Analysis of Liquid-Phase Adsorption Fractionation in Fixed

Louisiana State UniversityLSU Digital Commons

LSU Historical Dissertations and Theses Graduate School

1965

Analysis of Liquid-Phase Adsorption Fractionationin Fixed Beds.Elmer Lawrence Morton JrLouisiana State University and Agricultural & Mechanical College

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Recommended CitationMorton, Elmer Lawrence Jr, "Analysis of Liquid-Phase Adsorption Fractionation in Fixed Beds." (1965). LSU Historical Dissertationsand Theses. 1019.https://digitalcommons.lsu.edu/gradschool_disstheses/1019

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T h is d i s s e r t a t io n h a s b een 65—6416 m ic r o f i lm e d e x a c t ly a s r e c e iv e d

M ORTON, J r . , E lm e r L a w r e n c e , 1 9 3 4 - ANALYSIS O F LIQUID—PHASE ADSO RPTIO N FRAC TIO NATIO N IN FIX E D B E D S.

L o u is ia n a State U n iv e r s i t y , P h .D ., 1965 E n g in e e r in g , c h e m ic a l

University Microfilms, Inc., Ann Arbor, M ichigan

ANALYSIS OF LIQUID-PHASE ADSORPTION FRACTIONATION

IN FIXED BEDS

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

S u b m it te d t o th e Graduate F a c u l t y o f th e L o u i s i a n a S t a t e U n i v e r s i t y and

A g r i c u l t u r a l and M e c h a n i c a l C o l l e g e i n 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 the d e g r e e o f

D o c to r o f P h i l o s o p h y

in

The D epartm ent o f C hem ica l E n g i n e e r i n g

byElmer Lawrence M orton , J r .

B. Ch. E . , G e o r g ia I n s t i t u t e o f T e c h n p l o g y , 1956 M . S . , L o u i s i a n a S t a t e U n i v e r s i t y , 1960

J a n u a r y , 1965

ACKNOWLEDGME NT S

The a u t h o r s i n c e r e l y w i s h e s t o thank Dr. P a u l M u r r i l l f o r

h i s many h e l p f u l s u g g e s t i o n s and s u p p o r t d u r i n g t h e m ajor p a r t o f

t h i s r e s e a r c h ; t o Dr. A dr ian E. J o h n so n f o r t h e i n i t i a l i n s p i r a t i o n

and en cou ragem e nt w hich led t o t h e i n i t i a l groundwork; t o Dr. James

B. C o r d i n e r , Dr. Frank R. Groves Dr. F r i t z A. McCameron, Dr. Ben E.

M i t c h e l l , and Dr. D a le U. Von R o se n b e r g o f t h e g r a d u a t e c o m m i t t e e ,

who p r o v id e d a s s i s t a n c e and c r i t i c i s m s ; t o th e e n t i r e f a c u l t y and

s t a f f o f t h e D ep ar tm en ts o f C hemica l E n g i n e e r i n g , M e c h a n i c a l En

g i n e e r i n g , and th e Computer R e s e a r c h C en ter f o r t h e i r i n t e r e s t ; t o

Dr. B i l l B. Townsend f o r a u t h o r i z a t i o n t o u s e th e f a c i l i t i e s o f the

Computer R e s e a r c h C e n te r f o r s e v e r a l hundred h ou rs o f m ach ine t im e ;

t o t h e C h a r l e s C o a tes Fund f o r f i n a n c i a l a s s i s t a n c e i n p r e p a r i n g th e

m a n u s c r i p t ; t o Mrs. Mary Mevers fo r t y p i n g t h e m a n u s c r i p t ; and t o

h i s w i f e f o r her c o n t i n u e d and e n t h u s i a s t i c moral su p p o r t and

a s s i s t a n c e .

DEDICATION

To my l a t e , b e l o v e d p a r e n t s .

Mr, and Mrs. Elmer L. Morton

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ........................................................................................................................... i i

DEDICATION.................................................................................................................................................i i i

LIST OF TABLES................................................................................................................................ v i

LIST OF FIG U R ES.................................................................................................................................v i i

ABSTRACT............................................................................................................................................... i x

Chapter

I . INTRODUCTION ............................................................................................................. 1

I I . BACKGROUND.................................................................................................................. 3

I I I . LITERATURE SURVEY................................................................................................... 5

IV. DEVELOPMENT OF A MODEL FOR LIQUID-PHASE ADSORPTION . . . 13

I n t r o d u c t i o n ....................................................................................................... 13D e r i v a t i o n o f e q u a t i o n s . ............................................................. 15Boundary c o n d i t i o n s ................................................................................. .. . 24

V. SOLUTION OF ADSORPTION MODEL BY NUMERICAL METHODS. . . . 27

N um erica l a n a l y s i s .............................................. 28S o l u t i o n o f a d s o r b e d - p h a s e ( d i f f u s i o n ) e q u a t i o n . . . . 30S o l u t i o n o f b u l k - p h a s e e q u a t i o n ........................................................ 32D i s c u s s i o n o f c o m p u t a t i o n a l problems ......................................... 37Computer program ............................................................................................ 39

V I . DISCUSSION OF RESULTS......................................................................................... 42

P r e s e n t a t i o n o f c a l c u l a t i o n s . . . 42P h y s i c a l s i g n i f i c a n c e o f t h e d i m e n s i o n l e s s

v a r i a b l e s A, H, and T ............................................................................. 43B e h a v io r o f computed a d s o r p t i o n c u r v e ......................................... 44The mass t r a n s f e r c o e f f i c i e n t , K ......................... 49Comparison o f computed and e x p e r i m e n t a l c u r v e s . . . . 51D e s i g n o f a f i x e d - b e d a d s o r b e r ......................................................... 57M oving-bed a d s o r p t i o n .................................................................................. 64

i v

V

C hap ter Page

V I I . CONCLUSIONS AND RECOMMENDATIONS......................................................................67

V I I I . BIBLIOGRAPHY.................................................................................................................... 70

A pp en d ix A - N u m e r ic a l M e t h o d s ............................................................................74A pp en d ix B - R e s u l t s .................................................................................................. 80Appendix C - Computer P r o g r a m .........................................................................1 1 6N o m e n c l a t u r e ....................................................................................................................161A u t o b i o g r a p h y ......................................................... 164

LIST OF TABLES

T a b l e Page

I . A c c u r a c y Check on I n crem en t S i z e ................................................................ 75

I I . Sunmary o f A d s o r p t i o n S y s te m s I n v e s t i g a t e d ...................................... 81

I I I . Wave F o rm a t io n T im e s , Bed Depth and S a t u r a t i o n ........................... 82

v i

LIST OF FIGURES

F i g u r e Page

1. E q u i l i b r i u m and O p e r a t i n g L i n e s f o r a T y p i c a l Case . . . . 44

2 . Comparison o f Computed and E x p e r i m e n t a l R e s u l t s ............................ 52

3 . F orm at ion o f t h e U l t i m a t e W a v e ........................... 60

4 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 1 . 0 ...................... 83

5 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 5 . 0 ...................... 84

6 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 10 . . . . . 85

7 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 2 0 86


9 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 50 . . . . . 88

10. M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 6 0 ...................... 89


12. B e n z e n e -n -H e x a n e on S i l i c a G e l , A - 0 . 0 1 ........................................... 91

13 . B e n z e n e -n -H e x a n e on S i l i c a G e l , A = 0 . 1 .......................................... 92

14. B e n z e n e -n -H e x a n e on S i l i c a G e l , A = 1 . 0 .......................................... 93

15. B e n z e n e -n -H e x a n e on S i l i c a G e l , A = 2 . 0 ......................................... 94

16. B e n z e n e -n -H e x a n e on S i l i c a G e l , A = 1 0 ......................................... . 95

17 . B e n z e n e -n -H e x a n e on S i l i c a G e l , A = 2 0 ............................................... 96

18 . B e n z e n e -n -H e x a n e on S i l i c a G e l , A ** 3 0 ............................................... 97

19. M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 1 0 .......................... 98

2 0 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A = 2 0 .......................... 99

2 1 . M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A ■ 3 0 . ....................... 100

v i i

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

M e t h y l c y c l o h e x a n e - T o l u e n e on S i l i c a G e l , A ■ 4 0 .....................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A • . 0 1 ..........................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A ■ .1 ..........................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A - 1 . 0 ..........................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A ■ 2 . 0 . . . . . .

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A ■ 3 . 0 ..........................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lum ina , A * 5 . 0 . . . . . .

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A * 10 . .....................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A * 2 0 . . . . . .

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m ina , A = 4 0 ..........................

M e t h y l c y c l o h e x a n e - T o l u e n e on A lu m in a , A ■ 8 0 ..........................

E f f e c t o f E q u i l i b r i u m Curve on Computed R e s u l t s .....................

E f f e c t o f P a r t i c l e S i z e on th e S ys tem : M e t h y l c y c l o h e x a n e -T o lu e n e on Alumina ..............................................................................................

Comparison o f J o h n s o n ' s Model w i t h t h e p r o p o s e d Model f o r Curves A and B ........................................................................................................

Comparison o f J o h n s o n ' s Model w i t h t h e P ro p o s ed Model f o r Curves C and D ..................... ...................................................................................

ABSTRACT

In t h o s e p r o c e s s e s i n v o l v i n g th e s e p a r a t i o n o f s o l u t i o n s ,

a d s o r p t i o n column d e s i g n has b een one o f t h e l a t e s t u n i t o p e r a t i o n s t o

u n d e r g o t h e o r e t i c a l t r e a t m e n t b e c a u s e o f t h e r e l a t i v e c o m p l e x i t y o f t h e

m echanism s i n v o l v e d . The p u r p o se o f t h i s i n v e s t i g a t i o n i s t o e s t a b l i s h

an improved method f o r s o l v i n g a d s o r p t i o n p r o b l e m s , e s p e c i a l l y where

i n t r a p a r t i c l e d i f f u s i o n i s a r a t e c o n t r o l l i n g mechanism. In p a r t i c u l a r

i t i s d e s i r e d t o d e r i v e a model d e s c r i b i n g f i x e d - b e d , l i q u i d - p h a s e

a d s o r p t i o n f r a c t i o n a t i o n i n a column i n term s o f c o n c e n t r a t i o n , bed

d e p t h , t im e and p a r t i c l e r a d i u s . The a p p r o p r i a t e boundary c o n d i t i o n s

f o r l i q u i d - p h a s e a d s o r p t i o n and a g e n e r a l n o n - l i n e a r e q u i l i b r i u m r e l a

t i o n a r e i n c l u d e d i n d e r i v i n g a s y s t e m o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s

d e s c r i b i n g t h e p r o c e s s .

The m a t h e m a t i c a l model c o n s i s t s o f a b u l k - p h a s e m a t e r i a l

b a l a n c e , an i n t r a p a r t i c l e d i f f u s i o n e q u a t i o n , and a r a t e e q u a t i o n f o r

i n t e r p h a s e t r a n s f e r a c r o s s t h e s u r f a c e f i l m ; t h e s e e q u a t i o n s must be

s o l v e d s i m u l t a n e o u s l y w i t h the b oundary c o n d i t i o n s and e q u i l i b r i u m

f u n c t i o n . The e q u a t i o n s a r e e x p r e s s e d i n term s o f c e r t a i n d i m e n s i o n l e s s

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

e r t i e s o f an y p a r t i c u l a r a d s o r b e n t b e d . T h i s n o n - l i n e a r s e t o f e q u a t i o n s

r e q u i r e s a s u i t a b l e n u m e r i c a l means o f s o l u t i o n . A n u m e r i c a l p r o c e d u r e

i s d e v e l o p e d and d i s c u s s e d f o r u s e on a h i g h - s p e e d c o m p u te r . The program

u s e d t o s o l v e t h e prob lem on an IBM 7040 computer i s d e s c r i b e d .

i x

X

The r e s u l t s o b t a i n e d i n t h i s work a r e s e t s o f computed c u r v e s

f o r f o u r l i q u i d s y s t e m s , e a c h s y s t e m r e q u i r i n g a s e p a r a t e computer s o l u

t i o n b e c a u s e o f c h a n g e s i n c o n c e n t r a t i o n or e q u i l i b r i u m r e l a t i o n s h i p .

The computed c u r v e s a r e matched w i t h e x p e r i m e n t a l d a t a on t h e s e s y s t e m s

t o d e t e r m i n e mass t r a n s f e r c o e f f i c i e n t s , w h i c h u l t i m a t e l y a r e u s e d f o r

d e s i g n p u r p o s e s . I t i s shown t h a t , f o r t h r e e o u t o f f o u r s y s t e m s

t e s t e d i n t h i s r e s e a r c h , e x p e r i m e n t a l d a t a c o r r e l a t e w i t h computed r e

s u l t s b e t t e r th an i n e a r l i e r work. T h i s f a c t l e n d s su p p o r t t o t h e

v a l i d i t y o f t h e p r o p o s e d m o d e l . Moreover t h e c o e f f i c i e n t s c a l c u l a t e d

from t h e i n v e s t i g a t i o n a r e w i t h i n an o r d e r o f m a g n i tu d e o f v a l u e s found

i n an i n d e p e n d e n t d e t e r m i n a t i o n , a g a i n g i v i n g some c r e d e n c e t o t h e

p r o p o s e d m o d e l . A l t h o u g h t h e s o l u t i o n o f t h i s model has b een c a r r i e d

out f o r p a r t i c u l a r e x a m p l e s , i t s h o u l d be a p p l i c a b l e i n t h e c a s e o f o t h e r

l i q u i d - p h a s e s y s t e m s f o l l o w i n g t h e mechanism s d e s c r i b e d , a s w e l l a s t h e

assumpticns i n v o l v e d in d e r i v i n g t h e m o d e l .

A d i s c u s s i o n i s p r e s e n t e d f o r t h e p u rp o se o f a p p l y i n g t h e

r e s u l t s t o an a c t u a l co lum n. For p u r p o s e s o f c o m m erc ia l d e s i g n e q u a t i o n s

a r e d e v e l o p e d t o compute wave f o r m a t i o n t im e and c o r r e s p o n d i n g bed d e p t h ,

d e g r e e o f s a t u r a t i o n , and l e n g t h o f th e u l t i m a t e wave i n f i x e d - b e d

a d s o r b e r s .

The r e s u l t s o f t h i s p r o j e c t have l e d t o a number o f c o n c l u

s i o n s and rec o m m e n d a t io n s f o r f u t u r e c o n s i d e r a t i o n . In p a r t i c u l a r i t has

b een found t h a t a s o l u t i o n can be found f o r a d e r i v e d model w i t h o u t t h e

n e c e s s i t y o f making o v e r s i m p l i f y i n g a s s u m p t i o n s . An improved c o r r e l a

t i o n b e tw e e n e x p e r i m e n t and t h e o r y has b een shown, but a t t h e same t im e

t h e r e i s need t o c o n s i d e r o t h e r m echanisms such a s l o n g i t u d i n a l

x i

d i f f u s i o n , m u l t i - c o m p o n e n t s y s t e m s , e f f e c t o f c o n c e n t r a t i o n on t h e r a t e

c o e f f i c i e n t , and n o n - i s o t h e r m a l o p e r a t i o n . In t h e f u t u r e i t i s p o s s i b l e

t h a t n u m e r i c a l m eth od s may be d i s c o v e r e d t o h a n d l e s i m u l t a n e o u s l y some

or a l l o f t h e s e f a c t o r s w i t h f a s t e r co m p u te r s than a r e p r e s e n t l y a v a i l

a b l e .

I . INTRODUCTION

T h i s d i s s e r t a t i o n i s th e r e s u l t o f r e s e a r c h on a d s o r p t i o n

f r a c t i o n a t i o n o f b i n a r y l i q u i d s . The pr im ary a n a l y s i s c o n c e r n e d th e

e f f e c t o f i n t r a p a r t i c l e d i f f u s i o n on the k i n e t i c s o f f i x e d - b e d

a d s o r p t i o n c o lu m n s . A t h e o r e t i c a l a p p r o a c h was used in w h ich p a r t i a l

d i f f e r e n t i a l e q u a t i o n s were d e r i v e d t o s i m u l a t e column o p e r a t i o n .

A c t u a l e q u i l i b r i u m r e l a t i o n s f o r r e a l s y s t e m s were i n c o r p o r a t e d i n t o

th e n u m e r i c a l s o l u t i o n s w hich were o b t a i n e d on a h i g h - s p e e d e l e c t r o n i c

c o m p u t e r .

Much o f th e work i n t h i s p r o j e c t i s an e x t e n s i o n o f r e s e a r c h

by numerous i n v e s t i g a t o r s a s o u t l i n e d in a s u r v e y o f p a p e r s on

a d s o r p t i o n column o p e r a t i o n . A l a r g e m a j o r i t y o f e a r l i e r s t u d i e s

were l i m i t e d to s i m p l e r c a s e s , su ch a s l i n e a r e q u i l i b r i u m or s t e a d y -

s t a t e o p e r a t i o n , w h ich c o u l d be s o l v e d e a s i l y . In t h i s p r o j e c t the

p a r t i c u l a r boundary c o n d i t i o n s u s e d , th e g e n e r a l e q u i l i b r i u m f u n c

t i o n , and t r a n s i e n t o p e r a t i o n a l l r e q u i r e d th e s o l u t i o n be o b t a i n e d

n u m e r i c a l l y . Both f i l m r e s i s t a n c e and i n t e r n a l d i f f u s i o n a l r e s i s

t a n c e s were i n c o r p o r a t e d i n th e m a t h e m a t i c a l d e s c r i p t i o n o f th e

p r o c e s s .

A s u c c e s s f u l c o r r e l a t i o n o f e x p e r i m e n t a l d a t a o b t a i n e d in

e a r l i e r s t u d i e s w i t h t h e t h e o r e t i c a l model d e v e l o p e d h e r e i n i s c o n

s i d e r e d a u s e f u l c o n t r i b u t i o n t o th e knowledge o f a d s o r p t i o n column

k i n e t i c s . A p r o c e d u r e f o r s o l v i n g t h e m o d e l , a l o n g w i t h a computer

program, i s o u t l i n e d . A l s o p r e s e n t e d were means o f a p p l y i n g the

s o l u t i o n t o t h e d e s i g n o f an a d s o r p t i o n co lum n , t a k i n g i n t o a c c o u n t

t h e t r a n s i e n t ph ase a f t e r s t a r t u p , and t h e c o r r e s p o n d i n g l e n g t h o f

bed r e q u i r e d f o r f o r m a t i o n o f th e s t e a d y - s t a t e w a v e . E q u a t i o n s f o r

c a l c u l a t i n g th e n e t d e g r e e o f s a t u r a t i o n i n an o p e r a t i n g f i x e d - b e d

co lumn were d e v e l o p e d f o r p u r p o s e s o f e c o n o m ic c o n s i d e r a t i o n s . The

a p p l i c a t i o n o f f i x e d - b e d d a t a t o m o v in g -b e d p r o c e s s e s was d i s c u s s e d

b r i e f l y .

I t i s hoped t h a t t h i s work w i l l s e r v e , a t l e a s t i n p a r t , a s

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

I I . BACKGROUND

The m a t h e m a t i c s o f a r i g o r o u s model o f a d s o r p t i o n p r o c e s s e s

i n column o p e r a t i o n has u n t i l r e c e n t t i m e s b een t o o com plex f o r e x a c tc

s o l u t i o n . S im p le c a s e s have b een t r e a t e d from a t h e o r e t i c a l s t a n d

p o i n t , but i n such i n s t a n c e s a c t u a l p r o c e s s e s d id n o t g e n e r a l l y

f o l l o w t h e r e s u l t i n g t h e o r y . Some i n v e s t i g a t o r s have d e r i v e d m o d e l s ,

p r i m a r i l y f o r g a s - p h a s e o p e r a t i o n , i n w h ic h s i m p l i f y i n g a s s u m p t i o n s

such a s l i n e a r e q u i l i b r i u m r e l a t i o n s h i p s and no i n t e r n a l d i f f u s i o n a l

r e s i s t a n c e s w i t h i n th e a d s o r p t i o n bed have b een made.

The b a s i c i n s p i r a t i o n f o r t h i s r e s e a r c h p r o j e c t e v o l v e d from

an i n t e r e s t in n u m e r i c a l a p p l i c a t i o n s f o r t h e s o l u t i o n o f e n g i n e e r i n g

p r o b le m s , where t h e number o f s i m p l i f y i n g a s s u m p t i o n s i s m in im i z e d a t

t h e e x p e n s e o f more d i f f i c u l t c o m p u t a t i o n a l t e c h n i q u e s . The need f o r

a d d i t i o n a l work i n th e s u b j e c t a r e a o f a d s o r p t i o n i s summarized by

e x c e r p t s from Joh n so n ( 3 0 ) :

S i n c e t h e computed s o l u t i o n s o f t h i s work do not y i e l d an e x a c t f i t w i t h d a t a o f l a r g e p a r t i c l e s i z e a d s o r b e n t , the n e x t l o g i c a l improvement i n th e method o f a n a l y s i s w hich was u s e d h e r e would be t o i n c l u d e i n th e b a s i c e q u a t i o n s a m a t h e m a t i c a l e x p r e s s i o n f o r th e i n t r a p a r t i c l e r e s i s t a n c e .

The a d d i t i o n o f an e x t r a unknown p a r a m e t e r , D. and an a d d i t i o n a l i n d e p e n d e n t v a r i a b l e , r , makes t h e prob lem a much more d i f f i c u l t one than was s o l v e d i n t h i s w ork . I t i s b e l i e v e d , h o w e v e r , t h a t t h e t e c h n i q u e s d e m o n s t r a t e d h e r e w i l l be a p p l i e d i n t h e f u t u r e , u s i n g f a s t e r and l a r g e r c o m p u ters i f n e c e s s a r y , t o a p p ro a ch more c l o s e l y t h e e x a c t s o l u t i o n to a d s o r p t i o n f r a c t i o n a t i o n p r o b le m s .

With th e p o s s i b i l i t y o f f u r t h e r im p r o v in g t h e t h e o r e t i c a l

4

a p p r o a c h t o a d s o r p t i o n , i t was d e c i d e d t o s e a r c h f o r a r e a l i s t i c model

w h ic h i n v o l v e d t h e a d d i t i o n a l mechanism o f i n t e r n a l p a r t i c l e d i f f u s i o n ,

h o p in g t h a t a more a d e q u a t e m a t h e m a t i c a l s t a t e m e n t o f th e b e h a v i o r o f

an a d s o r p t i o n column c o u l d be d e v e l o p e d . A n u m e r i c a l s o l u t i o n t o th e

improved model was a n t i c i p a t e d , r e q u i r i n g t h e u s e o f a v e r y h i g h - s p e e d

co m p u ter . S i n c e t h e e s t i m a t e d com p u t in g c a p a c i t y was a v a i l a b l e a t the

t im e t h i s r e s e a r c h was b e g u n , i t was deemed p r a c t i c a l t o u n d e r t a k e t h e

p r o j e c t f o r t h i s d i s s e r t a t i o n . I n i t i a l l y th e c o m p u t a t i o n s were t o be

done a t a remote l o c a t i o n u n t i l s i m i l a r f a c i l i t i e s a t L . S . U . were

i n s t a 1 le d .

T h i s r e s e a r c h h as c e n t e r e d around a model i n w hich f o u r major

v a r i a b l e s a r e c o n s i d e r e d s i m u l t a n e o u s l y : t i m e , bed d e p t h , p a r t i c l e

r a d i u s , and c o n c e n t r a t i o n , a l o n g w i t h a g e n e r a l e q u i l i b r i u m f u n c t i o n .

The l i q u i d - p h a s e was c h o s e n t o d e m o n s t r a t e th e model b e c a u s e t h e r e has

been l e s s work i n t h i s a r e a , and t h e boundary c o n d i t i o n s a r e more com

p l i c a t e d . No r e s t r i c t i o n was to be put on t h e e q u i l i b r i u m r e l a t i o n .

I I I . LITERATURE SURVEY

In t h i s s e c t i o n , a b r i e f s u m m a r iza t io n w i l l be g i v e n t o t h o s e

p r e v i o u s i n v e s t i g a t i o n s d e a l i n g w i t h the s u b j e c t o f a d s o r p t i o n , e s p e

c i a l l y t o p apers d e a l i n g w i t h column o p e r a t i o n . A l a r g e s t o r e h o u s e

o f l i t e r a t u r e e x i s t s f o r the many s e p a r a t e a s p e c t s o f a d s o r p t i o n , ea ch

o f w hich i s i n v o l v e d i n th e o v e r a l l o p e r a t i o n o f a p r o c e s s f o r th e

s e p a r a t i o n o f m i x t u r e s . E a r l y r e s e a r c h e r s were co n c e r n e d w i t h the

m e c h a n ic s o f s u r f a c e p r o c e s s e s , 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 form

o f i s o t h e r m s , a c t i v a t i o n e n e r g i e s , and thermodynamics o f a d s o r p t i o n in

g e n e r a l . The i n c o r p o r a t i o n o f t h e s e v a r i o u s mechanisms i n t o the d e s i g n

o f a p r o c e s s f o r the f r a c t i o n a t i o n o f m i x t u r e s i s , h ow ever , r e l a t i v e l y

r e c e n t , due to th e c o m p l e x i t y o f the m a th e m a t ic s when t h e s e mechanisms

a r e combined . The p a p ers summarized in t h i s s e c t i o n are t h o s e p r im a r

i l y i n v o l v e d i n the a t t e m p t t o r e l a t e the many known v a r i a b l e s i n t o a

u s a b l e model from which a p r o f i t a b l e p l a n t may be d e s i g n e d .

The e a r l y a t t e m p t s t o c o r r e l a t e r e s u l t s o f a d s o r p t i o n e x p e r i

ments were u s u a l l y h i g h l y e m p i r i c a l . The v e l o c i t y o f a d s o r p t i o n o f

carb on t e t r a c h l o r i d e on c h a r c o a l was measured by Hernad i n 1920 ( 2 0 ) .

A r e s u l t i n g e m p i r i c a l , but s a t i s f a c t o r y , c o r r e l a t i o n in w hich the r a t e

was assumed t o be p r o p o r t i o n a l t o an e x p r e s s i o n o f t h e form, Ae” ‘t ,

was u s e d , but no e x p l a n a t i o n o f th e r e a s o n s u n d e r l y i n g t h i s approach

were g i v e n . There was some s p e c u l a t i o n t h a t d i f f u s i o n p o s s i b l y

a f f e c t e d th e r a t e . Most o f th e work o f t h i s p e r i o d was s t i l l on e q u i

l i b r i u m i s o t h e r m s .

6

Other p a p e r s i n l i n e w i t h th e work o f Hernad were g i v e n by

R o g en sk y ( 4 2 ) , Schumann, (47 ) C o n s t a b l e ( 6 ) , and T a y l o r ( 4 9 ) . The

l a t t e r a u t h o r was one o f t h e f i r s t t o a t t r i b u t e • c o n c e p t s o t h e r th a n

a c t i v a t i o n e n e r g i e s a s r a t e - c o n t r o l l i n g m e c h a n is m s . Im p or tant t h e o

r e t i c a l a s p e c t s o f a d s o r p t i o n phenomena were d e s c r i b e d . Not o n l y

a c t i v a t i o n e n e r g i e s , but I n t e r n a l d i f f u s i o n a s a p o s s i b l e r a t e -

d e t e r m i n i n g f a c t o r were d i s c u s s e d i n T a y l o r ' s p a p e r . He o b t a i n e d d a t a

and t r e a t e d q u a l i t a t i v e l y t h e e f f e c t s o f a d s o r b e n t m a t e r i a l , c o n c e n

t r a t i o n , s u r f a c e c h a r a c t e r i s t i c s , h e a t o f a d s o r p t i o n , and t e m p e r a t u r e .

An e a r l y paper w hich i n v o l v e d t h e d ynam ics o f a g a s - s o l i d

s y s t e m was w r i t t e n by Furnas ( 1 2 ) . M a t h e m a t i c a l e q u a t i o n s f o r a h e a t

t r a n s f e r p rob lem , r e l a t i n g th e v a r i a b l e s o f t im e and t e m p e r a t u r e w i t h

d i s t a n c e were s o l v e d . E x p e r i m e n ta l d a t a were c o r r e l a t e d , u s i n g th e

t h e o r e t i c a l r e s u l t s o f h i s m o d e l .

