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Optimization of multi-stage absorption systems Lucas, James Peter 1968-12-31

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OPTIMIZATION OF MULTI-STAGE ADSORPTION SYSTEMS by JAMES PETER LUCAS B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1964 M.S., U n i v e r s i t y of Michigan, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Chemical Engineering We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard Members of the Department of Chemical Engineering THE UNIVERSITY OF BRITISH COLUMBIA June, 1968 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by h.ils representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemical Engineering The University of Brit ish Columbia Vancouver 8, Canada Date June 2k, 1968 ABSTRACT M u l t i - s t a g e a d s o r p t i o n c o n f i g u r a t i o n s w i t h e q u a l i t y and I n e q u a l i t y c o n s t r a i n t s are optimized and comparisons are made between t h e i r r e l a t i v e e f f e c t i v e n e s s . For a given number of stages countercurrent flow i s always s u p e r i o r , as expected. Although c r o s s f l o w i s g e n e r a l l y second best, i t becomes i n f e r i o r t o c e r t a i n a l t e r n a t i v e networks f o r c e r t a i n ' ranges of the magnitude o f the ad s o r p t i o n isotherm exponent. Gener a l l y , the order o f the e f f e c t i v e n e s s o f var i o u s networks i s b e l i e v e d to be according t o how much the networks resemble countercurrent flow. An a l g o r i t h m i s deri v e d and used to solve the N-stage c r o s s f l o w network. A l l other c o n f i g u r a t i o n s had t o be t r e a t e d as problems i n constra i n e d o p t i m i z a t i o n . For t h i s purpose, a theorem of Courant and a method of C a r r o l l f o r e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s r e s p e c t i v e l y are used t o formulate the problem. Three d i f f e r e n t o p t i m i z a t i o n methods were considered w i t h the d e f l e c t e d gradient method of F l e t c h e r and Powell p r o v i n g t o be the s u p e r i o r f o r t h i s p a r t i c u l a r c l a s s of o p t i m i z a t i o n problems. i i i , TABLE OF CONTENTS Page INTRODUCTION 1 THEORY 4 A. ADSORPTION CASCADES 4 B. FORMULATION OF CONSTRAINT PROCEDURES 8 1. Method o f Courant , 8 2. Method o f C a r r o l l 17 3 . G e n e r a l F o r m u l a t i o n * 18 C. OPTIMIZATION PROCEDURES B. 19 D. OPTIMIZATION METHODS 21 1. P a t t e r n S e a r c h „ 21 2. D e f l e c t e d G r a d i e n t 21 3. Conjugate D i r e c t i o n s 22 PROCEDURE 24 A. OPTIMIZATION METHODS 2^ 1. P a t t e r n S e a r c h . . . . „ 24 2. Conjugate D i r e c t i o n s ». 24 3 . D e f l e c t e d G r a d i e n t 24 B. FORMULATION OF ADSORPTION PROBLEMS 24 1. G e n e r a l Two Stage Systems 25 2. C r o s s f l o w Systems. 25 3 . C o u n t e r c u r r e n t Systems 26 4. Three Stage Systems 26 5. Four Stage Systems „ 26 C. VARIATON OF PENALTY FACTORS 26A 1. E q u a l i t y C o n s t r a i n t s . 26A 2. I n e q u a l i t y C o n s t r a i n t s 26A Iv. RESULTS 27 A. OPTIMIZATION METHODS... 27 1. Basic Methods 27 a. Pattern Search 27 b. Conjugate Directions ..... 27 c. Deflected Gradient 28 2. One-dimensional Minimization 28 3. Computation Time 28 a. Pattern Search 28 b. Conjugate Directions ...... 28 c. Deflected Gradient 29 d. One-dimensional Minimization 29 B. ADSORPTION SYSTEMS 29 1. Two Stage System..... „.... . 29 2. Countercurrent and Crossflow Systems 29 j5. Three Stage Systems . 31 4. Four Stage Systems . 31 5. E f f e c t of Number of Stages on Crossflow Systems ............... 3^ CONCLUSIONS 37 RECOMMENDATIONS FOR FURTHER STUDY 39 NOMENCLATURE. 40 LITERATURE CITED 4 2 APPENDIX I - Table of Results.... 1-1 APPENDIX II - FORTRAN IV L i s t For Pattern Search Method2-1 APPENDIX I I I - Deflected Gradient Optimization Method..3-1 APPENDIX IV - Conjugate Direction Method 4-1 V . APPENDIX V - Pibonnaci Sub-Program For One-dimensional Minimization . . .5-1 APPENDIX VI - One Dimensional Pattern Search Sub-program....6-1 APPENDIX VII - . . . . 7 - 1 1. Derivation of Crossflow Algorithms . . . 7 - 2 2. Formulation of Derivatives For Deflected Gradient Method 7-5 APPENDIX VIII - Equations of Isotherms Used 8-1 LIST OF ILLUSTRATIONS Figure Page 1. General Two Stage System....... < 5 2. Koble-Corrigan Isotherms Used i n Two Stage System 6 3. N Stage Crossflow With S p l i t Adsorbent 7 •4. N Stage Countercurrent 9 5. Three Stage System A 10 .6. Three Stage System B 10 7. Three Stage System C . 11 8 . Three Stage System D 11 9 . Combination of Three Stage Systems B and D 12 10. Four Stage System A 13 11. Four Stage System B 0 13 12. Four Stage System C 14 13. Four Stage System D .14 14. Four Stage System E ..15 15. Comparison of Countercurrent and Crossflow, With S p l i t Adsorbent Stream, Systems 30 16. Comparison of Three Stage Systems 32 17. Comparison of Four Stage Systems 0.......33 18. E f f e c t of Freundlich Exponent on Crossflow, With S p l i t Adsorbent Stream, Systems 35 v i i . ACKNOWLEDGEMENT I wish to thank Dr. D.A. Ratkowsky of the Depart ment of Chemical Engineering f o r h i s guidance i n helping to carry out t h i s project. I also wish to thank the Computing Centre of the uni v e r s i t y of B r i t i s h Columbia f o r use of the IBM 7044 i n doing the necessary c a l c u l a t i o n s . I also wish to thank the National Research Council of Canada and the Univer s i t y of B r i t i s h Columbia f o r f i n a n c i a l assistance. INTRODUCTION 1 An e a r l i e r study of Lerch (7) investigated the r e l a t i v e effectiveness of various two-stage contacting oper ations involving adsorption systems. The c r i t e r i o n f o r effectiveness i s based upon the minimum a l l o c a t i o n of adsorb ent f o r a f i x e d quantity of solute-free solvent that i s required to reduce the outgoing solute concentration to a given f r a c t i o n of the i n i t i a l concentration. Lerch (7) found that i r r e s p e c t i v e of the adsorption isotherm considered, countercurrent operation was always superior to crossflow operation and that between the two types of crossflow opera t i o n the one i n which the adsorbent i s s p l i t into two portions was always superior to the one i n which the s o l u t i o n i s s p l i t into two portions. The isotherms considered by Lerch (7), the Preundlich and the Koble-Corrigan, have also been used i n the present study. The equations of these isotherms are given i n APPENDIX VIII. These p a r t i c u l a r isotherms were employed because they both have r e l a t i v e l y simple formulae and they c l o s e l y approximate isotherms that are observed i n p r a c t i c a l systems. The present study extends the work of Lerch (7) to configurations having a higher number of stages. Although Lerch (7) demonstrated the s u p e r i o r i t y of countercurrent flow to crossflow, i t was hoped that the following could be determined: 2 . i . The number of crossflow stages necessary to compete i n effectiveness with two-stage counter- current . i i . The type of three and four-stage networks that are superior to three and four-stage eros'sflow respectively, i f any. Since countercurrent flow i s always superior to any a l t e r n a t i v e , the use of other systems must be J u s t i f i e d . Treybal (9) J u s t i f i e s the use of networks other than counter- current only i n small scale processing, where there may be an appreciable time l a g between stages. The crossflow network may be more p r a c t i c a l i f the adsorbent deteriorates during inter-stage storage, since fresh adsorbent i s employed i n every crossflow stage. In addition to providing an insight into the e f f e c t  iveness of various adsorption networks, the configurations considered provided a means by which several d i f f e r e n t methods that optimize systems subject to constraints may be studied. In the course of optimizing networks subject to constraints, i t was found that the pattern search technique of Hooke and Jeeves (6) as modified by Weisman, Wood, and R i v l i n ( 1 0 ) , which had been used e a r l i e r with success by Lerch (7 ) on his more simple configurations, was not adequate f o r the present class of problems. Therefore, other optimization methods were studied, i n p a r t i c u l a r , the "deflected gradient" and "conjugate d i r e c t i o n " techniques of Fletcher and Powell (5) and Powell (8) respectively, and compared with the modified pattern search technique. Since these methods were o r i g i n a l l y 3 . proposed f o r use on problems not involving constraints i t was necessary to modify the objective function to be optimized by the addition of the constraint r e l a t i o n s . The equality constraints were treated by the app l i c a t i o n of a theorem of Courant ( 3 ) , while the ine q u a l i t y constraints, a r i s i n g from the necessity of having c e r t a i n l i m i t s on the range of the variables, were treated by a procedure suggested by C a r r o l l ( 2 ) . THEORY 4. A. ADSORPTION NETWORKS. The adsorption networks to be solved are formulated i n the notation of Treybal ( 9 ) , except that the isotherms are dimensionless. The Preundlich isotherm i s used almost e n t i r e l y since i t i s very simple and covers a l l ranges of ease of s e p a r a b i l i t y . The Koble-Corrlgan Isotherm Is used only i n the general two-stage network given by f i g . 1. Typical shapes of these isotherms are shown i n f i g . 2. To understand the notation used i n the present study, reference should be made to a t y p i c a l network such as the general N-stage crossflow network i n f i g . 3. The streams L s and G_ r e f e r to the quantities of adsorbent and solvent respectively, both streams being on a solute-free basis. The quantities X and Y are dimensionless concentrations of solute i n the adsorbent and solvent streams, respectively. The sub s c r i p t s on these concentrations are derived from the stage number from which t h e i r respective streams are leaving. The sole exception to t h i s r u l i are the i n i t i a l concentrations, which have two subscripts, these being zero and stream number. It i s assumed i m p l i c i t l y throughout the entire study that the s o l i d adsorbent i s completely insoluble with the solvent which could, i n p r i n c i p l e , be either a l i q u i d or a gas. The general N-stage crossflow network, with s p l i t adsorbent stream, i s shown i n f i g . 3. Since t h i s network has been shown by Lerch (7) to be always superior to the crossflow network i n which the solution stream i s s p l i t , the L s i ^2^81"^ " G S 2 (I - k2) G s l Y. ' S I kgGLj+6 L 32 l<l'-Sl^ l'T*LS2^2  K|LS|+LS2 S 2 k26stY2+GS2Y02 K 2 G S I + G S 2 FIGURE G E N E R A L TWO STAGE S Y S T E M 6 Y 0.0 0.2 0.4 0.6 0.8 1.0 X F I G U R E 2. K 0 B L E - CORRIGAN I S O T H E R M S U S E D IN T W O S T A G E S Y S T E M l a t t e r configuration has not been considered. The network shown i n f i g . 3 i s not subject to any constraints; hence an algorithm f o r i t s s o l u t i o n can be determined (see APPENDIX VII). A l l the other adsorption problems, i . e . the general two stage configuration shown i n f i g . 1, the N-stage counter- current network shown i n f i g . 4 and the various three and four-stage networks shown i n f i g s . 5 -9 and 10-14 r e s p e c t i v e l y had to be treated as problems i n constrained optimization. Some of the configurations, such as three and four-stage net works of figures 5-14 are subject to the constraint of having to blend two solvent streams to y i e l d the required f i n a l concentration. The four-stage networks of f i g s . 10-14 possess an a d d i t i o n a l constraint due to the f a c t that only three of the four i n d i v i d u a l adsorbent-solvent r a t i o s ( L s-j/G s.j , i <= 1,2; J = 1,2) can be f i x e d independently. The N-stage countercurrent system of f i g . 4 i s subject to a t o t a l of N-l constraints, these deriving from the f a c t that the adsorb ent-solvent r a t i o must be the same for each of the N stages. The general two-stage system of f i g . 1 requires two constraints, these being due to the blending of the two solvent streams and the necessity that L/G be the same f o r each stage. These constraints are a l l of the equality type and can be treated using the method of Courant (3) discussed i n section B. B. FORMULATION OP CONSTRAINT PROCEDURES 1. Method of Courant Courant (3) has proved a theorem fo r convergence of a sequence of functions to the optimum subject to an equality FIGURE 4. N STAGE C O U N T E R C U R R E N T '02 S2 ' S 2 THREE STAGE S Y S T E M A Y, 'si X L si V \1/ FIGURE 6. T H R E E S T A G E S Y S T E M B X oi L SI Y oi 'SI Y5 ' S2 V Al/ FIGURE 7. T H R E E STAGE S Y S T E M C F IGURE ' 8 . T H R E E STAGE S Y S T E M D 12. F I G U R E 9 . COMBINATION OF T H R E E STAGE S Y S T E M S B AND D. FIGURE II. FOUR STAGE SYSTEM B Y, si S 2 FIGURE 12. FOUR STAGE SYSTEM Y, 'si Y. S2 FIGURE 13. FOUR S T A G E S Y S T E M 15. FIGURE 14. FOUR STAGE S Y S T E M E 16. constraint: Theorem: In a space S of r e a l functions where convergence i s defined, i f : i . t ! ; ( X ^ , . . » , X n ) and $ (X^ , . . .^X^) are lower semi-continuous functions, i i . ISJ(X 1, . . . jX^) > 0 f o r a l l X ± i n S and there exists some X ± i n S such that ^(X^ . . . , X ^ ) = 0 . i i i > P C X ^ , . • • , x n k ) = f ( x l k , . . . , x n ] k ) + t k * ( x l k , . . ^ X ^ ) , k = 1,2,... where t ^ i s a p o s i t i v e r e a l number and P i s a minimum f o r any k . I f I 11m tK = oo (1.1) k_» oo l i m *(X l k,... JX n k) = 0 (1.2) k_» o o l i m F ( X L K , . . 0 , X N F C ) = ^ X ^ . o . ^ ) (1.3) k_» c<a Since • •'3 Xnk^ i s a s e ( l u e n c e °f minimum functions on S > ®( Xlk** * *> Xnk^ i s t h e m i n l m u m subject to the constraint *(Xlk-» " ° >Xnk^ = 0 * T h e P r o o f > m a y be found i n ( 3 ) . Weisman, Wood, and R i v l i n (10) have generalized the theorem to be: Corollary: I f M functions * m ( X l 3 .. o,^) > 0 , m = l,...,M for a l l X 1 i n S and there exist X± such that the then: 17. * m( xl» • • • jXfc) = 0 simultaneously, and F ( X l k , . o .^X^) = M S^lk'-"''*!!*;) + * tmk*m( xlk'**"' xnk)' > K = 1 * 2 ^ " * w h e r e m=l the t ^ are p o s i t i v e r e a l numbers and F i s a minimum f o r any k , then equations ( l . l ) and ( 1 . 