Analysis of Distillation Column Dynamics.

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LSU Historical Dissertations and Theses Graduate School

1979

Analysis of Distillation Column Dynamics.Kurt W. KominekLouisiana State University and Agricultural & Mechanical College

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KOMINEK, KURT W.

ANALYSIS OF DISTILLATION COLUMN DYNAMICS

The Louisiana State U niversity andA g r ic u ltu r a l and Mechanical C o l . PH.D. 1979

University Microfilms

intern St ionel 300 N. Zeeb Road, Ann Arbor, MI 48106 18 Bedford Row. London WCIR 4EJ, England

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KOMINEK, KURT W.

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ANALYSIS OF

DISTILLATION COLUMN DYNAMICS

A D i s s e r t a t i o n

Submitted t o t h e G raduate F a c u lty o f t h e L o u is ia n a S t a t e U n iv e rs i ty and

A g r i c u l t u r a l and M echanical C o l le g e in p a r t i a l f u l f i l l m e n t o f th e r e q u i r e m e n t s f o r t h e degree o f

D o c to r o f Ph ilosophy

in

The D epartm en t o f Chemical E n g in e e r in g

by

K u r t W. KominekB .S ., C o lo r a d o S ta te U n iv e rs i ty , 1972

M .S ., Case W e s te rn R eserve U n iv e r s i ty , 1974December 1979


ACKNOWLEDGEMENT

The a u t h o r w ish es t o th a n k Dr. C e c i l L. Sm ith f o r

h i s gu idance i n t h i s r e s e a r c h e f f o r t . W ithou t Dr. P . A.

B r y a n t s 's s u g g e s t i o n s and b a s i c s t e a d y s t a t e c a l c u l a t i o n

program t h i s p r o j e c t would have been much more d i f f i c u l t .

Dr. A. E. J o h n s o n 's s u g g e s t io n s have a l s o been h e l p f u l in

d e v e lo p in g t h e s t e a d y s t a t e program. H aze l LaC oste has

been a g r e a t a i d i n c o m p le t in g the c o r r e c t fo rm s, i n

k e e p in g th e a u t h o r a t th e t e n n i s c o u r t s and in g e n e r a l

c o n s u l t a t i o n .

The a u t h o r w ish es to th a n k the f o l lo w in g f a c u l t y

members f o r s e r v i n g on h i s com m ittee an d g iv in g t h e i r

t im e , e f f o r t a n d s u g g e s t io n s ; Dr. P . A. B ry a n t ,

Dr. A. B. C o r r i p i o , Dr. F . R. Groves, D r. D. P. H a r r is o n ,

Dr. A. E. J o h n s o n , Dr. W. G. Rudd, and D r. C. L. Sm ith .

The a u t h o r w ish es t o th a n k th e L .S .U . Chem ical

E n g in e e r in g D ep artm en t f o r fund ing t h i s r e s e a r c h and the

C o a te s F o u n d a t io n fo r p r o v id in g funds t o re p ro d u c e t h i s

d i s s e r t a t i o n .

11


TABLE OF CONTENTS

P age

Acknow ledgem ent................................................................................... i i

L i s t o f T a b l e s ................................................................................... i v

L i s t o f F i g u r e s ................................................................................... v

A b s t r a c t ....................................................................................................... v i i i

C h a p te r I - I n t r o d u c t i o n ............................................................. 1

C h a p te r I I - S teady S t a t e C a l c u l a t i o n ................................ 13

C h a p te r I I I - Equim olal O verflow M o d e l ........................... 20

C h a p te r IV - E n tha lpy B a lance M o d e l .................................... 56

C h a p te r V - P r e s e n t a t io n o f System Dyanmics................... 108

C h a p te r VI - A p p l i c a t i o n ............................................................. 117

C h a p te r V II - A n a ly s is o f System I n t e r a c t i o n s . . . . 163

C h a p te r V I I I - C o n c lu s io n ............................................................. 201

R e f e r e n c e s ............................................................................................. 203

Appendix

A U s e r 's Guide f o r S te a d y S t a t e Program . . . 207

B Program L i s t i n g o f S te a d y S ta te Model . . . 219

C Program L i s t i n g o f E q u im o la lO verflow Model ............................................................. 246

D U s e r 's Guide f o r Dynamic Model.................... 257

E Program L i s t i n g o f E n th a lp y M o d e l ........... 264

F Program L i s t i n g o f O u tp u t R o u t in e s ........... 283

G D e s c r ip t io n o f E th y le n e Column.................... 308

H ' D e r iv a t io n o f 1 /g ^ , M^........................ 314

I D e s c r ip t io n o f BTX and PBP M o d e l s ........... 318

V ita ...................................................................................................... 329

111


LIST OF TABLES

T ab le Page

1-1 N ecessa ry E x p re s s io n s f o r a CompleteDynamic Model...................................................................... 4

1-2 Common A s s u m p t io n s .......................................................... 5

1-3 Secondary E f f e c t s Which ImproveModel R e s p o n s e .................................................................. 8

2 -1 G e n e ra l Overview o f S te a d y S t a t e Program . . 17

2-2 L im i ta t io n s , C o n v e n tio n s andAssum ptions f o r S te a d y S t a t e Program . . . . 18

3 -1 Dynamic Model C ondenser O p t io n s ............................. 23

3-2 Dynamic Model R e b o i l e r O p t io n s ............................. 24

3-3 C a lc u la t io n P ro c e d u re s f o r E quim olalO verflow M o d e l .................................................................. 46

5-1 G e n e ra l Tuning P ro c e d u re f o r a Two In p u t-Two Output P ro c e s s U s in g EP^^ and EPg2 • • • 113

5-2 Types o f D e c o u p le rs .......................................................... 116

6-1 E s t im a t io n o f L iq u id L e v e l C o n tro lLoop Gain................................................................................ 119

6-2 Sample In p u t Data f o r L iq u id T ra n s fe rF u n c t io n s ............................................................................... 160

6-3 L iq u id T ra n s fe r F u n c t i o n s ........................................... 161

6-4 Feedforw ard C o n t r o l l e r s ................................................ 162

7-1 P a ram e te rs Used f o r Dynamic M odel........................ 185

7-2 ' C o n t r o l Systems - P a i r i n g o f Measureda n d M anipulated V a r i a b l e s ......................................... 191

7-3 Measurement S e t ................................................................... 192

I V


LIST OF FIGURES

F ig u r e Page

3 .1 G en e ra liz ed E q u i l ib r iu m S t a g e ................................... 25

3 .2 Condenser - L iq u id L ev e l M an ip u la te sD i s t i l l a t e L iq u id F low .................................................... 25

3 .3 Condenser - L iq u id L ev e l M an ip u la te sReflux F l o w ............................................................................ 34

3 .4 Condenser - S e l f - R e g u l a t i n g ....................................... 34

3 .5 R e b o ile r - L iq u id L eve l M an ip u la te sBottoms L iq u id F l o w ......................................................... 37

3 .6 R e b o ile r - L iq u id L eve l M an ip u la te sVapor Flow................................................................................. 37

3 .7 Flow C hart o f S o l u t i o n Techniquef o r E quim olal O verflow M odel...................................... 44

3 .8 D i s t i l l a t i o n Column w i th O ptionsC.2 o r C.4 and R . l ......................................................... 48

3 .9 Flow C hart o f O v e r a l l A n a ly s is Package . . . . 50

4 .1 Flow C h a rt o f S o l u t i o n P ro c e d u re .............................. 70

4 . 2 - T ra n s fe r F u n c t io n s f o r BTX Example.........................74-894 .1 7

4 .1 8 - T ra n s fe r F u n c t io n s f o r PBP Example.........................91-1064 .3 3

5 .1 Closed Form T r a n s f e r F u n c t io n E q u a t io n s . . . . 109

5 .2 C h a r a c t e r i s t i c E q u a t i o n ................................................ 112

5 .3 Block Diagram and T r a n s f e r F u n c tio n f o r EP^^. 112

5 .4 Block Diagram f o r C o n t r o l System U singD eco u p le rs ................................................................................. 115

6 .1 E thy lene C om po sit io n Used to M an ip u la teD i s t i l l a t e V a p o r .................................................................. 123

6 .2 E thy lene C om position Used to M an ipu la teS id es tream L i q u i d ............................................................. 124

V


LIST OF FIGURES (co n tin u ed )

F igu re Page

6 .3 - P l o t s o f Column Dyanmics-Base C a s e ............................ 125-1306 .8

6 .9 - P l o t s o f Column Dynamics-6.14 l / g ^ = gg /G i = 0 133-138

6 .15- P l o t s o f Column Dynamics-6.20 L a rg e C o n d e n s e r s ....................................................................143-148

6 .21 - P l o t s o f Column Dynamics-6.26 L a rg e R e b o i l e r ................................. 149-154

6.27 Time Response o f P g g ................................................................156

7 .1 T r a n s f e r F u n c t io n Block D iagram o f Two In p u t-Two O u tp u t System w ith C o n v e n t io n a l C on tro l C o n f i g u r a t i o n ........................................................................... 165

7 .2 P l o t o f C h a r a c t e r i s t i c E q u a t io na . U n s ta b le P r o c e s s .............................................................. 168b . S t a b l e P r o c e s s ...................................................................168

7 .3 B lo c k Diagram o f Two Inpu t-T w o O u tp u tS ys tem w i th o u t C o n t r o l l e r s ................................................ 169

7.4 The E f f e c t s o f K o f E P ^ ^ ...................................................... 173

7 .5 P o l a r P l o t o f K - F a v o rab le I n t e r a c t i o n . . . . 175

7 .6 P o l a r P l o t o f K - P o s s ib le U n fa v o ra b leI n t e r a c t i o n .....................................................................................175

7 .7 P o l a r P l o t o f K - U n fav o rab le I n t e r a c t i o n . . . 177

7 .8 P o l a r P l o t o f K - F av o rab le I n t e r a c t i o n . . . . 177

7 .9 P o l a r P l o t o f K - P o s s ib le U n fa v o ra b leI n t e r a c t i o n .......................................................... 178

7.10 P o l a r P l o t o f K - U n fav o rab le I n t e r a c t i o n . . . 178

7.11 P o l a r P l o t o f K PggcL ..........................................................180

V I


LIST OF FIGURES (con tinued )

F ig u re P ag e

7.12 S te a d y S t a t e O pera ting C o n d i t i o n s fo rE th y l e n e Tower ...................................................................... 186

7.13 C o n t r o l C o n f ig u ra t io n sa . f o r Systems 1 , 4 , 7 , 1 0 ................................................. 188b . f o r Systems 2 , 5 , 8 , 1 1 ................................................. 189c . f o r Systems 3 , 6 , 9 , 1 2 ................................................. 190

P lo t s

7 .1 - P o l a r P lo t s o f K fo r Exam ple 1 ................................. 194-7.12 197

7.13 P o l a r P l o t o f K fo r Exam ple 2 ................................. 200

V I 1


ABSTRACT

To a n a ly z e th e dynam ics o f a d i s t i l l a t i o n colum n,

i t i s n e c e s s a ry to have a dynamic m odel as w e l l a s an

a n a l y s i s te c h n iq u e . The dynamic m odel g e n e ra te s t r a n s f e r

f u n c t i o n s by s im u l ta n e o u s ly s o l v i n g th e dynamic m a t e r i a l

and e n e rg y b a la n c e s and e q u i l i b r i u m r e l a t i o n s h i p s in the

f r e q u e n c y domain. T his l e a d s t o t h e use o f a f r e q u e n c y

dom ain method as th e b a s i s o f th e a n a l y s i s t e c h n iq u e , i . e .

Bode and N y qu is t m ethods. In a d d i t i o n to th e s e t e c h n iq u e s

t h e a u th o r has e x p lo i t e d t h e R i j n s d o r p / B r i s t o l i n t e r a c t i o n

in d e x f o r th e i n v e s t i g a t i o n o f sy s te m i n t e r a c t i o n s . The

a b o v e c o n c e p ts a re combined in o r d e r t o an a ly ze t h e e f f e c t

o f d o u b le -e n d com position c o n t r o l on a d i s t i l l a t i o n column.

The r e s u l t s o f t h i s r e s e a r c h l e d in s e v e r a l d i r e c t i o n s .

F i r s t , an i n t e g r a t e d a n a l y s i s p ack ag e was d e v e lo p e d . T h is

p a c k a g e a l lo w s one to i n v e s t i g a t e t h e e f f e c t s o f th e s t e a d y

s t a t e o p e r a t in g c o n d i t i o n s , v e s s e l s i z e , t r a y t y p e , and

c o n t r o l c o n f ig u r a t io n upon th e dynam ics o f a d i s t i l l a t i o n

c o lu m n . Second, th e a n a l y s i s o f t h e system i n t e r a c t i o n

d e m o n s t r a te s t h a t c o n t r o l sy s tem d e s ig n based o n l y on

s t e a d y s t a t e d a ta can be m i s l e a d i n g . The R i j n s d o r p / B r i s t o l

i n t e r a c t i o n index p ro v id e s a b e t t e r gu ide to c o n t r o l sys tem

d e s i g n . A lthough t h i s in d e x i s an improvement o v e r p r e v io u s

i n t e r a c t i o n m easures , i t i s n o t d e f i n i t i v e .

V l l l


CHAPTER I

INTRODUCTION

The a n a l y s i s o f d i s t i l l a t i o n co lum n dynam ics r e q u i r e s

a model a s w e l l a s an a n a l y s i s t e c h n iq u e . The com bination

o f t h e s e f a c t o r s a l lo w s one to i n v e s t i g a t e th e in f lu e n c e

o f equ ipm ent s e l e c t i o n and s te ad y s t a t e o p e r a t i n g c o n d i t io n s

upon v a r io u s t y p e s o f c o n t r o l sy s te m s . F o r u s e r convenience

th e m odeling and a n a l y s i s tech n iq u e s h o u l d be combined in to

an i n t e g r a t e d p a c k a g e .

The m o deling p a r t o f th e a n a l y s i s p ack ag e g e n e ra te s th e

dynamic re s p o n se o f th e column. T his r e s p o n s e i s r e p r e s e n te d

by Bode p l o t s . The model in c lu d e s t h e b a s i c dynamic m a te r i a l

and e n e rg y b a l a n c e s a s w e ll a s t r a y , l i q u i d and v ap o r e f f e c t s .

The model p r o v id e s f o r d i f f e r e n t column c o n f i g u r a t i o n s , e q u i

l ib r iu m r e l a t i o n s h i p s and c o n t r o l s y s te m s .

The i n t e r p r e t a t i o n o f th e dynam ics r e l i e s on th e c l a s

s i c a l Bode and N y q u i s t methods as w e l l a s e x p l o i t i n g the

R i j n s d o r p / B r i s t o l i n t e r a c t i o n index . V a r io u s f req u en cy

domain g r a p h i c a l m ethods a re i n c o r p o r a t e d i n t o t h e package.

But t o u n d e rs ta n d t h e i n t e r a c t i o n s o f d o u b le - e n d com position

c o n t r o l , i . e . i n t e r a c t i o n in a tw o-by -tw o p r o c e s s , th e

c u r r e n t th e o ry h a d t o be ex tended . The i n t e r p r e t a t i o n

o f th e R i j n s d o r p / B r i s t o l i n t e r a c t i o n i n d e x p ro v id e s

i n s i g h t i n to t h e d e s ig n o f a c o n t r o l s y s te m .


There w ere t h r e e phases in t h e d e v e lo p m e n t o f t h i s

r e s e a r c h . The f i r s t was the d e v e lo p m e n t o f an equ im ola l

overflow dynam ic m odel. This p r o v id e d an o p p o r tu n i ty to

dem onstra te t h e s o l u t i o n techn ique a n d th e u s e f u ln e s s

o f v a r io u s o p t i o n s and assu m p tio n s . The secon d phase was

th e developm ent o f an en th a lp y b a l a n c e dynamic model.

T h is ex tended t h e m odeling te c h n iq u e b u t f a i l e d to develop

an adequate s o l u t i o n method. The f i n a l p h a se was the

t h e o r e t i c a l a n a l y s i s o f system i n t e r a c t i o n s . T h is i d e n t i

f i e d two f a c t o r s which govern sy s te m i n t e r a c t i o n s . These

d ev e lop m en ta l p h a s e s a r e recoun ted i n c h a p t e r s I I I , IV,

and VII, r e s p e c t i v e l y . Chapter I I p r e s e n t s t h e s te ad y

s t a t e c a l c u l a t i o n program . C hap te r V r e v ie w s the c l a s s i c a l

frequency dom ain a n a l y s i s methods. C h a p te r VI a p p l ie s

th e above c o n c e p t s t o an i n d u s t r i a l s c a l e e th y le n e column.

L i t e r a t u r e R ev iew - Dynamic S im u la t io n o f a C ontinuous

D i s t i l l a t i o n Column

The l i t e r a t u r e can be d iv id e d i n t o two p a r t s , one

p a r t which d e a l s w i th th e models a n d t h e o t h e r which d e a ls

w i th s o l u t i o n t e c h n i q u e s . A lthough an y p a p e r in the

f i e l d would u s u a l l y in c lu d e both t o p i c s , i t i s conven ien t

t o d ea l w i th them s e p a r a t e l y . T h is r e v ie w f o c u s e s on the

v a r io u s m odels p r e s e n t l y in u se . Of t h e two c l a s s e s o f

s o lu t io n t e c h n i q u e s , t im e domain a n d f r e q u e n c y domain,

only the l a t t e r w i l l be reviewed s i n c e i t h a s a d i r e c t


b e a r in g on t h i s work. Rademaker e t a l . [ 1 ] p ro v id e a l i s t

o f r e f e r e n c e s which d i s c u s s time dom ain s o l u t i o n t e c h n iq u e s .

Models

A u s e f u l way t o d i s c u s s models i s t o c l a s s i f y them

based on th e e q u a t io n s which th e y encom pass. The l i s t o f

e q u a t io n s (T ab le 1 -1 ) p re s e n te d by W il l ia m s [2] w i l l be

used in t h i s r e v ie w . S i m p l i f i c a t i o n s o f th e s e e q u a t io n s

have been made t o r e d u c e the c o m p u ta t io n a l e f f o r t r e q u i r e d

f o r a s o l u t i o n . Complex models a l s o r e q u i r e a d d i t i o n a l

p a ra m e te rs w hich a r e o f t e n d i f f i c u l t t o o b t a in . S in ce

Rademaker e t a l . [1] have r e c e n t l y p r e s e n t e d a d e t a i l e d

rev iew , t h i s r e v ie w w i l l h i g h l i g h t t h e m odeling e f f o r t so

as to p r o v id e a background and p o i n t o f r e f e r e n c e . Addi

t i o n a l r e v ie w a r t i c l e s have been w r i t t e n by Archer and

R o th fuss [3 ] and W ill iam s [2] .

The more common assum ptions u se d t o s im p l i f y th e

e q u a t io n s o f T ab le 1 -1 a re g iven by W il l ia m s (Table 1 -2 )

[2 ] . Not m en tioned i n th e s e l i s t s a r e r e b o i l e r s and con

d e n s e r s . T hese i te m s commonly have t h e a ssu m p tio n s o f

Table 1-2 a p p l i e d t o them. T h e r e f o r e , d e t a i l e d dynamic

h e a t t r a n s f e r m odels a r e r a r e l y d e v e lo p e d f o r r e b o i l e r s

and c o n d e n s e r s .

The a s su m p t io n s o f Table 1-2 r e f l e c t th e a t t i t u d e o f

m odelers t h a t c o m p o s it io n dynamics a r e o f p r im ary im

p o r ta n c e . T h e r e f o r e , th e y re a so n t h a t t h e r e s u l t i n g


TABLE 1-1

N ecessary E x p re s s io n s f o r a Complete Dynamic M odel(quoted from Williams 1.2] ]

1 . A m a t e r i a l - b a la n c e e q u a t io n f o r each com ponent l e s s one , f o r e a c h w e l l -m ix e d r e g io n , ( t h a t i s , w h e re c lo s e c o n ta c t o f l i q u i d and vapor and u n ifo rm l i q u i d c o m p o s it io n o c cu rs ) on each t r a y . T h is m ust i n c l u d e e x p re s s io n s f o r th e v a p o r - l i q u i d e q u i l i b r iu m s e x i s t i n g .

2 . An e n th a lp y b a la n c e e q u a t io n f o r each w e l l -m ix e d r e g io n on e ac h t r a y r e l a t i n g l i q u i d and v a p o r f lo w and h e a t lo s s o r g a in .

3 . An e q u a t io n r e l a t i n g t h e l i q u i d dynamics i n e a c h w e l l - mixed r e g io n on each t r a y .

4 . An e q u a t io n r e l a t i n g t h e vapor dynamics i n e a c h w e l l - mixed r e g io n on eac h t r a y .

5 . An e q u a t io n r e l a t i n g t h e flow and p r e s s u r e dyn am ics o f th e v ap o r betw een t r a y s .

6 . An e n th a lp y -b a la n c e e q u a t io n r e l a t i n g any h e a t g a in o r lo s s i n th e r e g i o n betw een t r a y s t o t h e r e s u l t i n g l i q u i d and v ap o r f lo w r a t e s .

7 . An e q u a t io n r e l a t i n g t h e f l u i d dynamics o f t h e l i q u i d f lo w in g betw een t r a y s .

8 . An e q u a t io n f o r e ac h w e ll-m ix e d r e g io n on e a c h t r a y r e l a t i n g t h e r a t e o f a t t a in m e n t o f e q u i l i b r i u m betw een l i q u i d and v a p o r i n t h a t r e g io n .


TABLE 1-2

Common Assumptions (Quoted from Williams (.21 )

1. Equilibrium between liq u id and vapor is in stan taneously rea lized . This assum ption is almost u n iv e r a lly used. The remaining con sid era tion s of column dynamics are r e a lly flu id dynamic considerations, flow, m ixin g , and such.

2 . There is no holdup o f vapor on the trays, th u s r e moving any need fo r tr a y vapor dynamics, item 4 (Table 1 -1 ), This i s a lso almost u n iv ersa lly used by investigators o f stagew ise operation dynam ics.

3 . The column operates a d ia b a tic a lly , that i s , no heat i s gained or lo s t through the column w alls. T his elim inates item 6 above from consideration. Although i t i s almost always u sed , i t may be one of th e le s s valid of those norm ally used when one co n sid ers the actual f ie ld operation of d is t i l la t io n system s.

4 . There is l i t t l e or no holdup of vapor between tra y s. This assumption, w hich i s almost as popular a s number 2, elim inates item 5 above.

5. The mixture involved ex h ib its the phenomenon o f "equal-molal overflow ", that i s , there is no enthalpy unbalance between th e components of the m ixture and thus no drastic change in the flow rates of l iq u id and vapor across th e tray . Item 2 is not n ecessa ry i f th is assumption i s v a lid . This i s another very common assumption.

6. A ll of the liquid and vapor on a tray are w e l l mixed. Thus only one w ell-m ixed region i s required p er tray and the requirements o f item 1 in the equation l i s t are considerably r e la x e d ,

7 . There is no holdup o f liq u id between trays. T h is is an assumption which o ften is not used p a r t ic u la r ly in re la tio n to the r e b o i le r . I f used, the eq u ation of item 7 is not n ecessa ry .

8 . There i s no change in holdup on a tray. T his assumption has two ra m ifica tio n s .


a) There i s no delay in a flow surge moving across the tray, th a t i s , such a surge i s in s ta n tly transm itted to the tray below.

b) There i s no change in the actual p h ysica l amount o f liq u id and vapor on the tray , th a t i s , the value of holdup does not require adjustment as the sim ulation proceeds.


so lu tio n should be close to r e a lity i f th e major compo

s i t io n equation (Table 1 -1 , item 1) i s included in the

model. To a cer ta in degree th is philosophy has been v er i

f ied by experim ental work (Armstrong and Wilkinson [4],

Stainthrop [5 ], and Stainthrop and Searson [6]), But

Rademaker e t a l . [1] and Rosenbrock [73 have indicated

that secondary e f fe c ts can be important in some columns.

They derive and describe secondary e f f e c t s which they

b e lie v e have been neglected by modelers. Buckley e t a l.

[243 has re la ted these secondary tray e f f e c t s to the

column's p h y sica l structure. With increased computing

power i t i s now fea s ib le to include th ese e f fe c ts .

Rademaker e t a l . [13 in d ica te that the secondary e f fe c t s

l i s t e d in Table 1-3 should be included to make the model

b etter f i t a c tu a l operating u n its .

The models using the frequency domain solution

technique g en era lly adhere to the assumptions in Table

1-2 . The binary component model employed by Lamb e t a l.

[83 conforms to Table 1-2 assumptions excep t for item 8,

i . e . they a llow variable holdup. In a d d itio n , they have

included a tray e ff ic ie n c y fa cto r . The next s ig n if ica n t

sim ulation attem pt was made by Cadman and Carr [ 93 •

Although th e ir equations were the same as Lamb et a l . ,

Cadman and Carr employed a d iffer e n t so lu t io n technique

and extended th e method to multi-component columns.

Wood's [iQj multi-component model extends Cadman and


8

TABLE 1 -3

S e c o n d a ry E f f e c t s Which Im p ro v e Model Response

1) V a r ia b le ho ldup , i . e . f lo w la g s

2) In f lu e n c e o f vapor r a t e on l i q u i d r a t e

3) E f f e c t o f P re s s u re F l u c t u a t i o n s

4) S e l f - r e g u l a t i n g c o n d e n s e r and r e b o i l e r


C a r r ’ s w ork by a l lo w in g l i q u i d f lo w p e r t u r b a t i o n s t o be

c au se d b y a change i n th e v a p o r r a t e ( i . e . same e f f e c t

a s b o i l e r s w e l l ) . The Shun t a and Luyben [11] b i n a r y model

e x te n d s t h e work o f Lamb e t a l . by rem oving th e e q u im o la l

o v e r f lo w r e s t r i c t i o n (Table 1 - 2 , i te m 5 ) . They in c l u d e

an e n t h a l p y b a lan ce b u t do n o t i n c l u d e te m p e ra tu re o r

b o i l e r s w e l l as i n h e r e n t p a r t s o f th e model.

The above models a re n o t f l e x i b l e enough to h a n d le

d i f f e r e n t ty p e s o f colum ns. F o r exam ple, th e Cadman and

C a rr m odel was developed f o r a t o t a l co n d en se r , s i n g l e

fe e d s t r e a m and two l i q u i d p r o d u c t s t re a m , i . e . d i s t i l l a t e

and b o t to m s . None o f th e above m odels had p r o v i s i o n s f o r

m u l t i p l e f e e d s or s id e s t r e a m s . T h ese models a l s o depended

upon s i m p l i f i e d s te a d y s t a t e c a l c u l a t i o n s to f in d t h e

o p e r a t i n g p o i n t . For exam ple, i t i s common to f i n d s t e a d y

s t a t e e q u im o la l o v e rf lo w c a l c u l a t i o n s made w ith c o n s t a n t

v o l a t i l i t i e s . These models w ere l im i t e d i n t h e i r u s e f u l

n e s s f o r c o n t r o l s t u d i e s . A l th o u g h t r a n s f e r f u n c t i o n s

can be g e n e r a te d w i th many d i f f e r e n t measured v a r i a b l e s ,

on ly r e f l u x and bo ttom s vapor f lo w s could be used a s

e x t e r n a l l y m an ip u la ted v a r i a b l e s .

F re q u en c y Domain S o lu t io n T ec h n iq u e

The f i r s t a u th o r s to d e v e lo p th e complex f r e q u e n c y

domain s o l u t i o n te c h n iq u e were Lamb e t a l . T h e i r a p p ro a c h

was t o l i n e a r i z e th e d i f f e r e n t i a l e q u a t io n s and th e n


10

app ly L ap lace t r a n s f o r m s to th e s e e q u a t i o n s . T h is p ro

cedure c o n v e r t s a s e t of n o n - l in e a r d i f f e r e n t i a l e q u a t io n s

in th e t im e domain i n t o a s e t of l i n e a r a l g e b r a i c

e q u a t io n s i n th e L ap lace domain. N ext a s u b s t i t u t i o n

of v a r i a b l e s c o n v e r t s th e L ap lacean o p e r a t o r , s , i n t o the

complex f re q u e n c y domain, J u t . Now t h e e q u a t io n s can be

so lved a t d i s c r e t e v a lu e s o f f re q u e n c y , w . The r e s u l t i n g

s o l u t io n y i e l d s t r a n s f e r f u n c t i o n s . The above p rocedu re

c o n c e p tu a l iz e s t h e complex freq u en cy dom ain s o l u t i o n

te c h n iq u e . P o i n t s o f d i f f e r e n c e be tw een a u th o r s occu r a t

two j u n c t u r e s , when the b a s i c time dom ain e q u a t io n s a re

w i t t e n and when t h e l i n e a r a lg e b r a i c e q u a t io n s a re

a c t u a l l y so lv e d .

The f re q u e n c y domain s o l u t i o n t e c h n iq u e s can be

perform ed w ith e i t h e r an a lo g or d i g i t a l h a rd w are . Lamb

e t a l . and Rademaker e t a l . deve loped e l e c t r i c a l a n a lo g

c i r c u i t s which r e p r e s e n t th e e q u a t io n s o f t h e model. By

i n j e c t i n g d i f f e r e n t f req u en cy s i n u s o i d a l waves i n t o th e

c i r c u i t s th e y w ere a b le t o deve lop th e t r a n s f e r f u n c t i o n s .

Cadman and C a r r , Wood, and Shun t a and Luyben , on th e o th e r

hand, to o k th e d i g i t a l app roach . They form ed th e e q u a t io n

in to a m a t r ix o f complex numbers. A t e ac h f req u e n cy t h i s

m a tr ix was i n v e r t e d , th e re b y , y i e l d i n g one p o in t o f th e

t r a n s f e r f u n c t i o n . I t e r a t i v e c a l c u l a t i o n s were n o t

r e q u i r e d w ith t h e above te c h n iq u e s b e c a u s e o f th e p a r t i c u l a r

model e q u a t io n s t h a t the a u th o r s c h o se .


11

At p r e s e n t th e s t a t e of t h e a r t f i n d s th e modeling

f r o n t i e r p u sh e d fo rw ard in s e v e r a l a r e a s b u t no one m odel

c o n ta in i n g a l l f e a t u r e s . Radem aker e t a l . have demon

s t r a t e d t h a t a model can be w r i t t e n in th e frequency

domain w h ic h in c lu d e s a l l known dynam ic b e h a v io r of a

d i s t i l l a t i o n column. E x p e r im e n ta l work has shown t h a t

d i f f e r e n t se co n d a ry e f f e c t s can s i g n i f i c a n t l y in f lu e n c e

c o m p o s i t io n r e s p o n s e s .

L i t e r a t u r e Review - I n t e r a c t i o n I n d e x

In t h e developm ent of an i n t e r a c t i o n in d ex th re e

d i f f e r e n t m easu res have evo lved . D avison [1 2 ] , and Tung

and E dgar [ I 3 ] have developed d i f f e r e n t m easures w hile

s t a r t i n g f ro m th e same p o in t . W i t c h e r and McAvoy [14]

show, i n a r e c e n t re v ie w a r t i c l e , t h a t B r i s t o l [ I 5 ] and

R i jn s d o rp [16 ] a r r i v e d a t the same m easure a f t e r s t a r t i n g

from d i f f e r e n t p o i n t s . In a l l c a s e s a u th o r s a re q u a n t i f y

ing the e f f e c t of c o u p lin g b e tw ee n p ro c e s s e s in a m u l t i

in p u t , m u l t i - o u t p u t « iv iro n m en t .

D a v i s o n , Tung and Edgar d e r i v e d t h e i r r e s u l t s u s i n g

a s t a t e s p a c e app ro ach . Davison m a n ip u la te d the e ig e n

v a lues o f t h e c lo sed loop system i n t o a co n ven ien t in d e x

form ula . T he r e s u l t i n g index i s t h e n i n t e r p r e t e d by a

ta b le l o o k u p . Tung and Edgar, on t h e o th e r hand, use t h e

e ig e n v a lu e s t o o b ta in the time r e s p o n s e o f a p rocess o u t

pu t to a c o n t r o l l e r i n p u t . T h is r e s p o n s e i s p re sen te d a s


12

a s e r i e s o f exponents and s i n u s o i d a l s . The c o e f f i c i e n t s

and p o w e r s o f e must th en be compared t o de te rm ine w h ich

o u tp u t i s most s e n s i t i v e t o a g iv e n in p u t .

W i t c h e r and McAvoy show t h a t B r i s t o l ' s index i s

i d e n t i c a l to R i jn s d o r p 's . B r i s t o l d e v e lo p s h i s i n d e x by

c a l c u l a t i n g the r a t i o of open lo o p to c lo s e d loop g a i n s .

R i j n s d o r p c o n s id e r s the r a t i o o f open lo o p to c lo s e d lo op

t r a n s f e r f u n c t io n s . A f te r s e v e r a l m a n ip u la t io n s , he

i s o l a t e s a term which depends o n ly on th e p ro c e ss t r a n s f e r

f u n c t i o n s . R i jn sd o rp th e n d r o p s th e dynamics t e r m s ,

r e t a i n i n g only s te a d y s t a t e g a i n s . T h is he c a l l s t h e

m easure o f i n t e r a c t i o n . B o th i n d i c e s a r e i n t e r p r e t e d by

means o f a t a b l e lookup.

Few a u th o r s have u n d e r t a k e n t o e x p la in the m ea n in g of

the d y n a m ic e x te n s io n to t h e R i j n s d o r p / B r i s t o l i n d e x .

R i j n s d o r p b e l i e v e s the dy n am ics t o be an i n s i g n i f i c a n t

i n f l u e n c e on the p ro c e ss i n t e r a c t i o n s . R e c e n t ly , B r i s t o l

[ 17] a t t e m p t e d t o e x p la in t h e dynamic e x te n s io n o f t h i s

in d ex . W itche r and KcAvoy i n d i c a t e t h a t th e dynamic

i n f l u e n c e can be s i g n i f i c a n t . They ap p ly the s t e a d y s t a t e

c r i t e r i a f o r i n t e r a c t i o n a t e a c h f re q u e n c y to d e te r m in e the

s e v e r i t y o f i n t e r a c t i o n . T hey s u s p e c t t h a t the p h a s e

a s s o c i a t e d w ith th e index m e a su re s th e tim e i t t a k e s a

d i s t u r b a n c e to p ro p ag a te from lo o p to lo o p . V /itcher and

KcAvoy, D av ison , and Tung and S d g a r c i t e examples w h e re

s te a d y s t a t e g a in s i n d i c a t e l i t t l e i n t e r a c t i o n b u t t h e

dynam ics show s i g n i f i c a n t i n t e r a c t i o n .


CHAPTER I I

STEADY STATE CALCULATIONS

The c a l c u l a t i o n o f s te a d y s t a t e c o n d i t i o n s i s one of

th e c o rn e r s to n e s upon w hich t h i s r e s e a r c h e f f o r t i s based.

The l i n e a r i z e d dynamic model r e l i e s on a com ple te s teady

s t a t e p r o f i l e o f t h e d i s t i l l a t i o n co lum n. The dynamic

model would become awkward to use and l o s e i t s economic

v a lu e i f t h e s t e a d y s t a t e p r o f i l e s c o u ld n o t be e a s i l y

c a l c u l a t e d and t r a n s f e r r e d t o th e dynamic m odel.

To a cc o m p lish t h i s c a l c u l a t i o n i t i s n e c e s s a r y to

have a s te a d y s t a t e program which can s o lv e th e g e n e ra l

h e a t and m a t e r i a l b a la n c e e q u a t io n s f o r any g iv e n s e t of

p h y s i c a l p r o p e r ty d a t a . S e v e ra l ty p e s o f program s can

pe rfo rm t h i s f e a t f o r d i s t i l l a t i o n co lu m n s. The b a s ic

Wang/Henke b ubb le p o i n t p ro ced u re was c h o se n becau se i t

was m ost r e a d i l y a v a i l a b l e t o th e a u t h o r . D uring t h i s

r e s e a r c h e f f o r t many improvements have b een made t o t h i s

b a s i c program to re d u c e t h e r e q u i r e d co m p u te r memory s iz e

and e x e c u t io n t im e , t o im prove th e c o n v e rg e n c e c h a r a c t e r

i s t i c s and t o i n c r e a s e f l e x i b i l i t y in t e r m s o f program

o p t io n s . T h is p rogram h a s been t e s t e d w i t h v a r io u s

sample prob lem s and w i th s e v e r a l i n d u s t r i a l a p p l i c a t i o n s .

13


14

D escrip tion of Program

This program performs rating c a lc u la tio n s for a complex

multi-component d is t i l la t io n column (m ultiple feed s,

products, and in terstage heaters o r c o o le r s ) . The m ixture

to be sep arated can have any d egree of liq u id phase non

id e a li ty . The vapor phase i s assumed to be id ea l. S ince

enthalpy d ata can be supplied, i t i s not necessary to

assume equim olal overflow. Three condenser options are

p ossib le: t o ta l , p a r tia l, or none. No provisions have

been made fo r a decanting condenser. The column must have

a p a r t ia l r e b o ile r . The number o f equilibrium sta g es,

the column pressure p ro file , the fe e d rate, feed composi

tion and feed temperature must be g iv en . In addition, th e

flow r a te o f each product stream and the L/D ratio must

be s p e c i f i e d .

The b a s ic so lu tion method i s based on the Wang/Henke

procedure w ith convergence a cce lera ted by Holland's

Theta method and a simple p a r tia l su b stitu tio n approach.

B riefly , th e Wang/Henke method works as follow s:

1) Read the data.

2) Obtain in i t ia l estim ates o f X's, L's , and V s .

3) U sing the individual component balances, so lve

th e trid iagonal matrix f o r a new s e t of X's.

4) C orrect these X's with th e Theta procedure.

5) Obtain new T's, K's and Y 's using a bubble


15

point procedure.

6) Modify the T's by means o f p artia l su b s t itu t io n .

7) Simultaneously so lv e the heat balance and o v er a ll

m aterial balance on each stage for new L 's and

V s .

8) Modify L's and V s by means of p a r tia l su b stitu

tio n method.

9) I f the re la tiv e change in T's, X's and V s from

the la s t ite r a t io n i s too large, return to

step 3.

10) P rin t and p lo t the r e s u lt s .

11) C all the procedure to generate the dynamic

r e s u lt s .

C a p a b ilit ie s

Table 2-1 provides an overview of the program's

c a p a b i l i t ie s . These are on par w ith sim ilar programs in

in d u stry . In order to minimize the program s i z e , no

gen era l data base is supplied w ith the program. Although

some common physical property ca lcu la tion rou tin es are

p a rt o f the program, the user can provide h is own. Table

2-2 l i s t s the lim ita tio n s , conventions, and assumptions

a sso c ia ted with the program. A complete u ser 's guide

which explains in d e ta il how to use the program i s in

Appendix A. A program l i s t in g i s provided in Appendix B.


16

N um erical C o n s id e r a t io n s

The n u m e r ic a l c h a r a c t e r i s t i c s o f th e s te a d y s t a t e

e q u a t io n s have been examined by many a u th o r s . A summary

o f t h e i r f i n d in g s a s w e l l as a l i s t o f r e f e r e n c e s i s

p r e s e n te d i n C h a p te r 13, pages 3 0 -3 2 , o f th e Chemical

E n g in e e r s ' Handbook, f i f t h e d i t i o n .


17

TABLE 2-1

G enera l Overview o f S tead y S ta te Program

A ssu m p tio n s : E q u i l i b r iu m Stages C om plete M ixing Per S tage A d i a b a t i c S tage Two P h a s e s No H e a t o f Mixing I d e a l V a p o r Phase

Type o f Program: M ulti-C om ponen t M u l t i - F e e d Streams M u l t i - P r o d u c t Streams M u l t i - I n t e r s t a g e H e a te r s /C o o le r s R a t in g Program - R equ ires t h e Number

o f S t a g e s and F inds t h e S e p a r a t i o n

M ethod: Wang-Henke (Convergence A ided b y th e T h e t a Method and P a r t i a l S u b s t i t u t i o n )

P h y s i c a l P ro p e r ty C a l c u l a t i o n s :

E n th a lp y - User S upp lied L iq u id P h a se A c t iv i ty C o e f f i c i e n t -

U s e r S u p p lied Pure L i q u i d F ug ac ity - User S u p p l i e d Vapor P h a s e F ug ac ity C o e f f i c i e n t -

I d e a l


18

TABLE 2-2

L im i ta t io n s , C o n v e n tio n s and A ssu m p tio n s

F o r S t e a d y S t a t e Program

Heat added i s p o s i t i v e .

S tages a r e numbered t o p down.

A p a r t i a l r e b o i l e r i s a lw a y s p r e s e n t .

No h e a t o f m ix ing .

A l l flow r a t e s a r e p o s i t i v e .

Stage 1 i s the to p m o s t s t a g e w hether i t i s a c o n d en se r

o r n o t .

The r e b o i l e r i s t h e b o t to m most s t a g e .

The number o f s t a g e s m u s t be g r e a t e r t h a n one and l e s s

th an 126.

The number o f com ponen ts m ust be g r e a t e r th a n one and le s s

th an s i x .

The number o f p r o d u c t s t r e a m s must be l e s s th an s i x .

No d ia g n o s t i c e r r o r m essa g es a re g e n e r a t e d .

A p ro d u c t s tream c a r d i s n o t needed f o r t h e bo ttom stream .

The flow r a t e f ro m t h e r e b o i l e r i s c a l c u l a t e d to

c lo s e th e o v e r a l l m a t e r i a l b a la n c e .


19

V a r ia b le s Used in C h a p te r I I

K K v a lu e s

L l i q u i d r a t e s (m olar)

T te m p e ra tu re

V v a p o r r a t e s (m olar)

X l i q u i d c o m p o s itio n (mole f r a c t io n )


CHAPTER I I I

EQUIMOLAL OVERFLOW MODEL

The developm ent o f t h e equ im ola l o v e rf lo w m odel and

th e a s s o c ia te d s o l u t i o n te c h n iq u e w i l l b e d is c u s s e d i n

t h i s c h a p te r . T h is m odel was developed i n o r d e r to

i l l u s t r a t e an im proved d i g i t a l s o lu t io n te c h n iq u e an d to

p ro v id e a means o f c o m p a riso n fo r th e e n th a lp y m odel o f

C h a p te r IV. I t s d e v e lo p m e n t a ls o i l l u s t r a t e s th e

v e r s a t i l i t y which i s n e c e s s a r y in a model t o make i t

p r a c t i c a l fo r i n d u s t r i a l u s e . The f i r s t s e c t i o n o f t h i s

c h a p te r d e sc r ib e s t h e a ssu m p tio n s and t h e i r i n f lu e n c e on

th e m odel. The m odel e q u a t io n s a re d e v e lo p ed in th e

second se c tio n . The t h i r d s e c t io n d e s c r ib e s th e n u m e ric a l

c o n s id e r a t io n s . The f o u r t h s e c t io n p r e s e n t s th e v a r io u s

m odeling o p tio n s w h ich a r e p o s s ib le .

A ssum ptions

The m u lti-co m p o n en t eq u im o la l o v e rflo w m odel o f t h i s

c h a p te r i s the same a s t h a t o f Wood [1 0 ] . The fo l lo w in g

e q u a t io n s and a s s u m p tio n s a r e c o n s id e re d in th e same o rd e r

a s th o s e o f W illiam s [2] (T ab le 1 -1 ) .

1) The com ponent m a t e r i a l ba lance i s w r i t t e n f o r

each component. No a t t e m p t i s made to f o r c e th e sum o f

th e com position p e r t u r b a t i o n s to equal z e r o . K v a lu e s

20


21

a re assumed to be in d e p e n d e n t o f c o m p o s itio n and

te m p e ra tu re . The v a p o r and l iq u id on a t r a y a r e w e ll

m ixed.

2) The e n th a lp y b a la n c e i s s im p l i f i e d by u s in g th e

e q u im o la l overflow a s s u m p tio n . T h is e l im in a te s th e need

to c a lc u la t e t e m p e r a tu r e s . T h is a ssu m p tio n a l lo w s vapor

r a t e p e r tu r b a t io n s t o b e t r a n s m it te d i n s t a n t a n e o u s l y up

th e colum n.

3) There i s one w e l l mixed l iq u id r e g io n on a t r a y .

Flow from one t r a y t o th e n e x t i s in f lu e n c e d b y v ap o r

r a t e and l iq u id h o ld u p p e r tu r b a t io n .

4) Vapor p h a se h o ld u p i s n e g le c te d , th u s th e r e a re

no vapor phase d y n a m ic s . T h is assum ption r e s t r i c t s th e

u se o f th e model t o colum ns whose l iq u id h o ld u p i s much

g r e a te r th a n t h e i r v a p o r h o ld u p .

5) P re s s u re t r a n s i e n t s a re assumed t o be f a s t

compared to c o m p o s it io n e f f e c t s , t h e r e f o r e , th e y a re

n e g le c te d .

6) The column o p e r a te s a d i a b a t i c a l l y .

7) There a r e n o r e g io n s betw een s t a g e s . A l l

th e l iq u i d in th e downcomer i s combined w i th th e l iq u i d

on th e s ta g e .

8) E q u il ib r iu m b e tw een l iq u id and v a p o r i s

in s ta n ta n e o u s ly r e a l i z e d . Each t r a y i s an e q u i l ib r iu m

s ta g e , i . e . th e e f f i c i e n c y f a c to r i s one. F o r colum ns

w ith e f f i c i e n c i e s l e s s th a n one, th e num ber o f e q u il ib r iu m


22

s ta g e s i s v a r i e d u n t i l a match w i t h th e p l a n t d a ta i s

fo u n d . For exam ple, an a c tu a l t e n t r a y colum n w ith an

e f f i c i e n c y o f 0 .7 w ould be m odeled w ith seven e q u il ib r iu m

s ta g e s w ith e ac h s ta g e c o n ta in in g 1 .0 /0 .7 t im e s th e l i q u i d

m ass o f an a c t u a l t r a y . This r e d u c e s th e num ber o f flo w

and c o m p o s itio n la g s i n s e r i e s . The la r g e r t h e column

t h e le s s s e r io u s t h i s r e s t r i c t i o n .

Model E q u a tio n s

Three s e t s o f e q u a t io n s c o m p rise th e e q u im o la l o v e rf lo w

dynam ic m o d e l. The g e n e ra l iz e d s t a g e e q u a t io n s a llo w f o r

l i q u i d and v a p o r fe e d s and s id e s t r e a m s on e a c h e q u il ib r iu m

s t a g e . They a ls o in c lu d e th e i n f lu e n c e o f h o ld u p and v a p o r

f lo w upon t h e l iq u id le a v in g th e s t a g e . The second s e t o f

e q u a t io n s c o n s i s t s o f t h e v a r io u s co n d en se r c o n f ig u r a t io n s

w i th fo u r o p t io n s to ch o o se from . (Table 3 -1 ) The t h i r d

s e t o f e q u a t io n s c o n s i s t s o f th e v a r io u s r e b o i l e r c o n f ig u r

a t i o n s w ith two o p tio n s to choose from . (T ab le 3 -2 ) .

The fo l lo w in g tim e domain e q u a t io n s a r e w r i t t e n f o r

t h e g e n e r a l iz e d s ta g e o f F igu re 3 . 1 . The l i q u i d flow

e f f e c t s a re b a sed on t h e app roach o f Rademaker e t a l . [1]

p ag e 131.


23

TABLE 3 -1

Dynamic Model C o n d en se r O ptions

C .l ) F looded condenser ( i . e . s e l f - r e g u la t in g ) w i thl iq u id d i s t i l l a t e a s t h e w ild stream

C .2) F looded condenser ( i . e . s e l f - r e g u la t in g ) w i thr e f lu x a s th e w ild s t r e a m

C .3) N on-flooded c o n d e n se r w ith accum ulato r l i q u i dle v e l m a n ip u la tin g l i q u i d d i s t i l l a t e

C .4) N on-flooded c o n d e n se r w ith accum ulato r l i q u i dle v e l m a n ip u la tin g r e f l u x


24

TABLE 3-2

Dynamic Model R e b o i le r O p tions

R . l ) R e b o ile r l i q u i d l e v e l m a n ip u la tin g vapor b o i l u p

R .2) R e b o ile r l i q u i d l e v e l m a n ip u la tin g bo ttom sl iq u i d flow


Figure 3.1

G eneralized Equilibrium Stage

2 5

'n-1

Stage n

V,n+1

Figure 3 .2

CondenserLiquid Level Manipulates D is t i l la t e Liquid Flow

L ieStage 1

SSL


26

E q u ilib r iu m e q u a t io n :

’'i n = « i n ’' i n

L iqu id f lo w e f f e c t s :

M a te r ia l b a la n c e :

dM„ar = V i + V i - V®“ n> - 'V®S’n> + I-n + ^n

Component b a la n c e :

dX. M„dO in - 1 n -1 in+1 in + 1 n+1

- ’' i n ' V = : \ ' - ’' i n ‘'in < V ® = ''n > + “ in^^'n

+ ZV. PV_ i n n

Vapor f lo w :

V. = V i + ^n - “ ’'n

27

Feed s p l i t e q u a t io n s :

PLn = FT^QP^

FV„ = FT (1-QF„) n n n

FT = PL + FV n n n

L i n e a r i z i n g th e above e q u a t io n s and c o n v e r tin g them

to th e c o m p le x fre q u e n c y domain y i e l d s th e fo llo w in g

e q u a t io n s . The low er c a se s c r i p t l e t t e r s r e f e r to th e

same v a r ic d ) le a s th e upper ca se b l o c k l e t t e r s b u t r e f e r

to th e c o m p le x freq u e n cy domain.

E q u il ib r iu m :

^ in ^ in ^ i n

L iquid f lo w e f f e c t s ;


28

w here

3l 3l„

n n+1


■'“"n “ V l * "n+l '

- F[ '’n+l' = V l + "n+1 ’

- ( i /^ + i4 W „ + + S v ^

“ '" n + l ‘^ + ■*"’ * ^ n -1 ' ' " " " '^ " n

+ + < % ] / ( ! + BY i«)

Component B a lan ce

h J ”% * V'"*in = *tn-lVl + in-l n-1

■*■ in+l*in+l n+l '"' in+l in+l"n+l " *in*V®® n’

29

In c lu d in g th e m a te r i a l b a la n c e y ie ld s :

-Vl^in-l + 'V'" +

- ^ i n + l V l * i n - f l = - 5 = i „ tW ''n + l - < ^ n + “ ^n)

+ X i n - i V l

i n

™ i n + Â n + l “n + l "

V ap o r flow :

% ■ "n + l + ^ '’n - ■‘•‘ ‘’n

30

F eed s p l i t e q u a tio n s :

QP, + K P .„ (1 -Q P „ ) n i n n

The fo llo w in g t im e dom ain e q u a tio n s a r e w r i t t e n f o r

t h e c o n d en se r shown i n F ig u r e 3 .2 . T his c o n d e n s e r h a s th e

l i q u i d l e v e l c o n t r o l l e r m a n ip u la tin g th e d i s t i l l a t e

l i q u i d flo w (SSL^). T he c o n d en se r i s c o n s id e re d s ta g e 1 .