The d e v e lo p m e n t o f more s o p h i s t i c a t e d t h e o r i e s o f a d s o r p t i o n

f r a c t i o n a t i o n p r o c e s s e s i n a column began in t h e l a t e 1 9 3 0 ' s and e a r l y

1 9 4 0 ' s l a r g e l y i n t h e f i e l d o f c h ro m a to g ra p h y . Some h i g h l y c o m p l i c a t e d

m a t h e m a t i c a l e q u a t i o n s were d e r i v e d , however in many i n s t a n c e s t h e y

vjere e i t h e r o v e r l y s i m p l i f i e d or not s o l v e d f o r want o f a p r a c t i c a l

means o f s o l u t i o n . W i l s o n ( 5 3 ) a p p l i e d a d s o r p t i o n t o th e s t u d y o f r a t e

p r o c e s s e s i n c h r o m a t o g r a p h ic c o lu m n s , making t h e a s s u m p t i o n o f i n s t a n

ta n e o u s e q u i l i b r i u m b e tw e e n th e f l u i d and a d s o r b e n t p h a s e s , w i t h

n e g l i g i b l e d i f f u s i o n . T h ese a s s u m p t i o n s r e s u l t e d i n a s t r a i g h t f o r w a r d

a n a l y t i c a l s o l u t i o n f o r a two-component s y s t e m . T h i s a u t h o r r e c o g n i z e d

th e p r o b a b l e , but a d m i t t e d l y c o m p le x , e f f e c t o f i n t r a p a r t i c l e d i f f u s i o n .

D e v a u l t (7 ) c o n t i n u e d W i l s o n ' s work i n c h r o m a t o g r a p h ic a n a l y s i s

by d e r i v i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s w hich c o n s i d e r e d m u l t i p l e

components a l t h o u g h t h e y were not s o l v e d . The c h a r a c t e r i s t i c sh apes

o f waves formed i n ch r o m a to g r a p h ic columns were d i s c u s s e d q u a l i t a t i v e l y .

G lu e c k a u f and C oates ( 1 6 , 1 7 , 1 8 ) , i n a s e r i e s o f p a p e r s , d e s c r i b e d

s t e a d y - s t a t e a d s o r p t i o n f o r both a s i n g l e s o l u t e and a b in a r y s o l u t i o n

a s r e l a t e d t o chrom atography . Thomas (50 ) d e v e l o p e d a k i n e t i c approach

t o th e a n a l y s i s o f chromatography for a s i n g l e s o l u t e . Cases f o r b o th

i n i t i a l l y dry and i n i t i a l l y s a t u r a t e d beds a re c o n s i d e r e d u s i n g e q u i

l i b r i a o f l i n e a r or Langmuir t y p e s . Improved agreement between th e o r y

and e x p e r im e n t r e s u l t e d o v e r t h a t shown where i n s t a n t a n e o u s e q u i l i b r i u m

was assumed .

Other work, c a r r i e d out d u r in g the wart im e p e r i o d , i n c l u d e d

s t u d i e s by Gamson, e t a l . (13 ) who d e v e l o p e d e m p i r i c a l mass t r a n s f e r

r a t e c o r r e l a t i o n s f o r v a r i o u s a d s o r p t i o n s y s t e m s ; and by Thomas ( 5 1 ) ,

who a p p l i e d a s e c o n d - o r d e r r a t e e q u a t i o n t o th e s u r f a c e r e a c t i o n s t e p

w hich was assumed t o c o n t r o l .

In the p e r i o d f o l l o w i n g World War I I , the s t u d y o f a d s o r p t i o n

p r o c e s s e s in a column expanded t r e m e n d o u s ly . In g e n e r a l most a u t h o r s

r e c o g n i z e d a t l e a s t q u a l i t a t i v e l y , t h r e e r a t e mechanisms which were

thought Lo i n f l u e n c e column o p e r a t i o n : s u r f a c e r e a c t i o n , i n t r a p a r t i c l e

d i f f u s i o n , and e x t e r n a l d i f f u s i o n throu gh a f i l m . Most o f the work

c e n t e r e d on g a s - s o l i d s y s t e m s and w i t h l i n e a r i s o t h e r m s . Some v e r y

e l e g a n t m od e ls were proposed and a n a l y t i c a l s o l u t i o n s were found fo r

r e s t r i c t e d c a s e s , not o n l y i n a d s o r p t i o n , but a l l o f the u n i t o p e r a

t i o n s , a s d em o n str a ted by M a r s h a l l and P i g f o r d ( 3 7 ) . T h e o r e t i c a l

i n v e s t i g a t i o n s began t o e x p l a i n th e phenomena o f i o n - e x c h a n g e , which

8

i s a l l i e d i n many r e s p e c t s t o a d s o r p t i o n . L a r g e - s c a l e com m erc ia l

a p p l i c a t i o n s fo r n ew ly d e v e l o p e d a d s o r p t i o n o p e r a t i o n s b eg a n , a s e x

e m p l i f i e d by th e Hypersorb and A rosorb p r o c e s s e s f o r s e p a r a t i o n o f

p e t r o le u m f e e d - s t o c k s ( 4 , 2 4 , 2 7 ) .

Hougen and M a r s h a l l (25 ) d e v e l o p e d e q u a t i o n s d u r in g t h i s

p e r i o d f o r g as a d s o r p t i o n under both f l o w and n o n - f l o w c o n d i t i o n s .

L i n e a r e q u i l i b r i u m i n an i s o t h e r m a l bed was assumed, f o r which a n a l y t

i c a l s o l u t i o n s were found. G r a p h ic a l methods f o r the c a s e o f non

l i n e a r i s o t h e r m s were g i v e n , and a l s o f o r n o n - i s o t h e r m a l beds i n which

c h e m ic a l r e a c t i o n o c c u r s . E f f e c t s o f p a r t i c l e s i z e on th e k i n e t i c s

was not t r e a t e d q u a n t i t a t i v e l y . A s t a g e - w i s e approach was made by

M a ir , e t a l . ( 3 6 ) , f o r th e s e p a r a t i o n o f two components i n a f a s h i o n

a n a lo g o u s t o d i s t i l l a t i o n . Lap id u s and Rosen (35 ) proved the e x i s

t e n c e o f th e c o n s t a n c y o f the u l t i m a t e w a v e - s h a p e i n a s u f f i c i e n t l y

l on g co lumn, p r o v id e d the c u r v a t u r e o f the e q u i l i b r i u m f u n c t i o n i s

n e g a t i v e . Barrer (3 ) o b t a in e d r a t e d a ta f o r b a t c h w i s e f l o w o f ammonia

and o t h e r g a s e s and c a l c u l a t e d s u r f a c e d i f f u s i o n c o e f f i c i e n t s , u s i n g a

H e nr y 's Law e q u i l i b r i u m r e l a t i o n . E a g le and S c o t t (8 ) p u b l i s h e d a

co m p reh en s iv e paper g i v i n g k i n e t i c and e q u i l i b r i u m d a ta a l o n g w i t h

o t h e r s i g n i f i c a n t m e a su rem en ts , f o r a number o f l i q u i d - p h a s e a d s o r p t i o n

s y s t e m s .

The a c t u a l s o l u t i o n o f e q u a t i o n s i n s i m p l e c a s e s fo r which

a n a l y t i c a l methods were u n a v a i l a b l e or o v e r l y c o m p l i c a t e d began i n the

e a r l y 1 9 5 0 ' s p a r a l l e l i n g th e i n t r o d u c t i o n o f h i g h - s p e e d , s t o r e d - p r o g r a m ,

e l e c t r o n i c computing m a c h i n e s , which c o u ld o f t e n p r o v id e a c c u r a t e

n u m e r ic a l r e s u l t s . An e a r l y paper d e a l i n g w i t h s o l u t i o n o f a d s o r p t i o n

9

k i n e t i c s p rob lem s on th e computer was a u t h o r e d by R o s e , Lombardo,

and W i l l i a m s ( 4 3 ) , i n a s i m p l e s t a g e w i s e a p p r o a c h t o f r a c t i o n a t i o n o f

a b i n a r y , l i q u i d m i x t u r e . I n s t a n t a n e o u s e q u i l i b r i u m a t t h e p a r t i c l e

s u r f a c e was a s su m e d , n e g l e c t i n g pore d i f f u s i o n r e s i s t a n c e s . Computed

s o l u t i o n s were compared w i t h e x p e r i m e n t a l d a t a o b t a i n e d from s t e a d y -

s t a t e ru n s u s i n g a m i x t u r e o f b en zen e and hexane on f i n e p a r t i c l e s o f

s i l i c a g e l . F a i r l y good agree m en t was o b t a i n e d b e c a u s e o f t h e f i n e n e s s

o f th e a b s o r b e n t , s o t h a t i n t e r n a l d i f f u s i o n was l a r g e l y , but not

e n t i r e l y , e l i m i n a t e d . S i m i l a r t r e a t m e n t was g i v e n t o l i q u i d - s o l i d

a d s o r p t i o n s y s t e m s i n t h e s t e a d y - s t a t e by S c h m e l z e r , e t a l . ( 4 6 ) , who

showed q u a l i t a t i v e l y by a s e r i e s o f i n t e r r u p t e d runs t h a t i n t e r n a l

d i f f u s i o n v a s a s i g n i f i c a n t m echanism . In t h e s e t e s t s th e f e e d r a t e

was c u t o f f d u r i n g t h e o p e r a t i o n o f an a d s o r p t i o n e x p e r i m e n t . The

p r e s e n c e o f a v e r t i c a l , downward d i s p l a c e m e n t o f th e e f f l u e n t c o n c e n

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

Amundson ( 1 , 2 ) was one o f th e e a r l i e s t i n v e s t i g a t o r s t o a t t a c k

th e prob lem o f i n t e r n a l d i f f u s i o n q u a n t i t a t i v e l y in o v e r a l l column

o p e r a t i o n . He c o n s i d e r e d t h e p r o c e s s i n term s o f an i r r e v e r s i b l e m echan

ism where t h e r a t e o f mass t r a n s f e r i s p r o p o r t i o n a l t o f l u i d s t r e a m g a s

c o n c e n t r a t i o n and t o t h e q u a n t i t y o f a d s o r b a t e c o n t a i n e d i n t h e b e d .

He a l s o found s o l u t i o n s f o r t h e r a t e where a k i n e t i c t h e o r y was

a s su m e d , and s e p a r a t e l y f o r t h e c a s e o f r a d i a l d i f f u s i o n c o n t r o l l i n g ,

K a s te n and Amundson ( 3 2 ) s t u d i e d a d s o r p t i o n f r a c t i o n a t i o n from a t h e o

r e t i c a l s t a n d p o i n t i n f l u i d i z e d b e d s . S t e a d y - s t a t e o p e r a t i o n w i t h a

l i n e a r i s o t h e r m and a k i n e t i c r a t e mechanism was a s s u m e d , but w i t h

q u a n t i t a t i v e t r e a t m e n t o f i n t e r n a l d i f f u s i o n . The r a t e o f s e p a r a t i o n

10

o f t h e f e e d I n t o two s t r e a m s was b a s ed on a p r o b a b i l i t y t h e o r y . Other

p a p e r s i n v o l v e d w i t h s t u d y o f d i f f u s i o n a l and o t h e r r a t e p r o c e s s e s a r e

t h o s e by E d e s k u ty and Amundson ( 1 0 ) , i n l i q u i d - p h a s e a d s o r p t i o n and i n

t h e f i e l d o f i o n e x c h a n g e by L a p id u s and Amundson ( 3 3 ) . In b o th c a s e s

l i n e a r e q u i l i b r i u m was as su m ed . As y e t no s u b s t a n t i a l t r e a t m e n t o f

t h e f o r m a t i o n t im e r e q u i r e d f o r t h e u l t i m a t e , or s t e a d y - s t a t e a d s o r p

t i o n wave i n f i x e d bed s had b een g i v e n . O th er im p o r ta n t p a p ers i n

t h i s c a t e g o r y i n c l u d e work by J u ry and L i c h t ( 3 1 ) i n t h e d r y i n g o f

g a s e s , by G e ser and C anjar (1 4 ) i n a d s o r p t i o n o f m e th a n e , by H i e s t e r

and V erm eulen (22 ) i n i o n - e x c h a n g e , and by Rosen ( 4 4 ) .

One o f t h e f i r s t p a p ers w here th e r e s t r i c t i o n o f l i n e a r e q u i

l i b r i u m c o u l d be l i f t e d was by H i e s t e r e t a l . ( 2 1 ) . In t h i s r e p o r t

d e a l i n g w i t h i o n e x c h a n g e , m eth od s were p r e s e n t e d f o r c o r r e l a t i n g

e x p e r i m e n t a l r e s u l t s when b o th e x t e r n a l and i n t e r n a l d i f f u s i o n a l r e s i s

t a n c e o c c u r s i m u l t a n e o u s l y . S t e a d y - s t a t e o p e r a t i o n was assumed a s

w e l l a s a r a t e o f r a d i a l d i f f u s i o n p r o p o r t i o n a l t o a c o n c e n t r a t i o n

d r i v i n g f o r c e . S e l k e e t a l . (48 ) o b t a i n e d d i f f u s i v i t y d a t a f o r i o n -

e x c h a n g e columns u s i n g s h a l l o w - b e d e x p e r i m e n t s , and a g e n e r a l e q u i l i b

r ium. They s u g g e s t e d d i v i d i n g a column i n t o a s e r i e s o f s h a l l o w bed s

f o r a s t e p w i s e s o l u t i o n . No c o n s i d e r a t i o n f o r wave f o r m a t i o n t im e or

bed d e p t h was g i v e n . G i l l i l a n d and Baddour (1 5 ) t e s t e d a s t e a d y - s t a t e

i o n - e x c h a n g e s y s t e m , u s i n g an o v e r a l l c o e f f i c i e n t o f mass t r a n s f e r f o r

a l l r e s i s t a n c e s com bined . The v a r i a b l e s c o n s i d e r e d were p a r t i c l e s i z e ,

bed h e i g h t and d i a m e t e r , f e e d r a t e and c o n c e n t r a t i o n . Column d im en

s i o n s were found not t o a f f e c t r e s u l t s o v e r th e ran ge o f v a r i a b l e s

i n v e s t i g a t e d . E b e r l y and S p e n c e r ( 9 ) , a l s o u s i n g a lumped r e s i s t a n c e

11

o b t a i n e d a d s o r p t i o n r a t e c o n s t a n t s by means o f a p u l s e d - f l o w e x p e r i

m e n t . I n t h i s r e s e a r c h , t h e sh ape c h a n g e s i n t h e p u l s e were a t t r i b u t e d

t o a d s o r p t i o n e f f e c t s .

J o h n so n (30) e x t e n d e d th e r e s e a r c h i n t o l i q u i d - p h a s e a d s o r p

t i o n f r a c t i o n a t i o n i n f i x e d bed s by a c o m p l e t e n u m e r i c a l s o l u t i o n t o

s y s t e m s w h ich i n v o l v e d s i m u l t a n e o u s l y th e v a r i a b l e s o f bed d e p t h , t i m e ,

and c o n c e n t r a t i o n . The e x t e r n a l f i l m was assumed t o c o n t r o l , w i t h

i n s t a n t a n e o u s e q u i l i b r i u m a t th e i n t e r f a c e b e tw een a d s o r b e d and non

a d s o r b e d p h a s e s . I n t e r n a l d i f f u s i o n was n e g l e c t e d , a l t h o u g h d i s c u s s e d

q u a l i t a t i v e l y . No r e s t r i c t i o n s were put on th e e q u i l i b r i u m , e x c e p t

t h a t i t s h o u l d not c r o s s t h e o p e r a t i n g l i n e . P a r t i c u l a r l y i n t e r e s t i n g

was a c o m p le t e r e - e v a l u a t i o n o f boundary c o n d i t i o n s used by p a s t r e

s e a r c h e r s and a r e s u l t a n t m o d i f i c a t i o n f o r u s e in l i q u i d s y s t e m s . H is

work d e m o n s t r a t e d t h a t s o l u t i o n s t o c e r t a i n p rob lem s f o r w h ich no known

a n a l y t i c a l t e c h n i q u e s e x i s t may be found by n u m e r ic a l means on a com

p u t e r , and t h a t th e c o m p l e x i t y o f th e problem w h ich c o u ld be s o l v e d

d ep en d s l a r g e l y on t h e c a p a b i l i t i e s o f th e computer i t s e l f .

More r e c e n t l y Masamune and Sm ith (38 ) , r e c o g n i z i n g t h a t d i f f u

s i o n a l r e s i s t a n c e s w i t h i n th e p o ro u s bed a r e o f im p o r ta n c e have d e r i v e d

and s o l v e d e q u a t i o n s r e l a t i n g th e v a r i a b l e s o f c o n c e n t r a t i o n , t i m e ,

l e v e l , and p a r t i c l e r a d i u s f o r v a p o r - p h a s e a d s o r p t i o n . A t t e n t i o n was

g i v e n t o th e s p e c i a l c a s e s o f s u r f a c e a d s o r p t i o n c o n t r o l l i n g , pore

d i f f u s i o n c o n t r o l l i n g and e x t e r n a l d i f f u s i o n c o n t r o l l i n g . These i n

v e s t i g a t o r s assumed l i n e a r e q u i l i b r i u m and boundary c o n d i t i o n s su ch t h a t

an a n a l y t i c a l s o l u t i o n was o b t a i n e d f o r e a c h c a s e . U s i n g a n i t r o g e n -

h e l i u m vapor m i x t u r e on v y c o r p a r t i c l e s the r e s u l t s o f e x p e r i m e n t a l

12

t e s t s compared f a v o r a b l y w i t h t h e o r y o n l y when t h e mechanism o f pore

d i f f u s i o n was i n c l u d e d .

IV. DEVELOPMENT OF A MODEL FOR LIQUID-PHASE ADSORPTION

1 . I n t r o d u c t i o n

In o r d e r t o d e f i n e a p r a c t i c a l m odel i t i s n e c e s s a r y t o s t a t e

t h e b a s i c a s s u m p t i o n s a s t o th e n a t u r e o f t h e a d s o r p t i o n p r o c e s s .

S i n c e f i x e d - b e d l i q u i d p h ase b i n a r y a d s o r p t i o n i s under c o n s i d e r a t i o n ,

a l l a s s u m p t i o n s w i l l be b a s e d on t h i s t y p e o f p r o c e s s .

a . The f l o w o f l i q u i d th r o u g h th e s y s t e m i s assumed t o be

p l u g - l i k e f l o w i n g upward from bo t to m o f t h e s y s t e m . No

c o n c e n t r a t i o n or v e l o c i t y g r a d i e n t s e x i s t i n a r a d i a l

d i r e c t i o n from t h e d i r e c t i o n o f f l o w . In d e r i v i n g th e

d i f f e r e n t i a l e q u a t i o n s t h e s e term s c o u l d be i n c l u d e d when

th e f i e l d o f f l u i d d ynam ics p r o d u c e s more k n ow ledge o f

r a d i a l e f f e c t s . In a d d i t i o n , l o n g i t u d i n a l , or a x i a l ,

d i f f u s i o n i s c o n s i d e r e d n e g l i g i b l e .

b . A c o n s t a n t f e e d c o m p o s i t i o n w i l l be as su m ed , a s t h i s i s

th e normal mode o f o p e r a t i o n , u n l e s s c h a n g in g from one

o p e r a t i n g l e v e l t o a n o t h e r .

c . The f e e d i s a b i n a r y or p s e u d o - b i n a r y s y s t e m o f c o m p o n e n t s ,

one o f w h ic h i s p r e f e r e n t i a l l y a d s o r b e d , th u s e f f e c t i n g a

f r a c t i o n a t i o n i n t o two p h a s e s , one a n o n - a b s o r b e d , or b u l k ,

p h a se and t h e o t h e r p h a s e b e i n g t h e l i q u i d s o l u t i o n c o n

t a i n e d w i t h i n th e a d s o r b e n t .

d . An i n i t i a l l y dry s y s t e m i s u s e d . In v a p o r a d s o r p t i o n the

13

14

e n t e r i n g va p o r s tr e a m d i s p l a c e s t h e g a s e o u s f l u i d found

i n t h e v o i d s p a c e s around t h e p a r t i c l e s o f a d s o r b e n t ,

but n o t th e f l u i d c o n t a i n e d w i t h i n the p a r t i c l e s them

s e l v e s . In l i q u i d p r o c e s s e s , i t i s assumed t h a t a s th e

f e e d r i s e s t h r o u g h t h e b e d , t h e e n t e r i n g l i q u i d f i l l s

b o th t h e v o i d s p a c e s and a t t h e same t im e e n t e r s and f i l l s

t h e p ore vo lume a lm o s t i n s t a n t a n e o u s l y . I n e f f e c t i v e sp a c e

c o n t a i n i n g t r a p p e d v ap or i s not c o n s i d e r e d a p a r t o f th e

pore volume i n w h i c h a d s o r p t i o n o c c u r s . As w i l l be

p o i n t e d ou t t h i s d i f f e r e n c e b e tw e e n v ap or and l i q u i d phase

m echanism s c a u s e s a m ajor d i f f i c u l t y i n t h e l a t t e r c a s e

b e c a u s e o f t h e r e s u l t i n g boundary c o n d i t i o n s . T h i s assum p

t i o n p r e c l u d e s a s t u d y o f t h e e f f e c t o f a s t e p chan ge i n

f e e d c o m p o s i t i o n a t a l a t e r moment o f t i m e , w i t h o u t m o d i

f y i n g t h e e n t i r e m o d e l .

e . The s y s t e m w i l l be c o n s i d e r e d t o o p e r a t e i s o t h e r m a l l y ; a l l

h e a t e f f e c t s a r e e i t h e r n e g l i g i b l e or c a n c e l e a c h o t h e r

o u t , a s i s t y p i c a l o f l i q u i d - p h a s e p r o c e s s e s .

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

and p o s s e s s an e f f e c t i v e r a d i u s , c o n s t a n t th r o u g h o u t th e

b e d . A l t h o u g h i t i s known t h r o u g h m i c r o s c o p i c s t u d y t h a t

th e p a r t i c l e s a r e h i g h l y i r r e g u l a r , i t i s i m p o s s i b l e to

d e s c r i b e them e x c e p t by means o f f a m i l i a r g e o m e t r y . The

i n t e r i o r p o re s p a c e , by means o f w h ic h d i f f u s i o n o c c u r s

i n e a c h p a r t i c l e i s h i g h l y t o r t u o u s but w i l l be a c c o u n t e d

f o r by an e f f e c t i v e i n t e r n a l o v e r a l l d i f f u s i v i t y , a s su m in g

15

o n l y r a d i a l d i f f u s i o n . There a r e s e v e r a l i n t e r n a l d i f f u

s i o n mechanism s t h o u g h t t o o c c u r , bu t t h e s e a r e u s u a l l y

lumped t o g e t h e r and d e s c r i b e d by a s i n g l e e f f e c t i v e d i f f u

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

g . The o n l y f a c t o r c o n t r o l l i n g a d s o r p t i o n w i t h i n t h e pore

s p a c e i s d i f f u s i o n i t s e l f . G e n e r a l l y t h i s has b een shown

t o be the c a s e , a l t h o u g h i t i s p o s s i b l e t h a t th e a d s o r p

t i o n s t e p a t t h e i n t e r f a c e b e tw e e n f l u i d and s o l i d may

c o n t r o l i n some i n s t a n c e s . There i s no r e a s o n why t h i s

m echanism c o u ld not a l s o be i n c l u d e d , p r o v id e d t h e r a t e

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

h . There i s no e f f e c t o f c o n c e n t r a t i o n on th e f i l m c o n s t a n t

or d i f f u s i o n c o n s t a n t . T h i s a s s u m p t i o n i s n e c e s s i t a t e d by

t h e f a c t t h a t t h e s e e f f e c t s a r e v e r y d i f f i c u l t t o e v a l u a t e .

i . Eq u im olar c o u n t e r - d i f f u s i o n i s a s su m ed . There i s a n e t

f l o w o f l i q u i d o n to th e a d s o r p t i v e s u r f a c e s , but t h i s i s

s m a l l compared t o th e t o t a l q u a n t i t y o f l i q u i d i n w h ich

d i f f u s i o n o c c u r s .

2 . D e r i v a t i o n o f E q u a t i o n s

C o n s i d e r a v e r t i c a l tower packed w i t h a d s o r b e n t m a t e r i a l , f ed

by a b i n a r y s tr e a m o f c o n s t a n t c o m p o s i t i o n e n t e r i n g a t th e b o t tom and

f l o w i n g upward in p l u g - l i k e f l o w . T aking a h o r i z o n t a l s l i c e o f t h i c k

n e s s , d z , one may w r i t e an o v e r a l l b u l k - p h a s e m a t e r i a l b a l a n c e on the

more a d s o r b e d com pon en t , i n terms o f v o l u m e t r i c f l o w r a t e s , a s f o l l o w s ;

16

I n p u t s t r e a m : Qx

Qx +

f t 3 / h r

Output s tream : dz f t 3 / h r 0 , r

f t 3 / h rOutput from b u lk p h a s e t o a d s o r b e n t : W

A c c u m u la t io n i n b u lk p h a s e :

w here x = volume f r a c t i o n o f t h e more a d s o r b e d component Q = t o t a l f e e d f l o w r a t e , f t ^ / h r , assumed c o n s t a n t z * bed d e p t h , f t .0 = e l a p s e d t i m e , h r .S = c r o s s s e c t i o n a l a r e a , f t ^ f v = f r a c t i o n v o i d s i n th e bed r = p a r t i c l e r a d i u s , f t

(The a b ove l i s t e d d i m e n s i o n s a r e f o r i l l u s t r a t i v e p u r p o s e s ; any o t h e r

c o n s i s t e n t s e t o f u n i t s may be u s e d . )

Combining t h e s e e x p r e s s i o n s and h o l d i n g Q c o n s t a n t , g i v e s an o v e r a l l

b a l a n c e :

The q u a n t i t y , W, i s e v a l u a t e d by means o f a term e x p r e s s i n g mass

t r a n s f e r from t h e b u l k p h ase a c r o s s t h e o u t e r s u r f a c e o f t h e a d s o r b e n t

p a r t i c l e s and i n t o t h e i n t e r i o r o f t h e p a r t i c l e s , a s f o l l o w s ;

a = e f f e c t i v e p a r t i c l e r a d i u s , f t

2 . The s u r f a c e a r e a , Ag , i s computed from the s u r f a c e a r e a per

( 4 - 1 )

1. The v o l u m e t r i c t r a n s f e r r a t e o f th e more a d s o r b a b l e com

ponent per c u b i c f o o t o f a d s o r b e n t i s AgD

where Ag = s u r f a c e a r e a a c r o s s w h ich d i f f u s i o noc c u r 3 on t h e s u r f a c e o f t h e s p h e r i c a l p a r t i c l e s , f t 2 / f t 3 bed

D = e f f e c t i v e d i f f u s i v i t y , f t 2 / hr

= c o n c e n t r a t i o n g r a d i e n t i n a d s o r b e d ph ase a t t h e s u r f a c e

17

s p h e r e t i m e s a term e x p r e s s i n g t h e f r a c t i o n o f s u r f a c e

open t o t h e d i f f u s i n g s u b s t a n c e , t i m e s t h e t o t a l number o f

s p h e r e s p e r u n i t vo lume o f bed;

7 7S u r f a c e a r e a per s p h e r e = 4 n a f t

Number o f s p h e r e s p e r f t 3 * 3 C .~£y)4 jt ad

Pore volume p e r u n i t V_Pbvolume o f s p h e r e s = — f t 3 / f t 3

v

where Vp = p o r e v o lu m e , f t 3 / l ba b s o r b e n t

Pb = d e n s i t y o f a b s o r b e n t ,l b / f t 3 bed

The l a t t e r term i s a f a c t o r w h i c h a c c o u n t s f o r t h e s o l i d , non-

p o ro u s m a t e r i a l , and i s assumed t o be t h e same on a s u r f a c e

b a s i s a s on a volume b a s i s .

Combining t h e s e term s g i v e s :

A - 4 , . 2 . 3 ( 1 - f y ) . v p P b f t 2 / f t 3 beds 4 ii a3 ! _ f v

The t o t a l d i f f u s i o n , W, i n a v o l u m e t r i c s e c t i o n , S d z , i s t h u s :

W = 3Vp p b . Sdz ( o y \ D f t 3 / h r ( 4 - 2 )a I 3 r l 9 , z , r = a

S u b s t i t u t i n g f o r W i n t o e q u a t i o n ( 4 - 1 ) g i v e s an o v e r a l l b u l k - p h a s e

m a t e r i a l b a l a n c e :

-Q ( - 1 ^ dz + 3.Q.CV-P. PP2 sd z (J 1 Z \ = fv ( - | ^ SdzyrHzj Qp a \ ^ r ) 9»z>r=a l d 0 y z , r

+ 3P(Vp p b) / _ d j r \ = f / _ | x \ ( 4 - 3 )\ b z J 9 , r a ^ r ] 9 , z , r « a ^ j z , r

or - QS

An a d s o r b e d - p h a s e , or i n t r a p a r t i c l e m a t e r i a l b a l a n c e m a y ' a l s o

be d e r i v e d , u s i n g a s p h e r i c a l s h e l l . Making t h e a s s u m p t i o n t h a t

18

d i f f u s i o n o c c u r s o n l y i n a r a d i a l d i r e c t i o n one may w r i t e t h e f o l l o w i n g

t e r m s , n o t i n g t h a t t h e d i r e c t i o n o f d i f f u s i o n i s o p p o s i t e t o t h e d i r e c

t i o n o f change o f r a d i u s .