3 ) of the theorem s t i l l h o l d and (1«2) becomes: l i m *m( XIk* 0"> xnk) = 0 m = 1>"tVL C1*4) k_» oo 2. Method of C a r r o l l C a r r o l l (2) has proposed the f o l l o w i n g technique f o r s o l u t i o n of problems w i t h M* i n e q u a l i t y constraints: i . Write a l l the I n e q u a l i t y c o n s t r a i n t s so that they are p o s i t i v e : l-m( Xlk' °°^ Xnk ) ^ 0 = M+l, „. .,M+M« "(2.1); i i . Choose a set of p o s i t i v e constants . t mO m = M+l, „. ,,M+M' i l l . Choose the s t a r t i n g independent v a r i a b l e s (X 1 Q,...,X n Q) so that a l l the c o n s t r a i n t s are s a t i s f i e d o The general system may be w r i t t e n as f o l l o w s : M+M' t ^ F ( X l k * ° °'>Xnk^ = * ( X l k * 0 * °'Xnk) + £ A lit "TP T ^ 1 K ^ m = M + l V^lk' 0 0 0 ' ^ ' ( 2 . 2 ) The f u n c t i o n F(X l k,...^X^) i s minimized s u c c e s s i v e l y f o r i n c r e a s i n g k , The t ^ are decreasing w i t h k so that: 18. l i m t , = 0 m = M+l, . .«,M+M' ( 2 . 3 ) k - . o o l l m F ( X l k > . . . , X n k ) = » ( X 1 J T , . . . , X F L K ) (1.5) k _ » o o Since the F(X-yc, .. .^X^) axe successive minima the l i m i t i n g s o l u t i o n i s that s(X l k,-.. . j X ^ ) i s a minimum subject to the constraints * m ( x i k * 0 ' ,-'Xnk^ — °> m = M + l * • • • 3» General Formulation Using both the methods of (2) and (4) the general system becomes: M p ( x l k > •••* x nk)'= * ( x i k ' " - ' x n k ) + \ tmk*m^ xlk' , , ,' xnk^ + m=l m=M+l M Xlk> •••» Xnk* ( 3 . 1 ) wher@ the *m,m = 1, ...-,M are the equality constraints squared to s a t i s f y condition i i . of Courant's Theorem. I f r (oo m = x3 ..., M l l m t n f c = f . then: k_ o o J U ^ (0 ra = M+l,.. .,M+M» 0 m = 1, .. .,M M X l k - - ^ X n k ) i •(«> ~* °° 1 > 0 m = M+l, .0<!,M+M' li m F(X l k,...,X n k) = fCXj^-,...,^) ( 1 . 3 ) In t h i s , manner FCx^ ,. ^  . j X ^ ) , the constrained optimum, may be determined. 19. C. OPTIMIZATION PROCEDURES The adsorption networks Involving constraints, discussed i n section A, were examined using several d i f f e r e n t optimization methods. The f i r s t method to be tested i n the present study was a modification of the pattern search method of Hooke and Jeeves (6). Weisman, Wood, and R I v l l n (10) combined the pattern search procedure with the theorem of Courant (3) to solve problems involving equality constraints. However, when t h i s procedure was applied to the four-stage network shown i n f i g . 10 i t was found to give solutions highly dependent upon the s t a r t i n g parameters. In p a r t i c u l a r , the values of the acceleration and deceleration factors suggested by Weisman, Wood, and R i v l l n (10) did not always r e s u l t i n the same solution (see table'1). However, by u t i l i z i n g d i f f e r  ent values of these factors, the method could be made to y i e l d the correct r e s u l t s . The network shown i n f i g . 10 proved to be very suitable f o r the evaluation of the various optimization methods since under the conditions of Y Q 1 » Y Q 2 = 1.0 and X01 = X02 = 0 * ^ can be seen that t h i s network becomes i d e n t i c a l to the two-stage countercurrent flow configuration previously solved by Lerch (7). His re s u l t s are shown i n the l a s t column of table 1 f o r comparison. Since the pattern search method proved to be unrel i a b l e , other methods of optimization were investigated. It was shown by Box ( l ) , who made a study of a wide v a r i e t y of optimization methods, that f o r problems with sharp v a l l e y s , the ideas of Davidon (4), as formulated by Fletcher and 20. P o w e l l ( 5 ) , were the most s u c c e s s f u l although knowledge o f the f i r s t d e r i v a t i v e s i s r e q u i r e d . He a l s o showed that the method of Powell ( 8 ) was a s u c c e s s f u l procedure when f i r s t d e r i v a t i v e s are not known. Therefore, the methods of F l e t c h e r and Powell ( d e f l e c t e d g r a d i e n t ) and Powell (conjugate d i r e c t  i o ns) were s t u d i e d as a l t e r n a t i v e s t o the p a t t e r n search tech nique. Both of the above procedures were modified to d e a l w i t h e q u a l i t y c o n s t r a i n t s by employing the same theorem of Courant (3) as was used by Weisman, Wood, and R i v l i n ; ( 1 0 ) . In a d d i t i o n , the I n e q u a l i t y procedure o f C a r r o l l (2) was used when i t was necessary t o have l i m i t s on the range of v a r i a b l e s . For example, the dimensionless concentrations X and Y had to be r e s t r i c t e d t o the range between 0 and 1 and the adsorbent-solvent r a t i o s f o r each stage were r e q u i r e d t o be non-negative. The general f o r m u l a t i o n f o r m i n i m i z a t i o n w i t h M e q u a l i t y and M' i n e q u a l i t y c o n s t r a i n t s i s given by equation ( 3 . 1 ) . For the problems i n t h i s study the o b j e c t i v e f u n c t i o n , $(Xlk-» * * ** Xnk^ > l s ^ ^ Y 8 equal t o the r a t i o of t o t a l E L s i a l l o c a t i o n of adsorbent t o s o l v e n t : L/G = —w== . I t i s s G s i t h i s f u n c t i o n which must be minimized subject to the c o n s t r a i n t s on the system. F ( x l i c * • • • * x n k) i n equation ( 3 . 1 ) Is the constrai n e d o b j e c t i v e f u n c t i o n . The c o e f f i c i e n t s t , are mk o f t e n r e f e r r e d t o as p e n a l t y f a c t o r s . The theorem of Courant (3) i m p l i e s t h a t i f the p e n a l t y f a c t o r s are increased w i t h each i t e r a t i o n (index k) , the e q u a l i t y c o n s t r a i n t s are s a t i s f i e d 21. more and more c l o s e l y . However, the i n e q u a l i t y constraints have t h e i r penalty factors decreased with each i t e r a t i o n . I n i t i a l high values of the penalty factors on the i n e q u a l i t y constraints prevent them from being v i o l a t e d while r e s t r i c t  ing the variables to a narrow range. As the penalty factors are decreased, i n subsequent i t e r a t i o n s , the variables be come fr e e r to assume a wide range of values. It then follows from the theorem of Courant (3) and the method of C a r r o l l (2) that equation (3.1) should converge to the constrained optimum, within cer t a i n prescribed tolerances. The use of equation (3.1) remained the same irregard- l e s s of which of the three methods; pattern search, deflected gradient, or conjugate d i r e c t i o n s were employed. The methods themselves are discussed i n more d e t a i l i n the following section. D. OPTIMIZATION METHODS 1. Pattern Search Hooke and Jeeves (6) pattern search method i s a type of d i r e c t search procedure, i . e . a sequential examination of a f i n i t e set of t r i a l values of the function under study, incorporating a simple strategy f o r f i n d i n g the various t r i a l points. The pattern search program of Weisman, Wood, and R i v l i n (10), modified to FORTRAN IV, i s given i n APPENDIX I I . 2. Deflected Gradient Fletcher and Powell (5) modified the method o r i  g i n a l l y derived by Davldon (4), making i t more e f f i c i e n t , and proving quadratic convergence. A more r e a d i l y understand able description i s given by Wilde and Beightler (11). The flow diagram and FORTRAN TV l i s t i n g are i n APPENDIX I I I . The method I t s e l f i s such that the search moves i n the d i r e c t i o n of l o c a l l y improving values of the objective function, but seldom exactly along the gradient, thereby explaining the name "deflected gradient". It i s necessary with each step to f i n d the minimum of the objective function along t h e deflected gradient. Fletcher and Powell ( 5 ) , contending that the method of obtaining the minimum was not cen t r a l to the theory, used a cubic i n t e r p o l a t i o n technique given by Davidon ( 4 ) , and found i t to be s a t i s f a c t o r y although i t did not locate the minimum along the "deflected" gradient very accur at e l y . However, f o r the class of problems considered In the present study, i t was found that i t i s necessary to have a more precise estimate of the true one-dimensional minimum i n order f o r the method to converge r a p i d l y . Other one-dimensional minimization techniques considered were: i . The parabolic approximation of Powell (8). i i . The one-dimensional pattern search technique of Hooke and Jeeves ( 6 ) . The flow diagram and FORTRAN IV l i s t are found i n APPENDIX VI. i i i . The c l a s s i c a l Fibonacci search (see Wilde and Beightler (11)). The flow diagram and FORTRAN IV l i s t are found i n APPENDIX V. 3. Conjugate Directions. The method of conjugate direc t i o n s , formulated by Powell (8), has quadratic convergence l i k e the deflected gradient method but does not require c a l c u l a t i o n of der i v a t i v e s . 23. The method i s a simple v a r i a t i o n of the c l a s s i c a l method of mini m i z i n g a f u n c t i o n o f s e v e r a l v a r i a b l e s by changing one v a r i a b l e at a time. Since the o b j e c t i v e f u n c t i o n i s the most important v a r i a b l e under c o n s i d e r a t i o n an a d d i t i o n a l convergence c r i t e r i o n was added t o stop the program when l i t t l e change occurs. The f l o w diagram and FORTRAN IV l i s t  i n g are i n APPENDIX IV. In the present study, the conjugate d i r e c t i o n s method was improved by i n c o r p o r a t i n g the one-dimensional p a t t e r n search technique i n preference t o the p a r a b o l i c i n t e r p o l a t i o n technique o r i g i n a l l y used by Powell ( 8 ) . PROCEDURE 24 A. OPTIMIZATION METHODS When = X Q 2 and Y Q 1 = Y Q 2 the system shown i n f i g . 10 becomes equivalent to a two-stage countercurrent system f o r which a known solution exists (7). Hence, the optimization procedures were tested on t h i s s i m p l i f i e d four- stage system using an exponent of 1 f o r the Freundlich Isotherm to determine the best method. 1. Pattern Search The s t a r t i n g values of the independent variables, the i n i t i a l step size and the acceleration and decelerations r a t i o s are varied. The solutions that r e s u l t are tabulated and compared. 2. Conjugate Directions A l l four methods f o r determining a one-dimensional minimum: cubic and parabolic i n t e r p o l a t i o n , Fibonacci and one-dimensional pattern searches, were used and compared. 3. Deflected Gradient A l l four methods f o r determing a one-dimensional minimum were used and compared. B. FORMULATION OF ADSORPTION PROBLEMS As previously mentioned i n the THEORY, the adsorption systems a l l have ce r t a i n s i m i l a r c h a r a c t e r i s t i c s i n t h e i r form u l a t i o n . S p e c i f i c d e t a i l s with respect to various methods are discussed here. A l l the independent variables must be between 0 and 1 and a l l the adsorbent r a t i o s f o r each stage must be greater than or equal to 0 . 25. 1. General Two Stage System The general two-stage system, as shown i n f i g a 1 i s a combination of the f o l l o w i n g networks: i . Two-stage countercurrent. i i . Two-stage crossflox? w i t h s p l i t adsorbent stream. I i i . Two-stage c r o s s f l o w * r i t h s p l i t solvent stream. The network produces the normal p a i r o f independent v a r i a b l e s (X i) expressing the con c e n t r a t i o n i n the adsorbent streams l e a v i n g the r e s p e c t i v e stages and the f o l l o w i n g : k 1 - F r a c t i o n of the adsorbent stream l e a v i n g stage one and going to stage two; 0 < k^ _< 1 . kg - F r a c t i o n of solvent stream l e a v i n g stage two and going to stage one; 0 _< kg _< 1 . p-^ - R a t i o of solvent streams i^s^^sl^'' ^1 — 0 * 3 2 - R a t i o of adsorbent streams ( ^ g ^ ^ s l ^ ^2 — 0 * I t i s e a s i l y seen that when k^ = kg = 1 and ^1 ~ &2 ~ ® t h e s y s ^ e m becomes two-stage countercurrent; when k^ = 0 3 kg •= 1 } fi^ = 0 } and P 2 — 0 t t l e s y s - t e m i s two - stage c r o s s f l o w w i t h s p l i t adsorbent stream; and when k^ = 1 } kg = 0 t F>-^  _> 0 9 and pg = 0 the system i s two-stage cross- f l o w w i t h s p l i t s o lvent stream. The Koble-Corrigan Isotherms shown i n f i g . 2 and F r e u n d l i c h isotherms, were used i n t h i s system. 2. Crossflow Systems Since Lerch ( 7 ) showed that two-stage c r o s s f l o w w i t h 26. s p l i t adsorbent stream i s superior to two-stage crossflow with s p l i t s olution stream, only systems of the f i r s t type were solved.(see f i g . 3). 3. Countercurrent Systems As was previously noted i n the THEORY, an N-stage countercurrent network (see f i g . 4) formulates into a problem of N-l variables and R - l equality constraints. 4. Three-stage Systems Pour three-stage configurations (see f i g s . 5-8) with no j o i n i n g or separating streams are optimized. They a l l have three independent variables and one equality constraint (the f i n a l s o l u t i o n concentration i s constant). A combination of systems B and D ( f i g s . 6 and 8) gives the system i n figure 9, i n which the adsorbent streams from stages one and two are Joined before entering stage three. The Joining of the two streams creates a new independent variable k (^L=^BX/^JB2^* which i s the r a t i o of the adsorbent streams. 5. Pour-stage Systems Five four-stage systems are created following the example of Treybal (9), (see f i g s . 10-14). They have four independent variables and two equality constraints. The f i r s t constraint i s the f i x e d value of the f i n a l solution concent r a t i o n , and the second involves the adsorbent-solution r a t i o s . A thorough discussion of the constraints i s found i n APPENDIX VII where the system of f i g . 10 has sample c a l c u l a t i o n of derivatives f o r the Fletcher-Powell method. 26A C. VARIATION OF PENALTY FACTORS 1. E q u a l i t y C o n s t r a i n t s I n i t i a l l y , t he p e n a l t y f a c t o r s a r e s e t v a l u e s o f the o r d e r o f 10. When o n l y one c o n s t r a i n t o c c u r s i n t h e pr o b l e m the p e n a l t y f a c t o r i s i n c r e a s e d by a f a c t o r o f 10 a f t e r each o m i n i m i z a t i o n u n t i l i t rea c h e s a v a l u e o f around 10 . The program i s t h e n t e r m i n a t e d . When more t h a n one e q u a l i t y con s t r a i n t o c c u r s the p e n a l t y f a c t o r c o r r e s p o n d i n g t o the c o n s t r a i n t f u n c t i o n o f l a r g e s t a b s o l u t e v a l u e i s i n c r e a s e d by a f a c t o r o f 10. Each o f t h e o t h e r p e n a l t y f a c t o r s i s m u l t i p l i e d by 10 times t h e r a t i o o f i t s c o r r e s p o n d i n g c o n s t r a i n t f u n c t i o n t o i the l a r g e s t c o n s t r a i n t f u n c t i o n . The program i s t h e n t e r m i n a t e d o when one p e n a l t y f a c t o r becomes a p p r o x i m a t e l y 10 . T h i s p r o c e d u r e t e n d s t o keep the v a l u e s o f the c o n s t r a i n t f u n c t i o n s a p p r o x i m a t e l y e q u a l . 