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The simulation and optimization of a gasoline polymerization plant Friedman, Paul 1971

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THE SIMULATION AND OPTIMIZATION OF A GASOLINE POLYMERIZATION PLANT by PAUL FRIEDMAN B . C h . E . , C i t y C o l l e g e of New Y o r k , 1959 A T H E S I S SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILMENT OF FOR THE DEGREE OF PHILOSOPHY i n the Department of CHEMICAL ENGINEERING We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BR I T I S H COLUMBIA D e c e m b e r , 1971 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of C V ^ e,>^~tc.es.\ ^ ^Q^C-^<2^a-<rC^v^ The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date a_c_«_ Supervisor: Dr. K. L. Pinder ABSTRACT A m o d e l o f a c h e m i c a l p r o c e s s p l a n t w i t h r e c y c l e s t r e a m s was s i m u l a t e d on a d i g i t a l c o m p u t e r u s i n g t h e CHESS e x e c u t i v e p r o g r a m . T h i s m o d e l was t h e n o p t i m i z e d by means o f an a u t o m a t i c o p t i m i z a t i o n a l g o r i t h m . The c h e m i c a l p l a n t c h o s e n f o r t h e s t u d y was t h e g a s o l i n e p o l y m e r i z a t i o n u n i t o f t h e S h e l l b u r n r e f i n e r y , S h e l l C a n a d a L i m i t e d , i n B u r n a b y , B . C . M o d i f i c a t i o n s w e r e made t o t h e f o l l o w i n g o p t i m i z a -t i o n t e c h n i q u e s so t h a t t h e y c o u l d be u t i l i z e d f o r t h e s t a t e d p r o b l e m : D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e , P a t t e r n S e a r c h , C o m p l e x M e t h o d . I t was f o u n d t h a t t h e M o d i f i e d C o m p l e x M e t h o d w o r k e d w e l l and d i d n o t r e q u i r e an e x c e s s i v e a m o u n t o f c o m p u t e r t i m e . A s t r a t e g y o f m o d e l b u i l d i n g was u t i l i z e d t o r e d u c e t h e c o m p u t e r c a l c u l a t i o n t i m e f o r a s i m u l a t i o n t o a m i n i m u m a n d a t t h e same t i m e c r e a t e a m o d e l t h a t w o u l d a d e q u a t e l y r e p r e s e n t t h e i m p o r t a n t a s p e c t s o f p l a n t o p e r a t i o n . The r e s u l t s o f t h e p l a n t m o d e l s i m u l a t i o n w e r e w i t h i n t h e r a n g e o f r e p o r t e d p l a n t d a t a . In o r d e r t o be a b l e t o d e v e l o p a r e a s o n a b l e m o d e l o f t h e p o l y m e r i z a t i o n r e a c t o r , an e x p e r i m e n t a l k i n e t i c s t u d y was made o f t h e p o l y m e r i z a t i o n o f m i x e d o l e f i n s t o t h e i r d i m e r i z e d p r o d u c t s u t i l i z i n g t h e U . O . P . s o l i d p h o s p h o r i c a c i d c a t a l y s t . A g e n e r a l i z e d r a t e e x p r e s s i o n was d e v e l o p e d w h i c h f i t t e d t h e e x p e r i m e n t a l d a t a v e r y w e l l ; r = k c l ^ l i ° ( l + x ) 2 a n d , -7540 k = 2 . 8 7 x 1 0 5 e R T w h e r e r = r a t e o f r e a c t i o n , m o l e o l e f i n / h r / c c c a t a l y s t , k = r e a c t i o n r a t e c o n s t a n t , c c o l e f i n / h r / c c c a t a l y s t , x = f r a c t i o n o f o l e f i n s c o n v e r t e d i n t o p r o d u c t . Gas f i l m r e s i s t a n c e e f f e c t s w e r e n o t o b s e r v e d b u t p o r e d i f f u s i o n r e s i s t a n c e was p r e s e n t . The o p t i m i z a t i o n r e s u l t s s h o w e d t h a t h i g h e r t o t a l o l e f i n c o n v e r s i o n s c o u l d be a c h i e v e d i f s t r i c t e r t e m p e r a t u r e c o n t r o l s on t h e c a t a l y s t b e d s w e r e m a i n t a i n e d by means o f t h e l i q u i d p r o p a n e q u e n c h . t l i TABLE OF CONTENTS Page ABSTRACT i i L I S T OF TABLES v i i i L I S T OF FIGURES x i ACKNOWLEDGEMENTS x i v C h a p t e r 1 INTRODUCTION 1 A. P u r p o s e and S c o p e 1 B. D i g i t a l C o m p u t e r S i m u l a t i o n and O p t i m i -z a t i o n o f C h e m i c a l P r o c e s s e s 2 1. C h e m i c a l P r o c e s s S i m u l a t i o n S y s t e m s . . . 2 2. CHESS — C h e m i c a l E n g i n e e r i n g S i m u l a -1 a t i o n S y s t e m . . 7 a. S y s t e m d e s c r i p t i o n . . . 7 b. S y s t e m e x e c u t i v e p r o g r a m s 9 c. S y s t e m s u p p o r t i n g p r o g r a m s 10 d. P r o c e s s r e p r e s e n t a t i o n 19 e. M a t e r i a l and h e a t b a l a n c i n g 20 3. P r o c e s s S i m u l a t i o n s R e p o r t e d i n t h e L i t e r a t u r e 21 4. C h e m i c a l P r o c e s s O p t i m i z a t i o n R e p o r t e d i n t h e L i t e r a t u r e 22 i v C h a p t e r Page C. The U.O.P. C a t a l y t i c P o l y m e r i z a t i o n P r o c e s s 2 9 1. H i s t o r y 2 9 2. P r o c e s s D e s c r i p t i o n 31 3. P r o c e s s V a r i a b l e s 34 D. The K i n e t i c s o f t h e P o l y m e r i z a t i o n o f O l e f i n s - P r e v i o u s Work 4 0 2 THE K I N E T I C S OF THE POLYMERIZATION OF O L E F I N S IN THE PRODUCTION OF POLYMER GASOLINE. . . . 4 7 A. I n t r o d u c t i o n 4 7 B. E x p e r i m e n t a l A p p a r a t u s 4 8 C. E x p e r i m e n t a l P r o c e d u r e 5 2 D. E x p e r i m e n t a l R e s u l t s 5 3 E. D i s c u s s i o n 7 0 F. C o n c l u s i o n s 7^ 3 THE IMPLEMENTATION OF CHESS FOR THE SIMULATION OF THE POLYMERIZATION PLANT 73 A. S t r a t e g y f o r t h e D e v e l o p m e n t o f t h e S i m u l a t i o n Model 73 B. C o m p l e t e P l a n t M o d e l s 76 1. P r e l i m i n a r y M o d e l 76 2. I n t e r m e d i a t e M o d e l . 85 a. S y s t e m C o n v e r g e n c e E r r o r . . . . . 91 3. F i n a l M o d e l 101 v C h a p t e r Page C. P r o c e s s E q u i p m e n t M o d e l s 104 1. P o l y m e r i z a t i o n R e a c t o r 104 2. D e p r o p a n i z e r - D e b u t a n i z e r 113 3. S i m p l e H e a t E x c h a n g e r 131 D. D i f f i c u l t i e s i n CHESS I m p l e m e n t a t i o n . . . . 132 4 OPTIMIZATION OF THE POLYMERIZATION PLANT SIMULATION MODEL 136 A. The O p t i m i z a t i o n P r o b l e m 136 B. The D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e M e t h o d . 140 1. The D e f l e c t e d G r a d i e n t M e t h o d f o r U n c o n s t r a i n e d N o n l i n e a r O p t i m i -z a t i o n 140 2. The C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e . . 146 3. I m p l e m e n t a t i o n o f S e a r c h M e t h o d 152 4. R e s u l t s w i t h I n t e r m e d i a t e P l a n t M o d e l . . 158 C. C o n s t r a i n e d P a t t e r n S e a r c h 177 1. The P a t t e r n S e a r c h M e t h o d f o r U n c o n -s t r a i n e d N o n l i n e a r O p t i m i z a t i o n . . . . 177 2. C o n s t r a i n e d P a t t e r n S e a r c h 183 3. R e s u l t s w i t h I n t e r m e d i a t e P l a n t M o d e l . . 189 D. The M o d i f i e d Complex M e t h o d 200 1. The Complex M e t h o d 200 2. M o d i f i c a t i o n s o f t h e C o m p l e x M e t h o d . . . 206 v i C h a p t e r Page 3. R e s u l t s w i t h I n t e r m e d i a t e P l a n t M odel 210 E. C o m p a r i s o n o f O p t i m i z a t i o n M e t h o d s 220 F. O p t i m i z a t i o n o f F i n a l P l a n t Model 223 5 CONCLUSIONS 234 6 RECOMMENDATIONS 236 L I S T OF REFERENCES 2 3 8 NOMENCLATURE 2 4 5 APPENDICES: A P o l y m e r i z a t i o n R e a c t o r P l a n t T e s t s 249 B C a l i b r a t i o n C u r v e s and E x p e r i m e n t a l E q u i p m e n t S p e c i f i c a t i o n s 258 C C a l c u l a t i o n s o f Mass T r a n s f e r E f f e c t . . . . 264 D C o m p u t e r P r o g r a m s o f C o m p l e t e P l a n t and P r o c e s s M o d e l s 272 E CHESS O u t p u t f o r F i n a l P o l y m e r i z a t i o n P l a n t Model 281 F E s t i m a t i o n o f H e a t o f R e a c t i o n f o r P o l y m e r i z a t i o n o f O l e f i n s 294 G C h a n g e s Made i n CHESS S o u r c e P r o g r a m f o r u s e w i t h WATFIV C o m p i l e r . 308 H C o m p u t e r P r o g r a m s o f O p t i m i z a t i o n M e t h o d s . . 316 v i i L I S T OF TABLES T a b l e Page 1-1 S i m u l a t i o n S y s t e m s f o r C h e m i c a l P r o c e s s e s . . . . 4 1-2 D e s c r i p t i o n o f CHESS E x e c u t i v e P r o g r a m s 12 1-3 D e s c r i p t i o n o f CHESS S u p p o r t i n g F u n c t i o n s . . . . 15 1-4 P u r e Components i n CHESS T h e r m o d y n a m i c P a c k a g e . . 18 1-5 Range o f F e e d C o m p o s i t i o n s 33 1-6 S p e c i f i c a t i o n s and T y p i c a l C o m p o s i t i o n s o f P r o d u c t s 35 2-1 O l e f i n C o n v e r s i o n a t D i f f e r e n t F l o w R a t e s and T e m p e r a t u r e s 54 2-2 N u m e r i c a l V a l u e s f o r I n t e g r a t e d R a t e E q u a t i o n s 5 7 2-3 R e a c t i o n R a t e C o n s t a n t o f E q u a t i o n ( 2 - 2 ) a t D i f f e r e n t T e m p e r a t u r e s 61 2-4 O l e f i n C o n v e r s i o n s a t V a r y i n g F l o w R a t e s and T e m p e r a t u r e s a t C o n s t a n t V/F R a t i o s . . . . 63 2-5 O l e f i n C o n v e r s i o n as a F u n c t i o n o f P a r t i c l e S i z e 66 2-6 C o n v e r s i o n o f I n d i v i d u a l O l e f i n s 68 3-1 P o l y F e e d S p e c i f i c a t i o n f o r P r e l i m i n a r y P o l y m e r i z a t i o n P l a n t Model 80 3-2 E q u i p m e n t P a r a m e t e r S p e c i f i c a t i o n s f o r P r e l i m i n a r y P o l y m e r i z a t i o n P l a n t M o d e l . . . . 8 2 v m T a b l e Page 3-3 O v e r a l l M a t e r i a l B a l a n c e f o r P r e l i m i n a r y P o l y m e r i z a t i o n M o d e l 84 3-4 E f f e c t o f Number o f R u n g e - K u t t a C a l c u l a t i o n S t e p s on P r o c e s s C a l c u l a t i o n s 87 3-5 V a l u e s o f I n d e p e n d e n t V a r i a b l e s f o r T a b l e 3-6 . . 93 3-6 C o n s t r a i n t and O b j e c t i v e F u n c t i o n V a l u e s f o r R e p e a t e d S i m u l a t i o n S y s t e m C a l c u l a t i o n s . . . . 94 3-7 E f f e c t o f T e m p e r a t u r e o f S t r e a m 13 on P r o c e s s C a l c u l a t i o n s . 1 0 5 3-8 F i t o f R e a c t o r Model t o P l a n t D a t a 1 1 2 3-9 P r o d u c t S p e c i f i c a t i o n s and T y p i c a l O p e r a t i n g C o n d i t i o n s f o r D e p r o p a n i z e r and D e b u t a n i z e r 127 3-10 R e s u l t s o f T e s t s on D e p r o p a n i z e r -D e b u t a n i z e r M o d e l 129 4-1 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t M o d e l U s i n g D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e M e t h o d - 1 160 4-2 " " " " -2. .169 4-3 " " " " - 3 . .170 4-4 " " " " -4. .173 4-5 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t M odel U s i n g t h e C o n s t r a i n e d P a t t e r n S e a r c h M e t h o d - 1 192 4-6 " " " -2. .195 4-7 " " " - 3 . .198 4-8 M o d i f i c a t i o n s o f t h e C o m p l e x M e t h o d 207 i x T a b ! e Page 4-9 The Use o f t h e M o d i f i e d C o m p l e x M e t h o d f o r t h e O p t i m i z a t i o n o f t h e I n t e r m e -d i a t e P o l y P l a n t Model - 1 214 4-10 " " " " " -2 216 4-11 " " " " " -3 218 4-12 C o m p a r i s o n o f O p t i m i z a t i o n M e t h o d s T e s t e d . . . . 221 4-13 C o m p a r i s o n o f R e s u l t s w i t h I n t e r m e d i a t e and F i n a l P l a n t M o d e l s 226 4-14 R e s u l t s o f O p t i m i z a t i o n U s i n g T o t a l M o l e s o f P o l y m e r G a s o l i n e as O b j e c t i v e F u n c t i o n . . . . 228 4-15 V a l u e s o f D i f f e r e n t O b j e c t i v e F u n c t i o n s i n F i n a l P l a n t M o d e l O p t i m i z a t i o n 230 4-16 A f f e c t o f T e m p e r a t u r e o f S t r e a m 4 on Optimum C o n v e r s i o n 232 A - l D a t a f r o m P o l y m e r i z a t i o n R e a c t o r P l a n t s T e s t s . . 253 F - l O l e f i n s i n F e e d and P o s s i b l e P r o d u c t s 295 F-2 S t a n d a r d H e a t s o f F o r m a t i o n and C o m b u s t i o n f o r Some O l e f i n s 297 x L I S T OF FIGURES F i g u r e Page 1-1 CHESS - O v e r a l l P r o g r a m L i n k a g e 11 1-2 F l o w D i a g r a m o f S h e l l b u r n P o l y m e r i z a t i o n U n i t . . 32 1-3 O l e f i n C o n v e r s i o n v s . S p a c e V e l o c i t y . . . . . . 38 1-4 O l e f i n C o n v e r s i o n v s . T o t a l P r o d u c t i o n 39 1-5 E f f e c t s o f P h o s p h o r i c A c i d C o n c e n t r a t i o n and T e m p e r a t u r e on R e a c t i o n R a t e C o n s t a n t 41 1-6 E f f e c t s o f Q u a r t z P a r t i c l e S i z e and P r o p e n e / N - B u t e n e R a t i o on R e a c t i o n R a t e C o n s t a n t . . 42 2-1 E x p e r i m e n t a l A p p a r a t u s . 2-2 E f f e c t o f F l o w R a t e and T e m p e r a t u r e on O l e f i n C o n v e r s i o n 2-3 L i n e a r i t y T e s t s f o r D i f f e r e n t R a t e E q u a t i o n s . 2-4 F i t o f E q u a t i o n ( 2 - 2 ) t o E x p e r i m e n t a l D a t a 2-5 E f f e c t o f T e m p e r a t u r e on R e a c t i o n R a t e C o n s t a n t 62 2-6 T e s t f o r Mass T r a n s f e r E f f e c t 64 2-7 C o n v e r s i o n o f I n d i v i d u a l O l e f i n s a t D i f f e r e n t T o t a l O l e f i n C o n v e r s i o n s 69 49 55 58 59 x i F i g u r e Page 3-1 P r e l i m i n a r y P o l y m e r i z a t i o n P l a n t M o d e l 78 3-2 I n t e r m e d i a t e P o l y m e r i z a t i o n P l a n t M o d e l 90 3-3 O b j e c t i v e F u n c t i o n V a l u e s f o r R e p e a t e d C a l c u l a t i o n s 99 3-4 F i n a l P o l y m e r i z a t i o n P l a n t Model 102 3-5 P o l y m e r i z a t i o n R e a c t o r Bed T e m p e r a t u r e v s . T i m e 109 3-6 C o m p u t e r F l o w D i a g r a m f o r P o l y m e r i z a t i o n R e a c t o r - ADD3 3-7 Maximum O v e r a l l C o n v e r s i o n o f O l e f i n s v s . T i m e 3-8 E x i t Bed T e m p e r a t u r e v s . Time f o r Maximum O l e f i n C o n v e r s i o n 3-9 R e c y c l e F l o w S p l i t s v s . Time f o r Maximum O l e f i n C o n v e r s i o n 3-10 C o m p u t e r F l o w D i a g r a m f o r D e p r o p a n i z e r -D e b u t a n i z e r - ADD4 3-11 C o m p u t e r F l o w D i a g r a m f o r S i m p l e H e a t E x c h a n g e r - ADD5 4-1 C o m p u t e r F l o w D i a g r a m o f D e f l e c t e d G r a d i e n t M e t h o d 147 4-2 C r e a t e d R e s p o n s e S u r f a c e 150 4-3 FP P r o g r a m L i n k a g e 153 4-4 C o m p u t e r F l o w D i a g r a m f o r P a t t e r n S e a r c h M e t h o d 180 4-5 C o m p u t e r F l o w D i a g r a m f o r C o n s t r a i n e d P a t t e r n S e a r c h 184 x i i F i g u r e Page 4-6 CONPAT P r o g r a m L i n k a g e 186 4-7 E x a m p l e s o f C o n s t r a i n e d P a t t e r n S e a r c h 188 4-8 C o m p u t e r Flow D i a g r a m o f t h e Complex M e t h o d 204 4-9 C o m p u t e r Flow D i a g r a m o f t h e M o d i f i e d Compl ex M e t h o d 211 4-10 M0C0MP P r o g r a m L i n k a g e 212 A - l F i r s t Bed o f E a s t P o l y m e r i z a t i o n R e a c t o r -S h e l l b u r n R e f i n e r y 251 A-2 P l a n t R e a c t o r R a t e T e s t - L i q u i d P h o s p h o r i c A c i d Model 256 A-3 P l a n t R e a c t o r R a t e T e s t - F i r s t O r d e r M e c h a n i s m 257 B - l M i l r o y a l D C a l i b r a t i o n C u r v e 259 B-2 R o t a m e t e r C a l i b r a t i o n 260 B-3 E x p e r i m e n t a l R e a c t o r . . . . . 261 B-4 E x p e r i m e n t a l R e a c t o r I n t e r n a l s 262 B-5 T y p i c a l C h r o m a t o g r a m o f F e e d S ample 263 x i i i r ACKNOWLEDGEMENTS I w i s h t o t h a n k D r . K e n n e t h P i n d e r , u n d e r whom t h i s i n v e s t i g a t i o n was c o n d u c t e d f o r h i s g u i d a n c e ' i n h e l p i n g t o c a r r y o u t t h i s p r o j e c t . I a l s o w i s h t o t h a n k t h e s t a f f o f S h e l l b u r n R e f i n e r y , S h e l l O i l Company o f C a n a d a who have been v e r y h e l p f u l i n s u p p l y i n g i n f o r m a t i o n as w e l l as c a t a l y s t and f e e d s t o c k f o r t h e e x p e r i m e n t s . I am i n d e b t e d t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Ca n a d a and t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r t h e i r f i n a n c i a l s u p p o r t o f t h i s r e s e a r c h . I w o u l d a l s o l i k e t o t h a n k my p a r e n t s , Mr. and M r s . Max F r i e d m a n , f o r t h e i r c o n t i n u o u s s u p p o r t t h r o u g h o u t t h i s w o r k . I w o u l d l i k e t o e x p r e s s my d e e p e s t g r a t i t u d e t o my w i f e I s a b e l f o r a l l t h e s a c r i f i c e s s h e has made so t h a t I c o u l d u n d e r t a k e and c o m p l e t e t h i s p r o j e c t . x i v 1 C h a p t e r 1 INTRODUCTION A. P u r p o s e and S c o p e The p u r p o s e o f t h i s work was t o s t u d y t h e u t i l i -z a t i o n o f a u t o m a t i c m e t h o d s o f o p t i m i z a t i o n on a d i g i t a l c o m p u t e r s i m u l a t e d model o f a c h e m i c a l p r o c e s s p l a n t . An e x e c u t i v e c h e m i c a l p r o c e s s s i m u l a t i o n s y s t e m was u s e d f o r t h e s i m u l a t i o n o f t h e p r o c e s s p l a n t so t h a t t h e m e t h o d -o l o g y e v o l v e d i n t h e s t u d y w o u l d be o f g e n e r a l u s e . The p r o c e s s p l a n t c h o s e n f o r t h e s t u d y was t h e g a s o l i n e p o l y m e r i z a t i o n u n i t o f t h e S h e l l b u r n R e f i n e r y , S h e l l C a n a d a L i m i t e d , i n B u r n a b y , B.C. A l t h o u g h , a l a r g e amount o f p l a n t i n f o r m a t i o n was a v a i l a b l e on t h i s u n i t , i t was n e c e s s a r y t o u n d e r t a k e an e x p e r i m e n t a l i n v e s t i g a t i o n o f t h e k i n e t i c s o f t h e p o l y m e r i z a t i o n r e a c t i o n so t h a t t h e s i m u l a t e d model m i g h t g i v e an a d e q u a t e r e p r e s e n t a t i o n o f t h e p l a n t d a t a . E ven t h o u g h e x a c t d a t a were n o t a v a i l a b l e on c e r t a i n a s p e c t s o f p l a n t o p e r a t i o n , s u c h as c a t a l y s t p o i s o n i n g , i t was f e l t t h a t t h e m o d e l s d e v e l o p e d a d e q u a t e l y d e s c r i b e d t h e m o st i m p o r t a n t a s p e c t s o f t h e p l a n t . 2 In t h e p a s t , p l a n t m o d e l s , s i m u l a t e d on c h e m i c a l p r o c e s s s i m u l a t i o n s y s t e m s , have b e e n u t i l i z e d f o r p r o c e s s o p t i m i z a t i o n by means o f c a s e s t u d i e s . A u t o m a t i c o p t i m i z a -t i o n m e t h o d s have n o t bee n u s e d b e c a u s e o f t h e l o n g com-p u t e r c a l c u l a t i o n t i m e n e c e s s a r y f o r t h e p r o c e s s s i m u l a t i o n . In t h i s w o r k , a s t r a t e g y o f model d e v e l o p m e n t was u t i l i z e d t o m i n i m i z e c o m p u t e r t i m e and t h u s a l l o w e d t h e u s e o f a u t o -m a t i c o p t i m i z a t i o n m e t h o d s , w i t h o u t e x c e s s i v e u s e o f c o m p u t e r t i m e . T h r e e o p t i m i z a t i o n m e t h o d s were t e s t e d : 1. D e f l e c t e d . G r a d i e n t - C r e a t e d Response S u r f a c e Method, 2. C o n s t r a i n e d P a t t e r n S e a r c h , 3. M o d i f i e d Complex Method. E x t e n s i v e m o d i f i c a t i o n s w e re made t o t h e o p t i m i z a t i o n a l g o r -i t h m s so t h a t t h e y c o u l d be s u c c e s s f u l l y u s e d f o r t h e c h e m i c a l p l a n t model o p t i m i z a t i o n . B. D i g i t a l C o m p u t e r S i m u l a t i o n o f C h e m i c a l P r o c e s s e s 1. C h e m i c a l P r o c e s s S i m u l a t i o n S y s t e m s In r e c e n t y e a r s , s y s t e m s h a v e been d e v e l o p e d f o r t h e s i m u l a t i o n o f t h e s t e a d y - s t a t e o p e r a t i o n o f c h e m i c a l p r o c e s s e s w h i c h may c o n s i s t o f many i n t e r c o n n e c t e d p r o c e s s -i n g u n i t s w i t h r e c y c l e s t r e a m s . E v a n s et al. ( 1 ) r e v i e w s s e v e r a l o f t h e s e p r o g r a m s w h i c h a r e among t h o s e l i s t e d i n T a b l e 1-1. Many o f t h e s e s y s t e m s c o n t a i n common c h a r a c -t e r i s t i c s w h i c h w i l l now be d i s c u s s e d b r i e f l y . A more d e t a i l e d d e s c r i p t i o n w i l l t h e n be made o f CHESS, t h e s y s t e m u t i l i z e d i n t h i s s t u d y . A c h e m i c a l p r o c e s s s y s t e m may be d i v i d e d i n t o s e v e r a l p a r t s . The f i r s t i s t h e e x e c u t i v e s y s t e m w h i c h t r a n s m i t s i n f o r m a t i o n t h r o u g h t h e p r o c e s s s t r e a m s and s t o r e s t h e c a l c u l a t e d r e s u l t s . The e x e c u t i v e w i l l r e a d t h e i n p u t d a t a , c a l l t h e u n i t c o m p u t a t i o n s i n a c c o r d a n c e w i t h some c a l c u l a t i o n s e q u e n c e , u t i l i z e c o n v e r g e n c e r o u t i n e s f o r m a t e r i a l and h e a t b a l a n c i n g , and p r i n t o u t t h e r e s u l t s . The s e c o n d p a r t o f a s i m u l a t i o n s y s t e m c o n s i s t s o f t h e u n i t c o m p u t a t i o n s , w h i c h d e s c r i b e t h e p i e c e s o f e q u i p m e n t i n t h e p l a n t . Some s y s t e m s a l s o c o n t a i n p r o g r a m s f o r t h e c a l c u l a t i o n o f t h e p h y s i c a l p r o p e r t i e s o f t h e s t r e a m s The a b o v e m e n t i o n e d p a r t s o f t h e s y s t e m a r e o f g e n e r a l a p p l i c a b i l i t y . F o r a g i v e n c a s e , a s e t o f d a t a must be s u p p l i e d d e s c r i b i n g t h e i n i t i a l s t r e a m v a r i a b l e s and t h e e q u i p m e n t p a r a m e t e r s t o be u s e d i n t h e u n i t compu-t a t i o n s . A c o m p a r i s o n i s made b e t w e e n some o f t h e c h e m i c a l p r o c e s s s i m u l a t i o n s y s t e m s l i s t e d i n T a b l e 1-1 by S t e w a r d ( 1 4 ) and E v a n s , et al. ( 1 ) . An e x c e l l e n t d i s c u s s i o n on t h e u s e o f a s i m u l a t i o n s y s t e m , i s g i v e n by C r o w e , et al. ( 1 5 ) . TABLE 1-1 S i m u l a t i o n S y s t e m s f o r C h e m i c a l P r o c e s s e s S h o r t Name F u l l Name I n s t i t u t i o n ( s ) w h e r e D e v e l o p e d R e f e r e n c e s S o u r c e 1 L a n g u a g e APACHE A p p l i c a t i o n P a c k a g e f o r C h e m i c a l E n g i n e e r s G e n e r a l E l e c t r i c 16 CHEOPS C h e m i c a l E n g i n e e r i n g O p t i m i z a t i o n S y s t e m S h e l l D e v e l o p m e n t Co. 2 FORTRAN I I , FAP CHESS C h e m i c a l E n g i n e e r i n g Simu-l a t i o n S y s t e m U n i v . o f H o u s t o n 3 FORTRAN IV C h e v r o n S y s t e m G e n e r a l i z e d H e a t and M a t e r i a l B a l a n c i n g S y s t e m C h e v r o n R e s e a r c h Co. 4 FORTRAN I I . FAP CHIPS C h e m i c a l E n g i n e e r i n g I n f o r -m a t i o n P r o c e s s i n g S y s t e m S e r v i c e B u r e a u C o r p . 5,6 FORTRAN DISCOSSA D i s c O r i e n t a t e d O r e g o n S t a t e S y s t e m s A n a l y z e r O r e g o n S t a t e U n i v . 7 FLOWPACK ICI P r o c e s s F l o w s h e e t i n g P a c k a g e I m p e r i a l C h e m i c a l I n d . L t d . 8 FORTRAN FLOWTRAN M o n s a n t o 16 GEMCS G e n e r a l E l e c t r i c - M c M a s t e r S i m u l a t o r M c M a s t e r U n i v . 11 FORTRAN T a b l e 1-1 i s a m o d i f i e d f o r m o f T a b l e 1 o f R e f e r e n c e 1. TABLE 1-1 C o n t i n u e d S h o r t Name F u l l Name I n s t i t u t i o n ( s ) w h e r e D e v e l o p e d R e f e r e n c e s S o u r c e L a n g u a g e G I F S 2 G e n e r a l i z e d I n t e r r e l a t e d F l o w S i m u l a t i o n S e r v i c e B u r e a u C o r p . 1 GPFS G e n e r a l i z e d P r o c e s s F l o w S i m u l a t o r Sun O i l Co. 9 FORTRAN F l e x i b l e F l o w s h e e t None M. W. K e l l o g g Co. 13 MACSIM 3 M c M a s t e r S i m u l a t o r S y s t e m M c M a s t e r U n i v . 11 FORTRAN MAEBE M a t e r i a l a nd E n e r g y B a l a n c e E x e c u t i on U n i v . o f T e n n e s s e e 1 FORTRAN IV PACER P r o c e s s A s s e m b l y C a s e E v a l u a t o r R o u t i n e P u r d u e U n i v . and D a r t m o u t h C o l l e g e 10 FORTRAN I I PACER ( M A D ) 4 None U n i v . o f H o u s t o n 1 MAD S L E D 5 S i m p l i f i e d L a n g u a g e f o r E n g i n e e r i n g D e s i g n U n i v . o f M i c h i g a n 1 MAD GIFS has bee n s u p e r s e d e d by CHIPS and now CHIPS-2. PACER u s e d as e x e c u t i v e s y s t e m . PACER was f i r s t m e c h n i c a l l y t r a n s l a t e d f r o m FORTRAN t o MAD u s i n g t h e MADTRAN p r o g r a m . T h e n e x t e n s i v e r e v i s i o n s were made. F o r t h i s r e a s o n i t has been d e s i g n a t e d as a s e p a r a t e s y s t e m , PACER (MAD). SLED has n o t y e t been made a v a i l a b l e f o r g e n e r a l u s e . TABLE 1-1 C o n t i n u e d S h o r t Name F u l l Name I n s t i t u t i o n ( s ) where D e v e l o p e d R e f e r e n c e s S o u r c e L a n g u a g e SPEED- U P 6 S i m u l a t i o n P r o g r a m f o r t h e E c o n o m i c E v a l u a t i o n and D e s i g n o f U n s t e a d y S t a t e P r o c e s s e s I m p e r i a l C o l l e g e , L o n d o n , E n g l a n d 12 Not Imp!emented UOS U n i t O p e r a t i o n s S i m u l a t o r B o n n e r and Moore E n g i n e e r i n g A s s o c . 1 SPEED-UP i s a p r o p o s a l f o r a s y s t e m t h a t has n e v e r been f u l l y i m p l e m e n t e d . 7 M o s t o f t h e s y s t e m s l i s t e d i n T a b l e 1 - 1 w e r e d e v e l o p e d by l a r g e c o m p a n i e s e n g a g e d i n c o n s t r u c t i o n and o p e r a t i o n o f c h e m i c a l p r o c e s s e s and a r e f o r company us e o r f o r s a l e . I t was p o s s i b l e , h o w e v e r t o o b t a i n c o p i e s o f PACER and CHESS. CHESS was c h o s e n f o r t h i s s t u d y s i n c e i t c o n t a i n s a c o m p r e h e n s i v e p h y s i c a l p r o p e r t i e s c a l c u l a t i o n p a c k a g e , a f e a t u r e w h i c h was n o t a v a i l a b l e i n PACER. 2. CHESS - C h e m i c a l E n g i n e e r i n g S i m u l a t i o n S y s t e m a. S y s t e m d e s c r i p t i o n The C h e m i c a l E n g i n e e r i n g S i m u l a t i o n S y s t e m , CHESS was d e v e l o p e d a t t h e U n i v e r s i t y o f H o u s t o n and i s b a s e d on t h e e a r l i e r v e r s i o n o f PACER ( 1 0 ) and a MAD l a n g u a g e v e r s i o n c a l l e d PACER (MAD). I t u t i l i z e s t h e c o n c e p t o f m o d u l a r i t y w h i c h a l l o w s c h a n g e s t o be made i n t h e o r d e r i n g o f u n i t s i n d e p e n d e n t l y o f what t h e u n i t s a r e , p r o v i d e d t h a t i n f o r m a t i o n i s p a s s e d b e t w e e n t h e u n i t s i n a s t a n d a r d f o r m a t . The u n i t m o d u l e s (may a l s o be c a l l e d u n i t c o m p u t a t i o n s o r p r o c e s s m o d u l e s ) , must u t i l i z e d a t a f r o m o t h e r p a r t s o f t h e s y s t e m and n o r m a l l y t h i s e x c h a n g e w o u l d r e q u i r e t h e u s e o f s u b r o u t i n e c a l l i n g s e q u e n c e s i n t h e c o m m u n i c a t i o n o f d a t a f r o m one s u b r o u t i n e t o a n o t h e r . CHESS, and most o t h e r l a r g e c o m p u t e r s i m u l a -t i o n s y s t e m s make u s e o f p r o g r a m COMMON d a t a d e c l a r a t i o n s 8 w h i c h make a l a r g e p o o l o f d a t a u s e d i n t h e s i m u l a t i o n a v a i l -a b l e t o a l l t h e s y s t e m s u b r o u t i n e s . In t h e p h y s i c a l p r o p e r t i e s p r e d i c t i o n p a c k a g e a v a i l a b l e i n CHESS, b a s i c p h y s i c a l c o n s t a n t s a r e s t o r e d f o r 62 common s u b s t a n c e s , m o s t o f w h i c h a r e h y d r o c a r b o n s . P h y s i c a l p r o p e r t i e s c a n a l s o be c a l c u l a t e d f o r w i d e - b o i l i n g p e t r o l e u m f r a c t i o n s i f t h e i r s p e c i f i c g r a v i t y , m o l e c u l a r w e i g h t and mean a v e r a g e b o i l i n g p o i n t a r e known ( 1 7 ) . P h y s i c a l p r o p e r t i e s c a n a l s o be c o m p u t e d f o r c o m p o n e n t s whose b a s i c p r o p e r t i e s a r e n o t s t o r e d , by s u p p l y i n g t h e s y s t e m w i t h t h e s e c o n s t a n t s . One o f t h e m o st d i f f i c u l t p r o b l e m s i n c h e m i c a l p r o c e s s s i m u l a t i o n i s t h e s o l u t i o n o f t h e m a t e r i a l and h e a t b a l a n c e when r e c y c l e e x i s t s . What i s r e a l l y i n v o l v e d i s t h e s o l u t i o n o f a s e t o f s i m u l t a n e o u s n o n l i n e a r e q u a t i o n s r e p r e s e n t i n g t h e p r o c e s s . Much work has b e e n done t o d e v e l o p e f f i c i e n t s o l u t i o n s t o t h i s p r o b l e m ( 1 8 , 19, 20, 2 1 , 22, 2 3 , 24, 2 5 , 26, 2 7 ) . M o s t s i m u l a t i o n s y s t e m s , i n c l u d i n g CHESS use t h e t e c h n i q u e o f s u c c e s s i v e s u b s t i t u t i o n . PACER (1 0 ) c o n t a i n s an a l g o r i t h m t o d e t e r m i n e t h e s h o r t e s t l i s t o f u n i t c o m p u t a t i o n s n e c e s s a r y t o c a l c u l a t e t h e f l o w s h e e t w h i l e DISCOSSA ( 7 ) c o n t a i n s a method t o m i n i m i z e t h e t o t a l number o f r e c y c l e p a r a m e t e r s ( 2 8 , 2 9 ) . The a u t h o r s o f CHESS u t i l i z e an a l g o r i t h m d e v e l o p e d by W e g s t e i n ( 3 0 ) f o r 9 c o n v e r g e n c e f o r c i n g . T h e y f e e l t h a t t h e u s e o f c o m p l e x o r d e r i n g a l g o r i t h m s and m a t e r i a l b a l a n c e s o l u t i o n m e t h o d s i s n o t j u s t i f i e d a t t h i s t i m e and t h u s t h e y r e l y on t h e "common s e n s e " o f t h e p r o g r a m m e r i n c h o o s i n g t h e c a l c u l a -t i o n o r d e r a n d t h e c o n v e r g e n c e f o r c i n g s t r e a m s . CHESS c a n u t i l i z e two t y p e s o f i n p u t d a t a f o r -m a t t i n g . T he f i r s t i s t h e FORTRAN NAMELIST t y p e w h i c h r e -q u i r e s some c a r e on t h e p a r t o f t h e p r o g r a m m e r . The s e c o n d t y p e i s a f r e e - f o r m f o r m a t w h i c h i s much e a s i e r f o r t h e u n i n i t i a t e d p r o g r a m m e r , b u t t h i s s u b p r o g r a m o c c u p i e s much more c o r e s p a c e and i s o n l y s u i t a b l e f o r l a r g e r c o m p u t e r s . I t s h o u l d be n o t e d t h a t t h e CHESS s y s t e m i s q u i t e l a r g e . I t c o n t a i n s a b o u t 6,000 s o u r c e l a n g u a g e c a r d s and o c c u p i e s a p p r o x i m a t e l y 40,000 w o r d s o f IBM 360 c o r e s t o r a g e . I t was p o s s i b l e t o r u n t h e c o m p l e t e p r o g r a m a s a s i n g l e l o a d m o d u l e on t h e U.B.C. IBM 3 6 0 / 6 7 . A l s o , by t h e u s e o f c h a i n i n g o r o v e r l a y t e c h n i q u e s t h e c o r e r e s i d e n c y r e q u i r e m e n t s c a n be r e d u c e d so t h a t t h e s y s t e m c o u l d be r u n on a s m a l l e r c o m p u t e r o f o n l y 128,000 b y t e s . b. S y s t e m e x e c u t i v e p r o g r a m s The e x e c u t i v e p r o g r a m s c o n t r o l t h e s y s t e m f u n c t i o n s o f i n p u t , o u t p u t , r e c y c l e c a l c u l a t i o n , and c o n v e r g e n c e f o r c i n g . A v e r y b r i e f MAIN c a l l i n g p r o g r a m i s p r o v i d e d 10 f o r t h e m a j o r s u b r o u t i n e s o f t h e s y s t e m , w h i c h may i n t u r n c a l l s e c o n d l e v e l s u b r o u t i n e s as n e e d e d . F i g u r e 1 - 1 d e s c r i b e s t h e o v e r a l l p r o g r a m l i n k a g e w h i l e T a b l e 1-2 g i v e s a b r i e f d e s c r i p t i o n o f t h e s y s t e m e x e c u t i v e p r o g r a m s and t h e i r f u n c t i o n s . c. S y s t e m s u p p o r t i n g f u n c t i o n s The s y s t e m p r o v i d e s t h e u s e r w i t h some s t a n d a r d e q u i p m e n t s u b r o u t i n e s f o r t h e most commonly u s e d b a s i c c h e m i c a l p r o c e s s u n i t s and a t h e r m o d y n a m i c p r o p e r t i e s e v a l -u a t i o n p r o g r a m . I n d i v i d u a l u s e r s may add t h e i r own s u b -r o u t i n e s f o r p r o c e s s m o d u l e s i f so d e s i r e d . A b r i e f d e s c r i p t i o n o f t h e p r o c e s s m o d u l e s a n d t h e r m o d y n a m i c s u b r o u t i n e s i s g i v e n i n T a b l e 1 - 3 . The t h e r m o d y n a m i c c o n s t a n t s f o r 62 p u r e com-p o n e n t s a r e s u p p l i e d by KHZT. T a b l e 1 - 4 l i s t s t h e s e com-p o n e n t s . The s y s t e m w i l l h a n d l e a l l o f t h e s e c o m p o n e n t s i n s i n g l e o r two p h a s e m i x t u r e s w i t h t h e e x c e p t i o n o f w a t e r w h i c h c a n o n l y be u s e d as a c o n s t i t u e n t o f a v a p o r m i x t u r e . O t h e r p u r e c o m p o n e n t s s u p p l i e d by t h e u s e r c a n a l s o be h a n d l e d by KHZT i f t h e f o l l o w i n g c o n s t a n t s a r e s u p p l i e d : C r i t i c a l p r e s s u r e A c e n t r i c f a c t o r C r i t i c a l t e m p e r a t u r e S o l u b i l i t y p a rameter C r i t i c a l volume C h a r a c t e r i s t i c m o l ar volume M o l e c u l a r w e i g h t C o e f f i c i e n t s i n z e r o p r e s s u r e heat c a p a c i t y - 4 FIGURE 1-1 CHESS - Overall Program Linkage 1 1 CLEAN ZERO TRANSF DREAD DPRINT TPRINT DCHECK INIT COMPID EQUIP REQUIP SCAN SUBSET RCYCLE TEST EQCALL KHZT: ENTH ZDENS KVAL TSUBH BUBTP/DEWTP EQUIPMENT SUBROUTINES: DVDR, MIXR, DIST, ADBF, HXER, MSEQ/ ABSR, VALV , FHTR/ REAC, PUMP, ADDl AD19 i 12 TABLE 1-2 Description of CHESS Executive Programs Program MAIN DREAD COMPID (in KHZT) Called By MAIN MAIN Function INIT DCHECK MAIN MAIN DPRINT SUBSET MAIN MAIN Main control program of the system. Reads input data such as i n i t i a l stream values, control con-stants, process matrix and equipment parameters. Selects the necessary subset of thermodynamic data cor-relation constants in KHZT subroutine from component identification numbers sup-plied as data in COMPNT vector. Calculates enthalpy values for a l l feed streams in the process. Compares feed and product streams as obtained from an analysis of the process matrix (KPM) with the stream status code in the stream matrix. If any discrepancy has been detected the control w i l l set up a return code to the main ca l l i n g program which causes the current problem to be can-celled and to start a new problem. Prints a l l process structure data, control constants and input status of stream and equipment matrices. This subroutine f i r s t c a l l s SCAN to flag status of streams and equipments. Then i t c a l l s 13 SCAN SUBSET EQUIP SUBSET or RCYCLE EQCALL SUBSET or RCYCLE REQUIP SUBSET or RCYCLE RCYCLE SUBSET EQUIP/ EQCALL and REQUIP to perform a u n i t c a l c u l a t i o n i f a l l inputs are known. I f un i t s s t i l l remain to be com-puted and inputs are unknown, SUBSET c a l l s RCYCLE to com-mence re c y c l e loop c a l c u l a t i o n s . Flags feed streams to +1, . unused equipment and stream members to -1 and others to 0 (zero). Transfers s p e c i f i c stream data associated with an equipment module to be c a l c u l a t e d from the system stream data matrices to the equipment module input/ output stream matrices. Stream data w i l l appear i n the module matrices i n the order i n which they appear i n the process matrix. C a l l s equipment or process module f o r a s p e c i f i c u n i t . Restores input and output stream data from the equipment module stream matrices to the system stream data matrices a f t e r each u n i t c a l c u l a t i o n . This subroutine i s c a l l e d when SUBSET finds that process c a l -c u l a t i o n s are incomplete and cannot proceed due to the presence of recycle streams. The u n i t c a l c u l a t i o n s c a l l i n g sequence - EQUIP, EQCALL, TEST and REQUIP, i s repeated f o r each u n i t i n the recycle loop according to the order of the u n i t i d e n t i f i c a t i o n numbers s p e c i f i e d i n KE2 or KE3. vectors. A f t e r each i t e r a t i o n loop t h i s routine also deter-mines whether recycle c a l c u l a t i o n s have reached the given termina-t i o n c r i t e r i a of convergence tolerance or maximum number of loops. 14 TEST RCYCLE TPRINT: PTPRNT BIPRNT PTEQPT DPUNCH DPRINT or RCYCLE MAIN DPRINT or MAIN MAIN In a recycle loop of unit calculations TEST determines whether the output stream values are within given con-vergence tolerances. TEST also performes the control function adjustments whenever a controlling stream i s identi-fied (see section on Convergence Promotion). Prints input or intermediate status of stream matrix. Prints summary of f i n a l stream matrix. Prints input or f i n a l status of equipment matrix. Punches out f i n a l status of a l l input data in reloadable NAMELIST block data formats. This deck may be used to con-tinue the recycle calculations i f the f i n a l result has not converged within the specified number of iterative loops or to restart a problem for sub-sequent case studies. When punch output i s requested, the SUBSET subroutine also produces a reloadable NAMELIST deck for the status of stream and equipment flags at the beginning of recycle calculations. This permits one to restart calculation at the recycle loop, and enables a l l product streams to update after completion of the restart calculations. Three additional system service routines are available, namely, CLEAN, to reset entire system data region to zero; ZERO, to zero any specified variables; and TRANSF; to transfer system arrays to output arrays printout. ... 15 T A B L E 1-3 D e s c r i p t i o n o f CHESS S u p p o r t i n g F u n c t i o n s P r o g r a m F u n c t i o n D V D R - D i v i d e r D i v i d e s one i n p u t s t r e a m i n t o a s p e c i f i e d number o f o u t p u t s t r e a m s . DIST - S i m p l e D i s t i l l a t i o n A s i m p l e l i n e a r s p l i t by c o m p o n e n t i s made o f t h e i n p u t s t r e a m t o t h e t o p and b o t t o m o u t p u t s t r e a m s . The i n p u t t e m p e r a t u r e and p r e s s u r e w i l l be d i r e c t l y t r a n s f e r r e d t o t h e o u t p u t s t r e a m s and t h e s t r e a m e n t h a l p y w i l l be c a l c u l a t e d f o r t h e o v e r h e a d ( a s v a p o r ) and t h e b o t t o m s ( a s l i q u i d ) . MIXR - M i x e r M i x e s a s p e c i f i e d number o f i n p u t s t r e a m s t o p r o d u c e one o u t -p u t s t r e a m . The p r e s s u r e o f t h e o u t p u t s t r e a m i s t h a t o f t h e l o w e s t p r e s s u r e i n p u t s t r e a m . The o u t p u t s t r e a m t e m p e r a t u r e i s c a l c u l a t e d a n d , i f n e c e s s a r y , an a d i a b a t i c f l a s h c a l c u l a -t i o n i s p e r f o r m e d t o f i n d t h e t e m p e r a t u r e and v a p o r f r a c t i o n o f t h e o u t p u t s t r e a m . ADBF - A d i a b a t i c F l a s h D e t e r m i n e s i f s t r e a m i s a l l l i q u i d o r a l l v a p o r o r t w o - p h a s e m i x t u r e . T e m p e r a t u r e , e n t h a l p i e s and v a p o r and l i q u i d q u a n -t i t i e s a r e d e t e r m i n e d . May be r u n i n 3 modes: 1. A d i a b a t i c f l a s h w i t h p h a s e d e t e r m i n a t i o n by dew p o i n t and b u b b l e p o i n t , 2. C o n s t a n t t e m p e r a t u r e f l a s h , 3. A d i a b a t i c f l a s h w i t h no p h a s e d e t e r m i n a t i o n by dew p o i n t and b u b b l e p o i n t . REAC - S i m p l e R e a c t o r P r o d u c t d i s t r i b u t i o n i s c a l c u l a t e d g i v e n a s e t o f s t o i c h i o -m e t r i c numbers and t h e c o n v e r s i o n o f a key c o m p o n e n t . The o u t p u t s t r e a m e n t h a l p y i s c o m p u t e d a t t h e i n l e t t e m p e r a t u r e and p r e s s u r e and a t p r o d u c t o r o u t p u t c o m p o s i t i o n w i t h no a d j u s t m e n t f o r h e a t o f r e a c t i o n . 16 TABLE 1-3 ( C o n t i n u e d ) P r o g r a m F u n c t i o n VALV - D o w n s t r e a m P r e s s u r e C o n t r o l l e r C a l c u l a t e s t h e t e m p e r a t u r e and v a p o r f r a c t i o n o f t h e o u t p u t s t r e a m a t t h e e n t h a l p y o f t h e i n l e t s t r e a m and t h e s p e c i f i e d o u t p u t s t r e a m p r e s s u r e . HXER - H e a t E x c h a n g e r O u t p u t t e m p e r a t u r e s a r e c o m p u t e d when two i n p u t s t r e a m s u n d e r g o c o u n t e r c u r r e n t h e a t e x c h a n g e f o r a g i v e n g e o m e t r i c c o n f i g u r a -t i o n . I f o n l y one i n p u t s t r e a m i s s p e c i f i e d , t h e d e s i r e d o u t p u t t e m p e r a t u r e may be o p t i o n a l l y s p e c i f i e d o r t h e s e c o n d i n p u t s t r e a m i s a s s u m e d t o be c o o l i n g w a t e r a t 5 5 0 ° R w i t h a t e m p e r a -t u r e r i s e o f 1 5 ° R . PUMP - Pump o r C o m p r e s s o r The work n e c e s s a r y t o pump a l i q u i d o r c o m p r e s s a f l u i d t o a d e s i r e d p r e s s u r e i s c a l c u l a t e d . ABSR - A b s o r b e r The amount o f e a c h c o m p o n e n t a b s o r b e d f r o m a r i c h gas s t r e a m by a h y d r o c a r b o n l e a n o i l i s c a l c u l a t e d u s i n g t h e a b s o r p t i o n f a c -t o r m e t h o d . MSEQ - M u l t i - S t a g e E q u i l i b r i u m A m u l t i - s t a g e e q u i l i b r i u m p r o c e s s o f s p e c i f i e d number o f t h e o -r e c t i c a l s t a g e s i s t r e a t e d as a s e r i e s o f a d i a b a t i c f l a s h c a l -c u l a t i o n s w h i c h a r e r e c y c l e d u n t i l c o n v e r g e n c e i s o b t a i n e d . FHTR - D i r e c t F i r e d H e a t e r F u e l gas c o n s u m p t i o n r e q u i r e d t o h e a t a s t r e a m t o a d e s i r e d o u t l e t t e m p e r a t u r e i s c a l c u l a t e d . KHZT - T h e r m o d y n a m i c P r o p e r t i e s P a c k a g e s e v e n e n t r i e s and i s c a l l e d upon w h e n e v e r a r e n e e d e d . The s e v e n e n t r i e s a r e t h e T h i s s u b r o u t i n e has p h y s i c a l p r o p e r t i e s f o l 1 o w i ng : 17 TABLE 1-3 ( C o n t i n u e d ) COMPID S e t s up t h e t h e r m o d y n a m i c d a t a c o r r e l a t i o n c o n s t a n t s f o r t h e s p e c i f i e d c o m p o n e n t s . ENTH Computes t h e e n t h a l p y o f t h e d e s i g n a t e d s t r e a m KVAL Computes t h e v a p o r i z a t i o n e q u i l i b r i u m r a t i o s o f t h e d e s i g -n a t e d s t r e a m . DEWTP C a l c u l a t e s t h e dew p o i n t t e m p e r a t u r e o f a d e s i g n a t e d s t r e a m . BUBTP C a l c u l a t e s t h e b u b b l e p o i n t o f a d e s i g n a t e d s t r e a m . TSUBH Computes t h e t e m p e r a t u r e o f t h e d e s i g n a t e d s t r e a m a t t h e e n t h a l p y o f t h a t s t r e a m . ZDENS Computes t h e c o m p r e s s i b i l i t y f a c t o r o f t h e d e s i g n a t e d s t r e a m w h e t h e r l i q u i d o r v a p o r . 18 TABLE 1-4 Pure Components in CHESS Thermodynamic Package 1. HYDROGEN 32. 2-METHYL-l-BUTENE 2. METHANE 33. 3-METHYL-l-BUTENE 3. ETHANE 34. 2-METHYL-2-BUTENE 4. PROPANE 35. 1-HEXENE 5. I-BUTANE 36. CYCLOPENTANE 6. N-BUTANE 37. METHYLCYCLOPENTANE 7. I-PENTANE 38. CYCLOHEXANE 8. N-PENTANE 39. METHYLCYCLOHEXANE 9. NEO-PENTANE 40. BENZENE 10. N-HEXANE 41. TOLUENE 11. N-HEPTANE 42. O-XYLENE 12. N-OCTANE 43. M-XYLENE 13. N-NONANE 44. P-XYLENE 14. N-DECANE 45. ETHYLBENZENE 15. N-UNDECANE 46. NITROGEN 16. N-DODECANE 47. OXYGEN 17. N-TRIDECANE 48. CARBON MONOXIDE 18. N-TETRADECANE 49. CARBON DIOXIDE 19. N-PENTADECANE 50. HYDROGEN SULFIDE 20. N-HEXADECANE 51. SULFUR DIOXIDE 21. N-HEPTADECANE 52. 2-METHYL-C5 22. ETHYLENE 53. 3-METHYL-C5 23. PROPYLENE 54. 2/2-DI-Cl-C4 24. 1-BUTENE 55. 2,3-DI-Cl-C4 25. CIS-2-BUTENE 56. 1-HEPTENE 26. TRANS-2-BUTENE 57. PROPADIENE 27. I-BUTENE 58. 1,2-BUTADIENE 28. 1,3-BUTADIENE 59. C2-CYCLO-C5 29. 1-PENTENE 60. C2-CYCLO-C6 30. CIS -.2 - PENTENE 61. ISOPRENE 31. TRANS-2-PENTENE 62. WATER 19 d. P r o c e s s r e p r e s e n t a t i o n The f i r s t s t e p i n r e p r e s e n t i n g t h e c h e m i c a l p r o c e s s t o be s i m u l a t e d i n t h e CHESS s y s t e m i s t o d e v e l o p an i n f o r -m a t i o n f l o w d i a g r a m f r o m t h e p r o c e s s f l o w d i a g r a m . The i n f o r m a t i o n f l o w d i a g r a m i s v e r y s i m i l a r t o t h e p r o c e s s f l o w d i a g r a m ( s e e C h a p t e r 3 ) . A l t h o u g h t h e r e i s u s u a l l y a d i r e c t c o r r e s p o n d e n c e b e t w e e n t h e p r o c e s s m o d u l e s and t h e p i e c e o f p l a n t e q u i p m e n t t h i s i s n o t a l w a y s t h e c a s e . The p r o c e s s m a t r i x c o n s i s t s o f a l i s t o f t h e u n i t c o m p u t a t i o n s c o r r e s p o n d i n g t o t h e s y m b o l s and numbers o f t h e i n f o r m a t i o n f l o w d i a g r a m . The i n p u t ( p o s i t i v e ) and o u t -p u t ( n e g a t i v e ) s t r e a m numbers a r e l i s t e d f o r e a c h u n i t com-p u t a t i o n ( s e e A p p e n d i x E ) . The i n f o r m a t i o n on t h e s t r e a m c o n n e c t i o n s g i v e s c o n s i d e r a b l e i n s i g h t i n t o t h e r e c y c l e s t h a t e x i s t i n t h e p r o c e s s and CHESS s t o r e s t h i s i n f o r m a t i o n i n a s e p a r a t e s y s t e m p r o c e s s m a t r i x . E q u i p m e n t p a r a m e t e r v e c t o r s a r e p r o v i d e d f o r t h e i n p u t o f t h e p h y s i c a l c h a r a c t e r i s t i c s f o r s p e c i f i c p i e c e s o f e q u i p m e n t w h i c h w i l l be r e p r e s e n t e d i n a more g e n e r a l p r o c e s s m o d u l e . The v e c t o r s a r e s t o r e d i n t h e e q u i p m e n t p a r a m e t e r m a t r i x . The s t r e a m d a t a a r e s t o r e d i n e x t e n s i v e and i n t e n -s i v e p r o p e r t y s t r e a m v e c t o r s w h i c h a r e o n c e a g a i n c o m b i n e d i n t o s t r e a m p r o p e r t y m a t r i c e s . 20 The c o m p o n e n t s o f t h e i n p u t s t r e a m s must a l s o be s p e c i f i e d , and t h e a p p r o p r i a t e t h e r m o d y n a m i c c o n s t a n t s s u p -p l i e d i f t h e y a r e n o t f o u n d i n t h e l i s t o f s t a n d a r d compo-n e n t s . e. M a t e r i a l and h e a t b a l a n c i n g As m e n t i o n e d p r e v i o u s l y , t h e d e s c r i p t i o n o f r e -c y c l e l o o p s i s l e f t t o t h e u s e r . V e r y l i t t l e p r o v i s i o n i s made f o r n e s t e d l o o p s b e c a u s e t h e a u t h o r s o f CHESS ( 3 ) f e e l t h a t t h e u s e r ' s e x p e r i e n c e and i n s i g h t i n t o t h e p r o c e s s c a l c u l a t i o n s w i l l a l l o w him t o p i c k a r e c y c l e s e q u e n c e t h a t i s as g o o d as t h o s e c h o s e n by some r e c y c l e l o o p a l g o r i t h m . The s e q u e n c e o f u n i t numbers f o r m i n g t h e r e c y c l e scheme a r e s u p p l i e d as an i n p u t t o t h e s y s t e m and s t o r e d i n a r e c y c l e v e c t o r . The CHESS s y s t e m u s e s t h e method o f s u c c e s s i v e s u b s t i t u t i o n f o r t h e c a l c u l a t i o n o f r e c y c l e l o o p s . A l t h o u g h t h i s m e t h o d i s s i m p l e i t s r a t e o f c o n v e r g e n c e i s q u i t e s l o w . F o r t h i s r e a s o n , CHESS u t i l i z e s W e g s t e i n s c o n v e r g e n c e a c c e l -e r a t i o n m ethod ( 3 0 ) i n t h e s y s t e m e x e c u t i v e p r o g r a m c a l l e d T E S T . The m e t h o d a l l o w s c o n v e r g e n c e a c c e l e r a t i o n a t e a c h i t e r a t i o n s t e p and i s a p p l i e d t o e a c h c o m p o n e n t as i f i t were t h e o n l y v a r i a b l e i n t h e l o o p . The u s e r c a n s p e c i f y t h e s t r e a m s t o w h i c h t h e c o n v e r g e n c e a c c e l e r a t i o n m e t h o d s h o u l d be a p p l i e d . 21 3. P r o c e s s S i m u l a t i o n s R e p o r t e d i n t h e L i t e r a t u r e M o s t o f t h e u s e s t h a t h a v e b e e n made o f t h e chem-i c a l p r o c e s s s i m u l a t i o n s y s t e m s h a v e been i n t h e d e s i g n r a t h e r t h a n i n t h e o p e r a t i o n o f a c h e m i c a l p l a n t . CHEOPS has been u s e d f o r t h e d e s i g n o f p e t r o l e u m r e f i n e r i e s . H u g h e s , et al. ( 2 ) show t h a t i t i s p o s s i b l e t o r e d u c e t h e r e f i n e r y p r o b l e m t o a number o f s m a l l e r u n i t s u b p r o b l e m s by a j u d i c i o u s c h o i c e o f v a r i a b l e s . I t was m e n t i o n e d t h a t CHEOPS had b e e n a p p l i e d t o t h e d e s i g n o f a number o f c h e m i c a l p l a n t s p r o d u c i n g s u c h p r o d u c t s as i s o p r e n e , e t h y l e n e o x i d e , p o l y p r o p y l e n e , and e t h y l a l c o h o l . Hughes ( 4 4 ) went i n t o g r e a t d e t a i l i n an e x a m p l e o f model b u i l d i n g f o r p r o c e s s d e s i g n o p t i m i z a t i o n i n an A . I . C h . E . T o d a y S e r i e s C o u r s e . The p r o c e s s model was f o r m u l a t e d and t h e n m a t c h e d t o an o p t i m i z a t i o n a l g o r i t h m . S e v e r a l p r o c e s s e s were s i m u l a t e d as e x a m p l e s i n t h e u s e o f : G I F S - E t h y l y n e d i c h l o r i d e p l a n t , m u l t i f l a s h c o n d e n s a t i o n (1) CHIPS - M u l t i f l a s h c o n d e n s a t i o n (5,6) GPFS - Gas p l a n t a b s o r b e r - s t r i p p e r (9) F l e x i b l e F l o w s h e e t was u s e d t o e v a l u a t e v a r i o u s a l t e r n a t i v e s i n t h e d e s i g n o f a p e t r o l e u m r e f i n e r y gas r e -c o v e r y p i a n t ( 1 3 ) . 22 The m o s t c o m p l e t e d e s c r i p t i o n s o f p l a n t s i m u l a -t i o n s h a v e b e e n g i v e n by t h e g r o u p a t M c M a s t e r U n i v e r s i t y ( 1 5 , 3 1 , 32, 33, 3 4 ) . The s i m u l a t i o n o f a c o n t a c t s u l p h u r i c a c i d p l a n t u s i n g t h e PACER s y s t e m i s d e s c r i b e d i n g r e a t d e t a i l ( 3 1 , 3 2 ) . PACER i s a l s o u s e d i n t h e s i m u l a t i o n o f a s y n -t h e t i c r u b b e r p l a n t ( 3 4 ) . MACSIM and GEMCS were u s e d i n t h e s i m u l a t i o n o f an a l k y l a t i o n u n i t a t a p e t r o l e u m r e f i n -e r y ( 3 3 ) . CHESS was u t i l i z e d i n t h e m o d e l i n g and p a r a m e t r i c s t u d y o f a v a p o r r e c o m p r e s s i o n d i s t i l l a t i o n s y s t e m ( 3 5 ) . 