Gas-Phase, Pressurized CSTR Nonisothermal CSTR Single-Component Vaporizer Multicomponent Flash Drum Batch Reactor Reactor With Mass Transfer Ideal Binary Distillation Column Multicomponent Nonideal Distillation Column Batch Distillation With Holdup pH Systems 3.14.1 Equilibrium-Constant Models 3.14.2 Titration-Curve Method Problems Computer Simulation 4 0 40 4 1 43 44 45 46 5 1 54 57 62 64 70 72 74 74 75 77 4 Numerical Methods 89 4.1 Introduction 8 9 4.2 Computer Programming 90 4.3 Iterative Convergence Methods 9 1 4.3.1 Interval Halving 93 4.3.2 Newton-Raphson Method 96 4.3.3 False Position 100 4.3.4 Explicit Convergence Methods 101 4.35 Wegstein 1 0 3 4.3.6 Muller Method 1 0 3 4.4 Numerical Integration of Ordinary Differential Equations 1 0 5 4.4.1 Explicit Numerical Integration Algorithms 1 0 6 4.4.2 Implicit Methods 1 1 3 Problems 1 1 4 5 Simulation Examples 5 .1 Gravity-Flow Tank 5.2 Three CSTRs in Series 5.3 Nonisothermal CSTR 5.4 Binary Distillation Column 5.5 Multicomponent Distillation Column 5.6 Variable Pressure Distillation 5.6.1 Approximate Variable-Pressure Model 5.6.2 Rigorous Variable-Pressure Model 1 1 6 1 1 6 1 1 9 1 2 4 1 2 9 1 3 2 1 4 1 1 4 1 1 4 2 CONTENTS xv 10.3 10.4 Performance Specifications 350 10.3.1 Steadystate Performance 350 10.3.2 Dynamic Specifications 3 5 1 Root Locus Analysis 3 5 3 10.4.1 Definition 3 5 3 10.4.2 Construction of Root Locus Curves 3 5 7 10.4.3 Examples 3 6 3 Problems 3 6 7 11 11.1 11.2 11.3 11.4 11.5 Laplace-Domain Analysis of Advanced Control Systems Cascade Control 11.1.1 Series Cascade 11.1.2 Parallel Cascade Feedforward Control 11.2.1 Fundamentals 11.2.2 Linear Feedforward Control 11.2.3 Nonlinear Feedforward Control (Ipenloop Unstable Processes 11.3.1. Simple Systems 11.3.2 Effects of Lags 11.3.3 PD Control 11.3.4 Effects of Reactor Scale-up On Controllability Processes With Inverse Response Model-Based Control 11.51 Minimal Prototype Design 11.52 Internal Model Control Problems Part V Frequency-Domain Dynamics and Control 376 376 3 7 7 3 8 2 3 8 3 3 8 3 384 3 8 9 3 9 1 392 397 397 3 9 8 3 9 8 402 402 404 407 1 2 Frequency-Domain Dynamics 12.1 Definition 12.2 Basic Theorem 12.3 Representation 12.3.1 Nyquist Plots 12.3.2 Bode Plots 12.3.3 Nichols Plots 12.4 Frequency-Domain Solution Techniques Problems 415 415 417 420 4 2 1 427 440 442 452 1 3 Frequency-Domain Analysis of Closedloop Systems 455 13.1 Nyquist Stability Criterion 456 13.1.1 Proof 456 13.1.2 Examples 460 13.1.3 Representation 468 . xvi coNTENls 13.2 13.3 ii4 Closedloop Specifications in the Frequency Domain 470 13.2.1 Phase Margin 470 \’ 13.2.2 Gain Margin 470 13.2.3 Maximum Closedloop Log Modulus (Ly) 472 Frequency Response of Feedback Controllers 478 13.3.1 Proportional Controller (P) 478 13.3.2 Proportional-Integral Controller (PI) 479 13.3.3 Proportional-Integral-Derivative Controller (PID) 481 Examples 4 8 1 13.4.1 Three-CSTR System 4 8 1 13.4.2 First-Order Lag With Deadtime 488 13.4.3 Openloop Unstable Processes 490 Problems 493 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Process Identification 502 Purpose 502 Direct Methods 503 14.2.1 Time-Domain (L Eyeball ” Fitting of Step Test Data 5 0 3 14.2.2 Direct Sine-Wave Testing 5 0 5 Pulse Testing 507 14.3.1 Calculation of Gu,, From Pulse Test Data 5 0 8 14.3.2 Digital Evaluation of Fourier Transformations 512 14.3.3 Practical Tips on Pulse Testing 515 14.3.4 Processes With Integration 516 Step Testing 5 1 8 ATV Identification 519 14.51 Autotuning 520 14.52 Approximate Transfer Functions 522 Least-Squares Method 525 State Estimators 529 Relationships Among Time, Laplace, and Frequency Domains 530 14.8.1 Laplace to Frequency Domain 530 14.8.2 Frequency to Laplace Domain 530 14.8.3 Time to Laplace Domain 530 14.8.4 Laplace to Time Domain 530 14.85 Time to Frequency Domain 532 14.8.6 Frequency to Time Domain 532 Problems 5 3 3 Part VI Multivariable Processes 1 5 Matrix Properties and State Variables 537 15.1 Matrix Mathematics 5 3 7 15.1.1 Matrix Addition 5 3 8 15.1.2 Matrix Multiplication 538 comwrs xvii 151.3 Transpose of a Matrix 5 3 9 151.4 Matrix Inversion 5 4 0 15.2 Matrix Properties 5 4 1 15.2.1 Eigenvalues 5 4 2 15.212 Canonical Transformation 5 4 3 152.3 Singular Values 546 15.3 Representation of Multivariable Processes 5 4 8 15.3.1 Transfer Function Matrix 5 4 8 15.3.2 State Variables 5 5 1 15.4 Openloop and Closedloop Systems 554 15.4.1 Transfer Function Representation 5 5 4 15.4.2 State Variable Representation 5 5 6 15.5 Computer Programs For Matrix Calculations 559 Problems 5 6 1 1 6 Analysis of Multivariable Systems 16.1 Stability 16.1.1 Openloop and Closedloop Characteristic Equations 16.1.2 Multivariable Nyquist Plot 16.1.3 Characteristic Loci Plots 16.1.4 Niederlinski Index * 16.2 Resiliency 16.3 Interaction 16.3.1 Relative Gain Array (Bristol Array) 16.3.2 Inverse Nyquist Array (INA) 16.3.3 Decoupling 16.4 Robustness 16.4.1 Trade-off Between Performance and Robustness 16.4.2 Doyle-Stein Criterion 16.4.3 Skogestad-Morari Method Problems 1 7 Design of Controllers For Multivariable Processes 17.1 17.2 17.3 17.4 17.5 17.6 17.7 Problem Definition Selection of Controlled Variables 17.2.1 Engineering Judgment 17.2.2 Singular Value Decomposition Selection of Manipulated Variables Elimination of Poor Pairings BLT Tuning Load Rejection Performance Multivariable Controllers 17.7.1 Multivariable DMC 17.7.2 Multivariable IMC Problems 5 6 2 562 562 564 5 6 8 5 7 2 573 5 7 5 576 5 7 9 5 8 1 5 8 4 5 8 5 5 8 5 5 8 8 5 9 1 594 594 5 9 6 5 9 6 5 9 6 5 9 8 5 9 8 5 9 9 6 0 5 606 606 609 6 1 1 . PREFACE The first edition of this book appeared over fifteen years ago. It was the first chemical engineering textbook to combine modeling, simulation, and control. It also was the first chemical engineering book to present sampled-data control. This choice of subjects proved to be popular with both students and teachers and of considerable practical utility. During the ten-year period following publication, I resisted suggestions from the publisher to produce a second edition because I felt there were really very few useful new developments in the field. The control hardware had changed drastically, but the basic concepts and strategies of process control had under- gone little change. Most of the new books that have appeared during the last fifteen years are very similar in their scope and content to the first edition. Basic classical control is still the major subject. However, in the last five years, a number of new and useful techniques have been developed. This is particularly true in the area of multivariable control. Therefore I feel it is time for a second edition. In the area of process control, new methods of analysis and synthesis of control systems have been developed and need to be added to the process control engineer’s bag of practical methods. The driving force for much of this develop- ment was the drastic increase in energy costs in the 1970s. This led to major redesigns of many new and old processes, using energy integration and more complex processing schemes. The resulting plants are more interconnected. This increases control loop interactions and expands the dimension of control prob- lems. There are many important processes in which three, four, or even more control loops interact. As a result, there has been a lot of research activity in multivariable control, both in academia and in industry. Some practical, useful tools have been devel- oped to design control systems for these multivariable processes. The second edition includes a fairly comprehensive discussion of what I feel are the useful techniques for controlling multivariable processes. xxi . Xxii P R E F A C E Another significant change over the last decade has been the dramatic increase in the computational power readily available to engineers. Most calcu- lations can be performed on personal computers that have computational horse- power equal to that provided only by mainframes a few years ago. This means that engineers can now routinely use more rigorous methods of analysis and synthesis. The second edition includes more computer programs. All are suitable for execution on a personal computer. In the area of mathematical modeling, there has been only minor progress. We still are able to describe the dynamics of most systems adequately for engi- neering purposes. The trade-off between model rigor and computational effort has shifted toward more precise models due to the increase in computational power noted above. The second edition includes several more examples of models that are more rigorous. In the area of simulation, the analog computer has almost completely dis- appeared. Therefore, analog simulation has been deleted from this edition. Many new digital integration algorithms have been developed, particularly for handling large numbers of “stiff” ordinary differential equations. Computer programming is now routinely taught at the high school level. The-second edition includes an expanded treatment of iterative convergence methods and- of numerical integra- tion algorithms for ordinary differential equations, including both explicit and implicit methods. The second edition presents some of the material in a slightly different sequence. Fifteen additional years of teaching experience have convinced me that it is easier for the students to understand the time, Laplace, and frequency tech- niques if both the dynamics and the control are presented together for each domain. Therefore, openloop dynamics and closedloop control are both dis- cussed in the time domain, then in the Laplace domain, and finally in the fre- quency domain. The z domain is discussed in Part VII. There has been a modest increase in the number of examples presented in the book. The number of problems has been greatly increased. Fifteen years of quizzes have yielded almost 100 new problems. The new material presented in the second edition has come from many sources. I would like to express my thanks for the many useful comments and suggestions provided by colleagues who reviewed this text during the course of its development, especially to Daniel H. Chen, Lamar University; T. S. Jiang, Uni- versity of Illinois-Chicago; Richard Kerrnode, University of Kentucky; Steve Melsheimer, Cignson University; James Peterson, Washington State University; and R. Russell Rhinehart, Texas Tech University. Many stimulating and useful discussions of multivariable control with Bjorn Tyreus of DuPont and Christos Georgakis of Lehigh University have contributed significantly. The efforts and suggestions of many students are gratefully acknowledged. The “ LACEY” group (Luyben, Alatiqi, Chiang, Elaahi, and Yu) developed and evaluated much of the new material on multivariable control discussed in Part VI. Carol Biuckie helped in the typing of the final manuscript. Lehigh undergraduate and graduate classes have contributed to the book for over twenty years by their questions, CHAPTER 1 INTRODUCTION This chapter is an introduction to process dynamics and control for those stu- dents who have had little or no contact or experience with real chemical engi- neering processes. The objective is to illustrate where process control fits into the picture and to indicate its relative importance in the operation, design, and devel- opment of a chemical engineering plant. This introductory chapter is, I am sure, unnecessary for those practicing engineers who may be using this book. They are well aware of the importance of considering the dynamics of a process and of the increasingly complex and sophisticated control systems that are being used. They know that perhaps 80 percent of the time that one is “on the plant” is spent at the control panel, watching recorders and controllers (or CRTs). The control room is the nerve center of the plant. 