2. I n e q u a l i t y C o n s t r a i n t s i The p e n a l t y f a c t o r s on the i n e q u a l i t y c o n s t r a i n t s a r e i n i t i a l l y s e t t o be o f the o r d e r o f 10. A f t e r each m i n i m i z a t i o n the f a c t o r s a re m u l t i p l i e d by p r e - a r r a n g e d c o n s t a n t f a c t o r s v a r y i n g f r o m 0.1 t o 0.5 depending on the imp o r t a n c e g i v e n t o t h e i r r e s p e c t i v e c o n s t r a i n t s . ' C o n s t r a i n t s t h a t t e n d t o be v i o l a t e d have t o have t h e i r p e n a l t y f a c t o r s d e c r e a s e d l e s s r a p i d l y . 27. RESULTS The i n i t i a l c oncentrations i n the adsorbent and so l v e n t streams are 0.0 and 1.0 r e s p e c t i v e l y ^ and the f i n a l s o l u t i o n c o n c e n t r a t i o n i s 0.1 i n a l l cases. The F r e u n d l i c h isotherm i s used f o r a l l systems w i t h the exponent v a r y i n g from 0.3 t o 3.0 . The Koble-Corrigan isotherm i s used o n l y I n the general two-stage system. A. OPTIMIZATION METHODS Four-stage system A (see f i g . 10) w i t h a F r e u n d l i c h exponent of 1 i s used as a comparison of each method's e f f e c t  iveness. 1* B a s i c Methods a. P a t t e r n Search The most s e r i o u s l i m i t a t i o n of the P a t t e r n Search procedure i s i t s i n a b i l i t y t o give i d e n t i c a l r e s u l t s f o r d i f f  erent i n i t i a l and op e r a t i n g c o n d i t i o n s . Table 1 shows that although the o b j e c t i v e f u n c t i o n i s the same except f o r one case, to 3 s i g n i f i c a n t f i g u r e s , the independent v a r i a b l e s are not, and d e v i a t e q u i t e markedly from the expected s o l u t i o n of two stage countercurrent, which i s shown i n the l a s t column of t a b l e 1. b. Conjugate D i r e c t i o n s As shown i n t a b l e 2, the s o l u t i o n determined by Con- Jugate D i r e c t i o n method approaches the tru e optimum f o r moderate values of the p e n a l t y f a c t o r s , but then veers away as the f a c t o r s are increased. The d i v e r g i n g p r o p e r t y i s probably due t o two t h i n g s : 28. 1. The creation of a great many conjugate d i r e c t  ions could cause the loss of an independent v a r i a b l e . Zangwill (12), i n a very recent paper, has shown that t h i s can occur, and when i t does the method breaks down. i i . The large penalty factors create very sharp bends i n the v a l l e y that the method i s following. This could cause the method to appear as i f i t has found an optimum and stop. c. Deflected Gradient In contrast to the previous procedures, the optimum (see table 16), determined by the Deflected Gradient method, Is very close to the expected with very l i t t l e v a r i a t i o n due to d i f f e r e n t s t a r t i n g values. 2 . One-dimensional Minimization, When the deflected gradient and conjugate directions methods were t r i e d using parabolic and cubic interpolation^ no solu t i o n was o b t a i n e d h e a r t h e e x p e c t e d v a l u e . •• - The Fibonacci and One Dimensional Pattern searches give almost equivalent r e s u l t s that are quite acceptable when used with the deflected gradient method. 3. Computation Time a. Pattern Search The computation time (see table 1) varies consider ably from run to run. It does seem to be somewhat dependent on how accurate the so l u t i o n i s j the longer the time, the closer the answer to the expected s o l u t i o n . b. Conjugate Directions The computation time i s dependent on the accuracy of the f i n a l s o l u t i o n . For low accuracy the method gives 29. r e s u l t s f a s t e r than pattern search. However, f o r high accuracy, the pattern search time Is l e s s . c. Deflected Gradient The computation time i s about 20 seconds f o r four- stage system A, a good deal faster than the best time of the \ other methods. d. One-dimensional Minimization Although approximately 1 . 5 times f a s t e r than one- dimensional pattern search, Fibonacci does not give quite as good r e s u l t s , implying that to get the best r e s u l t s a thorough search f o r the optimum must be undertaken. B. ADSORPTION SYSTEMS The r e s u l t s discussed below were a l l determined using the modified deflected gradient method of Fletcher and Powell ( 5 ) , using either Fibonacci or one-dimensional pattern searches to determine the one-dimensional minima. 1 . Two-stage Systems The optimum generalized two-stage system becomes equivalent to the two-stage countercurrent system f o r isotherms of both the Freundlich and Koble-Corrigan v a r i e t y (see tables 3 and 4 ) . These r e s u l t s are i n good agreement with Lerch (7) f o r two-stage countercurrent (see table 5 ) . 2 . Countercurrent and Crossflow Systems A comparison, i n f i g . 1 5 , i s made between two, three, and four-stage countercurrent systems and two, f i v e , and f o r t y - stage crossflow, with s p l i t adsorbent stream, systems. The re s u l t s are found i n tables 5 , 8 , and 15 f o r countercurrent and tables 6 , 7, 1 4 , and 20 f o r crossflow. I t i s apparent that f o r easy s e p a r a b i l i t y as determined by a high exponent 3 0 . n 31. value on the Preundllch isotherm, countercurrent procedures are f a r superior. However, as the exponent value decreases, the effectiveness of the countercurrent systems decreases f a s t e r than that of the crossflow systems. Over the range of Preundlich isotherms tested, the three-stage countercurrent i s superior i n a l l cases to forty-stage crossflow. It i s doubted whether t h i s s u p e r i o r i t y would remain f o r lower Preundlich exponents. 3. Three-stage Systems Three-stage countercurrent i s the most e f f e c t i v e of a l l three-stage systems over the range of Preundlich isotherms tested (see f i g . 16). Systems A and B (see f i g s . 5 and 6) have almost I d e n t i c a l objective functions, s l i g h t l y better than the i d e n t i c a l p a i r C and D (see f i g s . 7 and 8) . Systems A and B are markedly i n f e r i o r to three-stage crossflow f o r low Preundlich exponents but become superior at an exponent value of 2.4. The r e s u l t s f o r three-stage systems are i n tables 7 to 13. The systems shown i n f i g . 9 i s found to give solutions equivalent to systems B (see f i g . 6), implying that an optimum combination of the two systems does not ex i s t f o r the isotherms tested. The solution i s i n table 13. 4. Four-stage Systems * F i g . 17 shows that countercurrent i s superior to a l l other four-stage systems over the range of Freundlich isotherms tested. For Freundlich exponents below 0.4 crossflow i s super i o r to a l l of systems A - D (see f i g s . 10 - 13). However, systems A and B become superior to crossflow at exponent values of 0.4 and 1.2 respectively« It should be noted that the 3 4 . points of crossover mentioned are only v a l i d for the conditions used. Systems A, B, C, and D are e f f e c t i v e i n descending order over the range of isotherms tested. System E (see f i g . 14) does not have a f e a s i b l e solution. It can be sliown that the fe a s i b l e values for the independent variables are «= X 2 = X^ = X^ which contradict the I n i t i a l conditions of the problem. A possible explanation f o r the r e l a t i v e effectiveness may l i e i n the resemblances of the various configurations to counter- current flow. System A , equivalent to two-stage counter- current when X Q 1 = X Q 2 and Y Q 1 = Y Q 2 , i s the most e f f e c t  ive. The effectiveness of systems B, C, and D decrease i n that order, which i s also the order of decreasing s i m i l a r i t y of these systems to countercurrent flow. Some of the solutions f o r system D may be i n doubt since there was d i f f i c u l t y i n obtaining consistent r e s u l t s . The re s u l t s f o r four-stage systems are i n tables 1 4 - 1 9 . 5. E f f e c t of Number of Stages on Crossflow Systems F i g . 18 shows the general e f f e c t of stages on e f f e c t  iveness i n crossflow systems. For large values of the Freundlich exponent the effectiveness i s nearly independent of the number of stages. The separation i s easy enough so that the f i r s t few stages accomplish i t . As the exponent decreases the scoftber of stages has more and more e f f e c t on the effectiveness of the system. The near independence of the effectiveness ofl the number of stages i s shown a n a l y t i c a l l y i n APPENDIX VII fo r the case when the Freundlich isotherm exponent i s 1 . I — . — I — • • • — ' — ' — ' — ' — ' — ' — ' — 1 — ' — ' — ' — ' — ' — * - 2 4 6 8 10 12 14 16 18 2 0 22 2 4 2 6 28 3 0 S T A G E S 3 6 . The l i m i t i n g L/G f a c t o r as the number of stages approach i n f i n i t y i s the n a t u r a l l o g a r i t h m of 10 , f o r Y p = 0.10 , which i s the o n l y value considered i n t h i s study. CONCLUSIONS 3 7 . The deflected gradient method of Fletcher and Powell (5) using eit h e r Fibonacci or pattern search one- dimensional minimization subroutines i s found to be a success f u l method of optimizing constrained adsorption systems. On a test problem the solutions obtained by t h i s method are more accurate and can be computed f a s t e r than e i t h e r the pattern search or conjugate d i r e c t i o n methods. Therefore, the deflected gradient method was used on a l l remaining problems. A general two-stage network, converges to the counter- current network at optimum, confirming the s u p e r i o r i t y of countercurrent over other two-stage networks. Countercurrent networks are superior to any others having the same number of stages f o r Freundlich isotherms between 0 . 3 and 3 . 0 . In three-stage systems, excluding countercurrent, crossflow i s superior to a l l others f o r low exponent values of the Freundlich isotherm but becomes i n f e r i o r to some of the other networks at very high exponent values. A combination of two types of three-stage networks gives solutions i d e n t i c a l to the superior system of the two. In four-stage networks, excluding countercurrent, the crossflow was most e f f e c t i v e at low Freundlich exponent, but became i n f e r i o r to some of the others as the exponent increased. The other systems are believed to be e f f e c t i v e according to t h e i r resemblance to countercurrent networks. The e f f e c t of varying the number of stages i n cross- flow networks i s only appreciable f o r small Freundlich exponent. For exponent values over 2.0 there i s v i r t u a l l y no change i n effectiveness a f t e r a few stages. 38. I n a l l problems c o n s i d e r e d i n t h i s s t u d y the f i n a l s o l u t e c o n c e n t r a t i o n was t a k e n t o be o n e - t e n t h o f the i n i t i a l c o n c e n t r a t i o n . This r e s u l t e d i n " c r o s s - o v e r " p o i n t s where one c o n f i g u r a t i o n becomes s u p e r i o r o r i n f e r i o r t o a n o t h e r a t c e r t a i n s p e c i f i c v a l u e s o f the F r e u n d l i c h exponent. F o r o t h e r v a l u e s o f the f i n a l c o n c e n t r a t i o n t h e s e c r o s s - o v e r p o i n t s would u n d o u b t e d l y be s h i f t e d . RECOMMENDATION FOR FURTHER STUDY 39 A more p r a c t i c a l problem would involve the optimi zation of a r e a l adsorption system taking into account the following considerations: I. Cost of adsorbent, i i . Operation of stages at d i f f e r e n t temperatures, i l l . Equipment and maintenance cost, i v . P u r i t y of f i n a l product. Zangwill's (12) modification to Powell's (8) procedure should be investigated to see i f the modified method becomes suitable f o r constrained adsorption systems. NOMENCLATURE 40. The following symbols hold i n a l l sections except Appendices II to VT. F constrained objective function quantity of solute-free solvent i n solvent stream J, l b . G E G S t j 3 t o t a l quantity of solute-free solvent, l b . L g i quantity of solute-free adsorbent i n adsorbent stream i , l b . L £ L g i , t o t a l quantity of solute-free adsorbent, l b . m parameter In Koble-Corrigan isotherm M number of equality constraints M' number of inequality constraints n number of Independent variables (section A of Theory only) n parameter i n Freundlich and Koble-Corrigan isotherms N t o t a l number of stages i n a configuration S space of r e a l lower semi-continuous functions *mk c o n s t r a i n t penalty factor X, dimensionless concentration of solute i n adsorbent stream leaving stage i X Q i dimensionless i n i t i a l concentration of solute i n adsorbent stream i Y^ dimensionless concentration of solute i n solvent stream leaving stage i Y Q 1 dimensionless i n i t i a l concentration of solute i n solvent J stream J Subscripts F index i n d i c a t i n g f i n a l concentration i index f o r a stage or a stream J index f o r a stream k index f o r number of i t e r a t i o n s Continued.... Subscripts (Continued) Index denoting constraint number index i n d i c a t i n g i n i t i a l concentration Greek solvent-adsorbent r a t i o objective function constraint function LITERATURE CITED 42. 1. Box, M.J.j Comp. J., 9 , 67 ( 1 9 6 6 ) . 2. C a r r o l l , C.W.; Operations Research, £, 169 ( 1 9 6 1 ) . 3. Courant, R.; "Calculus of Variations and Supplementary and Exercises", New York University I n s t i t u t e of . Mathematical Sciences, New York ( 1 9 5 6 ) . 4 . Davidon, W.C.; A.E.C. Research and Development Report ANL-5990 (Rev), ( 1 9 5 9 ) . . 5. Fletcher, R., and Powell, M.J.D.; Comp. J . , j6, 163 ( 1 9 6 3 ) . 6 . Hooke, R., and Jeeves, F.A.; J. Assoc. Comp. Mach., 8 , 119 ( 1 9 6 1 ) . 7. Lerch, R.G.; "Optimization Studies i n Adsorption Systems", (B.A.Sc. t h e s i s ) , Department of Chemical Engineer ing, University of B r i t i s h Columbia, Vancouver 8 , B.C., Canada; see also Lerch. R.G. and Ratkowsky, D.A.; I. and E.C. Fund. 6, 308 (1967); Erratum, i b i d . , 6 , 480 ( 1 9 6 7 ) . 8. Powell, M.J.D.; Comp. J., 7, 155 ( 1 9 6 4 ) . 9. Treybal, R.E.; "Mass Transfer Operations", McGraw-Hill, New York ( 1 9 5 5 ) . 10. Weisman, J., Wood, C.F., and R i v l i n , L.j Chem. Eng. Symp. Series, 61^ No. 55, 50 ( 1 9 6 5 ) . 