4. C h e m i c a l P r o c e s s O p t i m i z a t i o n R e p o r t e d i n t h e  L i t e r a t u r e In most o p t i m i z a t i o n s t u d i e s a s u i t a b l e o b j e c t i v e f u n c t i o n i s a v a i l a b l e , t h e m i n i m i z a t i o n ( o r m a x i m i z a t i o n ) o f w h i c h w i t h r e s p e c t t o t h e i n d e p e n d e n t v a r i a b l e s , and s u b j e c t t o t h e i m p o s e d c o n t r a i n t s , y i e l d s t h e optimum s o u g h t . I t i s n o t a l w a y s e a s y t o s e t up s u c h a f u n c t i o n , f o r i t s h o u l d t a k e i n t o a c c o u n t t h e w h o l e e c o n o m i c e n v i r o n m e n t w i t h i n w h i c h t h e p r o c e s s o p e r a t e s . In t h e c a s e o f o p t i m i z -i n g t h e o p e r a t i o n o f an e x i s t i n g p l a n t , t h e o b j e c t i v e f u n c -t i o n w o u l d u t i l i z e t h e c o s t and p r i c e d a t a g e n e r a t e d by a q u a n t i t a t i v e e c o n o m i c model o f t h e e n t i r e company. The i n d e p e n d e n t v a r i a b l e s i n v o l v e d w o u l d be p l a n t o p e r a t i n g c o n d i t i o n s s u c h as t e m p e r a t u r e , p r e s s u r e , f l o w 23 r a t e , e t c . The c o n s t r a i n t s i n v o l v e d w o u l d be t h o s e due t o p r o d u c t s p e c i f i c a t i o n s and e q u i p m e n t l i m i t a t i o n s . S e v e r a l o p t i m i z a t i o n m e t h o d s have been u t i l i z e d i n t h e o p t i m i z a t i o n o f c h e m i c a l p r o c e s s e s . As i n t h e c a s e o f p r o c e s s s i m u l a t i o n , most o f t h e s t u d i e s r e p o r t e d i n t h e l i t e r a t u r e r e f e r t o t h e d e s i g n r a t h e r t h a n t h e o p e r a t i o n o f c h e m i c a l p r o c e s s e s . I f t h e o b j e c t i v e f u n c t i o n and a l l t h e c o n s t r a i n t s w e r e l i n e a r w i t h r e s p e c t t o t h e i n d e p e n d e n t v a r i a b l e s t h e n t h e s t a n d a r d l i n e a r p r o g r a m m i n g t e c h n i q u e s ( 3 6 ) c o u l d be u s e d . T h e s e t e c h n i q u e s a r e p r o b a b l y t h e most p o w e r f u l o n e s a v a i l a b l e . I f i t i s p o s s i b l e t o a d a p t t h e p r o b l e m t o make them a p p l i c a b l e , t h e n t h e r e i s e v e r y i n c e n t i v e t o do s o . A t t h e same t i m e , c a r e must be u s e d n o t t o f o r c e t h e l i n e a r i z a t i o n o f a s e v e r e l y n o n l i n e a r p r o b l e m s i n c e t h e r e s u l t a n t e r r o r s i n c o m p u t a t i o n c a n be v e r y l a r g e . When t h e c o n s t r a i n t s and t h e o b j e c t i v e f u n c t i o n a r e f o r m e d by t h e sum o f l i n e a r and n o n l i n e a r t e r m s s u c h t h a t t h e l a t t e r a r e f u n c t i o n s o f o n l y a s i n g l e v a r i a b l e , t h e t e c h n i q u e o f s e p a r a b l e p r o g r a m m i n g may be u s e d . Hwa ( 3 7 ) j s e s t h i s t e c h n i q u e i n t h e f o r m u l a t i o n and o p t i m i z a t i o n o f h e a t e x c h a n g e r n e t w o r k s . G r i f f i t h and S t e w a r t ( 3 8 ) d e v e l o p e d an a l g o r i t h m f o r t h e r e p e t i t i v e s o l u t i o n o f l i n e a r p r o g r a m m i n g m a t r i c e s i n d i f f e r e n t p o r t i o n s o f t h e f e a s i b l e r e g i o n o f t h e 24 c o n s t r a i n t s p a c e so as t o a c c o m p l i s h t h e n o n l i n e a r o p t i m i -z a t i o n . Hens ( 3 9 ) u s e d t h e a b o v e method c a l l e d MAP, m ethod o f a p p r o x i m a t i o n p r o g r a m m i n g , f o r t h e o p t i m i z a t i o n o f s e v e r a l n o n l i n e a r p l a n t m o d e l s . The a u t h o r s t a t e s t h a t t h e m o d e l s t e s t e d v a r i e d f r o m t h o s e w i t h 10 i n d e p e n d e n t v a r i a b l e s up t o one w i t h 175 i n d e p e n d e n t v a r i a b l e s and 130 c o n s t r a i n t s . T h i s s t u d y i s o f s p e c i a l i n t e r e s t s i n c e i t i s one o f t h e o n l y r e f e r e n c e s f o u n d i n t h e l i t e r a t u r e f o r t h e s i m u l a t i o n and o p t i m i z a t i o n o f a model o f an e x i s t i n g p r o c e s s . ( C a s e s o f o n - l i n e p r o c e s s o p t i m i z a t i o n h a v e b e e n r e p o r t e d . ) When r e l a t i v e l y few o f t h e c o n s t r a i n t s a r e n o n -l i n e a r o r i n v o l v e d i s c r e t e - v a l u e d v a r i a b l e s t h e p r o b l e m may be d e c o m p o s e d i n t o l i n e a r and n o n l i n e a r o r i n t e g e r p r o b l e m s . One o f t h e p r o c e d u r e s w h i c h u t i l i z e s t h i s p r i n c i p l e i s p a r t i t i o n p r o g r a m m i n g , PPNL w h i c h was a p p l i e d t o a r e f i n e r y s c h e d u l i n g o p e r a t i o n by O r n e a an d E l d r i d g e ( 4 0 ) . S e v e r a l m e t h o d s f o r n o n l i n e a r c o n s t r a i n e d o p t i -m i z a t i o n u s e d w i t h s u c c e s s by t h e S h e l l D e v e l o p m e n t Co. h ave b e e n c i t e d by S i n g e r ( 4 1 ) . T h e y a r e t h e g r a d i e n t p r o j e c t i o n m e t h o d , Maze, L e a p and C r e e p , and d e f l e c t e d a s c e n t . The g r a d i e n t p r o j e c t i o n m ethod was d e v e l o p e d by R o s e n ( 4 2 , 4 3 ) . The m ethod was o r i g i n a l l y d e s i g n e d f o r 25 p r o b l e m s w h e r e t h e o b j e c t i v e f u n c t i o n may be n o n l i n e a r b u t a l l o f t h e c o n s t r a i n t f u n c t i o n s must be l i n e a r . I t was l a t e r e x t e n d e d t o c o r r e c t f o r t h e c u r v a t u r e o f t h e c o n -s t r a i n t s . Hughes ( 4 4 ) r e p o r t s on t h e c o m p a r i s o n s made o f t h e S h e l l D e v e l o p m e n t m e t h o d s . The t e s t s w e re n o t e n t i r e l y c o n c l u s i v e s i n c e one method m i g h t h a n d l e a p a r t i c u l a r p r o b l e m b e t t e r t h a n t h e o t h e r s and f a i l t o s o l v e a n o t h e r p r o b l e m w h i c h c o u l d be s o l v e d by a n o t h e r m e t h o d . The c h o i c e o f s t a r t i n g p o i n t was f o u n d t o be c r u c i a l and none o f t h e me t h o d s were s a t i s f a c t o r y f o r c a s e s w h e r e more t h a n two m u l t i v a r i a b l e c o n s t r a i n t s were a c t i v e a t one t i m e . F i a c c o and M c C o r m i c k ( 4 5 , 46, 47) d e v e l o p e d SUMT ( S e q u e n t i a l U n c o n s t r a i n e d M i n i m i z a t i o n T e c h n i q u e ) f o r t r a n s f o r m i n g a c o n s t r a i n e d o p t i m i z a t i o n p r o b l e m i n t o an u n c o n s t r a i n e d p r o b l e m . B r a c k e n and M c C o r m i c k ( 4 8 ) a p p l i e d t h e m e t h o d t o t h e o p t i m i z a t i o n o f an a l k y l a t i o n p r o c e s s w h e r e t h e model i s f o r m u l a t e d as a s e t o f n o n l i n e a r a l g e -b r a i c e q u a t i o n s . B e a u d u i n ( 4 9 ) u t i l i z e d SUMT i n t h e o p t i m i z a t i o n o f a s i m u l a t e d model o f a h y d r o d e a l k y l a t i o n u n i t . T h e s i m u l a t i o n was q u i t e s p e c i f i c s i n c e t h e model was w r i t t e n i n FORTRAN and a s s u m p t i o n s were made t o e l i m i n a t e t h e r e -c y c l e c a l c u l a t i o n s so t h a t t h e c o m p u t a t i o n t i m e p e r p r o c e s s e v a l u a t i o n c o u l d be r e d u c e d t o a b o u t 8 s e c o n d s on an IBM 26 7094 c o m p u t e r . An o p t i m i z a t i o n r u n m i g h t u t i l i z e 60 m i n u t e s o f C.P.U. t i m e w i t h a b o u t 400 p r o c e s s e v a l u a t i o n s t o r e a c h an o p t i m u m . Umeda ( 5 0 ) u t i l i z e d t h e Complex M e t h o d d e v e l o p e d by Box ( 5 1 ) i n t h e o p t i m a l d e s i g n o f an a b s o r b e r - s t r i p p e r s y s t e m . The s y s t e m c o n t a i n e d f i v e i n d e p e n d e n t d e s i g n v a r i -a b l e s and c o u l d be m o d e l e d by a s e t o f n o n l i n e a r a l g e b r a i c e q u a t i o n s . Umeda and I c h i k a w a ( 5 2 ) m o d i f i e d t h e Complex M e t h o d and c o m p a r e d i t s u s e w i t h t h a t o f t h e o r i g i n a l C omplex M e t h o d i n t h e a b o v e m e n t i o n e d a b s o r b e r - s t r i p p e r d e s i g n p r o b l e m . K omatsu ( 5 3 ) u t i l i z e d t h e l i n e a r i z a t i o n m e t h o d t o s i m p l i f y t h e s i m u l a t i o n o f a h y d r o d e a l k y l a t i o n p r o c e s s and t h e n a p p l i e d t h e Complex M e t h o d t o o p t i m i z e t h e p r o c e s s d e s i g n . A l t h o u g h a s i z e a b l e number o f o p t i m i z a t i o n m e t h o d s have b e e n a p p l i e d t o t h e p r o c e s s d e s i g n p r o b l e m , t h e r e i s v e r y l i t t l e c o m p a r i s o n o f me t h o d s f o r t h i s t y p e o f p r o b l e m i n t h e l i t e r a t u r e . B a r n e s o n et al. ( 5 4 ) compare s i x n o n l i n e a r o p t i -m i z a t i o n t e c h n i q u e s a p p l i e d t o two d i f f e r e n t p r o c e s s d e s i g n p r o b l e m s . The m e t h o d s t e s t e d w e r e : 1. P a t t e r n S e a r c h d e v e l o p e d by Hooke and J e e v e s (55) and m o d i f i e d by Weisman et al. ( 5 6 ) , 27 2 . The a c c e l e r a t e d d i r e c t s e a r c h method o f P o w e l l ( 5 7 ) , 3 . The S i m p l e x t e c h n i q u e o f Spendley et al. ( 5 8 ) as m o d i f i e d by N e l d e r and Mead ( 5 9 ) , 4 . The method o f P a r a l l e l T a n g e n t s , P a r t a n , o f Shah et al. ( 6 0 ) , 5 . The V a r i a b l e M e t r i c T e c h n i q u e , V a r m i t , w h i c h was b a s i c a l l y d e v e l o p e d by Davidon ( 6 l ) , and 6 . The P r o c e s s O p t i m i z a t i o n Program, POP, an IBM program s i m i l a r t o MAP ( 3 8 ) . The a u t h o r s c o n c l u d e d t h a t no one o p t i m i z a t i o n m e t h o d was b e s t i n a l l c a s e s b u t t h a t i n g e n e r a l t h e POP and P a t t e r n S e a r c h m e t h o d s g a v e t h e b e s t r e s u l t s . T h e P a t t e r n S e a r c h was t h e e a s i e s t t o p r o g r a m and d e b u g , t h e b e s t t o u s e f o r a s i m p l e u n c o n s t r a i n e d p r o b l e m and t h e b e s t t o u s e f o r p r o b l e m s w i t h i n t e g e r v a r i a b l e s . POP, w h i l e b e i n g more c o m p l i c a t e d , was t h e most p o w e r f u l o f t h e m e t h o d s and was b e s t f o r u s e on c o n s t r a i n e d p r o b l e m s . The g r a d i e n t m e t h o d s s u c h as P a r t a n and V a r m i t moved t o t h e o p t i m a l r e g i o n q u i t e q u i c k l y b u t e x p e r i e n c e d d i f f i c u l t i e s , b e c a u s e o f model n o i s e , a t t h a t p o i n t . F i n a l l y t h e a u t h o r s recommended t h e u s e o f two o p t i m i z a t i o n t e c h n i q u e s , p r e f e r a b l y h a v i n g d i f f e r e n t a p p r o a c h e s , t o t h e p r o c e s s d e s i g n p r o b l e m . 28 T h e r e a r e s e v e r a l c o n c l u s i o n s t h a t c a n be made f r o m t h e s u r v e y o f l i t e r a t u r e i n t h e f i e l d o f p r o c e s s s i m u -l a t i o n and o p t i m i z a t i o n . The f i r s t i s t h a t o n l y a few p u b l i c a t i o n s r e p o r t f u l l - s c a l e d e t a i l e d p r o c e s s m o d e l s e i t h e r f o r s i m u l a t i o n o r f o r d e s i g n o f c h e m i c a l p l a n t s ( 3 1 , 32, 33, 34, 3 5 ) . Q u i t e a b i t o f e f f o r t was p u t i n t o t h e s e s t u d i e s i n s p i t e o f t h e u s e o f e s t a b l i s h e d s i m u l a t i o n s y s t e m s s u c h as PACER and CHESS. The s u l p h u r i c a c i d p l a n t s t u d y ( 3 1 , 32) was c a r r i e d o u t by 15 s t a f f members and g r a d u a t e s t u d e n t s , w i t h s u p p o r t i n g h e l p f r o m t h e p l a n t p e r s o n n e l . B e c a u s e o f t h e c o m p l e x i t y o f s u c h a model w h i c h i n v o l v e s 44 u n i t s and 69 s t r e a m s and w h i c h b a s i c a l y r e p r e -s e n t s a b o u t 500 e q u a t i o n s , many n o n l i n e a r , w h i c h r e l a t e a b o u t 1000 s t r e a m v a r i a b l e s and 200 e q u i p m e n t p a r a m e t e r s , a s i m u l a t i o n r u n , o n c e programmed r e q u i r e s a b o u t 7 t o 8 m i n u t e s on an IBM 7040 c o m p u t e r . The o n l y o p t i m i z a t i o n s t u d i e s t h a t h a v e b e e n made on s u c h a model a r e manual ; s e v e r a l c a s e s t u d i e s may be r u n f o r d i f f e r e n t v a l u e s o f t h e i n d e p e n d e n t v a r i a b l e s . A s e c o n d c o n c l u s i o n t h a t one a r r i v e s a t i s t h a t m o s t o f t h e p r o c e s s o p t i m i z a t i o n s t u d i e s a r e f o r t h e d e s i g n o f c h e m i c a l p l a n t s and i n o n l y one c a s e ( 3 9 ) i s an a u t o m a t i c o p t i m i z a t i o n p r o c e d u r e a p p l i e d t o an e x i s t i n g p l a n t . The a u t h o r d o e s n o t make c l e a r , h o w e v e r , i f t h e model was s i m u -l a t e d u s i n g an e x i s t i n g s i m u l a t i o n s y s t e m . 29 A t h i r d c o n c l u s i o n i s t h a t t h e p r o c e s s e s o r e q u i p ment u n i t s whose d e s i g n s have been o p t i m i z e d a r e u s u a l l y r e p r e s e n t e d by e i t h e r a s e r i e s o f n o n l i n e a r a l g e b r a i c e q u a -t i o n s o r by r e l a t i v e l y s i m p l e FORTRAN m o d e l s and a r e q u i t e s p e c i f i c i n n a t u r e . A f i n a l c o n c l u s i o n i s t h a t e x c e p t f o r t h e s t u d y o f B a r n e s o n et al. ( 5 4 ) , no c o m p a r i s o n has been made o f t h e r e l a t i v e e f f i c i e n c y o f t h e n o n l i n e a r c o n s t r a i n e d o p t i -m i z a t i o n m e t h o d s a v a i l a b l e . C. The U.O.P. C a t a l y t i c P o l y m e r i z a t i o n P r o c e s s 1 . H i s t o r y Due t o t h e g r e a t i n c r e a s e i n t h e v o l u m e o f h y d r o -c a r b o n g a s e s p r o d u c e d i n p e t r o l e u m r e f i n e r i e s b e c a u s e o f e n l a r g e d c r a c k i n g f a c i l i t i e s , d e v e l o p m e n t work was s t a r t e d on t h e u t i l i z a t i o n o f t h e s e g a s e s a r o u n d 1925. In 1931 a t h e r m a l p o l y m e r i z a t i o n p l a n t was s t a r t e d up by t h e P u r e O i l Company a t i t s r e f i n e r y i n T o l e d o , O h i o ( 6 2 ) . O t h e r c o m p a n i e s a l s o s t a r t e d s i m i l a r p l a n t s t h a t o p e r a t e d u n d e r h i g h p r e s s u r e s and t e m p e r a t u r e s ( 8 0 0 - 3 0 0 0 p . s . i . a . , 900-1 1 0 0 ° F ) . In 1 9 3 4 , V l a d i m i r I p a t i e f f and c o - w o r k e r s o f t h e U n i v e r s a l O i l P r o d u c t s Company d e v e l o p e d a c a t a l y t i c p o l y m e r i z a t i o n p r o c e s s u s i n g l i q u i d p h o s p h o r i c a c i d as a c a t a l y s t ( 6 3 , 64, 6 5 ) . One y e a r l a t e r a s o l i d p h o s p h o r i c 30 a c i d c a t a l y s t was d e v e l o p e d and t h e f i r s t c o m m e r c i a l p l a n t was p u t i n t o o p e r a t i o n a t t h e E a s t C h i c a g o r e f i n e r y o f t h e S h e l l O i l Company ( 6 6 , 67, 6 8 ) . T h i s p r o c e s s o p e r a t e d a t t e m p e r a t u r e s and p r e s s u r e s (200 p . s . i . a . , 2 8 0 - 4 7 5 ° F ) much l o w e r t h a n t h o s e e m p l o y e d i n t h e t h e r m a l p r o c e s s e s . The u s e o f c a t a l y t i c p o l y m e r i z a t i o n s p r e a d t h r o u g h o u t t h e f o l l o w i n g d e c a d e u n t i l a l m o s t e v e r y r e f i n e r y w i t h c r a c k i n g f a c i l i t i e s e m p l o y e d p o l y m e r i z a t i o n o f t h e c r a c k e d g a s e s . In 1 9 5 1 , t h e r e were more t h a n 150 U n i v e r s a l O i l P r o d u c t s (U.O.P.) c a t a l y t i c p o l y m e r i z a t i o n p l a n t s i n o p e r a t i o n ( 6 9 ) . O t h e r p o l y m e r i z a t i o n p r o c e s s e s were a l s o d e v e l o p e d . In 1934 t h e S h e l l D e v e l o p m e n t Company i n t r o d u c e d t h e c o l d a c i d p r o c e s s w h i c h s e l e c t i v e l y p o l y m e r i z e s i s o b u t e n e u s i n g s u l f u r i c a c i d as a c a t a l y s t . The h o t a c i d p r o c e s s was a l s o d e v e l o p e d by S h e l l and d i f f e r e d f r o m t h e c o l d a c i d p r o c e s s as i t p o l y m e r i z e d a l l C^ o l e f i n s . S o l i d c o p p e r p y r o p h o s p h a t e was d e v e l o p e d f o r u s e as a c a t a l y s t i n a p r o c e s s d e v e l o p e d by t h e C a l i f o r n i a R e s e a r c h C o r p o r a t i o n ( 7 0 ) . In t h e f i r s t U.O.P. p o l y m e r i z a t i o n p l a n t s a number o f c a t a l y s t t o w e r s were u s e d i n s e r i e s . The c a t a l y s t was r e g e n e r a t e d , u s u a l l y one t o w e r a t a t i m e , w h i l e t h e r e m a i n -i n g o n e s c o n t i n u e d on s t r e a m . The r e g e n e r a t i o n was accom-p l i s h e d by o x i d i z i n g t h e c o k e and h e a v y p o l y m e r d e p o s i t s w i t h a c o n t r o l l e d c o n c e n t r a t i o n o f o x y g e n i n an i n e r t g a s , 31 f o l l o w e d by s t e a m i n g o f t h e c a t a l y s t f o r t h e p u r p o s e o f r e -s t o r i n g t h e w a t e r o f h y d r a t i o n n e c e s s a r y f o r h i g h c o n v e r -s i o n and e x t e n d e d c a t a l y s t l i f e . By 1939 i t was f o u n d ( 7 1 ) t h a t when o p e r a t i n g t h e p o l y m e r i z a t i o n r e a c t i o n a t p r e s s u r e s o f a b o u t 500 p . s . i . a . a s u f f i c i e n t amount o f f e e d r e m a i n e d i n a " d e n s e p h a s e " w h i c h w a s h e d t h e c a t a l y s t c l e a r o f m o st o f t h e h e a v y p o l y m e r , t h u s e x t e n d i n g c a t a l y s t l i f e and e l i m i n a t i n g c a t a l y s t r e g e n e r a t i o n . 2. P r o c e s s D e s c r i p t i o n The c a t a l y t i c p o l y m e r i z a t i o n u n i t a t t h e S h e l l b u r n r e f i n e r y , S h e l l O i l Co. o f C a n a d a L t d . i n B u r n a b y , B.C. i s t y p i c a l o f t h e U.O.P. u n i t s . A f l o w d i a g r a m o f t h e u n i t i s shown i n F i g u r e 1-2 ( 7 2 ) . The f e e d t o t h e u n i t i s t h e t r e a t e d p r o d u c t f r o m t h e gas r e c o v e r y u n i t s t a b i l i z e r . I t i s t r e a t e d f o r r e m o v a l o f h y d r o g e n s u l f i d e , m e r c a p t a n s and b a s i c n i t r o g e n com-pounds b e f o r e b e i n g c h a r g e d t o t h e p o l y m e r i z a t i o n r e a c t o r s . A t y p i c a l r a n g e i n f e e d c o m p o s i t i o n i s shown i n T a b l e 1-5. The p o l y f e e d c h a r g e pump r e c e i v e s t h e f e e d a t a b o u t 300 p . s . i . a . and 9 0 ° F and pumps t h e s t r e a m t o a p r e s -s u r e o f a b o u t 550 p . s . i . a . The s t r e a m p a s s e s t h r o u g h t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r and t h e p o l y f e e d h e a t e r , b e f o r e e n t e r i n g t h e p o l y m e r i z a t i o n r e a c t o r s a t a t e m p e r a t u r e FRESH FEED FROM POLY FEED TREATER POLY FEED HEATER POLY FEED EFFLUENT EXCHANGER POLYMERIZATION REACTORS I z o UJ o RECYCLE S - W T a }^ • •••1 • • • • • • • • '{St* mm • • • • •mi mi l DEPROPANIZER DEBUTANIZER CONDENSOR CONDENSOR PRESSURE CONTROL VALVE c ACC. FRESH FEED PUMP RECYCLE-QUENCH PUMP ) -6«-4 DEPROP. REFLUX PUMP 1 az_D ACCUMULATOR 1 COOLER COOLER DEBUT. REFLUX PUMP F I G U R E 1 - 2 F L O W D I A G R A M O F S H E L L B U R N P O L Y M E R I Z A T I O N U N I T CaCI 2 PROPANE DRIERS PROPANE TO STORAGE t»-POLY GAS. TO STORAGE b~ BUTANE TO STORAGE CO 33 TABLE 1-5  Range o f F e e d C o m p o s i t i o n s m o l e % E t h a n e 0 .1 0.9 P r o p a n e 21 - 24 P r o p e n e 14 - 20 I s o b u t a n e 23 - 26 N - B u t a n e 10 .5 20 I s o b u t e n e + 1 - B u t e n e 6 - 7.2 T r a n s - 2 - B u t e n e 4 .2 5.8 C i s - 2 - B u t e n e 2 .8 4.3 I s o p e n t a n e 3 .3 7.2 N - P e n t a n e 0 .1 0.2 1 - P e n t e n e 0 .1 0.2 2 - M e t h y l - 1 - B u t e n e 0 .2 0.3 T r a n s - 2 - P e n t e n e 0 .1 0.2 2 - M e t h y l - 2 - B u t e n e 0 .1 0.2 TOTAL OLEFINS 28 36 / 34 o f a b o u t 3 6 0 ° F and a p r e s s u r e o f 515 p . s . i . a . The t e m p e r a -t u r e o f t h e r e a c t i o n t a k i n g p l a c e i n t h e c a t a l y s t b e ds i s c o n t r o l l e d by t h e i n t r o d u c t i o n o f l i q u i d p r o p a n e f r o m t h e d e p r o p a n i z e r , w h i c h s e r v e s as a q u e n c h b e t w e e n t h e c a t a l y s t b e d s . I f t h e c o n c e n t r a t i o n o f o l e f i n s i n t h e f e e d i s t o o h i g h , t h e p r o p a n e s t r e a m c a n a l s o be u s e d as a r e c y c l e t o d i l u t e t h e f e e d . The r e a c t o r e f f l u e n t t h e n p a s s e s t h r o u g h t h e f e e d / e f f l u e n t e x c h a n g e r , and a p r e s s u r e r e d u c t i o n v a l v e , and e n t e r s t h e d e p r o p a n i z e r where p r o p a n e i s r e m o v e d as t h e t o p p r o d u c t . The p r o p a n e i s c o n d e n s e d and a p a r t o f i t i s r e c y c l e d t o t h e r e a c t o r w h i l e t h e r e s t i s d r i e d a n d s e n t t o s t o r a g e . The d e p r o p a n i z e r b o t t o m p r o d u c t i s s e n t t o t h e d e b u t a n i z e r w h e r e b u t a n e i s r e m o v e d as t h e t o p p r o d u c t . The b u t a n e i s c o n d e n s e d , c o o l e d and s e n t t o s t o r a g e . The b o t t o m p r o d u c t o f t h e d e b u t a n i z e r i s p o l y m e r g a s o l i n e w h i c h i s c o o l e d and s e n t t o s t o r a g e . T a b l e 1-6 shows some t y p i c a l c o m p o s i t i o n s o f t h e p r o d u c t s t r e a m s a l o n g w i t h t h e p r o d u c t s p e c i f i c a t i o n s . 3. P r o c e s s V a r i a b l e s The m o st i m p o r t a n t p r o c e s s v a r i a b l e s a r e t h o s e w h i c h a p p l y t o t h e p o l y m e r i z a t i o n r e a c t o r s . T h e y a r e : c a t a l y s t bed t e m p e r a t u r e , f e e d c o m p o s i t i o n , f e e d w a t e r c o n -t e n t , s p a c e v e l o c i t y , and c a t a l y s t a g e . 35 TABLE 1-6 S p e c i f i c a t i o n s and T y p i c a l C o m p o s i t i o n s o f P r o d u c t s PROPANE S p e c i f i e a t i o n s P r o p e n e P r o p a n e + p r o p e n e Typical Corn-positions E t h a n e P r o p a n e + p r o p e n e I - b u t a n e 10% maximum 9 5 % minimum Mol% ~ 0 7 T 98.6 0 . 0 Mol% 0.0 99.4 0.6 BUTANE S p e c i f i c a t i o n s P r o p a n e + p r o p e n e P e n t a n e s B u t a n e + b u t e n e Typical Compositions P r o p a n e + p r o p e n e I - b u t a n e N - b u t a n e B u t e n e s I - p e n t a n e N - p e n t a n e POLYMER GASOLINE S p e c i f i c a t i o n s R e i d V a p o r p r e s s u r e Typical Composition I - b u t a n e N - b u t a n e B u t e n e s I - p e n t a n e N - p e n t a n e P e n t e n e s 5% maximum 2% maximum 9 3 % minimum Mol% ~3TT 31 .5 40.8 10.4 14.0 0.1 Mol% 10.2 38.4 40.0 10.4 0.9 0.0 10 p . s . i . g . maximum Mol% 9.6 2.8 9.2 1 .8 2.8 71 .7 36 The c a t a l y s t bed t e m p e r a t u r e i s t h e m o s t i m p o r t a n t v a r i a b l e i n t h e p r o c e s s . I t w i l l , w i t h i n c e r t a i n l i m i t a -t i o n s , d e t e r m i n e t h e o l e f i n c o n v e r s i o n and t h e p r o d u c t i o n o f p o l y m e r g a s o l i n e . The t e m p e r a t u r e must be m a i n t a i n e d b e t w e e n a minimum o f 2 8 0 ° F ( b e l o w t h i s t e m p e r a t u r e t h e o l e f i n s c o m b i n e w i t h t h e p h o s p h o r i c a c i d t o p r o d u c e s t a b l e e s t e r s i n l i q u i d f o r m w h i c h s o f t e n t h e c a t a l y s t a nd c a u s e p l u g g i n g ) and a maximum o f 4 5 0 - 4 7 5 ° F . Above t h i s t e m p e r a -t u r e t h e p o l y m e r i z a t i o n r e a c t i o n w i l l be e x t e n d e d t o f o r m h i g h e r m o l e c u l a r w e i g h t h y d r o c a r b o n s , t a r l i k e m a t e r i a l s , w h i c h c o a t t h e c a t a l y s t s u r f a c e t h u s r e d u c i n g t h e t o t a l c a t a l y s t a c t i v i t y . The f e e d c o n c e n t r a t i o n i s i m p o r t a n t b e c a u s e i f t h e p r o p o r t i o n o f o l e f i n s i n t h e f e e d i s t o o h i g h ( o v e r 3 5 % ) , t h e t e m p e r a t u r e r i s e i n t h e f i r s t b e d , due t o t h e h e a t o f r e a c t i o n , c o u l d c a u s e a v e r y h i g h bed t e m p e r a t u r e and o v e r p o l y m e r i z a t i o n o f t h e o l e f i n s c o u l d c a u s e an i n -c r e a s e d r a t e o f c o k e and t a r d e p o s i t on t h e c a t a l y s t s u r f a c e . The w a t e r c o n t e n t o f t h e f e e d must be m a i n t a i n e d w i t h i n c e r t a i n l i m i t s s i n c e t h e c a t a l y s t i s a m i x t u r e o f a K i e s e l g u h r b a s e and p h o s p h o r i c a c i d w h i c h i s p r e s e n t i n c o m b i n a t i o n w i t h a c e r t a i n amount o f w a t e r . The amount o f w a t e r i n c o m b i n a t i o n i n f l u e n c e s t h e r e a c t i o n r a t e . When t h e c a t a l y s t i s h e a t e d i t t e n d s t o l o s e w a t e r by e v a p o r a -t i o n . I f t h e o r t h o and p y r o s t a t e s o f t h e p h o s p h o r i c a c i d 37 a r e d e h y d r a t e d t o t h e meta s t a t e , v o l a t i l i z a t i o n o f t h e p h o s p h o r i c a c i d and l o s s o f c a t a l y s t a c t i v i t y o c c u r s ( 6 7 ) . In o r d e r t o p r e v e n t t h i s l o s s and t h u s m a i n t a i n t h e c a t a l y s t a t a d e s i r e d a c i d c o n c e n t r a t i o n , t h e f e e d s h o u l d c a r r y s u f f i c i e n t w a t e r t o m a i n t a i n t h e p a r t i a l p r e s s u r e i n t h e f e e d e q u a l t o t h e v a p o r p r e s s u r e o f w a t e r i n t h e c a t a l y s t . T h i s i s v e r y d i f f i c u l t t o do i n an a c t u a l r e a c t o r w h e r e t h e r e i s a l a r g e v a r i a t i o n i n t e m p e r a t u r e . In p r a c t i c e , t h e amount o f w a t e r i n j e c t e d i n t o t h e f e e d as i t e n t e r s t h e r e a c t o r i s r e l a t e d t o some a v e r a g e b e d t e m p e r a t u r e . The s p a c e v e l o c i t y i s g e n e r a l l y c h o s e n t o g i v e c o n v e r s i o n s o f 8 0 - 9 0 % o f t h e f e e d o l e f i n s . H i g h e r c o n v e r -s i o n s a r e p o s s i b l e b u t i f t h e s p a c e v e l o c i t y i s t o o low o v e r p o l y m e r i z a t i o n and a r e d u c t i o n i n c a t a l y s t l i f e may o c c u r . J o n e s ( 7 3 ) showed some t y p i c a l s p a c e v e l o c i t y -c o n v e r s i o n d a t a i n a g r a p h r e p r o d u c e d i n F i g u r e 1-3. The c a t a l y s t l o s e s i t s a c t i v i t y w i t h t i m e ( o r t o t a l p r o d u c t i o n ) b e c a u s e o f t h e f o r m a t i o n o f h i g h p o l y m e r p r o d u c t s on i t s s u r f a c e . The r a t e o f f o r m a t i o n o f t h e s e p r o d u c t s d e p e n d on many f a c t o r s s u c h as t e m p e r a t u r e , f e e d w a t e r c o n t e n t , and s p a c e v e l o c i t y . F i g u r e 1-4 shows some t y p i c a l p l a n t d a t a f o r t h e d e c r e a s e i n c o n v e r s i o n w i t h t i m e o r a c c u m u l a t e d t o t a l p r o d u c t i o n . The o t h e r p r o c e s s v a r i a b l e s o f i n t e r e s t a r e t h e r e f l u x r a t i o s and r e b o i l e r l o a d s o f t h e d e p r o p a n i z e r and » 100 F I G U R E 1 - 3 O L E F I N C O N V E R S I O N V S . S P A C E V E L O C I 1001 i 1 r 9 0 h 80 h 70 h 60 -50 I 1 1 I , I 0 2 0 4 0 60 80 100 120 TOTAL PRODUCTION (gal/lb catalyst) F I G U R E 1 - 4 O L E F I N C O N V E R S I O N V S . T O T A L P R O D U C T I O N 40 d e b u t a n i z e r a l o n g w i t h t h e d e p r o p a n i z e r p r e s s u r e . T h e s e p r o c e s s v a r i a b l e s d e t e r m i n e t h e p r o d u c t s p l i t s i n t h e d i s -t i l l a t i o n t o w e r s and a r e t h u s r e l a t e d t o t h e p r o d u c t s p e c i f i c a t i o n s . D. T h e K i n e t i c s o f t h e P o l y m e r i z a t i o n o f O l e f i n s -P r e v i o u s Work V e r y l i t t l e i s known a b o u t t h e c a t a l y t i c p o l y -m e r i z a t i o n r e a c t i o n . S e v e r a l q u a l i t a t i v e m e c h a n i s m s h a v e b e e n p r o p o s e d ( 7 4 , 7 5 ) . W h i t m o r e ' s c a r b o n i u m i o n t h e o r y p o s t u l a t e s t h e a d d i t i o n o f a c a r b o n i u m i o n t o an o l e f i n t o f o r m a h i g h e r m o l e c u l a r w e i g h t c a r b o n i u m i o n w h i c h t h e n y i e l d s t h e o l e f i n p o l y m e r by e l i m i n a t i o n o f a p r o t o n . W i t h a c i d c a t a l y s t s , t h e i n i t i a l c a r b o n i u m i o n i s f o r m e d by a d d i t i o n o f a h y d r o g e n i o n f r o m t h e a c i d t o t h e e x t r a e l e c t r o n p a i r i n t h e d o u b l e b o n d o f t h e o l e f i n . A c c o r d i n g t o I p a t i e f f ( 6 4 ) , t h e p o l y m e r i z a t i o n o f o l e f i n s on p h o s p h o r i c a c i d i s p r e -c e e d e d by e s t e r i f i c a t i o n . T h e p o l y m e r i s t h e n f o r m e d by i n t e r a c t i o n o f two e s t e r m o l e c u l e s . The f r e e e n e r g y o f p o l y m e r i z a t i o n o f n o r m a l o l e f i n s ( a b o v e e t h e n e ) may be e x p r e s s e d by t h e f o l l o w i n g e q u a t i o n ( 7 5 ) : 41 92 96 100 104 108 112 116 ( b ) ACID CONCENTRATION (% H 3 P 0 4 ) F I G U R E 1 - 5 E F F E C T S O F P H O S P H O R I C A C I D C O N C E N T R A T I O N A N D T E M P E R A T U R E O N R E A C T I O N R A T E C O N S T A N T (b) 0 4 0 8 0 120 RECIPROCAL D I A M E T E R , l/D (inches"1) F I G U R E 1 - 6 . E F F E C T S O F Q U A R T Z P A R T I C L E S I Z E A N D P R O P E N E / N - B U T E N E R A T I O O N R E A C T I O N R A T E C O N S T A N T 43 C 2 n H 2 n = C 2 n H 4 n A F° = -20,320 + 33.26 T ( 1 - 1 ) E q u a t i o n ( 1 - 1 ) shows t h a t t h e l o s s o f f r e e e n e r g y and t h e p o l y m e r i z a t i o n t e n d e n c y o f o l e f i n s d e c r e a s e w i t h i n c r e a s i n g t e m p e r a t u r e . On t h e c o n t r a r y , h i g h p r e s s u r e s f a v o r p o l y m e r i z a t i o n as shown by A F = - R T l n P + A F° ( 1 - 2 ) L i t t l e has b e e n p u b l i s h e d on t h e k i n e t i c s o f t h e p o l y m e r i z a t i o n r e a c t i o n . I p a t i e f f ( 6 4 ) showed t h a t t h e r a t e s o f p o l y m e r i z a t i o n o f t h e Cg-C^ o l e f i n s i n c r e a s e d i n t h e f o l l o w i n g o r d e r : p r o p e n e , 1 - b u t e n e , 2 - b u t e n e , i s o -b u t e n e . The m o st c o m p l e t e s t u d y on t h e p o l y m e r i z a t i o n o f o l e f i n s was done by L a n g l o i s and W a l k e y ( 7 0 ) . T h e y p r e s e n t a r a t e e q u a t i o n f o r c o m p u t i n g t h e c o n v e r s i o n o f m i x t u r e s o f p r o p e n e and n - b u t e n e as a f u n c t i o n o f f e e d r a t e when u s i n g a l i q u i d p h o s p h o r i c a c i d on q u a r t z c a t a l y s t . The r e a c t i o n r a t e c o n s t a n t i s a f u n c t i o n o f t e m p e r a t u r e , p h o s p h o r i c a c i d c o n c e n t r a t i o n , q u a r t z p a r t i c l e s i z e , and p r o p e n e / n - b u t e n e r a t i o ( F i g u r e s 1-5, 1 - 6 ) . 44 The r e a c t i o n r a t e e q u a t i o n d e r i v e d by t h e s e a u t h o r s and shown b e l o w was o b t a i n e d e m p i r i c a l l y by t h e m o d i f i c a t i o n o f a r a t e e q u a t i o n w h i c h r e p r e s e n t e d an a s s u m e d m e c h a n i s m . S i n c e t h e m o d i f i c a t i o n s were c o n s i d e r a b l e t h e a u t h o r s s t a t e t h a t t h e f i n a l e q u a t i o n has no s u b s t a n t i a l t h e o r e t i c a l s i g n i f i c a n c e and s h o u l d be c o n s i d e r e d t o be o n l y an e m p i r i c a l r e p r e s e n t a t i o n o f t h e r a t e d a t a . f x (1+e x ) 2 dx . 2 = K ( 1 _ 3 ) ( 1 - x ) 2 + 0 . 3 x ( l - x ) 5 £ = Y m m m - 1 w h e r e : Y m = mole f r a c t i o n monomer i n f e e d m M = m o l e c u l a r w e i g h t o f f e e d m 3 Mp = m o l e c u l a r w e i g h t o f p o l y m e r x = f r a c t i o n a l c o n v e r s i o n o f monomer S = s p a c e r a t e i n g a s v o l u m e s a t r e a c t o r t e m p e r a t u r e and p r e s s u r e p e r v o l u m e o f c a t a l y s t v o i d s p e r h o u r , h r _ l ( v o i d v o l u m e a s s u m e d t o be 42% o f b u l k v o l u m e ) k = s p e c i f i c r e a c t i o n r a t e c o n s t a n t , h r ~ ^ 45 I t was n o t known w h e t h e r t h e l i q u i d p h o s p h o r i c a c i d model m e n t i o n e d a b o v e w o u l d a l s o d e s c r i b e t h e U.O.P. p r o c e s s w h e r e t h e p o l y m e r i z a t i o n r e a c t i o n o c c u r s on t h e s o l i d k i e s e l g u h r - p h o s p h o r i c a c i d c a t a l y s t . In a d i s c u s s i o n a t t h e T h i r d W o r l d P e t r o l e u m c o n g r e s s i n 1951 ( 7 6 ) q u e s t i o n s were r a i s e d as t o w h e t h e r t h e two c a t a l y s t s a nd t h u s t h e two p r o c e s s e s w ere n o t r e a l l y t h e same. Dr. E g l o f f , one o f t h e d e v e l o p e r s o f t h e U.O.P. p r o c e s s s t a t e d t h a t i t was U.O.P.'s c o n v i c t i o n t h a t some s i l i c a t e p h o s p h a t e i s t h e a c t i v e body i n t h e i r c a t a l y s t and a c c o r d i n g l y o n l y s p e c i f i c t y p e s o f k i e s e l g u h r c o u l d be u s e d i n t h e p r e p a r a t i o n o f t h e s o l i d c a t a l y s t . I t was a l s o started t h a t l i q u i d p h o s p h o r i c a c r d c o u l d be u s e d on o t h e r c a r r i e r s , b e s i d e s q u a r t z , i f t h e y had a n o n p o r o u s s t r u c t u r e . The c u r r e n t b e l i e f i s t h a t no r e a c t i o n t a k e s p l a c e b e t w e e n t h e l i q u i d p h o s p h o r i c a c i d and t h e q u a r t z , and i n a g r e e m e n t h e r e w i t h , t h e q u a r t z c a r r i e r c a n be r e g e n e r a t e d an i n d e f i n i t e number o f t i m e s . Dr. E g l o f f made t h e f o l l o w i n g s t a t e m e n t i n a n s w e r t o a q u e s t i o n on t h e c o m p a r i s o n o f t h e two t y p e s o f c a t a l y s t s : The U.O.P. polymerization catalyst referred to as the ' s o l i d phosphoric acid' catalyst is a combination of kieselguhr and phosphoric acid and in t h i s form the catalyst as pre-pared for use is a s o l i d 3 dry3 noncorrosive material. The phosphoric acid is present in varying combinations with the s i l i c a present in the c a r r i e r and in t h i s respect the U.O.P. 46 catalyst varies considerably from the phosphoric acid on quartz catalyst de-scribed in the paper of Langlois and Walkey. Therefore no direct comparison can be made regarding the overall state of the acid in both c a t a l y s t s . In another sense a s i m i l a r i t y does exist in that for optimum a c t i v i t y the catalyst hydration is controlled by regulating the water content of the feed to maintain a certain e q u i l i b -rium between water vapor content of the feed and the vapor pressure of the acid in the c a t a l y s t . The r a t e e q u a t i o n p r o p o s e d by L a n g l o i s a n d W a l k e y i s t h e o n l y one p u b l i s h e d i n t h e l i t e r a t u r e f o r t h e p o l y -m e r i z a t i o n o f o l e f i n s t o m o t o r f u e l . H o w e v e r , i t was n o t c l e a r as t o w h e t h e r i t c o u l d be u t i l i z e d f o r t h e U.O.P. p r o c e s s w h e r e a s o l i d p h o s p h o r i c a c i d c a t a l y s t was e m p l o y e d . The a n s w e r t o t h e a b o v e q u e s t i o n c o u l d o n l y be a r r i v e d a t by t e s t i n g t h e l i q u i d p h o s p h o r i c a c i d model w i t h t h e d a t a p r o d u c e d by an e x p e r i m e n t a l s t u d y u t i l i z i n g t h e s o l i d p h o s p h o r i c a c i d c a t a l y s t . 47 C h a p t e r 2 THE K I N E T I C S OF THE POLYMERIZATION OF OLEFINS IN THE PRODUCTION OF POLYMER GASOLINE A. I n t r o d u c t i o n In o r d e r t o be a b l e t o s u c c e s s f u l l y model t h e p o l y m e r i z a t i o n p l a n t i t i s n e c e s s a r y t o a c c u r a t e l y d e s c r i b e t h e most i m p o r t a n t u n i t p r o c e s s o f t h a t p l a n t , t h e p o l y -m e r i z a t i o n r e a c t o r . T h i s c a n n o t be a d e q u a t e l y done w i t h -o u t some k n o w l e d g e o f t h e k i n e t i c s o f t h e p o l y m e r i z a t i o n r e a c t i o n . I t i s n o t n e c e s s a r y i n t h i s c a s e t o d e r i v e a m e c h a n i s t i c model b u t r a t h e r t o e s t a b l i s h an o v e r a l l r a t e e x p r e s s i o n w h e r e t h e m a i n p r o c e s s v a r i a b l e s a r e r e l a t e d t o t h e r a t e o f r e a c t i o n . T h i s has b e e n done f o r a l i q u i d p h o s p h o r i c a c i d c a t a l y s t ( 7 0 ) b u t t h e r e were d o u b t s as t o w h e t h e r t h e r e s u l t s w o u l d a p p l y i n t h e c a s e o f t h e U.O.P. s o l i d p h o s p h o r i c a c i d c a t a l y s t . A number o f p l a n n e d e x p e r i m e n t s i n t h e p l a n t were made i n t h e h o p e ' t h a t t h e d a t a o b t a i n e d f r o m t h e s e t e s t s 48 c o u l d be u s e d t o t e s t t h e l i q u i d p h o s p h o r i c a c i d model o r p e r h a p s some o t h e r g e n e r a l i z e d r a t e e x p r e s s i o n . The d a t a f r o m t h e s e t e s t s i s p r e s e n t e d i n A p p e n d i x A. B e c a u s e o f t h e n a t u r e o f a c h e m i c a l p r o c e s s p l a n t i t i s n o t u s u a l l y p o s s i b l e t o i s o l a t e t h e e f f e c t s o f t h e i n d e p e n d e n t v a r i a b l e s . T h i s was t h e c a s e f o r t h e p o l y m e r i z a t i o n r e a c t o r t e s t s f r o m w h i c h no s i g n i f i c a n t r e s u l t s c o u l d be o b t a i n e d . T h e , n e e d t o c o n s t r u c t an e x p e r i m e n t a l r e a c t o r f r o m w h i c h s i g n i f i c a n t d a t a c o u l d be o b t a i n e d became o b v i o u s . B. E x p e r i m e n t a l A p p a r a t u s A s c h e m a t i c f l o w d i a g r a m o f t h e e x p e r i m e n t a l e q u i p m e n t i s shown i n F i g u r e 2-1. The f e e d was o b t a i n e d f r o m t h e S h e l l b u r n r e f i n e r y and i s t y p i c a l o f t h e f e e d s t o s u c h i n d u s t r i a l u n i t s ( s e e T a b l e 1 - 5 ) . The f e e d c y l i n d e r , w h i c h was f i l l e d o r i g i n a l l y w i t h l i q u i d t o a b o u t 80% o f i t s v o l u m e was h e l d i n an i n -v e r t e d p o s i t i o n t o a l l o w t h e w i t h d r a w a l o f a l i q u i d s t r e a m o f a l m o s t c o n s t a n t c o m p o s i t i o n . To i n c r e a s e t h e f e e d p r e s -s u r e , t h e c y l i n d e r was h e a t e d by an e x t e r n a l c o p p e r - s h i e l d e d e l e c t r i c h e a t i n g c a b l e w h i c h was m a n u a l l y c o n t r o l l e d by a V a r i a c . To p r e v e n t f l a s h i n g i n t h e m e t e r i n g pump, t h e f e e d was c o o l e d i n two s t e p s , o n c e b e f o r e t h e n e e d l e c o n t r o l PROPANE CYLINDER FEED CYLINDER FLOWMETER COOLING BATH HEATER HEATER ° o oo o o ° o O Q Q O ~ TO " C D REACTOR -<j> ^ PRODUCT |XH SAMPLING TO M PNEUMATIC CONTROL VALVE METERING PUMP FEED SAMPLING VALVE COOLING BATH LIQUID SAMPLING VALVE CHROMATOGRAPH AND RECORDER I G U R E 2 - 1 E X P E R I M E N T A L A P P A R A T U S 10 50 v a l v e and r o t a m e t e r and a g a i n a f t e r t h e p o i n t o f m i x i n g w i t h p r o p a n e . P r o p a n e c o u l d be u s e d as a f e e d d i l u e n t . The c o n t r o l o f t h e p r o p a n e f l o w was s i m i l a r t o t h a t o f t h e f e e d . A M i l t o n R o y - M i l r o y a l D c o n t r o l l e d v o l u m e pump was u s e d t o f o r c e t h e m i x e d f e e d t h r o u g h two h e a t e r s and a r e a c t o r . The c a l i b r a t i o n c u r v e s f o r t h e pump and r o t a -m e t e r s a r e shown i n A p p e n d i x B. The h e a t e r s were made o f s t a i n l e s s s t e e l p i p e and were 1 i n c h i n d i a m e t e r and 16 i n c h e s l o n g . The e x i t t e m p e r a t u r e s o f t h e f e e d were m e a s u r e d w i t h i r o n - c o n s t a n t a n t h e r m o c o u p l e s and c o n t r o l l e d by v a r y i n g t h e v o l t a g e t o a 500 w a t t i m e r s i o n h e a t e r i n e a c h v e s s e l by means o f a V a r i a c . The r e a c t o r was a s t a i n l e s s s t e e l p i p e , 2 i n c h e s i n d i a m e t e r and 12 i n c h e s l o n g , f l a n g e d a t e a c h e n d . The l e n g t h o f t h e r e a c t o r was d i v i d e d i n t o a number o f s e c t i o n s by s c r e e n s w h i c h were h e l d i n p o s i t i o n by s p a c e r s ( s e e F i g u r e s B-3, B - 4 ) . The f i r s t s e c t i o n s e r v e d as a f i n a l h e a t e r . E v e n h e a t d i s t r i b u t i o n was o b t a i n e d by a p a c k i n g o f 1/4 i n c h c o p p e r r i n g s . The c a t a l y s t was s u p p o r t e d b e t w e e n s c r e e n s i n t h e l a s t t h r e e s e c t i o n s o f t h e r e a c t o r . S i x t h e r m o c o u p l e i n l e t s were s p a c e d a l o n g t h e r e a c t o r so t h a t s h i e l d e d t h e r m o c o u p l e s c o u l d be i n s e r t e d i n t o t h e c e n t e r o f t h e gas s p a c e b e t w e e n t h e c a t a l y s t b e d s . 51 Good t e m p e r a t u r e c o n t r o l w i t h i n t h e r e a c t o r was o b t a i n e d by u s i n g an A . P . I . Model 712 p r o p o r t i o n a l p l u s r e s e t c o n t r o l l e r t o a d j u s t t h e i n p u t t o a s h i e l d e d h e a t i n g c a b l e w h i c h was wound u n d e r t h e i n s u l a t i o n o f t h e r e a c t o r . The c o n t r o l l e r was c o n n e c t e d t o t h e i r o n - c o n t a n t a n t h e r m o -c o u p l e l o c a t e d a t t h e e n t r a n c e t o t h e c a t a l y s t s e c t i o n . The o t h e r t h e r m o c o u p l e s were r e a d a t r e g u l a r t i m e i n t e r v a l s u s i n g a L e e d s and N o r t h r u p Model 8686 m i l l i v o l t p o t e n t i o -m e t e r . A c o n s t a n t p r e s s u r e (515 p . s . i . a . ) was h e l d i n t h e s y s t e m by a b a c k p r e s s u r e c o n t r o l v a l v e and a F o x b o r o p n e u m a t i c p r e s s u r e c o n t r o l l e r w i t h p r o p o r t i o n a l p l u s r e s e t modes. The g a s e s l e a v i n g t h e r e a c t o r p a s s e d t h r o u g h t h i s v a l v e t o a p u r g e l i n e . P r o d u c t s a m p l e s were o b t a i n e d i m m e d i a t e l y a f t e r t h e r e a c t o r t h r o u g h a s a m p l e l i n e w h i c h was c o n n e c t e d t o a Beckman GC-2 gas c h r o m a t o g r a p h . The s a m p l e s were c o o l e d t o c o n d e n s e t h e g a s e s , t h e n were m e a s u r e d d i r e c t l y i n t o t h e i n s t r u m e n t by a M i c r o t e k h i g h p r e s s u r e l i q u i d s a m p l i n g v a l v e . The gas c h r o m a t o g r a p h o u t p u t was i n t e g r a t e d d i r e c t l y on a S a r g e n t M o d e l SR r e c o r d e r w i t h Model 204 d i s c c h a r t i n t e g r a t o r . 52 C. E x p e r i m e n t a l P r o c e d u r e A t t h e b e g i n n i n g o f e a c h d a y ' s r u n s t h e r e a c t o r was f i l l e d w i t h new c a t a l y s t . The s y s t e m was t h e n f i l l e d w i t h l i q u i d f e e d a t t h e c y l i n d e r p r e s s u r e . T h e n t h e p r e -h e a t e r s and r e a c t o r h e a t e r w ere t u r n e d o n . When t h e d e s i r e d t e m p e r a t u r e was r e a c h e d t h e m e t e r i n g pump was s t a r t e d a nd t h e f l o w r a t e was a d j u s t e d by means o f t h e m e t e r i n g s c r e w on t h e pump. When m i x t u r e s o f p r o p a n e and f e e d were u s e d , t h e i r r a t i o was s e t by a d j u s t i n g t h e n e e d l e v a l v e s l o c a t e d b e f o r e t h e r o t a m e t e r s . Once t h e t e m p e r a t u r e i n s i d e t h e r e a c t o r had come t o s t e a d y s t a t e , w h i c h n o r m a l l y o c c u r r e d i n a b o u t one h o u r , s a m p l e s o f t h e f e e d and p r o d u c t were f e d t o t h e gas c h r o m a t o g r a p h . (A t y p i c a l c h r o m a t o g r a m i s shown i n F i g u r e B-5.) C o n d i t i o n s were t h e n a d j u s t e d f o r t h e n e x t r u n . The dew p o i n t f o r a t y p i c a l f e e d i s a b o u t 2 5 0 ° F , and s o t h e f e e d was c o m p l e t e l y v a p o r i z e d b e f o r e e n t e r i n g t h e c a t a l y s t s e c t i o n . S t e a d y s t a t e i n t h e r e a c t o r was shown by c o n s t a n t v a l u e s o f t e m p e r a t u r e on a l l r e a c t o r t h e r m o -c o u p l e s . Runs were made a t t e m p e r a t u r e s o f 3 0 0 ° , 3 4 0 ° , 3 8 0 ° , 4 2 0 ° , and 4 6 0 ° F and a t a s e r i e s o f f e e d f l o w r a t e s w h i l e h o l d i n g t h e f e e d c o n c e n t r a t i o n and t h e v o l u m e and p a r t i c l e s i z e o f c a t a l y s t c o n s t a n t . 53 A n o t h e r s e t o f r u n s were made t o c h e c k t h e p o s -s i b i l i t y t h a t t h e gas f i l m d i f f u s i o n r a t e was c o n t r o l l i n g . F e e d f l o w r a t e s were v a r i e d b u t t h e s p a c e t i m e , V / F , was m a i n t a i n e d c o n s t a n t . Runs were made a t v a r i o u s t e m p e r a t u r e l e v e l s and f i x e d p a r t i c l e s i z e . A f i n a l s e r i e s o f r u n s was made t o c h e c k f o r t h e e f f e c t o f p o r e d i f f u s i o n . In t h i s c a s e t h e t e m p e r a t u r e , t h e f l o w r a t e , and t h e c o m p o s i t i o n were m a i n t a i n e d c o n s t a n t w h i l e two d i f f e r e n t s i z e f r a c t i o n s o f c a t a l y s t p a r t i c l e s , o b t a i n e d by s i e v i n g were u s e d ; 4.76-6.73 mm. and 0.42-0.595 mm. D. E x p e r i m e n t a l R e s u l t s The t o t a l o l e f i n c o n v e r s i o n s , o b t a i n e d e x p e r i m e n t a l l y a t d i f f e r e n t f e e d f l o w r a t e s and t e m p e r a t u r e s a r e l i s t e d i n T a b l e 2-1 and shown i n F i g u r e 2-2. An i n t e g r a l a n a l y s i s was u s e d t o d e t e r m i n e a r a t e e x p r e s s i o n w h i c h w o u l d s a t i s f a c t o r i l y f i t t h e d a t a . A r a t e e x p r e s s i o n was s e l e c t e d and i n s e r t e d i n t o t h e p l u g f l o w r e a c t o r e q u a t i o n V/F = X dx r ( 2 - D 54 TABLE 2-1 O l e f i n C o n v e r s i o n a t D i f f e r e n t F l o w  R a t e s and T e m p e r a t u r e s F l o w R a t e o f F e e d O l e f i n C o n v e r s i o n m o l . / h r . 4 6 0 ° F 4 2 0 ° F 3 8 0 ° F 3 4 0 ° F 3 0 0 ° F 7.62 25.06 19 .24 14.49 11 .00 8.20 6.35 28.23 21 .72 17.54 13.00 10.74 5.08 33.05 24.53 20.33 15.50 13.96 3.81 35.89 29.70 23.86 19.00 15.70 2.54 45.73 38.91 31 .02 25.76 21 .30 1 .27 52.35 50.44 42.05 36.68 31 .59 F I G U R E 2 - 2 E F F E C T O F F L O W R A T E A N D T E M P E R A T U R E O N O L E F I N C O N V E R S I O N cn cn 56 The r i g h t - h a n d s i d e o f E q u a t i o n ( 2 - 1 ) was i n t e -g r a t e d n u m e r i c a l l y f o r e a c h e x p e r i m e n t a l r u n and a l i n e a r i t y t e s t was t h e n p e r f o r m e d t o c h e c k t h e s e l e c t e d r a t e e q u a t i o n s . The e m p i r i c a l r a t e e q u a t i o n o f L a n g l o i s and W a l k e y ( 7 0 ) as w e l l as a f i r s t o r d e r and s e c o n d o r d e r r a t e e q u a t i o n were t e s t e d . T a b l e 2-2 l i s t s t h e n u m e r i c a l v a l u e s f o r t h e i n t e g r a t e d e q u a t i o n s a t d i f f e r e n t f l o w r a t e s and t e m p e r a -t u r e s . I t c a n be s e e n i n F i g u r e 2-3 t h a t none o f t h e s e t h r e e e q u a t i o n s f i t t h e d a t a v e r y w e l l . An e m p i r i c a l e q u a -t i o n was t h e n d e v e l o p e d w h i c h g a ve a r e a s o n a b l y g o o d f i t ( s e e F i g u r e 2 - 4 ) . r = kc ( 1 ~ x > 2 ( 2 - 2 ) 0 ( 1 + x ) 2 I t was f e l t t h a t a more s o p h i s t i c a t e d p r o c e d u r e o f model d e v e l o p m e n t and d i s c r i m i n a t i o n was n o t j u s t i f i e d i n t h i s work f o r two r e a s o n s : 1. The b a s i c g o a l o f t h i s s t u d y was t o o b t a i n a r a t e e q u a t i o n w h i c h would a d e q u a t e l y d e s c r i b e t h e r e s u l t s from an i n d u s t r i a l r e a c t o r r a t h e r t h a n one whi c h would be used i n a m e c h a n i s t i c s t u d y . 2. S t a t i s t i c a l d i f f e r e n t i a t i o n between more com-p l e x models would p r o b a b l y f a l l w i t h i n the l i m i t s o f the e x p e r i m e n t a l e r r o r . 57 TABLE 2-2  N u m e r i c a l V a l u e s f o r I n t e g r a t e d  R a t e E q u a t i o n s T e m p e r a t u r e o f V/F c c . c a t a l y s t N u m e r i c a l V a l u e o f I n t e g r a l ( 2 ) ( 4 ) ( 3 ) ( 1 ) mol o l . / h r . 300 14.8 .0851 .0971 .0883 .0871 17.8 .1127 .1342 .1184 .1164 22.3 .1488 .1872 .1589 .1552 29.1 .1688 .2188 .1819 .1770 44.6 .2358 .3375 .2619 .2519 89.4 .3706 .6444 .4388 .4114 340 14.8 .1156 .1382 .1216 .1194 17.8 .1379 .1707 .1466 .1434 22.3 .1665 .2151 .1792 .1745 29.1 .2078 .2854 .2279 .2203 44.6 .2921 .4541 .3332 .3172 89.4 .4442 .8560 .5452 .5034 380 14.8 .1549 .1966 .1658 .1618 17.8 .1904 .2548 .2071 .2008 22.3 .2239 .3149 .2473 .2384 29.1 .2678 .4017 .3019 .2887 44.6 .3627 .6236 .4278 .4017 89.4 . 5278 1.1406 .6756 .6125 420 14.8 .2107 .2906 .2313 .2235 17.8 .2409 .3475 .2683 .2578 22.3 .2763 .4197 .3128 .2986 29.1 .3445 .5773 .4029 .3796 44.6 .4781 .9655 .5968 .5471 89.4 .6739 1.7675 .9305 .8146 460 14.8 .2831 .4343 .3215 .3066 17.8 .3247 .5288 .3761 .3558 22.3 .3912 .7002 .4678 .4368 29.1 .4324 .8199 .5277 .4884 44.6 .5893 1.3831 .7785 .6958 89.4 .7104 1.9529 1 .0000 .8674 ( l ) - L i q u i d P h o s p h o r i c A c i d , ( 2 ) - F i r s t O r d e r I r r e v e r s i b l e , ( 3 ) - S e c o n d O r d e r I r r e v e r s i b l e , ( 4 ) - E q u a t i o n ( 2 - 2 ) V / F (cc. catalyst/mole o l e f i n / h r ) F I G U R E 2 - 3 L I N E A R I T Y T E S T S F O R D I F F E R E N T R A T E E Q U A T I O N S c n OO 0 15 30 45 6 0 75 9 0 105 V / F (cc. catalyst/mole o le f in/hr ) F I G U R E 2 - 4 F I T O F E Q U A T I O N ( 2 - 2 ) T O E X P E R I M E N T A L D A T A <J0 60 The r a t e c o n s t a n t d e f i n e d by E q u a t i o n ( 2 - 2 ) was c a l c u l a t e d by m a k i n g a l e a s t s q u a r e s f i t o f t h e d a t a t o t h a t e q u a t i o n . The v a l u e s a r e l i s t e d i n T a b l e 2-3. An A r r h e n i u s p l o t o f t h e r a t e c o n s t a n t s a t f i v e t e m p e r a t u r e s ( F i g u r e 2-5) was made and t h e b e s t s t r a i g h t l i n e was f i t t e d by t h e l e a s t s q u a r e s m e t h o d . The e q u a t i o n o f t h i s l i n e i s : -7540 k = 2.87 x 1 0 5 e R T ( 2 - 3 ) The v a l u e o f 7,540 c a l . p e r gram m o l e i s c l o s e t o t h e v a l u e o f 7,920 c a l . p e r gram mole r e p o r t e d by L a n g l o i s and W a l k e y ( 7 0 ) f o r a l i q u i d p h o s p h o r i c a c i d on q u a r t z c a t a l y s t s y s t e m . T h e i r A r r h e n i u s p l o t i s a l s o shown on F i g u r e 2-5. Thomas ( 7 8 ) s u g g e s t e d an e s t i m a t e d v a l u e o f a b o u t 5,000 c a l . p e r gram-mole w h i c h d i f f e r s by a p p r o x i m a t e l y 50% f r o m t h e e x p e r i m e n t a l v a l u e f o u n d . To c h e c k w h e t h e r mass t r a s f e r t o t h e c a t a l y s t was e f f e c t i v e i n c o n t r o l l i n g t h e r a t e o f r e a c t i o n , r u n s were made a t s e v e r a l f l o w r a t e s h o l d i n g t h e r a t i o V/F c o n -s t a n t . The r e s u l t s shown i n T a b l e 2-4 and F i g u r e 2-6 show t h a t t h e c o n v e r s i o n was i n d e p e n d e n t o f f l o w r a t e , h e n c e i t i s a s s u m e d t h a t u n d e r t h e c o n d i t i o n s o f t h e s e e x p e r i -ments gas f i l m d i f f u s i o n i s u n i m p o r t a n t . 