L e t t h e i n n e r r a d i u s o f a s p h e r i c a l s h e l l be r , th e n t h e o u t e r

r a d i u s i s r + A r .

M a t e r i a l e n t e r i n g a t r + A r i s th u s e q u a l t o

4*» ( V d (a

M a t e r i a l l e a v i n g a

n r2 D ( - +V r) e , » 1 - f v

t r i s 4 ji r 2 P / d y \ Vp

e , z '

4 jtr2 VP . / d j A dr1 - f v r . z

dr

9 , zv

and a c c u m u l a t i o n i s

E q u a t in g term s g i v e s t h e i n t r a p a r t i c l e m a t e r i a l b a l a n c e :

or D

fa [ r2D(-tf ) ]L d r

[■’ ( - & ) * (■ &

r 2 /jL * \\ riQ I r *:

- ) 2r ' e , z

= r^ 0

r ,z

w h i c h l e a d s t o :

D d 2^h r 2

+ 2 f-4-o r

0 , z

o ( 4 - 4 )

0,z r , z

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

o n l y , i t w i l l be n e c e s s a r y t o f i n d a p a r t i c u l a r s o l u t i o n t o e v a l u a t e

t h e boundary c o n d i t i o n s n eed ed t o s o l v e e q u a t i o n ( 4 - 1 ) .

The c o n d i t i o n s a t t h e i n t e r f a c e b e t w e e n b u l k - p h a s e and

a d s o r b e n t s u r f a c e n eed t o be d e s c r i b e d . I t h as b een shown by s e v e r a l

i n v e s t i g a t o r s t h a t t h e r e e x i s t two r e s i s t a n c e s , a f i l m r e s i s t a n c e

19

e x t e r n a l t o t h e a b s o r b e n t p a r t i c l e s a s w e l l a s the i n t r a p a r t i c l e

r e s i s t a n c e i t s e l f . The r a t e o f more a d s o r b a b l e component f l o w i n g

a c r o s s t h e i n t e r f a c e b e tw een t h e s e r e s i s t a n c e s i s :

K ( x - x * ) - D ( 4 - 5 )\ r / 0 , z , r = a

where K = f i l m c o e f f i c i e n t o f mass t r a n s f e r ,f t / s e c

x - x * = c o n c e n t r a t i o n d i f f e r e n c e p r o p o r t i o n a l t o t h e d r i v i n g f o r c e g i v i n g r i s e t o movement o f more a d s o r b a b l e m a t t e r a c r o s s th e phase b o u n d a r i e s .

I n ord er t o compute th e d r i v i n g f o r c e , x - x , an e q u i l i b r i u m

r e l a t i o n s h i p i s r e q u i r e d . The s i m p l e s t , but a t t h e same t im e most

i n e f f i c i e n t , method i s t o assume l i n e a r e q u i l i b r i u m , w h ic h may i n

c e r t a i n c a s e s p r o v i d e a n a l y t i c a l s o l u t i o n s . However i t i s w e l l known

t h a t l i n e a r e q u i l i b r i u m r e l a t i o n s a r e good o n l y f o r d i l u t e g a s e s or a

v e r y narrow range o f c o n c e n t r a t i o n . S in c e i t i s t y p i c a l t o c o v e r a

w ide r a n g e , p a r t i c u l a r l y i n l i q u i d - p h a s e a d s o r p t i o n , where the e q u i l i b

rium i s u s u a l l y a c o m p l i c a t e d or n o n - l i n e a r f u n c t i o n a s o l u t i o n t o th e

model w i l l be so u g h t f o r any ty p e o f e q u i l i b r i u m c u r v e . Even i f a

m a t h e m a t i c a l r e l a t i o n i s unknown i t s h o u ld be p o s s i b l e by a n u m e r ic a l

p r o c e d u r e t o u s e an i n t e r p o l a t i o n formula on a t a b l e o f measured e q u i

l i b r i u m d a t a . The e q u i l i b r i u m e q u a t i o n w i l l be g i v e n i n a most g e n e r a l

form; y = f ( x * )

For p u r p o s e s o f d e m o n s t r a t i o n t h e e q u i l i b r i u m f u n c t i o n may be

changed t o o b t a i n s o l u t i o n s f o r a number o f s y s t e m s , w i t h o u t a l t e r i n g

th e o v e r a l l method o f s o l u t i o n .

E q u a t io n s ( 4 - 3 , 4 - 4 , 4 - 5 , and 4 - 6 ) s i m u l t a n e o u s l y d e s c r i b e the

20

model o f t h e a d s o r p t i o n p r o c e s s ch o s en f o r t h i s i n v e s t i g a t i o n . These

e q u a t i o n s , a l o n g w i t h th e boundary c o n d i t i o n s t o be d i s c u s s e d i n a

l a t e r s e c t i o n , must be s o l v e d . S i n c e a g e n e r a l e q u i l i b r i u m i s assum ed,

th e e q u a t i o n s w i l l r e q u i r e a n u m e r ic a l ap p roach t o th e s o l u t i o n . As

w i l l be shown l a t e r , one o f th e boundary c o n d i t i o n s , a s f a r a s i t i s

known, a l s o p r e c l u d e s any but a n u m e r ic a l s o l u t i o n . I t i s hoped t h a t

a c o n v e r g e n t , s t a b l e n u m e r ic a l method may be found t o p erm it an a c c u r a t e

and p r e c i s e r e p r e s e n t a t i o n o f t h e t r u e p r o c e s s .

i t w i l l be h i g h l y c o n v e n i e n t t o f i n d a s e t o f d i m e n s i o n l e s s p a ra m eters

whose s u b s t i t u t i o n f o r t h e r e a l v a r i a b l e s w i l l p r o v i d e a s o l u t i o n i n

d ep en den t o f th e p h y s i c a l p r o p e r t i e s o f th e p r o c e s s . F i n d i n g a p p r o

p r i a t e t r a n s f o r m a t i o n e q u a t i o n s i s l a r g e l y a m a t t e r o f t r i a l and e r r o r .

I n tr o d u c e t h r e e new p a r a m e te r s T, H, and R w hich a r e r e l a t e d t o the

o r i g i n a l v a r i a b l e s 0 , z , and r by t h e s e e q u a t i o n s :

B e f o r e p r o c e e d i n g d i r e c t l y t o a s o l u t i o n o f t h e fo u r e q u a t i o n s ,

R = r . K , or i f r = a , A = aD D

( 4 - 7 )

(Q/S)a( 4 - 8 )

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

(4 - 10 )

2 1

From e q u a t i o n ( 4 - 7 ) , and s i n c e R i s a f u n c t i o n o f r ,

( H ) „ , KD ( 4 - U )

Then

T rD \ b K l H ,T ,1

( 4 - 1 2 )0 , z , r = a \ / H,T,R»A

S u b s t i t u t i o n o f e q u a t i o n ( 4 - 1 2 ) i n t o e q u a t i o n ( 4 - 5 ) g i v e s th e f o l l o w i n g

d i m e n s i o n l e s s form:

V ( 4 - 1 3 )x - x

f a ) H.T.R,

The i n t r a p a r t i c l e e q u a t i o n ( 4 - 4 ) may be t ra n s fo rm ed by th e

f o l l o w i n g p r o c e d u r e . The term , y , i s f i r s t put i n t o d i m e n s i o n l e ;3 r2

form:

3 ri9 . 2

. r» H f ) m,z 0 , z

S i n c e 0 and d R = K , 3 r D

9 , 2KD

M3 r

\ o R /3 R *)

t h e n

( § ]

e , z ( 4 - 1 4 )

= K

0 ,zD2 \ 3 R‘i

T , H

The term 3 y i s r e a d i l y changed t o d i m e n s i o n l e s s form;

(fe) riZ ■ _(fi) (t |) + (tr) (t I) + (th) ( i i) ]r>i

2 2

The l a s t two term s on th e r i g h t o f t h i s e q u a t i o n a r e z e r o and

( - I f ) ^ - # , . . _ ■ $ ( - u ) r _ - ' « - . »S u b s t i t u t i n g e q u a t i o n s ( 4 - 7 ) , ( 4 - 1 4 ) , and ( 4 - 1 5 ) i n t o e q u a t i o n ( 4 - 4 )

r e s u l t s I n | o 2 y + 2 __dj; | _ l-&l\ 14-161Ian2 R aRJ H,T \ 9T/ H,R

E q u a t i o n ( 4 - 1 6 ) e x p r e s s e s t h e i n t r a p a r t i c l e d i f f u s i o n i n term s

o f t h e d i m e n s i o n l e s s p a r a m e t e r s , R and T, t h u s making i t s s o l u t i o n com

p l e t e l y in d e p e n d e n t o f t h e p h y s i c a l c h a r a c t e r i s t i c s o f any g i v e n p r o

c e s s .

The b u l k - p h a s e e q u a t i o n i s changed by com p u t in g t h e p a r t i a l

d e r i v a t i v e s , Z_dx\ . / 5 x \ , and / o y \ i n term s o f t h eI d z j r 0 I d e l I\ / r , y \ ' r , z ' ' 0 , z , r = a

d i m e n s i o n l e s s p a r a m e t e r s R ,T , and H:

[ 4 ~ \ = f d x dT j. 9 x 3H fix S R lV / r , 9 + ^ i ^ l j r . 0 ( '

/ 5 x \ _ r a x 5T . 5 x d H , 9 x Br"1 ,, . 0.[ S o l L 5 T a e aR a 0 J ( 4 - 1 8 )\ / r , z r , z

( j L l \ = K / d j A (4 - 1 2 )\ r/ 9 , z , r = a D \ 5R / H,T,R-A

But s i n c e R i s a f u n c t i o n o f r o n l y and H i s a f u n c t i o n o f z o n l y ,

e q u a t i o n s ( 4 - 1 7 ) and ( 4 - 1 8 ) become

( d x \ _ f dx ^T _dx dH~| .\ S z / r , Q L a T d z dH d z j r > 9

\&9j l ^ t d e j' f x , z r , ;

( 4 - 2 0 )

23

The p a r t i a l d e r i v a t i v e s a r e e v a l u a t e d

from e q u a t i o n s ( 4 - 8 ) and ( 4 - 9 ) :

/ 3T \ = K_ £vC ^ / r . G = D Q/Sr , 0 D Q/S

K v ( 4 - 2 1 )

\ / r , 0 (Q /S)a( 4 - 2 2 )

( 4 - 2 3 )

S u b s t i t u t i o n o f e q u a t i o n s ( 4 - 1 2 ) , ( 4 - 1 9 ) , ( 4 - 2 0 ) , ( 4 - 2 1 ) , ( 4 - 2 2 ) , and

( 4 - 2 3 ) i n t o e q u a t i o n ( 4 - 3 ) w i l l r e s u l t i n th e f o l l o w i n g t r a n s fo r m e d

m a t e r i a l b a l a n c e fo r th e b u l k - p h a s e ;

E q u a t io n s ( 4 - 1 3 ) , ( 4 - 1 6 ) and ( 4 - 2 4 ) i n v o l v e o n l y t h e d i m e n s i o n

l e s s p aram eter R ,T , and H, and w i t h t h e p r o p e r boundary c o n d i t i o n 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 w i l l c o n s t i t u t e t h e m a t h e m a t i c a l model o f th e

p r o c e s s u n der s t u d y . I t i s t h e s e e q u a t i o n s f o r w h ic h a n u m e r ic a l

t e c h n i q u e must be foun d , a s w i l l be d i s c u s s e d i n th e n e x t c h a p t e r . To

summarize, the e q u a t i o n s a r e r e p e a t e d ;

( 4 - 2 4 )

x - x . * ( 4 - 1 3 )

2 R r> R(1 R

d 2 y + 2 d y ( 4 - 1 6 )

( 4 - 2 4 )

24

3 . Boundary C o n d i t i o n s

One o f t h e m ajor a r e a s o f d i f f i c u l t y i n s o l v i n g such m o d e l s

a s d e r i v e d i n t h e l a s t s e c t i o n i s t h a t o f d e r i v i n g t h e p r o p e r boundary

c o n d i t i o n s . In v a p o r - p h a s e work t h e a s s u m p t io n i s g e n e r a l l y made t h a t

a t t h e s t a r t , t h e bed i s f i l l e d w i t h f l u i d h a v in g c o n c e n t r a t i o n , y ,

f o r a l l l e v e l s . As t h e f e e d s tr e a m e n t e r s and d i s p l a c e s t h i s o r i g i n a l

f l u i d t h e pore c o n c e n t r a t i o n i n c o n t a c t w i t h f e e d i s a lw a y s yD.

M a t h e m a t i c a l l y t h i s c o n d i t i o n i s g i v e n ( i n term s o f t h e d i m e n s i o n l e s s

p a r a m e t e r s ) by t h e f o l l o w i n g s t a t e m e n t :

a t T ■ 0 , y * y Q, f o r a l l H

In l i q u i d - p h a s e a d s o r p t i o n , i t i s more r e a s o n a b l e t o s t a t e t h a t th e

bed i n i t i a l l y c o n t a i n s no f l u i d , i . e . , t h a t t h e bed i s c o m p l e t e l y d r y .

A c t u a l l y t h e r e I s some g a s e o u s f l u i d in th e bed a t the s t a r t , but l i q u i d

i s assumed t o f i l l th e p ore s p a c e a s w e l l a s t h e v o i d volume a l m o s t

i n s t a n t a n e o u s l y and t h e r e f o r e t h e f e e d s tream i s now i n c o n t a c t w i t h

t h e p o r e f l u i d o f t h e same c o m p o s i t i o n a s t h a t o f t h e f e e d . T h i s con

d i t i o n h o l d s f o r t h e f r o n t o f l i q u i d a s i t moves up t h e b e d . T h e r e f o r e

a t th e l i q u i d f r o n t , x = y . J o h n s o n , i n h i s s t u d y ( 3 0 ) , found t h a t

t h e wave f r o n t was d e f i n a b l e m a t h e m a t i c a l l y by th e e q u a t i o n H = T.

However when th e a d d i t i o n a l v a r i a b l e , p a r t i c l e r a d i u s , i s i n t r o d u c e d ,

i t can be shown t h a t t h i s boundary e q u a t i o n i s no l o n g e r a p p l i c a b l e

and a new one has t o be o b t a i n e d . I t may s t i l l be assumed t h a t th e

p o r e s a t t h e l i q u i d f r o n t a r e f i l l e d w i t h l i q u i d o f b u lk c o m p o s i t i o n

x , i . e . , y * x a t t h e wave f r o n t ( f o r a l l r ) . In o r d e r t o f i n d the

c o r r e c t e q u a t i o n d e s c r i b i n g t h e wave f r o n t i n terms o f t h e new d im en

s i o n l e s s p a r a m e t e r s , an e q u a t i o n d e s c r i b i n g t h e t im e e l a p s e d when th e

25I

bed i s f i l l e d t o d e p t h z i s s t a t e d by:

* - - q7 ^ - <*v + VP P b) <*-25)

The te r m , ( f v + Vp P b ) > g i v e s th e t o t a l s p a c e o c c u p i e d by l i q u i d per

c u b i c f o o t o f bed d e p t h . R earran ge e q u a t i o n ( 4 - 2 5 ) a s f o l l o w s :

f V Po _ v z = D b z0 STS q 7 s '

2M u l t i p l y through by K_ :

D

0 _ f v z = VP r2 zD D ' (Q/S) Q/S ' D

From e q u a t i o n ( 4 - 9 ) , t h e l e f t hand s i d e i s T:

T = vp Pb . R2 zQ/S D

S i n c e H = K v p P h x , t h e r i g h t hand s i d e may be w r i t t e n i n terms o f (Q/S)a

H:

T = H .D

Let aK = A, where A i s the d i m e n s i o n l e s s p a r t i c l e r a d iu s a t D

the sphere s u r f a c e . Then

T = HA ( 4 - 2 6 )

E q u a t io n ( 4 - 2 6 ) i s a s t a t e m e n t o f th e wave f r o n t i n th e d i m e n s i o n l e s s

v a r i a b l e s a l o n g w i t h t h e boundary c o n d i t i o n ; a t T = HA, x ■ y f o r a l l

R. An a n a l y t i c a l s o l u t i o n u s i n g a boundary c o n d i t i o n o f t h i s t y p e i s

p r o b a b l y v e r y com plex i f n o t i m p o s s i b l e . No s o l u t i o n s u s i n g t h i s ty p e

o f boundary c o n d i t i o n s have b een s e e n i n l i t e r a t u r e .

The o t h e r boundary c o n d i t i o n , a t th e f e e d i n l e t i s s t r a i g h t

forw ard : a t z = 0 , x ■ Xp f o r a l l 0 > O , or i n terms o f t h e v a r i a b l e s

26

T and H:

a t H ■ 0 , x * Xj, f o r a l l T ^ 0

Now t h e p rob lem may be s o l v e d p r o v i d e d a s u i t a b l e n u m e r i c a l

method can be f o u n d . The f o l l o w i n g c h a p t e r w i l l i l l u s t r a t e a means o f

a c c o m p l i s h i n g a s o l u t i o n th r o u g h n u m e r i c a l a n a l y s i s .

V. SOLUTION OF ADSORPTION MODEL BY NUMERICAL METHODS

The n u m e r i c a l s o l u t i o n o f a s y s t e m o f n o n - l i n e a r d i f f e r e n t i a l

e q u a t i o n s i s o f t e n a r a t h e r t e d i o u s , t i m e - c o n s u m in g t r i a l - a n d - e r r o r

p r o c e d u r e . There has b e e n l i t t l e s u c c e s s i n th e n u m e r i c a l a n a l y s i s o f

n o n - l i n e a r s y s t e m s , and t h e f i n a l s o l u t i o n , i f one can be f o u n d , r e s t s

upon c o n s i d e r a b l e c o m p u t a t i o n a l e x p e r i m e n t a t i o n ( 1 1 ) , In t h i s work

th e s o l u t i o n , r e p r e s e n t e d by a s e t o f c u r v e s r e l a t i n g t h e d i m e n s i o n l e s s

p a r a m e t e r s A,H, and T, t o l i q u i d c o n c e n t r a t i o n , was e f f e c t e d by t a k i n g

a d v a n t a g e , i n p a r t , o f c e r t a i n w e l l - e s t a b l i s h e d t e c h n i q u e s f o r l i n e a r

e q u a t i o n s . The c r i t e r i a e n s u r i n g c o n v e r g e n c e and s t a b i l i t y f o r l i n e a r

e q u a t i o n s , ca n n o t u s u a l l y be g u a r a n t e e d f o r a n o n - l i n e a r s y s t e m su ch

a s t h a t e n c o u n t e r e d i n t h i s r e s e a r c h p r o j e c t . However, i t can be

r e a s o n a b l y e x p e c t e d t h a t i n t h o s e p rob lem s r e p r e s e n t i n g p h y s i c a l p r o

c e s s e s a s u f f i c i e n t l y a c c u r a t e s o l u t i o n w i l l have b een r e a c h e d i f t h e

v a r i a t i o n o f in c r e m e n t s i z e and p r e c i s i o n o v e r a c o n s i d e r a b l e range

r e s u l t s i n e s s e n t i a l l y t h e same a n sw er ( 4 0 ) . S e v e r a l t r i a l run s w ere

made w i t h d i f f e r e n t v a l u e s o f t h e f i n i t e d i m e n s i o n l e s s t im e in c r e m e n t s

and w i t h p r e c i s i o n s o f e i g h t and s i x t e e n d i g i t s t o c h e c k t h e a c c u r a c y

o f th e computer s o l u t i o n s , A c o m p a r i s o n o f r e s u l t s u s i n g f o u r in c r e m e n t

s i z e s and a t b o th p r e c i s i o n s i s g i v e n i n T a b le I o f Appendix A. In a l l

c a s e s an a t t e m p t was made t o y i e l d an a p p r o x i m a t i o n o f t h e t r u e s o l u

t i o n t h a t was a c c u r a t e t o w i t h i n one p e r c e n t , w h i c h i s s u f f i c i e n t f o r

t h e t y p e o f p r o c e s s under c o n s i d e r a t i o n .

27

2 8

1. N u m e r ic a l A n a l y s i s

The t y p i c a l n u m e r i c a l a p p r o x i m a t i o n p r o c e d u r e i n v o l v e s s c a n n i n g

a g r i d o f s u f f i c i e n t l y s m a l l mesh s i z e and c a l c u l a t i n g enough s i n g l e

p o i n t s t o c o v e r t h e r a n g e o f p a r a m e t e r s i n v o l v e d . Many u n f o r e s e e n

p r o b l e m s , n o t o n l y o f a m a t h e m a t i c a l n a t u r e , bu t p r a c t i c a l o n es a s w e l l ,

can be e x p e c t e d t o a r i s e i n t h e i n i t i a l a t t e m p t t o s o l v e an y s y s t e m o f

d i f f e r e n t i a l e q u a t i o n s . Even i f a c o n v e r g e n t , s t a b l e n u m e r i c a l method

i s known, t h e r e a r e a l s o p ro b lem s o f computer s t o r a g e and s p e e d t o be

o v e r c o m e . I t i s n o t d i f f i c u l t t o t a x t h e c a p a b i l i t i e s o f t h e l a r g e s t

and f a s t e s t m a c h in e s now i n e x i s t e n c e . The m a t h e m a t i c a l p rob lem t o be

s o l v e d i n t h i s r e s e a r c h i s d e f i n i t e l y r e s e r v e d f o r a f a s t c o m p u te r , s i n c e

s e v e r a l m i l l i o n a r i t h m e t i c o p e r a t i o n s a r e e x e c u t e d i n e v e n t h e s i m p l e s t

c a s e .

In f i n d i n g a s o l u t i o n t o t h e a d s o r p t i o n m o d e l , th e f o l l o w i n g

a p p r o a c h was t a k e n . The d e s i r e d c o m p l e t e s o l u t i o n i s a v a l u e o f x ,

l i q u i d - p h a s e c o n c e n t r a t i o n , f o r a l l p o s s i b l e v a l u e s o f t h e d i m e n s i o n

l e s s v a r i a b l e s , A,H, and T. In t h i s c a s e a t h r e e - d i m e n s i o n a l g r i d i s

n e c e s s a r y . I n p r a c t i c e i t i s n e c e s s a r y t o c o n s i d e r o n l y one v a l u e o f

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

compute a c o m p l e t e s e t o f c u r v e s f o r v a r i o u s H and T. At a g i v e n A,

t h e r e s u l t i n g t w o - d i m e n s i o n a l g r i d can be shown t o have a t r i a n g u l a r

s h a p e , due t o t h e n a t u r e o f t h e boundary c o n d i t i o n s ( 3 0 ) . To

i l l u s t r a t e t h i s a s k e t c h o f a p a r t i a l g r i d a t some c o n s t a n t A i s g i v e n

b e lo w ;

29

H, D im e n s to n le s s Bed L e v e l

Boundary (Wave Front ) AH = T

a)*i-4H

C

H

i

The p o s i t i o n o f th e l i n e r e p r e s e n t i n g th e l i q u i d f r o n t a s i t

r i s e s through the a d s o r p t i o n column depends upon t h e v a lu e o f A. A l l

a r ea t o the r i g h t o f the boundary l i n e r e p r e s e n t s u n f i l l e d a d so rb en t

ahead o f th e l i q u i d f r o n t .

To o b t a i n a s o l u t i o n , a s y s t e m a t i c method must be d e r iv e d

u s i n g a l l a v a i l a b l e i n f o r m a t i o n . At the o u t s e t , when th e l i q u i d f i r s t

e n t e r s a t th e bottom o f the bed , the c o n c e n t r a t i o n i s t h a t o f the f e e d ,

x f . Thus = X f , The p o i n t s co rr e sp o n d in g t o the bottom o f th e bed

a t a l l l a t e r T a re p o s i t i o n e d a lo n g th e f i r s t column o f th e s k e t c h

(j = 1) , and th u s have a v a l u e , x j ]_ = x £ , a s s p e c i f i e d by the boundary

c o n d i t i o n . The nu m erica l v a l u e o f a l l rem ain ing p o i n t s on the g r i d ,

e x c e p t fo r column 1, must be de term ined by nu m erica l means, and a s w i l l

be s e e n , the procedure fo r o b t a i n i n g t h e s e p o i n t s i n v o l v e s c o n s i d e r a b l e

co m p u ta t io n a l e f f o r t . Furthermore, c e r t a i n p o i n t s r e q u i r e s p e c i a l

t r e a t m e n t . A b r i e f d i s c u s s i o n o f th e s t a r t o f the s o l u t i o n i s g i v e n in

th e f o l l o w i n g s e c t i o n . I t would be i m p o s s i b l e t o i n c l u d e a l l o f the

30

c o m p u t a t i o n a l d e t a i l s h e r e , a l t h o u g h i t s h o u l d be p o s s i b l e f o r one who

d e s i r e s t o u s e t h i s a p p ro a ch t o o b t a i n a l l t h e n e c e s s a r y i n f o r m a t i o n

from t h e computer program and f l o w d ia g r a m .

A l th o u g h t h e b u l k - p h a s e c o n c e n t r a t i o n i s o f pr im ary i n t e r e s t ,

i t i s n e c e s s a r y a t e a c h s t a g e o f c o m p u t a t i o n t o compute a s o l u t i o n f o r

t h e a d s o r b e d - p h a s e c o n c e n t r a t i o n . The p u r p o se o f t h e a d s o r b e d - p h a s e

s o l u t i o n i s t o p r o v i d e a v a l u e o f y , i n e q u i l i b r i u m w i t h x , a t t h e

i n t e r f a c e . The e q u i l i b r i u m r e l a t i o n g i v e s a v a l u e o f x , w h ich i s

n e c e s s a r y i n t h e b u l k - p h a s e e q u a t i o n ,

2 . S o l u t i o n o f A d s o r b e d - P h a s e ( D i f f u s i o n ) E q u a t io n

The most w i d e l y - u s e d a p p r o a c h t o t h e n u m e r i c a l s o l u t i o n o f t h e

d i f f u s i o n e q u a t i o n i s t h e s e l e c t i o n o f a s u i t a b l e f i n i t e - d i f f e r e n c e

e q u a t i o n . The p r o c e d u r e s e l e c t e d f o r t h i s s t u d y i s an i m p l i c i t f i n i t e

d i f f e r e n c e m eth o d , w h ic h a s t h e name i m p l i e s , i n v o l v e s t h e s o l u t i o n o f

a s e t o f s i m u l t a n e o u s e q u a t i o n s . However , in t h e c a s e o f l i n e a r

p a r t i a l d i f f e r e n t i a l e q u a t i o n s , t h i s method i s b o th c o n v e r g e n t and

s t a b l e ( 5 , 1 1 ) , and t h e c o m p u t a t i o n a l a s p e c t s a r e r e l a t i v e l y s i m p l e . The

boundary c o n d i t i o n s f o r t h i s e q u a t i o n r e q u i r e s p e c i a l t r e a t m e n t a t th e

p ore o p e n in g and a t t h e c e n t e r o f t h e s p h e r e . The d e r i v a t i o n o f t h e

f i n i t e d i f f e r e n c e e q u a t i o n s i s shown i n A p p en d ix A.

For an i n t e r i o r p o i n t a l o n g t h e pore r a d i u s th e r e s u l t i n g d i f

f e r e n c e e q u a t i o n i s ;

m ( l - l / i ) y + m ( l + l / i ) y + y y L ----------------------------------- i ± ! -------- i ( 5 - 1 )

w here y = a d s o r b e d p h ase c o n c e n t r a t i o n a t t i m e , T + AT

31

y 1 ■ a d s o r b e d p h ase c o n c e n t r a t i o n a t t i m e , T i ■ s u b s c r i p t d e n o t i n g p o s i t i o n a l o n g pore r a d i u s

m ■ r a t i o , A T / ( A R ) ^

The c o r r e s p o n d i n g e q u a t i o n f o r t h e c e n t e r o f a s p h e r e (a t R - 0 , i * 0)

i s :

6 my, + y '

' - r V s r <5- 2 >

The b oundary p o i n t i n v o l v e s t h e c o n c e n t r a t i o n g r a d i e n t a t t h e e n t r a n c e

t o t h e p ore and i s g i v e n by:

v = 2myi _ 1 + y^ + 2 ( x - x* )m . ( i + i / n ) A R

y" ---------------------- ( T T S o ----------------------------------------------- ( 5 ‘ 3 >

“ftwhere x - x = c o n c e n t r a t i o n d i f f e r e n c e i n b u lk p h a se p r o p o r t i o n a l t o g r a d i e n t a t pore o p e n in g

n = number o f f i n i t e i n t e r v a l s i n t o w hich p ore r a d i u s i s d i v i d e d

AR = l e n g t h o f one i n t e r v a l

The maximum e r r o r bounds f o r e q u a t i o n s o f t h i s t y p e have b een

2shown t o be o f t h e o r d e r o f (AR) , or l e s s , f o r any g i v e n t im e l e v e l

and o f ( ^ T ) , or l e s s , f o r c h a n g e s i n t im e ( 3 3 ) . No e x a c t l i m i t s on th e

e r r o r can be g i v e n , a l t h o u g h i t can be assumed t h a t s i n c e th e t r u e

s o l u t i o n r e p r e s e n t s a r e a s o n a b l y w e l l - b e h a v e d p r o c e s s , the h i g h e r o r d e r

d e r i v a t i v e s a r e s m a l l , t h e r e f o r e t h e a c t u a l e r r o r i s l i k e w i s e s m a l l .