11. Wilde, D.J., and Beightler, C.S.; "Foundations of Optimi zation", Prentice-Hall, Englewood C l i f f s , N.J. ( 1 9 6 7 ) . 12. Zangwill, W.I.j Comp. J., 10, 293 ( 1 9 6 7 ) . APPENDIX I 1.1 T a b l e s o f R e s u l t s The F r e u n d l i c h I s o t h e r m i s used f o r t h e r e s u l t s i n a l l t a b l e s e x c e p t t a b l e 4 where th e K o b l e - C o r r i g a n I s o t h e r m i s used. The D e f l e c t e d G r a d i e n t p r o c e d u r e has t e s t r e s u l t s i n t a b l e 16. T a b l e 1. Four Stage System A - P a t t e r n S e a r c h Method. 2. Four Stage System A - Conjugate D i r e c t i o n Method. 3. G e n e r a l Two Stage System - F r e u n d l i c h I s o t h e r m Used. 4. G e n e r a l Two Stage System - K o b l e - C o r r i g a n I s o t h e r m Used. 5. Two Stage C o u n t e r c u r r e n t . 6. Two Stage C r o s s f l o w - A l g o r i t h m S o l u t i o n . 7. Three Stage C r o s s f l o w - A l g o r i t h m S o l u t i o n . 8. Three Stage C o u n t e r c u r r e n t . 9- Three Stage System A . 10. Three Stage System B. 11. Three Stage System C. 12. Three Stage System D. 13 • C o m b i n a t i o n o f Three Stage Systems B and D. 14. Four Stage C r o s s f l o w - A l g o r i t h m S o l u t i o n . 15. Four Stage C o u n t e r c u r r e n t . 16. Four Stage System A. 17. Four Stage System B . 18. Four Stage System C . 19. Four Stage System D . 20. L/G F o r M u l t i - S t a g e C r o s s f l o w Systems. ' TABLE 1* Pour Stage System A - Pattern Search Method Preundlich exponent = 1 . 0 I n i t i a l F i n a l Y p Y Q 1 - 1 .0 = 0 . 1 Y 0 2 ~ l o 0 X 0 1 = 0 . 0 x 0 2 ° ° ' 0 Acceleration Factor 1.1111 2.0000 2.0000 2.0000 2.0000 2.5000 True Optimum Deceleration Factor 0,1000 0 . 1 0 0 0 0 . 1 0 0 0 0 . 2 0 0 0 0 . 2 5 0 0 0.2500 I n i t i a l Step Size 0.1000 OolOOO 0.1500 0.1500 0.1500 0.2000 X l 0 . 8 0 0 0 0 . 8 0 0 0 0 .8000 0 . 8 0 0 0 0.8000 0.5100 S t a r t i n g X 2 0 .2000 0.2000 0,2000 0.2000 0,2000 0.4900 Variables X3 0 . 8 0 0 0 0 . 8 0 0 0 0.8000 0 . 8 0 0 0 008000 0.5100 X4 0 . 2 0 0 0 0 . 2 0 0 0 0 . 2 0 0 0 0,2000 0 . 2 0 0 0 0 . 4 9 0 0 X l 0.4667 0,3844 0.3868 0.3551 0.3646 0.3548 0.3541 F i n a l X 2 0.1565 0.1046 0.1014 0.1023 0.1020 0.1031 OolOOO Variables h 0.3899 0.3291 0.3276 0.3548 0.3451 O o 3 5 4 l 0.3541 x 4 0.1467 0.0971 0.1004 0 . 0 9 8 6 0 . 0 9 8 6 0.0973 0.1000 L/G 2.0380 2.5416 2.5454 2.5341 2.5376 2.5383 2,5410 Time (sec.) 6 5 . 6 99.6 80.4 73.5 108 ? 4 9 2 . 1 1.3 TABLE 2. Pour Stage System A - Conjugate D i r e c t i o n Method Preundlich exponent = 1 .0. Penalty factors increase with each i t e r a t i o n . D i f f e r e n t end tolerances used. I n i t i a l P i n a l X 0 1 ~ x 0 2 ~ O o ° Y P - 0.1 L01 Y 0 2 = l o ° I t e r a t i o n P* time Start 1 2 I I 7 101.0444 23.3618 5.1705 2.89820 2.59301 2.56189 2.56506 2.55866 0.40000 0.70008 0.50429 0.34915 0.33728 0.33628 0.33628 0.33611 0.20000 O.36728 0.18060 0.11800 O.IO749 0.10661 0.10660 0.10650 0.40000 0.70043 0.46426 0.39351 0.37351 0.37177 0.37176 0.37155 0.20000 0 .37317 0.17954 0.10044 0.09355 O.O9295 0.09295 0.09286 32.38 sec, Start 1 2 3 4 5 6 101.04444 23.34164 5.17166 2.87804 2.58010 2.55011 2.54746 0.40000 0.70071 0.50608 0.37632 0.36511 0.36389 0.36369 0.20000 0.36772 0.18068 O.II583 0.10654 0o10556 Oo10541 0.40000 0.69997 0.46279 0.36117 0.34536 0.34367 0.34341 0.20000 0.37310 0.17959 0.10055 0.09468 0.09403 0.09393 32.05 sec, Start 1 2 \ \ I 101.04444 23.34069 5.16573 2.86796 2.57100 2.54491 2.54991 2.58468 2.89342 9.72685 0.40000 0.70327 0 .48847 0.37215 0.35988 0.35735 0.35737 0.35768 0.35785 0.35796 0.20000 0.37079 0 .17921 Oo10673 0 .10023 O.O9859 0.09846 0.09859 O.O9877 O.O9885 0.40000 0.70022 0.48567 0.56702 0.35451 0.55154 0.35102 0.55H5 0.55136 0.55147 0.20000 0.57327 0.18509 0.11082 0.10308 0.10163 0.10161 0.10166 0.10169 0.10172 90.5 sec. Start 1 2 I 5 6 101.04444 23.54085 5.16540 2.86755 2.57105 2o54473 2.57575 3.27035 12.44869 0.40000 0.70235 0 .48447 0.57118 0.55869 0.55551 O.556OQ 0.55644 0.55670 0.20000 0.57056 Oo 17958 0.10822 0.10085 0.09925 0.09943 0.09957 0.09962 0.40000 0.69955 0.48250 0.36715 O.55445 0.55519 0.55545 0.55360 0.55566 0.20000 0.57175 0.17968 0.10875 0.10179 0.10100 0.10122 0.10135 0.10145 65*5 sec. *P i s the constrained objective function (= L/G plus constraints) TABLE 3. General Two Stage System F r e u n d l i c h Isotherm Used I n i t i a l Y A, = 1.0 Y r t o = 1.0 X~, = 0 F i n a l Y p = 0.1 n 1.0 0.4 X x 0.09999 0.00316 x 2 0..354OS 0.02326 k x 0.99994 0.99973 k 2 0.99995 0.99988 3 X 0.00006 0.00008 3 2 0 . 0 0 0 2 7 0.00021 Y X 0.09999 0.10002 Y 2 0.35406 0.22216 ( L / G ) 1 2.54149 38.62495 ( L / G ) 2 2.54149 38.69969 L/G 2.54204 38.68355 TABLE 4 . G e n e r a l Two Stage System K o b l e - C o r r i g a n I s o t h e r m Used I n i t i a l Y Q 1 = 1 . 0 P i n a l n m (3, Y p = 0 . 1 0 . 3 1.7 1.00000 L 02 = 1 . 0 xQ1 = 0 . 0 0 . 2 2.0 1.00000 1.00000 1.00000 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0.00000 0 . 2 1.9 0.00318 0.00033 0.00026 0.02959 Oo00438 0.00342 1.00000 1.00000 0.00000 0.00000 Y 1 0.100000 0.10000 0.10000 Y 2 O . 1 9 6 6 5 0.16784 0.16781 (L/Q)1 30.42287 205.7613 263.1578 ( L / G ) 2 30.42287 205.7613 263.1578 L/G 30.42055 205.62225 263.26908 TABLE 5. 1.6 Two Stage Countercurrent F r e u n d l i c h Isotherm Used (The Values marked w i t h a * are due to Lerch) I n i t i a l Y„, = 1 . 0 X r t n = 0 . 0 F i n a l Y = 0.1 h X l X 2 Y l Y 2 L/G 3.0 0.84170 Oo46416 0.59631 0.10000 1 . 0 6 9 2 7 2.8 0.82336 0.43940 0.58030 0.10000 1 . 0 9 3 0 8 2.6 0.80176 0.41246 0 . 5 6 3 0 0 0.10000 1 . 1 2 2 5 3 2,>4 0.77611 0 . 3 8 3 1 2 0.54427 0.10000 1 . 15963 2.2 0.74542 0.35H2 0.52393 0.10000 1 . 2 0 7 3 8 2.0* 0 . 7 0 8 4 0.3162 0.5018 0.10000 1 .271 1.8* 0 . 6 6 3 3 0 . 2 7 8 3 0 . 4 7 7 6 0.1000 1 . 3 5 7 1 . 6 * 0.6080 0 . 2 3 7 1 0.4510 0.1000 1.480 1.4* 0 . 5 3 9 9 0.1931 0.421.9 0.1000 1 . 6 6 7 1.2* 0.4560 0 . 1 4 6 7 0 . 3 8 9 7 0.1000 1 .974 1.0* 0.3541 0.1000 0.3541 0.1000 2 . 5 4 1 0.8*', 0.2357 O .0562 0.3147 0.1000 3 . 8 1 8 0.6* O . I I 3 4 0.0215 0 . 2 7 0 9 0.1000 7 . 9 3 4 0.4* 0.0233 0.0032 0*2222 Oo10000 3 8 . 6 5 5 0.3 0.00436 0.00046 0.1958J. 0.10000 206.42484 0.2 0.00014 0.00001 O . I 6 9 7 6 0.10000 6 3 3 9 . 2 8 3 1 2 TABLE 6 . Two Stage C r o s s f l o w - A l g o r i t h m S o l u t i o n F r e u n d l i c h I s o t h e r m Used I n i t i a l Y A, = 1 .0 X f t, = 0 . 0 Xn„ = 0 . 0 F i n a l Y F = 0 . 1 n X l X 2 L/G 3/Cp 0.72146 0 .46416 1 .45920 2.8 0.70315 0 .43940 1.51302 2 .6 O.68236 0 .41246 1 .57809 2.4 O.65854 0.38312 1 .65809 2 . 2 0 .63101 0 .35112 1.75873 2 . 0 0.59891 0 .31623 I . 8 8 8 8 5 1.8 O . 5 6 I I O 0 .27826 2.06306 1 . 6 O.51606 0 .23714 2 .30692 1 . 5 0 .49025 0.21544 2.46874 1 .4 0.46186 0.19307 2 .66930 1 . 3 0 .43059 0.17013 2 .92362 1 . 2 0.39610 0.14678 3.25475 1 . 1 0.35807 0.12328 3 .70016 1.0 0.31623 0.10000 4.32456 0 . 9 0.27048 0.07743 5.24724 0 . 8 0 .22106 0.05623 6.70930 0 . 7 0.16893 0 .03728 9.25827 0 . 6 0.11624 0.02154 14.35702 0 . 5 0.06717 0.01000 26 .94636 0 . 4 O .02820 0.00316 71 .20493 0 . 3 0.00607 0.00046 379.57169 TABLE 7 . 1.8 Three Stage Crossflow - Al g o r i t h m S o l u t i o n F r e u n d l i c h Isotherm Used I n i t i a l Y 0 1 = 1.0 x01 = 0 . 0 x02 = 0 . 0 F i n a l Y F = 0 . 1 n X l X 2 H L/G 3 . 0 0.81507 0.63564 0.46416 1 . 3 4 8 2 7 2.8 0.80170 0.61386 0c43940 1.39028 2 . 6 0.78630 0.58943 Oo 4 1 2 4 6 1.44085 2.4 0.76836 O .56185 0.38312 1.50276 2 . 2 0.74722 0.53055 0 . 3 5 H 2 1.58022 2 . 0 0.72199 0.49478 0.31623 1.67976 1.8 0.69139 0 . 4 5 3 6 8 0.27826 1.81199 1.6 0.65362 0.40619 0.23714 1 . 9 9 5 4 2 1.5 0.63127 0.37970 0.21544 2.11616 1.4 0 . 6 0 6 0 6 0.35119 0.19307 2.26494 1.3 0 . 5 7 7 4 4 0 . 3 2 0 5 4 0.17013 2 . 4 5 2 1 7 1.2 0 . 5 4 4 8 0 0.28767 0 . 1 4 6 7 8 2.69402 1.1 0.50734 0.25258 0.12328 3.01631 1 .0 0.46416 0.21544 0.10000 3 . 4 6 3 3 1 0 . 9 0 . 4 1 4 2 3 O.17665 0 . 0 7 7 4 3 4.11549 0 . 8 0.35657 0.13702 O .05623 5.13341 0 . 7 0.29053 0.09799 0.03728 6.87596 0 „ 6 0.21674 0.06184 0.02154 10.28372 0 . 5 0.13886 0.03180 0 . 0 1 0 0 0 18.46101 0 . 4 0.06664 0.01132 Oo00316 46 .15715 0 . 3 0.01708 0.00190 0 . 0 0 0 4 6 229.56334 TABLE 8. Three Stage Countercurrent Freundlich Isotherm Used I n i t i a l X Q = 0 . 0 Y Q = 1 .0 F i n a l Y p = 0 . 1 n 3.0 2.6 2.2 1.8 1.6 1.4 1.2 X x 0.95373 0.93379 0.90110 0.84442 0 .80052 0.74026 0.65687 X 2 0.81332 0.76453 O . 6 9 6 I O 0.59820 0.53410 0.45761 0.36779 X ^ 0.46416 0.41246 0.35112 0.27826 0.23714 0.19367 0.14678 Y x 0.86751 O . 8 3 6 8 6 0.79525 0.73757 0.70048 0.65636 0 .60392 Y 2 0 .53801 0.49754 0.45061 0.39657 0.36661 .0.33473 0.30111 Y^ 0.10000 0.10000 0.10000 0.10000 Oo10000 0 .10000 0.10000 L/G 0.94367 0.96381 0.99878 1 .06582 1.12427 1.21578 1.37013 Continued TABLE 8. (CONTINUED) Three Stage C o u n t e r c u r r e n t F r e u n d l i c h I s o t h e r m Used 1.10 I n i t i a l X 0 = 0 . 0 Y r t = 1.0 F i n a l v 0 F 0 . 1 n 1.0 0.8 0 . 6 0.5 0.4 0.3 X± 0.54191 O.38856 0.20512 O . H 6 7 6 0.04733 0.00952 x 2 0.26608 0.15950 0.06531 0.03136 0.01026 0.00156 X^ 0.10000 0.05623 0 .02154 0.01000 0.00316 0.00046 Y 1 0.54191 0.46943 O.38655 0.34170 0.29515 0.24753 Y 2 0.26608 0.23025 0.19453 0 .17708 0.16014 0.14388 Y^ 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 L/G 1 .66080 2.31622 4.38763 7.08010 19.01685 91.50349 1.11 TABLE 9- Three Stage System A Preundlich Isotherm Used I n i t i a l Y Q 1 = 1 .0 Y Q 2 = 1 .0 X Q 1 = 0 . 0 X Q 2 = 0 . 0 P i n a l Y p = 0 . 1 n 3 . 0 2.6 2 . 2 2 . 0 1.8 1.6 1.4 X x 0.43813 0.37943 0.31740 0.27268 0.23849 0.20247 0.16378 X 2 0.75904 0.71776 0.66244 0.62876 0.58743 O.53861 0.47946 X 3 0.48394 0.43341 0.36948 0.33590 0.29493 0.25051 0.20324 Y x 0.08410 0.08049 0 .08008 0.07436 0 .07576 0.07765 0.07943 Y 2 0.43732 0.42222 0.40413 0.39534 O.38381 0.37156 0.35732 Y^ 0.1133^ 0.11375 0.11187 0.11283 0.11105 0.10917 0.10745 t s l / G s l 2 .09047 2.42341 2.89829 3.39454 3.87551 4.55560 5.62081 L g l / G s 2 1.75334 1.70773 I . 7 2 6 9 6 1 .69808 I .76587 1 .86957 2.03537 L s 2 / G g 2 0 .66948 0.71173 0 .79100 0 .84107 0 .92484 1.04744 1.22939 L/G 1.31767. 1.41931 1.57782 1.69251 1.84845 2.06823 2.39705 Continued.... 1.12 TABLE 9- (CONTINUED) Three Stage System A F r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q 1 = 1.0 Y Q 2 = 1.0 X Q 1 = 0.0 X Q 2 = 0.0 F i n a l Y p = 0.1 n X, X- 1.2 1.0 0.8 0.6 0.5 0.4 0.3 0.12400 0.08415 0.04718 0.01806 0.00839 0.00266 0.00039 0.40835 0.32271 0.22265 0.11508 0.02739 0.02739 0.00692 0.15376 0.10414 0.05814 0.02209 0.01020 0.00322 0.00047 0.08168 0.08415 0.08690 O.08995 0.09161 0.09161 0.09507 0.3^138 0.32271 0.30067 0.27325 0.25655 0.23714 0.21372 0.10573 0.10414 ,0.10276 0.10150 0.10100 0.10068 0.,10031 L s l / G s l L s l / G s 2 7.40576 10.8837 19.3498 50.4032 103.225 340 .136 2325-58 2.31620 2.8391 3.9857 7.4906 12.947 30.855 144.51 1.53259 2.0933 3.4033 7.7767 15-246 42.427 241 .55 L/G 2.93187 3.91637 6.12855 13.2913 25.1315 67.1987 363.461 TABLE 10. Three Stage System E F r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q 1 = 1.0 Y Q 2 = 1 . 0 X Q ] = 0.0 X Q 2 = 0 . 0 F i n a l Y p = 0.1 n 3.0 2 .6 2.2 2.0 1.8 1.6 1.4 X l 0.74603 0.70617 0.65314 O.61962 0.57931 0.53232 0.47458 x2 0.37246 0.33950 0. 29140 O.26205 0.23051 O.19630 0.16020 X 3 0.62704 0.56921 O.49718 O.45500' 0.40754 0.35444 0.29518 Y l 0.41521 0.40472 O.39176 0.33392 0.37490 0.36466 0.35224 Y 2 0.05167 0.06028 O.O6636 0.06867 0.07126 0.07390 0.07701 Y 3 0.24654 0.23105 0.21494 0.20703- 0.19875 0.19022 0.18119 W G s l O.78387 0.84297 0.93125 0.994 23. 1.07812 1.19353 1.36490 L s 2 / G s l O.97605 1.01456 1.11669 1.20303 1.31723 1.48117 1.71805 L s 2 / G s 2 2.95953 3.34749 3.81507 4.10964 4.52610 5.12058 6.06621 L/G 1.32345 1.42549 1.58428 1.69975 1.85538 2.07461 2.40252 Continued TABLE 10. (CONTINUED) I n i t i a l F i n a l Three Stage System B F r e u n d l i c h I s o t h e r m Used Y 0 1 = 1 ' ° Y 0 2 = 1 ' ° X 01 - ° ' ° X 0 2 = °-° 0.1 n L s l / G s l L s 2 / G s l L s V G s2 1.2 1.0 O.4058Q 0.32280 0.12233 0.08409 0.23005 0.16109 0.33391 0.32230 0.08036 0.08409 0.17147 0.16109 1.62874 2.09789 2.11353' 2.83938 7.68112 10.8920 0.8 0.6 0.5 0.4 0.3 0.22443 0.11751 0.06690 0.02331 0.00602 0.04798 O.OI838 0.00827 0.00287 0.00042 0.09335 0.03774 0.01872 0.00555 O.OOO87 0.30265 0.27672 O.25865 0.24031 0.21572 0.08307 0.09090 0.09096 0.09622 0.09708 0.15000 0.13998 0.13684 0.12523 0.12057 3.10645 6.15521 11^0816 26.8312 130.264 4.47168 10.1124 20.2666 50.1756 282.136 13.7195 44.4134 82.5970 326 .797 1960.78 L/G 2.93559 3.91637 6.11352 13.2506 25.1718 66.7575 360.841 TABLE 11 Three Stage System C P r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q 1 = 1.0 Y Q 2 = 1.0 X Q 1 = 0.0 X Q 2 = 0.0 P i n a l Y„ = 0.1 F n 3 . 0 2 .6 2 . 2 2 .0 1.8 1.6 1.4 X± 0.00660 0.00310 0.00025 0.00021 0.00011 0.00015 0.00011 X 2 0.73175 0.69150 O.63651 0.59947 0.56557 O .516H 0."46208 X ? 0.46481 0.41280 0 . 3 5 H 5 0.31626 0 .27327 0.23716 0.19309 Y1 0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Y 2 0.39183 0.38323 0.3701-'! 0.359 3 6 0.35849 0.3470 5 0.33931 0 .10042 0.10021 0.10002 0.10002 0.10001 0.10002 0.10001 L o/G - 151.515 322.580 4000 .00 4761.90 9090.90 6666.67 9 0 9 0 . 9 0 S c. S i L , / G 0 0.83112 0.89193 0.98956 1.06368 1.13427 1.26515 1.42981 S J. E c L „ o / G 0 0.63597 0.69079 0.76979 0.82059 0.92925 1.04227 1 .24001 S d. S c. L / G 1.46096 1.57933 1.75901 1.88894 2.06330 2.30706 2.66948 C o n t i n u e d . TABLE 11. (CONTINUED) Three Stage System C F r e u n d l i c h Isotherm Used I n i t i a l Y Q 1 = 1 .0 Y Q 2 = 1 .0 X Q 1 = 0 . 0 X Q 2 = 0 . 0 F i n a l Y p = 0 . 1 n 1.2 1.0 0 . 8 0 . 6 0 . 5 0 . 4 0 . 3 x x 0.00009 0 . 0 0 0 0 6 0 . 0 0 0 0 3 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0.00000 0 . 0 0 0 0 0 X 2 0 .39619 0.31623 0 .22045 0.11623 0.06718 0.02823 0 . 0 0 6 0 5 X^ 0.14680 0.10001 0.05624 0.02155 0.01000 0.00317 0 .00046 Y X 0 . 0 0 0 0 1 0 . 0 0 0 0 6 0 .00027 0 .00122 0.00241 O.OO526 0.01434 Y 2 O.32922 0.31623 0 .29830 0 .27492 0.25920 0.24005 0.21604 Y ^ 0.10001 0.10001 0.10001 0.10001 0.10001 0.10008 0.10003 L s 2 / G s l 16666.7 33333.3 100000. 100000. ___*___ *These f i g u r e s exceeded the output format L s l / G s 2 1.69307 2 .16225 3.18299 6.23830 I I . 0 2 6 5 26.9179 129 .533 L s 2 / G s 2 1.56232 2.16314 3.52771 3 .12215 15.9261 44.1391 250.ooq i L/G 3 . 2 5 4 9 3 4.32483 6.70987 14 .3586 26.9493 71.1088 379-556 o> TABLE 12. Three Stage System D F r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q 1 = 1 . 0 Y Q 2 = 1 . 0 X Q 1 = 0 . 0 X Q 2 = 0 . 0 F i n a l Y p = 0.1 n 3.0 2.6 2.2 2.0 1.3. 