1.1 EXAMPLES OF THE ROLE OF PROCESS DYNAMICS AND CONTROL Probably the best way to illustrate what we mean by process dynamics and control is to take a few real examples. The first example describes a simple process where dynamic response, the time-dependent behavior, is important. The second example illustrates the use of a single feedback controller. The third example discusses a simple but reasonably typical chemical engineering plant and its conventional control system involving several controllers. Example 1.1. Figure 1.1 shows a tank into which an incompressible (constant- density) liquid is pumped at a variable rate F, (ft3/s). This inflow rate can vary with 1 2 PROCESS MODELING, SIMULATION. AND CONTROL FOR CHEMICAL ENGINEERS FIGURE 1.1 Gravity-flow tank. time because of changes in operations upstream. The height of liquid in the vertical cylindrical tank is h (ft). The flow rate out of the tank is F (ft’/s). Now F, , h, andf will all vary with time and are therefore functions of time t. Consequently we use the notation Foe,), I+,), and F(,, . Liquid leaves the base of the tank via a long horizontal pipe and discharges into the top of another tank. Both tanks are open to the atmosphere. Let us look first at the steadystate conditions. By steadystate we mean, in most systems, the conditions when nothing is changing with time. Mathematically this corresponds to having all time derivatives equal to zero, or to allowing time to become very large, i.e., go to infinity. At steadystate the flow rate out of the tank must equal the flow rate into the tank. In this book we will denote steadystate values of variables by an overscore or bar above the variables. Therefore at steady- state in our tank system Fe = F. For a given F, the height of liquid in the tank at steadystate would also be some constant Ii. The value of h would be that height that provides enough hydrau- lic pressure head at the inlet of the pipe to overcome the frictional losses of liquid flowing down the pipe. The higher the flow rate F, the higher 6 will be. In the steadystate design of the tank, we would naturally size the diameter of the exit line and the height of the tank so that at the maximum flow rate expected the tank would not overflow. And as any good, conservative design engineer knows, we would include in the design a 20 to 30 percent safety factor on the tank height. Actual height specified_________------------------- Maximum design height F FIGURE 1.2 Steadystate height versus flow. INTRODUmION 3 Since this is a book on control and instrumentation, we might also mention that a high-level alarm and/or an interlock (a device to shut off the feed if the level gets too high) should be installed to guarantee that the tank would not spill over. The tragic accidents at Three Mile Island, Chernobyl, and Bhopal illustrate the need for well- designed and well-instrumented plants. The design of the system would involve an economic balance between the cost of a taller tank and the cost of a bigger pipe, since the bigger the pipe diameter the lower is the liquid height. Figure 1.2 shows the curve of !i versus F for a specific numerical case. So far we have considered just the traditional steadystate design aspects of this fluid flow system. Now let us think about what would happen dynamically if we changed Fe. How will he) and F$) vary with time? Obviously F eventually has to end up at the new value of F,. We can easily determine from the steadystate design curve of Fig. 1.2 where h will go at the new steadystate. But what paths will h(,, and Fgb take to get to their new steadystates? Figure 1.3 sketches the problem. The question is which curves (1 or 2) rep- resent the actual paths that F and h will follow. Curves 1 show gradual increases in h and F to their new steadystate values. However, the paths could follow curves 2 where the liquid height rises above its final steadystate value. This is called “overshoot.” Clearly, if the peak of the overshoot in h is above the top of the tank, we would be in trouble. Our steadystate design calculations tell us nothing about what the dynamic response to the system will be. They tell us where we will start and where we will end up but not how we get there. This kind of information is what a study of the dynamics of the system will reveal. We will return to this system later in the book to derive a mathematical model of it and to determine its dynamic response quantita- tively by simulation. Example 1.2. Consider the heat exchanger sketched in Fig. 1.4. An oil stream passes through the tube side of a tube-in-shell heat exchanger and is heated by condensing steam on the shell side. The steam condensate leaves through a steam trap (a device 6 PROCESS MODELING. SIMULATION, AND CONTROL FOR CHEMICAL ENGINEERS Thermodynamics and mass transfer. Operating pressure, number of plates and reflux ratio in the distillation column; temperature profile in the column; equi- librium conditions in the reactor But how do we decide how to control this plant? We will spend most of our time in this book exploring this important design and operating problem. All our studies of mathematical modeling, simulation, and control theory are aimed at understanding the dynamics of processes and control systems so that we can develop and design better, more easily controlled plants that operate more efficiently and more safely. For now let us say merely that the control system shown in Fig. 1.5 is a typical conventional system. It is about the minimum that would be needed to run this plant automaticalljl without constant operator attention. Notice that even in this simple plant with a minimum of instrumentation the total number of control loops is IO. We will find that most chemical engineering processes are multivariable. 1.2 HISTORICAL BACKGROUND Most chemical processing plants were run essentially manually prior to the 1940s. Only the most elementary types of controllers were used. Many operators were needed to keep watch on the many variables in the plant. Large tanks were employed to act as buffers or surge capacities between various units in the plant. These tanks, although sometimes quite expensive, served the function of filtering out some of the dynamic disturbances by isolating one part of the process from upsets occurring in another part. With increasing labor and equipment costs and with the development of more severe, higher-capacity, higher-performance equipment and processes in the 1940s and early 195Os, it became uneconomical and often impossible to run plants without automatic control devices. At this stage feedback controllers were added to the plants with little real consideration of or appreciation for the dynamics of the process itself. Rule-of-thumb guides and experience were the only design techniques. In the 1960s chemical engineers began to apply dynamic analysis and control theory to chemical engineering processes. Most of the techniques were adapted from the work in the aerospace and electrical engineering fields. In addi- tion to designing better control systems, processes and plants were developed or modified so that they were easier to control. The concept of examining the many parts of a complex plant together as a single unit, with all the interactions included, and devising ways to control the entire plant is called systems engineer- ing. The current popular “buzz” words artificial intelligence and expert systems are being applied to these types of studies. The rapid rise in energy prices in the 1970s provided additional needs for effective control systems. The design and redesign of many plants to reduce energy consumption resulted in more complex, integrated plants that were much more interacting. So the challenges to the process control engineer have contin- ued to grow over the years. This makes the study of dynamics and control even more vital in the chemical engineering curriculum than it was 30 years ago. INTRODUCTION 7 13 PERSPECTIVE Lest I be accused of overstating the relative importance of process control to the main stream of chemical engineering, let me make it perfectly clear that the tools of dynamic analysis are but one part of the practicing engineer’s bag of tools and techniques, albeit an increasingly important one. Certainly a solid foundation in the more traditional areas of thermodynamics, kinetics, unit operations, and transport phenomena is essential. In fact, such a foundation is a prerequisite for any study of process dynamics. The mathematical models that we derive are really nothing but extensions of the traditional chemical and physical laws to include the time-dependent terms. Control engineers sometimes have a tendency to get too wrapped up in the dynamics and to forget the steadystate aspects. Keep in mind that if you cannot get the plant to work at steadystate you cannot get it to work dynamically. An even greater pitfall into which many young process control engineers fall, particularly in recent years, is to get so involved in the fancy computer control hardware that is now available that they lose sight of the process control objectives. All the beautiful CRT displays and the blue smoke and mirrors that computer control salespersons are notorious for using to sell hardware and soft- ware can easily seduce the unsuspecting control engineer. Keep in mind your main objective: to come up with an effective control system. How you implement it, in a sophisticated computer or in simple pneumatic instruments, is of much less importance. You should also appreciate the fact that fighting your way through this book will not in itself make you an expert in process control. You will find that a lot remains to be learned, not so much on a higher theoretical level as you might expect, but more on a practical-experience level. A sharp engineer can learn a tremendous amount about process dynamics and control that can never be put in a book, no matter how practically oriented, by climbing around a plant, talking with operators and instrument mechanics, tinkering in the instrument shop, and keeping his or her eyes open in the control room. You may question, as you go through this book, the degree to which the dynamic analysis and controller design techniques discussed are really used in industry. At the present time 70 to 80 percent of the control loops in a plant are usually designed, installed, tuned, and operated quite successfully by simple, rule- of-thumb, experience-generated techniques. The other 20 to 30 percent of the loops are those on which the control engineer makes his money. They require more technical knowledge. Plant testing, computer simulation, and detailed con- troller design or process redesign may be required to achieve the desired per- formance. These critical loops often make or break the operation of the plant. I am confident that the techniques discussed in this book will receive wider and wider applications as more young engineers with this training go to work in chemical plants. This book is an attempt by an old dog to pass along some useful engineering tools to the next generation of pups. It represents over thirty years of experience in this lively and ever-challenging area. Like any “expert,” I’ve learned 8 PROCESS MODELING, SIMULATION, AND CONTROL FOR CHEMICAL ENGINEERS from my successes, but probably more from my failures. I hope this book helps you to have many of the former and not too many of the latter. Remember the old saying: “If you are making mistakes, but they are new ones, you are getting smarter.” 