A co m p a r i s o n o f computed s o l u t i o n s o f t h e d i f f u s i o n e q u a t i o n f o r d i f

f e r e n t v a l u e s o f AR and AT i n T a b le I o f A ppendix A show t h a t the

a c t u a l e r r o r i s much l e s s th an t h e l i m i t s a l l o w e d by n u m e r i c a l a n a l y s i s .

T h is p e r m i t s u s e o f l a r g e r in c r e m e n t s i z e s w i t h o u t a f f e c t i n g t h e o v e r

a l l a c c u r a c y o f t h e r e s u l t , w i t h i n one per c e n t .

To s o l v e t h e d i f f u s i o n e q u a t i o n i t i s n e c e s s a r y t o o b t a i n a

s e t o f v a l u e s , y Q, y^ , - - - , yn _j_, yn > a l o n g t h e p ore r a d i u s . A c t u a l l y

32

t h e o n l y p o i n t o f i n t e r e s t i s th e v a l u e o f yn a t t h e p ore s u r f a c e , a s

v e i l a s t h e g r a d i e n t a t t h e s u r f a c e , ( ^ / & R ) r» a . T h i s p o i n t i s u s e d

t o o b t a i n x from an e q u i l i b r i u m r e l a t i o n , w h ich g i v e s t h e d r i v i n g

p o t e n t i a l x - x need ed i n t h e s o l u t i o n o f t h e b u l k - p h a s e e q u a t i o n .

S o l u t i o n o f t h e s e t o f n+1 s i m u l t a n e o u s e q u a t i o n s (Eq A - 16) w h i c h were

d e r i v e d f o r t h e p o re i s r e l a t i v e l y s t r a i g h t f o r w a r d , e x c e p t t h a t a v a l u e

o f x must be as su m ed . As w i l l be shown l a t e r , t h i s p r o c e d u r e i n v o l v e s

c o n s i d e r a b l e i t e r a t i o n a t e v e r y g r i d p o i n t , i n w h ic h t h e s e t o f p ore

e q u a t i o n s must be r e - s o l v e d s e v e r a l t i m e s .

3 . S o l u t i o n o f B u lk - P h a s e E q u a t io n

The f i n a l s o l u t i o n o f th e prob lem f o r a g i v e n p a r a m e t e r , A, i s

o b t a i n e d from t h e o v e r a l l b u l k - p h a s e m a t e r i a l b a l a n c e e q u a t i o n . A

n u m e r i c a l s o l u t i o n may be c o n s i d e r e d c o m p le t e when th e a d s o r p t i o n wave

r e a c h e s s t e a d y s t a t e ; h o w e v e r , in o r d e r t o compare e x p e r i m e n t a l r e s u l t s

w i t h computed r e s u l t s d o e s not r e q u i r e t h e c o m p l e t e s o l u t i o n i f t h e

a c t u a l e x p e r i m e n t d o e s n o t r e a c h s t e a d y s t a t e . In t h e work o f Joh n son

(30 ) the d a t a w ere o b t a i n e d b e f o r e s t e a d y s t a t e was r e a c h e d . Other

e x p e r i m e n t e r s c i t e d a l l o w e d t h e s y s t e m t o r e a c h s t e a d y s t a t e ( 4 3 , 1 , 2 ,

2 2 , 1 6 , 1 7 , 1 8 ) .

The p r o c e d u r e s f o r s o l v i n g th e b u l k - p h a s e e q u a t i o n w i l l be

e x p l a i n e d b r i e f l y . P o i n t s a l o n g the d i a g o n a l , AH = T, and a l o n g co lumns

one and two o f t h e t r i a n g u l a r g r i d r e q u i r e a m o d i f i c a t i o n o f th e g e n e r a l

methods u s e d f o r an i n t e r i o r g r i d p o i n t .

In column o n e , w h ic h c o r r e s p o n d s t o f e e d e n t e r i n g t h e a d s o r p

t i o n b e d , t h e v a l u e s o f b u lk c o n c e n t r a t i o n a r e known: xj = x^ .

33

However i t i s n e c e s s a r y t o have t h e c o n c e n t r a t i o n , y n , a t t h e s u r f a c e

o f t h e p a r t i c l e i n o r d e r t o o b t a i n t h e s o l u t i o n a t th e n e x t bed l e v e l ,

i . e . , a t i « 2 . The p r o c e d u r e f o r o b t a i n i n g yn a t the f e e d i n l e t f o r

a l l l a t e r t ime v a l u e s can be suntnarized a s f o l l o w s :

(a ) Assume i n i t i a l e s t i m a t e o f s u r f a c e c o n c e n t r a t i o n , y ^ ° ^

(b) C a l c u l a t e x* from e q u i l i b r i u m f u n c t i o n : x* » f ( y n ° ^ )

( c ) C a l c u l a t e x^ - x

(d) S o lv e d i f f u s i o n e q u a t i o n u s i n g th e above e s t i m a t e o f

X f - x . T h is g i v e s a new e s t i m a t e , y ^ ^

( e ) Compare the assumed y n w i t h t h a t c a l c u l a t e d in s o l u t i o n o f d i f f u s i o n e q u a t i o n . I f th e two f i g u r e s a r e s u f f i c i e n t l y c l o s e , th e s o l u t i o n a t t h i s one p o i n t i s c o m p l e t e .

( f ) I f th e two f i g u r e s do not b a l a n c e , a d j u s t th e e s t i m a t e o f y n by a c o n v e r g e n c e f a c t o r , q:

yf l2 = yrS°'> + < y it°^ " ( 5 - 4 )

Repeat the computat ions from s te p ( b ) , u s in g y ^ ^ y ^ ° ^ ■

At the c o m p l e t i o n o f t h e above a p p r o x im a t io n f o r yn (a t p o i n t

j , l ) , th e e n t i r e p ro ced u re must be r e p e a t e d fo r y n ( a t p o i n t j + 1 , 1 )

w hich c o r r e s p o n d s t o th e n e x t t im e l e v e l . In o r d e r t o c a l c u l a t e yn a t

any l a t e r t im e s t e p i t i s n e c e s s a r y t o s a v e th e e n t i r e d i f f u s i o n cu rv e

(as a s e q u e n c e o f p o i n t s ) from t h e p r e c e d i n g t im e l e v e l , a s e a c h o f

t h e s e v a l u e s i s n e c e s s a r y i n s o l v i n g th e d i f f u s i o n e q u a t i o n . As w i l l

be s e e n l a t e r , i t i s a l s o n e c e s s a r y t o s ave the q u a n t i t i e s x and x fo r

u s e a t h i g h e r bed l e v e l s . The e x t e n t t o w h ich th e s o l u t i o n s t e p s a r e

c a r r i e d ou t a l o n g t h e T - a x i s depends upon th e n a t u r e o f t h e s y s t e m b e in g

s t u d i e d . The computer program was s o d e s i g n e d t o d i s c o n t i n u e the above

p r o c e d u r e a f t e r yn approached i t s e q u i l i b r i u m v a l u e w i t h x^ a t any

g i v e n d i m e n s i o n l e s s bed l e v e l , H, w i t h i n a p r e - d e t e r m i n e d t o l e r a n c e .

34

At t h i s p o i n t i t can be assumed t h a t t h e a d s o r b e d ph ase c o n c e n t r a t i o n

w i l l r em ain c o n s t a n t a t i t s e q u i l i b r i u m v a l u e .

I t i s n e x t p o s s i b l e t o e s t i m a t e t h e b u lk p h a se c o n c e n t r a t i o n

a l o n g co lumn two o f t h e t r i a n g u l a r g r i d . T h i s c o r r e s p o n d s t o t h e

s e c o n d d i m e n s i o n l e s s bed l e v e l o v e r a ran ge o f t im e s t a r t i n g w i t h

T = A T . The p r o c e d u r e f o r e v a l u a t i n g X2 2 i s f ll s o i t e r a t i v e , but

s l i g h t l y d i f f e r e n t from th e p r e c e d i n g d i s c u s s i o n f o r t h e f i r s t bed

l e v e l . F i r s t an e s t i m a t e o f X2 2 * s made by f i t t i n g a f i r s t - o r d e r

e q u a t i o n o v e r t h e i n t e r v a l from H = 0 t o H = AH, w i t h e r r o r bound o f

m a g n i tu d e O^H) • T h i s e s t i m a t e i s o b t a i n e d from t h e b u l k - p h a s e

m a t e r i a l b a l a n c e ;

‘ " 3 ( l5 r) ( 4 _ 2 4 )T,R \ / H,T,R=A

The r i g h t hand term i s e q u a l t o 3 (x - x ) . The f i r s t - o r d e r a p p r o x im a

t i o n i s t h u s ;

= X2 > 1 + 3 ( x - x* ) 2 lAH ( 5 - 5 )

U s i n g x2 °^ a b e t t e r e s t i m a t e , x 2 ^2» can be o b t a i n e d by means o f a

s e c o n d - o r d e r f o r m u la , w i t h e r r o r o f o r d e r (AH) . T h i s form ula can be

a p p l i e d r e p e a t e d l y u n t i l x2 2 d e t e r m i n e d w i t h i n a s p e c i f i e d

t o l e r a n c e . The s e c o n d - o r d e r e q u a t i o n r e q u i r e s a v a l u e o f <3x a t b o thdH

g r i d p o i n t s ( 2 , 1 ) and ( 2 , 2 ) . U s i n g t h e e s t i m a t e d x2 2 f rom t h e f i r s t

o r d e r e q u a t i o n , an e s t i m a t e o f x 2 , 2 i® iaade , s i n c e i t i s known t h a t

x 2 2 * ^ 2 2 from t h e m o d e l , and t h a t x* 2 2 = ^ ( ^ 2 2 ^» ^rom e q u i l i b r i u m

d a t a . Now i t i s p o s s i b l e t o g e t a new e s t i m a t e o f x2 2 :

4 2} - - ! [ ( - f i ) + ( H i ) 2 J35

AH( 5 - 6 )

or

* x 2 , 1 + 1 * 5 [ (x* " X>2 , 1 + (x* " X ^ 2 ^2 J AH 5 ' 7 ^

The new e s t i m a t e o f X£ 2 may be u8e< t o r e f i n e t h e t e r m , (x - x ) 2 2 >

and r e p e a t c a l c u l a t i o n s u n t i l no f u r t h e r improvement i n X2 2 o c c u r s .

D uring t h i s s t a g e i t i s n o t n e c e s s a r y t o s o l v e th e d i f f u s i o n e q u a t i o n

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

p o i n t . In o r d e r t o compute a d i s t r i b u t i o n a t t h e n e x t s t e p ( t im e l e v e l

3 ) , t h e i n i t i a l d i s t r i b u t i o n w i t h i n t h e pore must be d e f i n e d ; i . e . ,

yk = x2 ,2 (k = 0 , 1 , - - - , n ) .

The r e m a i n i n g g r i d p o i n t s o f column two o f t h e t r i a n g u l a r g r i d

may be s o l v e d by a p r o c e d u r e , w h ic h t u r n s ou t t o be d o u b l e t r i a l - a n d -

e r r o r . T h i s happens b e c a u s e two unknowns, b o t h x; 9 and y = 9 must beJ > ^ J ? **

d e t e r m i n e d . B a s i c a l l y b o th Xj 2 an ^ Yj 2 a r e e s t i n i a t e d . For the

e s t i m a t e d yj 2 » t h e c o r r e s p o n d i n g Xj 2 i s computed by e s s e n t i a l l y th e

same means a s f o r p o i n t ( 2 , 2 ) . T h i s v a l u e o f x f 9 i s th e n u sed i nJ

s o l v i n g t h e d i f f u s i o n e q u a t i o n t o o b t a i n a b e t t e r e s t i m a t e o f y^ 9 a sJ J ^

was done f o r column o ne ,w hich in tu rn r e f i n e s x . 0 . A f t e r both x . 0J >2 j , 2

and yj 2 compare f a v o r a b l y w i t h p r e v i o u s e s t i m a t e s , t h e p r o c e d u r e i s

r e p e a t e d f o r p o i n t (j + 1 , 2 ) , and c a r r i e d out f o r s u c c e s s i v e j ' s u n t i l

y i s s u f f i c i e n t l y c l o s e t o y , i n e q u i l i b r i u m w i t h , At e a c h h i g h e r

t im e l e v e l , t h e p r e c e d i n g a d s o r b e d - p h a s e d i s t r i b u t i o n must be s a v e d f o r

ka t l e a s t one a d d i t i o n a l t im e l e v e l . L i k e w i s e t h e term x - x f o r e a c h

p o i n t a l o n g column one must have b een p r e v i o u s l y s a v e d . As c a l c u l a t i o n s

p r o g r e s s c o l u m n - w i s e , o l d e r term s may be r e p l a c e d by newer o n es a s

n e e d e d . In terms o f computer s t o r a g e , i t i s n e c e s s a r y t o c o n s e r v e

s p a c e , s o t h a t o n l y t h o s e q u a n t i t i e s w h ich must be k ep t f o r l a t e r u s e

a r e s a v e d . For i n s t a n c e when th e s o l u t i o n f o r column t h r e e h as b een

c o m p l e t e d , i t i s no l o n g e r n e c e s s a r y t o s ave t h e c o n t e n t s o f column

o n e . As w i l l be shown i t w i l l be n e c e s s a r y t o r e t a i n o n l y co lumns two

and t h r e e t o f i n d t h e s o l u t i o n t o column f o u r .

o n a l e l e m e n t s , f o l l o w s a g e n e r a l p r o c e d u r e , d e s c r i b e d b r i e f l y b e lo w .

The d i a g o n a l s a r e o b t a i n e d by a method s i m i l a r t o t h a t f o r p o i n t

(2 , 2 ) , e x c e p t t h a t from now on a t h i r d - o r d e r method i s u s e d , w i t h e r r o r

p u t e r s t o r a g e . The t h i r d o r d e r i n t e g r a t i o n form u la used was a s f o l l o w s

(where 3 < i ) :

employed i n tu r n on t h e r i g h t hand s i d e u n t i l t h e d e s i r e d b a l a n c e i s

r e a c h e d . As i n t h e c a s e o f g r i d p o i n t ( 2 , 2 ) t h e p o r e - p h a s e d i s t r i b u

t i o n i s c o n s t a n t a t t h e d i a g o n a l : £ - y k (.k = 0 , 1 , 2 , - - - , n ) .

The c a l c u l a t i o n o f r e m a i n i n g g r i d p o i n t s , i n c l u d i n g t h e d i a g -

o f o r d e r (MI)^. U s in g a t h i r d o r d e r i n t e g r a t i o n form ula r e q u i r e s t h a t

*||fx and x from two p r e c e d i n g co lumns must have b een r e t a i n e d i n com-

( 5 - 8 )

w h ich b e c o m e s , a f t e r s u b s t i t u t i o n o f e q u a t i o n ( 1 3 ) :

Co j +I n the ab ove e q u a t i o n i t i s n e c e s s a r y t o g u e s s x.' { , g e t (x - x ) - j■L j 1 j 1

from e q u i l i b r i u m r e l a t i o n s , th en compute x ^ ? . T h i s v a l u e may be

The r e m a i n i n g p o i n t s a l o n g column t h r e e and a l l h ig h e r -n u m b e r e d

37

columns a r e s o l v e d by d o u b l e t r i a l - a n d - e r r o r i n a manner s i m i l a r t o

t h a t d e s c r i b e d f o r column two. The o n l y d i f f e r e n c e i s t h a t a t h i r d -

o rd er i n t e g r a t i o n formula i s a p p l i e d , r e q u i r i n g th e t e r m s , (x - x ) ,

from two p r e c e d i n g c o l u m n s . The g e n e r a l p roced u re i s th e same: t o

, (o ) (o )g u e s s both x j a n d y j , i , compute a r e f i n e d v a l u e o f x j , i , th en s o l v e

th e d i f f u s i o n e q u a t i o n f o r a new yj These two new v a l u e s o f x and

y may be used a s i n i t i a l e s t i m a t e s fo r f u r t h e r i t e r a t i o n u n t i l a

b a la n c e i s a c h i e v e d .

The o v e r a l l s o l u t i o n , u s i n g t h e a b o v e - d e s c r i b e d methods i s

o b t a i n e d c o lu m n w is e , t h a t i s , th e e n t i r e h i s t o r y o f th e bed a t a g i v e n

d i m e n s i o n l e s s bed l e v e l i s computed. The bed l e v e l i s in c r e m e n te d by

and th e same p r o c e s s r e p e a t e d . The n e t r e s u l t i s a f a m i l y o f

c u r v e s o f x v e r s u s T f o r v a r i o u s H. A t y p i c a l s o l u t i o n i s shown in

F ig u r e 4 , i n Appendix B. The c o m p le t e s o l u t i o n t o the problem i s

t h r e e - d i m e n s i o n a l and would be a f a m i l y o f f i g u r e s , e a c h one f o r a

v a l u e o f th e param eter A. The r e s u l t s o f t h e s e c a l c u l a t i o n s w i l l be

d i s c u s s e d i n th e n e x t c h a p t e r . B e f o r e p r o c e e d i n g w i t h th e r e s u l t s , how

e v e r , a few s a l i e n t f e a t u r e s o f th e n u m e r ic a l p roced u re s h ou ld be

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

e s t , p a r t i c u l a r l y th e c o m p u t a t i o n a l e f f o r t r e q u i r e d .

4 . D i s c u s s i o n o f C om pu ta t ion a l Problems

The amount o f computer t im e r e q u i r e d t o s o l v e th e problem i s

o f prim ary c o n c e r n , and i t i s a lm o s t e x a c t l y p r o p o r t i o n a l t o th e number

o f i t e r a t i o n s ; th u s e v e r y e f f o r t n e e d s t o be made t o keep t h e t o t a l

number o f i t e r a t i o n s a t a minimum, y e t produce the r e q u i r e d a c c u r a c y i n

t h e s o l u t i o n . In a d d i t i o n t h e number o f i n t e r v a l s c h o s e n f o r th e

d i f f u s i o n e q u a t i o n a f f e c t s o v e r a l l c om p u t in g t i m e . At f i r s t i t m ig h t

a p p e a r t h a t an i n c r e a s e i n s t e p - s i z e i n e i t h e r th e H - d i r e c t i o n or t h e

T - d i r e c t i o n would r e d u c e t h e t o t a l e f f o r t i n v o l v e d . However i t has

b e e n o b s e r v e d t h a t i f s t e p - s i z e i s i n c r e a s e d , t h e number o f i t e r a t i o n s

i n t h e t r i a 1 - a n d - e r r o r p r o c e d u r e may a l s o i n c r e a s e , s o t h a t v e r y l i t t l e

n e t s a v i n g s i n t im e i s r e a l i z e d . One o f t h e most c r i t i c a l v a l u e s

a f f e c t i n g t h e r a t e o f c o n v e r g e n c e , i s t h e damping f a c t o r u s e d t o a d j u s t

t h e e s t i m a t e d v a l u e o f y f o r i t e r a t i o n p u r p o s e s . I t has b een found

t h a t f o r some v a l u e s o f t h i s f a c t o r , an e x c e s s i v e number o f i t e r a t i o n s

i s r e q u i r e d , and e v e n i n some c a s e s d i v e r g e n c e was o b s e r v e d . I t i s

m o s t l y a m a t t e r o f t r i a l - a n d - e r r o r t o f i n d an optimum f a c t o r , w h ic h i f

t o o s m a l l , c o u l d i n c r e a s e t h e number o f i t e r a t i o n s , and i f t o o l a r g e

c a u s e d i v e r g e n c e . G e n e r a l l y a f a c t o r b e tw e e n 0 . 2 and 1 . 0 was s u f f i c i e n t

t o p ro d u ce r e l a t i v e l y r a p i d c o n v e r g e n c e . However i n t h e r e g i o n o f most

r a p i d mass t r a n s f e r ( a l o n g t h e s t r a i g h t e s t and s t e e p e s t p o r t i o n s o f t h e

a d s o r p t i o n w a v e ) b e t w e e n t e n and t h i r t y i t e r a t i o n s was u s u a l l y r e q u i r e d .

In a d d i t i o n t o t h e damping f a c t o r , th e number o f i t e r a t i o n s i s a f f e c t e d

t o some e x t e n t by t h e c l o s e n e s s o f q s t i m a t e r e q u i r e d f o r b o t h x and y

a t e a c h g r i d p o i n t . I t was c o n c l u d e d t h a t t h e s e t o l e r a n c e s s h o u l d a l l

be s u c h t h a t t h e o v e r a l l s o l u t i o n s h o u l d be s i g n i f i c a n t t o w i t h i n one

p e r c e n t .

Even w i t h i t e r a t i o n t o l e r a n c e s s m a l l en ou gh t o b a l a n c e t h e

e s t i m a t e s o f x and y t o many s i g n i f i c a n t d i g i t s , th e t r u n c a t i o n e r r o r

o f t h e n u m e r i c a l i n t e g r a t i o n fo r m u la s and f i n i t e d i f f e r e n c e e q u a t i o n s

must be c o n s i d e r e d . A l t h o u g h th e maximum bounds f o r l i n e a r s y s t e m s (1 1 )

39

a r e s t a t e d , t h e y a r e assumed t o b e v a l i d f o r t h i s p a r t i c u l a r a d s o r p

t i o n m o d e l . The p r o c e s s i s o b s e r v e d t o p ro d u ce r e l a t i v e l y smooth

v a r i a t i o n s i n e a c h o f t h e p a ra m eters , and h e n c e , f o l l o w s t o some e x t e n t

t h e bounds g i v e n f o r a t r u l y l i n e a r p r o c e s s . I t was found t h a t th e

a c t u a l t r u n c a t i o n e r r o r was much l e s s th an t h e maximum; t h i s i s p r o b

a b l y due t o a r e l a t i v e l y s m a l l h i g h e r d e r i v a t i v e c o r r e s p o n d i n g t o t h e

o r d e r o f t h e e r r o r . Only n u m e r i c a l e x p e r i m e n t a t i o n can s u p p o r t th e

v a l i d i t y o f t h e r e s u l t s , and t h i s was done f o r a f a i r l y w id e ran ge o f

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

That t h e method i s s t a b l e and c o n v e r g e n t can be presumed, s i n c e

t h e r e s u l t s behaved a s e x p e c t e d , a s w e l l a s th e f a c t t h a t t h e s o l u t i o n

c o n v e r g e d a s th e i n c r e m e n t s AH, AR, and AT w ere d e c r e a s e d . The o n l y

i t e m found t o be c r i t i c a l i n p r o d u c in g a s t a b l e r e s u l t i s t h e damping

f a c t o r d e s c r i b e d p r e v i o u s l y . I t i s c o n c e i v a b l e t h a t some v e r y w id e

v a r i a t i o n i n p a r a m e t e r s or u n u s u a l e q u i l i b r i u m c o u ld make t h e p r o c e s s

u n s t a b l e , a l t h o u g h su ch was n o t t h e c a s e f o r any o f t h e runs made in

t h i s s t u d y , p r o v i d e d a s u i t a b l e damping f a c t o r was u s e d .

The computer t im e r e q u i r e d f o r the s o l u t i o n o f t h e model f o r

a g i v e n A was n o t e d t o be r o u g h l y p r o p o r t i o n a l t o th e m a g n i tu d e o f A.

The s h o r t e s t t i m e s were a p p r o x i m a t e l y one hour f o r v a l u e s o f A =* 1 or

l e s s , and i n c r e a s e d t o a maximum o f 6 - 8 h o u r s f o r t h e l a r g e s t v a l u e s

o f A s t u d i e d .

5 . Computer Program

The a p p l i c a t i o n o f a computer i n an y s i t u a t i o n r e q u i r e s t h a t ,

on ce a p rob lem has b een d e f i n e d m a t h e m a t i c a l l y , t h e n u m e r i c a l p r o c e d u r e

be s t a t e d i n term s o f m a c h in e o p e r a t i o n s . The b a s i c com puter o p e r a

t i o n s , known a s m ach in e la n g u a g e i n s t r u c t i o n s , may be o r g a n i z e d i n t o a

l o g i c a l a r r a n g e m e n t , o r program, t o s o l v e t h e n u m e r i c a l p r o b le m . I n

s t e a d o f c o d i n g t h e m a ch in e l a n g u a g e program, w h i c h i s a v e r y a r d u o u s

t a s k , an a l g e b r a i c a l l y - o r i e n t e d l a n g u a g e , c a l l e d FORTRAN, was u s e d ( 2 9 )

The computer p r o c e d u r e s may be s t a t e d i n FORTRAN and t r a n s l a t e d i n t o

t h e b a s i c computer i n s t r u c t i o n s . The m a jo r a d v a n t a g e o f t h i s m eth od ,

i s t h a t once t h e prob lem i s w r i t t e n i n FORTRAN, a l m o s t any com puter

may be u s e d w i t h o u t t h e n e c e s s i t y o f r e -p r o g r a m m in g . Fu rtherm ore i t i s

much s i m p l e r f o r one t o s c a n a FORTRAN-coded program i n making any

d e s i r e d c h a n g e s or c o r r e c t i o n s . There a r e many more p e r s o n s w i t h a

r e a s o n a b l e k now ledge o f FORTRAN th an o f b a s i c m a ch in e l a n g u a g e , p a r t i c u

l a r l y i n t h e s c i e n c e s . Even th o u g h i t i s r e l a t i v e l y s i m p l e t o c o d e a

n u m e r i c a l p r o c e d u r e i n FORTRAN, t h e c o m p l e x i t y o f t h e program u s e d i n

t h i s r e s e a r c h i s c o n s i d e r e d g r e a t enough t o be i n c l u d e d f o r r e f e r e n c e

p u r p o s e s , i n Appendix C. One o f t h e m ain f e a t u r e s o f t h i s program i s

t h e p r o c e d u r e by w hich t h e i n s t r u c t i o n s f o r com p u t in g e q u i l i b r i u m r e l a

t i o n s a r e h a n d l e d . The e q u i l i b r i u m c a l c u l a t i o n s a r e s e t a p a r t from t h e

m ain n u m e r i c a l p r o c e d u r e i n a " s u b r o u t i n e . " Any t im e a chan ge i n th e

e q u i l i b r i u m f u n c t i o n i s d e s i r e d ( s u c h a s c h a n g in g a d s o r b e n t m a t e r i a l )

i t i s a s i m p l e m a t t e r t o r e p l a c e o n l y t h o s e FORTRAN i n s t r u c t i o n s p e r

t a i n i n g t o t h e e q u i l i b r i u m f u n c t i o n , w i t h o u t d i s t u r b i n g t h e main

p o r t i o n o f t h e program. Many r e f e r e n c e b ook s and m an u a ls g i v e b a s i c

i n s t r u c t i o n i n e f f i c i e n t o r g a n i z a t i o n o f a computer program ( 2 8 , 3 9 , 4 1 ) .

The com p u ter u s e d f o r s o l v i n g t h e n u m e r i c a l p r o c e d u r e d e

s c r i b e d i n p r e c e d i n g s e c t i o n s was an IBM 7040 d a t a p r o c e s s i n g s y s t e m

41

a t t h e Computer R e s e a r c h C e n t e r o f L o u i s i a n a S t a t e U n i v e r s i t y . I t was

w i t h t h e a n t i c i p a t e d u s e o f a l a r g e - s c a l e computer t h a t t h i s r e s e a r c h

p r o j e c t was u n d e r t a k e n . A b r i e f d e s c r i p t i o n o f t h e m achine c o n f i g u r a

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

A p p en d ix C.

VI. DISCUSSION OF RESULTS

1 . P r e s e n t a t i o n o f C a l c u l a t i o n s

Computed s o l u t i o n s fo r the proposed model for four b i n a r y ,

l i q u i d - p h a s e .sys tems have been o b t a i n e d . The r e s u l t s a r e b a s i c a l l y

a s e t o f t a b l e s fo r each run made, a sample o f w hich i s found i n

Appendix C. However th e t a b u l a r form i s much t o o b u lk y t o p r e s e n t

h e r e , b e in g o f th e order o f a thousand p a g e s ; i n s t e a d the s o l u t i o n s

a r e g i v e n as a s e t o f c u r v e s in Appendix B, F i g u r e s 4 through 32 .