1.6 1.4 X± 0.6110 5 0.60231 0.56919 0 . 59 562 0 . 54669 O.51560. 0.44863 X 2 0.39116 0.36421 0.31416 0.31438 0.26846 0.23707 O.19265 X^ 0.69324 0.66340 O.61587 0.59830 0.59994 0.51570 0.44924 Y x 0.21118 0.26763 0.28945 0.35477 0.33724 0.34650 O.32557 Y 2 0.05985 0.07236 0.07829 0.09884 0.09375 0.09995 0.09970 Y^ 0 . 3 4 0 4 3 0.34405 0.34426 0.35796 0.35209 0.34661 0.32619 L s l / G s l 1.26315 1-21593 1.24836 I.O8329 1.21232 1.26745 1.50330 L s 2 / G g l 0.43026 0.53614 0.67212 0.8l408 0.90698 1.03997 1.17248 L s 2 / G s 2 7.56428 1 0 . 7 3 7 1 1 4 . 0 4 7 0 2 3 9 - 8 o 8 43.9017 6700.00 1114. 1 1 L/G I.45109 1.57384 1.76374 1 . 88884 2.06803 2.30699 2.67217 Continued. H TABLE 12. (CONTINUED) Three Stage System D F r e u n d l i c h I s o t h e r m Used I n i t i a l Y = 1.0 Y Q 2 = 1.0 X Q 1 = 0 . 0 X Q 2 = 0 . 0 F i n a l Y p = 0 . 1 n 1.2 1.0 0 . 8 . 0 .6 0 . 5 0.4 0 . 3 X x 0.39590 0.31603 0 . 2 2 0 9 6 0.11622 0.06712 0.02827 0.00604 X 2 0.14675 0.09998 0.05623 0.02154 0.01000 0.00317 0.00046 X 5 0.39590 0.31606 0.22097 0.11623 0.06712 0.02827 0.00604 Y x O.32893 O.31603 0 . 29884 0 . 27490 0 . 2590 7 0 . 24019 0 . 21595 Y 2 0 .09997 0.09998 0.09999 0.09999 0.09999 0.10006 0.10001 Y ? 0.32898 0.31606 -O.29886 0.2749 1 0.25908 0.24019 0.21595 L 1.69505 2.16422 3.17228 6.23833 11.0393 26.8744 129.776 s i s i L s 2 / G g l I . 5 6 O I 8 2.16094 3.53669 8.12033 15.9103 44.2477 249.700 L,/G „ 14000.0 25000.0 50000.0 89314.8 188745. ------- ---- *These figures exceeded * d the output format. L/G 3.25486 4.32475 6.70954 14.3582 26.9481 71.1232 379-465 CO 1.19 TABLE 13. C o m b i n a t i o n o f Three Stage Systems B and D Streams from Stages 1 and 2 a r e J o i n e d f o r s t a g e 3 F r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q 1 = 1.0 Y Q 2 = 1.0 X Q 1 = 0.0 tXQ2 = 0.0 F i n a l Y p = 0.1 n 1.0 *?2 X- X, 0.32256 0.08403 0.16113 0.00010 0.32256 .0.08403 0.16118 W G s l L s 2 / G s l L s 2 / G s 2 2.10013 2.33849 10.87547 L/G 3.91651 1.20 TABLE 14. Pour Stage C r o s s f l o w - A l g o r i t h m S o l u t i o n F r e u n d l i c h I s o t h e r m Used I n i t i a l Y 0 1 = 1.0 X 01 = ° ' ° X 0 2 = ° - ° X 0 3 = ° J F i n a l Y F = 0 . 1 n X l X 2 X 3 X4 L/G 3 . 0 0 . 8 6 2 0 5 0.72624 0.59326 0.46416 1.29968 2.8 0.85164 0.70807 0.57021 0.43940 1.33668 2 . 6 0.83957 0.68738 O.54453 0.41246 1.38110 2.4 0.82533 0.66362 O.51578 0.38312 1.43535 2.2 0.80851 0.63612 0.48344 0.35112 1.50309 2.0 0.78814 0.60396 0.44690 O.31623 1.58987 1.8 0.76306 0 . 56 592 0.40543 O.27826 1.70471 1.6 0.73154 0.52044 0.35828 0.23714 1.86336 1-5 0.71258 0.49428 0.33234 0.21544 1.96740 1.4 O.69089 0.46544 0.30471 0.19307 2.09519 1.3 O.66588 0.43356 0.27537 0.17013 2.25549 1.2 0.63682 0.39830 0.24433 0.14678 2.46174 1.1 0.60272 0.35928 0.21172 0.12328 2.73546 1 .0 0.56234 0.31623 0.17783 0.10000 3.11312 0 . 9 0.51412 0.26903 0.14317 0.07743 3.66094 0 . 8 0.45611 0.21798 0.10864 0.05623 4.50997 0 . 7 0.38622 0.16422 0.07564 0.03728 5.95130 0 . 6 0.30289 0.11031 0.04618 0.02154 8.74140 0.5 0.20729 0.06110 0.02277 0.01000 15.35217 0.4 0.10863 0.02374 0.00768 0.00316 37.38514 0 . 3 0.03126 0.00441 0.00120 0.00046 180.26419 0 xQh = 0 . 0 1.21 TABLE 15. ; 1 Four Stage C o u n t e r c u r r e n t F r e u n d l i c h I s o t h e r m Used I n i t i a l Y Q = 1 .0 X Q = 0 . 0 F i n a l Y„ = 0 . 1 n 3 . 0 2.6 2 . 2 1.3 1.4 1.2 X l 0.98629 0.97751 0.96076 0.92622 0.84888 0.77679 X 2 0.94134 0.91516 0.87078 0.79361 0.65555 0.55110 h 0.80597 0.75390 0.68060 0.57592 0.42789 0.33591 H 0.46416 0.41246 0 . 3 5 H 2 0.27826 0.19307 0.14678 Y i 0.95943 0.94259 0.91571 0.87114 0.79503 0 .73852 Y 2 0.83546 0.79412 0.73756 0.65961 0.55366 0.48919 h 0.52355 O.47976 0.42891 0.37033 0.30470 0.27006 Y4 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 L/G 0.91249 0.92209 0.93675 0.97169 1.06023 1.15862 C o n t i n u e d TABLE 15. (CONTINUED) Four Stage Countercurrent Freundlich Isotherm Used I n i t i a l Y Q = 1.0 X Q = 0 . 0 F i n a l Yra = 0 . 1 n 1 .0 0.8 0.6 0 . 5 0 . 4 0.3 X l 0.66549 O.49920 0.27753 0.16243 0.06774 0.01403 X 2 6.41814 0.26270 0.11207 0.05468 0.01811 0.00278 X 3 0.23524 0.13491 0.05210 0.02415 0 . 0 0 7 6 0 0.00111 0 . 1 0 0 0 0 O.05623 0.02154 0.01000 0.00316 0.00046 Y l 0.66549 0.57362 0.46343 0.40301 0.34066 0.27805 Y 2 0.4l8l4 0.34322 0.26896 0.23384 0.20100 0.17100 h 0.23524 0.20138 O .I6987 0.15541 0.14202 0.12978 Y 4 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 L/G 1.35239 1.80287 3.24293 5.54132 13.28698 64.14546 1.23 TABLE 16. Pour Stage System A F r e u n d l i c h I s o t h e r m Used I n i t i a l Y 0 1 ~ 1 * 0 Y 0 2 = 1.0 x Q 1 = 0.0 X 0 2 = ° F i n a l Y p = 0. 1 n 210 ) 1.8 1.6 1.4 1.2 : : X 1 0.70837 0.66392 0.60803 O.55283 0.45611 X 2 0.31632 0.27833 0.23695 0.19446 0.14689 h 0.70836 0.66262 0.60792 0.52706 O.45586 0.31613 0.27319 0.23733 0.19163 0.14667 Y i 0.50178 0.47841 0.45111 0.43614 0.38984 ' Y 2 0.10006 0.10005 0.09987 0.10101 0.10009 h 0.50178 0 . 4 7 6 7 3 0.45097 0.40795 O.38958 Y 4 0.09994 0.09995 0.10013 O.09896 0.09991 W ^ l 1.27019 1.35222 1.43069 1.56017 1.97182 L s l / G s l 1.26998 1.35939 1 .43234 1.72339 1.97256 W G s 2 1.27084 I .36165 I.47998 1.73007 1.97566 L S 2 / G S 2 1.27112 1 . 35440 1.47828 1.61243 1.97498 L/G 1.27053 1.35692 1 . 4 3 0 3 3 1.66881 1.97375 C o n t i n u e d . , . . . 1.24 TABLE 16. (CONTINUED) Four Stage System A Freundlich Isotherm Used I n i t i a l Y01 - 1 ' ° Y02 - 1.0 x Q 1 = 0.0 X02 = °' 0 F i n a l Y p - 0 . 1 n 1.0* 1.0* 1.0* 1.0* 0 .8 0 . 6 X l 0.35320 0.35462 0.35394 0.35421 0.23551 0.11372 X2 o.09987 0.09999 0.09999 0 .10001 O.05618 0.02164 h 0 .35503 0.35366 0.35434 0.35407 0.23592 0.11319 x 4 0 .10013 0.10001 0.10001 0 .09999 O.05628 0.02145 Y l 0 .35320 0.35462 0.35394 0.35421 0.31449 0.27133 Y2 0 .09987 0.09999 0.09999 0.10001 0.09994 0 .10028 h 0 .35508 0.35366 0.35434 0.35407 0.31493 0.27058 Y 4 0.10013 0 .10001 0 .10001 0.09999 0 .10006 0 .09975 W G S 1 2.55581 2.53478 2.54404 2.54028 3.82467 7.89764 L s l / G s l 2.53659 2.54655 2.53975 2.54175 3.81810 7.90263 L s l / G s 2 2 .52702 2.54796 2.53847 2.54243 3.81170 7.96749 L s ^ G s 2 2.54619 2.53625' 2.54325 2.54105 3.81825 7.96241 L/G 2.54139 2.54138 2.54138 2.54138 3.8l8l6 7.93237 *Test solutions for Conjugate D i r e c t i o n method. 1.25 TABLE 17. Four Stage System B F r e u n d l i c h I s o t h e r m Used I n i t i a l Yoi ~ -1* 0 Y 0 2 = 1.0 x01 = = 0.0 X 0 2 = °- 0 F i n a l 1 n 3.0 2 . 6 2.2 1.8 1.4 1 . 0 X l 0 . 6 3 6 5 7 0.59419 0.54007 0. 46779 0 . 3 7 1 0 4 0. 23331 X 2 0.31504 0.26937 0 . 2 2369 0. 17184 0 . 1 1 6 3 4 0. 05702 X 3 0.83753 0.80307 0.76300 0.69557 0.58875 0 . 4 1 8 9 9 X 4 0.60456 0 . 5 4 5 3 6 0 . 4 7 2 4 3 0. 3 8 1 7 5 0 . 2 7059 0. 14324 Y l 0.25795 0.25835 0 . 2 5 7 8 6 0. 25474 0.24957 0. 2 3 8 3 1 Y 2 0.03127 .0.03303 0 . 0 3709 0. 04200 0 . 049 21 0. 0 5 7 0 2 h 0.53760 0.57460 0.55151 0. 52025 0.47633 0. 4l890 Y 4 0.22097 0.20672 0 . 1 9 2 1 1 0. 17668 0 . 1 6 0 4 1 0. 14324 W G s l I.I657O 1.24315 1.37415 1. 59314 2 . 02249 3. 1 9 6 2 2 L s 2 / G s l 0 . 7 1 9 5 3 0 . 8 3 6 4 8 0 . 9 8 6 9 5 1. 23805 1 . 72217 3. 1 7 9 4 1 L s l / G s 2 2 . 0 5 1 6 1 I .98898 2 . 0 1178 2. 10619 2.40534 3. 21569 L s 2 / G s 2 I .26635 1 . 33294 1 . 44494 1. 63674 2 . 0 4817 3. 19714 L/G 1.20217 I .28085 1 . 40288 1. 61192 2 . 0 3 4 2 2 3. 19733 C o n t i n u e d TABLE 17.-(CONTINUED) Pour Stage System B F r e u n d l i c h I s o t h e r m Used I n i t i a l Y 01 = 0 Y 0 2 = 1.0 x 0 1 = 0.0 x 0 2 = o F i n a l Y p = 0. 1 n 0.8 0.6 0.5 0.4 0.3 X l 0.16209 0.08062 0.04655 0.01819 0.00377 0 . 0 3 4 0 1 0 . 0 1 2 4 8 0.00584 0.00172 0.00025 X 3 0.29073 0.15712 0.09327 0.03924 0.00873 x 4 0.08120 0.03138 0.01460 0.00463 0.00068 Y l 0.23324 0.22073 0.21576 0.20135 0.18741 Y 2 0.06688 0.07208 0 . 0 7 6 4 0 0.07347 0.08249 h 0.37222 0.32941 0.30541 0.27523 0.24113 Y 4 0.13416 0.12532 0.12082 O . H 6 5 1 0.11215 L s l / G s l 4.73059 9 . 666 se  1 6 . 8 4 7 1 43.8981 215.715 L82^ 08l 4.88615 ll.9 0 7 5 23 .8748 71 .2250 420 . 9 0 6 L s l / G s 2 4 . 8 7 9 7 1 8.76577 14 .8659 33.6360 153.032 L s 2 / G s 2 5.04489 1 0 .7979 21.0671 54.5553 298 . 593 L/G 4.88510 1 0 .2599 19 .0890 49.9436 264.201 1.27- TABLE 18. Pour Stage System C Freundlich Isotherm Used I n i t i a l Y 0 1 = 1,0 Y Q 2 = 1.0 X Q 1 = 0.0 X Q 2 = 0.0 F i n a l Y p =0.1 n 3.0 2.6 2.2 1.8 1.4 1.0 X l 0.44077 0.46250 0.54338 0.48971 0.38599 0.25371 X 2 0.73662 0.68899 0 .74031 0.68228 0.56404 0.39242 0.45652 0 .42968 0.39185 O .31623 0.22307 0.11839 x 4 0.47139 O .38501 0.25769 0.19904 0.13976 0.07262 Y l O.O8563 0.13467 0.26135 O .27663 0.26376 0.25371 Y 2 0 .39970 0.37962 0.40694 0 .50249 0.44858 0.39242 h 0.09514 • 0.11122 0.12731 O.12589 0.12241 O . I I 8 3 9 Y4 0.10475 O . 0 8 3 6 I 0.05063 O .05471 O.O636I 0 .07262 W G s l 2.07477 1 .87097 1.359 36 1.47712 1.90745 2.94151 L s 2 / G s 2 2 .02905 2 .73905 2.45736 2.58357 3.09693 4.38022 L s l / G s l 0.63970 0.52519 0.99913 I . 2 8 6 2 3 1.69649 2.95652 L s 2 / G s 2 O .62569 O .76885 1.80619 2.24971 2.75444 4.40622 L/G 1.34207 1.42369 1.51850 1.75816 2.23029 3.52885 Continued TABLE 18. (CONTINUED) Four Stage System C Preundlich Isotherm Used. I n i t i a l Y o i - i - 0 Y 0 2 = 1.0 x 0 1 = 0 . 0 X 0 2 = < F i n a l Y F = 0 . ,1 n 0.8 0.6 0.5 0.4 0 . 3 x i 0.17550 0.09228 0.05333 0.02171 0.00481 X 2 0.28037 0.15014 0 . 0 8 8 2 2 0.03707 0.00809 X 3 0.06771 0.02654 0.01247 0.00399 0.00060 x 4 0.04072 0.01517 0.00700 0.00224 0.00031 Y l 0.24885 0.23936 0.23093 0.21610 0.20162 Y 2 0.36156 0.32055 0.29702 0.26796 0.23572 h 0.11601 0.11333 0.11168 0.10974 0.10785 0.07724 0 . 0 8 1 0 2 O.O8369 O.O8718 0.08902 W Q 8 1 4.28192 8.24334 14.4208 36.1141 166.113 L S l / G S 2 6 .08753 11.7426 20.1470 47.6644 232.730 W°sl 4.91136 11.0840 21.8048 60.9384 332.405 W G S 2 6 .98275 15.7903 30.4631 80.4505 465.719 I/O 5.39708 11.3557 21.1132 55.2124 290.892 1.29 TABLE 19. Pour Stage System D F r e u n d l i c h Isotherm Used I n i t i a l Y 0 1 - X - 0 Y 0 2 = 1.0 x Q 1 = 0 . 0 X 0 2 * ° ' ,0 P i n a l Y p = 0 0 ,1 n • 3 . 0 2.6 2 .2 1.8 1.4 1.0 0.59502 0.57171 0.50460 0.42470 O.29692 0.17512 x2 0.21381 0.30613 0.25256 0.19321 0 . 1 1 7 9 7 0.07211 0.62500 0.59456 0.52526 0.43907 0.31612 0.18049 0.67508 0.66699 0.60469 0.50967 0.42809 0.25342 0.21067 0.23370 0.22207 0.21413 O.18268 0.17512 Y 2 0.00977 0 . 0 4 6 0 6 0 . 0 4 8 4 4 0.05186 O.05017 0.07211 h 0 . 2 4 4 1 4 0.25877 0.24257 0.22728 0.19943 0.18049 0.30765 0.34891 0.33065 0.29725 0.30489 0.25342 W G s l 1.32655 1.34035 1.54168 I . 8 5 O I 3 2.75269 4.71031 L S 2 / G B 1 0.93958 0.61295 0.68749 0.83982 1.12325 1.42853 W Qs2 2.11915 3.94535 4.26261 4.89214 5.49285 13.5759 W Gs2 1.50098 1.80424 1.90086 2.22067 2.24138 4.11760 L/G 1.39370 1.45798 1.63708 1.95181 2.58199 4.55766 Continued TABLE 19. (CONTINUED) Pour Stage System D Preundlich Isotherm Used I n i t i a l Y o i - i -•° Y02 = 1.0 X 01 = °'° X02 = C P i n a l Y p = 0. ,1 n 0 .8 0.6 0 .5 0.4 0 . 3 X l 0 .07955 0.02402 0 .05211 0 .00749 0.00000 x2 0 .03306 0.00674 0.00771 0.00133 0.00000 h O.O9234 O.O3285 0.05136 0 .00766 0.00046 X4 0.26427 0.14753 0.01241 0.01213 0 . 0 0 7 9 2 Y l 0.13198 0.10676 0 .22829 0.14117 0.01023 Y 2 O.Q6537 0.04980 0.08779 0.07071 0 . 0 0 3 0 2 h 0.14869 0 .12881 0.22664 0.14246 0.10002 0.34486 0.31720 0.11140 0.17119 0.21642 L s l / G s l 10.9115 37.1804 14.8079 114.701 * W G s l 2.01500 8.45133 18.2289 53.0049 L s l / G s 2 15.3437 21.3355 153.489 166.130 250.760 L s 2 / G s 2 2.83349 4.84969 188.948 76.7702 128.764 L/G 7.55434 16.6378 30.1230 9 9 . 2 0 8 7 379.501 *These figures exceeded the output format. TABLE 20. L/Q For Multi-Stage Crossflow Systems Freundlich Isotherm Used I n i t i a l Y Q = 1.0 X Q i = 0.0 , i = 1,... ,40 F i n a l Y F = 0 .1 Stages n=3.0 n=2.8 n=2.6 n=2.4 n<=2.2 2 1.45920 1.51302 1 .57809 I . 6 5 8 0 9 1.75873 3 1.34827 I . 3 9 0 2 8 1.44085 1.50276 1.58022 4 1.29968 1.33668 1.38110 1.43535 1.50309 5 1.27248 1.30674 1.34774 1.39782 1.46022 6 1.25515 I . 2 8 7 6 3 1.32654 1.37392 1.43298 8 1.23427 1.26464 1.30099 1.34529 1.40036 10 1.22220 1.25138 1 .28628 1.32875 1.38150 12 1.21433 1.24270 1.27662 1.31794 1 .36922 14 1.20874 1.23661 I . 2 6 9 8 5 1.31037 1.36058 16 1.20463 1.23203 1.26485 I . 3 0 4 7 0 1.35421 18 1.20147 1.22859 I . 2 6 0 9 8 1.30037 1.34927 20 1.19893 1.22581 1 .25790 I . 2 9 6 9 4 1.34535 24 1.19520 1.22164 1 .25333 I .2918O 1.33953 28 1.19249 1.21875 1 .25007 1.28817 1.33541 32 1.19051 1.21657 1.24765 I .28544 1.33231 36 1 .18895 1.21486 1.24577 1.28337 I . 3 2 9 9 4 40 1.18773 1.21350 1.24429 I . 2 8 1 6 6 1.32804 Continued TABLE 20. (CONTINUED) L/G For Multi-stage Crossflow Systems Freundlich Isotherm Used I n i t i a l Y Q = 1. 0 X 0 i = 0.0 , 1=1,.. .,40 F i n a l Y p = 0. 1 Stages n=2.0 n=1.8 n=1.6 n=1.5 n=1.4 2 1 .88885 2.06306 2 .30692 2.46874 2.66930 3 1.67976 1.81199 1.99542 2.11616 2,26494 4 1 .58987 1.70471 1.86336 1.96740 2.09519 5 1.54004 1.64547 1.79069 I . 8 8 5 6 9 2.00224 6 1.50840 1 .60793 1.74475 1.83418 1.94370 .8 1.47058 1.56307 1.69005 1.77284 1.87410 10 1.44876 1.53725 I . 6 5 8 5 5 1.73761 1.83422 12 l c43457 1.52048 1.63814 1.71473 1.80832 14 1 .42458 1.50870 1.62381 I . 6 9 8 6 9 1.79019 16 1.41722 1.49996 I .61318 1.68684 1.77678 18 1.41151 1.49322 I . 6 0 5 O I 1.67770 1.76645 20 1 .40696 1.48789 1.59851 1 .67045 1 .75826 24 1.40024 1.47994 I . 5 8 8 8 9 1.65970 1.74609 28 1.39546 1.47433 1 .58204 1.65206 1.73748 32 1.39192 1.47012 1.57696 1.64638 1.73107 36 1.38916 1.46689 1.57302 1.64198 1.72608 40 1.38697 1.46430 I . 5 6 9 8 9 I .6385O 1.72215 Continued TABLE 20. (CONTINUED) 1. L/Q For Multi-stage Crossflow Systems Freundlich Isotherm Used Initial Y Q 1 = 1.0 X Oi = ° ' ° i = 1,...,40 Final Y F = 0.1 Stages n=1.3 n=1.2 n=l.l n=1.0 n=0.9 n=0.8 2 2.92362 3.25475 3.70016 4.32456 5.24724 6.70930 3 2.45217 2.69402 3.01631 3.46331 4.11549 5.13341 4 2.25549 2.46174 2.73546 3.11312 3.66094 4.50997 5 2.14183 2.33544 2.58338 2.92445 3.41756 4.17878 6 2.08066 2.25620 2.48822 2.80679 3.26634 3.97398 8 2.00057 2.16238 2.37578 2.66816 3.08881 3.73451 10 1.95469 2.10873 2.31164 2.58926 2.98805 3.59910 12 1.92498 2.07401 2.27018 2.53834 2.92314 3.51208 14 ' 1.90419 2.04973 2.24119 2.50276 2.87787 3.45148 16 1.88880 2.03177 2.21978 2.47652 2.84447 3.40684 18 I .87695 2.01795 2.20333 2.45634 2.81883 3.37260 20 1.86755 2.00700 2.19028 2.44037 2.79856 3.34552 24 I .85362 1.99074 2.17090 2.41666 2.76844 3.30537 28 1.84376 1.97923 2.15723 2.39990 2.74720 3.27704 32 1.83641 1.97069 2.14703 2.38745 2.73140 3.25599 36 I .83072 1.96407 2.13915 2.37782 2.71919 3.23974 40 I .82617 1.95879 2.13287 2.37016 2.70947 3.22679 Continued TABLE 20. (CONTINUED) L/G F o r M u l t i - s t a g e C r o s s f l o w Systems F r e u n d l i c h I s o t h e r m Used I n i t i a l Y 0 1 ~ 1 * 0 x 0 i = 0 . 