1.4 MOTIVATION FOR STUDYING PROCESS CONTROL Some of the motivational reasons for studying the subjects presented in this book are that they are of considerable practical importance, they are challenging, and they are fun. 1. Importance. The control room is the major interface with the plant. Automa- tion is increasingly common in all degrees of sophistication, from single-loop systems to computer-control systems. 2. Challenging. You will have to draw on your knowledge of all areas of chemical engineering. You will use most of the mathematical tools available (differential equations, Laplace transforms, complex variables, numerical analysis, etc.) to solve real problems. 3. Fun. I have found, and I hope you will too, that process dynamics is fun. You will get the opportunity to use some simple as well as some fairly advanced mathematics to solve real plant problems. There is nothing quite like the thrill of working out a controller design on paper and then seeing it actually work on the plant. You will get a lot of satisfaction out of going into a plant that is having major control problems, diagnosing what is causing the problem and getting the whole plant lined out on specification. Sometimes the problem is in the process, in basic design, or in equipment malfunctioning. But sometimes it is in the control system, in basic strategy, or in hardware malfunctioning. Just your knowledge of what a given control device should do can be invaluable. 1.5 GENERAL CONCEPTS I have tried to present in this book a logical development. We will begin with fundamentals and simple concepts and extend them as far as they can be gain- fully extended. First we will learn to derive mathematical models of chemical engineering systems. Then we will study some of the ways to solve the resulting equations, usually ordinary differential equations and nonlinear algebraic equa- tions. Next we will explore their openloop (uncontrolled) dynamic behavior. Finally we will learn to design controllers that will, if we are smart enough, make the plant run automatically the way we want it to run: efficiently and safely. Before we go into details in the subsequent chapters, it may be worthwhile at this point to define some very broad and general concepts and some of the terminology used in dynamics and control. I N T R O D U C T I O N 11 FIGURE 19 Stability. Time stability if it oscillates, even when undisturbed, and the amplitude of the oscil- lations does not decay. Most processes are openloop stable, i.e., stable with no controllers on the system. One important and very interesting exception that we will study in some detail is the exothermic chemical reactor which can be openloop unstable. All real processes can be made closedloop unstable (unstable when a feedback controller is in the system) if the controller gain is made large enough. Thus stability is of vital concern in feedback control systems. The performance of a control system (its ability to control the process tightly) usually increases as we increase the controller gain. However, we get closer and closer to being closedloop unstable. Therefore the robustness of the control system (its tolerance to changes in process parameters) decreases: a small change will make the system unstable. Thus there is always a trade-off between robustness and performance in control system design. 1.6 LAWS AND LANGUAGES OF PROCESS CONTROL 1.6.1 Process Control Laws There are several fundamental laws that have been developed in the process control field as a result of many years of experience. Some of these may sound similar to some of the laws attributed to Parkinson, but the process control laws are not intended to be humorous. (1) FIRST LAW. The simplest control system that will do the job is the best. Complex elegant control systems look great on paper but soon end up on “manual” in an industrial environment. Bigger is definitely not better in control system design. (2) SECOND LAW. You must understand the process before you can control it. 12 PROCESS MODELING, SIMULATION, AND CONTROL FOR CHEMICAL ENGINEERS No degree of sophistication in the control system (be it adaptive control, Kalman filters, expert systems, etc.) will work if you do not know how your process works. Many people have tried to use complex controllers to overcome ignorance about the process fundamentals, and they have failed! Learn how the process works before you start designing its control system. 1.6.2 Languages of Process Control As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give func- tions of time is a “time domain” technique. Use of Laplace transforms to charac- terize the dynamics of systems is a “Laplace domain” technique. Frequency response methods provide another approach to the problem. All of these methods are useful because each has its advantages and dis- advantages. They yield exactly the same results when applied to the same problem. These various approaches are similar to the use of different languages by people around the world. A table in English is described by the word “TABLE.” In Russian a table is described by the word “CTOJI.” In Chinese a table is “ $ 5.” In German it is “der Tisch.” But in any language a table is still a table. In the study of process dynamics and control we will use several languages. English = time domain (differential equations, yielding exponential time function solutions) Russian = Laplace domain (transfer functions) Chinese = frequency domain (frequency response Bode and Nyquist plots) Greek = state variables (matrix methods applies to differential equations) German = z domain (sampled-data systems) You will find the languages are not difficult to learn because the vocabulary that is required is quite small. Only 8 to 10 “words” must be learned in each lan- guage. Thus it is fairly easy to translate back and forth between the languages. We will use “English” to solve some simple problems. But we will find that more complex problems are easier to understand and solve using “Russian.” AS problems get even more complex and realistic, the use of “Chinese” is required. So we study in this book a number of very useful and practical process control languages. I have chosen the five languages listed above simply because I have had some exposure to all of them over the years. Let me assure you that no political or nationalistic motives are involved. If you would prefer French, Spanish, Italian, Japanese, and Swahili, please feel free to make the appropriate substitu- tions! My purpose in using the language metaphor is to try to break some of the psychological barriers that students have to such things as Laplace transforms and frequency response. It is a pedagogical gimmick that I have used for over two decades and have found it to be very effective with students. PART MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS In the next two chapters we will develop dynamic mathematical models forseveral important chemical engineering systems. The examples should illus- trate the basic approach to the problem of mathematical modeling. Mathematical modeling is very much an art. It takes experience, practice, and brain power to be a good mathematical modeler. You will see a few models developed in these chapters. You should be able to apply the same approaches to your own process when the need arises. Just remember to always go back to basics : mass, energy, and momentum balances applied in their time-varying form. 13 16 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 3. Plant operation: troubleshooting control and processing problems; aiding in start-up and operator training; studying the effects of and the requirements for expansion (bottleneck-removal) projects; optimizing plant operation. It is usually much cheaper, safer, and faster to conduct the kinds of studies listed above on a mathematical model than experimentally on an operating unit. This is not to say that plant tests are not needed. As we will discuss later, they are a vital part of confirming the validity of the model and of verifying impor- tant ideas and recommendations that evolve from the model studies. 2.1.2 Scope of Coverage We will discuss in this book only deterministic systems that can be described by ordinary or partial differential equations. Most of the emphasis will be on lumped systems (with one independent variable, time, described by ordinary differential equations). Both English and SI units will be used. You need to be familiar with both. 2.1.3 Principles of Formulation A. BASIS. The bases for mathematical models are the fundamental physical and chemical laws, such as the laws of conservation of mass, energy, and momentum. To study dynamics we will use them in their general form with time derivatives included. B. ASSUMPTIONS. Probably the most vital role that the engineer plays in mod- eling is in exercising his engineering judgment as to what assumptions can be validly made. Obviously an extremely rigorous model that includes every phe- nomenon down to microscopic detail would be so complex that it would take a long time to develop and might be impractical to solve, even on the latest super- computers. An engineering compromise between a rigorous description and getting an answer that is good enough is always required. This has been called “optimum sloppiness.” It involves making as many simplifying assumptions as are reasonable without “throwing out the baby with the bath water.” In practice, this optimum usually corresponds to a model which is as complex as the avail- able computing facilities will permit. More and more this is a personal computer. The development of a model that incorporates the basic phenomena occurring in the process requires a lot of skill, ingenuity, and practice. It is an area where the creativity and innovativeness of the engineer is a key element in the success of the process. The assumptions that are made should be carefully considered and listed. They impose limitations on the model that should always be kept in mind when evaluating its predicted results. C. MATHEMATICAL CONSISTENCY OF MODEL. Once all the equations of the mathematical model have been written, it is usually a good idea, particularly with FUNDAMENTALS 17 big, complex systems of equations, to make sure that the number of variables equals the number of equations. The so-called “degrees of freedom” of the system must be zero in order to obtain a solution. If this is not true, the system is underspecified or overspecified and something is wrong with the formulation of the problem. This kind of consistency check may seem trivial, but I can testify from sad experience that it can save many hours of frustration, confusion, and wasted computer time. Checking to see that the units of all terms in all equations are consistent is perhaps another trivial and obvious step, but one that is often forgotten. It is essential to be particularly careful of the time units of parameters in dynamic models. Any units can be used (seconds, minutes, hours, etc.), but they cannot be mixed. We will use “minutes” in most of our examples, but it should be remem- bered that many parameters are commonly on other time bases and need to be converted appropriately, e.g., overall heat transfer coefficients in Btu/h “F ft’ or velocity in m/s. Dynamic simulation results are frequently in error because the engineer has forgotten a factor of “60” somewhere in the equations. D. SOLUTION OF THE MODEL EQUATIONS. We will concern ourselves in detail with this aspect of the model in Part II. However, the available solution techniques and tools must be kept in mind as a mathematical model is developed. An equation without any way to solve it is not worth much. E. VERIFICATION. An important but often neglected part of developing a math- ematical model is proving that the model describes the real-world situation. At the design stage this sometimes cannot be done because the plant has not yet been built. However, even in this situation there are usually either similar existing plants or a pilot plant from which some experimental dynamic data can be obtained. The design of experiments to test the validity of a dynamic model can sometimes be a real challenge and should be carefully thought out. We will talk about dynamic testing techniques, such as pulse testing, in Chap. 14. 