Tab le I I summarizes the c h a r a c t e r i s t i c s o f each sys te m s t u d i e d . The

main d i s a d v a n t a g e o f a n u m e r ic a l s o l u t i o n i s t h a t no c o n c i s e a l g e b r a i c

formula r e l a t i n g a l l v a r i a b l e s can be o b t a i n e d . T h is means t h a t i t

i s n e c e s s a r y t o i n t e r p o l a t e or r e - r u n th e problem each t ime a change

in one o f th e v a r i a b l e s i s made. The problem o f h a n d l in g the l a r g e

volume o f o u tp u t from a computer has been met in p art by th e use o f

modern, h i g h l y compact s t o r a g e d e v i c e s such a s m a g n e t ic t a p e s or d i s k s .

The o b j e c t o f t h i s work i s t o s e e k a s a t i s f a c t o r y comparison

b etw een t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s w i t h o u t n e c e s s a r i l y c o v e r

i n g a wide range o f p a r a m e te r s . The c u r v e s p r e s e n t e d i n Appendix B

r e f l e c t a number o f t r i a l runs made t o s e a r c h f o r th e optimum s o l u t i o n .

As w i l l be d i s c u s s e d l a t e r th e optimum i s reached when an e x p e r i m e n t a l

s e t o f d a ta f i t s c l o s e l y th e computed r e s u l t s . In F i g u r e s 4 through

3 2 , th e computer s o l u t i o n s are s o l i d l i n e s , w h i l e r e a l data, c o n v e r t e d

t o d i m e n s i o n l e s s form, f o r the c o r r e s p o n d i n g s y s t e m a r e p l o t t e d a s

42

p o i n t s f o r c o m p a ra t iv e p u r p o s e s . In o rd er t o c l a r i f y t h e co m p a r i s o n s

some d i s c u s s i o n o f th e p h y s i c a l meaning o f the d i m e n s i o n l e s s p a r a

m e t e r s A, H, and T i s need ed .

2 . P h y s i c a l S i g n i f i c a n c e o f th e D i m e n s i o n l e s s V a r i a b l e s A, H, and T

The u s e o f th e d i m e n s i o n l e s s v a r i a b l e s , A, H, and T, sh ou ld

be made w i t h some u n d e r s t a n d i n g o f t h e i r r e l a t i o n t o th e t r u e v a r i a b l e s ,

p a r t i c l e r a d i u s , bed h e i g h t and t im e e l a p s e d from s t a r t u p , a s s t a t e d

i n E q u a t io n s ( 4 - 7 ) , ( 4 - 8 ) , and ( 4 - 9 ) .

The d i m e n s i o n l e s s parameter A i s d i r e c t l y p r o p o r t i o n a l to the

r e a l p a r t i c l e r a d i u s , a . I f a p a r t i c u l a r sys te m i s c o n s i d e r e d , th en A

i s f i x e d f o r a l l o p e r a t i n g c o n d i t i o n s , a ssu m ing c o n s t a n c y o f K and D.

The d i m e n s i o n l e s s param eter T i s r e l a t e d t o a c t u a l t i m e , a s

£ zf o l l o w s . The term , — , i s s e e n t o be th e v o id volume ( t h i s d o es notQ/S

i n c l u d e pore vo lum e) d i v i d e d by a f lo w r a t e , which i s t h e same a s the

t im e r e q u i r e d t o f i l l th e v o id volume o f bed t o d ep th z . T h i s term ,

s u b t r a c t e d from t o t a l e l a p s e d t i m e , 0 , g i v e s th e t ime w hich e x c e e d s

th a t r e q u ir e d t o f i l l th e v o id volume t o d ep th z . T i s p r o p o r t i o n a l

t o t h i s e x c e s s t i m e , th e p r o p o r t i o n a l i t y f a c t o r b e in g K /D.

The d i m e n s i o n l e s s bed d e p t h , H, i s d i r e c t l y p r o p o r t i o n a l t o

bed l e v e l , z . The f a m i l y o f c u r v e s a t c o n s t a n t H thus p r o v id e a h i s t o r y

o f c o n c e n t r a t i o n a t any bed d e p t h , when th e p r o p o r t i o n a l i t y f a c t o r ,KV p b— 1—-— , i s known. The H-curve c o r r e s p o n d i n g t o t o t a l bed d ep th sh ou ld a ( Q / S ) r *

th u s c o r r e l a t e w i t h th e e x p e r i m e n t a l measurement o f c o n c e n t r a t i o n in

th e e f f l u e n t over a p e r i o d o f t i m e . A key p o i n t to remember i s , t h a t

i f th e p roposed model h o l d s , t h e f i r s t drop o f e f f l u e n t i s a l s o a t the

l i q u i d f r o n t . Th is means t h a t the i n i t i a l c o n c e n t r a t i o n measurement

4 4

s h o u ld f a l l on t h e boundary c u r v e . Hence th e minimum p o s s i b l e v a l u e s

o f T and H a r e d e f i n e d .

3 . B e h a v io r o f Computed A d s o r p t i o n Curve

The computed s o l u t i o n s w hich a r e shown in F i g u r e s 4 th rough

32 a l l have a s i m i l a r a p p e a r a n c e , a s would be e x p e c t e d from t h e o r e t i c a l

c o n s i d e r a t i o n s . The r e a s o n s u n d e r l y i n g th e p a r t i c u l a r shape o f t h e s e

c u r v e s m e r i t some q u a l i t a t i v e d i s c u s s i o n . Comparison o f a c t u a l w i t h

t h e o r e t i c a l c u r v e s w i l l be t r e a t e d i n a l a t e r s e c t i o n . A s k e t c h , a s

shown i n F ig ur e 1, o f t h e c o n c e n t r a t i o n r e l a t i o n s h i p s w i l l a s s i s t in

e x p l a i n i n g t h e dynamic p r o c e s s e s i n v o l v e d from i n i t i a l s t a r t - u p t o

s t e a d y - s t a t e o p e r a t i o n .

x=y

r=a

F ig u r e 1. E q u i l i b r i u m and O p e r a t i n g L i n e s f o r a T y p i c a l C ase .

The computed c u r v e s , a s w e l l a s t h e e x p e r i m e n t a l r e s u l t s , a re

a l l n o te d t o f o l l o w a g e n e r a l p a t t e r n i n w hich th e f i r s t few c u r v e s a re

45

s t r o n g l y c o n c a v e toward t h e T - a x i s , f o l l o w e d by a s e t o f c u r v e s showing

a s l i g h t c o n v e x i t y a t low c o n c e n t r a t i o n s , and f i n a l l y , a t s t e a d y - s t a t e

a p ronounced ” S" sh ape i n w h ich t h e o n l y ch an ge i s p h y s i c a l d i s p l a c e

ment o f t h e u l t i m a t e c u r v e , or w a v e , a l o n g t h e T - a x i s . I f th e s k e t c h

i s o b s e r v e d , s e v e r a l i m p l i c a t i o n s may be made. I t i s o b v i o u s t h a t th e

s t e e p n e s s o f t h e c u r v e s a t any g i v e n p o i n t i s d i r e c t l y r e l a t e d t o the

r a t e o f mass t r a n s f e r t h r o u g h th e b u l k , or u n a d s o r b e d , p h a s e . T h is

r a t e i s p r o p o r t i o n a l t o t h e t e r m s , x - x * , w hich can be shown on th e

s k e t c h a b ove a s a s e r i e s o f h o r i z o n t a l l i n e s . The l i n e , x = y , i s

shown, w h ich g i v e s t h e s t a t u s o f th e wave f r o n t a t any i n s t a n t .

O p e r a t i n g l i n e s , a s f o r e x a m p l e , l i n e s , KA, K.C, KE, KI and KL, may be

drawn f o r an y d e p th p o s i t i o n w i t h i n an a d s o r b e n t b ed . I t has been

shown f o r a p r o c e s s o f t h i s t y p e t h a t t h e o p e r a t i n g l i n e s a r e s t r a i g h t

( 3 0 ) .

At i n i t i a l s t a r t - u p ( t h a t i s , when f e e d f i r s t c o n t a c t s t h e

b o t to m o f t h e a d s o r b e n t b ed ) th e maximum d r i v i n g p o t e n t i a l i s c r e a t e d

a t th e bed i n l e t , or H “ 0 , a s shown by th e l i n e , AB, b e tw e e n Xf and

x T h i s r e l a t i o n s h i p would h o ld f o r a p o i n t p l o t t e d on t h e upper

l e f t h a n d c o r n e r o f t h e computed c u r v e s , a t x = X£ and T = 0 .

As t im e p r o g r e s s e s t h e b u lk p h ase c o n c e n t r a t i o n a t H = 0

r e m a in s a t X £ , but t h e a d s o r b e d ph ase c o n c e n t r a t i o n , y , r i s e s and

a p p r o a c h e s y * , t h e e q u i l i b r i u m v a l u e w i t h r e s p e c t t o X£. The h o r i

z o n t a l d i s t a n c e b e tw e e n t h e v e r t i c a l o p e r a t i n g l i n e f o r H = 0 and the

e q u i l i b r i u m c u r v e ( l i n e GH) s l o w l y d e c r e a s e s u n t i l th e p o r e s are com-

p l e t e l y s a t u r a t e d w i t h t h e e q u i l i b r i u m c o n c e n t r a t i o n , y . F u r th e r

d i f f u s i o n w i l l not o c c u r w i t h i n t h e a d s o r b e n t m a t e r i a l a t t h e bed

i n l e t , and the l i q u i d c o n c e n t r a t i o n w i l l remain a t x f .

As a d s o r p t i o n p roceeds the w a v e - f r o n t c o n c e n t r a t i o n d i m i n i s h e s

w it h T, as shown by the en v e lo p e curve convex toward the o r i g i n a t a l l

t i m e s . The space to the l e f t o f t h i s curve r e p r e s e n t s th a t p o r t i o n o f

th e bed not y e t reached by the l i q u i d fron t a s i t t r a v e l s up the

column. This curve i s u s e f u l in comparing r e s u l t s a t d i f f e r e n t p a r

t i c l e s i z e s , a s w i l l be d i s c u s s e d l a t e r . The e n v e lo p e curve a l s o may

be used a s an i n d i c a t o r o f the approach to s t e a d y - s t a t e , a s i t has

been observed t h a t by the time t h i s curve r ea c h e s a low v a lu e o f x , in

e v e r y c a se the a d s o r p t i o n wave i s f u l l y d eve lop ed i n t o i t s u l t i m a t e

form .

At s l i g h t l y h ig h er bed l e v e l s , r e p r e s e n t e d by cu r v es a t co n

s t a n t H, the d r i v i n g fo r c e a t the wave f r o n t , g i v e n by l i n e CD, i s

l a r g e , which r e s u l t s i n a very s t e e p a d s o r p t i o n c u r v e . The approach

to e q u i l i b r i u m , as t r a n s f e r o f m a t e r i a l from bulk to adsorbed phase

t a k e s p l a c e , i s shown as the curve bends over t o the r i g h t and a p

proaches the l i n e , x = x f , a s y m p t o t i c a l l y . The d r i v i n g p o t e n t i a l , as

r e p r e s e n t e d by h o r i z o n t a l l i n e s between the o p e r a t i n g l i n e s and e q u i

l ib r iu m curve g r a d u a l l y d i m i n i s h e s and become z e r o when the bulk co n

c e n t r a t i o n has reached x f . The ad sorb en t bed i s s a t u r a t e d a t t h i s

l e v e l and f r a c t i o n a t i o n has l i k e w i s e s to p p ed .

The b e h a v io r d e s c r i b e d above c o n t i n u e s fo r some d i s t a n c e i n t o

the bed , but the cu r v es become l e s s and l e s s s t e e p . This i s e x p l a i n e d

by th e s h o r t e r h o r i z o n t a l d i s t a n c e s between the o p e r a t i n g l i n e fo r a

p a r t i c u l a r bed l e v e l and the e q u i l i b r i u m l i n e . At some p o s i t i o n in

the bed an i n t e r e s t i n g change i s n o t e d . The a d s o r p t i o n waves b e g in to

show a s l i g h t c o n v e x i t y n e a r t h e boundary c u r v e . Note t h a t t h e s l o p e

o f t h e e q u i l i b r i u m c u r v e a t p o i n t J i s l a r g e r th an t h e o p e r a t i n g l i n e

c o r r e s p o n d i n g t o t h i s bed l e v e l . I n i t i a l l y a much s m a l l e r d r i v i n g

p o t e n t i a l (shown by l i n e I J ) i s n o t e d . However , a s a d s o r p t i o n o c c u r s ,

c a u s i n g h i g h e r b u lk c o n c e n t r a t i o n s , th e l e n g t h o f t h e h o r i z o n t a l l i n e s

b e tw e e n t h e o p e r a t i n g l i n e and e q u i l i b r i u m i s s e e n t o i n c r e a s e , th u s

more r a p id t r a n s f e r w i l l o c c u r . T h i s r e s u l t s i n a r e l a t i v e l y s m a l l

s l o p e i n i t i a l l y , t h e n a s t e e p e r s l o p e d u r i n g th e p e r i o d o f h ig h t r a n s

f e r r a t e , f o l l o w e d by a c u r v a t u r e toward t h e r i g h t a s th e p r o c e s s

n e a r s c o m p l e t i o n . E v e n t u a l l y t h e o p e r a t i n g l i n e KL, a t i n f i n i t e bed

d e p t h , w i l l be r e a c h e d ; t h i s c o r r e s p o n d s t o s t e a d y - s t a t e , i n w h ich a l l

c u r v e s w i l l be t h e same. In p r a c t i c e t h e f i n a l o p e r a t i n g l i n e i s

re a c h e d a t some f i n i t e d i s t a n c e i n t o th e b e d , and c o r r e s p o n d i n g l y , t h e

a d s o r p t i o n c u r v e s have r e a c h e d an u l t i m a t e , or u n v a r y i n g s h a p e . T h i s

f i n a l wave sh ape can be o b s e r v e d t o t r a v e l down t h e bed i n t a c t , th e

w i d t h o f t h e wave d e n o t i n g t h a t p o r t i o n o f a d s o r b e n t i n w hich mass

t r a n s f e r i s a c t u a l l y o c c u r r i n g .

I t s h o u ld be r e a d i l y a p p a r e n t from th e f o r e g o i n g d i s c u s s i o n

t h a t th e e q u i l i b r i u m cu rv e I s a m ajor f a c t o r i n c o n t r o l l i n g th e a d s o r p

t i o n p r o c e s s . As w i t h any r a t e p r o c e s s , th e c l o s e r t h e e q u i l i b r i u m

l i n e t o t h e o p e r a t i n g l i n e , th e more d i f f i c u l t i s t h e j o b o f s e p a r a t i n g

two co m p o n en ts . P r e v i o u s a t t e m p t s t o s t u d y th e k i n e t i c s o f a d s o r p t i o n

u s i n g s t r a i g h t e q u i l i b r i u m r e l a t i o n s h i p s can be s e e n t o g i v e r e s u l t s

f a r d i f f e r e n t from t h e s e p r e s e n t e d h e r e i n . As shown i n F i g u r e 3 3 ,

computer runs u s i n g t r u e e q u i l i b r i u m c u r v e s f o r t h r e e d i f f e r e n t s u b

s t a n c e s were o b s e r v e d t o be q u i t e d i f f e r e n t w i t h r e s p e c t t o th e c a l

c u l a t e d r e s u l t s , f o r A = 1 and H = 2 .

48

I f t h e e f f e c t o f p a r t i c l e s i z e upon a d s o r p t i o n i s t o be

a n a l y z e d , s e v e r a l v a l u e s o f th e d i m e n s i o n l e s s r a d i u s p a r a m e t e r , A,

must be c o n s i d e r e d s i m u l t a n e o u s l y . F i g u r e 34 shows how p a r t i c l e s i z e ,

w h ic h i s p r o p o r t i o n a l t o A, i n f l u e n c e s t h e r a t e o f a d s o r p t i o n , h en ce

t h e r e l a t i v e d i f f i c u l t y o f s e p a r a t i o n . Comparison o f s l o p e s shows t h a t

t h e c u r v e s a r e f l a t t e r t h e h i g h e r th e v a l u e o f A, w h ich i s d i r e c t l y

p r o p o r t i o n a l t o p h y s i c a l d i a m e t e r o f a d s o r b e n t p a r t i c l e s . As e x p e c t e d ,

w here i n t r a p a r t i c l e d i f f u s i o n i s found t o a f f e c t r a t e , i n o t h e r w o r d s ,

a t h i g h e r p a r t i c l e r a d i i , th e l o n g e r d i f f u s i o n p a th r e q u i r e s more t im e

t o come t o e q u i l i b r i u m , h e n c e t h e f l a t t e n e d c u r v e s . Note t h a t i t r e

q u i r e s c o n s i d e r a b l y l o n g e r f o r t h e wave f r o n t c u r v e t o f a l l t o a

s p e c i f i e d l e v e l b e c a u s e t h e r e i s much l e s s t e n d e n c y f o r i n t e r n a l d i f f u

s i o n t o produce e q u i l i b r i u m . As a r e s u l t one must a n t i c i p a t e a much

w i d e r a d s o r p t i o n zon e f o r l a r g e p a r t i c l e s . I t i s r e a l i z e d t h a t t h e r e

must be an e c o n o m ic b a l a n c e b e tw e e n th e h ig h p r e s s u r e drop i n s m a l l

p a r t i c l e beds and t h e e x t r a a d s o r b e n t n eed ed f o r t h e more d i f f i c u l t

s e p a r a t i o n s i n b ed s o f l a r g e p a r t i c l e s i z e . W hatever p a r t i c l e s i z e i s

s e l e c t e d f o r u s e , i t w i l l be s e e n t h a t th e r a d i u s v a r i a b l e w i l l be

t a k e n i n t o a c c o u n t when e x p e r i m e n t a l d a t a a r e compared w i t h t h e o r e t i c a l

r e s u l t s .

I f t h e d i m e n s i o n l e s s p a ra m eter A i s d e c r e a s e d , a s shown by

F i g u r e s 23 and 2 4 , i t i s o b s e r v e d t h a t t h e a d s o r p t i o n w aves a p p r o a c h a

c o n s t a n t shape when one p l o t i s compared w i t h a n o t h e r . T h i s phenomenon

s h o u l d be a n t i c i p a t e d h o w e v e r , when t h e q u a l i t a t i v e n a t u r e o f t h e p r o

c e s s i s r e c a l l e d . At low er and lo w e r p a r t i c l e d i a m e t e r s th e d i f f u s i o n

mechanism p l a y s a p r o g r e s s i v e l y s m a l l e r r o l e , and i n t h e l i m i t (a t A = 0)

t h e r e i s no d i f f u s i o n o c c u r i n g , t h e e n t i r e r e s i s t a n c e b e i n g due t o

t h e s u r f a c e f i l m . However t h e model b r e a k s down m a t h e m a t i c a l l y a t

A “ 0 , and o n l y t h e a p p ro a ch t o z e r o r a d i u s may be s t u d i e d . At low

enough r a d i u s , t h e p r e s e n t m odel s h o u l d become s i m i l a r t o t h e one p r o

p o s e d by J o h n s o n . Furtherm ore one s h o u l d not e x p e c t any improvement

o f th e new model o v e r t h a t o f Joh n so n a t low r a d i i , t h e l a t t e r model

a s s u m in g an i n f i n i t e l y r a p id i n t r a p a r t i c l e d i f f u s i o n r a t e . A l s o , t h e

p h y s i c a l p r o p e r t i e s o f an y s y s t e m un der i n v e s t i g a t i o n would d e t e r m i n e

what a c t u a l p a r t i c l e s i z e would be s u f f i c i e n t l y s m a l l f o r t h i s e f f e c t

t o be n e g l e c t e d , s o t h a t a model su ch a s t h a t p r o p o s e d by J o h n s o n ,

c o u ld be u s e d .

4 . The Mass T r a n s f e r C o e f f i c i e n t , K

One o f t h e p r i n c i p a l o b j e c t i v e s o f t h i s s t u d y i s t h e d e t e r m i n a

t i o n o f a s u i t a b l e mass t r a n s f e r c o e f f i c i e n t s o t h a t a p r a c t i c a l e s t i

m a t i o n o f o p e r a t i n g c o n d i t i o n s , and bed h e i g h t , can be g i v e n . The

pr im ary p u r p o se h e r e i s t o d e m o n s t r a t e an improved method f o r t a k i n g

i n t o a c c o u n t a d d i t i o n a l f a c t o r s not p r e v i o u s l y c o n s i d e r e d i n e v a l u a t i n g

K. The u l t i m a t e g o a l o f any su ch work i s t o red u c e or e l i m i n a t e t h e

n e c e s s i t y o f t i m e - c o n s u m i n g and e x p e n s i v e d e v e l o p m e n t a l or p i l o t - s c a l e

p r o j e c t s . In p a r t i c u l a r , i t i s a n t i c i p a t e d t h a t a s m a l l f i x e d - b e d

p l a n t w h ich i s c h e a p e r , e a s i e r t o b u i l d and o p e r a t e , and c a p a b l e o f

more a c c u r a t e m e a s u r e m e n ts , can be u s e d t o p r e d i c t d e s i g n f a c t o r s fo r

a m o v i n g - b e d , c o u n t e r c u r r e n t p r o c e s s . Even f o r com m erc ia l f i x e d - b e d

d e s i g n , a few t e s t s w i t h b e n c h - s c a l e eq u ip m ent s h o u l d s u f f i c e t o p r e

d i c t p e r fo r m a n c e o f t h e l a r g e r p l a n t . O th er f a c t o r s , such a s t im e to

50

r e a c h s t e a d y - s t a t e a f t e r s t a r t - u p , bed d e p th r e q u i r e d t o r e a c h s t e a d y -

s t a t e , and p e r c e n t o f unused a d s o r b e n t a t p o i n t o f b r e a k t h r o u g h , s h o u l d

be d e t e r m i n a b l e from b e n c h - s c a l e t e s t s . N a t u r a l l y t h e v a l i d i t y o f su ch

p r o c e d u r e s d ep en d s n o t o n l y on p r e v i o u s l y n o t e d a s s u m p t i o n s , but on

s c a l e - u p c o n s i d e r a t i o n s a s w e l l . P rob lem s o f f e e d d i s t r i b u t i o n , w a l l

e f f e c t s , and c h a n n e l l i n g , w h ic h a l t e r t h e a s s u m p t i o n o f a p l u g - l i k e

v e l o c i t y p r o f i l e , rem ain t o be c o n s i d e r e d ; t h e s e e f f e c t s g e n e r a l l y a r e

more s e v e r e i n t h e c a s e o f moving b e d s . The e x i s t e n c e o f a x i a l

d i f f u s i o n i s a l s o a f a c t o r w hich may a f f e c t t h e l a r g e r b ed .

I t has b e e n f a i r l y w e l l e s t a b l i s h e d t h a t t h e mass t r a n s f e r c o

e f f i c i e n t i s s t r o n g l y d e p e n d e n t on numerous p h y s i c a l and g e o m e t r i c

p r o p e r t i e s and c a r e must be ta k e n i n i t s u s e . Even w i t h t h e i n c l u s i o n

o f i n t r a p a r t i c l e d i f f u s i o n i n th e p r e s e n t m o d e l , t h e r e a r e many u n

knowns y e t t o be fo u n d . I f th e ran ge o f c o n d i t i o n s c o v e r e d d u r in g

b e n c h - s c a l e s t u d i e s i s not t o o f a r d i f f e r e n t from t h o s e t o be e n c o u n

t e r e d i n t h e l a r g e p l a n t , i t h as f r e q u e n t l y b een found t h a t a c t u a l

p e r fo r m a n c e o f t h e l a r g e p l a n t i s r e l a t i v e l y c l o s e t o p r e d i c t i o n s .

F u r t h e r t h e o r e t i c a l t r e a t m e n t s h o u l d make t h e s e d e s i g n e s t i m a t e s e v e n

more r e l i a b l e . F o r t u n a t e l y i n l i q u i d a d s o r p t i o n the r e l a t i v e v e l o c i t y

o f t h e l i q u i d and s o l i d p h a s e s rem ain s c o n s t a n t . The d e p e n d e n c y o f K

on v e l o c i t y i s th u s e l i m i n a t e d i f the b e n c h - s c a l e and l a r g e p l a n t a r e

o p e r a t e d a t t h e same r a t e s . L i k e w i s e t h e e f f e c t o f p a r t i c l e s i z e

a n d / o r g e o m e tr y can be e l i m i n a t e d i f th e same m a t e r i a l i s u sed f o r b o th

s m a l l and l a r g e p l a n t s . Any v a r i a t i o n o f K w i t h c o n c e n t r a t i o n w i l l be

more d i f f i c u l t t o h a n d l e . As t h e r e has been no s a t i s f a c t o r y q u a n t i t a

t i v e a n a l y s i s o f c o n c e n t r a t i o n e f f e c t s on K, i t s h o u l d be r e q u i r e d t h a t

51

K e i t h e r n o t v a r y g r e a t l y o v e r t h e r a n g e s o f c o n c e n t r a t i o n i n v o l v e d or

t h a t b o t h t h e e x p e r i m e n t a l and f u l l - s i z e p l a n t s be o p e r a t e d o v e r s i m i

l a r r a n g e s . The a c t u a l d e t e r m i n a t i o n o f K from computed and r e a l d a t a

w i l l be e l a b o r a t e d upon i n t h e f o l l o w i n g s e c t i o n .

5 . Comparison o f Computed and E x p e r i m e n t a l Curves

The t h e o r e t i c a l s o l u t i o n s found i n v o l v e t h e p a r a m e t e r s A, H,

and T a s d e f i n e d p r e v i o u s l y . To c o n v e r t t h e s e r e s u l t s i n t o t h e terms

o f a c t u a l t i m e , p a r t i c l e r a d i u s , and bed d e p th r e q u i r e s knowledge o f

t h e mass t r a n s f e r c o e f f i c i e n t , K, and t h e e f f e c t i v e i n t e r n a l d i f f u s i o n

c o n s t a n t , D. J o h n so n (3 0 ) showed t h a t th e v a l u e o f K may be d e t e r m i n e d

i n a t r i a l - a n d - e r r o r p r o c e d u r e o f m a t c h in g e x p e r i m e n t a l w i t h t h e o r e t i

c a l c u r v e s . To do t h i s a v a l u e o f K was s e l e c t e d s o t h a t a c u r v e o f

T v e r s u s X c o u l d be c o n v e r t e d i n t o 0 v e r s u s x by means o f th e t r a n s

f o r m a t i o n e q u a t i o n . That v a l u e o f K w h i c h p roduced t h e c l o s e s t match

o f t h e o r e t i c a l w i t h e x p e r i m e n t a l c u r v e s was s e l e c t e d a s th e p r o p e r mass

t r a n s f e r c o e f f i c i e n t f o r t h e s y s t e m under i n v e s t i g a t i o n . The c r i t e r i o n

o f b e s t f i t , u sed by J o h n s o n , was t o f i n d a K s o t h a t th e p o s i t i o n o f

t h e two c u r v e s was t h e same on a p l o t o f x v e r s u s t im e (or e q u i v a l e n t l y ,

e f f l u e n t v o l u m e ) . In t h e o r y t h e shape o f t h e t h e o r e t i c a l and r e a l

c u r v e s s h o u l d have b een i d e n t i c a l a t t h e same p o s i t i o n . In o t h e r words

t h e r e s h o u l d be some K w hich g i v e s one c u r v e r e p r e s e n t i n g b o th e x p e r i

m e n t a l and t h e o r e t i c a l r e l a t i o n s . As an e x a m p le , i n F i g u r e 2 , would

be th e s e l e c t e d v a l u e f o r t h e i d e a l s y s t e m f o l l o w i n g J o h n s o n ' s m o d e l .

In t h i s i l l u s t r a t i v e s k e t c h w hich d o e s n o t r e p r e s e n t any p a r t i c u l a r

s y s t e m , t h e o r e t i c a l c u r v e s a r e s o l i d l i n e s , and e x p e r i m e n t a l d a t a

52

p l o t t e d a s c i r c l e s . The t h r e e e x p e r i m e n t a l c u r v e s shown i n F i g u r e 2

c o u ld r e p r e s e n t th e same s y s t e m , p l o t t e d f o r d i f f e r e n t t r i a l v a l u e s o f

K.

0Time

F i g u r e 2 . Method I l l u s t r a t i n g Comparison o f Computed and E x p e r im e n ta l R e s u l t s (Not A c t u a l D a t a ) .

In F i g u r e 2 th e l i n e s o f c o n s t a n t K a r e a l s o l i n e s o f c o n

s t a n t H, b e c a u s e th e two a r e d i r e c t l y p r o p o r t i o n a l t o e a c h o t h e r . The

l i n e m a tch in g e x p e r i m e n t a l d a t a w i t h t h e o r y i s t h e r e f o r e a r e p r e s e n t a

t i o n o f th e a d s o r p t i o n wave a t th e o u t l e t .