0 , i = 1 , . . . , 4 0 F i n a l Y p = 0 . 1 Stages n=0.7 n=0.6 n=0.5 n=0.4 n=0.3 2 9 .25827 14.35702 26 .94636 71 .20493 379.57169 3 6 .87596 10 .28372 18 .46101 46.15715 229 .56334 4 5.95130 8 .74140 15 .35217 37 .38514 180 .26419 5 . 5.46492 7 .94046 13.76512 33 .01127 156.48170 6 5 .16593 7.45199 12.80748 30 .41069 142.63016 8 4.81826 6.88790 11 .71204 27 .47484 127.28229 10 4.62258 6 .57240 11 .10458 25 .86619 119 .01552 12 4.49721 6 .37107 10 .71891 24.85265 113.86353 14 4 .41007 6 .23146 10 .45251 24.15611 110.34996 16 4 .34598 6.12901 10 .25757 23 .64819 107.80231 18 4 . 29688 6 .05062 10.10862 23.26154 105.87124 20 4 .25805 5 .98872 9 .99123 22 .95736 104.35749 24 4.20058 5 .39719 9 .81795 22 .50956 102.13763 28 4.16006 5 .83276 9 . 6 9 6 2 1 22 .19580 IOO.58859 32 4 .12998 5 .78497 9 . 6 0 6 0 1 21 .96376 99 .44643 36 4.10676 5 .74809 9 .53649 21 .78520 98 .56959 40 4.08829 5.71879 9 .48128 21 .64355 97.87524 APPENDIX II FORTRAN IV L i s t For Pattern Search Method P L E A S E RETURN TP THE CHEMICAL ENGINEERING BUILDING JOB NUMBER 1 6 0 9 6 CATEGORY F U S E R ' S NAME- J P LUCAS * * * * * T H E COMPUTING CENTRE WILL BE CLOSED F R I . , S A T . , S U N . , AND H O N . , JOB START 15HRS 57MIN 0 9 . 4 S E C V9M011  2. 2 U S E R ' 5 A P R I L 1 2 , 1 3 , O F F - L I $JOB 1 6 0 9 6 $T IME $FORTRAN 5 6 7 10 11 12 J . P . LUCAS 1 1 COMMON X , X M A X t X M I N , S N , T O L , A L P H A , B E T A , D , S C , C , D E L , N , L A , K , K K , L T , L S N , 1 N C T , A K I , N S 2 DIMENSION X ( 5 0 ) , Y ( 5 0 ) , X M A X ( 5 0 } , X M I N ( 5 0 ) , C ( 50 ) , 0~( 50 ) , AK I ( 5 0 ) , A L P ( 9) 3 1 FORMAT ( 2 I 2 , 2 E 1 0 . 5 , 3 F 1 0 . 5 ) 4 4 FORMAT ( 8 F 1 0 . 5 )  READ ( 5 , 1 ) N,NC , T O , R I , A L P H A , B E T A , D E L READ ( 5 , 4 ) Q , Y 0 1 , Y 0 2 , Y F , X 0 1 , X 0 2 25 READ ( 5 , 4 ) ( X ( I ) , I = 1 , N ) READ ( 5 , 4 ) ( X M I N ( I ) , X M A X ( I ) , I = 1 , N ) WRITE ( 6 , 6 0 ) WRITE : ( 6 , 5 1 ) N , N C , T O , A L P H A , B E T A , D E L 13 14 15 WRITE ( 6 , 5 6 ) ( X ( I ) , X M I N ( I ) , X M A X ( I ) , I = 1 ,N ) DATE D E S C R I P T I O N 60 FORMAT ( 1 H 1 , 9 0 H RUN NO. 1 ) 51 FORMAT ( 1 H 0 , 2 4 H INDEPENDENT V A R I A B L E S = , I 2 / / 1 4 H CONSTRAINTS = , 1 2 , / 1/20H I N I T I A L TOLERANCE = , E 1 2 . 5 / / , 2 2 H ACCELERAT ION FACTOR = , F 6 . 3 / / 1 29H REDUCTION FACTOR = , F 6 . 3 / / 5 0 H FRACT ION OF INTERVAL USED FOR IN IT 3 I A L STEP S I Z E = , F 6 . 3 ) 16 56 F O R M A T S 1 H 0 , 4 3 H I N I T I A L VALUE M I N . VALUE 1 4 , 7 X , F 8 . 4 , 6 X , F 8 . 4 ) ). MAX . V A L U E / / ( 4 X , F 9 . 17 R = RI 20 RR = RI 21 TOL = TO 22 LSW = 1 23 NS = 1 24 LA = 1 2 5 LSN = 0 26 LT - 0 27 11 DO 10 I = 1 , N 30 10 Y ( I ) = X ( I ) * * Q 31 A L P ( l ) = ( X ( l ) - X ( 4 ) ) / ( Y 0 1 - Y d ) ) 32 A L P ( 2 ) = ( X ( 2 ) - X 0 1 ) / ( Y ( 1 ) - Y ( 2 ) ) 3 3 A L P ( 3 ) = ( X ( 3 ) - X ( 2 ) ) / ( Y 0 2 - Y ( 3 ) ) 34 A L P ( 4 ) = ( X ( 4 ) - X 0 2 ) / ( Y ( 3 ) - Y ( 4 ) ) 35 AL1 = l . / ( A L P ( l 1 + A L P ( 4 ) ) 36 C0N1 = YF - ( ALP (1 ),*Y ( 2) + ALP ( 4 ) *Y ( 4 ) ) *AL 1 37 C0N2 = A L P U ) / A L P ( 2 ) - ALP ( 4 )/ALP ( 3 ) 4 0 CC = ( 1 . + A L P ( 1 ) / A L P ( 2 ) ) * A L 1 41 SN = CC + R*C0N1**2 + RR*C0N2**2 42 CALL S T E P 5 0 43 GO TO ( 1 1 , 4 9 , 5 0 ) , NS 44 5 0 WRITE ( 6 , 1 9 ) C C S N 45 WRITE ( 6 , 2 2 ) ( X ( I ) , I = 1 , N ) 46 . WRITE ( 6 , 2 3 ) ( Y ( I ) , I = 1 ,N ) 47 WRITE . 1 6 , 2 0 ) R , C 0 N 1 , R R , C 0 N 2 50 WRITE i 6 , 2 1 ) NC T 51 52 33 54 2.3 19.FORMAT ( 1 H 0 , 1 6 H OPTIMAL - L/G = , F 1 4 . 5 , 6H SN = , F 1 4 . 5 / ) 20 FORMAT (20H PENALTY FACTOR C I = , E 1 5 . 5 , 4 H C 1 = , E 1 5 . 5 , 2 0 H PENALTY FAC 1T0R C2 = , E 1 5 . 5 , 4 H C 2 = , E 1 5 . 5 / ) 21 FORMAT ( 1 H 0 , 2 6 H RETURNS TO MAIN PROGRAM = , 1 5 / / ) 22 FORMAT (38H INDEPENDENT V A R I A B L E S AT OPTIMUM ARE , 8 F 1 0 . 6 / ) 5 5 56 57 60 61 62 23 FORMAT (38H DEPENDENT V A R I A B L E S AT OPTIMUM ARE , 8 F 1 0 . 6 / C I = A B S ( C O N l ) C2 = ABS(CON2) C3 = C 2 / C 1 IF ( C 1 . G T . C 2 ) GO TO 2 0 6 RR = R R * 1 0 . 63 64 65 66 67 70 IF ( C 3 . L T . 1 0 . O ) R =. 1 0 . * R / C 3 GO TO 207 206 R = R * 1 0 . IF ( C 3 - . G T . 0 . 1 ) RR = 1 0 . * R R * C 3 2 0 7 I F ( R . G T . I . E 8 . A N D . R R . G T . 1 . E 8 ) GO TO 49 NS = 1 71 7 2 73 74 LA = 1 GO TO 11 49 STOP END 75 2A SUBROUTINE S T E P 5 0 76 COMMON X , X M A X , X M I N , S N , T O L , A L P H A , B E T A , D , S C , C , D E L , N , L A , K , K K , L T , L S N , 1 N C T , A K I , N S 77 DIMENSION X ( 5 0 ) , XMAX(50) , X M I N ( 5 0 ) , C { 5 0 ) , P ( 5 0 ) , D ( 5 0 ) , A K I ( 5 0 ) ICO GO TO ( 1 0 0 , 2 8 0 , 4 6 0 , 5 8 0 , 5 1 0 ) , L A C F I R S T ENTRY TO SUBROUTINE C I N I T I A L I Z E V A R I A B L E S 101 100 SP = SN 102 NCT = 1 103 SC = SN 104 M l = l 105 M2 = l 106 NPF = 0 107 K = l 110 KK=1 111 IF ( T O L . L E . O . O ) TOL = l . E - 5 112 IF ( A L P H A . L E . 0 . 0 ) ALPHA = 1 . 1 1 1 1 1 1 1 113 IF ( B E T A . L E . O . O ) BETA = 0 . 1 114 IF ( D E L . L E . 0 . 0 ) DEL = 0 . 1 115 DO 180 I = 1 , N C SET INDEPENDENT V A R I A B L E S AT LAST BASE POINT 116 C ( I )=X ( I ) C SET I N I T I A L STEP S I Z E S 117 180 D ( I ) = D E L * ( X M A X ( I ) - X M I N ( I ) ) C SET SWITCH FOR F I R S T SET OF PATTERN MOVES - PREVENTS STEP S I Z E FROM c BECOMING TOO LARGE 120 190 JSW = 1. c JSW = 1 I N I T I A L L Y AND AFTER EACH TIME THROUGH THE LAST V A R I A B L E WITH c NO IMPROVEMENT WITHIN TOLERANCE OF LAST PATTERN MOVE OVER OLD BASE c P O I N T . 121 2 0 0 LA = 2 122 . ._ IF (D(K).EQ.O.O) GO TO 4 9 0 123 X ( K ) = X ( K ) + D (K ) c MAKES F I R S T PATTERN MOVE c CHECK TO SEE IF V A R I A B L E S ARE WITHIN BOUNDARIES 124 2 3 0 IF ( X ( K ) . L E . X M A X ( K ) . A N D . X M I N ( K ) . L E . X ( K ) ) GO TO 2 7 0 125 GO TO ( 5 0 0 , 3 6 0 , 4 8 0 , 5 0 0 , 5 0 0 ) , LA c ADD TO SUBROUTINE ENTRY COUNTER 126 270 NCT = NCT + 1 127 RETURN c CHECK TO SEE IF PATTERN MOVE WAS S U C C E S S F U L 130 280 IF I S N . G E . S P ) GO TO 3 6 0 c SUCCESSFUL PATTERN MOVE c IF . I N I T I A L BASE POINT REMAINS DO NOT INCREASE STEP S I Z E 131 IF ( J S W . E Q . 2 ) D (K ) = D(K) * ALPHA c SET SP TO BE BEST VALUE FOUND ON T H I S SEARCH 132 3 0 0 SP = SN 133 NPF = 0 134 M2=l 1 3 5 . Ml = l c GO TO NEXT V A R I A B L E 136 3 0 5 K = K + 1 c I F PREVIOUS V A R I A B L E WAS THE LAST - RETURN TO F I R S T 137 IF ( K . G T . N ) K = 1 c LT = 1 FOR TRUNCATED SEARCH 140 "c IF ( L T . G T . O ) GO TO 340 ADD TO COUNTER OF V A R I A B L E S STUDIED S INCE LAST TEST FOR BASE POINT 141 KK = KK + 1 142 IF ( K K . L E . N ) GO TO 2 0 0 275 C IF KK EXCEEDS THE NO. OF V A R I A B L E S DO TOLERANCE CHECK ON LAST BASE PT 143 3 4 0 IF (SP + T O L * A B S ( S C ) . G E . S C ) GO TO 4 0 0 C IF THE O B J E C T I V E FUNCTION ON THE LAST PATTERN MOVE IS GREATER THAN C THE O B J E C T I V E FUNCTION FOR THE LAST BASE POINT OR IS L E S S THAN A C TOLERANCE OF THE BASE POINT - CHECK TO SEE IF LAST V A R I A B L E HAS C BEEN USED - OTHERWISE SET THE CURRENT VALUES TO BE THE NEW BASE PT 144 IF ( J S W . E Q . 2 ) GO TO 3 5 3 145 352 LA = 5 C DO BASE POINT C A L C U L A T I O N 146 Ml=l 147 GO TO 270  C IF THE NUMBER OF S U C C E S S I V E PATTERN MOVES FOLLOWED BY F A I L U R E OF C INDIV IDUAL MOVES EXCEEDS 5 CHECK TO SEE IF LAST V A R I A B L E HAS BEEN C USED - OTHERWISE SET CURRENT VALUES TO BE NEW BASE POINT 150 3 5 3 I F ( N P F - 5 ) 3 5 2 , 4 0 0 , 4 0 0 151 360 LA = 3 152 X ( K ) = X ( K ) - 2 . » D ( K )  153 GO TO 230 154 4 0 0 IF ( L T . G T . O ) KK = KK + 1 155 IF ( K K . L E . N ) GO TO 200 156 * IF ( J S W . N E . 2 ) GO TO 425 157 SP = SC C SET THE CURRENT VALUES TO BE NEW BASE POINT  160 DO 420 I = 1 „ N 161 4 2 0 X ( I ) = C ( I ) 162 NPF=0 163 4 4 0 KK = 1 164 Ml=.l 165 M2=l  166 GO TO 190 167 4 2 5 JSW = 2 170 IF ( M l . L E . N ) GO TO 4 4 0 C DURING THE SECOND SWEEP THROUGH THE V A R I A B L E S WITH NO IMPROVEMENT IN . 0 THE O B J E C T I V E FUNCTION THE COUNTER Ml IS TESTED TO SEE I F IT C EXCEEDS THE NUMBER OF V A R I A B L E S - IF IT HAS GO TP END OF JOB AND C PRINT OUT LOCAL OPTIMUM - IF NOT SET M l = 1 , K K = 1 AND RETURN FOR C ANOTHER SWEEP THROUGH V A R I A B L E S 171 DO 1015 1=1 ,N 172 1015 X ( I ) - C ( I ) 173 NS = 3 174 RETURN  175 4 6 0 IF ( S N . G E . S P ) GO TO 4 8 0 C IF THE* O B J E C T I V E FUNCTION HAS IMPROVED OVER THE ORIG INAL PATTERN C POINT BY A REVERSE MOVE CHANGE THE SIGN OF THE STEP SO THAT JHE C F I R S T MOVE ON THE V A R I A B L E DURING THE NEXT PATTERN MOVE WILL BE IN C THE SAME D I R E C T I O N AS THE CURRENT SUCCESS 176 D ( K ) = - D ( K )  C THE STEP S I Z E I S NOT INCREASED ON A REVERSE SUCCESS - ONLY WHEN THE C F I R S T MOVE SUCCEEDS IS THE STEP S I Z E INCREASED BY ALPHA FACTOR 17.7_ GO TO 300 200 4 8 0 X ( K ) = X ( K ) + D ( K ) 2 0 1 D ( K ) = D ( K ) * B E T A C PREVENTS INTERVALS FROM BECOMING TOO SMALL  202 DX =AB*S(X(K)/D(K.)*TOL) C WHEN DX R I S E S ABOVE UNITY THE MINIMUM STEP S I Z E HAS BEEN REACHED 2 0 3 IF ( l . - D X ) 4 8 1 , 4 8 2 , 4 8 4 2 0 4 " 4 8 1 D ( K ) =DIK)*DX 2 0 5 4 8 2 DX= ABS ( 1 . E - 3 0 / D ( K ). ) 2 0 6 IF ( 1 . - DX ) 4 8 5 , 4 9 0 , 4 9 0 ; 207 4 8 4 2.6 D X = . A B S U . E - 3 0 / D ( K ) ) C ADDIT IONAL CHECK TO PREVENT THE STEP S I Z E S FROM GOING BELOW E - 3 0 210 I F ( l . - D X ) 4 8 5 , 4 9 0 , 4 9 2 2 1 1 4 8 5 0 ( K ) =.D(K)*DX C AS SOON AS THE STEP S I Z E BECOMES MINIMUM THE COUNTER M l BEGINS TO c FUNCTION 212 4 9 0 Ml = M l + 1 213 4 9 2 IF { J S W . E O . 1 ) GO TO 3 0 5 2 1 4 4 9 3 M2 = M2 + 1 215 IF ( M 2 . L E . N ) GO TO 305 216 M2 = 1 217 NPF=NPF+1 2 2 0 GO TO 305 2 2 1 500 W R I T E ( 6 , 1 0 0 1 ) 2 2 2 1001 FORMAT (52H PROGRAM HAS CRAPPED OUT - RESET STARTING V A R I A B L E S ) 2 2 3 NS = 2 2 2 4 RETURN 2 2 5 510 KK = 1 2 2 6 IF ( L S N . G T . O ) SP = SN 227 530 SC = SP c MAKES PATTERN MOVE JUDGING FROM LAST RUN THRU ALL V A R I A B L E S 2 3 0 LA=4 2 3 1 DO 570 1 = 1 , N c STORE LAST BASE POINT V A R I A B L E S IN P ARRAY 232 PI I K U ) c SETS V A R I A B L E S FROM LAST PATTERN MOVE TO BE NEW BASE POINT 233 C( I } = X.t I ) c NEW INDEPENDENT VARA I B L E S FORMED FOR NEXT PATTERN MOVE BY INCREASING c THE V A R I A B L E S BY AN EQUAL AMOUNT IN THE D IRECT ION OF THEIR LAST c SUCCESS 234 X ( I ) = 2 . * X ( I ) - P ( I ) c CHECK TO SEE THAT V A R I A B L E S REMAIN WITHIN BOUNDS 2 3 5 IF(,X( I ) - X M A X ( I ) ) 5 5 0 , 5 7 0 , 5 4 0 236 5 4 0 X ( I ) = X M A X U ) 23 7 GO TO 570 240 550 IF (XM I N ( I ) . G T » X ( I ) ) X ( I ) = X M I N ( I ) 2 4 1 5 70. CONTINUE ^242 GO TO 270 243 5 8 0 SP = SN 2 4 4 JSW = 2 2 4 5 GO TO 200 246 END $ENTRY APPENDIX I I I Deflected Gradient Optimization Method (Fletcher-Powell) 1. Nomenclature 2 . Flow Diagram 3. FORTRAN IV L i s t NOMENCLATURE 3.2 Dirac bra - ket notation i s used as applied to r e a l vectors. |x> Column vector <x| Row vector <x|x> Scalar product |x><x| Matrix operator < X | M | X > Quadratic form, M matrix of c o e f f i c i e n t s I Identity matrix H Matrix determining d i r e c t i o n of search |G> Gradient Vector determined before search for minima |S> D i r e c t i o n of search f o r minima CH Magnitude of |s> |o> Vector between s t a r t i n g point and minima P Step along |S> to minima |GI> Gradient vector determined a f t e r minima found |Y> Gradient difference vector A Matrix computed to make an improvement on H B Matrix computed to correct i n i t i a l guess of H K KK = 1 move i n p o s i t i v e |S> d i r e c t i o n KK = 2 move i n negative |S> d i r e c t i o n R,RR Penalty factors on equality constraints R1,R2 Penalty factors on inequality constraints TOL Minimum magnitude of |S> TAU Maximum f i n a l value f o r equality penalty factors FLETCHER - POWELL MAIN PROGRAM I N I T I A L I Z E V A R I A B L E S 100 H = I C A L C U L A T E |G> N O 1 |o> . P | S > |G«> = |G> C A L C U L A T E |G> A T |X> + |CT> h>= |G> - | G I > A - \0> <CT| <o"IX> H | Y > < Y | H B " < Y | H | Y > H = H + A, + B w — { |s> = - H | G > C H =y<SlS> 1 0 9 W R I T E O P T I M U M Y E S I N C R E A S E R . R R D E C R E A S E RI.R2 E N D P L E A S E RETURN TO THE CHEMICAL ENGINEERING BUILDING 3.4 JOB NUMBER 1 6 0 9 6 CATEGORY F U S E R ' S NAME- J P LUCAS U S E R ' S *****THE COMPUTING CENTRE WILL BE CLOSED F R I . , S A T . , S U N . t AND M O N . , - A P R I L 1 2 , 1 3 , JOB START 15HRS 13MIN 4 9 . 3 S E C V9M0.U O F F - L J $JOB 1 6 0 9 6 J . P . L U C A S $T I ME 1 $FORTRAN C DESCENT METHOD FOR M I N I M I Z A T I O N 1 ._. COMMON G , R , R 1 , R 2 , R R , W , Y F , X O 1 , X 0 2 , A L P , S , S O , J J , Q , Y 0 1 , Y 0 2 , P , CONI ,CON2 1 , C C , Z , K C T 2 DIMENSION H ( 2 0 , 2 0 ) t A ( 2 0 , 2 0 ) , B ( 2 0 , 2 0 ) , X ( 2 0 ) , S ( 2 0 ) , S I G ( 2 0 ) , Y ( 2 0 ) » G ( 2 1 0 ) , G I ( 2 0 ) , Y T 1 ( 2 0 ) , Y T 2 ( 2 0 ) , A L P ( 2 0 ) , W ( 5 0 ) , S O ( 2 0 ) , Z ( 2 0 ) 3 READ ( 5 , 1 ) MM,N , T O L , R I 4 1 FORMAT ( 2 I 2 , 2 E 1 0 . 5 ) 5 J J = 1 6 DO 4 0 0 M = 1,MM 7 READ ( 5 , 2 ) ( X ( I ) , I = 1 ,N ) 10 READ ( 5 , 2 ) X 0 1 , X 0 2 , Y 0 1 , Y 0 2 , Y F , Q 11 READ ( 5 , 2 ) (W(I ) , 1 = 1 , 1 3 ) , R 1 , R 2 12 WRITE ( 6 , 3 ) 13 WRITE ( 6 , 4 ) X 0 1 , X 0 2 14 WRITE ( 6 , 5 ) Y 0 1 , Y 0 2 15 WRITE ( 6 , 6 ) YF 16 WRITE ( 6 , 7 ) Q 17 2 FORMAT ( 8 F 1 0 . 5 ) 20 3 FORMAT ( 1 H 1 , 2 0 H FOUR STAGE SYSTEM A//) 21 4 FORMAT (48H I N I T I A L CONCENTRATION IN ADSORBENT - STREAM 1 = , . F 1 2 . 5 , 113H , STREAM 2 = , F 1 2 . 5 / ) 22 5 FORMAT (47H I N I T I A L CONCENTRATION IN SOLUTION - STREAM 1 = , F 1 2 . 5 , 1 13H , STREAM 2 = , F 1 2 . 5 / ) 23 6 FORMAT (34H F I N A L CONCENTRATION IN SOLUTION = , F 1 2 . 5 / ) 24 7 FORMAT (37H VALUE OF N FOR FREUNDLICH ISOTHERM = , F 1 2 . 5 / / ) 25 R = RI 26 RR = R l 27 KCT = 1 30 KK = 1 31 100 WRITE ( 6 , 1 0 1 ) ( X ( I ) , I = 1 , N ) 32 101 FORMAT (12H V A R I A B L E S = , 1 0 F 1 0 . 5 ) 33 DO 105 J = 1 , N 34 DO 105 I = 1 , N 35 105 H( I , J ) = 0 . 0 36 DO 106 I = 1 , N 37 106 H ( I , I ) = 1 . 0 40 CALL GRAD (X , N ) 41 109 DO 110 I = I , N 42 S( I ) = 0 . 0 43 DO 110 J = 1 , N 44 n o S ( I ) = S ( I ) - H ( I , J ) * G ( J ) 45 CHK = 0 . 0 46 DO 111 I = 1 , N 47 111 CHK = CHK + S( I )**2 50 CH = SQRT(CHK) 51 IF ( C H . L T . T O L ) GO TO 2 0 0 3.5 |52 64 CALL P A R M I N ( X , N ) '53 GO TO ( 6 5 , 2 1 0 ) , K C T 54 65 KK = 1 55 DO 115 I = 1 , N 56 115 S I G ( I ) = P * S O ( I ) 157 DO 107 I = 1 , N 60 107 GI ( I ) = G ( I ) 61 CALL GRAD ( X , N ) 62 120 DO 121 I = 1 , N 63 121 Y ( I ) = G ( I ) - G U I ) 64 DEN 1 = 0 . 