2.2 FUNDAMENTAL LAWS In this section, some fundamental laws of physics and chemistry are reviewed in their general time-dependent form, and their application to some simple chemical systems is illustrated. 2.2.1 Continuity Equations A. TOTAL CONTINUITY EQUATION (MASS BALANCE). The principle of the conservation of mass when applied to a dynamic system says 18 MATHEMATICAL MODEL-S OF CHEMICAL ENGINEERING SYSTEMS The units of this equation are mass per time. Only one total continuity equation can be written for one system. The normal steadystate design equation that we are accustomed to using says that “what goes in, comes out.” The dynamic version of this says the same thing with the addition of the world “eventually.” The right-hand side of Eq. (2.1) will be either a partial derivative a/at or an ordinary derivative d/dt of the mass inside the system with respect to the inde- pendent variable t. Example 2.1 Consider the tank of perfectly mixed liquid shown in Fig. 2.1 into which flows a liquid stream at a volumetric rate of F, (ft3/min or m3/min) and with a density of p,, (lb,,,/ft’ or kg/m’). The volumetric holdup of liquid in the tank is V (ft3 or m”), and its density is p. The volumetric flow rate from the tank is F, and the density of the outflowing stream is the same as that of the tank’s contents. The system for which we want to write a total continuity equation is all the liquid phase in the tank. We call this a macroscopic system, as opposed to a micro- scopic system, since it is of definite and finite size. The mass balance is around the whole tank, not just a small, differential element inside the tank. F, p,, - Fp = time rate of change of p V The units of this equation are lb&nin or kg/min. (2.2) (AL)(!!) - (EJ!!) = ‘“3zf3) Since the liquid is perfectly mixed, the density is the same everywhere in the tank; it does not vary with radial or axial position; i.e., there are no spatial gradients in density in the tank. This is why we can use a macroscopic system. It also means that there is only one independent variable, t. Since p and V are functions only of t, an ordinary derivative is used in E,q. (2.2). 4Pv-=F,p,-Fp dt Example 2.2. Fluid is flowing through a constant-diameter cylindrical pipe sketched in Fig. 2.2. The flow is turbulent and therefore we can assume plug-flow conditions, i.e., each “slice” of liquid flows down the pipe as a unit. There are no radial gra- dients in velocity or any other properties. However, axial gradients can exist. Density and velocity can change as the fluid flows along the axial or z direc- tion. There are now two independent variables: time t and position z. Density and F U N D A M E N T A L S 21 The minus sign comes from the fact that A is being consumed, not produced. The units of all these terms must be the same: moles of A per unit time. Therefore the Y/K, term must have these units, for example (ft3)(min-‘)(moles of A/ft3). Thus the units of k in this system are min- ‘ . d(VCA)Time rate of change of A inside tank = 7 Combining all of the above gives WCJ- = F,C,, - FC, - VkC, dt We have used an ordinary derivative since t is the only independent variable in this lumped system. The units of this component continuity equation are moles of A per unit time. The left-hand side of the equation is the dynamic term. The first two terms on the right-hand side are the convective terms. The last term is the gener- ation term. Since the system is binary (components A and B), we could write another component continuity equation for component B. Let CB be the concentration of B in moles of B per unit volume. WC,)- = F, C,, - FCB + VkCA dt Note the plus sign before the generation term since B is being produced by the reaction. Alternatively we could use the total continuity equation [Eq. (2.3)] since C, , CB , and p are uniquely related by MAC,++ M,Cg=p (2.11) where MA and M, are the molecular weights of components A and B, respectively. Example 2.4. Suppose we have the same macroscopic system as above except that now consecutive reactions occur. Reactant A goes to B at a specific reaction rate k,, but B can react at a specific reaction rate k, to form a third component C. A-B-k2 c Assuming first-order reactions, the component continuity equations for com- ponents A, B, and C are 4VCJ-=F,C,,-FC,-Vk,C, dt WC,)-=F,,CBO--FCB+VklCA-Vk2CB dt (2.12) WC,)-=F,&-FC,+Vk,C, dt 22 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS The component concentrations are related to the density i MjCj=p j=A (2.13) Three component balances could be used or we could use two of the component balances and a total mass balance. Example 2.5. Instead of fluid flowing down a pipe as in Example 2.2, suppose the pipe is a tubular reactor in which the same reaction A h B of Example 2.3 takes place. As a slice of material moves down the length of the reactor the concentration of reactant CA decreases as A is consumed. Density p, velocity V, and concentration CA can all vary with time and axial position z. We still assume plug-flow conditions so that there are no radial gradients in velocity, density, or concentration. The concentration of A fed to the inlet of the reactor at z = 0 is defined as C Act. 0) - CAO(r) The concentration of A in the reactor ehluent at z = L is defined as C A& L) - CAL(r) (2.14) We now want to apply the component continuity equation for reactant A to a small differential slice of width dz, as shown in Fig. 2.4. The inflow terms can be split into two types: bulk flow and diffusion. Diffusion can occur because of the concentration gradient in the axial direction. It is usually much less important than bulk flow in most practical systems, but we include it here to see what it contributes to the model. We will say that the diffusive flux of A, N, (moles of A per unit time per unit area), is given by a Fick’s law type of relationship where fD, is a diffusion coefficient due to both diffusion and turbulence in the fluid flow (so-called “eddy diffusivity”). fDA has units of length’ per unit time. The terms in the general component continuity equation [Eq. (2.9)] are: Molar flow of A into boundary at z (bulk flow and diffusion) = vAC, + AN, (moles of A/s) z I + dz z=L z=o FIGURE 2.4 Tubular reactor. FUNDAMENTALS 23 Molar flow of A leaving system at boundary z + dz = @AC,, + AN,) + a(uACA + ANA) dzaZ Rate of formation of A inside system = -kc,.4 dz Time rate of change of A inside system = d(A dz CA) at Substituting into Eq. (2.9) gives a(A ; “) = @AC, + AN,) - ( uAC, + AN, + d(uAC, + AN,) aZ dz > - kCA Adz $$+ @CA + NA) + kC aZ A = o Substituting Eq. (2.16) for N,, a(ucA)f&A+- aZ +kC,=; ‘BA$$ ( > (2.17) The units of the equation are moles A per volume per time. 2.2.2 Energy Equation The first law of thermodynamics puts forward the principle of conservation of energy. Written for a general “open” system (where flow of material in and out of the system can occur) it is Flow of internal, kinetic, and I[ flow of internal, kinetic, and potential energy into system - potentia1 energy out of system by convection or diffusion by convection or diffusion 1 heat added to system by I[ work done by system on+ conduction, radiation, and - surroundings (shaft work andreaction PV work) time rate of change of internal, kinetic, = and potential energy inside system 1 (2.18) Example 2.6. The CSTR system of Example 2.3 will be considered again, this time with a cooling coil inside the tank that can remove the exothermic heat of reaction 1 (Btu/lb . mol of A reacted or Cal/g. mol of A reacted). We use the normal convention that 1 is negative for an exothermic reaction and positive for an endothermic reac- tion. The rate of heat generation (energy per time) due to reaction is the rate of consumption of A times 1. Q. = -RVC,k (2.19) 245 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS and replacing internal energies with enthalpies in the time derivative, the energy equation of the system (the vapor and liquid contents of the tank) becomes d@,KH+pV,h) d t = F,p,h, - Fph - F,p,H + Q - IVkCA (2.27) In order to express this equation explicitly in terms of temperature, let us again use a very simple form for h (h = C, T) and an equally simple form for H. H = C, T + 1, (2.28) where 1, is an average heat of vaporization of the mixture. In a more rigorous model A, could be a function of temperature TV, composition y, and pressure P. Equation (2.27) becomes 0, KW, T + 4) + PV, C, T3 d t = F,p,C,T, - F&T - F,pdC, T + I,) + Q - WkC, (2.29) Example 2.8. To illustrate the application of the energy equation to a microscopic system, let us return to the plug-flow tubular reactor and now keep track of tem- perature changes as the fluid flows down the pipe. We will again assume no radial gradients in velocity, concentration, or temperature (a very poor assumption in some strongly exothermic systems if the pipe diameter is not kept small). Suppose that the reactor has a cooling jacket around it as shown in Fig. 2.7. Heat can be transferred from the process fluid reactants and products at temperatur<Zlto the metal wall of the reactor at temperature TM. The heat is subsequently transferred to the cooling water. For a complete description of the system we would need energy equations for the process fluid, the metal wall, and the cooling water. Here we will concern ourselves only with the process energy equation. Looking at a little slice of the process fluid as our system, we can derive each of the terms of Eq. (2.18). Potential-energy and kinetic-energy terms are assumed negligible, and there is no work term. The simplified forms of the internal energy and enthalpy are assumed. Diffusive flow is assumed negligible compared to bulk flow. We will include the possibility for conduction of heat axially along the reactor due to molecular or turbulent conduction. TV (1. 