Computer Applications to Chemical Engineering

Computer Applications to Chemical Engineering Process Design and Simulation

Publication Date: May 30, 1980 cloi: 10. 1021/bk- 1980-0124.fw001

Robert G. Squires, EDITOR Purdue University G. V. Reklaitis, EDITOR Purdue University Based on a symposium sponsored by the Division of Industrial and Engineering Chemistry at the 178th Meeting of the American Chemical Society, Washington, D. C., September 11-13, 1979.

ACS SYMPOSIUM SER1ES124

AMERICAN CHEMICAL SOCIETY WASHINGTON, D. C.

1980

In Computer Applications to Chemical Engineeriniz Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

Publication Date: May 30, 1980 I doi: 10. 1021/bk-1980-0124Sw00I

Library of Congress Or Data Computer applications to chemical engineering process design and simulation. (ACS symposium series; 124 ISSN 0097-6156) Includes bibliographies and index. 1. Chemical process control—Congresses. 1. Squires, Robert G., 1935. II. Reklaitis, G. V., 1942. III. American Chemical Society. Division of Industrial and Engineering Chemistry. IV. Series. American Chemical Society. ACS symposium series 124. TP155.75.C65 ISBN 0-8412-0549-3

660.2'81 ACSMC8

79-27719 124 1-511 1980

Copyright © 1980 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of the first page of each article in this volume indicates the copyright owner's consent that reprographic copies of the article may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating new collective works, for resale, or for information storage and retrieval systems. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto.

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1155Chemical 16th S.twitilei..goG4uires, N. W. R. el al.: litglonvnilcaMociAu.uoviiington. DC. 1980.

Publication Date: May 30, 1 980 cloi: 1 0. 1 021/ bk- 1 980-01 24.fw001

FOREWORD The ACS SYMPOSIUM SERIES was founded in 1974 to provide a medium for publishing symposia quickly in book form. The format of the Series parallels that of the continuing ADVANCES IN CHEMISTRY SERIES except that in order to save time the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable since symposia may embrace both types of presentation.

In Computer Applications to Chemical Engineering; Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

PREFACE

Publication Date: May 30, 1 980 I doi: 1 0. 1021/bk- 1980-01 24.pr001

his volume contains selected papers from a five-session symposium T -I- on "Computer Applications to Chemical Engineering Process Design and Simulation" sponsored by the I&EC Division of ACS held in Washington, D.C. in September of 1979. Although shorter symposia on special topics in chemical engineering computation have been held under the auspices of the AIChE, this was the first symposium devoted to the entire field to be held in the United States. The European Federation of Chemical Engineers has held regular symposia on Computer Applications in Chemical Engineering but the proceedings of these meetings have enjoyed only limited circulation in the United States. This volume thus represents the only collection of works on computer applications to appear in the United States since the C.E.P. Symposium Series volumes on computational topics which appeared in the middle sixties. The papers comprising this volume are subdivided into four categories: reviews of four major areas of computation research, reviews of several key computational topics within these areas, papers discussing specific new advances in methodology, and papers demonstrating the effective use of computation in modeling, design, and control covering a broad range of applications. The first of the broad computational area reviews discusses the general direction of research in steady-state process simulation and summarizes the new ideas in computational architecture to have emerged since 1975. This is followed with a review of the main thrusts in control theory and on evaluation of the relevance to chemical engineering applications. Next the significant developments in numerical methods for minimizing nonlinear constrained and unconstrained functions are traced. The new developments in recursive quadratic programming methods for general nonlinear programs should be of particular interest to chemical engineers since they appear to offer a significant advance over the generalized reduced gradient techniques that have dominated the field for some ten years. Finally, research in computer-aided synthesis is appraised, and a summary is given of the significant results in six problem areas: heat exchanger networks, separation systems with and without heat integration, reaction paths, total flowsheets, and control systems. xi

In Computer Applications to Chemical Engineering Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