61 TABLE 2-3 R e a c t i o n R a t e C o n s t a n t o f E q u a t i o n ( 2 - 2 )  a t D i f f e r e n t T e m p e r a t u r e s T e m p e r a t u r e °F 1000/T o K - l R e a c t i o n R a t e C o n s t a n t , k c c . f e e d / h r . / c c . c a t a l y s t 300 2.369 35.0 340 2.252 48.4 380 2.141 69.0 420 2.044 109.0 460 1 .956 169.0 REACTION RATE CONSTANT (CC. FEED/HR/CC. CATALYST) 0 N> OJ ^ 01 CD -si co ID O ro oi * 01 cn ->i oo coO 1 i I I I I I I I | I I I I I M i l 63 TABLE 2-4 O l e f i n C o n v e r s i o n s a t V a r y i n g Flow R a t e s and  T e m p e r a t u r e s a t C o n s t a n t V/F R a t i o s V/F c . c . c a t . / m o l o l . / h r . F mol o l . / h r . V C . C . C d . O l e f i n C o n v e r s i o n 3 0 0 ° F 3 8 0 ° F 460 °F 89.1 .381 33.95 31 .59 42.05 52.35 .762 67.90 30.58 39.91 50.35 1 .143 101.85 32.09 41 .87 49.80 44.6 .762 33.95 21 .30 31 .02 45.73 1 .524 67.90 23.49 32.24 47.34 2.286 101.85 22.54 30.48 44.61 29.7 1 .143 33.95 15.70 23.86 35.89 2.286 67.90 17.10 23.46 37.05 3.429 101.85 16.30 25.15 35.96 0 - 5 0 1 1 r V/F = 29-7 cc. catalyst/mole olef in/hr 0 - 4 0 2 O CO > O o 0-20 UJ IT t r 460° F TT 380° F A 3 0 0 ° F 0-10 -0 - 0 0 I 2 OLEFIN FLOW R A T E , F (mole/hr) F I G U R E 2 - 6 T E S T F O R M A S S T R A N S F E R E F F E C T 4^  65 The r u n s w h i c h w ere made i n t h e i n i t i a l s t u d i e s u s e d t h e c o m m e r c i a l c a t a l y s t s c r e e n f r a c t i o n b e t w e e n 4.76 and 6.73 mm. T h i s f r a c t i o n c o n t a i n e d t h e m a j o r p a r t o f t h e c a t a l y s t as u s e d i n i n d u s t r i a l r e a c t o r s . Thomas ( 7 8 ) had s u g g e s t e d t h a t t h e i n d u s t r i a l r e a c t i o n m i g h t be d i f f u s i o n c o n t r o l l e d . To t e s t t h e p o s s i b i l i t y t h a t p o r e d i f f u s i o n was an i m p o r t a n t f a c t o r , a number o f r u n s w ere c a r r i e d o u t w i t h a s c r e e n f r a c t i o n b e t w e e n 0.420 a n d 0.595 mm. I f p o r e d i f f u s i o n i s n e g l i g i b l e t h e r a t e o f r e -a c t i o n s h o u l d be t h e same f o r b o t h s i z e s o f p a r t i c l e s . F o r s t r o n g p o r e d i f f u s i o n c o n t r o l , t h e r a t e o f r e a c t i o n v a r i e s i n v e r s e l y as t h e p a r t i c l e s i z e ( 7 7 ) . I t c an be s e e n i n T a b l e 2-5 t h a t p o r e d i f f u s i o n d o es a f f e c t t h e r a t e o f r e a c t i o n . The r a t i o o f c o n v e r s i o n s v a r i e d f r o m 1.5 t o 3.7 f o r a p a r t i c l e s i z e r a t i o o f a b o u t e l e v e n . F o r t h e s m a l l p a r t i c l e s , o v e r t h e r a n g e o f f l o w r a t e s u s e d i n t h e s e t e s t s , t h e R e y n o l d s number v a r i e d f r o m a b o u t 0.3 t o 2. ( s e e A p p e n d i x C f o r mass t r a n s f e r c a l c u l a -t i o n s ) . S i n c e t h e c a l c u l a t e d mass t r a n s f e r c o e f f i c i e n t s w e r e o f t h e same o r d e r o f m a g n i t u d e as t h e c a l c u l a t e d r e -a c t i o n r a t e c o n s t a n t s , i t i s e x p e c t e d t h a t f o r t h e s m a l l p a r t i c l e s ( l o w e r p a r t i c l e R e y n o l d s n u m bers) mass t r a n s f e r d o e s p l a y a r o l e i n c o n t r o l l i n g t h e r a t e o f r e a c t i o n . A t t h e h i g h e s t f l o w r a t e , t h e R e y n o l d s number a p p r o a c h e s t h e l o w e s t v a l u e u s e d w i t h t h e l a r g e s t p a r t i c l e s ; t h e r e g i o n 66 Table 2 - 5 Olefin Conversion as a Function of Particle Size Olefin Flow. Conversion Rate 300°F 3*40 °F 0 380 °F moles/hr. L S S/L L • S L S S/L 2.29 8.20 30.27 3.69 11.00 30.79 2.79 25.06 - -1.90 ID.7k 32.61 3.03 13.00 33.56 2.58 28.23 76.68 2.71 1.52 13.96 - - 15.50 39.07 2.53 33.05 77.00 2.32 1.1k 15.70 38.87 2.47 19.00 1+3-78 2.31 35.89 79.99 2.22 0.76 21.30 - - 25.76 50.1+0 I.96 1+5.73 87.1+8 1.92 O.38 31.59 1+9-78 1.58 36.68 55.06 1.51 52.35 Catalyst particle size : L U.76 - 5.73 H E B * S = 0.1+20 -0.595 mm. 67 where i t had p r e v i o u s l y b e e n shown t h a t mass t r a n s f e r was n o t a l a r g e f a c t o r . T h e r e f o r e , t h e l i m i t i n g c o n v e r s i o n r a t i o f o r t h e two p a r t i c l e s i z e s a p p e a r s t o be a b o u t 4. Thus t h e e f f e c t o f p o r e d i f f u s i o n i s q u i t e a p p r e c i a b l e e v e n t h o u g h i t i s n o t e q u i v a l e n t t o t h a t e x p e c t e d i n t h e r e g i o n o f s t r o n g p o r e d i f f u s i o n . S i n c e t h e c a t a l y s t p a r t i c l e s i z e u s e d i n t h e i n i t i a l e x p e r i m e n t s was t h e main f r a c t i o n o f t h e c o m m e r c i a l c a t a l y s t , and a l s o s i n c e t h e c o n v e r s i o n v a r i a t i o n w i t h p a r -t i c l e s i z e i s n o t c o n t r o l l i n g , t h e p o s t u l a t e d r a t e e q u a t i o n s h o u l d be a good model f o r t h e r e a c t i o n i n an i n d u s t r i a l r e a c t o r . S t u d i e s ( 6 4 , 70) h a v e b e e n made o f t h e r e a c t i o n r a t e s f o r t h e d i m e r i z a t i o n o f i n d i v i d u a l o l e f i n s . The r e -a c t i o n r a t e i n c r e a s e s i n t h e o r d e r : p r o p e n e , 1 - b u t e n e , 2 - b u t e n e , i s o b u t e n e . I t was a l s o shown t h a t i s o b u t e n e has an e f f e c t on t h e r a t e o f p o l y m e r i z a t i o n o f p r o p e n e and n - b u t e n e , t h a t i s , i n t e r a c t i o n e f f e c t s a r e a p p r e c i a b l e . An e s t i m a t e has be e n made o f t h e r e l a t i v e r e a c t i o n r a t e s o f t h e i n d i v i d u a l o l e f i n s by a v e r a g i n g t h e i n d i v i d u a l c o n -v e r s i o n d a t a f o r s e v e r a l r a n g e s o f o v e r a l l c o n v e r s i o n ( T a b l e 2-6, F i g u r e 2 - 7 ) . C o n s i d e r i n g o n l y t h e m a i n o l e f i n s i n t h e f e e d , i . e . p r o p e n e and t h e v a r i o u s b u t e n e i s o m e r s , t h e o r d e r o f i n c r e a s i n g r e a c t i v i t y w o u l d b e : T - 2 - b u t e n e , C - 2 - b u t e n e , p r o p e n e , i s o b u t e n e + 1 - b u t e n e . The l a c k o f 68 TABLE 2-6 C o n v e r s i o n o f I n d i v i d u a l O l e f i n s O v e r a l l O l e f i n 10-20 21-30 31-40 41-55 75-85 C o n v e r s i o n - % Individual Olefin Conversion-% P r o p e n e 5 17 27 41 84 I s o b u t e n e + 1 - B u t e n e 30 50 62 69 83 T - 2 - B u t e n e -7 -10 -5 6 61 C - 2 - B u t e n e 2 9 12 20 68 P e n t e n e - 1 30 40 55 65 85 2 - M e t h y l - 1 - B u t e n e 65 70 75 70 28 T - P e n t e n e - 2 -12 -25 -30 -25 27 2 - M e t h y l - 2 - B u t e n e 1 V a l u e s o f -100 t o -300 were f o u n d . TOTAL CONVERSION (PERCENT) F I G U R E 2 - 7 C O N V E R S I O N O F I N D I V I D U A L O L E F I N S A T D I F F E R E N T T O T A L O L E F I N C O N V E R S I O N S \ 70 a g r e e m e n t o f t h i s o r d e r w i t h t h a t o f t h e p r e v i o u s l y m e n t i o n e d w o r k e r s i s p r o b a b l y due t o t h e i s o m e n " z a t i o n r e a c t i o n s w h i c h a r e o c c u r r i n g . T h i s w o u l d a l s o e x p l a i n t h e n e g a t i v e c o n -v e r s i o n s shown i n F i g u r e 2-7. E. D i s c u s s i o n A number o f a s s u m p t i o n s were made i n t h e p r e s e n t s t u d y and w i l l now be e x a m i n e d . C o n s t a n t c a t a l y s t - b e d t e m p e r a t u r e s h a v e b e e n a s -sumed. H o w e v e r , s m a l l t e m p e r a t u r e d i f f e r e n c e s were f o u n d b e t w e e n t h e b e d s . A t 3 0 0 ° F a d i f f e r e n c e o f 1-3°F was f o u n d w h i l e a t 4 6 0 ° F a d i f f e r e n c e o f f r o m 1 0 - 1 3 ° F o c c u r r e d . Be-c a u s e o f t h e low a c t i v a t i o n e n e r g y f o r t h e r e a c t i o n , t h e r e s u l t a n t maximum e r r o r s i n t h e r e a c t i o n r a t e w o u l d be o n l y 2 t o 3% a t 3 0 0 ° F and a b o u t 10% a t 4 6 0 ° F . The o v e r a l l e r r o r i n t h e c a l c u l a t i o n o f t h e c o n v e r s i o n w o u l d be l e s s t h a n t h i s s i n c e e a c h v a l u e r e p o r t e d i s t h e a v e r a g e o f s e v e r a l r u n s . C o n s t a n t c a t a l y s t a c t i v i t y has been a s s u m e d d u r i n g a d a i l y s e r i e s o f r u n s (8-12 h o u r p e r i o d ) . S i n c e f r e s h c a t a l y s t was u s e d e a c h d a y t h i s s h o u l d be a f a i r l y g ood a s s u m p t i o n . As a c h e c k , r e p e a t r u n s were made a t t h e s t a r t and f i n i s h of a s e r i e s o f r u n s . Good a g r e e m e n t was a l w a y s f o u n d . 71 L a n g l o i s and W a l k e y ( 7 0 ) h a v e shown t h a t t h e p h o s -p h o r i c a c i d c o n c e n t r a t i o n o f t h e c a t a l y s t i s i m p o r t a n t i n d e t e r m i n i n g i t s a c t i v i t y . In i n d u s t r i a l r e a c t o r s w a t e r i s a d d e d t o t h e f e e d b e f o r e e n t e r i n g t h e r e a c t o r t o m a i n -t a i n t h e c a t a l y s t i n t h e same s t a t e o f h y d r a t i o n . In t h i s work no w a t e r was a d d e d . H o w e v e r , as p o i n t e d o u t p r e v i o u s l y , no c h a n g e i n c a t a l y s t a c t i v i t y was f o u n d b e t w e e n t h e s t a r t and t h e f i n i s h o f a d a y ' s r u n . C a l c u l a t i o n s o f t h e r a t e c o n s t a n t s were made a s -s u m i n g p l u g f l o w t h r o u g h t h e c a t a l y s t b e d s . R e s i d e n c e t i m e d i s t r i b u t i o n s t u d i e s were n o t made on t h e r e a c t o r , b u t b e c a u s e o f t h e s h o r t l e n g t h o f e a c h bed and t h e c a r e f u l - p a c k i n g o f t h e c a t a l y s t b e t w e e n t h e s c r e e n s u p p o r t s t h i s a p p e a r s t o be t h e b e s t a s s u m p t i o n . A c h e c k was made on t h e p o s s i b i l i t y o f t h e r m a l p o l y m e r i z a t i o n o c c u r r i n g i n t h e h e a t e r s b e f o r e t h e c a t a l y s t b e d s . Runs were made a t t h e l o w e s t f l o w r a t e s t h r o u g h t h e empty r e a c t o r . A t t h e maximum p r e h e a t e r o u t l e t t e m p e r a t u r e o f 2 5 0 ° F t h e r e d i d n o t a p p e a r t o be any t h e r m a l p o l y m e r i z a -t i o n . F. C o n c l u s i ons The e x p e r i m e n t a l k i n e t i c d a t a o b t a i n e d f o r t h e p o l y m e r i z a t i o n o f a c o m m e r c i a l o l e f i n m i x t u r e was c o r r e l a t e d w e l l by an e m p i r i c a l r a t e e q u a t i o n , w h e r e a s t h e r a t e e q u a t i o n 72 p o s t u l a t e d by L a n g l o i s and W a l k e y (7 0 ) f o r p o l y m e r i z a t i o n on a l i q u i d p h o s p h o r i c a c i d c a t a l y s t d i d n o t r e p r e s e n t t h e d a t a a d e q u a t e l y . H o w e v e r , t h e a c t i v a t i o n e n e r g y r e p o r t e d by L a n g l o i s and W a l k e y f o r t h e r e a c t i o n on a l i q u i d p h o s -p h o r i c a c i d c a t a l y s t a g r e e d q u i t e c l o s e l y w i t h t h e a c t i v a -t i o n e n e r g y f o u n d i n t h i s work w i t h a c o m m e r c i a l s o l i d p h o s p h o r i c a c i d c a t a l y s t . S i n c e t h e t e s t c o n d i t i o n s were q u i t e d i f f e r e n t ( o l e f i n i c m i x t u r e s i n t h i s work c o m p a r e d t o p u r e o l e f i n s i n L a n g l o i s and W a l k e y ' s w o r k ) i t w o u l d h a v e b e e n f o r t u i t o u s 1 i f t h e r a t e e q u a t i o n s had a g r e e d more c l o s e l y . The s i m i l a r i t y i n t h e v a l u e s o f t h e a c t i v a t i o n e n e r g i e s i n t h e two c a s e s w o u l d seem t o i n d i c a t e t h a t t h e r e a c t i o n s w h i c h o c c u r d u r i n g t h e p o l y m e r i z a t i o n o f o l e f i n s a r e v e r y s i m i l a r w i t h t h e two t y p e s o f c a t a l y s t . I t was f o u n d t h a t p o r e d i f f u s i o n a f f e c t e d t h e o v e r a l l c o n v e r s i o n w i t h t h e c o m m e r c i a l c a t a l y s t . H o w e v e r , t h e k i n e t i c model p r o p o s e d i n t h i s w o r k , w h i c h a d e q u a t e l y f i t s t h e c o n v e r s i o n d a t a o b t a i n e d w i t h t h e m o st r e p r e s e n t a -t i v e s i z e f r a c t i o n i n t h e i n d u s t r i a l c a t a l y s t s h o u l d h o l d w e l l f o r m o d e l i n g a f u l l s c a l e p l a n t e v e n i f t h e r e i s a s m a l l amount o f c a t a l y s t s h a t t e r i n g d u r i n g t h e d u mping an d l e v e l i n g o f t h e b e d s . 73 C h a p t e r 3 THE IMPLEMENTATION OF CHESS FOR THE SIMULATION OF THE POLYMERIZATION PLANT A. S t r a t e g y f o r t h e D e v e l o p m e n t o f t h e S i m u l a t i o n Model A d e s c r i p t i o n o f t h e s t r a t e g y f o r t h e s i m u l a t i o n o f c h e m i c a l p l a n t s has been v e r y a b l y p r e s e n t e d by C r o w e , et al. i n t h e i r r e c e n t book ( 1 5 ) . T h e r e f o r e , o n l y a b r i e f d i s c u s s i o n o f t h e m e t h o d o l o g y u s e d i n t h i s work w i l l now be p r e s e n t e d . P e r h a p s t h e most i m p o r t a n t . q u e s t i o n t h a t must be a n s w e r e d b e f o r e t h e s i m u l a t i o n o f a c h e m i c a l p l a n t i s t o be a t t e m p t e d i s : Why s h o u l d t h i s p l a n t be s i m u l a t e d ? The a n s w e r t o t h i s q u e s t i o n s h o u l d d e t e r m i n e t h e s t e p s t o be t a k e n i n t h e d e v e l o p m e n t and u s e o f t h e s i m u l a t e d m o d e l . In t h e p r e s e n t work t h e a n s w e r t o t h e a b o v e q u e s t i o n i s : The p l a n t s h o u l d be s i m u l a t e d so t h a t t h e s i m u l a t i o n model c a n t h e n be u s e d t o o p t i m i z e t h e p l a n t o p e r a t i o n . O p t i m i -z a t i o n o f p l a n t o p e r a t i o n means f i n d i n g t h e s e t o f i m p o r -t a n t i n d e p e n d e n t p r o c e s s v a r i a b l e s w h i c h w i l l m a x i m i z e ( o r 74 m i n i m i z e ) some p r e d e t e r m i n e d o b j e c t i v e f u n c t i o n ( s u c h as c o n v e r s i o n , c o s t , p r o d u c t i o n o f a p r o d u c t o r p r o d u c t s , o r p r o f i t ) . W i t h t h e a b o v e m e n t i o n e d g o a l i n mind s e v e r a l c r i t e r i a f o r t h e s i m u l a t i o n c a n be s e t : 1. The i m p o r t a n t p r o c e s s v a r i a b l e s must be l o -c a t e d . These a r e t h e independent p r o c e s s v a r i a b l e s t h a t might a p p r e c i a b l y a f f e c t t h e v a l u e o f t h e o b j e c t i v e f u n c t i o n . F o r example; i f t h e o b j e c t i v e f u n c t i o n i s c o n v e r s i o n i n a r e a c t o r , t h e amount of c o o l a n t used i n t h e r e a c t o r might be an i m p o r t a n t v a r i a b l e w h i l e t h e p r e s s u r e a c h i e v e d by a pump t o send t h e p r o d u c t t o s t o r a g e would p r o b a b l y be o n l y o f a v e r y minor s i g n i f i c a n c e . 2 . A second f a c t o r w h i c h must always be kept i n mind i s t h e equipment l i m i t a t i o n s . Because o f t h e s e l i m i -t a t i o n s , t h e method used f o r t h e s i m u l a t i o n o f an o p e r a t i n g p l a n t i s q u i t e d i f f e r e n t from t h a t used f o r a s i m u l a t i o n f o r t h e d e s i g n o f a new p l a n t . The equipment d e s i g n c h a r -a c t e r i s t i c s ( s u c h as heat exchanger a r e a , number o f p l a t e s i n a d i s t i l l a t i o n column, e t c . ) a r e f i x e d i n an e x i s t i n g p l a n t . Thus the o p e r a t i n g range i s much more l i m i t e d t h a n i n t h e d e s i g n c a s e . S i n c e t h e equipment l i m i t a t i o n s d e f i n e t h e bounds o f some o f the p r o c e s s v a r i a b l e s , t h i s s h o u l d be c o n s i d e r e d i n t h e development o f t h e p r o c e s s equipment models. 75 F o r example, I f i t i s known t h a t a c e r t a i n heat exchanger u s i n g steam a t a g i v e n p r e s s u r e can o n l y heat a f e e d s t r e a m t o a c e r t a i n maximum t e m p e r a t u r e because o f equipment and p r o c e s s l i m i t a t i o n s , t h e n perhaps a s i m p l e model f i x i n g t h i s t e m p e r a t u r e might be used r a t h e r t h a n a complex o v e r a l l heat exchanger model. 3. A v e r y i m p o r t a n t f a c t o r t h a t must be c o n s i d e r e d i n t h e development o f a p l a n t model f o r o p t i m i z a t i o n i s computer c a l c u l a t i o n t i m e . S i n c e t h e s i m u l a t e d model may be c a l l e d upon hundreds (and perhaps thousands) o f t i m e s f o r t h e c a l c u l a t i o n o f t h e o b j e c t i v e f u n c t i o n and c o n s t r a i n t s , t h e computer c e n t r a l p r o c e s s i n g u n i t (C.P.U.) c a l c u l a t i o n t i m e f o r each p l a n t s i m u l a t i o n must be kept as s m a l l as p o s s i b l e . On t h e o t h e r hand the s i m u l a t e d model must r e p r e -s e n t t h e o p e r a t i n g p l a n t i n an adequate manner i f t h e r e s u l t s o f t h e o p t i m i z a t i o n a r e t o be o f any v a l u e . P l a n t s i m u l a t i o n t o a g r e a t e x t e n t i s s t i l l an a r t and t h e s u c c e s s or f a i l u r e o f t h e s t u d y may o f t e n depend on t h e "good judgement" and e x p e r i e n c e o f t h e s i m u l a t o r . W i t h t h e a b o v e m e n t i o n e d p o i n t s i n m i n d , t h e p o l y -m e r i z a t i o n p l a n t was m o d e l e d i n s e v e r a l s t e p s . F i r s t , a p r e l i m i n a r y p l a n t model was d e v e l o p e d i n w h i c h t h e p r o c e s s m o d u l e s were k e p t v e r y s i m p l e . W i t h t h i s p r e l i m i n a r y model v a r i o u s f a c t o r s c o u l d be s t u d i e d s u c h a s : o r d e r o f p r o c e s s 76 m o d u l e c o m p u t a t i o n s , c o n v e r g e n c e f o r c i n g , and p e r h a p s e v e n ' more i m p o r t a n t , w h e t h e r t h e s i m u l a t i o n s y s t e m w o r k e d c o r -r e c t l y on t h e U.B.C. IBM 360/67 c o m p u t e r . An i n t e r m e d i a t e p l a n t model was t h e n d e v e l o p e d w here some o f t h e s i m p l e p r o c e s s m o d u l e s were r e p l a c e d by u s e r - d e v e l o p e d p r o c e s s m o d u l e s t h a t were more c o m p r e h e n s i v e and t h u s d e s c r i b e d t h e p r o c e s s more a d e q u a t e l y . As t h e s e new m o d e l s were i n t r o d u c e d , t h e C.P.U. t i m e was c a r e f u l l y w a t c h e d a n d , w h e r e v e r p o s s i b l e , s t e p s were t a k e n t o l o w e r t h e c o m p u t a t i o n t i m e . The t h r e e c h o s e n o p t i m i z a t i o n m e t h o d s were t e s t e d on t h e i n t e r m e d i a t e p l a n t model so t h a t t h e most e f f e c t i v e m ethod c o u l d t h e n be u t i l i z e d on t h e f i n a l p l a n t m o d e l . S e v e r a l more p r o c e s s m o d u l e s were r e p l a c e d by more e x a c t m o d e l s t o a r r i v e a t t h e f i n a l p o l y p l a n t m o d e l . T h i s model was t h e n u s e d f o r a more c o m p l e t e o p t i m i z a t i o n s t u d y . B. C o m p l e t e P l a n t M o d e l s 1. P r e ! i m i n a r y M o d e l The r e a s o n s f o r t h e d e v e l o p m e n t o f t h e p r e l i m i -n a r y p l a n t model were t o : 1. 'Check the o p e r a t i o n o f CHESS on t h e U.B.C. IBM 360/67 d i g i t a l computer. 77 2. Get an i d e a o f t h e convergence o f t h e m a t e r i a l and energy b a l a n c e s w i t h v a r y i n g c a l c u l a t i o n o r d e r s and convergence f o r c i n g s t r e a m s . 3. See i f t h e model g e n e r a l l y d e s c r i b e d t h e p l a n t and o b t a i n e x p e r i e n c e f o r t h e development o f more comprehensive models. x The i n f o r m a t i o n f l o w d i a g r a m f o r t h e p r e l i m i n a r y p o l y p l a n t model i s shown i n F i g u r e 3-1. I t i s v e r y s i m i l a r t o t h e f l o w d i a g r a m o f t h e S h e l l b u r n p o l y m e r i z a t i o n u n i t ( F i g u r e 1 - 2 ) . Some d i f f e r e n c e s a r e t h a t t h e i n f o r m a t i o n f l o w d i a g r a m u t i l i z e s u n i t c o m p u t a t i o n s f o r f l o w d i v i d e r s and f l o w m i x e r s , and t h a t t h e p o l y m e r i z a t i o n r e a c t o r s a r e d i v i d e d i n t o f o u r s e p a r a t e r e a c t i o n u n i t s and f o u r m i x e r u n i t s . The a c c u m u l a t o r s a r e n o t i n c l u d e d i n t h e i n f o r m a -t i o n f l o w d i a g r a m s i n c e t h e s i m u l a t i o n i s f o r a s t e a d y s t a t e o p e r a t i o n . The p r o p a n e d r i e r s were a l s o n o t i n c l u d e d and t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r was d i v i d e d i n t o two s i m p l e h e a t e x c h a n g e r s . E x c e p t f o r t h e ADD2 p r o c e s s m o d u l e , a l l t h e o t h e r m o d u l e s a r e i n c l u d e d i n t h e CHESS s y s t e m . The ADD2 m o d u l e was i n c l u d e d e s s e n t i a l l y t o g a i n e x p e r i e n c e i n t h e u s e o f a u s e r s u p p l i e d p r o c e s s m o d u l e . The r e p r e s e n t a t i o n o f t h e i n f o r m a t i o n f l o w d i a g r a m as i n p u t t o t h e CHESS s y s t e m i s shown i n A p p e n d i x D. The p r e l i m i n a r y p o l y p l a n t i n p u t d a t a F I G U R E 3 - 1 P R E L I M I N A R Y P O L Y M E R I Z A T I O N P L A N T M O P E L 79 i s shown i n two f o r m s . The f i r s t f i l e , POLY, c o n t a i n s t h e i n p u t d a t a i n t h e f r e e - f o r m f o r m a t w h i l e t h e s e c o n d f i l e , POLYNL, c o n t a i n s t h e same d a t a i n t h e NAMEL 1ST f o r m a t . The f r e e - f o r m f o r m a t i s a b i t e a s i e r t o u s e f o r t h e u n i n i t i a t e d p r o g r a m m e r b u t i t s u s e r e q u i r e s a l a r g e r p r o g r a m f o r t h e r e a d i n g o f d a t a t h a n d o e s t h e NAMELIST f o r m a t . The f i r s t v e r s i o n o f CHESS ( A p r i l 1968) u s e d i n t h i s s t u d y has f a c i l i t i e s f o r b o t h t y p e s o f i n p u t . How-e v e r , t h e u p d a t e d v e r s i o n o f CHESS u s e d ( V e r s i o n 1, Mod. 9, J u n e 1970) f o r t h e i n t e r m e d i a t e and f i n a l p o l y p l a n t m o d e l s was d e s i g n e d f o r u s e on s m a l l e r c o m p u t e r s and t h u s u t i l i z e d t h e more e f f i c i e n t NAMELIST o p t i o n o n l y . The i n p u t s t r e a m 1, p o l y f e e d , was s p e c i f i e d a s shown i n T a b l e 3-1. The c o m p o s i t i o n c h o s e n was w i t h i n t h e r a n g e o f t y p i c a l f e e d c o m p o s i t i o n s shown i n T a b l e 1-5. The s m a l l a mounts o f 2 - m e t h y l - 1 - b u t e n e and 2 - m e t h y l - 2 - b u t e n e were i n c l u d e d w i t h t h e T - 2 - p e n t e n e . The r e p r e s e n t a t i o n o f t h e d i m e r i z e d p r o d u c t o f t h e o l e f i n p o l y m e r i z a t i o n i s v e r y c o m p l e x b e c a u s e o f t h e l a r g e number o f i s o m e r s t h a t c a n be p r o d u c e d . H o w e v e r , f o r t h i s s t u d y , 1 - h e p t e n e was c h o s e n t o r e p r e s e n t t h i s p r o d u c t . T h i s s h o u l d be a r e a s o n a b l e e s t i m a t e s i n c e h e x e n e s , h e p t e n e s , and o c t e n e s a r e f o r m e d so t h a t 1 - h e p t e n e a p p e a r s t o r e p r e s e n t an a v e r a g e c o m p o s i t i o n . S i n c e t h e s i m p l e s t p r o c e s s m o d u l e s were u s e d , t h e s t r e a m p r o p e r t i e s a r e g e n e r a l l y f i x e d , r a t h e r t h a n c a l c u l a t e d TABLE 3-1 P o l y F e e d S p e c i f i c a t i o n f o r P r e l i m i n a r y  P o l y m e r i z a t i o n P l a n t Model M o l e s / h o u r P r o p a n e 23 I s o b u t a n e 20 N - B u t a n e 18 I s o p e n t a n e 3 N - P e n t a n e 1 P r o p e n e 17 C - 2 - B u t e n e 4 T - 2 - B u t e n e 3 I s o b u t e n e 10 T - 2 - P e n t e n e 1 1 - H e p t e n e 0 TOTAL FLOW 100 T e m p e r a t u r e - 5 5 0 ° R P r e s s u r e - 300 p . s . i . a . V a p o r F r a c t i o n - 0.0 81 by t h e p r o c e s s m o d u l e s . A l i s t o f t h e p r o c e s s m o d u l e s and t h e i r e q u i p m e n t p a r a m e t e r s i s shown i n T a b l e 3-2. The o r d e r o f p r o c e s s m o d u l e c a l c u l a t i o n s u s e d was 1, 2, 3, 36, 4, 5, 6, 7, 8, 9, 10, 1 1 , 12, 13, 14, 1 5 , 16, 17, 18, 19, 20, 2 1 , 38, 3 7 , 22, 2 3 , 24, 2 5 , 26, 27, 28. S t u d i e s were made o f c o n v e r g e n c e and C.P.U. t i m e . I t was f o u n d t h a t c o n v e r g e n c e f o r c i n g w i t h s t r e a m 15 g a v e t h e b e s t r e s u l t s and f o r a c o n v e r g e n c e t o l e r a n c e o f 0.001, c o n v e r -g e n c e was r e a c h e d a f t e r 5 l o o p c a l c u l a t i o n s w i t h a t o t a l C.P.U. t i m e ( l o a d i n g o f o b j e c t p r o g r a m and c a l c u l a t i o n s ) o f 36 s e c o n d s . ( T h e m a t e r i a l and e n e r g y b a l a n c e c a l c u l a -t i o n s c o n v e r g e when t h e s t r e a m v a r i a b l e s o f a l l t h e s t r e a m s show a f r a c t i o n a l c h a n g e b e t w e e n s u c c e s s i v e l o o p s l e s s t h a n a s p e c i f i e d c o n v e r g e n c e t o l e r a n c e .) I n i t i a l v a l u e s were e n t e r e d f o r s t r e a m 15 ( i n s t e a d o f 0.0) i n t h e hope t h a t t h i s w o u l d r e d u c e t h e number o f r e c y c l e c a l c u l a t i o n s , b u t t h e r e s u l t s o b t a i n e d were t h e same as a b o v e . The v a l u e o f t h e c o n v e r g e n c e t o l e r a n c e was v a r i e d b u t o n c e a g a i n t h e number o f r e c y c l e l o o p s r e m a i n e d t h e same as shown b e l o w . The o v e r a l l m a t e r i a l b a l a n c e i s shown i n T a b l e 3-3. TABLE 3-2 E q u i p m e n t P a r a m e t e r S p e c i f i c a t i o n s f o r P r e l i m i n a r y  P o l y m e r i z a t i o n P l a n t Model E q u i p. P r o c e s s S t r e a m s S t r e a m s F u n c t i o n o f E q u i p m e n t Number M o d u l e In Out P a r a m e t e r 1 PUMP 1 2 p r e s s = 515 p . s . i . a . 2 HXER 2 3 temp = 7 3 5 ° R 3 HXER 3 4 temp = 8 1 0 ° R 36 MIXR 4,51 52 4 DVDR 52 5,6 s p l i t 0.5,0.5 5 MIXR 6,50,33 7 6 ADD2 7 8 c o n v . i s o b u t e n e = 0.90 7 MIXR 8,32 9 8 ADD2 9 10 c o n v . p r o p e n e = 0.60 9 MIXR 10,31 11 10 ADD2 11 12 c o n v . p r o p e n e = 0.75 11 MIXR 12,30 13 12 ADD2 13 14 c o n v . C - 2 - b u t e n e = 0.95 13-20 (same f u n c t i o n s as 5-12) 38 HXER 15 24 temp = 7 1 5 ° R 37 VALV 24 55 p r e s s = 265 p . s . i . a . 00 ro TABLE 3-2 ( C o n t i n u e d ) E q u i p. Number P r o c e s s M o d u l e S t r e a m s In S t r e a m s Out F u n c t i o n o f E q u i p m e n t P a r a m e t e r 22 DIST 55 25,42 . f r a c . t o p s : p r o p a n e = 0.98 I - b u t a n e = 0.05 p r o p e n e = 0.98 23 HXER 25 26 temp = 5 6 0 ° R 24 DVDR 26 27,39 s p l i t = 0.6,0.4 25 PUMP 27 28 p r e s s = 530 p . s . i . a . 26 DVDR 28 29,34,51 s p l i t = 0.5,0.5,0.0 27 DVDR 29 30,31,32,33 s p l i t = .16,.16,.33,.33 28 DVDR 34 35,36,37,38 s p l i t = .33,.33,.16,.16 29 PUMP 39 40 p r e s s = 300 p . s . i . a . 31 DIST 42 45,43 f r a c . t o p s : p r o p a n e = 1.0 I - b u t a n e = 0.99 N - b u t a n e = 0.95 I - p e n t a n e = 0.6 N - p e n t a n e = 0.6 p r o p e n e = 1.0 b u t e n e s = 0.95 32 HXER 43 44 temp = 5 7 0 ° R 33 HXER 45 46 temp = 5 6 0 ° R 34 PUMP 46 . 47 p r e s s = 100 p . s . i . a . 35 HXER 47 48 temp = 5 6 0 ° R CO 84 TABLE 3-3 O v e r a l l M a t e r i a l B a l a n c e f o r P r e l i m i n a r y  P o l y m e r i z a t i o n Model M o l e s / h r . F e e d P r o p a n e B u t a n e P o l y G a s o l i n e P r o p a n e 23 21 .9 1.1 0.0 I - B u t a n e 20 0.40 19.4 0.19 N - B u t a n e 18 0.0 17.1 0.90 I - P e n t a n e 3 0.0 1 .8 1 .2 N - P e n t a n e 1 0.0 1.0 0.0 P r o p e n e 17 0.79 0.03 0.002 C - 2 - B u t e n e 4 0.0 0.19 0.01 T - 2 - B u t e n e 3 0.0 2.85 0.15 1 - B u t e n e 10 0.0 0.0 1 .0 T - 2 - P e n t e n e 1 0.0 0.0 1 .0 1 - H e p t e n e 0 0.0 0.0 14.48 TOTAL 100.0 23.09 43.48 18.94 85 C o n v e r g e n c e R e c y c l e C.P.U. s e c . T o l e r a n c e L o o p s ( A p p r o x i m a t e ) 0.001 5 40 0.01 5 40 0.05 5 39 Some p r o g r a m d i f f i c u l t i e s were e x p e r i e n c e d w i t h t h e CHESS HXER m o d u l e . The HXER m o d u l e w o u l d n o t c o n v e r g e f o r t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r and c o u l d n o t be u s e d f o r h e a t e x c h a n g e b e t w e e n s t r e a m s . I t was u s e d as a s i m p l e t e m p e r a t u r e s e t t e r b u t e v e n i n t h e s i m p l e mode, p r o b l e m s a r o s e i n t h e d i s t i l l a t i o n c o l u m n c o n d e n s e r s , 23, 33 when t h e p r o d u c t s t r e a m s were c a l c u l a t e d as a l l v a p o r i n s t e a d o f a l l l i q u i d . 2. I n t e r m e d i a t e Model Once a l l t h e i n f o r m a t i o n t h a t c o u l d be o b t a i n e d w i t h t h e p r e l i m i n a r y model was f o u n d , t h e n e x t s t e p became e v i d e n t , t h a t i s , t h e u s e o f more c o m p r e h e n s i v e p r o c e s s m o d u l e s i n t h e o v e r a l l p l a n t m o d e l . A p r o c e s s m o d u l e , ADD3, was d e v e l o p e d f o r t h e p o l y m e r i z a t i o n r e a c t o r s u s i n g t h e o v e r a l l r e a c t i o n r a t e m e c h a n i s m o b t a i n e d f r o m t h e e x p e r i m e n t a l k i n e t i c s t u d y ( s e e C h a p t e r 2 ) . 86 Two b a s i c c h a n g e s were made t o t h e p r e l i m i n a r y m o d e l . The f i r s t was t h e r e p l a c e m e n t o f t h e ADD2 by t h e more c o m p r e h e n s i v e ADD3 r e a c t o r m o d u l e . The s e c o n d c h a n g e was t h a t t h e t o t a l f l o w r a t e was r a i s e d t o 250 m o l e s / h o u r f r o m 100 m o l e s / h o u r . T h i s was done t o a d j u s t t h e model f l o w r a t e s t o t h e a c t u a l p l a n t f l o w r a t e s . ( T h e f e e d c o m p o s i -t i o n was n o t c h a n g e d . ) The CHESS i n p u t f o r t h e i n t e r m e d i a t e p l a n t m o d e l , POLYR, i s shown i n A p p e n d i x D. S e v e r a l p r o b l e m s became a p p a r e n t when t h e i n t e r m e -d i a t e model was u t i l i z e d . The m a j o r p r o b l e m e n c o u n t e r e d was t h e l a r g e r i s e i n C.P.U. t i m e as c o m p a r e d t o t h e p r e l i m i n a r y p o l y p l a n t m o d e l . T h i s was due t o t h e i n c r e a s e d c a l c u l a t i o n t i m e r e q u i r e d by t h e ADD3 p r o c e s s m o d u l e . In ADD3 a s e t o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i s s o l v e d by t h e R u n g e - K u t t a method w h i c h o p e r a t e s on t h e c o n t i n u o u s f u n c t i o n s i n a s t e p -w i s e m a n n e r . T h u s , t h e a c c u r a c y o f t h e m e t h o d i s r e l a t e d t o t h e c a l c u l a t i o n s t e p s i z e u s e d . H o wever, t h e s m a l l e r t h e s t e p s i z e ( s t e p s i z e = bed l e n g t h / n u m b e r o f s t e p s ) , t h e g r e a t e r t h e C.P.U. t i m e r e q u i r e d f o r t h e s o l u t i o n . T h i s i s shown i n T a b l e 3-4. The d i f f e r e n c e s i n c o n v e r s i o n (shown by m o l e s o f 1 - h e p t e n e p r o d u c e d ) i s v e r y s m a l l when t h e number o f c a l c u l a -t i o n s s t e p s i s r e d u c e d f r o m 50 ( m o s t a c c u r a t e v a l u e ) t o 5 ( o n l y a b o u t 0 . 0 5 % ) . H owever, when t h e number o f s t e p s i s 87 TABLE 3-4 E f f e c t o f Number o f R u n g e — K u t t a C a l c u l a t i o n  S t e p s on P r o c e s s C a l c u l a t i o n s Convergence Tolerance = 0. 001 No. o f R e c y c l e C.P.U. M o l e s 1 - H e p t e n e S t e p s L o o p s T i m e , s e c . Bed 1 Bed 2 Bed 3 Bed 4 50 9 175 3. 5768 8.9543 14.2283 19.1988 20 9 107 3. 5713 8.9400 15.2126 19.1905 10 9 77 3. 5658 8.9256 15.1968 19.1821 5 9 61 3. 5729 8.9477 15.2283 19.2029 3 9 55 3. 6268 9.1150 15.4610 19.3510 1 3 29 10. 5783 25.6807 34.1481 96.8733 88 r e d u c e d t o 3 t h e d i f f e r e n c e i s much g r e a t e r ( a b o u t 0.8%) and t h e r e s u l t s a r e c o m p l e t e l y e r r a t i c when o n l y one c a l c u l a t i o n s t e p i s u s e d . H o w e v e r , e v e n when t h e number o f c a l c u l a t i o n s t e p s i s 3, t h e C.P.U. t i m e i s a b o u t 50% g r e a t e r t h a n t h a t f o r t h e p r e l i m i n a r y p l a n t model (55 s e c . v s . 36 s e c ) . C h a n g e s were t h e n made i n t h e c o n v e r g e n c e t o l e r a n c e i n an a t t e m p t t o r e d u c e t h e C.P.U. t i m e as shown b e l o w . No. o f R-k s t e p s = 3 Convergence R e c y c l e C.P.U. Time T o l e r a n c e Loops Seconds 0.001 9 55 0.01 7 49 0.05 6 46 A c o m p a r i s o n was t h e n made o f t h e most e x a c t c a l c u -l a t i o n p r o c e d u r e ( l a r g e s t number o f R-k c a l c u l a t i o n s t e p s and l o w e s t c o n v e r g e n c e t o l e r a n c e ) and a more a p p r o x i m a t e one i n o r d e r t o c h o o s e a c o n v e r g e n c e t o l e r a n c e a nd a number o f R-k s t e p s f o r w h i c h t h e r e s u l t s w o u l d a d e q u a t e l y r e p r e s e n t t h e a c t u a l p l a n t and a t t h e same t i m e u t i l i z e a minimum o f C.P.U. t i m e . 89 More e x a c t L e s s E x a c t No. R-k s t e p s 50 3 Conv. t o l e r a n c e 0.001 0.05 R e c y c l e l o o p s 8 6 C P . U . t i m e , s e c . 160 43 A f t e r bed 4 Moles 1-heptene 20.11 20.24 Temperature, °R 944 945 The d i f f e r e n c e o f o n l y 0.5% b e t w e e n t h e more e x a c t c a l c u l a t i o n and t h e l e s s e x a c t d i d n o t m e r i t a C P . U . t i m e a l m o s t f o u r t i m e s as l a r g e f o r t h e more e x a c t c a s e and t h u s i t was d e -c i d e d t o u t i l i z e t h e p a r a m e t e r s o f t h e l e s s e x a c t c a s e . ( T h i s seemed q u i t e r e a s o n a b l e s i n c e t h e e r r o r s i n v o l v e d i n a p p r o x i m a t i n g t h e p l a n t u n i t s by t h e p r o c e s s m o d u l e s a r e much l a r g e r . H o wever, t h i s c a l c u l a t i o n d i f f e r e n c e p r o v e d t o be q u i t e i m p o r t a n t when o p t i m i z a t i o n m e t h o d s were u t i l i z e d as i s p o i n t e d o u t l a t e r on i n t h i s w o r k . ) A b a s i c c h a n g e was t h e n made i n t h e i n t e r m e d i a t e p o l y p l a n t model i n o r d e r t o r e d u c e t h e C P . U . t i m e e v e n f u r t h e r . The two p o l y m e r i z a t i o n r e a c t o r u n i t s were c o m b i n e d i n t o one u n i t ( F i g u r e 3 - 2 ) . I t was f e l t t h a t t h i s was q u i t e r e a s o n a b l e s i n c e t h e two u n i t s w o u l d n o r m a l l y be o p e r a t i n g u n d e r s i m i l a r c o n d i t i o n s . In t h e c a s e where i t m i g h t be F I G U R E 3 - 2 I N T E R M E D I A T E P O L Y M E R I Z A T I O N P L A N T MODEL O 91 d e s i r e d t o s t u d y b o t h u n i t s i n d i v i d u a l l y ( i f t h e i r o p e r a t i n g c h a r a c t e r i s t i c s were d i f f e r e n t ) t h e p l a n t model w i t h t h e two p a r a l l e l r e a c t o r u n i t s c o u l d be u s e d . T h i s c h a n g e s i m p l i f i e d t h e c a l c u l a t i o n s ( s e e i n p u t d a t a , P O L Y - I , i n A p p e n d i x D ) . Thus f o r a c o n v e r g e n c e t o l e r a n c e o f 0.05 and w i t h no. R-k s t e p s = 3 t h e r e d u c e d s y s t e m now c o n v e r g e d i n 6 r e -c y c l e l o o p s u t i l i z i n g 32 C P . U . s e c , a b o u t 25% l e s s t h a n w i t h t h e two r e a c t o r t r a i n s . The c a l c u l a t i o n o r d e r u t i l i z e d was 1 , 2 , 3 , 5 , 6 , 7 , 8 , 9 , 1 0, 1 1 , 12 , 38 , 37 , 22 , 23 , 24, 25, 2 6, 11, 12 and s t r e a m 12 was u s e d f o r c o n v e r g e n c e f o r c i n g . The i n t e r m e d i a t e p o l y p l a n t model was u t i l i z e d f o r t h e c o m p a r i s o n o f s e v e r a l o p t i m i z a t i o n m e t h o d s ( s e e C h a p t e r 4 ) . E a r l y i n t h a t s t u d y , i t became c l e a r t h a t t h e d i f f e r e n c e b e t w e e n t h e more e x a c t c a l c u l a t i o n and t h e l e s s e x a c t c a l -c u l a t i o n p r e v i o u s l y a c c e p t e d as b e i n g r e a s o n a b l e was n o t a d e q u a t e f o r t h e u s e o f t h e s i m u l a t e d s y s t e m as a b l a c k box c a l c u l a t o r f o r an o p t i m i z a t i o n m e t h o d . T h u s a f u r t h e r s t u d y was made o f c o n v e r g e n c e e r r o r w i t h t h e g o a l o f a r r i v i n g a t a "maximum a c c e p t a b l e " e r r o r w i t h a minimum C P . U . t i m e . a. S y s t e m C o n v e r g e n c e E r r o r An o p t i m i z a t i o n m e t h o d t r e a t s t h e s i m u l a t e d s y s t e m l i k e a b l a c k b ox. C e r t a i n i n d e p e n d e n t v a r i a b l e s a r e s e t and t h e b l a c k box c a l c u l a t e s t h e o b j e c t i v e f u n c t i o n and t h e 92 c o n s t r a i n t s . T h e s e v a l u e s a r e t h e n e x a m i n e d by t h e o p t i m i -z a t i o n p r o g r a m w h i c h t h e n c h a n g e s t h e v a l u e o f t h e i n d e p e n d e n t p a r a m e t e r s i n a way w h i c h w i l l e v e n t u a l l y l e a d t o an optimum v a l u e f o r t h e o b j e c t i v e f u n c t i o n w i t h i n t h e c o n s t r a i n e d a r e a . The i n d e p e n d e n t v a r i a b l e s u s e d i n t h e o p t i m i z a t i o n s t u d i e s on t h e i n t e r m e d i a t e model were t h e s p l i t p a r a m e t e r s o f d i v i d e r s 24 and 26. F o r d i v i d e r 24, t h e s p l i t o f s t r e a m 17 was t r e a t e d as an i n d e p e n d e n t v a r i a b l e w i t h t h e c o n s t r a i n t b e i n g t h a t t h e sum o f t h e s p l i t s o f s t r e a m s 17 and 23 be u n i t y . In t h e c a s e o f d i v i d e r 26, t h e s p l i t s o f s t r e a m s 19, 20, 2 i were v a r i e d i n d e p e n d e n t l y w h i l e t h e s p l i t o f s t r e a m 22 was c o n s t r a i n e d so t h a t t h e t o t a l s p l i t be u n i t y . The o b j e c t i v e f u n c t i o n c o n s i d e r e d was t h e t o t a l amount o f p o l y m e r p r o d u c t p r o d u c e d . The p r o b l e m was a l s o c o n s t r a i n e d by l i m i t s on t h e s t r e a m t e m p e r a t u r e s l e a v i n g t h e r e a c t o r u n i t s ; 6, 8, 10, and 12. The e f f e c t s o f R-k s t e p s i z e , and c o n v e r g e n c e t o l e r a n c e on t h e a c c u r a c y o f t h e o b j e c t i v e f u n c t i o n c a l c u -l a t i o n s w i t h d i f f e r e n t b a s e c o n d i t i o n s i s shown i n T a b l e s 3-5 and 3-6. T a b l e 3-5 l i s t s t h e i n d e p e n d e n t v a r i a b l e s f o r e a c h r u n . Run 1 s t a r t s w i t h a l l p r o c e s s s t r e a m s ( e x c e p t t h e p o l y f e e d i n p u t ) a t z e r o . In t h e f o l l o w i n g r u n s ( 2 - 8 ) t h e f i n a l v a l u e s o f t h e p r o c e s s s t r e a m v a r i a b l e s f r o m t h e p r e -c e e d i n g r u n a r e u t i l i z e d as t h e i n i t i a l v a l u e s i n t h e new TABLE 3-5  V a l u e s o f I n d e p e n d e n t V a r i a b l e s f o r T a b l e 3-6 I n d e p e n d e n t Run Number V a r i a b l e 1 2 3 4 5 6 7 8 DVDR 24 s p l i t s t r e a m 17 0 .60 0 .62 0 .60 0 .70 0.60 0 .61 0 .60 0 .60 DVDR 26 s p l i t s t r e a m 19 0 .20 0 .05 0 .20 0 .10 0.20 0 .21 0 .20 0 .20 20 0 .30 0 .31 0 .30 0 .30 0.30 0 .31 0 .30 0 .30 21 0 .30 0 .41 0 .30 0 .40 0.30 0 .31 0 .30 0 .30 TABLE 3-6 C o n s t r a i n t and O b j e c t i v e F u n c t i o n V a l u e s f o r R e p e a t e d S i m u l a t i o n S y s t e m C a l c u l a t i o n s No. of Runge Kutta Steps = S Convergence Tolerance = 0.05 RUN 1 2 3 4 5 6 7 8 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 855 908 940 945 40.065 6 885 940 945 943 40.637 3 853 940 934 939 39.829 3 864 909 916 919 39.186 6 844 885 913 921 38.825 3 846 889 917 925 39.145 2 851 898 927 933 39.661 3 851 899 927 934 39.665 1 Convergence Tolerance = 0.001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 854 905 936 942 39.944 13 884 938 943 941 40.560 8 853 904 934 939 39.871 5 863 906 913 916 39.047 12 847 890 918 925 39.130 10 849 894 923 930 10 851 896 929 935 39.685 7 852 902 931 937 39.785 6 Convergence Tolerance = 0.0001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 853 905 936 941 39.941 18 884 938 943 941 40.558 13 853 904 934 939 39.868 9 863 906 914 916 39.043 17 847 890 918 925 39.133 15 849 894 923 930 39.422 14 851 900 929 935 39.687 12 852 902 932 938 39.788 11 T a b l e 3-6 ( C o n t i n u e d ) No. of Runge-Kutta Steps = 10 Convergence Tolerance = 0.05 RUN 1 2 3 4 5 6 7 8 Te m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 853 905 937 944 6 883 936 942 942 40.387 3 852 900 931 937 39.556 3 864 907 915 919 38.969 5 843 882 910 919 38.528 3 844 886 914 924 38.840 2 850 895 924 932 39.398 3 850 896 925 932 39.384 1 Convergence Tolerance = 0.001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 852 902 933 940 39.673 13 883 934 940 940 40.308 8 852 901 931 937 39.600 5 861 903 910 914 38.735 12 845 887 915 923 38.818 10 847 891 920 928 39.114 10 850 896 926 933 39.412 7 851 898 938 936 39.513 6 Convergence Tolerance = 0.0001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 852 902 933 940 39.671 17 883 934 940 940 40.305 13 852 900 931 937 39.596 9 861 903 909 914 38.731 17 845 887 914 923 38.821 15 847 891 920 928 39.117 14 850 897 926 933 39.414 12 851 899 929 936 39.515 11 T a b l e 3-6 ( C o n t i n u e d ) No. of Runge-Kutta Steps = 50 Convergence Tolerance = 0.05 RUN 1 2 3 4 5 6 7 8 Te m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 854 905 937 943 39.816 6 883 936 942 942 40.410 3 852 901 931 937 39.582 3 864 908 915 919 39.002 5 843 883 910 919 38.564 3 845 886 914 924 38.873 2 850 895 924 932 39.424 3 850 896 925 932 39.410 1 Convergence Tolerance = 0.0001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 852 902 933 940 39.696 18 883 935 940 940 40.328 13 852 901 931 937 39.623 8 861 903 910 915 38.760 17 845 887 915 923 38.854 15 848 891 920 929 39.148 15 850 897 926 933 39.441 11 857 899 929 936 39.541 10 vo T a b l e 3-6 ( C o n t i n u e d ) No. of Runge-Kutta Steps = 5 Convergence Tolerance = 0.01 RUN 1 2 3 4 5 6 7 8 Te m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 853 903 934 941 39.726 9 883 935 941 941 40.363 4 852 901 931 937 39.620 4 862 904 910 915 38.795 8 845 886 915 923 38.821 7 847 891 919 928 39.116 5 850 896 925 933 39.401 4 851 898 928 935 39.518 3 Convergence Tolerance = 0.001 T e m p - S t r e a m 6 8 10 12 M o l e s p o l y m e r R e c y c l e l o o p s 852 902 933 940 39.707 13 883 935 940 940 40.340 8 852 901 932 938 39.634 5 861 903 910 915 38.773 12 845 887 915 923 38.856 10 847 891 920 928 39.152 10 850 897 926 933 39.446 7 851 899 929 936 39.547 6 98 r e c y c l e c a l c u l a t i o n s . The i n d e p e n d e n t v a r i a b l e s were c h a n g e d f o r e a c h r u n w h i c h i s what an o p t i m i z a t i o n m ain p r o g r a m w o u l d do. W h i l e most o f t h e i n d e p e n d e n t v a r i a b l e c h a n g e s b e t w e e n r u n s are q u i t e s m a l l , a l a r g e c h a n g e was i n s e r t e d b e t w e e n r u n s 3 and 4 t o t e s t t h e s y s t e m ' s r e s p o n s e . T a b l e 3-6 shows t h a t when t h e number o f R-k s t e p s i s 3 o r 5, t h e c a l c u l a t i o n s f o r c o n v e r g e n c e t o l e r a n c e s o f 0 . 0 0 1 and 0 . 0 0 0 1 a r e e s s e n t i a l l y t h e same a l t h o u g h t h e r e i s a s i g n i f i c a n t d i f f e r e n c e when t h e c o n v e r g e n c e t o l e r a n c e i s 0 . 0 5 . The r e s u l t s a r e a l s o v e r y c l o s e f o r t h e c a s e s where t h e number o f R-k s t e p s i s 1 0 and 50, b u t b o t h d i f f e r a p p r e -c i a b l y f r o m t h e c a s e where t h e number o f R-k s t e p s i s 3. F u r t h e r t e s t s were t h e n made w i t h t h e number o f R-k s t e p s a t 5 and t h e c o n v e r g e n c e t o l e r a n c e a t 0 . 0 1 . Even t h o u g h s m a l l d i f f e r e n c e s i n t h e r e s u l t s were shown i t was f e l t t h a t t h e r e d u c t i o n i n C P . U . t i m e was o f g r e a t e r i m p o r t a n c e . F o r r u n 8, t h e p e r c e n t a g e d i f f e r e n c e b e t w e e n t h e o b j e c t i v e f u n c t i o n v a l u e s i s o n l y 0 . 0 6 % f o r t h e f o l l o w i n g c a s e s : No. R-k s t e p s 5 0 5 Conv. T o l e r a n c e 0.0001 0.01 Moles polymer 39-541 39-518 Some o f t h e r e s u l t s o f T a b l e 3-6 a r e p l o t t e d i n F i g u r e 3-3 and c l e a r l y show t h a t t h e c a l c u l a t e d r e s u l t s 99 4 0 0 39-5 o Li. UJ > o IxJ CQ O 39-0 38 -5 NO. R - K CONVERSION STEPS TOLERANCE o 3 0 0 0 1 , 0-0001 10 0 - 0 0 1 , 0 0001 0 50 0 0 0 0 1 Q 5 0 0 0 1 V 5 0 0 1 9 9 3 5 RUN NUMBER 8 F I G U R E 3 - 3 O B J E C T I V E F U N C T I O N V A L U E S FOR R E P E A T E D C A L C U L A T I O N S 100 a r e v e r y s i m i l a r f o r t h e number o f R-k s t e p s >_ 5, and c o n -v e r g e n c e t o l e r a n c e <_ 0.01. The c o n v e r g e n c e s t u d y p o i n t s o u t a v e r y s e r i o u s d i f f i c u l t y i n t h e u s e o f t h e s i m u l a t e d model as a b l a c k box c a l c u l a t o r w i t h i n an o p t i m i z a t i o n p r o g r a m . T h i s i s , t h a t t h e v a l u e c a l c u l a t e d f o r t h e o b j e c t i v e f u n c t i o n , f o r t h e same s e t o f i n d e p e n d e n t v a r i a b l e s , w i l l d e p e n d on t h e s t a r t i n g v a l u e s o f t h e p r o c e s s s t r e a m v a r i a b l e s . In t h i s r e s p e c t , t h e s i m u l a t i o n s y s t e m m i g h t be c o n s i d e r e d more l i k e a " s t o c h a s t i c - t y p e " model t h a n l i k e a " d e t e r m i n i s t i c - t y p e " m o d e l . H o w e v e r , due t o t h e f a c t t h a t t h e o p t i m i z a t i o n m e t h o d s , i n t h e a r e a n e a r t h e op t i m u m , make o n l y v e r y s m a l l c h a n g e s i n t h e v a l u e s o f t h e i n d e p e n d e n t v a r i a b l e s b e t w e e n s i m u l a t i o n r u n s , t h e o b j e c t i v e f u n c t i o n c a l c u l a t i o n s a r e more ^ e x a c t b e c a u s e o f t h e r e p e a t e d c a l c u l a t i o n s made a t v e r y s i m i l a r v a l u e s o f t h e i n d e p e n d e n t v a r i a b l e s . The i m p o r t a n c e o f e s t a b l i s h i n g t h e maximum a c c e p t -a b l e t o l e r a n c e s i s made c l e a r by t h e r e s u l t s l i s t e d b e l o w . Case No. R-k Steps Approx.C.P.U. S e c . / c y c l e Conv. t o l . T o t a l C y c l e s Runs 1-8 T o t a l C.P.U. s e c . ( a p p r o x ) 1 3 3 0.05 27 81 2 5 3.7 0.01 44 163 3 10 — 0.001 71 — 4 50 11 0.0001 107 1177 101 W i t h t h e t o l e r a n c e s c h o s e n ( C a s e 2) t h e c o m p u t e r t i m e i s a b o u t t w i c e as l o n g as o b t a i n e d p r e v i o u s l y ( C a s e 1) b u t t h e c a l c u l a t e d r e s u l t s a r e q u i t e s i m i l a r t o t h o s e o f C a s e 4 w h i c h , w h i l e b e i n g a b i t more e x a c t , r e q u i r e s a b o u t 7 t i m e s t h e C P . U . t i m e . 3. F i n a l M odel Two more u s e r - d e v e l o p e d p r o c e s s m o d u l e s were a d d e d t o t h e i n t e r m e d i a t e p o l y p l a n t model t o a r r i v e a t t h e f i n a l p o l y p l a n t m o d e l . The ADD4 p r o c e s s m o d u l e r e p l a c e d t h e DIST p r o c e s s m o d u l e and t h e ADD5 p r o c e s s m o d u l e r e p l a c e d t h e HXER m o d u l e . The ADD4 m o d u l e i s a more c o m p r e h e n s i v e model f o r a d i s t i l l a t i o n c o l u m n ( s e e C h a p t e r 3-C f o r more d e t a i l s ) . The ADD5 m o d u l e i s a s i m p l e t e m p e r a t u r e s e t t e r and was u s e d b e c a u s e o f d i f f i c u l t i e s w i t h t h e CHESS s u p p l i e d HXER m o d u l e . The i n f o r m a t i o n f l o w d i a g r a m i s shown i n F i g u r e 3-4 and t h e CHESS i n p u t , POLY-F, i s l i s t e d i n A p p e n d i x D. A l i s t i n g o f a s a m p l e CHESS o u t p u t f o r t h e f i n a l p o l y m e r i z a -t i o n p l a n t model i s shown i n A p p e n d i x E. The CHESS s o u r c e p r o g r a m was o r i g i n a l l y c o m p i l e d u t i l i z i n g t h e FORTRAN-G c o m p i l e r . The WATFIV c o m p i l e r was t h e n u t i l i z e d f o r d e b u g g i n g p u r p o s e s when p r o b l e m s a r o s e i n t h e e x e c u t i o n o f CHESS u n d e r an o p t i m i z a t i o n m a i n p r o g r a m ( s e e s e c t i o n D o f t h i s c h a p t e r ) . The WATFIV d e b u g g e d s o u r c e v e r s i o n o f CHESS was t h e n c o m p i l e d u s i n g t h e FORTRAN-H \ 103 c o m p i l e r . A l t h o u g h t h e l o a d i n g t i m e f o r t h e c o m p i l e d o b j e c t p r o g r a m i s a b i t l o n g e r t h a n t h a t f o r t h e FORTRAN-G c o m p i l e d o b j e c t p r o g r a m t h e e x e c u t i o n t i m e i s a p p r e c i a b l y f a s t e r and t h e r e c y c l e l o o p c a l c u l a t i o n t i m e was r e d u c e d by a b o u t 3 0 % , f r o m 3.5 s e c . / l o o p u s i n g FORTRAN-G t o 2.5 s e c . / l o o p u s i n g FORTRAN-H. A l t h o u g h t h e ADD4 p r o c e s s m o d u l e u s e s a d i r e c t c a l c u l a t i o n p r o c e d u r e t h e C.P.U. s e c . / l o o p was i n c r e a s e d o n c e a g a i n t o 3.5 when t h e ADD4 m o d u l e was i m p l e m e n t e d . A c h e c k was t h e n made t o s e e i f i t w o u l d be n e c e s -s a r y t o d e v e l o p a p r o c e s s m o d u l e f o r t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r w h i c h was r e p r e s e n t e d by two t e m p e r a t u r e s e t t e r s , 2 and 38. Due t o l i m i t a t i o n s i n t h e a c t u a l p l a n t t h e p o l y f e e d c a n n o t be h e a t e d a b o v e 8 1 5 ° R i n t h e p o l y f e e d h e a t e r ( s e e F i g u r e 1 - 2 ) . Thus t h e t e m p e r a t u r e o f t h e p o l y f e e d l e a v i n g t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r i s o f l i t t l e i m p o r t a n c e t o t h e p r o c e s s as a w h o l e . H o w e v e r , t h e t e m p e r a t u r e o f t h e e f f l u e n t l e a v i n g t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r c a n i n f l u e n c e t h e o p e r a t i o n o f t h e d e p r o p a n i z e r . In t h e f i n a l p o l y p l a n t model t h e d e p r o p a n i z e r i s r e p r e s e n t e d by t h e ADD4 p r o c e s s m o d u l e . T h i s m o d u l e u t i l -i z e s F e n s k e ' s ( 8 1 ) method f o r t h e c a l c u l a t i o n o f minimum s t a g e s and minimum r e f l u x r a t i o , and G i l l i l a n d ' s ( 8 2 ) c o r -r e l a t i o n f o r t h e c a l c u l a t i o n o f t h e r e f l u x r a t i o . T h e r e f o r e t h e t e m p e r a t u r e o f t h e i n c o m i n g s t r e a m w o u l d i n f l u e n c e t h e 104 c a l c u l a t i o n o f t h e a v e r a g e r e l a t i v e v o l a t i l i t y f o r t h e c o l u m n . T e s t s were made t o c h e c k t h e e f f e c t o f u s i n g d i f f e r e n t t e m p e r -a t u r e s f o r s t r e a m 13 and t h e r e s u l t s a r e shown i n T a b l e 3-7. The e f f e c t on t h e s t r e a m s l e a v i n g t h e d e p r o p a n i z e r , s t r e a m s 15 and 2 5 , i s n e g l i g i b l e a nd i n e s s e n c e , t h e v a r i a t i o n i n i n c o m i n g s t r e a m t e m p e r a t u r e o r i n t h e s t r i c t e r s e n s e , e n t h a l p y , w o u l d be c o m p e n s a t e d by c h a n g e s i n t h e r e b o i l e r l o a d . C. P r o c e s s E q u i p m e n t M o d e l s 1 . P o l y m e r i z a t i o n R e a c t o r The p o l y m e r i z a t i o n r e a c t o r s a r e t h e most i m p o r t a n t p r o c e s s u n i t s i n t h e p o l y m e r i z a t i o n p l a n t and t h u s t h e p r o -c e s s m o d u l e d e v e l o p e d f o r t h e p o l y m e r i z a t i o n r e a c t o r s must be more c o m p r e h e n s i v e t h a n t h o s e u s e d f o r t h e o t h e r p r o c e s s u n i t s . The ADD3 p r o c e s s m o d u l e u t i l i z e s t h e g e n e r a l i z e d r a t e e x p r e s s i o n ( E q u a t i o n 2-2) d e v e l o p e d f r o m t h e e x p e r i -m e n t a l k i n e t i c s t u d y . I t a l s o t a k e s i n t o a c c o u n t t h e p o i -s o n i n g o f t h e c a t a l y s t and t h e i n c r e a s e i n p r e s s u r e d r o p t h r o u g h t h e b e d , b o t h as f u n c t i o n s o f t i m e . S i n c e t h e f i r s t c a t a l y s t bed i s d e a c t i v a t e d much more r a p i d l y t h a n t h e f o l l o w i n g beds a d i f f e r e n t r e l a t i o n s h i p b e t w e e n t h e c a t a l y s t e f f e c t i v e n e s s f a c t o r and t i m e must be u s e d f o r i t f r o m t h a t used f o r the other three beds. C a l c u l a t i o n of the c o n v e r s i o n i n e a c h c a t a l y s t bed was made by m u l t i p l y i n g t h e p l u g f l o w e q u a t i o n by a 105 TABLE 3-7  E f f e c t o f T e m p e r a t u r e o f S t r e a m 13 on P r o c e s s C a l c u l a t i o n s T e m p e r a t u r e - S t r e a m 1 3 , ° R 675 715 755 S t r e a m 13 V a p o r F r a c t i o n 0.0 0.43 0.87 P r e s s u r e , p . s . i . a . 504.9 504.9 504.9 E n t h a l p y ( B T U x l O 6 ) 1 .18 2.15 3.13 S t r e a m 14 V a p o r F r a c t i o n 0.29 0.64 0.92 T e m p e r a t u r e 642.6 673.2 718.7 P r e s s u r e 265.0 265.0 265.0 E n t h a l p y 1.18 2.15 3.13 M o l e s : p r o p a n e 141.64 141.64 141.65 I - b u t a n e 50.84 50.84 50.84 N - b u t a n e 45.05 45.05 45.06 p r o p e n e 6.21 6.21 6.22 S t r e a m 15 V a p o r F r a c t i o n 1.0 1 .0 1 .0 T e m p e r a t u r e 586.5 586.5 586.5 P r e s s u r e 265.0 265.0 265.0 E n t h a l p y 0.83 0.83 0.83 M o l e s : p r o p a n e 140.24 140.24 140.24 I - b u t a n e 1 .40 1 .40 1 .40 N - b u t a n e 0.085 0.088 0.094 p r o p e n e 6.19 6.19 6.19 S t r e a m 25 V a p o r f r a c t i o n 0.0 0.0 0.0 T e m p e r a t u r e 713.8 713.8 713.8 P r e s s u r e 265. 