(o p t io n C .3)

E q u il ib r iu m e q u a tio n :

p a r t i a l c o n d en se r

t o t a l condenser = 0, = 0

L iq u id flo w e f f e c t :

SSL- = K H, 1 C l


31


dM-— - = V, - L. - SSLi - SSV, + PL. 1 + FV, ^ 0 ^ 1 i 1 1 1


dX. , M.- a r - V l 2 ^ 2 - X ii(L^+SSL^) - X^iK^iSSV^

Vapor flo w - no e q u a t io n

Feed s p l i t e q u a tio n - same a s g e n e r a l i z e d s ta g e

Geometry o f c o n d e n se r P

L in e a r iz in g t h e above e q u a tio n s and c o n v e r t in g them

to th e com plex f re q u e n c y domain y i e l d s th e fo l lo w in g

e q u a tio n s :

E q u il ib r iu m e q u a tio n :

p a r t i a l c o n d e n se r = * l l * i l

t o t a l c o n d en se r ^ i l “ ^


32

L iqu id f lo w e f f e c t :


p^A.yw— 6 U j _ = V

c

V - 66 V.+|(Z.+j(v,

(1 + )^c

Note: T he tim e c o n s ta n t fo r th e l i q u i d l e v e l c o n tro l

loop i s P ^ /^c •


= \ 2 ^ i 2 ^ 2 ■*“ ^12^12^2

- X^j^(fj^+66fj^) - % ii (L^+SSL^^) -

- « i iK iiS S ^ l + + z « iiF V i+ Z V ii(v i


33

S u b s t i t u t i n g th e m a te r ia l b a l a n c e f o r th e te rm

y ie ld s

*11 l « l J “ + - *12^12^2

= + Ki2*i2'^2

+ + ZV il ^ 1 +

T h e fo l lo w in g tim e domain e q u a t io n s a re w r i t t e n f o r

th e c o n d e n s e r shown in F ig u re 3 . 3 . T h is condenser h a s th e

l iq u id l e v e l c o n t r o l l e r m a n ip u la t in g th e r e f lu x (L^)

(o p tio n 4) . The o n ly d i f f e r e n c e be tw een condenser m o d e ls

i s t h a t d u e to th e l iq u id f lo w e f f e c t s , i . e . = K^H^.

T h e r e s u l t i n g eq u a tio n s i n t h e com plex fre q u e n c y

domain a r e :

L iq u id f lo w e f f e c t :

- V i

M a te r i a l b a la n c e :


34

Figure 3*3

Condenser Liquid Level Manipulates Reflux Flow

SSVFVi'FLn

LflOJStage 1

SSL

Figure 3*4

Condenser - S e lf-r eg u la tin g

Stage 1FL,

SSL


35

= Vg - + jjv.

Pi A, JÇ = Vg - - 44^2 - 44V^ + + j(v^

V 2 ~ 4 4 £ j “ 4 4 V V

^ 1 PiAiiW(1 + i ^ )

c

The tim e c o n s t a n t fo r th e l iq u id l e v e l c o n t r o l loop i s t h e

same a s b e f o r e , a lth o u g h d i f f e r e n t m a n ip u la te d v a r ia b le s

a re u se d . The component b a la n c e i s th e same a s fo r o p t io n

C .3 .

The fo l lo w in g tim e domain e q u a t io n s a r e w r i t te n f o r

th e c o n d e n se r shown in F ig u re 3 .4 . T h is i s a flooded

( s e l f - r e g u l a t i n g ) condenser w ith e i t h e r th e r e f lu x (o p t io n

C.2) o r th e d i s t i l l a t e (o p tio n C . l ) v a ry in g a s a fu n c t io n

o f l iq u id l e v e l . The o n ly e q u a t io n which i s changed from

th e p re c e d in g m odels i s th e flow e f f e c t e q u a t io n .

H,Q, = UATA. (1 -

^ ^ “ imax ^ ^

o r in th e com plex frequency dom ain

UATA.


36

M a t e r i a l b a la n c e :

p.y«;v,X H,- ■ o i § - ' - - - = “2 - - 44^1 - + ( " i

i f £ ^ i s th e w ild s tre a m , th e n (o p t io n C .2)

= V g(l + i'w - ) - 44^1 - 44Vi + 6^1 + dv.

i f 44Z^ i s th e w ild s tream , t h e n (o p tio n C .l )

4 4 ^ 1 = Vg (1 + /w —■ - 44V^ + 6-^1 +

The com ponent b a la n c e i s th e sam e as p re v io u s c o n d e n s e r s .

T he fo llo w in g tim e dom ain e q u a t io n s a r e w r i t t e n f o r

th e r e b o i l e r shown in F ig u re 3 . 5 . T h is r e b o i l e r h a s t h e

l i q u i d l e v e l c o n t r o l m a n ip u la t in g th e bo ttom s l iq u id f lo w

( o p t io n R .2 ) . The r e b o i l e r i s c o n s id e re d s ta g e j .

E q u i l ib r iu m e q u a tio n :

- ■'ij

L iq u id f lo w e q u a tio n :


Figure 3*5

37

R eboiler

Liquid Level Manipulates Bottoms Liquid Flow

j - 1

FL Stage j mm

Lie

Figure 3*6

R eboiler

Liquid L ev el Manipulates Vapor Flow

from R eboiler In d irec tly by D ire c tly

Manipulating Heat Flow to R eboiler

j-1

,10Stage j


38

M a te r ia l B a lan ce

dM .■ ' ' j ■ ■ S S L j - SSV. + F L j + P V j


- V i j ( V j + S S V j )

“ X ^j (L j+SSL j ) + ZL j F L j + ZVj^jFVj

Vapor f lo w - no e q u a tio n

Feed s p l i t e q u a tio n i s th e same a s th e g e n e ra liz e d s t a g e .

Geom etry o f r e b o i l e r :

Mj = P jA jH j

To a c c o u n t f o r r e b o i l e r s w e l l th e fo llo w in g a p p ro a c h

can be u s e d . I t i s b a sed on a s i m i l a r approach ta k e n b y

B uckley e t a l . [24] f o r th e s w e l l e f f e c t s seen on a t r a y .

The F f a c t o r i s in tro d u c e d b e c a u s e i t has been shown t o

c o r r e l a t e w e l l w ith th e d e n s i ty o f a e ra te d l iq u id o r

f r o th f o r a t r a y .


39

dM. _ dH. _ dp .a r = & j» i â T + * j H j 5 T

a n d

dPj _ d p j jp dV j _ dVj d5 dF dV 3S ^ d0

dM _ dH^ _ dV^Ï Ï T " ^ jP j d0” ■*■ V j ^ d0

V.F = ' (a c h a r a c t e r i s t i c number i n t r a y d esig n )

dF _ 1dVz

j

A lthough no p lo ts o f P j v s F e x i s t f o r a r e b o i l e r , th e

s lo p e o f th e curve c o u l d be in f e r r e d to be n e g a t iv e from

t e s t s o f t r a y s . The r e l a t i o n s h i p betw een Pj a n d F shou ld

b e m ainly governed by r e b o i l e r ty p e and g e o m e try i f i t

beh av es s im i l a r to a t r a y . P r e s e n t ly , th e o n ly way to

o b ta in d P j/d F i s th r o u g h p la n t t e s t s o f a c t u a l colum ns.

L in e a r iz in g th e aibove e q u a tio n s and c o n v e r t in g to

t h e complex freq u en cy dom ain y i e ld s th e f o l lo w in g r e s u l t s .


40

E q u il ib r iu m e q u a t io n :

y t j - % i f i j

L iqu id f lo w e q u a tio n :


A jPji'w fcj + AjHjYVjjW = - 44V j

+ iJ- j + (JVj

A jp j/w ZK, + A j H j Y V j J W = Z j _ 2 " V j - - 4 4 ^ j - 4 4 V j

+ + jîVj

/ • - V ( 1 + A ^ H ^ Y y w) + Z 2 , - 4 4 Z ^ - 4 4 V z + d f ^ + d v .e . — J J J d_____ i____ i I sL^ A jPj

(1 + iW)

Note: The tim e c o n s ta n t fo r t h e l i q u i d l e v e l c o n tro l

loop i s

IjP ]Kr


41


“ X y K j^j ( v j + 4 4 V j ) ~ X j^ jK j^ j C V j+ S S V j )

” Xj j ( Z j+ 4 4 Z j ) — Xj j (L j+SSLj) + Z ^ j FL j

+ ZLij iJ-6 j + ZVj_j FVj +

X y [Mj/W + K^j (V j+SSVj) + Lj + S S L j ]

■ * i j - X ^ j - l “ ■ * i j ■ "y -

+ ij-^j + iJVjl + ^ i j - l '^ j - 1 ” ij ^ i j

- X jjC C j+aaZ j) + zZ ^ jF L j + Z L^jd^j

+ 2 V i jF V . + Z V . . 4 V .

The fo llo w in g t im e dom ain e q u a tio n s a r e w r i t t e n f o r

th e r e b o i l e r shown i n F ig u re 3 .6 . T h is r e b o i l e r has th e

l i q u i d le v e l c o n t r o l m a n ip u la tin g th e v a p o r b o i lu p .

(o p t io n R .l) The t im e dom ain e q u a tio n s a r e t h e same a s

i n o p t io n R .2 e x c e p t f o r th e vapor and l i q u i d flo w

e q u a t io n s .


42

L iq u id flow - n o e q u a tio n

Vapor flow:

Vj = ^ Q j Oj = V j '

th e r e f o r e ,

''j ' ^ V i

L in e a r i z in g th e above e q u a tio n s a n d c o n v e r t in g to

th e complex f r e q u e n c y dom ain y ie ld s t h e fo l lo w in g r e s u l t s ,

E q u ilib riu m e q u a t io n :

( /ij “ K i j * i j

Vapor flow:

X


AjPj/w f i j + AjHjYVji'w = - Vj - - a a Z j - aaVj


43

A jpjiW

xfV j + . - V . - Z . - a i , - 4 4 V .J J J J ] - l ] ] ] ]

jf "4 4 £ j - 4 4 V j +IÎ £ j +<5 V j

' . A4F4I + (AjHjY + yw

x f ^ R

N ote: The tim e c o n s ta n t f o r th e l i q u i d l e v e l c o n t r o l

loop i s

A jP jAjHjY + /

XT Kr

N um erical C o n s id e ra t io n s

The f lo w c h a r t o f F ig u re 3 .7 d e p i c t s th e g e n e ra l

s o lu t io n te c h n iq u e f o r th e e q u im o la l o v e rf lo w m odel. I t i s

c h a r a c t e r i s t i c o f t h i s m odel t h a t t h e l i q u i d and v a p o r

flow r a t e s a r e in d e p e n d e n t o f t h e c o m p o s itio n (same a s th e

s te a d y s t a t e c a l c u l a t i o n ) . T h is in d ep e n d en c e e l im in a te s

th e need f o r a s im u lta n e o u s s o l u t i o n o f th e m a te r i a l ,

energy and c o m p o s itio n b a la n c e e q u a t io n s .

44

Figure 3*7

Flow Chart o f Solution Technique for Equilm olal Overflow Model

(Subrou tine I/IODEQM)

y es f I <26

noEND

fo r frequency w(l)

Save desired re su lts f o r frequency I

For each component s e t up and invert

tr id ia g o n a l matrix (component balance)

Calculate 1^ and v^ for each

stage from m ateria l balance

45

The d i f f e r e n t r e b o i l e r and c o n d e n se r o p t io n s d e te rm in e

th e method u s e d f o r s o lv in g th e m a t e r i a l b a la n c e e q u a t io n .

T hree d i f f e r e n t c a l c u l a t i o n p ro c e d u re s (T a b le 3-3) have

been d e v e lo p ed f o r th e v a r io u s o p t io n s .

The EQMLVl p ro c e d u re r e q u i r e s an i t e r a t i v e method to

f in d L j and V j . T h is can be shown t o b e n e c e s s a ry b e ca u se

th e m a te r ia l b a la n c e f o r R .l g iv e s an e x p re s s io n fo r Vj

as a f u n c t io n o f L j_ ^ . But L j_^ d ep en d s on Lj_g and V j.

As an i n i t i a l g u e s s V j i s assumed to b e z e r o . Then th e

vapor flow p e r t u r b a t i o n s a re p a ssed u p t h e column u s in g

th e vapor f lo w e q u a t io n . The m a t e r i a l b a la n c e o f th e

condenser i s u s e d to c a lc u la t e p e r t u r b a t i o n s in th e

d i s t i l l a t e l i q u i d f lo w . Next th e l i q u i d flo w p e r tu rb a

t io n s a re c a l c u l a t e d go ing down th e c o lu m n . At th e bo tto m

i t i s p o s s ib le t o c a lc u la te Vj from t h e m a te r i a l b a la n c e .

I f t h i s new V j e q u a ls th e p re v io u s g u e s s o f V j, th en th e

i t e r a t i o n s a r e s to p p e d . The c o n v e rg en c e m ethod i s

d i r e c t s u b s t i t u t i o n . T his p ro ce d u re h a s b een observed

to be s t a b l e an d g e n e ra l ly converges i n s i x i t e r a t i o n s .

P ro c e d u re s EQMLV2 and EQMLV4 a r e s i m i l a r in t h a t no

i t e r a t i o n s a r e r e q u i r e d . T h is i s b e c a u s e th e m a te r ia l

b a la n c e o f R .2 d o e s n o t c a lc u la t e a v a p o r f lo w . Both

p ro ced u res s t a r t by p ro p a g a tio n o f t h e v a p o r flow

p e r tu r b a t io n s up th e column. Next t h e c o n d e n se r m a te r ia l

b a la n c e i s u s e d t o c a lc u la t e e i t h e r t h e r e f l u x flow

re sp o n se (EQMLV4) o r th e l iq u id d i s t i l l a t e flow resp o n se


46

TABLE 3-3

C a lc u la t io n P ro c ed u re s f o r E qu im o la l O verflow M odel

O p tio n s Name o f C a lc u la t io n P ro c e d u re

C .l o r C .3 and R . l EQMLVl

C .l o r C.3 and R .2 EQMLV2

C.2 o r C.4 and R . l n o t im plem ented

C .2 o r C.4 and R .2 EQMLV4


47

(EQMLV2). F i n a l l y , th e l iq u id flow p e r tu r b a t io n s a re

c a lc u la te d down t h e colum n.

O ptions C .2 o r C .4 and R .l w ere n o t im plem ented b ecau se

C .4 and R .l i s n o t a p h y s ic a l ly r e a l i z a b l e a l t e r n a t i v e .

T h is o p tio n d o e s n o t a llo w fo r flow p e r tu r b a t io n s to le a v e

th e column. A n o th e r way o f v iew ing t h i s o p t io n i s shown in

F ig u re 3 .8 . The l i q u i d d i s t i l l a t e f lo w and l iq u id bottom

flow a re m a n ip u la te d by c o m p o sitio n c o n t r o l s . T h is does n o t

a llo w f o r an o v e r a l l m a te r ia l b a la n c e t o be m a in ta in ed in

th e column.

The p ro c e d u re which s o lv e s th e com ponent b a la n c e

e q u a tio n s f o r t h e co m position p e r t u r b a t i o n s i s a t r id i a g o n a l

m a tr ix in v e r s io n r o u t in e . T his p a r t o f th e s o lu t io n te c h

n iq u e i s a m a jo r im provem ent o v e r Cadman and C a r r 's [9 ]

te c h n iq u e w hich c o n s i s t s o f i n v e r t in g a d ense m a tr ix o f

complex num bers. A f te r th e flow p e r t u r b a t i o n s have been

found , th e o n ly unknowns rem ain ing i n t h e component b a la n c e

a re th e l i q u i d com ponent p e r tu r b a t io n s . The vapor component

p e r tu r b a t io n s a r e found by u sin g t h e e q u i l ib r iu m e q u a tio n .

The n u m e r ic a l te c h n iq u e s used i n t h i s model c o n s i s t o f

t r id i a g o n a l m a t r ix in v e r s io n and th e d i r e c t s u b s t i t u t i o n

convergence m eth o d . The in v e rs io n a lg o r i th m i s based on th e

s ta n d a rd fo rw ard e lim in a tio n /b a c k w a rd s u b s t i t u t i o n m ethod.

A dap ting t h i s m ethod to complex num bers d id n o t r e q u ir e

m o d if ic a t io n o f t h e a lg o r ith m ; i t o n ly r e q u i r e d th e use o f


48

Figure 3.8

D is t i l la t io n Column with

Options C .2 or C.4 and R .l

Feed

%C

Lie

Lie:

cc

->.SSL^


49

com plex a r i th m e t ic . The d i r e c t s u b s t i t u t i o n c o n v e rg en c e

a lg o r i th m was chosen f o r i t s s im p l i c i ty . A d raw back o f

t h i s a lg o r ith m i s t h a t i s has p o o r co n v erg en ce c h a r a c t e r

i s t i c s [2 9 ] .

M odeling O ptions

The in t e g r a t i o n o f th e model (MODEQM) i n t o a u s e f u l

p ack ag e i s covered i n t h i s s e c t io n . The f lo w c h a r t o f

F ig u re 3 .9 shows th e m a jo r e le m e n ts o f th e p a c k a g e and

t h e i r i n t e r r e l a t i o n s h i p s . The s te a d y s t a t e c a l c u l a t i o n

p rogram has a lr e a d y b e e n d is c u s s e d in C h a p te r I I . The

o u tp u t r o u t in e s w i l l b e p re s e n te d in C h ap te r V. The ite m s

t h a t rem ain to be d i s c u s s e d in t h i s s e c t io n a r e : th e

g e n e ra t io n o f t r a n s f e r f u n c t io n s , ty p e s o f t r a n s f e r

f u n c t io n s which can b e g e n e ra te d , and th e d a t a r e q u i r e d

by th e dynamic m odel. The program l i s t i n g s a r e p ro v id e d

i n Appendix C.

Two ty p e s o f t r a n s f e r f u n c t io n s can be g e n e r a te d :

th o s e f o r fe e d d i s tu r b a n c e in p u ts and th o se f o r m a n ip u la te d

i n p u t s . The t r a n s f e r fu n c t io n r e l a t e s th e s e i n p u t s t o

m easured v a r i a b le s . The m easured v a r ia b le s in c lu d e l i q u i d

o r v apo r c o m p o s itio n s and l iq u id o r vapor f lo w s from any

s t a t e . The c la s s o f d i s tu r b a n c e in p u ts i n c l u d e s fe e d r a t e ,

f e e d c o m p o s itio n , and fe e d q u a l i t y . The c l a s s o f m anip

u l a t e d in p u ts in c lu d e s th e fo llo w in g flo w s: r e f l u x , v a p o r

b o i lu p , d i s t i l l a t e l i q u i d o r v a p o r , bottom s l i q u i d and any


50

Figure 3»9

Flow Chart o f O verall Analjrsis Package

S ta r t

(!> IRead steady s ta te data

-------------------

Calculate steady s ta te operating poin t

) Read data fo r dynamic model

Set conditions fo r ca lcu lation o f feed stream disturbance

tra n sfer functions

C all MODEQM to find tra n sfer functions

Call output routines I Bode p lo ts and parameter

f i t rou tines

Increment to next feed stream

Have a l l feed streams been used?

yes

Set conditions fo r ca lcu la tion o f manipulated input tra n sfer functions

C all MODEQM to find tra n sfer functions

C all output routines Bode p lo ts and parameter f i t s


51

Increment to nextmanipulatedinput

No Have a l l manipulated inputs been used?

C a ll output routines: Nyquist p lo ts and

decoupler p lots

Is th e r e another se t of d a ta for the dynamic model?

Is th e r e another se t of d a ta fo r the steady s ta te ca lcu la tio n s?


52

l i q u id o r v a p o r s id e s t re a m . The m o d e l and program m ing

package a r e f l e x i b l e enough to g e n e r a t e a t r a n s f e r f u n c t io n

f o r any c o m b in a tio n o f in p u ts and m easu red v a r i a b l e s . I t

i s assumed t h a t th e v a lv e s , m ea su rin g d e v ic e s , and t r a n s

m i t t e r s a s s o c i a t e d w ith a given t r a n s f e r fu n c t io n a re

l i n e a r .

The g e n e r a t i o n o f t r a n s f e r f u n c t i o n s c o n s i s t s o f t h r e e

s t e p s : s e l e c t i o n , e x e c u tio n and s t o r a g e . The s e le c t io n

s te p f i r s t s e t s a l l fe e d d is tu r b a n c e and m a n ip u la te d in p u ts

t o z e ro . N e x t , th e d e s i r e d in p u t i s s e t t o o n e . I f a f e e d

d is tu rb a n c e h a s been s e le c te d , t h e f e e d s p l i t e q u a tio n s a r e

u sed to d i s t r i b u t e th e e f f e c t s o f t h e d is tu r b a n c e . A v a lu e

o f one f o r a n in p u t in th e complex f re q u e n c y domain i s

e q u iv a le n t t o g e n e ra t in g a u n i t im p u ls e in th e tim e dom ain .

The u n i t im p u ls e g e n e ra te s a r e a c t i o n c u rv e . T h is r e a c t i o n

cu rv e in th e f re q u e n c y domain w ould b e th e t r a n s f e r f u n c t i o n .

The s e le c te d i n p u t i s th e n m a in ta in e d a t i t s i n i t i a l v a lu e

w h ile th e m o d el i s e x e c u te d . At e a c h fre q u e n c y th e m odel

c a lc u la t e s a n o th e r com plex value f o r th e m easurem ent v a r i

a b le . T hese com plex num bers a re s a v e d and l a t e r p l o t t e d .

Three t y p e s o f d a ta b e s id e s t h a t f o r th e s te a d y s t a t e

c a lc u la t io n a r e r e q u ir e d f o r th e dynam ic m odel ( fo r a d d i

t i o n a l d e t a i l s , e .g . fo rm a ts , see A ppend ix D) . The f i r s t

s e t o f d a ta f u l f i l l s th e a d m in i s t r a t iv e needs o f th e

package. T h is in c lu d e s item s such a s th e t i t l e , o u tp u t


53

o p t io n s , and p l o t t i n g s c a le f a c t o r s . The s e c o n d s e t

d e s c r ib e s w hich t r a n s f e r fu n c tio n s a r e to b e g e n e ra te d .

Codes f o r th e m easu red v a r ia b le s i n d i c a t e t h e s ta g e

number and s ta g e v a r i a b l e to be r e c o rd e d . I n a d d i t io n ,

th e g a in , dead t im e , a n d tim e c o n s ta n t o f t h e m easu ring

d e v ic e can be s p e c i f i e d . This o p t io n a llo w s o n e to

i n v e s t i g a t e th e e f f e c t s o f m easurem ent dynam ics upon th e

column c o n t r o l sy s te m . The codes f o r th e m a n ip u la te d

in p u ts i n d i c a t e th e s t a g e number and in p u t v a r i a b l e .

T h is c a rd a l s o c o n ta in s param ete rs f o r s p e c i f i c a t i o n o f

co n d en se r o p t io n and l i q u i d le v e l c o n t r o l l e r t im e c o n s ta n t .

The l a s t s e t o f d a ta c o n ta in s th e m ole h o ld u p on each s ta g e

and th e 1 /6 ^ and g g /g ^ p a ram ete rs f o r each s t a g e .


54

V a r ia b le s Used in C h a p te r I I I

c r o s s - s e c t i o n a l a r e a , s t a g e n

PL^ l i q u i d fee d to s ta g e n

FT^ t o t a l o f l iq u id and v a p o r fe e d to s ta g e n

FV^ v a p o r fee d to s ta g e n

^Imax maximum h e ig h t o f l i q u i d i n condenser

h e ig h t o f l iq u id in s t a g e n

HF^ t o t a l e n th a lp y o f f e e d t o s ta g e n

HL^ l i q u i d e n th a lp y le a v in g s t a g e n

HV v a p o r e n th a lp y le a v in g s t a g e n

j = / - 1

Kg c o n t r o l l e r g a in f o r l i q u i d l e v e l in c o n d en se r

K v a lu e o f component i , s t a g e n

Kjj c o n t r o l l e r g a in f o r l i q u i d l e v e l in r e b o i l e r

l i q u i d flow from s ta g e n

h o ld u p on s ta g e n

Q h e a t f lu x in to (+) o r o u t ( - ) o f s ta g e n(o n ly used in e n th a lp y m odel)

QF^ l i q u i d f r a c t io n o f f e e d t o s ta g e n

SSL^ s id e s t r e a m l iq u id l e a v in g s ta g e n

SSV^ s id e s t r e a m vapor l e a v in g s t a g e n

U o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t

v a p o r flow from s ta g e n

l i q u i d mole f r a c t io n com ponen t i , s ta g e n

v a p o r mole f r a c t io n com ponen t i , s ta g e n


55

Z in t o t a l mole f r a c t i o n i in f e e d to s ta g e n

ZLj^^ l i q u i d mole f r a c t i o n i in fe e d to s ta g e n

v ap o r mole f r a c t i o n i in fe e d to s ta g e n

AT te m p e ra tu re drop a c r o s s co n d en se r

l a t e n t h e a t o f v a p o r i z a t i o n on s ta g e n

X l a t e n t h e a t o f v a p o r i z a t i o n f o r h e a t in g v a p o r( f o r exam ple, l a t e n t h e a t o f steam )

w freq u e n cy

l i q u i d d e n s ity on s t a g e n

e tim e

b a r in d ic a te s s t e a d y s t a t e v a lu e s

N o te : Lower c a se s c r i p t l e t t e r s r e f e r to th e same

v a r i a b le as th e u p p e r case b lo c k l e t t e r s b u t

r e f e r t o th e com plex freq u e n cy domain.


CHAPTER IV

ENTHALPY BALANCE MODEL

The deve lopm en t o f th e e n th a lp y b a la n c e model and

i t s a s s o c ia te d s o lu t io n te c h n iq u e w i l l be d isc u sse d in

t h i s c h a p te r . T h is model was d e v e lo p e d to d e te rm in e

w hether a dynam ic energy b a la n c e i s n eed ed fo r a dynam ic

column' m odel. T h is e v a lu a t io n i s p e rfo rm e d by com paring

th e r e s u l t s o f th e e q u ilm o la l o v e rf lo w model to th e

e n th a lp y b a la n c e model f o r s e v e r a l ty p e s o f s e p a r a t io n s .

The f i r s t s e c t i o n d e s c r ib e s th e a ssu m p tio n s and t h e i r

in f lu e n c e upon th e m odel. The m odel e q u a t io n s a re d e v e l

oped in th e seco n d s e c t io n . The n u m e r ic a l c o n s id e r a t io n s

a re d is c u s s e d in th e t h i r d s e c t io n . V a rio u s m odeling

o p tio n s a r e d is c u s s e d in s e c t io n f o u r .

A ssum ptions

The m u lti-co m p o n en t e n th a lp y b a la n c e model o f t h i s

c h a p te r e x te n d s th e e q u ilm o la l o v e r f lo w model by a llo w in g

th e dynamic f l u c t u a t i o n o f h e a t s t o r a g e . The fo llo w in g

e q u a tio n s and assu m p tio n s a r e c o n s id e re d in th e same

o rd e r as th o s e o f W illiam s [ 2 ] . (T a b le 1-1)

1) The com ponent m a te r ia l b a la n c e i s w r i t te n f o r

each com ponent. The l iq u i d c o m p o s itio n p e r tu rb a t io n s

a re fo rc e d t o sum to z e ro . By means o f a bubb le p o in t

c a l c u l a t i o n , th e vapor co m p o s itio n p e r tu r b a t io n s sum to

z e ro . P e r tu r b a t io n s in th e K v a lu e s a r e fu n c tio n s o f

56


57

l i q u i d co m position and te m p e ra tu re p e r t u r b a t i o n s . The

K v a lu e i s in d ep en d en t o f t h e v ap o r p h ase . The v a p o r and

l i q u i d on a t r a y a re w e l l m ix e d .

2) The e n th a lp y b a la n c e around each s ta g e g e n e r a te s

v a p o r flow p e r tu r b a t io n s d u e t o th e te m p e ra tu re g r a d i e n t in

t h e co lum n . T h is a llo w s t h e in t r o d u c t io n o f t e m p e r a tu r e

e f f e c t s . The h e a t e f f e c t s due to th e m eta l o f t h e t r a y and

co lum n can a ls o be in c lu d e d .

3) T here i s one w e l l m ixed l iq u id re g io n on a t r a y .

F low from one t r a y to th e n e x t i s in f lu e n c e d b y v a p o r r a t e

an d l i q u i d ho ldup p e r t u r b a t i o n .

4) Vapor phase h o ld u p i s n e g le c te d ; th u s t h e r e a re

no v a p o r phase dynam ics. T h is assum ption r e s t r i c t s th e

u s e o f th e model to colum ns w hose l iq u id h o ld u p i s much

g r e a t e r th a n t h e i r v ap o r h o ld u p .

5) P re s s u re t r a n s i e n t s a r e assumed to be f a s t

com pared to co m p o sitio n e f f e c t s ; th e r e f o r e , t h e y a r e

n e g le c t e d .

6) Due to th e e n th a lp y b a la n c e , th e dynam ic e f f e c t o f

i n t e r s t a g e h e a t e r / c o o le r s c a n b e a s s e s s e d .

7) T here a re no r e g i o n s betw een th e s t a g e s . A l l th e

l i q u i d in th e downcomer i s com bined w ith th e l i q u i d on th e

s t a g e .

8) E q u ilib r iu m b e tw ee n l i q u i d and vapor i s

i n s t a n t a n e o u s ly r e a l i z e d . E ach t r a y i s an e q u i l i b r iu m

s t a g e , i . e . th e e f f i c i e n c y f a c t o r i s one . The num ber


58

o f e q u il ib r iu m s t a g e s i s v a r ie d u n t i l a m a tc h w ith th e

p l a n t d a ta i s fo u n d . For exam ple, an a c t u a l te n tra y

column w ith an e f f i c i e n c y o f 0 .7 would be m odeled w ith

seven e q u il ib r iu m s t a g e s w ith each s ta g e c o n ta in in g

1 ,0 /0 .7 tim es th e l i q u i d mass o f an a c t u a l t r a y . T his

re d u c e s th e number o f flow and co m p o s itio n l a g s in s e r i e s .

The l a r g e r th e co lum n th e l e s s s e r io u s t h i s r e s t r i c t i o n .

Model

T here a re t h r e e s e t s o f e q u a tio n s w h ic h com prise

t h e e n th a lp y b a la n c e m odel. The g e n e r a l i z e d s ta g e

e q u a t io n s a llo w f o r l i q u i d and v ap o r fe e d s an d s id e -

s tre a m s on each e q u i l i b r iu m s ta g e and an i n t e r s t a g e

h e a t e r / c o o le r . The seco n d and t h i r d s e t s o f eq u a tio n s

c o n ta in th e v a r io u s c o n d en se r and r e b o i l e r o p t io n s ,

r e s p e c t iv e ly . The o p t io n s a re th e same a s th o s e in

T a b le s 3-1 and 3 -2 .

The fo llo w in g t im e domain e q u a tio n s a r e w r i t te n

f o r th e g e n e r a l iz e d s t a g e o f F ig u re 3 .1 .

E q u il ib r iu m e q u a t io n :

^ in = « in ^ in


59

Summation e q u a tio n :

NC NC

i L " i . = ^ '• " i n = ^

L iq u id flow e f f e c t s :

M a te r i a l b a la n c e :

dM„W r = V l + \ + l - ( t n + S S L .I - (V „+ S S V „) + PL^ + FV„

E n th a lp y b a la n c e :

d(HL M +HM )”d8“ ° " = “ n-1 V l + " n+1 ''n+1

-H L „a„+ S S L „)-H V „ (V„+SSV„) +

+ aPL^ FL^ +HPV PV

NC 2 3“ x i+ “ 2 l V “ 3 i ’^ n ^ 4 i ’'n> =‘i n

NC 2 3HV„ = (* li+ °2 iT n + * 3 iT n + * 4 i ’'n> ’' i n

™ n = " l + + *3?n


6o

Component b a l a n c e :

dX . M- a r = X i n - l V l + ■ 'in + A n + l’^n-fl ' ’= in < V

- Kin% in < ) + Z L .„ P L .„ + ZV.^PV.^


F = f l ^ + PV n n n

HP P = HPL^PL + HPV PV

L in e a r iz in g t h e above e q u a tio n s and c o n v e r t in g them to

th e complex f re q u e n c y dom ain y ie ld s t h e fo llo w in g equa

t io n s :

E q u ilib r iu m e q u a t io n :

^in ^ i n ^ i n + ^ in ^ in

Summation e q u a t io n s :

NC NC

"in - =

6 1

B ubble p o i n t :

7 k . NC ? K .

NC NC W .

i = i ^ ^

NC NC ÏÏK. NC% ! l " A i f ' V ^ in > \ l , h n h n - 0

3H

t h e r e f o r e .

NC NC NC ^- : <V i „ > -

L iq u id f lo w e f f e c t (see e q u im o la l o v e rf lo w model f o r

d e r i v a t i o n )

1 hm_ = 7:— Z — T— V'n - ^n - BY "n+l

M a te r i a l b a la n c e (see e q u im o la l o v e rf lo w model f o r

d e r i v a t i o n ) :

■ " n + l ' l * e f * 51

“ V l - “n -

62

E n th a lp y b a la n c e :

HL j i m + M j w h l + M j w k m n n n n n n

+ HFI.^

+ HPV„<v„ +

In c lu d in g th e l i q u i d flo w e f f e c t s y i e l d s :

^ “n + l ' ' « V l V l + "^n+1% +1

- a j » i h m ^ + + '‘“n + x V l ' “ n ‘V = ® ^ n ’

- f c v „ ( v s § v „ ) +

+ HPV„|!v„ + fc<v„PV„

63

S o lv in g th e e q u a tio n f o r and y ie ld s :

g

- '" n + l t '^ n + l+ ^ n ^ =

+ “ n-lVl - ‘ V'^^n

- + '■“ n + l V i - '■ « n 'V ^ n »

- (V„ + SSV„) + + h p l„< £ ^ + H t ^ F L ^

+ HPV„<V„ + H o ^ F V ^

whereNC 4 . .

- j , “jA ' >*in +

NC 4 . .

i=l S=2 >*in

NC 4 _ . _ , NC 4 _ 2= j , ( .! /j i ) in + n

and

and

K - ' “ 2 +

Com ponent b a lan ce :

hJ'^n * V “ in = ’‘in-lV l + *in-lVl

■*■ ^ i n + l * i n + l \ + l * * in + l* in + l^ n + l * ^ in+ 1 * i n + l ''n+X

&*

z ii„F L „ + Z L ^ J t ^ + :V in + 2V in< "n

In c lu d in g th e m a t e r i a l b a la n c e y ie ld s :

- V l ‘in-1 + 'V'“ + K.„(V„+SSV„) + (VSSL„)1*1in

- ( ^ n - l + ' 'n + r '

* * in - l '^ n - l * ’' i n + l ' ' i n + l “n+l * ^ in + l* in + l^ n + l

- * in ^ i n < V s ® V +

+ zVi„FV„ +

65


a l l l i q u id fe e d F o r a l l v a p o r

= < n . K ‘ <nz t = z zv = zn n n n

* M ^n = '‘<n

F o r com bined l iq u id /v a p o r fe e d :

" i n = K finZ ^in

N ote: The te rm has b een dropped th u s n e g le c t in g

th e c o m p o s itio n o r te m p e ra tu re in f lu e n c e on

V i n + =in<n =

0 # • V i " + - ZVin<»n

f " F L „ + K F .„ F V „

66

The r e b o i l e r o p t io n s R .l and R .2 u s e th e same e q u a t io n s

as th e e q u im o la l o v e rf lo w model b u t w ith th e a d d it io n o f t h e

e n th a lp y b a la n c e . T h e re fo re , o n ly th e e n th a lp y b a la n ce

w i l l be p r e s e n te d h e r e .

Time dom ain e n th a lp y b a la n c e :

- H L j(L j+ S SL j) - HVj(Vj+SSVj)

+ HPLj FLj + HPVj PVj

When th e e f f e c t o f r e b o i l e r s w e l l i s i n c lu d e d , th e tim e

d e r iv a t iv e b eco m es:

^ dHM _ dHL. _ dH. dV .

"d0“ ■*’ ” d 0^ de de ^

In th e f r e q u e n c y domain th e e n th a lp y b a la n c e i s :

[Mfcm. + M . k t . + A.HL. ( p . f i . + H . y v . ) ] i W

f c £ , ( L . + S S L . ) - H L . U . + 4 4 - e . ) ] ] ] ] ] ]


67

- k v . ( V . + S S V . ) - H V . ( V . + 4 4 V . ) J J ] J J J

+ k d ^ jP L j + HFLjd^j

+ kdVjPVj + HPVjdVj

P o r o p t i o n R . l = &j/K^

P or o p t i o n R .2 = q j/K ^

The c o n d en se r e q u a tio n s a r e s im i la r to th e e q u im o la l

o v e r f lo w m odel. The o n ly m o d i f i c a t io n i s th e i n c l u s i o n

o f t h e e n th a lp y b a la n c e . The fo l lo w in g b a la n c e i s

w r i t t e n f o r o p tio n s C.3 and C . 4 . Condenser o p tio n s C . l

and C .2 w ere n o t im plem ented f o r th e e n th a lp y m odel.

Tim e dom ain e n th a lp y b a la n c e

dCHL^M.+HM-M.) y 1 = HVgVg + - HV^SSV^

- HL^(L^+SSL^) + HPL^PL^ + HPV^FV^

C o n v e r t in g t h i s e q u a tio n to t h e freq u e n cy domain y i e l d s ;


68

q ^ + HVgVg + kVgVg - W^(L^ + SSL^)

- HL^(£^ + 4 4 ^ ) - HV^44Vj - bv^SSV^

+ + HFL^dZ^ + h i v ^ F V ^ + HFV^dv^

For o p t io n C.3 “ 44^/K ^

F or o p t io n C.4

N um erical C o n s id e ra tio n s

B ecause th e e n th a lp y b a la n c e i s now p a r t o f th e dynam ic

m odel, th e i n t e r n a l flow p e r tu r b a t io n s o f th e colum n become a

fu n c t io n o f c o m p o sitio n p e r tu r b a t io n s .^ T his means t h a t th e

flow p e r tu r b a t io n s can no lo n g e r be d e te rm in e d b e fo r e th e com

p o s i t io n p e r t u r b a t i o n s , a s in th e e q u i lm o la l o v e rf lo w m odel.

T h e re fo re , i t i s n e c e s s a ry t o so lv e th e m a t e r i a l , sum m ation,

e q u il ib r iu m and h e a t e q u a t io n s s im u lta n e o u s ly . S in c e a l l t h e

e q u a t io n s have been l i n e a r i z e d , two a p p ro a c h e s can be ta k e n

^T his s i t u a t i o n a l s o o c c u rs w ith th e s te a d y s t a t e e n th a lp y m odel.


69

i n s o lv in g th e s e e q u a tio n s : m a tr ix in v e r s io n o r a n i t e r a

t i v e a p p ro a c h . The l a t t e r was chosen and a s t r a t e g y was

d e v e lo p e d w h ich i s s im i la r t o th e Wang/Henke b u b b le p o i n t

p ro c e d u re [ 1 8 ] . This m ethod was ch o sen because i t h a s been

u se d t o c o n v e rg e a s e t o f n o n - l i n e a r s te a d y s t a t e e q u a t io n s

h a v in g th e same form as t h e dynam ic e q u a tio n s . T he m a t r ix

i n v e r s io n ap p ro ach would r e q u i r e th e in v e rs io n o f a b lo c k

d ia g o n a l m a t r ix w ith two o f f - d i a g o n a l te rm s . T h ese o f f -

d ia g o n a l te rm s a re caused by th e c o n t r o l o p tio n s . The

o r d e r o f t h e m a tr ix would b e e q u a l to NS (NC+3).

The f lo w c h a r t o f F ig u r e 4 .1 d e p ic t s the g e n e r a l s o lu

t i o n te c h n iq u e fo r the e n th a lp y m odel. At each f r e q u e n c y

t h e m odel i t e r a t e d to a c o n v e rg e d s o l u t io n . C o n vergence i s

a id e d by a p a r t i a l s u b s t i t u t i o n m ethod. The e q u a t io n s f o r

t h e d i f f e r e n t o p tio n s a r e s o lv e d s i m i l a r l y and no a d d i t i o n a l

c o n v e rg e n c e lo o p s are r e q u i r e d . T r id ia g o n a l i n v e r s io n was

u se d t o o b t a in th e com ponent p e r t u r b a t i o n s . T e m p e ra tu re

p e r t u r b a t i o n s a r i s e from t h e b u b b le p o in t p ro c e d u re . The

m a t e r i a l an d energy b a la n c e s a r e s o lv e d s im u lta n e o u s ly by

m eans o f a b lo c k t r id i a g o n a l r o u t in e f o r new v a lu e s o f th e

l i q u i d and v a p o r flow p e r t u r b a t i o n s .

The s o l u t io n te c h n iq u e i s s e n s i t i v e to s te a d y s t a t e

c o n d i t i o n s and to the c o n t r o l o p t io n s . Moderate c h a n g e s

i n t h e s te a d y s t a t e c o n d i t io n s w ould produce a d y n am ic

m odel w hich w ould no t c o n v e rg e . N ot a l l c o n tro l o p t io n s


70

Figure 4.1

Flow Chart o f Solution Procedure fo r Enthalpy Model Subroutine (MODENT)

1 = 0I n i t i a l guess for a l l

perturbations i s zero

1 = 1 + 1 frequency = w (l)

V Using tr id ia g cn a l inversion find

Force perturbation o f on sta g e n to sum to zero

Using bubble point procedure f in d t^ , k ^ ,

Solving m aterial and enthalpy b a la n ces , find 1^, v^


71

Are the r e la t iv e changes in the temperature

perturbations le s s than the to lerance?

Yes

Save d esired re su lts

Yes I <26 7

No

End

72

w ould p ro v id e a c o n v e rg e d s o lu t io n f o r e a c h c o lu m n . The

a u th o rs o f s te a d y s t a t e m odels have overcom e s i m i l a r

problem s by ch an g in g th e o rd e r in which t h e e q u a t io n s were

so lv e d .

Modeling O p tio n s

The i n t e g r a t i o n o f th e model (MODENT) i n t o an a n a ly s is

package fo llo w s th e sam e approach as th e e q u i l m o l a l o v e r

flo w model. The f lo w c h a r t o f F ig u re 3 .9 s t i l l a p p l i e s .

T h e re fo re , t h i s s e c t i o n w i l l o n ly m ention t h e a d d i t i o n a l

m odeling o p t io n s .

The a d d i t io n o f t h e e n th a lp y b a la n c e i n c r e a s e s th e

ty p e o f t r a n s f e r f u n c t io n s w hich can be g e n e r a t e d an d take

i n to account a n o th e r se co n d a ry e f f e c t . S in c e te m p e ra tu re

i s an i n t e g r a l p a r t o f th e m odel, i t i s now p o s s i b l e to

u se i t as a m easu red v a r i a b l e . I n t e r s t a g e h e a t e r s / c o o l e r s

can be used a s m a n ip u la te d v a r i a b le s . As a n o p t io n th e

m e ta l mass o f each e q u i l ib r iu m s ta g e may b e p r o v i d e d .

T h is adds more la g i n th e t r a n s f e r o f h e a t fro m one s ta g e

t o th e n e x t. T h is l a g can be s i g n i f i c a n t i n h i g h p re s s u re

columns w hich r e q u i r e th ic k w a lls o r in c o lu m n s w h ich have

a sharp te m p e ra tu re ch an g e o v e r a few s t a g e s . T he h e a t

c a p a c ity d a ta u sed f o r th e e n th a lp y c a l c u l a t i o n i s th e

same as t h a t u sed i n t h e s te a d y s t a t e c a l c u l a t i o n s . The

m e ta l h e a t c a p a c i ty i s t h a t o f 316 s t a i n l e s s s t e e l .


73

Because o f th e summation equations, a change in the feed

composition o f one component now gen era tes feed disturb

ance tra n sfe r functions for the o th e r components.

Comparison o f Equimolal and Enthalpy Models

Two h y p o th e tica l separation p ro cesses were used to

determine i f there i s a d ifference in the model r e su lts .

A benzene-toluene-xylene system (BTX) was used because

i t s steady s t a t e separation i s t r u ly equimolal overflow.

A propane-butane-pentane (FBP) was used because i t

represented a steady s ta te separation which required an

enthalpy b a la n ce; d e ta ils of the system s can be found

in Appendix I . The comparison was made by gnerating

transfer fu n c tio n s for each system w ith both dynamic

models.

The t r a n s fe r function for the BTX system indicates

l i t t l e d if fe r e n c e between the dynamic models. This con

clusion was reached a fte r observing th e Bode p lo ts for

the three c o n tr o l schemes. Figures 4 .2 -4 .1 7 represent

the equim olal overflow model ( 4 .2 -4 .9 ) and enthalpy

balance model (4 .10-4 .17) transfer fun ction s for the

bottoms vapor and liq u id d i s t i l l a t e con tro l scheme.

Transfer fu n c tio n s for the other c o n tr o l schemes were

not presented because they gave s im ila r r e su lts . The

phase p lo ts show that the order o f th e dynamics generated


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by both models i s the same. This ind icates that th e

underlying dynamic equations are s im ilar . The magnitude

p lo ts rev ea l that the break p o in ts are e s s e n t ia l ly th e

same fo r a l l transfer fu n ctio n s. The steady s ta te ga in s

of the tran sfer functions are d iffe r e n t but are o f th e

same order o f magnitude and have the same sign .

The tran sfer functions o f the F3P system in d ic a te

larger d ifferen ces between the dynamic models than those

shown by the BTX system. F igures 4,18 to 4.33 rep resen t

the equim olal overflow model (4 .1 8 to 4.25) and th e

enthalpy balance model (4 .26 to 4 .3 3 ) tran sfer fu n ctio n s

for the bottoms vapor and liq u id d i s t i l la t e com position

control scheme. Transfer fu n ctio n s for other co n tro l

schemes were not presented because the enthalpy balance

model would not converge. The d ifferen ces mainly occur

among th e feed disturbance tr a n s fe r functions (F igures

4.18 to 4 .21 and 4.26 to 4 .29 )* Steady s ta te g a in s ,

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orders o f magnitude d iffe r e n t . Transfer function break

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show th a t the order of sev era l tra n sfer functions has

changed. The manipulated input tran sfer functions

(Figures 4 .22 to 4,25 and 4 ,2 0 to 4 ,33) show a high

degree o f s im ila r ity . The ga in s and break points have

not s h if te d s ig n if ic a n tly although the order of tr a n s fe r

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c .. c o e f f i c i e n t o f v a p o r e n th a lp y p o ly n o m ia lf o r component i


CHAPTER V

PRESENTATION OF SYSTEM DYNAMICS

System dynamics can be presented in several u sefu l

forms. C lassic frequency domain techniques are used to

present the dynamics o f both the equilm olal overflow model

and the enthalpy balance model. This approach follow s

n atu rally from the so lu tio n technique which generates the

magnitude and phase s h i f t of a tran sfer function as a

function o f frequency. Therefore, Bode p lo ts , a n a ly tica l

expressions, polar p lo t s , Nyquist p lo ts and various com

b inations of tra n sfer functions can be e a s i ly d isp layed .

This chapter presents a review o f these methods. The

computer routines used to generate these r e su lts can be

found in Appendix F.

The c la ss ic grap h ica l method for presenting dynamics

i s with Bode p lo ts . These p lo ts represent the two compo

nents o f a transfer fun ction , magnitude and phase s h i f t ,

as a function of frequency. These p lo ts com pletely define

the input/output re la tio n sh ip between manipulated and

measured variab les.

The second method for presenting th e dynamic

ch a ra c ter is tic s o f a system i s by means o f closed form

tran sfer functions. As an option the two equations o f

Figure 5*1 are f i t to the data generated by the dynamic

model. I f the dead tim e parameter i s n eg a tiv e , then the

equations are r e f i t w ith dead time se t to zero. A pattern

108


F ig u re 5 .1

C losed Form T r a n s f e r F u n c tio n E q u a tio n s

Model 1 P =T y w + 1

G e'djWModel 2 P = 2

2 5w y w + i - ^"n

109


110

search method is used to obtain the parameters. The

search routine minimizes the sum of the ab so lu te d ifference

between the equation and the model generated data for both

th e rea l and imaginary p a r ts of the transfer fu n ction .

The third type of output is polar p lo ts o f

Pgj» and P22' Although t h i s same information i s presented

w ith Bode p lo ts for th e se transfer fun ction s, i t is

sometimes convenient to have polar p lo ts . These p lo ts and

the follow ing ones are provided as an option.

The fourth type o f output characterizes the closed

loop response of the column. This requires a d d itio n a l

information such as the co n tro ller se ttin g s and the

sp e c if ic a t io n of the p ro cess transfer fu n ctio n s . The

la t t e r information is obtained by considering only the

f i r s t two manipulated in p u ts and the f ir s t two measured

outputs sp ec ified on th e data cards. The former information

must be supplied by the u ser in the form of tun ing constants

fo r both PID co n tr o lle r s . This routine p lo ts the follow ing:

the R ijnsdorp/B ristol in te r a c tio n index as a function of

frequency, a Nyquist diagram of the c h a r a c te r is t ic equa

t io n , two e ffe c t iv e p ro cess transfer fu n ctio n s, and de

coupler transfer fu n ction s for s ix decoupler configura

t io n s . A ll but the decouplers are presented on polar

p lo ts . The decouplers a re presented as Bode magnitude

p lo ts . Chaper VII in c lu d es a discussion o f the sig n ifica n ce

o f these p lo ts .


I l l

N yquist D iagram o f C h a r a c t e r i s t i c E q u a tio n

This p lo t i s used to analyze the s t a b i l i t y of the

closed loop con tro l system. The equation o f Figure 5*2

i s the c h a r a c te r is tic equation fo r a two input-two output

system. I t i s a function of the co n tro ller se ttin g s as

w ell as the process transfer fu n ction . A working, but

not com pletely general, statement of the Nyquist s t a b i l i t y

cr iter io n in d ica tes that i f th is polar p lo t encloses the

(-1 ,0 ) p o in t, the system w ill be unstable when the co n tro l

loops are c lo sed . This p lot provides a convenient check

as to whether the proposed co n tro l systems and tuning

constants w i l l provide a stable system.

E ffective Process Transfer Functions

These p lo ts present the e f f e c t iv e process tran sfer

functions generated by closin g one control loop and

leaving the other open. Figure 5*3 i l lu s t r a te s the Block

diagram and e f fe c t iv e process tra n sfer function EP^ . EP^

can be used to design the co n tro lle r C , s in ce EP i s

the process which con tro ller w i l l see. H is to r ic a lly ,

P^2 has been used as the design b a s is for but Schwanke

et a l [2Ô) poin t out that th is lead s to unstable c losed

loop system s. A con tro ller tuning method based on EP^

and EPg2 i s i l lu s tr a te d in Table 5-1» The procedure i s

ite r a tiv e and depends greatly on the d esign er's " feel for

the process" to obtain an answer.


112

F igure 5*2

C h a ra c te r i s t ic Equation

0 s 1 + + CgPgg; + ^1^2^11^22 ( ^ ^ )

F igure 5.3

Block D iagram and T ran sfe r F u n ctio n f o r E P u

^11

V ^21 /

^1?

+22

'2 e R,

■ ^11L : + V i i J


113

TABLE 5 -1

G en era l Tuning P ro c e d u re fo r

Two Input-T w o O u tpu t P ro c e s s U sing EP^^ and EPgg

1) O b ta in e s t im a te s o f tu n in g p a ra m e te rs b ased on P , _ a n d Pgg (M artin e t a l . [ 2 2 ] )

2) U s in g th e l a s t s e t o f t u n i n g p a ra m e te rs , g e n e r a te t h e N y q u is t p l o t o f th e c h a r a c t e r i s t i c e q u a tio n a n d th e p o la r p lo t s o f E P ._ and EP-g» I f th e c lo s e d lo o p system i s s t a b le and t h e g a in and phase m a rg in a r e s a t i s f a c t o r y , s to p t h i s p ro c e d u re .