2) -. Water c FIGURE 2.7 Jacketed tubular reactor. FUNDAMENTALS 27 Flow of energy (enthalpy) into boundary at z due to bulk flow : VA&, T lb Btu with English engineering units of -&fP 2 - “R = Btu/min ft3 lb,,, “R Flow of energy (enthalpy) out of boundary at z + dz: vApC P T + a(vApCpT) dz a2 Heat generated by chemical reaction = -A dz kCA 1 Heat transferred to metal wall = -h&tD dz)(T - TM) where h, = heat transfer film coefficient, Btu/min ft2 “R D = diameter of pipe, ft Heat conduction into boundary at z = qZ A where qZ is a heat flux in the z direction due to conduction. We will use Fourier’s law to express qr in terms of a temperature driving force: q, = -k, E (2.30) where k, is an effective thermal conductivity with English engineering units of Btu/ft min “R. Heat conduction out of boundary at z + dz = qZ A + 7WA) dz Rate of change of internal energy (enthalpy) of the system = a(pA dz C, 79 at Combining all the above gives tic, T) + WPC, 7.7 + IkC at az A + %D (T - TM) = aCkA;;‘az)l (2.31) 2.2.3 Equations of Motion As any high school student, knows, Newton’s second law of motion says that force is equal to mass times acceleration for a system with constant mass M. &!! (2.32) CL where F = force, lbr M = mass, lb,,, a = acceleration, ft/s2 gc = conversion constant needed when English engineering units are used to keep units consistent = 32.2 lb,,, ft/lbr s2 28 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS This is the basic relationship that is used in writing the equations of motion for a system. In a slightly more general form, where mass can vary with time, i q = jiI Fji ge (2.33) where Ui = velocity in the i direction, ft/s Fji = jth force acting in the i direction Equation (2.33) says that the time rate of change of momentum in the i direction (mass times velocity in the i direction) is equal to the net sum of the forces pushing in the i direction. It can be thought of as a dynamic force balance. Or more eloquently it is called the conservation ofmomentum. In the real world there are three directions: x, y, and z. Thus, three force balances can be written for any system. Therefore, each system has three equa- tions of motion (plus one total mass balance, one energy equation, and NC - 1 component balances). Instead of writing three equations of motion, it is often more convenient (and always more elegant) to write the three equations as one vector equation. We will not use the vector form in this book since all our examples will be simple one-dimensional force balances. The field of fluid mechanics makes extensive use of the conservation of momentum. Example 2.9. The gravity-flow tank system described in Chap. 1 provides a simple example of the application of the equations of motion to a macroscopic system. Referring to Fig. 1.1, let the length of the exit line be L (ft) and its cross-sectional area be A, (ft’). The vertical, cylindrical tank has a cross-sectional area of A, (ft’). The part of this process that is described by a force balance is the liquid flowing through the pipe. It will have a mass equal to the volume of the pipe (APL) times the density of the liquid p. This mass of liquid will have a velocity v (ft/s) equal to the volumetric flow divided by the cross-sectional area of the pipe. Remember we have assumed plug-flow conditions and incompressible liquid, and therefore all the liquid is moving at the same velocity, more or less like a solid rod. If the flow is turbulent, this is not a bad assumption. M = A,Lp F (2.34)v=- 4 The amount of liquid in the pipe will not change with time, but if we want to change the rate of outflow, the velocity of the liquid must be changed. And to change the velocity or the momentum of the liquid we must exert a force on the liquid. The direction of interest in this problem is the horizontal, since the pipe is assumed to be horizontal. The force pushing on the liquid at the left end of the pipe is the hydraulic pressure force of the liquid in the tank. Hydraulic force = A,ph f c (2.35) FUNDAMENTALS 31 FIGURE 2.9 Laminar flow in a pipe. The rate of change of momeritum of the system is + ; (2~ dz dr pv,) c Combining all the above gives The aP/az term, or the pressure drop per foot of pipe, will be constant if the fluid is incompressible. Let us call it AP/L. Substituting it and Eq. (2.41) into Eq. (2.44) gives (2.45) 2.2.4 Transport Equations We have already used in the examples most of the laws governing the transfer of energy, mass, and momentum. T ese transport laws all have the form of a flux (rate of transfer per unit area) bein i proportional to a driving force (a gradient in temperature, concentration, or velo ,‘ty). The proportionality constant is a physi- cal property of the system (like thermal conductivity, diffusivity, or viscosity). For transport on a molecular vel, the laws bear the familiar names of Fourier, Fick, and Newton. + ‘\\Transfer relationships of a more m croscopic overall form are also used; for kexample, film coefficients and overall toe rcients in heat transfer. Here the differ- ence in the bulk properties between two locations is the driving force. The pro- portionality constant is an overall transfer eoeficient. Table 2.1 summarizes some to the various relationships used in developing models. 32 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS TABLE 2.1 Transport laws Quantity Heat Mass Momentum Flux 4 N.4 Molecular transport aT Driving force z Law Fourier’s Fick’s Property Thermal Diffusivity conductivity b 9, Overall transport Driving force AT AC,t Relationship q=h,AT N, = k, A C , au, dr Newton’s Viscosity P AP $ t Driving forces in terms of partial pressure.s and mole fractions are also commonly used. t The most common problem, determining pressure drops through pipes, uses friction factor correlations,f = (g, D AP/L)/Zpv*. 2.2.5 Equations of State To write mathematical models we need equations that tell us how the physical properties, primarily density and enthalpy, change with temperature, pressure, and composition. Liquid density = pL =&r, r, Xi, Vapor density = pv = j&, r, yi, . . Liquid enthalpy = h = j&, r, Xi) Vapor enthalpy = H = &., r, uij (2.46) Occasionally these relationships have to be fairly complex to describe the system accurately. But in many cases simplification can be made without sacrificing much overall accuracy. We have already used some simple enthalpy equations in the examples of energy balances. h=C,T H=C,T+& The next level of complexity would be to make the C,‘s functions of temperature: h = =Cpo, dT s (2.48) T O A polynomial in T is often used for C, . C pm = 4 + A2 T FUNDAMENTALS 33 Then Eq. (2.48) becomes [ T2 = h = A,T+A211 To =A,(T-To)++(T2-T;) (2.50) Of course, with mixtures of components the total enthalpy is needed. If heat-of- mixing effects are negligible, the pure-component enthalpies can be averaged: “c” xjhjMj h = ‘=ic (2.51) jzl Xj”j where xi = mole fraction ofjth component M, = molecular weight of jth component hj = pure-component enthalpy of jth component, energy per unit mass The denominator of Eq. (2.51) is the average molecular weight of the mixture. Liquid densities can be assumed constant in many systems unless large changes in composition and temperature occur. Vapor densities usually cannot be considered invariant and some sort of PVT relationship is almost always required. The simplest and most often used is the perfect-gas law : PV = nRT where P = absolute pressure (lb,/ft2 or kilopascals) V = volume (ft3 or m3) (2.52) n = number of moles (lb * mol or kg. mol) R = constant = 1545 lbt ft/lb. mol “R or 8.314 kPa m3/kg .mol K T = absolute temperature (OR or K) Rearranging to get an equation for density pv (lb,,,/ft3 or kg/m3) of a perfect gas with a molecular weight M, we get nM M P P”=y=E (2.53) 2.2.6 Equilibrium The second law of thermodynamics is the basis for the equations that tell us the conditions of a system when equilibrium conditions prevail. A. CHEMICAL EQUILIBRIUM. Equilibrium occurs in a reacting system when (2.54) where vi = stoichiometric coefficient of the jth component with reactants having a negative sign and products a positive sign ,lj = chemical potential of jth component 36 M A T H E M A T I C A L M O D E L S O F C H E M I C A L E N G I N E E R I N G S Y S T E M S In a binary system the relative volatility c( of the more volatile com- pone\nt compared with the less volatile component is Y/X IX = (1 - y)/(l - x) Rearranging, ax y = 1 + (a - 1)x (2.66) 3. K values. Equilibrium vaporization ratios or K values are widely used, partic- ularly in the petroleum industry. Kj =t (2.67) The K’s are functions of temperature and composition, and to a lesser extent, pressure. 4. Activity coefficients. For nonideal liquids, Raoult’s law must be modified to account for the nonideality in the liquid phase. The “fudge factors” used are called activity coefficients. NC P = c xjp,syj j=l (2.68) where yj is the activity coefficient for the jth component. The activity coefi- cient is equal to 1 if the component is ideal. The y’s are functions of composi- tion and temperature 2.2.7 Chemical Kinetics We will be modeling many chemical reactors, and we must be familiar with the basic relationships and terminology used in describing the kinetics (rate of reaction) of chemical reactions. For more details, consult one of the several excel- lent texts in this field. A. ARRHENIUS TEMPERATURE DEPENDENCE. The effect of temperature on the specific reaction rate k is usually found to be exponential : k = Cle.-E/RT (2.69) where k = specific reaction rate a = preexponential factor E = activation energy; shows the temperature dependence of the reaction rate, i.e., the bigger E, the faster the increase in k with increasing tem- perature (Btu/lb * mol or Cal/g * mol) T = absolute temperature R = perfect-gas constant = 1.99 Btu/lb. mol “R or 1.99 Cal/g. mol K FUNDAMENTALS .37 This exponential temperature dependence represents one of the most severe non- linearities in chemical engineering systems. Keep in mind that the “apparent” temperature dependence of a reaction may not be exponential if the reaction is mass-transfer limited, not chemical-rate limited. If both zones are encountered in the operation of the reactor, the mathematical model must obviously include both reaction-rate and mass-transfer effects. B. LAW OF MASS ACTION. Using the conventional notation, we will define an overall reaction rate Yt as the rate of change of moles of any component per volume due to chemical reaction divided by that component’s stoichiometric coefficient. (2.70) The stoichiometric coefficients vj are positive for products of the reaction and negative for reactants. Note that 3 is an intensive property and can be applied to systems of any size. For example, assume we are dealing with an irreversible reaction in which components A and B react to form components C and D. Then k v,A + v,B - V, c + v,j D The law of mass action says that the overall reaction rate ZR will vary with tem- perature (since k is temperature-dependent) and with the concentration of reac- tants raised to some powers. 3. = k&*)“G)b where CA = concentration of component A Cr, = concentration of component B (2.72) The constants a and b are not, in general, equal to the stoichibmetric coefficients v, and vb . The reaction is said to be first-order in A if a = 1. It is second-order in A if a = 2. The constants a and b can be fractional numbers. As indicated earlier, the units of the specific reaction rate k depend on the order of the reaction. This is because the overall reaction rate % always has the same units (moles per unit time per unit volume). For a first-order reaction of A reacting to form B, the overall reaction rate R, written for component A, would have units of moles of A/min ft3. 3. = kCA If C, has units of moles of A/ft3, k must have units of min-‘. 