0 265.0 265.0 E n t h a l p y 0.99 0.99 0.99 M o l e s : p r o p a n e 1 .40 1 .40 1 .40 I - b u t a n e 49.43 49.44 49.44 p r o p e n e 0.02 0.02 0.02 D e p r o p a n i z e r R e f l u x R a t i o 1 .37 1 .27 1 .55 1 06 f a c t o r t o c o r r e c t f o r t h e n o n i d e a l i t y o f t h e f l o w i n t h e a c t u a l r e a c t o r s . A c t u a l p l a n t c o n v e r s i o n s and t e m p e r a t u r e s were u s e d t o f i t a f l o w e f f i c i e n c y f a c t o r t o t h e t h e o r e t i c a l model so t h a t t h e p r o c e s s m o d u l e w o u l d a d e q u a t e l y r e p r e s e n t t h e p l a n t u n i t . The b a s i c p l u g f l o w r e a c t o r e q u a t i o n i s d x _ r A c ( 3 _ 1 } dz F y o 1 The r e a c t i o n r a t e i s r e p r e s e n t e d by t h e e x p e r i -m e n t a l l y d e v e l o p e d r e a c t i o n r a t e e x p r e s s i o n ( e q u a t i o n 2-2) and m u l t i p l i e d by t h e e f f e c t i v e n e s s f a c t o r , E , and t h e f l o w e f f i c i e n c y f a c t o r F g t o o b t a i n , ^ = W k C o 0^x11 ( 3 _ 2 )  d z F y o 1 ( 1 + x ) 2 The e x p e r i m e n t a l r e l a t i o n s h i p f o r t h e v a r i a t i o n o f t h e r e a c t i o n r a t e c o n s t a n t w i t h t e m p e r a t u r e i s i n t r o d u c e d g i v i n g -68A0/T . F r dj<_ = 2.87 x 1 0 s e rtcre11 ( 1 - x ) 2 ^ 3 _ 3 j d Z " Q ( 1 + x ) 2 where T = °R 107 E q u a t i o n 3-3 r e p r e s e n t s t h e c h a n g e i n t o t a l o l e f i n c o n v e r s i o n w i t h bed d e p t h and i s t h e f i r s t b a s i c e q u a t i o n f o r t h e m o d e l . A s e c o n d r e l a t i o n s h i p c a n be d e r i v e d f r o m an e n t h a l p y b a l a n c e . a r = BT D E L H - hw V T - V ( 3 - 4 ) The p r e s s u r e d r o p t h r o u g h t h e bed i s m o d e l e d by an e q u a t i o n d e r i v e d by E r g u n ( 8 3 ) . dP dz 150 + 1 .75 (p V 0 2 ) ( l - e ) 144 D g„ e 3 p 3 c ( 3 - 5 ) where Re. _ p o  p ~ v ( 1 - e ) E q u a t i o n s 3-3, 3-4, 3-5 f o r m t h e b a s i s o f t h e ADD3 p r o c e s s m o d u l e . H o w e v e r , b e f o r e t h e s e e q u a t i o n s c o u l d be u t i l i z e d i n t h e model i t was n e c e s s a r y t o e s t i m a t e t h e h e a t o f r e a c t i o n as a f u n c t i o n o f o l e f i n c o m p o s i t i o n and t h e r e -l a t i o n s h i p s b e t w e e n t h e e f f e c t i v e n e s s f a c t o r and bed p o r o s i t y w i t h t i m e . J o n e s ( 7 3 ) and Thomas ( 7 8 ) have r e p o r t e d some e s t i m a t e d v a l u e s f o r t h e h e a t o f r e a c t i o n , b u t t h e s e v a l u e s were 108 c h e c k e d by s e v e r a l e s t i m a t i o n m e t h o d s and f o u n d t o be q u i t e low ( s e e A p p e n d i x F f o r t h e e s t i m a t i o n o f h e a t s o f r e a c t i o n ) . The method u s e d f o r e s t i m a t i n g t h e a v e r a g e h e a t o f r e a c t i o n i s shown b e l o w . DELH = AH 3 y 3 F + A H „ ( y „ + y 5 ) F ( 3 - 6 ) where AH 3 = H e a t o f p o l y m e r i z a t i o n f o r p r o p e n e = -38,000 BTU/mole AHi» = H e a t o f p o l y m e r i z a t i o n f o r b u t e n e s and p e n t e n e s = -28,000 BTU/mole V e r y l i t t l e i s known a b o u t t h e p o i s o n i n g o f t h e c a t a l y s t w i t h t i m e . M o s t o f t h e p l a n t d a t a a v a i l a b l e shows t h e d r o p i n c o n v e r s i o n t h r o u g h t h e w h o l e r e a c t o r (4 c a t a l y s t b e d s ) w i t h t i m e ( s e e F i g u r e 1 - 4 ) . I t i s known t h a t t h e f i r s t c a t a l y s t bed i s p o i s o n e d much more r a p i d l y t h a n t h e f o l l o w i n g b e ds s i n c e i t a c t s as a f i l t e r i n t h e s e n s e t h a t i t r e m o v e s most o f t h e m a t e r i a l ( o t h e r t h a n t h e h i g h p o l y m e r p r o d u c t f o r m e d i n t h e r e a c t i o n ) t h a t m i g h t a f f e c t t h e c a t -a l y s t a c t i v i t y . Some p l a n t d a t a has been p l o t t e d i n t h e f o r m shown i n F i g u r e 3-5. The t e m p e r a t u r e r i s e t h r o u g h t h e bed i s an i n d i c a t i o n o f t h e amount o f r e a c t i o n t a k i n g p l a c e , and i n the c a s e o f bed no. 1, i t i s an i n d i c a t i o n o f t h e c a t a l y s t a c t i v i t y . I t c a n be s e e n t h a t bed no. 1 becomes J I I I L 0 1 2 3 4 MONTHS OF OPERATION F I G U R E 3 - 5 P O L Y M E R I Z A T I O N REACTOR BED T E M P E R A T U R E V S , T I M E no i n a c t i v e a f t e r a b o u t 3 months o f o p e r a t i o n . I t i s more d i f -f i c u l t t o j u d g e t h e a c t i v i t y o f beds 2, 3, and 4 . b e c a u s e o f t h e e f f e c t o f t h e v a r i a b l e amount o f p r o p a n e q u e n c h u s e d . H o wever, p l a n t d a t a a l s o shows t h a t t h e o v e r a l l c o n v e r s i o n d r o p s t o a b o u t 7 0 - 7 5 % a f t e r a p p r o x i m a t e l y 6 months o f o p e r a -t i o n , and t h i s p i e c e o f i n f o r m a t i o n was u s e d t o e s t i m a t e t h e e f f e c t i v e n e s s f a c t o r - t i m e r e l a t i o n s h i p f o r beds 2, 3, and 4; F o r bed no. 1 E = 1.0 - 0.010 D s ( 3 - 7 ) beds no. 2, 3, 4 E = 1 .0 - 0.0015 D s ( 3 - 8 ) where D s = number o f d a y s on s t r e a m The i n c r e a s e i n p r e s s u r e d r o p t h r o u g h t h e c a t a l y s t beds i s due t o c o a t i n g o f t h e c a t a l y s t p a r t i c l e s w i t h t h e c o k e and t a r l i k e h i g h e r p o l y m e r s f o r m e d d u r i n g t h e p o l y -m e r i z a t i o n r e a c t i o n . T h i s f o r m a t i o n r e s u l t s i n a d e c r e a s e d p o r o s i t y f o r t h e b e ds and an i n c r e a s e d p r e s s u r e d r o p . How-e v e r , t h e r e i s a l s o a p r e s s u r e d r o p due t o t h e c a t a l y s t s u p p o r t s , e t c . w h i c h i s a l w a y s p r e s e n t and w h i c h i s a b o u t 10 p . s . i . A c t u a l p l a n t o p e r a t i n g d a t a was u s e d t o d e v e l o p t h e f o l l o w i n g r e l a t i o n s h i p o f bed p o r o s i t y w i t h t i m e , I l l e = 0.40 - 0.0015 D g ( 3 - 9 ) Thus a f t e r 200 d a y s o f o p e r a t i o n , t h e bed c a l c u -l a t e d p o r o s i t y i s 0.10. T h i s means t h a t t h e v o i d v o l u m e d i f f e r e n c e b e t w e e n s t a r t - u p and t h e 2 0 0 t h day i s t a k e n up by t h e a s p h a l t l i k e p r o d u c t . The c a l c u l a t i o n s b e l o w show an i n d e p e n d e n t c h e c k o f t h e a b o v e r e l a t i o n s h i p , Volume o c c u p i e d by heavy polymer = Volume o f c a t a l y s t x ( d i f f e r e n c e i n p o r o s i t i e s ) = 143(0.4 - 0.1) = 42 . 7 f t 3 Weight o f a s p h a l t formed z 75 l b / f t 3 (42 .7) = 3 ,200 l b s . Dimer produced = (50,000 l b / d a y - r e a c t o r ) ( 1 8 0 days) ( 0 . 8 ) = 7.2 x 10 6 l b s P e r c e n t a s p h a l t produced = 0.044$ The a b o v e v a l u e i s l o w e r t h a n e x p e c t e d and t h u s t h e p o r o s i t y - t i m e r e l a t i o n s h i p a p p e a r s t o be r e a s o n a b l e . A v a l u e was f o u n d f o r t h e f l o w e f f i c i e n c y f a c t o r by f i t t i n g t h e p l u g f l o w c a l c u l a t i o n t o t h e p l a n t d a t a . The o v e r a l l c o n v e r s i o n a t s t a r t - u p i s a b o u t 8 5 - 9 0 % . H o w e v e r , t h e t e m p e r a t u r e s l i s t e d i n T a b l e 3-8 a r e r e a l l y n o t t h o s e a t t h e b e g i n n i n g and end o f t h e c a t a l y s t beds s i n c e t h e t h e r -m o c o u p l e s a r e i n s e r t e d 4-6 i n c h e s i n t o t h e c a t a l y s t b e d s . Thus t h e t e m p e r a t u r e s a c t u a l l y e n t e r i n g t h e beds s h o u l d be l o w e r and t h o s e l e a v i n g t h e beds somewhat h i g h e r . I t was 112 TABLE 3-8 F i t o f R e a c t o r Model t o P l a n t D a t a Typical Plant Data at Start-up T e m p e r a t u r e , °R Bed Number In Out 1 350-360 390-400 2 360-370 400-410 3 400-420 430-450 4 430-440 440-450 Model Data at Start-up Flow E f f i c i e n c y Factor F =0.60 Bed Number T e m p e r a t u r e , ° R C u m u l a t i ve In Out C o n v e r s i o n 1 330 395 0.20 2 360 445 0.35 3 410 475 0.55 4 455 480 0.88 113 f o u n d t h a t w i t h a f l o w e f f i c i e n c y f a c t o r o f 0.6 t h e p r o c e s s m o d u l e a d e q u a t e l y r e p r e s e n t e d t h e p l a n t d a t a ( s e e T a b l e 3 - 8 ) . The c o m p u t e r f l o w d i a g r a m f o r t h e p o l y m e r i z a t i o n r e a c t o r p r o c e s s m o d u l e , ADD3, i s shown i n F i g u r e 3-6 and t h e c o m p u t e r l i s t i n g i s f o u n d i n A p p e n d i x D. A p a r a m e t r i c s t u d y was made o f t h e p o l y m e r i z a t i o n r e a c t o r by m a n u a l l y s e a r c h i n g f o r t h e "optimum" c o m b i n a t i o n o f r e c y c l e s p l i t s t o a c h i e v e a maximum c o n v e r s i o n a t d i f -f e r e n t t i m e s d u r i n g t h e c a t a l y s t ' s l i f e . The d a t a a r e p l o t t e d i n F i g u r e 3-7, 3-8, and 3-9. T h i s p r e l i m i n a r y p a r a -m e t r i c s t u d y a l s o s e r v e d t o g i v e some i d e a o f t h e r e s p o n s e s u r f a c e , a k n o w l e d g e o f w h i c h w o u l d be v e r y u s e f u l i n t h e o p t i m i z a t i o n s t u d i e s t o f o l l o w . The d i s c o n t i n u i t i e s a t 100 d a y s o f o p e r a t i o n a r e due t o t h e d i s c o n t i n u i t y o f t h e e f f e c -t i v e n e s s f a c t o r c o r r e l a t i o n f o r c a t a l y s t bed 1, w h i c h r e a c h e s a v a l u e o f 0.0 a t 100 d a y s o f o p e r a t i o n . F i g u r e s 3-7, 3-8, 3-9 show t h a t t o a c h i e v e a max-imum o l e f i n c o n v e r s i o n a t any one p o i n t d u r i n g t h e c a t a l y s t bed l i f e c y c l e a s p e c i f i c c o m b i n a t i o n o f r e c y c l e f l o w s p l i t s must be u s e d and t h a t t h i s c o m b i n a t i o n o f f l o w s c h a n g e s w i t h t i m e . 2. D e p r o p a n i z e r - D e b u t a n i z e r M o s t o f t h e m e t h o d s u s e d i n d i s t i l l a t i o n c o l u m n s i m u l a t i o n and d e s i g n a r e c o v e r e d i n t h e t e x t s by Hanson 114 CALCULATE E' E » 1 0 - 0 0 1 D, IF IF E < 0 0 , E = 0 0 Z > l - 6 8 , E = 1 0 - 0 0015 De CALCULATE v. VOLUMETRIC FLOW RATE T, TOTAL OLEFIN MOLAR FLOW RATE r v = V / 2 0 G = 6 / 2 0 h w " 2 h w <4-RUNGE-KUTTA INITIALIZATION SET H.N.M Yd) « 0 0 Y(2) = 0 0 Y(3) « SIENTH(I) SOTEMP(I) "SITEMP(I) 1 ERGUN'S PRESSURE  DROP EQUATION € = 0-40 - 00015 0 S DELP = Z*-(I50/Re p + 1-75) ' G*V0*( I - C) \ * 144 * Dp *. g c *e3 SOPRES(I) = SIPRES(I) - DELP S0VPFR(I) = SIVPFR(I) SOCOMP(PAR) = SICOMP(PAR) DELH = AH. 5 y 3 F A H 4 ( y 4 • y 5 ) F F I G U R E 3 - 6 COMPUTER FLOW D I A G R A M FOR P O L Y M E R I Z A T I O N R E A C T O R - A D D 3 115 INITIALIZATION CALL AUXRK CALCULATION F(2) = 2-87 x 10 * e SOTEMP(I) * ( | - Y ( 2 ) ) ^ * E * F„ * A . /V ( ltY(2))2 e c F(3) = F(2) * D E L H - h w * A R *(SOTEMP(l) - T w ) SOCOMP(OL) = (l-Y(2)) * SICOMP(OL) SOCOMP(POL) = SICOMP(POL) + T * Y ( 2 ) / 2 0 SOMOLE(I) = SIMOLE(I) - T * Y ( 2 ) / 2 0 SOENTH(I) = Y(3) CALL TSUBH F I G U R E 3 - 6 ( C O N T I N U E D ) 0 0 0-95 h or UJ > •z. o o 0 - 9 0 _l CO _i z: * U J 0 - 8 5 o ° u_ So 2 0 - 8 0 x < 0 - 7 5 50 REACTOR INLET TEMPERATURE = 815 °R 100 150 2 0 0 DAYS ON STREAM 2 5 0 F I G U R E 3 - 7 MAXIMUM O V E R A L L C O N V E R S I O N OF O L E F I N S V S . T I M E 950 DAYS ON STREAM F I G U R E 3 - 8 E X I T B E D T E M P E R A T U R E V S . T I M E FOR MAXIMUM O L E F I N C O N V E R S I O N 10 1 REACTOR ! INLET TEMPERATURE 1 = 815 °R I i I I I 1 I 0 50 100 150 200 250 3 0 0 DAYS ON STREAM F I G U R E 3 - 9 R E C Y C L E FLOW S P L I T S V S . T I M E FOR MAXIMUM O L E F I N C O N V E R S I O N 119 et al. ( 8 5 ) , H o l l a n d ( 8 6 ) , and S m i t h ( 8 7 ) . S e v e r a l r i g o r o u s c a l c u l a t i o n m e t h o d s a r e d e s c r i b e d s u c h as t h o s e o f L e w i s -M a t h e s o n , T h i e l e - G e d d e s , and A m u n d s e n - P o t i n e n . H o w e v e r , e v e n t h o u g h t h e s e m e t h o d s h a v e had w i d e u s a g e and a r e q u i t e a c c u r a t e , t h e y consume l a r g e a mounts o f c o m p u t e r t i m e and s t o r a g e and a r e t h u s n o t a c c e p t a b l e f o r t h i s work where a p r o c e s s m o d u l e c a l c u l a t i o n may be r e p e a t e d h u n d r e d s o f t i m e s . The most w i d e l y u s e d s h o r t c u t method i s t h e F e n s k e -G i 1 1 i 1 a n d - U n d e r w o o d method w h i c h u s e s F e n s k e ' s ( 8 1 ) t o t a l r e f l u x e q u a t i o n , U n d e r w o o d ' s (88) e q u a t i o n s f o r minimum r e f l u x and G i l l i l a n d ' s ( 8 2 ) p l a t e - r e f l u x c o r r e l a t i o n . The E r b a r - M a d d o x (8 9 ) method i s s o m e t i m e s u s e d i n s t e a d o f t h a t o f G i l l i l a n d . The m ethod c h o s e n i n t h i s s t u d y u t i l i z e s F e n s k e ' s e q u a t i o n s ( 8 1 ) f o r b o t h minimum s t a g e s and minimum r e f l u x . The r e f l u x r a t i o i s t h e n c a l c u l a t e d by u s i n g a f i t o f G i l l i l a n d ' s c o r r e l a t i o n made by a g r o u p a t M c M a s t e r s ( 3 3 ) who u t i l i z e d a s i m i l a r t y p e o f m o d e l . L i d d l e ' s ( 9 0 ) f i t o f G i l l i l a n d ' s d a t a was a l s o u t i l i z e d t o c o v e r a s e c t i o n o f t h e d a t a n o t r e p r e s e n t e d by t h e f i r s t f i t . F e n s k e ' s ( 8 1 ) e q u a t i o n f o r t h e minimum number o f t h e o r e t i c a l s t a g e s i s 11 SM a l k ^hk a h k y h k d . * ( 3 -120 The a b o v e e q u a t i o n c a n a l s o be e x p r e s s e d i n t e r m s o f m o l a r f1ows , a l k a hk SM l k l k . hk. hk In SM = l k d LX hk. hk ( 3 - 1 1 ) a In a l k hk I f t h e s p l i t o f t h e l i g h t key c o m p o n e n t i s s p e c i f i e d , t h e m o l e s o f t h e l i g h t key c o m p o n e n t i n t h e t o p p r o d u c t c a n be c a l c u l a t e d ; F l k f F i k = r ~ ( 3 " 1 2 ) d h l k v  1 + F d I f one a s s u m e s t h a t t h e t o p p r o d u c t i s c o m p o s e d o f o n l y t h e l i g h t key and h e a v y key c o m p o n e n t s and i f t h e m o l e f r a c t i o n o f t h e l i g h t key c o m p o n e n t i n t h e t o p p r o d u c t i s s p e c i f i e d , t h e n t h e s p l i t o f t h e h e a v y key c o m p o n e n t c a n be 121 c a l c u l a t e d , Fhk, F l k J d d i - y l k ( 3 - 1 3 ) F = F - F hk, hk,. hk-b i d ( 3 - 1 4 ) A l l t h a t i s n e e d e d now t o s o l v e E q u a t i o n 3-11 f o r t h e minimum number o f s t a g e s a r e t h e r e l a t i v e v o l a t i l i t i e s . A r e f e r e n c e compound i s c h o s e n and kv a l k kv l k kv 7 ' r e f a hk hk kv ( 3 - 1 4 ) r e f The r e l a t i v e v o l a t i l i t i e s a r e a v e r a g e d t h r o u g h -o u t t h e c o l u m n a s , a = (a,. * a, * a, ) r d b ( 3 - 1 5 ) The minimum number o f t h e o r e t i c a l s t a g e s c a n now be c a l c u l a t e d . E q u a t i o n 3-11 i s r e w r i t t e n as 122 hk, hk a ^ SM l k a hk fF ( 3 - 1 6 ) l k . l k + 1 .0 E q u a t i o n 3-16 i s now u s e d t o c a l c u l a t e t h e s p l i t s o f t h e r e m a i n i n g c o m p o n e n t s i n t h e m i x t u r e . The minimum r e f l u x i s c a l c u l a t e d by u t i l i z i n g a r e l a t i o n s h i p d e r i v e d by F e n s k e ( 8 1 ) , RM = a l k a hk l k l k , a l k a hk hk hk, ( 3 - 1 7 ) The r e f l u x r a t i o c a n t h e n be c a l c u l a t e d u t i l i z i n g t h e l i n e a r i z a t i o n o f G i l l i l a n d ' s d a t a by t h e g r o u p a t M c M a s t e r U n i v e r s i t y ( 3 3 ) E q u a t i o n ( 3 - 1 8 ) o r t h a t o f L i d d l e ( 9 0 ) --- E q u a t i o n ( 3 - 1 9 ) . I f SS S-SM S+l F o r SS < 0.60 123 1 + 1.16 RM - 1.7 SS 1.7 SS + 0.16 ( 3 - 1 8 ) F o r SS > 0.60 18.5715 RM - SS + 1.0 17.5715 + SS ( 3 - 1 9 ) The c o m p u t e r f l o w d i a g r a m f o r t h e ADD4 p r o c e s s m o d u l e i s shown i n F i g u r e 3-10 and t h e c o m p u t e r l i s t i n g i s g i v e n i n A p p e n d i x D. a r e t h e number o f r e a l s t a g e s , a v e r a g e c o l u m n e f f i c i e n c y , e s t i m a t e d t e m p e r a t u r e o f t o p and b o t t o m p r o d u c t s , f r a c t i o n o f l i g h t key c o m p o n e n t i n t h e t o p p r o d u c t , s p l i t o f l i g h t key c o m p o n e n t , and w h e t h e r t h e c o l u m n i s t h e d e p r o p a n i z e r o r d e b u t a n i z e r . The r e l a t i v e v o l a t i l i t i e s a r e c a l c u l a t e d f r o m t h e i r r e s p e c t i v e k - v a l u e s . Even t h o u g h CHESS c o n t a i n s a c o m p r e h e n s i v e t h e r m o d y n a m i c s u p p o r t p a c k a g e w h i c h i n c l u d e s a k - v a l u e c a l c u l a t i o n i t was n o t p o s s i b l e t o u s e t h i s t o t a l p a c k a g e b e c a u s e o f p r o g r a m d i f f i c u l t i e s . I t was p o s s i b l e , h o w e v e r , t o u t i l i z e t h e k - v a l u e s u b r o u t i n e t o c a l c u l a t e k - v a l u e s i n t h e t e m p e r a t u r e r a n g e o f i n t e r e s t . The e q u i p m e n t p a r a m e t e r s t h a t must be s p e c i f i e d V READ: NO. REAL STAGES AVG. STAGE EFFICIENCY EST. T(TOP) T(BOT) FRACTION LK(TOP) SPLIT LK DEPROP. OR DEBUT. CALCULATION OF RELATIVE VOLATILITIES' K (I) = W(I) + X(I) * Tp + Y ( I )*T2 KK(I)= W(I)+ X ( I ) * T T + Y( I )*T| K K K(I ) = W(I) + X ( I ) * T B Y ( I ) * t | A d ) = K(I) / K( N - BUTANE) AA(I)= KK( I )/KK(N - BUTANE) AAA(I) = KKK(I)/ KKK( N-BUTANE) A(I) = [A ( I )*AA( I ) f A A A ( l ) ] 0-333 YES IS UNIT NO 1 ^ D E P R O P A N I Z E R LK = PROPANE HK = I-BUTANE LK • C 4 TOTAL HK = I - PENTANE f 1 CALCULATION OF MINIMUM STAGES: LK(TOP) = LK(FEED) 1 0 + SPLIT LK HK (TOP) = LK(TOP)* (1-0 - FRAC LK(TOP)) LOG SM = r uK] I LK (TOP) „ HK(BOT)" (BOT) HK(TOP) LOG f A( LK) "1 L A (H K) J CALCULATION OF COMPONENT SPLITS: X ( I ) , LK (FEED) |"A(LK)]S L A ( I ) J -SM LK(BOT) , * L K ( T O P ) + l ° xd)B= xd) F X(I) , F I G U R E 3 - 1 0 COMPUTER FLOW D I A G R A M FOR D E P R O P A N I Z E R -D E B U T A N I Z E R - A D D 4 125 CALCULATION OF M I N I M U M REFLUX R A T I O -LK(TOP) A (LK) HK(TOP) LKiFEEb) AIHKJ * HK(FEED) D M „ A (LK) A (HK) - 10 L I N E A R I Z A T I O N O F  G I L L I L A N D C O R R E L A T I O N ' S = S ( R E A L ) * A V G . S T A G E E F F S S - S ~ S M S +- 1 0 F I G U R E 3 - 1 0 ( C O N T I N U E D ) 126 The a v e r a g e o p e r a t i n g p r e s s u r e s o f t h e c o l u m n s were u s e d i n t h e c a l c u l a t i o n s . Even i f s m a l l c h a n g e s i n p r e s s u r e t a k e p l a c e , t h e c h a n g e i n r e l a t i v e v o l a t i l i t i e s w o u l d be v e r y s m a l l . The r e s u l t s were t h e n f i t t e d t o a s e c o n d o r d e r p o l y n o m i a l and t h e c o e f f i c i e n t s were i n c l u d e d i n t h e ADD4 s u b r o u t i ne. N - b u t a n e i s u s e d as t h e r e f e r e n c e compound f o r t h e c a l c u l a t i o n o f t h e r e l a t i v e v o l a t i l i t i e s . The l i g h t and h e a v y key c o m p o n e n t s f o r t h e d e p r o p a n i z e r a r e p r o p a n e and i s o b u t a n e . T h i s i s r e a s o n a b l e s i n c e i n p r a c t i c e t h e t o p p r o d u c t i s more t h a n 9 8% p r o p a n e and p r o p e n e and o n l y c o n -t a i n s s m a l l amounts o f i s o b u t a n e . The c h o i c e o f l i g h t and h e a v y key c o m p o n e n t s f o r t h e d e b u t a n i z e r i s n o t as s t r a i g h t f o r w a r d . H e r e t h e t o t a l o f a l l c o m p o n e n t s i s c h o s e n as t h e l i g h t key c o m p o n e n t . H o w e v e r , s i n c e t h e o l e f i n s a r e p r e s e n t o n l y i n v e r y s m a l l q u a n t i t i e s t h e l i g h t k e y i s e s s e n t i a l l y N - b u t a n e and i s o b u t a n e . The h e a v y key i s i s o p e n t a n e . The r e l a t i v e v o l a t i l i t y o f t h e l i g h t key i s c a l c u l a t e d by a l k ct 2 * ct 2 * ct I - b u t a n e N-butane B u t e n e s ( 3 - 2 0 ) Some t y p i c a l p l a n t o p e r a t i n g d a t a and p r o d u c t s p e c i f -i c a t i o n s a r e shown i n T a b l e 3-9. 127 TABLE 3-9 P r o d u c t S p e c i f i c a t i o n s and T y p i c a l O p e r a t i n g  C o n d i t i o n s f o r D e p r o p a n i z e r and D e b u t a n i z e r D e p r o p a n i z e r D e b u t a n i z e r No. o f T r a y s 42 24 F e e d T r a y 23 12 Column D i a m e t e r 4' 0" 4' 0" Column H e i g h t 9 5 1 9" 4 6 ' - 4 " O p e r a t i n g P r e s s u r e 265 p . s . i . a . 90 p . s . i . a . S u g g e s t e d E f f i c i e n c y 7 5 % 5 5 - 6 5 % R e f l u x R a t i o 2.5 1.15 O p e r a t i n g T e m p e r a t u r e s F e e d 2 0 5 ° F T o p s 123 1 2 0 ° F B o t t o m s 244 270 P r o d u c t i on T o p s T o p s S p e c i f i c a t i ons P r o p a n e + P r o p e n e > 9 7 % B u t a n e + B u t e n e > 9 8 % P r o p e n e < 10% P e n t a n e s < 1% P r o p a n e + P r o p e n e < l % B o t t o m s B o t t o m s Must c o n t a i n p r o -pane + p r o p e n e i n s u f f i c i e n t l y low amounts so t h a t s p e c i f i c a t i o n s o f t o p p r o d u c t i n d e b u t a n i z e r w i l l n o t be v i o l a t e d . R e i d V . P<1 Op . s. i .g. 128 The two i n d e p e n d e n t v a r i a b l e s t h a t must be s p e c i -f i e d f o r t h e d i s t i l l a t i o n c o l u m n s i m u l a t i o n a r e t h e s p l i t o f t h e l i g h t key com p o n e n t and t h e mole f r a c t i o n o f t h e l i g h t key c o m p o n e n t i n t h e t o p p r o d u c t . B e c a u s e o f p r o d u c t s p e c -i f i c a t i o n s and e q u i p m e n t l i m i t a t i o n s t h e f o l l o w i n g bounds were f i x e d f o r t h e d e p r o p a n i z e r , 0.95 < y < 0.99 propane, F propane 0.001 < p - < 0.01 propane. and t h e d e b u t a n i z e r , 0.98 < y < 0.99 d 0.01 < C it. < 0.08 S e v e r a l r u n s were made v a r y i n g t h e p r o d u c t s p l i t s and p u r i t i e s and t h e r e s u l t s a r e shown i n T a b l e 3-10. F o r TABLE 3-10 R e s u l t s o f T e s t s on D e p r o p a n i z e r - D e b u t a n i z e r Model D e p r o p a n i z e r Mole P e r c e n t Propane i n Tops 0 .990 0 .990 0 .995 0 .995 0 .999 Propane S p l i t -Bottom/Top 0 .01 0 .005 0 .001 0 .005 0 .0001 R e f l u x R a t i o 1 .38 1 .53 2 .24 2 .51 4 .83 T e m p e r a t u r e : Top 586 .5 586 .5 585 .9 585 .9 585 .5 B o t t o m 713 .7 714 .7 715 .4 715 .5 715 .5 M o l e s : P r o p a n e - Top 140 .24 141 .75 143 .18 143 .48 143 .66 P r o p a n e - B o t t o m 1 .40 0 .71 0 .14 0 .07 0 .01 I - B u t a n e - Top 1 .40 1 .42 0 .72 0 .72 0 .14 I - B u t a n e - B o t t o m 49 .44 49 .43 49 .71 49 .71 49 .94 P e r c e n t : P r o p a n e + P r o p e n e - Top 98 .8 98 .9 99 .4 99 .4 99 .9 P r o p e n e - Top 4 .2 4 .2 4 .2 4 .2 4 .2 ro vo T a b l e 3-10 ( C o n t i n u e d ) D e b u t a n i z e r Mole P e r c e n t ( t o t a l ) i n Tops 0 .990 0 • 990 0 .990 0 • 990 0 • 990 Ci, ( t o t a l ) S p l i t - Bottom/Top 0 .070 0 .050 0 .020 0 .010 0 .030 R e f l u x R a t i o 0 .83 0 .90 1 .14 1 .42 1 .06 T e m p e r a t u r e : Top 589 .0 589 .5 589 .9 589 .9 589 .9 B o t t o m 723 .1 731 .4 746 .9 753 .1 741 .2 M o l e s : Cn ( t o t a l ) - Top 93 .10 95 .31 97 .96 99 .00 97 .24 Ci» ( t o t a l ) - B o t t o m 6 .59 4 .88 2 .14 1 .13 3 .00 I - P e n t a n e - Top 0 .12 0 .12 0 .10 0 .09 0 .11 I - P e n t a n e - B o t t o m 2 .38 2 .38 2 .40 2 .41 2 .39 P e r c e n t : C\ ( t o t a l ) - Top 97 .4 98 .1 98 .6 99 .0 98 .9 C 5 ( t o t a l ) - Top 1 .1 1 .1 1 .1 1 .0 1 .1 C 3 ( t o t a l ) - Top 1 .5 0 .7 0 .1 0 .0 0 .0 C- ( t o t a l ) - B o t t o m 11 .9 9 .1 4 .2 2 .2 5 .9 CO o 131 t h e d e p r o p a n i z e r , an i n c r e a s e i n r e f l u x r a t i o o c c u r s w i t h a d e c r e a s e o f t h e p r o p a n e s p l i t ( b o t t o m / t o p ) . O n l y s m a l l v a r i -a t i o n s were f o u n d f o r t h e c o m p o s i t i o n o f t h e t o p s and b o t t o m s and i n t h e r e s p e c t i v e dew p o i n t and b u b b l e p o i n t t e m p e r a -t u r e s . The r e f l u x r a t i o o f a b o u t 2.5 r e p o r t e d i n t h e p l a n t d a t a seems t o f i t i n t o t h e r a n g e o f v a l u e s f o u n d by t h e s i m u l a t i o n m o d e l . The d e b u t a n i z e r r u n s r e p o r t e d i n T a b l e 3-10 were made a t t h e same t i m e as t h e d e p r o p a n i z e r r u n s w h i c h a r e a b o v e . T h u s t h e Cg c o n t e n t i n t h e d e b u t a n i z e r t o p p r o d u c t d e p e n d s d i r e c t l y on t h e r e s u l t s f r o m t h e d e p r o p a n i z e r and i s n o t s i g n i f i c a n t . As i n t h e c a s e o f t h e d e p r o p a n i z e r , an i n c r e a s e i n r e f l u x r a t i o o c c u r s w i t h a d e c r e a s e i n t h e t o t a l s p l i t . The r e p o r t e d p l a n t r e f l u x r a t i o o f 1.15 f a l l s i n t o t h e r a n g e o f c a l c u l a t e d r e f l u x r a t i o s . The c o m p o s i t i o n and dew p o i n t o f t h e t o p p r o d u c t v a r y v e r y s l i g h t l y b u t t h e b u b b l e p o i n t t e m p e r a t u r e o f t h e b o t t o m p r o d u c t shows a g r e a t e r v a r i a t i o n b e c a u s e o f v a r i a t i o n s i n t h e t o t a l b o t t o m s c o m p o s i t i o n a t d i f f e r e n t s p l i t s . 3. S i m p l e H e a t E x c h a n g e r B e c a u s e o f p r o b l e m s w i t h t h e CHESS s u p p l i e d HXER h e a t e x c h a n g e r p r o c e s s m o d u l e a s i m p l e h e a t e x c h a n g e r ( o r t e m p e r a t u r e s e t t e r ) was p rogrammed. The c o m p u t e r f l o w d i a g r a m 132 f o r t h e ADD5 p r o c e s s m o d u l e i s shown i n F i g u r e 3-11 and t h e c o m p u t e r l i s t i n g c a n be f o u n d i n A p p e n d i x D. D. D i f f i c u l t i e s i n CHESS I m p l e m e n t a t i o n Some o f t h e d i f f i c u l t i e s e n c o u n t e r e d i n t h e i m p l e -m e n t a t i o n o f CHESS on t h e U.B.C. IBM 360/67 c o m p u t e r have a l r e a d y b e e n m e n t i o n e d . O t h e r a u t h o r s ( 3 5 ) ha v e a l s o r e p o r t e d i m p l e m e n t a t i o n p r o b l e m s . One o f t h e f i r s t p r o b l e m s e n c o u n t e r e d was w i t h t h e CHESS s u p p l i e d HXER p r o c e s s m o d u l e . A l t h o u g h many t r i a l s w ere made u n d e r d i f f e r i n g c o n d i t i o n s t h e r e were two main d i f -f i c u l t i e s . The f i r s t was t h a t when t h e HXER m o d u l e was u s e d f o r t h e p o l y f e e d / e f f l u e n t h e a t e x c h a n g e r , c o n v e r g e n c e was n o t a t t a i n e d . The s e c o n d p r o b l e m was t h a t i n some c a s e s t h e v a p o r f r a c t i o n o f t h e e x i t s t r e a m was n o t c a l c u l a t e d c o r -r e c t l y . As m e n t i o n e d p r e v i o u s l y , a s i m p l e h e a t e x c h a n g e r , o r t e m p e r a t u r e s e t t e r p r o c e s s m o d u l e , ADD5 was d e v e l o p e d and u t i l i z e d i n s t e a d o f t h e HXER m o d u l e . D i f f i c u l t i e s were a l s o f o u n d i n t h e u s e o f t h e t h e r m o d y n a m i c p r o p e r t i e s p a c k a g e . B e c a u s e o f t h i s p r o b l e m , k - v a l u e s had t o be c o r r e l a t e d o v e r a c e r t a i n t e m p e r a t u r e r a n g e and a t an a v e r a g e p r e s s u r e f o r t h e ADD4 m o d u l e w h i c h d e s c r i b e s t h e d e p r o p a n i z e r and d e b u t a n i z e r . P e r h a p s t h e most i m p o r t a n t d i f f i c u l t y a r o s e when CHESS was u s e d a l o n g w i t h an o p t i m i z a t i o n main p r o g r a m . 133 S U B R \ \ A D D 5 J T S A V E 3 = E Q P A R ( 2 , N E ) S I T E M P ( I ) = S A V E 3 S A V E 4 = S I E N T H ( 1 ) E 0 P A R ( 2 , N E ) = 1 0 ( C A V A D B F / E Q P A R ( 2 , N E S I E N T H ( 1 ) ) = S A V E 3 = S A V E 4 ^ E T U R N J F I G U R E 3 - 1 1 C O M P U T E R F L O W D I A G R A M F O R S I M P L E H E A T E X C H A N G E R - A D D 5 134 A l t h o u g h t h e t o t a l s o u r c e p r o g r a m was c o m p i l e d w i t h o u t e r r o r on t h e FORTRAN-G c o m p i l e r , e r r o r s i n e v i t a b l y o c c u r r e d d u r i n g e x e c u t i o n . I t was i m p o s s i b l e t o l o c a t e t h e s e e r r o r s b e c a u s e t h e y c h a n g e d when s i m p l e c h a n g e s were made i n t h e m a i n o p t i m i z a t i o n p r o g r a m . I t was t h e n d e c i d e d t o u t i l i z e t h e WATFIV c o m p i l e r s i n c e i t s d i a g n o s t i c c a p a b i l i t i e s a r e much g r e a t e r t h a n t h o s e o f t h e p r e v i o u s l y u s e d FORTRAN-G c o m p i l e r . A number o f p r o g r a m m i n g e r r o r s were f o u n d by t h e WATFIV c o m p i l e r , some o f w h i c h m i g h t have c a u s e d t h e d i f f i -c u l t i e s w h i c h were b e i n g e n c o u n t e r e d . The n e c e s s a r y c o r r e c t i o n were made ( s e e A p p e n d i x G f o r a c o m p l e t e l i s t ) and t h e WATFIV v e r s i o n o f CHESS was r u n . O t h e r c h a n g e s had t o be made i n t h e i n p u t d a t a b e c a u s e o f NAMELIST I/O d i f f i c u l t i e s b e f o r e a s u c c e s s f u l r u n was a c h i e v e d . The d e b u g g e d s o u r c e p r o g r a m was t h e n c o m p i l e d a g a i n w i t h t h e FORTRAN-G c o m p i l e r and CHESS was r u n s u c c e s s f u l l y t h e r e -a f t e r . I t s h o u l d be p o i n t e d o u t t h a t t h e WATFIV c o m p i l e r was o n l y u s e d as a d i a g n o s t i c a i d s i n c e t h e e x e c u t i o n t i m e o f t h e c o m p i l e d p r o g r a m i s much l o n g e r t h a n t h a t o f t h e FORTRAN-G v e r s i o n . P r e v i o u s a t t e m p t s had been made t o c o m p i l e t h e CHESS s o u r c e p r o g r a m u s i n g t h e FORTRAN-H c o m p i l e r b e c a u s e o f i t s more e f f i c i e n t o b j e c t c o d e . H o w e v e r , t h e s e e a r l i e r a t t e m p t s had n o t been p u r s u e d b e c a u s e o f t h e l a r g e number 135 o f e r r o r m e s s a g e s and p r o g r a m i n t e r r u p t s o b t a i n e d . H o wever, t h e WATFIV d e b u g g e d v e r s i o n o f CHESS c o u l d be c o m p i l e d w i t h t h e FORTRAN-H c o m p i l e r and t h e r e s u l t i n g o b j e c t c o d e b r o u g h t a 3 0% r e d u c t i o n i n t h e c a l c u l a t i o n t i m e as c o m p a r e d t o t h e FORTRAN-G o b j e c t p r o g r a m . 136 C h a p t e r 4 OPTIMIZATION OF THE POLYMERIZATION PLANT SIMULATION MODEL A. The O p t i m i z a t i o n P r o b l e m The o p t i m i z a t i o n p r o b l e m c o n s i s t s o f m a x i m i z i n g ( o r m i n i m i z i n g ) some c h o s e n o b j e c t i v e f u n c t i o n w h i l e s a t i s f y -i n g some s e t o f c o n s t r a i n t s . When t h e o b j e c t i v e f u n c t i o n a n d / o r t h e c o n s t r a i n t s a r e n o n l i n e a r f u n c t i o n s o f t h e i n d e -p e n d e n t ( o r d e c i s i o n ) v a r i a b l e s , t h e n t h e p r o b l e m i s one o f n o n l i n e a r o p t i m i z a t i o n . f u n c t i o n may be p r o f i t , c o n v e r s i o n , p r o d u c t i o n , e t c . and t h e c o n s t r a i n t s may be p r o d u c t p u r i t i e s , p r o c e s s v a r i a b l e l i m i t a t i o n s , and e q u i p m e n t d e s i g n l i m i t a t i o n s . F o r t h e o p t i m i z a t i o n o f an o p e r a t i n g p l a n t t h e o b j e c t i v e f u n c t i o n m i g h t be t h e p r o f i t w h e r e , In t h e c a s e o f an o p e r a t i n g p l a n t t h e o b j e c t i v e P r o f i t = S a l e s - C o s t s ( 4 - 1 ) and C o s t s = Raw m a t e r i a l s + U t i l i t i e s + Labor + Maintenance + Overhead ( 4 - 2 ) 137 When o n l y s m a l l c h a n g e s a r e made i n t h e p r o c e s s v a r i a b l e s t h e l a b o r , m a i n t e n a n c e , and o v e r h e a d c o s t s c a n be assumed c o n s t a n t . The u t i l i t i e s c o s t s f o r t h e s p e c i f i c c a s e o f t h e p o l y m e r i z a t i o n p l a n t do n o t show much v a r i a t i o n f o r t h e s m a l l c h a n g e s b e i n g made and so f o r any g i v e n raw m a t e r i a l ( p o l y f e e d ) , t h e o b j e c t i v e f u n c t i o n c a n be c o n -s i d e r e d as t h e t o t a l amount o f p r o d u c t s s o l d . T h e r e a r e f o u r p r o d u c t s f r o m t h e p o l y m e r i z a t i o n p l a n t ; p o l y m e r g a s o l i n e , b u t a n e , p r o p a n e , and f u e l g a s . The r a t i o o f t h e i r s a l e s v a l u e ( e s t i m a t e d ) i s shown b e l o w . polymer g a s o l i n e 2 butane 1 propane 1.1 f u e l gas 0.8 T h u s one m i g h t c o n s i d e r an o b j e c t i v e f u n c t i o n o f t h e f o r m , O.F. = 2 x Moles polymer g a s o l i n e + (1-a) Moles butane + 1.1 (1-b) Moles propane + 0.8 [ ( a ) Moles (4 butane + (b) Moles propane] where a =• f r a c t i o n o f propane p r o d u c t i o n used as f u e l gas b = f r a c t i o n o f butane p r o d u c t i o n used as f u e l gas On t h e o t h e r h a n d , t h e o p e r a t i n g p r i n c i p l e f o r t h e p o l y m e r i z a t i o n p l a n t i s t o m a x i m i z e t h e p o l y m e r g a s o l i n e 138 p r o d u c t i o n , so one c o u l d u s e t h e p r o d u c t i o n o f p o l y m e r g a s o -l i n e as t h e o b j e c t i v e f u n c t i o n . S i n c e t h e main c o m p o n e n t o f t h e p o l y m e r g a s o l i n e i s t h e d i m e r i z e d p r o d u c t o f t h e p o l y -m e r i z a t i o n r e a c t i o n , t h e s i m p l e s t o b j e c t i v e f u n c t i o n w h i c h r e p r e s e n t s t h e o p e r a t i n g p h i l o s o p h y o f t h e p o l y m e r i z a t i o n p l a n t i s t h e amount o f d i m e r i z e d p r o d u c t p r o d u c e d . T h i s was t h e o b j e c t i v e f u n c t i o n u s e d i n t h e m a j o r p a r t o f t h i s s t u d y ( s e e C h a p t e r 4-F f o r f u r t h e r d i s c u s s i o n ) . The i n t e r m e d i a t e p o l y p l a n t model was u s e d f o r t h e c o m p a r i s o n o f t h e o p t i m i z a t i o n m e t h o d s t e s t e d . The i n d e p e n -d e n t v a r i a b l e s c o n s i d e r e d i n t h i s model were t h e r e c y c l e f l o w s p l i t s . T hus t h e r e were 4 i n d e p e n d e n t v a r i a b l e s , one i n d e -p e n d e n t s p l i t i n DVDR24 and t h r e e i n d e p e n d e n t s p l i t s i n DVDR26. The e x p l i c i t c o n s t r a i n t s a r e t h a t t h e sum o f t h e ^ s p l i t s i n e a c h DVDR must e q u a l u n i t y . L i m i t s were a l s o s e t on t h e t e m p e r a t u r e s o f t h e s t r e a m s l e a v i n g t h e ADD3 u n i t s . T h i s a c c o u n t e d f o r 4 i m p l i c i t c o n s t r a i n t s . T hus t h e n o n l i n e a r c o n s t r a i n e d o p t i m i z a t i o n p r o b l e m c o n s i s t e d o f m a x i m i z i n g t h e p r o d u c t i o n o f t h e d i m e r i z e d p r o d u c t by v a r y i n g t h e r e c y c l e f l o w s , s a t i s f y i n g t h e e x p l i c i t c o n s t r a i n t s t h a t t h e sum o f t h e r e c y c l e s p l i t s f o r any DVDR be e q u a l t o u n i t y and t h e i m p l i c i t c o n s t r a i n t s t h a t t h e tem-p e r a t u r e s o f t h e s t r e a m s l e a v i n g t h e r e a c t o r b e ds must be l e s s t h a n some s p e c i f i e d l i m i t . 1 39 In t h e c a s e o f t h e f i n a l p o l y m e r i z a t i o n p l a n t model f o u r more i n d e p e n d e n t v a r i a b l e s were c o n s i d e r e d ; t h e s p l i t o f t h e l i g h t key c o m p o n e n t , and t h e mole f r a c t i o n o f t h e l i g h t k e y c o m p o n e n t i n t h e t o p p r o d u c t , f r o m t h e d e p r o p a n i z e r and t h e d e b u t a n i z e r . U p p e r and l o w e r bounds ( o r e x p l i c i t c o n -s t r a i n t s ) on t h e a b o v e i n d e p e n d e n t v a r i a b l e s were a l s o i n t r o -d u c e d . In o r d e r t o s o l v e t h e i n t e r m e d i a t e p o l y p l a n t o p t i m i z a t i o n p r o b l e m 3 m e t h o d s were u t i l i z e d . The f i r s t m e t h o d was t h e D e f l e c t e d G r a d i e n t M e t h o d o f F l e t c h e r and P o w e l l ( 9 1 ) b a s e d on t h e e a r l i e r work o f D a v i d o n ( 6 1 ) . T h i s m e t h o d f o r u n c o n s t r a i n e d n o n l i n e a r o p t i m i z a t i o n c a n be com-b i n e d w i t h t h e C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e o f C a r r o l l ( 9 2 ) f o r c o n s t r a i n e d n o n l i n e a r o p t i m i z a t i o n . S e v e r a l a u t h o r s ( 9 3 , 94) c o n s i d e r t h e a b o v e m e t h o d s t o be among t h e most p o w e r f u l t e c h n i q u e s a v a i l a b l e f o r t h e s o l u t i o n o f c o n s t r a i n e d n o n l i n e a r o p t i m i z a t i o n p r o b l e m s . The s e c o n d method u t i l i z e d i s a d i r e c t s e a r c h m ethod d e v e l o p e d by Hook and J e e v e s ( 5 5 ) c a l l e d t h e P a t t e r n S e a r c h . I t was u s e d b e c a u s e o f i t s s i m p l i c i t y o f p r o g r a m m i n g and u s e and t o c o m p a r e a d i r e c t s e a r c h method w i t h t h e p r e v i o u s l y m e n t i o n e d g r a d i e n t m e t h o d . The t h i r d t e c h n i q u e u s e d was t h e Complex M e t h o d d e v e l o p e d by Box ( 5 1 ) . The Complex M e t h o d i s a d i r e c t s e a r c h 140 method w h i c h a c c o r d i n g t o i t s a u t h o r c a n h a n d l e c o n s t r a i n t s v e r y w e l l . S e v e r a l a u t h o r s ( 9 3 . 94) c o n s i d e r t h e Complex M e t h o d t o be one o f t h e most u s e f u l i n t h e s o l u t i o n o f c o n -s t r a i n e d n o n l i n e a r o p t i m i z a t i o n p r o b l e m s . B. The D e f l e c t e d G r a d i e n t - C r e a t e d Response S u r f a c e M e t h o d 1 . The D e f l e c t e d G r a d i e n t M e t h o d f o r U n c o n s t r a i n e d  N o n l i n e a r O p t i m i z a t i o n In 1963 F l e t c h e r and P o w e l l ( 9 1 ) d e v e l o p e d a method o f n o n l i n e a r o p t i m i z a t i o n f o l l o w i n g t h e p r e v i o u s work o f D a v i d o n ( 6 1 ) . A b r i e f d e s c r i p t i o n o f t h e method w i l l now be g i v e n . D i r a c b r a - k e t n o t a t i o n f o r r e a l v e c t o r s w i l l be u s e d w h e r e t h e c o l u m n v e c t o r i s w r i t t e n as |x> and t h e row v e c t o r w i t h t h e same e l e m e n t s as <x|. The s c a l a r p r o d u c t o f <x| and |y> i s w r i t t e n <x|y>. <x i i X > However | X X X | = [ X i « " X n ] — — X i = • • • • n |— 2 X l X 2 X 1 X X 1 n X 1 X n X 1 X 2 141 A l so N N < x | [ H ] | x > = I I H.. x. x i = l j = l J J The s t a n d a r d q u a d r a t i c f o r m o f t h e o b j e c t i v e f u n c -t i o n i n N d i m e n s i o n s i s , N N N f = f 0 + I a. x. + \ I I G.. x. x. 0 i = l 1 1 i = l j = l J where ( 4 - 4 ) r - 9 2 f - n  b i j 8x. dx. ~ 9 i j w h i c h i n t h e D i r a c n o t a t i o n i s , f = f + <a|x> + 3^<x|[G]|x> ( 4 - 5 ) The g r a d i e n t i s , | v f > = |a> + [ G ] | x > ( 4 - 6 ) |x> = [ G " 1 ] | V f > - [ G - 1 ] 1 a > |x . > = - [ G " 1 ] | a > ( 4 - 7 ) 1 min 1 T h u s t h e s t e p t o w a r d s t h e minimum i s , l X m i n > " l X > = " L G ~ 1 ] | V f > ( 4 - 8 ) 142 The m a t r i x [ G _ 1 ] i s n o t e v a l u a t e d d i r e c t l y . A m a t r i x [H] i s u s e d w h i c h may i n i t i a l l y be any p o s i t i v e d e f i n i t e s y m m e t r i c m a t r i x . T h i s m a t r i x i s m o d i f i e d a f t e r t h e n t h i t e r -a t i o n u s i n g t h e i n f o r m a t i o n g a i n e d by m o v i n g down t h e d i r e c -t i o n , |a > = |x > - |x _> = - a [H _ ] | v f _> ( 4 - 9 ) 1 n 1 n 1 n - l m n - l 1 n - l The v a l u e o f a must be c h o s e n by some u n i v a r i a t e m J s e a r c h p r o c e d u r e so t h a t x n i s t h e minimum p o i n t a l o n g t h e d i r e c t i o n [H I ^ n - i * • T n e a p p r o x i m a t i o n t o t h e i n v e r s e m a t r i x o f t h e s e c o n d d e r i v a t i v e s i s u p d a t e d u s i n g t h e r e l a t i on [ H n ] = [ H n - l ] + [ A n ] + [ B n ] ( 4 " 1 0 ) S u c h t h a t [ H n ] = [ G " 1 ] N I n=l I [ A n ] = [ G " 1 ] ( 4 - 1 1 ) f [B 1 = - [ H I ( 4 - 1 2 ) i n o n = 1 143 [H ] = [H ,] + [A ] + [A .] + [B ] + [B J ( 4 - 1 3 ) n n - 1 n n - 1 n n - l J ' [H ] = [H ] + I [A ] + f [B ] n o L, n L ^ n n=l n=l [H ] = [ H I + [ G " 1 ] - [ H i ( 4 - 1 4 ) n o o [ H n ] = [ G " 1 ] ( 4 - 1 4 ) The m a t r i x [A ] i s f o r m e d as f o l l o w s : A g r a d i e n t d i f f e r e n c e v e c t o r i n t h e n t h i t e r a t i o n i s d e f i n e d a s , |y > = |Vf > - |Vf ..> (4-15) iJn 1 n 1 n - 1 | V f n > = |a> + [ G ] | x n > |y > = [ G ] ( | x n > - \ \ _ ± > ) ( 4 - 1 6 \o > = [ l ] \ o > 1 n L 1 n [ I ] = [ G - ^ C G ] 144 f[A.] = [ G ~ 1 ] i = l |a n> = [ G ~ 1 ] [ G ] | a n > ( 4 - 1 7 ) a i s r e l a t e d t o y o n l y i n t h e n t h s t e p n n [ A . ] | y n > = 0 , i = 1,2 .... n - l ( 4 - 1 8 ) a > = [A ] | y > ( 4 - 1 9 ) <a y > 1 n n <a |y > n 1 J n <a y > a > = 1 a > n <a y > 1 n n' 1 7 n l a ><a I [A ] = — 2 - , ( 4 - 2 0 ) L n J <a y > n 1 n The m a t r i x [B ] e n s u r e s t h a t e a c h e s t i m a t e o f [H ] L n J n J i s p o s i t i v e d e f i n i t e . I t c a n be shown t h a t , 145 [ H n ] [ G ] | 0 . > = \a±> i = l, •••• n ( 4 - 2 1 ) [ H n ] | y n > = [ H n ] [ G ] | a n > ( 4 - 2 2 ) ] [ G ] | a n > = C H n _ 1 ] | y n > + [ A n ] | y n > + [ B n ] | y n > ( 4 - 2 3 ) EBJ|yn> = - C H n _ 1 ] | y n > ( 4 - 2 4 ) R e j e c t t h e t r i v i a l s o l u t i o n , [B ] = - [H . ] L n J L n - l J I f |z> i s any r e a l v e c t o r , 1 " <y |z> * n 1 - [H , ] | y ><z|y > [B ]|y > = n ^ - ( 4 - 2 5 ) L n J | J n < y n | Z> 146 - [ H n _ 1 3 | y n > < y n | [ H n _ 1 ] ( 4 - 2 6 ) The c o m p u t e r f l o w d i a g r a m f o r t h e a b o v e d e s c r i b e d m e t h o d i s shown i n F i g u r e 4-1. The a l g o r i t h m was a m o d i f i e d f o r m o f t h a t w r i t t e n by L u c a s (95) w h i c h u s e s a g o l d e n s e c -t i o n o n e - d i m e n s i o n a l s e a r c h . 2. The C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e In 1 9 6 1 , C a r r o l l ( 9 2 ) p r o p o s e d t h a t a " c r e a t e d r e s p o n s e s u r f a c e " be d e f i n e d a s , M P ( x , K ) = f ( x ) + K I w i / c i ( x ) i = l ( 4 - 2 7 ) w h e r e f ( x ) i s t h e o b j e c t i v e f u n c t i o n K i s a p e n a l t y f a c t o r w. a r e w e i g h t i n g f a c t o r s c ± ( x ) a r e c o n s t r a i n t s o f t h e f o r m C. ( x ) > 0 i = l , 2 M i The minimum o f t h i s s u r f a c e i s t h e n f o u n d and u s e d as a s t a r t i n g p o i n t f o r t h e m i n i m i z a t i o n o f a n o t h e r r e s p o n s e s u r f a c e c o r r e s p o n d i n g t o a r e d u c e d v a l u e o f K. The method i s 147 G R A D I E N T C A L C U L A T I O N I v O = o > + [ G ] | X O > o — |s>= - [H]|Vf> O N E - D I M E N S I O N A L S E A R C H D E T E R M I N E a m ki>" - amCH]IVf> F I G U R E 4 - 1 C O M P U T E R F L O W D I A G R A M O F D E F L E C T E D G R A D I E N T M E T H O D 1 h> = |Vfi>-|Vfi_,> \ ° i > < o i \ <°. 1 y, > i - l ] 1 » l > < » i 1 KG N + [M F I G U R E 4 - 1 ( C O N T I N U E D ) 149 r e p e a t e d f o r a s e q u e n c e o f r e s p o n s e s u r f a c e s c o r r e s p o n d i n g t o s u c c e s s i v e l y s m a l l e r v a l u e s o f K and f i n a l l y f o r a t e r -m i n a l v a l u e o f K=0. Thu s a s i n g l e c o n s t r a i n e d m i n i m i z a -t i o n has been r e p l a c e d by a s e q u e n c e o f u n c o n s t r a i n e d m i n i -m i z a t i o n s . F i a c c o and M c C o r m i c k ( 4 5 , 46) have s u g g e s t e d t h a t t h e w e i g h t i n g f a c t o r s ( w i S i = l , 2 •••• M) be s e t t o u n i t y and have p r o v e d t h a t u n d e r c e r t a i n c o n d i t i o n s t h e m i n i m a o f t h i s s e q u e n c e o f r e s p o n s e s u r f a c e s c o n v e r g e t o t h e r e q u i r e d c o n s t r a i n e d minimum. The me t h o d h a s , h o w e v e r , b e e n s u c c e s s f u l l y a p p l i e d t o p r o b l e m s f o r w h i c h i t i s n o t p o s s i b l e t o p r o v e t h a t c o n v e r g e n c e i s a s s u r e d ( p r o b l e m s w i t h n o n - c o n v e x f e a s i b l e r e g i o n s ) . The c h o i c e o f t h e i n i t i a l v a l u e o f K i s v e r y im-p o r t a n t . I f t o o s m a l l a v a l u e i s u s e d , t h e c r e a t e d r e s p o n s e f u n c t i o n P ( x , K ) w i l l c l o s e l y a p p r o x i m a t e t h e u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n , f ( x ) , and t h e c o n v e r g e n c e t o e a c h r e -s p o n s e s u r f a c e minimum may be v e r y s l o w . The s e a r c h may a l s o s k i p o v e r t h e c o n s t r a i n t s and e n t e r t h e i n f e a s i b l e r e g i o n . On t h e o t h e r hand i f t h e i n i t i a l v a l u e o f K i s t o o h i g h , t h e m i n i m a o f t h e f i r s t few r e s p o n s e s u r f a c e s w i l l be f o r c e d w e l l away f r o m t h e c o n s t r a i n t s and w i l l o n l y d e p e n d v e r y s l i g h t l y on t h e o b j e c t i v e f u n c t i o n . F i g u r e 4-2 shows how a r e s p o n s e s u r f a c e i s c h a n g e d by t h e u s e o f C a r r o l l ' s t e c h n i q u e . F i a c c o and M c C o r m i c k ( 4 6 ) have s u g g e s t e d two m e t h o d s f o r c o m p u t i n g t h e i n i t i a l v a l u e o f K w h i c h make u s e I— o ZD UJ > t— O UJ -> CD o CONSTRAINT-F E A S I B L E REGION NON FEASIBLE REGION UNCONSTRAINED CONSTRAINED (K SMALL) CONSTRAINED (K LARGE) INDEPENDENT VARIABLE F I G U R E 4 - 2 C R E A T E D R E S P O N S E S U R F A C E <JI O 151 o f t h e d e r i v a t i v e s o f t h e o b j e c t i v e f u n c t i o n and t h e p e n a l t y t e r m . H o w e v e r , t h e c a l c u l a t i o n i s t i m e c o n s u m i n g and a c -c o r d i n g t o Box et al. ( 9 3 ) t h e method d o e s n o t seem t o be c o m p l e t e l y r e l i a b l e . Box et al. ( 9 3 ) s u g g e s t u s i n g an e m p i r i c a l a p p r o a c h and a d v i s e t h a t a l o w e r l i m i t o f K = 0.0001 be a p p l i e d f o r h a n d l i n g i n i t i a l p o i n t s c l o s e t o c o n s t r a i n t s . F i a c c o and M c C o r m i c k ( 4 6 ) h a v e a l s o f o u n d t h a t t h e r a t e o f r e d u c t i o n o f K has l i t t l e e f f e c t on t h e t o t a l e f f o r t i n v o l v e d i n f i n d i n g t h e c o n s t r a i n e d minimum s i n c e t h e l a r g e r t h e r e d u c t i o n f a c t o r , t h e s m a l l e r t h e number o f r e s p o n s e s u r f a c e s t h a t n e e d be m i n i m i z e d b u t t h e l o n g e r t h e t i m e r e q u i r e d f o r e a c h m i n i m i z a t i o n . R e d u c t i o n f a c t o r s o f f o u r and t e n have b e e n s u g g e s t e d ( 4 9 , 9 3 ) . Box et al. ( 9 3 ) h a v e f o u n d t h a t t h e c o n v e r g e n c e i s more r a p i d and c o n s i s -t e n t when a l l t h e v a r i a b l e s and c o n s t r a i n t s a r e s c a l e d so as t o be o f t h e same o r d e r o f m a g n i t u d e . T h e y a l s o f o u n d t h a t i f t h e c r e a t e d s u r f a c e f u n c t i o n was r e d e f i n e d as M P ( x , K ) = f ( x ) + K I w./c. 2 ( x ) ( 4 - 2 8 ) i = l 1 1 t h e method was more l i k e l y t o l e a d t o t h e g l o b a l r e s u l t f o r p r o b l e m s i n w h i c h two o r more l o c a l o p t i m a e x i s t . A n o t h e r s u g g e s t i o n o f t h e a b o v e a u t h o r s i s t h a t o n l y t h e c o n s t r a i n t s 152 w h i c h a r e v i o l a t e d n e e d be i n c l u d e d i n t h e c r e a t e d r e s p o n s e s u r f a c e f u n c t i o n , t h u s s a v i n g some c o m p u t i n g e f f o r t . The m ethod was e x t e n d e d t o c o v e r e q u a l i t y c o n -s t r a i n t s by F i a c c o and M c C o r m i c k ( 4 7 ) . In t h e c a s e where c . ( x ) > 0 i = l. 2 m' i c . ( x ) = 0 i=m'+l •••• m i P ( x , K ) = f ( x ) + K I w . / c . ( x ) + l I c 2 ( x ) ( 4 - 2 9 ) i - 1 1 1 K 2 i=m'+l 1 3. I m p l e m e n t a t i o n o f S e a r c h M e t h o d The D e f l e c t e d G r a d i e n t M e t h o d and R e s p o n s e S u r f a c e T e c h n i q u e were c o m b i n e d a l o n g w i t h CHESS by means o f a m a i n c a l l i n g p r o g r a m ( s e e F i g u r e 4-3 f o r p r o g r a m l i n k a g e and A p p e n d i x H f o r p r o g r a m l i s t i n g s ) . The m a i n c a l l i n g p r o g r a m c a l l s t h e r e s p e c t i v e CHESS s u b r o u t i n e s f o r r e a d i n g i n t h e CHESS i n p u t d a t a ( p o l y m e r i z a t i o n p l a n t m o d e l ) and i n i t i a l i z i n g t h e CHESS p r o g r a m . The i n p u t d a t a i s t h e n r e a d i n f o r t h e o p t i m i z a t i o n s u b r o u t i n e FP w h i c h i s t h e n c a l l e d . The e v a l u a t i o n o f t h e o b j e c t i v e f u n c t i o n i s h a n d l e d by s u b r o u t i n e FTN w h i c h c a l l s on t h e CHESS s u b r o u t i n e , C A L L S U B S E T S C A L L C O N M I N I M U M P O I N T A L O N G S E A R C H D I R E C T I O N I S D E T E R M I N E D . F T N C A L L E D F O R P O I N T C A L C U L A T I O N S M A K E G R A D I E N T S T E P , D E L G P(x,K) • f(x) + ECJ(X) R E T U R N 1 fG ( I ) = S N I - S N D E L G ^ E T U R N | an • K(I)/ ( 9 4 5 - T ) I F C ( I ) < O O , ' C ( I ) - - 1 0 0 C ( I ) 1 r i R E T U R N . M A T E R I A L A N D E N E R G Y B A L A N C E C A L C U L A T I O N S . U T I L I Z E S O T H E R C H E S S S U B R O U T I N E S F I G U R E 4-3 ( C O N T I N U E D ) 155 SUBSET t o c a r r y o u t t h e m a t e r i a l and e n e r g y b a l a n c e c a l c u -l a t i o n s u t i l i z i n g any o f t h e o t h e r CHESS and u s e r s u p p l i e d ADD s u b r o u t i n e s n e c e s s a r y t o c a r r y o u t t h e a b o v e m e n t i o n e d c a l c u l a t i o n . S i n c e t h e t e m p e r a t u r e c o n s t r a i n t s a r e im-p l i c i t t h e y c a n o n l y be c a l c u l a t e d a f t e r t h e m a t e r i a l and e n e r g y b a l a n c e has c o n v e r g e d . T h i s i s done by s u b r o u t i n e CON. The m e t h o d u s e d i s t h a t s u g g e s t e d by C a r r o l l ( 9 2 ) w i t h t h e r e c o m m e n d a t i o n o f Box et al. ( 9 3 ) t h a t d i f f e r e n t w e i g h t i n g f a c t o r s be u s e d . To p r e v e n t t h e s e a r c h f r o m s k i p p i n g o v e r t h e i m p l i c i t c o n s t r a i n t ( t e m p e r a t u r e < 9 4 5 ) , i f t h e c o n s t r a i n t f u n c t i o n , K ( I ) / 9 4 5 - T ( I ) , becomes n e g a t i v e t h e n t h i s c o n s t r a i n t f u n c t i o n i s m u l t i p l i e d by a l a r g e n e g a t i v e number b e f o r e b e i n g a d d e d o n t o t h e v a l u e o f t h e u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n . ( I n t h i s c a s e t h e s e a r c h i s a m i n i m i z a t i o n , and t h u s t h e optimum i s t h e l a r g e s t n e g a t i v e number.) T h i s s e r v e s t o d r i v e t h e s e a r c h i n t o t h e f e a s i b l e r e g i o n . O n e o f t h e p r o b l e m s i n v o l v e d i n t h e o p t i m i z a t i o n o f a s i m u l a t e d p l a n t model i s t h a t t h e c a l c u l a t i o n o f t h e o b j e c t i v e f u n c t i o n and t h e i m p l i c i t c o n s t r a i n t s r e q u i r e s a c o m p l e t e m a t e r i a l and e n e r g y b a l a n c e c a l c u l a t i o n w h i c h i s q u i t e t i m e c o n s u m i n g . T h u s i t i s n e c e s s a r y t o keep t h e o b j e c t i v e f u n c t i o n a n d / o r c o n s t r a i n t f u n c t i o n c a l c u l a t i o n s t o a minimum. H o w e v e r , b e c a u s e t h e g r a d i e n t s o f s u c h a s y s t e m c a n n o t be c a l c u l a t e d d i r e c t l y , a n u m e r i c a l c a l c u l a t i o n 156 m e t h o d must be u s e d f o r t h e g r a d i e n t c a l c u l a t i o n s . T h u s o i n t h e c a s e o f t h i s p r o b l e m w i t h 4 i n d e p e n d e n t v a r i a b l e s , s u b r o u t i n e 6RAD must c a l l f o r f i v e o b j e c t i v e f u n c t i o n c a l -c u l a t i o n s e a c h t i m e t h e g r a d i e n t i s e v a l u a t e d . I t s h o u l d a l s o be n o t e d t h a t s u b r o u t i n e FIBO c a l l s f o r a minimum o f 4 o b j e c t i v e f u n c t i o n c a l c u l a t i o n s a n d may u t i l i z e 10-20 f u n c t i o n c a l c u l a t i o n s u n d e r c e r t a i n c o n d i t i o n s . The s u b r o u t i n e FP c o n t i n u e s i t s c a l c u l a t i o n p r o c e d u r e s u n t i l t h e m a g n i t u d e o f t h e g r a d i e n t i s s m a l l e r t h a n a f i x e d t o l e r a n c e . When t h i s o c c u r s , t h e u n c o n -s t r a i n e d minimum has be e n r e a c h e d , and a new s e t o f p e n a l t y f a c t o r s a nd t o l e r a n c e s a r e u t i l i z e d . T h i s p r o c e s s c o n -t i n u e s u n t i l t h e m e t h o d f a i l s o r t h e p r o g r a m m e r f e e l s t h a t no f u r t h e r i m p r o v e m e n t c a n be o b t a i n e d . The s u c c e s s o f t h e a b o v e m e n t i o n e d m e t h o d i s a l m o s t c o m p l e t e l y d e p e n d e n t on t h e c h o i c e o f p a r a m e t e r s f o r a g i v e n r u n . I f t h e v a l u e c h o s e n f o r t h e minimum m a g n i t u d e o f t h e d i r e c t i o n v e c t o r TI were t o o s m a l l t h e n t h e s e a r c h w o u l d go on i n d e f i n i t e l y . On t h e o t h e r hand i f t h e v a l u e c h o s e n w ere t o o l a r g e t h e n t h e s e a r c h w o u l d h a v e t o be r e i n i t i a t e d f r e q u e n t l y w i t h t h e s u b s e q u e n t l o s s o f p a s t i n f o r m a t i o n . I t was f o u n d t h a t i n t h i s s t u d y v a l u e s o f TI = 0.01-0.10 g a v e t h e b e s t r e s u l t s . The i n i t i a l s t e p s i z e f o r t h e g o l d e n s e c t i o n s e a r c h , DEL, must a l s o be c h o s e n c a r e f u l l y , s i n c e i f i t were i i 1 57 t o o s m a l l , t h e n many f u n c t i o n e v a l u a t i o n s w o u l d be r e q u i r e d b e f o r e a minimum were r e a c h e d . On t h e c o n t r a r y , i f i t were t o o l a r g e t h e s e a r c h m i g h t jump o v e r a c o n s t r a i n t i n t o t h e n o n f e a s i b l e r e g i o n . In t h i s s t u d y , v a l u e s o f DEL=0.005-0.01 were u s e d d e p e n d i n g on t h e e s t i m a t e d d i s t a n c e o f t h e s t a r t -i n g p o i n t f r o m t h e o p t i m u m . The minimum s t e p s i z e f o r t h e g o l d e n s e c t i o n s e a r c h T2 c a n a l s o a f f e c t t h e c a l c u l a t i o n s . I f t h e v a l u e c h o s e n were t o o s m a l l , t h e n t h e number o f f u n c t i o n e v a l u a t i o n s n e c e s s a r y t o r e a c h t h e o n e - d i m e n s i o n a l minimum w o u l d be g r e a t e r t h a n r e q u i r e d w h i l e i f t h e v a l u e c h o s e n were t o o l a r g e t h e s e a r c h c o u l d end q u i t e f a r f r o m t h e minimum. I t was f o u n d t h a t v a l u e s o f T2 = 0.001-0.005 were a c c e p t a b l e . A g r a d i e n t s t e p s i z e , DELG = 0.01, was u s e d i n a l l t h e s e a r c h e s b e c a u s e i t was f o u n d t h a t s m a l l e r v a l u e s w o u l d g i v e r e s u l t s i n w h i c h t h e s y s t e m c o n v e r g e n c e e r r o r s ( s e e C h a p t e r 3-B) were o f t h e same o r d e r o f m a g n i t u d e as t h e d i f f e r e n c e s b e t w e e n t h e v a l u e s o f t h e o b j e c t i v e f u n c -t i o n s a t t h e b a s e and p e r t u r b e d p o i n t s . I t i s c l e a r t h e n t h a t i n o r d e r f o r t h e s e a r c h m e thod t o work s u c c e s s f u l l y i t i s o f u t m o s t i m p o r t a n c e t o u s e "good v a l u e s " f o r t h e s e a r c h p a r a m e t e r s as d e s c r i b e d a b o v e . I t i s , h o w e v e r , e v e n more i m p o r t a n t t o u s e "good v a l u e s " f o r t h e p e n a l t y f a c t o r s . I f t h e p e n a l t y f a c t o r s a r e t o o low t h e n t h e c o n s t r a i n t s a r e h a r d l y c o n s i d e r e d i n 158 t h e c a l c u l a t i o n o f t h e c o n s t r a i n e d o b j e c t i v e f u n c t i o n and t h i s m i g h t a l l o w t h e s e a r c h t o e n t e r i n t o t h e n o n f e a s i b l e r e g i o n . I f t h e p e n a l t y f a c t o r s a r e t o o h i g h t h e n t h e s e a r c h w i l l t a k e p l a c e a l o n g way f r o m t h e c o n s t r a i n t s ( a n d p o s -s i b l y t h e optimum as i n t h e c a s e b e i n g s t u d i e d ) t h u s u s i n g e x c e s s i v e c o m p u t e r c a l c u l a t i o n t i m e . S i n c e t h e r e s p o n s e s u r f a c e i n t h e p r e s e n t s t u d y was f a i r l y w e l l known f r o m p r e v i o u s t e s t s ( s e e C h a p t e r 3-C) i t was n o t t o o d i f f i c u l t t o s e l e c t r e a s o n a b l e v a l u e s f o r t h e p e n a l t y f a c t o r s . I t i s n o t known h o w e v e r , i f t h e v a l u e s s e l e c t e d were t h e b e s t t o a r r i v e a t an optimum s o l u t i o n w i t h t h e minimum c o m p u t e r c a l c u l a t i o n t i m e . 