3) M o d ify th e tu n in g p a ra m e te r s so as to improves t a b i l i t y o f th e g a in and p h a s e m arg in . The m ethod o f M a r t in e t a l . [22] c a n b e used i n t h i s s te p t o f i n d and from EP^^ a n d EPgg*

4) Go t o 2.


114

D e c o u p le rs

D eco u p le rs a re p ro v id e d as an a l t e r n a t i v e t o d e s ig n in g

c o n t r o l l e r s f o r th e i n t e r a c t i n g s y s te m . The s i x b a s i c

d e c o u p le r d e s ig n s o f W a l le r 21 a r e u se d . These d e s ig n s

c o m p le te ly decoup le t h e p ro c e s s , i . e . th e e f f e c t i v e d ia g o n a l

t e r m s a re z e r o . F ig u re 5*4 i l l u s t r a t e s th e B lock d iag ram

f o r a c o n tr o l system em p loy ing d e c o u p le rs and fe e d b a c k con

t r o l l e r s . T ab le 5»2 l i s t s th e fo rm u la s used to c a l c u l a t e

t h e d i f f e r e n t d e c o u p le r ty p e s . The e q u a tio n s d e f in in g

a n d EPgg w ere used to o b t a i n th e e f f e c t i v e p ro c e s s t r a n s f e r

f u n c t i o n s .

C hoosing th e c o r r e c t d e c o u p le r ty p e i s a t r i a l and

e r r o r p ro c e s s . G e n e r a l ly , on ly two o u t o f th e s i x d e c o u p le rs

w i l l be r e a l i z a b l e . T he o th e r f o u r w i l l r e q u i r e t r a n s f e r

f u n c t io n s whose n u m e ra to rs a re o f h ig h e r o rd e r th a n t h e i r

d e n o m in a to rs . Once t h e form o f th e d e c o u p le r h a s b een

c h o s e n , i n i t i a l tu n in g c o n s ta n ts f o r th e d e c o u p le r c a n be

o b ta in e d from th e Bode p l o t o f th e d e c o u p le r . The d e c o u p le rs

s h o u ld th e n be tuned o n - l i n e open lo o p . I f th e r e s u l t s ,

p ro v e s a t i s f a c t o r y , t h e lo o p s can b e c lo s e d and th e con

t r o l l e r s tu n e d in d e p e n d e n t ly o f one a n o th e r .


Figure 5» 4

Block Diagram fo r

Control System Using Decouplers

115

11

21

12

22

D 's in d icate decoupler tr a n s fe r functions C*s in d icate con tro ller tra n sfer functions P 's ind icate process tr a n s fe r functions R 's are setp o in ts


116

TABLE 5-2

Types o f D ecoup lers

Type °11 °12 °21 »22

1 1 .0 - P 1 2 / P 1I ”^ 2 1 /^ 2 2 1 .0

2 1 .0 1 .0 "^21^^22 "^11^^123 " ^ 2 3 /^ 2 1 “ ^12^^11 1.0 1 .0

4 " * 2 2 /^ 2 1 1 .0 1 .0 ^11^^125 P 2 2/DEN "P l2 /°B N -Pgi/DEN P^l/DEN

6 Pll*P22/DBN - ^1 2 * ^2 2 /°^N -P 2 i *Pi i /DEN *11*P22/D '

DEN = ^11*^22 - ^ 12*^21

BPll = P l l * » l l + ^12*°21

^^22 = ^21*°12 + ^22*°22


CHAPTER VI

APPLICATION

A p p l ic a t io n o f th e e q u im o la l o v e rf lo w dynamic m odel

t o an i n d u s t r i a l s c a le e th y le n e colum n i l l u s t r a t e s th e

u s e fu ln e s s o f th e a n a ly s i s p ro ce d u re a s w e l l as p ro v id in g

in fo rm a tio n a b o u t th e p a ra m e te rs w hich g o v e rn column dynam ics.

The t r a n s l a t i o n o f o p e ra t in g o r d e s ig n c o n d i t io n s i n t o a com

p le t e s te a d y s t a t e p r o f i l e i s th e f i r s t s t e p in t h i s a n a ly s i s

p ro c e d u re . D e te rm in in g th e p a ra m e te rs r e q u i r e d f o r th e dynam

i c model c o m p le te s th e model s p e c i f i c a t i o n p h a se . T r a n s f e r

fu n c t io n s sh o u ld be g e n e ra te d u s in g ex trem e v a lu e s o f th e

above p a ra m e te rs i n o rd e r t o check th e s e n s i t i v i t y o f th e

colum n. T hese t r a n s f e r fu n c t io n s show how column p a ra m e te rs

in f lu e n c e colum n dynam ics.

Developm ent o f th e S teady S ta te P r o f i l e

In o rd e r t o d e v e lo p th e s te a d y s t a t e p r o f i l e ^ , i t i s

n e c e ssa ry t o know th e p la n t o p e ra t in g o r d e s ig n c o n d i t io n s

and th e p h y s ic a l p r o p e r ty d a ta . The o p e r a t in g o r d e s ig n

c o n d it io n s c o n s i s t o f fe e d and p ro d u c t r a t e s , c o m p o s itio n s

and te m p e ra tu re s a s w e ll a s th e r e f l u x r a t i o and colum n

p r e s s u r e . The p h y s ic a l p r o p e r ty d a ta p ro v id e s in fo rm a tio n

a b o u t th e i n d iv i d u a l com ponents and t h e i r i n t e r a c t i o n s .

The p ro ce d u re u se d to d ev e lo p th e s te a d y s t a t e p r o f i l e s

^These p r o f i l e s a r e th e te m p e ra tu re , c o m p o s it io n s , and l iq u id /v a p o r flo w r a t e s f o r each e q u i l ib r iu m s ta g e .

117


118

in v o lv e s a t r i a l and e r r o r m atching o f th e g iv e n column

c o n d i t io n s w ith th o se c a l c u l a t e d by th e m odel. The equ i

l ib r iu m s ta g e s o f th e m o d e l , feed and p ro d u c t s t r e a m lo ca

t i o n s a r e a d ju s te d u n t i l t h e d a ta c o n d it io n s a r e m atched .

The above p ro ced u re was u s e d by Wu [23] to d e v e lo p th e

s te a d y s t a t e p r o f i l e s u s e d in t h i s r e s e a r c h .

D e te rm in a tio n o f Dynamic P a ra m e te rs

The dynamic p a ra m e te r s can be d iv id e d i n t o tw o groups;

1) th o s e a s s o c ia te d w i th t h e r e b o i l e r and c o n d e n s e r and

2) th o s e a s s o c ia te d w i th t h e t r a y s . The p a ra m e te r s a s s o c i

a te d w ith th e r e b o i l e r a n d condenser in c lu d e a c c u m u la tio n

h o ld u p s and l iq u id l e v e l c o n t r o l tim e c o n s ta n t s . The

a c c u m u la to r ho ldups can b e o b ta in e d from a re v ie w o f the

d e s ig n r e p o r t and d i s c u s s i o n w ith th e u n i t o p e r a t o r s . The

t im e c o n s ta n ts shou ld b e o b ta in e d from p la n t t e s t i n g o f th e

c o n t r o l lo o p . W hile th e colum n i s in th e d e s ig n s t a g e , th e

l i q u i d l e v e l tim e c o n s t a n t can be e s t im a te d u s in g th e

e q u a t io n o f T abel 6 -1 . T he t r a y p a ra m e te rs a r e d e r iv e d

from th e d e ta i l e d t r a y d e s i g n r e p o r t . Appendix G c o n ta in s

th e e q u a t io n s fo r o b t a i n i n g 1 /g ^ , and m o la r ho ldup .

The m e ta l mass p e r s ta g e c a n be used in th e d y nam ic en tha lpy

b a la n c e m odel. I t i s o b t a in e d by d iv id in g th e m e ta l mass

p e r t r a y s e c t io n by th e f r a c t i o n a l e f f i c ie n c y f o r t h a t

s e c t i o n o f th e column.


119

TABLE 6-1

E s tim a tio n o f L iq u id L evel C o n tro l L oop Gain

K = g a in o f c o n t r o l l e r / s p a n o f l e v e l t r a n s m i t t e r x

maximum flow r a t e

A v a lv e w ith l i n e a r l y i n s t a l l e d c h a r a c t e r i s t i c s i s assum ed


120

S e n s i t i v i t y A n a ly s is

A t t h i s p o in t a l l th e in fo rm a tio n fo r th e dynam ic

model h a s been c o l l e c t e d . T h is in fo rm a tio n r e p r e s e n t s

th e n o rm al o p e ra t in g p a ra m e te rs o f th e column, i . e . most

l i k e l y v a lu e s . S e v e ra l ru n s o f th e dynam ic model shou ld

be made v a ry in g th e p a ra m e te rs a b o u t t h i s norm. The

m oles in th e r e b o i l e r and c o n d en se r and on the s t a g e s

shou ld be t e s t e d a t +30'^ o f t h e i r n o rm a l v a lu e . The

r e b o i l e r and co n d en se r l iq u id l e v e l t im e c o n s ta n ts due

to th e c o n t r o l l e r g a in can ta k e on much w ider v a lu e s so

th e y can be v a r ie d by a f a c t o r o f f i v e . The h y d r a u l ic

tim e c o n s t a n t and a e r a t io n f a c t o r s h o u ld be t e s t e d f o r

v a lu e s o f z e ro a s w e ll a s +30# o f n o rm a l o p e ra t io n .

In a d d i t i o n t o t e s t i n g p a ra m e te r s e n s i t i v i t y , th e

s e n s i t i v i t y t o s te a d y s t a t e o p e ra t in g c o n d it io n s sh o u ld

be c h e c k e d . T h is w i l l c o n s i s t o f v a ry in g feed c o n d i t io n s ,

L/D r a t i o and p ro d u c t r a t e s . T hese r a t e s should encompass

th e e x tre m e s o f norm al o p e ra t io n s . C are must be ta k e n to

en su re t h a t th e c o r r e c t s ta g e e f f i c i e n c i e s a re u se d when

th e i n t e r n a l l i q u i d and vapo r r a t e s a r e changed. S in ce th e

h y d ra u l ic tim e c o n s ta n t and a e r a t i o n f a c t o r depend on

th e i n t e r n a l l i q u i d and v ap o r r a t e , i t m ight be

n e c e s s a ry to r e c a l c u l a t e them fo r th e d i f f e r e n t o p e ra t in g

c o n d i t io n s .

A lso a s p a r t o f th e s e n s i t i v i t y a n a ly s i s d i f f e r e n t

c o n t r o l o p t io n s sh o u ld be t r i e d . T h is c o n s is ts o f t r y in g


121

re a s o n a b le c o m b in a tio n s o f m easured and m a n ip u la te d

v a r i a b l e s . The t e s t s o f d i f f e r e n t c o n t r o l system s sh o u ld

b e gu ided by th e o v e r a l l s e p a r a t io n o b j e c t i v e s of th e

colum n. M easurem ent la g s and dead tim e s h o u ld be in c lu d e d

i f s i g n i f i c a n t . I t i s recommended t h a t v a p o r phase

co m p o s itio n s b e m easu red s in c e t h i s i s w h a t a gas

ch rom atograph w i l l r e c o r d .

I n t e r p r e t a t i o n o f S e n s i t i v i t y T e s t

The o b j e c t i v e o f th e i n t e r p r e t a t i o n o f th e s e n s i t i v i t y

t e s t i s t o i d e n t i f y t r e n d s and e x p la in t h e i r m eaning. The

t r e n d s a re i d e n t i f i e d by o b se rv in g th e B ode and p o la r

p l o t s as th e d i f f e r e n t p a ram ete rs a re c h a n g e d . Changes in

th e freq u e n cy a t w h ich th e Bode p l o t c h a n g e s s lo p e a re

o b se rv ed to e s t a b l i s h th e t r e n d s , i . e , how th e break p o in ts

change w ith p a ra m e te r v a r i a t i o n s . A n o th e r key c h a r a c te r i s

t i c o f th e Bode p l o t i s th e s lo p e o f th e c u rv e between

b re a k p o i n t s . The r e c ip r o c a l o f th e f r e q u e n c y a t th e

b re a k p o in t i s a p ro c e s s tim e c o n s ta n t . T hus th e t r e n d in

th e movement o f t h e b re a k p o in ts i n d i c a t e s how p ro cess

t im e c o n s ta n ts v a r y . Changes i n th e s lo p e o f th e Bode

m agnitude p l o t i n d i c a t e changes i n th e o r d e r o f the

d i f f e r e n t i a l e q u a t io n s which govern th e p r o c e s s . For

exam ple, a change fro m a n e g a tiv e s lo p e t o a p o s i t iv e

s lo p e would i n d i c a t e t h a t th e p ro c e s s l a g s have been


122

d o m in a te d by p ro c e ss le a d s o v e r th e same freq u en cy r a n g e .

The b a se case fo r th e e t h y l e n e column c o n s i s t s o f

th e p a ra m e te r s o f Appendix G a n d th e two c o n tr o l a l t e r

n a t i v e s o f F ig u re s 6 .1 and 6 .2 . These c o n tr o l s t r a t e g i e s

w ere c h o se n f o r s tu d y b ecau se t h e y r e p re s e n t th e m o s t

r e a l i s t i c c o n t r o l a l t e r n a t i v e s . F ig u re s 6 .3 and 6 .4

c o n ta in t h e base ca se p lo ts f o r fe e d d is tu rb a n c e s . T hese

p l o t s a r e a lw ays i d e n t i c a l f o r b o th c o n tro l c o n f ig u r a t io n s

s in c e t h e l i q u i d le v e l c o n t r o l c o n f ig u r a t io n s a re t h e same

f o r b o th c o n t r o l a l t e r n a t i v e s . T h e re fo re , o n ly one s e t o f

fe e d d i s tu r b a n c e p lo t s w i l l b e p r e s e n te d . F ig u re 6 .5 and

6 .6 c o n ta in th e base case p l o t s f o r th e m an ip u la ted

v a r i a b l e s . The resp o n se f o r b o t h c o n tro l a l t e r n a t i v e s

h as b e en superim posed on one p l o t . The re sp o n se i n d i

c a te d by c h a r a c te r 2 i s common t o bo th c o n tr o l a l t e r n a

t i v e s . C h a ra c te r 1 r e p r e s e n ts t h e com position r e s p o n s e

t o a c h an g e in th e s id e s tre a m w h i le c h a r a c te r 3 r e p r e s e n t s

th e c o m p o s it io n resp o n se to a c h an g e in th e vapo r d i s t i l

l a t e . O c c a s io n a l ly , th e c h a r a c t e r 3 w i l l be p r i n t e d o v e r

c h a r a c t e r 1 . T h e re fo re , th e t r a n s f e r fu n c t io n s w h ic h a r e

l a b e le d and Pgg a re a c t u a l l y P^^ and Pg^, r e s p e c t i v e l y ,

f o r th e v a p o r d i s t i l l a t e c o n t r o l a l t e r n a t i v e . But f o r

c o n v e n ie n c e th e y w i l l be r e f e r r e d to a s P^g and P g^ . T h is

i s th e o n ly c o n fu s io n t h a t a r i s e s from su p e rim p o sin g p l o t s .

F ig u r e s 6 .7 and 6 .8 c o n ta in t h e R i jn s d o r p /B r is to l i n t e r a c t i o n


Figure 6,1

Ethylene Composition Used

to Manipulate D is t i l la t e Vapor

123

Lie

CC

L i e


124

Pigure 6 .2

Ethylene Composition Used

to Manipulate Sidestream l iq u id


125

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ETHVLENE TOWER T C F |« O C P S IA l« F I.O H C M O L E S /M IN I S .S # METHOD EQUIM O LAL OVERFLOW RASE CASE

A NEPRESEN1S THE TRANSFER FU N C TIO N P I 2 A I 5 REPRESENTS THE TRANSFER FU N C TIO N P f 2 51

THERE IS NO IN P U T/O U T P U T R E L A T IO N S H IP P I 2 6 }THERE IS NO IN P U T /O U T P U T R E L A T IO N S H IP P I 2 71

R REPRESENTS THE TRANSFER FU N C TIO N P I 2 n l

llODF MAGNITUDE PLOTS FOR F F F D P IS T U P R A N C F S

WHICH RELATES FEED RATE WHICH RELATES FEED Q U AIL TV WHICH RELATES FEED METHANE WHICH R FLA TFS FEED E TH V l FNE WHICH RELATES FEED ETHANE

FEED STAGE = -T.T TO ETHANE VAPOR TO ETHANE TO FTMANF TO F THANE TO FTHANT

THERE IS NO IN P U T/O U TP U T R E L A T IO N S H IP P I 2 91 WHICH RELATES FEED PROPYLFN TO ETHANE

VAPORVAPORVAPO?VAPORVAPOR

I H AD/UN IT T IM F 1 IL O C -L O G l C O M PO SITIO N ON STAGE 4 8 C O M PO SITIO N ON STAGC 4R

48 4 8 4 8 4 8

C O M PO SITIO N ON STAGE C O M PO SITIO N ON STAGE CO M PO SITIO N ON STAGE C O M PO SITIO N ON STAGE0.0

3 .2 0 0 E - 0 I I

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5 . I2 0 E 4 0 0

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- 5 .7 6 0 C 4 0 0

6 .0 B 0 E 4 0 0

6 .4 Û 0 E 4 0 0

- 6 .7 2 0 E 4 0 0

7 .0 4 O H 4 0 0

7 .3 6 0 E 4 0 0

7 .6 6 0 E 4 0 0

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131

in d e x . Since t h e s c a l e s on th e s e p l o t s a r e ex trem ely

d i f f e r e n t , th e y w i l l n o t be superim posed .

Comparison o f t h e c o n tr o l a l t e r n a t i v e s in d ic a te s

d i f f e r e n c e s i n o n ly one t r a n s f e r f u n c t i o n . b re a k s

down w ith a g r e a t e r s lo p e th an T h is i n d ic a t e s

t h a t th e i n t e r a c t i o n s w i l l be more s e v e r e i f th e s id e

s tre a m i s used t o c o n t r o l com position t h a n i f th e vapor

d i s t i l l a t e i s u s e d t o c o n tr o l c o m p o s itio n . T h is con

c lu s io n i s r e i n f o r c e d when th e i n t e r a c t i o n p l o t s a re

o b se rv e d . At low f r e q u e n c ie s th e i n t e r a c t i o n p lo ts

a r e o f s im ila r m a g n itu d e and d i r e c t io n o f r o t a t i o n .

Up to frequency P (10** ( -1 .2 ) rad /m in) t h e i n t e r a c t io n s

a r e th e same. A f t e r t h a t freq u en cy , t h e i n t e r a c t i o n s

o f th e s id e s tre a m c o n t r o l jump to v a lu e s o u t s i d e o f

th e u n i t c i r c l e . T h e r e fo r e , fo r low f r e q u e n c y d i s t u r b

a n ce s th e i n t e r a c t i o n s and b eh av io r o f b o t h c o n tro l

a l t e r n a t iv e s a r e t h e sam e. At h igh f r e q u e n c i e s th e

s id e s tre a m c o n t r o l becom es h ig h ly i n t e r a c t i v e due to

th e r a p id f a l l o f f o f P ^^ . P ^ f a l l s o f f f a s t e r th a n

Pj 2 because a c h a n g e i n l iq u id r a t e m ust p ro p a g a te down

th e column 43 s t a g e s t o th e r e b o i l e r and t h e n back up

w h ile a change i n v a p o r d i s t i l l a t e d i r e c t l y in f lu e n c e s

th e l iq u id flow t o s t a g e f i v e , i . e . th e l i q u i d change

need on ly t r a v e l f i v e s ta g e s down from t h e c o n d en se r .

Thus th e re i s l e s s l a g .


132

The f i n a l o b se rv a tio n a b o u t th e b ase case i s th e

shape o f These c u rv e s in d ic a te a p o s i t iv e

s lo p e f o r m id -ra n g e f r e q u e n c ie s . A f te r in v e s t i g a t in g two

o th e r co lum ns and d i f f e r e n t c o n t r o l a l t e r n a t i v e s , i t was

found t h a t t h i s re sp o n se i s c h a r a c t e r i s t i c o f th e way in

which th e l i q u i d le v e l c o n t r o l l e r s a re a rra n g e d . In

a d d i t io n shows a n e g a t iv e and th e n a p o s i t iv e s lo p e .

The p o la r p l o t s o f Pgg in d ic a te i t i s a fo u r th o rd e r

t r a n s f e r f u n c t i o n w ith num era to r o r le a d term . P^g# on

th e o th e r h a n d , shows no n u m era to r o r le a d term on th e

p o la r p l o t t o acco u n t f o r i t s bump. P^^g shows th e

c h a r a c t e r i s t i c s o f a n in th or t e n t h o rd e r p ro c e s s . As th e

a n a ly s i s p r o g r e s s e s t h i s c h a r a c t e r i s t i c w i l l be expanded.

The c o m p o s itio n re sp o n se i s c o m p lic a te d by h y d ra u l ic

tim e c o n s ta n t and a e r a t io n f a c t o r . T h e re fo re , th e p l o t s

o f F ig u re s 6 . 3 to 6.8 a re now r e p e a te d in F ig u re s 6 .9 to

6.14 w ith l /@ 2 " @2 /^ 1 ” T hese p l o t s in d ic a te how th e

column r e a c t s i f th e re a re no l i q u i d flow l a g s , i . e .

a l l l iq u id c h a n g e s a re made s im u lta n e o u s ly on e v e ry t r a y .

Comparing F ig u r e s 6.9 and 6.3 show s l i t t l e change i n th e

t r a n s f e r f u n c t i o n . T h is i n d ic a t e s t h a t feed d is t r u b a n c e s

to s ta g e f i v e a r e n o t p ro p ag a ted by th e l iq u id flow and

th u s a re n o t in f lu e n c e d by l /g ^ and g g /P i" T h is i s

s u b s ta n t i a t e d when l /p ^ and in d iv id u a l ly s e t to

z e ro . I t s h o u ld be remembered t h a t th e feed i s a l l v a p o r .


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E'THTLBHB TOBEB T ( P | , P ( P S » ) , PLOK (n O L B S /H t R) S . S . flETHOD EQDIHOLRL OTBRPLflUf t P O R T I R E C0N5TRHT = 0 . 0 ; UTRDkULlC T I R E CONSTANT => 0 . 0 PEED STARE ^ 31# REPRESENTS THE TRAHSPER FUNCTION P ( 1 N) HRICn RELATES PEED RATE TO ETHTIENE TAPOR5 REPRESENTS TUB TRANSFER FUNCTION P ( 1 5) WHICH RELATES FEED O H A ILT I TO ETHTLENE TAPOR

THERE I S NO IN P O T /3 O T P U T R EL AT IO N SH IP P ( 1 6 ) WHICH RELATES PEED METHANE TO ETHTLENE TAPOR7 REPRESENTS THE TRANSFER FONCTION P ( 1 7 ) WHICH RELATES PEED ETHTLENE TO ETHTLENE TAPOR

THERE I S NO IN PU T /0O T PO T R ELATIONSHIP P ( 1 8 ) WHICH RELATES FEED ETHANE TO ETHTLENE TAPOR

BODE MACNITODE PLOTS FOR FEED DISTORBANCES (B A D /U N IT TIME) (LOG-LOO)

COMPOSITION ON STAGE COMPOSITION ON STAGE COMPOSITION ON STAGE COMPOSITION ON STAGE COMPOSITION ON STAGE

THERE I S NO I N P O T /3 OTPOT R ELATIONSHIP P ( 1 9 ) WHICH RELATES FEED PROPTLEN TO ETHTLENE TAPOR 0.0 COMPOSITION ON STA

- 3 . 2 0 0 E - 0 I

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- 2 . 2 « 0 E * 0 0

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- 5 . 4 4 0 8 * 0 0

- 5 . 7 6 0 8 * 0 0

- 6 . 0 8 0 8 * 0 0

- 6 . 4 0 0 8 * 0 0

- 6 . 7 2 0 8 * 0 0

- 7 . 3 6 0 B * 0 0

- 7 . 6 8 0 8 * 0 0

fi.00E*00 -5 .2 0 8 * 0 0 -4 .4 0 8 * 0 0 -3 .6 0 8 * 0 0 -2 .8 0 8 * 0 0 -2 .0 0 8 * 0 0 -1 .2 0 8 * 0 0 -4 .00E -O J 4 .00 8 -0 1 1.208*00 2.008*00

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BTnTLCIIB TOaRR T ( F ) . P ( P S 1 » ) .PLO-I { n O L B S /n lH I s . t k P O B T i n s CONSTANT = 0 . 0 ; HTRDAULICN REPRESENTS THE TRANSFER FONCTION P ( 2 A) 5 REPRESENTS THE TRANSFER FONCTION P ( 2 S)

THESE I S NO THPÜT/OOTPUT RELATIONSHIP P ( 2 6 ) THESE I S NO IH P U T /0O T P U T R ELATIONSHIP P ( 2 7 )

8 BBPRESENTS THE TRANSFER FUNCTION P ( 2 H) THERE I S NO INPUT/OUTPUT R EL AT IO N SH IP P ( 2 9 )

s . NET HOD EQUINOLAL OWE RFI ON TIME CONSTANT 0 . 0 WHICH RELATES FEED RATE TOWHICH RELATES FEED QOAILTT TOWHICH RELATES FEED HETHANE TOWHICH RELATES FEED ETHTLENE TOWHICH RELATES FEED ETHANE TO ETHANEWHICH RELATES FEED PROPTLEN TO ETHANE

BODE HACNITUDE PLOTS FOR FEED DISTORBANCESFEED STAGE

ETHANE ETHANE ETHANE ETHANE

3 3 (RAD/O NIT T i n s I (LOR-LOG)VAPOR COMPOSITION ON STAGE A8 VAPOR CONPOSITION OH STAGE VAPOR COMPOSITION ON STAGE VAPOR COMPOSITION ON STAGE VAPOR COMPOSITION ON STAGE TAPOR COMPOSITION ON STAGE

AOASA8«81 8

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- 1 . 6 0 0 B » 0 0

- 1 . 9 2 0 E * 0 0

- 2 . 2 9 0 8 * 0 0

- 2 . 5 6 0 E * 0 0

- 2 . 8 8 0 R » 0 0

- 3 . 5 2 0 E » 0 0

- 3 . 8 4 0 E » 0 0

- A . i e 0 E * 0 0

-A .A O O E 'O O

- 4 . 8 0 0 K * 0 0

- 5 . 1 2 0 E » 0 0

- 5 . A A 0 E » 0 0

- 5 . 7 6 0 E » 0 0

- 7 . 0 A 0 B * 0 0

- 7 . If iORtOO

- 7 . 6 8 0 E « 0 0

-B .O O O E *O 0- e . 0 O E * 0 0 - 9 . 2 0 E + 0 0 - A . A 0 E * 0 0 - 3 . 6 0 E * 0 0 - 2 . 8 0 E * 0 0 - 2 . 0 0 E » 0 0 - I . 2 0 E » 0 0 - A . O O E - 0 1 A .O O E -0 1 1 . 3 0 E * 0 0 2 . 0 0 E * 0 0

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BTIlILBIie TOHBR T (r t . P ( P S I * | ,FLnR(nOLGS/HIN) S. r t P O R T I N S CONSTANT - 0 . 0 ; HTRDAOLICI RBFRESENTS THE TRANSFER FUNCTION P ( 1 1)

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S . NETHOD BQUinOLAL O T B R F L O BODB NAGNITUOE PLOTS FOR NANIPIILATEP TARIABLESTIME CONSTANT = 0 . 0 MAGNITUDE VS FREQUENCT (RAO/O NIT TIM E ) (LOG-LOG)WHICH BELATES S . S . L IQ U ID FLOW TO ETHTLENE TAPOR COMPOSITION ON STAGE 5

5 5

2 REPRESENTS TUB TRANSFER FUNCTION P f 1 2 ) WHICH RELATES BOTTOMS TAPOR3 REPRESENTS THE TRANSFER FUNCTION P ( 1 3) WHICH RELATES D IS T IL L A T E

TO ETHTLENE TAPOR COMPOSITION ON STAGE APOR TO ETHTLENE TAPOR COMPOSITION ON STAGE

0.0

- 3 . 2 0 0 E - 0 1

- 6 . T 0 0 E - 0 1

- 9 . 6 0 0 E - 0 1

1 . 2 8 0 E * 0 0

- 3 . 2 0 0 E * 0 0

- 3 . 5 2 Q E » 0 0

- 3 . 8 < » 0 E * 0 0

- W . 1 6 0 E * 0 0 Ov

- N . N 8 0 E * 0 0

- W . 8 0 0 E * 0 0

- 5 . i m 0 E * 0 0

- 6 . 7 2 0 8 * 0 0

- 7 . 0 « 0 e * 0 0

- 7 . 3 6 0 E * 0 0

- 7 . 6 8 0 E * 0 0

- S . 0 0 0 E * 0 0- 6 . 0 0 E * 0 0 - 5 . 2 0 E » 0 0 -N .W O E * 0 0 - 3 . 6 0 E * 0 0 - 2 . 8 0 E * 0 0 - 2 . 0 0 E * 0 0 - l . 2 0 E * 0 0 - « . O O E - 0 1 A-.OOE-Ol 1 . 2 0 E * 0 0 2 . 0 0 E * 0 0

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ETHYLENE TOHEB T ( F | . P (P S 1 K | , FLOB V»POF TIME CONSTANT = 0 . 0 1 REPRESENTS THE TRANSFER PO

- 3 . 2 0 0 E - 0 I

M OLES/NI N) S .: HTROAIILIC

C I ION P ( 2 1)

S . HBTHOO EQUINOLAL OVERFLOB RODE HACNITUDE PLOTS FOR MANIPULATED VARTARLESTIM E CONSTANT = 0 - 0 MACNITODE VS FREQUENCT (R AD/ÜNIT TIME) (I .OC-LOC)BHICH RELATES S . S . L IQ U ID FLOU TO ETHANE VAPOR COMPOSITION ON ST ACE « 0BHICn RELATES BOTTOMS VAPOR TO ETHANE VAPOR COMPOSITION ON STAKE 1 0

COMPOSITION I 1 -------- 1 -

REPRESENTS THE TRANSFER FUNCTION P IREPRESENTS THE TRANSFER CTION P I 2

I 1BHICU RELATES D IS T IL L A T E I 1-------- 1-------- 1---------1

APOR TO ETHANE I 1 -------- 1

l.200B»00l . 6 0 0 E * 0 0

I 1 ------------- 1 -------------- 1 --------------1 -------------- 1 -------------- 1

♦ 0 0 - 2 . 0 0 E f O O - 1 . 2 0 E 4 0 0 - A . O O E - 0 1 « . 0 0I 1 ------------- 1

- 0 1 1 .2 0 F .fO O 2 . 0 0

BTAOE « 0

- 6 . « O O E - 0 1

- 9 . 6 0 0 E - 0 1

- 1 . 0 2 0E4 0 0

- 2 . 2 « O E f O O

- 2 . 5 6 0 E ^ 0 0

- 2 . 8 B O E f O O

-3. 200Ef00-3 .520E100

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139

Comparing F ig u re s 6.10 and 6.4 i s a d i f f e r e n t

m a t t e r . The o n ly t r a n s f e r fu n c tio n to re m a in th e same

i s t h e one r e l a t i n g e th a n e fee d c o m p o s itio n to th e

e th a n e c o m p o s itio n on s ta g e 48. and Pg^ show th e

same low fre q u e n c y b re a k p o in ts in b o th c a s e s , b u t the

h ig h e r f r e q u e n c ie s a re d i s t i n c t l y d i f f e r e n t .

I n c o r p o r a t in g th e h y d ra u l ic and vapor tim e c o n s ta n ts in to

th e c a l c u l a t i o n c a u se s th e s e t r a n s f e r f u n c t io n s to f a l l

o f f q u ic k e r , th u s ad d in g more la g t o th e sy s te m . S e t t in g

th e v a p o r tim e c o n s ta n t t o z e ro w h ile l e a v in g th e

h y d r a u l ic tim e c o n s ta n t a t i t s norm al v a lu e in d ic a te s

t h a t th e a d d i t i o n a l la g i s accoun ted f o r by th e h y d ra u lic

tim e c o n s ta n t .

Figures 6 .11 and 6.12 indicate the same s itu a tio n

which caused in tera ctio n in the base ca se . P^ r o l ls

o ff fa s te r than thus in d icating th a t composition

co n tro l with sidestream flow is more in te r a c tiv e at

high frequencies than composition co n tro l with d i s t i l la t e

vapor. Comparing the p lo ts of K (Figure 6 .14 and 6.13 )

at frequencies above R (10**(-.4 ) rad/min) confirms the

above premise. In addition these p lo ts in d ica te an

alm ost v e r tic a l l in e from frequencies X and beyond. I t

i s in t ic a t iv e of numerator dominace. This in teraction

e f f e c t can again be accounted for by 1/# s in c e , when

i t i s returned to i t s normal value and @g/g^ se t to zero,

the base case in tera ctio n p lo ts are duplicated for both

con tro l a lte r n a tiv e s . A comparison of Figures 6.11


140

to 6.5 and Figures 6.12 to 6.6 shows the high frequency

influence o f l/p ^ and

One s ig n if ic a n t low frequency e f f e c t i s seen in

I t no lon ger has a negative slope a t 10**(-2) rad/min.

The polar p lo t o f confirms the change in the equations

governing I t now has a lead term and the order o f

the process has been reduced from ten th to fourth. Addi

t io n a l s tu d ie s show that the bump i s caused by the in te r

action o f th e Bg/Pi 1/^2' The la c k o f high frequency

r o l l o f f in Pg^ and i s caused by se tt in g l/g^ to zero.

Again Pg and Pg are id en tica l showing that the d i s t i l l a t e

vapor flow h as the same influence on the ethane o f stage 48

as does the sidestream liquid flow . Comparing polar p lo ts

ind icates th a t the other manipulated variab le transfer

functions have had th e ir phase la g reduced by 90 degrees

or one order.

A nalysis o f l/g ^ and shows that their in fluence

generally appears at high freq u en cies. Additional s tu d ie s

have shown th a t ± 30^ changes in th e se parameters do not

cause major s h i f t s in the transfer fu n ctio n s. The in flu en ce

o f 1/^2 cannot be ignored in the a n a ly s is o f high frequency

in te r a c tio n s . le s s s ig n if ic a n t parameter o f

the two. These observations should not be taken as a

general gu ide to a l l columns but o n ly for th is ethylene

column.


I4l

T h e re a re two p a r t s t o th e in v e s t ig a t io n o f t h e h o ld

up; t h e ho ldup on th e s t a g e s and th e holdup o f th e ac

c u m u la to r s . T h is d i s t i n c t i o n i s made because v a r i a t i o n in

s ta g e h o ld u p i s sm a ll com pared to t h a t o f the a c c u m u la to r s .

S ta g e h o ld u p s were i n v e s t i g a t e d a t o f n o rm a l. The

h o ld u p o f th e accu m u la to rs was in v e s t ig a te d a t i t s

n o rm a l v a lu e (which i s v e ry s m a ll ) and w ith 24 h o u r s o f

h o ld u p .

The column dynamics a r e i n s e n s i t i v e to v a r i a t i o n s in th e

s te a d y s t a t e s ta g e ho ldup . L i t t l e change is found due to

a 6 0 % o v e r a l l change in s t a g e h o ld u p . Most im p o r ta n t ly ,

th e s lo p e s o f th e p lo ts show no change . Thus th e u n d e r ly in g

e q u a t io n s which govern th e r e s p o n s e have no t c h a n g e d . T h is

m a g n itu d e o f change has n o t c au se d any b reak p o i n t s to

ch an g e s i g n i f i c a n t l y . Even th e i n t e r a c t io n p l o t s a r e

re m a rk a b ly s im i la r . These o b s e r v a t io n s in d ic a te t h a t th e

v a r i a t i o n in s te a d y s t a t e s t a g e ho ldup has l i t t l e in f lu e n c e

on co lum n dynam ics.

The changes in column dynam ics caused by v a r i a t i o n s in

s te a d y s t a t e accum ulato r h o ld u p can be s i g n i f i c a n t . T h is

i s b e c a u s e th e r e a re no s t a n d a r d s f o r th e s iz in g o f accumu

l a t o r s . The e th y le n e colum n n o rm a lly o p e ra te s w i th 0 .25

m in u te s o f ho ldup in th e c o n d e n s e r and th re e m in u te s in th e

r e b o i l e r based on th e r e f l u x flo w and bottom s l i q u i d flow ,

r e s p e c t i v e l y . For an i n d u s t r i a l column o f t h i s s i z e and

im p o r ta n c e , th e s e holdups a r e v e ry s m a ll . A more t y p i c a l holdup


142

tim e would be i n t h e ra n g e o f s i x to tw e lv e h o u rs ; h o ld u p

as h ig h as 48 h o u rs h a s b e e n o b se rv e d . T h e re fo re , 24

h ou rs w o rth o f h o ld u p was ch o sen a s th e t e s t v a lu e on th e

h ig h s i d e . For t h e r e b o i l e r 24 h o u rs w o rth o f ho ldup i s

12,000 m oles o r an i n c r e a s e in s i z e by 480 tim e s . F o r

th e c o n d en se r 24 h o u rs w o rth o f h o ld u p i s 220,000 m oles

o r an in c r e a s e i n s i z e by 5 ,500 t im e s . The p lo ts o f

F ig u re s 6 .15 to 6 .2 0 r e p r e s e n t th e colum n dynam ics w ith a

la r g e co ndenser and b a se c a s e r e b o i l e r w h ile F ig u re s 6 .2 1

to 6 .2 6 a re fo r a b a s e c a s e c o n d e n se r and a la rg e r e

b o i l e r .

In c re a s in g t h e c o n d e n se r h o ld u p o n ly in f lu e n c e s th e

e th y le n e co m p o s itio n on s t a g e f i v e . The t r a n s f e r fu n c

t io n s f o r th e e th a n e o f s t a g e 48 a r e n o t a f f e c te d . The

feed d is tu rb a n c e t r a n s f e r f u n c t io n s f o r e th y le n e r e t a i n

th e same shape a s t h e b a se c a s e a t h ig h f re q u e n c ie s b u t a t

low f re q u e n c ie s d ro p s l i g h t l y and th e n l e v e l o u t.

i n d ic a t e th e same low fre q u e n c y c h a r a c t e r i s t i c , w h ile

Pg2 r i s e s s l i g h t l y a t low f r e q u e n c ie s . A t h ig h f r e q u e n c ie s

th e s lo p e o f P^g i s i n i t i a l l y th e same a s P^^ b u t th e s e

d e v ia te from each o th e r m ore a s th e f r e q u e n c ie s i n c r e a s e .

T h e re fo re , th e i n t e r a c t i o n o f b o th c o n t r o l a l t e r n a t i v e s

w i l l be th e same f o r a w id e r ran g e o f f r e q u e n c ie s .

Comparing i n t e r a c t i o n p l o t s c o n firm s t h i s b u t shows t h a t

th e i n i t i a l shape o f th e p o l a r p l o t s h a s changed. Now

in s te a d o f i n i t i a l l y s p i r a l i n g inw ard i t lo o p s o u tw ard .


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ETHTLBMB TOMER T |F ) . P ( P S I i ) , PLÜM (nOLES/K T H) I.RRCB COHDERSEB HOLDUP

« REPRESENTS THE TRANSFER PONCTION P ( 1 «1 S REPRESENTS THE TRANSFER FUNCTION P ( 1 S)

THERE I S NO IN PUT/OUTPUT R ELATIONSHIP P ( 1 6 ) 7 REPRFSEHTS THE TRANSFER FUNCTION P ( 1 7 )

THERE I S NO INPUT/OUTPUT R ELATIONSHIP P ( 1 R) THERE I S NO INPUT/O UTPUT R ELATIONSHIP P ( 1 9 )

S . S . METHOD EQUinOLAL OTERPLOH

WHICH RELATES FEED RATE WHICH RELATES FEED QUATLTT WHICH RELATES FEED METHANE WHICH RELATES PEED ETHILEHE WHICH RELATES FEED ETHANE WHICH RELATES FEED PROFILER

RODE MAGNITUDE PLOTS FOR FEED DISTURRANCES FEED STAGE = 13 (R A D /U N IT TIME) (LOG-LOG)

TO ETHILEHE VAPOR TO ETHYLENE VAPOR TO ETHYLENE VAPOR TO ETHYLENE VAPOR TO ETHYLENE VAPOR TO ETHYLENE VAPOR

COMPOSITION ON STAGE 5COMPOSITION ON STAGE 5COMPOSITION ON STAGE SCOMPOSITION ON STAGE 5COMPOSITION ON STAGE 5COMPOSITION ON STAGE 50.0

- 3 . 2 0 0 E - 0 1

- 9 . 6 0 0 B - 0 1

- I . 2 8 0 E + 0 0

- 1 . 6 0 0 B * 0 0

- 1 . 9 2 0 B » 0 0

- 2 . 2 N 0 E » 0 0

- 2 . S 6 0 E * 0 0

- 3 . 5 2 0 E * 0 0

- 3 . 8 9 0 E * 0 0

- 4 . If iOEtOO

- 4 . 4 8 0 B * 0 0 Vjl

- 4 . 8 0 0 E * 0 0

- 5 . 1 2 0 B » 0 0

- 5 . 4 4 0 B * 0 0

- 5 . 7 6 0 K » 0 0

- 6 . 0 B 0 B » 0 0

- 7 . 0 4 0 E + 0 0

- 7 . 360B»OQ

- 7 . 6 8 0 E » 0 0

- 8 . 0 0 0 E + 0 0- 6 . 0 0 8 * 0 0 - S . 2 0 B « 0 0 - 4 . 4 0 B * 0 0 - 3 . 6 0 K » 0 0 - 2 . R 0 R » 0 0 - 2 . 0 0 8 * 0 0 - 1 . 2 0 8 * 0 0 - 4 . 0 0 E - 0 1 4 . 0 0 B - 0 1 1 . 2 0 8 * 0 0 2 . 0 0 E * 0 0

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eTHTLBNE TOHEB T ( F | , P ( P S I A ) , PLOW (M O L E S/N I# ) LARGE CONDENSER HOLDOP

N REPRESENTS TtlB TRANSFER FUNCTION P (6 REPRESENTS THE TRANSFER FONCTION P (

THERE I S NO INPUT/O UTPUT R ELATIONSHIP P ( THERE I S NO INPUT/O UTPUT RELATIONSHIP P (

8 REPRESENTS THE TRANSFER FUNCTION P ( THERE I S NO INPUT/OUTPUT RELATIONSHIP P (

0.0I

- 3 . 2 0 0 E - 0 1 I I-6.N00B-01 X I

- 9 . 6 0 0 8 - 0

- 1 . 2 8 0 8 * 0 0

- 1 . 6 0 0 B * 0 0

- l . 9 2 0 E * 0 0

S . S . METHOD EQUINOLAL OFERFLOH

WHICH RELATES FEED RATE WHICH RELATES FEED QOAILT WHICH RELATES FEED NRTHAR WHICH RELATES FEED ETHTLB WHICH RELATES FEED ETHANE

BODE MAGNITUDE PLOTS FOR FEED DFEED STAGE = 33 (RAO/U NIT TIME) (LOG-

TO ETHANE TAPOR COHPOSITION ON STAGE 1 8TO ETHANE VAPOR COMPOSITION ON STAGE 1 8TO ETHANE VAPOR COMPOSITION ON STAGE 1 8

8 TO ETHANE VAPOR COMPOSITION ON STAGE 1 8TO ETHANE VAPOR COMPOSITION ON STAGE 1 8

N TO ETHANE VAPOR COMPOSITION ON STAGE 1 8

- 2 . 2 N 0 E + 0 0

- 2 . 5 6 0 8 * 0 0

- 2 . 8 8 0 8 * 0 0

- 3 . 2 0 0 8 * 0 0

- 3 . 5 2 0 8 * 0 0

- 3 . 8 1 0 8 * 0 0

- 1. 1 6 0 8 * 0 0

- 1 . 1 8 0 8 * 0 0

- 1 . 8 0 0 8 * 0 0

- 5 . 1 2 0 8 * 0 0

-S .1 1 0 E * 0 0

- 5 . 7 6 0 8 * 0 0

- 6 . 0 8 0 6 * 0 0

- 6 . 1 0 0 8 * 0 0

- 6 . 7 2 0 8 * 0 0

- 7 . 0 1 0 E * 0 0

- 7 . 3 6 0 8 * 0 0

- 7 . 6 8 0 8 * 0 0

- 8 . 0 0 0 8 * 0 0- 6 . 0 0 8 * 0 0 - 5 . 2 0 R * 0 0 - 1 . 1 0 E * 0 0 - 3 . 6 0 6 * 0 0

STURBANCTES

o \

k

I 1 --------- 1------- 1 --------1 ---------1 --------1---------1 - -------1 ---------1 H I 1 -- 2 . 8 0 8 * 0 0 - 2 . 0 0 8 * 0 0 - 1 . 2 0 8 * 0 0 - 1 . 0 0 8 - 0 1 1 . 0 0 E - 0 I 1 . 2 0 E » 0 0 | 3 . 0 0 6 * 0 0

1^5

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Figure 6.17

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ETHILRMB TOHRR T ( F ) , P { P S I I ) , r iO H (HOLRS/HIM) LARGE CnnOENSER HOLDUP

1 REPRESENTE THE TRANSFER FUNCTION P ( 22 REPRESENTE TUE TRANSFER FUNCTION P ( 2

S . S . NETHOD E Q U in O L A L OTRRFLOH BODE RA GNITU DR PLOTS FOR MANIPULATED VARIABLESMAGNITUDE VS FREOUENCV (RAD/U NIT TIME) (LOG-LOG)

1) HHICH RELATES S . S . L IQ U ID FLOW TO ETHANE VAPOR COMPOSITION ON STAGE « 82) WHICH RELATES BOTTOMS VAPOR TO ETHANE VAPOR COMPOSITION ON STAGE 4 8

J REPRESENTS THE TRANSFER FUNCTION P ( 2 3) WHICH.RELATES D IS T IL L A T E VAPOR TO ETHANE VAPOR COMPOSITION ON STAGE 4 80 .0

- 3 . 2 0 0 E - 0 1

- 6 . 4 0 0 E - 0 1

- 9 . 6 0 0 B - 0 1

- 1 . 2 8 0 K * Q 0

- 1 . 6 0 0 8 * 0 0

- 1 . 9 2 0 8 * 0 0

- 2 . 2 4 0 8 * 0 0

- 2 . B 8 0 B * 0 0

- 9 . 1 6 0 8 * 0 0

- 9 . 9 8 0 8 * 0 0 cx>

- 9 . 8 0 0 8 * 0 0

- 9 . 1 2 0 8 * 0 0

- 9 . 9 9 0 8 * 0 0

- 7 . 1 6 0 8 * 0 0

- 7 . 6 8 0 8 * 0 0

- 6 . 0 0 8 * 0 0 - 5 . 2 0 8 * 0 0 - 9 . 9 0 8 * 0 0 , - 3 . 6 0 8 * 0 0 - 2 . 8 0 8 * 0 0 - 2 . 0 0 8 * 0 0 - 1 . 2 0 8 * 0 0 - 9 . 0 0 8 - 0 1 9 . 0 0 8 - 0 1 1 . 2 0 8 * 0 0 2 . 0 0 8 * 0 0

14?

Figure 6.19

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148

Figura 6.20

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e O o O e e a e e e e e O o O o e e a a o o ae e e e o e o o o o e e e o e e e O o o e a a e e •

1 1 ♦u h i h i h a h i b ) h : M u X X X X X X X X X X X X X X X X 1o e e e e o o e e e o o o e e o o e o e a a a o oe o o « o <N » VO a o n # X W9 œ e «N 9 VO X e p g 9 VO CD o<N 9 <N \ 0 C ' fM in a N in e 9 o 9 pm o 9 pm a m VO o

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eriIT LBR B TORER T ( F ) , P <P SIR ) . PLOH (H O L E S /H l N} LARGE REBOILER HOLDUP

4 REPRESENTS TilE TBRRSPER FUNCTION P (5 REPRESENTS THE TRANSFER FURCTIOH P (

THERE I S NO IN PUT/O UTPUT REL AT IO N SH IP P ( THERE I S NO INPUT/O UTPUT RELATIONSHIP P (

a REPRESENTS TilE TRANSFER FUNCTION P( THERE IS NO INPUT/OUTPUT RELATIONSHIP P(

0 .0

-3 .200B-01

- 6 . 4 0 0 E - 0 1

- 9 . 6 0 0 E - 0 1

S . S . TETHOD E3UIN0LAL OTERFLOR BODE RAGRITUDE PLOTS FOR FEED DISTURBANCESFEED STAGE = 31

ETHANE VAPOR(R A D /U N IT T IH E ) (LOG-LOG)

COMPOSITION ON STAGE 4 0 COMPOSITION ON STAGE

- 1 . 2 8 0 E » 0 0

- 1 . 6 0 0 E » 0 0

- t . 9 2 0 E » 0 0

- 2 . 2 4 0 E * 0 0

- 2 . S 6 0 E » Q 0

-2.S80R*00

-3.200R»00

-3.520B»00

- 3 . 8 4 0E*0G

- 4 . 1 6 0 E * 0 0

- 4 . 4 8 0 E » 0 0

- 4 . 8 0 0 E » 0 0

- 5 . 1 2 0 E * 0 0

- S . 4 4 0 E » 0 0

- 5 . 7 6 0 E » 0 0

- 6 . 0 8 0 E » 0 0

-6.400EADO

-6.720B*00

- 7 . 0 4 0E»00

- 7 .3 6 0 B » 0 0

-7 .e B O E * 0 0

- 8 . 0 0 0 E * 0 0

COMPOSITION ON STA COMPOSITION ON STA COMPOSITION ON STAGE COMPOSITION ON STAGE

I 1-------1

WHICH RELATES FEED RATE WHICH RELATES FEED UUAILT WHICH RELATES PEED MET BAN WHICH RELATES FEED ETRTLE WHICH RELATES FEED ETHANE

ETHANE ETHANE ETHANE ETHANE ETHANE

I 1

VAPOR

8II1CH RELATES FEED PROPTLER TO I 1 Yc 1------ 1

VAPOR

\

4 8484 84848

Vj\o

- f i . O 0 E » O 0 - S . 2 0 E » 0 0 - 4 . 4 0 E » 0 0 - 3 . 6 0 E » 0 0I 1--------1 -------- 1-------- 1 -------- 1 - ------1 -------->1— • - ! -

- 2 . 8 0 E * 0 D - 2 . 0 0 E » 0 0 - 1 . 2 0 E » 0 0 - 4 . 0 8 E - 0 1 4 . 0 0 E - 0 1 1 . 2 0 E * 0 0 2 . 0 0 E * 0 0

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BTHTLBRB TOBEB T (rt . P ( P S I X ) .PLOW (H O LRS/HIH ) LKR3B BEBOILEB HOLnOP

1 BEPRBSENTS THE TBXN5PEB PONCTION P ( 12 REPBESBNTS TUB TBARSPBR PONCTION P ( 1

S . S . METHOD BQOIHOLAL OVBRPLOH BODE HARNITODE PLOTS FOB MANIPULATED VARIABLESMAGNITUDE VS FPEQOENCV (RAD/O NIT T IM E | (LOG-LOG)

1) WHICH RELATES S . S . L I Q U I D FLOW TO BTHTLERE VAPOR COMPOSITION ON STAGE S2 ) WHICH RELATES BOTTOMS VAPOR TO BTHTLENB VAPOR COMPOSITION ON STAGE 5

53 REPRESENTS THE TRANSFER FUNCTION P ( 1 3) WHICH RELATES D IS T IL L A T E VAPOR TO BTHTLENB VAPOR COMPOSITION ON STAGE0.0- 3 . 2 0 0 E - 0 1 I

- e . H O O E - O I I

9 . 6 0 0 R - 0 1 I

- 1 . 2 S 0 E » 0 0 I

- 1 . 6 0 0 E * 0 0 t

- 3 . 2 0 0 E » 0 0

- 3 . 5 2 0 E » 0 0

- 3 . 8 B 0 E * 0 0

- a . I 6 0 E » 0 0 0»

VJ

- S . 1 2 0 B » 0 0

- S . 4 a O B * 0 0

- 5 . 7 6 0 B » 0 0

- 6 . 0 8 0 B * 0 0

- 6 . 7 2 0 8 * 0 0

- 7 . 0 4 0 8 * 0 0

6 . 0 0 B * 0 0 - 5 . 2 0 E * 0 0 - 4 . 4 0 B * 0 0 - 3 . 6 0 E * 0 0 - 2 . B 0 E * 0 0 - 2 - 0 0 E * 0 0 - 1 . 2 0 F . * 0 0 - 4 . 0 0 F . - 0 ) 4 . 0 0 - 0 1 1 . 2 0 E * 0 0 2 . 0 0 E * 0 0

VAH

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T h is i n c r e a s e s th e i n t e r a c t i o n s a t lower f r e q u e n c ie s f o r

b o th c o n t r o l a l t e r n a t i v e s . By i n s t a l l i n g l a r g e accu m u la

t o r s m o s t d e s ig n e rs a r e hop ing t o avo id s e v e re i n t e r a c t i o n s .