38 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS If the overa% reaction rate for the system above is second-order in A, 3% = kc; 3X still has units of moles of A/min ft3. Therefore k must have units of ft3/min mol A. Consider the reaction A + B -+ C. If the overall reaction rate is first-order in both A and B, 3% still has units of moles of A/min ft3. Therefore k must have units of ft3/min mol B. PROBLEMS 2.1. Write the component continuity equations describing the CSTR of Example 2.3 with: (a) Simultaneous reactions (first-order, isothermal) k1 A-B h A - C (b) Reversible (first-order, isothermal) 2.2. Write the component continuity equations for a tubular reactor as in Example 2.5 with consecutive reactions occurring: kl kz A - B - C 2.3. Write the component continuity equations for a perfectly mixed batch reactor (no inflow or outflow) with first-order isothermal reactions: (a) Consecutive (b) Simultaneous (c) Reversible 2.4. Write the energy equation for the CSTR of Example 2.6 in which consecutive first- order reactions occur with exothermic heats of reaction 1, and 1,. kl kz A - B - C 2.5. Charlie Brown and Snoopy are sledding down a hill that is inclined 0 degrees from horizontal. The total weight of Charlie, Snoopy, and the sled is M. The sled is essen- tially frictionless but the air resistance of the sledders is proportional to the square of their velocity. Write the equations describing their position x, relative to the top of the hill (x = 0). Charlie likes to “belly flop,” so their initial velocity at the top of the hill is u,, . What would happen if Snoopy jumped off the sled halfway down the hill without changing the air resistance? 26. An automatic bale tosser on the back of a farmer’s hay baler must throw a 60-pound bale of hay 20 feet back into a wagon. If the bale leaves the tosser with a velocity u, in EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 41 strategies and procedures so that you can apply them to your specific problem. Remember, just go back to basics when faced with a new situation. Use the dynamic mass and energy balances that apply to your system. In each case we will set up all the equations required to describe the system. We will delay any discussion of solving these equations until Part II. Our purpose at this stage is to translate the important phenomena occurring in the physical process into quantitative, mathematical equations. 3.2 SERIES OF ISOTHERMAL, CONSTANT-HOLDUP CSTRs The system is sketched in Fig. 3.1 and is a simple extension of the CSTR con- sidered in Example 2.3. Product B is produced and reactant A is consumed in each of the three perfectly mixed reactors by a first-order reaction occurring in the liquid. For the moment let us assume that the temperatures and holdups (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup). Density is assumed constant throughout the system, which is a binary mixture of A and B. With these assumptions in mind, we are ready to formulate our model. If the volume and density of each tank are constant, the total mass in each tank is constant. Thus the total continuity equation for the first reactor is 4PV;)-=pF,-pF, =o dt (3.1) Likewise total mass balances on tanks 2 and 3 give F, = F, = F, = F, = F (3.2) where F is defined as the throughput (m3/min). We want to keep track of the amounts of reactant A and product B in each tank, so component continuity equations are needed. However, since the system is binary and we know the total mass of material in each tank, only one com- ponent continuity equation is required. Either B or A can be used. If we arbi- trarily choose A, the equations describing the dynamic changes in the amounts of - - Fo * “I Fl * “2 - F2 t “3 ’ F3 t k, ’ k2 4 CA0 CA1 CA2 _ (3 FIGURE 3.1 Series of CSTRs. 42 MATHEMATICAL MODELS OF CHEMICAL ENOINEERING SYSTRMS reactant A in each tank are (with units of kg - mol of A/min) v dCA1 - = F(cAO - CAlI - VlklCA1’ dt y dCAz- = F(C,, ’ dt - cA2) - v, k2 cA2 I (3.3) v dCA3- = F(cA2 3 dt - CA3) - v3 k, CA3 The specific reaction rates k, are given by the Arrhenius equation k, = CIe-E/RTm n = 1, 2, 3 (3.4) If the temperatures in the reactors are different, the k’s are different. The n refers to the stage number. The volumes V, can be pulled out of the time derivatives because they are constant (see Sec. 3.3). The flows are all equal to F but can vary with time. An energy equation is not required because we have assumed isothermal operation. Any heat addition or heat removal required to keep the reactors at constant temperatures could be calculated from a steadystate energy balance (zero time derivatives of temperature). The three first-order nonlinear ordinary differential equations given in Eqs. (3.3) are the mathematical model of the system. The parameters that must be known are Vi, V, , V, , kl, k2, and k, . The variables that must be specified before these equations can be solved are F and CA,. “Specified” does not mean that they must be constant. They can be time-varying, but they must be known or given functions of time. They are theforcingfunctions. The initial conditions of the three concentrations (their values at time equal zero) must also be known. Let us now check the degrees of freedom of the system. There are three equations and, with the parameters and forcing functions specified, there are only three unknowns or dependent variables: CA1, CA2, and cA3. Consequently a solution should be possible, as we will demonstrate in Chap. 5. We will use this simple system in many subsequent parts of this book. When we use it for controller design and stability analysis, we will use an even simpler version. If the throughput F is constant and the holdups and tem- peratures are the same in all three tanks, Eqs. (3.3) become ~ dt & dt dCA3 dt (3.5) EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENOINEERING SYSTEMS 43 where r = V/F with units of minutes. There is only one forcing function or input variable, CA0 . 3.3 CSTRs WITH VARIABLE HOLDUPS If the previous example is modified slightly to permit the volumes in each reactor to vary with time, both total and component continuity equations are required for each reactor. To show the effects of higher-order kinetics, assume the reaction is now nth-order in reactant A. Reactor 1: dV’=F -F dt ’ ’ f (v,c,,) = Fo CAO - FICAI - V,kl(CAl) (3.6) Reactor 2: dVZ=F -F dt ’ 2 f (v, cA2) = FICA, - F, CAZ - v, k,(C,,) (3.7) Reactor 3 : dV3- = F, - F3 dt f 05 cA3) = F2 cA2 - F, CAS - v, k,(CA,) (3.8) Our mathematical model now contains six first-order nonlinear ordinary differential equations. Parameters that must be known are k,, k,, k,, and n. Initial conditions for all the dependent variables that are to be integrated must be given: CA1, cA2, CA3, VI, V2, and V, . The forcing functions CAo(r) and Focr, must also be given. Let us now check the degrees of freedom of this system. There are six equa- tions. But there are nine unknowns: C*i, CA,, C,, , VI, V,, V,, F,, F,, and F,. Clearly this system is not sufficiently specified and a solution could not be obtained. What have we missed in our modeling? A good plant operator could take one look at the system and see what the problem is. We have not specified how the flows out of the tanks are to be set. Physically there would probably be control valves in the outlet lines to regulate the flows. How are these control valves to be set? A common configuration is to have the level in the tank con- trolled by the outflow, i.e., a level controller opens the control valve on the exit 46 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS The concentration of reactant in the reactor is with units of moles of A per unit volume. The overall reaction rate for the forward reaction is The overall reaction rate for the reverse reaction is With these fundamental relationships pinned down, we are ready to write the total and component continuity equations. Total continuity : Component A continuity: p&&c dt o A0 - FCA - 2Vk,(C*)‘.5 + 2Vk, CR The 2 in the reaction terms comes from the stoichiometric coefficient of A. There are five equations [Eqs. (3.14) through (3.18)] that make up the math- ematical model of this system. The parameters that must be known are V, C,, k,, kz, R, MA, and MB. The forcing functions (or inputs) could be P,, po, Fo, and CA,. This leaves five unknowns (dependent variables): C, , p, P, F, and y. 3.6 NONISOTHERMAL CSTR In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component conti- nuity equations. Energy equations were not needed because we assumed isother- mal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. k A - B The reaction is nth-order in reactant A and has a heat of reaction A (Btu/lb. mol of A reacted). Negligible heat losses and constant densities are assumed. To remove the heat of reaction, a cooling jacket surrounds the reactor. Cooling water is added to the jacket at a volumetric flow rate F, and with an EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 47 FJ TJ TJ v T FJ - ‘J CA TJO I F CA T FIGURE 3.3 k A - B Nonisothermal CSTR. inlet temperature of I”, . The volume of water in the jacket V, is constant. The mass of the metal walls is assumed negligible so the “thermal inertia” of the metal need not be considered. This is often a fairly good assumption because the heat capacity of steel is only about 0.1 Btu/lb,“F, which is an order of mag- nitude less than that of water. A. PERFECTLY MIXED QlCU.ING JACKET. We assume that the temperature e in the jacket is TJ. The heat transfer between the process at tem- rature T and the cooling water at temperature TJ is described by an cuuxall heat transfer coefficient. Q = f-J4rV’ - ‘I’ where Q = heat transfer rate U = overall heat transfer coefficient AH = heat transfer area In general the heat transfer area c* vary with the J&&p in the reactor if some area was not completely-d with reaction mass liquid at all times. The equations describing the system are: Reactor total continuity: Reactor component A continuity : d(vcA)~ = Fo CA0 - FCA - Vk(CJ dt Reactor energy equation : p y = p(Foho - Fh) - LVk(CJ” - UA,,(T - TJ) (3.20) 48 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS Jacket energy equation : PJ v, % = F.,PJ@,, - h,) + UAAT - TJ) where pJ = density of cooling water h = enthalpy of process liquid h, = enthalpy of cooling water The-n of constant densities makes C, = C, and permits us to use en- thalpies in the time derivatives to replace internal energies, A hydraulic- between reactor holdup and the flow out of the reactor is also needed. A level controller is assumed to change the outflow as the volume in the tank rises or falls: the higher the volume, the larger the outflow. The outflow is shut off completely when the volume drops to a minimum value Vm i n * F = K,(V - Vrni”) (3.22) The level controller is a proportional-only feedback controller. Finally, we need enthalpy data to relate the h’s to compositions and tem- peratures. Let us assume the simple forms h=C,T and h,=CJT, where C, = heat capacity of the process liquid C, = heat capacity of the cooling water (3.23) Using Eqs. (3.23) and the Arrhenius relationship for k, the five equations that describe the process are dv=F -F dt ’ (3.24) d(VCd- = F,-, CA0 - FCA - V(CA)“ae-E’RT dt (3.25) 4W PC, d t - = pC,(F, To - FT) - ,lV(C,)“ae-E’RT - UA,,(T - TJ) (3.26) PJ YrC, $ = FJPJCGJO - G) + U&V - Trl (3.27) F = I&@ - Vmin) (3.28) Checking the degrees of freedom, we see that there are five equations and five unknowns: V, F, CA, T, and TJ. We must have initial conditions for these five dependent variables. The forcing functions are To, F, , CA,, and F, . The parameters that must be known are n, a, E, R, p, C,, U, A,, pJ, V,, CJ, TJo, K,, and Vmin. If the heat transfer area varies with the reactor holdup it EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 51 pM = density of metal wall CM = heat capacity of metal wall V, = volume of metal wall Ai = inside heat transfer area A, = outside heat transfer area 3.7 SINGLE-COMPONENT VAPORIZER aoiling systems represent some of the most interesting and important operations in chemical engineering processing and are among the most dificult to model. To describe these systems rigorously, conservation equations must be written for both the vapor and liquid phases. The basic problem is finding the rate of vapor- ization of material from the liquid phase into the vapor phase. The equations used to describe the boiling rate should be physically reasonable and mathemati- cally convenient for solution. Consider the vaporizer sketched in Fig. 3.6. Liquefied petroleum gas (LPG) is fed into a pressurized tank to hold the liquid level in the tank. We will assume that LPG is a pure component: propane. Vaporization of mixtures of com- ponents is discussed in Sec. 3.8. The liquid in the tank is assumed perfectly mixed. Heat is added at a rate Q to hold the desired pressure in the tank by vaporizing the liquid at a rate W, (mass per time). Heat losses and the mass of the tank walls are assumed negligi- ble. Gas is drawn off the top of the tank at a volumetric flow rate F,. F, is the forcing function or load disturbance. A. STEADYSTATE MODEL. The simplest model would neglect the dynamics of both vapor and liquid phases and relate the gas rate F, to the heat input by P,F,W, - ho) = Q (3.34) where I-Z, = enthalpy of vapor leaving tank (Btu/lb, or Cal/g) h, = enthalpy of liquid feed @&u/lb, or Cal/g) FIGURE 3.6 LPG vaporizer. 