In p r a c t i c e Johnson d i s c o v e r e d t h a t no e x a c t match o f the

a d u a l w i t h t h e r e t i c a l c u r v e s c o u ld be fou n d . In g e n e r a l , th e e x p e r i

m e n t a l c u r v e s w h ich gave t h e b e s t f i t were somewhat f l a t t e r than the

t h e o r e t i c a l c u r v e s . I t was b e c a u s e o f t h i s f a c t t h a t Johnson c o n c lu d e d

53

i n t r a p a r t i c l e d i f f u s i o n a f f e c t e d t h o s e a d s o r p t i o n s y s t e m s w hich he

i n v e s t i g a t e d and f u r t h e r recommended t h a t a d d i t i o n a l t h e o r e t i c a l t r e a t

ment i n v o l v i n g t h i s mechanism was n e c e s s a r y .

When i n t r a p a r t i c l e d i f f u s i o n i s c o n s i d e r e d , a s i m i l a r m atch in g

p r o c e d u r e can be em p loyed , but i t i s c o n s i d e r a b l y more i n v o l v e d b eca u s e

o f t h e added p a r a m e te r , A. The f o l l o w i n g d i s c u s s i o n d e s c r i b e s b r i e f l y

how t h e v a l u e o f K i s d e term in e d i n t h e model p roposed i n t h i s s t u d y .

S t a t e d b r i e f l y , th e p roced ure i n v o l v e s f i n d i n g t h e b e s t m a tch , not

o n l y on one s e t o f c u r v e s but from a l l c u r v e s u s i n g any c o m b in a t io n o f

H, T, and A. The problem becomes e s s e n t i a l l y t h r e e - d i m e n s i o n a l in

s c o p e .

I n i t i a l l y a s o l u t i o n f o r a s e l e c t e d c h o i c e o f A i s computed .

The b e s t match i s th e n formed b etw een t h e o r e t i c a l and e x p e r i m e n t a l r e

s u l t s . S i n c e th e r i g h t c h o i c e o f A i s not known, i t s h o u ld not be

e x p e c t e d t h a t a v e r y good match w i l l be o b t a i n e d on th e f i r s t t r y .

F ig u r e 4 shows th e c l o s e s t match f o r th e v a l u e o f A = 1 f o r th e s y s t e m ,

m e t h y l c y c l o h e x a n e - t o l u e n e on s i l i c a g e l a t a f e e d c o n c e n t r a t i o n ,

X£ = .5 (Curve A ) . I t i s o b v io u s t h a t th e f i t i s u n s a t i s f a c t o r y and

o t h e r v a l u e s o f A s h ou ld be t r i e d . Whether to i n c r e a s e or d e c r e a s e A

i s l a r g e l y a m a t t e r o f c o n j e c t u r e . When A = 5 was t r i e d , the r e s u l t i n g

computer s o l u t i o n was s e e n t o f i t th e e x p e r i m e n t a l d a t a b e s t a t K *

0 . 8 2 x 10” c m / s e c . I t was n o ted t h a t th e two c u r v e s matched somewhat

b e t t e r than i n t h e p r e v i o u s c a s e . With t h i s i n f o r m a t i o n i n mind, A

was i n c r e a s e d and th e proulem r e - r u n s e v e r a l t i m e s . Up t o a c e r t a i n

p o i n t i t was o b s e r v e d t h a t th e computer c u r v e s matched a c t u a l d a ta

b e t t e r w i t h i n c r e a s i n g A, but a f t e r a p o i n t , f u r t h e r i n c r e a s e o f A

54

y i e l d e d p o o r e r c o m p a r i s o n s . H e n c e , t h e p r o p e r c h o i c e o f K would be

from t h e s e t o f c u r v e s f o r t h a t v a l u e o f A p r o d u c i n g t h e c l o s e s t r e l a

t i o n s h i p b e tw e e n t h e o r y and e x p e r i m e n t , p r o v i d e d t h e m od e l h o l d s . For

th e c a s e o f m e t h y l c y c l o h e x a n e - t o l u e n e on s i l i c a g e l a t 0 . 5 f e e d c o n c e n

t r a t i o n , t h e b e s t m atch was found t o - b e n e a r A * 5 0 , w i t h a v e r a g e

_ K ■ 2 . 5 4 x 10 c m / s e c , a s shown i n F i g u r e 9 . Other ru n s a t A ■ 0 . 0 1 ,

0 . 1 , 1 0 , 2 0 , 4 0 , and 6 0 were made i n a t t e m p t i n g t o f i n d t h e b e s t f i t .

S u b s t a n t i a l improvement i n m a t c h in g c u r v e s i s s e e n by com paring F i g u r e

9 (A = 5 0 ) w i t h F i g u r e s 35 and 3 6 , o b t a i n e d from J o h n s o n ’ s work ( 3 0 ) .

Of p a r t i c u l a r i n t e r e s t i s t h e c o n s t a n c y o f K w i t h a ch an ge i n

bed h e i g h t . An a d d i t i o n a l e x p e r i m e n t a l run a t a d i f f e r e n t bed l e v e l

d e m o n s t r a t e d t h a t i t i s p o s s i b l e t o o b t a i n a p p r o x i m a t e l y t h e same K

th r o u g h a s i m i l a r m a t c h i n g p r o c e d u r e (Curve B i n F i g u r e s 4 - 1 1 ) .

F i g u r e 9 shows r e s u l t s a t th e s e l e c t e d optimum f i t , f o r b o t h r u n s .

An e x p e r i m e n t o f t h i s t y p e t e n d s t o s u p p o r t t h e a s s u m p t i o n n e g l e c t i n g

l o n g i t u d i n a l ( a x i a l ) d i f f u s i o n , a t l e a s t b e tw e e n th e two bed h e i g h t s

i l l u s t r a t e d , and f o r t h e s y s t e m c o n s i d e r e d . Over much w i d e r l e n g t h s

o f bed s u c h an a s s u m p t i o n may n o t h o l d , s i n c e i t i s a l m o s t c e r t a i n

a x i a l d i f f u s i o n can and d o e s o c c u r . One o f t h e s y s t e m s s t u d i e d , a s

w i l l be m e n t io n e d l a t e r , may have d i s p l a y e d su ch l e n g t h w i s e d i f f u s i o n .

F i lm c o e f f i c i e n t s o b t a i n e d by m a t c h in g e x p e r i m e n t a l d a t a w i t h

com p u ter r e s u l t s f o r a l l o f t h e f o u r c a s e s s t u d i e d a r e g i v e n i n T a b le

I I , a l o n g w i t h th e c o r r e s p o n d i n g d i m e n s i o n l e s s p a r a m e t e r , A, g i v i n g

th e b e s t f i t f o r e a c h p a r t i c u l a r ru n . C o e f f i c i e n t s a s s o c i a t e d w i t h

e a c h p l o t o f e x p e r i m e n t a l d a t a i n F i g u r e s 4 - 3 2 a r e found on the a p p r o

p r i a t e d r a w i n g s .

55

The f i l m c o e f f i c i e n t s o b t a i n e d i n t h i s s t u d y may be compared

w i t h t h o s e c a l c u l a t e d from r a t e d a t a g i v e n by E a g l e and S c o t t ( 8 ) .

A l t h o u g h d i r e c t c o m p a r i s o n s f o r i d e n t i c a l s y s t e m s a r e n o t a v a i l a b l e , i t

may be s e e n t h a t t h e r e s u l t s compare w i t h i n an o r d e r o f m a g n i t u d e .

For i n s t a n c e i n t h e a d s o r p t i o n o f t o l u e n e - o c t a n e on 1 0 - 1 4 mesh s i l i c a

- 4g e l v a l u e s o f K a r e n o t e d t o be i n t h e r a n g e , 1 . 2 7 - 1 . 6 9 x 10 c m / s e c ,

u s i n g t h e d a t a o f E a g l e and S c o t t , w h ic h compare w i t h th e v a l u e s o f

2 . 5 4 x 1 0 " 3 and 1 . 1 2 x 10" 3 o b t a i n e d i n t h i s s t u d y w i t h m e t h y l c y c l o -

h e x a n e - t o l u e n e on s i l i c a g e l . A l t h o u g h th e two f i g u r e s d i f f e r by a

f a c t o r o f a b ou t t e n , t h i s i s n o t t o t a l l y u n e x p e c t e d b e c a u s e o f t h e

m eth od s u s e d and t h e d i f f e r e n c e i n s y s t e m s . S i m i l a r c o m p a r i s o n s a r e

made f o r a d s o r p t i o n on a l u m i n a . Data o b t a i n e d from E a g l e and S c o t t

y i e l d a v a l u e o f K « .85 x 10 ^ f o r t h e a d s o r p t i o n o f t o l u e n e - o c t a n e ,

_ Ocompared w i t h 0 . 7 4 x 10 i n t h i s s t u d y f o r m e t h y l c y c l o h e x a n e - t o l u e n e

on a l u m i n a .

The o n l y s y s t e m i n w h ic h a good c o r r e l a t i o n b e tw e e n e x p e r i

m e n t a l d a t a and computed c u r v e s i s n o t o b t a i n e d i s t h a t o f m e t h y l c y c l o

h e x a n e - t o l u e n e on a lu m in a (S ys tem Number 3 i n T a b le I I ) . A c h o i c e o f

A = 3 was made b e c a u s e t h e a c t u a l d a t a f e l l on b o th s i d e s o f t h e com

p u te d c u r v e , w h e r e a s f o r o t h e r v a l u e s o f A th e a c t u a l d a t a l a y w e l l t o

one or t h e o t h e r s i d e o f t h e computed c u r v e . The l a c k o f p e r f e c t

a g r e e m e n t i s p r o b a b l y a t t r i b u t a b l e t o s e v e r a l f a c t o r s , e a c h one or a l l

c o n t r i b u t i n g t o t h e d i s c r e p a n c y . The n a t u r e o f th e a d s o r b e n t i s v e r y

l i k e l y i n v o l v e d , a s t h e l i q u i d s o l u t i o n , m e t h y l c y c l o h e x a n e - t o l u e n e was

s u c c e s s f u l l y t e s t e d i n two o t h e r c a s e s . The p ore volume o f a lu m in a

i s s m a l l compared t o t h a t o f t h e o t h e r s y s t e m s s t u d i e d , and h e n c e th e

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i n t e r n a l d i f f u s i o n a l mechanism s a r e p r o b a b l y i n v o l v e d t o a l e s s e r

d e g r e e , e v e n th ou gh a m o d e r a t e l y c o a r s e m a t e r i a l ( 8 - 1 4 mesh) i s u s e d .

For a l l v a l u e s o f A c h e c k e d , t h e a c t u a l c u r v e s a r e s t e e p e r th an t h e

computed c u r v e s a t t h e lo w e r c o n c e n t r a t i o n s , s u g g e s t i n g t h a t t h e d i f

f u s i o n m echanism w i t h i n th e p a r t i c l e i s b e i n g o v e r r i d d e n by a n o t h e r

r a t e c o n t r o l l i n g s t e p . The c a s e o f e x t e r n a l f i l m r e s i s t a n c e s o l e l y

c o n t r o l l i n g i s d o u b t f u l , b e c a u s e a good c o r r e l a t i o n s h o u l d be found

a t low A, w h ic h i s n o t th e c a s e f o r t h i s s y s t e m . There i s a p o s s i

b i l i t y t h a t t h e mechanism w h ic h d o e s a f f e c t r a t e s t r o n g e s t ch a n g es

d u r i n g th e c o u r s e o f a change i n c o n c e n t r a t i o n from h i g h t o low . The

e q u i l i b r i u m c u r v e a l s o a f f e c t s t h e sh ape o f t h e w a v e , h en ce t h e f u n c

t i o n e x p r e s s i n g e q u i l i b r i u m r e l a t i o n s h i p s must be c o r r e c t l y d e t e r m i n e d .

There i s some p o s s i b i l i t y t h a t t h e e q u i l i b r i u m measured i n t h e l a b o r a

t o r y i s not t h e t r u e e q u i l i b r i u m under p r o c e s s c o n d i t i o n s .

A m ajor f a c t o r i s t h e p o s s i b l e e f f e c t o f l o n g i t u d i n a l d i f f u

s i o n i n some c a s e s w here i t i n f l u e n c e s th e r a t e o f a d s o r p t i o n . I f

l o n g i t u d i n a l d i f f u s i o n i s c o n s i d e r e d , i t s e f f e c t w i l l be to change t h e

shape o f th e a d s o r p t i o n w ave . The maximum c o n c e n t r a t i o n g r a d i e n t

g i v i n g r i s e t o a x i a l , or l o n g i t u d i n a l , d i f f u s i o n o c c u r s i n the s t e e p e s t

p o r t i o n o f t h e a d s o r p t i o n w a v e . At t h e low er end o f t h e wave s m a l l

g r a d i e n t s in th e a x i a l d i r e c t i o n w i l l no t produ ce an e x t e n s i v e t r a n s f e r

o f m a t e r i a l a x i a l l y , t h e r e f o r e t h e shape o f t h e cu rv e s h o u l d n o t be

changed s u b s t a n t i a l l y . However a t t h e upper end o f t h e wave l a r g e

g r a d i e n t s have o c c u r r e d d u r i n g t h e p a s s a g e o f t h e w a v e , t e n d i n g to

r e d u c e t h e c o n c e n t r a t i o n i n t h e b u lk p h ase a t a g i v e n l e v e l . C onse

q u e n t l y a f l a t t e n i n g o f t h e wave a t t h e upper end s h o u ld be e x p e c t e d ;

57

t h i s f l a t t e n i n g i s a c t u a l l y o b s e r v e d f o r t h e a lu m in a b e d , w here a r e l a

t i v e l y good c o r r e l a t i o n o f d a t a w i t h t h e o r y i s o b s e r v e d a t t h e lo w er

e n d . A p o s s i b l e n e x t s t e p i n t h e o r e t i c a l t r e a t m e n t would be th e i n c l u

s i o n o f a term i n t h e b a s i c model f o r a x i a l d i f f u s i o n . C e r t a i n p ro b lem s

would a r i s e , h o w e v e r , w h ich s h o u l d be c o n s i d e r e d . An a d d i t i o n a l p a r a

m e t e r , t h e d i f f u s i v i t y c o n s t a n t i n th e l o n g i t u d i n a l d i r e c t i o n would

have t o be d e t e r m i n e d . The a d d i t i o n o f a n o t h e r term i n t h e model would

o f c o u r s e r e q u i r e i n c r e a s e d c o m p u t a t i o n .

In t h e f i n a l a n a l y s i s i t must be remembered t h a t t h e p r o p o s e d

model i s n o t a l l i n c l u s i v e o f the many known f a c t o r s i n f r a c t i o n a t i o n

o f any s o l u t i o n . The a d s o r b e n t i t s e l f i s f a r from b e i n g o f t h e g e o m e t r y

p r o p o s e d . The b a s i c a s s u m p t i o n s o u t l i n e d a t t h e b e g i n n i n g a r e a l l

a p p r o x i m a t i o n s t o t h e t r u e s y s t e m , f o r m a t h e m a t i c a l s i m p l i c i t y . I t i s

m ost e n c o u r a g i n g t o n o t e , h o w e v e r , t h a t t h e e f f e c t o f i n t r a p a r t i c l e

d i f f u s i o n , t h e major f a c t o r t o be c o n s i d e r e d i n t h i s p r o j e c t , has been

found q u a n t i t a t i v e l y i n t h r e e o f t h e f o u r c a s e s s t u d i e d , and h as r e

s u l t e d i n an improved c o r r e l a t i o n o v e r p r e v i o u s m o d e l s .

6 . D e s i g n o f a F i x e d -B e d A dsorber

A c o m p l e t e s o l u t i o n to th e e q u a t i o n s f o r f i x e d - b e d a d s o r p t i o n

can p r o v i d e s u f f i c i e n t d a t a f o r d e s i g n i n g an e c o n o m i c a l a d s o r p t i o n

co lum n, p r o v i d e d p h y s i c a l p r o p e r t i e s and o p e r a t i o n a l d a t a f o r t h e s y s

tem a r e known. S im p le r t e c h n i q u e s may be u s e d t o f i n d th e c h a r a c t e r i s

t i c s o f t h e u l t i m a t e a d s o r p t i o n w a v e , or mass t r a n s f e r z o n e , f o r t h e

s t e a d y s t a t e a l o n e . However w i t h th e l a t t e r p r o c e d u r e , n o t h i n g can be

s t a t e d a b o u t t h e f o r m a t i o n t im e o f t h e s t e a d y - s t a t e w a v e , or th e

amount o f a d s o r b e n t r e q u i r e d * f o r t h i s p u r p o s e . The u s u a l a s s u m p t i o n

58

i s t o n e g l e c t or g u e s s t h e h e i g h t o f bed r e q u i r e d f o r f o r m a t io n o f t h e

u l t i m a t e wave; however t h i s i s c e r t a i n l y not a l w a y s a v a l i d p r o c e d u r e .

I t i s h i g h l y p o s s i b l e t h a t f o r a g i v e n t h r o u g h p u t , b r e a k th r o u g h may

o c c u r b e f o r e t h e d e s i r e d c o n c e n t r a t i o n o f e f f l u e n t i s r e a c h e d , or in

o t h e r w o r d s , a column c o u ld c o n c e i v a b l y be d e s i g n e d i n which s t e a d y -

s t a t e i s n e v e r r e a c h e d . In t h i s s i t u a t i o n th e bed would be t o o s h a l l o w

t o a c h i e v e t h e d e s i r e d p u r i t y o f e f f l u e n t . Where i n t r a p a r t i c l e d i f f u

s i o n c o n t r o l s , th e u l t i m a t e wave i s more d i f f i c u l t t o r e a c h , and i n

su ch c a s e s knowledge o f t h e t r a n s i e n t s t a t e i s v e r y i m p o r t a n t , In

beds c o n t a i n i n g l a r g e p a r t i c l e s , pore p r o c e s s e s may c a u s e a f l a t t e n i n g

o f t h e a d s o r p t i o n c u r v e s , th u s i n c r e a s i n g t im e t o r e a c h s t e a d y - s t a t e a s

w e l l a s th e amount o f a d s o r b e n t m a t e r i a l r e q u i r e d t o produce t h e u l t i

mate wave. T h is i n f o r m a t i o n i s o f c o u r s e n e c e s s a r y t o p r e d i c t a c c u

r a t e l y th e t o t a l p r o c e s s t im e r e q u i r e d t o a c h i e v e a g i v e n s e p a r a t i o n

a t some p a r t i c u l a r t h r o u g h p u t . M oreover , t h e most e c o n o m ic a l o v e r a l l

c y c l e t im e f o r both p r o c e s s i n g and r e g e n e r a t i o n s t e p s i s dep en d ent

upon a d s o r p t i o n zone c h a r a c t e r i s t i c s .

The computed s o l u t i o n f o r th e s y s t e m , m e t h y l c y c l o h e x a n e - t o l u e n e

on s i l i c a g e l , w i l l be used t o i l l u s t r a t e how d e s i g n d a ta f o r a f i x e d -

bed a d s o r b e r may be o b t a i n e d . The c u r v e s o f x v e r s u s T may be e a s i l y

c o n v e r t e d t o a c u r v e o f x v e r s u s a c t u a l t i m e , 9 , by u s e o f th e t r a n s

f o r m a t io n e q u a t i o n

T = M ( 0 J b d ) ( 4 - 9 )a Q/S

As d e s c r i b e d i n an e a r l i e r s e c t i o n , t h e v a l u e s o f A and K may be ob

t a i n e d th rou gh a com p ar ison o f e x p e r i m e n t a l and t h e o r e t i c a l c u r v e s .

59

The p a r a m e t e r H i s c o n v e r t e d t o a c t u a l bed l e v e l , z , by E q u a t io n ( 4 - 8 ) .

F i g u r e 35 i s a p l o t o f b o t h r e a l and t h e o r e t i c a l r e s u l t s f o r a t y p i c a l

s y s t e m (Joh nson Run F - l ) ( 3 0 ) . The minimum t im e t o r e a c h s t e a d y s t a t e

i s c o n s i d e r e d t o be t h e p o i n t w here t h e e n v e l o p e c u r v e c o r r e s p o n d i n g t o

t h e wave f r o n t d r o p s r e a s o n a b l y c l o s e t o z e r o . The bed l e v e l c o r r e

s ponds t o t h a t c u r v e i n t e r s e c t i n g t h e boundary c u r v e a t j u s t t h i s p o i n t

I t i s c o n c e i v a b l e t h a t a co lumn c o u l d be d e s i g n e d w i t h a d e p t h j u s t

e q u a l t o t h e l e v e l a t w h i c h s t e a d y s t a t e i s r e a c h e d . However t h a t p o r

t i o n o f bed u n s a t u r a t e d w i t h th e more a d s o r b a b l e component i s r e l a

t i v e l y l a r g e i n c o m p a r i s o n t o t h e t o t a l b e d . The p r o p o r t i o n o f u n s p e n t

a d s o r b e n t may be r e d u c e d by i n c r e a s i n g bed h e i g h t . The l i m i t a t i o n upon

bed h e i g h t i s an e c o n o m ic o n e , s i n c e o p e r a t i n g and f i x e d c o s t s i n c r e a s e

w i t h h e i g h t .

The amount o f u n s a t u r a t i o n a t any bed l e v e l , w h e t h e r t h e u l t i

m ate wave has b een e s t a b l i s h e d or n o t , may be d e t e r m i n e d a s d e s c r i b e d

b e l o w . C o n s i d e r F i g u r e 3 , a s k e t c h o f e f f l u e n t c o n c e n t r a t i o n a s a

f u n c t i o n o f r e a l t i m e . The s t e a d y s t a t e i s c o n s i d e r e d t o o c c u r a t

9 = 9 S . At a n y g i v e n l e v e l t h e t im e r e q u i r e d f o r a wave t o p a s s a

p a r t i c u l a r p o i n t i s g i v e n by

ew = e2 - 0 i ( 6 - i )

T h i s i s a l s o t h e t im e r e q u i r e d f o r t h e measurement o f a b r e a k t h r o u g h

wave a s i t p a s s e s o u t o f t h e a d s o r p t i o n co lum n. The q u a n t i t y o f

l i q u i d c o l l e c t e d a s t h i s wave p a s s e s ou t o f t h e co lumn i s e x p r e s s e d

v o l u m e t r i c a 1 l y by t h e form u la

Vw - Q0w ( 6 - 2 )

where i s t h e t o t a l vo lume o f e f f l u e n t l i q u i d , f t .

60

0

Time

F i g u r e 3 . Form at ion o f the U l t i m a t e Wave

IThe maximum q u a n t i t y o f component A, V , t h a t c o u ld be c o n t a i n e d in

t h e l e n g t h o f a wave a t s a t u r a t i o n i s e x p r e s s e d by th e f o l l o w i n g

f o r m u l a :

Va ' = Q = Q x f (02 - 0 L) ( 6 - 3 )

The amount o f u n adsorbed component A a t b r e a k th r o u g h can be computed

by c o n s i d e r i n g a d i f f e r e n t i a l s e c t i o n o f w ave , a s shown in th e above

s k e t c h . At s t e a d y s t a t e , the q u a n t i t y o f component A i n t h e t r a n s f e r

zone i s g i v e n by

dVa = Q xd9 ( 6 - 4 )

The t o t a l q u a n t i t y o f component A i n t h e e f f l u e n t w ave , w hich i s th e

same a s th e t o t a l amount o f A not a d s o r b e d , can be o b t a i n e d by i n t e g r a

t i o n b e tw een l i m i t s , 9^ and 0 2 ;

The volume u n adsorb ed i s th e a r e a under t h e a d s o r p t i o n c u r v e m u l t i p l i e d1

by o v e r a l l l i q u i d f l o w r a t e . The f r a c t i o n a d s o r b e d , f a , i s t h e n g i v e n

by:

x d 0

V - Vf ’ = — ------ j-3- = 1 --------±--------- (6 - 6 )

V <«2 - 0 x) x f

The l i m i t s o f i n t e g r a t i o n , 0^ and ©2 , must be s e l e c t e d j u d i

c i o u s l y , b e c a u s e t h e t h e o r e t i c a l l i m i t s a r e o v e r an i n f i n i t e r a n g e .

The a c t u a l l i m i t s a r e c h o s e n a r b i t r a r i l y t o i n c l u d e t h e p o r t i o n o f

a d s o r p t i o n zo n e w h e r e , f o r a l l p r a c t i c a l p u r p o s e s , t h e s l o p e o f t h e

c u r v e i s n o n - z e r o , i n d i c a t i n g t h e r e g i o n o f a c t i v e mass t r a n s f e r . For

t h e s y s t e m , m e t h y l c y c l o h e x a n e - t o l u e n e on s i l i c a g e l ( a t x^ = . 5 ) , a> i

s t e a d y - s t a t e v a l u e o f f a o f .372 was o b t a i n e d . The m a g n i tu d e o f f a

d e p e n d s , o f c o u r s e , on t h e sh ape o f t h e a d s o r p t i o n w ave .

The n e t d e g r e e o f s a t u r a t i o n f o r t h e e n t i r e column s h o u l d be

t h e t o t a l q u a n t i t y a d s o r b e d , d i v i d e d by th e maximum a d s o r b a b l e c a p a c i t y

o f t h e b ed . The l e n g t h , z , o f t h e zo n e b e tw e e n l i m i t s , 0^ and ^ >

i s g i v e n by

z. _ Q®w( 6 - 7 )w fvS+VpPbS

The r e m a in d e r o f t h e column i s s a t u r a t e d , s o t h a t t h e o v e r a l l d e g r e e

o f s a t u r a t i o n , or a d s o r p t i o n , f a , i s e x p r e s s e d by t h e r e l a t i o n

or i n term s o f bed w e i g h t and d e n s i t y ,

f * 1 - b , i _ f '-v( V p P b+ f v)W C 8 (6-®>

I n o r d e r t o u s e E q u a t i o n ( 6 - 9 ) t h e v a r i a b l e , Qw , must be computed.

R e a r r a n g i n g E q u a t io n ( 4 - 9 ) g i v e s

9 = 1 + ^ <6 - 10>

S u b s t i t u t i n g f o r z from E q u a t i o n ( 4 - 8 ) g i v e s

0 = I* + f y a -.H ( 6 - 1 1 )AK KVp P b

S i n c e H = T/A,

9 = (1 + T?------- 5 <6_12>AK V n ,p H b

S i n c e 9^ = 8 3 “ >

0w = <T 2 - Tl> fr? < 1 + <6 - ! 3 )AK vp P b

At s t e a d y s t a t e , 0 ■ 0 g , and

« s - — ( 1 + — ------- ) ( 6 - 1 4 )AK v p P b

A c c o r d in g t o E q u a t io n ( 6 - 1 4 ) t h e t im e o f f o r m a t i o n o f t h e u l t i m a t e wave

i s in d e p e n d e n t o f o p e r a t i n g c o n d i t i o n s .

In t h e e v e n t a s t e a d y s t a t e wave has n o t b e e n form ed , i . e . , i f 0 < 0 S ,

63

t h e e f f e c t o f t h e boundary c u r v e must be c o n s i d e r e d i n com p u t in g Vf l .

E q u a t i o n ( 6 - 1 5 ) must be m o d i f i e d by s u b t r a c t i n g t h e c o n c e n t r a t i o n a t

t h e b o u n d a ry , w h ic h c o r r e s p o n d s t o t h e wave f r o n t :

6":

9 < 0 S ( 6 - 1 5 )

w here x, i s t h e i n s t a n t a n e o u s wave f r o n t c o n c e n t r a t i o n , b

The d e g r e e o f s a t u r a t i o n , f a , a t 0 g may be o b t a i n e d by draw-s

i n g a v e r t i c a l l i n e from t h e 0 - a x i s a t 0 S , The p o i n t a t w h ic h t h i s

l i n e c r o s s e s t h e f e e d l i n e i s t h e u pper l i m i t , ©2 , o f t h e c u r v e r e p r e

s e n t i n g c o n d i t i o n s i n t h e column j u s t a s pure e f f l u e n t a p p e a r s . The

l o w e r l i m i t i s t h e i n t e r s e c t i o n o f t h i s c u r v e w i t h th e boundary c u r v e .

I n T a b le I I I a r e g i v e n v a l u e s o f f o r m a t i o n t i m e , 0 g , and c o r r e

s p o n d in g bed l e n g t h f o r e a c h s y s t e m c o n s i d e r e d , u s i n g t h e most r e a s o n

a b l e f i l m c o e f f i c i e n t a s d e t e r m i n e d by m a t c h in g e x p e r i m e n t a l and

t h e o r e t i c a l c u r v e s . A l s o l i s t e d i n t h e t a b l e a r e v a l u e s o f f„ anda sI

f a f o r e a c h s y s t e m a t t h e b e g i n n i n g o f s t e a d y - s t a t e ; i n a d d i t i o n , v a l u e s

o f f 3 w > d e g r e e o f s a t u r a t i o n w i t h i n a z o n e o f u n v a r y i n g s h a p e , a r e

l i s t e d .