The ab o v e o b s e rv a t io n i n d i c a t e s t h a t th e d e s ig n e rs ' p r e m is e

i s n o t c o r r e c t fo r t h i s colum n.

I n c r e a s in g th e r e b o i l e r h o ld u p m ainly in f lu e n c e s t h e

e th a n e c o m p o s itio n on s ta g e 48 . T h is p a ra m e te r v a r i a t i o n

c a u s e s t h e b rea k p o in t o f th e f e e d d is tu rb a n c e s to s h i f t

l e f t by h a l f o f a d ecad e w ith no s i g n i f i c a n t change i n

s lo p e . and have th e sam e shape a s th e base c a s e .

Pj 2 now h a s a new bump. F u r th e r i n v e s t i g a t i o n r e v e a l s

t h i s t o b e caused by 1 /8 ^ . The b r e a k p o in t o f and

s h i f t s t o th e l e f t one d e c a d e and m a in ta in s th e sam e

s lo p e a s i n th e b ase c a s e . The r i s e in ? 22 I s n o t as

g r e a t a s i n th e b ase c a s e th u s i n d i c a t i n g th e s i g n i f i c a n c e

o f th e l e a d term h as d e c re a s e d . R e b o ile r h o ld u p i s th e

f i r s t p a ra m e te r to s i g n i f i c a n t l y in f lu e n c e Pg2 a t low

f r e q u e n c i e s . A t y p i c a l tim e d o m ain re sp o n se o f 9 ^ 2 i s

shown i n F ig u re 6 .2 7 . The o v e r s h o o t would be le s s w i th a

l a r g e r e b o i l e r m ass. An im p u lse o f vapor w ould i n c r e a s e

th e e th a n e c o n c e n tr a t io n to a v a lu e g r e a te r th a n i t s f i n a l

s te a d y s t a t e . As th e v ap o r i s r e f l e c t e d b ack down th e

co lum n, t h e e thane c o n c e n t r a t io n d e c re a se s u n t i l i t

r e a c h e s s te a d y s t a t e . The Scime mechanism in f lu e n c e s P ^ 2

b u t i s confounded by v apo r and l i q u i d tim e c o n s ta n ts .

C om parison o f th e i n t e r a c t i o n p l o t s r e v e a ls no change .


156

F ig u re 6.27

Time Response o f Pgg

§4»#4m

Iotim e


157

Thus a l a r g e r e b o i l e r g r e a t ly in f lu e n c e s th e in d iv id u a l

t r a n s f e r f u n c t io n , b u t th e s e in f lu e n c e s a re c a n c e l le d

when th e i n t e r a c t i o n in d e x i s c a l c u l a t e d .

Combining a l a r g e r e b b i l e r and a l a r g e condenser

shows t h a t th e e f f e c t s f o r in d iv id u a l t r a n s f e r fu n c t io n s

a re a d d i t iv e . Thus t r a n s f e r f u n c t io n s r e l a t e d to

e th y le n e ta k e on th e c h a r a c t e r i s t i c s g iv en t o them by th e

c o n d en se r, w h ile th e e th a n e t r a n s f e r f u n c t io n s ta k e on

c h a r a c t e r i s t i c s g iv e n to them by t h e r e b o i l e r . The i n t e r

a c t io n in d ex assum es th e c h a r a c t e r i s t i c g iv e n i t by a

la r g e c o n d e n se r , i . e . th e s p i r a l i n g outw ard a t low

f r e q u e n c ie s . A t h ig h freq u e n cy i t r e v e r t s b a c k to i t s

b a se c a se b e h a v io r .

The l i q u i d l e v e l c o n t r o l tim e c o n s ta n ts a re th e

e a s i e s t p a ra m e te rs to m odify a f t e r c o n s t r u c t io n . They

depend on equ ipm en t s i z e a s w e ll a s c o n t r o l l e r g a in s .

Thus to a c e r t a i n d e g re e th e s e tim e c o n s ta n ts can be

tuned o r s e l e c t e d . The m agnitude o f th e s e c o n s ta n ts

d e te rm in es th e am ount o f l iq u id l e v e l and l i q u i d o u t l e t

s tream w i l l f l u c t u a t e . High g a in s f o r th e p r o p o r t io n a l

o n ly c o n t r o l l e r o r a s m a ll tim e c o n s ta n t a llo w l i t t l e

f lu c tu a t io n i n th e l e v e l and p a ss o n flow d is tu rb a n c e un

a tte n d e d . Low g a in s a l lo w th e l e v e l to r i s e and f a l l t o

abso rb flo w d i s t u r b a n c e s . Thus a low g a in s e t t i n g a llo w s

th e o u t l e t s tre a m o f th e a cc u m u la to r to rem ain r e l a t i v e l y

c o n s ta n t . The o b j e c t i v e o f a l e v e l c o n t r o l l e r fo r th e s e


158

a c c u m u la to rs sh o u ld b e to sm ooth o u t flow d i s t u r b a n c e s

w i th o u t l e t t i n g th e a c c u m u la to r ru n d ry o r o v e r f lo w . Thus

g a in s g r e a t e r th a n 1 .0 a r e r e q u i r e d . On th e h ig h s i d e a

g a in o f 1 0 .0 w i l l b e u se d to r e p r e s e n t t i g h t l e v e l c o n t r o l .

A lth o u g h th e l i q u i d l e v e l tim e c o n s ta n ts w e re exam ined

over a w id e range o f v a lu e s , t h e i r e f f e c t was n e g l i g i b l e .

The o n ly tre n d to b e e s t a b l i s h e d concerned and t h e

c o n d e n se r tim e c o n s ta n t . As t h e condenser tim e c o n s t a n t

i s i n c r e a s e d th e bump in 6aear 10**{-2.4} r a d /m in in

base c a s e ) d i s a p p e a r s . T h is t r a n s f e r fu n c t io n th e n

assum es a shape s i m i l a r to Pgg* S ince th e r e b o i l e r l e v e l

c o n t r o l l e r fo r th e s e c o n t r o l a l t e r n a t i v e s d o es n o t r e f l e c t

flow p e r t u r b a t i o n s b ack i n t o t h e column, th e r e b o i l e r tim e

c o n s t a n t d id no t in f lu e n c e t h e colum n.

M is c e lla n e o u s C a lc u la t io n s f o r a C on tro l System

H a v in g e s ta b l i s h e d th e p r o c e s s t r a n s f e r f u n c t i o n s , i t

i s p o s s i b l e to c a l c u l a t e th e fe e d fo rw a rd and fe e d b a c k

c o m p e n sa to r . T hese c a l c u l a t i o n s a re p re s e n te d to

i l l u s t r a t e a d d i t io n a l u se s o f t h e dynamic m o d e lin g p a c k

age .

S m ith e t a l . [25] have d e v e lo p e d a c o n t r o l sy s te m

which r e q u i r e s l i q u i d flow t r a n s f e r f u n c t io n s . One

t r a n s f e r fu n c tio n r e l a t e s a c h a n g e in r e f lu x to t h e change

in l i q u i d flow e n te r in g th e f e e d s ta g e . The se c o n d fu n c


159

t i o n r e l a t e s th e c h an g e i n l iq u id flow le a v in g t h e fe e d

s ta g e to th e l iq u id f l o w e n te r in g th e r e b o i l e r . To o b ta in

th e s e fu n c t io n s , s p e c i f y th e l iq u id flow o f t h e d e s i r e d

s ta g e s a s m easured v a r i a b l e s in a d d i t io n to t h e com posi

t i o n m easurem ents. A ls o s p e c ify th e c o n t r o l o p t io n which

h a s r e f lu x and b o tto m s v a p o r as th e m a n ip u la te d v a r i a b l e s .

T he sample in p u t d a ta o f T ab le 6-2 i s f o r t h e e th y le n e

colum n. T ra n s fe r f u n c t i o n co rre sp o n d s t o S m i th 's

GT w h ile c o r re s p o n d s to GB. I f th e i d e n t i f i c a t i o n

o p t io n i s s p e c i f i e d , t h e package w i l l f i t a t r a n s f e r

f u n c t io n e q u a tio n to t h e r e s u l t s g e n e ra te d by t h e m odel.

T a b le 6-3 c o n ta in s t h e p a ra m e te rs g e n e ra te d u s i n g th e

exam ple d a ta .

Feedforw ard t r a n s f e r fu n c t io n s f o r fe e d d i s tu r b a n c e s

c a n be d e r iv e d from t h e f e e d d is tu rb a n c e and m a n ip u la te d

v a r i a b l e t r a n s f e r f u n c t i o n s . Luyben [26] i l l u s t r a t e s th e

developm en t o f th e f e e d fo rw a rd t r a n s f e r f u n c t i o n i n h i s

exam ple 1 3 -1 . GL r e l a t e s th e d is tu rb a n c e to t h e m easured

v a r i a b l e s . GM r e l a t e s t h e m an ip u la ted v a r i a b l e t o th e

m easured v a r i a b l e . The r e s u l t i n g fe e d fo rw a rd t r a n s f e r

f u n c t io n i s F = -GL/GM. F o r th e exam ple o f t h e e th y le n e

co lum n, a fee d fo rw a rd c o n t r o l l e r would be com posed o f

GL = Pj^ and GM = P ^ o r P^^g depending on th e c o n t r o l

a l t e r n a t i v e . T his c o n t r o l l e r would r e l a t e f e e d f lo w to th e

c o n c e n tr a t io n o f e th y l e n e i n th e s id e s tre a m . T he r e s u l t i n g

fe e d fo rw a rd e q u a tio n a p p e a r s in T ab le 6 -4 .


160

TABLE 6 -2

Sample I n p u t Data f o r L iqu id T r a n s f e r F u n c tio n s

M easurem ent Card:

005Y02048YQ3032L 047L

C o n t r o l l e r card :

Ref EVA .52 .0473


161

TABLE 6-3

Liquid T ransfer Functions o f Smith e t a l .

a t- * 9,9*5 :i , * i2.95 2.95

O B(s) .(1 + 8 ' 1 0 - '4 ) 8

Note; The negative s ig n and numerator term fo r GB appear because the feed i s a l l vapor. The term in the numerator i s due to . The equation fo r GB was derived by hand.


162

TABIE 6 -4

Feedforward C on tro llers

- 0 . 0 2 6 e ' l G . O sP ( l l ) = ---- ; ------------------------ ;-------

s2 2 • 0 .9 2 6+ 8 + 1

0.04l^ 0 .0 4 1

- 3 7 .7 sP ( 1 3 ) -------------------*

P(14)

(37.9 8 + 1)

0.022

2 * 0.026 + 8 + 1

0.042% 0 .0 4 2

P ( l ) a -P (1 4 ) /P (1 1 ) « 0 . 8 4 e"3'% ®

1 .16 (37.9s + 1 )P(2) = -P(14)/P(13) « ---- 2---------------------------------

j a l _ + J -L 2 i 926 3 ^ 1

0.042% 0.042

Note I An adjustment should be made to P(2) sin ce i t co n ta in s an unrealizable p re d ic tio n term.


CHAPTER VII

A nalysis o f System In teraction

To analyze system in te ra c tio n s in a two-by-two

p ro cess i t i s u sefu l to consider the closed loop

p ro cess transfer fu n c tio n s . This analysis r e v ea ls

th a t two types o f in te ra c tio n s combine to form the

com plete system in te r a c tio n , one component o f which

can be measured by the Ri jnsd orp /B risto l in te ra c tio n

in d e x . These concepts are applied to a d i s t i l la t io n

column in drder to aid in the design o f i t s con tro l

system .

Measure of In teraction

System in tera ctio n s a r is e from the p rocess, but

are influenced by the co n tro l system. The in te ra c tio n

in h eren t in a process can be measured by the Ri jnsdorp/

B r is t o l index. The in flu en ce from the control system

i s in d ica ted by in d iv id u a l closed loop tran sfer

fu n c tio n s . The product o f these factors i s the to ta l

co u p lin g exhibited by the system.

A measure o f process in tera c tio n i s o f in te r e s t

because i t i s independent o f tuning and of the con tro l

l e r ty p e . I t only depends on the process and co n tro ller

co n figu ra tion . These p rop erties allow i t to be used

as a screening to o l which can elim inate undesirable

163


164

p rocesses and control configiarations during the e a r ly

d esign s ta g e s .

The two types o f in te r a c tio n s can best be viewed

by examining (equation 7«1)» which is the e f f e c t iv e

^ - P u ( 1 - P ggciK ) (Eg. 7 .1 )

p ° 2 ? 2 2 ____ K ,

22CL 1 + CgPpp ' ^11^22

process tra n sfe r function obtained when control lo o p 2

i s c lo sed and control loop 1 i s open (Figure 7*1).

This i s the process which c o n tr o lle r 1 must r e g u la te .

The in te r a c tio n term which m od ifies P^ is composed o f

the product o f the process in te r a c tio n term (R ijn sd orp /

B r is to l index or K) and the c lo sed loop c h a r a c te r is t ic s

of th e other loop (^2201^* the product ^22CL c lo se to zero , then no in te r a c tio n s e x is t . This can be

achieved by proper s e le c t io n o f the process and co n tro l

con figu ration s or by tuning co n tro lle r 2 with very low

g a in s . This decoupled co n d itio n allows co n tro ller 1 to

be tuned w ithout regard fo r P g» ^21 ^22* Thus

system s t a b i l i t y , i . e . ga in and phase margins, would

only be based on the tuning o f con tro ller 1. Because

of th e symmetry in a two-by-two process, the above a lso

a p p lies to EP22'

The in fluence of the p rocess in teraction upon the

s t a b i l i t y o f the system can be seen by examining the


165

Figure 7'1

Transfer Function

Block Diagram of Two Input-Two Output System

with Conventional Control C onfiguration

22

12

21

11

C*s in d ic a te co n tro ller tran sfer functions P's in d ic a te process transfer fu n ctio n s R's are s e tp o in ts X's are ou tp utsM's are p ro cess inputs and c o n tr o lle r outputs


166

c h a r a c te r is t ic equation (equation 7 , 2 )# This equation i s

-1 = + CgPgg + (1 -K) (Sq. 7.2)

obtained by developing process tra n sfer functions

(equation 7 *3) with both control loops c losed .

^ 1 C iP ii ( 1 + ^2^22 )

S P i (1 + ( 1 + CgPgg ) - CiPiiCgPggK(Sq. 7 . 3 )

When K i s zero, the tran sfer fu n ction , X^/SP^, does not

include co n tro ller 2 . This aga in shows the influence

and d e s ir a b i l i t y of having process in teraction s as low

as p o s s ib le . For the system to be stab le the right h a lf

of eq u ation 7.2 must have a magnitude le s s than one when

i t s phase crosses the - 180° p o in t.^

T his statement o f s t a b i l i t y leads to the d e f in it io n

of favorab le and unfavorable in te r a c tio n s . I f the phase

sh if t a sso c ia ted with the system in tera ctio n causes th e

magnitude o f the ch a ra c ter is tic equation to decrease a t

the -1 8 0 ° point, then the in te ra c tio n i s favorable.

This i s s im ila r to rotating the p lo t of the ch a ra c ter is tic

^Although th is i s not a complete d e f in it io n , i t w i l l serve to i l l u s t r a t e the b asic idea .


167

equation counterclockwise about i t s stead y sta te value.

The bandwidth o f the closed loop process i s increased

by th is a d d itio n o f phase lead . Unfavorable in teraction s

would cause the magnitude to in crea se . For il lu s tr a t io n

a constant magnitude o f in te r a c tio n a t a l l frequencies

i s assumed. Figure 7 .2a shows an unstab le process. I f

in tera ctio n s were favorable, then the system would be

s ta b iliz e d (Figure 7«2b). The e f f e c t o f favorable

in tera ctio n s i s to s h i f t the frequency at which the

-180® point i s crossed to a frequency which has a sm aller

magnitude. Unfavorable in tera c tio n s would have the

reverse e f f e c t .

D iscussion o f R ijnsdorp /B risto l In te r a c tio n Index

The development o f the Ri jn sd orp /B risto l in teraction

index follow ed two separate paths which were united by

Witcher and McAvoy. B r is to l's steady s ta te index related

the steady s ta te gains of the open loop process to the

steady s ta te ga in s of the closed loop process. For a

two input, two output process (Figure 7»3)» equation 7.4

gives the r e su lt in g index.

(^h\\3 mJ “ 2 = 0/AXi'X( smJ X2 = 0

(Eq. 7.4)

Evaluating t h is expression in terms o f transfer functions

y ie ld s equation 7 •5»


Figure ?,2a

168

imaginaryA.. +j

-1 +1real

Unstable Process - P lo t o f C haracteristic Equation

F igure 7.2b

imaginaryiI• +j

y. . .

1fStab le Process - P lo t o f C haracteristic Equation


Figure 7.3

169

M,

12

22

^2 " **1^21 ■*■ ^2^22

Block Diagram of Two Input-Two Output System

without C ontrollers


170

^11 - P 2 2

w here K =^ 2 2 ^ 1 1

The K o f equation 7#5 i s th e measure which R ijn sd o rp

id e n t i f i e d . By examining th e r a t io o f open loop to

c lo sed loop t r a n s f e r fu n c tio n s , R ijnsdorp found K to

be a commonly appearing g roup which was independen t o f

c o n t r o l l e r s e t t in g s . Throughout the r e s t of th e t e x t

K w i l l be used a s th e in te r a c t io n measure. This cho ice

was made because of the d is c o n tin u i ty in a t K = +1.

Extending th e R ijn sd o rp /B r is to l index beyond th e

o r ig in a l steady s t a t e in te r p r e t a t io n re q u ire s an ap p ro

p r i a t e ex ten sio n o f the c r i t e r i a used to e v a lu a te i t .

S ince K is a v e c to r valued fu n c tio n , the c r i t e r i a must

a llow in te r p r e ta t io n of i t s magnitude and phase. In

a d d i t io n the c r i t e r i a must p ro v id e a l im itin g freq u en cy

beyond which in te ra c t io n need no t be co n sid ered . T his

l a t t e r c r i t e r i a i s provided because p rocesses do n o t

respond to high frequency e x c i ta t io n s . The e v a lu a t io n

o f th e magnitude and phase' m ust lead to an o rd e r in g o f

p ro c e sse s or c o n tro l c o n fig u ra tio n s by t h e i r d eg ree o f

in te r a c t io n and th u s by t h e i r degree o f d e s i r a b i l i t y .

The in te ra c t io n index K = (^12^21^^^ 11^22^ i s the


171

r a t io o f the d iag o n a l t r a n s f e r fu n c tio n s to th e s tr a ig h t

through t r a n s f e r fu n c tio n s (Figure 7»3) and , as such, is

a function o f fre q u en cy . I t i s a v e c to r v a lu ed function

which rev ea ls th e r e l a t i v e magnitude and r e l a t i v e phase

angle and, th e r e fo r e , i t in d ic a te s which s e t o f tra n s fe r

functions i s dom inant. P o lar p lo ts a re a conven ien t way

of d isp lay in g bo th m agnitude and phase in fo rm a tio n .

T herefore , in o rd e r to in te r p r e t K i t i s n e c e ssa ry to

id e n tify the c h a r a c t e r i s t i c s of the p o la r p lo t o f K and

then to r e la te th e se c h a r a c te r i s t ic s to p ro c e ss behavior.

As in d ic a te d by R ijn sd o rp , the meaning o f th e

magnitude of K as a fu n c tio n of frequency i s s im ila r to

th a t o f the s te a d y s t a t e in te rp r e ta t io n . R ijn sd o rp 's

observation th a t l i t t l e in te ra c tio n e x i s ts f o r K close

to zero i s s t i l l v a l id . Furtherm ore, i f th e magnitude

o f K i s c lose to one, th e n ^21 as much

in flu en ce on th e o u tp u t as P^^ and Pg2 » T h is i s the

most severe case o f p ro c e ss in te ra c t io n . I t i s possib le

th a t fo r a segment o f th e frequency range a p rocess could

have a la rg e K v a lu e w h ile a t o ther f re q u e n c ie s the

process would have sm a ll K values. I d e a l ly , th e p lo t of

K should be co n ta in ed in s id e a c i r c le o f sm a ll rad iu s

whose cen ter i s a t th e o r ig in . As in th e s te a d y s ta te

a n a ly s is , i f th e m agnitude of K i s g r e a te r th a n one fo r

a l l freq u en c ies , th e n re v e rs in g tbe p a ir in g o f manipulated

and measured v a r ia b le s w i l l map K w ith in th e u n i t c i rc le .


172

The e f f e c t s o f the magnitude o f K can be seen by

viewing e q u a t io n 7 «I» Assuming fo r i l l u s t r a t i v e purposes

th a t th e m agnitude o f P22CL equal to +1 fo r a l l

f req u en c ies , th e n a l l v a r ia t io n s in system in te ra c tio n s

w il l be due to th e process. F igure shows the four

b asic d i r e c t io n s which K can tak e when i t c ro sses the

u n it c i r c l e . I f th e imaginary p a r t o f K i s ze ro , then

the gain o f w i l l e i th e r double o r change s ig n . I f

th e locus o f K proceeds along th e im aginary a x is , then

th e magnitude w i l l inc rease and th e re w i l l be a s i g n i f i

can t phase c o n tr ib u t io n . I f the p lo t o f K ever crosses

the v e r t i c a l l i n e through +1, th e s ig n o f EP^^ w ill

change. T h e re fo re , fo r a c e r ta in range o f frequencies

th e c o n t ro l le r w i l l have p o s it iv e feedback and fo r o th e r

frequencies i t w i l l have negative feedback . This l a t t e r

behavior o f K i s in d ic a tiv e o f system s which could become

unstab le i f one c o n tro l le r rem ains on autom atic while th e

o ther i s p u t on manual.

The phase an g le of K can be used to t e l l whether

process in te r a c t io n s w ill be fav o rab le o r unfavorab le .

A re la te d in t e r p r e t a t i o n o f th e phase ang le rev ea ls the

r e la t iv e tim e i t tak es fo r a d is tu rb a n c e to propagate

through th e sy stem . A d d itio n a lly , th e phase angle i s th e

only way th e r e l a t i v e dead time can be determ ined. The

in te r p r e ta t io n o f th e phase angle of K re q u ire s th a t th e


173

Figure 7*4

The E f f e c ts o f K on EP.11

im aginary

in c re a s in g m agnitude w ith phase la g

g a in o f EP. doubles

> r e a l

‘ g a in o f EP^^ changes s ig n

+1-1

in c re a s in g m agnitude with phase le ad


174

polar plot be la b e le d as a function o f frequency. This

allows one to s e e the ro ta tion of K as i t sp ir a ls , and

i t allows id e n t i f ic a t io n o f the steady s t a t e (w«=0) and

'WTsinfinity v a lu es o f K.

The phase a n g le of K ind icates th e d ifferen ce in

phase between ^12^22 ^21^12 th u s accounts for

the dead time and time lag d ifferen ce . I t ind icates

whether ^12^21 lead or lag ï*22^11* d irectionof ro ta tion th a t th e p lot o f K takes on in d ica tes which

term w il l have phase angle dominance. For example, i f

K was dominated by the dead time o f ^12^21 ' then the

polar p lot o f K would be a c ir c le cen tered at the orig in

revolving in the clockw ise d irection . Disturbances

would then propagate fa ster through ^^2^22 ^han through

^12^21* the dead time o f ^22^11 S’ ®siter than

that o f 1*22 21 • th e p lot would rotate counterclockwise.

Favorable o r unfavorable process in te r a c tio n s are

determined by th e ro ta tion o f K. Favorable in teractions

add phase lead when the ch a ra cter istic equation crosses

-180°. Unfavorable in teractions add phase lag when the

ch a ra cter istic equation crosses -180°. For a p o sitiv e

steady sta te v a lu e o f K, Figure 7.5 y ie ld s favorable

in teraction s. I f K continues to s p ir a l in to quadrants

one and two (F igure 7 . 6) , then the in te r a c t io n can be

unfavorable. Unfavorable in teractions would arise i f


Figure 7.5

175

imaginary

+1

Polar P lo t of K - Favorable In teraction

Figure 7*6

imaginary

+1

Polar P lo t o f K - Possib le Unfavorable Interaction


176

th e ch a ra c ter is tic equation crossed the -180° p o in t while

K was in quadrants one and two. Another form o f unfavor

ab le in tera ctio n i s shown in Figure 7»7* For a negative

stead y sta te value o f K Figure 7*8 shows favorab le

in te r a c tio n s . I f K sp ir a ls through quadrants one and two

as above, in te ra c tio n can be unfavorable (Figure 7 .9 ) ,

w h ile in tera c tio n w il l d e f in i t e ly be unfavorable fo r p lo ts

s im ila r to Figure 7*10. The fo llow in g i s a gen era l ru le

o f thumb for va lu es o f K which l i e in the plane to the

l e f t o f a v e r t ic a l l in e through +1. I f the c h a r a c te r is t ic

equation crosses the -180° poin t while K i s below the rea l

a x is , then in tera c tio n s are favorab le . Above th e rea l

a x is in teraction s are unfavorable. For values o f K to

the r ig h t of the +1 lin e the above i s reversed , i . e .

in tera c tio n s are favorable above the rea l a x is and

unfavorable below the r e a l a x is .

The cu to ff frequency fo r K i s the frequency a t which

the ch a ra c ter is tic equation passes the -180° p o in t . Any

process dynamics beyond t h is point usually w i l l not

in flu en ce the s t a b i l i t y . U nfortunately, th is frequency

cannot be determined u n t il a f te r the system has been

tuned. An approximation to t h is frequency would be the

h ig h e st frequency a t which e ith e r or 22 ®^osses the

-180° point. In tera ctio n beyond th is frequency should

not be considered when comparing control con figu ra tion s.


177

Figxire 7*7

imaginary

+14— r e a l

Polar P lo t o f K - Unfavorable In te r a c tio n

Figure 7*8

imaginary

+14—4 r e a l

Polar P lo t o f K - Favorable In te r a c tio n


178

Figure 7.9

imaginary

+1

Polar Plot o f K - P ossib le Unfavorable In teraction

Figure 7*10

imaginary

+1

Polar P lo t o f K - Unfavorable In tera c tio n


179

Vector D escrip tion o f System In tera c tio n

Another d escrip tion o f system in tera ctio n can he

developed based on a vector diagram. This approach s t i l l

views the system in tera ctio n s as a product of the process

in tera c tio n (K) and one o f the c lo sed loop tran sfer

functions (e ith er or ^2201^ ' The product,

i s p lo tted on polar coordinates as a function o f frequency

(see Figure 7 «11)» To analyze the system in tera c tio n a t

a given frequency, a v ec to r i s drawn from the frequency o f

in te r e s t to +1 on the r e a l a x is . The length of the v ec to r

i s the magnitude o f the system in te r a c tio n . The angle

which the vector forms with the h o rizo n ta l i s the phase

component o f the in tera c tio n . This vector i s the grap h ica l

equivalent o f ( 1 - K P^iCL g iven frequency.

In terp reta tion o f the system in tera ctio n vector

requires id e n t if ic a t io n o f the s ig n if ic a n c e o f i t s

magnitude and phase. I f the magnitude i s c lose to one,

then EF^i w i l l equal (see equation 7*1) • Thus fo r a

vector magnitude o f one, the system in teraction s w i l l be

n e g lig ib le . The angle determines whether the system

in tera c tio n i s favorable or unfavorable. I f the angle

i s p o s it iv e , then EP^ w i l l lead P^ and the in te r a c tio n

i s favorab le. This s itu a tio n i s deemed favorable because

i t widens the bandwidth o f the e f f e c t iv e process thus

allowing higher co n tro lle r ga in s.


180

Figure 7•11

P olar P lo t o f KP22CL

im ag in ary

+1* re a l


181

In order to d es ig n a control system i t i s desirab le

to measure the system in teraction s before th e con tro llers

are tuned. This i s im possib le, but the system in teraction s

can be approximated by the process in te r a c tio n , K. For

th is approximation to be u sefu l the f in a l tuning parameter

should make and ^2201 c lose to one w ith zero phase

s h if t . I f th is approximation i s a ccep tab le , then the above

discussion can be ap p lied to a polar p lo t o f K.

Diagonal Dominance

Diagonal dominance is a concept used in the s t a b i l i t y

c r ite r ia of Rosenbrock’s Inverse Nyquist Array. I t i s

used to examine the in teraction s of the system , e .g . process

and controller in te r a c t io n s . I f a process and i t s con tro ller

are diagonally dominant, then in teraction s are "small".

Because of the sm all in tera c tio n s , o v era ll system s ta b i l i t y

can be determined by only examining the d iagonal elem ents.

To r e la te the In verse Nyquist Array technique to the

R ijnsdorp/Bristol in te r a c tio n index i t i s necessary that

the process and the co n tro llers have the same structure

as in Figure 7»1« Equation 7 .6 then g iv e s the expression

for Q < s), Q ( s ) , R ( s ) . ('

be dropped for c l a r i t y ) .

_ A Afor Q ( s ) , Q (s) , R ( s ) . (The Laplace an op erator, s , w il l

Q = ■«u * 12 '^11 ^12 •Ol 0

«21 ^22 ^21 ^22 0 Oj(Eq. 7 .6)

182

Ù. - “1Q = Q = f S u $12 ■ A

; R = R = [$11 $12' +1 0

$21 ^22 $21 ^22 0 1

The system i s d iagonally dominant i f the fo llow ing

conditions are met.

i $11 ^ 1 2i i i

A

« 1 1> A

« 1 2

i i® 2 2 > $21 iv

« 2 2>

A

« 2 1

Applying these conditions to equation 7*6 y ie ld s the

follow ing;

i ) ^22 > ^12 i i i ) 2«22"*" 1 2 ^«11«22"«12«21^ > ^2«12

i i ) ^11 > «21 iv ) V l l ’*’ 1^2^«ll«22“«12«21^ > °1«21

Even though Cg and can cel in i i i and iv , respec

t iv e ly , these conditions ( i i i and iv ) depend not only on

the process but a lso on the c o n tr o lle r s and Cg,

r e sp e c t iv e ly . The la t te r two con d ition s insure diagonal

dominance when the ind ividual co n tro l loops are c lo sed .

Comparing the f i r s t two con d ition s to the R ijnsdorp/

B r is to l in tera c tio n index re v ea ls an in te r e s tin g r e la t io n

sh ip . For n on-in teraction the R ijn sd orp /B risto l index

requires | 11^22 ^12^2l|" Ri jnsdorp /B risto l index

can in d ica te that the process i s non-in ter active even

though conditions i and i i in d ic a te that the process i s not


183

d ia g o n a lly dominant. Although sim ilar in purpose, th e

concepts measure d iffe r e n t in tera c tio n s . I f the p rocess

^11^22 ^12^21 I Ss a t i s f i e s conditions i and i i , then

a lso s a t is f ie d hut the converse i s not true. In t h i s

sen se diagonal dominance i s a more stringent requirem ent

fo r non-in teraction .

Conditions i through iv a lso reveal that in some

in sta n ces the system in te r a c tio n s can be m odified w ithout

elab ora te decouplers. For example, i f P 221

11

P12

^ 2 1 and F2 2 ^21 » then by m ultip lying Pgg and

P21 by a fraction , the co n d itio n s i and i i can be

s a t i s f i e d . and Cg would be adjusted to f u l f i l l

con d ition s i i i and iv .

A n alysis of In teraction - Example 1

The developnent o f a dual composition con tro l scheme

fo r an ethylene column i l lu s t r a t e s the use o f the in t e r

a c t io n concepts. I n i t i a l l y , there are twelve p o ss ib le

combinations of measured and manipulated v a r ia b le s , i . e .

co n tro l systems, to choose from. By examining the

Ri jnsdorp /B risto l in te r a c tio n index for each co n tro l

system , some can be d iscarded . Additional process

considerations w il l e lim in ate several more con tro l

system s leaving sev era l f e a s ib le schemes to be te s te d

on the actual column.


184

D escrip tion o f Ethylene Column

The process o b je c t iv e s are to produce high p u rity

ethylene with a minimum o f lo s s . Figure 7 .12 shows the

steady s ta te design which f u l f i l l s these o b je c t iv e s .

Table 7-1 contains the parameters required for th e

dynamic model. Equimolal overflow and id ea l liq u id /vap or

phases were used in th e modeling. The actual column

contained small q u a n tit ie s o f hydrogen but th is component

was neglected for the stu d y .

Control A lternative

Due to process o b je c t iv e s the column d esign allov/s

fo r three dual com position control co n fig u ra tio n s. This

r e s tr ic t io n was due to th e lack of a liq u id d i s t i l l a t e

stream. Since the column only has a vapor d i s t i l l a t e , i t

would be awkward to use t h i s stream to adjust th e condenser

liq u id le v e l . Therefore, the vapor d i s t i l l a t e i s used2

on ly for composition or pressure control. V/hen the

vapor d i s t i l la t e i s used fo r composition co n tr o l, the

pressure i s maintained by the condenser coolant r a te .

Otherwise, th e coolant r a te i s flow con tro lled . The

above combination means th a t reflux must be ad justed

to maintain the condenser le v e l and the bottoms liq u id

stream must be adjusted to maintain the re b o iler l e v e l .

The la t te r control loop provides an o v era ll l iq u id

inventory balance or m a ter ia l balance con tro l. The

2In teraction s which in v o lv e pressure control were not studied.


185

TABIE 7-1

Parameters Used for Dynamic Model

R eboiler T.C. 0.52 min.

Condenser T.C. 0.0473 min.

Reboiler mass ^24.7 moles

Condenser mass ^38.8 moles

- Liquid T.C. * 1 ( a l l stages)

0.0630 min.

- Aeration Factor ( a l l stages)

0.0307 min.

Holdup^77.9Stages 2-4 moles

Stage 5 ^29.2 moles

Stsiges 6-32 ^74.3 moles

Stage 33 4*26.4 moles

Stages 34-47 4*58.5 moles

*T.C. = l e v e l co n tro ller time constant

^Obtained from Wu (23)


186

Figure 7.12

Steady S ta te Operating Conditions

for Ethylene Tower

Methane 0 .14 Ethylene 37*87 Ethane 7•91 Propylene 0.34

Vapor (saturated)

Stage 5 T= -12°F

S ta g e 33 2®F

S ta g e 48 T = 31°F

9 P = 323 psia

Methane 0 . l4 Ethylene 0.35

— »

S ta g e 1T = -3.4°F P = 315 p sia

A ll flows are in m oles/m in.

Methane 0 .3 5 Ethylene 1 4 5 .0 Ethane 0 .0 Propylene 0 .0

Liquid Sidestream

Methane 0 .0Ethylene 37*5 Ethane 0 .0 1 Propylene 0 .0

Methane 0 .0 Ethylene 0 .0 2 Ethane 7 • 90 Propylene 0 .3 4


187

remaining fea sib le co n tro l configurations a re shown in

Figures 7.13a, 7*13b and 7«13c.

For each con figu ration i t i s possib le to measure

e ith er the major component or the impurity o f the stream.

Twelve control system s (Table 7~2) were generated by

combining the co n tro l configurations of F igure 7.13

with various combinations of measured v a r ia b le s . T hree-

s e ts of measurements are shown in Table 7-3» Measurement

s e t three was used tw ic e , once with the p a ir in g s as

shown in Figure 7 .13 and once with the p a ir in g s reversed.

Comparative In te ra c tio n Analysis

Based on the Ri jn sd orp /B risto l in te r a c tio n index

i t i s possib le to s e l e c t a control system fo r the ethylene

column. The twelve con tro l systems of Table 7-2 are the

in i t i a l members o f th e fea s ib le l i s t . By applying the

Ri jnsdorp/B ristol in te r a c tio n index, some co n tro l

systems can be e lim in ated from th is l i s t . A dditional

considerations such as type o f un its loca ted upstream

and downstream and th e range and accuracy o f ava ilab le

analyzers w ill e lim in a te other members o f th e l i s t .

This procedure w il l lea v e severa l control system s which

can then be tested on the column to determine th e ir

e ffec tiv e n e ss .

The f i r s t con sid eration i s to elim inate control

systems whose p lo t o f K crosses the v e r t ic a l l in e


188

Figure 7 .13a

Control Configuration for Systems 1,4 ,7 ,10

LC

CC

CC

reversed pairing


189

Figure 7»13‘b

Control Configuration for Systems 2 ,5 ,8 ,1 1

LC

CC

CC

reversed p a ir in g


Figure 7 «13 c

Control Configuration for Systems 3 ,6 ,9 ,1 2

1 9 0

LC

CC

CC

reversed p a ir in g


191

table 7 -2

Control Systems

P airing o f Measured and Manipulated Variables

ControlSystem

Control Loon 1

Control Loon 2

Figu7 .1

1 (Cg (5 / . S S L 5 ) (Cg(48)^, BVA) a

2 (€2“ (5 ) , DVA) (C g(48), SSL5) b

3 (Cg" (5 ) , DVA) (C g(48), BVA) 0

4 (C] (5 ) . SSL5) (Cg“ (4 8 ), BVA) a

5 (C (5 ) . DVA ) (C g'(48),SSL 5) b

6 {C (5 ) , DVA) (Cg"(48), BVA) 0

7 (Cg” (5),S SL 5) (Cg‘ (4 8 ), BVA) a

8 (Cg (5 ) , DVA ) (Cg“ (4 8 ), SSL5) b

9 (Cg (5 ) , DVA) (Cg“ (4 8 ), BVA) 0

10 (C] (5 ) , BVA) (Cg“ (48),SSL 5)«

a

11 (C] (5 ).SSL 5) (Cg“ (4 8 ), DVA )*

b

12 (C] (5 ) , BVA) (Cg“ (4 8 ), DVA)#

c

SSL5 1 sidestream liq u id stage 5 C, t methaneDVA 1 d i s t i l l a t e vapor Gg ' ethaneBVA 1 bottoms vapor C2-. ethylene# in d ica te s th a t number in parentheses i s stage on which

measurement was made* reversed pairing


TABLE 7-3

M easurem ent S e t

S ta g e 5 <=2 c

S ta g e 48 =2 =2- =2=

192

193

through +1. For system s 2 ,4 ,5 ,6 ,1 0 ,1 1 and 12 th e cross

over occurs within the frequency of in tere st ( s e e p lo ts

7 * 1 -7 .1 2 ). These con tro l system s can he tuned fo r stable

operation but i f one loop i s switched to manual, the

remaining loop can become unstab le. Current engineering

and operating p ractice f in d s th is event unacceptable.

From a process point o f view configurations 4 ,5 * 6 ,1 0 ,1 1

and 12 are d i f f ic u l t to make work because they requ ire

the condenser to l iq u ify methane. Therefore, th e above

co n tro l systems w i l l not be given any further con sid era tion .

Control systems 3 ,8 and 9 a l l have small p rocess in ter

a ctio n s at high freq u en c ie s , i . e . at frequencies where the

ch a ra c ter is tic equation cr o sse s the -180° p o in t. This

property of the column should allow large c o n tr o lle r gains

to be used without ad versely influencing closed lo o p

s t a b i l i t y . Ifhen th ese co n tro l systems were tun ed , the

above observation was shown to be true. Each system was

found to be m arginally s ta b le when both loops were closed

and the respective in d iv id u a l loop ultim ate g a in s were

used. The individual loop ultim ate gain is the h ig h est

fe a s ib le gain s e t t in g which could be used.

Control systems 1 and 7 tuned sim ilar ly to 3 ,8 and 9*

but th e ir process in te r a c t io n s were greater than one near

the frequency at which the ch a ra c ter is tic equation

cro sses the -180° p o in t. This behavior only happens


194

P lot 7.1

P lo t 7.2

P lo t 7.3

Output

c / (5)

C2’ (4 8 )

Input

SSL5

BVA

Gain Matrix

- 2 . 6 2 X 10"^ 1.186X10"!

Kgs= -0 -4 3 8

7 .2 8 X 1 07.19X 10

-6-5

Giu= -126

G2u= 1890

Output

C2" (5 )

C2’ (4 8 )

Input

DVA

SSL5

Gain Matrix

-1.88X10"% -2.62x10"%1.186X10

Kss = l '3 9

-1 1.186X10

Giu= -65

G2u= 175

-1

Output

02= (5)

C g - f te )

Input

DVA

BVA

Gain Matrix

-1 .8 8 X 10"% 7.281X10"*1.186X10 -1 7 . 1 9 x 10 -5

Kgs = "0-637 G,, = -65luG2u= 1890


195

P lo t 7 .5

P lo t 7 .6

Output

(5)C2=(48)

Input

SSL5

BVA

Gain Matrix\*8- 3 . 9 x 1 0 - " - 5 . 8 x 1 0

- 6 . 8 x 10 ’ ^ - 5 . 5 x 1 0 - 5

Kss = 1830 G ]^^=-1.5X 107

G2u= -2 8

Output

Cl (5)C2=(48)

Input

DVA

SSL5

-4 ,1 X 1 0- 6 .8 X 1 0

Gain Matrix -8-4

- 3 . 9 x 1 0- 6 .8 X 1 0

—8- 4

Kgg = 0 .9 6 6 -3 X 10?

®2u* - 1 ^ 10^

Output

Cl (5)C2“ (48)

Input

DVA

BVA

Gain Matrix .-8 *—6

- 4 . 1 X 1 0 - 5 . 8 X 1 0

- 6 .812 X I O '4 - 5 . 5 1 5 X 1 0 - 5

Kgg = 1770 Gi^=-1.5X10?

®2u* - 5 0


196

P lo t 7 .7

II

P lo t 7.8

P lo t 7 .9

Output

Cg- (5)

C2=(48)

Input

SSL5

BVA

Gain Matrix4 .8 X 1 0 " 5

- 6 .8 X 1 0 ”^-8 .8X 10" *- 5 . 5 x 1 0 “^

K g g - - 2 .2 3 7 . 5 x 1 0 ?

G 2u=-1*0X10^

Output

Cg- (5)

C2=(48)

Input

DVA

SSL5

Gain Matrix

5 .1 X 1 0 " ^ —6*8 X 10 ^

Kgg =0 .9 5 2

4 .8 X 1 0 ” - 6 .8 X 10 ^

G iy= 6 X 10*

G2u= -9 9 0 0

Output

3/ (5)C2’ (48)

Input

DVA

BVA

Gain Matrix

-1 .8 8 X iO " 2 1.186X 10"!

7.281X10"*7 . 1 9 x 1 0 "^

K g g --0.637 G lu= -6 5

^2 u * !®90


197

Output Input

BVA

SSL5

Gain Matrix

- 3 .9 X10 -6 .8 X 1 0

= .0005 G, =-2.2X1088P lo t 7.10

Output Input

SSL5

DVA

Gain Matrix

-4 .1 X 1 0-3 .9 X 1 0

- 6 . 8 X10 -6 .8 X 1 0

88P lo t 7.11

Output Input

BVA

DVA

Gain Matrix

-4 .06X 10-6 .8X 10"

-5 .8 X 1 0- 5 . 5 x 10

K__ =.00056 G,, , = -2.2X10P lo t 7.12


198

i f ^2201 ^ IIC L sm a ll m a g n itu d e s in th e re g io n

w here K i s l a r g e . P lo ts o f ’2 2 0 1 ^ ’ ^ ^IIC L t h i s to

he t r u e .

Summary - E xam ple 1

The above a n a ly s i s o f dynam ic i n t e r a c t i o n in d ic a te s

a n eed to go bey o n d s te a d y s t a t e i n t e r a c t i o n c o n s id e ra

t i o n s i n c h o o s in g a c o n tro l schem e. I f th e s te a d y s t a t e

i n t e r a c t i o n in d e x had been th e d e s ig n b a s i s , th e n c o n tro l

sy s tem 10 o r 12 w ould have been c h o s e n . Im p lem en ta tion

o f th e s e schem es c o u ld have le d t o u n s ta b le o p e ra t io n

as in d ic a t e d by th e p lo t o f K c r o s s i n g th e v e r t i c a l l in e

th ro u g h +1. The im p o rta n t sy s tem i n t e r a c t i o n o c cu rs a t

th e f re q u e n c y w h ere th e c h a r a c t e r i s t i c e q u a tio n c ro s se s

th e - 1 8 0 ° p o i n t . The m agnitude o f t h i s i n t e r a c t io n

in f lu e n c e s t h e lo o p g a in s and th u s t h e system re sp o n se .

The e f f e c t o f r e v e r s in g th e p a i r i n g o f m an ip u la ted

and c o n t r o l l e d v a r i a b le s can be s e e n i n p lo ts 4 ,5 ,6 and

1 0 ,1 1 , 12 . The f re q u e n c ie s o u t s id e o f th e u n i t c i r c l e

a re mapped w i t h i n th e u n i t c i r c l e and v i s a v e r s a when

th e p a i r i n g s a r e r e v e r s e d . T h e r e f o r e , i f th e p ro c e ss

i n t e r a c t i o n c r o s s e s th e v e r t i c a l l i n e th ro u g h +1, th en

r e v e r s in g th e p a i r i n g w i l l n o t im p ro v e th e s i t u a t i o n .

R e v e rs in g th e p a i r i n g w i l l re d u c e p r o c e s s i n t e r a c t io n s i f

th e p l o t o f K i s co m p le te ly o u t s id e o f th e u n i t c i r c l e

f o r a l l f r e q u e n c i e s . T his r e d u c t io n i n p ro c e ss i n t e r a c t i o n s


199

does not guarantee a reduction in system in teraction s

since the previous off-d iagonal term s are now used in

^IICL ^22CL*

Analysis o f In teraction - Example 2

Schwanke et a l , (20) examine th e design o f a column

control system based on approximated closed form tra n sfer

functions. These transfer fu n ctio n s were obtained by

testin g a p i l o t plant column. A pp lication o f the

R ijnsd orp /B risto l in teraction in d ex to these transfer

functions shows that the process in te r a c tio n s are

dominated by dead time beyond 0 .3 2 rad/m inute. (Plot 7 .1 3 )

V/hen th is frequency i s atta ined , a l l the numerator and

demoninator dynamics of the tr a n s fe r function have been

cancelled o u t. This leads to u n r e a lis t ic in teraction a t

frequencies higher than 0.32 rad/m inute.

This phenomenon o f in te r a c tio n s caused by pure dead

time i s n o t observed in example 1 but can be induced.

V/hen the frequency responses ob tained by rigorous calcu

la tion s are approximated by low o rd er transfer functions,

the above phenomenon becomes apparent. This d istortion

of the c h a r a c te r is t ic equation a t th e c r i t ic a l -180°

crossover p o in t can be detrim ental to a con tro l study.


Plot 7 .1 3

200

3 2 r a d / i n i n


CHAPTER V III

CONCLUSION

The a n a ly s i s o f d i s t i l l a t i o n colum n dynam ics le d to

th e e x te n s io n o f a m odeling tec h n iq u e an d b e t t e r u n d er

s ta n d in g o f sy s te m i n t e r a c t i o n s . The c o m b in a tio n o f the

above item s p ro v id e s a c o n v e n ie n t to o l f o r th e in v e s t ig a

t i o n o f column d y n am ics .

The model and s o lu t io n tec h n iq u e h a v e d em o n s tra ted

th e f e a s i b i l i t y o f th e freq u e n cy domain a p p ro a c h . A lthough

th e model was l im i t e d to s e le c te d c o n t r o l c o n f ig u r a t io n s ,

o th e r s cou ld be a d d e d . Convergence d i f f i c u l t i e s w ith th e

e n th a lp y b a la n c e m odel i n d ic a t e a need f o r an im proved

s o lu t io n te c h n iq u e . These m odels p ro v id e a h ig h e r degree

o f accu racy (due t o few er assum ptions) th a n p re v io u s m odels.

The t o t a l sy s te m i n t e r a c t io n i s th e p ro d u c t o f two

te rm s : th e p r o c e s s i n t e r a c t i o n r e p r e s e n te d by th e

R i jn s d o r p /B r i s to l i n t e r a c t i o n index and t h e c o n t r o l system

i n t e r a c t i o n r e p r e s e n te d by th e in d iv id u a l c lo s e d loop

t r a n s f e r f u n c t io n s . Because o f th e m u l t i p l i c a t i v e p ro p e r ty

o f th e i n t e r a c t i o n s , i t i s d i f f i c u l t t o s e l e c t a c o n tro l

c o n f ig u r a t io n b a se d s o le ly on th e dynam ic R i jn s d o r p /B r is to l

i n t e r a c t i o n in d e x . In a d d i t io n , th e s e i n t e r a c t i o n s in d ic a te

t h a t c o n tr o l sy s te m d e s ig n b ased on a s t e a d y s t a t e i n t e r a c

t i o n index can be m is le a d in g .

201


202

F u tu re r e s e a r c h sh o u ld be d i r e c t e d tow ards im proving

the dynam ic m odel based on the f r e q u e n c y domain s o lu t io n

te c h n iq u e . T h is improvement cou ld b e d iv id e d i n to th re e

a re a s : c o n t r o l o p t io n s , column c h a r a c t e r i s t i c s , and s o lu

tio n p r o c e d u r e . A d d it io n a l c o n tro l o p t io n s would in c lu d e

PI and n o n - l i n e a r c o n tr o l o f a c c u m u la to r le v e ls and th e

in f lu e n c e o f sam p lin g r a t e on a n a ly z e r co m p o sitio n fe e d

back c o n t r o l . A d d it io n a l column c h a r a c t e r i s t i c s would

re q u ire i n c l u d i n g sim ple p re s su re e f f e c t s , r e b o i l e r .

dynam ics, i . e . therm osiphon vs. k e t t l e and condenser

dynam ics. A w o rk ab le s o lu t io n p r o c e d u r e fo r th e e n th a lp y

model c o u ld b e b ased on a com plete m a t r i x in v e r s io n p ro

cedure o r a n o th e r s te a d y s t a t e s o l u t i o n p ro c e d u re .