52 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS B. LIQUID-PHASE DYNAMICS MODEL. A somewhat more realistic model is obtained if we assume that the volume of the vapor phase is small enough to make its dynamics negligible. If only a few moles of liquid have to be vaporized to change the pressure in the vapor phase, we can assume that this pressure is always equal to the vapor pressure of the liquid at any temperature (P = P, and WV = pu F,). An energy equation for the liquid phase gives the temperature (as a function of time), and the vapor-pressure relationship gives the pressure in the vaporizer at that temperature. A total continuity equation for the liquid phase is also needed, plus the two controller equations relating pressure to heat input and liquid level to feed flow rate F, . These feedback controller relationships will be expressed here simply as functions. In later parts of this book we will discuss these functions in detail. Q =fim Fo =fz(YL) (3.35) An equation of state for the vapor is needed to be able to calculate density pv from the pressure or temperature. Knowing any one property (T, P, or p,) pins down all the other properties since there is only one component, and two phases are present in the tank. The perfect-gas law is used. The liquid is assumed incompressible so that C, = C, and its internal energy is C, T. The enthalpy of the vapor leaving the vaporizer is assumed to be of the simple form: C, T + I,. Total continuity : Energy : c ? p d(l/, T) -=PoC,FOTO-~,F,(C,T+~,)+Qdt State : (3.38) Vapor pressure : lnP=$+B Equations (3.35) to (3.39) give us six equations. Unknowns are Q, F,, P, V’, pv, and T. C. LIQUID AND VAPOR DYNAMICS MODEL. If the dynamics of the vapor phase cannot be neglected (if we have a large volume of vapor), total continuity and energy equations must be written for the gas in the tank. The vapor leaving the tank, pv F, , is no longer equal, dynamically, to the rate of vaporization W, . EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 53 The key problem now is to find a simple and reasonable expression for the boiling rate W,. I have found in a number of simulations that a “mass-transfer” type of equation can be conveniently employed. This kind of relationship also makes physical sense. Liquid boils because, at some temperature (and composi- tion if more than one component is present), it exerts a vapor pressure P greater than the pressure P, in the vapor phase above it. The driving force is this pres- sure differential W” = K&P - P”) (3.40) where K,, is the pseudo mass transfer coefficient. Naturally at equilibrium (not steadystate) P = P,. If we assume that the liquid and vapor are in equilibrium, we are saying that KM, is very large. When the equations are solved on a com- puter, several values of K,, can be used to test the effects of nonequilibrium conditions. The equations describing the system are: Liquid phase Total continuity: pZ=p,F,-- W” Energy : P d(v, UL)-=poFOh,- W,H,+Q dt Vapor pressure : Vapor phase Total continuity: Energy : State : p = @/T+B WC P,)-= w,--P”F” dt MP, ” = RT, (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) where UL = internal energy of liquid at temperature T HL = enthalpy of vapor boiling off liquid U, = internal energy of vapor at temperature T, Hv = enthalpy of vapor phase 56 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS Thermal properties : 4 = Axe,. To) h =“&j. T) H =-f&j, T, 0 (3.55) The average molecular weights Ma’ are calculated from the mole fractions in the appropriate stream [see Eq. (3.50)]. The number of variables in the system is 9 + 2(NC- 1): Pv, F,, M?, Yl, Y,, . . . . YNC-1, PL, FL, MY’, XI, xz, . ..> XNc-r, T, h, and H. Pressure P and all the feed properties are given. There are NC - 1 component balances [Eq. (3.52)]. There are a total of NC equilibrium equations. We can say that there are NC equations like Eq. (3.53). This may bother some of you. Since the sum of the y’s has to add up to 1, you may feel that there are only NC - 1 equations for the y’s. But even if you think about it this way, there is still one more equation: The sum of the partial pressures has to add up to the total pressure. Thus, what- ever way you want to look at it, there are NC phase equilibrium equations. Total continuity Energy Component continuity Vapor-liquid equilibrium Densities of vapor and liquid Thermal properties for liquid and vapor streams Average molecular weights Number of Equation equations (3.5 1) 1 , (3.54) 1 (3.52) “NC-1 (3.53) N C (3.48) and (3.49) 2 (3.55) 2 (3.50) 2 2NC+7 The system is specified by the algebraic equations listed above. This is just a traditional steadystate “equilibrium-flash” calculation. B. RIGOROUS MODEL. Dynamics can be included in a number of ways, with varying degrees of rigor, by using models similar to those in Sec. 3.7. Let us merely indicate how a rigorous model, like case C of Sec. 3.7, could be developed. Figure 3.8 shows the system schematically. An equilibrium-flash calculation (using the same equations as in case A above) is made at each point in time to find the vapor and liquid flow rates and properties immediately after the pressure letdown valve (the variables with the primes: FL, F’, , yi , xi, . . . shown in Fig. 3.8). These two streams are then fed into the vapor and liquid phases. The equations describing the two phases will be similar to Eqs. (3.40) to (3.42) and (3.44) to (3.46) with the addition of (1) a multi- component vapor-liquid equilibrium equation to calculate P, and (2) NC - 1 component continuity equations for each phase. Controller equations relating V, to F, and P, to F, complete the model. EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 57 FIGURE 3.8 Dynamic flash drum. C. PRACTICAL MODEL. A more workable dynamic model can be developed if we ignore the dynamics of the vapor phase (as we did in case B of Sec. 3.7). The vapor is assumed to be always in equilibrium with the liquid. The conservation equations are written for the liquid phase only. Total continuity : 4 v, ~3-=PoFo-P,F,-PLFLdt Component continuity : d PO Fo P” f’, PLFL dt =~XOj-M:‘Yj-px’My ’ (3.58) Energy : , dK',p,h) dt = PoFoho - P”F,H - pLF,h The NC vapor-liquid equilibrium equations [Eqs. (3.53)], the three enthalpy relationships [Eqs. (3.55)], the two density equations [Eqs. (3.48) and (3.49)], the two molecular-weight equations [Eq. (3.50)], and the feedback controller equa- tions [Eqs. (3.56)] are all needed. The total number of equations is 2NC + 9, which equals the total number of variables: P,, V,, p”, F,, My, y,, y,, . . ., YN~-~,P~,F~,MBLV,X~,X~,...,~N~-~,T,~,~~~H. Keep in mind that all the feed properties, or forcing functions, are given: Fo, PO, ho, Xoj, and JG"'V.,,,,t i. 3.9 BATCH REACT& Batch processes offer some of the most interesting and challenging problems in modeling and control because of their inherent dynamic nature. Although most s MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS Reactants charged initially water outlet sensor T.U 4 I Temperature transmitter , f Steam , V-l I , v-2 3 Cooling water inlet -I- A Condensate w, Products withdrawn finally FIGURE 3.9 Batch reactor. large-scale chemical engineering processes have traditionally been operated in a continuous fashion, many batch processes are still used in the production of smaller-volume specialty chemicals and pharmaceuticals. The batch chemical reactor has inherent kinetic advantages over continuous reactors for some reac- tions (primarily those with slow rate constants). The wide use of digital process control computers has permitted automation and optimization of batch processes and made them more efficient and less labor intensive. Let us consider the batch reactor sketched in Fig. 3.9. Reactant is charged into the vessel. Steam is fed into the jacket to bring the reaction mass up to a desired temperature. Then cooling water must be added to the jacket to remove the exothermic heat of reaction and to make the reactor temperature follow the prescribed temperature-time curve. This temperature profile is fed into the tem- perature controller as a setpoint signal. The setpoint varies with time. First-order consecutive reactions take place in the reactor as time proceeds. EXAMPLES OF MATHEMATICAL MODELS OE CHEMICAL ENGINEERING SYSTEMS 61 follow the setpoint. Valve v-3 must be opened and valve V-4 must be shut when- ever cooling water is added. We will study in detail the simulation and control of this system later in this book. Here let us simply say that there is a known relationship between the error signal E (or the temperature setpoint minus the reactor temperature) and the volumetric flow rates of steam F, and cooling water F, . F, = h(E) Fw =.fm (3.66) To describe what is going on in the jacket we may need two different sets of equations, depending on the stage: heating or cooling. We may even need to consider a third stage: filling the jacket with cooling water. If the cooling-water flow rate is high and/or the jacket volume is small, the time to fill the jacket may be neglected. A. HEATING PHASE. During heating, a total continuity equation and an energy equation for the steam vapor may be needed, plus an equation of state for the steam. Total continuity: (3.67) where pJ = density of steam vapor in the jacket V, = volume of the jacket pS = density of incoming steam WC = rate of condensation of steam (mass per time) The liquid condensate is assumed to be immediately drawn off through a steam trap. Energy equation for steam vapor: v W,p,) J -=FF,p,H,-h,A,(T,-T,)- W,h,dt (3.68) where UJ = internal energy of the steam in the jacket H, = enthalpy of incoming steam h, = enthalpy of liquid condensate The internal energy changes (sensible-heat effects) can usually be neglected com- pared with the latent-heat effects. Thus a simple algebraic steadystate energy equation can be used w = ho ‘%tTJ - Th4) e Hs - k (3.69) 62 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS The equations of state for steam (or the steam tables) can be used to calcu- late temperature TJ and pressure P, from density pJ . For example, if the perfect- gas law and a simple vapor-pressure equation can be used, (3.70) where M = molecular weight of steam = 18 A, and B, = vapor-pressure constants for water Equation (3.70) can be solved (iteratively) for TJ if pJ is known [from Eq. (3.67)]. Once TJ is