0 65 DO 125 I = 1 , N . 66 125 DEN 1 = DEN1 + S I G ( I )*Y( I ) 67 DO 1 3 0 . 1 = 1 , N 70 DO 130 J = 1 , N 71 130 A ( I , J ) •= S IG ( I )*S IG( J J / D E N 1 72 DO 135 J = 1 , N 73 Y T 1 ( J ) = 0 . 0 74 DO 135 I = 1 , N 75 135 Y T K J ) = Y ( I ) * H ( I , J ) + Y T K J ) 76 DEN2 = 0 . 0 7 7 DO 140 I = 1 , N 100 140 DEN2 = DEN2 + Y T 1 ( I ) * Y ( I ) 101 DO 145 I = 1 , N 102 YT2( I ) = 0 . 0 103 DO 145 J = 1 , N 1 0 4 145 Y T 2 ( I ) = Y T 2 ( I ) + H ( I » J ) * Y ( J ) 105 DO 150 I = 1 , N 106 DO 150 J = 1 , N 107 150 B ( I , J ) = - Y T 2 ( I ) * Y T H J ) / D E N 2 110 DO 155 I = 1 , N 111 DO 155 J = 1 , N 112 " 1 5 5 " H ( I , J ) = H ( I , J ) + A ( I , J ) + B ( I , J ) 1 1 3 GO TO 109 114 200 WRITE ( 6 , 2 0 1 ) ( X ( I ) , I = 1 , N ) 115 WRITE ( 6 , 2 0 2 ) ( Z ( I ) , I = 1 , N ) 116 WRITE 1 6 , 2 0 3 ) CC 117 WRITE ( 6 , 3 0 3 ) ( A L P ( I ) , I = 1 ,N ) 120 WRITE ( 6 , 3 0 5 ) R ,RR 121 WRITE ( 6 , 3 0 6 ) C 0 N 1 , C 0 N 2 122 2 0 1 FORMAT ( 1 H 0 , 2 7 H V A R I A B L E S AT OPTIMUM ARE =, 1 0 F 1 0 . 5 ) 123 2 0 2 FORMAT (35H DEPENDENT V A R I A B L E S AT OPTIMUM A R E , 8 F 1 2 . 5 ) 124 2 0 3 FORMAT ( 1 H 0 , 2 4 H O B J E C T I V E FUNCTION IS , F 1 2 . 5 ) 125 3 0 3 FORMAT ( 1 H 0 , 9 H ALPHAS = , 1 0 F 1 2 . 5 ) 126 305" F O R M A T ( 1 H 0 , 3 3 H PENALTY FACTORS - CONSTRAINT 1 = , E 1 5 . 5 , 1 5 H CONSTRAI INT 2 = , E 1 5 . 5 / ) 127 3 0 6 FORMAT (25H CONSTRAINT VALUES - C I = , E 1 5 . 5 , 5H C2 =, E 1 5 . 5 / ) 130 DO 310 I = 1 , N 13:1 310 A L P ( I ) = 1 . / A L P ( I ) 132 WRITE ( 6 , 3 1 1 ) ( A L P ( I ) , I = 1 , N ) 133 3 1 1 FORMAT ( 1 H 0 , 6H L/G = , 1 0 F 1 2 . 5 / ) 1 3 4 C I = A B S ( C O N l ) 135 C2 = A B S ( C 0 N 2 ) 136 C3 = C 2 / C 1 137 IF ( C l . G T . C 2 ) GO TO 206 140 RR = R R * 1 0 . 141 IF ( C 3 - L T . 1 0 . 0 ) R = 1 0 . * R / C 3 1 4 2 GO TO 207 143 2 0 6 R =. R * 1 0 . 3.6 144 IF ( C 3 . G T . 0 . 1 ) RR = 1 0 . * R R * C 3 145 2 0 7 IF ( R . L T . l . E 9 . A N D . R R . L T . 1 . E 9 ) GO TO 100 146 GO TO 4 0 0 147 2 1 0 IF ( K K . G T . 2 ) GO TO 200 150 KK = KK + 1  151 DO 2 1 1 I = 1 , N 152 2 1 1 S ( I ) = - S ( I ) 153 WRITE ( 6 , 2 1 2 ) 1 5 4 2 1 2 FORMAT ( 1 H 0 , 3 2 H REVERSE DIRECTIONS ON PARAMETER/) 155 KCT = 1 156 GO TO 64  157 4 0 0 CONTINUE 160 STOP 161 END 3.7 162 SUBROUTINE GRAD ( X , N ) 163 COMMON ' G , R r R l f R 2 , R R , W , Y F , X 0 1 , X 0 2 f A L P , S , S O , J J , 0 , Y O 1 , Y 0 2 , P , C 0 N 1 , C 0 N 2 l t C C , Y , K C T 164 DIMENSION G ( 2 0 ) , A L P ( 2 0 ) , W ( 5 0 ) , X ( 2 0 ) , Y ( 2 0 ) , S O ( 2 0 ) , S ( 2 0 ) 165 DO 5 I = 1 , N 166 5 Y ( I ) = X ( I ) * * Q 167 A L P ( l ) = ( X ( l ) - X ( 4 ) ) / ( Y 0 1 - Y ( l ) ) 170 A L P ( 2 ) = ( X ( 2 ) - X 0 2 ) / ( Y ( 1 ) - Y ( 2 ) ) 171 A L P ( 3 ) = ( X ( 3 ) - X ( 2 ) ) / ( Y 0 2 - Y(3)}~ 172 A L P ( 4 ) - ( X ( 4 ) - X 0 1 ) / ( Y ( 3 ) - Y ( 4 ) ) 173 A L 1 = l . / ( A L P ( l ) + A L P ( 4 ) ) 174 AL2 = A L 1 * * 2 175 C0N1 = YF - ( A L P d ) * Y ( 2 ) + A L P { 4 ) * Y ( 4 ) ) * A L 1 176 C0N2 = A L P ( 2 ) / A L P d ) - ALP ( 3 ) /ALP (4 ) y 177 DFDA1 = A L 2 * ( A L P ( 4 ) / A L P ( 2 ) - 1. + 2 . * R*C0N1* A L P ( 4 ) * ( Y ( 4 ) - Y ( 2 ) ) > - 1 ( 2 . * R R * C 0 N 2 * A L P(2) + R2 ) / A L P (1)**2 ; 200 DFDA2 = - ( A L P ( 1 . ) * A L 1 + R 2 ) / A L P ( 2 ) * * 2 + 2 . * R R * C 0 N 2 / A L P ( 1 ) 201 DFDA3 = - 2 . * R R * C 0 N 2 / A L P ( 4 ) - R 2 / A L P ( 3 ) * * 2 202 DFDA4 = - A L 2 * ( 1 . + A L P ( 1 ) / A L P ( 2 ) + 2 . * R * C 0 N 1 * A L P ( 1 ) * { Y < 4 ) - Y ( 2 ) ) ) 1 + ( 2 . * R R * C 0 N 2 * A L P ( 3 ) - R 2 ) / A L P ( 4 ) * * 2 , 20 3 DFDY2 = - 2 . * R * C 0 N 1 * A L P { 1 ) * A L 1 204 DFDY4 = D F D Y 2 * A L P ( 4 ) / A L P ( 1 ) ' 205 DA1DX1 = 1 . / (Y01 - Y d ) ) 206 DA1DX4 = - D A 1 D X 1 207 DA2DX2 = l . / ( Y ( : l ) - Y ( 2 ) ) 210 DA3DX2 = - l . / ( Y 0 2 - Y ( 3 ) ) , 211 DA3DX3 = - D A 3 D X 2 212 DA4DX4 = l . / ( Y ( 3 ) - Y ( 4 ) ) 2 1 3 DA1DY1 = A L P ( 1 ) * D A 1 D X 1 214 DA2DY1 = - A L P ( 2 ) * D A 2 D X 2 2 1 5 DA2DY2 = - D A 2 D Y 1 2 1 6 DA3DY3 = A L P ( 3 ) * D A 3 D X 3 217 DA4DY3 = - A L P ( 4 ) * D A 4 D X 4 2 2 0 DA4DY4 = - D A 4 D Y 3 ' 2 2 1 DO 10 J = 1 , N 222 J2 = 2*J 223 10 G U ) = R l * ( W ( J 2 ) / t l . - X ( J ) ) * * 2 - W ( J2-.1 )/X ( J ) **2 > 224 DX = Q * X d ) * * ( Q - 1. ) 2^5 G ( l ) = G ( l ) + DFDA1 * (DA1DX1 + DA'1DY1*DX) + DFDA2*DA2DY1*DX 2 2 6 OX = Q*X(2)**(Q - l . ) 7 2 2 7 G ( 2 ) = G ( 2 ) + DFDA2*(DA2DX2 + DA2DY2*DX) + DFDA3*DA3DX2 + DFDY2*DX ; 2 3 0 DX = Q*X(3)**(Q - 1 . ) ! 2 3 1 G ( 3 ) = G ( 3 ) + DFDA3*(DA3DX3 + DA3DY3*DX) + DFDA4*DA4DY3*DX 2p2 DX = Q*X ( 4)**(Q - 1 . ) ! 2 3 3 G ( 4 ) = G ( 4 ) + DFDA l *DA 1DX4 +• DFDA4* ( DA4DX4 + DA4DY4*DX) + DFDY4*DX , 2 3 4 RETURN T 2 3 5 END > APPENDIX IV Conjugate D i r e c t i o n Method (Powell) Nomenclature Flow Diagram FORTRAN IV L i s t 4.2 NOMENCLATURE JK JK = 1 Determine i n i t i a l minima JK = 2 Determine second minima JK « 3 Determine possible minima between previous p a i r JL JL = 1 Normal variable minimization JL = 2 Minimize along conjugate d i r e c t i o n PMAX Maximum distance moved by any variable i n a set of variable moves K Variable index N Number of variables Q Factor on minimum step size TOL (< l ) TOL Minimum step size XX Coordinates of i n i t i a l minima XY Coordinates of second minima QQ Factor on minimum step s i z e (> l ) RPRR Penalty factors on equality constraints R1,R2 Penalty factors on inequality constraints TAU Maximum penalty factor on equality constraints POWELL MAIN PROGRAM^ JK YES FORM NEW CONJUGATE DIRECTION REPLACE DIRECTION OF BEST PREVIOUS MOVE BY CONJUGATE DIRECTION XX = MINIMA A ADD QQ-TOL TO ALL VARIABLES JK : = 2 ( ? ) INITIALIZE VARIABLES ® FIND MINIMUM ALONG ONE DIRECTION 1,2 JK - 3 J 2 JL = 1 I YES 3- SET (A-B) TO BE NEW CONJUGATE DIRECTION JK = 2 XY = MINIMA B DETERMINE VECTOR ( x x - XY) JK = 3 INCREASE R,RR DECREASE RI, R2 POWELL I S N SOURCE STATEMENT FORTRAN SOURCE L I S T IC 0 * S I B F T C POWELL * * C POWELL M I N I M I Z A T I O N WITHOUT D E R I V I T I V E S 1 COMMON A L P , Y F , Y 0 1 , Y 0 2 , X O 1 , X 0 2 , W , R , R R , R l , R 2 , Q , C 0 N 1 , C 0 N 2 2 DIMENSION X I ( 2 0 , 2 0 ) , X 0 ( 2 0 ) , X ( 2 0 ) , X J ( 2 0 ) , S ( 2 0 ) , X N ( 2 0 ) , A L P ( 2 0 ) , W ( 5 0 ) * * 1 , X X ( 2 0 ) , X Y ( 2 0 ) * 1 ' 3 READ ( 5 , 4 ) N f R I 5 * 4 FORMAT ( I 2 , E 1 0 . 5 ) * 6 * READ ( 5 , 1 ) ( X ( I ) , I = 1 , N ) * 13 * READ ( 5 , 1 ) X 0 1 , X 0 2 , Y 0 1 , Y 0 2 , Y F , Q 14 READ ( 5 , 1 ) ( W ( I ) , I 1 , 1 2 ) , R 1 , R 2 * 21 * 1 FORMAT ( 8 F 1 0 . 5 ) * 22 R = RI * 23 * RR = RI 2 4 * READ ( 5 , 3 ) T O L , T A U , D E L * 2 5 3 FORMAT ( 2 E 1 0 . 5 , F 1 0 . 5 ) * 26 * STL = T 0 L * 1 0 . 27 * FAC = 5 . 0 30 * NP1 = N + 1 31 NM1 = N - I 32 98 J L = 1 * 33 * JK = 1 , 34 * FAC I = l . / F A C 3 5 * DO 5 J = l , N 36 * DO 5 I = 1 ,N 37 * 5 X I ( J , I ) = 0 . 0 * 42 * DO 6 I = 1 ,N * 43 * 6 XI ( I , I ) = 1 . 0 45 * SN = F ( X , N ) * 46 * WRITE ( 6 , 1 1 ) S N , ( X ( I ) , I = 1 , N ) * 53 11 FORMAT ( 1 H 0 , 2 2 H STARTING VALUES - F = , F 1 0 . 5 , 1 2 H V A R I A B L E S = , 8 F 1 0 . 5 * 1/) 54 2 SO - F ( X , N ) 55 K = 1 * 56 * 10 DO 15 I = 1 , N * 57 15 X N ( I ) = X { I ) * 61 DELTA = 0 . 0 * 62 * PMAX = 0 . 0 * 63 7 P = 0 . 0 * 64 * DO 14 I = i , N * 65 * 14 X O ( I ) = X ( I ) * 6 7 * LA = 1 70 KY = I * 71 * DELT = DEL 72 * 9 DO 8 I = 1 , N * 73 8 X ( I ) = X O ( I ) + P * X I ( K , I ) * 75 * GO TO ( 1 0 0 , 1 2 0 , 1 4 0 , 1 6 0 , 1 8 0 , 2 0 0 , 2 4 0 ) , L A * 76 * 100 FX = F I X , N ) 7 7 FF = FX ICO * P = DELT * 101 * LA = 2 * 102 GO TO 9 * 103 * 120 FXP = F ( X , N ) * 104 IF ( F X P . G T . F F ) GO TO 125 107 FF = FXP * 110 * DELT = D E L T * 1 . 6 1 8 4.5 POWELL FORTRAN SOURCE L I S T POWELL 10 . ISN SOURCE STATEMENT 111 * P = P + DELT * 1112 * KY = 2 +_ 113 * GO TO 9 * 114 * 125 GO TO ( 1 3 0 . 1 3 5 ) , K Y * 115 * 130 P = - D E L T * 116 * LA = 3 * 117 * PF = DELT * |l20 * GO TO 9 * _ 121 * 135 PO = 0 . 0 * 122 * PM = 0 . 3 8 l * P * 1 2 3 * PD = 0 . 6 1 9 * P * 124 * GO TO 150 * 125 * 140 FXMP = F ( X , N ) * 126 • IF ( F X M P . L T . F F ) GO TO 1 5 5 * _ 1 3 1 * PO = - D E L T * 132 * PP = 2 . * D E L T * 133 * PM = 0 . 3 8 1 * P P + PO * 134 * PD = 0 . 6 1 9 * P P + PO * 1 3 5 * 150 LA = 5 * 136 * P = PM +_ 137 * GO TO 9 * 140 * 155 LA = 4 * 141 * DELT = - D E L T * 142 * 156 DELT = D E L T * 1 . 6 1 8 * 1 4 3 * P = P + DELT * 144 * GO TO 9 +_ 1 4 5 * 160 FXP = F ( X , N ) * 146 * IF ( F X P . L T . F F ) GO TO 156 * 151 * PO = P * 152 * PF = 0 . 0 * 153 * PM = 0 . 6 1 9 * P 0 * 154 • PD = 0 . 3 8 1 * P 0 * _ 155 * GO TO 150 * 156 * 180 FPM = F ( X , N ) * 1 5 7 * P = PD * 16*0 * LA = 6 * 161 * GO TO 9 * 162 * 2 0 0 FPD = F t X i N ) * _ 163 * 2 0 5 I F ( F P D . G T . F P M ) GO TO 2 1 0 * 166 * PO = PM * 167 * PM = PD * 170 * FPM = FPD * 171 * FPO = FPM * 172 * PD = 0 . 6 1 9 * ( P F - PO) + PO * _ 173 * P = PD * 174 * LA * 6 * 175 * GO TO 215 * 176 * 2 1 0 PF = PD * 177 * PD = PM * 2 0 0 * FPO = FPM . * _ 2 0 1 * FPF = FPD * 2 0 2 * PM = 0 . 3 8 1 * ( P F - PO) + PO * 203 * P = PM * 204 * LA = 7 " " " " * 2 0 5 * 2 1 5 IF ( ( A B S ( P F - P O M . G T . T A U ) GO TO 9 * POWELL ISN SOURCE STATEMENT FORTRAN SOURCE L I S T POWELL .4.6 10 2 1 0 GO TO 260 * 2 1 1 * 2 4 0 FPM = F ( X , N ) 2 1 2 * GO TO 205 * 2 1 3 * 2 6 0 WRITE ( 6 , 2 7 0 ) S N , ( X ( I ) , I = 1 , N ) * 2 2 0 * 2 7 0 FORMAT (23H MINIMUM ON PARAMETER = , F 1 0 . 5 , 1 2 H V A R I A B L E S = , 8 F I 0 . 5 ) * 2 2 1 * S ( K ) = SN 2 2 2 GO TO ( 3 9 9 , 3 9 9 , 4 9 0 ) , J K * 2 2 3 * 399 GO TO ( 1 3 , 1 2 ) , J L * 2 2 4 12 JL = 1 * 2 2 5 GO TO 2 2 2 6 * 13 IF ( K . E Q . 1 ) GO TO 4 0 0 * 2 3 1 * DELT = S(K - 1) - S ( K ) * 2 3 2 * GO TO 4 1 0 2 3 3 * 400 DELT = SO - S ( K ) * 2 3 4 4 1 0 IF ( D E L T . L T . D E L T A ) GO TO 4 2 0 * 2 3 7 * DELTA =• DELT * 2 4 0 * M = K 241 4 2 0 PP = ABS ( X ( K ) - X O ( K ) ) 2 4 2 * K = K + 1 2 4 3 IF ( P M A X . L T . P P ) PMAX = PP 2 4 6 IF ( K . L E . N ) GO TO 7 2 5 1 IF ( P M A X . L T . T O L * F A C I ) GO TO 4 5 0 2 5 4 DO 4 3 0 I = 1 , N * 2 5 5 * 4 3 0 X J ( I ) = 2 . * X ( I ) - X N ( I ) * 2 5 7 F l = SO * 2 6 0 F2 = SN * 2 6 1 * F3 = F ( X J , N ) * 262 FF •= ( F l - 2 . * F 2 + F 3 ) * ( F 1 - F 2 - D E L T A ) * * 2 2 6 3 * FG = 0 . 5 * D E L T A * ( F 1 - F 3 ) * * 2 * 2 6 4 * IF ( F 3 . G E . F 1 . 0 R . F F . G E . F G ) GO TO 2 * 2 6 7 * DO 4 4 0 I = 1 , N 2 7 0 * 4 4 0 X I ( M , I ) = X ( I ) - X N ( I ) * 2 72 * SUM = 0 . 0 * 2 7 3 * DO 4 4 1 I - 1 , N * 2 7 4 4 4 1 SUM = SUM + XI ( M , I ) * * 2 * 2 7 6 * DO 4 4 2 I = 1 , N * 2 7 7 4 4 2 X I ( M , I ) = X I ( M , I ) / S Q R T ( S U M ) * 301 * IF ( M . E Q . N ) GO TO 82 * 3 0 4 DO 79 I = 1 , N * 3 0 5 79 X I ( N P 1 , 1 ) = X I ( M , I ) * 3 0 7 DO 80 KK = M,N * 3 1 0 * DO 80 I = 1 , N * 3 1 1 * 80 X I ( K K t l ) = X I ( K K + l , I ) 3 1 4 82 K = N * 315 JL = 2 * 316 * GO TO 7 3 1 7 * 450 GO TO ( 4 5 5 , 4 7 0 ) , J K 320 * 455 DO 4 6 0 I = I , N 321 * X X ( I ) = X ( I ) * 3 2 2 * SA = SN * 323 4 6 0 X ( I ) = X ( I ) + FAC*TOL 3 2 5 JK = 2 326 * GO TO 2 327 4 7 0 PMAX = 0 . 0 * 3 30 DO 471 I = 1 , N * POWELL ISN SOURCE STATEMENT FORTRAN SOURCE LIST POWELL 4.7 i o 331 PP = ABS(XX<I ) - X ( I ) ) * 332 * 471 IF (PP.GT.PMAX) PP = PMAX * 336 * IF (PMAX.LT.TOL*FACI) GO TO 500 * 341 * DO 475 I = 1,N * 342 XY(I) = X( I) 343 * SB = SN * 344 * IF (ABSI(SA-SB) / SB).LT.STL) GO TO 500 347 * 475 x i i N P i , ! ) = x i i ) - x x m * 351 SUM = 0.0 * 352 * DO 480 I = U N * 3 53 * 480 SUM = SUM + XI(NP1,I)**2 355 DO 481 I = U N 356 481 X I ( N P U I ) = XI (NP.1? I )/SQRT(SUM) * 360 * JK = 3 * 361 * GO TO 10 362 490 PMAX = 0.0 * 363 * DO 491 I = U N * 364 PP = ABS(XX(I) - X ( I )) * 365 * IF (PP.GT.PMAX) PMAX = PP 370 * PP - ABS (XY(I) - X(I)) t 371 * 491 IF (PP.GT.PMAX) PMAX = PP * 375 * IF (PMAX.LT.T0L*0.25) GO TO 500 400 * SC = SN * 401 * IF (ABS((SA-SC)/SC).LT.STL.OR.ABS((SB-SC)/SC).LT.STL) GO TO 500 404 * JK = 1 405 K = 2 * 406 DO 492 I = U N * 407 * 492 XI (1 , I ) = XI (NPU I ) * 411 GO TO 10 412 500 WRITE (6,501) SO,(X(I),I = 1,N) * 417 501 FORMAT(1H0,10H OPTIMUM =,F10.5,12H VARIABLES =t8F10.5/) * 420 CC = (1. + ALP(1)/ALP(2))/(ALP(l) + ALP(4)) * 421 * WRITE (6,502) CC * 422 * 502 FORMAT (6H L/G =,F12.5/) * 423 * WRITE (6,503) C0NUC0N2 * 424 503 FORMAT (4H C1=,E12.5,5H C2 =,E12.5/) - 425 * FAC = FAC*0.8 426 * DO 505 I = 1,12 427 * 505 W ( I ) = W(I)*0.1 * 431 CI = ABS(CONl) * 432 C2 = ABS(C0N2) 433 * C3 = C2/C1 * 434 * IF (C1.GT.C2) GO TO 206 * 437 RR = RR*10. * 440 IF (C3.LT.10.0) R = 10.*R/C3 * 443 * GO TO 207 4 44 * 206 R = R*10. 445 * IF (C3.GT.0.1) RR = 10.*RR*C3 * 450 * 207 IF (R.LT.1.E8.AND.RR.LT.1.E8) GO TO 98 453 * STOP 454 * END * 4. 8 17? FUNCTION F(X,N) 173 COMMON ALPtYF,Y01tY02tX01tX02,WtR»RR , R l,R2,Q , C O N l , C O N 2 174 DIMENSION X ( 2 0 ) , Y ( 2 0 ) , W ( 5 0 ) , A L P ( 2 0 ) 175 DO 5 I = i,N 176 5 Y d ) = X I I )**Q 177 A L P d ) = ( X l l ) - X ( 4 ) ) / ( Y 0 1 - Y ( l ) ) 200 A L P ( 2 ) = (X 12 ) - X 0 1 ) / ( Y ( 1 ) - Y ( 2 ) ) 201 A L P ( 3 ) = ( X ( 3 ) - X ( 2 ) ) / ( Y 0 2 - Y ( 3 ) ) 202 A L P ( 4 ) = ( X ( 4 ) - X 0 2 ) / ( Y ( 3 ) - Y ( 4 ) ) 203 CC = ( 1 . + A L P ( 1 ) / A L P ( 2 ) ) / ( A L P ( l ) + A L P ( 4 ) ) 204 C0N1 •= YF - ( A L P ( 1 ) * Y ( 2 ) + ALP ( 4) *Y ( 4 ) ) / ( ALP ( 1 ) + A L P ( 4 ) ) 205 CON2 = A L P ( 2 ) / A L P d ) - ALP ( 3 ) /ALP ( 4 ) 2C6 F = CC + R*C0N1**2 + RR*CON2**2 + R1*(W( 1 ) / X( 1 ) + W ( 2 ) / ( l . -X ( I ) ) _. _ _ 1+ W ( 3 ) / X ( 2 ) + W ( 4 ) / ( l . - X ( 2 ) ) + W ( 5 ) / X ( 3 ) + W ( 6 ) / ( l . - X ( 3 ) ) + W( 2 7 ) / X ( 4 ) + W ( 8 ) / ( l . - X ( 4 ) ) ) + R2* ( W (9 ) / A L P d ) + W ( 1 0 ) / A L P ( 2 ) + w ( i 3 1 ) / A L P ( 3 ) + W ( 1 2 ) / A L P ( 4 ) ) 207 RETURN 210 END . - - - SENTRY - - - - . - - - . APPENDIX V Pibonnaci Sub-program f o r One-dimensional Minimization 1. Nomenclature 2. Plow Diagram 3. FORTRAN IV L i s t 5.2 NOMENCLATURE LA LA = 1 Set s t a r t i n g point LA = 2 Set end point so that a minima occurs between the two points LA = 3 Determine objective function at points PP and PP i n i n t e r v a l . LA » 4;,5 Shrink i n t e r v a l u n t i l minimum step size i s reached SN Current value of objective function FX Value of objective function at s t a r t i n g point FF Minimum value of objective-function during end point search DELT Step size P Step taken from i n i t i a l point MS MS = 1 Bounds of problem have not been exceeded MS = 2 Bounds of problem have been exceeded FXP Objective function calculated a distance P from start PF End point of i n t e r v a l for Fibonacci search PM Point within i n t e r v a l closest to i n i t i a l point PD Point within Interval closest to end point FPM Value of objective function at PM FPD Value of objective function at PD PO End point of i n t e r v a l nearest i n i t i a l point POO Star t i n g point of i n t e r v a l LA = I SUBROUTINE F I B O ( X . N ) INITIALIZE VARIABLES 9 CALCULATE OBJECTIVE FUNCTION LA = 2 LA= 3 LA = 4 F X = SN F F : = F X P = DELT LA • = 2 YES YES FF = FXP DELT =1.618-DELT P = P + DELT PF = P PM = .381-PF PD =.619- PF P = PM LA = 3 DELT = . 