REFERENCES

1. R adem aker, 0 . , R ijn s d o rp , J . E. and A. M a rr le v e ld ,

D ynam ics and C o n tro l o f C o n tin u o u s D i s t i l l a t i o n

U n i t s , E ls e v ie r S c i e n t i f i c P u b l i s h in g C o ., New Y ork ,

1 9 7 5 .

2. W il l ia m s , T. J . , "The S t a t u s o f S tu d ie s o f th e

D ynam ics o f M a ss -T ra n s fe r O p e r a t io n s " , Chem. E ng .

P r o g r . , Synposium S e r ie s 4 £ , 1-8 (1 9 6 3 ) .

3. A r c h e r , D. H. and R. R. R o th fu s s , "Dynamics o f

D i s t i l l a t i o n U n its and O th e r Mass T ra n s f e r E q u ip m en t" ,

Chem. Eng. P r o g r . , Symposium S e r ie s 36, 2 -1 9 (1961).

4. A rm stro n g , W. D. and W. L . W ilk in so n , "An I n v e s t ig a t i o n

o f t h e T ra n s ie n t Response o f a D i s t i l l a t i o n Colum n,

P a r t I I . E x p erim en ta l Work and Com parison W ith T heo ry",

T r a n s . I n s tn . chem. E n g rs . 15 , 352-367 (1957) .

5. S t a in t h o r p , F . P . , "The Dynam ics o f F r a c t io n a t in g

C olum ns, P a r t I I - A R educed Model f o r R esponses to

V a p o r Flow C hanges", T ra n s . I n s tn . Chem. E n g rs . 5 1 ,

5 1 -5 5 (1973).

6 S t a in t h o r p , F . P . and H. M. S e a rso n , "The Dynam ics

o f F r a c t io n a t in g Columns, P a r t I - A Reduced M odel

f o r R esponses t o R eflux F low C hnges", T rans. I n s t n .

Chem. E ngrs. 51 , 42-50 (1 9 7 3 ) .

7. R o sen b ro ck , H. H ., "The C o n tro l o f D i s t i l l a t i o n

C o lu m n s" , T ra n s . I n s tn . Chem. E n g rs . 40, 35-53 (1 9 6 2 ).

203


204

8 . Lamb, D. E . , P ig f o r d , R. L. and D. W. T . R ip p in ,

"Dynamic C h a r a c t e r i s t i c s and A nalogue S im u la tio n o f

D i s t i l l a t i o n C olum ns", Chem ical E n g in e e r in g P ro g re s s ,

Symposium S e r i e s 132-147 (1961) .

9 . Cadman, T , W. and N. L. C a rr, "F eed fo rw ard C o n tro l

S y n th e s is f o r M ulticom ponent D i s t i l l a t i o n . I . Theor

e t i c a l D evelopm ent o f Tower T r a n s f e r F u n c t io n s " , ISA

T ra n s a c t io n s (4) 386-394 (O c to b e r 1 9 66 ).

10 . Wood, R. M ., "The F requency R esponse o f M ulticom ponent

D i s t i l l a t i o n C olum ns", T rans. I n s t n . Chem. E n g rs . 45,

T190-T194 (1 9 6 7 ) .

1 1 . S h u n ta , J . P . , and W. L. Luyben, "Dynamic E f f e c t s o f

T em p era tu re C o n tro l T ray L o c a tio n I n D i s t i l l a t i o n

C olum ns", A. I . Ch. E . Jo u rn a l 1 7 , (1) 92-96 (January

1971) .

1 2 . D av ison , E . J . , "A Nonminimum P h a se In d ex and I t s

A p p l ic a t io n to I n t e r a c t i n g M u l t iv a r i a b le C o n tro l

S y s tem s" , A u to m a tlc a 791-799 (1 9 6 9 ) .

13 . Tung, L . S . , and T . F . Edgar, "Dynamic I n t e r a c t io n

A n a ly s is and I t s A p p lic a tio n to D i s t i l l a t i o n Column

C o n tro l" , P ro c e e d in g s o f th e 1977 IEEE C onfe rence of

D e c is io n an d C o n t r o l , 1107-1112 (1 9 7 7 ).

14 . W itc h e r , M. F . , and T . J . McAvoy, " I n t e r a c t i n g C o n tro l

S ystem s. S te a d y S t a t e and Dynamic M easurem ent o f

I n t e r a c t i o n " , ISA T ra n s a c t io n s , (3) 35-41 (1977).


205

15. B r i s t o l , E . H ., "On a New M easu re o f I n te r a c t io n f o r

M u l t iv a r i a b l e P ro c ess C o n tro l" , IEEE T ra n sa c tio n s on

A u to m atic C o n tr o l , AC-11, 133-134 (Ja n u a ry 1966) .

16 . R i jn s d o rp , J . E . , " I n te r a c t io n i n T w o-V ariab le C o n tro l

System s f o r D i s t i l l a t i o n Columns - I and I I " ,

A u to m a tic a , 1 , 15-51 (1965).

17. B r i s t o l , E . H ., "Dynamic E f f e c t s o f I n te r a c t io n " ,

FP4, P ro c e e d in g s o f th e 1977 IEEE C o nference o f

D e c is io n an d C o n tro l, 1 , 1096-1100 (1977) .

18 . Wang, J . C . and G. E. Henke, " T r id ia g o n a l M atrix fo r

D i s t i l l a t i o n " , Hydrocarbon P r o c e s s in g , £5 , 155

(A ugust, 1 9 6 6 ) .

19 . H o lla n d , C . D ., M ulticom ponent D i s t i l l a t i o n , P re n t ic e

H a l l , E nglew ood C l i f f s , N. J . , 1 9 6 3 .

20. Schw anke, C . 0 . , E dgar, T. F . a n d Hougen, J . 0 . ,

"D evelopm ent o f M u lt iv a r ia b le C o n t r o l S t r a te g ie s f o r

D i s t i l l a t i o n Columns", ISA T r a n s a c t i o n s , (4)

69-81 (1 9 7 7 ) .

21. W a lle r , K u r t V. T . , "D ecoupling i n D i s t i l l a t i o n " ,

A .I .C h .E . J o u r n a l , 20, (3) 5 9 2 -5 9 4 (May 1974).

22. M a r tin , J a c o b , J r . , C o r r ip io , A . B. and Sm ith , C. L . ,

" C o n t r o l l e r Tuning from Sim ple P r o c e s s M odels",

I n s t r u m e n ta t io n Technology, (12) 39-44 (1975).

23. Wu, C heng-T sung Rex, S im u la tio n o f a D i s t i l l a t i o n

Column, M. S . T h e s is , L o u is ia n a S t a t e U n iv e rs i ty ,

A ugust 1 9 7 8 .


2 0 6

2 4 . B u c k le y , P . S . , Cox, R. K. a n d D. L. R o l l in s , " I n v e r s e

R esp o n se i n D i s t i l l a t i o n C o lum ns", p re s e n te d a t A . I .

C h .E . m e e tin g , H ouston, T e x a s , March 1 6 -2 0 , 1975.

25 . S m ith , D. E . , S te w a r t , W. S . , and D. E . G r i f f i n ,

" D i s t i l l W ith C om position C o n t r o l" , H ydrocarbon

P ro c e s s in g * 5 2 . (2 ) 98 - 104- (F e b ru a ry , 1978).

2 6 . L uyben , W. L . , P ro cess M o d e lin g S im u la tio n and C o n tr o l

F o r C hem ica l E n g in e e rs , M cG raw -H ill C o ., New Y ork ,

1973 .

27 . Van W in k le , M atthew , D i s t i l l a t i o n , C hem ical E n g in e e r in g

S e r i e s , M cG raw -H ill, New Y o rk , 1967.

2 8 . S m ith , B uford D ., D esign o f E q u il ib r iu m S tag e P ro

c e s s e s , M cGraw-Hill C o ., New Y ork , 1963.

29. Franks, R.G.E., Modeling and Sim ulation in Chemical

Engineering, McGraw-Hill C o . , New York, 196?.


APPENDIX A

USER'S GUIDE FOR STEADY STATE PROGRAM

207


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w INPUT DATA MANUAL FOR STEADY STATE DISTILLATION CALCULATIONS BY K.W.KQMINtK 15' AUGUST 30,1979 2Q 1) DESCRlPTION/INTROruCTION 3n: THIS PROGRAM PERFORMS RATING CALCULATIONS FOR A COMPLEX MULTI-COMPONENT 4m disti llati on column (MULTIPLE FEEDS, PRODUCTS, AND INTERSTAGE HEATERS OR 58 COOLERS). THE MIXTURE TO BE SEPARATED CAN HAVE ANY DEGREE OF LIQUID PHASE 6 NONIDEALITY. THE VAPOR PHASE IS ASSUMED TO BE IDEAL. SINCE ENTHALPY DATA IS 7

c5- REQUIRED, IT IS NOT NECESSARY TO ASSUME EQUIMQLAL OVERFLOW. THREE TYPES OF 8- CONDENSERS ARE POSSIBLE : TOTAL, PARTIAL, OR NONE. NO PROVISIONS HAVE BEEN MADE 9i FOR A DECANTING CONDENSER. THE COLUMN MUST HAVE A PARTIAL REBOILER. THE NUMBER 10

m OF equilibrium stages , the column pressure p r o f i l e , the feed rate, FEED 11

COMPOSITION AND FEED QUALITY MUST BE GIVEN. IN ADDITION THE RATE OF EACH 12c PRODUCT AND THE L/D RATIO MUST BE SPECIFIED. 13g- THE BASIC SOLUTION METHOD IS BASED ON THE WANG/HENKE METHOD (HYDROCARBON 14

PROCESSING, VOL. 45, NO. S, AUGUST 1966) . 15THIS METHOD REQUIRES AN INITIAL COLUMN PROFILE OF TEMPERATURES, LIQUID AND VAPOR 16

S RATES. AND LIQUID AND VAPOR COMPOSITIONS. THE CONVERGENCE OF THIS ITERATIVE 17C PROCEDURE DEPENDS ON HOW GOOD THE INITIAL PROFILES ARE. CONVERGENCE IS AIDED BY IB§ THE THETA-METHOD (HOLLAND) AND A PARTIAL SUBSTITUTION DAMPING FACTOR. 19= 20

Z 2) INTRODUCTION TO DATA TYPES 22S THE JOB DECK IS COMPOSED OF JCL (JOB CONTROL LANGUAGE) CAROS, USER SUPPLIED 23^ SUBROUTINES AND PROGRAM DATA CARDS. THE JCL CARDS SUPPLY THE INFORMATION TO THE 24I COMPUTER SYSTEM SPECIFYING WHICH PROGRAM TO RUN, MEMORY S^ACE REQUIRED, COMPUTER 25

0 TIME ALLOWED AND ACCOUNTING INFORMATION. THE USER SUPPLIED FORTRAN SUBROUTINES 26CALCULATE PHYSICAL PROPERTY INFORMATION. THE PROGRAM DATA CARDS SUPPLY 27

INFORMATION ABOUT THE PROBLEM TO BE SOLVED. 28

1I 3) INPUT DATA : JCL CARDS ?1

PRESENTLY THIS PROGRAM RESIDES IN A FILE ON THE LSU/SnCC COMPUTER SYSTEM. 32this IS the most EFFICIENT MEANS OF HANDLING THIS LARGE OF PROGRAM. THEREFORE, 33THE JCL CARDS ARE STRUCTURED FOR THIS FILE SYSTEM. 54

55 M/ / (JOB CARD) 36 °

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/ /EX2 EXEC FûRTGCLG,LIB=*CSMITH.KWKLDAD’ , PA RM. LKED=’ NQXREF.NOLIST* ,/ / REGION.GO=IOOK,TIME.SO-3//FCRT.SYSIN DD -

ANY FORTRAN SUBROUTINE///LKEO.SYSIN DD *

INCLUDE SYSLIB(KAIN)ENTRY MAIN

/ V

//GO.SYSIIm CO *PROGRAM DATA CARDS

/ -/ /

THE ABOVE CAROS WHICH BEGIN WITH A / ARE JCL CARDS. THESE CARDS MUST APPEAR IN THE ORDER SHOWN FOR EVERY SUBMISSION OF THE CARD DECK TO THE COMPUTER. JCL CAROS ARE NOT CHANGED OR OMITTED IF FORTRAN SUBROUTINES ARE NOT PRESENT.

INITIAL ESTIMATES FOR MEMORY SPACE, CPU TIME AND LINES PRINTED SHOULD BE ?OOK, 1 MIN AND 1000 LINES, RESPECTIVELY.

4) INPUT DATA : PROGRAM DATA CARDS

ESTIMATES OF INITIAL VALUES, IF ANY, ARE GIVEN IN PARENTHESES.

A,B,C ARE ONLY REQUIRED FOR THE FIRST SET OF DATA.

A? TITLE CARD : THIS IS A HEADER CARD FOR THE PHYSICAL PROPERTY DATA SET. THE FIRST TWO COLUMNS CONTAIN THE NUMBER OF COMPONENTS (RIGHT JUSTIFIED) THENEXT 76 COLUMNS wiLL BE PRINTED AT THE TOP OF THE FIRST PAGE. THIS CARD SHOULD IDENTIFY THE COMPONENTS AS WELL AS INDICATE ANY SPECIAL ATTRIBUTES OF THE COMPONENT SYSTEM.

PROPERTY DATA CARD SET EACH COMPONENT MUST HAVE

CARD 1 FORMATC?A4,2X,7E1C.C )ITS OWN SET OF TWO CARDS.

3738394041

424344454647484950515253545556

5758 54to616263646566676869707172 oVO

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FIELD 1 : NAF.E OF COMPONENTFIELD 2 : MOLECULAR WEIGHT FIELDS 3 TO 8 : CHEFS FOR CALCULATIONS

CARD 2 FORKAKSEIO.O )FIELDS I TO 4 : CQEFS FOR CALCULATIONS

FIELDS 5 TO 8 : COEFS FOR CALCULATIONS

OF VAPOR PRESSURES OR K VALUES

OF LIQUID ENTHALPY

OF VAPOR enthalpy

C) INITIAL ESTIMATES FOR THE TOP AND BOTTOM LIQUID COMPOSITIONS (10E8.0) (Two CARDS) FIRST CARD CONTAINS TOP COMPOSITION, SECOND CARD CONTAINS BOTTOM COMPOSITION MOLE FRACTIONS. THESE TWO CARDS ARE ONLY REQUIRED FOR THE FIRST RUN. IF TWO ELANK CARDS ARE SUPPLIED, ALL COMPOSITIONS ARE ASSUMED EQUAL.

D) TITLE CARD : THIS IS A HEADER CARD. ALL 80 COLUMNS WILL Br PRINIEDAT THE TOP OF EACH NEW PAGE. THIS CARD SHOULD IDENTIFY THE PROBLEM AS WELL AS INDICATE THE UNITS OF TEMPERATURE, PRESSURE, ENERGY, MASS AND TIME.

E ) DATA CARD 1 : CONTROL CARD F0RMAT(7I5)FIELD 1 : NUMBER OF STAGESFIELD 2 : NUMBER OF COMPONENTSFIELD 3 : MAXIMUM NUMBER OF ITERATIONS (50)FIELD 4 : FLAG FOR EQUIMQLAL OVERFLOW

= 0 EQUImCLAL OVERFLOW IS ASSUMED 1 enthalpy BALANCE IS USED

FIELD 5 : FLAG FOR INTERSTAGE HEATER OR COOLERS = 0 NONE = 1 ONE OR MORE

FIELD 6 : FLAG FOR PRINT OPTION OF FINAL SOLUTION- 0 ONLY PRINT STAGES WITH FEEDS, PRODUCTS = I PRINT INFORMATION FOR EACH STAGE

FIELD 7 : FLAG FOR PLOT OPTION (C) 3 NO PLOTS

= 1 PLOT VAPOR AND LIQUID COMPOSITION (M.F. ) VS FIELD 8 : FLAG FOR THETA METHOD

= 0 USE THETA METHOD TO AID CONVERGENCE = 1 DO NOT USE THETA METHOD

(0 )OR HEATERS/COOLERS

STAGES (SEMI-LOG)

7374757677 767980 81 628384 65 868788 69 90 «1929394 C'59697 96 99

100101132103104105106 137 108 MMO

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F) DATA CARD 2 : INITIALIZATION CARD FORMAK8 £ 1 0 . 0 )FIELDFIELDFIELDFIELDFIELD

FIELDFIELDFIELD

FIELDFIELDFIELD

12345

G ) 1 ?3

456

field 1

FIELDFIELD

REFLUX RATIO (L/D) ( RE F LUX/( DIST LÎQ + DIST VAPOR))STAGE 1 PRESSUREREBCILER PRESSUREESTIMATE OF STAGE 1 TEMPERATUREESTIMATE OF REBOILER TEMPERATURE

0< D .F . <.0 . 1 0<D.F .<0.1

FIELD 7

DATA CARD 3 : CONVERGENCE CARD FORMAT( SE 10.0 )DAMPING FACTOR : TEMPERATURE DAMPING FACTOR : LIQUID VAPOR RATES RELATIVE CHANGE TOLERANCE : TEMPERATURE

RELATIVE CHANGE TOLERANCE : VAPOR FLOWRELATIVE CHANGE TOLERANCE : LIQUID M.F.MINIMUM TEMPERATURE (IF ZERO THEN DEFAULT VALUE OF STAGE 1 TEMPERATURE-100.)MAXIMUM TEMPERATURE (IF ZERO THEN DEFAULT VALUE IS INITIAL ESTIMATE OF REBOILER TEMPE RATURE+100 . )

( 0 . 9 )( 0 . 9 )( 0 . 0 1 )( 0 . 0 1 )(3 .01 )IS INITIAL ESTIMATE

H) FEED STREAM DATA CARD FüRMAT( 11 , 1 3 , A1 , F 5 .0 ,1 OFT.0)FLAG TO INDICATE ANOTHER FEED STREAM DATA CARD

= 0 THERE IS ANOTHER CARD= 1 THERE ARE NO MORE FEED STREAM DATA CARDS AFTER THIS

STAGE NUMBER WHERE FEED ENTERS TYPE OF FEED

O N I

L.V,CARD

AND F CAN ONLY bE USED IF ENTHALPY BALANCE IS SPECIFIED ONL . *

- L LIQUID FEED = V VAPOR FEED= F flashes FEED AT GIVEN TEMPERATURE AND FEED STAGE PRESSURE = C USED ONLY IF EQUILMQLAL OVERFLOW IS SPECIFIED ON CARD E.

FIELD 4 : TEMPERATURE OF FEED IF TYPE OF FEED IS L,V, OR F , OTHERWISE FRACTION LIQUID OF FEED IF TYPE OF FEED IS C. fields 5 TO 14 : FEED RATE OF EACH COMPONENT (MOLES)

T> PRODUCT stream CARDS FÜRMAT( I I , 1 9 , 2E10.0 )NOTE : IT IS NOT NECESSARY TO SUPPLY & PRODUCT STREAM CARD FOR THE LIQUi!

109 1 10 111 112113114115116117118119

120 121 122 123 1 24125126127128129130131 1 32133134

135136137 139 13 9 1 40

1 41 1 42 1 43 144

IN)

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FIELD 2 FIELD 3 FIELD 4

FLAG TO INDICATE ANOTHER PRODUCT STREAM CARD = 0 THERE IS ANOTHER CARD= ] There are no more product stream data cards after this one

STAGE NUMBER OF PRODUCT STREAM LIQUID FLOW RATE (MOLES)VAPOR FLOW RATE ( MOLES )

145146147 Î 48 149 1 50151152

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J) INTERSTAGE HEATER/COOLER DATA CARD ( I 1 , I 9 , E 1 0 . 0 )IF THE FLAG ON DATA CARD 1 FIELD 5 IS G, DO NOT SUPPLY THESE CARDS

FIELD 1 : FLAG TO INDICATE ANOTHER INTERSTAGE HEATER/COOLER DATA CARD = 0 THERE IS ANOTHER CARD= 1 THERE ARE NO MORE INTERSTAGE HEATED/COOLER DATA CARDS AFTER

1 54 155 1 56157158

THIS ONE. 159

FIELD 2 : STAGE NUMBER OF INTE RSTAGE HEATER/COOLER 1 60p i ElD 3 : DUTY FOR THE INTERSTAGE HEATER/COOLER 161

162K) EXAMPLE SET OF PROGRAM DATA CARDS ( PP DATA USES ROUTINES IN SECTION 5) 1 63

THE LETTERS AT THE FAR RIGHT REFER TO THE ABOVE SECTIONS. 1641 65

03 EXAMPLE CHAPTER 4 A 166

PROPANE 44.09 6 . 5 620. 212. B. l 167

4750. 23 .5 .045 12100. 11.67 B .2 1 68N-3UTANE 58.12 6 .79 904 .8 232. B . l 1695775 . 32.675 .01875 15900. 19.75 B.2 170N-PENTANE 72.15 6.85 1041.57 227.1 B. l 1716800. 39.5 .02 19400. 27.5 8 . 2 172

C. l 173C.2 174

EXAMPLE CHAPTER 4 u 1756 3 200 1 5 176

2. 250. 250. 121. 275.5 F 1771 . 1. .0001 .0001 .0001 G 178

2V160. 5. 5 . H.l 179

10C5L225. 23. 37. 40. H.2 180toHto

213

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i - SET OF UNITS. PHYSICAL PROPERTY DATA CAN BE SUPPLIED VIA CUEFV AND COEFL 2175- OR A DATA STATEMENT IN THE USERS ROUTINE. BOTH LIQUID AND VAPOR ENTHALPIES 2180 MUST BE CALCULATED EACH TIME THE ROUTINE IS CALLED. 219Z: ARGUMENT LIST : ENT H ( NC, J , T1 , E NL , E NV, X, Y , IL , 1 R ) 223m NOMENCLATURE : 2218 NC IS THE NUMBER OF COMPONENTS 2225 J IS THE CURRENT STAGE BEING PROCESSED 223

T1 IS 1 HE TEMPERATURE OF STAGE J 224- ENL IS THE LIQUID ENTHALPY 2251 ENV IS THE VAPOR ENTHALPY 226g X IS THE MOLE FRACTION OF LIQUID 227 Y IS THE MOLE FRACTION OF THE VAPOR 228

^ IL AND IR ARE USED IN THE DIMENSION STATEMENT ONLY 2293 example : THE FOLLOWING ROUTINE IS USED IF THE USER DOES NOT SUPPLY ONE 230

^ NOTE : THIS ROUTINE USES A THIRD ORDER POLYNOMIAL TO CALCULATE THE ENTHALPY 231232

SUBROUTINE ENTH( NC, J , T1 , ENL, ENV, X, Y, IL , IR) 233DIMENSION X(IL, IR) ,Y(IL, IR) 234COMMON/PPOATA/COEFV(5 , 4 ) , COEFL( 5 , 4 ) ,CONVAP(5 , 6 ) ,NAME(5,2) ,AMW(5) 235T2=T1*T1 236T3-T2-T1 237ENV=0.0 238ENL=C.O 23900 500 1 = 1 ,NC 2 40ENV = (COEFV(1, 1) + C0EFV( I , 2 ) -T 1+ COE F V( I ,3 )<T2 + C0EFV( 1 , 4 ) T3) 2 41

1#Y( I,J)+ENV 242ENL = ( COEFL (I , 1 )+COEFL(I ,2 )’'T 1+CÜEFL ( I , 3 )*T2+C0EFL(I ,4 )--T3 ) 243

1*X( I,J)fENL 244500 CONTINUE 245

RETURN 2 46END 247

2488) SUBROUTINE GAMMA 249

PURPOSE : TO CALCULATE LIQUID PHASE ACTIVITY CHEF 2 50METHOD : USER SUPPLIED 251REQUIREMENTS AND LIMITATIONS : USER ACCEPTS RESFONSIBLITY FOR USING A CONSISTENT 252 M

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SET OF UNITS. AN ACTIVITY COcF MUST BE CALCULATED FOR EACH COMPONENT. 253ARGUMENT SEQUENCE : GAMMA(T, J , M, GAM ) 254

= NOMENCLATURE : 255:: T IS THE TEMPERATURE OF STAGE J 256g" J IS THE CURRENT STAGE BEING PROCESSED 257g NC IS THE NUMBER OF COMPONENTS 2585 GAM(I) IS THE LIQUID ACTIVITY COEF FOR COMPONENT I 259g- EXAMPLE : THE FOLLOWING ROUTINE IS USED IF THE USER DOES NOT SUPPLY ONE 2603 261I SUBROUTINE GAMMA(T, J , NC, GAM ) 262g c This assumes an ideal liquid phase 263

" COMMON X( 5 , 125 ) 264DIMENSION GAM(NC) 265DO 10 1= 1 , NC 266

10 GAM(I)=1.0 267

5 END 269g- 270

C) SUBROUTINE VAPP 2715 PURPOSE : TO CALCULATE THE PURE LIQUID FUGACITY 272I METHOD : USER SUPPLIED 273Z ARGUMENT LIST : VAPP( T, NC, P , VAP) 274S REQUIREMENTS AND LIMITATIONS : USER ACCEPTS R6SP0NSIBLITY FOR USING A CONSISTENT 275

2- SET OF UNITS. PHYSICAL PROPERTY DATA CAN BE SUPPLIED VIA CÜEFVAP 276I OR A DATA STATEMENT IN THE USERS ROUTINE. 277o NOMENCLATURE : 278^ T IS THE TEMPERATURE 279(D NC IS THE NUMBER OF COMPONENTS 260i. F IS THE PRESSURE 281M VAP(I) IS THE PURE LIQUID FUGACITY OF COMPONENT I ( I . E . VAPOR PRESSURE) 282§ EXAMPLE : THE FOLLOWING ROUTINE IS USED IF THE USER DOES NOT SUPPLY ONE 283

264SUBROUTINE VAPP(T, NC, P, VAP) 285

C » THIS ROUTINE ASSUMES : 286C 1) THAT THE VAPOR PRESSURE CORRELATION REQUIRES TEMPERATURE IN 287 ^C DEGRESS C AND THAT THE ROUTINE RECEIVES TEMPERATURE IN DEGRESS F 288 M

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FROM THE REBDILER (STAGE N) IS CALCULATED TO CLOSE THE OVERALL MATERIAL BALANCE.

7) GENERAL OVERVIEW OF PROGRAM

ASSUMPTIONS

THE

TYPE OF PROGRAM

EQUILIBRIUM STAGES COMPLETE MIXING PER STAGE ADIABATIC STAGE Two PHASES NO HEAT OF MIXING IDEAL VAPOR PHASE

THESE ASSUMPTIONS LEAD TO THE MESH EQUATIONS. MATERIAL BALANCE EQUILIBRIUM RELATIONSHIPS SUMMATIONS EQUATIONS HEAT BALANCE

PROGRAM RIGOROUSLY SOLVES THE MESH EQUATIONS.

MULTI-COMPONENTMULTI-FEED STREAMSMULTI-PRODUCT STREAMSMULTI-INTERSTAGE HEATERS/COOLERSRATING PROGRAM - REQUIRES THE NUMBER OF

FINDS THE SEPARATION. METHOD - WANG/HENKE (CONVERGENCE AIDED

STAGES AND

BY THE THETA METHODAND PARTIAL SUBSTITUTION)

PHYSICAL PROPERTY CALCULATIONS -ENTHALPY - USER SUPPLIEDLIQUID PHASE ACTIVITY COEFFICIENT - USER SUPPLIED PURE LIQUID FUGACITY - USER SUPPLIED VAPOR PHASE FUGACITY COEFFICIENT - IDEAL

3C1362363364365

3 66367368369 373371372373374375376377378379380

381382383384385386387388 369390391392393394

M00

APPENDIX B

PROGRAM LISTING OF STEADY STATE MODEL

219


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c * * * * * MAIN PROGRAM OVERLAY R E A D D S . L O O P O S . O U T D S OOOOOOlO 1C * * * * * P U T MAIN AND EN TH I N THE ROOT 0 0 0 2 0 2C * * * * * F E E D F L A S H R O U T I N E U S E S STAGE 1 S E E R O U T I N E S KVAL AND F E E D 3

REAL L , K 0 0 0 3 0 4C O M M O N / P P D A T A / C O E F V ( 5 # 4 ) , C O E F L ( 5 , 4 ) , C 0 N V A P ( 5 . 6 ) , N A M E ( 5 , 2 ) , AM W (5 ) 0 0 0 4 0 5COMMON X ( 5 t 1 2 5 ) , Y ( 5 , I 2 5 ) . K ( 5 , 1 2 5 ) , S M F L ( 5 , 1 2 5 ) • S M F V ( 5 , 1 2 5 ) 0 0 0 SO 6COMMON T ( | 2 5 ) , P ( 1 2 5 ) • V ( 1 2 5 ) , L ( 1 2 5 ) * 0 ( 1 2 5 ) . U ( 1 2 5 ) , W ( 1 2 5 ) . H L ( 1 2 5 ) , 0 0 0 6 0 7

* H V ( 1 2 5 ) , F L ( 1 2 5 ) , F V ( 1 2 5 ) , H F L ( t 2 5 > • H F V ( 1 2 5 ) . I T I T ( 2 0 ) , F Q ( 1 2 5 ) . I F C O N 7 0 8COMMON N S . N C , R . T M I N . T M A X . I T M A X . T L T . t l f . t l x . i e q m o l . n i t t . i p r i n t . 0 8 0 9

* A L P H A T , A L P H L V . I C T H 0 0 0 9 0 1 0E Q U I V A L E N C E ( U ( I ) . D I S T L ) . ( V ( 1 ) . D I S T V ) 0 0 0 1 0 0 11D I M E N S I O N I S Y ( 5 ) 1 1 0 1 2DATA I S Y / * 1 ' . * 2 ' , ' 3 * . * 4 ' , ' 5 ' / 1 2 0 1 3DATA I S T A R T / 1 / 0 0 0 1 3 0 1 4

1 0 C O N T I N U E 0 0 0 1 4 0 1 5CALL R E A D D S C I S T A R T • I P L O T ) 1 5 0 1 6CALL L O O P D S 0 0 0 1 6 0 17CALL OUTDS 0 0 0 1 7 0 1 8CALL EXTRA 1 8 0 1 9I F ( I P L O T . E Q . O ) G O TO 1 0 1 9 0 2 0P R I N T l O O . I T I T 2 0 0 2 1P R I N T 1 0 2 . ( ( N A M E ! I . J ) . J = 1 . 2 ) . I S Y ( I ) . I = 1 . N C ) 2 1 0 2 2CALL P Q L C O M ( X , N S . N C * I S Y ) 2 2 0 2 3P R I N T l O l . I T I T 2 3 0 2 4P R I N T 1 0 2 , ( ( N A M E ( I , J ) , J = I , 2 ) , I S Y ( I ) , 1 = 1 , N C ) 2 4 0 2 5CALL P Q L C O M ( Y , N S , N C , I S Y ) 2 5 0 2 6GO TO 1 0 0 0 0 2 6 0 2 7

1 0 0 F O R M A T ! * 1 l i q u i d C O M P O S I T I O N ( M . F . ) VS S T A G E S ( S E M I - L O G ) ' . 5 X . 2 0 A 4 ) 2 7 0 281 0 1 F O R M A T ! • 1 VAPOR C O M P O S I T I O N ( M . F . ) VS S T A G E S ( S E M I - L O G ) ' . 5 X . 2 0 A 4 ) 2 8 0 2 91 0 2 F O R M A T ( S ( 4 X . 2 A 4 . ' = ' . A l ) ) 2 9 0 3 0

END 0 0 0 3 0 0 31S U B R O U T I N E KVALUECN C .J , T • P , S U M Y ) 0 0 3 1 0 32REAL K 0 0 0 3 2 0 3 3COMMON X ( 5 . 1 2 5 ) . Y ( 5 . 1 2 5 > . K ( 5 . 1 2 5 ) 0 0 0 3 3 0 3 4

C * * * * * G I V E N X AND T F I N D S K 0 0 0 3 4 0 3 5C * * * * * MAKE SURE C O N S I S T A N T TE M PE R A T U R E S AND P R E S S U R E S U N I T S ARE USEDOOO 3 5 0 3 6

D I M E N S I O N G A M ( 5 ) , V A P ( S ) 0 0 0 3 6 0 3 7CALL V A P P ( T . N C . P . V A P ) 3 7 0 3 8CALL G A M M A ! T . J . N C . G A M ) 0 0 3 8 0 3 9S U M Y = - I . O 0 0 0 3 9 0 4 0DO 5 0 0 1 = 1 . NC 0 0 4 0 0 4 1K ( I , J ) = G A M ( I ) * V A P ( I ) / P 0 0 0 4 1 0 4 2

5 0 0 S U M Y = S U M Y + K ( I , J ) * X ( I , J ) 0 0 0 4 2 0 4 3RETURN 0 0 0 4 3 0 4 4END 0 0 0 4 4 0 4 5S U B R O U T I N E V A P P ! T . N C . P . V A P ) 4 5 0 4 6

C 1 ) THAT TH E VAPOR P R E S S U R E C O R R E L A T I O N R E Q U I R E S TEMPE RA TURE I N 4 6 0 4 7 roroo

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C O M M O N / P P D A T A / C O E F V ( 5 . 4 ) . C O E F L ( 5 . 4 ) . C O N V A P ( 5 . 5 ) . NAME( 5 . 2 ) . AMM( 5 ) 0 5 0 0 5 1D I M E N S I O N V A P ( N C ) 5 1 0 5 2T I N = ( T + 4 6 0 . 1 / 1 . 8 - 2 7 3 . 5 2 0 5 3DO 1 0 1 = 1 , NC 5 3 0 5 4

10 V A P ( 1 ) = 1 0 . 0 * * ( C O N V A P ( I . 1 ) - C O N V A P ( I . 2 ) / ( C O N V A P ( I . 3 ) + T I N ) ) / 5 1 . 7 3 5 8 7 5 4 0 5 5RETURN 5 5 0 5 6END 5 6 0 5 7S U B R O U T I N E E N T H f N C . J • T 1 . E N L . E N V . X . Y . I L • I R > 5 7 0 5 8D I M E N S I O N X ( I L . I R ) . V ( I L . I R ) 5 8 0 5 9C O M M O N / P P D A T A / C O E F V ( 5 . 4 ) « C O E F L ( 5 . 4 ) . C O N V A P ( 5 . 6 ) . NAME( 5 . 2 ) . AMM( 5 ) OOO 5 9 0 6 0

c * * * * * P E R F O R M S EN T H A L PY C A L C U L A T I O N 000 6 0 0 6 1c * * * * * CO EFV C O N T A I N S VAPOR E N THALPY C O E F S 000 6 1 0 6 2c * * * * * C O E F L C O N T A I N S L I Q E N THA LPY C O E F S 000 6 2 0 6 3

T 2 = T 1 * T 1 000 6 3 0 6 4T 3 = T 2 * T I 0 00 640 65E N V = 0 .0 000 650 66E N L = 0 .0 0 00 660 67DO 500 1 = 1 .NC 00 670 6 8E N V = ( C O E F V ( 1 . 1 ) + C O E F V ( I . 2 ) * T l + C O E F V ( I . 3 ) * T 2 + C O E F V ( I . 4 ) * T 3 ) 000 6 8 0 6 9

1 * Y ( I . J ) + E N V 000 6 9 0 7 05 0 0 • E N L = ( C O E F L ( I . l ) + C O E F L ( I . 2 ) * T l + C O E F L ( I . 3 ) * T 2 * C O E F L ( I . 4 ) * T 3 ) 000 7 0 0 7 1

1 * X ( 1 , J ) + E N L 000 7 1 0 7 2RETURN 000 7 2 0 7 3END 000 7 3 0 7 4S U B R O U T I N E GAMMA( T . J . N C . GAM) 740 7 5

C * * * * * T H IS ASSUMES AN IDEAL L IQ U ID PHASE 7 5 0 7 6c * * * * * J I S S TAGE NC I S NUMBER OF COMPO NETS 7 6 0 7 7

COMMON X ( 5 . 1 2 5 ) 7 7 0 7 8D I M E N S I O N G A M (N C ) 7 8 0 7 9DO 1 0 1 = 1 . NC 7 9 0 8 0

10 GAM( 1 ) = 1 . 0 8 0 0 8 1RETURN 8 1 0 8 2END 820 8 3SUBROUTINE READDS(I START• IP L O T ) 940 84REAL L .K 000 9 5 0 8 5COMMON/PPDA T A / C 0 E F V ( 5 . 4 ) . C 0 E F L ( 5 . 4 ) . C ONVAP( 5 . 6 ) , NAME( 5 . 2 ) . AMW( 5 ) 0 0 0 9 6 0 86COMMON X ( 5 . 1 2 S ) . Y ( 5 . I 2 5 ) . K ( 5 . I 2 S ) . S M F L ( 5 . 1 2 5 ) . S M F V ( 5 . I 2 5 ) 000 9 7 0 8 7COMMON T ( 1 2 S ) . P ( 1 2 5 ) . V ( 1 2 5 ) . L ( 1 2 5 ) . 0 1 1 2 5 ) , U ( 1 2 5 ) * M ( 1 2 5 ) . H L ( 1 2 5 ) . 000 9 8 0 8 8

* H V ( 1 2 5 ) . F L ( 1 2 5 ) , F V ( 1 2 5 ) . H F L ( 1 2 5 ) , H F V ( 1 2 5 ) • I T I T ( 2 0 ) . F Q ( 1 2 5 ) • I F C O N 9 9 0 8 9COMMON N S . N C . R . T M I N . T M A X . I T M A X . T L T . t l f . t l x . I E Q M O L . N I T T . I P R I N T . 0 1000 9 0

* A L P H A T . A L P H L V . I C T H 0 0 0 1010 9 1E Q U I V A L E N C E ( U ( 1 ) . D I S T L ) . ( V ( 1 ) . D I S T V ) 0 00 1020 9 2D I M E N S I O N S F ( 1 0 . 1 ) 1 0 3 0 9 3DATA C V . C L . C F / • V • . » L • . » F • / 1 0 4 0 9 4 |N>

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c lE Q M O L I S TH E F L A G THAT D E T E R M I N E S I F EOUI MO LAL OVERFLOW I S ASSOOO 1 0 5 0 9 5c I F lEO MOL I S 0 THEN E O U IM O L A L OVERFLOW I S ASSUMED OOO 1 0 6 0 9 6c I F lE Q M O L I S 1 E N THALP Y B A LANCE S ARE PE RFO RM ED I N G E T T I N G NEW LOOO 1 0 7 0 9 7c * * * * * N S I S THE NUMBER OF S T A G E S 0 0 1 0 8 0 9 8c * * * * * NC I S THE NUMBER OF COMPENENTS 0 0 1 0 9 0 9 9c * * * * * P O S I T I V E O I S HEAT ADDED * * * * * 0 0 0 1 1 0 0 1 0 0c * * * * * L ( N S ) I S BOTTOMS FLOW RATE 0 0 1 1 1 0 1 0 1c * * * * * D I S T L I S U ( l ) * * * * * D I S T V I S V { 1 ) 0 0 0 1 1 2 0 1 0 2c T R I D I A G O N A L MATRIX D I S T I L L A T I O N PROGRAM 0 0 0 1 1 3 0 1 0 3c W R I T T E N BY K .M .W O N G AND P . A . B R Y A N T { D E C . 1 9 7 5 ) 0 0 0 1 1 4 0 1 0 4c U S I N G METHOD OF WANG AND HENKE (HYDROCARBON P R O C E S S I N G . 0 0 0 1 1 5 0 1 0 5c V O L . 4 5 . N O . 8 . A U G U S T . 1 9 6 6 ) 0 0 0 1 1 6 0 1 0 6c T R I - D I A G O N A L METHOD BY BOST ON AND S U L L I V A N ( C A N D I A N JO URNAL 0 0 0 1 1 7 0 1 0 7c OF CH EM IC AL ENG V O L . 5 0 . O C T . 1 9 7 2 ) 0 0 0 1 1 8 0 1 0 8c * * * * * R E W R I T T E N BY K . W . K O M I N E K MAY 1 9 7 7 0 0 0 1 1 9 0 1 0 9c * * * * * PROGRAM R E Q U I R E S 2 0 0 K B Y T E S IBM OF CORE 1 2 0 0 1 1 0c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 2 1 0 1 1 1c * * * * * READ AND P R I N T P H Y S I C A L P R O P E R T Y DATA * * * * * 0 0 0 1 2 2 0 1 1 2

I F ( I S T A R T . N E . l ) G O TO 2 0 5 0 0 0 1 2 3 0 1 1 3READ 1 0 3 . N C . I T I T 1 2 4 0 1 1 4P R I N T 1 7 3 . N C . I T I T 1 2 5 0 1 1 5DO 2 1 0 1 = 1 . NC 0 0 1 2 6 0 1 1 6READ 2 1 1 . ( N A M E ! I . J ) • J = t . 2 ) . A M W ( I ) . ( C O N V A P C I . J ) . J = 1 . 6 ) 0 0 0 1 2 7 0 1 1 7READ 3 0 0 . ( C O E F L ( I . J ) . J = 1 . 4 ) . ( C O E F V ( I • J ) . J = l . 4 ) 0 0 0 1 2 8 0 1 1 8P R I N T 2 2 0 . I . ( N A M E ( I . J ) . J = 1 . 2 ) • A M W ( I ) 1 2 9 0 1 1 9P R I N T 2 2 3 . ( C O N V A P ( I . J ) . J = 1 . 6 ) 0 0 0 1 3 0 0 1 2 0P R I N T 2 2 1 . ( C O E F L ( 1 . J ) . J = 1 . 4 ) 0 0 0 1 3 1 0 1 2 1

2 1 0 P R I N T 2 2 2 . ( C 0 E F V ( I . J ) , J = 1 , 4 ) 0 0 0 1 3 2 0 1 2 2C * * * * * READ AND P R I N T I N I T I A L VALUES OF TOP AND BOTTOMS C O M P O S I T I O N 1 3 3 0 1 2 3

P R I N T 2 9 9 1 3 4 0 1 2 4DO 2 0 9 J = 1 . 2 1 3 5 0 1 2 5READ 3 0 1 . ( X ( I . J ) . I = 1 . N C ) 1 3 6 0 1 2 6

2 0 9 P R I N T 3 0 2 . ( X f I . J ) . I = 1 . N C ) 1 3 7 0 1 2 7C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 3 8 0 1 2 8C * * * * * r e a d T I T L E AND DATA CARD S 1 . 2 . 3 1 3 9 0 1 2 92 0 5 READ f 5 . 1 0 1 . E N D - 1 0 0 0 ) I T I T 1 4 0 0 1 3 0

READ 1 0 0 . N S . N C . 1 T M A X . l E O M O L . I F Q . I P R I N T . I P L O T . I F C O N 1 4 1 0 1 3 1READ 3 0 0 . R . P T O P . P B O T T . T T O P . T B O T 0 0 0 1 4 2 0 1 3 2READ 3 0 0 . A L P H A T . A L P H L V . T L T • T L P . T L X . T M I N . T M A X 0 0 0 1 4 3 0 1 3 3I F I T M I N . E Q . 0 . 0 ) T M I N = T T O P - 1 0 0 . 0 1 4 4 0 1 3 4I F I T M A X . E O . 0 . 0 ) T M A X = T B O T + 1 0 0 . 0 1 4 5 0 1 3 5I F ( A L P H L V . G T . 2 . 0 ) A L P H L V = 2 . 0 0 0 0 1 4 6 0 1 3 6I F ( A L P H L V . L T . 0 . 1 ) A L P H L V = 0 . 1 0 0 0 1 4 7 0 1 3 71 F ( A L P H A T . G T . 2 . 0 ) A L P H A T = 2 . 0 0 0 0 1 4 8 0 1 3 8I F ( A L P H A T . L T . O . 1 ) A L P H A T = 0 . 1 0 0 0 1 4 9 0 1 3 9P R I N T 1 0 2 . I T I T 1 5 0 0 1 4 0P R I N T 1 7 0 . N S . N C . I T M A X . l E Q M O L . I F O . I P R I N T . I P L O T . I F C O N 0 1 5 1 0 1 4 1 roro

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P R I N T I 7 2 , R . P T O P * P B O T T . T T O P . T B O T 0 0 0 I S 2 0 1 4 2P R I N T ! 7 9 , A L P H A T . A L P H L V , T L T , T L F . T L X . T M I N , T M A X 0 0 0 1 5 3 0 1 4 3I F ( N S . L T . 2 . 0 R . N S . G T . 1 2 S ) S T 0 P 9 9 7 7 1 4 4I F ( N C . L T . 2 . 0 R . N C . G T . S ) S T 0 P 9 9 7 6 1 4 5N M 1 = N S - 1 0 0 1 5 6 0 1 4 6E = C P B O T T - P T O P ) / N M l 1 5 7 0 1 4 7DO 1 J = I . N S 0 0 1 S 8 0 1 4 8

C * * * * * G E N E R A T E S A L I N E A R P R E S S U R E P R O F I L E 1 5 9 0 1 4 9P U ) = P T O P ♦ E * C J - l ) 1 6 0 0 1 5 0F L f J ) = 0 . 0 0 0 0 1 6 1 0 1 5 1F V ( J ) = 0 « 0 0 0 0 1 6 2 0 1 5 2F O ( J ) = 0 . 0 1 6 3 0 1 5 3U ( J ) = 0 . 0 0 0 0 1 6 4 0 1 5 4V l J i = 0 . 0 0 0 0 I 6 S 0 1 5 5Q ( J ) = 0 . 0 OOO 1 6 6 0 1 5 6H F L ( J ) = 0 . 0 0 0 0 1 6 7 0 1 5 7H F V C J ) = 0 . 0 OOO 1 6 8 0 1 5 8DO 1 1 = 1 . N C 0 0 1 6 9 0 1 5 9S H F L ( I . J ) = 0 . 0 0 0 0 1 7 0 0 1 6 0

1 S M F V I I . J ) = 0 . 0 0 0 0 1 7 1 0 1 6 1C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 7 2 0 1 6 2C * * * R E A D I N F E E D CAR DS WITH COMPONENT FLOWS I N EACH F E E D STREAM 0 0 0 1 7 3 0 1 6 3C * * * C A L C U L A T E TOTAL F E E D AND TOTAL E N T H A L PY OF EACH F E E D STREAM 0 0 0 1 7 4 0 1 6 4

P R I N T 9 6 6 . ( I . 1 = 1 . N C ) 0 0 1 7 5 0 1 6 5SUMF = 0 . 0 0 0 0 1 7 6 0 1 6 6

2 READ l a O . N S T O P . N F S . F T Y P E . F T E M P . S F 1 7 7 0 1 6 7I F < I E Q M O L . £ Q . O ) C A L L F E E D O ( N F S . F T E M P , S F . S U M F ) 1 7 8 0 1 6 8I F ( l E O M O L . E Q . O ) G O TO SO 1 7 9 0 1 6 9I F ( F T Y P E . E Q . C V ) C A L L F E E D ( N F S . F T Y P E . F T E M P . S F . 1 8 0 0 1 7 0

* F V . S M F V . H F V . S U M F . N C . C V . C L ) 1 8 1 0 1 7 1I F ( F T Y P E . E O . C D C A L L F E E D ( N F S . F TY PE . F T E M P . S F . 1 8 2 0 1 7 2

* F L . S M F L . H F L . S U M F . N C . C V . C L ) 1 8 3 0 1 7 3I F ( F T Y P E . E O . C F ) C A L L F E E D F L I N F S . F T Y P E . F T E M P . S F . C V , C L . S U M F ) 1 8 4 0 1 7 4

SO I F ( N S T O P . E O . O ) GO TO 2 0 0 0 1 8 5 0 1 7 5P R I N T 2 7 . SUMF 0 0 0 I 8 6 0 1 7 6

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 8 7 0 1 7 7C * * * * * R E A D A N D P R I N T P R O D U C T F L O W R A T E S 0 0 0 1 8 8 0 1 7 8

S U M P = 0 . 0 0 0 0 1 8 9 0 1 7 9P R I N T 8 0 3 0 0 0 1 9 0 0 1 8 0

3 READ 2 0 0 . N S T O P . N T . E N H F L . E N H F V 0 0 0 1 9 1 0 1 8 1P R I N T 8 0 6 . NT . E N H F L . E N H F V 0 0 0 1 9 2 0 1 8 2U ( N T ) = U ( N T ) + E N H F L 0 0 0 1 9 3 0 1 8 3W(NT ) = W ( N T ) + E N H F V 0 0 0 1 9 4 0 1 8 4S U M P = S U M P + E N H F L + E N H F V 0 0 0 1 9 5 0 1 8 5I F ( N S T O P . E O . O ) GO TO 3 0 0 0 1 9 6 0 1 8 6D I S T V = W ( 1 ) 0 0 0 1 9 7 0 1 8 7W ( 1 ) = 0 . 0 0 0 0 1 9 8 0 1 8 8 ro

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L ( N S ) = S U M F - S U M P 0 0 1 9 9 0 1 8 9P R I N T 8 0 6 t N S * L ( N S ) 0 2 0 0 0 1 9 0I F f L ( N S ) . L T . O . O I S T O P 7 7 7 8 0 0 2 0 1 0 1 9 1

c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2 0 2 0 1 9 2C * * * R E A D S HEAT L O S S E S ON TRAYS 0 0 0 2 0 3 0 1 9 3

I F ( I F Q . E Q . O ) GO TO 1 2 2 0 4 0 1 9 4P R I N T 1 6 0 2 0 5 0 1 9 5

9 READ 2 0 0 . N S T O P . N T . E 2 0 6 0 1 9 6P R I N T 8 0 6 . N T . E 2 0 7 0 1 9 70 ( N T ) = 0 ( N T ) + E 2 0 8 0 1 9 8I F ( N S T O P . E Q . O ) G O TO 9 2 0 9 0 1 9 9

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2 1 0 0 2 0 0C * * * * * I N I T I A L I Z E COLUMN P R O F I L E S F O R X . Y . L . V . T 2 1 10 2 0 1I Z L f 1 ) - R * ( D I S T L + D I S T V ) OOO 2 1 2 0 2 0 2

I F I R . E Q . O . O ) L ( 1 > = F L ( I ) 0 0 0 2 1 3 0 2 0 3V ( 2 ) = W ( I ) + V ( I ) + U ( I ) + L ( I ) - F V ( I ) - F L ( I 1 0 0 0 2 1 4 0 2 0 4I F ( I S T A R T . N E . 1 ) G 0 TO 6 5 2 1 5 0 2 0 5c * * * * * I N I T I A L I Z A T I O N PROCE DURE FO R F I S R T RUN 2 1 6 0 2 0 6

c * * * * * I N I T I A L I Z E L AND V WITH EO U IM O LA L OVERFLOW A S S U M P T I O N 2 1 7 0 2 0 7DO 3 2 J = 2 , N M I 2 1 8 0 2 0 8A L = F L ( J ) 2 1 9 0 2 0 9A V = F V ( J » 2 2 0 0 2 1 0I F ( l E O M O L . E Q . O ) A V = ( 1 . 0 - F Q ( J ) ) * F V ( J ) 2 1 1I F ( l E O M O L . E Q . O >A L = F Q ( J ) * F V ( J ) 2 1 2

L 1 J ) =LC J - 1 ) - U I J ) +AL 2 2 3 0 2 1 33 2 V ( J + I ) = V ( J ) + W ( J ) - A V 2 2 4 0 2 1 4C * * * * * I N I T I A L I Z E T E M PE RA TURE WITH A L I N E A R P R O F L I E 2 2 5 0 2 1 5

E = I T B O T - T T O P ) / N M I 0 0 0 2 2 6 0 2 1 6DO 5 1 J = I . N S 2 2 7 0 2 1 7

S I T ( J ) = T T D P + E * < J - i ) 2 2 8 0 2 1 8C * * * I N I T : A L : Z E Y AND X P R O F I L E 0 0 0 2 2 9 0 2 1 9

S U M - 0 . 0 2 3 0 0 2 2 0DO 5 2 1 = 1 . NC 2 3 1 0 2 2 1X ( I « N S ) = X ( I . 2 ) 2 3 2 0 2 2 2

5 2 S U M = S U M + X ( I . I ) + X ( I . N S ) 2 3 3 0 2 2 3E = 0 . 0 2 3 4 0 2 2 4I F C A B S ( S U M ) . G T . I . O E - 5 ) G O TO 5 4 2 3 5 0 2 2 5DO 5 6 1 = 1 , NC 2 3 6 0 2 2 6

5 6 X ( 1 , 1 ) = | . 0 2 3 7 0 2 2 75 4 DO 5 5 1 = 1 . NC 2 3 8 0 2 2 8

I F ( X ( I . l ) . L T . 1 . 0 E - 4 ) X ( I . l ) = 1 . 0 E - 4 2 3 9 0 2 2 9I F ! A B S f S U M ) . G T . t . 0 E - 5 ) E = ( X ( I . N S ) - X ( l . t ) ) / N M : 2 4 0 0 2 3 0DO 5 5 J = 1 . N S 2 4 1 0 2 3 1

5 5 XI I . J ) = X ( I . I ) + E * ( J - 1 ) 2 4 2 0 2 3 2DO 5 7 J = 1 . N S 2 4 3 0 2 3 3S U M = 0 . 0 2 4 4 0 2 3 4DO 5 8 1 = 1 . NC 2 4 5 0 2 3 5 roMf-

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s a S U M = S U M + X I I . J ) 2 4 6 0 2 3 6DO 5 7 1 = 1 . NC 2 4 7 0 2 3 7X I I . J ) = X f I . J ) / S U M 2 4 8 0 2 3 8

5 7 Y ( I . J ) = X ( I . J ) 2 4 9 0 2 3 9GO TO 6 6 2 5 0 0 2 4 0

c * * * * * I N I T I A L I Z A T I O N PRO CEDURE FO R EACH RUN A F T E R TH E F I R S T 2 5 1 0 2 4 16 5 I F I N S . L E . I S T A R T I GO TO 6 7 2 5 2 0 2 4 2C * * * * * EXTEND P R O F I L E S TO I N C L U D E NEW S T A G E S 2 5 3 0 2 4 3

DO 6 0 J = I S T A R T . N S 2 5 4 0 2 4 4T ( J ) = T I I S T A R T ) 2 5 5 0 2 4 5V ( J ) = V C I S T A R T ) 2 5 6 0 2 4 6I F ( J . N E . N S ) L ( J 1 = L ( I S T A R T - 1 ) 2 5 7 0 2 4 7DO 6 0 1 = 1 , NC 2 5 8 0 2 4 8X ( I . J ) = X < I . I S T A R T ) 2 5 9 0 2 4 9

6 0 Y ( 1 . J ) = Y I I . I S T A R T ) 2 6 0 0 2 5 06 7 I F ( l E Q M O L . E Q . 1 ) GO TO 6 6 2 6 1 0 2 5 1C * * * * * EO UIMO LAL OVERFLOW I S A S S U M E D . GEN E R A T E A NEW L . V P R O F I L E 2 6 2 0 2 5 2C * * * * * FQ I S F R A C T I O N L I Q U I D 2 5 3

DO 3 4 J = 2 . N M 1 2 6 3 0 2 5 4L I J ) = L ( J - 1 ) - U ( J ) + F Q ( J ) * F V { J ) 2 6 4 0 2 5 5

3 4 V I J + l ) = V I J ) + W ( J ) - ( 1 . 0 - F Q ( J ) ) * F V ( J ) 2 6 5 0 2 5 66 6 I S T A R T = N S 2 6 6 0 2 5 7c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2 6 7 0 2 5 8C * * * P R I N T I N I T I A L I Z E P R O F I L E S 0 0 0 2 6 8 0 2 5 9

I F ! I P R I N T . E Q . O ) G O TO 7 7 9 0 0 0 2 6 9 0 2 6 0P R I N T 1 0 4 0 0 0 2 7 0 0 2 6 1DO 1 0 7 3 = 1 . NS 0 0 2 7 1 0 2 6 2

1 0 7 P R I N T 1 0 8 . J . T ( J ) . P ( J ) . L C J ) . V f J ) . Q ( J ) 0 0 0 2 7 2 0 2 6 37 7 9 C O N T I N U E 0 0 0 2 7 3 0 2 6 4

P R I N T 1 5 0 0 0 0 2 7 4 0 2 6 5RETURN 0 0 0 2 7 5 0 2 6 6

1 0 0 0 S T O P 0 0 0 2 7 6 0 2 6 7C * * * * * FORMATS US ED FOR I N P U T * * * * * 2 7 7 0 2 6 81 0 0 F 0 R M A T I 8 I 5 ) 2 7 8 0 2 6 91 0 1 F O R M A T ! 2 0 A 4 ) 2 7 9 0 2 7 01 0 3 F O R M A T ! I 2 . 1 9 A 4 . A 2 ) 2 8 0 0 2 7 1I S O F O R M A T ! I 1 . I 3 . A 1 . F 5 . 0 . 1 0 F 7 . 2 ) 2 8 1 0 2 7 22 0 0 F O R M A T ! I 1 . I 9 . 2 E 1 0 . 0 ) 0 0 0 2 8 2 0 2 7 32 1 1 F O R M A T I 2 A 4 . 2 X . 7 E 1 0 . 0 ) 0 0 0 2 8 3 0 2 7 43 0 0 F O R M A T I B E I O . O ) 0 0 0 2 8 4 0 2 7 53 0 1 F O R M A T ! 1 0 F 8 . 0 ) 2 8 5 0 2 7 6C * * * * * FORMATS USED FOR OU TP UT * * * * * 2 8 6 0 2 7 72 7 FORMAT! / S I X . ‘ SUM OF F E E D S = • . F I 1 . 3 . • M O L S * ) 2 8 7 0 2 7 81 0 2 F O R M A T ! M T I T L E CARD 0 • . 2 0 A 4 ) 2 8 8 0 2 7 91 0 4 F O R M A T I / l O X , * I N I T I A L VA LUES F O R S TA GE V A R I A B L E S ' / I O X . 2 8 9 0 2 8 0

* • S T A G E • . 5 X . " T E M P ' . 9 X . ' P R E S S ' . 7 X . ' L I Q U I D ' . 7 X , ' V A P O R ' . 4 X . 2 9 0 0 2 8 1* ' I N T E R S T A G E H E A T E R / C O O L E R ' ) 2 9 1 0 2 8 2 ro

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1 0 8 F O R M A T ! l O X . I 3 . S F I 3 . 3 11 5 0 F 0 R M A T ( / 1 5 X , 1 0 ( ' * * ) . ' MAXIMUM R E L A T I V E CHANGE ' . I 0 ( ' * ' )

* / • I T E R A T I O N * . 9 X , * T E M P * , 1 3 X , ' V A P F L O W ' . I O X . ' L I O M . F . ' ,* lO X . 'B U B I T ' . 9 X . ' I C T H ' )

1 6 0 F O R M A T ! / ' I N T E R S T A G E H E A T E R / C O O L E R D U T I E S ' I1 7 0 F O R M A T ! • DATA CARD I ' / I O X . ' N U M B E R OF S T A G E S = ' . I 3 . 5 X .