The p a r a m e t e r s d e s c r i b e d a b o v e a r e i m p o r t a n t t o t h e e s t a b l i s h

ment o f th e p r o p e r o p e r a t i n g c o n d i t i o n s and s i z i n g o f f i x e d bed eq u ipm ent

Note t h a t t h e s e v a r i a b l e s have b een d e t e r m i n e d i n d e p e n d e n t l y o f th e

p o s i t i o n o f t h e a c t u a l b r e a k t h r o u g h c u r v e w i t h r e s p e c t t o th e s t e a d y

s t a t e . I n d e s i g n i n g f i x e d bed eq u ip m en t c a r e must be t a k e n t h a t a bed

o f s u f f i c i e n t d e p t h f o r t h e d e s i r e d t h r o u g h p u t i s p r o v i d e d . The

64

r e s i d e n c e t im e i n t h e bed must be a t l e a s t e q u a l t o 9 S ( c o r r e s p o n d i n g

t o a l e n g t h , z Q) t o remove one component c o m p l e t e l y from t h e main

s t r e a m . The l e n g t h o f bed i n e x c e s s o f t h e minimum i s d e p e n d e n t on

e c o n o m ic c o n s i d e r a t i o n s , and t h e s a t u r a t i o n f a c t o r , f a , must be i n c l u d e d

i n an o v e r a l l e c o n o m ic e s t i m a t e . For t a l l c o l u m n s , f a a p p r o a c h e s 100

p e r c e n t f o r an e f f i c i e n t a d s o r b e n t u t i l i z a t i o n , b u t t h e c y c l e t im e i n

c r e a s e s , a s w e l l a s f i x e d c o s t s . Some e c o n o m ic b a l a n c e b e tw e e n c o m p l e t e

a d s o r b e n t u t i l i z a t i o n and p l a n t c o s t r e l a t i v e t o o p e r a t i o n i s t h e r e f o r e

n e c e s s a r y .

7 . Moving Bed A d s o r p t i o n

Data from s m a l l f i x e d - b e d a d s o r p t i o n e x p e r i m e n t s may be e x

p e c t e d t o p r e d i c t p e r fo r m a n c e o f m o v in g - b e d s y s t e m s , a s w i l l be e x p l a i n e d

i n t h e f o l l o w i n g b r i e f d e s c r i p t i o n . In f i x e d - b e d a d s o r b e r s t h e s t e a d y -

s t a t e wave i s c o n s i d e r e d a s m oving th r o u g h t h e bed a t a f i x e d v e l o c i t y ,

t h e l e n g t h and sh ap e o f t h e wave r e m a i n i n g f i x e d . I f however t h e bed

i s made t o move a t a c e r t a i n v e l o c i t y , t h e n t h e a d s o r p t i o n wave may be

s e e n t o rem ain f i x e d r e l a t i v e t o th e co lum n. I f t h e co lumn o f a d s o r b e n t

i s j u s t t h e same l e n g t h a s t h e w a v e , th e n t h e c o n d i t i o n s a t b o th e n d s

o f t h e co lumn c o r r e s p o n d t o t h e c o n d i t i o n s a t e a c h end o f t h e w ave .

That i s , i f f e e d i s i n t r o d u c e d a t t h e b o t to m o f a downward m oving b e d ,

t h e a d s o r b e n t em er g es from t h e b o t to m s a t u r a t e d w i t h t h e more a d s o r b a b l e

component; a t t h e up p e r end o f t h e co lum n , c o r r e s p o n d i n g t o t h e wave

f r o n t i n t h e f i x e d - b e d c a s e , t h e f l u i d s t r e a m e m e r g e s from t h e column

f r e e o f t h e more a d s o r b a b l e com p on en t , w h i l e f r e s h a d s o r b e n t i s b e i n g

i n t r o d u c e d . The o p e r a t i n g l i n e f o r such a c a s e w here t h e f i x e d bed

and moving bed p r o c e s s e s c o r r e s p o n d a s j u s t d e s c r i b e d , i s t h e s t r a i g h t

l i n e on th e e q u i l i b r i u m diagram c o n n e c t i n g p o i n t s L and K i n F ig u r e 1.

I n r e a l i t y th e h e i g h t o f column ( e q u i v a l e n t t o th e l e n g t h o f th e s t e a d y -

s t a t e wave) r e q u i r e d f o r 100 p e r c e n t s e p a r a t i o n b e tw een p o i n t s L and K

would be i n f i n i t e , but by a c c e p t i n g a c o n c e n t r a t i o n ran ge s l i g h t l y l e s s

than t h i s , s a y 9 9 . 5 p e r c e n t o f th e t o t a l , a p r a c t i c a l l e n g t h o f th e

column may be d e t e r m i n e d .

I f an o p e r a t i n g l i n e o t h e r th an l i n e L-K i s c h o s e n b e c a u s e

o f p r o c e s s demands, i t i s s t i l l p o s s i b l e t o p r e d i c e th e column h e i g h t

r e q u i r e d f o r a moving b e d . In t h i s c a s e t h e t r a n s f e r u n i t method may

be u t i l i z e d , a l t h o u g h t h e a s s u m p t i o n s r e l a t i v e t o t h e c o n s t a n c y o f th e

mass t r a n s f e r c o e f f i c i e n t must be c o n s i d e r e d . The r a t e o f t r a n s f e r

from a l i q u i d f e e d t o t h e a d s o r b e n t o ver a d i f f e r e n t i a l s e c t i o n i s

g i v e n by Qdx, and i s a l s o e q u i v a l e n t t o th e r a t e o f i n t e r p h a s e t r a n s

f e r a s d e s c r i b e d i n an e a r l i e r s e c t i o n ;

Qdx = Ka (x - x ) Sdz ( 6 - 1 6 )

I n t e g r a t i o n o f t h i s e q u a t i o n be tw een a p p r o p r i a t e l i m i t s g i v e s ;

dx = KaSz = N7 ( 6 - 1 7 )

T h i s e q u a t i o n d e f i n e s the number o f t r a n s f e r u n i t s , N^, based on the

b u l k - p h a s e , l i q u i d - f i l m c o e f f i c i e n t , K. The h e i g h t o f a t r a n s f e r u n i t

b a s ed on t h e same c o e f f i c i e n t i s d e f i n e d by

66

A ssuming t h a t column o p e r a t i o n i s s u c h t h a t i s th e same a s f o r t h e

f i x e d - b e d e x p e r i m e n t t h e n t h e h e i g h t o f bed r e q u i r e d t o f r a c t i o n a t e a

f e e d b e tw e e n c o n c e n t r a t i o n s o f x^ and X2 may be p r e d i c t e d . I f co lumn

f e e d r a t e s a r e known i n a d v a n c e t h e n i t i s a s i m p l e m a t t e r t o run f i x e d -

bed e x p e r i m e n t s under s i m i l a r l i q u i d f l o w r a t e s . O t h e r w is e t h e v a r i a

t i o n o f Ka w i t h f l o w must be p r e d i c t e d by a s u i t a b l e number o f in d e p e n d e n t

t e s t s . The e f f e c t o f c o n c e n t r a t i o n on K i s t o be c o n s i d e r e d , a l t h o u g h

t h e r e i s l i t t l e q u a n t i t a t i v e k now led ge on t h i s s u b j e c t .

V I I . CONCLUSIONS AND RECOMMENDATIONS

As a r e s u l t o f r e s e a r c h on th e s u b j e c t o f l i q u i d - p h a s e ,

b i n a r y a d s o r p t i o n f r a c t i o n a t i o n , t h e f o l l o w i n g c o n c l u s i o n s can be

m ade:

1 . A m a t h e m a t i c a l d e s c r i p t i o n o f a d s o r p t i o n f r a c t i o n a t i o n o f b i n a r y

l i q u i d s i n a column h a s b een d e v e l o p e d w h ic h c o n s i d e r s s i m u l t a n e o u s l y

th e e f f e c t o f t h e v a r i a b l e s , t i m e , bed d e p t h , p a r t i c l e r a d i u s , and

c o n c e n t r a t i o n .

2 . A s u f f i c i e n t l y a c c u r a t e n u m e r i c a l s o l u t i o n has b een found f o r

t h e s y s t e m o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s , d e r i v e d f o r t h e o p e r a

t i o n o f an a d s o r p t i o n co lum n, t o g e t h e r w i t h a g e n e r a l e q u i l i b r i u m

f u n c t i o n and t h e p r o p e r boundary c o n d i t i o n s . O t h e r w i s e a s o l u t i o n

c o u l d p r o b a b ly have n o t b een o b t a i n a b l e .

3 . A s u b s t a n t i a l improvement i n t h e c o r r e l a t i o n b e tw e e n e x p e r i m e n t a l

d a t a and t h e o r y h as b e e n d e m o n s t r a t e d f o r t h r e e o u t o f f o u r s y s t e m s

i n v e s t i g a t e d , p r o v i n g t h a t i n t r a p a r t i c l e d i f f u s i o n s t r o n g l y i n f l u e n c e s

t h e d yn am ics o f a d s o r p t i o n , a t l e a s t i n t h o s e s y s t e m s s t u d i e d .

4 . The d e c i d e d e f f e c t o f e q u i l i b r i u m on t h e sh ap e o f t h e a d s o r p t i o n

has b een i l l u s t r a t e d , showing t h a t i t i s im p o r t a n t t o u s e t h e p r o p e r

e q u i l i b r i u m r e l a t i o n i n s o l v i n g t h e m a t h e m a t i c a l m o d e l .

5 . A computer program i n t h e FORTRAN - 4 l a n g u a g e has b een p r e p a r e d ,

s o t h a t t h e p rob lem may be r e - s o l v e d on a l a r g e number o f c o m p u ters

w i t h o u t e x c e s s i v e re -p rogram m in g e f f o r t .

67

6 8

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

f i x e d - b e d a d s o r b e r , w here t h e f o r m a t i o n t im e o f t h e u l t i m a t e w a v e , bed

d e p t h r e q u i r e d f o r t h e u l t i m a t e wave t o form, d e g r e e o f s a t u r a t i o n , and

l e n g t h o f t h e s t e a d y - s t a t e w a v e , h a s b een made. E q u a t i o n s r e l a t i n g

f i x e d - b e d d a t a t o m o v in g -b e d a d s o r b e r s were d e v e l o p e d .

The f o l l o w i n g p r o c e d u r e s o r a r e a s o f i n v e s t i g a t i o n a r e rec o m

mended f o r f u t u r e work:

1 . The i n c l u s i o n o f l o n g i t u d i n a l d i f f u s i o n a s a r a t e - d e t e r m i n i n g

f a c t o r may be n e c e s s a r y f o r c e r t a i n s y s t e m s , a s p o s s i b l y e v i d e n c e d by

t h e m e t h y l c y c l o h e x a n e - t o l u e n e - a l u m i n a s y s t e m . The low p o r e vo lum e o f

t h i s s y s t e m p o s s i b l y r e s u l t s i n a g r e a t e r e f f e c t o f l o n g i t u d i n a l

d i f f u s i o n .

2 . A more s i m p l i f i e d n u m e r i c a l o r a n a l y t i c a l a p p r o a c h y i e l d i n g t h e

same s o l u t i o n i s n e e d e d , s o t h a t a d d i t i o n a l mechanisms may be i n c l u d e d .

3 . A s i m i l a r i n v e s t i g a t i o n i n t h e c a s e o f more th an two components

would make t h i s p r o c e d u r e more u s e f u l i n p e t r o l e u m p r o c e s s e s u s i n g

a d s o r p t i o n .

4 . N o n - i s o t h e r m a l o p e r a t i o n , in w h ich t h e h e a t o f a d s o r p t i o n i s

s i g n i f i c a n t , i s o f p r im ary c o n c e r n i n g a s - p h a s e s y s t e m s , but may

h a v e t o be c o n s i d e r e d i n some l i q u i d - p h a s e a d s o r p t i o n p r o c e s s e s .

3 . S i t u a t i o n s i n v o l v i n g c h e m i c a l r e a c t i o n s h o u l d be s t u d i e d , i f

t h e r e i s a ch a n g e in t h e p r o p e r t i e s o f t h e s y s t e m or i n t h e number

o f com ponents i n v o l v e d .

6 . Of p a r t i c u l a r i n t e r e s t i n co lum n o p e r a t i o n i s t h e e f f e c t o f a sudden

ch a n g e i n any o p e r a t i n g c o n d i t i o n . W ith o u t making e x t e n s i v e c h a n g e s

t h e p r e s e n t p r o c e d u r e s c o u l d p o s s i b l y be r e d e v e l o p e d to show t h e e f f e c t

o f a change i n f e e d c o m p o s i t i o n or r a t e upon column o p e r a t i o n .

7 . V a r i a t i o n o f t h e f i l m c o e f f i c i e n t and i n t e r n a l d i f f u s i o n c o n s t a n t

w i t h c o n c e n t r a t i o n s h o u ld be s t u d i e d .

8 . A more a d e q u a t e d e s c r i p t i o n o f f l o w p a t t e r n s i n t h e v o i d s p a c e

b etw een p a r t i c l e s i s n e e d e d .

9 . The o p e r a t i o n o f u n i t p r o c e s s e s , i n c l u d i n g b o t h f i x e d and moving

bed a d s o r b e r s , under computer c o n t r o l i s a r e l a t i v e l y new f i e l d w hich

s h o u ld be o f i n t e r e s t i n f u t u r e r e s e a r c h p r o j e c t s . E x p e r i m e n ta t i o n

i n t h i s a r e a i s r e l a t i v e l y complex b e c a u s e o f prob lem s i n s e n s i n g

and t r a n s m i t t i n g s i g n a l s t o t h e com pu ter , and th e n back t o th e

p r o c e s s t o a c h i e v e c l o s e d lo o p c o n t r o l .

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APPENDIX A

NUMERICAL METHODS

TABLE I

ACCURACY CHECK ON INCREMENT SIZE (At A ■ 10 )

AT 8 - D i g i t P r e c i

.1 0 . 4 9 0 9 4

.25 .49052

.5 0 .490551 . 0 0 .49016

.1 0 . 4 9 0 4 4

.25 . 4 9 0 7 9

.5 0 .490551 . 0 0 .49015

.1 0 . 4 9044

.25 . 4 9079

.50 .490551 . 0 0 .49015

Bulk C o n c e n t r a t i o n At H - . 1 , T * 10

lion 1 6 - D i g i t P r e c i s i o n

. 4 9 0 9 4

.4 9084

.4 9084

. 4 9 0 8 4

76

(A- 1)

DERIVATION OF IMPLICIT FINITE DIFFERENCE EQUATIONS FOR THE INTRAPARTICLE DIFFUSION EQUATION

[ d h . + 1 = f d x \LaR 2 r s r J \ S T /

T R

F i n i t e D i f f e r e n c e Formulas ( R e f e r e n c e 1 1 ) :

& L - .+ y i + 1 + 0 (A x ) 2 (A-2)SR2 <£tft) 2

» y i + 1 ~ V i ~ 1 + 0 ( ^ x ) 2 (A-3)JR 2AR

a * _ y j - y±BT AT

+ 0 ( ^ t ) (A-4)

o 2U sin g A = the d i f f u s i o n e q u a t i o n b ecom es , upon s u b s t i t u t i o n o f

R iAR

(A-5)

f i n i t e d i f f e r e n c e f o r m u l a s , n e g l e c t i n g e r r o r t e r m s ;

yj-1 - 2yj + yj+i + 2 (yj+i - yi - i ) = yj - yj & R ) 2 i ( ^ R ) 2 2 AT

Let m = — and s o l v e f o r y , :( ^ R ) 2

- m (1 - j ) y i _ 1 + (2"> + l ) y i - m (1 + j ) y i + 1 = y^ (A-6 )

T h i s e q u a t i o n e x p r e s s e s th e c o n c e n t r a t i o n , , i n terms o f t h r e e

n e i g h b o r i n g p o i n t s , as shown i n th e s k e t c h b e lo w :

, Vi- 1 y i y i +1i • — ■ « — ■■ %

y i ' I *- R

At th e c e n t e r p o i n t o f a s p he re (a t R =* 0 , i = 0) th e above

f i n i t e d i f f e r e n c e e q u a t i o n becomes u n d e f i n e d , and a m o d i f i e d formula

must be d e r i v e d . T h i s i s done by a p p l y i n g La H o p i t a l ' s r u l e a t the

c e n t e r o f t h e s p h e r e :

Thus t h e d i f f u s i o n e q u a t i o n becomes

3 . & - EXl x = dj3r2 dT (at R - 0) (A-8)

The c o r r e s p o n d i n g s u b s t i t u t i o n o f f i n i t e d i f f e r e n c e f o r m u la s g i v e s

t h e f o l l o w i n g :

3 - y l - 2 y o + y l = y o - y o ( A - 9)C^R)Z AT

S o l v i n g f o r y^:

( 1 + 6m) y 0 - 6my i = y^ ( A - 10)

There i s one a d d i t i o n a l form ula n e e d e d , t h e f i n i t e d i f f e r e n c e

fo rm u la s t a t i n g c o n d i t i o n s a t t h e p o re s u r f a c e (a t R = A, i « n) .

T h i s form u la w i l l p r o v i d e t h e l i n k a g e t o t h e b u l k - p h a s e e q u a t i o n by

u s e o f t h e c o n c e n t r a t i o n g r a d i e n t a t th e pore o p e n i n g . A T a y l o r ' s s e r i e s

e x p a n s i o n a b o u t t h e p o i n t y n , u s i n g t h e f i r s t t h r e e term s t o e s t i m a t e

yn - l , g i v e s

V . i = v _ - (*l\ AR + (£x\ . £* n - l - ( | W f e ) ' ^

' 'R=A / R=;

S o l v e f o r t h e term , d_ X~dR

A

2 ,

- & n - l - (A_12>R=A v R=A

S u b s t i t u t i o n o f t h e ab ove e x p r e s s i o n i n t o t h e d i f f u s i o n e q u a t i o n g i v e s

(yn _ i - y n ) - 2. - + • — + I l & \ - f e A ( A - 13)Wn 1 yn ; ^ A ^ r ; r _a ^ t ; a

At th e p ore o p e n in g t h e g r a d i e n t , ( 5 y / A R ) , i s a l s o g i v e n by t h e term,

(x - x * ) . S u b s t i t u t i o n o f t h i s term and th e f i n i t e d i f f e r e n c e form ula

78

f o r f S i j r e s u l t s i n

-FI:.1 + (x - x * ) i _ + 1 = ^ - _ y n ( A - 14)^ R ) Z ; AR A AT

1S o l v i n g fo r yn .

y„ * "2m yn - 1 + (1 + 2m)yn - 2m ( x - x * ) ( l + ( A - 15)

An e x a m i n a t i o n o f e q u a t i o n s ( 1 ) , ( 2 ) , and (3 ) shows t h a t

t h e r e can be w r i t t e n n + 1 e q u a t i o n s , and t h e r e a r e n+1 v a l u e s o f y t o

be d e t e r m i n e d . A ssuming t h a t t h e n + 1 v a l u e s o f y ' c o r r e s p o n d i n g t o

t h e p r e c e d i n g t im e l e v e l a r e known, a s w e l l a s an e s t i m a t e o f (x - x ) ,

t h e r e a r e th u s n + 1 e q u a t i o n s w h i c h may be s o l v e d s i m u l t a n e o u s l y fo r

a c o n c e n t r a t i o n d i s t r i b u t i o n a t t h e n e x t h i g h e r t im e s t e p .

The m a t r i x c o r r e s p o n d i n g t o t h e s y s t e m o f e q u a t i o n s as

d e s c r i b e d above may be s e e n to be t r i - d i a g o n a l , hen ce s o l v a b l e by t h e

Thomas m eth o d . The g e n e r a l l a y o u t o f tViis s y s t e m a p p e a r s a s f o l l o w s :

(1 + 6m) y Q - 6myi =

Ia y o + b y j + Cy 2 = y x

a y i + by 2 + c y 3 = y2

■2myn _ 1 + byn = y^ + w ( A - 16)

w here a = -m (1 - —)i

b = (2m +1)

c = -m (1 + —) i

w = 2m (x - x * ) (1 +A

79

A computer s u b r o u t i n e , named TD, was w r i t t e n t o p r o v i d e i n s t r u c t i o n s

f o r s o l v i n g t h e a b o v e s y s t e m o f e q u a t i o n s by e l i m i n a t i o n d u r i n g e a c h

i t e r a t i o n .

APPENDIX B

RESULTS

TABLE I I

SUMMARY OF ADSORPTION SYSTEMS INVESTIGATED

System and R e fe r e n c e

CurveId en t

(F ig s

Feed Column Bed L iquid 71 ad Pore F r a c t i o n M. T. D imension- KangF r a c t i o n , Diam. Wt. Rate D e n s i t y V ol . Voids Coeff.* l e s s P ara - (xlO*) Xf K(xlO^) m e t e r , A

(Best f o r__________ _______ ___ ____________________ _______ ______________________System) ________

4 - 3 2 ) (cm) (gm) ( c c / s e c ) ( g / c c ) ( c c / g ) (c ia / sec )

1. MCH-Toluene A 0 .5 2 . 4 7 195 0 .0 7 8 6 0 .6 7 9 0 .4 0 2 0 .2 9 3 2 .4 2on

6 -1 2 Mesh B 0 . 5 2 . 4 7 95 0 .0 7 8 6 0 . 6 7 9 0 .4 0 2 0 .2 9 3 2 . 4 3S i l i c a Gel

(30) C 0 . 5 2 . 4 7 191 0 . 0 5 0 0 . 6 7 9 0 .402 0 .2 9 3 2 . 6 1

D 0 . 5 2 . 4 7 95 0 . 0 5 0 0 . 6 7 9 0 .4 0 2 0 .2 93 2 .70

2 . MCH-Toluene on E 0 .1 2 . 4 7 96 0 .1 7 4 0 .6 7 9 0 .4 0 2 0 .2 9 3 1 .076-12 MeshS i l i c a G e l (30) F 0 . 1 2 . 4 7 45 0 . 1 7 4 0 .6 7 9 0 .4 0 2 0 . 2 9 3 1 .17

3 . MCH-Toluene on G 0 .5 2 . 4 7 268 0 .0 6 2 5 0 .8 8 3 0 . 1 8 9 0 .4 2 5 0 .8 1 68 - 1 4 MeshAlumina (30) H 0 .5 2 . 4 7 130 0 .0 6 2 5 0 .8 8 3 0 .1 8 9 0 .4 2 5 0 . 7 1 1

I 0 .5 2 . 4 7 63 0 .0 6 2 5 0 .8 8 3 0 . 1 8 9 0 .4 2 5 0 .6 9 7

4 . Benzen e-n - J 0 .5 1 . 9 20 0 .00406 0 .7 1 2 0 .3 5 7 0 . 5 2 8 0 .0442Hexane on

Thru-200-Mesh S i l i c a Gel

(43)

50

30

20

( c m /s e c )

2 . 5 4

1.12

0 .7 4 1

0 .0 4 4 2

oo

TABLE I I I

WAVE FORMATION TIMES, BED DEPTH, AND SATURATION

ICurve f a s â s

Constant wave

System Ident z s » cm ©s , s e c E q (6 -6 ) E q (6 -9 ) 6 „ , s e c

1. MCH-Toluene- A,B 8 4 . 2 2900 0 . 2 1 1 0 . 2 6 4830 0 .3 7 2S i l i c a Gel

(x = . 5 ) C,D 5 3 . 5 2900 0 .2 1 1 0 . 2 6 4830 0 .372

2 . MCH-Toluene- E , F 95 .5 1489 0 .1 8 3 0 . 2 0 3870 0 . 3 6 9S i l i c a Gel

(x = . 1 )

3 . MCH-Toluene- G,H,I 1 3 6 . 0 6170 0 . 3 0 1 0 . 5 4 5930 0 . 4 3 4Alumina

(x = .5 )

4 . B enzene-n-H exane- J 2 7 . 5 2645 0 . 1 5 9 0 . 2 4 4020 0 . 3 5 0S i l i c a Gel

(x = .5 )

d ata

z^ , cm

1 4 0 .5

8 9 . 4

248

1 3 1 .0

4 1 . 6

00Ni

8 3

3 n JL

2 . 3V.O

0 =. 0B o . u V sC 0 . 1 ’ 1X)

6 - H

8

—■ _______•

/•••

<7 I

0

0°

-•

)

M • c o

° o£ *

O n -----/*.--------- a -------ft

• • • • •

3 = H 1 1i

<?]

0°

/ cP J

----- D"6 "o o o c b T o o o ° cfi

u~

V

ct f

o

00

r-

O

C?

C3

5

1 . 2 5

2 . 2 5

3 . 2 5

4 . 2 5

5 . 2 5

6 . 2 5

8 . 2 5

&

in

3" “

X ru —

Data KCurve (xlC)3)

c m /s ec

A 1 . 3 0B 1 .2 6D 1 .62

o250.00.00

FIGURE 7. fiETHYLCYCLOHEXANE-TOLUENE ON SILICA GEL, A = 2000O'

Oata K 3urve (xlO^)

500 .0

FIGURE 8. METHYLCYCLOHEXANE-TOLUENE ON SILICA GEL A = 4000-vj

6 f t

Oo

Data KCurve (xlCp)

cm /s ec

A 2 . 7 5B 3 . 0 4C 2 . 7 0D 3 . 2 2

150(10 2000011500

FIGURE 10. METHYLCYCLOHEXANE-TOLUENE ON SILICA GEL A = 60

Data KCurve (xlO3)

cm /sec

A 3 . 3 6B 3 .9 5D 4 . 0 1

1 1 5 1 0IOODLO 1500.0

FIGURE 11. METHYLCYCLOHEXANE-T0LUENE ON SILICA GEL, A - 80

91

a .o01

Ct

o

9a

.5

H = 0 .3 7 5

CD

o

r °o

CO

1 0 . 5

o

^ . s w

C6

- nC l

1 1 . 5

1 3 . 5

1 5 . 5

rw1 7 . 5

1 9 . 5

r o

co

^ 6

95

e»

C3

C3

ru

9 fe

91

u»

Data KCurve (xlO 3 ) H_

cm /sec E 0 . 9 9 0 . 9 4F 1 . 0 6 0 . 4 7oo

o -

to

o -

ruq -

oa12.5 T5JD

T

FIGURE 19. METHYLCYCLOHEXflNE-TOLUENE ON SILICR GEL R = 10 00

o

250.0 3125

Data KCurve (xlCp)

cm /sec

E 1 .03F 1 .1 7

315.0 H3X5

0 . 9 80 .5 2

SJOiJT

FIGURE 20. METHYLCYCLOHEXflNE-TOLUENE ON SILICA GEU fl = 20VOvO

T .

- O~ o - o —

Data KCurve ( x l ( P )

c m /s ec

E 1 .2 9F 1 .32

3 J

1 . 220 . 5 9

62.5

T

FIGURE 22. METHYLCYCLOHEXflNE-TOLUENE ON SILICA GEL, A = 40

101

.5

-n

O

o

o

1 0 .5

*-* 7lO

o

z o i

.0

5

it

au

t.o

ON

X

DataC u 0 £ .