known, P, can be calculated from the vapor-pressure equation. It is usually necessary to know PJ in order to calculate the flow rate of steam through the inlet valve since the rate depends on the pressure drop over the valve (unless the flow through the valve is “critical”). If the mass of the metal surrounding the jacket is significant, an energy equation is required for it. We will assume it negligible. In most jacketed reactors or steam-heated reboilers the volume occupied by the steam is quite small compared to the volumetric flow rate of the steam vapor. Therefore the dynamic response of the jacket is usually very fast, and simple algebraic mass and energy balances can often be used. Steam flow rate is set equal to condensate flow rate, which is calculated by iteratively solving the heat- transfer relationship (Q = UA AT) and the valve flow equation for the pressure in the jacket and the condensate flow rate. B. COOLING PHASE. During the period when cooling water is flowing through the jacket, only one energy equation for the jacket is required if we assume the jacket is perfectly mixed. PJ vJ cJ z = Fw CJ PATJO - TJ) + ho 4Ui.t - q) (3.7 1) where TJ = temperature of cooling water in jacket pJ = density of water C, = heat capacity of water TJ, = inlet cooling-water temperature Checking the degrees of freedom of the system during the heating stage, we have seven variables (C, , CB , T, TM, TJ , p J, and W,) and seven equations [Eqs. (3.61), (3.62), (3.64), (3.65), (3.67), (3.69), and (3.70)]. During the cooling stage we use Eq. (3.71) instead of Eqs. (3.67), (3.69), and (3.70), but we have only TJ instead OfT,,p,,and WC. 3.10 REACTOR WITH MASS TRANSFER As indicated in our earlier discussions about kinetics in Chap. 2, chemical reac- tors sometimes have mass-transfer limitations as well as chemical reaction-rate limitations. Mass transfer can become limiting when components must be moved EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 63 FL Liquid product Enlarged view of bubble surface FB 0 ’ “. ’ 0 0 0 1 L, ,kmLiquid\ D pB 0 0 0 c 0 BO oooc* Liquid feed oo”o I Gas feed cnc .-..>m*r FA PA ‘V’y FIGURE 3.11 Gas-liquid bubble reactor. from one phase into another phase, before or after reaction. As an example of the phenomenon, let us consider the gas-liquid bubble reactor sketched in Fig. 3.11. Reactant A is fed as a gas through a distributor into the bottom of the liquid-filled reactor. A chemical reaction occurs between A and B in the liquid phase to form a liquid product C. Reactant A must dissolve into the liquid before it can react. A+B:C If this rate of mass transfer of the gas A to the liquid is slow, the concentration of A in the liquid will be low since it is used up by the reaction as fast as it arrives. Thus the reactor is mass-transfer limited. If the rate of mass transfer of the gas to the liquid is fast, the reactant A concentration will build up to some value as dictated by the steadystate reaction conditions and the equilibrium solubility of A in the liquid. The reactor is chemical-rate limited. Notice that in the mass-transfer-limited region increasing or reducing the concentration of reactant IS will make little difference in the reaction rate (or the reactor productivity) because the concentration of A in the liquid is so small. Likewise, increasing the reactor temperature will not give an exponential increase in reaction rate. The reaction rate may actually decrease with increasing tem- perature because of a decrease in the equilibrium solubility of A at the gas-liquid interface. 6 6 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS A further assumption we will make is that of equimolal overflow. If the molar heats of vaporization of the two components are about the same, whenever one mole of vapor condenses, it vaporizes a mole of liquid. Heat losses up the column and temperature changes from tray to tray (sensible-heat effects) are assumed negligible. These assumptions mean that the vapor and liquid rates through the stripping and rectifying sections will be constant under steadystate conditions. The “operating lines” on the familiar McCabe-Thiele diagram are straight lines. However, we are interested here in dynamic conditions. The assumptions above, including negligible vapor holdup, mean that the vapor rate through all F z Cooling water Mu XD LC?!iii?D xD FIGURE 3.12 Binary distillation column. EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 67 trays of the column is the same, dynamically as well as at steadystate. Remember these V’s are not necessarily constant with time. The vapor boilup can be manipulated dynamically. The mathematical effect of assuming equimolal overtlow is that we do not need an energy equation for each tray. This- is quite a significant simplification. The liquid rates throughout the’ column will not be the same dynamically. They will depend on the fluid mechanics of the tray. Often a simple Francis weir formula relationship is used to relate the liquid holdup on the tray (M,) to the liquid flow rate leaving the tray (L,). F, = 3.33Lw(h,w)‘.5 (3.78) where F, = liquid flow rate over weir (ft3/s) L, = length of weir (ft) h,, = height of liquid over weir (ft) More rigorous relationships can be obtained from the detailed tray hydraulic equations to include the effects of vapor rate, densities, compositions, etc. We will assume a simple functional relationship between liquid holdup and liquid rate. Finally, we will neglect the dynamics of the condenser and the reboiler. In commercial-scale columns, the dynamic response of these heat exchangers is usually ,much faster than the response of the column itself. In some systems, however, the dynamics of this peripheral equipment are important and must be included in the model. With all these assumptions in mind, we are ready to write the equations describing the system. Adopting the usual convention, our total continuity equa- tions are written in terms of moles per unit time. This is kosher because no chemical reaction is assumed to occur in the column. Condenser and Rejlux Drum Total continuity: dM,--V-R-D dt Component continuity (more volatile component): d(MD xd dt = VYNT - (R + D)x, Top Tray (n = NT) Total continuity: dM,,- = R - LNT dt (3.80) (3.81) (3.82) 68 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS Component continuity : 4Mm XNT) dt = Rx, - LNT XNT + ~YNT- I - vYNT Next to Top Tray (n = N, - 1) Total continuity: dM,,-1 L - =dt NT - LNT-l Component continuity i (3.83) (3.84) d(MN,- 1 XNT- 1) = L dt NT x N T - LNT-lXNT-l + vYNT-Z - I/YNT-l (3*85) nth Tray Total continuity: i!LLL -L dt-“+I ’ Component continuity: d(Mnxn) _L dt n+lX,+l - JkX” + VY,-1 - VY” Feed Tray (n = NF) TotaI continuity: dM,,=L dt NF+l - LNF + F Component continuity : d(M,, XNF) = dt LNF+l XNF+l - LNF XNF + ~YNF - I First Tray (n = 1) Total continuity : dM,- = L, - L, dt Component continuity : (3.86) (3.87) (3.88) VyNF + FZ (3.89) (3.90) (3.91) EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 71 FIGURE 3.13 nth tray of multicomponent column. Energy equation (one per tray): hd&f, M _ L dt n+1 n+1+ F;4hf + ~,Y_Je-1+ I/,-lH,-, - ~‘,H,-L,h,-S;4h,-S,YH” (3.99) where the enthalpies have units of energy per mole. Phase equilibrium (NC per tray): Y,*j = Lj, Pm. T-1 (3.100) An appropriate vapor-liquid equilibrium relationship, as discussed in Sec. 2.2.6, must be used to find y~j. Then Eq. (3.96) can be used to calculate the ynj for the inefficient tray. The y,‘_ r, i would be calculated from the two vapors entering the tray: FL-, and V,-,. Additional equations include physical property relationships to get densities and enthalpies, a vapor hydraulic equation to calculate vapor flow rates from known tray pressure drops, and a liquid hydraulic relationship to get liquid flow TABLE 3.1 Streams on nth tray Nlllllber Flow rate Composition Temperature F: c 1 L“+I v, K-1 s: L” S.’ 72 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS rates over the weirs from known tray holdups. We will defer any discussion of the very real practical problems of solving this large number of equations until Part II. If we listed all the variables in this system and subtracted all the equations describing it and all the parameters that are fixed (all feeds), we would find that the degrees of freedom would be equal to the number of sidestreams plus two. Thus if we have no sidestreams, there are only two degrees of freedom in this multicomponent system. This is the same number that we found in the simple binary column. Typically we would want to control the amount of heavy key impurity in the distillate x,,, HK and the amount of light key impurity in the bottoms xg, LK. 3.13 BATCH DISTILLATION WITH HOLDUP Batch distillation is frequently used for small-volume products. One column can be used to separate a multicomponent mixture instead of requiring NC - 1 con- tinuous columns. The energy consumption in batch distillation is usually higher than in continuous, but with small-volume, high-value products energy costs seldom dominate the economics. Figure 3.14 shows a typical batch distillation column. Fresh feed is charged into the still pot and heated until it begins to boil. The vapor works its way up the column and is condensed in the condenser. The condensate liquid runs into FIGURE 3.14 Batch distillation. EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 73 the reflux drum. When a liquid level has been established in the drum, reflux is pumped back to the top tray in the column. The column is run on total reflux until the overhead distillate composition of the lightest component (component 1) xm reaches its specification purity. Then a distillate product, which is the lightest component, is withdrawn at some rate. Eventually the amount of component 1 in the still pot gets very low and the xm purity of the distillate drops. There is a period of time when the distillate contains too little of component 1 to be used for that product and also too little of component 2 to be used for the next heavier product. Therefore a “slop” cut must be withdrawn until xD2 builds up to its specification. Then a second product is withdrawn. Thus multiple products can be made from a single column. The optimum design and operation of batch distillation columns are very interesting problems. The process can run at varying pressures and reflux ratios during each of the product and slop cuts. Optimum design of the columns (diameter and number of trays) and optimum operation can be important in reducing batch times, which results in higher capacity and/or improved product quality (less time at high temperatures reduces thermal degradation). Theoretical trays, equimolal overflow, and constant relative volatilities are assumed. The total amount of material charged to the column is M,, (moles). This material can be fresh feed with composition zj or a mixture of fresh feed and the slop cuts. The composition in the still pot at the beginning of the batch is XBoj. The composition in the still pot at any point in time is xBj. The instanta- neous holdup in the still pot is M,. Tray liquid holdup and reflux drum holdup are assumed constant. The vapor boilup rate is constant at V (moles per hour). The reflux drum, column trays, and still pot are all initially filled with material of . . composition x,ej. The equations describing the batch distillation of a multicomponent mixture are given below. Still pot: dM,-c-D dt 4MB XBjl dt = Rx,~ - VY, Uj XBj YBj = NC c ak xBk k=l Tray n : M, dx,j = R[x,+,, j dt - Xnjl + UYn- 1, j - Ynjl (3.101) (3.103) clj Xnj Ynj = NC c ak xnk k=l