* ' ( I N C L U D I N G COND ENSER AND R E B O I L E R ) '« / t O X . ' N U M B E R O F COMPONENTS = ' « 1 3 /* l O X . MT M AX = ' . I 5 . S X « ' I E O M O L = ' . I I , S X .♦ • I F Q = • • I I . 5 X . ' I P R I N T = ' . I I . S X . ' I P L O T = ' . I I . S X . ' I F C O N = ' . I I )

1 7 2 F O R M A T ! ' DATA CARD 2 ' / I OX. ' R E F L U X R A T I O = ' . F 1 0 . 4 . ' L / D ' /* l O X . 'COLUM N P R E S S U R E = ' . F 1 0 . 2 . ' T O ' . F 1 0 . 2 .* / I OX. ' I N I T I A L TE MP ERA TU RE P R O F I L E = ' . F 8 . 2 . ' TO ' . F 8 . 2 )

1 7 3 F O R M A T ! I H l . ' P H Y S I C A L P R O P E R T Y DATA NUMBER OF COMPONENTS = ' .* 1 3 / ' T I T L E CARD 0 ' . A 2 . 1 9 A 4 )

1 7 9 F O R M A T ! ' DATA CARD 3 ' / I O X . ' A L P H A T = ' . F 7 . 3 . ' AL PH LV = ' . F 7 . 3 / 1 0 X .4 ' M A X I M I U M R E L A T I V E CHANGE T O LE RA N CE 0 TEMPE RAT URE = ' . F 6 . 4 . 3 X .* ' FLOW = ' . F 6 . 4 . 3 X . ' L I Q MOLE F R A C T I O N = ' . F 6 . 4* / l O X . ' T M I N = ' . F I 0 . 3 . 5 X . ' T M A X = ' . F I 0 . 3 )

2 2 0 F 0 R M A T I / 5 X . I 2 . 3 X . 2 A 4 . 5 X . ' M O L E C U L A R WEIGHT = ' . F I 2 . 4 )2 2 1 F O R M A T ! l O X . ' L I Q U I D ENTH ' . I P 4 E I 5 . 5 )2 2 2 F O R M A T ! I O X . ' V A P O R ENTH ' . I P 4 E I 5 . 5 )2 2 3 F O R M A T ! I OX. 'V A P O R P R E S S . ' . 1 P 6 E I S . 5 )2 9 9 F O R M A T ! / ' I N I T I A L L I Q U I D C O M P O S I T I O N 0 T O P AND BOTTOM S T A G E S ' )3 0 2 F O R M A T ! I O F I O . S )8 0 3 F O R M A T ! 9 X . ' P R O D U C T S T R E A M S ' . 3 X . ' S T A G E ' . 6 X . ' L I Q U I D ' * 6 X . ' V A P O R ' .

* S X . ' ! M O L E S / U N I T T I M E ) ' )8 0 6 F O R M A T ! 2 8 X . I 3 . I X . 2 ! 2 X . G I 0 . 3 ) )9 6 6 F O R M A T ! / ' F E E D S T R E A M S ' . 3 7 X . ' C O M P O N E N T S ! M O L E S / U N I T T I M E ) ' . 2 6 X .

* ' T O T A L ' . 6 X .* ' E N T H A L P Y ' / I 0 3 X . ' M 0 L S ' . 7 X . ' E N E R G Y / M O L ' / ' S T A G E ' . 6 X , ' T E M P ' . 1 0 1 8 )

ENDS U B R O U T I N E F E E D Q ! N F S . F T E M P . S F . S U M F )REAL L . KCOMMON X ! S . I 2 S ) . Y ! S . I 2 S ) . K ! S . 1 2 S ) . S M F L ! 5 . 1 2 5 ) . S M F V ! 5 . 1 2 S )COMMON T ! 1 2 5 ) . P ! 1 2 5 ) . V ! 1 2 5 ) . L ! 1 2 5 ) . Q ! 1 2 5 ) . U ! 1 2 5 ) . W ! 1 2 5 ) . H L ! 1 2 5 ) .

* H V ! 1 2 5 ) . F L ! I 2 5 ) . F V ! 1 2 5 ) . H F L ! I 2 5 ) . H F V ! 1 2 5 ) • I T I T ! 2 0 ) . F O ! 1 2 5 ) • I F C O N COMMON N S . N C . R . T M I N . T M A X . I T M A X . T L T . T L F . T L X , I E Q M O L . N I T T . I P R I N T .

* A L P H A T . A L P H L V . I C T H D I M E N S I O N S F ! 1 0 . I )S U M = 0 . 0DO 1 6 0 1 = 1 . NC S U M = S U M + S F ! I . l )

1 6 0 S M F V ! I . N F S ) = S M F V ! I . N F S ) + S F ! I . l )C * * * * * FQ I S F R A C T I O N L I Q U I D

F Q ! N F S ) = ! F Q ! N F S ) * F V ! N F S ) + F T E M P * S U M ) / ! F V ! N F S ) ♦ S U M )SUMF=SUMF+SUM

2 9 2 0 2 8 30 0 0 2 9 3 0 2 8 40 0 0 2 9 4 0 2 8 50 0 0 2 9 5 0 2 8 6

2 9 6 0 2 8 72 9 7 0 2 8 8

0 0 0 2 9 8 0 2 8 90 0 0 2 9 9 0 2 9 00 0 0 3 0 0 0 2 9 1

3 0 1 0 2 9 20 0 0 3 0 2 0 2 9 3

3 0 3 0 2 9 40 0 0 3 0 4 0 2 9 5

3 0 5 0 2 9 63 0 6 0 2 9 7

0 0 0 3 0 7 0 2 9 80 0 0 3 0 8 0 2 9 90 0 0 3 0 9 0 3 0 00 0 0 3 1 0 0 3 0 1

3 I I 0 3 0 20 0 0 3 1 2 0 3 0 30 0 0 3 1 3 0 3 0 40 0 0 3 1 4 0 3 0 5

3 1 5 0 3 0 63 1 6 0 3 0 73 1 7 0 3 0 83 1 8 0 3 0 93 1 9 0 3 1 03 2 0 0 3 1 13 2 1 0 3 1 23 2 2 0 3 1 3

0 0 0 3 2 3 0 3 1 43 2 4 0 3 1 53 2 5 0 3 1 63 2 6 0 3 1 73 2 7 0 3 1 83 2 8 0 3 1 93 2 9 0 3 2 03 3 0 0 3 2 13 3 1 0 3 2 23 3 2 0 3 2 33 3 3 0 3 2 43 3 4 0 3 2 53 3 5 0 3 2 6

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F V ( N F S ) = F V ( N F S ) + S U M 3 3 8 0 3 3 0P R I N T 9 1 0 . N F S . F T E M P . S F . S U M 3 3 9 0 3 3 1RETURN 3 4 0 0 3 3 2

9 1 0 F O R M A T I I 5 . * 0 - • . F 7 . 4 . 3 X . 1 0 F 8 . 3 . F I 0 . 3 ) 3 4 1 0 3 3 3END 3 4 2 0 3 3 4S U B R O U T I N E FE E D * N F S . F T Y P E . F T E M P . S F . F . S M F . H F . S U M F . N C . C V . C l _ » 3 4 3 0 3 3 5D I M E N S I O N S F ( 1 0 . 1 > • S M F ( 5 • i 2 5 ) . H F < 1 2 5 ) . F f 1 2 5 ) 3 4 4 0 3 3 6S U M = 0 . 0 3 4 5 0 3 3 7DO 1 6 0 1 = 1 . NC 3 4 6 0 3 3 8S U M = S U M * S F ( I . l ) 3 4 7 0 3 3 9

1 6 0 S M F ( I . N F S ) = S M F ( I . N F S ) + S F ( I . 1 ) 3 4 8 0 3 4 0F ( N F S ) = F ( N F S ) + S U M 3 4 9 0 3 4 1SUMF=SUMF+SUM 3 5 0 0 3 4 2CALL E N T H I N C . 1 . F T E M P , E N L . E N V . S F . S F . 1 0 , 1 ) 3 5 1 0 3 4 3

E = E N V 3 5 2 0 3 4 4I F ( F T Y P E . E Q . C L ) E = E N L 3 5 3 0 3 4 5H F ( N F S ) = H F ( N F S ) + E 3 5 4 0 3 4 6I F l S U M . G T . l . 0 E - 1 0 ) G O TO 1 0 3 5 5 0 3 4 7E = 0 . O 3 5 6 0 3 4 8S U M = 0 . 0 3 5 7 0 3 4 9GO TO 2 0 3 5 8 0 3 5 0

1 0 E = E / S U M 3 5 9 0 3 5 12 0 P R I N T 9 1 0 . N F S . F T Y P E . F T E M P . S F . S U M , E 3 6 0 0 3 5 2

RETURN 3 6 1 0 3 5 39 1 0 F O R M A T ! I 5 . A 5 . F 5 . 1 . 4 X . 1 0 F 8 . 3 . F 1 0 . 3 . F 1 2 . 3 ) 0 0 0 3 6 2 0 3 5 4

END 3 6 3 0 3 5 5S U B R O U T I NE F E E D F L ( N F S . F T Y P E . F T E M P . S F . C V . C L . S U M F ) 3 6 4 0 3 5 6C O M M O N / F E E D F C / Z F . K F . X F . Y F , X F S . Y F S , A L P H A . F U N A . T E M P . P R E S , RELCHM 3 6 5 0 3 5 7

* . D A M P F . N C C . I F I N 3 6 6 0 3 5 8REAL Z F ( 1 0 ) . K F ( 1 0 ) . X F ( 1 0 ) . Y F ( 1 0 ) . X F S ( 1 0 ) . Y F S ( I O ) 3 6 7 0 3 5 9REAL L . K 0 0 0 3 6 8 0 3 6 0COMMON/PPDA T A / C O E F V ( 5 . 4 ) . C O E F L ( 5 . 4 ) . C O N V A P ( 5 . 6 ) . NAME( 5 . 2 ) . AMW( 5 ) 0 0 0 3 6 9 0 3 6 1COMMON X ( 5 . 1 2 5 ) . Y ( 5 . 1 2 5 ) . K ( 5 . 1 2 5 ) « S M F L ( 5 . I 2 5 ) . S M F V ( 5 . 1 2 5 ) 0 0 0 3 7 0 0 3 6 2COMMON T ( 1 2 S ) . P ( 1 2 5 ) » V ( 1 2 5 ) , L ( 1 2 5 ) , 0 ( 1 2 5 ) . U ( 1 2 5 ) . W ( 1 2 5 ) , H L ( 1 2 5 ) , 0 0 0 3 7 1 0 3 6 3

♦ H V ( 1 2 5 ) . F L ( 1 2 S ) . F V ( 1 2 5 ) . H F L ( 1 2 5 ) , H F V ( 1 2 5 ) . I T I T ( 2 0 ) . F Q ( 1 2 5 ) . I F C O N 3 7 2 0 3 6 4COMMON N S . N C . R . T M I N . T M A X . I T M A X . T L T . T L F . T L X . I E Q M O L . N I T T . I P R I N T . 0 3 7 3 0 3 6 5

♦ A L P H A T . A L P H L V . I C T H 0 0 0 3 7 4 0 3 6 6E Q U I V A L E N C E ( U ( 1 ) . D I S T L ) . ( V ( l ) . D I S T V ) 0 0 0 3 7 5 0 3 6 7D I M E N S I O N S F ( l O . l ) 3 7 6 0 3 6 8

C * * * * * FUNA VALUE G EN ERA TED BY S U B R O U T I N E F R G I V E N ALPHA 3 7 7 0 3 6 9C * * * * * Z F I S I N L E T STREAM C O M P O S I T I O N 3 7 8 0 3 7 0C * * * * * AL PHA I S THE F R A C T I O N OF THE I N L E T STREAM WHICH I S V A P O R I Z I E D 3 7 9 0 3 7 1C * * * * * RELCHM I S THE MAX R E L A T I V E CHANGE USED TO CONVERGE OUTER L O O P 3 8 0 0 3 7 2C * * * * * DAMPF I S THE DAMPIN G FA C TO R 0 0 . 0 ) D A M P F ) 1 . 0 3 8 1 0 3 7 3

NCC=NC 3 8 1 5 3 7 4P R E S = P ( N F S ) 3 8 2 0 3 7 5T EM P = F T E M P 3 8 3 0 3 7 6 ro

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D A M P F = 0 . 8 3 8 4 0 3 7 7R E L C M M = 2 . 0 E - S 3 8 5 0 3 7 8I F I N = 0 3 8 6 0 3 7 9

c * * * * * F I N D MOLE F R A C T I O N OF F E E D 3 8 7 0 3 8 0S U M = 0 . 0 3 8 8 0 3 8 1DO 3 5 1 = 1 , NC 3 8 9 0 3 8 2

3 5 S U M = S U M * S F ( I . l ) 3 9 0 0 3 8 3DO 4 0 1 = 1 . NC 3 9 1 0 3 8 4

4 0 Z F ( I ) = S F ( I . D / S U M 3 9 2 0 3 8 5P R I N T 9 1 0 . N F S . F T Y P E . F T E M P . S F . S U M 3 9 3 0 3 8 6

C * * * * * I N I T I A L G U E S S FO R X AND Y 3 9 4 0 3 8 7DO 1 0 1 = 1 . NC 3 9 5 0 3 8 8X F C I > = Z F « I ) 3 9 6 0 3 8 9Y F ( I ) = Z F ( I ) 3 9 7 0 3 9 0X F S ( I ) = Z F ( I ) 3 9 8 0 3 9 1

1 0 Y F S C I ) = Z F C I > 3 9 9 0 3 9 2c * * * * * I T E R A T I O N L O O P FOR C O NV ERG IN G C O M P O S I T I O N S 4 0 0 0 3 9 3c * * * * * F I N D V A P O R / L I Q U I D MOLE F R A C T I O N ( Y F AND X F ) 4 0 1 0 3 9 4I S C O N T IN U E 4 0 2 0 3 9 5

I T E R C = 0 4 0 3 0 3 9 6DO 2 0 1 = 1 . 3 0 4 0 4 0 3 9 7I T E R C = I T E R C + : 4 0 5 0 3 9 8CALL KVAL 4 0 6 0 3 9 9CALL F L V I K 4 0 7 0 4 0 0CALL CALXY 4 0 8 0 4 0 1CALL CHECKF 4 0 9 0 4 0 2I F ( I F I N . N E . O ) G O TO 2 5 4 1 0 0 4 0 3

2 0 C O N T IN U E 4 1 10 4 0 4I F ( D A M P F . L T . 0 . 1 5 ) G 0 TO 1 7 4 1 2 0 4 0 5DAMPF=DA M P F / 2 . 0 4 1 3 0 4 0 6P R I N T 1 0 1 . DAMPF 4 1 4 0 4 0 7GO TO 1 5 4 1 5 0 4 0 8

1 7 P R I N T 1 0 2 4 1 6 0 4 0 9S T O P 4 1 7 0 4 1 0

c * * * * * P R I N T R E S U L T S OF F L A S H 4 1 8 0 4 1 12 5 F R L = 1 . 0 - A L P H A 4 1 2

P R I N T 1 0 6 . I T E R C . F R L . F U N A 4 1 9 0 4 1 3P R I N T 1 0 7 . ( X F ( J ) . J = 1 . N C ) 4 2 0 0 4 1 4P R I N T l O a . I Y F C J ) . J = 1 . N C ) 4 2 1 0 4 1 5P R I N T 1 0 9 . ( K F ( J ) . J = 1 . N C ) 4 2 2 D 4 1 6I F ( A L P H A . E Q . 0 . 0 . A N D . F U N A . G T . 0 . 0 I P R I N T 1 0 3 4 2 3 0 4 1 7I F ( A L P H A . E Q • 1 • 0 . A N D . F U N A . L T . 0 . 0 I P R I N T I 0 4 4 2 4 0 4 1 8

C * * * * * S E T U P VAPOR F E E D 4 2 5 0 4 1 9DO 5 5 1 = 1 . NC 4 2 6 0 4 2 0

5 5 S F C l . 1 ) = S U M * A L P H A * Y F ( I ) 4 2 7 0 4 2 1CALL F E E D ( N F S . C V . T E M P . S F . F V . S M F V . H F V . S U M F . N C . C V . C L ) 4 2 8 0 4 2 2

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0 0 6 0 1 = 1 . NCS F f l . l ) = S U M * ( 1 . 0 - A L P H A ) * X F ( I )CALL F E E D f N F S . C L • T E M P • S F . F L • S M F L . H F L . S U M F • N C . C V • C L 1 RETURNF O R M A T ! • F E E O F L A S H O I D NOT CONVERGE I N 3 0 I T E R A T I O N S . ' /

* ' OAM PING FAC TOR HA S BEEN REOUCEO TO = ' . F S . 3 )F O R M A T ! • F E E O F L A S H O I D NOT C O N V E R G E . PROGRAM A B O R T E D - ' ) F O R M A T ! ' SUBCO OLEO L I Q U I D F E E D ' )F O R M A T ! ' S U P E R H E A T E D VAPOR F E E D ' )F O R M A T ! 1 OX. ' R E S U L T S OF F E E D F L A S H ' / 3 X . ' NUMBER OF I T E R A T I O N S = ' .

* 1 3 , ' F R A C T I O N L I Q U I D = ' * F 7 . 4 • ' CONVERGENCE ERROR = ' . F 7 . 4 ) F O R M A T ! 3 X . ' L I Q U I D M . F . ' • S X . 1 0 F 8 . 3 )F O R M A T ! 3 X , ' V A P O R M . F . ' . 6 X . 1 0 F 8 . 3 )

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F O R M A T ! 3 X . ' K V A L U E S ' . 8 X . 1 0 F 8 . 3 )F O R M A T ! I 5 . A S . F 5 . I . 4 X . 1 0 F 8 . 3 . F I 0 . 3 )ENDS U B R O U T I N E KVALC O M M O N / F E E D F C / Z F . K F . X F . Y F . X F S . Y F S . A L P H A . F U N A . T E M P . P R E S . R E L C H M

* . D A M P F . N C . I F I NREAL Z F ! 1 0 ) . K F ! 1 0 ) . X F C 1 0 ) . Y F ! I 0 ) . X F S ! 1 0 ) . Y F S I I O I . K

* * * * * I N T E R F A C E TO KVALUE R O U T I N E OF D I S T I L L A T I O N PROGRAM * * * * * T H I S R O U T I N E EXCHANGES X . Y . K V A LU ES V I A STAGE NUMBER 1

COMMON X ! S . 1 2 S ) . Y ! 5 . 1 2 5 ) . K ! 5 . 1 2 5 )DO 1 0 1 = 1 . NC X i I . l ) = X F ! I )Y ! I . l ) = Y F ! I )CALL K V A L U E I N C . 1 . T E M P . P R E S . S U M Y )DO 2 0 1 = 1 . NC K F 1 I ) = K ! I . 1 >RETURNENDS U B R O U T I N E F L V I K

* * * * * F L A S H F I N D S ALPHA G I V E N Z F . K F I . E . A L I Q U I D / V A P O R F L A S H W IT H * * * * * KF I N D E P E N D E N T OF C O M P O S I T I O N* * * * * U S E S A NEW TO N-RAPH ASO N PR OCEDUR E 0 W I L L ALWAYS CONVERGE

C O M M O N / F E E D F C / Z F . K F . X F . Y F . X F S . Y F S . A L P H A , F U N A . T E M P . P R E S . R E L C H M* . D A M P F . N C . I F I N

REAL Z F ! 1 0 ) . K F ! 1 0 ) . X F ! 1 0 ) . Y F ! 1 0 ) . X F S ! 1 0 ) . Y F S ! 1 0 )A L P H A = 1 . 0 CALL FRI F I F U N A . l t . O . O I R E T U R N A L P H A = 0 . O CALL FRI F ! F U N A . G T . 0 . 0 IR E T U R N DO 1 0 1 = 1 . 2 0 CALL FRI F ! A B S ! F U N A ) . L T . R E L C H M / 2 . 0 ) RETURN

4 3 0 04 3 1 04 3 2 04 3 3 04 3 4 04 3 5 04 3 6 04 3 7 04 3 8 04 3 9 04 4 0 04 4 1 04 4 2 04 4 3 04 4 4 04 4 5 04 4 6 04 4 7 04 4 8 04 4 9 04 5 0 04 5 1 04 5 2 04 5 3 04 5 4 04 5 5 04 5 6 04 5 7 04 5 8 04 5 9 04 6 0 04 6 1 04 6 2 04 6 3 04 6 4 04 6 5 04 6 6 04 6 7 04 6 8 04 6 9 04 7 0 04 7 1 04 7 2 04 7 3 04 7 4 04 7 5 04 7 6 0

4 2 44 2 54 2 64 2 74 2 84 2 94 3 04 3 14 3 24 3 34 3 44 3 54 3 64 3 74 3 84 3 94 4 04 4 14 4 24 4 34 4 44 4 54 4 64 4 74 4 84 4 94 5 04 5 14 5 24 5 34 5 44 5 54 5 64 5 74 5 84 5 94 6 04 6 14 6 24 6 34 6 44 6 54 6 64 6 74 6 84 6 94 7 0 MMVO

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1 0 A L P H A = A L P H A - F U N A / F R D E V ( A L P H A • Z F . K F , N C ) 4 7 7 0 4 7 1RETURN 4 7 8 0 4 7 2END 4 7 9 0 4 7 3S U B R O U T I N E FR 4 8 0 0 4 7 4

c * * * * * C A L C U L A T E S TH E F U N C T I O N F R f A L P H A ) 4 8 1 0 4 7 5C O M M O N / F E E O F C / Z F « K F . X F . Y F . X F S . Y F S . A L P H A . F U N A . T E M P . P R E S . RELCHM 4 8 2 0 4 7 6

* . D A M P F . N C . I F I N 4 8 3 0 4 7 7REAL Z F ( I O ) . K F ( 1 0 ) . X F ( 1 0 ) . Y F ( 1 0 ) . X F S I 1 0 ) . Y F S I l O ) 4 8 4 0 4 7 8

3 0 F U N A = 0 . 0 4 8 5 0 4 7 9DO 1 0 1 = 1 . NC 4 8 6 0 4 8 0T ER M = K F( D - l . O 4 8 7 0 4 8 1D E M = 1 . 0 + A L P H A * T E R M 4 8 8 0 4 8 2I F ( A B S ( D E M ) . G T .1 . 0 E - 2 0 ) 6 0 TO 10 4 89 0 4 8 3PR IN T 2 0 0 4 9 0 0 4 8 4P R I N T I 0 0 . A L P H A . T E R M . D E M . F U N A . Z F . K F 4 9 1 0 4 8 5A L P H A = A L P H A - . 0 1 4 9 2 0 4 8 6GO TO 3 0 4 9 3 0 4 8 7

1 0 F U N A = F U N A - Z F C I ) * T E R M / D E M 4 9 4 0 4 8 8RETURN 4 9 5 0 4 8 9

1 0 0 F O R M A T ! 1 P 1 0 E 1 3 . S ) 4 9 6 0 4 9 02 0 0 F O R M A T ( 5 X . « M E S S A G E FROM S U B R O U T I N E F R 0 H E L P - ' ) 4 9 7 0 4 9 1

END 4 9 8 0 4 9 2F U N C T I O N F R O E V ( A L P H A . Z . K . N C ) 4 9 9 0 4 9 3

C * * * * * CA L C U L A T E S TH E D E R I V A T I V E OF F R WITH R E S P E C T TO ALPHA 5 0 0 0 4 9 4REAL Z I N C ) . K ( N C ) 5 0 1 0 4 9 5F R O E V = 0 . 0 5 0 2 0 4 9 6DO 1 0 1 = 1 . NC 5 0 3 0 4 9 7T E R M = K I I ) - l . 0 5 0 4 0 4 9 8

1 0 F R D E V = F H O E V + Z C I ) * T E R M * T E R M / ( 1 . 0 + A L P H A * T E R M > * * 2 SOSO 4 9 9RETURN 5 0 6 0 5 0 0END 5 0 7 0 5 0 1S U B R O U T I N E CALXY 5 0 8 0 5 0 2

C * * * * * n o r m a l i z e s X AND Y 5 0 9 0 5 0 3C O M M O N / F E E D F C / Z F . K F . X F . Y F . X F S . Y F S . A L P H A . F U N A . T E M P . P R E S . R E L C H M 5 1 0 0 5 0 4

* . D A M P F . N C . I F I N 5 1 1 0 5 0 5REA L Z F ! 1 0 ) . K F ! I O ) . X F ! 1 0 ) . Y F ! 1 0 ) . X F S ! 1 0 ) . Y F S ! 1 0 ) 5 1 2 0 5 0 6S U M Y = 0 . 0 5 1 3 0 5 0 7S U M X = 0 . 0 5 1 4 0 5 0 8DO 1 0 I = 1 . N C 5 1 5 0 5 0 9X F ! 1 ) = Z F ! I ) / ! 1 . 0 * A L P H A * ! K F ! I ) - 1 . 0 ) ) 5 1 6 0 5 1 0I F ! X F ! I ) . L T . 1 . 0 E - 4 0 ) X F ! I ) = 1 . O E - 4 0 5 1 1Y F ! I ) = K F ! I ) * X F ! I ) 5 1 7 0 5 1 2S U M X = S U M X + X F ! I ) 5 1 8 0 5 1 3

1 0 S U M Y = S U M Y + Y F ! I ) 5 1 9 0 5 1 4DO 2 0 1 = 1 . NC 5 2 0 0 5 1 5X F ! I ) = X F ! I ) / S U M X 5 2 1 0 5 1 6

2 0 Y F ! I ) = Y F ! I ) / S U M Y 5 2 2 0 5 1 7 üo

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RETURN 5 2 3 0 5 1 8END 5 2 4 0 5 1 9S U B R O U T I N E CHECKF 5 2 5 0 5 2 0C O M M O N / F E E D F C / Z F . K F . X F . Y F . X F S . Y F S . A L P H A . F U N A . T E M P . P R E S . R E L C H M 5 2 6 0 5 2 1

* . D A M P F . N C . I F I N 5 2 7 0 5 2 2REAL Z F ( 1 0 ) . K F ( 1 0 ) . X F ( 1 0 ) . Y F ( 1 0 ) . X F S ( 1 0 ) . Y F S ( I O ) 5 2 8 0 5 2 3

C * * * $ $ CHEC KS FOR CONVERGENCE 5 2 9 0 5 2 4DO 1 0 1 = 1 . NC 5 3 0 0 5 2 5I F ( A B S ( X F S ( I ) - X F ( I ) ) / X F S ( I ) . G T . R E L C H M ) G O TO 2 0 5 3 1 0 5 2 6I F C A B S f Y F S ( I ) - Y F ( I ) ) / Y F S I I ) « G T . R E L C H M l G O TO 2 0 5 3 2 0 5 2 7

1 0 C O N T IN U E 5 3 3 0 5 2 8I F I N = 1 5 3 4 0 5 2 9RETURN 5 3 5 0 5 3 0

2 0 CA LL I N C R E M I Y F S . Y F . N C . D A M P F ) 5 3 6 0 5 3 1CALL I N C R E M I X F S . X F . N C . D A M P F ) 5 3 7 0 5 3 2RETURN 5 3 8 0 5 3 3END 5 3 9 0 5 3 4S U B R O U T I N E I N C R E M ( A S A V E . A . N C . D A M P F ) 5 4 0 0 5 3 5

C * * * * * P E R F O R M S A P A R T I A L S U B S T I T U T I O N 5 4 1 0 5 3 6D I M E N S I O N A S A V E ( N C ) . A ( N C ) 5 4 2 0 5 3 7S U M = 0 . 0 5 4 3 0 5 3 8DO 3 0 1 = 1 . NC 5 4 4 0 5 3 9A N E W = ( 1 . 0 - O A M P F ) * A S A V E ( I ) + D A M P F * A ( I ) 5 4 5 0 5 4 0I F I A N E W . l t . 1 . 0 E - 4 0 ) A N E W = 1 . 0 E - 4 0 5 4 1SUM=SUM+ANEW 5 4 6 0 5 4 2A S A V E I I ) = A ( I ) 5 4 7 0 5 4 3

3 0 A ( I ) = A N E W 5 4 8 0 5 4 4DO 4 0 1 = 1 . NC 5 4 9 0 5 4 5

4 0 A ( I ) = A ( I ) / S U M 5 5 0 0 5 4 6RETURN 5 5 1 0 5 4 7END 5 5 2 0 5 4 8S U B R O U T I N E LO O P D S 0 0 0 5 5 3 0 5 4 9REAL L . K 0 0 0 5 5 4 0 5 5 0COMMON X I 5 . 1 2 5 ) . Y ( 5 . 1 2 S ) . K I S , 1 2 5 ) . S M F L ( 5 . 1 2 5 ) . S M F V I S . 1 2 5 ) 0 0 0 5 5 5 0 5 5 1COMMON T ( 1 2 S ) . P ( 1 2 5 ) . V ( 1 2 S ) . L ( 1 2 5 ) . 0 ( 1 2 5 ) . U C 1 2 5 ) . W I 1 2 5 ) . H L ( 1 2 5 ) . 0 0 0 5 5 6 0 5 5 2

* H V ( 1 2 5 ) . F L ( 1 2 5 ) . F V ( 1 2 5 ) . H F L ( 1 2 5 ) . H F V ( 1 2 5 ) . I T I T C 2 0 ) . F Q ( 1 2 5 ) • I F C O N 5 5 7 0 5 5 3COMMON N S . N C . R . T M I N . T M A X . I T M A X . T L T , T L F . T L X . I E Q M O L . N I T T . I P R I N T . 0 5 5 8 0 5 5 4

* A L P H A T , A L P H L V . I C T H 0 0 0 5 5 9 0 5 5 5C O M M O N / F P C P / F P . C P 5 6 0 0 5 5 6E Q U I V A L E N C E ( U ( 1 ) . D I S T L ) . ( V ( 1 ) . D I S T V ) 0 0 0 5 6 1 0 5 5 7D I M E N S I O N V S A V E I 1 2 5 ) . T S A V E ( 1 2 5 ) . X S A V E < 5 . 1 2 5 ) 5 6 2 0 5 5 8DOUBLE P R E C I S I O N Q L . Q V . B P P . B P . C F R . F P ( 1 2 5 ) . C P ( 1 2 5 ) . S . D B L E 0 0 0 5 6 3 0 5 5 9N I T T = 0 0 0 0 5 6 4 0 5 6 0

1 0 1 N I T T = N I T T ♦ 1 0 0 0 5 6 5 0 5 6 1DO 1 J = 1 . N S 0 0 5 6 6 0 5 6 2

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* H V f 1 2 5 ) . F L ( 1 2 5 ) . F V ( 1 2 5 ) . H F L ( 1 2 5 ) . H F V ( 1 2 5 ) . I T I T ( 2 0 ) . F Q ( 1 2 5 ) . I F C O N 7 5 8 0 7 5 8COMMON N S . N C . R . T M I N . T M A X . I T M A X . T L T . T L F . T L X . I E Q M O L . N I T T . I P R I N T . 0 7 5 9 0 7 5 9

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C * * * * * CORRECTS X VALUES USING THETA METHOD * * * * * 0 0 0 7 6 2 0 7 6 2c * * * * * TO I N C R E A S E THE NUMBER OF COMPS 0 0 0 7 6 3 0 7 6 3c * * * * * I N C R E A S E THE S I Z E OF THE FO LLO W IN G D I M E N S I O N ED V A R I A B L E S 0 0 0 7 6 4 0 7 6 4

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D I M E N S I O N X O O T ( S ) . P G ( 5 . 5 ) • A I N V ( 5 . 5 ) . D E L T H ( 5 ) . T H E T A ( 5 ) • L O C S S ( 5 ) 0 0 0 7 6 8 0 7 6 8D I M E N S I O N B B B B I S . S ) 7 6 9 0 7 6 9

c * * * * * P G . A I N V . B B B B MUST BE THE SAME S I Z E ARRAYS 7 7 0 0 7 7 0c * * * * * ALSO WHEN I N C R E A S E I N G THE NUMBER OF S T A G E S WITH PRODUCT 0 0 0 7 7 1 0 7 7 1c * * * * * STREA MS I N C R E A S E THE ARRAYS I N S U B R O U T I N E ----------- S I M E Q 0 0 0 7 7 2 0 7 7 2c * * * * * FOR THE D I M E N S I O N E D V A R I A B L E R A T I O THE F I R S T S U B S C R I P T I S 0 0 0 7 7 3 0 7 7 3c * * * * * THE NUMBER OF COMPS THE SECOND I S THE NUMBER OF S T A G E S W IT H 0 0 0 7 7 4 0 7 7 4c * * * * * PRODUCT STREA MS * * * * * 0 0 0 7 7 5 0 7 7 5

D I M E N S I O N R A T I O I 5 . 5 ) 0 0 0 7 7 6 0 7 7 6c * * * * * NO VA LU ES ARE P A S S E D J U S T S P A C E S A V E D . 7 7 7 0 7 7 7c * * * * * THE FO LLOW ING CARDS WERE ADDED I N AN ATTEMPT T O REDUCE THE 7 7 8 0 7 7 8c * * * * * AMOUNT OF MEMORY R E Q U I R E D . 7 7 9 0 7 7 9

C O M M O N / F P C P / F P . C P 7 8 0 0 7 8 0DOUBLE PR E C IS IO N F P ( 1 2 5 ) * C P ( I 2 5 ) 7 6 1 0 7 8 1EQUIVALENCE ( F P ( I ) • R A T I 0 ( 1 . 1 ) ) . ( C P I 1 ) . P G ( 1 . 1 ) ) 7 8 2 0 7 8 2

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2 C O N T I N U E 0 0 0 7 9 6 0 7 9 61 0 C O N T IN U E 0 0 0 7 9 7 0 7 9 7c * * * * * F I N D ALL THE R A T I O S * * * * * 0 0 0 7 9 8 0 7 9 8

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P R I N T 7 0 0 5P R I N T 2 0 3 « S U M V « S U M L . T U M . O R . a C , T W M RETURNF O R M A T ! • CO NDENSE R ----- I N T E R N A L L I Q STREAM I S R E F L U X I N T E R N A L * •

* * VAPOR STREAM I S VAPOR P R O D U C T * )F O R M A T ! / * S T A G E NO . 1 3 . ' TEMPERAT URE = * , F 8 . 2 ,

♦ • P R E S S U R E = ' , F 8 . 2 . ' I N T E R S T A G E H E A T E R / C O O L E R ! E N E R G Y / U N I T T I M E ) = 1 P E I S . 5 )

F O R M A T ! « R E B O I L E R — I N T E R N A L L I O STREAM I S BOTTOMS P R O D U C T * ) F O R M A T ! I 3 X . 2 A 4 . 4 E I 2 . 4 . 2 X . 2 ! F 1 3 . 5 . F 1 3 . 3 ) )FORMAT! 8 X . • F E E D S • . 8 X . • P R O D U C T S • . 7 X • • I N T E R S T A G E •

4 6 X , ' C O N D E N S E R ' . 7 X . ' I N - O U T • / I P 6 E 1 S . 3 )6 0 0 F 0 R M A T ! 2 6 X . * L I O A C T * . 2 X . #------------MOL F R A C T I O N S --------- • • 8 X . * K * . 8 X i&$

&/ 4 0 X . * X * . 1 1 X , * V * , 2 0 X ,

— MOLS— *» • — MASS— — MOLS— — — MASS—— *6 0 3 F O R M A T ! 4 6 X . * T O T A L F L O W • • I S X . 2 ! F I 3 . S . F I 3 . 3 ) )6 0 6 F O R M A T ! * TOTAL F E E D S TO S T A G E * . 5 X . * Q ! F R A C T I O N L I Q U I D ) = * . F 8 . 4 )

6 0 8 FORMAT! * IN T E R N A L STREA MS L E A V I N G ST AGE* )6 0 9 FORMAT! • PRODUCT STREA MS L E A V I N G S T A G E * )

6 1 0 F O R M A T ! 1 3 X . 2 A 4 . 1 2 X . 2 E 1 2 . 4 , 1 4 X . 2 I F 1 3 . 5 . F 1 3 . 3 ) >9 0 4 F O R M A T ! 4 6 X . *E N T H A L P Y . E N E R G Y / U N I T F L O W " . 2 ( F 1 3 . S . F 1 3 . 3 )

6 0 0 1 F O R M A T ! / / S X . * OVERALL COLUMN COMPONENT M A TE RI A L BALANCE ! M O L E S / ♦ T I M E ) * / 3 I X . * E N T E R I N G * , 1 I X , * L E A V I N G * . 8 X . * D I F F E R E N C E * . 5 X .« * ! R E L A T I V E E R R O R * )

7 0 0 1 F O R M A T ! 1 3 X , 2 A 4 . 1 P 3 E 1 8 . 3 . C P F l 8 . 3 )7 0 0 5 F O R M A T ! / / 5 X . * OVERALL COLUMN ENERGY BALANCE ! E N E R G Y / U N I T7 0 2 0 F O R M A T ! 1 H 1 . 2 0 A 4 . • * * * EQ U IM O LA L OVERFLOW ASSUMED * * * * / )7 0 2 1 F O R M A T ! 1 H 1 . 2 0 A 4 . * * * * E N THALPY BALANCE U S E D I N * .

» * CALCULAT ION P R O C E D U R E ' / )ENDS U B R O U T I N E PQ LCO M! X . L A S T . M O S T . I S Y M B O )D I M E N S I O N I G R A P H ! 1 0 1 . 2 6 ) , Z X ! 1 1 ) . X ! 5 . 1 2 5 ) . I S Y M B O ! M O S T )DATA I B O R D E . I B L A N K . I D / * I * . • * . * - * /

C * * * * * * * * * * 5 1 L I N E S FOR P L O T * * * * * * * * * *X S = - 5 . 0 X S 1 0 = 1 0 . 0 * * X S L I N E S = 2 6 J S = 1 I B = 1X S C A L E = - X S / 1 0 0 . 0 Y S C A L E = ! L A S T - 1 ) / 5 0 . 0 DO 7 0 0 I C = 1 . 2 DO 2 1 = 1 . L I N E S DO 3 J = 1 . 1 0 1

3 I G R A P H ! J . I ) = I B L A N KDO 2 3 = 1 . 1 0 1 . 2 0

)

T I M E ) * )

0 0 0 1 0 6 3 0 1 0 8 20 0 0 1 0 6 4 0 1 0 8 30 0 0 1 0 6 5 0 1 0 8 40 0 0 1 0 6 7 0 1 0 8 50 0 0 1 0 6 8 0 1 0 8 6

1 0 6 9 0 1 0 8 7= 1 0 7 0 0 1 0 8 8

1 0 7 1 0 1 0 8 90 0 0 1 0 7 2 0 1 0 9 00 0 0 1 0 7 3 0 1 0 9 10 0 0 1 0 7 4 0 1 0 9 20 0 0 1 0 7 5 0 1 0 9 3

1 0 7 6 0 1 0 9 41 0 7 7 0 1 0 9 5

0 0 0 1 0 7 8 0 1 0 9 6* 1 0 0 0 1 0 7 9 0 1 0 9 7

1 0 8 0 0 1 0 9 81 0 8 1 0 1 0 9 9

0 0 0 1 0 8 3 0 1 1 0 00 0 0 1 0 8 4 0 1 1 0 10 0 0 1 0 8 5 0 1 1 0 2

1 0 8 6 0 1 1 0 3I T 1 0 8 7 0 1 1 0 4

0 0 0 1 0 8 8 0 1 1 0 51 0 8 9 0 1 1 0 61 0 9 0 0 1 1 0 7

• ) 1 0 9 1 0 1 1 0 81 1 0 91 1 1 0 n i l

0 0 0 1 0 9 2 0 1 1 1 20 0 0 1 0 9 3 0 1 1 1 3

1 0 9 4 0 1 1 1 40 0 0 1 0 9 5 0 1 1 1 50 0 0 1 0 9 6 0 1 1 1 60 0 0 1 0 9 7 0 1 1 1 70 0 0 1 0 9 8 0 1 1 1 80 0 0 1 0 9 9 0 1 1 1 90 0 0 1 1 0 0 0 1 1 2 00 0 0 1 1 0 1 0 1 1 2 10 0 0 1 1 0 2 0 1 1 2 20 0 0 1 1 0 3 0 1 1 2 30 0 0 1 1 0 4 0 1 1 2 40 0 0 1 1 0 5 0 1 1 2 50 0 0 1 1 0 6 0 1 1 2 60 0 0 1 1 0 7 0 1 1 2 70 0 0 1 1 0 8 0 1 1 2 8 |N>

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I G R A P H C J , I ) = I B O R O E 0 0 7 7 J = J S . L I N E S . 1 0 0 0 7 7 K I = 2 . 1 0 0 . 2 0 N = K I + 1 8 0 0 7 7 I = K I . N I G R A P H * I . J ) = I O OO 7 9 1 = 1 . 1 0 1 . 5 1 G R A P H * I . I B ) = I BORDEOO 7 0 1 N O = l .M O S T OO 9 1 = 1 . L A S T1 F * X * N 0 . D . L T . X S I O I G O TO 91X=* ALO G IO * X l N O . l n - X S ) / X S C A L E + 1 . 5l Y = * L A S T - I ) / Y S C A L E + 0 . 51 Y = 5 1 - 1 YI F * 1 Y . L E . 2 6 . A N 0 . I C . E Q . 2 ) G 0 TO 9 I F * 1 Y . G T . 2 6 . A N 0 . I C . E O . 1 ) G 0 TO 9 I F * 1 Y . G T . 2 6 . A N D . I C . E Q . 2 ) I Y = I Y - 2 6 1 G R A P H * I X . I Y ) = I S Y M B O * N O )C O N T I N U EC O N T IN U EOO 1 5 1 = 1 , L I N E SC H E C K = ( - : ) * * II F * CHECK . G T . O I G O TO 1 4Y E S = 1 • + Y S C A L E * t I - 1 + 2 6 4 * I C - 1 > )P R I N T 1 7 . Y E S . * I G R A P H * J . I ) . J = 1 . 1 0 1 )GO TO 1 5P R I N T 1 8 . * I G R A P H * J . I > . J = 1 . 1 0 1 )C O N T IN U EL I N E S = 2 5J S = 51 8 = 2 5DO 1 2 J = l . 1 1Z X * J ) = 1 0 . 0 * 4 * 1 0 . 0 4 * J - 1 ) 4 X S C A L E 4 X S )P R I N T 2 0 . ZXRETURNFORMAT* F I 1 . 3 . 1 X . 1 0 1 A 1 )F O R M A T * 1 2 X . l O l A l )F O R M A T * 6 X , I 1 F 1 0 . 5 )END

0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 000 1 0001 0001 0001 000 1 0001 0001 0001 0001

1 0 9 011001 1 1 011201 1 3 01 1 4 01 1 5 01 1 6 01 1 7 01 1 8 01 1 9 01200121012201 2 3 01 2 4 01 2 5 01 2 6 01 2 7 01 2 8 01 2 9 01 3 0 01 3 1 01 3 2 01 3 3 01 3 4 01 3 5 01 3 6 01 3 7 01 3 8 01 3 9 01 4 0 01 4 1 01 4 2 01 4 3 01 4 4 01 4 5 01 4 6 01 4 7 0

1 1 2 91 1 3 01 1 3 11 1 3 21 1 3 31 1 3 41 1 3 51 1 3 61 1 3 71 1 3 81 1 3 91 1 4 01 1 4 11 1 4 21 1 4 31 1 4 41 1 4 5 1 1 4 61 1 4 71 1 4 81 1 4 911501 1 5 11 1 5 21 1 5 31 1 5 41 1 5 51 1 5 61 1 5 71 1 5 81 1 5 91 1 6 0 1 1 6 1 1 1 6 21 1 6 31 1 6 41 1 6 51 1 6 6 1 1 6 7

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APPENDIX CPROGRAM LISTING OF EQUIMOLAL OVERFLOW MODEL

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S U B R O U T I N E T F E Q ME X T E R N A L E Q M L V I « E Q M L V E . E Q M L V 4C O M P L E X L « V « S A V ER E A L M A G M A X . M A G M I NR E A L L B A R . K B A RC O M M O N X B A R < S . 1 2 S ) . V B A R f 5 * 1 2 5 ) * K B A R ( 5 * 1 2 5 ) • Z X B A R f 5 * 1 2 5 ) •

* Z Y B A R ( 5 « 1 2 5 )C O M M O N T B ARC 1 2 5 ) . P B A R C 1 2 5 ) * V B A R f 1 2 5 ) . L B A R f I 2 S ) . Q I N B A R { 1 2 5 ) .