Publication Date: May 30, 1 980 I doi: 1 0, 1 021/bk- 1 980-01 24.pr001

These wide-ranging reviews are followed by analyses of progress in several specialized problem categories: chemical and physical equilibrium computations; vapor—Iiquid equilibrium computations including single and multistage VLE separations, multiliquid phase systems, and VLE systems with reaction and electrolytes; treatment of measurement errors in process networks and computations of choking flows in gas pipe networks. Next follows a series of reports on important developments in computational methods or program packages incorporating novel computational features. Finally, the volume is capped with papers discussing computer applications involving modeling, design, and control spanning a wide range from microbial conversion to industrial reactor modeling to drug therapy control. The scope and quality of these contributions have made the symposium a milestone in chemical engineering computation and ensure that this volume will be of permanent significance to those involved or interested in this area. Both the papers and the symposium as a whole have benefitted substantially from the anonymous contributions of a large number of conscientious referees for whose efforts we are indebted. Finally, the organization and smooth functioning of the symposium as well as the successful assembly of this volume are in large part due to the commendable work of the following session chairmen: D. A. Mellichamp and R. G. Rinker of the University of California—Santa Barbara, G. Blau of Dow Chemical USA, and J. Zemaitis of OLI Systems, Inc. G. V. REICLAITIS R. G. SQUIRES

School of Chemical Engineering Purdue University West Lafayette, IN 47907 October, 1979

xii

In Computer Applications to Chemical Engineering; Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

2 A Review of Optimization Methods for Nonlinear Problems R. W. H. SARGENT

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Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, London SW7, England

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8 07 ct.

The field of optimization is vast and all-embracing. Relevant papers are published at a rate of more than 200 per month, spread over more than 30 journals, without counting the numerous volumes of conference proceedings and special collections of papers. The Tenth International Symposium on Mathematical Programming held in August this year has alone added 450 papers to the list. Applications are equally varied and widespread. This review cannot therefore hope to be comprehensive and its scope is firmly restricted to general methods for dealing with nonlinear problems, both with and without constraints, since these are the most common in chemical engineering applications. Integer programming methods are not reviewed, since most of the mathematical developments are concerned with mixed integer-linear problems which are of limited interest to chemical engineers. Branch-and-bound techniques are still the basic tools for nonlinear integer problems, and since heuristics play such an important role the techniques can only be considered in relation to specific applications. Many specialized techniques exploiting particular problem structures are ignored, and fields which involve considerations outside the question of the optimization techniques themselves are also excluded. Thus for example the whole field of function approximation and model parameter fitting has been left out. Although there have been significant theoretical advances in recent years, particularly in connection with stability, sensitivity and convergence analysis, these also are largely ignored. The emphasis is on algorithmic developments because to the user the theoretical advances are of no account until they are embodied in implementable algorithms.

0-8412-0549-3/80/47-124-037805.00/0 © 1980 American Chemical Society In Computer Applications to Chemical Engineering: Squires, R., el al.: ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING

Unconstrained Minimization. The quasi-Newton or variable-metric methods introduced by Davidon {1} have now become the standard methods for finding an unconstrained minimum of a differentiable function f(x), and an excellent review of the basic theory has been given by Dennis and Morel {2}. These are iterative methods of the form = xk - ak Sk gk

Publication Datc: May 30, 1980doi: 10. 1 02 1/bk - 1 980-0124.ch002

Sk+l = S(Sk, pk+1, cik4.1)

) (2.1) '

where x is an n-vector, {xk} k=0,1,2, ... is a sequence of iterates with an arbitrary starting point x , g is the gradient o k xk, qk+1= = xic+3. of the function f(x) at xk, and S is a local approximation to the inverse of the Hessian k matrix of f(x). Classically, the scalar a is chosen to minimize k the function f(xk -aSk g ) with respect to a. k. The methods differ in the formula used to generate the sequence S,, k=0,1,2, ...., and after Fletcher and Powell's {3} analysis of Davidon's method a whole spate of formulae were invented in the sixties. Broyden {4} introduced some rationalization by identifying a one-parameter family, and recommended a particular member, now commonly referred to as the BFGS (BroydenFletcher-Goldfarb-Shanno) formula. Huang {5) widened the family, but by the end of the sixties numerical experience was producing a consensus that the BFGS formula was the most robust of the formulae available. The formula is T = + p (T p q ) - S q )p k+1 k+1 k+1-S k k+1 I, -k a {p1+1 k k+1 k+1 Sk+1 k+1 (2.2) T where So=S0, ak+1=pk+iqk.o., Tk4.1 = pk+ok.o./qk.41.SKqk+1. A turning point came with a theorem of Dixon {6}, who showq ed that all quasi-Newton formulae (those for which k+1 k+1 = p) in Huang's family generate identical steps even for general k+1 functions, and this directed attention to a choice based on numerical stability rather than on theoretical properties, such as maintenance of positive-definiteness of the Sk {7}. In fact ed at the Broyden (4), Fletcher {8} and Shanno {9} all arriv choice of the BFGS formula from consideration of conditioning of the resulting matrices. Shanno and Kettler {10} specifically considered a quantitative criterion for optimal conditioning, while Fletcher {8} was the first to suggest varying the update formula from step to step in the light of such a criterion. The idea was further developed by Davidon {11} and by Oren and