4. R e s u l t s w i t h I n t e r m e d i a t e P l a n t Model Some o f t h e r e s u l t s o b t a i n e d w i t h t h e D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e a r e shown i n T a b l e s 4-1, 4-2, 4-3, and 4-4. The X ( I ) a r e t h e i n d e p e n d e n t v a r i a b l e s ( s e e F i g u r e 3-1 f o r i n t e r m e d i a t e p o l y p l a n t m o d e l ) . X ( l ) = S t r e a m s p l i t - DVDR 24 X ( 2 ) = " - DVDR 26 X ( 3 ) = " - DVDR 26 X ( 4 ) = " " - DVDR 26 1 59 The C ( I ) a r e t h e c o n s t r a i n t f u n c t i o n s , T K ( I ) / ( 9 4 5 - T ( I ) ) where t h e T K ( I ) a r e t h e p e n a l t y f a c t o r s and t h e T ( I ) a r e t h e t e m p e r a t u r e s o f t h e s t r e a m s l e a v i n g t h e ADD3 r e a c t o r m o d u l e s ( s t r e a m s 6, 8, 10, 1 2 ) . P ( x , K ) i s t h e c o n s t r a i n e d o b j e c t i v e f u n c t i o n , f ( x ) t h e u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n , SS i s t h e mag-n i t u d e o f t h e d i r e c t i o n o f s e a r c h v e c t o r and S ( l ) a r e t h e g r a d i e n t d i r e c t i o n s . ( W h e n e v e r an a s t e r i s k a p p e a r s i t d e n o t e s a p o i n t i n t h e n o n f e a s i b l e r e g i o n . ) The s t a r t i n g p o i n t i n T a b l e 4-1 i s q u i t e f a r f r o m t h e c o n s t r a i n t s and t h e optimum. W i t h t h e p e n a l t y f a c t o r s c h o s e n i n t h i s t e s t , t h e d i f f e r e n c e b e t w e e n t h e c o n s t r a i n e d and u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n s i s o n l y 0.03. T h u s , t h e i n i t i a l g r a d i e n t i s n o t a f f e c t e d v e r y much by t h e c o n s t r a i n t s . The g o l d e n s e c t i o n s e a r c h p r o c e e d s a l o n g t h e c h o s e n d i r e c t i o n and a p p r o a c h e s one c o n s t r a i n t , T ( 3 ) . The minimum f o u n d i s a p p r e c i a b l y b e t t e r t h a n t h e i n i t i a l p o i n t . The c o n s t r a i n t s now p l a y a l a r g e r p a r t i n t h e c a l c u l a t i o n s as shown by a d i f f e r e n c e o f 0.36 b e t w e e n t h e two o b j e c t i v e f u n c t i o n v a l u e s . N o t e a l s o t h a t t h e g r a d i e n t i s much s m a l l e r t h a n t h e i n i t i a l g r a d i e n t s i n c e p o i n t no. 1 i s much c l o s e r t o t h e op t i m u m . P o i n t no. 2 shows an i m p r o v e m e n t o v e r p o i n t no. 1 b u t t h e r a t e o f im-p r o v e m e n t i s now much s l o w e r . In f a c t , p o i n t no. 3 i s n o t 160 TABLE 4-1 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t Model  U s i n g D e f l e c t e d G r a d i e n t - C r e a t e d  R e s p o n s e S u r f a c e M e t h o d - 1 S t a r t i n g P a r a m e t e r s G r a d i e n t S t e p S i z e DELG = 0.01 M i n . M a g n i t u d e o f D i r e c t i o n V e c t o r TI = 0.10 M i n . S t e p S i z e f o r G o l d e n S e c t i o n S e a r c h T2 = 0.005 I n i t . " " DEL = 0.01 P e n a l t y F a c t o r s T K ( I ) = 1,0, 1.0, 1.0, 1.0 P t . No. 0 Time = 0 C P . U . s e c . X ( I ) = .700 .100 .300 .400 C ( I ) = .009 .009 .009 .009 T ( I ) = 835 844 833 842 P ( x , K ) = -33.061 f ( x ) = -33.098 G r a d i e n t s : SS = 4.03 S ( I ) = -3.6 -1.4 -0.8 -0.4 G o l d e n S e c t i o n x(D X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .691 .096 .298 .399 -34.05 -34.10 .676 .091 .295 .397 -35.93 -35.98 .652 .081 .289 .395 -38.08 -38.16 .614 .066 .281 .391 -36.04 -40.35 .667 .087 .293 .396 -36.95 -37.01 .647 .079 .288 .394 -38.49 -38.58 .634 .074 .285 .393 -39.26 -39.40 .626 .071 .284 .392 -39.61 -39.82 .621 .068 .283 .391 -39.77 -40.09 .619 .067 .282 .391 -39.74 -40.22 .624 .069 .283 .392 -39.76 -40.04 161 T a b l e 4-1 ( C o n t i n u e d ) P t . No. 1 Time = 263 s e c . X ( I ) = .621 .068 .283 .391 C ( I ) = .014 .075 .191 .080 T ( I ) = 877 932 940 933 P ( x , K ) = -39.79 f ( x ) = -40.15 G r a d i e n t s : SS = 0.365 P t . No. 2 Time = 363 s e c . X ( I ) = .611 .078 .292 .399 C ( I ) = .014 .076 .266 .148 T ( I ) = 876 932 941 938 P ( x , K ) = -39.77 f ( x ) = -40.25 G r a d i e n t s : SS = 0.294 S ( I ) = -0.09 0.06 0.15 0.22 G o l d e n S e c t i o n S ( I ) = -0.19 0.19 0.18 0.15 G o l d e n S e c t i o n X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .615 .074 .287 .396 -39.81 -40.23 .607 .082 .296 .402 -39.79 -40.24 .615 .074 .288 .395 -39.67 -40.28 .612 .077 .291 .398 -39.79 -40.24 .610 .079 .293 .399 -39.79 -40.23 .609 .080 .294 .400 -39.78 -40.24 X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .608 .080 .297 .406 -39.79 -40.23 .602 .084 .306 .418 -39.75 -40.22 .607 .081 .297 .407 -39.73 -40.27 .506 .082 .301 .411 -39.76 -40.25 .604 .083 .303 .414 -39.76 -40.24 .606 .081 .299 .409 -39.75 -40.26 162 T a b l e 4-1 ( C o n t i n u e d ) P t . No. 3 Tim e = 461 s e c . X ( I ) = .605 .082 .301 .412 C ( D = .014 .069 . 181 .223 T ( I ) = 876 931 939 941 P ( x , K ) = -39.76 f ( x ) = -40.25 G r a d i e n t s : SS < 0.1 New P a r a m e t e r s DELG = 0.01 TI = 0.05 T2 = 0.001 DEL = 0.005 T K ( I ) = 0.5 0.5 0.5 0.5 P t . No. 3A C ( I ) = .007 .035 .093 .115 P ( x , K ) = -40.00 f ( x ) = -40.25 G r a d i e n t s : SS = 0.110 S ( I ) = -0.08 -0.07 -0.02 -0.03 G o l d e n S e c t i o n X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) .602 .079 .300 .414 -39.95 .604 .081 .301 .413 -39.96 .603 .080 .301 .413 -39.96 .603 .080 .301 .414 -39.97 .603 .081 .301 .413 -39.96 f ( x ) •40.31 •40.29 •40.30 •40.30 •40.30 163 T a b l e 4-1 P t . No. 4 Time = 540 s e c . X ( I ) = .603 .080 C ( I ) = .007 .039 T ( I ) = 877 932 P ( x , K ) = -39.96 G r a d i e n t s : SS = 0.051 S ( I ) = 0.003 G o l d e n S e c t i o n X ( l ) X ( 2 ) X ( 3 ) .603 .078 .301 .604 .075 .303 .604 .069 .305 .605 .606 .309 .607 .045 .315 .610 .022 .323 .605 .058 .309 .607 .044 .315 .608 .036 .318 .607 .049 .313 .608 .041 .316 .608 .039 .317 .608 .037 .317 .608 .039 .316 .608 .038 .317 .608 .039 .317 .608 .038 .317 P t . No. 5 Time = 690 s e c . X ( I ) = .608 .039 C ( I ) = .008 .081 T ( I ) = 886 939 P ( x , K ) = -40.08 ( C o n t i n u e d ) .301 .413 .131 .163 941 942 ) = -40 .30 0. 008 0.04 X ( 4 ) P ( x , K ) f ( x ) .417 -39.97 -40.30 .424 -39.99 -40.31 .436 -40.02 -40.31 .455 -40.05 , -40.30 .486 -40.07 -40.31 .535 -40.02 -40.32 .459 -40.06 -40.28 .489 -40.08 -40.31 .506 -40.07 -40.31 .477 -40.08 -40.29 .495 -40.08 -40.30 .499 -40.08 -40.31 .502 -40.07 -40.30 .498 -40.08 -40.30 .500 -40.08 -40.31 .498 -40.08 -40.30 .499 -40.08 -40.30 .317 .499 .040 .096 933 940 f ( x ) = -40.30 164 T a b l e 4-1 ( C o n t i n u e d ) G r a d i e n t s : SS = 0.103 S ( I ) = 0.005 -0.05 -0.02 0.08 G o l d e n S e c t i on X ( l ) X ( 2 ) X ( 3 ) X(4) P ( x , K ) f ( x ) .608 .036 .318 .503 -40.08 -40.31 .609 .032 .320 .510 -40.08 -40.32 .608 .036 .318 .503 -40.07 -40.30 .608 .035 .319 .506 -40.08 -40.31 .608 .034 .319 .507 -40.08 -40.31 .608 .035 .319 .505 -40.08 -40.30 .608 .035 .319 .506 -40.08 -40.31 .608 .035 .319 .505 -40.08 -40.30 P t . No. 6 T i m e = 887 s e c . X ( I ) = .608 .035 .319 .506 C ( I ) = .008 .090 .038 .093 T ( I ) = 877 939 932 940 P ( x , K ) = -40.08 f ( x ) = -40.31 New P a r a m e t e r s DELG = 0.01 TI = 0.10 T2 = 0.001 DEL = 0.005 T K ( I ) = 0.05 0.05 0.05 0.05 P t . No. 6A C ( I ) = .008 .009 .004 .011 P ( x , K ) = -40.30 f ( x ) = -40.32 G r a d i e n t s : SS = 0.44 S ( I ) = -0.33 -0.25 -0.13 -0.05 165 T a b l e 4-1 ( C o n t i n u e d ) G o l d e n S e c t i o n x ( i ) X ( 2 ) X ( 3 ) .604 .032 .317 .598 .028 .315 .604 .032 .317 .602 .303 .316 .600 .029 .316 .599 .029 .316 .600 .029 .316 .600 .029 .316 P t . No. 7 T i m e = 1001 s e c . X ( I ) = .601 .029 C ( I ) = .0009 .041 T ( I ) = 888 944 P ( x , K ) = -40.37 G r a d i e n t s : SS < 0.10 New P a r a m e t e r s DELG = 0.01 TI = 0.10 T2 = 0.001 DEL = 0.005 T K ( I ) = 0.05 0.01 P t . No. 7A C ( I ) = .0009 .008 P ( x , K ) = -40.44 G r a d i e n t s : SS = 0.34 S ( I ) = -0.26 X ( 4 ) P ( x , K ) f ( x ) .505 -40.35 -40.39 .504 -40.07 -40.51 .505 -40.36 -40.41 .505 -40.39 -40.44 .508 -40.39 -40.46 .505 -40.36 -40.48 .505 -40.37 -40.47 .504 -40.37 -40.48 .316 .505 .007 .048 938 944 f ( x ) = -40.07 0.05 0.01 .007 .009 f ( x ) = -40.47 -0.18 -0.08 -0.017 166 T a b l e 4-1 ( C o n t i n u e d ) G o l d e n S e c t i o n x(i) .597 .599 .598 .599 .600 .600 X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .027 .028 .208 .029 .029 .029 .315 .316 .315 .316 .316 .316 .505 .505 .505 .505 .505 .505 -36.01 -34.43 2000. -40.42 -40.45 -40.45 - 4 0 . 5 3 * - 4 0 . 5 1 * - 4 0 . 5 1 * -40.49 -40.48, - 4 0 . 4 8 1 B e s t P o i n t ( I n l a s t G o l d e n S e c t i o n ) X ( I ) = .599 .029 .316 .505 C ( I ) = .0009 .019 .008 .048 T ( I ) = 888 944 939 945 P ( x , K ) = -40.42 f ( x ) = -40.493 D i r e c t i o n s r e v e r s e d , s e a r c h s t o p p e d T i m e = 1090 C.P.U. s e c . 167 much b e t t e r t h a n p o i n t no. 2. However, l a r g e a mounts o f c o m p u t e r t i m e a r e b e i n g u s e d i n d e t e r m i n i n g t h e g r a d i e n t and i n t h e o n e - d i m e n s i o n a l s e a r c h c a l c u l a t i o n s . The s e a r c h s t o p s a t p o i n t no. 3 b e c a u s e t h e m a g n i t u d e o f t h e s e a r c h d i r e c t i o n v e c t o r i s l e s s t h a n t h e c h o s e n TI = 0.10. New p a r a m e t e r v a l u e s a r e now c h o s e n . S i n c e p o i n t 3 i s much c l o s e r t o t h e optimum t h a n t h e i n i t i a l p o i n t , s m a l l e r v a l u e s a r e c h o s e n f o r most o f t h e p a r a m e t e r s . N o t e t h a t t h e v a l u e s o f t h e two o b j e c t i v e f u n c t i o n s a r e now c l o s e r . The s e a r c h c o n t i n u e s w i t h p o i n t no. 4 s h o w i n g f u r t h e r i m p r o v e m e n t . P o i n t s 5 and 6 i m p r o v e o n l y s l i g h t l y and t h e s e a r c h i s i n t e r r u p t e d o n c e more so t h a t t h e p e n a l t y f a c t o r s c a n be r e d u c e d f u r t h e r . V a l u e s o f T K ( I ) = 0.05 a r e c h o s e n and t h e two o b j e c t i v e f u n c t i o n s a r e v e r y s i m i l a r . P o i n t no. 7 i s a c l e a r i m p r o v e m e n t o v e r t h e p r e v i o u s p o i n t and l i e s n e a r two c o n s t r a i n t s , T ( 2 ) and T ( 4 ) . When t h e g r a d i e n t f a l l s b e l o w T-j = 0.10 t h e p e n a l t y f a c t o r s a r e r e d u c e d e v e n f u r t h e r , b u t t h i s c a u s e s t h e s e a r c h t o e n t e r t h e n o n f e a s i b l e r e g i o n and no f u r t h e r p r o g r e s s c a n be made. The b e s t p o i n t f o u n d i n t h e s e a r c h i s q u i t e c l o s e t o t h e optimum and l i e s on two o f t h e c o n s t r a i n t s . ( T h e optimum i s on t h e i n t e r s e c t i o n o f t h r e e c o n s t r a i n t s . ) In s p i t e o f t h e a b i l i t y o f t h i s t e c h n i q u e t o move c l o s e t o t h e c o n s t r a i n e d optimum s e v e r a l d i f f i c u l t i e s h ave become e v i d e n t . The f i r s t i s t h a t t h e c o m p u t e r c a l c u l a t i o n 168 t i m e o f 1090 C P . U . s e c . i s f a i r l y h i g h . The s e c o n d and m o s t i m p o r t a n t p o i n t i s t h a t a g r e a t d e a l o f t i m e was s p e n t on c h o o s i n g p r o p e r v a l u e f o r t h e s e a r c h p a r a m e t e r s and p e n a l t y f a c t o r s . T h i s was done w i t h some p r i o r k n o w l e d g e o f t h e r e s p o n s e s u r f a c e . I f t h i s p r i o r k n o w l e d g e had n o t e x i s t e d , p e r h a p s t h e r e s u l t s shown i n T a b l e 4-1 c o u l d n o t h a v e been o b t a i n e d . T a b l e 4-2 shows a c a s e b e f o r e t h e a d d i t i o n a l c o n s t r a i n i n g t e c h n i q u e o f m u l t i p l y i n g a v i o l a t e d c o n s t r a i n t f u n c t i o n by a l a r g e n e g a t i v e number was i m p l e m e n t e d . I f C ( I ) = T K ( I ) / ( 9 4 5 - T ( I ) ) < 0.0 C ( I ) = - 100 C ( I ) T h i s r e s u l t s i n a l a r g e p o s i t i v e number w h i c h when a d d e d on t o t h e u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n ( t h e method m i n i m i z e s ) c a u s e s t h e method t o move away f r o m t h e c o n s t r a i n t . T a b l e 4-3 shows how t h e s e a r c h p r o g r e s s e d a b i t f u r t h e r when t h e a b o v e - d e s c r i b e d c o n s t r a i n i n g t e c h n i q u e was i m p l e m e n t e d . The e x a m p l e shown i n T a b l e 4-3 i l l u s t r a t e s some o f t h e d i f f i c u l t i e s w i t h t h e D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e . One o f t h e m a j o r d i f f i c u l t i e s i s t h a t t h e s e a r c h c a n e n t e r t h e n o n f e a s i b l e r e g i o n and c o n t i n u e t o o p e r a t e i n t h a t r e g i o n i f t h e s e a r c h p a r a m e t e r s and p e n a l t y f a c t o r s a r e n o t c h o s e n " c o r r e c t l y . " The c h o i c e i s v e r y d i f f i c u l t . A n o t h e r d i f f i c u l t y , p e r h a p s e v e n more 169 TABLE 4-2 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t Model  U s i n g D e f l e c t e d G r a d i e n t - C r e a t e d  R e s p o n s e S u r f a c e M e t h o d - 2 S t a r t i n g P a r a m e t e r s DELG = 0.01 TI = 0.10 T2 = 0.005 DEL = 0.01 T K ( I ) = 1 . 0 1.0 1.0 1.0 P t . No. 0 Tim e = 0 C.P.U . s e c . X ( I ) = 0.675 0.175 0.225 0.187 C ( I ) = 0.008 0.012 0.024 0.015 T ( I ) = 827 863 904 879 P ( x , K ) = -36.41 f ( x ) = -36.47 G r a d i e n t s : SS = 2.58 S ( I ) = -2.4 -0.79 -0.46 -0.25 G o l d e n S e c t i o n XH!X(1)XJL21 M i l M i l P(x,K) f (x) .666 .1 72 .223 1.-86 -37.1 6 -37.23 .651 .167 .220 .184 -38.33 -38.46 .626 .158 .215 .181 -39.91 - 3 9 . 9 4 * 1 ^ S e a r c h c o n t i n u e s f o r 380 s e c . w i t h o u t r e s u l t . 170 TABLE 4-3 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t Model  U s i n g D e f l e c t e d G r a d i e n t - C r e a t e d  R e s p o n s e S u r f a c e M e t h o d - 3 S t a r t i n g P a r a m e t e r s DELG = 0.01 TI = 0 . 1 0 T2 = 0.005 DEL = 0.01 T K ( I ) = 1 . 0 1.0 1.0 1.0 P t . No. 0 Tim e = 0 C.P.U. s e c . X ( I ) = 0.675 0.175 0.225 0.187 C ( I ) = 0.008 0.012 0.024 0.015 T ( I ) = 827 863 904 879 P ( x , K ) = -36.41 f ( x ) = -36.47 G r a d i e n t s : SS = 2.58 S ( I ) = -2.4 -0.79 -0.46 -0.25 G o l d e n S e c t i o n X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .666 .172 .223 .186 -37.16 -37.23 .651 .167 .220 .184 -38.33 -38.46 .626 .159 .215 .182 -33.07 - 3 9 . 9 4 * .656 .168 .221 .185 -38.00 -38.10 .645 .165 .219 .184 -38.61 -38.86 .638 .163 .218 .183 3.29 - 3 9 . 2 9 * .649 .166 .219 .184 -38.47 -38.65 .642 .164 .219 .184 -38.53 -39.01 P t . No. 1 T i m e = 224 s e c . X ( I ) = 0.646 0.164 0.219 0.184 C ( I ) = 0.009 0.019 0.172 0.029 171 T a b l e 4-3 ( C o n t i n u e d ) T ( I ) 843 894 939 910 P ( x , K ) = -38.58 f ( x ) = -38.81 G r a d i e n t s : SS = 0.473 S ( I ) = -0.277 0.241 0.218 0.204 G o l d e n S e c t i o n T i m e = 4 2 7 s e c . X ( I ) = 0.631 0.179 0.230 0.193* C ( I ) = 0.010 0.023 700.1 0.044 T ( I ) = 847 901 945.1 922 P ( x , K ) = 660.8 f ( x ) = -39.39 x(D X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .641 .170 .224 .188 -38.72 -38.98 .631 .179 .231 .194 -37.62 -39.38 .641 .170 .224 .188 -38.68 -39.06 .637 .174 .227 .191 -38.76 -39.06 .634 .176 .229 .193 -38.73 -39.22 .639 .173 .226 .190 -38.34 -39.22 P t . No. 2 Time = 346 s e c . X ( I ) = 0.636 0.174 0.228 0.192 C ( I ) = 0.010 0.021 0.393 0.037 T ( I ) = 845 898 942 918 P ( x , K ) = -38.72 f ( x ) = -39.18 G r a d i e n t s : SS = 0.242 S ( I ) = -0.17 0.15 0.08 0.03 G o l d e n S e c t i o n X(D X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .629 .181 .231 .193 34.67 - 3 9 . 4 7 * .633 .176 .228 .192 248.35 - 3 9 . 3 6 * .632 .178 .229 .194 -35.05 -39.35 P t . No 172 T a b l e 4-3 ( C o n t i n u e d ) G r a d i e n t s : SS = 0.144 S ( I ) = -0.006 G o l d e n S e c t i o n x(D X ( 2 ) X ( 3 ) .630 .187 .229 .631 .182 .229 .631 .184 .229 .631 .181 .229 108 -0.009 -0.095 X ( 4 ) P ( x , K ) f ( x ) .186 177.53 - 3 9 . 3 8 * .190 68.94 - 3 9 . 4 3 * .189 69.82 - 3 9 . 4 3 * .191 61 .38 -39.431 B e s t P o i n t ( I n P r e v i o u s G o l d e n S e c t i o n ) X ( I ) = 0.632 0.178 0.229 0.193 C ( I ) = 0.010 0.022 4.22 0.043 T ( I ) = 847 901 944.7 922 P ( x , K ) = -35.05 f ( x ) = -39.35 D i r e c t i o n s r e v e r s e d , s e a r c h s t o p p e d T i m e = 494 C P . U . s e c . 173 TABLE 4-4 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t Model  U s i n g D e f l e c t e d G r a d i e n t - C r e a t e d  R e s p o n s e S u r f a c e M e t h o d - 4 S t a r t i n g P a r a m e t e r s DELG = 0.01 TI = 0.01 T2 = 0.001 DEL = 0.005 T K ( I ) = 0.10 0.10 0.10 0.10 P t . No. 0 T i m e X ( I ) = C ( D = T ( I ) = P ( x , K ) = 0 s e c . 0.618 0.002 884 -40.25 0.046 0.300 0.400 0.013 0.091 0.011 938 944 936 f ( x ) = -40.37 G r a d i e n t s SS 0.674 S ( I ) = -0.664 -0.065 -0.021 -0.091 G o l d e n S e c t i o n X ( D X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .613 .045 .300 .400 -39.91 -40.41 .616 .046 .300 .400 -39.98 -40.40 .615 .045 .300 .400 -40.12 -40.40 .614 .046 .300 .400 -40.12 -40.40 .615 .046 .300 .400 -39.88 -40.40 P t . No. 1 T i m e X ( D C ( I ) T O ) P ( x , K ) = 96 s e c = 0.614 = 0.002 884 = -39.96 0.046 0.300 0.400 0.015 0.414 0.013 938 944.7 937 f ( x ) = -40.40 174 T a b l e 4-4 ( C o n t i n u e d G r a d i e n t s : SS = 0.567 S ( I ) = -0.454 0.111 0.192 0.256 G o l d e n S e c t i o n X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .611 .047 .301 .403 -39.83 -40.41 .604 .048 .305 .406 -33.59 - 4 0 . 4 7 * .611 .047 .302 .403 -36.26 - 4 0 . 4 8 * .608 .047 .303 .404 -33.59 - 4 0 . 4 6 * .612 .046 .301 .402 -34.39 - 4 0 . 4 6 * .609 .047 .302 .403 -29.64 -40.44* .611 .047 .302 .402 -31.82 - 4 0 . 4 5 * .610 .047 .302 .403 -30.70 - 4 0 . 4 4 * P t . No. 2 Tim e = 191 s e c . X ( I ) = 0.611 0.047 0.302 0.403* C ( I ) = 0.002 0.016 9.85 0.017 T ( I ) = 884 939 946.0 939 P ( x , K ) = -30.54 f ( x ) = -40.44 G r a d i e n t s : SS = 0.051 S ( I ) = -0.043 0.016 0.018 0.009 G o l d e n S e c t i o n x ( i ) X ( 2 ) X ( 3 ) X ( 4 ) P ( x , K ) f ( x ) .607 .048 .303 .403 -30.42 - 4 0 . 4 4 * .599 .050 . 306 .405 -37.59 - 4 0 . 5 2 * .588 .055 .311 .407 -36.54 - 4 0 . 6 1 * .602 .049 .305 .404 -38.22 - 4 0 . 5 4 * .597 .052 .308 .405 -37.13 - 4 0 . 5 3 * .605 .049 .304 .404 -37.79 - 4 0 . 5 2 * .600 .050 .306 .405 -37.19 - 4 0 . 5 1 * .604 .049 .305 .404 -37.77 - 4 0 . 5 2 * .602 .050 .306 .405 -37.51 - 4 0 . 5 1 * .603 .049 .305 .404 -37.64 -4 0 . 5 2 * 175 T a b l e 4-4 ( C o n t i n u e d ) P t . No. 3 1 T i m e = 307 s e c . X ( I ) = 0.602 0.050 0.305 0.404* C ( I ) = 0.002 0.019 2.74 0.065 T ( I ) 883 940 948.6 943 P ( x , K ) = -37.69 f ( x ) = -40.52 B e s t P o i n t ( P r e v i o u s G o l d e n S e c t i o n ) X ( I ) = 0.611 0.047 0.302 0.403 C ( I ) = 0.002 0.015 0.542 0.014 T ( I ) = 884 938 944.8 938 P ( x , K ) = -39.83 f ( x ) = -40.41 S e a r c h s t o p p e d b e c a u s e p o i n t s i n n o n f e a s i b l e r e g i o n . 176 i m p o r t a n t , i s t h a t t h e c a l c u l a t e d g r a d i e n t s may d i r e c t t h e s e a r c h away f r o m t h e o ptimum. T h i s i s due t o two r e a s o n s : ( 1 ) The g r a d i e n t c a l c u l a t i o n s a r e n o t e x a c t and i f t h e r e s p o n s e s u r f a c e i s f a i r l y f l a t ( t r u e n e a r t h e o ptimum) t h e s y s t e m c o n v e r g e n c e e r r o r may be o f t h e same o r d e r o f m a g n i t u d e as t h e c a l c u l a t e d g r a d i e n t s ; ( 2 ) The p e n a l t y f a c t o r s g r e a t l y i n f l u e n c e t h e v a l u e o f t h e c o n -s t r a i n e d o b j e c t i v e f u n c t i o n ( e s p e c i a l l y n e a r t h e c o n s t r a i n t s ) and t h e r e b y d i r e c t l y i n f l u e n c e t h e v a l u e o f t h e c a l c u l a t e d g r a d i e n t s . I t i s n o t e a s y t o p r e d i c t how t h e p e n a l t y f a c t o r s a f f e c t t h e c a l c u l a t e d g r a d i e n t s and t h u s t h e s e a r c h may be d i r e c t e d away f r o m t h e c o n s t r a i n e d o p t i m u m . T h i s o b s e r v a t i o n i s i l l u s t r a t e d o n c e more i n T a b l e 4-4 where t h e i n i t i a l s t a r t i n g p o i n t l i e s on one c o n s t r a i n t , T ( 3 ) . O n l y a v e r y s m a l l i m p r o v e m e n t c a n be o b t a i n e d b e f o r e t h e s e a r c h e n t e r s t h e n o n f e a s i b l e r e g i o n . In c o n c l u s i o n , i t was f o u n d t h a t t h e d e f l e c t e d g r a d i e n t - r e s p o n s e s u r f a c e t e c h n i q u e i s v e r y s e n s i t i v e t o t h e v a l u e s c h o s e n f o r t h e s e a r c h p a r a m e t e r s and t h e p e n a l t y f a c t o r s . I f t h e s e v a l u e s a r e c h o s e n " c o r r e c t l y " t h e n good r e s u l t s c a n be o b t a i n e d . H o wever, no m e t h o d , o t h e r t h a n p r i o r e x p e r i e n c e i s a v a i l a b l e f o r t h e c h o i c e o f t h e s e param-e t e r s . In a d d i t i o n , t h e m ethod e n c o u n t e r s p r o b l e m s n e a r t h e c o n s t r a i n t s b e c a u s e o f t h e i n a c c u r a c i e s i n t h e g r a d i e n t 177 c a l c u l a t i o n s and b e c a u s e o f t h e unknown e f f e c t o f t h e p e n a l t y f a c t o r s i n t h e c a l c u l a t i o n o f t h e s e a r c h d i r e c t i o n . C. C o n s t r a i n e d P a t t e r n S e a r c h 1. The P a t t e r n S e a r c h M e t h o d f o r U n c o n s t r a i n e d  N o n l i n e a r O p t i m i z a t i o n Hooke and J e e v e s ( 5 5 ) d e v e l o p e d an a c c e l e r a t e d c l i m b -i n g t e c h n i q u e w i t h r i d g e f o l l o w i n g p r o p e r t i e s , s o m e t i m e s r e -f e r r e d t o as t h e P a t t e r n S e a r c h M e t h o d . The me t h o d i s b a s e d on a l o c a l e x p l o r a t i o n i n t h e d i r e c t i o n s p a r a l l e l t o t h e c o -o r d i n a t e a x e s a t p o i n t s a g i v e n d i s t a n c e f r o m t h e b a s e p o i n t . A l a r g e r move i n a d i r e c t i o n d e t e r m i n e d by t h e s e t e s t s i s t h e n a t t e m p t e d . I f t h i s move i s s u c c e s s f u l , and i f a new l o c a l e x p l o r a t i o n i n d i c a t e s t h a t t h e d i r e c t i o n i s a good o n e , a f u r t h e r move i s made w i t h a much l a r g e r s t e p s i z e . So e v e n t h o u g h t h e t e c h n i q u e s t a r t s c a u t i o u s l y w i t h s h o r t s t e p s t h e s t e p s i z e i n c r e a s e s w i t h r e p e a t e d s u c c e s s e s s i n c e i t i s a s s u m e d t h a t t h e p r e v i o u s moves a r e w o r t h r e p e a t i n g . I f t h e o b j e c t i v e f u n c t i o n d o e s n o t i m p r o v e a l o n g a g i v e n d i r e c t i o n s m a l l e r s t e p s a r e u s e d a g a i n and i f t h i s i s n o t s u c c e s s f u l , a new s t a g e and a new p a t t e r n i s s t a r t e d . L e t us assume t h a t t h e s e a r c h b e g i n s a t an a r b i t r a r i l y c h o s e n b a s e p o i n t , b ^ . A s t e p s i z e i s t h e n c h o s e n f o r e a c h i n d e p e n d e n t v a r i a b l e x . ( i =1 ,2 •••• N ) . The o b j e c t i v e f u n c t i o n i s e v a l u a t e d by b, and t h e n a t b,+D, 178 w h i c h i s c a l l e d t-j-| w here t h e d o u b l e s u b s c r i p t shows t h a t t h e f i r s t v a r i a b l e has been p e r t u r b e d i n t h e f i r s t p a t t e r n . I f t>.|+D.j i s n o t as good as b-j t h e n t r y b^-D^. I f t h i s new p o i n t i s b e t t e r t h a n b-j i t i s d e s i g n a t e d as t h e t e m p o r a r y h e a d . ( I n v i s u a l i z i n g t h e p a t t e r n one may c o n s i d e r an a r r o w , i t s b a s e a t one end and i t s head a t t h e o t h e r . ) A summary o f t h e s e r u l e s ( f o r m a x i m i z i n g a f u n c t i o n ) i s shown b e l o w . '11 b 1 + D ] i f y ( b 1 + D 1 ) > y ( b n ) 1 " 1 b ] - D 1 i f y ( b 1 + D ] ) > y ( b ] ) > y ( b ] + D ] ) b 1 i f y ( b n ) > m a x [ y ( b 1 + D 1 ) , y ( b 1 - D ] ) The n e x t i n d e p e n d e n t v a r i a b l e , x 2 , i s p e r t u r b e d i n t h e same manner b u t t h i s t i m e a b o u t t h e t e m p o r a r y h e a d t-| -j i n s t e a d o f t h e o r i g i n a l b a s e b-j . In g e n e r a l t h e j t h t e m p o r a r y head f o r t h e i t h p a t t e r n i s o b t a i n e d as f o l l o w s , 179 + ° j 1 f y ( t i , j - l + D j > > y ( t i , j - l > ' l . J - l 1 f y ( t i , j - l ' > m a x [ y ( t 1 , j - l + D j ) ' y ( t i , j - T D d ) " When a l l t h e v a r i a b l e s have been p e r t u r b e d t h e l a s t t e m p o r a r y h e a d , p o i n t t ^ i s d e s i g n a t e d as t h e s e c o n d b a s e p o i n t ; b^ = The o r i g i n a l b a s e p o i n t and t h e n e w l y d e t e r m i n e d b a s e p o i n t b^ t o g e t h e r e s t a b l i s h t h e f i r s t p a t t e r n . One t h e n a s s u m e s t h a t i f a s i m i l a r e x p l o r a t i o n were c o n -d u c t e d f r o m b^ t h e r e s u l t s w o u l d p r o b a b l y be t h e same, s o t h e s e e x p l o r a t i o n s a r e c u r t a i l e d and t h e a r r o w i s e x t e n d e d f r o m b-j t h r o u g h b^ d o u b l i n g i t s l e n g t h and e s t a b l i s h i n g a t e m p o r a r y head t^^ f o r t h e s e c o n d p a t t e r n b a s e d a t b^, t20 = b 2 + ( b 2 " b l } = 2 b 2 " b l L o c a l e x p l o r a t i o n s a r e now c a r r i e d o u t a r o u n d t^Q t o c o r r e c t t h e t e n t a t i v e p a t t e r n i f n e c e s s a r y . R e p e a t e d s u c c e s s i n one d i r e c t i o n c a u s e s t h e p a t t e r n t o grow e v e n 180 Z P R E V I O U S B A S E P O I N T W C U R R E N T B A S E P O I N T X C U R R E N T P O I N T Y O O B J E C T I V E F U N C T I O N A T C U R R E N T B A S E P O I N T Y O B J E C T I V E F U N C T I O N A T C U R R E N T P O I N T Y Y O B J E C T I V E F U N C T I O N B E F O R E C U R R E N T M O V E D S T E P S I Z E L I M L I M I T I N G S T E P S I Z E R S T E P S I Z E R E D U C T I O N F A C T O R Y = F U N X ( I ) = X ( I ) + D ( I ) Y Y = Y 0 V X ( L ) = X ( I ) -2DU) I N I T I A L I Z A T I O N Y O = F U N Y Y =• Y O I X ( I ) = W ( I ) t X ( I ) = X ( I ) + D ( I ) Y = F U N © E X P L O R A T I O N © F I G U R E 4 - 4 C O M P U T E R F L O W D I A G R A M F O R P A T T E R N S E A R C H M E T H O D I G U R E 4-4 ( C O N T I N U E D ) 182 f u r t h e r . When t h e c a s e o c c u r s t h a t t h e p e r t u r b a t i o n s a b o u t a b a s e p o i n t do n o t show any more i m p r o v e m e n t t h e n t h e p e r t u r b a t i o n s t e p s i z e s a r e r e d u c e d and a new p a t t e r n i s s t a r t e d . F i g u r e 4-4 shows t h e c o m p u t e r f l o w d i a g r a m f o r t h e p a t t e r n s e a r c h m e t h o d . Weisman et al. (56) u t i l i z e d a m o d i f i e d p a t t e r n s e a r c h i n a m u l t i v a r i a b l e c o n s t r a i n e d o p t i m i z a t i o n p r o b l e m . The p a t t e r n s e a r c h was u s e d t o o p t i m i z e a s e r i e s o f c o n -s t r a i n e d o b j e c t i v e f u n c t i o n s w h i c h c o n s i s t e d o f t h e o r i g i n a l o b j e c t i v e f u n c t i o n p l u s a p e n a l t y f u n c t i o n . The c h i e f d i f f e r e n c e b e t w e e n t h e i r t e c h n i q u e and t h a t o f Hooke and J e e v e s i s t h e m e t h o d u s e d f o r c o n t r o l l i n g s t e p s i z e s . In t h e o r i g i n a l p r o c e d u r e t h e s t e p s i z e f o r a l l i n d e p e n d e n t v a r i a b l e s i s c h a n g e d a t t h e same t i m e and by t h e same r a t i o , w h i l e i n t h e m o d i f i e d m e t h o d , t h e s t e p s i z e s a r e c h a n g e d d e p e n d i n g on t h e s u c c e s s e s and f a i l u r e s o f t h e p e r t u r b e d moves f o r a g i v e n v a r i a b l e . The a u t h o r s c l a i m t h a t t h e m o d i f i e d v e r s i o n i s somewhat f a s t e r t h a n t h e o r i g i n a l m e t h o d . K l i n g m a n and H i m m e l b l a u ( 9 6 ) u t i l i z e d Hooke and J e e v e s ' m ethod c o u p l e d t o t h e i r n e w l y d e v e l o p e d M u l t i p l e -G r a d i e n t s Summation T e c h n i q u e f o r t h e s o l u t i o n o f a c o n -s t r a i n e d n o n l i n e a r p r o g r a m m i n g p r o b l e m . In t h e i r m e t h o d , when a c o n s t r a i n t i s r e a c h e d , a new s u c c e s s f u l d i r e c t i o n , NSD, f o r t h e c o n t i n u a t i o n o f t h e s e a r c h i s d e f i n e d by t h e v e c t o r sum o f t h e n o r m a l i z e d g r a d i e n t s o f t h e c o n t a c t e d 183 c o n s t r a i n t and t h e o b j e c t i v e f u n c t i o n . The p a t t e r n s e a r c h i s m o d i f i e d s l i g h t l y so t h a t when a p a t t e r n move r e s u l t s i n a n o n f e a s i b l e p o i n t , t h e p o i n t i s r e j e c t e d and a n o t h e r e x p l o r a t o r y move i s made f r o m t h e l a s t f e a s i b l e p o i n t . 2. C o n s t r a i n e d P a t t e r n S e a r c h The o r i g i n a l P a t t e r n S e a r c h M e t h o d o f Hooke and J e e v e s ( 5 5 ) has been u t i l i z e d i n t h i s s t u d y and c o m b i n e d w i t h a d d e d l o g i c t o h a n d l e t h e c o n s t r a i n t s . A c o m p a r i s o n o f F i g u r e 4-4 and 4-5 shows t h e a d d i t i o n s made t o t h e o r i g -i n a l t e c h n i q u e ( s e e A p p e n d i x H f o r c o m p u t e r l i s t i n g ) . In t h e e x p l o r a t o r y s e g m e n t o f t h e s e a r c h e a c h new p o i n t i s i n i t i a l l y c h e c k e d f o r v i o l a t i o n o f t h e e x p l i c i t c o n s t r a i n t s . I f t h i s o c c u r s , t h e i n d e p e n d e n t v a r i a b l e w h i c h v i o l a t e d t h e c o n s t r a i n t i s p l a c e d j u s t w i t h i n t h e c o n s t r a i n t b o u n d a r y . T h i s i s f o l l o w e d by a c h e c k o f t h e " a b o r t i v e " i m p l i c i t c o n s t r a i n t s w h i c h when v i o l a t e d ( a s i n t h e c a s e o f a n e g a t i v e f l o w s p l i t ) may c a u s e t h e s i m u l a t i o n p r o g r a m t o a b o r t . ( C o n v e r g e n c e t e s t w i l l n o t work i n t h e c a s e o f n e g a t i v e f l o w s . ) I f t h i s t y p e o f c o n s t r a i n t i s v i o l a t e d , t h e p o i n t i s i m m e d i a t e l y r e j e c t e d . I f t h e a b o v e m e n t i o n e d c o n s t r a i n t s a r e s a t i s f i e d t h e s i m u l a t i o n p r o g r a m i s c a l l e d upon ( s e e F i g u r e 4-6 f o r p r o g r a m l i n k a g e ) and t h e o b j e c t i v e f u n c t i o n a l o n g w i t h t h e i m p l i c i t c o n s t r a i n t s a r e c a l c u l a t e d . I f t h e i m p l i c i t c o n -s t r a i n t s a r e v i o l a t e d , t h e p o i n t i s r e j e c t e d . YES <— Y * FUN F I G U R E 4 - 5 C O M P U T E R F L O W D P A T T E R N S E A R C H • I A G R A M F O R C O N S T R A I N E D F I G U R E 4-5 ( C O N T I N U E D ) 186 M A I N P R O G R A M CALL2 C A L L D R E A D C A L L C O M P I D N O r E X P L I C I T C O N S T R A I N T V I O L A T E D C A L L I N I T [ F U N C T I O N | D E F I N E I M P L I C I T V A R I A B L E S A N D C O N S T R A I N T S ' 1 C A L L D C H E C K R E A D C O N P A T I N P U T D A T A U P P E R B O U N D S O F I N D E P E N D E N T V A R I A B L E S A N D C O N S T R A I N T S [ F U N C T I O N ) U P P E R B O U N D S O F I N D E P E N D E N T V A R I A B L E S A N D C O N S T R A I N T S F I G U R E 4-6 CONPAT P R O G R A M L I N K A G E 187 In t h e c a s e o f t h e p a t t e r n move, t h e e x p l i c i t c o n s t r a i n t s a r e c h e c k e d as b e f o r e . The " a b o r t i v e " i m p l i c i t c o n s t r a i n t s a r e t h e n c h e c k e d , and i f v i o l a t e d , t h e p a t t e r n s t e p t e m p o r a r y p o i n t i s r e j e c t e d and an e x p l o r a t i o n i s made a b o u t t h e p r e s e n t b a s e p o i n t . I f t h e " a b o r t i v e " i m p l i c i t c o n s t r a i n t s a r e n o t v i o l a t e d t h e s i m u l a t i o n p r o -gram i s c a l l e d upon and t h e o b j e c t i v e f u n c t i o n and i m p l i c i t c o n s t r a i n t s a r e c a l c u l a t e d . I f t h e i m p l i c i t c o n s t r a i n t s a r e v i o l a t e d , i . e . ; t h e p a t t e r n s t e p p o i n t i s n o n f e a s i b l e , t h e n t h e p r e s e n t b a s e p o i n t i s u s e d f o r t h e c o m p a r i s o n o f o b j e c t i v e f u n c t i o n v a l u e s w i t h t h e e x p l o r a t o r y moves a b o u t t h e n o n f e a s i b l e p a t t e r n s t e p p o i n t . ( E v e n t h o u g h t h e p a t t e r n s t e p p o i n t may be n o n f e a s i b l e , t h e e x p l o r a t o r y s t e p s may r e - e n t e r t h e f e a s i b l e r e g i o n . ) I f t h e p a t t e r n s t e p p o i n t i s f e a s i b l e , t h e o b j e c t i v e f u n c t i o n v a l u e o f t h a t p o i n t i s u s e d f o r c o m p a r i s o n w i t h t h e e x p l o r a t o r y s t e p s . The C o n s t r a i n e d P a t t e r n S e a r c h T e c h n i q u e as d e s c r i b e d a b o v e c a n n o t a l w a y s c o p e w i t h c o n s t r a i n t s . F i g u r e 4 - 7 ( a ) i l l u s t r a t e s t h e c a s e where t h e P a t t e r n S e a r c h c a n c o n t i n u e t o move t o w a r d t h e optimum by a s e r i e s o f e x p l o r a -t o r y and p a t t e r n moves. I f t h e s e a r c h i s m o v i n g i n t h e d i r e c t i o n o f t h e v e c t o r shown, t h e p a t t e r n move f r o m t h e f e a s i b l e p o i n t A r e a c h e s B i n t h e n o n f e a s i b l e r e g i o n . S i n c e t h e s t e p i s a f a i l u r e , t h e m ethod r e t u r n s t o b a s e p o i n t A 188 (b) FAILURE F I G U R E 4 - 7 E X A M P L E S O F C O N S T R A I N E D P A T T E R N S E A R C H 189 where an e x p l o r a t i o n i s made. An i m p r o v e m e n t i s o b t a i n e d i n t h e x-j d i r e c t i o n a t p o i n t C b u t n o t a t p o i n t s E and D i n t h e x^ d i r e c t i o n . H o w e v e r , i n t h i s c a s e , a new p a t t e r n s t e p i s made t o F and t h e s e a r c h c o n t i n u e s t o w a r d t h e o p t i m u m . F i g u r e 4 - 7 ( b ) i s a l m o s t i d e n t i c a l t o t h a t o f F i g u r e 4 - 7 ( a ) e x c e p t t h a t t h e c o n s t r a i n t i s o f a s l i g h t l y d i f f e r e n t d i r e c t i o n . As i n t h e p r e v i o u s c a s e , t h e p a t t e r n move t h r o u g h A r e s u l t s i n a n o n f e a s i b l e p o i n t B. However i n t h i s c a s e t h e e x p l o r a t i o n a b o u t t h e b a s e p o i n t A f a i l s b e c a u s e p o i n t s D and E r e s u l t i n l o w e r f u n c t i o n v a l u e s and p o i n t s C and F a r e n o n f e a s i b l e . The s t e p s i z e w i l l t h e n be r e d u c e d and t h e s e a r c h m i g h t be a b l e t o r e a c h p o i n t P b e f o r e t h e s t e p s i z e i s r e d u c e d b e l o w t h e s p e c i f i e d minimum s i z e . In t h i s c a s e t h e s e a r c h has f a i l e d s i n c e p o i n t P i s q u i t e d i s t a n t f r o m t h e optimum. 3. R e s u l t s w i t h I n t e r m e d i a t e P l a n t Model In o r d e r t o c o m p a r e t h e o p t i m i z a t i o n m e t h o d s t e s t e d , t h e same i n i t i a l p o i n t s were u s e d i n e a c h c a s e ( s e e C h a p t e r 4-E f o r a c o m p a r i s o n o f t h e r e s u l t s ) . The i n i t i a l p o i n t s u s e d i n t h e t e s t s shown i n T a b l e 4-5 and 4-6 a r e r e l a t i v e l y f a r f r o m t h e optimum w h i l e t h e i n i t i a l p o i n t i n T a b l e 4-7 i s c l o s e t o t h e optimum and l i e s on one c o n s t r a i n t . 190 The f o l l o w i n g t e r m s a r e u s e d i n T a b l e s 4-5, 4-6, -7 : N -• Number o f i n d e p e n d e n t v a r i a b l e s M -• Number o f c o n s t r a i n t s R -• R e d u c t i o n f a c t o r f o r s t e p s i z e r e d u c t i o n LIM -• L i m i t i n g s t e p s i z e ITER -• Maximum number o f i n t e r a t i o n s p e r m i t t e d DEL -- V a r i a b l e s t e p s i z e A -• S t e p a c c e p t e d R -• S t e p r e j e c t e d BP -• Ba s e p o i n t CV -• C o n s t r a i n t v i o l a t i o n x i ± -• E x p l o r a t o r y s t e p P a t -- P a t t e r n s t e p SR -- S t e p s i z e r e d u c t i o n T a b l e 4-5 shows a c a s e i n w h i c h t h e i n i t i a l s t a r t -i n g p o i n t i s f a i r l y d i s t a n t f r o m t h e o p t i m u m . The me t h o d moves q u i t e w e l l f o r t h e f i r s t t h r e e e x p l o r a t i o n ; t h e f i r s t two p a t t e r n s t e p s a r e s u c c e s s f u l and t h r e e new b a s e p o i n t s a r e f o u n d . H o w e v e r , t h e t h i r d p a t t e r n s t e p ( 27) e n t e r s t h e n o n f e a s i b l e r e g i o n . An e x p l o r a t i o n a b o u t t h e p a t t e r n s t e p r e m a i n s i n t h e n o n f e a s i b l e r e g i o n . An e x p l o r a t i o n a b o u t t h e b a s e p o i n t i s t h e n made ( 3 6 ) and f a i l s . T h i s c a u s e s a s t e p s i z e r e d u c t i o n and f u r t h e r e x p l o r a t i o n ( 4 4 ) . A new b a s e p o i n t ( 4 7 ) i s f o u n d and a p a t t e r n move made ( 5 2 ) b u t 191 t h e moves a r e now q u i t e s m a l l . E x p l o r a t o r y s t e p s ( 5 3 - 6 0 ) a r e made a b o u t t h e p a t t e r n p o i n t b u t a r e n o t f e a s i b l e . F o u r s e t s o f e x p l o r a t o r y moves ( 6 1 - 6 8 , 6 9 - 7 6 , 7 7 - 8 4 , 8 5 - 9 2 ) a r e t h e n made a b o u t t h e c u r r e n t b a s e p o i n t ( 4 7 ) and f a i l , w i t h t h e s t e p s i z e b e i n g r e d u c e d a f t e r e a c h f a i l u r e . The m e t h o d f i n a l l y s t o p s when t h e s t e p s i z e i s r e d u c e d b e l o w t h e v a l u e o f t h e l i m i t i n g s t e p s i z e . The c o n s t r a i n e d p a t t e r n s e a r c h d i d n o t h a n d l e t h e c o n s t r a i n t s v e r y w e l l . The o n l y p r o g r e s s a c h i e v e d was when t h e p o i n t s were d i s t a n t f r o m t h e c o n s t r a i n t s . E s s e n t i a l l y , t h e i m p r o v e m e n t shown was i n t h e f i r s t 22 s t e p s s i n c e s t e p ( 4 7 ) was o n l y s l i g h t l y b e t t e r t h a n ( 2 2 ) . In T a b l e 4-6 t h e s t a r t i n g p o i n t u s e d i s a l s o f a i r l y d i s t a n t f r o m t h e o p t i m u m . Once a g a i n , good p r o g r e s s i s made d u r i n g t h e e a r l y s t a g e s o f t h e s e a r c h as two new b a s e p o i n t s a r e r e a c h e d ( 1 3 ) . V e r y l i t t l e p r o g r e s s i s t h e n a c h i e v e d b u t t h e s e a r c h c o n t i n u e s u n t i l s t e p (99) b e c a u s e two s l i g h t l y b e t t e r b a s e p o i n t s ( 4 0 ) , ( 6 8 ) a r e f o u n d . As i n t h e p r e v i o u s c a s e , t h e m e t h o d f a i l e d n e a r t h e n o n f e a s i b l e r e g i o n . The s t a r t i n g p o i n t i n T a b l e 4-7 i s q u i t e c l o s e t o t h e optimum and l i e s on one o f t h e c o n s t r a i n t s . V e r y l i t t l e p r o g r e s s i s a c h i e v e d a l t h o u g h a s l i g h t l y b e t t e r b a s e p o i n t ( 2 7 ) i s f o u n d . 192 TABLE 4-5 O p t i m i z a t i o n o f t h e I n t e r m e d i a t e P o l y P l a n t  M odel U s i n g t h e C o n s t r a i n e d P a t t e r n S e a r c h M e t h o d - 1 N = 4 LIM = 0.001 M = 10 ITER = 20 R = 0. 5 D E L ( I ) = 0.01 No. S t e p X ( D X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 0 O r i g . .70 .10 .30 .40 33.124 BP 1 X1 + .71 31.888 R 2 XI - .69 34.110 A 3 X2 + .11 33.951 R 4 X2- .09 34.562 A 5 X3+ .31 34.392 R 6 X3- .29 34.733 A 7 X4+ .41 34.648 R 8 X4- .69 .09 .29 .39 34.781 A,BP 9 P a t .68 .08 .28 .38 36.148 A 10 X1 + .69 35.450 R 11 X I - .67 37.009 A 12 X2 + .09 36.858 R 13 X2- .07 37.265 A 14 X3+ .29 37.144 R 15 X3- .27 37.380 A 16 X4 + .39 37.320 R 17 X4- .67 .07 .27 .37 37.414 A,BP 18 P a t .65 .05 .25 .35 39.193 A 19 X1 + .66 38.755 R 20 X I - .64 39.706 A 21 X2 + .06 39.616 R 22 X2- .64 .04 .25 .35 39.837 A,BP 23 X3+ .26 39.769 R 24 X3- .24 39.902 CV 25 X4+ .25 .36 39.804 R 26 X4- .34 39.862 CV 27 P a t .61 .01 .23 .33 41.069 CV 28 X1 + .62 41.002 CV 29 X I - .60 41 .158 CV 30 X2 + .61 .02 41 .039 CV 193 T a b l e 4-5 ( C o n t i n u e d ) No. S t e p X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) Res u l 31 X2- .61 .01 .23 .33 41 .153 CV 32 X3+ .24 41.044 CV 33 X3- .22 41.125 CV 34 X4+ .23 .34 41.067 CV 35 X4- .33 41 .102 CV 36 X1 + .65 .04 .25 .35 39.382 R 37 X I - .63 40.249 CV 38 X2 + .64 .05 39.767 R 39 X2- .03 39.954 CV 40 X3+ .26 39.765 R 41 X3- .24 39.903 CV 42 X4+ .25 .36 39.804 R 43 X4- .34 39.862 CV,SR 44 X1 + .645 .04 .25 .35 39.629 R 45 X I - .635 40.000 CV 46 X2 + .640 .045 39.827 R 47 X2- .640 .035 .250 .350 39.899 A,BP 48 X3+ .255 39.855 R 49 X3- . .245 39.918 CV 50 X4+ .250 .355 39.870 R 51 - X4- .345 39.898 CV 52 P a t .640 .030 .250 .350 39.943 CV 53 X1 + .645 39.739 R 54 X I - .635 40.094 CV 55 X2 + .640 .035 39.926 CV 56 X2- .025 39.997 CV 57 X3+ .030 .255 39.901 CV 58 X3- .245 39.967 CV 59 X4+ .250 .355 39.919 CV 60 X4- .345 39.947 CV 61 X1 + .645 .035 .250 .350 39 .682 R 62 X I - .635 40.047 CV 63 X2 + .640 .040 39.877 R 64 X2- .030 39.948 R 65 X3+ .035 .255 39.851 R 66 X3- .245 39.918 CV 67 X4+ .250 .355 39.870 R 68 X4- .345 39.898 R,SR 69 X1 + .6425 .0350 .2500 .3500 39.805 R 70 X I - .6375 39.968 CV 71 X2 + .6400 .0375 39.887 R 72 X2- .0325 39.919 CV 73 X3+ .0350 .2525 39.869 R 194 T a b l e 4-5 ( C o n t i n u e d ) No. S t e p X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 74 X3- .2475 39.907 CV 75 X4+ .2500 .3525 39.874 R 76 X4- .3475 39.893 CV,SR 77 X1 + .6413 .0350 .2500 .3500 30.891 CV 78 X I - .6386 39.898 R 79 X2 + .6400 .0363 39.884 R 80 X2- .0338 39.908 CV 81 X3+ .0350 .2513 39.874 R 82 X3- .2488 39.877 R 83 X4+ .2500 .3513 39.896 CV 84 X4- .3488 39.880 R,SR 85 X1 + .6406 .0350 .2500 .3500 39.889 R 86 XI - .6394 39.871 R 87 X2 + .6400 .0356 39.879 R 88 X2- .0344 39.894 R 89 X3+ .0350 .2506 39.879 R 90 X3- .2494 39.880 R 91 X4+ .2500 .3506 39.890 R 92 X4- .3494 39.882 Rl S e a r c h t e r m i n a t e d s i n c e s t e p s i z e l e s s t h a n l i m i t i n g s t e p s i z e . C P . U . T i m e = 812 s e c . 195 TABLE 4-6 O p t i m i z a t i o n o f I n t e r m e d i a t e P o l y P l a n t M odel U s i n g t h e C o n s t r a i n e d P a t t e r n S e a r c h M e t h o d - 2 N = 4 LIM = 0.001 M = 10 ITER = 20 R = 0.5 D E L ( I ) = 0.01 No. S t e p x ( i ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l 0 O r i g .675 .175 .225 .187 36.491 BP 1 X1 + .685 35.593 R 2 X I - .665 37.226 A 3 X2 + .185 37.088 R 4 X2- .165 37.462 A 5 X3+ .235 37.343 R 6 X3- .215 37.576 A 7 X4+ .197 37.513 R 8 X4- .665 .165 .215 .177 37.614 A,BP 9 P a t .655 .155 .205 .167 38.546 A 10 X1 + .665 38.022 R 11 X I - .645 39.137 CV 12 X2 + .655 .165 38.496 R 13 X2- .655 .145 .205 .167 38.743 A,BP 14 X3 + .215 38.626 R 15 X3- .195 38.808 CV 16 X4+ .205 .177 38.667 R 17 X4- .157 38.754 CV 18 P a t .645 .125 .195 .157 39.579 CV 19 X1 + .655 39.161 CV 20 X I - .635 40.043 CV 21 X2 + .645 .135 39.533 CV 22 X2- .115 39.731 CV 23 X3 + .125 .205 39.526 CV 24 X3- .185 39.680 CV 25 X4 + .195 .167 39.567 CV 26 X4- .147 39.637 CV 27 X1 + .665 .145 .205 .167 38.119 R 28 X I - .645 39.251 CV 29 X2 + .655 .155 38.630 R 30 X2- .135 38.872 CV 31 X3+ .145 .215 38.617 R 196 T a b l e 4-6 ( C o n t i n u e d ) No. S t e p X ( D X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 32 X3- .655 .145 .195 .167 38.807 CV 33 X4+ .205 .177 38.667 R 34 X4- .157 38.754 R.SR 35 X1 + .660 .145 .205 .167 38.476 R 36 X I - .650 38.928 CV 37 X2 + .655 .150 38.703 R 38 X2- .140 38.799 CV 39 X3 + .145 .210 38.667 R 40 X3- .655 .145 .200 .167 38.761 A,BP 41 X4+ .172 38.733 R 42 X4- .162 38.775 CV 43 P a t .655 .145 .195 .167 38.801 CV 44 X1 + .660 38.539 R 45 X I - .650 39.005 CV 46 X2 + .655 .150 38.786 CV 47 X2- .140 38.881 CV 48 X3 + .145 .200 38.750 R 49 X3- .190 38.843 CV 50 X4+ .195 .172 38.774 CV 51 X4- .162 38.816 CV 52 X1 + .660 .145 .200 .167 38.519 R 53 X I - .650 38.968 CV 54 X2 + .655 .150 38.745 R 55 X2- .140 38.840 CV 56 X3 + .145 .205 38.709 R 57 X3- .195 38.802 CV 58 X4+ .200 .172 38.732 R 59 X4- .162 38.775 CV,SR 60 X1 + .6575 .1450 .2000 .1670 38.652 R 61 X I - .6525 38.863 CV 62 X2 + .6550 .1475 38.761 A l 63 X3 + .2025 38.712 R 64 .1975 38.757 R 65 X4+ .200 .1695 38.709 R 66 X4- .1645 38.735 R,SR 67 X1 + .6563 .1450 .2000 .1670 38.727 R 68 XI - .6538 .1450 .2000 .1670 38.762 A,BP 69 X2+ .1463 38.789 CV 70 X2- .1438 38.835 CV 71 X3+ .1450 .2013 38.837 CV B e c a u s e o f p r o g r a m l o g i c , a c c e p t e d as c u r r e n t p o i n t b u t s i n c e e q u a l t o b a s e p o i n t , r e j e c t e d a f t e r e x p l o r a t i o n . 197 T a b l e 4-6 ( C o n t i n u e d ) No. S t e p X(D X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 72 X3- .1988 38.838 CV 73 X4+ .200 .1683 38.837 CV 74 X4- .1658 38.833 CV 75 P a t .6526 .1450 .2000 .1670 38.833 CV 76 X1 + .6538 38.860 CV 77 X I - .6513 38.845 CV 78 X2 + .6526 .1463 38.880 CV 79 X2- .1438 38.922 CV 80 X3+ .1450 .2013 38.918 CV 81 X3- .1988 38.914 CV 82 X4+ .2000 .1683 38.912 CV 83 X4- .1658 38.907 CV 84 X1 + .6550 .1670 38.803 CV 85 X I - .6525 38.839 CV 86 X2 + .6538 .1463 38.827 CV 87 X2- .1438 38.857 CV 88 X3 + .1450 .2013 38.851 CV 89 X3- .1988 38.843 CV 90 X4+ .2000 .1683 38.841 CV 91 X4- .1658 38.834 CV,SR 92 X1 + .6544 .1670 38.834 CV 93 X I - .6531 38.811 CV 94 X2 + .6538 .1456 38.836 CV 95 X2- .1444 38.823 CV 96 X3+ .1450 .2006 38.841 CV 97 X3- .1994 38.823 CV 98 X4+ .2000 .1676 38.836 c v n 99 X4- .1664 38.826 CV' S e a r c h t e r m i n a t e d s i n c e s t e p s i z e l e s s t h a n l i m i t i n g s t e p s i z e . C.P.U. Time = 766 s e c . 198 TABLE 4-7 O p t i m i z a t i on o f I n t e r m e d i a t e P o l y P l a n t Model U s i n g t h e C o n s t r a i n e d P a t t e r n S e a r c h M e t h o d - 3 N = 4 LIM = 5 .E-4 M = 10 ITER = 20 R = 0.5 D E L ( I ) = 0.005 No. S t e p X ( l ) X ( 2 ) X(3) X(4) f ( x ) R e s u l t 1 O r i g .618 .046 .300 .406 40.368 BP 2 X1 + .623 40.173 R 3 X I - .613 40.383 A 4 X2 + .051 40.372 R 5 X2- .041 40.453 CV 6 X3+ .046 .305 40.390 A 7 X4+ .411 40.382 R 8 X4- .613 .046 .305 .401 40.403 A,BP 9 P a t .608 .046 .310 .396 40.422 CV 10 X1 + .613 40.413 CV 11 X I - .603 40.509 CV 12 X2 + .608 .051 40.439 CV 13 X2- .041 40.501 CV 14 X3+ .046 .315 40.438 CV 15 X3- .305 40.483 CV 16 X4+ .310 .401 40.450 CV 17 X4- .391 40.470 CV 18 X1 + .618 .046 .305 .401 40.368 R 19 X I - .608 40.453 CV 20 X2 + .613 .051 40.380 R 21 X2- .041 40.445 CV 22 X3 + .046 .310 40.379 R 23 X3- .300 40.426 CV 24 X4+ .305 .406 40.392 R 25 X4- .396 40.412 R,SR 26 X1 + .6155 .0460 .3050 .4010 40.376 R 27 X I - .6105 .0460 .3050 .4010 40.407 A,BP 28 X2 + .0485 40.404 R 29 X2- .0435 40.455 CV 30 X3+ .0460 .3075 40.425 CV 31 X3- .3025 40.455 CV 32 X4+ .6105 .0460 .3050 .4035 40.429 CV 199 T a b l e 4-7 ( C o n t i n u e d ) No. S t e p x(D X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 33 X4- .3985 40.445 CV 34 P a t .6080 .0460 .3050 .4010 40.454 CV 35 X1 + .6105 40.463 CV 36 X I - .6055 40.480 CV 37 X2 + .6080 .0485 40.457 CV 38 X2- .0435 40.493 CV 39 X3+ .0460 .3075 40.461 CV 40 X3- .3025 40.489 CV 41 X4+ .3050 .4035 40.464 CV 42 X4- .3985 40.479 CV 43 X1 + .6130 .0460 .3050 .4010 40.422 CV 44 X I - .6080 40.446 CV 45 X2 + .6105 .0485 40.422 CV 46 X2- .0435 40.458 CV 47 X3+ .0460 .3075 40.426 CV 48 X3- .3025 40.455 CV 49 X4+ .3050 .4035 40.429 CV 50 X4- .3985 40.445 CV.SR 51 X1 + .6118 .0460 .3050 .4010 40.443 CV 52 X I - .6093 40.428 CV 53 X2 + .6105 .0473 40.428 CV 54 X2- .0448 40.453 CV 55 X3+ .0460 .3063 40.429 CV 56 X3- .3038 40.447 CV 57 X4+ .3050 .4023 40.433 CV 58 X4- .3998 40.441 CV,SR 59 X1 + .6111 .0460 .3050 .4010 40.440 CV 60 X I - .6099 40.433 CV 61 X2 + .6105 .0466 40.433 CV 62 X2- .0454 40.445 CV 63 X3+ .0460 .3056 40.433 CV 64 X3- .3044 40.435 CV 65 X4+ .3050 .4016 40.442 CV 66 X4- .4004 40.436 CV.SR 67 X1 + .6108 .0460 .3050 .4010 40.439 CV 68 X I - .6102 40.435 CV 69 X2 + .6105 .0463 .3050 .4010 40.439 CV 70 X2- .0457 40.441 CV 71 X3+ .0460 .3053 40.440 CV 72 X3- .3047 40.436 CV 73 X4+ .3050 .4013 40.440 CV 74 X4- .4007 40.437 c v i S e a r c h t e r m i n a t e d s i n c e s i z e . C P . U . t i m e = 447 s e c . s t e p s i z e l e s s t h a n l i m i t i n g s t e p 200 D. The M o d i f i e d Complex Meth o d 1. The C o m p l e x M e t h o d The C o m p l e x M e t h o d d e v e l o p e d by Box ( 5 1 ) i n 1965 i s an e x t e n s i o n o f t h e S i m p l e x M e t h o d , i n t r o d u c e d by S p e n d l e y e t al. ( 5 8 ) i n 1962 and m o d i f i e d by N e l d e r and Mead ( 5 9 ) i n 1 9 6 4 , t o s o l v e c o n s t r a i n e d o p t i m i z a t i o n p r o b l e m s . I t h a n d l e s t h e c o n s t r a i n t s by t h e u s e o f a f l e x i b l e f i g u r e o f more t h a n n+1 v e r t i c e s (n i s t h e number o f i n d e -p e n d e n t v a r i a b l e s ) w h i c h c a n e x p a n d o r c o n t r a c t i n a l l d i r e c t i o n s . V e r t i c e s a r e r e j e c t e d and g e n e r a t e d as i n t h e S i m p l e x M e t h o d b u t no a t t e m p t i s made t o p r e s e r v e a r e g u l a r f i g u r e i n w h i c h e a c h v e r t e x i s e q u i d i s t a n t f r o m a l l o t h e r p o i n t s . U p p e r and l o w e r bounds must be p l a c e d on e a c h o f t h e i n d e p e n d e n t v a r i a b l e s ( g ^ < x^ < h ^ ) . T h e s e l i m i t s t h e n f o r m a r e c t a n g u l a r s p a c e i n w h i c h t h e s e a r c h t a k e s p l a c e . A f i r s t p o i n t i s c h o s e n and i t s f e a s i b i l i t y i s c h e c k e d . I f f e a s i b l e , t h e a d d i t i o n a l k c - l p o i n t s o f t h e c o m p l e x a r e g e n -e r a t e d . - g.) ( 4 - 3 0 ) i = 1,2, • • • • n j = 1 ,2, k c 201 Where r ^ . i s a p s u e d o - r a n d o m d e v i a t e r e c t a n g u l a r l y d i s t r i b u t e d o v e r t h e i n t e r v a l [ 0 , 1 ] . The g e n e r a t e d p o i n t s w i l l a l l l i e w i t h i n t h e s e a r c h r e g i o n b u t may v i o l a t e an i m p l i c i t c o n s t r a i n t . I f t h i s o c c u r s t h e t r i a l p o i n t i s moved h a l f w a y b a c k t o t h e c e n t r o i d o f t h e e x i s t i n g v e r t i c e s ; x.. = %{cA . - xA .) ( 4 - 3 1 ) The c e n t r o i d w i l l a l w a y s e x i s t b e c a u s e a f e a s i b l e p o i n t i s c h o s e n t o s t a r t t h e s e a r c h . I f t h e r e g i o n i s c o n v e x a s a t i s f a c t o r y new p o i n t w i l l a l w a y s be f o u n d by c o n t i n u i n g t h i s c o n t r a c t i o n s t e p . S i n c e t h e b o u n d s on e a c h o f t h e i n d e p e n d e n t v a r i a b l e s have been s u p p l i e d , t h e s t e p s t o g e n e r a t e t h e o r i g i n a l c o m p l e x a r e a u t o m a t i c a l l y s c a l e d and t h e s e a r c h r e g i o n i s t h u s a d e q u a t e l y c o v e r e d . The o b j e c t i v e f u n c t i o n i s e v a l u a t e d a t e a c h v e r t e x and t h e v e r t e x W y i e l d i n g t h e p o o r e s t f u n c t i o n v a l u e i s r e j e c t e d and r e p l a c e d by a p o i n t w h i c h i s l o c a t e d a d i s t a n c e a e t i m e s as f a r f r o m t h e c e n t r o i d C o f t h e r e m a i n i n g p o i n t s as t h e d i s t a n c e f r o m t h e c e n t r o i d t o t h e r e j e c t e d v e r t e x i n a d i r e c t i o n d e f i n e d by a v e c t o r p o i n t i n g f r o m t h e r e j e c t e d p o i n t t o t h e c e n t r o i d . 