6 I 9 D E L T P = DELT LA = 4 F P M = SN P = P D LA = • 4 YES YES WRITE OPTIMUM FPD = SN PF = P O PD = PM FPD : = FPM PM = PO + • 38l-(PF-PO) P = P M LA = 5 Y E S P O = P M PM = PD FPM = FPD PD = PO + .6I9-(PF-P0) P = PD LA = 4 RETURN 5.4 363 SUBROUTINE FIBO(X,N) 364 COMMON G,R,R1,R2,RR,W,YF,X01,X02,ALP,S,SO,JJ,Q,Y01,Y02,P,C0N1,C0N2 1,CC,Y,KCT 365 DIMENSION G ( 2 0 ) , A L P ( 2 0 ) , W ( 5 0 ) , X ( 2 0 ) , Y ( 2 0 ) , S 0 ( 2 0 ) , X I ( 2 0 ) , S ( 2 0 ) 366 GO TO (2,1) , J J 367 2 JJ= 2 370 READ (5,3) TOL,DEL,XMAX,XMIN,ALPMN 371 3 FORMAT (E10.5,4F10.5) 372 1 LA = 1 373 MS = 1 374 DELT = DEL 375 POO = 0.0 376 PO = 0.0 377 P = PO 400 DO 4 I = 1, N 401 4 X I ( I ) = X(I ) 402 SS = 0.0 403 DO 7 I = 1, N 404 7 SS = SS + S ( 1 ) * * 2 405 SS = SORT(SS) 406 DO 8 I = 1,N 407 8 SO(I) = S ( I ) / S S 410 9 00 10 .1 = If N 411 10 X ( I ) = XI ( I ) + P*SO(I) 412 DO 11 I = 1,N 413 IF (X(I).GT.XMIN.AND.XlI).LT.XMAX) GO TO 11 414 MS = 2 415 GO TO 99 416 11 CONTINUE 417 DO 5 I = 1,N 420 5 Y ( I ) = X ( I ) * * 0 421 A L P ( l ) = ( X ( l ) - X ( 4 ) ) / ( Y 0 1 - Y d ) ) 422 ALP(2) = ( X ( 2 ) - X 0 2 ) / ( Y d ) - Y ( 2 ) ) 423 ALP(3) = (X(3) - X ( 2 ) ) / ( Y 0 2 - Y ( 3 ) ) 424 ALP(4) = (X(4) - X 0 1 ) / ( Y ( 3 ) - Y ( 4 ) ) 425 DO 14 I = 1,N 426 IF (ALP(I).GT.ALPMN) GO TO 14 427 MS = 2 430 GO TO 99 431 14 CONTINUE 432 AL1 = l . / ( A L P ( l ) + A L P ( 4 ) ) 433 CC = (1. + A L P ( 1 ) / A L P ( 2 ) ) * A L 1 434 C0N1 = YF - ( A L P ( 1 ) * Y ( 2 ) + A L P ( 4 ) * Y ( 4 ) ) * A L 1 435 C0N2 = A L P ( 2 ) / A L P ( 1 ) - A L P ( 3 ) / A L P ( 4 ) 436 SI = 0.0 437 S2 = 0.0 440 DO 15 I = 1,N 441 SI = SI + l . / X d ) + l . / d . - X d ) ) 442 15 S2 = S2 + 1./ALP(I) 443 SN = CC + R*C0N1**2 + RR*C0N2**2 + R1*S1 + R2*S2 444 99 GO TO (200,250,300,350,400),LA 445 200 FX = SN 446 FF = FX 447 P = DELT 450 LA = 2 451 GO TO 9 452 250 IF (MS.E0.2) GO TO 260 453 FXP = SN 454 IF (FXP.GE.FF) GO TO 25 455 FF = FXP ^ J 456 DELT = 1.618*DELT 457 p = P + DELT 460 GO TO 9 461 25 PF = P  462 PM = 0.381#PF 463 PD = 0.619*PF 464 P = PM 465 LA = 3 466 GO TO 9 467 260 DELT = 0.619»DELT  470 P = DELT 471 MS = 1 472 GO TO 9 473 300 FPM = SN 474 P = PD 475 LA = 4  476 GO TO 9 477 350 FPD = SN 500 GO TO 410 501 400 FPM = SN 502 410 IF (FPD.GT.FPM) GO TO 45 503 PO = PM  504 PM = PD 505 FPM = FPD 506 PD = 0.619*(PF - PO) + PO 507 P = PD 510 LA = 4 511 40 IF ((PF - PO).LT.TOL) GO TO 100  512 GO TO 9 513 45 PF = PD 514 PD = P» 515 FPD = FPM 516 PM = 0.381*(PF - PO) + PO 517 P = PM  520 LA = 5 521 GO TO 40 522 100 IF ((PO - POO).GT.l.E-20) GO TO 105 523 RETURN 524 105 WRITE (6,320) CC , S N , ( X ( I ) , I = 1,N) 52 5 320 FORMAT (6H L/G =,F10.5,6H SN =,F14.5,12H VARIABLES = t8F10.5) 526 RETURN 527 END SENTRY 6.1 APPENDIX VI. One Dimensional Pattern Search Sub-program 1. Nomenclature 2. Flow Diagram 3. FORTRAN IV L i s t 6.2 NOMENCLATURE LA = 1 Make forward step. LA = 2 Make backward step I f forward step unsuccessful. I f forward step successful increase step s i z e . LA = 3 Shrink step size i f both forward and backward steps are f a i l u r e s . Return to forward step procedure I f a success occurs i n an immediately previous forward or backward step. Current value of objective function. Minimum value of objective function from previous ca l c u l a t i o n s . Step from i n i t i a l point that corresponds to SP. Step from i n i t i a l point that corresponds to SN. I n i t i a l step s i z e . Step size a f t e r i n i t i a l i z a t i o n . Step size acceleration factor. Step size deceleration factor. Minimum step **ftotal move r a t i o ? . : Normalized step - t o t a l move r a t i o . 6.3 SUBROUTINE PARMIN(X.N) INITIALIZE VARIABLES ) * CALCULATE OBJECTIVE FUNCTION LA = LA = 2 SP = SN C = P D = DEL LA = 2 P = P + D NO SP = SN C = P D = [ )• a LA = 1 STOP Y E S LA= 3 p = p - 2 0 LA = 3 LA = 2 P P - 2 D LA = 3 NO YES SP = SN C = P D = - D WRITE OPTIMUM _ NO RETURN 236 SUBROUTINE P ARM IN (X,N) 237 COMMON G,R,R1,R2,RR,W,YF,XO1,X02,ALP,S,SO,JJ,Q,YO1,Y02,P,CON1,CON2 1,CC,Y,KCT 240 DIMENSION G(20) ,A L P ( 2 0 ) , W ( 5 0 ) , X ( 2 0 ) , Y ( 2 0 ) , S O { 2 0 ) , X I ( 2 0 ) , S ( 2 0 ) 241 GO TO (2,1) , J J  242 2 JJ= 2 243 READ (5,3) TOL,ALPHA,BETA,DEL,XMAX,XMIN,ALPMN 244 3 FORMAT (E10.5,6F10.5) 245 1 LA = 1 246 DO 4 I = 1, N 247 4 XI ( I ) = X ( I )  250 P = 0 251 ' SS = 0.0 252 DO 7 I = 1,N 253 7 SS = SS + S ( I ) * * 2 254 SS = SQRT(SS) 255 DO 8 I = 1,N  256 8 SO(I) = S ( I ) / S S 257 9 DO 10 I = 1,N 260 10 X ( I ) = X I ( I ) + P*SO(I) 261 DO 11 I = 1,N 262 IF (X-(I).GT.XMIN.AND.XU).LT.XMAX) GO TO 11 263 SN = 1.E20  264 GO TO 99 265 11 CONTINUE 266 DO 5 I = 1,N 267 5 Y ( I ) = X ( I ) * * Q 270 A L P ( l ) = ( X ( l ) - X ( 4 ) ) / ( Y 0 1 - Y ( 1 ) ) 271 ALP(2) = (X(2) - X 0 2 ) / ( Y ( 1 ) - Y ( 2 ) )  272 ALP(3) = (X(3) - X ( 2 ) ) / ( Y 0 2 - Y ( 3 ) ) 273 ALP(4) = (X(4) - X 0 1 ) / ( Y ( 3 ) - Y ( 4 ) ) 274 DO 14 I = 1,N 275 IF (ALPd).GT.ALPMN) GO TO 14 276 SN = 1.E20 277 GO TO 99  300 14 CONTINUE 301 ALI = l . / ( A L P ( l ) + A L P ( 4 ) ) 302 CC = ( 1 . + ALP( D / A L P 1 2 ) )*AL1 303 CONl = YF - ( A L P ( l ) * Y ( 2 ) + A L P ( 4 ) * Y ( 4 ) ) * A L 1 304 C0N2 = A L P ( 2 ) / A L P ( 1 ) - A L P ( 3 ) / A L P ( 4 ) 305 SI = 0.0  306 S2 = 0.0 307 DO 15 I = 1,N 310 SI - SI + l . / X ( I ) + l . / ( l . - X ( I ) ) 311 15 S2 = S2 + l . / A L P d ) 312 SN = CC + R*C0N1**2 + RR*C0N2**2 + R1*S1 + R2*S2 313 99 GO TO ( 100,200,300), LA  314 100 SP = SN 315 C = P 316 D = DEL 317 110 LA = 2 320 P = P + D 321 120 IF (P.GT.O.O) GO TO 9  322 GO TO (121,122,310),LA 323 121 STOP 324 122 P = P - 2.*D 325 GO TO 120 326 200 IF (SN.LT.SP) GO TO 220 327 P = P - 2.*D  6.5 330 LA = 3 331 GO TO 120 332 220 SP - SN 333 C = P 334 D = 0* ALPHA  335 GO TO 110 336 300 IF (SN.GE.SP) GO TO 310 337 SP = SN 340 C - P 341 D = -D 342 GO TO 110  343 310 P = P + D 344 D = 0*8ETA 345 DX = ABS(P*TOL/D) 346 IF (l.O.LT.DX) GO TO 311 347 DX = ABS(1.E-20/D) 350 IF (l.O.LT.DX) GO TO 312  351 GO TO 110 352 312 WRITE (6,313) 353 313 FORMAT - (1H0,27HST0P OCCURS ON DX UNDERFLOW/) 354 KCT = 2 355 RETURN 356 311 SN = SP  357 WRITE (6,320) CC,SN, (X(I),I=1,N) 360 320 FORMAT(6H L/G =,F10.5,6H SN =,F14.5,12H VARIABLES =,8F10.5) 361 RETURN 362 END 7.1 APPENDIX VII. 1. Derivation of Crossflow Algorithm. 2. Formulation of Derivatives f o r Deflected Gradient Method. 1. Derivation of Crossflow Algorithm. 7.2 Referring to f i g . 3 and using the notation of Treybal (9) the adsorbent-solvent r a t i o s f o r an N stage crossflow network may be written as follows: L s i _ Y i - l ~ Y i ^ s ~ X i " X O i i = 1,...,N (Al) and the t o t a l adsorbent-solvent r a t i o as: N L s i H Y i - r Y i y 1=1 A l A 0 i (A2) Providing an equilibrium r e l a t i o n (isotherm) exists between the and the X i , the following equations hold at optimum: dY, dP _ aF . >F i _ n - i - i M I Wl " + T5T_ cLXT " 0 i - 1, • • • ,N-1 (A3) Performing the above operations on (A2) and writing i n terms of Y ± : Y i - Y i - i + <V*oi> 1 - x r x o i X i + l ~ X O i + l dY_ dX, ( A U ) i = 1,...,N-1 (A4) i s the most general form of the crossflow, algorithm f o r an N stage system. I f a l l i n i t i a l adsorbent streams have zero solute, then (A4) s i m p l i f i e s to: x i Y I = Y i - l + x i 1 - 7.3 i + i •g^i i = 1,...,N -1 ( A 5 ) I f the Freundlich Isotherm: Y I = , i = 1 ,. ..,N , i s used then ( A 5 ) may he written i n terms of X^ , xi+i> x i + 2 : i = 0,l,...,N - 2 (A6) where X Q i s a pseudo - concentration i n equilibrium xfith the YQ of the entering solvent stream. Knowing the f i n a l concentra t i o n i n the solvent stream^ Yp , and hence the concentration XJJ i n equilibrium with Yp , the X I , i = 0 , 1 , . . . , N - 3 can be computed i n the following manner: i . Guess an appropriate value for XJJ_^ ( X N _ I > X N ) * . i i . Using (A6) compute x J J_ 2 • U s e t n e equation again to compute X N _ -^ . Continue i n t h i s manner u n t i l XQ i s computed, i i i . I f the computed X Q i s less than a c e r t a i n tolerance of the expected value assume that the solution i s correct and go to i v . I f not, reset X N - 1 according to whether the computed X Q was larger or smaller than the expected value. Return to i i . and r e - i t e r a t e u n t i l a new X i s computed. i v . Determine F from ( A 2 ) where XQ^ = 0 , lA For the s p e c i a l case when n = 1 an a n a l y t i c a l s o l u t i o n f o r the optimum may be determined. (A6) becomes: X 2 X , a -pi±i i = o,l, ...,N-2 (A7) with YQ = XQ and Y P = XJJ . It can be shown that (A7) may be written i n terms of Y P and Y Q as follows: 1 N-i X i = ^F " Y 0 B r i = 0 , l , . . . , N (A8) Substituting (A8) into (A2) and simplifying, noting that a l l i n i t i a l adsorbent streams have zero solute: yN P = P N = N ^ - 1 (A9) F (A9) gives the optimum value of F for any value of N . Tak ing the l i m i t as N approaches i n f i n i t y : Y o l i m P w = i n ^ (A10) N-* OG F When YQ = 1.0 and Y P = 0.1, the l i m i t i n g optimum i s ln(10) . 2. Formulation of Derivatives For Deflected Gradient Method 7.5 Using the nomenclature of Treybal (9) and r e f e r r i n g to f i g . 10 the mass balances for each stage are written as follows: Y 0 1 G s l + V s 2 - Y l G s l + X l L s 2 (Bl) i i . Y l G s l + X Q 2 L s l - Y 2 G S 1 + X 2 L s l i i i . Y Q 2 G s 2 + X 2 L s l - Y 3 G s 2 + X 3 L s l i v . Y 3 G s 2 + X Q 1 L s 2 = Y 4 G s 2 + X 4 L g 2 Rearranging equations (Bl) as solvent - adsorbent r a t i o s i t -is possible to obtain: , „ G s l V X4 1 = L s 2 = Y^r YT G s l X2" X02 i l . a 2 _ - T - Z T — m « G s 2 V X 2 1 1 1 . a., = y - — = -w w - 5 L s l Y02 h i v a - _^2 _ X4- X01 Since i t i s desired that the average concentration i n the oujbput solution be a fix e d quantity the f i r s t constraint i s : s l s2 (B2) By using equations (B2) and rearranging, (Bj5) becomes: 7.6 ^Y^a^Y, Yf-, - t I r , = C0N1 (B4) (B4) i s the f i r s t equality constraint and i s denoted as C0N1 . The second equality constraint comes from the i n t e r - r e l a t i o n of equations (B2): ao °h of " % " C 0 N 2 ^ (^ 5) i s denoted as C0N2 . Inequality constraints are of a more ar b i t r a r y nature, depending on the user's d i r e c t i o n . However, to prevent absurd solutions the following contraints are necessary: a_ > 0 i = 1,...,4 (B6) 0 < X i < 1 i = 1,.. .,4 (;B7) Occasionally i t is necessary to put a bound on the ct^ to prevent the numerator and denominator from changing s i g n at + o©. The standard function to be minimized i s the adsorbent- solvent r a t i o : L _ L s l 4 " L s 2 (vn\ ( B 8 ) using equations (B2), (B8) may be written i n terms of : k = a - ^ r ( B 9 ) According to the' theorem of Courant (3) and the method prescribed by C a r r o l l (2) the t o t a l objective function i s written: F = § + R-C0N1 2 + RR-C0N2 2 +.R1.- 4 7.7 - 1 - T T - / + R2* v -±- (BIO) where R,RR,R1 , and ' R2 are p e n a l t y f a c t o r s , the f i r s t p a i r t o be v a r i e d a c c o r d i n g t o t h e theorem, o f Co u r a n t , and t h e second p a i r a c c o r d i n g t o t h e method p r e s c r i b e d by C a r r o l l . U s i n g the c h a i n r u l e the g e n e r a l f o r m u l a t i o n of t h e g r a d i e n t v e c t o r i s .as f o l l o w s : ' dF dX *F 1 k-1 a a k -t-aX, }Y, dX d Y i \ + >F dY *Y. d X ± 1 , >F + *X, i = 1. • 3 ( B l l ) where the dF/dX^. are the components o f t h e g r a d i e n t v e c t o r , For t h e pr o b l e m o f f i g . 10 most o f the components o f ( B l l ) v a n i s h . The p a r t i a l d e r i v a t i v e s o f F w i t h r e s p e c t t o t h e a r e : ^a, i T2 1 (a ±+a k) c a2~ 1 + 2-R-C0Nl.a i,.(Y J +-Y 2) 2«RR.C0N2.a 2-R2 ^F r>Ct, cc, 1 ( 7 7 % r - + R2> f 2RRC0N a 2 ' ^ a l + a 4 a 1 2-RR-C0N2 _ R2 a (B12) (c^+c^)' a. 1 + ~ + 2«R-C0Nla 1.(Y^-Y 2) 2-RR.C0N2.ct -R2 + — ^ 2 a t The p a r t i a l s o f F w i t h r e s p e c t t o the Y. a r e : 2'R.CONl'Q^ ?»Y2 = a 1+a i | " -3Y 4 " o^+a^ *Y 2 a x The other ?»P/^Yi are zero. The p a r t i a l s of the respect to the are _ _ L _ 1 _ * X 1 Y 0 1 " Y 1 * X4 Y 0 1 " Y 1 "^1 . a 2 " Y 1 " Y 2 * X 2 Y 0 2 ~ Y 3 * X 3 Y 0 2 _ i 3 "'"2 > x 4 Y 3 ~ Y i | The other xcu^/^X^ are zero. The p a r t i a l s of the respect to the Y. are: "^ T " ( Y Q 1 - Y 1 ) 2 " a i ^ X l 7.9 TSOL, X2~ X02 ( Y R Y 2 ) " a 2 PCJ ^ a 2 * Y ^ X2" X02 ( Y R Y 2 ) 2 * Y 1 x r x s ( Y Q 2 - Y 3 ) a. 3 (B15) ^ 4 X 4 " X 0 1 _ ( Y 3 - V X 4~ X 01 The other ^ ( \ / ^ Y i are zero. The p a r t l a l s of F with respect to the X^ are 7sF = Rl- ( l " X i ) ' 1 o i = l , . . . , 1 * - (B16) Eliminating the elements of equations (B12) to (Bl6) that are zero, equations ( B l l ) become: dF _ ?F aT7_ ~ ^ fcct^ ?,a1 dY 1 >a2 d Y l + V 3 2 TF7_ dT7_ d|_ = M _ + >F_ dX 2 ^X 2 ?»a2 * a2 d Y 2 *x2 + * Y 2 "ax; + *F_ ^ , >F  d Y 2 aa^ *X 2 TTZ dX 2 (B17) dF = ^F dX^ pT7 + aa-j s o u d y > a 4 *Y?; dX^ 7 . 1 0 dF _ *F + _____ ? a l + _____ aajj dY^ ^ X i , "^ YT dX\. Equations (B17) now are the components of the gradient vector, When the Freundlich; isotherm i s used: 1 - v n ~ l •ax" = n * x i (B18) When the Koble - Corrigan isotherm i s used: d Y i n-Y i(l+m n) _ __ X_(l+m n(l-X 1)) (B19) APPENDIX VIII Equations of Isotherms Used. 8.2 P r e u n d l i c h Isotherm The general equation f o r the F r e u n d l i c h Isotherm i s w r i t t e n : 5 - c n where » = c o n c e n t r a t i o n o f s o l u t e i n f l u i d , l b . s o l u t e / l b . s o l v e n t . n = c o n c e n t r a t i o n of adsorbate, l b . s o l u t e adsorbed/lb. adsorbent. n = exponent (n > 0) I f n' = max ri , then = max F . The F r e u n d l i c h Isotherm may then be w r i t t e n i n the dimensionless form: Y = I , - sifL = X n < c r , ' n where: Y = c o n c e n t r a t i o n of s o l u t e In f l u i d , dimensionless (° < Y < 1 ) • X = c o n c e n t r a t i o n of adsorbate, dimensionless (0 < X < 1). Koble - Corrigan Isotherm Applying the same procedure as f o r the F r e u n d l i c h isotherm, the dimensionless Koble '- Corrigan isotherm becomes: _n Y = where: _ l + m 1 / n ( l - x)_ m = parameter (tt > 0) n = exponent (n > 0) 

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