In Computer Applications to Chemical Engineering; Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

Publ icat ion Date: May 30, 1 980 doi: 10. 1021ibk- 1980-0124.ch002

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Optimization Methods

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Spedicato {12}, but later Spedicato {13} noted that the criteria used by these authors were identical. Clearly Sk is related to the function f(x), and in particular it must be scaled in inverse proportion to any scaling of f(x). This led Oren and Luenberger {14} to investigate the symmetric members of Huang's family for which Sk .q. =11) ti k+1 k1 Pk+1$ with the scalar p chosen to adjust the seating ofS k+1 kAl This "self-scaling idea i was further developed by Spedicato {15} who considered formulae which were invariant to a scalar nonlinear transformation of f(x), and this also generalizes other attempts to approximate f(x) using more general classes than quadratic functions {16,17,18,19}. Numerical comparisons of the optimal conditioning and selfscaling ideas with the classical formulae have been published by Spedicato {15,20}, Brodlie {20, Shanno and Phua {22}, Zang {23} and Schnabel {24}. The evidence is not conclusive, but it seems that the classical BFGS formula is hard to beat. Optimal conditioning involves more arithmetic at each iteration, which pays off only on seriously ill-conditioned problems. There seem to be special types of functions for which self-scaling gives a marked improvement but in general its performance is inferior, and the same seems to be true of the methods based on nonlinear transformations. The early analysis of Fletcher and Powell 01 interpreted Davidon's method as one which generates conjugate directions, which naturally gives rise to the idea of minimization along these directions. However it was soon realized that minimization to high precision is an unnecessary expense, and indeed is not implied if the formulae are interpreted as secant approximations to the inverse of the Hessian matrix. In fact true minimization must be abandoned in favour of a "descent test" to guarantee convergence in a practical algorithm {25}, and various step-length rules are given by Sargent and Sebastian {7} who showed how algorithms can be designed to ensure global convergence to a stationary point. Numerical experience also shows that the simple Armijo rule {26,25} coupled with a descent test is more efficient than minimization, provided that step-length expansion is also used if the test is satisfied immediately. For years everyone has been content with algorithms which produce a descent path to a stationary point, which can of course be a saddle-point rather than the desired local minimum. However McCormick {27} has put forward an idea, later developed by Mord and Sorensen {28}, for the use of directions of negative curvature coupled with descent directions to ensure convergence to a local minimum. The goal of achieving the global minimum rather than just a local minimum still has its attractions. Various approaches are given in the collections of papers edited by Dixon and Szeso{29},

In Computer Applications to Chemical Engineering: Squires, R., el al.: ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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while the recent "tunnelling algorithm" of Levy and Montalvo {30} seems to be an effective version of the function-modification approach to the problem. An excellent discussion of the issues and the different approaches is given by Griewank {31}. As computers become more powerful the problems tackled become ever larger, and inevitably storage problems arise. This has revived interest in the conjugate gradient methods, which require storage of only a few n-vectors rather than an nxn matrix, Powell {32} gives an interesting analysis yielding new insight into the working of these methods. He extends the work of Beale {33} and Calvert {34}, giving evidence for favouring a particular conjugate-gradient formula and providing an automatic test for restarting. Even so, conjugate-gradient methods remain less efficient and less robust than quasi-Newton methods, providing an incentive to apply sparse-matrix techniques to the latter. Now if the Hessian matrix is sparse its inverse is likely to be dense, so instead of (2.1), we use xk+1 " xk

ak sk (2.3)