202 = x. + a ( x . - x. ) e 1 ,c 1 ,w ( 4 - 3 2 ) k w h e r e x. 1 , C .L, x.. I f t h i s new p o i n t N i s a f e a s i b l e p o i n t t h e f u n c t i o n i s e v a l u a t e d t h e r e and t h e a b o v e d e s c r i b e d e x p a n -s i o n s t e p i s r e p e a t e d . I f , h o w e v e r , t h e f u n c t i o n v a l u e o f t h e new p o i n t i s w o r s e t h a n t h a t o f t h e p r e v i o u s l y d e t e r m i n e d w o r s t p o i n t o r i f an i m p l i c i t c o n s t r a i n t i s v i o l a t e d t h e v e r t e x i s moved h a l f w a y i n t o w a r d t h e c e n t r o i d u n t i l a v a l i d p o i n t i s o b t a i n e d . F o r a c o n v e x r e g i o n , t h e v e r t e x w i l l u l t i m a t e l y s a t i s f y t h e r e s t r i c t i o n s a l t h o u g h d i f f i c u l t i e s m i g h t a r i s e i n n o n - c o n v e x r e g i o n s , e s p e c i a l l y i f t h e c e n t r o i d l i e s o u t -s i d e o f t h e f e a s i b l e r e g i o n . In a d d i t i o n , i f a new p o i n t d o e s n o t s a t i s f y one o f t h e e x p l i c i t c o n s t r a i n t s on an i n -d e p e n d e n t v a r i a b l e , t h a t v a r i a b l e i s r e s e t a t a s u i t a b l e s m a l l d i s t a n c e i n s i d e t h e a p p r o p r i a t e b o u n d a r y t o y i e l d a = ^ ( x i , N + X. i , c ( 4 - 3 3 ) 203 f e a s i b l e p o i n t . The method w i l l c o n t i n u e u n t i l f i v e c o n -s e c u t i v e f u n c t i o n e v a l u a t i o n s ( o r c e n t r o i d e v a l u a t i o n s ) g i v e e q u a l v a l u e s t o w i t h i n a s p e c i f i e d c r i t e r i a . T h i s u s u a l l y o c c u r s a t t h e optimum when t h e c o m p l e x c o l l a p s e s i n t o i t s c e n t r o i d . A c o m p u t e r f l o w d i a g r a m f o r t h e C o m p l e x M e t h o d i s shown i n F i g u r e 4-8. Box ( 5 1 ) f o u n d t h a t t h e s p e e d o f c o n v e r g e n c e i s n o t v e r y s e n s i t i v e t o t h e e x p a n s i o n c o e f f i c i e n t a g o r t h e number o f c o m p l e x v e r t i c e s k . He recommends v a l u e s r c o f a g = 1.3 and k £ z 2 n . I f n i s g r e a t e r t h a n 5, l e s s p o i n t s may be u s e d . H o w e v e r , i f k c = n+1 i s u s e d , t h e c o m p l e x t e n d s t o c o l l a p s e and f l a t t e n a l o n g t h e f i r s t c o n -s t r a i n t w h i c h i s e n c o u n t e r e d . The u s u a l method f o r c h e c k -i n g t h a t t h e g l o b a l r a t h e r t h a n a l o c a l maximum has been f o u n d i s t o r e s t a r t t h e p r o g r a m f r o m d i f f e r e n t p o i n t s and i n f e r t h a t i f t h e y a l l c o n v e r g e t o t h e same s o l u t i o n t h e n a g l o b a l o p t imum has b e e n f o u n d . T h e r e i s no n e e d t o s t a r t w i t h a n o t h e r f e a s i b l e p o i n t s i n c e t h e same i n i t i a l p o i n t w i t h d i f f e r e n t p s u e d o - r a n d o m number i n i t i a t o r s w i l l i n i t i a t e t h e p r o b l e m w i t h a d i f f e r e n t c o m p l e x . Umeda (50 ) u s e d t h e Complex M e t h o d f o r t h e o p t i m a l d e s i g n o f an a b s o r b e r - s t r i p p e r s y s t e m . The s y s t e m c o n s i s t e d o f an a b s o r b e r , s t r i p p e r , and s e v e r a l h e a t e x c h a n g e r s . F i v e i n d e p e n d e n t d e s i g n v a r i a b l e s were c o n s i d e r e d a l o n g w i t h s e v e r a l i n e q u a l i t y c o n s t r a i n t s i n m i n i m i z i n g t h e p r o d u c t i o n 204 GENERATE INITIAL COMPLEX Xj « g . + RAND( hj- gj) EVALUATE FUNCTION AT COMPLEX POINTS © FIND WORST POINT IN COMPLEX I CALCULATE CENTROID OF ALL POINTS BUT WORST I REFLECT WORST POINT THROUGH CENTROID x. * c.-t-o (c. — X ) I 1 6 1 W o CALCULATE CENTROID OF EXISTING POINTS MOVE 1/2 WAY BACK TO CENTROID Xj =l/2(Cj + Xj) MOVE 1/2 WAY BACK TO CENTROID x i ' 1/2 (c NEW POINT AT CENTROID ALL POINTS BUT WORST FIND BEST YES POINT 3 EXPLICIT CONSTRAINTS VIOLATED NO IMPLICIT CONSTRAINTS sVIOLATED NO EVALUATE OBJECTIVE FUNCTION CALCULATE CENTROID OF ALL POINTS EVALUATE FUNCTION AT CENTROID YES F I G U R E 4-8 COMPUTER FLOW D I A G R A M OF T H E C O M P L E X METHOD 205 c o s t . He f o u n d t h a t t h e Complex M e t h o d was a p p l i c a b l e t o h i s p r o b l e m and t h a t t h e c o m p u t e r t i m e was r e a s o n a b l e . ( E a c h i t e r a t i v e c o m p u t a t i o n t o o k a b o u t 0.2 s e c o n d s on a CDC-1604 c o m p u t e r . ) He c h e c k e d t h e g l o b a l maximum by u s i n g f i v e s e t s o f s t a r t i n g c o n d i t i o n s and f o u n d t h a t t h e y a l l c o n v e r g e d t o t h e same p o i n t . C o mplex M e t h o d by m u l t i p l y i n g t h e c o o r d i n a t e s o f a l l b u t t h e w o r s t c o m p l e x p o i n t s by t h e f u n c t i o n v a l u e o f t h e s e p o i n t s r a i s e d t o some power when c a l c u l a t i n g t h e c e n t r o i d ( o f a l l b u t w o r s t c o m p l e x p o i n t s ) . T he c a l c u l a t i o n o f t h e c e n t r o i d o f a l l p o i n t s e x c e p t t h e w o r s t i s now Umeda and I c h i k a w a ( 5 2 ) m o d i f i e d t h e o r i g i n a l k c X . i , c • X . ( 4 - 3 4 ) i = 1,2, • • • • n j = 1,2, k c The r e a s o n f o r t h i s s t e p i s t o t a k e t h e f u n c t i o n v a l u e s i n t o a c c o u n t and t h u s t o s p e e d c o n v e r g e n c e t o an optimum p o i n t . The a u t h o r s f o u n d t h a t i t was a d v a n t a g e o u s 206 t o u s e t h e i r m o d i f i e d method f o r t h e f i r s t p a r t o f t h e s e a r c h and t o u s e t h e o r i g i n a l C omplex M e t h o d i n t h e s e c o n d p a r t o f t h e s e a r c h . A f u r t h e r m o d i f i c a t i o n was made w h e r e b y a new t r i a l p o i n t was n o t moved h a l f w a y t o w a r d t h e c e n t r o i d o f t h e r e -m a i n i n g p o i n t s when i t v i o l a t e d a c o n s t r a i n t b u t i n s t e a d i t was moved some f i x e d p r o p o r t i o n o f t h e d i s t a n c e , as d e t e r -m i n e d by a s p e c i f i e d p a r a m e t e r o r by a g o l d e n s e c t i o n s e a r c h . Umeda and I c h i k a w a s t u d i e d s e v e r a l p r o b l e m s t o t e s t t h e e f f e c t i v e n e s s o f t h e m o d i f i c a t i o n . I t was f o u n d t h a t an e x p a n s i o n f a c t o r a g = 1.62 and an e x p o n e n t v a l u e f o r t h e f u n c t i o n o f Z = 1.0 g a v e t h e b e s t r e s u l t s . The c o n v e r g e n c e b e h a v i o r o f t h e a b s o r b e r - s t r i p p e r s y s t e m men-t i o n e d p r e v i o u s l y was s u p e r i o r t o t h a t o b t a i n e d by t h e o r i g i n a l C o m p l e x M e t h o d . The u s e o f t h e g o l d e n s e c t i o n s e a r c h f o r h a n d l i n g i m p l i c i t i n e q u a l i t y c o n s t r a i n t s showed no a d -v a n t a g e o v e r t h e p r e v i o u s m ethod o f s t e p p i n g b a c k h a l f w a y t o w a r d t h e c e n t r o i d . 2. M o d i f i c a t i o n s o f t h e Complex M e t h o d A number o f m o d i f i c a t i o n s were made t o t h e o r i g i n a l C o mplex M e t h o d f o r t h i s s t u d y and a r e l i s t e d i n T a b l e 4-8. B e c a u s e o f t h e l a r g e amount o f C P . U . t i m e ( 2 . 5 -25 s e c o n d s ) t o e v a l u a t e t h e o b j e c t i v e f u n c t i o n ( c o m p l e t e 207 T A B L E 4-8  M o d i f i c a t i o n s o f t h e Complex M e t h o d I n i t i a l c o m p l e x i s n o t g e n e r a t e d b u t s u p p l i e d by u s e r . I n i t i a l f u n c t i o n v a l u e s a r e n o t c a l c u l a t e d b u t s u p p l i e d by u s e r . The f u n c t i o n v a l u e o f a new p o i n t i s c a l c u l a t e d b e f o r e t h e i m p l i c i t c o n s t r a i n t s a r e c h e c k e d . When i m p l i c i t c o n s t r a i n t s a r e v i o l a t e d , a f i x e d number o f s t e p s a r e made b a c k t o w a r d t h e c e n t r o i d . When t h e f i x e d l i m i t i s e x c e e d e d , a new p o i n t i s g e n e r a t e d by r e f l e c t i n g t h e c e n t r o i d t h r o u g h t h e b e s t p o i n t . When a new p o i n t i s i n f e r i o r t o t h e w o r s t p o i n t . . . . same as 4. The s t o p p i n g c r i t e r i a i s t h e c h a n g e i n t h e c o o r d i n a t e s o f t h e c e n t r o i d o f a l l p o i n t s r a t h e r t h a n t h e c h a n g e i n f u n c t i o n v a l u e o f t h e c e n t r o i d . 208 m a t e r i a l and e n e r g y b a l a n c e o f s i m u l a t e d m o d e l ) i t i s n e c e s -s a r y t o r e d u c e t h e number o f f u n c t i o n e v a l u a t i o n s t o a m i n i -mum. T h i s i s t h e r e a s o n why m o d i f i c a t i o n s 1, 2, and 6 were made. S i n c e t h e f e a s i b l e r e g i o n o f o p e r a t i o n i s known i n an o p e r a t i n g p l a n t and a l s o b e c a u s e a p l a n t t h a t has been i n p r o d u c t i o n f o r a r e a s o n a b l e amount o f t i m e u s u a l l y o p e r -a t e s f a i r l y c l o s e t o an o p t i m u m , one c a n c h o o s e t h e p o i n t s f o r an i n i t i a l c o m p l e x w i t h a g o o d d e g r e e o f c e r t a i n t y . T h i s s a v e s a l a r g e amount o f c o m p u t e r t i m e s i n c e i n t h e o r i g i n a l C o mplex M e t h o d p o i n t s a r e g e n e r a t e d t h r o u g h o u t t h e f e a s i b l e r e g i o n r a t h e r t h a n i n an a r e a c l o s e t o t h e optimum.. I t p r o b -a b l y w o u l d t a k e q u i t e a w h i l e f o r a r a n d o m l y c h o s e n c o m p l e x t o r e a c h t h e a r e a w h e r e a u s e r w o u l d s i t u a t e h i s s t a r t i n g c o m p l e x . S i n c e t h e u s e r has p r o b a b l y made some p r e l i m i n a r y manual e x a m i n a t i o n s o f t h e r e s p o n s e s u r f a c e b e f o r e a p p l y i n g t h e o p t i m i z a t i o n m e t h o d , he c a n u s u a l l y s u p p l y b o t h t h e i n i t i a l p o i n t s o f t h e c o m p l e x and t h e f u n c t i o n v a l u e s o f t h e s e p o i n t s . In t h e o r i g i n a l m ethod t h e s t o p p i n g c r i t e r i a i s a p p l i e d e a c h t i m e a new p o i n t i s g e n e r a t e d . T h i s r e q u i r e s t h e c a l c u l a t i o n o f t h e f u n c t i o n v a l u e o f t h e c e n t r o i d w h i c h g r e a t l y i n c r e a s e s t h e c a l c u l a t i o n t i m e . (Two o b j e c t i v e 209 f u n c t i o n c a l c u l a t i o n s were r e q u i r e d f o r e a c h new p o i n t ) . I f t h e c o o r d i n a t e s , r a t h e r t h a n t h e f u n c t i o n v a l u e , o f t h e c e n t r o i d a r e u s e d as a s t o p p i n g c r i t e r i a t h e n o n l y one f u n c -t i o n e v a l u a t i o n i s r e q u i r e d f o r e a c h new p o i n t and many f u n c t i o n c a l c u l a t i o n s c a n be e l i m i n a t e d . The r e a s o n f o r m o d i f i c a t i o n 3 i s c l e a r ; t h e im-p l i c i t c o n s t r a i n t s must be c a l c u l a t e d by t h e " b l a c k b o x " s i m u l a t i on m o d e l . In t h e o r i g i n a l C omplex M e t h o d , when a new p o i n t v i o l a t e s an i m p l i c i t c o n s t r a i n t o r i s i n f e r i o r t o t h e w o r s t p o i n t , t h e new p o i n t i s moved b a c k w a r d h a l f w a y t o t h e c e n -t r o i d and t h i s i s r e p e a t e d u n t i l t h e c o n s t r a i n t i s s a t i s f i e d , o r t h e p o i n t i s s u p e r i o r t o t h e w o r s t p o i n t , o r t h e l i m i t o f r e d u c t i o n s t e p s has been r e a c h e d . When t h e l a t t e r o c c u r s t h e c e n t r o i d i s t h e n c h o s e n as t h e new p o i n t and t h e me t h o d c o n t i n u e s o n c e m o r e. S i n c e t h e s u r f a c e i s assumed t o be c o n v e x , t h e c e n t r o i d must a l w a y s s a t i s f y t h e c o n s t r a i n t s and be s u p e r i o r t o t h e w o r s t p o i n t o f t h e c o m p l e x . H o w e v e r , due t o t h e s y s t e m c o n v e r g e n c e e r r o r ( s e e C h a p t e r 3-B) c a l -c u l a t i o n d i f f i c u l t i e s may a r i s e when t h e c o m p l e x p o i n t s a r e v e r y c l o s e t o g e t h e r o r when t h e i r f u n c t i o n v a l u e s a r e v e r y s i m i l a r ( r e s p o n s e s u r f a c e v e r y f l a t ) . In t h i s c a s e t h e f u n c t i o n v a l u e c a l c u l a t e d f o r t h e c e n t r o i d may be i n f e r i o r t o t h a t c a l c u l a t e d f o r t h e w o r s t p o i n t o f t h e c o m p l e x . T h i s 210 w i l l c a u s e t h e meth o d t o s t o p s i n c e a f u r t h e r r e f l e c t i o n o f t h e w o r s t p o i n t t h r o u g h t h e c e n t r o i d c a n n o t s u c c e e d , s i n c e t h e two p o i n t s a r e t h e same. F o r t h e a b o v e m e n t i o n e d r e a s o n s , m o d i f i c a t i o n 4 was made w h i c h g i v e s t h e c o m p l e x a n o t h e r c h a n c e t o p r o c e e d t o w a r d s t h e opti m u m . When t h e number o f r e d u c t i o n s t e p s i s e x c e e d e d t h e n a new t y p e o f r e f l e c t i o n s t e p i s t a k e n by r e f l e c t i n g t h e p r e v i o u s c e n t r o i d t h r o u g h t h e b e s t p o i n t i n t h e c o m p l e x . The c o m p u t e r f l o w d i a g r a m s f o r t h e m o d i f i e d C omplex M e t h o d and p r o g r a m l i n k a g e a r e shown i n F i g u r e s 4-9 and 4-10. The c o m p u t e r l i s t i n g c a n be f o u n d i n A p p e n d i x H. 3. R e s u l t s w i t h I n t e r m e d i a t e P l a n t Model T a b l e s 4-9, 4-10, and 4-11 l i s t t h e r e s u l t s o f s e v e r a l t e s t s made w i t h t h e i n t e r m e d i a t e p l a n t m o d e l . In e a c h c a s e t h e r u n s were made w i t h , complex e x p a n s i o n f a c t o r a e = l « 4 no. o f p o i n t s i n complex k c = 4 The t y p e s o f s t e p s l i s t e d and t h e r e s u l t s o f t h e s t e p s a r e a b b r e v i a t e d as f o l l o w s : FIND WORST POINT IN COMPLEX o MOVE 1/2 WAY BACK TO CENTROID X: I 1/2 ( C j + Xj) CALCULATE CENTROID OF REMAINING POINTS REFLECT WORST POINT THROUGH CENTROID FIND BEST POINT IN COMPLEX f REFLECT CENTROID THROUGH BEST POINT xi = X B + Qe V EVALUATE OBJECTIVE FUNCTION IMPLICIT CONSTRAINTS SATISFIED YES NEW POINT BETTER^ JHAN WORST> YES CALCULATE CENTROID OF ALL POINTS -H RETURN F I G U R E 4-9 CO M P U T E R F L O W D I A G R A M OF T H E M O D I F I E D C O M P L E X METHOD MAIN PROGRAM CALL1 FUNCTION! FTN CALL DREAD CALL SUBSET CALL COMPID RETURN CALL INIT CALL DCHECK 1 r •READ MOCOMP INPUT DATA I FUNCTION ] UPPER BOUNDS OF INDEPENDENT VARIABLES AND CONSTRAINTS ( CALL \ I MOCOMP J 212 IMPLICIT VARIABLES AND CONSTRAINTS DEFINED LOWER BOUNDS OF INDEPENDENT VARIABLES AND CONSTRAINTS F I G U R E 4 - 1 0 MOCOMP PROGRAM L I N K A G E 213 s t e p t y p e a b b r e v i a t i on o r i g i n a l p o i n t e x p a n s i o n s t e p c o n t r a c t i o n s t e p new e x p a n s i o n s t e p OP ES CS NS s t e p r e s u l t s a b b r e v i a t i o n o r i g i n a l p o i n t new p o i n t c o n s t r a i n t v i o l a t i o n w o r s t p o i n t OP NP CV WP The o r i g i n a l c o m p l e x shown i n T a b l e 4-9 i s f a i r l y f a r f r o m t h e o ptimum. The s e a r c h p r o g r e s s e s w e l l t o t h e s t a g e w h e r e t w e n t y new p o i n t s h a v e been f o u n d ( t h e maximum p e r m i t t e d i n t h i s s p e c i f i c r u n ) . H o w e v e r , s i n c e good p r o -g r e s s was b e i n g made, t h e s e a r c h was r e s t a r t e d w i t h t h e l a s t c o m p l e x and c o n t i n u e d f o r a n o t h e r t w e n t y new p o i n t s . The s e a r c h was s t i l l p r o g r e s s i n g t o w a r d t h e optimum when t e r m i -n a t e d a f t e r a t o t a l c o m p u t e r t i m e o f 912 C.P.U. s e c . o r i g i n a l c o m p l e x s p e c i f i e d i s v e r y c l o s e t o t h e e x p e c t e d o p t i m u m . The e f f e c t o f t h e "new" e x p a n s i o n s t e p i s shown h e r e s i n c e i n s t e p s ( 1 1 , 2 9 , 41) t h e s e a r c h i s a b l e t o c o n -t i n u e by p r o j e c t i n g t h e c e n t r o i d t h r o u g h t h e b e s t p o i n t . Once a g a i n , e v e n t h o u g h t h e s e a r c h was t e r m i n a t e d a f t e r 20 new p o i n t s had b e e n f o u n d , i t was s t i l l m o v i n g t o w a r d t h e o p t i m u m . H o w e v e r , i t was f e l t t h a t f u r t h e r p r o g r e s s c o u l d n o t be a c h i e v e d s i n c e t h e b e s t p o i n t l a y on t h r e e c o n s t r a i n t s A s e c o n d t e s t i s shown i n T a b l e 4-10 where t h e 214 TABLE 4-9 The Use o f t h e Mod i f i ed Complex Method 1 f o r t h e O p t i m i z a t i o n o f t h e I n t e r m e d i a t e P l a n t Model - 1 No. S t e p T y p e X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 1 OP .700 .200 .100 .100 35.03236 OP 2 OP .650 .150 .350 .250 38.24535 OP 3 OP .670 .100 .300 .200 36.64583 OP 4 OP .680 .250 .150 .200 35.29765 OP 5 ES .620 .120 .500 .380 38.49809 NP 6 ES .600 .000 .710 .384 39.22656 CV 7 CS .623 .062 .547 .330 38.79761 NP 8 ES .576 .125 .697 .488 38.60588 CV 9 CS .604 .118 .581 .404 38.71819 CV. 10 CS .617 .114 .524 .362 38.67307 NP 11 ES .579 .027 .766 .508 39.01697 CV 12 CS .599 .063 .645 .432 38.94684 CV 13 CS .609 .081 .584 .395 38.90208 CV 14 CS .615 .089 .553 .376 38.90340 CV 15 CS .618 .094 .539 .367 38.78432 NP 16 ES .618 .048 .587 .315 39 .05006 NP 17 ES .623 .003 .604 .303 39.18283 NP 18 ES .627 .000 .637 .246 38.89560 NP 19 ES .623 .000 .697 .228 38.73621 WP 20 CS .623 .008 .653 .258 38.90022 NP 21 ES .613 .047 . 585 .358 39.15172 NP 22 ES .613 .067 .506 .419 39.27415 NP 23 ES .613 .027 .535 .423 39.48074 NP 24 ES .619 .011 .498 .415 39.61548 NP 25 ES .603 .080 .384 .581 39.60030 CV 26 CS .609 .058 .449 .500 39.57466 CV 27 CS .612 .046 .481 .460 39.53099 NP 28 ES .619 .000 .503 .451 39.71317 NP 29 ES .622 .008 .436 .468 39.73019 NP 30 ES .631 .000 .476 .424 39.17491 WP 31 CS .626 .003 .477 .435 39 .35390 WP 32 CS .623 .005 .478 .439 39.47849 WP 33 CS .621 .005 .479 .442 39.51744 WP 34 CS .620 .006 .479 .444 39.56354 NP 35 ES .619 .007 .479 .446 39.64046 NP 36 ES .620 .000 .437 .512 39.77541 NP 37 ES .622 .000 .430 .419 39.74382 NP 215 T a b l e 4-9 ( C o n t i n u e d ) No. S t e p T y p e X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t ] 3 8 ES .626 .006 .339 .569 39.91081 NP 39 ES .623 .000 .354 .623 39.87735 NP 40 ES .623 .004 .302 .635 40.03844 CV 41 CS .623 .003 .339 .601 39.91145 NP 42 ES .629 .007 .215 .719 40.10217 CV 43 CS .626 .005 .279 .648 39.92477 CV 44 CS .625 .004 .312 .628 39.88878 NP 45 ES .626 .010 .296 .565 40.01633 NP 46 ES .625 .010 .343 .509 39.91858 NP 47 ES .624 .011 .308 .545 40.05746 NP 48 ES .628 .021 .282 .453 4 0 . 1 8 2 6 8 NP 49 ES .628 .020 .229 .538 40.24280 CV 50 CS .627 .017 .263 .529 40.12152 CV 51 CS .627 .016 .279 .525 40.08540 NP 52 ES .626 .023 .282 .428 40.23395 NP 53 ES .631 .033 .243 .362 40.33517 CV 54 CS .629 .026 .262 .41 5 40.19778 NP 55 ES .629 .035 .271 .301 40.34731 CV 56 CS .629 .029 .273 .367 4 0 . 2 5 1 4 8 CV 57 CS .628 .026 .275 .399 40.22023 NP 58 ES .627 .031 .259 .359 40.37178 CV 59 CS .628 .029 .266 .387 40.29794 CV 60 CS .628 .027 .269 .400 4 0 . 2 6 4 2 8 CV 61 CS .628 .026 .271 .407 40.24432 NP 62 ES .625 .024 .295 .406 40.22134 NP 63 ES .624 .022 .294 .433 40.24942 NP 64 ES .627 .024 .265 .446 40.28151 CV 65 CS .627 .024 .274 .435 40.24109 NP 66 ES .626 .025 .276 .422 40.27365 NP 67 ES .625 .025 .290 .401 40.27940 NP 68 ES .621 .020 .309 .435 40.30569 NP 69 ES .624 .025 .288 .399 40.38115 CV 70 CS .623 .024 .290 .409 4 0 . 3 3 1 8 8 NP 271 ES .619 .021 .324 .406 40.34270 NP Run was t e r m i n a t e d a f t e r t w e n t y new p o i n t s w ere f o u n d (536 C P . U . s e c . ) new r u n c o n t i n u e d w i t h l a s t c o m p l e x . 2 S e c o n d r u n s t o p p e d a f t e r 20 new p o i n t s f o u n d - 376 C P . U . s e c . 216 TABLE 4-10 The Use o f t h e Modi f i ed Complex Method 1 f o r t h e O p t i m i z a t i o n o f t h e I n t e r m e d i a t e P l a n t Model - 2 No. S t e p T y p e X (D X(2) X(3) X(4) f ( x ) R e s u l t 1 OP .622 .035 .298 .407 40.37755 OP 2 OP .617 .051 .299 .403 40.30988 OP 3 OP .613 .055 .298 .404 40.31703 OP 4 OP .620 .042 .304 .409 40.33990 OP 5 ES .620 .034 .301 .412 40.40701 NP 6 ES) .631 .012 .306 .417 40.03702 WP) 7 est .626 .024 .303 .413 40.13564 WP/ 8 CS .623 .030 .302 .411 40.19003 WP) 9 cs .622 .034 .302 .410 40.21880 WP 10 cs1 .621 .035 .301 .410 40.25356 WP' 11 NS .619 .030 .302 .415 40.38113 NP 12 ES .621 .020 .295 .415 40.45206 CV 13 CS .621 .027 .298 .413 40.38226 NP 14 ES .618 .024 .304 .422 40.45679 NP 15 ES .619 .026 .299 .416 40.46315 NP 16 ES .617 .029 .306 .422 40.40823 NP 17 ES .615 .015 .306 .432 40.52429 CV 18 CS .617 .021 .305 .426 40.48532 NP 19 ES .619 .015 .297 .420 40.54552 CV 20 CS .619 .019 .299 .421 40.49426 CV 21 CS .618 .021 .301 .421 40.47104 NP 22 ES .619 .021 .299 .420 40.49149 CV 23 CS .619 .022 .300 .421 40.47214 NP 24 ES .615 .016 .306 .432 40.50876 CV 25 CS .617 .019 .304 .427 40.50385 CV 26 CS .617 .020 .303 .425 40.49681 CV 27 CS .618 .021 .303 .424 40.49078 CV 28 CS .618 .021 .302 .423 40.48680 CV 29 NS .615 .020 .308 .431 40.48016 NP 30 ES .614 .021 .309 .432 40.48196 NP 31 ES .610 .018 .317 .443 40.49014 NP 32 ES .612 .019 .31 3 .437 40.51134 NP 33 ES .611 .019 .315 .440 40.50137 NP 34 ES .604 .016 .329 .459 40.49992 NP 35 ES .608 .018 .322 .450 40.52655 NP 36 ES .619 .022 .299 .418 40.50533 CV 217 T a b l e 4-10 ( C o n t i n u e d ) No. S t e p T y p e X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 37 CS .615 .020 .308 .430 40.47800 WP 38 CS .613 .019 .312 .436 40.47321 WP 39 CS .611 .019 .315 .439 40.48579 WP 40 CS .611 .019 .316 .441 40.49509 WP 41 NS .604 .017 .330 .460 40.51447 NP 42 ES .603 .017 .331 .461 40.51572 NP 43 ES .594 .014 .348 .484 40.54710 CV 44 CS .599 .015 .338 .471 40.56227 CV 45 CS .602 .016 .333 .464 40.56471 NP '46 ES .605 .018 .327 .456 40.55489 NP Run t e r m i n a t e d a f t e r 20 C.P.U. s e c . new p o i n t s c a l c u l a t e d - 441 218 TABLE 4-11 The Use o f t h e M o d i f i e d Complex Method f o r  t h e O p t i m i z a t i o n o f t h e I n t e r m e d i a t e P l a n t Model - 3 No. S t e p T y p e x(D X ( 2 ) X ( 3 ) X ( 4 ) f ( x ) R e s u l t 1 OP .622 .035 .298 .407 40.38274 OP 2 OP .617 .051 .299 .403 40.3052 OP 3 OP .613 .055 .298 .404 40.36169 OP 4 OP .620 .042 .304 .409 40.37192 OP 5 ES .620 .034 .301 .412 40.40701 NP 6 ES .631 .012 .306 .417 40.03702 WP 7 CS .626 .024 .303 .413 40.13564 WP 8 CS .623 .030 .302 .411 40.19003 WP 9 CS .622 .034 .302 .410 40.21880 WP 10 CS .621 .035 .301 .409 40.25356 WP 11 NS .619 .030 .302 .41 5 4 0 . 3 8 1 1 3 NP 12 ES .621 .020 .295 .414 40.45206 CV 13 CS .621 .207 .298 .413 40.38226 NP 14 ES .623 .035 .295 .403 40.28227 WP 15 CS .622 .033 .297 .407 40.28284 WP 16 CS .622 .033 .298 .409 40.30186 WP 17 CS .621 .032 .299 .410 40.31726 WP 18 CS .621 .032 .299 .410 40.32222 WP 19 NS .619 .037 .305 .413 40.31799 WP 20 CS .620 .035 .302 .412 40.35272 WP 21 CS .621 .033 . 300 .411 4 0 . 3 5 9 3 8 WP 22 c s .621 .033 .300 .411 40.35289 WP 23 c s .621 0.32 .299 .411 40.34686 WP S e a r c h s t o p p e d a f t e r new s t e p and f o u r c o n t r a c t i o n s t e p s f a i l e d t o g i v e b e t t e r p o i n t . 219 ( t h e f i r s t c o n s t r a i n t i s n o t a c t i v e ) . T h e r e f o r e , w i t h t h e s p e c i f i c s e t o f e q u i p m e n t p a r a m e t e r s and p r o c e s s s t r e a m s de-f i n e d i n t h e i n t e r m e d i a t e p o l y p l a n t m o d e l , t h e optimum seems t o b e : X ( l ) = EQPAR(2,24) = 0.602 X ( 2 ) = " ( 2 , 2 6 ) = 0.016 X ( 3 ) = " ( 3 , 2 6 ) = 0.333 X ( 4 ) = " ( 4 , 2 6 ) = 0.464 The c o n s t r a i n e d t e m p e r a t u r e s w h i c h must be l e s s t h a n 9 4 5 ° R a r e : 891 °R 944 944 945 And t h e o b j e c t i v e f u n c t i o n v a l u e , t h e number o f m o l e s o f d i m e r i z e d p r o d u c t , f ( x ) = 40.56471 H o w e v e r , t h e m o d i f i e d C omplex M e t h o d i s n o t f o o l -p r o o f and t h i s i s shown i n T a b l e 4-11 where t h e o r i g i n a l c o m p l e x s p e c i f i e d i s t h e same as t h a t f o r T a b l e 4-10. How-e v e r t h e s p e c i f i e d f u n c t i o n v a l u e s a r e s l i g h t l y d i f f e r e n t ; t e m p e r a t u r e s t r e a m : 6 8 " 1 0 " 1 2 220 t h e maximum d i f f e r e n c e b e i n g a b o u t 0.1%, w h i c h d o e s n o t a p p e a r t o be v e r y s i g n i f i c a n t . One n o t e s on f o l l o w i n g t h e c a l c u l a t i o n s t e p s t h a t t h e two c a s e s e x h i b i t s i m i l a r b e h a v i o r up t o p o i n t 14. A d i f f e r e n t s t e p i s t h e n t a k e n i n T a b l e 4-11 b e c a u s e p o i n t 13 i s t h e w o r s t i n t h e c o m p l e x , w h i l e i n T a b l e 4-10, p o i n t 1 i s t h e w o r s t . T h i s c a u s e s t h e method t o g e t s t u c k i n T a b l e 4-11. T h u s , a d i f f e r e n c e i n o b j e c -t i v e f u n c t i o n v a l u e o f o n l y 0.0052 o r 0.01% ( b e t w e e n p o i n t s 1 i n T a b l e 4-10 and 4-11) c a n c a u s e t h e method t o s u c c e e d i n one c a s e and f a i l i n a n o t h e r . T h e r e f o r e , i f one f i n d s t h a t t h e s e a r c h s t o p s a t a c e r t a i n p o i n t , t h e f u n c t i o n v a l u e s s h o u l d be c h a n g e d v e r y s l i g h t l y and t h e s e a r c h r e s u m e d t o c h e c k w h e t h e r t h e o p t i m i -z a t i o n m e t h o d m i g h t p r o c e e d f u r t h e r . E. C o m p a r i s o n o f O p t i m i z a t i o n M e t h o d s T a b l e 4-12 l i s t s t h e r e s u l t s o b t a i n e d w i t h e a c h o f t h e t h r e e o p t i m i z a t i o n m e t h o d s t e s t e d on t h e i n t e r m e d i a t e p i a n t m o d e l . In t h e f i r s t c a s e , where t h e s t a r t i n g p o i n t ( o r c o m p l e x ) i s q u i t e d i s t a n t f r o m t h e optimum t h e t h r e e m e t h o d s show i m p r o v e m e n t i n t h e v a l u e o f t h e o b j e c t i v e f u n c t i o n b u t t h e C o n s t r a i n e d P a t t e r n S e a r c h M e t h o d (C0NPAT) i s d e f i n i t e l y i n f e r i o r t o t h e o t h e r two. The D e f l e c t e d G r a d i e n t - C r e a t e d TABLE 4 - 1 2 C o m p a r i s o n o f O p t i m i z a t i o n M e t h o d s T e s t e d M e t h o d I n i t i a l P o i n t I n i t i a l F u n c t i o n V a l u e F i n a l P o i n t F i n a l F u n c t i o n V a l u e C P . U . T i m e , X ( l ) X ( 2 ) X ( 3 ) X ( 4 ) x(D X ( 2 ) X ( 3 ) X(4) s e c . DGCR 1 . 7 0 0 . 1 0 0 . 3 0 0 . 4 0 0 3 3 . 0 9 8 . 5 9 9 . 0 2 9 . 3 1 6 . 5 0 5 4 0 . 4 9 3 1 0 9 0 CONPAT 2 n II n H . 6 4 0 . 0 3 5 . 2 5 0 . 3 5 0 3 9 . 8 9 9 8 1 2 MOCOMP 3 . 7 0 0 . 6 5 0 . 6 7 0 . 6 8 0 . 2 0 0 . 1 5 0 . 1 0 0 . 1 2 0 . 1 0 0 . 3 5 0 . 3 0 0 .1 5 0 . 1 0 0 . 2 5 0 . 2 0 0 . 2 0 0 3 5 . 0 3 2 3 8 . 2 4 5 3 6 . 6 4 6 3 5 . 2 9 8 . 6 1 9 . 0 2 1 . 3 2 4 . 4 0 6 4 0 . 3 4 3 9 1 2 DGCR . 6 7 5 . 1 7 5 . 2 2 5 . 1 8 7 3 6 . 4 7 0 . 6 3 2 . 1 7 8 . 2 2 9 . 1 9 3 3 9 . 3 5 0 4 9 4 CONPAT n II it . 6 5 4 . 1 4 5 . 2 0 0 . 1 6 7 3 8 . 7 6 2 7 6 6 DGCR . 6 1 8 . 0 4 6 . 3 0 0 . 4 0 0 4 0 . 3 6 7 . 6 1 1 . 0 4 7 . 3 0 2 . 4 0 3 4 0 . 4 1 0 3 0 7 CONPAT n H n n n . 6 1 1 . 0 4 6 . 3 0 5 . 4 0 1 4 0 . 4 0 7 4 4 7 MOCOMP . 6 2 2 . 6 1 7 . 6 1 3 . 6 2 0 . 0 3 5 . 0 5 1 . 0 5 5 . 0 4 2 . 2 9 8 . 2 9 9 . 2 9 8 . 3 0 4 . 4 0 7 . 4 0 3 . 4 0 4 . 4 1 2 4 0 . 3 7 6 4 0 . 3 1 0 4 0 . 3 1 7 4 0 . 3 4 0 . 6 0 2 . 0 1 7 . 3 3 3 . 4 6 4 4 0 . 5 6 5 4 4 1 D e f l e c t e d G r a d i e n t - C r e a t e d R e s p o n s e S u r f a c e T e c h n i q u e C o n s t r a i n e d P a t t e r n S e a r c h M o d i f i e d C o m p l e x M e t h o d ro 222 R e s p o n s e S u r f a c e T e c h n i q u e (DGCR) a c h i e v e s good r e s u l t s b u t as e x p l a i n e d e a r l i e r , t h i s i s q u i t e f o r t u i t o u s as o n l y a p r i o r k n o w l e d g e o f t h e r e s p o n s e s u r f a c e , a l l o w s one t o c h o o s e v a l u e s f o r t h e s e a r c h p a r a m e t e r s and p e n a l t y f a c t o r s so t h a t t h e s e a r c h c a n p r o c e e d . The M o d i f i e d C omplex M e t h o d (MOCOMP) g a v e good r e s u l t s and was s t i l l p r o g r e s s i n g t o w a r d t h e op-timum when t h e s e a r c h was t e r m i n a t e d . The C P . U . t i m e r e -q u i r e d i s o f t h e same o r d e r o f m a g n i t u d e f o r e a c h o f t h e t h r e e m e t h o d s . ( C P . U . t i m e l o w e s t f o r CONPAT b e c a u s e s e a r c h f a i l s e a r l i e r t h a n o t h e r s . ) In t h e s e c o n d c a s e , t h e s t a r t i n g p o i n t i s s t i l l f a i r l y f a r f r o m t h e opt i m u m . The DGCR t e c h n i q u e i s shown t o be much s u p e r i o r t o CONPAT. However t h e DGCR t e c h n i q u e s t o p s q u i t e a d i s t a n c e away f r o m t h e opt i m u m . The i n i t i a l s t a r t i n g p o i n t ( o r c o m p l e x ) i n t h e t h i r d c a s e i s v e r y c l o s e t o t h e optimum and l i e s on one c o n s t r a i n t . B o t h t h e DGCR t e c h n i q u e and CONPAT do n o t p r o g r e s s v e r y f a r b e f o r e t h e y s t o p and o n l y MOCOMP seems t o work e f f e c t i v e l y n e a r t h e c o n s t r a i n t s . In f a c t , t h e optimum p o i n t c a l c u l a t e d by MOCOMP l i e s on t h e i n t e r s e c t i o n o f t h e t h r e e a c t i v e c o n -s t r a i n t s . The r e s u l t s d e m o n s t r a t e t h a t t h e C o n s t r a i n e d P a t t e r n S e a r c h i s n o t e f f e c t i v e n e a r t h e c o n s t r a i n t s ( w h ere t h e opt i m u m may be) and t h u s i s i n f e r i o r t o t h e o t h e r method t e s t e d . 223 The DGCR t e c h n i q u e c a n g i v e good r e s u l t s b u t o n l y i f t h e s e a r c h p a r a m e t e r s and p e n a l t y f a c t o r s a r e c h o s e n c o r r e c t l y . W i t h o u t a p r i o r k n o w l e d g e o f t h e r e s p o n s e s u r f a c e ( a n d e v e n w i t h t h i s k n o w l e d g e ) a g r e a t d e a l o f t e s t i n g i s r e q u i r e d . The m o d i f i e d C o m p l e x M e t h o d g a v e t h e b e s t r e s u l t s and w o r k e d q u i t e w e l l i n t h e r e g i o n n e a r t h e c o n s t r a i n t s . I t i s q u i t e s i m p l e t o u s e and t h e C P . U . t i m e u t i l i z e d i s n o t e x c e s s i v e . Due t o t h e s y s t e m c o n v e r g e n c e e r r o r , t h e o b j e c t i v e f u n c t i o n v a l u e s f o r t h e c o m p l e x v e r t i c e s s h o u l d be c h a n g e d s l i g h t l y t o c h e c k i f t h e s e a r c h c o u l d p r o c e e d f u r t h e r . F. O p t i m i z a t i o n o f F i n a l P l a n t Model The M o d i f i e d Complex M e t h o d was u s e d f o r t h e o p t i -m i z a t i o n o f t h e f i n a l p o l y m e r i z a t i o n p l a n t model ( s e e C h a p t e r 3-B f o r model d e s c r i p t i o n ) . The i n d e p e n d e n t v a r i - . a b l e s i n t h i s model and t h e i r l o w e r and u p p e r b o u n d s ( c o n -s t r a i n t s ) a r e : DVDR S p l i t s : 0.0 < EQPAR ( 2 , 2 4 ) < 1 .0 0.0 < EQPAR ( 2 , 2 6 ) < 1 .0 0.0 < EQPAR ( 3 , 2 6 ) < 1 .0 0.0 < EQPAR ( 4 , 2 6 ) < 1 .0 Column V a r i a b l e s : 0.95 < EQPAR ( 6 , 2 2 ) < 0.995 224 0.001 < EQPAR ( 7 , 2 2 ) < 0.01 0.98 < EQPAR (6,31 ) < 0.995 0.02 < EQPAR (7,31 ) < 0.08 The i m p l i c i t ( d e p e n d e n t v a r i a b l e s t h a t a r e c o n -s t r a i ned) a r e : DVDR S p l i t s : EQPAR ( 3 , 2 4 ) = 1 .0-EQPAR ( 2 , 2 4 ) EQPAR ( 5 , 2 6 ) = 1.0-EQPAR ( 2 , 2 6 ) - EQPAR ( 3 , 2 6 ) - E Q P A R ( 4 , 2 6 ) 0.0 < EQPAR ( 3 , 2 4 ) < 1 .0 0.0 < EQPAR ( 5 , 2 6 ) < 1 .0 Bed E x i t T e m p e r a t u r e s : 760.0 < SINTSV ( 4 , 6 ) < 945.0 760.0 < SINTSV ( 4 , 8 ) < 945.0 760.0 < SINTSV ( 4 , 1 0 ) < 945.0 760.0 < SINTSV ( 4 , 1 2 ) < 945.0 225 The o p t i m i z a t i o n p r o b l e m i s t h u s c o mposed o f 8 i n d e p e n d e n t v a r i a b l e s and 14 s e t s o f c o n s t r a i n t s o r b ounds (8 f o r t h e i n d e p e n d e n t v a r i a b l e s and 6 f o r t h e i m p l i c i t v a r i a b l e s ) . The m a i n c a l l i n g p r o g r a m , C A L L F , i s shown i n A p p e n d i x H. I t was f o u n d t h a t when t h e m o l e s o f d i m e r p r o d u c e d was u s e d as an o b j e c t i v e f u n c t i o n ( a s i n t h e i n t e r m e d i a t e p l a n t model o p t i m i z a t i o n ) t h e o p t i m i z a t i o n m ethod d i d n o t p r o g r e s s as r a p i d l y as i n t h e i n t e r m e d i a t e model o p t i m i z a -t i o n . T h i s was due t o t h e f a c t t h a t t h e f o u r new d i s t i l l a -t i o n c o l u m n v a r i a b l e s do n o t a f f e c t t h e p r o d u c t i o n o f d i m e r and t h u s t e n d t o s l o w up and i n t e r f e r e w i t h t h e s e a r c h . A c o m p a r i s o n o f t h e two r e s u l t s i s shown i n T a b l e 4-13. I t s h o u l d be n o t e d t h a t t h e m e t h o d d i d n o t r e a c h t h e optimum w i t h t h e f i n a l p l a n t m o d e l . When t h e t o t a l p r o d u c t i o n o f p o l y m e r g a s o l i n e i s u s e d as t h e o b j e c t i v e f u n c t i o n t h e r e s u l t s o f t h e s e a r c h a r e t h a t t h e s p l i t on t h e d e b u t a n i z e r i s p u s h e d t o t h e u p p e r l i m i t as i s t h e f r a c t i o n o f t o t a l i n t h e d e b u t a n i z e r t o p p r o d u c t . T h i s g i v e s a maximum o f d e b u t a n i z e r b o t t o m p r o d u c t , p o l y m e r g a s o l i n e . Once a g a i n , t h i s o b j e c t i v e f u n c -t i o n d o e s n o t g i v e a t r u e v i e w o f t h e p l a n t o p e r a t i n g p h i l -o s o p h y s i n c e t h e d e b u t a n i z e r v a r i a b l e s d o m i n a t e t h e s e a r c h and t h e r e a c t o r c o n v e r s i o n i s o n l y o f a s e c o n d a r y i m p o r t a n c e . 226 T A B L E 4-13  C o m p a r i s o n o f R e s u l t s w i t h I n t e r m e d i a t e and F i n a l P l a n t M o d e l s I n t e r m e d i a t e M o d e l F i n a l Model DVDR S p l i t s : EQPAR ( 2 , 2 4 ) .602 .589 ( 2 , 2 6 ) .017 .015 ( 3 , 2 6 ) .334 .557 ( 4 , 2 6 ) .465 .210 Column V a r i a b l e s : EQPAR ( 6 , 2 2 ) — .993 ( 7 , 2 2 ) — .002 ( 6 , 3 1 ) — .993 ( 7 , 3 1 ) — .080 Bed E x i t T e m p e r a t u r e : SINTSV ( 4 , 6) 891.02 899.2 ( 4 , 8) 944.10 908.3 ( 4 , 1 0 ) 943.49 944.0 ( 4 , 1 2 ) 944.81 943.2 M o l e s D i m e r 40.566 40.270 C.P.U. S e c o n d s 4 4 1 1 1 0 3 3 1 S t a r t i n g p o i n t s d i f f e r e n t . 227 T h i s i s c l e a r l y shown i n T a b l e 4-14 where t h e c a s e w i t h low c o n v e r s i o n has an o b j e c t i v e f u n c t i o n v a l u e w h i c h i s 10% h i g h e r t h a n t h e c a s e w i t h a h i g h e r c o n v e r s i o n . T h i s l a r g e e f f e c t i s c l e a r l y d o m i n a n t when c o m p a r e d t o f u n c t i o n d i f -f e r e n c e s o f o n l y 0.1-0.5% c a u s e d by c h a n g e s i n DVDR f l o w s p l i t s . T h r e e c a s e s o f a t o t a l p r o d u c t o b j e c t i v e f u n c t i o n w e r e a l s o t e s t e d . The o b j e c t i v e f u n c t i o n w e i g h t s t h e d i f -f e r e n t p r o d u c t s i n a c c o r d a n c e w i t h t h e r e l a t i v e s e l l i n g p r i c e o f t h e s e p r o d u c t s . In t h e c a s e o f b u t a n e and p r o p a n e , p a r t o f t h e p r o d u c t i o n may be u s e d f o r f u e l gas w h i c h has a l o w e r s e l l i n g p r i c e . The amount u s e d f o r f u e l gas d e p e n d s on t h e c u r r e n t m a r k e t demand f o r p r o p a n e o r b u t a n e . 0.F = 2 * M o l e s P o l y m e r G a s o l i n e + ( 1 - a ) * M o l e s B u t a n e + 1.1 * ( 1 - b ) * M o l e s P r o p a n e + 0.8 ( a * M o l e s B u t a n e + b * M o l e s P r o p a n e ) w h e r e : a f r a c t i o n o f b u t a n e u t i l i z e d as f u e l gas b f r a c t i o n o f p r o p a n e u t i l i z e d as f u e l g a s T h e c a s e s t e s t e d were (1) ( 2 ) ( 3 ) a = 0.0 a = 0.50 a = 0.0 b = 0.0 b = 0.0 b = 0.25 228 T A B L E 4-14 R e s u l t s o f O p t i m i z a t i o n U s i n g T o t a l M o l e s  o f P o l y m e r G a s o l i n e as  O b j e c t i v e F u n c t i o n H i g h C o n v e r s i o n Low C o n v e r s i o n DVDR S p l i t s : EQPAR ( 2 , 2 4 ) .602 .660 ( 2 , 2 6 ) .017 .200 ( 3 , 2 6 ) .334 .150 ( 4 , 2 6 ) .465 .400 Column V a r i a b l e s : EQPAR ( 6 , 2 2 ) .990 .950 ( 7 , 2 2 ) .007 .010 ( 6 , 3 1 ) .980 .995 ( 7 , 3 1 ) .020 .080 M o l e s D i m e r 40.53 38.94 M o l e s P o l y . G a s . 50.59 56.40 229 Some o f t h e r e s u l t s a r e shown i n T a b l e 4-15. The r u n s shown were c h o s e n t o d e m o n s t r a t e t h e a f f e c t o f t h e i n d e p e n d e n t v a r i a b l e s on d i f f e r e n t o b j e c t i v e f u n c t i o n s and a r e n o t n e c e s s a r i l y t h e end r e s u l t o f an o p t i m i z a t i o n s e a r c h . F o r t h i s r e a s o n , C.P.U. t i m e s a r e n o t r e p o r t e d . Once a g a i n t h e m o s t i m p o r t a n t v a r i a b l e s i n a l l o f t h e a b o v e c a s e s a r e t h e d e b u t a n i z e r s p l i t and t o p p r o d u c t p u r i t y . The p r o d u c -t i o n o f p r o p a n e i s i n c r e a s e d by m i n i m i z i n g t h e s p l i t o f p r o p a n e and t h e t o p p r o d u c t p u r i t y b u t s i n c e t h i s c h a n g e d e c r e a s e s t h e amount o f b u t a n e p r o d u c e d , t h e o v e r a l l r e s u l t i s a b o u t t h e same. The optimum f o r t h e f i n a l p l a n t model a p p e a r s t o be t h a t shown f o r r u n 7 o f t h i s t a b l e . I t s h o u l d be m e n t i o n e d t h a t p r o b l e m s were e n -c o u n t e r e d w i t h t h e CHESS c o n v e r g e n c e s u b r o u t i n e , T E S T , when u s i n g t h e f i n a l p o l y p l a n t m o d e l . S y s t e m i n t e r r u p t s o c c u r r e d f r e q u e n t l y and many r u n s had t o be c h a n g e d o r r e p e a t e d w i t h d i f f e r e n t c o n d i t i o n s . The c a u s e o f t h e s e i n t e r r u p t s has n o t b e e n a s c e r t a i n e d b u t M e n z i e s and J o h n s o n ( 3 5 ) h a v e i n c l u d e d t h e i r own s y s t e m c o n v e r g e n c e r o u t i n e i n CHESS b e c a u s e o f t h e i n a d e q u a c y o f t h e CHESS s u p p l i e d s u b r o u t i ne. A f u r t h e r s t u d y was made on t h e a f f e c t o f t h e t e m p e r a t u r e o f t h e s t r e a m e n t e r i n g t h e r e a c t o r on t h e o p t i -mum c o n v e r s i o n f o r d i f f e r e n t v a l u e s o f d a y s on s t r e a m . TABLE 4-15 V a l u e s o f D i f f e r e n t O b j e c t i v e F u n c t i o n s f o r F i n a l P l a n t Model O p t i m i z a t i o n RUN 1 2 3 4 5 6 7 DVDR S p l i t s : EQPAR ( 2 , 2 4 ) .700 .680 .610 .602 .602 .602 .550 ( 2 , 2 6 ) .000 .010 .000 .017 .017 .017 .050 ( 3 , 2 6 ) .400 .370 .350 .334 .334 .334 .350 ( 4 , 2 6 ) .300 .250 .400 .465 .465 .465 .450 Column V a r i a b l e s : EQPAR ( 6 , 2 2 ) .950 .970 .970 .990 .990 .950 .950 ( 7 , 2 2 ) .005 .009 .005 .007 .007 .001 .001 ( 6 , 3 1 ) .990 .980 .990 .995 .980 .980 .995 ( 7 , 3 1 ) .080 .020 .070 .080 .020 .020 .080 M o l e s D i m e r 39.15 39.17 40.22 40.52 40.53 40.30 40.60 M o l e s P o l y Gas. 55.84 49.34 55.96 57.80 50.59 50.41 57.79 M o l e s B u t a n e 92.40 100.92 93.08 92.60 99.81 96.70 89.15 M o l e s P r o p a n e 62.61 60.41 60.94 59.13 59.11 62.51 62.53 O.F. ( 1 ) 272.96 266.05 272.05 273.24 - 266.01 266.29 273.51 O.F. ( 2 ) 257.45 249.92 256.64 258.06 250.12 250.29 258.34 O.F. ( 3 ) 268.26 261.52 267.48 268.80 261.47 261.60 268.83 ro CO o 231 The c o l u m n v a r i a b l e s were f i x e d and two c a s e s were e x a m i n e d : 1. E n t e r i n g t e m p e r a t u r e r e s t r i c t e d t o 8 1 5 ° R ( P r e s e n t C a s e ) . 2. E n t e r i n g t e m p e r a t u r e , EQPAR ( 2 , 3 ) , a l l o w e d t o v a r y . ( I n c l u d e d as an i n d e p e n d e n t v a r i a b l e i n t h e o p t i m i z a -t i o n m e t h o d . ) T h e s e c a s e s were t e s t e d a t 0, 50, 100, and 200 d a y s on s t r e a m and t h e r e s u l t s a r e shown i n T a b l e 4-16. D i f f e r e n c e s i n c o n v e r s i o n o f a b o u t 0.3-0.5% c a n be o b t a i n e d i f h i g h e r e n t e r i n g t e m p e r a t u r e s a r e u s e d . The r e s u l t s a l s o show t h a t c o n v e r s i o n s o f o v e r 80% c a n be a c h i e v e d a f t e r more t h a n 6 months o f o p e r a t i o n . Of c o u r s e , t h i s r e s u l t i s b a s e d on t h e a p p r o x i m a t e c a t a l y s t p o i s o n i n g r e l a t i o n s h i p u s e d i n t h e r e a c t o r m o d u l e b u t i t d o e s p o i n t o u t t h a t a c l o s e c o n t r o l o f r e a c t o r t e m p e r a t u r e s t h r o u g h t h e u s e o f r e c y c l e c a n b r i n g a p p r e c i a b l e i n c r e a s e s i n o l e f i n c o n v e r -s i o n . W h i l e t h e a d d i t i o n o f new p l a n t f a c i l i t i e s t o p e r m i t t h e r e a c t o r e n t e r i n g s t r e a m t e m p e r a t u r e t o be i n -c r e a s e d may n o t be e c o n o m i c a l l y j u s t i f i e d i n t h i s c a s e , t h e m e t h o d o f a l l o w i n g some v a r i a b l e s , w h i c h a r e f i x e d by e q u i p -ment l i m i t a t i o n s , t o be f r e e l y c h a n g e d i n an o p t i m i z a t i o n scheme c a n be a v e r y u s e f u l o n e . ( I n a p r o c e s s w h e r e t h e p r o d u c t i s v e r y v a l u a b l e , a d i f f e r e n c e o f t h e same o r d e r o f m a g n i t u d e shown c o u l d be o f g r e a t i m p o r t a n c e . ) TABLE 4-16 E f f e c t o f T e m p e r a t u r e o f S t r e a m 4 on Optimum C o n v e r s i o n D e p r o p a n i z e r D e b u t a n i z e r P r o p a n e S p l i t = 0.01 C 4 S p l i t = 0.07 F r a c . P r o p a n e T o p s = 0.99 F r a c . C. T o p s = 0.99 Days on S t r e a m 0 50 100 200 1 2 1 2 1 2 1 2 Temp. E n t e r i n g R e a c t o r 815 .0 8'60 .0 815. 0 850 .0 815. 0 865 .0 815 .0 865 .0 DVDR S p l i t : EQPAR ( 2 , 2 4 ) .602 .645 600 .650 558 .630 .530 .620 ( 2 , 2 6 ) .017 .105 000 .000 000 .000 .000 .000 ( 3 , 2 6 ) .334 .505 205 .375 000 .230 .000 .170 ( 4 , 2 6 ) .465 .360 • 501 .515 636 .620 .560 .510 Bed E x i t T e m p e r a t u r e : SINTSV ( 4 , 6) 891 .0 941 .3 860. 5 908 .8 81 2 . 1 861 .7 812 .1 861 .7 ( 4 , 8) 944 .1 944 .3 944. 2 943 .6 9 3 5 . 9 944 .6 917 .7 943 .9 ( 4 , 1 0 ) 943 .5 942 .8 944. 5 941 .2 944 . 5 935 .0 943 .9 941 .5 ( 4 , 1 2 ) 944 .8 944 .8 940. 0 939 .8 944. 3 944 .9 944 .8 943 .8 M o l e s D i m e r 40 .56 40 .75 39. 68 39 .83 38. 33 38 .52 36 .66 36 .86 C o n v e r s i o n 92 .71 93 .14 90. 70 91 .04 8 7 . 61 88 .04 83 .79 84 .25 1 - E n t e r i n g t e m p e r a t u r e r e s t r i c t e d t o maximum o f 8 1 5 ° R . 2 - E n t e r i n g t e m p e r a t u r e n o t r e s t r i c t e d . ro oo ro 233 A l t h o u g h t h e r e s u l t s f r o m t h e o p t i m i z a t i o n o f t h e i n t e r m e d i a t e p l a n t model a r e v e r y s i m i l a r t o t h a t o f t h e f i n a l p o l y m e r i z a t i o n p l a n t m o d e l , t h e f i n a l model g i v e s f u r t h e r i n s i g h t s i n t o t h e p r o d u c t d i s t r i b u t i o n p o s s i b i l i t i e s . I t s h o u l d be m e n t i o n e d t h a t t h e r e s u l t s o f t h e f i n a l p l a n t model s i m u l a t i o n s f i t i n t o t h e r a n g e o f r e p o r t e d p l a n t r e s u l t s . S i n c e t h e o p e r a t i o n o f t h e a c t u a l p o l y m e r i z a t i o n p l a n t i s q u i t e v a r i a b l e i t i s n o t p o s s i b l e t o c o m p a r e e x a c t v a l u e s . I t i s f e l t t h a t t h e p l a n t model c a n be u t i l i z e d by t h e r e f i n e r y p e r s o n n e l i n f u r t h e r s t u d i e s w i t h d i f f e r i n g o b j e c t i v e f u n c t i o n s , f e e d c o m p o s i t i o n s , e t c . 234 C h a p t e r 5 CONCLUSIONS The c o n c l u s i o n s r e a c h e d a f t e r t h i s s t u d y a r e l i s t e d b e l o w : 1. I t i s p o s s i b l e t o u t i l i z e an a u t o m a t i c o p t i -m i z a t i o n method f o r t h e o p t i m i z a t i o n o f a c h e m i c a l p l a n t model w i t h r e c y c l e streams which has been s i m u l a t e d by means o f a c h e m i c a l p l a n t s i m u l a t i o n system. 2. I n o r d e r t o be a b l e t o u t i l i z e an a u t o m a t i c o p t i m i z a t i o n method w i t h o u t an e x c e s s i v e r e q u i r e m e n t o f com-p u t e r c a l c u l a t i o n t ime a s t r a t e g y o f model b u i l d i n g must be used t h a t s i m p l i f i e s t h e c a l c u l a t i o n s wherever p o s s i b l e but a t t h e same time s u p p l i e s r e a s o n a b l e e s t i m a t e s f o r t h e major d e c i s i o n v a r i a b l e s and c o n s t r a i n t s . 3 . The r e s u l t s o f t h e p l a n t model s i m u l a t i o n are w i t h i n t h e range o f r e p o r t e d p l a n t d a t a . 4. The p o l y m e r i z a t i o n o f o l e f i n s on t h e U.O.P. s o l i d p h o s p h o r i c a c i d c a t a l y s t can be r e p r e s e n t e d by t h e g e n e r a l i z e d r a t e e q u a t i o n , 235 r = k c. ( 1 - x ): ( l + x ) : where, -7540 k = 2.87 x 1 0 s e R T The r a t e e q u a t i o n d e v e l o p e d by L a n g l o i s and Walkey (70) does not f i t the e x p e r i m e n t a l d a t a but t h e e n e r g i e s o f a c t i v a t i o n c a l c u l a t e d f o r b o t h mechanisms a r e q u i t e c l o s e , l e a d i n g one t o the b e l i e f t h a t t h e r e a c t i o n mechanisms a r e q u i t e s i m i l a r . 5- The C o n s t r a i n e d P a t t e r n S e a r c h method does not h a n d l e t h e c o n s t r a i n t s v e r y w e l l and d i d not g i v e good r e s u l t s i n t h e p r e s e n t s t u d y . The D e f l e c t e d G r a d i e n t - C r e a t e d Response S u r f a c e Technique a c h i e v e d b e t t e r r e s u l t s but i s e x t r e m e l y s e n s i t i v e t o the v a l u e s chosen f o r t h e s e a r c h p a r a m e t e r s and p e n a l t y f a c t o r s and i s thus q u i t e d i f f i c u l t t o u s e . The M o d i f i e d Complex Method works q u i t e w e l l , even i n t h e r e g i o n c l o s e t o t h e c o n s t r a i n t s . I t i s v e r y s i m p l e t o use and does not r e q u i r e e x c e s s i v e amounts o f computer t i m e . 236 C h a p t e r 6 RECOMMENDATIONS As a r e s u l t o f t h e s i m u l a t i o n and o p t i m i z a t i o n s t u d y o f t h e p o l y m e r i z a t i o n p l a n t a t t h e S h e l l b u r n r e f i n e r y , s e v e r a l r e c o m m e n d a t i o n s c an be made t o t h e r e f i n e r y manage-ment: 1. The temperature of the streams l e a v i n g the c a t a l y s t beds should be s t r i c t l y c o n t r o l l e d by means of the propane r e c y c l e so as to achieve maximum o l e f i n conversions. 2. P l a n t t e s t s might be made i n order to o b t a i n data on c a t a l y s t p o i s o n i n g w i t h time at d i f f e r e n t tempera-tur e l e v e l s so that a more r e a l i s t i c c a t a l y s t bed tempera-t u r e l i m i t could be s e t . 3. The p o s s i b i l i t y of heating the stream e n t e r i n g the r e a c t o r to a temperature higher than that which i s p r e s e n t l y a t t a i n a b l e should be examined since higher o l e f i n conversions could be obtained. 4 . The s i m u l a t i o n model developed i n t h i s study could be used by the S h e l l b u r n r e f i n e r y to t e s t f u r t h e r cases such as d i f f e r i n g ; feed compositions, o b j e c t i v e f u n c t i o n s , e t c . 237 Some r e c o m m e n d a t i o n s f o r f u r t h e r s t u d y a r e : 1. More e x t e n s i v e t e s t i n g o f s i n g l e o p t i m i z a t i o n methods o r c o m b i n a t i o n o f methods on p l a n t s i m u l a t i o n models s h o u l d be u n d e r t a k e n . I t has r e c e n t l y come t o our a t t e n t i o n t h a t work o f t h i s k i n d was u n d e r t a k e n a t McMaster U n i v . ( 9 7 ) . 2. F u r t h e r s t u d i e s s h o u l d be made on t h e e f f e c t o f system convergence e r r o r , f o r models w i t h r e c y c l e s t r e a m s , on d i f f e r e n t o p t i m i z a t i o n methods, and means t o reduce t h e s e e r r o r s . 3 . 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S o m e r v i l l e , "Computa-t i o n o f M u l t i s t a g e S e p a r a t i o n P r o c e s s e s , " R e i n h o l d , N.Y. , 1962. 86. H o l l a n d , C D . , " M u l t i c o m p o n e n t D i s t i l l a t i o n , " P r e n t i c e H a l l , N . J . , 1 9 6 3 . 87. S m i t h , B.D., " D e s i g n o f E q u i l i b r i u m S t a g e P r o c e s s e s , " McGraw H i l l , N.Y. , 1963. 88. U n d e r w o o d , A . J . V . , "Chem. Eng. P r o g . , " 44, 6 0 3 , 1948. 89. E r b a r , J . H . , R.N. Maddox, " P e t r o l . R e f i n e r , " 4 0 , 183, 1961 . 90. L i d d l e , C . J . , "Chem. E n g . , " 7 5 , 137, O c t . 2 1 , 1968. 91 . F l e t c h e r , R., M.J.D. P o w e l l , " C o m p u t e r J . , " 6, 1 6 3 - 1 6 8 , 1963. 92. C a r r o l l , G.W., " J . Opn. R e s . S o c . Am.," 9, 1 6 9 - 1 8 4 , 1961 . 93. Box, M.J., D a v i e s , D., W. H. Swann, " N o n - L i n e a r O p t i m i -z a t i o n T e c h n i q u e s , " O l i v e r and B o y d , E d i n b u r g h , 1969. 94. B e v e r i d g e , G.S.G., R.S. S c h e c h t e r , " O p t i m i z a t i o n : T h e o r y and P r a c t i c e , " McGraw H i l l , 1970. 95. L u c a s , J . P . , Ma.Sc. T h e s i s , D e p t . o f Chem. E n g . , U . B . C , 1 968. 244 96. K l i n g m a n , W.R., D.M. H i m m e l b l a u , " J . A s s o c . Comp. Mach.," 1 1 , 400-41 5 , 1964. 9 7 . A n v a r i , M., S t e p h a n o p o u l o s , G., A . I . J o h n s o n , " S y s t e m O p t i m i z a t i o n W i t h i n t h e M o d u l a r A p p r o a c h , " M c M a s t e r U n i v . , H a m i l t o n , O n t . , 1971. 