* S S L B A R f 1 2 5 ) . S S V B A R f 1 2 5 ) . H L B A R f 1 2 5 ) . H V B A R f 1 2 5 ) . F L B A R ( 1 2 5 ) ,* F V B A R f 1 2 5 ) * H F L B A R ( 1 2 5 ) . H F V B A R ( 1 2 5 ) . I T I T f 2 0 ) . O B A R f 1 2 5 ) . I F C O N . N S . N C

C O M M O N / P P O A T A / C O E F V ( 5 . 4 ) . C O E F L f 5 . 4 ) . C O N V A P ( 5 . 6 ) . N A M E ( 5 . 2 ) . A M W ( 5 ) C 0 M M 0 N / P T F / S A V E ( 5 . 2 6 . 7 ) . F R Q ( 2 6 ) . F R 0 E X P C 2 6 ) . I T T T f 1 9 ) . M E A S ( 5 . 3 ) .

* M A N X ( 5 . 3 ) . F R M A X . F R M I N . M A G M A X . M A G M I N . N M E A S . N M A N I . N S M l . I C C C O M M O N / M O D 1 / L ( 1 2 5 ) . V f 1 2 5 ) . F V ( 1 2 5 ) . S S V f 1 2 5 ) . F L f 1 2 5 ) . S S L f 1 2 5 ) .

* Z X f 5 , 1 2 5 ) . Z Y f 5 , 1 2 5 ) . B E T A ( 1 2 5 ) . H B A R f 1 2 5 ) . A L P H A f 1 2 5 ) . F R E Q . I K K K 0 1 M E N S I O N A N T R f 5 ) . H M B A R f 1 2 5 ) . Z f 5 )D A T A A N T R / 1 . 0 . 1 . 5 . 2 . 5 , 4 . . 6 . /

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * ** * * * ** * * * ** * * * ** * * * ** * * * ** * * * ** * * * ******* * * * ****************

L f l ) I S THE R E F L U X W I S L f N S )NS I S THE NUMBER OF S T A G E S NC I S TH E NUMBER OF COMPONENTS BAR I N D I C A T E S STEADY S T A T E VALUESHBARBETA

I SI S

ALPHA I SZ I S TH EF I S THEL I S THEV I S THEX I S THE

THE STE ADY S T A T E H O L D - U P THE H Y D RA U LIC T I M E CONSTANT THE VAPOR T I M E CONSTANT

F E E D C O N C E N T R A T IO N F E E D RATE L I Q U I D R A TES VAPOR R A T E S L I Q U I D MOLE F R A C T I O N S

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * r e a d DYNAMIC PARAMETERS

N S M 1 = N S - 1R E A D f 5 . 3 0 1 . E N D = 9 9 9 9 ) N S T O P . l T T T P R I N T 3 0 T . I T I T . I T T T P R I N T 3 0 9READ 3 0 5 . F R M A X . M A G M A X . S C A L E F . F L A G I D I F ( S C A L E F . L T . 0 . 2 ) S C A L E F = 1 . 0 I F f S C A L E F . G T . 1 0 . 0 ) S C A L E F = 1 0 .P R I N T 3 0 8 . F R M A X . M A G M A X . S C A L E F . F L A G I D F R M I N = F R M A X - 5 . 0 * S C A L E F MAGMIN=MAGMAX-5. O * S C A L E F I C C = 0DO 9 9 9 I D E C - 1 . 5 DO 9 9 9 I N T R = 1 . 5 I C C = I C C * 1

12345678 9

10 11 121 31 4151 61 71 81 920 21 222 32 42 52 62 72 82 93 0313 23 33 43 53 63 73 83 94 04 14 24 34 44 54 64 7

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F R Q E X P f I C C ) = 1 0 . 0 4 4 f F R M I N « - ( I D E C - 1 4At_OG 1 0 f ANTR ( I N T R ) M * S C A L E F ) 4 89 9 9 F R Q ( I C C ) = A L O G 1 0 ( F R Q E X P { I C C ) ) 4 9

F R Q E X P l 1 ) = 0 . 0 SOF R Q E X P ( 2 6 ) = 1 0 . 0 * * F R M A X 5 1F R 0 I 2 6 ) = F R H A X 5 2CAL L R O M E A S ( M E A S t N M E A S ) 5 3CA LL R O M A N U M A N I . N M A N I f M A N l T Y ) 5 40 0 s o t 1 = 1 « 5 5 5I F ( M E A S ( I • 2 ) . E 0 . 3 ) P R I N T 5 0 0 5 6I F ( M E A S ( I , 2 ) . E 0 . 3 ) S T 0 P 2 3 4 5 7I F I M A N K I . 2 ) . E 0 . 3 ) P R I N T 5 0 2 5 8I F f M A N K 1 . 2 ) . E Q . 3 ) S T O P 2 3 4 5 9

S O I C O N T I N U E 6 0READ 3 0 5 . I H B A R I I ) • 1 = 1 . N S ) 6 1READ 3 0 5 . f B E T A C I ) . 1 = 1 . N S ) 6 2READ 3 0 5 . ( A L P H A ! I ) . 1 = 1 . N S ) 6 3READ 3 0 5 . ( H M B A R ( I ) . I = 1 . N S ) 6 4P R I N T 3 0 2 6 5DO 1 1 = 1 . NS 6 6

1 P R I N T 3 0 6 . I . H BA R( 1 ) . B E T A ( I ) . A L P H A ( I ) . H M B A R ( I ) 6 7C * * $ » * # * * # * * * * * * * * * * * * * $ * * * * * * # * $ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 6 8C 4 * * 4 * CALC ULATE T R A N S F E R F U N C T I O N S FOR D I S T U R B A N C E V A R I A B L E S 6 9

N U M T R F = N C * 2 7 0DO 1 2 3 4 5 N F S = 1 . N S 7 1I F ( ( F L B A R ! N F S ) + F V B A R ( N F S ) ) . E O . O . O ) G O TO 1 2 3 4 5 7 2DO 2 2 1 = 1 . 5 7 3DO 2 2 J = i . 2 6 7 4DO 2 2 N = 1 . 7 7 5

2 2 S A V E ( I . J . N ) = C M P L X ( 0 . 0 . 0 . 0 ) 7 6DO 1 2 3 4 I K K K = 1 . N U M T R F 7 7F = 0 . 0 7 80 = 0 . 0 7 9DO 4 0 I C = 1 . NC 8 0

4 0 Z ! 1 0 = 0 . 8 1C A L L C L E A R ! N S . N C ) 8 2I F ! I K K K . E Q . 1 ) F = 1 . 0 8 3I F ! I K K K . E Q . 2 ) 0 = 1 . 0 0 8 4I F ! I K K K . G T . 2 ) Z ! I K K K - 2 ) = 1 . 0 0 8 5DO 1 6 I C = 1 . N C 8 6Z Y ! I C . N F S ) = Z ! I C ) 8 7Z X ! I C . N F S ) = Z X B A R ! 1 C . N F S ) * ! Z ! I C ) - Q * ! Z X B A R ! I C . N F S ) - Z Y B A R ! I C . N F S ) ) ) 8 8

* / ( Z X B A R ! I C . N F S ) * Q B A R ! N F S ) - Z Y B A R ! I C . N F S ) * ! i . O - Q B A R ! N F S ) ) ) 8 9I F ! Z X B A R ( I C . N F S ) . N E . 0 . 0 ) 9 0

* Z Y ! I C . N F S ) = Z X ! I C . N F S ) 4 Z Y B A R ! I C . N F S l / Z X B A R ! I C . N F S ) 9 11 6 C O N T IN U E 9 2

F L ! N F S ) = Q * ( F L B A R ( N F S ) + F V B A R ! N F S ) ) + F 4 0 B A R ( N F S ) 9 3F V ! N F S ) = - Q $ ( F L B A R ( N F S ) + F V B A R ( N F S ) ) * F * ( 1 . O - Q B A R ( N F S ) ) 9 4 rof-

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I F C R E A L f V I N S ) ) . E O . O . O ) V ( N S ) = F V C N S ) - S S V f N S )I F C M A N I T Y . E Q . D C A L L MQOEQMCEQMLV11 I F C M A N l T Y « E Q . 2 ) C A L t . MODEOM< E Q M L V 2 1 1 F ( H A N 1 T Y . E Q . 4 ) C A L L MOOEQMfEQMLV4)C O N T I N U EC A L L P L O T F D I I T I T . N U M T R F . N F S . N C )I F C F L A G I D . G T . 2 . 0 ) C A L L 1 0 E N T ( N U M T R F , I T I T . O )

C O N T I N U E****************»»*********************$************$»******$*****»*$ * * * * * C A L C U L A T E T R A N S F E R F U N C T I O N S F O R M A N I P U L A T E D V A R I A B L E S

DO 4 2 1 = 1 , 5 DO 4 2 J = l , 2 6 DO 4 2 N = l . 7S A V E * I , J , N ) = C M P L X ( 0 . 0 , 0 , 0 )NUMTRF=NMAN1DO 4 2 3 4 I K K K s f , N U M T R FC A L L C L E A R ( N S . N C )

C * * * * * P I C K M A N I P U L A T E D V A R I A B L E S I M A N I = M A N I I I K K K , 2 )I F ( I M A N I . E Q . t ) S S L ( M A N I ( I K K K , I ) ) = 1 . 0 I F ( I M A N I . E Q , 2 ) S S V ( M A N I f I K K K , I ) ) = I . O I F ( I M A N I . E 0 . 3 ) S T 0 P 2 3 4 I F ( I M A N I . E Q . 4 ) L I I I = C M P L X ( 1 , 0 , 0 , 0 )I F I I M A N I , E O , 5 ) S S V ( I 1 = 1 . 0 I F ( I M A N I . E 0 . 6 ) S S L ( I ) = I . O I F ( I M A N I . E 0 . 7 ) V ( N S 1 = C M P L X ( I . 0 , 0 . 0 )I F ( I M A N I . E a . 8 ) L ( N S ) = C M P L X ( I . 0 , 0 . 0 )I F ( R E A L ! V I N S n . E O . O . O ) V ( N S I = F V ( N S ) - S S V ( N S )I F C M A N I T Y . E G . I I C A L L MODEQ MIE OML VI)I F ( M A N I T Y . E Q . 2 ) C A L L MODEQMIEQM LV2)I F ( M A N I T Y . E Q . 4 ) C A L L MODEQMIEQML V4>

4 2 3 4 C O N T IN U EC A L L P L O T F M I I T I T , N U M T R F )I F ( F L A G I D . G T . 2 . 0 ) C A L L I D E N T * N U M T R F . I T I T , I )I F C F L A G 1 D . G T . 0 . 0 ) C A L L R G C A L I I T I T )I F I F L A G I D . G T . 1 . 0 ) C A L L D C C A L f I T I T )

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I F C N S T O P . E Q . O ) G O TO 4 R E T U R N

9 9 9 9 S T O P3 0 1 F O R M A T d l , I 9 A 4 )3 0 2 F O R M A T ( / I 9 X , ' H O L D U P ' , 5 X , ' H Y D R A U L I C T C ' , 3 X , ' VAPOR T O3 0 5 F O R M A T ( B E I 0 . 0 )3 0 6 F O R M A T I 6 X , ' S T A G E = • , 1 3 , I P 4 E 1 3 . 4 )3 0 7 F O R M A T ! I H I , 2 0 A 4 / I X . 1 9 A 4 )3 0 8 F O R M A T ! 5 X , ' L O A G I O f F R E Q M A X ) = ' , F 7 . 2 , 5 X , • L O A G I 0 CMAG M A X ) = ' , F 7 . 2 ,

♦ 5 X , ' S C A L E F A C T O R S ' , F 7 . 2 , 5 X , ' F L A G FOR I D E N T I F A C T I O N = ' , F 7 . 2 )

9 59 69 79 899

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♦ . « I N THE EQUIM OL AL MODEL — J O B A B O R T E D * » 1 4 45 0 2 F O R M A T ( 5 X « « M A N I P U L A T E D V A R I A B L E T Y P E - Q I N - I S NOT A L L O W E D * , 1 4 5

♦ * I N T H E E Q U I M O L A L MODEL J O B A B O R T E D * ) 1 4 6END 1 4 7S U B R O U T I N E E X T R A 1 4 8R E A L L . K 1 4 9

5 C O M M O N / P P D A T A / C O E F V I 5 . 4 ) « C O E F L C 5 . 4 ) . C O N V A P f 5 . 6 ) • N A M E ! 5 • 2 ) • A M W f 5 ) 1 5 0=. C O M M O N X I 5 . 1 2 5 ) , Y I 5 . 1 2 5 ) , K ( 5 « 1 2 5 ) , S M E L ( 5 . 1 2 5 ) # S M F V ( 5 , 1 2 5 ) 1 5 1'g. C O M M O N T ( 1 2 5 ) , P f 1 2 5 ) , V < 1 2 5 ) , L ( 1 2 5 ) , 0 ( 1 2 5 ) , U ( 1 2 5 ) , W ( 1 2 5 ) , H L f 1 2 5 ) • 0 0 1 5 2^ ♦ H V ( 1 2 5 ) , F L ( l 2 5 ) , F V ( 1 2 5 ) , H F L ( 1 2 5 ) , H F V ( l 2 5 ) , I T I T ( 2 0 ) , F Q f 1 2 5 ) , I F C O N 1 5 3i C O M M O N N S . N C . R . T M I N . T M A X . I T M A X . T L T . T L E . T L X . I E Q M O L . N I T T , I P R I N T . 1 5 43 * A L P H A T . A L P H L V . I C T H 1 5 5g E Q U I V A L E N C E I U ( 1 ) • D I S T L } . ( V ( 1 1 , D I S T V ) 1 5 6

C * * * * * * * * * * E X T R A P R O V I D E S T R A N S W I T H S T E A D Y S T A T E I N F O R M A T I O N 1 5 7? D O 5 I S = I . N S 1 5 81. I F ( F L ( I S ) . E Q . O . O ) G O TO 2 1 5 9g DO 4 I C = 1 . N C 1 6 0- 4 S M F L f I C . I S ) = S M F L ( I C . I S ) / E L ( I S ) 1 6 1(D 2 I F ( F V ( I S ) . E Q . O . O I G O TO 5 1 6 2

DO 1 1 I C = 1 . N C 1 6 3R I t S M F V I I C . I S ) = S M F V ( 1 C . I S ) / F V ( I S ) 1 6 4

5 C O N T I N U E 1 6 5C * * * * * S M F L AND S M F V NOW C O N T A I N MOLE F R A C T I O N S * * * * * 1 6 6

° I F f l E Q M O L . E Q . l ) G O TO 3 1 6 7C ONLY F O R E Q U I M O L A L S T E A D Y S T A T E C A L C U L A T I O N S 1 6 8

DO 1 0 I S = 1 . N S 1 6 9I F I F V I I S ) . E Q . O . O I G O T O 1 0 1 7 0I F ( E Q ( I S ) . G E . 1 . 0 ) G O TO 4 0 1 7 1I F I F Q I I S ) . L E . O . O I G O T O 1 0 1 7 2

Q. C * * * * * M I X E D L I Q U I D / V A P O R F E E D 1 7 3t . DO 3 5 I C = 1 . N C 1 7 45 S M F L ( I C . I S ) = S M F V ( I C . I S ) / ( F Q ( I S ) + K ( I C . I S ) * ( 1 . 0 - F Q ( 1 S 1 ) ) 1 7 5o 3 5 S M F V I 1 C . I S ) = K f I C . I S ) * S M F L ( I C . I S ) 1 7 65- F L ( I S ) = F Q ( I S ) * F V ( I S ) 1 7 7^ F V ( I S ) = ( 1 . 0 - F Q f I S ) ) * F V f I S ) 1 7 8q GO TO 1 0 1 7 92. 4 0 DO 4 5 I C = 1 . N C 1 8 0M S M F L ( I C . I S ) = S M F V ( I C . I S ) 1 8 1o 4 5 S M F V I I C . I S ) = 0 . 0 1 8 2■ F L ( I S ) = F V | I S ) 1 8 3

F V ( I S ) = 0 . 0 1 8 41 0 C O N T I N U E 1 8 53 C O N T I N U E 1 8 6

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2 0 S M F V d • J ) = 0 . 0 1 9 3T ( J l = 0 . 0 1 9 4P ( J ) = 0 . 0 1 9 5V f J ) = 0 . 0 1 9 6L ( J ) = 0 . 0 1 9 7Q C J > = 0 . 0 1 9 8U f J ) = 0 . 0 1 9 9W C J ) = 0 . 0 2 0 0H L C J } = 0 « 0 2 0 1H V f J ) = 0 . 0 2 0 2FLC J } = 0 . 0 2 0 3F V ( J ) = 0 . 0 2 0 4H F L f J ) = 0 . 0 2 0 5H F V f J ) = 0 . 0 2 0 6

F Q ( J ) = 0 . 0 2 0 7CALL TFEQM 2 0 8RETURN 2 0 9END 2 1 0S U B R O U T I N E E Q M L V l f N S ) 2 1 1

C * * * * * E X T E R N A L L Y M A N I P U L A T E D V A R I A B L E S ARE L f l ) AND L f N S ) 2 1 2C * * * * * CONDENSER L E V E L M A N I P U L A T E S D I S T I L L A T E 2 1 3c * * * * * R E B O I L E R L E V E L M A N I P U L A T E S BOTTOMS VAPOR STREAM 2 1 4c * * * * * P R M A N I f 1 )= A R E A O F R E B O IL E R » G A M M A 2 1 5c * * * * * P R M A N I f 2 ) = A R E A * D E N S I T Y * L A T E N T H E A T 0 F 8 0 T T 0 M S / 2 1 6c * * * * * f L A T E N T H E A T O F S T E A M * G A I N O F R E B O I L E R L E V E L L O O P ) 2 1 7c * * * * * PRMANI f 3 ) = A R E A * D E N S I T Y / G A I N O F C O N D E N S E R L E V E L L O O P 2 1 8c * * * * * P R M A N I f 4 I N C O N S T A N T FOR S E L F - R E G U L A T I N G CONDENSER 2 1 9c * * * * * P R M A N I f S ) = I F GREA TER THAN ZERO USE S E L F - R E G U L A T I N G CONDENSER O P T I O N 2 2 0c * * * * * M A N I T Y = 1 2 2 1c O P T I O N C . 1 OR C . 3 AND R . i * * * E Q M L V l * * * 2 2 2

COMPLEX L . V . S A V E . V S A V E ( 2 0 ) 2 2 3C O M M O N / M O D 1 / L ( : 2 5 ) , V ( 1 2 5 ) . F V f 1 2 5 ) . S S V f 1 2 5 ) • F L C 1 2 5 ) . S S L f 1 2 5 ) . 2 2 4

* Z X f 5 . 1 2 5 ) « Z Y f 5 . 1 2 5 ) . B E T A f 1 2 5 ) • H B A R f 1 2 5 ) . A L P H A f 1 2 5 ) « F R E Q . I K K K 2 2 5C O M M O N / P T F / S A V E f 5 . 2 6 . 7 ) . F R 0 ( 2 6 ) , F R Q E X P ( 2 6 ) . I T T T f 1 9 ) . M E A S f 5 . 3 ) . 2 2 6

* M A N I f 5 . 3 ) . F R M A X . F R M I N . M A G M A X . M A G M I N . N M E A S . N M A N I . N S M l . I C C 2 2 7C O M M O N / M O D 5 / P R M A N I f 5 ) . P R M E A S f 5 . 3 ) 2 2 8DATA T O L / . O O O l / 2 2 9I C O U N T = 0 2 3 0

5 C O N T I N U E 2 3 1OO 3 J = 2 . N S M 1 2 3 2I = 2 - J + N S M l 2 3 3

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END 3 3 0S U B R O U T I N E MODEOM( C A L L V ) 3 3 1REAL MAGMAX.MAGMIN 3 3 2COMPLEX L . V . S A V E . C D I A A . C . C O I A U . X 333REAL LBAR.KBAR 334COMMON X B A R ( S . 1 2 5 ) . Y B A R ( 5 . l 2 S ) . K B A R ( 5 . i 2 5 ) . Z X B A R ( S . I 2 S ) . 3 3 5

* Z Y B A R C S . 1 2 5 ) 3 3 6o COMMON T B A R ( 1 2 5 ) . P B A R ( I 2 5 ) « V B A R f 1 2 5 ) . L B A R ( 1 2 S ) . O I N B A R C 1 2 5 ) • 3 3 7

* S S L B A R f 1 2 5 ) . S S V B A R f 1 2 5 ) « H L B A R f 1 2 5 ) . H V B A R f 1 2 5 ) , F L B A R < 1 2 5 ) . 3 3 8=. ♦ F V B A R f 1 2 5 ) « H F L B A R f 1 2 5 ) « H F V B A R f 1 2 5 ) « I T I T f 2 0 ) « O B A R f 1 2 5 ) « I F C O N . N S . N C 3 3 9

COMMON/MOD 1 / L f 1 2 5 ) « V f 1 2 5 ) « F V f 1 2 5 ) . S S V f 1 2 5 ) , F L f 1 2 5 ) « S S L f 1 2 5 ) . 3 4 0^ ♦ Z X f 5 . 1 2 5 ) , Z Y f 5 « 1 2 5 ) « a E T A f 1 2 5 ) , H B A R f 1 2 5 ) « A L P H A f 1 2 5 ) « F R E Q . I K K K 3 4 1g C O M M O N / P T F / S A V E f 5 . 2 6 « 7 ) « F R O f 2 6 ) « F R O E X P f 2 6 ) . 1 T T T f 1 9 ) . M E A S f 5 , 3 ) , 3 4 2

* M A N I f 5 . 3 ) , F R M A X , F R M I N , M A G M A X . M A G M I N , N M E A S , N M A N I . N S M 1 « I C C 3 4 3C O M M O N / M O D S / P R M A N I ( S ) « P R M E A S f 5 . 3 ) 3 4 4D I M E N S I O N X f 1 2 5 ) « C D I A A f 1 2 5 ) « C f 1 2 5 ) « C D l A U f 1 2 5 ) « D I A L f 1 2 5 ) « D I A U f 1 2 5 ) 3 4 5DO 9 9 9 I C C = 1 « 2 6 3 4 6

3 F R E Q = F R Q E X P f I C C ) 3 4 7^ C ♦ ♦ ♦ * ♦ ♦ ♦ ♦ ♦ ♦ C A L C U L A T E L I Q U I D R A T E T R A N S F E R F U N C T I O N S 3 4 8? C A L L C A L L V f N S ) 3 4 93 C * ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ S T A R T CO MP ONE NT L O O P 3 5 0^ DO 4 5 1 J K = 1 , N M E A S 351° I C - M E A S ( I J K . 3 ) 352C I F ( I C . N E . O G O TO 4 0 353^ DO 4 4 1 = 1 . NS 3 5 4° 4 4 X f I ) = 0 . 0 3 5 5-o GO TO 4 2 3 5 63 4 0 C O N T I N U E 3 5 75 C * * * * * * * * * * L O A D ARR AYS 3 5 8g DO 5 0 1 = 1 «N SM1 3 5 9CD I P = I + 1 3 6 0g- D I A L f I P ) = - L B A R ( I ) 3 6 1g 5 0 D I A U f I ) = - V B A R f I P ) « K 8 A R f I C « I P ) 3 6 23= DO 6 0 1 = 1 « N S 3 6 3° T E M P = L B A R f : ) + K B A R f I C « I ) * f V B A R f I ) + S S V B A R f I ) ) + S S L B A R f I ) 3 6 4^ 6 0 C D I A A f I ) = C M P L X f T E M P « F R E Q * H B A R f I ) ) 3 6 5■o C * * * * * * * * * * C A L C U L A T E C O N S T A N T S 3 6 6q OO 5 5 1 = 1 « N S 3 6 7i . I P = I * 1 3 6 8% I M = I - 1 3 6 95 C ( I ) = X B A R ( I C . I ) 4 ( l . 0 - K B A R ( I C . i n « ( S S V ( I ) 4 V ( I ) ) 370^ * + V ( I P ) * ( X 8 A R ( I C . I P ) * K B A R ( I C , I P ) - X B A R ( ! C , I ) ) 371

* 4 ( Z X B A R ( I C . I ) - X B A R ( I C . I M 4 F L ( I ) 4 Z X ( I C . I ) 4 F L B A R ( I ) 3 7 2* ♦ ( Z Y B A R ( 1 C , I ) - X B A R ( I C . I ) ) * F V ( I ) + Z Y ( I C . I ) * F V B A R ( 1 ) 3 7 3

I F f I « N E . 1 ) 3 7 4*CC I ) = f X B A R f I C « I M ) - X B A R f I C . I ) ) * L f I M ) 4 - C f I ) 3 7 5

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DO 7 1 1 = 1 , N S I M = I - 1I F ( I . E O . l I G O T O 6 1C D I A A f I ) = C D I A A < I » - D I A L { I ) * C D I A U f I M )C f I l = C f I ) - D I A L f : I * C f IM )

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X f N S ) = C f N S )I = N S

6 5 I P = I1 = 1 - 1I F ( I . L E . O ) G O TO 7 0X f l ) = C ( I ) - X ( I P ) * C D I A U C I )GO TO 6 5

7 0 C O N T I N U EC * * * * * * * * * * S T O R E R E S U L T S 4 2 C O N T I N U E

l S = M E A S f I J K , 1 )I M E A S = M E A S f I J K • 2 )GO T O f 1 0 0 1 . 1 0 0 2 . 1 0 0 3 , 1 0 0 4 . 1 0 0 5 ) . I M E A S

1 0 0 1 S A V E f I J K . I C C . I K K K ) = X f I S )GO T O 4 6

1 0 0 2 S A V E f I J K , I C C . I K K K ) = X f I S ) * K B A R I I C . 1 S )GO TO 4 6

1 0 0 3 S T O P 1 2 31 0 0 4 S A V E f I J K . I C C . I K K K ) = L ( I S )

GO TO 4 61 0 0 5 S A V E f I J K . I C C . I K K K ) = V f I S )4 6 S A V E f I J K . I C C , I K K K ) = S A V E f I J K . I C C . I K K K ) *

$ P R M E A S f I J K . 1 ) * C E X P f C M P L X f 0 . 0 , - P R M E A S f I J K , 2 ) * F R E O ) ) /S C M P L X f 1 . 0 . P R M E A S f I J K . 3 ) « F R E Q )

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APPENDIX D

USER'S GUIDE FOR DYNAMIC MODEL

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1 ) DESCRIPTION/INTRODUCTION 3ThE USERS' GUIDE PRESENTS THE DATA CARD SEQUENCE AND FORMATS REQUIRED TD 4

USE THE DYNAMIC D I S T I L L A T I O N COLUMN MODELS. A DET A I L E D D E S C R I P T I O N OF THE 5PROGRAM CAN BE FOUND I N CHAPTERS 3 AND 4 . 6

7't* X ^ ï j î ' i ' ^ s jt ^ sj! Î.Î «I* X* ^ X* 'X X" 'X “X *X X* X X* *** X* X* X^ 'X X^ X* X* X» X® X» X* 'X X ' X* 'X *X X X^ Î Î X* *X "X *X 'X X* 'I* X* X* V X^ 'I* 'I* X* 'X X* *X X* 'X 'X X^ 'X 'X 'X X ^

2 ) I NTRODUCTI ON TO DATA TYPES 90 THE JOB DECK I S COMPOSED OF JCL (JOB CONTROL LANGUAGE) CARDS, USER SUPPLI ED 101 SUBROUTINES AND PROGRAM DATA CARDS. THE JCL CARDS SUPPLY THE I NFORMATI ON TO THE 11? COMPUTER SYSTEM S P E C I F Y I N G WHICH PROGRAM TO KUN, MEMORY SPACE REQUI RED, COMPUTER 12-n T IME ALLOWED AND ACCOUNTING I NF ORMA T I ON . THE USER SU P P L I E D FORTRAN SUBROUTINES 13

CALCULATE PHYSICAL PROPERTY I NF ORM AT I ON. THE PROGRAM DATA CARDS SUPPLY 14^ i n f o r m a t i o n ABOUT THE PROBLEM TO BE SOLVED. 15

" P * : t O * : x * * * * ;|c * :}: * ; j î A * X: * X= - r » - f * :X " f > ■ i ' f Xt X« X= * Xs X: X« X= X= v X: X-' X: X= X= X« X- * X= X ' X= * * X= X- X- X= X= X- * X= X= X= v X= * - r - X; X: X£ » v X< X ' X= X: X : |t 1 7

§■ 3 ) INPUT DATA : JCL CARDS 18a PRESENTLY T H I S PROGRAM RESI DES I N A F I L E ON THE LSU/SNCC COMPUTER SYSTEM. 19§ THI S I S THE MOST E F F I C I E N T MEANS OF HANDLING T H I S LARGE OF PROGRAM. THEREFORE, 20■p the JCL CARDS ARE STRUCTURED FUR T H I S F I L E SYSTEM. 21§■ / / ( JOB CARD ) 22

/ / £ X 2 EXEC F 3 R T G C L G , L I B - ' C S M I T H . K v J K L 0 A D ' , P A R M . L K E 0 = ' N 0 X R £ P , N 0 L I S T ' , 23/ / R E G I O N . G G = 5 1 2 K , T I M E . G 0 = 3 24/ / F U R T . S Y S I N DD • 25

ANY FORTRAN SUBROUTINE 26/=: 2 7/ / L K E D . S Y S I N DD x 28

INCLUDE SYSLTB( ) 29t e n t r y MAIN 305 / 31^ / / G D . S Y S I N DD o 32

PROGRAM DATA CARDS 33/ - , ! 34/ / 35

THE ABOVE CARDS WHICH BEGIN WI TH A / ARE JCL CARDS. THESE CARDS MUST 36 roUx03

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1 APPEAR I N THE ORDER SHOWN FOR EVERY SUBMI SSI ON OF THE CARD DECK TO THE COMPUTER. 37% JCL CARDS ARE NOT CHANGED OR OMI TTED I F FORTRAN SUBROUTINES ARE NOT PRESENT. 38§ I N I T I A L ESTI MATES FOR MEMORY SPACE, CPU TIME AND L I N E S P RI NT ED SHOULD bE 39o 5 1 2 K , 1 MIN AND 5 0 3 0 L I N E S , R E S P E C T I V E L Y . 405 I F THE EQUIMOLAL OVERFLOW DYNAMIC MODEL I S TO BE RUN, THEN THE " I N C L U D E 410 S Y S L I B " CARD SHOULD CONTAIN M A I N S . FOR THE ENTHALPY MODEL THE " I N C L U D E S Y S L I B " 42■8 CARD SHOULD CONTAIN M A I N E . 43

10 4 ) INPUT DATA: PROGRAM DATA CARDS 461 THE PROGRAM DATA CARDS CONSI ST OF TWO SETS : 1) ONE WHICH CONTAI NS 47r INFORMATION FOR THE STEADY STATE PROBLEM AND 2 ) ONE WHICH CONTAINS I NFORMATI ON 48-n FOR t h e d y n a m i c MODEL . S I NCE THE FORMER DATA WAS DESCRI BED I N THE STEADY STATE 49i . USERS' G U I D E , I T S D E S C R I P T I O N WI LL NOT BE REPEATED HERE. THE LATTER SET OF DATA 60m I S I NDEPENDENT OF THE DYNAMIC MODEL WHICH I S USED. THE DATA FOR THE DYNAMIC 51^ MODEL I MMEDI AT ELY FOLLOWS THE LAST DATA CARD OF THE STEADY STATE PROBLEM. 52■o A ) T I T L E CARD: THI S I S A HEADER CARD AND I S USED FOR I D E N T I F Y I N G THE 53â PROBLEM BEING CONSI D ERED. THE F I R S T COLUMN OF THE CARD MUST BE BLANK OS CONTAIN 54o THE CHARACTER 1 . A BLANK I N D I C A T E S ANOTHER SET OF DYNAMIC DATA FOLLOWS T H I S SET ' 55I ' OF DYNAMIC DATA. THE CHARACTER 1 I N D I C A T E S THAT A SET OF STEADY STATE DATA 5673 FOLLOWS THI S SET OF DYNAMIC DATA. THE REMAI NI NG 79 COLUMNS WILL BE PRI NTED AT 570 THE TOP OF EACH PAGE. 58CT B ) A D M I N I S T R A T I V E I NFORMATI ON ( f l E l G . O ) 59§ F I E L D 1 : THE LOG 10 OF THE MAXIMUM FREQUENCY OF I NT EREST ( 2 . 0 ) 60g F I E L D 2 : THE LOG 1 0 OF The MAXIMUM MAGNITUDE OF I NT EREST I 1 . 0 ) 615 f i e l d 3 : s c a l e FACTOR USED TO DETERMINE MI NI MUM FREQUENCY AND MAGNITUDE ( 1 . 0 ) 621 F Î E L D 4 : FLAG FOR OUTPUT 63■o > Ü . 0 GENERATE POLAR PLOTS 643 > 1 . 0 CALCULATE DECOUPLERS 65C/)C/)

> 2 . 0 F I T MODEL RESULTS TO EQUATIONS 66F I c L D 5 : DAMPING FACTOR FOR PARTIAL S U B S T I T U T I O N METHOD ( 0 . 9 ) 67f i e l d 6 : RELATI VE CHANGE TOLERANCE FOR CONVERGENCE C R I T E R I A ( 0 . 0 0 1 ) 66F I E L D 7 : NOT USED 69F I E L D 8 : NOT USED 70NOTE : F I E L D S 5 AND 6 ARE ONLY REQUIRED FOR THE ENTHALPY MODEL. 71

C ) MEASURED VARIABLES ( 5 ( I 3 , A 1 , 1 2 ) , 5 ( 3 F 3 . 1 ) ) 72rovaVO

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FÛR GROUP 1 ( I 3 , A 1 , 1 2 ) 73F I E L D 1 : STAGE NUMBER OF MEASURED VARI ABLE 74F I E L D 2 : CODE TO I N D I C A T E TYPE OF MEASURED VARI ABLE 75

X = L I Q U I D COMPOSI T I ON 76Y = VAPOR COMPOSI T I ON 77T = TEMPERATURE (ONLY FOR ENTHALPY MODEL) 76L = L I Q U I D RATE 79V = VAPOR RATE 80

F I EL D 3 : COMPONENT NUMBER FOR MEASURED V A R I A B L E . I F ME ASURED VARIABLE I S T , L , 81OR V , LEAVE BLANK. 82

FOR GROUP 2 ( 3 F 3 . 1 ) 83F I E L D 1 : GAIN FOR MEASURING D E V I C E . I F ZERO, I T DEFAULTS TO 1 . 0 . 64F I E L D 2 : DEAD T I ME f-OR MEASURING DEVICE 85F I E L D 3 : T IME CONSTANT FOR MEASURING DEVI CE 86NOTE: ONLY A MAXIMUM OF 5 MEASURED VARI ABLES CAN BE S P E C I F I E D AT ON E T I M E . 87

D ) MANIPULATED VARI ABLES ( 5 ( 1 3 ,A3 ) , 5 E 1 C . 0 ) 88FOR GROUP T 5 ( 1 3 , A 3 ) 89

F I E L D 1 : STAGE NUMBER OF MANIPULATED VARI ABLE 9Cf i e l d 2 : CODE FOR MANIPULATED VARI ABLE 91

SSL SIDESTREAM L I Q U I D s t a g e NUMBER REQUIRED 92SSV SIDE STREAM VAPOR STAGE NUMBER REQUIRED 93QI N I NTERSTAGE HEATER/COOLER STAGE NUMBER REQUIRED 94REF REFLUX STAGE NUMBER NOT REQUIRED 95OVA D I S T I L L A T E VAPOR STAGE NUMBER NOT REQUIRED 96D L I D I S T I L L A T E L I Q U I D STAGE NUMBER NOT REQUIRED (=7BVA BOTTOMS VAPOR STAGE NUMBER NOT REQUIRED 9 8B L I BOTTOMS L I Q U I D STAGE NUMBER N O T RE QUI RED 99

FOR GROUP 2 5 E 1 0 . 0 1 0 0F I E L D ] : AERATION FACTOR FOR REBOILER 101F I E L D 2 : T IME CONSTANT FOR REBOILER LEVEL CONTROLLER 132F I E L D 3 : T IME c o n s t a n t FOR CONDENSER LEVEL CONTROLLER 1 0 3F I E L D 4 : I F GREATER THAN ZERO, USE FLOODED CONDENSER EGDATION 104F I E L D 5 : T I ME c o n s t a n t FOR FLOODED CONDENSER 105

c ) HOLDUP ( 8 E 1 0 . 0 ) 106EACH F I E L D WI LL CONTAIN THE MOLES OF HOLDUP FOR THE CORRESPONDING STAGE. 107

THE F I R S T F I E L D CONTAINS T h E CONDENSER HOLDUP, THE SECOND F I E L D CONTAINS THE 103 toOvO

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8 5) INPUT d a t a : s u b r o u t i n e s 1 8 61 A) SUBROUTINE BUBBLE (ONLY USED I N ENTHALPY MODEL) 167

PURPOSE: TO CALCULATE TEMPERATURE PERTURBATION 188METHOD : SEE CHAPTER 4 OF D I S S E R T A T I O N 189ARGUMENT L I S T : NONE 190REQUIREMENTS: MUST USE L I Q U I D / V A P O R E Q U I L I B R I U M ROUTINES AND DATA 191

WHICH ARE CONSI STENT WITH THE STEADY STATE PROGRAM. CHECK ROUTINES KYALUE, 192gamma, V APP, F KPT , a n d P X P X . 193

^ l i s t i n g : s ee a p p e n d i c e s S and E. 194B) FUNCTION DENIL AND DENTV (ONLY USED IN ENTHALPY MODEL) 195

p u r p o s e : TO CALCULATE LIQUID AND VAPOR ENTHALPY PERTURBATIONS, 196o R E S PE C T I V E L Y . 197cao

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METHOD: SEE CHAPTER 4 OF D I S S E R T A T I O N . 198ARGUMENT L I S T : D E N T L ( I S ) , D E N T V ( I S ) WHERE I S = STAGE NUMBER 1 9 9REQUIREMENT: MUST BE CONSI STENT WITH ENTH ROUTINE 2 0 0L I S T I N G : SEE APPENDIXES B AND E . 201

C ) SUBROUTINE EXTRA 202PURPOSE: TO L I NK STEADY STATE AND DYNAMIC MODEL R O U T I N E S . CONSULT 203

L I S T I N G S I N APPENDICES C AND E AND THE STEADY STATE USERS' GUIDE BEFORE 2 0 4% MODI FYI NG EXTRA. 2C53"0"OCD

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APPENDIX E

PROGRAM LISTING OF ENTHALPY MODEL

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***** READ AND PRINT CODES FOR MEASURED VARIALBES 97CALL RDMEASIMEAS.NMEAS) 98

***** READ AND PRINT CODES FOR MANIPULATED VARIALBES 9 9CALL R D H A N I fH A N I .N M A N 1 .M A N IT Y ) ICO

* * * * * * * * * * READ DYNAMIC PARAMETERS 101o READ 3 0 5 * ( H B A R t I ) » I = 1 » N S 1 102

READ 3 0 5 , ( B E T A ( I l . I - I . N S l 103READ 3 0 5 » I ALPHA I I I , 1 = 1 t N S ) 104READ 3 0 5 , ( H M B A R I I ) , I = 1 , N S 1 105PRINT 3 0 2 1060 0 1 1 = 1 , NS 107

1 PRINT 3 0 6 , I , HBAR111, B E T A ! I ) «ALPHA 1 1 1 ,HMBAR11) 108C * * * * * CALCULATE TRANSFER FUNCTIONS FOR DISTURBANCE VARIABLES 109

CALL G E N TFD fFLA GID ,M A N ITY ) 110C * * * * * CALCULATE TRANSFER FUNCTIONS FOR MANIPULATED VARIABLES 111

= CALL G ENTFM IFLAG ID .M AN ITY) 112^ IF IN S T O P .E Q .O IG O TO 4 113^ RETURN 114S 9 9 9 9 STOP 115-S 301 FORMATII l f l 9 A 4 l 116o 302 F 0 R M A T I / 1 9 X , 'H O L D U P * . 5 X , 'H Y D R A U L I C T C * , 3 X , » A R E A T I 0 N F A C T O R ' , 117g- * 5 X , 'M A S S OF METAL*)' 118a 305 F O R M A T !8 E 1 0 .0 ) 119Ô 3 0 6 F 0 R M A T (6 X . 'S T A G E = ' , I 3 , 1 P 4 E 1 3 . 4 ) 120= 3 0 7 F O R M A T ! 1 H I , 2 0 A 4 / 1 X , 1 9 A 4 ) 121^ 3 0 8 F 0 R H A T ( 5 X , 'L 0 G 1 0 I F R E Q N A X ) = ' , F 7 . 2 . 5 X , ' L 0 G 1 0 ( N A G NAX) = ' , F 7 . 2 t 122§. * 5 X , 'S C A L E FACTOR = ' , F 7 . 2 , 5 X , 'O U T P U T FLAG = ' , F 7 . 2 1235: $ / 5 X , ' D A M P I N G FACTOR FOR CONVERGENCE = ' , F 5 . 2 , 5 X . ' TOLERENCE : ' 124S $ , ' RELATIVE CHANGE IN TEMPERATURE PERTURBATIONS = ' , F 7 . 4 ) 1252. 3 09 F 0 R M A T I / 5 X , 1 0 I ' * ' I , ' ENTHALPY MODEL ' , 1 0 ! ' * ' ) / ) 126S END 127a- SUBROUTINE GENTFD! FL A G ID , MAN I T Y ) 128

EXTERNAL E N T L V l ,E N T L V 2 , ENTLV4 1 2 9

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COMPLEX X , Y , K , L , V , T , S A V E 130REAL MAGMAX,MAGMIN 131REAL LBAR,KBAR 132COMMON X B A R ( 5 , 1 2 5 ) , Y B A R ( 5 , 1 2 5 ) , K B A R ( 5 . 1 2 5 ) , Z X B A R ( 5 , 1 2 5 ) , 133

* Z Y B A R I 5 , 1 2 5 ) 134COMMON T B A R I 1 2 5 ) , P B A R I 1 2 5 ) , V B A R I 1 2 5 ) , L B A R I 1 2 5 ) . 0 1 N B A R I 1 2 5 ) , 135

* S S L B A R I 1 2 5 ) , S S V B A R I1 2 5 ) ,H L B A R ! 1 2 5 ) , HVBARI1 2 5 ) , FLBAR1 1 2 5 ) , 136* FVBAR!1 2 5 1 .H F L B A R !1 2 5 ) .H FVB A R !1 2 5 ) , I T I T ! 2 0 ) , Q B A R ! 1 2 5 ) , IF C O N ,N S , NC 137

C 0 M M 0 N / P T F / S A V E ( 5 , 2 6 , 7 ) , F R Q ! 2 6 ) . F R Q E X P ! 2 6 ) , I T T T ! 1 9 ) ,M E A S !5 , 3 ) , 138* M A N I!5 ,3 ) ,F R M A X ,F R M IN ,M A G M A X ,M A G M IN ,N M E A S .N M A N I ,N S M 1 , IC C 139

C G H M O N /M O D 1 /F V I1 2 5 ) , S S V I 1 2 5 ) , F L ! 1 2 5 ) .SSL ! 1 2 5 ) , Q I N ! 12 5 ) .HMBAR!1 2 5 ) 1 40* . Z X ! 5 . 1 2 5 ) , Z Y ! 5 , 1 2 5 ) . B E T A ! 1 2 5 ) . HBAR!1 2 5 ) .ALPHA 1 1 2 5 ) , DENTFL11 2 5 ) , 141 m

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V (N S } -F V C N S ) 198I F I M A N I T Y . E Q . D C A L L MODENTI EN T L V l I 199I F I N A N I T Y . EQ.21CALL MODENTIENTLV21 2 0 0I F I M A N I T Y . E Q . 4 I C A L L MODENTIENTLV4I 201

1 23 4 CONTINUE 202C * * * * * * * * * * p l o t r e s u l t s f o r e a c h TRANSFER FUNCTION 2 03

CALL P L O T F O I IT IT .N U M T R F .N F S .N C I 2 0 4C * * * * * * * * * * i d e n t i f y t r a n s f e r FUNCTIONS * * * * * * * * * * 2 05

I F ( F L A G I 0 . G T . 2 . 0 ) C A L L l O E N T IN U M T R F » I T I T t 0 ) 2 0 6123 45 CONTINUE 2 07

^ RETURN 2 08END 2 0 9SUBROUTINE G E N T F M IF L A G ID ,MANITY) 2 1 0EXTERNAL E N T L V l ,E N T L V 2 , ENTLV4 211

o' REAL MAGMAX.MAGMIN 2 1 4^ COMMON X B A R I 5 , 1 2 5 ) , Y B A R I 5 , 1 2 5 ) , KBARC5 , 1 2 5 ) tZ X B A R f5 , 1 2 5 1 , 215- * Z Y B A R I5 , 1 2 5 ) 2 1 6

COMMON T B A R I 1 2 5 ) , P B A R ( 1 2 5 ) , V B A R I 1 2 5 ) , L B A R I 1 2 5 ) . Q I N B A R I 1 2 5 ) , 2 1 7* S S L B A R I 1 2 5 ) . S S V B A R I 1 2 5 ) , H L B A R I 1 2 5 I , H V B A R 1 1 2 5 ) , F L B A R I 1 2 5 ) , 2 18* F V B A R I 1 2 5 ) . H F L B A R I 1 2 5 ) , H F V B A R I 1 2 5 ) , I T I T I 2 0 ) ,Q B A R I1 2 5 ) , IF C O N ,N S , NC 2 19

C O M M O N / P T F / S A V E I 5 , 2 6 , 7 ) , F R Q I 2 6 I , F R Q E X P ( 2 6 ) , I T T T I 1 9 ) ,M E A S ( 5 , 3 ) , 2 2 0* MANII 5 , 3 ) ,FRMAX,FRMIN ,M AGMAX,MAGMIN,NMEAS,NMANI,N SM1, ICC 221

C 0 M M 0 N / M 0 D 1 / F V I 1 2 5 ) , S S V I 1 2 5 ) , F L 1 1 2 5 ) .SSL 11 2 5 ) , Q I N I 1 2 5 ) ,HMBARf1 2 5 ) 2 2 2* . Z X I 5 , 1 2 5 ) , Z Y f 5 , 1 2 5 ) . B E T A I 1 2 5 ) . H B A R I 1 2 5 ) , ALPHA 11 2 5 ) , DENTFL11 2 5 ) , 2 2 3* D E N T F V I1 2 5 ) , FREQ,IKKK 2 2 4

C 0 M M 0 N / M 0 D 3 / X ( 5 , 1 2 5 ) , Y I 5 , 1 2 5 ) • K l 5 , 1 2 5 ) , L I 1 2 5 ) , V 11 2 5 ) , T 1 1 2 5 ) 2 2 5DO 22 1 = 1 , 5 2 26DO 22 J = 1 , 2 6 2 2 7DO 22 N = l , 7 228

22 S A V E ! I , J , N ) = 0 . 0 2 2 9NUMTRF=NMANI 2 3 0DO 1 234 IKKK=l ,NUM TRF 231CALL CLEAR(NS,N C) 2 3 2

C * * * * * PICK MANIPULATED VARIABLES 233I M A N I = M A N I I I K K K . 2 ) 2 34I F ! I M A N I . E Q . D S S L I M A N I I I K K K , I ) ) = 1 . 0

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i I F ( 1 M A N I . E Û . 2 ) S S V I N A N I C I K K K , 1 ) ) - 1 . 0 2 3 6^ I F ( I N A N I . E Q . 3 ) Q I N ( N A N f ( I K K K . U I ^ l . O 2 3 7I i l l° I F ( 1 N A N I . E Q . 6 ) S S L ( 1 ) = 1 . 0 2 4 03= I F ( I N A N l . E Q . 71 V IN S 1 = C H P L X ( 1 . 0 . 0 . 0 ) 241

I F { I M A N I . E Q . S l L I N S l ^ C M P L X f 1 . 0 , 0 . 0 1 2 4 2I IFCMA iÎ W y I e q Î i ÎC A L L ’ mÔd ENt I I n T L V I ) ' ’ ^ ^ '^ *^ ^ * 2 4 4^ I F ( M A N I T Y . E Q . 2 I C A L L MODENT( ENTLV2) 2 4 5“ I F ( M A N I T Y . E Q . 4 I C A L L MODENTIENTLV41 2 4 6- 1 23 4 CONTINUE 2 4 70 C ♦ ♦ ♦ ♦ ♦ * ♦ ♦ ♦ ♦ p l o t r e s u l t s f o r EACH TRANSFER FUNCTION 2481 CALL P L O T F M I IT IT .N U M T R F l 249CD C * * * * * * * * * * ID E N T IF Y TRANSFER FUNCTIONS * * * * * * * * * * 250" , I F I F L A 6 I 0 . G T . 2 . 0 I C A L L I D E N T f N U M T R F . I T I T . i l 251-n C * * * * * * * * * * SET UP AND PLOT RELATIVE GAIN 252

I F ( F L A G I D . G T . O . O I C A L L R G C A L I I T I T ) 253C * * * * * * * * * * SET UP AND PLOT DECOUPLERS 2 5 4

I F I F L A G I D . G T . l . O l C A L L D C C A L ( I T I T ) 2 55

END^*^^ 2 5 7SUBROUTINE C LE A R !N S .N C I 2 58COMPLEX X . Y . K . L . V . T 259

* U c N I r v t I f P R c Q t 1KKK 262COMMON/H0D3/X( 5 . 1 2 5 1 . Y f 5 . 1 2 5 ) , K 1 5 . 1 2 5 1 . L 1 1 2 5 1 . V 1 1 2 5 1 . T I 1 2 5 1 2 6 3NSS=NS*2 2 6 4DO 2 1 = 1 , NSS 2 65VI I 1 = C M P L X ( 0 .0 . 0 . 0 1 2 66

Q. L ( I ) = C M P L X ( 0 . 0 , 0 . 0 ) 2 67

I îiÆ.’îîïSiS'"-'’'®*”’ IIIs i?î^ SSVI 1 1 = 0 . 0 272

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SUBROUTINE DDDUMPf IN U M ,N S tN C I 2 8 45 COMPLEX X . Y , K , L » V , T , S A V E 2 8 5= C 0 M H 0 N / N Q 0 3 / X ( 5 » 1 2 5 ) , V ( 5 , 1 2 5 ) , K f 5 , 1 2 5 I t L ( 1 2 5 ) , V I 1 2 5 ) » T f 1 25 ) 2 8 6o DO 2 0 2 I S = 1 , N S 2 87g: 202 PRINT 1 9 9 f INUMt I S t ( X I I C t l S ) f I C = l « N C ) f f L ( I S ) 288m DO 2 0 3 I S = 1 , N S 289o 203 PRINT 1 9 9 t INUHt IS> l Y I I C f l S ) « I C = l t N C ) t V I I S ) 290-g DO 2 0 4 :S=1 ,NS 291^ 2 0 4 P R IN T 1 9 9 , I N U M , I S , I K I I C , I S ) , I C = 1 , N C ) , T I I S ) 2 9 2CD RETURN 293Z 199 FORMAT!2 1 4 , 1 P 1 2 E 1 0 . 2 I 2 9 40 END 2 9 5% SUBROUTINE BUBBLE 2 9 6D COMPLEX T E M P I ,T E M P 2 .T E M P 4 297^ REAL LBAR.KBAR 2 9 8-n COMPLEX X , Y , K , L . V , T 2 9 9