gk Hk.1.1 = H( k,p10.1,qk+1)

where Hk is an approximation to the Hessian matrix itself, and in order to solve for st, we store and update the triangular factors of Hk. The teciques for updating sparse triangular hn factors are given by Toint {35}. There has been little recent work on methods for differentiable functions which avoid explicit evaluation of derivatives. Powell's conjugate direction method {36) is still used, but the generally accepted approach is now to use standard quasi-Newton methods with finite-difference approximations to the derivatives. On the other hand there has been considerable interest in methods for nondifferentiable functions, as shown by the collection of papers edited by Balinski and Wolfe {37), in which the technique described by Lemarechal is of particular interest. Other contributions in this difficult field are due to Shor {38}, Goldstein {39}, Clarke {40}, Mifflin {taxa, Auslendei-{43} and Watson{44}. In general these problems are much more difficult to solve than those involving differentiable functions, but they are becoming increasingly relevant to optimum design problems involving tolerances {45,46}. Nonlinear Programming. The general nonlinear programming problem is

Minimize subject to

f(x) 0 41(x) tp(x) = 0

) ) )

(3.1)

In Computer Applications to Chemical Engineering Squires, R. el al.; ACS Symposium Series: American Chemical Society: Washington, DC, 1980.

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where f(x) is a scalar function of the n-vector x, cp(x) is an m-vector and *(x) is a q-vector. The state of the art in 1974 in dealing with such problems is admirably summarized in the collection of papers edited by Gill and Murray (47}. At that time the middle ground was held by feasible-point projection or reduced-gradient methods, with a strong challenge from augmented Lagrangian methods. Fletcher himself was disenchanted with his "exact penalty-function" method and tended to favour the augmented Lagrangian approach, and there were still strong protagonists for the original penalty-function approach. The classical penalty-function methods have now finally become part of history, the early promise of the augmented Lagrangian approach has faded, and there has been a coalescence of the approach used in the projection methods with the exact penaltyfunction approach. The classical penalty-function idea was to convert the original constrained problem into an unconstrained one by increasing the objective function artificially if the constraints were violated, adding a penalty term reflecting the magnitude of the constraint violations. The method originated with Frisch {48} and Carroll {49} but was mainly developed by Fiacco and McCormick {50}. Good reviews are given by Lootsma {51} and Ryan {47,pp175-190}. The difficulty with the approach is that it is by definition approximate, and to obtain good approximations the constraint violations must be heavily weighted in relation to the objective function, yielding an ill-conditioned unconstrained problem. The practical solution was to solve a sequence of unconstrained problems with steadily increasing weight of the constraint violations, and methods were devized for extrapolating the sequence to infinite weight. In 1968, Powell {52} likened the process to shooting at a target in a strong wind and suggested it was better to "aim off" rather than wheel up heavier and heavier guns; he therefore introduced a shifting parameter for each constraint, adjusted so that the minimum of the penalty function actually satisfied the constraint. A sequence of minimizations is still necessary to adjust the shifting parameters, but these subproblems are much easier to solve. The lbxact penalty-function" idea was to devize a penalty function which has an unconstrained local minimum exactly coinciding with the constrained minimum of the original problem (3.1). This goal seems to have been consciously sought independently by Fletcher {53} and Pietrzykowski {54}, but the idea was already implicit in the work of Arrow and Solow {55} and Zangwill (561. The Zangwill-Pietrzykowski penalty function for problem (3.1) is P(c,x)=f(x)+c{ E I Vi (x) I + E max (0, - (0(x)) }. (3.2) j=1 j=1

In Computer Applications to Chemical Engineering: Squires, R, el al.: ACS Symposium Series: American Chemical Society: Washington, DC. 1980.

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This function is indeed an exact penalty function for all values of the scalar c above a certain finite threshold value. However it is nondifferentiable, and hence its minimization presents even more severe difficulties than that of the classical penalty functions. The general methods for nondifferentiable functions referred to in Section 2 could be used, but specific methods for (3.2) have been proposed by Conn and his coworkers (57,58,59), Bertsekas (60) and Chung {61}. More recently Charambalous (62,63) has proposed the use of the more general 1 -norm for the penalty term instead of the 1 -norm used in (3.2),Pand points out some .1 advantages for a choice 1