245 NOMENCLATURE T y p i c a l U n i t s m A. A. c i ( x ) D °P D s DELH E f ( x ) F F_ E x p a n s i o n f a c t o r i n Complex Method D i s t a n c e t o U n i v a r i a t e minimum, Eq. ( 4 - 9 ) C r o s s s e c t i o n a l a r e a o f c a t a l y s t bed S u r f a c e a r e a o f c a t a l y s t p a r t i c l e E x t e r n a l s u r f a c e a r e a o f p o l y m e r i z a -t i o n r e a c t o r C e n t r o i d i n Complex M e t h o d C o n s t r a i n t f u n c t i o n I n i t i a l O l e f i n c o n c e n t r a t i o n Di f f u s i v i t y C a t a l y s t p a r t i c l e d i a m e t e r Number o f d a y s on s t r e a m A v e r a g e h e a t o f p o l y m e r i z a t i o n E f f e c t i v e n e s s F a c t o r O b j e c t i v e f u n c t i o n M o l a r f l o w r a t e F l o w e f f i c i e n c y f a c t o r L o w e r bound i n Complex M e t h o d f t 2 mm2 f t 2 m o l e s o l e f i n / c c f e e d c m 2 / s e c mm BTU/mole m o l e / h r 246 w g kv K M M N P P ( x , K ) Q r R R r R e p RM S C o n v e r s i o n f a c t o r , mass t o f o r c e u n i t s Mass f l o w r a t e U p p e r b o u n d i n Complex M e t h o d R a d i a t i o n h e a t t r a n s f e r c o e f f i c i e n t E n t h a l p y R e a c t i o n r a t e c o n s t a n t Number o f v e r t i c e s i n Complex M e t h o d Mass t r a n s f e r c o e f f i c i e n t V a p o r - l i q u i d e q u i l i b r i u m d i s t r i b u -t i o n c o e f f i c i e n t P e n a l t y f a c t o r , E q. ( 4 - 7 ) Number o f c o n s t r a i n t s M o l e c u l a r w e i g h t Number o f i n d e p e n d e n t v a r i a b l e s P r e s s u r e C r e a t e d r e s p o n s e s u r f a c e f u n c t i o n , E q . ( 4 - 2 7 ) V o l u m e t r i c f l o w r a t e R a t e o f r e a c t i o n U n i v e r s a l gas c o n s t a n t R e f l u x r a t i o P a r t i c l e R e y n o l d s Number Minimum r e f l u x r a t i o Number o f t h e o r e t i c a l s t a g e s T y p i c a l U n i t s gm/mi n-mnr B T U / h r - f t 2 _ o i c c o l / h r / c c c a t , mm/sec p. s . i . a . f t 3 / h r mol o l / h r / c c c a t 247 T y p i c a l U n i t s SM Minimum number o f t h e o r e t i c a l s t a g e s SS (S-SM) / ( S + l ) T T e m p e r a t u r e °K, °R T w A m b i e n t t e m p e r a t u r e °R V Volume o f C a t a l y s t c c V Q L i n e a r gas v e l o c i t y f t / h r V Volume o f c a t a l y s t p a r t i c l e mm3 w. W e i g h t i n g f a c t o r , Eq. ( 4 - 2 7 ) x I n d e p e n d e n t v a r i a b l e x F r a c t i o n o f o l e f i n s c o n v e r t e d i n t o p r o d u c t y M o l e f r a c t i o n z L e n g t h o f c a t a l y s t bed f t G r e e k L e t t e r s a R e l a t i v e v o l a t i l i t y A F F r e e e n e r g y o f p o l y m e r i z a t i o n c a l / m o l e A F° F r e e e n e r g y o f p o l y m e r i z a t i o n a t 1 atm. c a l / m o l e A H H e a t o f p o l y m e r i z a t i o n BTU/mole V f G r a d i e n t e V o i d f r a c t i o n e M F r a c t i o n a l m o l a r c h a n g e on r e a c t i o n p D e n s i t y gm/cm 3 248 G r e e k L e t t e r s T y p i c a l U n i t s a L e n n a r d - J o n e s f o r c e c o n s t a n t A 0 a n D i r e c t i o n o f s e a r c h , E q. ( 4 - 9 ) y V i s c o s i t y S u b s c r i p t s 3 C 3 o l e f i n s 4 Ci, o l e f i n s 5 C 5 o l e f i n s b b o t t o m p r o d u c t d t o p p r o d u c t f f e e d i i ndex j i ndex hk h e a v y key l k l i g h t k ey min mi n imum o l o l e f i n r e f r e f e r e n c e 249 APPENDIX A POLYMERIZATION REACTOR PLANT TESTS The r a t e o f p o l y m e r i z a t i o n i s a f u n c t i o n o f t h e f o l l o w i n g v a r i a b l e s : a) p r e s s u r e b) t e m p e r a t u r e c) f e e d c o m p o s i t i o n d) c a t a l y s t p h o s p h o r i c a c i d c o n c e n t r a t i o n e) c a t a l y s t p a r t i c l e s i z e f ) c a t a l y s t age In t h e p l a n t t e s t s some o f t h e s e v a r i a b l e s were h e l d c o n s t a n t s i n c e i t was i m p o s s i b l e t o c h a n g e some o f them ( c a t a l y s t p a r t i c l e s i z e ) and o t h e r s c o u l d o n l y be c h a n g e d by g r e a t l y e x t e n d i n g t h e t e s t i n g t i m e p e r i o d ( c a t a l y s t a g e , p r e s s u r e , c a t a l y s t p h o s p h o r i c a c i d c o n c e n -t r a t i o n ) . The f o l l o w i n g m e a s u r e s were t a k e n : a) The p r e s s u r e was h e l d c o n s t a n t . b) The t e m p e r a t u r e was v a r i e d i n o r d e r t o f i n d the t e m p e r a t u r e dependence o f t h e r a t e c o n s t a n t . c) The o v e r a l l o l e f i n c o n t e n t o f the f e e d c o u l d v a r y o r r e m a i n c o n s t a n t . However, the r a t i o o f t h e d i f f e r -ent o l e f i n s i n the f e e d was h e l d as c o n s t a n t as p o s s i b l e . 250 d) The w a t e r r a t e was c o n t r o l l e d w i t h r e s p e c t t o the t e m p e r a t u r e and f l o w r a t e o f the f e e d so t h a t the a c i d c o n c e n t r a t i o n o f t h e c a t a l y s t remained a p p r o x i m a t e l y c o n s t a n t . e) The c a t a l y s t p a r t i c l e s i z e remained c o n s t a n t . f ) The t e s t was made soon a f t e r s t a r t - u p o f a new c a t a l y s t and thus the f a c t o r o f c a t a l y s t a g i n g was e l i m i n a t e d . The i n t e g r a l r e a c t o r a p p r o a c h was u s e d t o p l a n t h e e x p e r i m e n t s . B e c a u s e t h e t e m p e r a t u r e and o l e f i n c o n c e n -t r a t i o n v a r i e d t h r o u g h o u t t h e bed t h e p o s s i b i l i t y o f u s i n g t h e d i f f e r e n t i a l r e a c t o r a p p r o a c h was e l i m i n a t e d . The p r o c e d u r e was: 1. A s e r i e s o f ru n s were made v a r y i n g t h e f l o w r a t e a t a f i x e d t e m p e r a t u r e so t h a t a wide range o f c o n v e r -s i o n s c o u l d be o b t a i n e d . 2. A r a t e e q u a t i o n was s e l e c t e d f o r t e s t i n g . By i n t e g r a t i o n of the p l u g f l o w d e s i g n e q u a t i o n , one o b t a i n s V/F = x dx ( A - l ) r S i n c e " r " i s t h e r a t e e x p r e s s i o n t o be t e s t e d , a n u m e r i c a l v a l u e f o r t h e R.H.S. o f t h i s e x p r e s s i o n c a n be o b t a i n e d f o r e a c h e x p e i m e n t a l r u n . 251 FEED FEED SAMPLE F R C - 2 7 TOTAL FLOW RATE 5 ^ TO WEST REACTOR F l FEED SPLIT •TI-29 TEMPERATURE ENTERING BED F l FEED SPLIT to T R - 8 TEMPERATURE = BEGINNING OF - B E D TJ-31 = TEMPERATURE ^ A T END OF BED SAMPLE OF EFFLUENT F I G U R E A-l F I R S T B E D OF E A S T P O L Y M E R I Z A T I O N REACTOR - S H E L L B U R N R E F I N E R Y 252 3. A l i n e a r i t y t e s t can t h e n be p e r f o r m e d t o t e s t t h e mechanism chosen. When v a r i o u s r a t e e q u a t i o n s a r e t e s t e d , t h e one t h a t b e s t f i t s the d a t a i s chosen. In t h e p l a n t t e s t s , f o u r d i f f e r e n t f e e d r a t e s and f o u r d i f f e r e n t e n t e r i n g t e m p e r a t u r e s w ere s e t . I t was hop e d t h a t t h e s e t e s t s w o u l d s u p p l y s u f f i c i e n t d a t a t o e s t a b l i s h an o v e r a l l r a t e e x p r e s s i o n and t o f i n d t h e t e m p e r -a t u r e d e p e n d e n c e o f t h e r e a c t i o n r a t e c o n s t a n t . The e a s t p o l y m e r i z a t i o n r e a c t o r was s t a r t e d up w i t h f r e s h c a t a l y s t on J u l y 11, 1968. The p l a n t t e s t s were made on J u l y 30, 3 1 , A u g u s t 1, 2, 1968. E a c h d a y a d i f f e r -e n t f l o w r a t e was s e t and d i f f e r e n t e n t e r i n g t e m p e r a t u r e s were f i x e d by c o n t r o l l i n g t h e s t e a m t o t h e f e e d h e a t e r . The t e m p e r a t u r e s and f l o w r a t e s were n o t e d and s a m p l e s were t a k e n o f t h e f e e d t o and e f f l u e n t f r o m t h e f i r s t c a t a l y s t b e d ( F i g u r e A - l ) , w h e r e a s p e c i a l s a m p l i n g d e v i c e had b e e n p r e v i o u s l y i n s t a l l e d . The s a m p l e p r o b e c o n s i s t e d o f a 3/4 i n c h d i a m e t e r t u b e , a b o u t 1 f o o t l o n g , s e a l e d a t one end and p e r f o r a t e d w i t h 1/4 i n c h d i a m e t e r h o l e s a l o n g i t s l e n g t h . The d a t a o b t a i n e d i s shown i n T a b l e A - l . I t c a n be s e e n t h a t t h e c o n d i t i o n s i n t h e bed a r e f a r f r o m b e i n g i s o t h e r m a l ; t h e t e m p e r a t u r e d i f f e r e n c e a c r o s s t h e b ed r a n g e s f r o m 4 0 - 5 5 ° F . H o w e v e r , an a r i t h m e t i c mean o f t h e t e m p e r a t u r e s e n t e r i n g and l e a v i n g t h e bed was c h o s e n i n o r d e r t o a t t e m p t an a n a l y s i s o f t h e d a t a . TABLE A-1 D a t a From P o l y m e r i z a t i o n R e a c t o r P l a n t T e s t s T i m e F l o w R a t e T e m p e r a t u r e °F M o l e F r a c t i o n O l e f i n s C o n v e r s i o n s b b l / d a y T i n T m i d T O U t T 1 avg In Out J u l y 30 8:00 a.m. 10:15 a.m. 11:45 a.m. 2:00 p.m. 1019 1036 1088 1024 356 344 328 327 372 361 339 343 404 391 350 360 380 368 339 344 30.3 29.7 29.2 29.7 21.5 21 .2 22.8 22.3 0.290 0.286 0.219 0.249 J u l y 31 8:30 a.m. 10:30 a.m. 11:45 a.m. 736 747 752 357 343 336 377 363 357 414 400 394 386 372 365 29.7 29.7 29.6 20.2 20.7 21 .0 0.320 0.303 0.291 A u g u s t 1 8:00 a.m. 10:00 a.m. 12:00 p.m. 2:15 p.m. . 854 876 875 881 356 341 334 318 373 362 353 336 407 395 386 366 382 368 360 342 29.2 29.6 29.8 30.3 21 .2 20.8 20.6 22.3 0.274 0.297 0.307 0.264 A u g u s t 2 8:00 a.m. 10:30 a.m. 11:10 a.m. 1:00 p.m. 537 549 575 576 358 343 330 315 377 367 354 340 416 406 394 352 387 377 362 344 30.3 31 .0 30.2 30.3 19.8 19.2 18.4 18.6 0.347 0.381 0.393 0.386 T. + l n o u t a v g ro cn co 254 The d a t a were o r d e r e d i n t o t e m p e r a t u r e " r a n g e s " o f 5 - 7 ° F . Once a g a i n t h i s a p p r o x i m a t i o n was o n l y made so t h a t t h e p l a n t d a t a c o u l d be a n a l y z e d . Two r a t e e q u a t i o n s were t e s t e d . The f i r s t was t h e l i q u i d p h o s p h o r i c model o f L a n g l o i s and W a l k e y ( 7 0 ) VC k = (1+e x ) 2 dx m ( 1 - x ) 2 + 0 . 3 x ( l - x ) ( A - 2 ) and t h e s e c o n d was an i r r e v e r s i b l e f i r s t o r d e r m e c h a n i s m VC. k = 1+e x H ^ x ( A - 3 ) I t i s p o s s i b l e t o c a l c u l a t e t h e v a l u e s o f t h e a b o v e i n t e g r a l s f o r d i f f e r e n t v a l u e s o f c o n v e r s i o n and s p a c e v e l o c i t y . An a v e r a g e v a l u e was c h o s e n f o r e , a s s u m i n g t h a t t h e r e a c t i o n i s e s s e n t i a l l y a d i m e r i z a t i o n . ( i t has been r e p o r t e d t h a t u n d e r n o r m a l o p e r a t i n g c o n d i t i o n s l e s s 255 t h a n 5% o f t h e p o l y m e r i z e d p r o d u c t o c c u r s as h y d r o c a r b o n s h e a v i e r t h a n t h e d i m e r s . ) F i g u r e s A - 2 a , A-3a show t h e p l o t s o f t h e c a l c u l a t e d r e s u l t s . T h e r e i s a g r e a t d e a l o f s c a t t e r i n t h e d a t a and t h e v a l u e s o f t h e r e a c t i o n r a t e c o n s t a n t s o b t a i n e d by a l e a s t s q u a r e s f i t o f t h e d a t a i n e a c h t e m p e r a t u r e r a n g e show l i t t l e v a r i a t i o n . The l i n e p l o t t e d i s t h e b e s t f i t t h r o u g h a l l t h e d a t a . F i g u r e s A - 2 b , A-3b show o n c e a g a i n t h a t no t e m p e r a -t u r e d e p e n d e n c e c a n be f o u n d f o r t h e r e a c t i o n r a t e c o n s t a n t . The v a l u e s p l o t t e d a r e f o r one a v e r a g e t e m p e r a t u r e and t h e d a s h e d l i n e drawn t h r o u g h t h e c a l c u l a t e d p o i n t s i s o f t h e same s l o p e as t h a t o f L a n g l o i s and W a l k e y ( 7 0 ) . The i n c o n c l u s i v e r e s u l t s a r e due t o t h e v a r i a t i o n o f t e m p e r a t u r e i n t h e bed and t h e i m p o s s i b i l i t y o f c o n v e r t -i n g t h e d a t a o b t a i n e d i n t o an " a v e r a g e i s o t h e r m a l " f o r m . An a t t e m p t was a l s o made t o s i m u l a t e t h e c a t a l y s t bed u t i l i z i n g t h e l i q u i d p h o s p h o r i c a c i d model and t o compare t h e t e m p e r a t u r e s and c o n v e r s i o n s o b t a i n e d by t h e s i m u l a t e d model w i t h t h e d a t a f r o m t h e p l a n t t e s t s . H o w e v e r , o n c e a g a i n t h e c a l c u l a t e d r e s u l t s d i d n o t c o n f o r m t o t h e e x p e r i -m e n t a l d a t a . 256 2 1 0 0 2-125 (b) 2 1 5 0 2-175 2 -200 1000/T ( 6 K"' ) 2-225 0 - 5 4 < CD LU ± 0 - 4 6 Ll_ O UJ 3 0 - 3 8 < > 0 - 3 0 (a) TEMPERATURE RANGE (°F ) K (LEAST SQUARES o 3 8 0 - 3 8 7 11-42 A 3 6 8 - 3 7 7 12-28 • 3 6 0 - 3 6 5 12-48 0 3 3 9 - 3 4 4 12-02 O V E R A L L 1 2 0 ! 2-8 3-2 3-6 V C 0 / F ( I0 2 ) (hr"1) F I G U R E A - 2 P L A N T R E A C T O R R A T E T E S T - L I Q U I D P H O S P H O R I C A C I D M O D E L 2-100 2125 (b) 2-150 2175 _ 2 -200 1000/T ( ° K H ) 2 -225 _i 0 * 4 8 < or L U O - 4 4 u. o UJ 3 0 - 4 0 0 - 3 6 < 0 - 3 2 0 - 2 8 0 - 2 4 (a) TEMPERATURE RANGE ( ° F ) K (LEAST SQUARES) o 380 - 3 8 7 9 - 7 2 368 - 377 1 0 - 3 4 • 360 - 365 10 -35 0 33 9 - 3 4 4 10 -26 OVERALL 1 0 1 6 2-4 2 -8 3-2 3-6 4 - 0 V C 0 / F ( I 0 2 ) (hr" 1) 4 - 4 F I G U R E A-3 P L A N T R E A C T O R R A T E T E S T - F I R S T O R D E R M E C H A N I S M 258 APPENDIX B CALIBRATION CURVES AND EXPERIMENTAL EQUIPMENT S P E C I F I C A T I O N S The c a l i b r a t i o n c u r v e f o r t h e M i l r o y a l - D c o n t r o l l e d v o l u m e pump i s shown i n F i g u r e B - l . The G i l m o n t No. 1 r o t a m e t e r c a l i b r a t i o n c u r v e s a r e shown i n F i g u r e B-2 f o r p o l y m e r f e e d and p r o p a n e . The e x p e r i m e n t a l r e a c t o r and t h e r e a c t o r i n t e r -n a l s a r e shown i n F i g u r e s B-3 and B-4 r e s p e c t i v e l y . F i g u r e B-5 i s an e x a m p l e o f a t y p i c a l c h r o m a t o g r a m o b t a i n e d f r o m a f e e d s a m p l e . 01 I I I I 1 0 2 0 4 0 60 80 100 PERCENT CAPACITY F I G U R E B-l M I L R O Y A L D C A L I B R A T I O N C U R V E 260 100 SCALE READING (b) PROPANE - DENSITY=0-5I g m / c c VISCOSITY =0-12 cp 161 1 0 20 4 0 6 0 8 0 SCALE READING (a) POLY F E E D - D E N S I T Y = 0 5 5 gm/cc VISCOSITY = 0 1 6 cp F I G U R E B-2 R O T A M E T E R C A L I B R A T I O N •0-7 i-o-mJP ro ro _L lO CD «—2-2 — 1/8" NPT 120 - t — 2 2 — o o o o o o o o 0 0 0 0 0 0 0 : 0 0 o o o o o o o o 0 0 0 0 0 0 0 1 0 o O O O O O O O O O O O O O O Q : 0 O A—2-2—+ 0-7-^1-0 COPPER RINGS CATALYST 1/4" NPT 10 CvJ CD EXPERIMENTAL REACTOR INTERNALS (STEEL) F I G U R E B - 3 E X P E R I M E N T A L R E A C T O R —1-68 —*~l — 1 - 9 3 — > NOTE : A L L PARTS STAINLESS STEEL F I G U R E B - 4 E X P E R I M E N T A L R E A C T O R I N T E R N A L S 264 APPENDIX C CALCULATIONS OF MASS TRANSFER E F F E C T I t has been shown e x p e r i m e n t a l l y t h a t no mass t r a n s f e r e f f e c t i s p r e s e n t w i t h t h e c a t a l y s t p a r t i c l e s o f 4.76-6.73 mm.; n o t e v e n a t t h e l o w e s t f l o w r a t e s ( T a b l e 2 - 4 ) . H o w e v e r , a mass t r a n s f e r e f f e c t may e x i s t f o r t h e s m a l l e r c a t a l y s t p a r t i c l e s ( 0 . 4 2 0 - 0 . 5 9 5 mm.) t e s t e d s i n c e t h e p a r t i c l e R e y n o l d s No. i s s m a l l e r . T h i s p o s s i b i l i t y i s a l s o s u p p o r t e d by t h e d a t a i n T a b l e 2-5. The r a t i o o f c o n -v e r s i o n s o f s m a l l c a t a l y s t p a r t i c l e s / l a r g e c a t a l y s t p a r -t i c l e s d e c r e a s e s w i t h d e c r e a s i n g f l o w r a t e . I f mass t r a n s -f e r r e s i s t a n c e p l a y e d a r o l e i n t h e r e a c t i o n r a t e f o r t h e s m a l l p a r t i c l e s t h i s m i g h t e x p l a i n t h e a b o v e r e s u l t s . F o r mass t r a n s f e r r e s i s t a n c e t o be i m p o r t a n t t h e mass t r a n s f e r c o e f f i c i e n t must be o f t h e same o r d e r o f mag-n i t u d e as t h e r e a c t i o n r a t e c o n s t a n t . P 9 P ( c - 1 ) 265 On t h e o t h e r hand i f k V„ << k„ A n ( C - 2 ) P 9 P t h e n t h e mass t r a n s f e r r e s i s t a n c e i s u n i m p o r t a n t . The U.O.P. c a t a l y s t u s e d i s an e x t r u d a t e i n c y -l i n d r i c a l f o r m whose d i a m e t e r and h e i g h t a r e b o t h a b o u t 6 mm. V = irr 2 h = 3 . 1 4 ( 3 ) 2 ( 6 ) = 170 mm.3 r r A „ = 2iTr h + 2irr 2 = 170 mm.2 P P P From t h e e x p e r i m e n t a l r e s u l t s ( T a b l e 2-3) k = 35.0 - 169.0 c c . f e e d / h r . / c . c . c a t a l y s t o r k ~ ( 1 - 5 ) x 1 0 " 2 s e c " 1 R e p - l a r g e p a r t i c l e s D p ~ 0.5 cm. 266 G ~ 0.05 gm/min-cm : n r ( ° - 5 cm) ( 0 . 0 5 -j*™ 2 ) D G v ' v min-cm^' R e = _J5_ = P v (1 .39 x 1 0 - V m g m o r ) x 60 i f £ cm-sec min R e p = 3 F o r G = 0.35 — ^ , Re = 2 1 ,2 P mi n-cnrF o r l a r g e p a r t i c l e s , i n t h e r a n g e o f f l o w r a t e s u s e d R e p = 3-21 w h i c h i s i n t h e i n t e r m e d i a t e f l o w r a n g e b e t w e e n l a m i n a r and t u r b u l e n t . Re - s m a l l p a r t i c l e s — p e  D p = 0.05 cm F o r s m a l l p a r t i c l e s , i n t h e r a n g e o f f l o w r a t e s u s e d R e p = 0.3-2.1. 267 The mass t r a n s f e r c o e f f i c i e n t c a n be e s t i m a t e d f r o m a c o r r e l a t i o n f o r g a s e s a t low R e y n o l d s Numbers by P e t r o v i c and T h o d o s ( 7 9 ) . . = 0.357 ( . t J D R e 0.359 {{- 6> F o r 3 < R e „ < 2,000 P w h e r e , 'D k p 9 P y pD 2A In o r d e r t o c a l c u l a t e k g f r o m E q u a t i o n ( C - 3 ) t h e d i f f u s i v i t y D must f i r s t be c a l c u l a t e d . The method u s e d i s f r o m t h e API T e c h n i c a l D a t a Book ( 8 0 ) . 0.04381 T 1.5 0.5 P ( a 1 2 ) 2 n. ( C - 4 ) w h e r e , a b s o l u t e t e m p e r a t u r e , °R m o l e c u l a r w e i g h t s o f two s p e c i e s t o t a l p r e s s u r e , p . s . i . a . c o l l i s i o n i n t e g r a l L eonard Jones f o r c e c o n s t a n t p r o p e n e + b u t e n e = 1 p r o p a n e + b u t a n e — 2 Mj * M 2 - 52 'k a p r o p a n e 371 5. 24 n - b u t a n e 375 5. 87 average ~ 372 5. 6 p r o p e n e 545 4. 67 n - b u t e n e 466 5. 50 i s o b u t e n e 765 4. 77 average « 57 5 5. 1 P = 515 p. S . i .a .T = 400 °F = 860 °R a i 2 = % ( 5 . 6 + 5.1) = 5.3 269 e 1 2 / k = (372 x 5 7 5 ) ° *5 = 462 kT = 860 e 1 2 462 = 1 .86 fiD = 1.10 (0 . 0 4 3 8 1 ) ( 8 6 0 ) 1-5 fJL + J j 52 52 0.5 515 ( 5 . 3 ) 2 ( 1 ' 1 0 ) D i 2 = 0.014 f + 2 / h r = 3.5 x 1 0 - 3 cm 2/sec The S c h m i d t No. c a n now be c a l c u l a t e d T P P y 4 0 0 ° F 515 p. s . i . a . 0.055 gm/cm 3 1 .39 x 1 0 - l t gm/cm-sec 270 , = _y_ = 1 .39 x 10~k  S C p D ( 5 . 5 x 1 0 " 2 ) ( 3 . 5 x T O " 3 ) NSC " M i F o r t h e c a s e o f l a r g e p a r t i c l e s , and a t t h e h i g h e s t f l o w r a t e , R e p = 21. S o l v i n g E q u a t i o n ( C - 3 ) 0.40 k ( 5 . 5 x 1 0 ~ 2 ) ( 6 0 ) , n -,,^'3 = 0.357 9 v u . / i ; . n o . 3 5 9 0.35 k g = 3.24 x 1 0 ~ 2 cm/sec F o r t h e c a s e o f t h e s m a l l p a r t i c l e s , and a t t h e l o w e s t f l o w r a t e , R e p = 0.3. S o l v i n g E q u a t i o n ( C - 3 ) k = 2.62 x 1 0 " 2 cm/sec g Now k A c a n be c o m p a r e d t o k\/ 9 P P 271 k V p = ( 1 - 5 ) x TO" 2 s e c - 1 x 170 mm3 k g A p = ( 2 5 - 3 5 ) x 1 0 " 2 mm/sec x 170 mm2 T h u s , t h e mass t r a n s f e r c o e f f i c i e n t a p p e a r s t o be a b o u t 10-20 t i m e s g r e a t e r t h a n t h e r e a c t i o n r a t e c o n s t a n t . H o w e v e r , w i t h t h e s m a l l e r p a r t i c l e s , t h e r e a c t i o n r a t e c o n -s t a n t s c a l c u l a t e d w o u l d be a b o u t 4 t i m e s g r e a t e r so t h a t t h e y w o u l d be i n t h e same o r d e r o f m a g n i t u d e as t h e mass t r a n s f e r c o e f f i c i e n t s . T h e r e f o r e , i t a p p e a r s t h a t t h e mass t r a n s f e r r e -s i s t a n c e d o e s p l a y a r o l e i n d e t e r m i n i n g t h e r e a c t i o n r a t e f o r t h e s m a l l c a t a l y s t p a r t i c l e s i n t h e e x p e r i m e n t a l r e a c t o r . H o w e v e r , t h i s mass t r a n s f e r r e s i s t a n c e i s unim-p o r t a n t i n t h e p l a n t r e a c t o r where t h e p a r t i c l e R e y n o l d s No. i s a b o u t 350 and t h e mass t r a n s f e r c o e f f i c i e n t a b o u t 2.25 mm/sec so t h a t k A * 100 x kV„ g P P 272 APPENDIX D C o m p u t e r P r o g r a m s f o r c o m p l e t e P l a n t M o d e l s and P r o c e s s M o d u l e s . ' : - ~56 EC PA=17., 57 EC F«»in . • .75 ,6. ,0 . ,0 . ,0 . . 0 . ,0 . , - I . , C , 0 . , C . t C. , . S i «8 EC PA=1S. , 59 rc PA - 2 c . , . T i . i . , c. , c . , c . , c. , c . , c . , - i . . 0 . . c . , o . . . 5 , * U S T FCLY 60 EC PA = 21. . 1 CLEAN 61 EC Ff l=22.,0.,0 . . . 98 , .0 5 , 0 . , 0 . , 0 . . . 9 8 ,  ? 2 P R F L I M N A K Y POL TKnTTDTTTCTi P T J KT S T F C T A T T C K -MKCh 1969 62 FCf "PA = ? 3. . 6 C . . 2V4C. , I. . 1. . I . .56T7T 3 EC NU=1,EQ Sl=PLKP.E0 EX=P28S,IK ST=l,OU ST=2, 63 EC P4= 2 4 . , . 6 , .4, * EC MJ=2 ,EC S U=MCH , E 0 EX = E162,[t< ST=2,MU ST*3, 6* EC P i =2 5 . , L . , 36000 . , 5 30 . , 5 EC NC = 3, EG SL=HXfR,CC EX=E161,IN STO.CU ST = 4, 65 EQ PA = 26 . , . 5 , . 5 , C . , • ' '• 6 EC N U = 3 6 , E U <(.« K 1 Xii ,E Q EX' F - 1,1 N 51 = 4,51,CI S I » 5 2 , 66 EC P « = 21 . , . 1 6 6 , . It I,. 2 3 3 . . 3 3 3 , i 7 EC MJ=4,EC SIJ=nvCH . f 0 EX = C- l . IN ST=52,OU ST=5,6, 67_ _ E C PA =28 . , .33 3 , . 33 3 , . 166, . 16 6. i S EC M = 5,CC' SL = M»«,EC EX* > - 2 , l N ST =6 , 5 0 . 3 3 , Cli sr = 7~; 66 E C P A = 29. , 1 . , I K C C. , J C C . , " 1 9 fo Nu=6 ,rc s i . '«rr.7 ,rQ F X » R- i , ir> s i = 7 . r u st^n, 69 EQ PA^ n . , o. , c . , I . ,. <;<:,. 95 , . 6, I .,. 95 , .95 , . 9 5 , 10 EC NU=7,EC SU=NIXX,fO EX = C-2.IN S1=e,32,OL ST = 9, 70 EC PA = 32 ., 80 . , 146 ., 2 . , 1 . , 1 . , 57 C . , 11 CC Nl!«f!,EC 5U»ACI.2,F.C EX» K - 2 . I N ST='),nu ST*1C, 71 EC P A . J 3 . ,125 . ,f40 . ,1 . ,1 .,6. ,560., ! 12 EO Nl = 9,EC Sl = M » K . E C EX = f - 4 , I H S T = 1 0 , 3 I , C L ST= 11 , 72 EO P A = 3 4 . , 1. , 1 3 C C C . , ICC. , I 13 EC ••lU'lO.LU = i ,EU E X - K - J . I N Sl=l I ,Ut; S I - 1 2 , 73 EC P A* 35 . , 8 0 . , I 3 C . . I . . 1 . . 1 . . 5 6 C , . i~4 EC KL = l l , E C iU = MXXTTC~TX = f-5,IN ST=12, VC, fjll ST=TTJ 74 EC PA=36., • 15 EC Nl=12,EC SU = *>CC2,EC EX = R - 4 , I N ST=13,CU S T 0 4 , 75 EQ PA = 37 . ,2 65. . | 16 TO NU-1.3.E0 SlTMIXH.rQ E X = F - 6 . I N SI-5,'11;. 35.CII ST"16, 76 EC P A» 3 E ., 16.5, 1561. , 1 . , 1. , 1. , 715. • j 17 EC MJ = 14,E0 SU=ACC2,EQ EX= R-5.IN ST06.CU S7=17, 77 Ml CF=11 ,C0 NU = 4 , 5 , 6 , 7 , 8 , 2 3,2 5,26, 2 7,31. '. 6, j 18 EC NL05.EC SL=hIX«,EC EX* * - 7 , I N STO7,26,0U S T O e . 7e ST L I * 2 , 3 ,4 , 5 ,6 , 7 , 8 ,9 . 10 , 11 , 12 .1 3 , 14 , 15, 16 , 1 7. 18. 1 S . 20. 2.1, 22 , 2 3 , 24 . 25 • I 19 EO NUO6.E0 Sl=Ar.r.5 ,FQ FX = R-6 . IN ST = l f l , C l STO-J, 79 26, 2 7 ,28,29,3C, 3 1, 3 2,2 3 .34 ,35 ,36 ,37 ,38.39,4 2,43.45 ,46.47.51,52.55, 1 20 "EC-MJ = 17,CK SU = IM / a , f G Ex = y-&, IN $1=19 , 3 7,00 5T=2C\ 80 ST"NU=i,TC fcO= 1 C C . ,"CC=2 3 .",'2C .',"18 VV 3., 1., 17V,4i,3.~ilC. i\V,C. ,FL*1". V -* 21 EC MJ = 18,EC SU = AF.C2,EQ EX = K -7 .IN 5 T = 20,0L' ST=21, 81 VA FR=0. ,TE = 550.,FP = 3QC., ! 22 EC N105.EQ SL'Mxa.EC EX = Sr = ? l , 3 f l , C l S T - 2 2 , 8 2 ST NL = 4 ' i , F L O . » ] 21 EC NC = 20.E'.J SU = Ai:u;,LQ LX- « - 0 , l N 51 = 22,CC Sl = 23 , 83 ST N U - 5 C F L - 1 . , 24 EC MJ*71 , E C SU'MXIl.FC EX= K-1C . I N ST-23,14,0U ST=15, 84 ST NU = 4 0,FL> 2 . , ': : 25 EQ NL = 3E , E 0 SU = HXFS,EC EX= IEFP , IN ST = 15,EU ST=?4, 65 ST Nl = <4 , F L - 2 . , j C C~Tju' 177 'CC _ 5t»V,\ t V Vfa " T H ' P H I 5; I K~ J T'-"2 4 , f Lr" F, T-"5 5 D7I SI N U - 4 C , FL - 2., I 27 EC KU=22,EC SU=C 1ST . EQ EX = V-<J6,IN ST = 55,CU ST=25,42 , 87 PR EC = 1, 2, 3, 36, 4 ,5 ,6 ,7 .8 ,10 ,11,12,13 ,14 ,15,16,17, l e , 1 9 , 2 0 , ! 23 EC Mj = 23,EC SU = I- n EH , EC EX*E166,IN SI = 2S ,CU S I = 26, 88 21 ,38 ,37 ,22 ,23,24,25,26 ,2 7,281 i 29 EQ NIJ = 24,FO S», = rMnK . TO EX= D-7.IN 51=76,CL S I = 77 ,39 , 89 WE ST«15 , 30 EC NU»2 i , tC SU^PLTP .LU L « = I' 342 , 1N S W N C U Sl = ^ t , 90 M*X NU-20, FK A L«C . C C 1 , » ' 31 EC NU=26,EC SU = CVCF,EQ EX= C-3.IN ST = 7fl,CU ST=29,34,51, D CF FILE  ' n ro~sc=2T;Eo"sL--avtP ,"EC _txs -u-4"; i rs i5? v <,cC"s 1^30,31,32,33^ : . 33 EC NU=28.EQ SU=nVRI>,FQ EX= 0 - 5 , IN ST = 34,rjl' S T = 35 ,36 ,37 ,38. 34 EC NU=29,EC SU = PU«F,EC EX = P292,IN ST=39,0U ST = 4C, 35 EQ NL«31,E0 SL=CIS1.EC EX**-28,IN ST»42,CU ST-45,43, OPY »SKIP 36 EC Nlj»37,EO SU=h:>ER,EQ FX = E-16,IN ST-43.CU ST-44, 37 EC MJ03.EC SU=hXEH.EQ EX=E-15,IN Sr = 45,CU ST=4t, •-71 tC M.= 34 ,EQ SL = FLI-F ,EiTTX"= P-25.IK 5T»46,CU ST = 47, : 39 EQ NU=3SEC Su=H*£R,EO EX-E168.IN SI*47,CW ST*48, 40 EC P A = 1., 1., 5CCCC, 515., 41 EC PJ=2. ,16 .5 ,1564 . ,1 . , 1 . , 1.,735., «2 EQ PA = 3 . , 1 5 . . 3 S 1 . , 1 . , 1 . , 1 . ,810. , . ' 43^  EC P 4 « 4 . , . 5 , . 5 , ' :EC 'P*'=5.', ' 5 EQ PA = 6 ..C . 9 , 9 . ,C. , C . ,C. ,0 . ,C. , 0 . , 0 . , 0 . , - l . , 0 . . . 5 . 46 EC PA=7., , 47 EC PA-8 . .0 . 6 , 6 . , C . , C . , 0 . ,0 .,0 . , - 1 . , C . , a . , C , C . , . 5, 4e EC F 4 . 9 . , 49 EO P A » 1 C . . . 7 5 . 6 . , C . , 0 . t C . . C . . C . . - l . . C . , O . . a . , 0 . , . S .  515 EC PA » l l . " i 51 EC P / W 2 . , . 9 5 , 7 . , 0 . , 0 . , 0 . , 0 . , 0.,C . , - 1 . , 0 . , 0 . , 0 . I . 5 , 52 EQ P A » 1 2 . , 53 EC P J » 1 4 . , 0 . 9 , S . , C . , C . , C . , C . , C . , C . i C . , C . , - l . , 0 . , . 5 > 54 EC F * » 1 5 . , , iS EQ PA« 16. . 0 .6 ,6 . , C . ,0. ,0 . . 0 . ,0 . , - 1 . ,0. ,0 . . Q . , 0 . . . 5 ,  \ 3^ U) i 0 I 5 6 E C P 2 C = 20 . , . 9 5 , 7 . , 6 » C . , - l . , 3 » C . , . 5 , l l » C , 57 E C P 2 2 = 2 2 . , 0 . , C . , . S B , . C 5 , 0 . , C . , C . , . 9 8 , 1 6 * 0 . , 5 8 E C P 2 2 = 2 3 . , 6 C . , 2 1 4 C . , 1 . , 1 . , 1 . . 5 6 C . , 1 8 * 0 . , 59 E C P 2 ' i ' ? « . . . 6 , « c . , J L I S T F C I Y M 6C E C P 2 5 = 2 5 . , 1 . , 3 6 0 0 9 . , 5 2 0 . , 2 1 * 0 . , 1 C L E A N , N A K E L I S T tv E C P 2 6 = 2 6 . . . 5 , . 5 , Z 2 * C . , > 2 P R E L I M I N A R Y POL YMER 17 A T1 ON P L A N ! S I H L H T l C N - f A f t C H 1 9 6 9 " 6 2 E CP 21 = 2 7 . , . 1 6 6 , . 1 6 6 , . 2 1 3 , . 2 2 3 . 2 C « C , 3 G F f L I S I 6 3 E C P 2 0 = 2 8 . , . 3 3 3 , . 3 3 3 , . 1 6 6 , . 1 6 6 , 2 C * C , K P K l = l , ' P L P P ' , , P 2 e 9 ' ,1 . - 2 . 5 * 0 , 64 E C P 2 9 = 2 9 . , l . , 1 3 C C C . , 3 C C . , 2 l * 0 . , 5 K P M 2 = 2 , ' I - X E * ' , ' E 1 6 2 ' , 2 , - 3 . 5 * C , 6 5 E C P 2 1 = 3 1 . , C . , C . , 1 . , . 5 9 , . 9 5 , . 6 , 1 . , . 5 5 . . 9 5 , . 9 5 , 1 4 * 0 . . 6 *?>•:: = . » , ' i - x t f , • u 6 i , , 3 , - 4 , ' j « o , 6 6 C C P 3 2 = 2 2 . , U 0 . , 1 4 6 . , 2 . , l . , l . , 5 7 C . , l ( J « d . , 7 K F f ? 6 = 3 6 , • H X3 • , ' l < - l • , 4 , 5 V , - 5 2 . 4 * 0 , 6 7 E C P 3 3 = 1 2 . . V 2 5 . , 7 4 C . , l . , l . , . 6 , 5 6 0 . , 1 8 * 0 . , a K P V ' = 4 , ' U V L H ' , 'C- 1 ' , 5 2 , - 5 , - 6 , 4 * 0 , 68 EUP 2 4 = 2 4 . , 1 . , 1 3 C C C , I C C , 2 1 * 0 . , t • K F P 5 , 5 , ' » ' I X R ' . ' M - ; , , 6 , 5 C . 3 3 , - 7 , 3 * C , 69 E C P 3 5 = 2 5 . , P 0 . , l l C . , l . , 1 . . 1 . . 5 6 C . , i e * 0 . , 1 0 K F C 6 = 6 , ' A L X 2 • , M '-2 ' , 7 , - 8 , 5 * C , 70 E C F 3 7 = 3 7 . , 2 6 5 . , 2 3 * 0 . , 11 KPM 7>7 . ' C I X « ' . • f - ! • «a . 3 2 , - 9 . 4 * 0 , 71 E C P 3 E - 3 8 . , 1 6 - . 5 . 1 5 6 4 . , 1 . , 1 . > 1 . | 7 1 5 . , 1 8 * 0 . . 1 2 K P M £ = P , ' A C C 2 ' , • « - ; • , 9 , - i c , 5*c. 7 2 £ E N C 1 3 KF*'9 s ' l . ' f M X H ' . ' M - 4 ' . l ' . ) . $ 1 . - 1 1 . 4 * 0 , 7 i t S t X L S l 1 * K F e i C s l C . ' / C l ; ? . ' , ' H - 3 ' , 1 1 , - 1 2 , 5 * 0 , 74 S E X 1 = 1 . , 0 . , l C C . , ? 3 . , 2 0 . , l 8 . . 3 . , l . , 1 7 . , 4 . , 3 . , 1 C . , 1 . . C . , S * C , 1 5 KPM 11 = 1 1 , 'MI XK ' , ' f - 5 ' , 1 2 , 3C , - 1 3 , 4 * C , 7 5 S h A P E = 2, 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 , 2 1 , 2 2 , 2 3 . 2 * . 2 5 , • 1 6 K F H 2 = l 2 . , A n n ? , , , P - 4 ' , 1 3 , - 1 4 , 5 * C , 7 6 2 6 , 2 7 , 21 , 2 9 , 3 C , 3 1 . 3 2 , 3 3 , 3 4 , 3 5 , 3 6 , 3 7 . 3 8 , 3 9 , 4 2 , 4 3 , 4 5 , 4 6 . 4 7 , 5 1 , 5 2 , 5 5 . 1 7 K F > ' 1 3 = 1 3 , , M X H , , ' K - 6 ' , 5 , 4 9 , 3 5 , - 1 6 , 3 * 0 , 7 7 El END i e K P f l i = 1 4 , ' A ( ) D 2 ' , ' P - 5 ' , 1 6 , - 1 7 , 5 * 0 , • 7e C S 1 N L S 1 1 19 KP" 1 5 = 1 5 , • « I Xii • , • f - 7 ' , 1 7 , 36 , - 1 8 . 4 « C , 7 9 S I N 1 = 1 . , 1 . , 0 . , 5 5 C . , 3 C C . , 5 * C , 20 K F M 6 = 1 6 , ' A C C 2 < , ' P - 6 * , 1 H , - 1 9 , 5 « C , 80 S 1 N 4 9 = 4 9 . , 1 . , U » 0 . , ; I K P M n - n . ' C I x ^ ' . ' K - B ' , 1 9 , 3 7 , - 2 C , 4 * 0 . 81 S I N 5 C = 5 C . , 1 . , E * C . , 22 K P H l E = l F , ' A r ) C 2 , , , R - 7 ' , 2 C , - 2 1 , 5 * C , 8 2 S I N 4 C = 4 C . , 2 . , 8 * C , 2 3 K f C 1 9 = 1 9 , ' M X H , , , ^ - 9 ' , 2 1 , 3 a , - 2 i 1 4 * C , 83 S l N 4 4 = « 4 . , 2 . , e * C . , 2 * K F f 2 C ' 2 C . ' A C t l 2 ' P - n ' , 2 2 , - 2 3 , 5 * 0 , 84 SI N48 « 4 8 . ,2. , 8 * 0 . , 2 5 K.PM21 = 2 1 , ' M I X K ' , ' M - 1 C ' , 2 3 . 1 4 , - 1 5 , 4 * C , r 85 S N A P E = 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 C , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 , 2 1 . 2 2 , 2 3 . 2 * . 2 5 , 26 KF »• 2 C • V : , * i X U < ' M M / " , 1 5 , - 2 4 , 5 4 0 , 0 6 2 6 , 2 7 , 2 8 . 2 9 , i i , 3 1 . 3 2 . 2 3 , 3 4 , 3 5 , 3 6 , 3 7 , 3 8 , 3 9 , 4 2 , 4 3 . 4 5 , 4 6 , 4 7 . 5 1 , 5 2 , 5 5 , 2 7 K F M 7 = 3 7 , ' V A L V ' , ' P I U 5 ' , 2 4 , - 5 5 , 5 * 0 , 8 7 £ £ N C 2e K P M ; 2 = 2 2 , , U I S 1 * , " V - « t * , 5 ' . - 2 5 , - 4 2 , 4 * 0 , ee t K Q L I S I 2 9 K PM 21 = 2 3 , , M E ' \ ' , , F I 6 6 , , ? 5 < - 2 6 1 5 * C , 89 K E 2 = 1 , 2 , 3 , 3 6 , 4 , 5 , 6 . 7 , E , 9 , l C . l l , 1 2 , 1 3 . 1 4 , 1 5 , I t , 1 7 . 1 8 , 1 9 , 2 0 , 2 1 . 3 8 , 3 7 . 2 2 . 3C Kf^24=2 4 , ' L Y U I t , , ' E - 2 , , 2 6 I - 2 7 , - 3 S , 4 « C , 9 0 2 3 , 2 4 . 2 5 . 2 6 , 2 7 , 2 0 , 1 9 * 0 , 31 K F V 2 ; = 2 5 , , P l l ' P ' , ' F 3 4 2 ' , 2 7 . - 2 8 , 5 * 0 , 91 L C C P S » 2 C , K E 4 « 1 5 , 9 » C . C E f l f > C R " C . C 0 l , 32 K F M 2 t = 2«\ "CvCA 1 , ' 0 - 3 ' f 2 8 , - 2 9 . - 3 4 , - 5 1 , 3 « C , 9 2 SEND 3 3 K F ( - 2 7 = ? 7 , ' C V C P , , , C - 4 ' , 2 9 , - 3 0 , - 3 1 , - 32 , - 3 3 , 2 « C , ' •C C F F I L E 34 K F K 2 E =2 e . 'UVL'H • , , C - 5 ' , 3 4 , - 3 5 , - 3 6 , - 3 7 , - 3 8 , 2 * 0 , 3 5 K P f 2 S = 2 < ; , ' P L M P ' , ' P 2 9 2 ' , 3 9 , - 4 3 , 5 * C , 3 6 K F M l = 3 1 , ' C i S T " , ' V - 2 E ' , 4 2 , - 4 5 . - 4 3 , 4 « 0 , 3 7 KF I '32 = 3 2 , ' P X E K ' , ' E - U > , , 4 3 , - 4 4 , 5 » 0 , C P Y * $ K I P 28 KF>< 2 2 = 1 3 , ' H X t R ' . ' E - 1 5 ' , 4 5 . - 4 6 , 5 * 0 . 3 9 K P M 2 4 = 3 4 , ' P U K P ' , ' P - 2 5 ' , 4 6 , - 4 7 , 5 * 0 , , 4 0 K F M 5 = 3 5 , ' l - X C M ' , , E l 6 n , , 4 7 f - 4 8 , 5 « 0 , 4 1 N C C C f P * 11 . C O H P M = 4 , 5 , 6 , 7 , 6 , 2 3 , 2 5 , 2 6 . 2 7 , 3 1 , 5 6 . 9 * 0 . 4 2 C E N C 4 3 6 E C L I ST 44 E f - A K . 3 6 , 5 , 7 , 9 , 1 1 , 1 3 , 1 5 , 1 7 , 1 9 , 2 1 , 4 0 * 0 . 4 5 E C P 1 = 1 . , 1 . , 5 0 C C C . , 5 1 5 . , 2 1 * C , 4 6 E C P 2 = 2 . , 1 6 . 5 , 1 5 6 4 . , l . , l . , l . , 7 3 5 . , i e » 0 . , 4 7 E C P 3 > 3 . , 1 5 . , 3 9 1 . , l . , l . , J . , f ! 1 9 . , l f l * C . , 4 8 E C ? 4 » 4 . , . 5 , . 5 , 2 2 » C . , 4 9 C C » 6 = 6 . , 0 . 9 , 9 . , l ; * C . , - l . , C . , . 5 . l l * C . , TC E C T 8 = 8 . , . 6 , 6 . , 5 » 0 . , - l . , 4 * 0 . , . 5 , l l » C T ; 51 E C P 1 C * 1 C . , . 7 5 , 6 . . 5 * 0 . , - 1 . , 4 * 0 . , . 5 , 1 1 * 0 . . 5 2 E C P 1 2 = 1 2 . , . 9 5 , 7 . , 6 * 0 . , - l . , 3 * 0 . , . 5 , 1 1 * 0 . , 5 3 E C P 1 4 . 1 4 . , . 9 , 9 . , e » C . , - l . , 0 . , . 5 , 1 1 * 0 . , 54 E C F U > 1 6 . , . 6 , 6 . , 5 * C . , - l . , 4 * C . , . 5 . 1 1 * 0 . , 5 5 E C P U - 1 6 . , . 7 5 . 6 . . 5 * 0 . , - 1 . , 4 » C . . . 5 . 1 1 * 0 . . \ . • I • o n rg « «%. o o * • -»o <r •-a; • • • • <-» C7 "g o f • • ^  "\ o I » • rsj «; u i o • • tn ^ • • • • • "u <o| »u r» » m • • f- m i • gj •> O <-0 »0 U"> • • N ifl ^  • fM > DO • Cr • •: •> M rg • • • O * •O « •-OCT' " « <-> • O o n o* • — —-• o • • o • • " o •* ^ "i i • r** • —» • ^ \ i— "Mo * * • in. *  • ^  j<N rg • , u r\ ' • u . • ^ C J OJ -< « • ( INJ LA ' • •> i • -4 rg i » o o in O -— m •  j> * > m •» w UJ <J » » . o g v - w «. .—t rg rsi rsJ i • cr u o ^  (j rg >r m • lAj r\ r\ f>g a. u- u- c~ rg !"g rg rg m «i m r-1 r>t r- CO CL a. 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U U a J lU J UJ gr . o.i_#mnwj/^g'pj/g'g'^  g* «r w> c_> <_> o " o " T • * » » gj gj gj gj «w gj 4 gj <0 HJ gj »*»,»• CJ o o o o o • • • • • u u o u u o u u o o «J o u o «-J t> o U CO O CD CO CO (% CM r\j rg rg rg u u o u o o u u o u u o o O (JU CJ w UJ UJ cc cu CU CO i» ifl m w m >-» O <-J O O O rg I\J rg rg rg rg u> g^  in m ir> \r\ CM rg rg rg rg rg «-> o o *-» o o » < • • • • • r> m m fn (*i m >T> m •-* ro *n • • • • • cj O rg •gr «o aj CP H M « M 1  o (M -r -c 1 •3  *-i n a a a a. a. uLi j u *J u UJ UJ UJ 1U Url ) Nm «| u« \ m  in m «•» •> n* • • tj- - • • n# o * » o* j\ (M j • t*> • <M * :r- » eg i • r~ — 'o ^ o* : * • oj . • • oj fS. rs» w . ; II II » 1/1 w r- —J • •X a. oj (j , •3J aj # # * sO 2U" UJ OJ (M CM H C I o. -TNJ -r h- II II |i li I M «M -* ei *r i Z rr; ci ~ -« Z Z 2 «n to \n ssi ui I'M C < • * ^  UJ 1/1 (\ U w a. * 0 U J ~* a CO CT r- *M •o » • *r VTi U J -n • • o - CN O* 1 » I ' H V ) • a. o (MOZ U J U U J 1 J u k • «• * • <j ir. ^  ui» ' •* » • i .m r\j o >r » i i r c •*»« . *. - —.; | i (\ » » * » N* U> ' » * —« » sf VP I * (T y M in i *— »M CO » l_> I IN' CT — a. cf x a i"n : 3 >• X — ^ • Q- — j a. :<t : • I  B M tl | I I | —* CM I* > u^i : vU p* H »- ». 1 J . u. a. u. u. u a. i at ^ * OD tr ii 'i )| H l_) _ CO IT —• —• *. x a: x U. U. CL -J-ac x j£ it »uc«>o * o i in IT | in f\ \r\ t . rn ui ^  : — — w r* —• v i • <-i -o o j * O O » oj * -» 1—. « • » in r m m m u> L"\ 1 . — • — I *w —* c  t I » I p I i • * 1 pn in «o co ! • o r\j N (\ \ N c* n I - u. I •j- a. • I U J . I — I C tL ' • - • - I * • - • IM U J >c m w -_p ; C OJ •—• * I <M I I l|| UJ ex > UJ U J c ( •JJ H J » o o • • (J o • «-> o • CJ c cu a> « i" I»I —« CD - «• • • • • ° M? * • " o o CN. » — OJ OJ • .wo • M • » _ —« 1 u1 <r • * • •> • — KJ O <_* —• • m y\ y u o * cr — : > *- e « a ct o. a tf ir o- x i •>uico.'Jo.uxiQaxi t it OJ « J x a; Li. U-ac X I  II I I  I m in _j oj ci >y -*J v* rj ci Jo sj- ki\ r> oj >M o. eg <M oj fv m i»* p> m cu.a,a.u.u. a u. u a c a a hi ov c< ^  co u <\ y r-u ]^ a a a a %f\n-o'*-AJC7,o'~*fN''*>,>fw^ _l ^  -^ OJ IM IM M OJ IM ^ j g i^ UJ U J i • i-' • «o r-<J * o • m * m » H U (\ u -r o • • o — u *» M Ct NJ- U\ O CJ» »M CI I  IM r\J (M OJ O) CI fl vJCI«J«JUU*JOvJ ^ m »o r* tu fi> sr >x «r «r • * • •» O oj «_> • » u ri il it ii it ct ct OCJ a a a a 4 U J VJ CJ \ J UJ VI LU U J LU LU V9 H I S T FCLY-1 e 1C n 12 l J "T4"~ 15 16 1 7 18 19 "25" 21 2 2 23 24 2! "2 6" 27 2S 29 30 31 "1T" 33 34 3 c 36 37 39 40 41 42 43 4 5 4 6 4 7 4 6 49 r -PfJL STP LTliVT--TTi'P"7EI'ilE R 1011" CLEAN .MAKELI SI ""FINAL SlMiAATICrt-G F>* L I ST K F f ' l = l , , F U C P ' , , F 2 a 9 , , l , - 2 , 5 » 0 , K P C 2 = 2 . ' A O O 5 ' . ' E 16 / ' . 2 , - 3 , 5 * C , K F K 2 = i . , ' ( ! L ; C 5 , , , E l t l , , 2 . - 4 , 5 ' C , K F C 5 = r . . • C [XH ' , " P - 1 • , 4 . 1 9 , - 5 , 4 » Q , _ K ' F K t - 6 , ' A u U 3 ' , ' W - l ' , 5 . - 6 ,S*C K P M 7 - 7 , "M1XK ' , •!»-• 2 • ,#• , ? C - 7 , 4*C » K F P 8 = 3 , , A E C 3 , , , l » - 2 , , 7 , - 8 , 5 * 0 , K P C S - , ; , , f I X ' < , . , C - 3 , , e . 2 1 . - 9 , 4 » C . KP*!1C=1C, ~K~FTT2~=T2~, K P P 3 £ = 3 8 , K P « 2 7 - 3 7 , K r-f 22 =22 , K ? H i 2 = 2 2 , KPM7*= 74 , ~K f f-'2 5 =2 5 , X F f 2 6 = 2 6 , K P M 2 5 = 2 ? , K F * 2 , = 2 9 , KFI- II - 3 1 , K F M 2 2 0 2 , " T ( P ^ 2 7 ' 3 3 , K F H 3 6 =36 , K P H 2 * * 3 4 , KP>125 = 3 5 » ^CCCl•P=^l t E \ C ACC3 ' , ' R - 3 1 . 9 , - 1C ,5 * C . M X S " , ' f - 4 ' , l C , 2 2 , - 1 1 . 4 « C TSTITP . « C - 4 " , l l . - 1 2 . ' > * C " ADD5• , • lECF",12 ,-13 .5*0, V A L V ' , ' P i i i ; ' , 1 3 , - 1 4 , 5 * 0 , ALC4•,"V-96",14,-15,-25,4*0, AUU5 " , '8166',15,-16,5*0, n v r . « , , M ) - l 1 , U , - 1 7 ,-2 3 ,-34, 3*_C_,^  PUCF•,"F342•, 17,-18,5<0, C v O ' , ' C-7' ,18,-19,-20,-21 ,-2 2,2*0. PLCP "P29?" .23 ,-24 ,5*C, V»LV * , ' V «26", 25 , - 35 , 5 « 0 . ACL4' , 'V-78",35, -28, -26,4*0, ADOS ' , • E- 16•.26 ,-2 7 , 5 « C , A C C D • , • z-1'.•;2ev- 2 <;,5*0; CVCR•, 'C-3",29, -30 , -31 ,4*C, PLI'P", , F-25',30 ,-32 ,5*0, AOC5","E-16",32 ,-33 ,5*C, CCMPNt = 4 , 5 , 6 , 7 , a , 2 3 , 2 5 , i 6 , 2 Y , 3 l , 5 6 , S » C , ~~EECLI"5T E N A M F - 5 , 7 , 9 . 1 1 . 4 6 » C , ECF1=1. ,1. ,50CC0.,515. .21*0. , ECP2=2.,71C.,23*C., ECP 2= 3 . , E 1 5 . . ? 2 * C 56 57 58 59 60 61 63 64 65 66 67 "6 8"' 69 70 71 72 73 CtNC CSEXLST SEX 1 = 1. C O . ,25C. ,5 7. 5 ,50. ,45. ,7 .5 ,2. 5. 4 2 . 5 . 1 C . 7 . 5 , 2 5 . ,2 .5 , C-.<;*C., S.N AM E = 2 , 3 . 4 . 5 . 6 , 7 , 8 . 9 . I C I I . 1 2 . 1 3 . 14,15.16.17.18.19,20.21,22,2 3 .24,2 5. 26,27,28, 29, 30.31,32,33,34,35,66*C _ C E N C C "S I NL S i SIM1=1 . ,1 . , 0 . , 55C. ,3CC. ,5*C. , S IN24=24 . ,2 . ,8*0 . , SIN2 7 = 27. ,2. , £ * C . , SIN21 = 2 1 . , 2 . , 8 « C . , S IN3 2 = 3 3 . ,? . , 8 * C . , |4 75 OF FILE "SIN 2* = 3 4 . , 2 . | E * C. , SNArF*/ ,2 ,4 ,5 ,6 ,7 ,8 ,9 ,1C,11,17,13,14.15,16,17,18,19,20,71,72.23,24.25, 26,27,28,29,3C,31,22,33,34,35.66*C. CENC SKELIST K E 2 ° i , < , 7 . e . 9 , l C , l i , 12,3 6 .37.22 , 2 3 , 2 4 . ? ; , 2 6 , 3 5 » C ,  LCCFS=20,KE4«12.9*0,CERRQR*0.01, CENG "TY~*TKTr  = 3 . , c l . , / 3 * ( - . , ECF6=6 .00,1 .67 ,5.0 , 1.0, 2.0, 520 . C , 3 e C C C . , 2 e C C C . , 0 . 6 , C . q . l . C , 1 4 * C . , TrPe = fi . C C ' , J . T T , 5 . C ,1 .C,2 .0 .5TC.C,38COO., 2 8 0 C O . , 0 . « , C . C , 1 . C , 1 4 « C . , C?1C= 1C., 5. CC, 5. C, 1. C.2. C S 2 C . C 3 8 C C 0 . , 2 8 C C C , 0 .6 ,0 .0 ,1 .0 ,14*0. , CP12=12. ,5.22,5.C,1.C,2.C,52C.C,38CC0.,2 6 C C C . , C . 6 , 0 . 0 , 1 . C , 1 4 * C , CF38=2a. . 715..23*C.. E E P- -E P38=2a. , . ,23*C E C P 2 7 = 3 7 . , 2 6 5 . , 2 3 » C . , 2 6 5 . , 2 3 » C . , FCP?2*27. ,47. ,C.75, 5 8 7 . C . 7 1 4 . C , C . S S , 0 . 0 1 . 0 . C t l . C i C . 0115*0.0, Kf23=23 .,552.,23«G., "" ECP24 = 2 4 . , C . 6 , C . 4 , C . C . 2 1 * 0 . » ECP 25=25. , 1 . , 5 C C C C . , 5 3 C 21* C , C(P26-2 6..0 . 2 . C . 3 , ' J . 3 , Q . 2 , 7 0 * C , ECP29 = 2 9 . , l . , 1 3 C C C . . 3 C C , 2 l » C . , E C P 2 S = 3 S . . S C . , 2 3 « C TO ETP31 = 2 1 . , 2 4 . , t . 6 C 5 ? C . C , 7 2 C . C , C . S S , 0 . 0 7 . C . C - l . C , C . C 1 5 * C . C 51 ECP3202. ,550..23*0 . , 52 ECP22-23. ,567. ,23*0., 5 3 E C P 2 6 » 2 6 . , l . , C . , 2 2 * C , 54 ECP34-34..1 . ,13000.,100.,21*0., 5 5 ECP35-35. , 5 5 0 . , 2 3 « C . ,  3 \ a. x o vi — w LM • Z U J c  — • a r- < -* oc <-» a. * — — cj  i a. 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' : ~ FIMAL 5 I ML LA TICK P O L Y PLANT— SEPIEceER 1971 ' P R O C E S S V E C T O R S ' E Q U I P M E N T S T R E A M N U M B E R S NCM8ER S C B R O L i I N E N A f E 1 FUMP P 2 8 9 1 - 2 0 a 0 2 A C C 5 E l 62 2 - 3 0 0 o • • • • • « • • • U N I V E R S I T Y C F H C U S T C N 3 A C C 5 E 1 6 1 3 - 4 c c C • • • • • • C H E M I C A L E N G I N E E R I N G S l f U A T I C N S Y S T E M . 5 M X R H - l "« 19 - 5 C c • • • • • • I 6 ADD 3 R - l 5 -t c 0 0 1 7 M X R M - 2 i 2C - 7 c c 1 8 A C D 3 R - 2 7 - 8 0 0 0 I 1 9 MIXR M - 3 8 2 1 -9 0 0 ( 10 A C C 3 R - 3 9 _ - 1 0 0 _ 0 _ C 11 MI >R M - 4 1 0 22 - 1 1 0 G I 12 AC C 3 R - 4 11 - 1 2 0 0 C 22 ADC4 V - 9 6 14 - 1 5 - 2 5 0 0 1 23 A D 0 5 E 1 6 6 15 - I t C 0 0 " - • 2 4 CVCR C - l 16 - 1 7 - 2 3 - 3 4 0 25 P L f P P 3 4 2 1 7 - 1 8 0 0 0 2 6 CVCR C - 2 16 - 1 5 - 2 0 - 2 1 - 2 2 — - ~ " " ~ 2 5 F U K P P 2 9 2 2 3 - 2 4 0 0 0 31 A O D 4 V - 2 8 35 - 2 6 - 2 6 0 0 32 A C C S E - 1 6 2 6 - 2 7 0 0 c 33 A C C 5 E - 1 5 2 8 - 2 5 0 0 0 34 PUMP P - 2 5 3C _ - 3 2 _ 0 0 0 35 A C C S E - 1 6 3 2 - 3 3 0 c 0 36 CVDR D - 3 29 - 3 C - 3 1 0 0 3 7 V A L V P R 1 5 13 - 1 4 0 0 c • 2e A C C S T E P P 12 - 1 3 0 0 0 39 V A L V V » 2 8 2 5 - 3 5 0 0 0 CO CV) 28 31 33 • STRE*C CCNhECTlCNS' ' 29 3 3 36 STREAK ECU IPMEN7 FRCP TC 30 36 34 ( 1 0 1 31 36 C 1 2 32 34 " 3 5 " ~" " ~ 3 2 3 33 35 C ' 4 3 5 34 24 0 5 5 t 35 39 31 J 6 6 7 7 7 8 1 8 8 9 | 5 9 10 i 10 10 11 j 11 11 12 12 12 38 13 38 37 1 14 37 22 15 22 23 16 23 24 17 24 * 25 " ' " " " • — - - • - • -18 25 26 19 26 5 2C 26 7 21 2ft 9 22 2' 11 23 24 29 24 29 0 25 22 39 26 31 32 27 32 0 \ ' F I N A L S I M U L A T I O N POLY p i A N T — SEPTEMBER 1971 ' I N P L T C A T A ' *CTHEP S Y S T E M V A R I A B L E S ' F I N A L J I M L L A T I C N P C L Y P K N T — S E P T E M B E R 1 9 7 1 NUMBER CF COMPONENTS 11 ' S T R E A M SUMMARY* ( NLMEER CF ITEMS IN R E C Y C L E LISTIKE2) 15 STREAM NLMGER 1 2 5 4 NUMBER CF I I C M S I N R E C Y C L E L I S K K E 3 ) 0 E Q U I P . C O N X I C N FR 0 TO 1 FR 1 TC 2 FP 2 IC 3 FR 3 T0 5 FR 5 TO 6 CCMFCNENT NUMEERS USED 4 , 5 , 6 . 5 6 , , 7 , 8 , 2 3 . 2 5 . 2 6 , 2 7 , 3 1 V A F C P F P A C T I O N T f M r f K A T L R C . fl 0 . 0 5 5 0 .OCOO C C o.o 0 . 0 C .c C O C C 0 . 0 C C P R E S S U R E , P S I A 3 0 C C C C C 0 . 0 0 . 0 0 . 0 C C R E C Y C L E L I S T K E 2 5 , 6 . 7 . 2 2 . 2 3 . 2 4 , B . 9 . 1 0 . 11 . 2 5 . 2 6 . , 1 2 , 3 6 , 3 7 E N T H A L P Y , BTU - 2 5 6 8 7 3 . G C C C C C c.c C O 0 . 0 S T R E A M S L S E C IN C O N V . ROUT 1 N £ ( K E 4 ) 1 2 . C O M P O S I T I O N , L 8 - M O L E S / H O U R PROPANE 5 7 . 5 C C C 0 . 0 0 . 0 0 . 0 C C T C L E P A N C E ? ' C E R R C K ' 0 . C 1 C Q l - l ' L 1 ANE 5 C . C C C C C C O . C C C 0 . 0 N - D L T f N E 45.OCCO 0 .0 C C C C C C M A X . L O O P S I N R E C Y C L E C A L C . 20 I - P E N T A N E N - P E . N I A N E 7 . 5 C C C 2 . 5 C C C 0 . 0 C C 0 . 0 0 . 0 C C 0 . 0 C C C C F R C F Y L E N E 4 2 . 5 C C C C C O . C C C C O C I S - 2 - B L T E N E 1 0 . O C O O 0 .0 0 . 0 C C c c T R A N S - 2 - B L T E N 7 . 5CCC C O 0 . 0 0 . 0 . cc I - E U T E N E T R - 2 - P E N T E N E 2 5 . 0 C C C 2 . 5 C 0 0 C C 0 . 0 O . C 0 . 0 C O C O C O c.c 1 - H E P T E N E C C 0 . 0 0 . 0 C C C O t TGTAL 2 5 C C C C C O . C O . C C . C 0.0 i i S T R E A M NUMBER 6 7 f 9 10 E C U I P . C C N X I C N FR 6 TO 7 FR 7 TO 8 FR 8 TO 9 FR 9 TO IC FR IC TC 11 VAPCR F R A C T I C N C O 0 . 0 0 . 0 C . C C C T E M P E R A T U R E , R C C C C 0 . 0 0 . 0 C O P R E S S U R E , P S I A 0 . 0 C .C O . C C O C . C E M H A L P Y , BTU 0 . 0 0 . 0 0 . 0 C . C C . C C C K F C S I T I C N , L B - M C L E S / F C L R FRC PANE 0 . 0 C . C C O C . C C . C I - O L T A N E 0 . 0 0 . 0 0 . 0 C . C C . C Ni -BL TANE C . C O . C 0 . 0 0 . 0 C . C I - F E N T A N E 0 . 0 C . C O . C . C C C O N - P C N T « N C 0 . 0 0 . 0 0 . 0 C . C C C P K O P Y L E N E C . C 0 . 0 0 . 0 0 . 0 C . C . C I S - 2 - P U T E N E C O ^ C C C C O 0 . 0 i T R A N S - 2 - R U T E N C O OTO CTC C O " Ct 1 I - B L T E N E C C 0 . 0 0 . 0 C . C C . C T R - 2 - P E N T E N E C . C C O 0 . 0 0 . 0 C . C i 1 - h E P T E N E 0 . 0 C . C O . C C C O . C ! T C T A L 0.0 0.0 O . C CO c. c r o 00 TCIAL c o oTo ; 6~io ere rnr S T R E A K K l i P E E R 11 12 13 H 15 S T R E A K N L K B E R 21 22 2 3 2* 2 5 E Q U I P . C Q N X I C N F R 11 TO 12 FR 12 TC 3F. FR 38 1C 3 7 FR 37 TC 22 FR 22 TO 23 V A F C P F R A C T I O N 0.0 C C O . C C O 0.0 E Q U I P . C O N X I C N FR 26 TO 9 FP 26 TC 11 FR 2A TC 2 9 FR 29 TO 0 FR 22 TO 39 TEC P E r U l U R E . R 0 . 0 0 . 0 o . c C O c. c V A F C P F P A C T I O N 0 . 0 C C O . C C C 0 . 0 P R E S S U R E . P S I A C O 0.0 0.0 0.0 C O T E M P E R A T U R E . P 0.0 0 .0 0.0 C C C C E N T H A L P Y , BTU C C C C O . C 0.0 0.0 P R E S SLRE, P S I A C C 0.0 0.0 0.0 C O E M F A L P Y , CTU C C C C 0.0 C O 0.0 C C M P O S I T I O N , L B -MCLES/HOLR C C M P O S I T I O M , L B - M C L E S/HOLR P R O P A N E C O 0 . 0 0.0 0 . 0 0 . 0 I - E L T A N E C C C C O . C 0.0 0.0 P R O P A N E O . C 0.0 0.0 0.0 0.0 N - E L T A N E 0.0 0.0 0.0 C C C O I - C U T A N E C C O . C O . C • C O 0.0 I - P E N T A N E C O 0.0 0 .0 C O C C N - B U T A N E 0.0 0-.0 0.0 C C 0.0 N - P E N T A N E C C C C 0.0 0.0 0.0 I-PENTANE C C 0.0 0.0 C C C C F R C F Y L E N E 0.0 C C O . C C C C C N-PENTANE C C C C 0.0 0.0 C O C I S - 2 - B L T E N E 0 . 0 0 . 0 0 .0 C C C . C " F K C P Y L E N E 0.0 C C O . C C O C C 1R AN S - 2 - B L TEN C C C C 0.0 0.0 0.0 C I S - 2 - B L T E N E 0.0 0.0 0.0 C C C C I -euTENE 0.0 C C . 0. 0 C O 0.0 T R A N S - 2 - B L T E N C C C C O . C 0.0 C C T R - 2 - P E N T E N E 0.0 0 .0 C O C C C O I - E U T E N E 0.0 C C O . C C O 0.0 1 - H E P T E N E C C 0.0 0.0 0.0 C O T R - 2 - P E N T E N E 0.0 0.0 0.0 C C O . C 1 - H E P T E N E C O 0.0 0.0 C C C O TOTAL C C C O 0.0 C O 0.0 T O T A L C C C O 0.0 C O 0.0 S T R E A M NUNEER " 16 17 le 19 20 S T R E A K NUMBER 26 27 2 f 2 9 30 E C U I P . C C N X I C N FR 23 TO 24 FR 24 TO 25 FR 25 TO 26 FR 2 6 TO 5 FR 26 TC 7 VAFCR F K A C II C N C O 0 . 0 0 . 0 C C C C E C U I P . C C N X I C N FR i t Yo 3 2 FR 32 TO 0 FR 31 TO 33 FR 53 TO 36 Fft it TC 34 T E M P E R A T U R E , R C O C O 0 . 0 0 . 0 0 . 0 VAPCR F R A C T I C N C C 0 . 0 0 . 0 C O C C P R E S S U R E , P S I A 0 . 0 C . C 0 . 0 0 . 0 0 . 0 T E M P E R A T U R E , R C C C O 0 . 0 0 . 0 0 . 0 E N T H A L P Y , BTU 0 . 0 0 . 0 0 . 0 0 . 0 C C P R E S S U R E , F S I A 0 . 0 C C 0 . 0 C C 0 . 0 . E N T H A L P Y , B T L 0 . 0 0 . 0 0 . 0 C C C C C C K F C S I 1 I C N , L B - F C L E S / H C L R C C K F C S I I I C N , 1 6 - P C L E S / F C L ' R FRC PANE 0 . 0 C O O . C C O C O I - B L TANE 0 . 0 0 . 0 0 . 0 C C C C PRCPANE 0 . 0 C C O . C C O C C N - 6 L T A N E C C 0 . 0 0 . 0 0 . 0 C O I - B L T A N E 0 . 0 0 . 0 0 . 0 C C C C I - P E N T A N E 0 . 0 C C O . C C O 0 . 0 N - B U T A N E C C C O O . C 0 . 0 C C N - P E . S T * N E 0 . 0 0 . 0 O . C C O C . C I - P E N T A N E o . o C C O . C C O C O P K U P Y L L - N E C C 0 . 0 0 . 0 0 . 0 o .u N - P E N t « N £ ! 0 . 0 0 . 0 O . C C C C C C I S - 2 - e u T E N E C C C C O . C C O 0 . 0 P R O P Y L E N E C C 0 . 0 0 . 0 0 . 0 O . C T R A N S - 2 - B U T E 0 0 . 0 . 0 . 0 0 . 0 C O C O C I S - 2 - E U T E N E C C C C O . C C O 0 . 0 I - 8 L T E N E 0 . 0 0 . 0 0 . 0 C O C C 1R A N S - 2 — B U T EK 0 . 0 0 . 0 O . C C O C O T R - 2 - P E N T E N E C C C C 0 . 0 0 . 0 O . C I - B L T E N E C O 0 . 0 0 . 0 C C C C 1 - h E P T E N E 0 . 0 C C O . C C O O . C T R - 2 - P E N T E N E C C C O 0 . 0 0 . 0 C C \ ro 00 : o : 1 - H E P T E N E c o cTc c c ~8T6" C7B • INPUT C<TA« T C T A l 0.0 0.0 0.0 C O C O F I N A L S I K U L A T I C N P C L V P L A N T — S E P T E M B E R 1 9 7 1 * E Q U I P M E N T SUMMARY - E Q U I P M E N T L I S T * E C . * E X T . NAME S U B . NAME - • " 1 P 2 8 9 F L M P "' STREAM NUMBER 31 32 3 3 34 35 2 E 162 A C 0 5 ECU I P . C C ' J X I O N FR 36 TO C FR 34 TO 35 FR 35 TC 0 FR 24 TO 0 FR 39 TO 31 3 E 1 6 1 A C C 5 ; V A P C P F P A C T I O N 0 . 0 C . C O . C C O 0 . 0 T E M P E R A T U R E . R 0 . 0 0 . 0 O . C C O C . C 5 M - l M1XR j P R E S S U R E , P S I A C C C O 0 . 0 0 . 0 C O j E N T H A L P Y , eTU C . C C . C O . C C O 0 . 0 6 « - l A C C 3 C O M P O S I T I O N , L B - M O L E S / H O L R 7 M-2 M I X R j PROPANE C O 0 . 0 0 . 0 0 . 0 C O 8 R - 2 A C C 3 I - P U T A N E C . C C . C C C C O • 0 . 0 • N - 6 L T A N E 0 . 0 0 . 0 O . C C . C C O 9 M - 3 MIXR I I - P E N T A N E C C 0 . 0 0 . 0 C C C . C N - P E N T A N E . . 0 C . C 0 . 0 0 . 0 0 . 0 10 R - 3 A D 0 3 • '; F R C P Y I E N E 0 . 0 c .c O . C C O C . C C I S - 2 - B L T E N E 0 . 0 0 . 0 0 . 0 C O C . C 11 M - 4 M 1 X X TP.AN S- 2 - B L T E N C . C C C 0 . 0 0 . 0 C O I - E U T E N E 0 . 0 C C O . C C O C O 12 R - 4 * C C 3 i T ^ - 2 - P E H E N E 0 . 0 0 . 0 o . c c.c C O 1 - H E P T E N E C . C 0 . 0 0 . 0 C . C C O 22 V - 9 6 A C C 4 T O T A L C C O.C 0.0 C O 0 . 0 2 3 E 1 6 6 A0D5 . | « 24 C - l CVCR 2 5 P 3 4 2 FLMF " " ' ' ' ~ 26 C-2 OVOR 2 9 P 2 9 2 PUMP 31 V - 2 8 A C C 4 " " " * " ' 32 £ - 1 6 A C 0 5 33 E - 1 5 A 0 0 5 34 P - 2 S PUMP . . . . . " " •' " " ~ " • 3 5 E - 1 6 AC CS 36 0 - 3 OVOR 3 7 PR15 V A L V 38 TE*P ACC5 39 v«2e V A L V \ f F I N A L S I P U L A T I C N - - - P I I L Y PLANT — S E P T t P O E R 1971 O l I L C T F H I . S . 5 1 5 . 0 0 0 0 530 . 0 0 0 0 3C0.CCCC l c c c c c e PCVEH T Y P E : C O 0 . 0 0 . 0 c c « E Q U P P E M S U P P A P Y - I N C I V I C U A L C E T A I L S * ( • ) - S T E A M (0 l - E L E C . l - > - F L E L G A S «. H - 0 U 1 L E T STEAM C C o . c 0.0 0.0 ? • • • C I W I C E K S *•* ( B I U / L C 1 F U E L USAGE 0 . 0 0 . 0 0 . 0 c c E C L I P P E N T N C . 24 26 3 6 _ ( P S C F / H R ) WATER USAGE C C 0 . 0 0 . 0 0 . 0 E X T E R N A L N A P E U - l C - 2 0 - 3 ( C A L / h R ) F R X N • * 1 O . 6 0 C C C 2 C C C l . C O O O STEAP USAGE 0.0 0.0 0.0 c c . 2 0.4CCC" C . 3 C C 0 0 . 0 ( P L B S / H R ) 3 C C C 3 C C C c c KW USAGE C C o . c 0 . 0 0.0 4 0 . 0 C 2 C C C O . C 5 0 . 0 0 . 0 o . c 1 6 C C 0 . 0 0 . 0 • • • M I X E R S »•• 1 » * * A 0 D 3 * * * i E C U I P P E M N O . 5 7 9 11 ECU IPMENT N O . 6 E 10 12 • E X T E R N A L NAPE P - l P - 2 H-3 M - 4 E X I E H N H NAPE M - l R - 2 R - 3 ~ R - 4 ' "' P A R A P E T E R S : 1 . 6 7 0 0 3 . 1 7 C 0 5 .COOO 5 . 3 3 C C 5 . C C C C 5 . C C C C 5 . 0 0 0 C 5.0000 1 1 .OOCC l . C C C C l . C C C C I . C C C C 2 .OCCO 2 .OCCO 2 . C C C C 2 . L C C C 5 2 C C U C 5 2 0 . C C C C 5 2 0 . C O O O 52C .0000 J i f C C - C C C C ' C n c c c c c c c 3 E C C 1 . C C C C 3 E C C C CCCC 2 8 0 0 0 . O C C C 2 R 0 0 0 .OCCC J E C C C . C C C C J K C C C C C C • * » P . C . V A L V E S *•* C 6 C C C C . 6 C C C 0 . 6 0 0 0 C . 6 C C C C C C O O . C 0 . 0 EQUIPMENT N O . 37 35 l . O C C C 1 . C C C C 1 . C C C C l . C C C C 0 . 0 0 .0 0 . 0 C C E X T E R N A L NAME P R 1 5 V»2E C O C O o . o 0 . 0 D C W N S T P . P 2 6 5 . 0 C O O 9 0 . 0 0 0 0 0 . 0 C C o . c C O 0 . 0 0 . 0 0 . 0 c c C O O . C 0 . 0 c c C C C O 0 . 0 0 . 0 ,., , , 0 . 0 C C 0 . 0 C O 0 . 0 c c 0 . c C O 0 . 0 0 . 0 c . c 0 . 0 0 . 0 C C o . c c c 0 . 0 0 . 0 c . c c c • * » P L P F S / C C P F R E S S C R $ * » * o . c 0 . 0 0.0 C O c c C C o . c c c E C U I P M E N 1 NO • 1 25 29 34 0 . 0 0 . 0 o . c C O E X T E R N A L NAME P 2 8 5 P 3 4 2 P 2 9 2 P - 2 5 C C P F . S T A G E S 1 . 0 0 0 0 l . O C C C l . O O C O l . C C C C WCRK C A P A C I T Y 5 C C 0 0.CCCC 5 C C C 0 . 0 C 0 0 1 3 0 0 0 . 0 0 0 0 13CCC.CCCC « 1 3 T U / H R ) i 1 j ro CO •-3 0 . 0 0 . 0 0 . 0 c c c . C O C C 0 . 0 0 . 0 c . 0 . 0 C C O . C C O O.i 0 . 0 0 . 0 O . C C C C . C O C O 0 . 0 0 . 0 c. . C C C . C O . C C C 0 . 1 O T O cr.~o o.^ ' c~c c * » » A C C 4 * « » C O 0 . 0 0 . 0 c c c . C . C C O 0 . 0 0 . 0 C.i E C U I F ' E M hC. 2 2 3 1 0 . 0 C . C O . C C C G . ' 0 . 0 0 . 0 O . C C . C C.i E X T E R N A L N A P E V - S 6 V - 2 8 C O 0 . 0 0.0 0 . 0 C P A R A M E T E R S s 4 2 . C C C C 2 4 . C C C C ~ C . C C . C O . C C . C O . i 0 . 7 5 C C C . f t C C C 0 . 0 0 . 0 0 . 0 C . C C.i 5 8 7 . C C C 0 5 9 0 . C C 0 C _ 0 . 0 0 . 0 0 . 0 C . C C.i 7 1 4 . C C C C 7 2 0 . - C C C C ' C . C C C 0 . 0 0 . 0 C . i 0 . 9 5 C C C . S S C C 0 . 0 C . C O . C C C C.i 0 . 0 1 C O 0 . 0 7 C 0 •  CTi3 0 7 3 1 . C C C C - l . C C C C "*.*/.GC5»** ' E Q U I P M E N T N C . 2 3 2 3 3 2 33 E X T E R N A L NAME E 1 6 2 E 1 6 1 E 1 H E - 1 6 E - 1 5 P A R A M E T E R S 8 7 1 0 . Q C Q 0 8 1 5 . 0 0 0 0 5 5 2 . O C C C 5 5 C . C C 0 C 5 6 7 . C O O P  O T c 0 T 5 OTQ " CTC 0 . 0 : C . C C C 0 . 0 C O 0 . 0 N C CF L C C P S TC C C N V E K C E > 8 0 . 0 0 . 0 0 . 0 C . C C C CC 0 . 0 0 . 0 CC C.C CC O . C 0 . 0 0 . 0 O . C o.o c j cc; eg o.o CO r>) o crfSI cj> •*> C J c  O CJ 1— • ri •o t«g CO I V m QC u. IM ri •o rg in >r >r •3D o «r o ro r- r-Pt •o CC u. p- r-CJ> <*"» ri CJ f L J O <r CJ in »— • ri o in CO m r-ac u» CO CJ fO CJ CJ VJ L J L J rg 1— • t gj CV r>» —* ac u. 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'ti LU f) < w | u f i . v i X 3 io- X U J t-cj 'a uj a ^  U J > i— a U J riO p* o ri CJ OJ O O o cr ;gj o o o <• gj o H o r> u CJ gf tco cj cr C J m o C J C J LT IgJ C J C J gj '•co L J U C J m !c7* cj 0» n gj Cu C C O *J C J O V I L J C J OJ U* C J O C J — g" C J r- C J r*i CJ m C J CO O C J g j ri » • ri -o z z * u 2 X ri Cu Z C J C  O *a UCt r O. LL Ul " U Q . 3 LL J^ CI > P- X U J CU C J i n C J U~ CJ •J- o rg gr - 3 l / l ri X U J 0* g-croojirmrliroo frgu^ ocj.'* OO^OO gr W- •« o CJ C J O C J O O rg rg C J C J u l« u o c/ c» <j frM^ oOCirv'fviWoi C P a J C J O C j r - i Q O P ^ O t * o o o o * O C J C J o I • • • • CJ o C J C J o « r g g j -* o C J co ri c j o C J gj C J C J o cj — • w • i_» v« ri P* O C J •O Og CJ L J ; J ' ^ H v* o VJ gr • /ui(_)u«uvj ' "OLi "WaouNouoou • •••*•••••• <roooc?«ooooo >r m O1 u u gj ri rg C J w g- >r in u u ri UJ O C J L J V o O C J C J C J gr CM C J C J -T rg co c j C J •j- o cu L J i_/ rg gr C J C J o • • a • • C J ri L J C J O o ri ri r- C J C J m--cow ri C J C J C J C J P* C J C J O C J C J • > • • • » • noooou r» <u <v ca C J C J - - - OJ C I O C U U p w U ri C J o O w cj • ••••• o C J o O C J C J i I Lua h y- f M «J> ri | I Z - J CM LU ri [ri ^  o o rg gjy**i-^*gj P - i r t N o o c O u r * gj cr o ~ r— ccjo^ cpi-j'cr'vrg o \-c o g» m ui<7< g-> r» J « 1 U O O "Ix o f ^ rorgcjOriCJ»nrigr«-n "-*r*—«cj"JcJirgg3r-«M ocjtnmujriajr-rgco rg g- o u - O irviiocjr o r~ t% o o o ifikufliflp"" *r o u «o ui\n H or> r  ri ri - I 0" -I iT. m gr m A. «> r- rg •co r~ rg co ro C J ga o «-» co g; r- cr cr *JJ r - ' i ' g i M O L j r i i n g j v r ' s u r i g*0,nmtncp(cr*»Jci»CJ LJ in r» rg tn r-m gj C J C J ri rg O u> u <r o 4J U U U r-« O cr, m «J/ LP u y\ r- CM gr r-z LU U U J U J U J I - C O H - U J ri Z ^ - -> I WJ A Z Z H < X < U L U C J i M Z U J U J — t- - J I I U J O-Z Z >- pg I— I 0. 3 - J I U L U U . i 7 j rg U J d:a.u.Ljwf<aJ I J> I | i c « ri i ac I 289 rsi VI Minirn o ro O -J-i C3 o CM IM m r-r— IN m u <o rn ~* cr r> cj N cr c u c c* * w rsi >r 0* CJ cu <J aj o > o fM M" m cr -  crCJ m LJ sr m o CJ •T ^ U~ ui LI CJ IM cr o rt "O • • m >- • • « « • • • • • • • • • • • * LJ ri CJ ~ CJ m LT »r r* CM o o f* cj crV cr M w M" •r M m m HI -4 CO oc . J u. tr rx m CJ CJ o u MJ «-» O u CO u» o CJ O o CJ a o o o o o o CJ o o o O o o • • »T m m o CM m CJ z? O CJ cj O CJ CJ O CJ CJ CJ o CJ cr o CM «» CM X "N, fX v> U_ UJ -1 a o X CM m O O CJ J  CD V "f r- m CM o  v in o rsi m o «*» o o m OJ — u cr o m (\j r* at O O <o CJ CO O CJ cr - J O rg«i W \ r» H « ' —o ajo  CJ CJ o o *j-crrMKrn(j.