COMMON X B A R ( 5 , 1 2 5 ) , Y B A R ( 5 , 1 2 5 I , K B A R ( 5 , 1 2 5 ) , Z X B A R ( 5 , 1 2 5 ) , 3 0 01 * Z Y B A R I5 , 1 2 5 ) 301^ COMMON T B A R I 1 2 5 ) , P B A R I 1 2 5 ) , V B A R I 1 2 S ) , L B A R I 1 2 5 ) . Q I N B A R I 1 2 5 ) , 3023 * SSLBARI1 2 5 ) . S S V B A R I 1 2 5 ) . H L B A R I 1 2 5 ) . H V B A R I 1 2 5 ) . F L B A R I 1 2 5 ) , 303-c * F V B A R I 1 2 5 ) t H F L B A R (1 2 5 ) ,H F V B A R I 1 2 5 ) . I T I T I 2 0 ) .Q B A R I1 2 5 ) , I F C O N , N S ,N C 3 0 40 C 0 M M 0 N /M 0 D 3 /X I5 , 1 2 5 ) , Y 1 5 , 1 2 5 ) , K I 5 , 1 2 5 ) , L I 1 2 5 ) , V 1 1 2 5 ) , T 1 1 2 5 ) 3 0 5g- DO 10 I S = 1 , N S 3 0 6a T E M P I= C M P L X IO .0 , 0 . 0 ) 3 0 75 TEMP2=CMPLXI0 . 0 , 0 . 0 ) 3 0 8= T E M P 3 = 0 .0 3 09^ DO 20 I C * 1 , N C 310° T E M P 1 = T E M P 1 + K B A R I I C , I S ) * X ( I C , I S ) 3111 C * * * * * FOR A NON-IDEAL L IQ U ID PHASE REMOVE THE C«S 312^ C * * * * * AND REWRITE FUNCTION PKPX 313S. C TE M P 4 = C M P L X I0 .0 , 0 . 0 ) 3 1 4g C DO 30 I C I = 1 , N C 315g C30 T E M P 4 = T E M P 4 * P K P X I IC I . I C , I S ) * X I I C I , IS ) 3 1 60 C T E M P 2= T E M P 2 + T E M P 4*X B A R I IC . IS ) 3 1 7£ 2 0 T E M P 3 = T E « P 3 + P K P T I I C , I S ) * X B A R t I C , I S I 3 18^ T I I S ) = - I T E M P 1 * T E M P 2 ) / T E M P 3 3 1 9g 0 0 4 0 IC = 1 ,N C 3201 T E M P 4= C M P L XI0 .0 , 0 . 0 ) 321% C DO 50 I C I = 1 , N C 3 225 C50 T E M P 4 = T E M P 4 + P K P X I I C I , I C , I S ) * X ( I C I , I S ) 323? K l I C , I S ) = P K P T I I C , I S ) * T f I S ) + T E M P 4 3 2 4

4 0 Y ( I C , I S ) = K ( I C , I S ) * X B A R I I C , I S 1 + K B A R I I C , I S ) * X I I C , I S ) 32510 CONTINUE 326

RETURN 327C FUNCTION P K P X I I C I . I C . I S ) 328C PARTIAL OF K ( I C ) WITH RESPECT TO XI I C I ) ON STAGE I S 329 to-'j

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DO 71 1 = 1 , NS 3 9 4IM= 1 - 1 395I F d . E Q . D G Ü 1 0 61 3 9 6

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COMPLEX X , Y , K , L , V , T , A 1 1 , A 1 2 , A 2 1 , A22 ,D ET ,TER M 5 35=r REAL LBARtKBAR 5 36^ COMMON X B A R I 5 , 1 2 5 ) , Y B A R C 5 ,1 2 5 ) , KBA RC5,1 2 5 1 , Z X B A R I5 , 1 2 5 1 , 537^ * Z Y B A R I5 , 1 2 5 ) 538% COMMON T B A R ( 1 2 5 ) , P B A R I 1 2 5 ) , V B A R I 1 2 5 ) , L B A R I 1 2 5 ) . Q I N B A R ( 1 2 5 ) , 539S * S S L B A R I 1 2 5 ) . S S V B A R I 1 2 5 ) , H L B A R I 1 2 5 ) , H V B A R ( 1 2 5 ) , F L B A R C 1 2 5 ) , 540g- * F V B A R I1 2 5 ) , HFLBAR 1 1 2 5 ) , HFVBARI 1 2 5 ) , I T I T ( 2 0 ) , Q B A R I 1 2 5 ) , I F C 0 N , N S , N C 541a C O M M O N /M O D I /F V I1 2 5 ) , S S V I 1 2 5 ) , F L f l 2 5 ) , S S L 1 1 2 5 ) , Q I N I 1 2 5 ) , HMBARI 1 2 5 ) 5425 * , Z X ( 5 , 1 2 5 ) , Z Y I 5 , 1 2 5 ) , B E T A ( 1 2 5 ) , H B A R ( 1 2 5 ) , A L P H A I 1 2 5 ) , 0 E N T F L ( 1 2 5 ) , 543= * DENTFV( 1 2 5 ) , FREQ, IKKK 5 44I C 0 M M 0 N / M 0 D 3 / X i 5 , l 2 5 ) , Y ( 5 , 1 2 5 ) ,K C 5 , 1 2 5 ) , L ( 1 2 5 ) , V I 1 2 5 ) , T ( 1 2 5 ) 545

COMMON/ M0D5/ PRMANI15 ) , PRMEAS ( 5 , 3 ) 546AREA*GAMMA/GAINOFREBOILERLEVELLOOP 5 4 7

AREA*DENSITY/GAINOFREBOILERLEVELLOOP 548g C * * * * * PRMANI1 3 ) AREA*DENSITY/GAINOFCONDENSERLEVELLOOP 5 4 9

' 550551

FOR REBOILER R . 2 552__________ DUTY IS STORED IN T I N S + 2 ) 553

-g C M = L ( N S M l ) f F V I N S ) + F L I N S ) - S S V I N S ) - S S L ( N S ) 5 5 4g CE=-CMPLX10 . 0 ,L BARIN S) *FR EQ*PRMAN1 1 2 ) ) *D E N T L IN S ) 555i $ -C M P L X (O .O .H M B A R (N S )*F R E Q )*D E N T M (N S ) 556% $ +HLBARINSM1 ) * L I N S M 1 ) +DENTLINSM1 ) *L B A R IN S M l ) 5575 $ - D E N T V I N S ) * V B A R IN S ) - D E N T L ( N S ) * L B A R I N S ) 5 58? $ ♦ H F L B A R IN S ) * F L IN S ) * D E N T F L ( N S ) * F L B A R ( N S ) 559

$ ♦ H F V B A R IN S ) * F V IN S ) * D E N T F V IN S ) * F V B A R IN S ) 560$ - H L B A R ( N S ) * S S L I N S ) - D E N T L IN S ) * S S L B A R I N S ) 561$ - H V B A R ( N S ) * S S V IN S ) - D E N T V ( N S ) * S S V B A R ( N S ) 562

A11=CMPLX( 1 .0 ,F R E Q *PR M A N I ( 2 ) ) 5 63A 1 2 = C M P L X ( l . 0 , F R E Q * L B A R ( N S ) * P R M A N I ( 1 ) ) 5 6 4 ^

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i A 2 1 = H L B A R I N S I * C M P L X I 1 . 0 , F R E Q * P R M A N H 2 I » 565% A 2 2 = C M P L X (H V B A R ( N S ) .F R E Q * H L B A R ( N S ) « L B A R fN S } * P R M A N l l l ) ) 5665 L < N S I = ( C H - V ( N S I * A 1 2 ) / A l l 567

T ( N S + 2 ) = H N S ) * A 2 1 + V I N S ) * A 2 2 - C E 568o CALL LVPENT 5 69g: c * * * * * EQUATIONS FOR CONDENSER C . 3 5 7 0CD C * * * * * D I S T IL L A T E IS STORED IN L ( N S * 2 1 5718 L I N S * 2 J = ( V ( 2 ) + F L I I ) + F V I l ) - L ( l ) - S S V ( l l J / C M P L X I l . 0 , P R M A N I ( 3 ) * F R E Q ) 572° C * * * * * CONDENSER IS STORED IN V I N S * 2 I 573^ V I N S * 2 » = C M P L X | 0 . 0 , F R E Q } * l 0 E N T L C l ) * H B A R I I ) * D E N T M ( l l * H M B A R ( l ) 574to $ + H L B A R C I ) * P R M A N I ( 3 I * L C N S * 2 ) ) - D E N T V ( 2 ) * V B A R 1 2 ) - H V B A R ( 2 ) * V C 2 ) 575- $ ^ D E N T L ( i l * ( L B A R ( l ) + S S L B A R I i n * H L B A R I I ) * I L 1 I ) * L I N S * 2 ) ) 5760 $ +HVBARI I I * S S V ( L l *D E N T V f 1 } * S S V B A R I I ) 5771 $ - H F L B A R ( l ) * F L f l l - O E N T F L ( l l * F L B A R ( l ) 578w $ -HFVBARI I ) * F V { I I -OEN TFVC 1 ) * F V B A R I 1 ) 5 79^ RETURN 5 80-n END 581= SUBROUTINE E N T L V 4 IN S M I I 582^ COMPLEX DENTM,OENTL,DENTV,CE,CM 5833 COMPLEX X , Y t K » L t V f T . A i l , A i 2 t A 2 1 t A 2 2 t O E T , T E R N 5 84

REAL LBAR.KBAR 585COMMON X B A R ( 5 , 1 2 5 } , Y B A R f 5 , 1 2 5 ) ,K B A R ( 5 , 1 2 5 ) t Z X B A R i S , 1 2 5 ) , 586

* Z Y B A R ( 5 , 1 2 5 ) 587COMMON T B A R I1 2 5 ) , PBAR1 1 2 5 ) , VBARI1 2 5 ) , L B A R I 1 2 5 ) . Q I N B A R I 1 2 5 ) , 5 88

a * S S L B A R I 1 2 5 ) , S S V B A R I 1 2 5 ) , H L B A R C 1 2 5 ) . H V B A R I 1 2 5 ) , F L B A R I 1 2 5 ) , 5 89* F V B A R I1 2 5 ) , H F L B A R I 1 2 5 ) , H F V B A R ( 1 2 5 ) , I T I T I 2 0 ) , Q B A R ( 1 2 5 ) , I F C O N , N S , N C 590

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* , Z X I 5 , 1 2 5 ) . Z Y I 5 . 1 2 5 ) . B E T A I 1 2 5 ) , H B A R I 1 2 5 ) , ALPHA 1 1 2 5 ) , DENTFL11 2 5 ) , 592* DENTFV11 2 5 ) , FREQ,IKKK 593

C 0 M M 0 N /M 0 D 3 /X I5 , 1 2 5 ) , Y I 5 , 1 2 5 ) , K I 5 , 1 2 5 ) , L I 1 2 5 ) , V 1 1 2 5 ) , T 1 1 2 5 ) 5 94C0MM0N/M0D5/PRMANI1 5 ) ,PRM EASI5 , 3 ) 595

Q. C * * * * * PRMANI I I ) AREA*GAMMA/GAINOFREBOILERLEVELLOOP 596 ....... .......... AREA*DENSITY/GAINOFREBOILERLEVELLOOP 5 9 7

AREA*DENSITY/GAINOFCONDENSERLEVELLOOP 5985996 0 0

^ C * * * * * EQUATIONS FOR REBOILER R .2 601** * * * * * REBOILER DUTY I S STORED IN T ( N S * 2 ) 6 0 2

CM=LI N S M 1 ) + F V I N S ) + F L I N S ) - S S V I N S ) - S S L I N S ) 6 03C E =-C M PL X I0 . 0 , LBAR I N S ) *FREQ*PRMAN11 2 ) ) *DE N T L IN S ) 6 0 4

$ - C M P L X I 0 . 0 , H M B A R I N S ) * F R E Q ) * 0 E N T M I N S ) 6 05$ + H L B A R IN S M 1 )*L IN S M 1 )+ D E N T L IN S M 1 )*L B A R IN S M 1 ) 6 0 6$ -DENTV I N S ) * V B A R I N S ) - D E N T L I N S ) * L B A R I N S ) 607$ + H F L B A R ( N S ) * F L IN S ) + D E N T F L IN S ) * F L B A R IN S ) 6 0 8% +HFVBARINS) * F V I N S ) +DENT F V ( N S ) *F V B A R IN S ) 6 0 9$ - H L B A R I N S ) * S S L IN S ) - D E N T L I N S ) * S S L B A R I N S ) 6 1 0i -H V B A R IN S ) * S S V IN S ) - D E N T V IN S ) * S S V B A R IN S ) 6 11 M

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i A l l ^ C H P L X f L .OtFREQ«FRNANI <2) 1 6 1 2^ A12=CMPLX( 1 .0 ,F R E Q * L B A R IN S ) * P R M A N I ( i n 6 1 35 A21=HLBARINS) *CM PLX( 1 . 0 , FREQ*PRMAN1 ( 2 ) 1 6 1 4^ A 2 2 = C M P L X (H V B A R ( N S ) .F R E Q « H L B A R (N S )4 L B A R (N S )« P R M A N I ( in 6 1 52, L ( N S I = ( C M - V C N S J * A 1 2 ) / A 1 1 6 165 T I N S + 2 ) = L ( N S I * A 2 1 + V I N S ) * A 2 2 - C E 6 1 7(D CALL LVPENT 6 18o C * * * * * EQUATIONS FOR CONDENSER C . 4 6 1 9° C * * * * * D IS T IL L A T E IS STORED IN L IN S + 2 1 6 2 0^ L ( N S + 2 ) = C M P L X I S S L I 1 1 , 0 . 0 ) 621“ L ( I J = ( V ( 2 ) - S S L ( 1 ) - S S V C I ) + F L ( 1 I * F V ( 1 J ) 6 2 2^ $ / C M P L X i l . 0 , F R E Q 4 P R M A N l ( 3 ) l 6 2 30 C * * * * * CONDENSER I S STORED I N V IN S + 2 ) 6 2 41 V ( N S * 2 l = C M P L X f O . O , F R E Q ) * ( D E N T L I l ) * H B A R I l } * D E N T N f 1 } * H M B A R ( I I 6252 $ + H L B A R I 1 ) * P R M A N I ( 3 I * L ( 1 } ) - D E N T V ( 2 ) *V B A R 1 21 -HVBAR1 2 ) * V ( 2 ) 6 26

$ *OENTL(l)*(LBARfl)+SSLBAR(in+HLBARf l l * ( L f l } * Lf N S* 2 ) ) 627^ $ + H V B A R I 1 I * S S V I 1 I * D E N T V ( 1 I * S S V B A R C 1 I 62Bi. $ -H F L B A R l 1 ) * F L I l l - D E N T F L I l l * F L B A R ( I I 6 2 9g- $ -HFVBAR11 1 * F V ( 1 l - D E N T F V I 1 1*FVBAR11 1 6 3 0- RETURN 631S END 632^ SUBROUTINE MODENTIENTLVl 633° COMPLEX X , Y , K , L . V , r , S A V E 6 34c I REAL MAG MAX. MAG MIN 6 3 5S. I REAL LBARtKBAR 6 3 6Ô COMMON X B A R I 5 . 1 2 5 I , Y B A R I 5 , 1 2 5 ) , K B A R I 5 , 1 2 5 ) , Z X B A R I 5 , 1 2 5 ) , 6 3 7^ * Z Y B A R I5 , 1 2 5 ) 638^ COMMON TBAR 1 1 2 5 1 , PBAR1125 I , V B A R 1 1 2 5 1 ,L B A R 1 1 2 5 ) . QINBAR11 2 5 ) , 6 3 9° * S S L B A R ( I 2 5 ) , S S V B A R ( 1 2 5 ) , H L B A R 1 I 2 5 ) . H V B A R I 1 2 5 ) , F L B A R ( 1 2 5 ) , 6 4 0

* F V B A R I1 2 5 ) . HFLB ARl1 2 5 ) , HFVBAR11 2 5 ) , I T I T ( 2 0 ) , Q B A R ( 1 2 5 ) , I F C O N , N S , N C 641^ C 0 M M 0 N / P T F / S A V E ( 5 , 2 6 , 7 ) , F R Q ( 2 6 ) ,F R Q E X P { 2 6 ) , I T T T ( 1 9 } .MEAS1 5 , 3 ) , 6 4 2g * M A N I I5 ,3 ) .F R M A X ,F R M IN ,M A G M A X .M A G M IN .N M E A S ,N M A N I , N S M l , I C C 6 4 3I C O M M O N /M O D l/FV I1 2 5 ) . SSV11 2 5 ) . F L I 1 2 5 ) .SSL 1125 I . Q I N I 1 2 5 ) . HMBAR11 2 5 ) 6 4 4I Î j Z X I 5 j l « ) 5 2 Y I | j l 2 5 l , - B E T A O 2 5 , . H B * R < 1 2 5 . , A L P H A ( U 5 ) . 0 E N T F t ( l 2 5 ) . 6 4 5

i C % 0 N % 0 5 / P R M % M l 5 ; T p R A È A s k ' ^ 3 Î ' ' ^ ^ ' t t lB PRINT 1 0 0 , IKKK,NS .NC 6 4 9I ‘"O IIÎ

DO 3 J = 2 .N S M 1 6 5 4I= 2 -J + N S M 1 655

3 V I I ) = VI I * 1 ) * F V I D - S S V I I ) 656DO 10 I = 2 , NSMl 657I P = I + 1 6 5 8 N

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* * * * * * * * * * STORE RESULTSCONTINUEDO 20 IJK =1 ,N M EA S I S = M E A S I I J K , 1 )IMEAS=MEAS1IJK.21 IC=MEASf I J K , 3 )GO TO ( 1 0 0 1 . 1 0 0 2 , 1 0 0 3 . 1 0 0 4 , 1 0 0 5 ) ,1HEAS SAVE( U K , IC C , IK K K )= X ( f C , I S )GO TO 4 6

1002 SAVE! I J K , I C C , I K K K ) = Y ( I C , I S)GO TO 46

S A V E ! I J K , I C C , I K K K ) = T ( I S )GO TO 4 6SAVE! I J K , I C C , I K K K ) = L I I S )GO TO 4 6SAVE! U K , IC C , IKKK ) = V ( I S )

S A V E ! I J K . I C C , I K K K ) = S A V E ( I J K . I C C , I K K K ) # $ PRMEAS ( U K , 1 ) *C E X PC CMPLX ( 0 . 0 , -PRMEASI I J K , 2 ) *F R E Q ) ) /$ C M P L X ( 1 . 0 , P R M E A S ( I J K , 3 ) * F R E Q )

CONTINUE CONTINUE RETURN ENDSUBROUTINE LVPENT COMPLEX YMTEMP,YETEHP COMPLEX DENTM,DENTL,DENTV COMPLEX X , Y , K , L , V , T , A 1 1 , A12 . A21, A 2 2 ,0 E T COMPLEX ABAR11.ABAR12, ABAR21, ABAR22, B M ,B E ,C M ,CE COMPLEX Y M ( 1 2 5 ) , Y E ( 1 2 5 ) , G M ( 1 2 5 ) , G E ( 1 2 5 ) , F M ( 1 2 5 ) , F E ( 1 25 )REAL LBAR.KBARCOMMON X B A R I 5 , 1 2 5 ) , Y B A R ( 5 , 1 2 5 ] , K 6 A R ( 5 , 1 2 5 ) , Z X B A R ( 5 , 1 2 5 ) ,

* ZYBARI5 , 1 2 5 )COMMON TBARI 1 2 5 ) ,PBAR 1 1 2 5 ) , V B A R U 2 5 ) ,LBAR 11 2 5 ) , Q lNBAR1125 ) ,

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* I N A M E l 1 3 . J ) » J s l . 2 ) • I M E A S N A l J . 1 2 ) • J = l » 5 ) , 1 1 2 4 28 2 0 0 C O N T I N U E 2 4 3

DO 8 1 0 : C = 1 « 2 2 4 43. I K K = 0 2 4 5

CQ OO 8 1 1 : K K K = 1 . N U M T R F 2 4 6K= IK K K + N M A N I 2 4 7

i I F f C A B S ( S A V E ( I J K , 1 , I K K K ) ) , E O » 0 , 0 ) G O TO 8 1 1 2 4 83 DO 8 1 3 1 = 1 , 2 6 2 4 9CD C H E C K = C A B S f S A V E f I J K . I , I K K K ) )

I F f C H E C K . G T . T E S T ) G O TO 6 0 02 5 02 5 1

"n A L O A D C I * = - A O . 2 5 2GO TO 8 1 3 2 5 3

3" 6 0 0 A L O A D C I ) = A L O G 1 0 ( C H E C K ) 2 5 4CD 8 1 3 C O N T I N U E 2 5 53 I K K = I K K + 1 2 5 6

■D CA L L P L 0 T Q ( F R Q , A L 0 A 0 * 2 6 , I S Y ( K ) « I K K «NP. MAGMI N^ MAGMA X, 2 5 7O $ F R M I N , F R M A X , ! C ) 2 5 8C B i t C O N T I N U E 2 5 92 8 1 0 C O N T I N U E 2 6 0o CALL P L O T P O f I T 1 T , N U M T R F , N F S , N C , I J K , NAM) 2 6 1

8 1 2 C O N T I N U E 2 6 23 RETURN 2 6 33" 2 0 S FORMAT ( S X , A l , • R E P R E S E N T S THE T R A N S F E R F U N C T I O N P C , 2 1 2 , 2 6 4cr ♦ • » WHICH R E L A T E S F E E D ' , 2 A 4 , * TO ' , 7 A 4 , ' O N S T A G E * , 1 4 1 2 6 5CD 3 0 5 F O R M A T ( 2 X , ' T H E R E I S NO I N P U T / O U T P U T R E L A T I O N S H I P P ( • . 2 1 2 , 2 6 6Q. ♦ *1 WHICH R E L A T E S F E E D * , 2 A 4 , * TO ' , 7 A 4 , ' O N S T A G E * , 1 4 1 2 6 7g 3 0 7 FO R M A T ( I H l , 2 0 A 4 , * BODE MAGNITUDE P L O T S F O R F E E D D I S T U R B A N C E S * / 2 6 83 ♦ 1 X . 1 9 A 4 , * F E E D STA GE = * , 1 3 , 5 X , * ( R A D / U N I T T I M E ) ( L O G - L O G l * 1 2 6 9O END 2 7 01—H S U B R O U T I N E P L O T F M ( I T I T . N U M T R F ) 2 7 1

T3 REA L MAGMIN.MAGMAX 2 7 23 COMPLEX SAVE 2 7 33 C O M M O N / P P O A T A / C O E F V ( 5 , 4 ) . C O E F L f 5 , 4 ) , C O N V A P ( 5 , 6 ) , N A M E ! 5 , 2 ) , AMWf5 ) 2 7 4w C 0 M M 0 N / P T F / S A V E ( 5 , 2 6 , 7 ) , F R Q ( 2 6 1 , F R Q E X P ( 2 6 1 , I T T T I 1 9 ) , M E A S C 5 , 3 1 , 2 7 55 ' * M A N I ( 5 , 3 1 , F R M A X , F R M I N , M A G M A X , M A G M I N , N M E A S , N M A N I , N S M I , I C C 2 7 63 C 0 M M 0 N / M 0 0 4 / I S Y ( 3 5 ) , M E A S N A C 5 , 5 1 , M A N I N A ( 4 , 8 1 , I B L A N K ( 2 )

D I M E N S I O N I T I T C 2 0 1 , A L 0 A D ( 2 6 )DATA T E S T / l . O E - 4 0 /DO 8 1 2 I J K = 1 , N M E A S P R I N T 3 0 7 , I T I T , I T T T N P = 0

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$ P R I N T 2 0 5 . I S Y ( K * , I J K . K . I M A N I N A I J . I K ) , J = l , 4 ) .* I B L A N K . I M E A S N A I J , I 2 ) . J = 1 . S ) . : 1

I F f I 3 . N E . 0 )« P R I N T 2 0 5 . I S Y f K l . I J K . K . f M A N I N A f J . I K ) , J = 1 . 4 ) ,* f N A M E C I 3 . J ) . J = 1 . 2 ) . < M E A S N A ( J , I 2 ) . J = 1 . S ) . I 1

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* P R I N T 3 0 5 . I J K . K . f M A N l N A f J . I K l • J = t . 4 ) •* I B L A N K , ( M E A S N A f J . I 2 1 . 3 = 1 . 5 1 . 1 1

I F f 1 3 . N E . 0 1* P R I N T 3 0 5 . 1 J K . K . f M A N I N A f J . I K l . J = 1 . 4 ) .* ( N A M E f 1 3 . 3 1 , 3 = 1 , 2 1 . f M E A S N A f J . I 2 ) . 3 = 1 . 5 1 . 1 1

C O N T I N U E0 0 8 1 0 I C = 1 . 2 I K K = 0OO 8 1 1 | K K K = 1 , N U M T R F K = IK K KI F f C A B S f S A V E f I 3 K , 1 . I K K K l l . E Q . O . O l G O TO 8 1 1 OO 8 1 3 1 = 1 . 2 6C H E C K = C A B S f S A V E f I 3 K . I . I K K K l l I F f C H E C K . G T . T E S T 1 GO TO 6 0 0 A L O A D f l l = - 4 0 .GO TO 8 1 3A L O A O f 1 1 = ALOG1 O f CHECK 1C O N T I N U EI K K = I K K + 1CA LL P L O T Q f F R Q . A L O A O . 2 6 . I S Y C K l . I K K . N P . M A G M I N . M A G M A X .

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* M A N I ( 5 , 3 ) . F R M A X . F R M I N . M A G M A X . M A G M I N . N M E A S . N M A N I . N S M I • I C C 3 3 7C O M M O N /M O D 4 /1 SY < 3 5 ) • MEASNA( 5 . 5 ) . MANI N A ( 4 . 8 ) . I BLANK ( 2 ) 3 3 8D I M E N S I O N I T I T ( 2 0 ) . A L O A D ( 2 6 ) . N A M ( 2 . 7 ) 3 3 9DATA P H A M I N . P H A M A X / - 3 6 0 . . + 9 0 . / 3 4 0P R I N T 3 0 7 . I T I T . I T T T . N F S 3 4 1N P = 0 3 4 2DO 2 0 0 : K K K = 1 . N U M T R F 3 4 3K= IK K K + N M A N I 3 4 4I I = M E A S ( I J K . l ) 3 4 5I 2 = M E A S ( I J K . 2 ) 3 4 6I 3 = M E A S ( 1 J K . 3 ) 3 4 7I F ( C A B S ( S A V E ( I J K . l . I K K K ) ) . E Q . O . O I G O TO 1 5 0 3 4 8N P = N P + 1 3 4 9I F ( I 3 . E O . O ) 3 5 0

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I F ( I 3 . N E . 0 ) 3 5 3« P R I N T 2 0 5 . I S Y ( K ) . I J K , K . ( N A M ( J . I K K K ) . J = l . 2 ) . 3 5 4« ( N A M E ! I 3 . J ) . J = I . 2 ) . ( M E A S N A ( J . 1 2 ) . J = 1 . S ) . 1 1 3 5 5

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I F ( I 3 . N E . 0 ) 3 6 0« P R I N T 3 0 5 . 1 J K . K , ( N A M ( J . I K K K ) . J = 1 . 2 ) . 3 6 1« ( N A M E ( 1 3 . J ) . J = 1 . 2 ) . ( M E A S N A I J . I 2 ) , J = 1 . 5 ) . 1 1 3 6 2

CO N T I N U E 3 6 3DO 8 1 0 I C = 1 . 2 3 6 4I K K = 0 3 6 50 0 8 1 1 I K K K = I . N U M T R F 3 6 6K=IK K K +N M A N I 3 6 7I F ( C A B S ( S A V E ( I J K . I . I K K K ) ) . E Q . O . O I G O TO 8 1 1 3 6 8CALL P H A S E ( S A V E . I J K . I K K K , A L O A D ) 3 6 9I K K = I K K + 1 3 7 0CALL P L O T Q ( F R Q . A L O A D . 2 6 . I S Y ( K ) , IKK . N P . P H A M I N . P H A M A X . 3 7 1

« F R M 1 N . F R M A X . I C ) 3 7 2C O N T I N U E 3 7 3C O N T I N U E 3 7 4RETURN 3 7 5F O R M A T C S X . A l • • R E P R E S E N T S THE T R A N S F E R F U N C T I O N P C . 2 1 2 , 3 7 6

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W ♦ WHICH R E L A T E S F E E D * , 2 A * , ' TO ' , 7 A 4 . ' O N S T A G E » , 1 4 » 3 7 75 3 0 5 F 0 R M A T C 2 X » " T H E R E I S NO I N P U T / O U T P U T R E L A T I O N S H I P P I * . 2 1 2 . 3 7 83 4 » l WHICH R E L A T E S F E E D » . 2 A 4 . • TO » . 7 A 4 . » O N S T A G E * . 1 4 } 3 7 9o 3 0 7 F O R M A T ! I H l . 2 0 A 4 . » BODE P H A S E P L O T S F O R F E E D D I S T U R B A N C E S ' / 3 8 0Z: 4 I X . I 9 A 4 . » F E E D ST A G E = ' . I 3 . 5 X . " I R A D / U N I T T I M E ! ( S E M I - L O G ) ' 1 3 8 1m END 3 8 2o S U B R O U T I N E P L O T P M I I T I T . N U M T R F . U K ) 3 8 3o R E A L MAGMIN.MAGMAX 3 8 45 COMPLEX SAVE 3 8 5g- COMMON/PPDATA/COEFVI5 . 4 ) . C 0 E F L C 5 . 4 ) . C Q N V A P ( 5 . 6 ) . N A M E ! 5 . 2 ) .AMW(S) 3863 COMMON/PTF/SAV E I 5 , 2 6 , 7 ) . F R O 1 2 6 ) , FROEXP( 2 6 ) , I T T T I 1 9 ) . M E A S C S . 3 ) • 3 8 70 4 M A N I ( 5 i 3 ) . F R M A X . F R M I N . M A G M A X . M A G M I N f N M E A S . N M A N I . N S M l . I C C 3 8 81 C 0 M M 0 N /M 0 D 4 / IS Y 13 5 ) , MEASNA( 5 . 5 ) , MANINA14 . 8 ) , I BLANK12 ) 389CD D I M E N S I O N I T I T ( 2 0 ) . A L O A O I 2 6 ) 3 9 0^ DATA P H A M I N . P H A M A X / - 3 6 0 . , * 9 0 . / 3 9 1-n P R I N T 3 0 7 . I T I T . I T T T 3 9 23. N P = 0 3 9 3=r I I = M E A S ! I J K . l ) 3 9 4^ I 2 = M E A S ( I J K . 2 ) 3 9 53 I 3 = M E A S ( U K . 3 ) 3 9 6-o DO 2 0 0 I K K K = 1 . NUMTRF 3 9 7o I K = M A N I I I K K K . 2 ) 3 9 8g- K = I K K K 3 9 9Q . I F I C A B S ! SA VEI U K . I . I K K K ) ) . E Q . O . O I G O TO I S O 4 0 0

N P = N P * I 4 0 1I F < 1 3 . E Q . O ) 4 0 2

-o * P R I N T 2 0 5 , I S Y I K ) . U K . K . I M A N I N A U . I K ) . J = 1 . 4 ) . 4 0 3o 4 I B L A N K , I M E A S N A I J . I 2 ) , J = 1 , 5 ) , I I 4 0 4~ I F I I 3 . N E . 0 ) 4 0 5^ 4 P R I N T 2 0 S . I S Y I K ) . I J K . K , I M A N I N A ( J . I K ) . J - I . 4 ) . 4 0 6o. 4 (NAM E! I 3 . J ) , J = 1 . 2 ) , ( M E A S N A ( J , I 2 ) , J = 1 , S ) , H 4 0 7g GO TO 2 0 0 4 0 8? 1 5 0 I F I I 3 . E Q . 0 ) 4 0 9° 4 P R I N T 3 0 5 . U K . K . I M A N I N A U , I K ) , J = 1 . 4 ) . 4 1 0

4 I B L A N K . I M E A S N A I J . 1 2 ) , J = l , 5 ) . 1 1 4 1 I■g I F ! 1 3 . N E . O ) 4 1 2q 4 P R I N T 3 0 5 . U K , K , ( MANI NA ( J , I K ) . J = 1 . 4 ) , 4 1 3p. 4 (NAME! 1 3 . J ) . J = I . 2 ) . (M EASN AI J . 1 2 ) , J = I . 5 ) . 1 1 4 1 4M 2 0 0 C O N T I N U E 4 1 55 DO 8 1 0 I C = I . 2 4 1 6P I K K = 0 4 1 7

DO 8 1 I I K K K = I . N U M T R F 4 1 8K = I K K K 4 1 9I F I C A B S I S A V E I U K . l . I K K K ) ) . E Q . O . O I G O TO 8 1 1 4 2 0C A L L P H A S E I S A V E . U K , : KKK . A LOA D ) 4 2 1I K K = I K K + I 4 2 2 , .C A L L P L O T O I F R O . A L O A D , 2 6 . I S Y I K ) . I K K . N P . P H A M I N . P H A M A X , 4 2 3 g

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* F R M I N . F R H A X . I C )C O N T IN U EC O N T IN U ERETURNFORMAT C5X« A l • • R E P R E S E N T S THE T R A N S F E R F U N C T I O N P C . 2 1 2 ,

* * 1 WHICHF O R M A T ( 2 X .

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4 A 4 , " T O • , 7 A 4 . NO I N P U T / O U T P U T4 A 4 ( TO

ON S T A G E * , 1 4 ) R E L A T I O N S H I P P I *

ON S T A G E * . 1 4 ), 2 1 2 ,

F O R M A T ! 1 H 1 . 2 0 A 4 . * BODE P H A S E P L O T S F O R M A N IPU L A T E D V A R I A B L E S * / ♦ I X . 1 9 A 4 , ' D E G R E E S VS F R E Q U E N C Y ( R A D / U N I T T I M E ) C S E M l - L O G ) » )

ENDS U B R O U T I N E P H A S E ( S A V E . I J K . I K K K , Q P H S D )COMPLEX S A V E ( 5 . 2 6 . 7 )D I M E N S I O N 0 P H S D I 2 6 )DATA T O L / 1 . O E - 3 0 /S I G N = + 1 . 0I F ( R E A L ( S A V E C I J K . 1 . I K K K ) ) . L T . 0 . 0 ) S I G N = - 1 . 00 I L A G = 0 . 0W R A P = 0 . 0DO 4 2 0 K = 1 . 2 6Q I s A I M A G C S A V E f I J K . K . I K K K ) ) * S 1 G N Q R = R E A L ( S A V E f I J K . K . I K K K ) 1 * S I G N I F f O R . G T . 0 . 0 ) GO TO 4 1 0I F f O I . L T . O . O . A N D . O I L A G . G E . O . O ) W R A P = W R A P » 1 . 0 I F f O I . G E . 0 . 0 . A N D . Q I L A G . l t . O . 0 ) W R A P = W R A P - 1 . O I F f C A B S f S A V E f I J K . K . I K K K ) ) . G T . T O L ) G O TO 4 0 0 Q P H S D ( K ) = Q P H S D ( K - 1 )GO TO 4 2 0

C O N T I N U E O I L A G = Q I

4 2 44 2 54 2 64 2 74 2 84 2 94 3 04 3 14 3 24 3 34 3 44 3 54 3 64 3 74 3 84 3 94 4 04 4 14 4 24 4 34 4 44 4 54 4 64 4 74 4 84 4 94 5 04 5 14 5 24 5 3

Q P H S D ( K ) = A T A N 2 C Q I , Q R ) * 5 7 . 2 9 5 8 4 3 6 0 . 0 * WRAP 4 5 4C O N T I N U E 4 5 5RETURN 4 5 6END 4 5 7BLOCK DATA 4 5 8C 0 M M 0 N / M 0 0 4 / I S Y ( 3 5 ) M E A S N A ( 5 . 5 ) . MAN IN A( 4 , 8 ) . I B L A N K C 2 ) 4 5 9DATA I B L A N K / • * * / 4 6 0DATA I S Y / * 1 * 4 6 1

♦ * A* * 8 * . *C * * o * . • E * , * F » • G * . *H* * I * . * J * * K * . * L * . * M * . * N * , 4 6 2* * o * • P * , *Q • * R * • * S * , T* , * U * . * V* . * W * . * X * . * Y * , * Z * / 4 6 3

DATA M E A S N A / L I Q * • U I D * . *COMP* . * O S I T * . * I O N * . 4 6 41 VAP* . OR • • •COMP* . * O S I T * . • I O N * . 4 6 52 THE* • TEM* . *P ER A * . * T U R E * . • * . 4 6 63 THE* . L I Q * . * U I D * . • F L O W * . * * . 4 6 74 THE* . VAP* • • O R * . * F L O W * . * • / 4 6 8

DATA M A N I N A / S . S . * • L I Q * • • U I D * , * F L O W * • 4 6 91 S . S . * . VAP* . *OR * . * F L O W * , 4 7 0 VJ

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ENDS U B R O U T I N E R D M A N K M A N f . N H A N I . HA N IT Y )I N T E G E R BLANKD I M E N S I O N I C H M C B l ( M A N U S . S )C O M M O N / M O D 4 / 1 S Y C 3 5 ) . M E A S N A I S t S I • M A N I N A f 4 > 8 ) • I B L A N K I Z f C O M M O N / M O D S / P R M A N I ( 5 ) • P R M E A S ( 5 , 3 *D A TA B L A N K / ' ' /D ATA I C H M / ' S S L ' • ' S S V • ' Q I N * • ' R E F ' , ' O V A ' • ' D L I • • ' B V A ' , ' B L I ' /

C ♦ ♦ • ♦ ♦ ♦ ♦ ♦ ♦ ♦ M A N I T Y * * * * * MO D E LS C * 4 * 4 4 4 4 4 4 4 I R E F B UC 4 4 4 4 * 4 4 4 4 4 2 R E F BVAC 4 4 4 4 4 4 4 4 4 4 3 D V A / D L I B L I NO T F E A S I B L E D R O P P E DC 4 4 4 4 4 4 4 4 4 4 4 D V A / O L I BVAC 4 4 4 4 4 M A N I < 1 , 1 } = S T A G E NUMBERC 4 4 4 4 4 M A N I ( 1 , 2 ) = T Y P E O F M A N I P U L A T E D V A R I A B L EC * * * * * H A N K I , 3 1 = N O T U S E DC 4 4 4 4 4 PRMANI C O N T A I N S P A R A M ET ER S FOR L I Q U I D L E V E L L O O P S

READ 3 I 0 « ( ( M A N K I , J ) « J = 1 , 2 ) , 1 = 1 . 5 ) . P R M A N I P R I N T 3 1 1 . PRMANI DO 5 0 1 = 1 . 5I F C M A N K I . 2 ) . E 0 . B L A N K ) G 0 TO 5 1N M A N I = IDO 1 0 K=1 , 8I F C H A N K 1 . 2 ) . E O . I C H M ( K ) ) G O TO 11

1 0 C O N T I N U E P R I N T 5 0 1 S T O P 2 3 4

1 1 M A N I ( I . 2 ) = KP R I N T 5 2 5 . ( M A N I N A ( J . K ) * J = 1 . 4 ) . M A N I ( I . 1 )

5 0 C O N T I N U E5 1 C O N T I N U E

I F I N M A N I . L T . 2 1 P R I N T 5 0 2 I F ( N M A N I . L T . 2 I S T O P 1 2 3 4 DO 6 0 1 = 1 .NMA NI I 2 = M A N I ( 1 , 2 )1 F ( I 2 . E Q . 4 ) G O TO 6 1 I F ( 1 2 . E Q . 5 ) G O TO 6 2 I F ( I 2 . E Q . 6 ) G 0 TO 6 2

6 0 C O N T I N U EP R I N T 5 0 3 S T O P 2 3 4 1

4 7 14 7 24 7 34 7 44 7 54 7 64 7 74 7 84 7 94 8 04 8 14 8 24 8 34 8 44 8 54 8 64 8 74 8 84 8 94 9 04 9 14 9 24 9 34 9 44 9 54 9 64 9 74 9 84 9 95 0 05 0 15 0 25 0 35 0 45 0 55 0 65 0 75 0 85 0 95 1 05 1 15 1 25 1 35 1 45 1 55 1 65 1 7 WOVa

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I 2 = M A N I f I * 2 l 5 2 2I F ( I 2 « E Q « 7 ) G O TO 6 3 5 2 31 F ( I 2 « E Q . 8 ) G 0 TO 6 4 5 2 4

o 6 5 C O N T I N U E 5 2 5

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;r 6 4 C O N T I N U E 5 2 91 P R I N T 5 2 6 . MANITY 5 3 0Q I F I M A N I T Y . N E . 3 ) RETURN 5 3 1^ P R I N T 5 0 3 5 3 2

S T O P 2 3 4 3 5 3 3^ 3 1 0 F O R M A T f 5 ( I 3 . A 3 > . 5 E 1 0 . 0 ) 5 3 43 3 1 1 F O R M A T ( / S X • • M A N I P U L A T E D V A R I A B L E S " , 5 X . ' P A R A M E T E R S = • . 5 G 1 2 . 4 » 5 3 5

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2 ♦ • --------- J O B A B O R T E D * 1 5 3 8_ 5 0 3 F O R M A T ( 5 X . " M A N I P U L A T E D V A R I A L B E S S P E C I F I E D DO NOT FORM A V A L I D * • 5 3 9Q . ♦ * C O M B I N A T I O N — J O B ABORTED* ) 5 4 0

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§ END 5 4 3-o S U B R O U T I N E RDMEAS ( ME AS. NMEAS ) 5 4 48 D I M E N S I O N 1 C H C 5 ) . M E A S ( S . 3 ) 5 4 5Z C O M M O N / P P D A T A / C O E F V ( 5 . 4 ) . C O E F L ( 5 . 4 ) • C 0 N V A P ( 5 . 6 ) . N A M E ( 5 . 2 ) . A M M ( 5 ) 5 4 6g C O M M O N / M O D 4 / I S Y ( 3 5 ) , M E A S N A ( 5 . 5 ) , M A N I N A ( 4 . 8 ) . I B L A N K I 2 ) 5 4 73 C O M M O N / M O D S / P R M A N I ( 5 1 . P R M E A S ( 5 . 3 1 5 4 8^ DATA I C H / * X * • * Y * . * T * . " L * . * V * / 5 4 9<. C * * * * * MEAS( I • 1 ) = S T A G E NUMBER 5 5 03= C * * * * * ME AS ( I . 2 ) = V A R I ABLE TO BE MEASURED 5 5 10 C * * * * * M E A S d .3 1 = C O M P O N M E N T TO BE MEASURED 5 5 2^ C * * * * * P R M E A S d . 1 1 = G A I N FOR MEASUREMENT D E V I C E 5 5 3■o C * * * * * P R M E A S I I . 2 ) = D E A D T I M E F O R MEASUREMENT D E V I C E 5 5 4q C * * * * * P R M E A S d . 3 ) = T I ME CON ST AN T FOR MEASUREMENT D E V I C E 5 5 51 P R I N T 3 0 0 5 5 6Z READ 3 0 4 . ( ( M E A S ( I . J ) , J = 1 . 3 ) . 1 = 1 . 5 * . ( ( P R M E A S d # J ) , J = 1 . 3 ) . I = 1 . 5 ) 5 5 7o DO 2 0 1 = 1 . 5 5 5 8

I I = M E A S ( l . l ) 5 5 9I F d l . E O . O l G O TO 2 4 5 6 0N M E A S = I . 5 6 1I 3 = M E A S ( 1 . 3 ) 5 6 2I F ( P R M E A S d . 1 ) . E G . 0 . 0 ) PRMEAS d . l 1 = 1 . 0 5 6 3DO 3 0 K = 1 . 5 5 6 4 WOo\

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I F ( M E A S ( l t 2 ) . E Q . I C H ( K ) ) G O TO 3 1 5 6 53 0 C O N T I N U E 5 6 6

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♦ ' T I M E CON ST ANT FO R MEASUREMENT D E V I C E * 1 5 7 8CD 3 0 4 F O R M A T ( 5 ( I 3 . A 1 . 1 2 1 a 1 5 F 3 . 1 ) 5 7 9^ 5 0 0 F 0 R M A T ( 5 X . 7 A 4 . * ON S T A G E • . 1 4 . 3 F 1 1 • 2 ) 5 8 0-n 5 0 1 FORMA T ( 5 X . ' U N R E C O G N I Z E D MEASUREMENT T Y P E J O B A B O R T E D * 1 5 8 1a END 5 8 23 "CD

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APPENDIX G

DESCRIPTION OF ETHYLENE COLUMN

D e r i v a t i o n o f P a ram e te r s f o r E th y le n e Column B a l l a s t Trays

G iven Data:

0 . 1 = 2«-83

P , = 2-35

= 63 .79 f t ^ (assumed t o be 90% A^)

A^ = 23.436 f t ^

W, = 54 l b / s e cm

= 71.7 I b y s e c

A ^ = 7 .81 f t ^ (= 1 / 3 Ag from Eq. 2 .43 o f [23 ] )

= .128 f tCM

AP = 4 .47 l b g / f t 2

= .167 f t

A = .005 f t

Vcd = °

= .295 f t

A t = 70 .88 f t ^

The c a l c u l a t i o n s f o r 1 / 8 ^ a n d B j /P^ w ere b a se d on d a t a

f o r t r a y f i v e .

308


309

Parameter Values Csed for S e n s it iv i ty Test

Low Normal High

Reboiler T.C.* 0 .39 0.52 2.6 min.

Condenser T.C. 0.0416 0.0473 0.0208 min.

Reboiler mass — #24.7 12,000 moles

Condenser mass —— #38.8 23,300 moles

1/6^ - Liquid T.C ( a l l stages) 0.0523 0.0630 0.0926 min.

” Vapor T.C ( a l l stages) 0.0255 0.0307 0.0451 min.

Holdup

Stages 2-4 54 .5 #77.9 101.3 moles

Stage 5 20.4 #29.2 38.0 moles

Stages 6-32 52.0 #74.3 98.6 moles

Stage 33 18.5 #26.4 34.3 moles

Stages 34-47 41.0 #58.5 76.1 moles

T.C. = le v e l c o n tr o lle r time constant

^Obtained from Wu [23]


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APPENDIX HDerivation of Model Parameters and M

Buckley et al. [24] have derived expressions for and 2 based on the tray hydraulic equations. This section repeats those expressions for the sake of completeness. The only addition to their method has been to include an analytical expression for the foam density as a function of tray type and P factor. This function alleviates the need to obtain dp/dF by graphical means. All of the following expressions can be evaluated once the detailed tray design has been completed.

t - . S.h 3 ^

®ow

where

0.1 A. W- T . = ---- S— ^(sec)

P c i ^dm

^ - '"a -57 ■ If + (sec)

3 A PFor valve trays with caps not fully lifted — z 0.

314

ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission.

315

Ap should be obtained from dry hole pressure drop equa

tion for the given tray type.

3P_ . j, ^ _ _1____T = 3 F - TRT-

The follow ing expression for was obtained from Figure

13.16, page 516 o f Van Winkle.

Pf “ ^ c l

where C = 1.50 for bubble cap trays

C = 2.35 for perforated trays

therefore.

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_ (At“Ad)Hci Pgi

^ c l ®0


316

T erm in o lo g y Used in A ppendix H

A a c t i v e t r a y a r e a (a re a w h e re b u b b l in go c c u r s ) ( f t2 )

A^ a r e a o f downcomer ( f t ^ )

A. minimum a re a th roug h w h ic h l i q u i d f low si n downcomer assem bly (ft% )

A t o t a l a re a o f t r a y , i . e . c r o s s s e c t i o n a la r e a o f column ( f t^ )

o v e r a l l e f f i c i e n c y ( f r a c t i o n )

% l » f o r c e / “ mass

H . a v e r a g e h e ig h t o f c l e a r l i q u i d on a t r a y( f t )

Hg foam h e ig h t ( f t )

H. = H +H + A t h e h e ig h t o f th e c l e a r l i q u i d j u s t downs t r e a m o f t h e i n l e t w e i r ( i . e . downcomer)

h e i g h t o f l i q u i d above w e i r ( f t )

w e i r h e ig h t ( f t )

M m o le s o f c l e a r l i q u i d on a n e q u i l i b r iu ms t a g e (m o le s /s tag e )

Mw 2 m o le c u la r w e ig h t o f c l e a r l i q u i d

volum e o f a e r a t e d l i q u i d on a c t i v e a r e a'c do f t r a y ( f t ^ ) = A^

volum e o f c e n t e r downcomer i n f r o t h , i f any ( f t 3)

W. s t e a d y s t a t e l i q u i d f lo w fro m t r a y(Ib jjj /sec)

s t e a d y s t a t e vapor f lo w th r o u g h t r a y(Ib jjj /sec)


317

A t h e g r a d i e n t o f c l e a r l i q u i d a c r o s s thet r a y ( f t )

P p r e s s u r e d ro p a c ro s s a t r a y ( I b ^ / f t ^ )

d e n s i t y o f c l e a r l i q u i d ( I b ^ / f t ^ )

Pg d e n s i t y o f f r o t h ( I b ^ / f t ^ )

p^ d e n s i t y o f vapor ( I b ^ / f t ^ )


APPENDIX I

DESCRIPTION OF BTX AND PBP MODEIS

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FEED STREAMS

2 1 .0 0 0 " ■ iH rx » J75.566 ........ ....... . ■■■■ ■COMPONENTS(MOLES/UN1T TIME) TOTAL ENTHALPY

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VITA

Kurt W. Komi nek i s the son o f Majorie and Henry

Komxnek of Wilmette, I l l i n o i s . He was born January 29,

1950 in Cleveland, Ohio.

Kurt graduated from New T rier East High School,

Winnetka, I l l in o i s , in June 1968. He entered Colorado

S ta te U niversity in September 1968 and graduated from

th e re with a B.S. in Engineering Science in June 1972.

Returning to Cleveland in the F a ll o f 1972, he entered

Case Western Reserve U n iversity where he worked under

Dr. M. D. Mesarovic on a world economic model. He

rece iv ed his M.S. in Systems Engineering from CWRU in

January 1974.

Kurt was employed as a Process Control Engineer by

Union Carbide in C harleston, West V irginia from February

1974 u n t il August 1976. At th a t time he entered Louisiana

S ta te U niversity where he i s p resen tly a candidate for the

d egree o f Doctor o f Philosophy in Chemical Engineering.


EXAMINATION AND THESIS REPORT

Candidate: Kurt W. Komlnek

Major Field: Chenlcal Engineering

Title of Thesis: Analysis of D is tilla tio n Column Dynamics

Approved:

Major Prdiessor and Chairman

^ Dean of the Graduate Sgkol

EXAMINING COMMITTEE:

Date of Examination:

August 27■1979


Analysis of Distillation Column Dynamics.

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Transcript of Analysis of Distillation Column Dynamics.