*ujus o o  • m • • • • • • • • • o in  o o o -j ~* r- o O O O M O I M o o IA o in mo OJ o O* ro I A I I r-u rx r*i u> t V) a a. X o o in o CJ c«* CJ C V * •  m rsi co CJ m o rsim CO o CJ m J —« CJ cr —« o m CM r- cu oo ir rt CJ o r*» CJ <M in rt CM rsi i u» -* CJ UJ « in a o o o r«> •TV rsi tr o —i cu cr CJ CJ m fM »- • • • •in en O f*  CJ CJ CJ o —' c> eg O CJ 'in m m <r -o CJ cr m m CM cc CM u. o u» CJ o o *t W UJ cr m in a o UJ VJ o o o o J O O C U O L> O o i— • • • in m O F** CJ O CJ o o CJ O O CJ O o CJ CJ o m m >o >o tr m m or -j z ce < z -j DC O CJ ZD UJ UJ JJ o. 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VI DC A u» 1 UJ X U ^ CJ U X X >o cr CJ o r-1 •o CJ "1 O O —r> p- —• CJ o ~+ m o m CJ o 1 OJ cr r» fM cj o m Si r— m o o "J P cj rt cj rj in oo u in CU — o cr r— CJ c tr ~J cr «J flOO ~ O U — <J o cu C fM O f» O CM LA r* CJ o o o o o u» o o o f OOOOCJ *4 oof O CO cr f  r»  * • > • • • OJ CD IN o o o o IM O O CJ CJ O Cr IM O O fSJ z —* r-o CO u» o O u m tn CO cr o u fM m r*j in m in O r— >- m — CL •T o u. 1 vt X CO ft o c X X X •*> cj cj y CJ r> -o o u LJ fsj fM V O O tn O CJ o m CM r- J cr cu o o CJ rt cj CJ fl cj cj m J CJ r- L» CJ CO in r* CJ CJ fM fM <o —• U CJ •T JCJ CJ O CJ CJ CJ rt CJ —' CJr- CJ CJ o CJ- cr CO —< O CJ CJ r- o CJ c* o O U O CJ CJ cr CJ tn r-rsj • r* CO fM CJ rsi o f*i CJ o O CJ LJ o o CJ o o p-CM O o cj rg J —• C9 in rt o v* rt fM m m aj rt Mt fM pi r* Ct X •O it- 1 X 1 r» er O CJ oj v> M" C CJ CJ cn CM «n cj m tfl «^ CM rg CJ CJ fM m rs/ to CJ CJ o CJ «n o CJ cn rt <r CJ f*» CJ CO in •o u u uj ui _< vj CJ O o u u V p» CJ CJ cu CJ •—1 CJ o O CJ CJ aj fM CM O O CJ -ouoou »o O O — O -H o m r-»- <o r» CM O fM CJ in m CJ o CJ ( J H  CJ o o o •o CM cj rt O Lrt O ~* T>>• <o in m o CM fM ^ fM cr cr CM o CJ a fM X M» u. 1 X •r z 2<i< z J^ z z oc <i CC CJ O « 3 UJ UJ UJ < QC U CJ « u UJ z — <c UJ vl ^  m W UJ 11 OJ UJ ^  LU U CU X I* IU Q. CU X Z o a LU UJ J i- cC w UJ »- X *= LJ C  - J a «a _j * • UJ UJ J£ _J | UJ A £ U<1 j • • UJ UJ CJ 3 H UJ > LU iC <l < UJ CO CM ^  UJ UJ <c U CC K UJ J - UJ <c z u. < cr n O. <t I— k-—J i i uiac CL «a cr a z < x CC J J < l- I- il z->. rsi vi u- 1 CL X • ct ~j—J < 1— c Z UJ fl 4 a : J _> UJ cu u» | rr - J fM cu • «a O. LX LU V) < CL _J -t UJ CJ U. VI x o O ; C L ' C L C L « - J V » < C U i x UJ —• CJ v> X CJ CO CD oc ZJ U, JC UJ ct l l 1 l i-rr l a | c  U U. X CJ OC | 1 r— Cl *3. UJ rx Z o. — *c 2 a o i- » • M 1— U < UJ x JC. i/> UJ > H- O. UJ V) LU > u. • i CM co o m o *lu w fg r- w o i (M CM <j/ m v " " O •< C  V* O • • • • • • • ) U CJ H O N (J -< vj rt INJ r* u> Q - "i fM - * -J- m —i • o —• °J o* o J O O H C J I N O • LTV rsj UJ u> m o * icj rt CM r- co <j 1 <M (V <-< -f Ul w >t> O - * O CM O i tn o «r m (M r4 3 cr CJ <l cr O CM J J*— VJ fl OJ M* CJ > rt <j **• O ~+ fi J CM <J U Q O O »im cj tfi rsj »-J Hi <J« \J CM J p, vJ PI tfj ^  LJ > «J —« o —« rn 5 [cj C J < J O L) L I UJ UJ UJ Z t-UJ 3 J hj  LU I— OJ : z 7Z 3 I UJ z I UJ CU fM LU • _J I I UJ CL ' Z >" CN VI | > LL: O. I Z J CM L  O Vl <J CO t — a. | ( • F I N A L R E S L L T S ' F I N A L J I H I L A T I C N P C L Y P L A N T - - S E F T E P B E R 1 9 7 1 F I N A L 51 P U L A T I CN P O L Y P L A M - - S £ P I fc'P C ER 1 9 7 1 * E Q U I P P E N T S L P M A R Y - I N C I V I C U A L D E T A I L S * * E Q U I P M E N T SUMMARY - E Q U I P M E N T L I S T * • • • D I V I D E R S E Q U I P M E N T N C . E X T E R N A L NAME F P X N . * 1 EC. f 1 2 E X T . N A P E P 2 8 5 . E U 2 F.161 M - l R - l S U E . N A P E P U P P A C 0 5 A H C5 P 1XR A C D 3 2 4 D- 1 0 . 6 0 C O ~ C . 4 C C C ~ O . C 0 . 0 0 . 0 o.c 2 * D - 2 0 . 2 0 0 0 " 0 . 3 0 0 0 " C . 3 C C C 0 . 2 0 C 0 0 . 0 O . C 3 6 D - 3 l . C C C O " 0 . 0 0 . c C . C o.c o.c M - 2 P - 2 C - 3 H I X R ACC3 f l XR 1 0 11 12 22 2 3 2 4 R - 3 P - 4 _ R - 4 _ V - 9 6 F 1 6 6 C - l A C 0 3 P IXR J>DD3_ A 0 0 4 A C D 5 CV0R • • • F I X E R S E C L ' I P P E N T N C . E X T E R N A L N A P E • •• b P - i 7 P - 2 9 M - 3 11 H - 4 2 5 2 6 2 9 P 3 4 2 C - 2 P2 92 P U P P CVOR P U P P • • • P . C . V A L V E S E C U I P P E N T N O . • •• 37 3 9 3 1 32 3 3 3 4 2 5 3 6 V - 2 8 E - 1 6 E - 1 5 P - 2 5 E - 1 6 C - 3 ACD4 ADD5 P U P P A C D 5 DVDR E X T E R N A L N A P E D C W N S T P . P P R 1 5 2 6 5 . C C C C V ^ 2 8 9 0 . 0 C 0 C • • • P U M P S / C O P P R E S S O R S " T C U I F P E N T N O . E X T E P N A L N A P E C C P P . S T A G E S WORK C A P A C I T Y ( E T L / H R ) 3 7 3 8 3 9 P P 1 5 T E P P V ^ 2 £ V A L V A C D 5 V A L V 1 P 2 8 9 l . C C C C E C C C C . C C C C 2 5 P 3 4 2 l . C C C C 5CCCC.CCCC 29 P 2 9 2 1 . 0 0 0 0 1 3 0 0 0 . O C C C 3 4 P - 2 5 l . C C C C 1 2 0 0 0 . 0 0 0 0 \ ro vo H t C L U E T F B E S . / PEWER T Y P E s ( • l - S T E / H ( 0 1 - E L E C . 5 1 5 . C C C C C.C 530 .CCCC C C 3 0 0 . 0 0 0 0 0 . 0 I C C . C C C C C O t - l - F L E L GAS y~CUTLET STEAM 0.0 C C O.C C O ( E I L / L E ) F L E L U S J C E C.C 0.0 0 .0 C C * * * AC C 4 * * * ( M S C F / H R ) WATER U S A G E O.C C . C O.C C O E C U I P M E N T NC. 22 31 ( G A l / t-P ) STEAM U S A G E C O 0.0 0 .0 C C E X T E R N A L NAME V - 9 6 V - 2 E (M L f tSVHR ) Kk U S A G E 5 . 3 5 6 5 2 . 1 9 3 4 0 . 1 9 3 1 C . 1 0 7 4 PARAMETERS > 4 2 . U C C C 0 . 7 5 0 0 5 0 7 . C C C C 7 1 4 . C C C C 0 . 9 9 0 0 C . C K C 2 4 . C C C C 0 . 6 C C O 5 9 C C C C C 7 2 C . C C C C 0 . 9 9 0 0 C . C y C C I. 3 7 E 8 l . O C C C 1 . 1 9 6 7 c o C . C 0.0 C E 3 4 3 - l . C C C C 0 . 7 C 7 3 0.0 C.C C C 0 . 0 " 0 . 0 C O 0 . 0 » « » A C G 3 * « « C . C C . C E Q U I P M E N T N O . 6 E IC 12 0 . 0 0 . 0 C O C . c o . o e 0 . 0 E X T E R N A L NAME P A R A M E T E R S s R - l 1 . 6 7 C C R - 2 3 . 1 7 C C R - 3 5 . O C C O R - 4 5 . 3 300 C . c 0 . 0 c c c c 5 . 0 C C C 5 . C C C C 5 . C C C C 5 . C C C C 0 . 0 0 . 0 1 . O C O O 1 . O C C C l . C C C C l . C C C C C O 0 . 0 2.C C C C ? . C C C O 2 . c o n e ? . C C C C c c C . C 5 2 C . C C C C 5 2 C . C C C C 5 2 0 . C C C C 5 2 0 . C C C O 0 . 0 c c 3 8 C O C . O C C C 2 f . C C C . T t C T : — J T O W / C T C t — 3 T C T C T C C C C 2 8 C 0 0 . 0 C C 0 2 8 C C 0 . C C C C 2 8 C C 0 . C G C O 2 E C C C . C C C C C . 6 C C C C . 6 C C C 0 . O C C C 0 . 6 0 0 0 0 . 0 C C 0 . C C O 1 . O C O O 1 . 0 0 0 0 0 . 0 I . C C C C 0 . 0 l . C C C C 0 . 0 0 . c C . C C . C O . C C . C 0 . 0 0 . 0 0 . 0 C O C C 0 . 0 0 . 0 C C c . c C C o . a 0 . 0 0 . 0 C . C O . C C . C 0 . 0 0 . 0 0 . 0 C . C C C C C 0 . 0 0 . 0 0 . 0 C C O . C C O 0 . 0 0 . 0 0 . 0 C O ECU IPMENT N O . 2 3 2 3 3 2 33 C O 0 . 0 0 . 0 C O c . c C . C 0 . 0 0 . 0 E X T E R N A L NAME E 1 6 2 E 161 E 166 E - 1 6 E - 1 5 0 . 0 C C O . C C C P A R A M E T E R S s 7 1 C . C C CC 8 1 5 . O C C O 5 5 2 . C O C O • 5 C . C C C C 5 6 7 . C C C C 0 . 0 0 . 0 0 . 0 C . C C . C 0 . 0 0 . 0 C . C 0 . 0 0 . 0 C C C . C O . C C C C . C 0 . 0 O . C 0 . 0 0 . 0 0 . 0 O . C O . C 0 . 0 O . C C . C C C C . C C . C 0 . 0 C . C C O 0 . 0 C . C c c ro vo ro \ \ 293 • O O Q O O Jo o W U o w j o o w o o u u o o u -J O O LJQOMOOOO , 0 0 0 0 0 0 O O O O U U ^ 0 0 0 0 0 0 0 0 0 0 L J O O O O O O O O O O 0 o o c o o |o o o o o o 'O CJ o o o o :0 CJ O O O CJ O O O CJ CJ o o o O CJ CJ o n 0 0 0 0 0 0 0 0 0 0 3 0 0 O Q O O 0 0 0 0 O O O O O C J O c o u o 3 0 0 Q O O ,0 0 0 0 0 O O O O O O OOCJOO a CJ o 0 c^  cj O O CJ O O -A. a. cj o cj o CJ cj o O O CJ o o r - u r i O c j o o o o o O O O O O O O O O O O O O O C J O Ul o o o o o o O U U O j O U U O U C O •0 1— zlu CL |X »- cc x -a Lb <X o CJ o O O cj LJ CJ o o o o O O CJ CJ o o O O CJ c j O O O CJ CJ o o o o CJ cj o o <-» 294 APPENDIX F E s t i m a t i o n o f H e a t o f R e a c t i o n f o r  P o l y m e r i z a t i o n o f O l e f i n s The e x a c t r e a c t i o n s t h a t t a k e p l a c e d u r i n g t h e p o l y m e r i z a t i o n o f o l e f i n s i n t h e p r o d u c t i o n o f p o l y m e r g a s o -l i n e a r e n o t known. The o l e f i n s p r e s e n t i n t h e f e e d s u s e d i n t h i s s t u d y and some p o s s i b l e p r o d u c t s a r e shown i n T a b l e F - l . I t can be s e e n t h a t t h e number o f p o s s i b l e p r o d u c t s i s v e r y l a r g e . E a c h r e a c t i o n , o f c o u r s e , w i l l h a v e a d i f f e r e n t h e a t o f r e -a c t i o n . S i n c e t h e e x a c t s e t o f r e a c t i o n s w h i c h c o u l d t a k e p l a c e i s a f u n c t i o n o f t h e f e e d c o m p o s i t i o n , t e m p e r a t u r e , p r e s s u r e , e t c . i t i s n o t p o s s i b l e t o d e t e r m i n e an e x a c t h e a t o f r e a c t i o n . T h e r e f o r e some t y p i c a l v a l u e s f r o m t h e l i t e r a t u r e w i l l be c o m p a r e d w i t h some v a l u e s c a l c u l a t e d f r o m b a s i c d a t a by a p p r o x i m a t e m e t h o d s t o a r r i v e a t " a v e r a g e " h e a t s o f r e -a c t i o n f o r t h e C g - o l e f i n s and C ^ - o l e f i n s . A c c o r d i n g t o J o n e s ( 7 3 ) t h e a v e r a g e h e a t o f r e a c t i o n i n t h e p o l y m e r i z a t i o n o f p r o p e n e t o p o l y m e r g a s o l i n e i s 670 TABLE F-1 O l e f i n s i n F e e d and P o s s i b l e P r o d u c t s F e e d D i m e r i z e d P r o d u c t C 3 H 3 n 6 P r o p e n e C tt H e C - 2 - B u t e n e T - 2 - B u t e n e I - B u t e n e 1 - B u t e n e } C 6 HI 2 1-Hexene + 16 I s o m e r s C 7 H I k 1 - H e p t e n e + 35 I s o m e r s C 8 H i 1 - O c t e n e + 91 I s o m e r s Not c o n s i d e r e d f o r e s t i m a t i o n o f h e a t o f r e a c t i o n s i n c e o n l y c o n s t i t u t e a b o u t 2-3% o f t o t a l o l e f i n c o n t e n t . 296 B T U / l b . w h i l e t h a t o f t h e b u t y l e n e s i s 400 B T U / l b . Thomas ( 7 8 ) r e p o r t s a v a l u e o f 45-54,000 B T U / l b . m ole o f d i m e r . I t was d e s i r e d t o c h e c k t h e s e v a l u e s by c a l c u l a t -i n g some t y p i c a l h e a t s o f r e a c t i o n f r o m s t a n d a r d h e a t s o f f o r m a t i o n and s t a n d a r d h e a t s o f c o m b u s t i o n . The v a l u e s u s e d a r e shown i n T a b l e F-2. Some t y p i c a l h e a t s o f r e a c t i o n w i l l be c a l c u l a t e d by t h r e e m e t h o d s : 1. from h e a t s o f f o r m a t i o n 2. from h e a t s o f combustion 3. from group c o n t r i b u t i o n s 1. H e a t s o f r e a c t i o n f r o m h e a t s o f f o r m a t i o n : A H = T A H _ y |_| R e a c t i o n ^ Form-Prod ^ Form-React 2 5 ° C 2 C 3 H 6 -> C 6 H 1 2 ( 1 - H e x e n e ) A H = - 9.96 - 2 ( 4 . 8 8 ) = - 19.72 K c a l / g - m o l e (-35.486 B T U / l b - m o l e ) R-2 5 2 C 3 H 6 -> C 6 H i 2 ( I s o m e r m i x t u r e ) A H = - 1 3 . 6 1 - 2 ( - 4 . 8 8 ) = -23.37 K c a l / g - m o l e (-42,062 B T U / l b - m o l e ) R2 5 297 TABLE F-2 S t a n d a r d H e a t s o f F o r m a t i o n and C o m b u s t i o n f o r Some O l e f i n s ( 8 9 ) i H e a t o f F o r m a t i o n a t 2 5 ° C , K c a l p e r m o l e H e a t o f C o m b u s t i o n o f t h e gas a t 2 5 ° C , K c a l p e r m ole C 3 H 6 ( q a s ) P r o p e n e 4.88 491.99 C^He ( g a s ) C - 2 - B u t e n e - 1 .67 647.81 T - 2 - B u t e n e - 2.67 646.81 I s o b u t e n e - 4.04 645.43 1 - B u t e n e - 0.03 649.45 C 6 H 1 2 ( g a s ) 1 - Hexene - 9.96 964.26 (16 o t h e r i s o m e r s ) - 1 3 . 6 1 1 9 6 1 . 0 8 1 C7Hllt ( g a s ) 1 - H e p t e n e -14.89 1121 .69 C 8 H i 6 ( q a s ) 1 - O c t e n e -19.82 1276.13 A v e r a g e v a l u e . 2 C \ H 8 ( C - 2 - B u t e n e ) C 8 H x 6 ( 1 - O c t e n e ) = -19.82 - 2 ( 1 . 6 7 ) = -16.48 K c a l / g - m o l e (-29,664 B T U / l b - m o l e ) 2 C ^ H 8 ( T - 2 - B u t e n e ) -> C 8 H x 6 (1 - O c t e n e ) = -19.82 - 2 ( - 2 . 0 7 ) = -14.48 K c a l / g - m o l (-26,064 B T U / l b - m o l e ) 2 C.,H8 ( I s o b u t e n e ) -> C 8Hi. 6 (1 - O c t e n e ) = -19.82 - 2 ( - 4 . 0 4 ) = -11.74 K c a l / g - m o l 5 (-21 ,132 B T U / l b - m o l e ) 2 C^He ( 1 - B u t e n e ) -»- C 8 H 1 6 (1 - O c t e n e ) 299 A H = -19.82 - 2 ( - 0 . 0 3 ) = -19.76 K c a l / g - m o l e K-2 5 (-35,568 B T U / l b - m o l e ) C3H6 + C \ H 8 ( A v g . M i x ) + C 7 H m ( l - H e p t e n e ) A H = -14.89 - 4.88 - (-2.20) = -17.57 K c a l / g - m o l e (-39,626 B T U / l b - m o l e ) The h e a t s o f r e a c t i o n c a l c u l a t e d f r o m h e a t o f f o r -m a t i o n d a t a show q u i t e a v a r i a t i o n . In t h e c a s e o f p r o p e n e , o n l y one r e a c t a n t i s o m e r e x i s t s b u t one o b t a i n s h e a t s o f r e a c t i o n o f 35,500 t o 42,000 B T U / l b - m o l e d e p e n d i n g on t h e p r o d u c t f o r m e d . F o r t h e b u t e n e s , t h e r a n g e i s f r o m 21,000 B T U / l b -m o l e f o r i s o b u t e n e t o 35,500 B T U / l b - m o l e f o r 1 - b u t e n e . T h i s d o e s n o t t a k e i n t o c o n s i d e r a t i o n any o f t h e o t h e r C8Hi6 i s o m e r s b e c a u s e o f l a c k o f d a t a . When p r o p e n e c o m b i n e s w i t h a b u t e n e t o f o r m 1 - h e p t e n e t h e h e a t s o f r e a c t i o n c a n r a n g e f r o m 28,000 t o 35,200 BTU/ l b - m o l e , o n c e a g a i n w i t h o u t c o n s i d e r i n g t h e o t h e r CyHii, i s o m e r s . 300 2. H e a t s o f r e a c t i o n f r o m h e a t s o f c o m b u s t i o n Reactxon Comb-React L Comb-Prod 2 5 ° C 2 C3H6->- C 6 H i 2 ( l - H e x e n e ) A H = 2(-49 1 .99) - ( 9 6 4 . 2 6 ) = -19.72 K c a l / g - m o l e R 2 5 (-35,486 B T U / l b - m o l e ) 2 C 3H 6-> C 6 H 1 2 ( I s o m e r m i x t u r e ) A H_ = 2 ( - 4 9 1 . 9 9 ) - ( 9 6 1 . 0 8 ) = -22.90 K c a l / g - m o l e R2 5 (-41 ,220 B T U / l b - m o l e ) 2 C ^ H o ( C - 2 - B u t e n e ) = C 8 H x 6 ( 1 - O c t e n e ) 301 A r L = 2 ( - 6 4 7 . 8 1 ) - (-1276.13) = -19.49 K c a l / g - m o l e R2 5 (-35,082 B T U / l b - m o l e ) 2 C \ H 8 ( T - 2 - B u t e n e ) = C 8 H 1 6 (1 - O c t e n e ) A H = 2 ( - 6 4 6 . 8 1 ) - (-1276.13) = -17.49 K c a l / g - m o l e R2 5 (-31 ,482 B T U / l b - m o l e ) 2 CitHs ( I s o b u t e n e ) = C 8 H 1 6 (1 - O c t e n e ) A H = 2 ( - 6 4 5 . 4 3 ) - (-1276.13) = -14.73 K c a l / g - m o l e R2 5 (-26,514 B T U / l b - m o l e ) 2 C t H 8 ( l - B u t e n e ) = C 8 H x 6 ( 1 - O c t e n e ) A H = 2 ( - 6 4 9 . 4 5 ) - (-1276.13) = -23.77 K c a l / g - m o l e R2 5 ' (-42,786 B T U / l b - m o l e ) 302 C 3 H 6 + C \ r l 8 ( A v g . M i x ) = C 7H i k (1 - H e p t e n e ) A H = -491.99 - 647.38 - ( 1 1 2 1 . 6 9 ) = -17.68 K c a l / g - m o l e = (-31 ,824 B T U / l b - m o l e ) The h e a t s o f r e a c t i o n c a l c u l a t e d f r o m h e a t s o f c o m b u s t i o n d a t a a r e s i m i l a r t o t h o s e c a l c u l a t e d f r o m h e a t o f f o r m a t i o n d a t a i n t h e c a s e s o f p r o p e n e , and p r o p e n e p l u s an a v e r a g e b u t e n e m i x t u r e . In t h e c a s e s o f t h e v a r i o u s b u t e n e i s o m e r s f o r m i n g l - 0 c t e n e , t h e v a l u e s c a l c u l a t e d f r o m t h e h e a t s o f c o m b u s t i o n a r e a b o u t 2 0 - 2 5 % h i g h e r t h a n t h o s e c a l c u l a t e d f r o m h e a t s o f f o r m a t i o n . In l i g h t o f t h e d i s c r e p e n c i e s shown i n t h e two p r e -v i o u s m e t h o d s , a t h i r d e s t i m a t i o n method was c h o s e n t o c a l c u l a t e t h e h e a t s o f r e a c t i o n . I t was a l s o u s e d t o c a l c u l a t e d t h e h e a t o f r e a c t i o n a t 5 5 0 ° K ( 4 4 0 ° F , i n t h e r a n g e o f r e a c t i o n t e m p e r a t u r e s ) t o s e e i f t h e r e was much o f a v a r i a t i o n w i t h t e m p e r a t u r e . The m e t h o d u s e d was t o c a l c u l a t e t h e h e a t o f r e a c t i o n f r o m h e a t s o f f o r m a t i o n c a l c u l a t e d by t h e a d d i t i v e - g r o u p e s t i m a -t i o n t e c h n i q u e f o r h y d r o c a r b o n s by S o u d e r s , M a t t h e w s , and Hurd as e x p l a i n e d by R e i d and S h e r w o o d ( 8 4 ) . 303 H e a t o f R e a c t i o n f r o m H e a t s o f F o r m a t i o n f r o m M e t h o d o f S o u d e r s , M a t t h e w s , and H u r d 2 9 8 ° K C 3H 6 No 1 1 1 G r o u p - C H 3 H 2C H C C o n t r i b u t i o n -10.05 5.80 9.28 No 1 3 1 1 A H° = 5.03 K c a l / g - m o l e ( l ) G r o u p - C H 3 - C H 2 H 2C H C A H° C 6 H i 2 ( 1 - H e x e n e ) C o n t r i b u t i o n -10.05 -14.95 5.80 9.28 •9.92 K c a l / g - m o l e 298°K 2 C 3 H 6 -> C 6 H 1 2 ( 1 - H e x e n e ) A H = -9.92 - 2 ( 5 . 0 3 ) = -19.98 K c a l / g - m o l e R 2 5 (-35,964 B T U / l b - m o l e ) 5 0 0 ° K C 3 H 6 C 6 H ! 2 ( 1 - H e x e n e ) I n t e g r a l I I n t e g r a l I No. G r o u p C o n t r i b u t i o n No. G r o u p C o n t r i b u t i on 1 - C H 3 -2.101 1 - C H 3 -2.101 1 H 2C = -1.478 3 - C H 2 - -3.199 1 H C = -0.447 1 H 2C = -1 .478 - 4 . 0 2 6 ( 2 ) 1 H C = -0.447 - 7 . 2 2 5 ( 2 ) 304 I n t e g r a l I I I V ( a ) 0.370 ( 3 ) 4 R ( T - 2 9 8 ) = 1.604 ( 4 ) A H° = ( l ) + ( 2 ) + ( 3 ) + ( 4 ) r 5 0 0 = 2.974 K c a l / g - m o l e I n t e g r a l I I t I V ( a ) 0.370 ( 3 ) 4 R ( T - 2 9 8 = 1.604 (4) A H ° r = ( l ) + ( 2 ) + (3) + (4) ' 5 0 0 = -15.18 K c a l / g - m o l e 500°K 2 C 3 H 6 •*• C 6 H 1 2 ( 1 - H e x e n e ) A H = -15.18 - 2 ( 2 . 9 7 4 ) = -21.13 K c a l / g - m o l e R 5 o o (-38,034 B T U / l b - m o l e ) 2 9 8 ° K C ^ H s ( T - 2 - B u t e n e )  No. G r o u p C o n t r i b u t i o n 2 - C H 3 -20.10 2 ( H C = K 1 7 - 4 0 t r a n s A H' No 1 5 1 1 - 2.70 K c a l / g - m o l e ( l ) C 8 H 1 6 ( 1 - O c t e n e ) G r o u p - C H 3 - C H 2 H 2 C H C A H' C o n t r i b u t i o n -10.05 -24.75 5.80 9.28 : = -19.72 K c a l / g - m o l e ( l ) 2 9 8 ° K 2 C^He -* C 8 H 1 6 ( l - 0 c t e n e ) 305 A H = -19.72 - 2 ( - 2 . 7 0 ) = -14.32 K c a l / g - m o l e R 2 5 (-25,776 B T U / l b - m o l e ) 5 0 0 ° K C ^ H 8 ( T - 2 - B u t e n e )  I n t e g r a l I  No. G r o u p C o n t r i b u t i o n 2 - C H 3 -4.202 HC = 0.894 -5.096 ( 2 ) No 1 5 1 2 CsHj, 6 (1 - O c t e n e ) I n t e g r a l I G r o u p - C H 3 - C H 2 -H 2C = - HC = C o n t r i b u t i on -2.101 -5.330 -1 .478 -0.447 -9.356 ( 2 ) I n t e g r a l I I I n t e g r a l I I 2 V ( b ) 0.470 ( 3 ) 1 V ( a ) 0.370 (3) 4 R ( t - 2 9 8 ) 1.604 4 R ( T - 2 9 8 ) 1.604 A H ° F = -5.722 K c a l / g - m o l e A H ° F = -27.102 K c a l / g - m o l e hoo r 5oo 3 5 0 0 ° K 2 C^H 8 ( T - 2 - B u t e n e ) -> C 8Hj. 6 ( 1 - O c t e n e ) A H = -27.102 - 2 ( - 5 . 7 2 2 ) = -15.658 K c a l / g - m o l e R s 0 0 (-28,200 B T U / l b - m o l e ) 306 In b o t h c a l c u l a t i o n s o f t h e h e a t o f r e a c t i o n by t h e a b o v e method one s e e s t h a t t h e c h a n g e i n h e a t o f r e a c -t i o n w i t h t e m p e r a t u r e i s a b o u t 5-10% b e t w e e n 2 9 8 ° K and 5 0 0 ° K . T h i s c h a n g e due t o t e m p e r a t u r e i s much l e s s t h a n t h e d i f f e r e n c e s b e t w e e n i s o m e r s . The v a l u e s o b t a i n e d by t h e l a s t m e t h o d were v e r y s i m i l a r t o t h o s e o b t a i n e d by t h e f i r s t m e t h o d . The r e s u l t s c a n be c l a s s i f i e d as f o l l o w s : c a l c u l a t e d h e a t s o f r e a c t i o n ( B T U / l b - m o l e ) : 3 0 0 ° K 2 C 3 H 6 -»• C 6 H i 2 ( l - H e x e n e ) -35 ,500 2 C 3 H 6 •* C 6 H i 2 ( A v g . I s o m e r s ) -42 ,000 C 3H6 + C i f H s ( A v g . I s o m e r s ) + C 7H i •» (1 - H e p t e n e ) -31,600 2 U H 8 ( C - 2 - B u t e n e ) -»• C 8H i e ( 1 - O c t e n e ) -29 ,600 ( T - 2 - B u t e n e ) -»• " -26 ,000 ( I s o b u t e n e ) -»• " -21,000 ( 1 - B u t e n e ) + " -35,000 5 0 0 ° K 2 C 3 H 6 -»• C 6 H i 2 (1 - H e x e n e ) 2 C\H 8 ( T - B u t e n e ) -»• C 8 H x 6 (1 - O c t e n e ) -38,000 - 28,200 307 In c o n c l u s i o n i t i s f e l t t h a t t h e e f f e c t o f tem-p e r a t u r e c a n be n e g l e c t e d when c o m p a r e d t o t h e d i f f e r e n c e s o b s e r v e d f o r t h e d i f f e r e n t i s o m e r s . A v e r a g e v a l u e s w i l l be u s e d f o r p r o p e n e and f o r t h e C\H 8 i s o m e r s . C 3 H 6 A H = -38,000 B T U / l b - m o l e R C\H 8 , C 5 H 1 0 A H = -28,000 B T U / l b - m o l e 1. APPENDIX G CHANGES MADE IN CHESS SOURCE PROGRAM FOR USE WITH WATFIV COMPILER Insert REAL*8 DABS,DEXP,DSIGN after ADB00200 308 ADB00201 2. Delete ADB00290 3. Insert: DO 1666 10=1,20 1666 EQR(IQ)=0D0 DO 1667 1=1,20 1667 0LEQ(l)=0D0 DO 1777 1=1,20 after ADB00280 ADB00281 ADB00282 ADB00283 ADB00281+ ADB00290 Insert REAL*8 DSQRT after CID00350 CID00351 Insert: REAL BETTA(62) EQUIVALENCE(BETTA(1),BBTA1(1)),(BETTA(32),BBTA 2( 1)) REAL GAMA(62) EQUI VALENCE (G AI1A (1), GMM1 (1) ) , (GAMA (31) , GMM 2( 1)) after CID00380 6. Delete: CIDOOl+12 CID00U20 CIDOOU9O CID01370 CIDOl^lO CID01H70 CID01510 CID01520 7. Insert: CIDOII461 CID01U62 CID01561 CID01562 EQUIVALEKCE( SCNAME(1), SANAME(1)), (SCNAME(89),SDNAME (1)), (SCNAME CIDOOl+12 1(157),SBHAME(l)) CID00H13 INTEGER SANAME(88)/ 1 #HYD' , 'ROGE' , 'N' , ' , 'TfiMET' , 'HANE' ,2* ' , 'tfET CID00):20 after CID00l»ll 8. Insert: 3 Q g L'EXAD' , 'ECM' , 'E' , 1 #N-H1 ,'EPTA' , 'DECA', 'NE' , 'tfETH1 ,'YLEN' , 'E' , , ')5 7 CIDOO^ QO INTEGER CDNAME(68)/ CID00i+91 after CIDOOWo 9. Insert REAL BBTA1 (31)/-. 0^576E-2,18.0)4hE-3,38.201E-3,57 • 195E-3,75. 2lk E-3, CID01370 after CID01360 10. Insert: U078E-2,7.22E-2,7• 702E-2,8.35E-2,101.^E-3,109.623E-3,99.696E-3/ CIDOlHlO REAL BBTA2(31)/ lOCIDOlllll after CIDOlllOO 1 1 . Insert REAL GMM1(30)/.09563E-5, -**3. E-7,-110.U9E-7,-175.33E-7,-237.3h E - 7 , - CIDOll+70 after CID0l!+60 12. Insert: Ji.-5, - 1 * • 07** E-5, -3. kOkE-5 ,-3.981E-5, -5.582E-5 ,-55*+. 27E-7 , - 6 0 3 . U 5 E-7/ CID01510 REAL G!#2(32)/ -582.63E-7CID01511 5 . , -57*+. OUE-7, -551.51E-7,-50k. 37E-7, - 6 7 U. 01E-7,7. hkl E - 5 , - 8 CID01520 13. Delete COAOOJ4OO 1 ^ . Insert If(K(J).EQ.COM) Go To 325 COAOOUOO 1 5 . Transfer: COA00590 C0A00600 C0A00610 C0A00620 COA00630 COA006U0 after COAOOl+OO 1 6 . Delete: DCX00090 DCX00160 310 17. Insert: 1=0 1 0 0 1=1+1 IF(I.GT.NSMAX) Go To 8 after D C X 0 0 0 8 0 18. Insert Go To 1 0 0 after D C X 0 0 1 5 0 19. Delete: D R E 0 0 5 1 0 DREOO68O D R E 0 1 2 1 0 20. Insert: 8. SEXU9(23) ,SEX50(23) ,SEX5l(23) ,SEX52(23) ,SEX5-l*(23) COMMON/STMA/ after D R E 0 0 5 0 0 D C X 0 0 0 9 0 D C X 0 0 0 9 1 D C X 0 0 0 9 2 DCX00160 DRE00510 DRE00510 21. Insert: 1 SIKU9(lO),SIN50(lO),SIN5l(lO),SIN52(lO),SIN53(lO),SIN5l*(lO) COMMON/STMA/ after DREOO67O 22. Insert INTEGER CHX(2)/,CLEA' , '1DW/ after DRE01200 23. Transfer: FIA01290 FIAOlUlO FIA0llt20 FIA01U30 FIAOlUUO FIA01U50 FIA012J+0 FIA01250 FIA01260 FIA01270 FIA01280 DRE00680 DRE00681 DRE01210 after FIA00280 2h. Transfer: FIA01350 FIAOll+60 F I A 0 1 U 7 0 F I A 0 1 U 8 0 F I A 0 1 U 9 0 FIA01500 FIA01300 FIA01310 FIA01320 FIA01330 FIA013U0 after FIA00350 25. Transfer: FIAOll+00 FIA01510 FIA01520 FIA01530 FIA0151+0 FIA01550 FIA01360 FIA01370 FIA01380 FIA01390 after FIA01210 26. Delete: INM00210 INM00353 IMM00351* 27. Insert: 7 1=0 IK1I00210 100 1=1+1 INM00211 IF(l.GT.ll) Go To 110 INM00212 after IJ1M00200 28. Insert: Go To 100 IHM00353 110 KERR=H IM0035U Go To 17 INM00355 29. Delete: 312 KHZ00280 KHZ00330 KIIZ003H0 KHZOIU90 KHZO1600 KIIZ01830 K11Z01990 KHZ020U0 KHZO2I1IO KHZO2U50 KHZ02600 KHZO2650 30. Insert DOUBLE PRECISION HBASE,HDEL ,DELHVL, Z,SDTA ,MPOLY ,MADDY KHZ00280 after KHZ00270 31. Insert: REAL«8 DLOG,DMAX1,DMIN1,DABS,DEXP KHZ00321 INTEGER COUNTT,COURT,COUN,COMPNT,VPFRAC KHZ00330 REAL KV(20),NEWX(20),X(20),ALD(20)5AV25(20) KHZ003^0 after KHZ00320 32. Insert INTEGER ARG KHZOO67I after KHZOO670 33. Transfer: KHZ00380 KHZ00390 KHZOOUOO KHZOOl+lO KHZ00U20 KHZOOi*30 KHZOOltl+0 KHZOOU50 KHZOOl+60 after KKZOO6U0 3 b . Insert REAL LIST(20) KHZ01071 after KHZ01070 35« Insert DOUBLE PRECISION DPKV(20) after KHZ01310 36. Insert: COUNT=0 53 C0UNT=C0UNT+1 IF(COUNT.GT.32) Go To 59 after KHZOIUTO 37. Insert: Go To 53 59 WRITE(6,57)NE after KHZ01590 38. Insert: COUNT=0 I67 C0UNT=C0UNT+1 IF(COUNT.GT.1+0) Go To l68 after KHZ01820 313 KHZ01311 KHZ01U90 KHZ01U91 KHZ0ll«92 KHZ01591 KHZ01600 KHZ01830 KHZ01831 KHZ01832 39. Insert: 1*1. 69 1=0 169 1=1+1 IF(I.GT.NOCOMP) Go To 6 0 1 after KHZOI98O kO. Insert: Go To I69 601 C0UN=C0UN+1 after KHZ02030 Insert: Go To 167 168 CONTINUE after KHZ02220 h2. Insert: KHZ01990 KHZ01991 KHZ01992 KHZ02031 KHZ020J|0 KHZ02221 KIIZ02222 C 7 3 IF(PRSS0R.GT.PCRIT.AND.NES.NE.NE)TOITE(6,729)PRSSUR,PCRIT,NE K H Z O 2 I 1 I O 7 3 CONTINUE KHZ021410 after KHZ02U00 1*3. Insert: COUNT=0 177 C0UNT=C0UNT+1 IF(COUNT.GT. 1+0) Go To 175 after KHZ02Ut0 hk. Insert: 79 1=0 178 1=1+1 IF(I.GT.NOCOMP) Go To 179 after KHZ02590 1*5. Insert: Go To 178 179 C0UN=C0UN+1 after KHZ0261+0 U6. Insert Go To 177 after KHZ02830 1*7- Delete KLIOOOliO 1*8. Insert INTEGER ICL(1)/ 1CLEA 7,INCL(1)/1NOCL1/,IIJNL(2)/'NAMEV LIST 7 1*9- Delete KN300030 50. Insert INTEGER BLN/' WW 7 , INCL (2) / * NOCL' , ' EAl^ 7, COM/1 , 7 51. Delete MEQ01990 52. Insert: 1=0 71 1=1+1 IF(I.GT .6) Go To 68 after MEQ01980 53. Insert Go To 71 after MEQ02070 5*4. Delete: RNMOO67O RNM00750 314 KJIZ02lt50 KJJZ02»l51 KHZ02ll52 KKZ02600 KHZ02601 KHZ02602 KHZ0261H KHZ02650 KHZ02831 KLIOOOliO KN300030 MEQ01990 MEQ01991 MEQ01992 MEQ02071 315 5 5 - Insert I H H = K I J M ( K ( J ) , R T ) R H M O O 6 7 O a f t e r RNMOO660 5 6 . Insert I H H = N U M ( K ( J ) , R T ) RNM00750 a f t e r R H W O 7 I 1 O 5 7 . Delete S E T O O 6 8 O 5 8 . Insert: _ I 9 = 0 S E T O O 6 8 O 8 8 1 9 = 1 9 + 1 . S E T 0 0 6 8 1 IF(I9.GT.NEMAX) Go To 6 7 SET00682 a f t e r SETOO670 5 9 . Insert Go To 8 8 SET00781 a f t e r SET00780 APPENDIX H C o m p u t e r P r o g r a m s o f O p t i m i z a t i o n M e t h o d s . 3 O O — K vl X Z z */> •M m — rg oc w 1 > u. vi L* »— X X • U J — t- c < r-Z U J a — . — VI z z »- Z |C o ' U U J i £ • L i i ? : 1 Z ZD ~ |C 3 : i - u, • j u <: ir i — I « TO v : - J o x a x c  x c  x zc rsi m *f m gJ r~ 'co <?• C J —• rg «"n Z CJ —' yi t - ! z o t LJ LU f -u *r x. < z a r- u u L p o a x e CC C J C J U J II U J OC O X o x II U J III u in -u 2 • Z . Z J -O J -O J r- (_ r- t— LU I) LU II UJ a C J cr cj c  X — U J * LU X — CO o i - z 1 • ^ <x • J * •> CM X •-. LU x vi • o «. r11 *~ < »- — rg JL -A. Z < as x -z — J » CL -1 1 Cu M • U J -T v» oC J J v» '.Z V ) t— >- X z m — »- < O U> V7 X » » » i— <: a o L J UJ U J rg 1 — LJ» C J a. X C J C J a H> d 1^ o *_> — V I CV X t U U-v C J o 1 1 x —• O " C J <£ z in o n • * 1 z u*\ »• o 1 C J n X m » o 1 •a C J «~ C J -~ C J J3 UJ C J —t r- rg rn g<l C J » LU UJ H X CN* m I CM rg X -j — LA C J gj C rg CJ rg O C J rg o •u LT\ —. rg O x _ J «t • —» CM — M* • < CL LP —4 re rg rg xj CV g-•T ^ c: tn H a- •* > X -f-» X oc ~- tv rg in v> C J X 1 A. X «i > w J> — > »- <a C J u U J u. C J zz X _ J X -c C J a O U- V I • 1 Z »M O - J J -c v* h- V I QC ct r-a -I I— «i C J U J C J _ J X z C J CL C J C J I - U l X 1 * • , CJ JZ. o »-1 1 *~ •» «L <M 3C CJ Z ( U X- CC _> «1 Z a c x rg C J o z a. CL CL CL t U -«« U J LL L J ( J • • v r-— - V u j _ — • • c* C J «r JL •"«. "N. - J UL in CL 4^ V I r-t V I USJ C3 U IU _ J ^ "N. *-» _> «* ^ < "N. — • V z C J U-r\ w • X Z w «l "N. II it Hn II II • Z h- — >- c_-VI LL' »-•s. JL a -* cu o C J C L < CL L J _> JL » c o «—• *• < Z U » JL c  X l/l v l wJ X X V I V l V J V l C J J>. V I C J X >— * — —> u tx C * J L •J" gj Hj gj gr «0 X' LT C V * _l C J C J >L _L > > > > y- -JJ z < CL LU ^- L I -7 C J ^ u- f ~ rg(M rc I>J rg vi v tr _l U r- i/t C J UJ X X X X o. ^1 V) V I V I VI ' V I C J LU X x —J CL gj II LA U CO LU X z O ^ • X *I J . CT -s . \ •v *V •V "M *S. •V. \ >>• -N — X rf L J L J < z * — - 1 » >- w _ J It o - - 'S . rg rg r> *T m - J *0 g3 U J L J X -* 7* *C ^C z z *T z Z Z *c z z L - l C J >— LA 1 t— z —1 VI z _r — %n w — V J a * c5 C J -1 c L J w C J L J _/ C J L J L J L J O O z X < O — ! l <l < (— U U nr EC 21 r  T LU U J X 1 "Z u* U J X X X X 2. X JJ. 2; X X X X X X X XXX Lu C J - J — J _ J LU L J L J i-i o — JC »— > LL o U J J L JL «i <i <t < iji-i-4 H - Z X J L J*. X *_ JL X X •A. X J L JL X X X X JL <i _ J _l <» «a < a z 7 U L ) i ' X ^C X a LL L L cc CL a _j CL •z • J z -J o C J L J L7 L J C J C J C J - J c C J CJ u o u u c — * U J < «4 < X W UJ U J C U J — C J C J - J I— Z _ J — • C J C J a C J L J C J L J C J < « ct VJ» _ J X « T L > C J O C J C J U ( J U C J v J u C J U o u C J C J C J -2 o C J C J O U J CC X u . X O X > LC o X V l C J u . C J C J C J LU UJ UJ LU C J o 3 J : LU IX M* vn %o <o cr • C J i m. _ J ' U J U-s — MT - H pi - *~ » L : CM so — J •—< w  ^> z z - z o o X O X -> L J I- • _» • _J X —« — I -« Q II II U o o x cr x c  o oc cj oc o c  X CV o C J — LL' • • X o cocro^rMrMringor^cfjcr mM*M-M*M,.>* -T «^  M" J/ i CM r*> *n . VI O -O ^ IN o IM <J \-Jf\|rsi — v r\ s* U i j s s i j t i_f «v r - E w . " ^ N u «l ^ 1 X 3 J * C ' U U X < I 2. 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UJ x ^ f\ c. -r u u «* i-t C L C J C J ri) • < ^ . n n t t i i p q f\j • Z. ^ — — C L h Z O U S . y C L c C r y j y IM " U u I K » < < k * l < < l o. a. a u. c. a « 1/1 cu — O >o — 3 J» • - 2i/i tO ri UJ < ^  t- X J - J i : z •-• a H _ J a o U J i j : u. cc cj U J ri rg m -q- tn in in tn in 319 o a o cc o at II ai II u. II tu l a J- * i a o o Z • Z • cc i/> cc a cc , J-T 3^  J'T J ? ' - U ,hC* t— 'j* O II Uj !i w u ai'l m -C i c c x a z a x c e u j «j cr N w-r m o f* co o* o it (\(\(\(MfM(\j<Nr>J'OC vl UJ 0 -o " r\i •vT * I — rg <> — 'vl C  |»- < !z o. ! — CJ 1 vl UJ \ »rg rvj — MTi cc cc r- -» < ^ n «i z a c • z • — C J U _ J CJ) |V) U J U J ' U_ •—• • w LL UJ •V T U 1 1 V •• v r rg y x i • * » x * f l o U -4) — ' w — V l w t— • L J L J CC I UJ «t : — H- • a. — i— Q- ^  — « ' 1 L » J CC O ! i 3 U J n j , u. O r- <X> a C J —< rg tfiirun ul <u 4 ) <«) X O £ — X v l < r-X z UJ • Z vl Z cj L0 • — C N ^ tf> -vi z z i — i — ^  o o -> o o C J U J i . 3 L |, t- • c ^ -c J - * - f —• i J-UUI U K I a UL UU O") cr x cc x a m <r in o f to v L I r g » gj gj gj -O ^ [•Ovi-f^p.r.r* -» ^ o <r • cl i — c 3 A. XoCXCCXoCXCCuju. IfiiCMUO'OHNfl'I-Ul -v i>»r*l^r~r*cocoujcijrjocv*' tr c cn v • . J • — J • *- o * U J H U J 1 cc L J a i I (VJ I r- » u H co Q J U J CT* CT . O cr rg r« gr in g: r- co cr vr cr- o* tr w v • z • z — l/l o c c o a i - z ; •u _> %o - J C J U J » r- |— r— »— L J 4. J L ii U J •« U J J > - ( O a o a w L G i OO—•rgrn-NTU>^5r-(X)v' o o - u u o o o u u o u O * O <* < • _J • _> z or <M ri < V I CL » X C J •a lit — • > » — </! 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L J X LO ro vj M *- Z <c « • • ri LJ o  z >© — -O ri 1- IJ ^5 Z z i-0 LO UJ «r ID < C  2 LJ o 1  rx. •— 3. > a u a J i tX LT O rv — >r CJ • w — — N O t -> * x — * CJ CJ CJ O CJ CJ Z :»» I - u i »«. — < -t o. : _i I— P- t - CJ cj . _j \a u* LO cj '.u : X JL X X j ; u u -j z < C J U J l «* 3C _ J J— - J * CJ C J K. ri UJ Z ri VO C J » < X a •> c? z OJ - » Z U - N X * -N. v ri »*^  z • U l —» z — LO X i X "*"» UJ » — U J z z O »• LO > X I ri t-i z z . _ > 1^1 —' ri U . i — •— Ld —t tx ri i « - u u i u cc •~ "^ r~-r-'~r-<oooLL»ooQjcL'C3coojccovc^  o* — -• ri CJ . ' OJ LJ f i rv n i ; vi to i . x ri , . x n < C •£ or U J O CD X Z -> r~ UJ wo [CJ UJ «f -^ x —J m • ac • cr C < u • - o • tsj Z Z it — or • — >- _j > z ' — U UJ > x u a , P > o ' (x V l \ 1 *3 r - ri . v » —• ri • C J —• — (M Z C J — ri C J * | V- I -j- ri — ri m , — cn L. v) C J cJ ) CL CJ U V i e_ CJ VJ *~ • taj x — «3 UJ c *-! LO r v *a > v . "v x -s. w o : u u .a ) J1 I/l c * • >- >• LU 1 1,0 • — _ J UJ <-< it m x x jx x : 1 LJ .T a w» u A. * - X * - • CJ CJ cj CJ W O IXXX *- JL X I U U u I CJ CJ CJVO O, U J X — X ( *L •-• z ti L' ;* u . r^ L- — c CJ LJ • 0 -r 1 U' O J J J J i * . X < _J _J _J ^ • CJ < «i < » o o or u CJ o cj i * < c. i " 5 a ' I t l C ^ L i I : x u. 3. u. i : o — » • _j m »t I tn ri LL ) O X _ J : •« c- -i j O J L J <T : or u. u » psi m >f tn *o r-co. cr o cs, r> • J l / \ C» N CO 0* |( 3C — — - > ^ X u0 X z CJ U j ri — Z CO PM C J * * X Z Psf — — L I CJ O • CJ m o m L0 —. m » X CJ r\j fi 1 X Psl — rsi *- or — ri (M z at < > ri Q . VO X X p -X • O 1-— ri 1 1 ri - U J X in CJ CJ z CJ N LL — — • • X ri t\J CO JC X X X ri ri ri • u. — <r -v. II 11 II n It fl X X o. •«» V* J » » X UJ C J X >r <u •J- -CI LL.' VO CJ * — •c £ L U t - re CM V . ri ri ri C2 X z » — V-- i V S i f V r\i rO ro tn _^ LU C — CJ Z) vo ?- *d — ri — — — LO V I C J rg + CJ Z O CJ If OC ar — 1 ^ rg a LU X X < «1 •4 < <a —J H _ J LO CM V ) X X JL CL a. U . CL Uv rsj - J H II -J ri o CJ vj C J C J C_l L J Z < - u z VO O U cj LU UJ Cv UJ U l LU CJ V I uo a j o o ^ r g r ^ f - r t n o p - o o o * » m «JT >»" 'j" «r sr -f <r T <r g 1 ~'.t 334 fCMM'/>l ('">'!>'0 • , f 7 . 2 , « SEC.*) 57 GO 10 165,21C),KCT 50 6 5 K K = 1 5<) C DEFINE STEP T C EE MACE ALCNG CIRECTION VFCTOR *LIST FP 6C 00 1 1 5 I = l,N 1 SL FJR0UTINE FF(G,X,N ,K,T1,T2,CEL.DELG,SN, SN 2,K S S, TK ) 6 1 115 S1G1 I ) = P « S C ( I ) 2 C WR 11 IE H (!Y J . P A LLC* S 62 c CcrrutS GRADIENT A F T C K MINIMUM REACHED C N CNE'0IHENSIONAL SEARCI-3 C CEPT. CHEM. ENG. , 1. 0. C. 6 3 DC 107 I - 1 ,N C CCCIFIEC EY PAUL FR I ECPAN 64 1C7 G I ( I ) * G i l ) 5 c DEFT. ChEP. ENG. , U . P.. C. 65 CALL GRACIG.X.N .fl.TK .CELG) 6 c JL.'.E , 11 M 6 6 nc u i I - I . N 7 01 " E N S ION h( ;C,2C) ,A(2C,2C) ,B<20,2C) ,Xt2C) ,SI20) ,SIGl20) ,Y[20), G12 67 1 1 9 c m = G 11) /SNC 8 10 ) ,G I t.?0 I .YT t 120 I.YT2 I 20 I.SOl 20 ), TK<2CI,C< 2C ) 60 C ciiMP-uiE » Mum 9 ' l-'X LOr.KlC. C) 69 120 DC 121 I = 1 , N IC CO 10 1 1.2 1.KSS 7C 121 Y ( I ) - G i l ) - GI(11 n c P U M INITIAL VARIABLES 71 DEN 1 = O.C 12 1 V . P I U 1 6 . 1 C ) N 72 DO 125 I = l . N 13 WK 1 It ( 6 , 1 71 1- 73 125 DEM - UEM • SI C 1 I ) »Y( I ) 14 WP 1TE16, I E ) C EL G 74 CU L 3 C I = l .N 15 WRITE ( 6 , i n n 75 CC 130 J = l.N ie kaiiE 16 .1 ;> 1? 7 6 130 A ( I , J) = S IG 1 I )*S IG I J l / C E M 17 WH ! T E ( 6 , 1 2 ) DEL 77 c CCr'PUIE H KA1R1X 18 2 C U L L F T M X,N , r.SNC , S N 2,C.TK) 78 DO 135 J = l.N IS V.RI IE (6 .151 SNC.1XI I ) < I -1 >N 1 79 YT 1 I J ) = 0.0 2C re F0*HAI<24H Nt.vn.Cl* OF VAlUADLES I S " , I 2 / ) " " " EC Ull 1 J5 I « I ,N 21 1 7 FCFCAT (••MJHE..' Of CONSTRAINTS IS « , I 2 / ) 81 135 YT11J) = Y l I ) * H (I,J ) • YT1IJ1 22 i e FCPyATl'GRACIENT STEP SIZE IS ' .E10.5 /I 82 DEN 2 = O .C 23 11 Fl»<MAI (42K PINIfLM PAGM 1 CUE CF DIMECIICN VECTOR IS .E10.5/1 83 DC 1 4 0 I - L.N 24 12 FC.P*ATt4f> HINIML'H STEP SIZE FOR ONE-DIPENSICNAL SEARCH I S.E1C. 5/1 E4 14C DEN2 * 0EN2 • Y T 1 I I ) * Y I I ) 2 5 1 3 FCRNATI48H IMTIAL STEP SIZE FOR ONE-0IHENSIONAL SEARCH I S.F1C. 5/1 85 DO 145 1 = l .N 2t 11 ( W A I C U I I . C F . - • . f 11.5 . * I M T . VflLCS. • 1 , 10F10 .5 I 06 Y I ? ( I ) * 0 .0 27 K S S = 2 E7 DO 145 J « l.N 2 3 KC1 - 1 68 145 Yl?( I ) • Y 121 I ) • H ( I . J) *Y I J) 2 9 . c SET H fAIRIX E COAL TC IDENTITY 8 9 CC 150 I = l . N 30 LU 1C 'j J - 1 »N 90 DC I5C J - I,N 31 DC 105 I * 1,N 9 1 15C B( I . J ) =-Y T 21 I)*YTllJ)/DEN2 32 1 0 5 H1I.JI = 0.0 9 2 c CCr- PUTE NEH H HAIR IX 33 DO 106 I =• l.N 93 DC 1 5 5 I = 1 ,N 34 K 6 M l i l l « l .C "54 UO 1 5 5 J » 1,N 35 c CALCULATE INITIAL GRADIENT 95 155 H (1 ,J ) ' H I . J I •» A1I.J) *• B I I . J 1 36 C A L L G R A C I O .N . f . l K . C E L G ) 9 6 c RETLRN FCR NEXT ITERATION 37 co i ce 1=1,N 97 GC 10 ICS 38 toa G U I = Gl I 1/SNO 98 c SOL01 ION Al GPlIfLP 39 c DEFINE DIRECTION TC EE SEARCF EC 9 9 2 0 0 WHITE (6 . 2 C I 1 (X (I ), l - I . N ) "40 K 9 0 0 11C I « l.N ICO WHITE ( 6 . 2 C 1 ) SN 41 S( I ) = O.C 1C 1 TINE=SCLCCK( 1) 42 OC 110 J = 1,N 102 KKITE( 6,< 44 > T IME 4 3 11C SI I) • Stl ) - HI I , J ) * G ( J ) 1C3 4 4 4 F C R f A T U / ' T I f E » ' , F 7 . 2 , « S E C . ' / / / ) ' 44 c 'CALCULATE -HAGNITL'DE OF SEAftCH'OIRTCTION"VCCTCR "104 2 C1 "FOi»HAT"'(lllC.37ir"VARI AEIC5 AT" CFTTHlf " m - . 1 CJ F 1 0 . 5 ) 4 5 CFK * 0 . 0 1C5 <C3 FORMAT (1HC24H OBJECTIVE FL'NC TI CN IS .F12.51 46 CC 111 I = 1 ,N 1 0 6 GC TO 4 0 0 4 7 111 OIK - CI'K 1 < 1 I ) **2 1C7 C HFVF.RSF SEARCH FCR CASES VFEN NO SUCCESS CCCUSSIN POSITIVE CIRECTlf 4 8 CF = SCRT (CHX I 108 ac IF ( K K . G T . 2 ) GO TO 2 C C 49 c I F THE CACMTUCE CF TFE SEAKCF OIRECTinN VECTOR IS LESS THAN 1 0 9 KK « KK « 1 it c P^ESCRlbED TCLEPANCE, ASSOE CFTIfLI- REACFEC no DC 2 1 1 I » l.N 51 IF ( C h . L T . T l l GO 10 2CC 111 . 1 1 S(I) - - S i l l 52 c CC CNE CICENSICNAL SEARCH 112 WRITE ( 6 , 2 1 2 ) S 3 64 CALL F 1 B C I X.N,C5 . S C ,12.C E L . S N . S . N 2,P,KCT,TK) 1 1 3 2 1 2 FCRfAT U H 0 . 3 2 H REVERSE CIRECTIONS ON PARAMETER/) 5 4 TIME-SCLCCMI> 114 KCT = I 55 WPITE.6,334) T I M E 1 1 5 GO TO 64 ITS 4 C 0 RETURN 116 35C F PD = SN U J F M . I ? 7 C S E A R C H U N T I L H I N I H L M 0 8 TAI NEC W I T H I N G I V E N T C I E R A N C E 118 SLR 1 U J L T1 N £ F I D O ! X . N , C , J , S C , T 2 , D E L , S N , S N 2 , P , K C T , T X ) 178 GC TO 4 1 0 U ' l C I I - l N i l C N C <2C ) , X I 2 C ) , < 0 ( 2 C I , £ ( 2 0 ) , X I (201 , T K ( 20) , C ( 20) 1 " 4CC fPH - SN 1 2 0 W R I T E I 6 . 3 3 3 ) 180 41C IF ( F P O . G I . F P M ) GO TO 4 5 121 222 f C U f M I / / / ' F I B O N A C C I S £ A H C t - » « « « » « ' / / / > 101 PC * Pf T 2 2 C FN ITTATTI? £~VA R t A R L E S ' — 182 PT* = P t 1 2 3 LA = I l t 3 F P M = F P U 124 DE L T » DEL 181 PC - C . 6 1 9 « ( P F - PO) • PO 125 PDfl = C . C 185 P = PC 1 2 6 PC = 0 . 0 1 8 * LA « 4 1 2 7 P = PC I C 7 i C _ I F ( I P F - P O . L T . T 2 ) CC TC I O C  T 2 e u r r i ^ T T i i lea wifntl if, s"fs") P'F ,Pc 129 4 X I I I ) • X I ) 189 S99 F C R M « r ( ' P F « « . G 1 5 . 7 , " P C - • , C 1 5 . 7 / / I 13C SS = 0 . 0 ISC GC 10 9 lil 1,N lc'l 15 P F - PD I B ? 7 5 5 » SS • S ( I ) * * 2 192 PC - PM 1 3 3 SS = i O i t l I S S ) i_» [ P L * t-Pf T54 CC~8 i = 1 , N TO P f = C . 3 £ 1 * I F I - - PC I • P t 125 E SCI I > « 5,11 I / S S 1^5 P = PM 13 6 K : i l l t l « , ' < « l < ! , l ! l n , l ' l i M 1 9 6 LA « 5 137 4 4 4 F C P I < A T ( ' S S = ' , G 1 5 . 7 , ' S ( I ) * ' , 8 G 1 5 . 7 ) 1 9 7 GC T C 4 0 138 5 CC 10 I » l . N 1^8 ICC IF ( ( P O - P C C 1 . G T . 1 . E - E ) GC TC 105 139 IC X ( I ) = X I I I ) • P * S a ( I ) 199 KCT = 2  T i o : axi -nN"i'x;NTM7s»;TSN"'2Tr;sTin "2x5^5 R T T U N 111 9 9 GC TC ( 2 C 0 , 2 5 0 , 3 0 0 , 3 5 0 , I C O ) , L A 201 C P R I N T C I M f U C V A L U E S 142 C MAKE I N I T I A L STEP ALCNG C I P E C 1 I 0 N VECTOR 2C2 I C E W R I T E ( t , 3 2 C > SN , ( X( I ) , I = 1 , N) 1 4 3 2 0 0 FX = SN 2 0 3 32C FORMAT I • 0 F « • , G 1 5 . 1 , / • V A H l A B L E S» «,eG15 .7) 144 FC - FX 2 0 4 RETURN 145 P = UELT 2 C 5 ENC ~m r s ~ " 7 tf>0 CF F I L E 147 • W » I T E ( 6 t J ; 5 l 146 555 F C H f A T C I M T S T E P — 2 C C f ) 149 GO TO 9 150 C I N C R E A S E STEP A L C N G G R A D I E N T * C C P V ' S H I P 151 25C F X P = SN IT? rr~irxp.Ge.FGi cc IC *s " 153 FC = FXP 154 C E L T = l . C i e * C E L l 1 5 5 P » P • CELT 156 W R I T E ( 6 , 6 6 6 ) 157 6 6 6 FCRI- AT I ' I N C STEP 2 5 C )  158 CC 1 0 1 — 159 25 PF = P 160 PM « C . 3 C l » P F 161 PC » 0 . 6 1 9 » P F 162 P = P * 163 LA * 3 164 W P I T C U . i 7 7 ) " '" 1 6 5 7 7 7 F C R C M ( ' W I T H I N RANGE 2 5 ' ) 166 GC TO 9 167 3 C 0 TPM • SN 1 6 8 C I M M P U I ' U T H I N R A N G E , SET FieCNACCI V A R I A B L E S 169 C C A L C L L A I E F L N C I I C N AT C . 3 0 1 CF V t C T C B D I S T A N C E , T T 5 P ~ » PD ' ~ ~ 171 LA = 4 172 KRITE(6 , e88) 173 £68 F C R f A T C F L N AT .381 3CC I 174 GO TO 9 1 7 5 C C A L C U L A T E F U N C T I O N AT 0 . 6 8 1 GF VECTOR C I S T A N C E . r o ( !6 1)0 9 9 J - 1.N2 57 WR11E16,E8) IP! I , J ) , 1 « 1 , N ) , Y U ) 5fl es FCPMATI8015.7) 59 99 CCNT1NLE f L 1ST MCCCMP 60 C MAIN LCOP BEGINS s 1 FUNCTION MCCCMPIP ,Y , M AX , N, N2 ,M, R,ITER,11TER,EPS,FUN,G,H,X) 61 CC 1000 11=1,1CCC ? 2 C 'KCC I r I'GC COMPLf.x MCThCl) P C R > CNLT NCATT CCNS T R A INtE OPT IMIZ AT ION " 62 ItCNT=.0 3 C SINGLE PRECISION VERSION. 63 MM = C 4 C REFERENCE: M . J . t U X . A NEW METFCD OF CONSTRAINED C P T I M 11 ATI C N ANC 64 C FINC A PUINT WITH A LARGE VALUE 5 C A CCKFARISCN WITH CTHER METHODS. "THE COMPUTER 65 RIG=Y(1) 6 C JCLRNAL'SE ,42-52. 66 LARCE-1 7 c WRITTEN: CENMS C RE ILLY 67 DC 3C I -2,N2 6 c CUMPL II NG CENTER U . E . C. " 60 I F I Y I D . L E . B 1G1G0T0 3C 9 c MODIFIED: PALL rnlFOHAN . 69 L A R C F = I 10 c CEPT. O E H . ENG. , L . 8. C . 7C BIG = Y { | ) 11 c j u N E . r m 71 30 CONTINUE t 12 c PURPOSE: TO FIND A L CCA L M M H I 1 CF A CCNSTRAINEC FUNCTION CF 72 C FINC CENTRCIC OF ALL eLT eiGEST L 3 c N VARIABLES. 73 UC 3 5 l » l , N 14 c CESCRIFT1CN Ct- PARAMETERS: ' 74 35 C ( I ) =C. C 15 c P=ARRAY DlfENSICNED AT LEAST IMXIN2). CN ENTRY COL 1 SHOULD 75 CO 40 1=1, N 2 1 16 c CONTAIN rSIMAIFD fll'-llllRINA Tl:5 OF T MC MINIMUM. CN EXIT COL 1 76 IF ( I .fC.IAHCE )CC TC 40 17 c SMOULC CONTAIN CO-LI* C IN AT E S CF OPTIMUM (HOPEFULLY). 77 DO 37 J = l » N ie c Y * V A L L E CF CEJECIIVE FUNCTION 78 37 C ( J ) - C ( J)«P ( J , I ) j 19 c HA X - T»HO LI D OF SI.PPLIFn AS CARDINAL FIRST DIMENSION OF ARRAY P 79 40 CCNTINUE ! 20 c MNLMtiER CF C e J E C T I V E FCNCIIGN VARIABLES " 80 DC 45 1=1, N • " " 21 c NCTE THAT 1 <= N <= ICC. E 1 C(I)=C(I) /TNFl i 22 c N2=DIMCNSICN CF SIMPLEX ( 2 < = N? <= 2C0 ) 82 C REFLECT WORST POINT THROUGH CENTROID. 1 23 C M=NLMHKR VARIABLES,GXPLICIT C IMPLICIT. 83 45 P ( I .LAKOt ) = C 1 1 )«H» ( C 1 I )-P( I,LARGE) ) 24 c ('NUMBER Cf CONSTRAINTS). E4 C CHECK TO SEE IT NEW FCINT IS CK £ IF IT ISN'T | 25 c NOTE Th A T N <= M 85 C TRY TO GENERA IE A NEW ONE. 26 c K - E X I - A N S I U N rAG 1 OR l :(,it U E F L E C M C N «)CP 0 6 C ( I ) T I ' C C K LXi'LlCIT CONSTRAINTS: 27 c ITER=MAX. NO ITERATIONS OR STEPS PERMITTED. e7 55 CC 60 J=1,N 23 c I I T E K - m n A l l C N LCG IS PRINTED EVE><Y IIIEK'IK STEP. 88 A MM J , LARGO 29 c EPJ=CONVERr,FNCE 10 OPTIMUM IS ASSUCEC WHEN FUNCT1CN VALUE REFUSES 89 GC = G(P( 1,LARGE 1 ,N,J) ' ' ! 30 c IC CHANCE BY IF1S ABSCLLIL AMULNI 5 UMtS IN SUCCLSS1CN. 90 . IF (A.OL.CG )GCIU 56 31 c 91 P( J .LARGE) = G G M . E-6 22 c THE FOLLCUNG DUMMY PARAMETERS CCRfESPCNC TC USER SUPPLIED ' 92 GO 10 6C 23 c AUXILIARY FUNCTION SUBPROGRAMS WHICH SHCULC EE DEC LAREO 93 56 HF=F I P 1 1 , L»RCE ) ,N,J) 34 c EXTERNAL Hi THE CALLING PROGRAM: 94 IF(A . L E.HF ) G C T C 6C 35 c 9 5 P ( J , L A R G E)=HH - l.E - 6 36 c F LN = F UNC TI O N 10 EVALLATE OBJECTIVE AT A GIVEN PCINT. 96 60 CCNT INUE 37 c G=FUNCTICN TO EVALUATE LCI<ER 8CUN0S CORRESPONDING TC 97 CI2) CHECK IMPLICIT CONSTRAINTS 33 c CONSTRAINTS 1,2,...,M. 98 C ""'EVALUATE FUNCTICN AT NEW f C I M 39 c H=FUriCTICN TC EVALUATE UPPER ECUNDS CORRESPOND ING TO 99 75 MOCOMI> = FUN(P( 1,LARGE),N) 40 c CIJCCTKAINTS 1.2... . . f . ICO CC 0 5 J - N P 1 , M 41 c X=FUNCTICN TC EVALUATE IMPLICIT VARIABLES. 1C1 A=X(P(1 .LARGE) ,N,J) 42 REAL MOCOMP 102 I F ( A.LI .G( P ( 1 .LARGE) ,N , J) . CR. A. C T. H { P U , IARGE) , N , J) )CCTC 220 43 DIMENSION P<MAX.l),G(ICC) ,CLO( 100) .YI2C0) 103 85 CCNTINUE 44 T=SCLCCK(C.CI "1C4 C'.3)'~ CHECK 10 SEE '! F ~ N E w ' I 5 ~ $ TIL L~ W C f ST " '" 45 TWCN=N2 1C5 IF(MOCOMP.GT.YlLARGE))G0 IC 220 46 NP1=.NM 106 Y(LARGE)=KCCCMP 47 IMP 1 = N 2 - 1 1C7 C NEW PC I NT ' *** 48 c SET NC. STEPS » 0 ice C FINISHED? 49 NS 1EPS = C 109 90 CC 95 K=1,N 5 0 c SET ''PRINT INTERMEDIATE STATISTICS?" FLAG. n c C(K)=P(K,1) 5-1 IN T E R-0 i l l DU 58 J=2 ,N2 52 c SET NC. CONSECUTIVE ECU AL FUNCTICN VALUES • I 112 98 C ( K ) * C < K ) « P ( K . J ) ! 53 ISTCPM U J 95 C(X)«C(K)/TWCN 54 URITEI 6, 2CC2 I 114 00 96 J - l . N 55 WPITE(6,2C07) 115 If 1ABS(C(J)-CL0( JD.GE.EPS) GC TO 120 ro f 116 i n 96 CCST1NUE I < 1CP = 1 S K P t l - 176 177 239 GO TCI lie CC 241 I - l .N 118 If( IS10P.NE.5jnO TO 14C 178 241 P ( I , L A R G E I > P ( I , L L ) * H » ( F ( I , L L ) - C ( I I J 119 C AT LASt WE AWE PINISFEC 1 79 ICKNI'C 12C N< 1EPS = NSTEFS*l 180 H>= 1 12 1 I K I STOP = 5 181 GC TU 75 12? f.t.tC' IftOl 1P2 2CCC F n i lH AT ( / ' TI K P =1 ,F7 .2 , ' SEC. IT ER . S C . ' , IS ) 123 118 W» ITEIj,2004 ) 183 2CC1 FORMA II / / / ' IMEKfEUIAlE STATISTICS:') 124 1>M IE (t ,?CC7I 184 2002 FCHPAT(// /• IN ITIAL COMPLEX:') 125 TIME'SELCCKI 1 ) 185 2CC3 FCPPAIC FLNCTICN V*LLE= « , G 1 5 . 7 / 126 UK 11 E(t,2CCC)I IME.NSTEPS 186 C ' CO-ORCINATES:•/(8(1x ,G15.7))) 127 Vi t t ITC(£ .2CC3>l 'CCCHP t (P ( I , l ) > I> l ,NI 187 70C4 FCPCATI / / / ' FINAL STATISTICS:') 128 FE IljSNv 188 2CC6 FCRKATC*. » 1 12* 120 LI) I2I|J>I,H 169 2CC7 IUHMAII •• • J 1 30 121 c i r ( j ) = c i j ) 190 RETURN 131 IS ILP = I 14C NSTfPS=N<UPS«l 191 ENO 132 ENO OF FILE 133 If ( N S r t P S . C C . ITER >CU TC 110 134 J-KfJlNSlEFS. I IT EH ) 135 IFIJ.NE.OGO TO ICCC I 36 If I IMGB.CC. 1 JGOTO 15CC $CCPY »SKIP 137 IMER=1 138 VRI IE (6 .2CC1I 139 WR ITE( 6.7CC6 1 14C " ' 15CC riKE = SCLCCK(T ) • 141 V.<UTE(6,2CCC>TIKE.NSTEFS • 142 on io i c c i 143 1600 Wfl I T L(6 .2C03(SMALL,IP(I,LEAST),1=1,Nl 1*4 ICCC CCMINLE 145 J=l 146 C SCAN FC P C-C-ST POINT SQ FAR 147 1CC1 SMLL = Y(1> LP. IS 1 = 1 149 Cfl 1C1C l = 2,N2 150 IFIYI I).GE.SMALL JGU TO 1C10 U l SCALL=YII) 152 L EA 5 T=I I 53 1010 CCNTINUE 154 IFIJ.EC.CJGCTC 1600 U S C INSERT BEST PCINT IN FIRST CCLLMN 156 CO 1020 1=1,N 157 1C20 F(I,1)=PII,LEAST) 1 E 8 MCCCMP=YILEAST) 159 IF! ISTCfp.EO.i IGOTO 11E 160 22C ICCM»VciNr«l 1(1 IFIICENT.GE.5) GC TC 235 162 00 2 3C.«.J=1,N 163 730 PI J.I AnGF) = <P<J,LAP.GFHC<J>>«C.5 164 ' " GC TC ,75 165 C SCAN FOR BEST PC I NT 166 167 235 LL=1 CC 240.?!=?,N2 168 IF(Y(lT .GE.Y(LL)l GC TC 24C 169 L L ' I i 170 • 171 240 CCNTINUE C REFLECT CENTRCUC TKMCUGH BEST POINT 172 IF(MM.EQ.C) GC TC 239 173 MOCCMP=Y<LL) 174 CC 1030 I«1,N 175 1C3C PI I , 1 ) « P ( I , L L ) 5 6 PI I ) = F H - l . E - 6 57 3 0 CC NT INUE 58 C EVALUATE O B J E C T I V E F L N C T I C N A T NEW P C I N T 5 9 CONPA 1•FLN(P » N I S L 1 S T C C N P A T 6 0 C CHECK IMPLICIT C O N S T R A I N T S 1 F U N C T I O N CCNPAT t N , M , R , E P S , I T E K , D E L , P . F U N , G , H , X ) 6 1 DC 40 J ^ N . C 2 C K C I F I F . D P A I I F . K N SEARCH FCP NCM. INTAR C C N S T R A IN E C O P T I M I Z A T I O N 62 A > X ( i' , N , J 1 3 C REFERENCE: K• HOIJKE, T . A . JEEVES. DIRECT SEARCH S O L U T I O N O F 6 3 GC=G(P,N,J ) 4 C NUMERICAL mi STATISTICAL P R O B L E M S . J . A . C . M . , 6 4 F H = F ( P , N , J > 5 C n , ? 1 2 - 2 ? 9 , U 9 6 X ) . 6 5 I F(A . I . T . C G.OP.A . C T.FH) GC TO 66 6 C V R I I T E N : P A L L F R I E L K A N 6 6 4C CCNIINUE 7 C DEPT. CI'EM. ENG. , L . B . C . 6 7 C COMPARE FIJNCMON V A L L E S 8 C JUNE , 19 7 1 6 8 IFICUNPAT.GT.YYIGC T C 66 9 c P U R P O S E : IU I I M J A L C C A L M M P U P cr A C C N S T M A I N E C F U N C T I O N 6 9 YY 'CriNI'AT I C C O F N V A R I A B L E S 70 GO TI) IC 1 1 C C E S C R I F T I C N CF P tRAMEI tuS : 71 66 L L ' L L « 1 1? C N « M J M R E M CF INCEPrNCFNT VARIABLES (N<50) 72 I F (LL.GT.1 I CC T C 77 13 C f » N U M B E R CF CCNSIBAIMS 73 P I I I - - N I )-2.C*BEL( 1) 14 C K ' R E X U C f l C N FACTOR FOR STEP S I Z E R E D U C T I O N 74 GU 1(1 5b 15 C EFS = L I M 1 ING STEP S I Z E 75 77 P( I > = P(I>*CEL(I) It C OEL-VAKI AI :LE STEP 3IZC 7 6 I C CCNIINUC 17 C I T E ^ M A M P L C NLMHER CF ITERAIICNS P E R M T T E C 77 IFIYY.LT.YOI GO IC 22 18 C ' F=CL'CRCIN*TES Of GIVEN PCINT ' 70 If IK.GT .1 1 GO TO 1 19 C FLN* F I.NCTICN TC FVAIUME TPJECTIVE AT A C I V E N P O I N T 79 C STEP SIZE RECUCTICN 20 C G = FU.MCTICN 10 E VALLA IE LCViEK KCUNOS CCRRESPCN'CING "8C DC SO I * 1 ,N 2 1 C TC CCNSTDAIi'lTS 1.2, ,M 81 IFICELI I) .IT . E P S ) G O TO 15 22 C H--FUNCTICN TC EVALUATE UPPER BOUNDS CORRESPONDING 82 9 0 CONTINUE ; 23 C 10 CCNS ISAI MS 1,2, M 83 GO 10 33 24 C »«FUNCI10N 10 EVALUATE IMPLICIT V A R I A B L E S 84 15 TIME3SCLOCKI 1 ) 25 CI P E N S ICN F (50 ) ,Z(50) ,W(5C ) ,0EL (50> 85 25 W F I T r ( 6 , I C C ) TIME.NSTEPS 2 6 T = SCLLCK(C.C ) 8 6 " U P I T ( . ( 6 , r ; G 0 ) YC, ( P ( I I, I - l . N ) 2 7 WRIIEI6,3CC)N,M ,R,FPS E7 5 C C FGP.PAK//•CPTIfl l - HAS B E E N P E A C H E C / / • C E J E C T IV E F U N C T I C N - ' .1G15. 2 8 WRITE ( 6 , 4CC ) (P ( I ), 1=1 , N 1 88 17//'VARIABLES • '.EG15.7) | 2 9 WB I T F (6,7001 C C F L ( I I, 1= 1 , N I 89 RETURN i 3 C 3 C C F 0 4 M A I ( / / ' N L P B E R CF V/PI ABLES « • , 12//• NUM BE P O F C O N S T R A I N T S « ' , S C 3 3 uo to 1*1 ,N ' i 21 1I2//'STEP SIZE REDUCTION F A C T C R = • , F 5 . 1 / / • L I C I T I NG S T E P S I Z E - 91 CEL I I) = CFL( I)*R 32 2E10.3) 92 5 0 CCMINUE 1 a 4 C C FCPPATI//'INITIAL V A R I A B L E S = • .8G15.7 , / / ) 93 WRITE(6 ,n00) (CEL ( I ) . 1 = 1,N ) 34 7 C C F ( J H M A I ( / / ' I N I T I A L S T E P S I Z E S » ' . 8 G 1 5 . 7 ) 9 4 6C C FORMAII/'SIKF SIZE REULCTICN'/ / •NEV S T E P S I Z E S » ' , 8 C 1 5 . 7 > 3 5 C IMT IAL IZ/T ION 95 GO TU 1 3 6 NSTEPS=0 9 6 2 2 TIME=SCLCCK(T) •37 NN=N*1 97 N<1EPS=NSIEPS«1 28 YU=FU:4(P.N) 98 I F ( N S t G P S . c e . I I E R I GO TO 2 5 , 3 9 YY = Y 0 99 WRITE(6 .1CC) TIME.NSTEPS • 4 0 DC 80 1=1,N ICO WRITL(6,<>C0> YY , (P ( 1 ), I" l . N ) 4 1 E C MI>=P(I> 1C1 6(C FORMAT!'OBJ. F U N . = ' , G 1 5 . 7 / ' V A R I A B L E S = ' . 4 C 1 5 . 7 ) 4 2 1 K=l 102 I C C FUR MA I ( / ' T I M E - ' , F 7 . 2 , ' S E C . I T E R . N C ' , 1 5 ) } 4 3 CC 180 1=1,N 1C3 c FMTERN CCVF r H l i t P ( 1 I = K I ) K 4 " " CC 70 1=1,N 4 5 C EXPLORATION 105 Z(I)=w ( I) : 4 6 2 CC 10 1=1, N 1 0 6 w(i)=pii) - ; 4 7 P ( I ) - P ( I ) « C E L ( I ) 107 7 0 CCNTINUE 4 6 C CHECK E X P L I C I T C C N S T P - M N T S 1 C 8 YC'YY j 4 9 L L - 0 109 CU 11C 1*1,N * 5 0 55 GG = r , ( P,N , I | 110 P( I) = 2 . 0 » F ( I ) - Z ( I ) ; 51 I F (PI I 1 . G E . C C ) G C T C 2 0 111 c CHECK CXPLIC1I C L N S I R A I N T S 52 P ( I > = GG*l .E-6 1 1 2 D C 12C 1=1,N ' ; 5 3 GC TO 3 0 1 1 3 C C « G ( P , N , I ) • 5 4 2 C H f . H ( P , N , I ) 1 1 4 IF ( P I I ) . C E . C C I CC T O 1 1 1 5 5 I F I P I I ).LE.HH) GC TC 3 C 115 PI I )=GG*l .E -6 i ro X LO ~ «_» o — — — fM rg L J o CM CJ gj »-» LTi — rg >- *- L J X —J 4 — »>J • «i IT rsi rg gr CL ~* in —• t <r X in CL X CT W ^ rg c  12. V) > a —» X 1 rvi X «~ • -= Z> w — • w *i L J CJ U J O 3C _ J 2 . Z O CL LJ> CL LO 1 z (M L J - J >- U . r - ^> *-*— •«a u L J U J u. O i £ - J JL z O CL L J C J gj U ' X 1 »J z <J> » -1 I a. Z j LU X _j •—• <1 z » tX LL' X rg <r <JJ r-CO O O • CL UJ " X LU C L U3 L J •N. • • —t <"- a. K. -•». Z — to UJ Z «. _ J CL c — - LO *— i— •~ t-gj C . LP _i i iC L J •N. »— V . >v ^ < L J < X a * X — «X "S. M II a n P | i i| B a R rg UJ ->» < CJ. c U . L J X L J _j c J- » £. z. X C J J* LO iO c ~ 2 . V i cn L O i/i o X L J L^. t— m L - • — — L J u> L J J i . vT gj gj o rg rg —* ~+ *r gj ITi - J — X t— L J X - * i X >- > J - L U «<C LL -J gj ijT> X L*J L J (V rg rg rg fM rg (*\ rg rg _1 c UJ tO L J JJ x 2 . X JL CL J l •-o "uO CO «J1 to tO L J LU UJ 2 . _« X - J a. o <r II in — U J X Z u L O CC CL cn •V. •N. ^ "v. \ v. a LJ z - J ? II L J "»s. rg rg <*-»o gj r-m tr« <I L J X Z z z z z *C z z Z z IS) L J LJO o SL _ J to z z - J UJo <_) a gj _3 c L J L J LJ 6 - a z 21 — •c L J 1 < — <: V-Z CJ O nc (X or X C£ X X a: TC C J ^ < •- 2 . JL. X JL X a. a. 2 . X X 2 . X 2 . 2 . 2 . X X LU O _j 1 _ J UJ C J 2 . gj C J X > CL L J L U 2_ X •c: < «£ «* <l < ^ . < -a U J z CV i. JL. 2_ 2 . JL JL 2 . 2_ 2 . X JL JL 2 . X 2 . JL <I < (X 1 <i cr Z CJ CJ 2 . a. x LL a CL a. CL UL UL o. a — LU l_) L J n L J L J a L J t_i w CJ L J L J L J o zz L J i—i ULt <L •< < X L U •j L J LU C UJ — O _ J <— Z Z3 •— C_l 3 LJJ <j C7 L J V J a 1_» L J L J o u-JL. L L VI L J u UO u O L J l_) ( J u o o O ( J O p L J a X L J ( J u UJ CC ac X X c X > X o U . V J UJ u. O C J O L U LU LU L U L J LU UJ U J U J LU rgmgru^Or-COJ'O—'rg^ u^^so p.co cr O ~* rg rn j-ipsjr-tDCfO-i • l LU U J . X ^ *i - — ii — -J . a. — •— —J — — ^  <i , LL — L J ^ . « - Q. <-> U J _) t— CL U M U a x « «. • i- — z ^~ C C J » - J Xnla • J U — - «Q - L U X — k- I U « U . r\j gr tn gj r-•i mn ^  rt — -^grgj-gr"'-*-**'*"'*'-*"-'' U o —i (NI f^ i ^ tn irt irt tn «> m ( 5 6 FUN = 1.C 116 13 G= 7 6 0 . 0 57 WR ITEI6,222) EOPAR(5,26) 117 RETURN 58 222 FORMAT(//• IMPLICIT CONSTRAINT V 1CLAT EO— F.CP AR ( 5, 26)* ',G15.7I l i e 14 G«7<0 .0 59 Pf. TERN 1 19 RETURN 60 77 CALL SLUSCI ' W PIT 0 ( 6 , 99) SINTSV(4,6),SINTSV(4 ,n).SINTSV(4,10),SINTSV(4 , l2) 120 121 ENO FUNCTION K T . N . J ) W R 1111 6 , V'S") EC F5 API? 7 2 41 ,ECPAR(2,?6),EOPAIU3,26), 122 OlMtNSlUN Ml) 63 64 I tCPAKI4 ,26 ) ,ECPA»(5 ,26 ) WRITE(6.S91ECPAR(6,22),EQPAR (7,22) ,E0PAR(6,31) ,ECPAR17,31) 123 124 1 C.n T i l l 1, 2, 3, « , 5 ,6 ,7 , f ,<;, 1C11,1? ,1 3 ,14) , J F = l . 0 65 FUN=-SEXTSV(14,27) 125 REURN 66 VRITC(6,S9)-FLN,<I1XTSV(3,?7),SEXTSVI3.33),SEXTSV(3,24I 126 2 H= 1 . C 67 99 FORMAT(3X,8FIC.5) 127 RETURN 68 REILWN I2B 3 H * 1 . 0 69 END 129 RETLRN 70 FLNC1 ION X ( 1,N,J) 130 4 H=1.C 71 OIMENSICN.T(I ) 131 RETURN 72 CCMKCN/ ECFA /ECPAR (25,5C 1 ,NEM.AX .MAXEQP 132 5 H=0.995 73 COMMON/SIM»/SEXTIV< 23.1CC),SINTSV(10,1C0),NSMAX,MAXSEX,PAXSIN 133 REURN 74 JI>N = J-N 134 6 H=C.Ol 75 CC 10(9,10,11,12,13,14),JMN 135 RETURN 76 9 X = 1 . 0 - T H ) 136 7 H=0.995 77 RETC<N 137 REURN 78 10 X = I . 0 - T (2 )-T (3)-T ( 4 ) 138 e H=C.C8 79 RETURN 139 RETURN ec 11 X = S I N I $ V ( 4 ,61 140 9 H = l.t> 81 RETURN 141 RETLRN 82 12 X=SINTSV(4,e) 142 10 H=1.C 83 RETURN 143 RETLRN 84 13 X*SINTSV(4,1C) 144 11 H = 945 .0 85 RE IU4N 145 RE URN 8 6 14 X ' ' , i f . r , i v ( 4 , i 2 ) 146 12 H=S45.C 87 RETURN 147 RETURN ee END 148 13 FU945.0 89 FUNCTION GIT.N.J) 149 RETLRN 90 OIMLNJILN T ( t ) 150 14 H-S45.C 91 GC T O ( l , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 ) , J 151 RETURN 92 1 G = C.C 152 ENC 93 R ETURN 0 OF F I L E 94 2 G * 0 . 0 S5 RE URN 96 i G=C.C 97 RETURN OPY »SKIP 96 . 4 G = 0 . 0 99 REURN 100 5 G'C.15 101 RETURN 1C2 6 G-=0 .001 1C3 RE URN IC4 " 7 C'C.',fc 105 Rt'T'.RN 1C6 8 G=0.U1 IC7 PE TURN 1C8 9 G=C.C 109 RETURN n o 10 G*0.0 U l RI: ILKN 112 11 G=76C.C 113 RETURN 114 12 G-760.0 " 5 RETLRN UA) 3 

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