Gft1

0-6 50.5&o . t f

3 .00 1 .2^ 0 - 5 0

\ 5

f iGUOt 2<V-

1 » * * * * * ' ’

^ L c v a o * * " *

\ &

Z t\

rtf u ~

• *

CL

2DJ3

C*l

3*

D ata

H

0 . 8 20 . 8 80 .^ 8

25J03D.0

ft

3 . 7 8 1 . 3 0 0 . 7 8

15JD

n G U ft f - 26-

VO 6

IT* .~ • o

a

0-1

w

\pfc

<3,

01

in

ru —*v

Data K Curve (x lC p ) H

cra/sec

200.0 mo100.0 150.050DT

FIGURE 30. METHVLCVCLOHEXRNE-TOLUENE ON ALUM I NR, fl = 20

109

Curve (xlO^) cm/sec

H

G 2.62 11.5H 2 .70 6 .0I 2.33 2 .5

fipn.O m . o

FIGURE 3U METHVL CYCLOHEXRNE-TOLUENE ON RLUMINfl, fl = 40

no

Data KCurve ( x ! Q 3 )

cm/sec

G 4 .14H 4.32I 3.91

1800.0 4800.0

FIGURE 32, METHYLCYCLOHEXflNE-TOLUENE ON ALUMINA, A = 80

in

* - th c c h a w p i o j ; l \ u r ■ u o C 4 i

c h oi - i ’ b? . T i o ii - Tw. '. o u **h ; 1 u I N C K

T

i

x , C oncentration o f More Adsorbable Component in E fflu en t

: r :■

s

0 .5

(Curves Represent C onditions a t Wave Front)

<h3

4J

6010 ; 2 0 30 . 40 . ,"!' I ; | : ;j " T, D im ensionless Time

Figure 34. E ffe c t o f P a r t ic le S iz e on the System: MCH-Toluene on Aluminaw

" T H T C H A M P I O N K. I " S’O C- iT

C F? O S fi S i C U O r i - * '0 LC.' L K H i _ ’J T u r . r. h

f ' •'

-1 - :


::Ur

S

•*TH C C H A M P I O N l . W T . " N O 6 4 3

c r o s s s t c t t o r ; * to c c v a k t s t o i ‘; c h


fin

m

S

APPENDIX C

COMPUTER PROGRAM

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o SOLUTI ON OF AOSORPTI ON MODEL - MAIN PROGRAM

DEFI NE ERROR FUNCTI ON

E ( A 1 , A 2 1 E R R O R ) = A B S ( ( A I - A 2 ) / A 1 ) - E R R 0 R

DI MENSI ON AND COMMON DECLARATIONS DI MENSI ON X ( 1 0 0 n , X S ! 1 0 0 1 ) , X B < 1 0 0 1 ) , XSB 1 1 0 0 11 COMMON Y 0 L D ( 2 0 1 ) , Y N E W ( 2 0 1 )COMMON I , J , K , M , A , X N R , N R t I S W, P T S W, Y T O L , NR 1 * I T P T , I T T O T , I T M A X , I NSW, KKCOMMON H F R E Q , T F R E Q , R F R E Q , Y S , D H , D TCOMMON K D I F , I P C H X , I P C H Y , 1 0 , JO, JMAXCOMMON D E L X , DE L XO, XF FCOMMON EW,MAXOUTLOGICAL I SW, PTSWREAL MINTEGER HF REQ, TFREQ. RFREQ

I NPUTS

1 REWIND1 REWIND3 PAUSE I

2 R E A D 3 , X F , Q Q , D T , A , X N R , P 1 , P 2 , P 3 , P 4 , P 5 , P 6 RE AD 3 , YT O L , E X , EY

3 F 0 R M A T ( 5 E 1 0 . 0 , 6 A 5 )R E A D 4 , J M A X , I T M A X , H F R E 0 , T F R E 0 , R F R E Q , K D I F , I P C H X , I P C H Y , M A X O U T

4 FORMAT ( 1 6 1 5 )

PRELI MI NARY CALCULATI ONS

M = D T * ( X N R / A ) * * 2NR«XNRNR I = NR + 1

117

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DH=DT/ AXFF=XF1=1

WRITE DATA IN HEADING

1 4 W R I T E ! L , 5 ) P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , X F , D H , D T , A , X N R , M W R I T E ( L , 6 ) Y T 0 L , E X , E Y WRI TE( L » 7 ) JMAXt I TMAX WR I T E ( L , 1 0 0 0 ) 0 0

1 0 0 0 F ORMAT ! I X , 1 4 HD AMP I NG FACTOR, F 4 6 . 5 )5 FORMAT( 1 H 1 , 7 H P R 0 6 1 E M , 1 X , 6 A 5 / / 1 X , 2 6 H I N L E T FEED COMPOS I TI ON, X F ,

1 2 3 X , F 1 1 . 5 / I X , 2 5 H I N C R E M E N T A L BED LEVEL, D H , 2 4 X , F 1 1 . 5 /2 IX,2 OHINCREMENTAL TI ME, D T , 3 0 X . F 1 0 . 5 /3 1 X . 2 3 H D I M E N S I 0 N L E S S R A D I U S , A , 2 6 X , F 1 1 . 5 /4 I X , 34HNUMBER OF I NTERVALS OF R A D I U S , XNR, 1 5 X , FI 1 . 5 /5 I X , 2 5 H R A T 1 0 OF DT TO ! A / X N R > * * 2 , F 3 5 . 5 )

6 F O R M A T ! 1 X 1 0 H T 0 L E R A N C E S / 6 X , 2 IHWAVE FRONT T E S T , YTOL, 2 3 X , 1 P E 1 5 . 5 /7 6 X , 3 8 HERR0 R IN CONCENTRATION - BULK P HAS E , E X , 6 X , 1 PE 1 5 . 5 /8 2 9 X 1 9 H - A D S O R B E D PHASE, E Y , 1 P E 1 7 . 5 )

7 FORMAT! 1X26HNUMBER OF TIME S T E P S , J M A X , 2 8 X , I 6 /9 I X, A3HMAXI MUM ALLOWED I TERATI ONS PER P O I N T , I T M A X , 1 1 7 )

I F D I F F U S I O N CURVES ARE TO BE OUTPUT, WRITE HEADING ON LOGICAL UNI T 3 .

L * 3I F ( K D I F . N E . 1 ) GO TO 1 7 K D I F = - 1 GO TO 1 4

GO TO FUNCTI ON EQ TO READ AND WRITE E QUI L I B RI UM DATA. SET INSW = 1 TO ENTER EQ I N I T I A L L Y .

17 L = 7

118

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1 F C I P C H X . N E . n GO TO 1 1 7 IPCHX = - 1 GO TO K

1 1 7 INSW= 1I P C H X = I A B S t I P C H X )D UMMY= EQ( . 5 )WR I T E < 1 , 8 )

8 F O R M A T ( / / / / 1 3 X , 6 H I T E R A - 2 9 X , 1 4 H C 0 N C E N T R A T I 0 N S / 4 X 1 H J 4 X 1 H I 3 X1 5 H T I 0 N S , 4 X 4 H T I M E , 5 X 5 H D E P T H 7 X 2 H X , 1 X 4 H B U L K 6 X 2 H Y , 1 X ,2 8 H A D S 0 R B E 0 / / )

I F ( K D I F . N E . O ) WRITE ( 3 , 9 19 F O R M A T ( / / / 1 X , 1 6 H D I F F U S I 0 N CURVES / / )

I T P T = 0

SET EXCES S I VE I TERATI ON SWITCH

1 8 KK=1

SET FREQUENCY OF OUTPUT PARAMETERS

10=1J 0 = 1

SET TOTAL I TERATI ON COUNTER

I TT QT = 0

SET ULTIMATE WAVE SWITCHES

KULT=1 KULTX=0 K QU ] T = JMAX

SET D I F F U S I O N CALCULATION SWI TCHES 119

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I S W = . F A L S E .P T S W = . F A L S E .

I N I T I A L I Z E TIME AND BED DEPTH I NDEXES

1 * 1J = 1

GRI D POI NT 1 , 1 ( APEX OF TRI ANGLE)

DEFI NE YOLD( K) AND I N I T I A L ESTI MATE OF Y AT SURFACE

D 0 1 0 K = 1 , NR1 YNEW( K) * XF

1 0 Y O L D ( K ) = X F D 0 1 1 K = 1 • JMAX

11 X ( K ) = X FCALL OUT( X F , XF )J = 2

GRI D POI NT J , 1 ( I NCREAS E TIME S T EP )

2 0 Y E S T = Y 0 L D ( N R 1 )3 0 X S ( J ) = E Q ( Y E S T )

D E L A = X S ( J ) - X F DELX=- DELA D E L X O = X F - X S ( J - l )CALL TDCALL C K ( 1 , D E L A , X S ( J ) , Y E S T )I F ( K K . N E . l ) GO TO 6 0

4 0 I F ( E < Y N E W I N R 1 ) , Y E S T , E Y ) ) 6 0 , 5 0 , 5 0 5 0 Y E S T = Y E S T + Q Q * ( Y N E W ( N R 1 ) - Y E S T )

GO T 0 3 0 6 0 YY=YNEW( N R 1 )

CALL OUT( XF , Y Y )

120

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SAVE VALUES OF X AND Y FOR I* EXT POI NT

D 0 7 0 K = 1 , NR1 7 0 YOLD( K ) = YNEW( K J7 2 J = J + l

I F C J . G T . J M A X ) GO TO 7 8 1 F I K K . N E . 3 ) GO TO 2 0 X S < J ) = X S ( J - l )GO TO 7 2

7 8 1 = 2

GRI D POI NT 2 , 2 ( BED LEVEL = DH)

GUESS CONCENTRATION ALONG BOUNDARY A*H=T

1=2 J = ID E L A = X S < J I - X C J )X B ( J ) = X ( J ) + D E L A * D H * 3 .

6 0 X S B ( J ) = E Q ( X B ( J ) )D E L B = X S B ( J ) - X B ( J )X N E W= X ( J ) ♦ DH*( DELA + D E L B ) * 1 - 5CALL C K ( 2 , D E L B , X B ( J ) , X S B ( J ) )I F ( K K , N E . l ) GO TO H O

9 0 I F ( E ( X N E W , X B ( J ) , E X ) ) 1 1 0 , 1 1 0 , 1 0 0 1 0 0 X B ( J ) =XNEW

GOT0 8 01 1 0 CALL O U T ( X B ( J ) , X B ( J ) )

KK= 1

SET YOLD( K) FOR NEXT TIME STEP

D 0 1 2 0 K = 1 , NR1 1 2 0 Y O L D ( K ) = X B ( J )

121

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GRI D POI NT J , 2 ( BED LEVEL = DH)

1 3 0 J = J + LI F ( J * G T . J M A X ) G O T 0 2 3 0 I F I K K . N E . 3 ) GO TO U O X B ( J ) = X B ( J - l )X S B I J ) = X S B ( J - l )GO TO 1 3 0

GUESS Y AT SURFACE OF SPHERE - NEXT TIME STEP

1 AO Y E S T = Y 0 L D I N R 1 )

GUESS X AT THE SAME POI NT

D E L A = X S ( J ) - X ( J )X B ( J ) = X ( J ) + D E L A * D H * 3 .

1 5 0 X S B ( J ) = E Q ( Y E S T )1 6 0 D E L B = X S B ( J ) - X B CJ)

X N E W= X ( J ) + DHMDE L A + D E L B > * 1 . 5CALL C M 3 , D E L B , X B ( J ) , Y E S T )I F ( K K . N E . 1 ) GO TO 1 8 0

1 7 0 I F ( E { X N E W , X B ( J ) , E X ) ) 1 8 0 , 1 7 5 , 1 7 5 1 7 5 X B ( J ) = XNEW

G 0 T 0 1 6 0 1 8 0 DE L X = X N E W- X S B ( J )

D E L X O = X B ( J - l ) - X S B ( J - l )CALL TOCALL C M A , D E L B , X B I J ) , Y E S T )G O T O ! 1 9 0 , 2 1 0 ) , KK

1 9 0 I F ( E ( Y E S T , Y N E W ( N R 1 ) , E Y ) ) 2 1 0 , 2 1 0 , 2 0 0 2 0 0 Y E S T = Y E S T + Q Q * ( Y N E W ( N R l ) - Y E S T )

G 0 T 0 1 5 0 2 1 0 YY=YNEW( N R l )

122

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CALL OUT( XNEWt YY)

220

2 3 0

2 3 2

2 3 4

2 4 0

SAVE X AND Y FOR NEXT TIME STEP

D 0 2 2 0 K = 1 , NR 1 Y 0 L D ( K ) =YNEW( K )GO TO 1 3 0

GRI D POI NT 1 , 1 (ALONG BOUNDARY A * H = T )

1 = 1 + 1I F ( I . G T . J M A X J G O T 0 3 9 0

TEST IF ULTIMATE WAVE WAS FOUND

I F ( I . G T . K Q U I T ) GO TO 3 9 0 I F I K U L T . E Q . O ) GO TO 4 0 0

TEST SS 4 FOR JOB INTERRUPT

CALL S S W T C H ( 4 , I I )I F ! I I . N E . 1) GO TO 2 4 0REWIND 1REWIND 3CALL ICKPTPAUSE2J = IKULT=0D E L A = X S ( J ) - X ( J 1 D E L B = X S B ( J ) - X B ( J )

ESTI MATE X AT THI S POI NT BY A SECOND DEGREE F I T , THEN THI RD DEGREE

X N E W= X B ( J ) + D H * ( DELB+ DELA1 * 1 . 5 XSNEW=EQ( XNEW)

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2 5 0 DELC=XSNEW-XNEWX N E W 2 = X B ( J ) + D H * ( 5 . * D E L C + 8 . * D E L B - D E L A ) * . 2 5XSNEW=EQ( XNEW2)CALL CK{ 5 , XN E W, X NE W2 , X S NE W)I F ( K K . N E . l ) GO TO 2 8 0

2 6 0 I F < E ( X N E W , X N E W 2 , E X ) ) 2 8 0 , 2 7 0 , 2 7 0 2 7 0 XNEW=XNEW2

G 0 T 0 2 5 0

R E - D E F I N E Y ALONG BOUNDARY

2 8 0 D 0 2 8 2 K = 1 , NR12 8 2 Y0 L D( K) = XN E W2

CALL OUT( XNEW2»XNEW2)KK= 1

R E - D E F I N E X ( J ) , X S ( J ) , X B ( J ) AND X S B ( J ) F 0 R USE AT NEXT BED LEVEL

2 9 2 X ( J ) =X B ( J )X S ( J > = XS B ( J )X B ( J ) = X N E W2 X S B ( J ) =XSNEW

2 9 5 J = J + 1I F ( J . G T * JMAX J G OT 0 2 3 0

I F AT UPPER END OF WAVE, S K I P I NTEGRATI ON ANDDEFI NE ALL X TERMS EQUAL TO LAST VALUE IN WAVE

1 F I K K . E Q . 3 ) GO TO 2 9 2

GRI D POI NT J , I ( T Y P I C A L P OI NT )

GUESS Y AT POI NT ( J , I )

YEST = YOLD( NR 1)

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ESTI MATE X AT POI NT t J , I )

XNEW = ( ( X S B ( J ) - X B ( J ) ) * 4 . 5 - ( X S ( J ) - X ( J ) ) * 1 . 5 ) * D H + X B ( J ) 3 0 0 XS NE W= E Q( YE S T )

DELA = XS C J ) - X f J >D E L B = X S B I J ) - X B ( J )

3 1 0 0ELC=XSNEW- XNEW3 2 0 XNEW2=XB( J ) + D H * ( 5 . » D E L C + 8 . * D E L B - D E L A ) * . 2 5

CALL C K ( 6 , X N E W , X N E W 2 , Y E S T )

I F AT LOWER END OF WAVE S K I P I NTEGRATI ON AND I NCREASE TIME UNTI L WAVE I S ENTERED.

I F ( K K . N E . l ) GO TO 3 8 0 3 3 0 I F ( E ( X N E W , X N E W 2 , E X ) ) 3 5 0 , 3 4 0 , 3 4 0 3 4 0 XNEW=XNEW2

G 0 T 0 3 I 0 3 5 0 DELX=XNEW2- XSNEW

D £ L X 0 = X B ( J - I ) - X S B ( J - l )CALL TOCALL C K ( 7 , X N E W , X N E W 2 , Y E S T )I F ( K K . N E . l ) GO TO 3 8 0

3 6 0 I F ( E ( Y E S T , Y N E W ( N R l ) , E Y ) ) 3 8 0 , 3 8 0 , 3 7 0 3 7 0 YEST = Y E S T + Q Q M Y N E W ( N R 1 ) - Y E S T )

G 0 T 0 3 0 03 8 0 CALL O U T ( X N E W2 , Y N E Wt N R I ) I

R E - D E F I N E Y FOR NEXT TIME STEP

3 8 1 DO 3 8 2 K * 1» NR 13 8 2 YOLD( K) = YNE W( K)

CHECK FOR ULTIMATE WAVE 125

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3 8 4 I F l A B S t ( X ( J ) - X B ( J ) ) - ( X B ( J ) - X N E W 2 ) ) . G T . Y T O L ) KUL T =1

WAIT UNTI L XNCW2 LESS THAN YTOL BEFORE PERMI TTI NG ULTIMATE WAVE TEST TO BE EFFECTI VE

I F ( X N E W 2 . G E . Y T O L ) KULT=1 GO TO 2 9 2

FI NAL COMPUTATIONS

3 9 0 WRITE ( 1 , 3 9 1 ) ITTOT3 9 1 F O R M A T ( / / 1 X 1 6 H T O T A L I T E R A T I O N S , I 1 0 )

G 0 T 0 1

ULTIMATE WAVE ROUTINETEST I F ULTIMATE WAVE ALREADY FOUND

4 0 0 I F ( K U L T X . E Q . 1 ) GO TO 2 3 2 H = D H * F L 0 A T ( 1 - 2 )I F t I P C H X . E O . 1 ) WRI T E( 7 , 4 0 1 ) H

4 0 1 FORMAT( 1 X, 2 0 HUL T I MAT E WAVE AT H = F 1 0 . 4 )W R I T E ! 1 , 4 0 1 ) HKULTX=K Q U I T = I 0 + H F R E Q * 2GO TO 2 3 2END

S U B R O U T I N E TDCC SOLUTI ON OF D I F F U S I O N EQUATION BY THOMAS METHODCC COMMON AND DI MENSI ON DECLARATI ONSC

DI MENSI ON T ( I 0 I ) , S ( 1 0 1 )COMMON Y 0 L D I 2 0 1 )* YNEWI 2 0 1)COMMON I , J , K , M , A , X N R , N R , I SW , PTSW, YTOL , NR I , I TP T , I TTOT, I TMAX , I N S>,, KKCOMMON H F R E Q , T F R E Q , R F R E Q , Y S , DH, DTCOMMON K O I F , I P C H V , I P C H Y , 1 0 , JO, JMAXCOMMON DE L X, DE L XO, XF FCOMMON EW, MAXOUTINTEGER HFREQ, TFREQ, RFREQLOGICAL I SW, PTSWREAL M

CC REDUCTIONC

I F ( I SW) G 0 T 0 2 I S W * . T R U E .B = 2 . * M + 1 .W = 2 . * M * A / X N R * ( l . + i . / X N R )XMSQ=M*M S I 1 ) = 6 . * M + 1 .S 1 2 ) = B D 0 1 K = 3 , NR

1 S I K ) = B - X M S Q / S ( K - I )S I N R l ) = B - 2 . * X M S Q » X N R / ( I X N R - 1 . ) * S I N R ) )

2 YE ND= Y0 LD( NR1 ) + DE LX* W I F I P T S W I G 0 T 0 4 P T S W= . T R U E .T i l ) = * Y OL D I 1 ) / S ( 1 )T 1 2 ) * Y O L D ( 2 ) / SI 2 )0 0 3 K = 3 , N R K

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Z = K - 13 T ( K ) = ( Y 0 L D ( K ) + ( M - M / Z ) * T { K - 1 ) > / S ( K )4 T ( N R 1 ) = I Y E N D + 2 . * M * T ( N R ) J / S I N R 1 )

BACK SOLUTI ON

YN E WI N R+ 1 J = T ( NR + 1 )K=NR

5 Z = K - 1Y N E W( K ) = T ( K ) + ( M+M/ Z) * YNEW( K + l ) / S ( K > K = K—1I F I K . G T . 1 J G0 T 0 5YNEWI 1 ) = T ( U + 6 . * M * Y N E W ( 2 ) / S ( 1 )RETURNEND

128

SUBROUTI NE OUT ( XX, Y Y )

I F I N S I D E WAVE OUTPUT SUBROUTI NE WRITES RESULTS AT SELECTED INTERVALS

DI MENSI ON AND COMMON DECLARATI ONS

COMMON Y 0 L D I 2 0 1 ) , YNEW( 2 0 1 )COMMON I , J , K , M , A , X N R , N R , I S W , PTSW, YTOL, NR1 , I T P T . ITTOT * I TMAX, I NS^ , KKCOMMON H F R E Q , T F R E Q , R F R E Q , Y S , D H , D TCOMMON K D I F , I P C H X , I P C H Y , I D , J O , J M A XCOMMON D E L X , DE L XO, XF FCOMMON EW, MAXOUTLOGICAL I SW, PTSWREAL MINTEGER HF REQ, TFREQ, RFREQ

KWRIT E = 0I F I I . E Q . l . A N D . J . E Q . l ) GO TO 21

TEST WHETHER Y I S CHANGING WITH TIF NOT, SET KK = 3 TO S KI P REMAINDER OF P OI NT S I N THI S LEVEL

I F ( A B S ( Y N E W ( N R 1 ) - Y 0 L D ( N R 1 ) J . L T . E W ) KK=3

I F AT UPPER END OF WAVE SET KK = 3 TO S KI P REMAINDER OF POI NTS I N THI S BED LEVEL.

I F { A B S ( Y S - Y Y J . L E . Y T O L ) KK=3 I F ( I . N E . I O ) GO TO 12 I F < J . L T . J M A X ) 1 F I K K - 3 ) 7 , 2 0 , 7

2 0 I 0 = IO+HFREQ J O = I O GO TO 3

7 I F ( J . N E . J O ) GO TO 12

129

2 1 JO = JO+TFREQ3 H = D H * F L 0 A T ( I — 1 )

T = D T * F L O A T ( J - l )I F t l A B S l I P C H X J . E O . l ) W R I T E ! 7 , 1 ) J , I , I TP T , T , H , XX, YY W R I T E ! 1 , 1 ) J , I » I T P T , T , H , X X , Y Y

1 FORMAT( I X , 2 I 5 , I 6 » F 1 1 . 4 , F 1 0 . 4 » 2 F 1 4 . 8 )I F ( K D I F ) 4 , 2 , A

4 I F ( I . N E . J ) GO TO 9D 0 8 K = 1 , N R , R F R E Q

8 YNEWI K) =XX YNEWt NRl J = XX

9 WRI T E! 3 » 6 ) J , I6 FORMAT( / 1 X 5 H P 0 I N T I 4 V 1 H , , I 4 )

I F ( I P C H Y . E Q . l ) W R I T E ! 7 , 6 ) J , II F ! I P C H Y . E O . 1 ) WRITE! 7 , 5 ) ! K f YNE W ( K ) , K= 1 , NR , RF REQ ) , NR 1 , YNE A ! SIR 1) WR I T E ( 3 » 5 ) { K , Y N E W ( K ) , K = l , N R , R F R E Q ) , NR 1 , YNEWI NR 1)

5 F O R M A T ! A ! I X , 1 7 , F 1 2 . 8 ) )2 I T P T = 0

P T S W = . F A L S E .11 RETURN1 2 I F ! I . L T . M A X O U T ) GO TO 3

GO TO 2END

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S U B R O U T I N E C K ( L , D , X X , Y Y )

C THI S SUBROUTI NE CHECKS FOR EXCESS I TERATI ONS AND CONTROLS OUTPUTC OUTPUT OF I NTERMEDI ATE RESULTS

C DI MENSI ON AND COMMON DECLARATI ONS

COMMON Y O L D ( 2 0 1 ) , Y N E W ( 2 0 1 )COMMON I , J , K , M , A , X N R , N R , I S W . P T S W , Y T 0 L . N R 1 , I T P T , I T T O T , I T K A X , I NSW,KKCOMMON H F R E Q , T F R E Q , R F R E Q , Y S , D H , D TCOMMON K D I F , 1 P C H X , I P C H Y , I D , J O , J M A XCOMMON DELX, DELXO, XFFCOMMON EW,MAXOUTINTEGER HFREQ, TFREQ, RFREQLOGICAL I SW, PTSWREAL M

CCALL S SWTCHI 6 , MM)GO TO ( 1 1 , 1 2 ) , MM

11 WRITE 1 ^ , 2 ) J , I M M - I - 1REWIND 2

1 2 KK=1 I TPT = I T P T + 1

WRITE OUTPUT IF S S 2 I S ON

CALL SSWTCHt 2 , MM)GOTO ( 1 , A ) , MMWRI T E ( 1 , 2 ) J , I , I T P T , D , X X , Y Y , L F O R M A T ! I X , 2 1 5 , 1 6 , 7 X , 3 F 1 4 . 8 , 2 X3 HCK= I 2 )CONTINUE

TEST FOR EXCES S I VE I TERATI ONS 131

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I F < I T P T - I T M A X ) 3 , 3 , 5

WRITE LOCATION OF EXCESSI VE I TERATI ONS AND SET KK TO EXI T LOOP

5 WR I T E * 1 . 6 ) J , I , ITPT7 KK = 23 RETURN

GO TO 36 FORMAT( I X , 2 1 5 , 1 6 , 5 0 X , 1 1 H I E X C L OOP S ) )

END

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F U N C T I O N E QMCH A{ Y )

METHYL CYCLOHl XANE - TOLUENE EQUI LI BRI UM ON ALUMINA ( JOHNSON)

DI MENSI ON AND COMMON DECLARATI ONS COMMON Y O L D I 2 0 I ) , YNEW( 2 0 1 )COMMON I , J » K » M, A , X N R , N R » I SW,PTSW,YTOL t N R l , 1 T P T , I T T 0 T , 1 T M A X , I N S W , K k

COMMON H F R E Q , T F R E Q , R F R E Q , Y S , D H , D TCOMMON K D I F , I P C H X , I P C H Y , 1 0 , JO, JMAXCOMMON DE L X , DE L XO, XF FCOMMON EW, MAXOUTLOGICAL I SW, PTSWREAL MINTEGER HFREQ, TFREQ, RFREQ I F ( I NSW) 7 , 7 , 1

1 I NSW=0

GET Y* IN EQUI LI BRI UM WITH FEED CONCENTRATION

YS = 2 . 7 1 8 M X F F / 1 I . - X F F ) ) * » . 7 6 YS = Y S / ( I . + Y S )WRI TE( 1 , 5 ) YS

5 FORMAT( I X » 4 2 H A D S 0 R B E D PHASE CONC IN EQUI LI BRI UM WITH X F , F 1 8 . 5 ) WRI T E ( 1 , 6 )

6 FORMAT( 1 5 X , 2 8 H ( ME T H Y L CYCLOHEXANE- ALUMI NA) )

GET X I N EQUI LI BRI UM WITH Y

7 I T T O T = 1 TTOT +1EQMCHA = ( Y / ( L . - Y ) » . 3 6 8 1 * * 1 . 3 1 6 EQMCHA = EQMCHA/ ( I . +EQMCHA)RETURNEND

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C BENZENE N- HEXANE EQUI LI BRI UMC

FUNCTI ON EQBZHX( Y)

DI MENSI ON AND COMMON DECLARATIONS

COMMON Y 0 L D I 2 0 1 ) . Y N E W I 2 0 1 )COMMON I , J , K , M , A , X N R , N R , I S W , P T S W, Y T OL , N R 1 , I T P T , I T T O T , I T M A X , I N S W, K K COMMON H F R E Q , T F R E Q , R F R E Q , Y S , DM, DT COMMON K D 1 F , I P C H X , I P C H Y , 1 0 , JO, JMAX COMMON DE L X . DE L XO, XF F COMMON E W, MAX OUT LOGICAL I SW, PTSW REAL MINTEGER HEREQ, TFREQ, RFREQ

IF { I NS W ) 6 , 6 , 1

GET Y* IN EQN WITH XF

INSW=0I F I X F F - . 2 2 6 ) 2 , 2 , 3 Y S = X F F / ( 1 . 1 3 5 4 * X F F + . 1 0 3 2 )GO TO AY S = X F F / ( . 9 3 9 8 * X F F + . 1 4 7 5 )W R I T E ! 1 * 5 ) YSF O R MA T ( I X , 4 2 H A D S 0 R B E D PHASE CONC IN EQUI LI BRI UM WITH X F , F 1 8 . 5 ) W R I T E ! 1 , 1 0 )FORMAT( 1 5 X , 1 8 H I BENZENE N- HEXANE ) )

GET X IN EQUI LI BRI UM WITH Y

6 I T T 0 T = I T T 0 T + 1 T F I Y - . 6 2 9 ) 7 , 7 , 8

7 E Q = . l 0 3 2 * Y / { l . - l . 1 3 5 4 * Y )

134

GO TO 98 E Q = . U 7 5 * Y / l l . - . 9 3 9 8 * Y )9 EQBZHX=EQ

RETURN END

SAMPLE L I S T I N G OF COMPUTED RESULTS I METHYLCYCLOHEXANE- TOLUENE- ALUMI NA, A = 4 0 » X = . 5 )

PROBLEM MCH- T O L - A L , A = 4 0 . » X= . 5 , 1 1 / 2

I NLET FEED COMPOSI TI ON, XF 0 . 5 0 0 0 0INCREMENTAL BED LEVEL, DH 0 . 0 5 0 0 0INCREMENTAL TI ME, DT 2 . 0 0 0 0 0DIMENS I ONLESS RADI US , A 4 0 . 0 0 0 0 0NUMBER OF INTERVALS OF RADIUS , XNR 2 5 . 0 0 0 0 0RATIO OF DT TO I A/ XNR ) « *2 0 . 7 8 1 2 5TOLERANCES

WAVE FRONT T E S T , YTOL 1 . 0 0 0 0 0 t - 0 5ERROR IN CONCENTRATION - BULK PHASE, EX 1 . OOOOOE-G5

- ADSORBED PHASE, EY 1 . 0 0 0 0 0 b - 0 5NUMBER OF TIME S T E P S , JMAX 4 0 1MAXIMUM ALLOWED I TFRATI ONS PER P OI NT , ITMAX 2 0 0

J I LOOPS T H X ( J , I ) Y I J , I )

1 1 0 0 . 0 . 0 . 5 0 0 0 0 0 0 0 0 . 5 0 0 0 0 0 0 011 11 4 2 0 . 0 0 0 0 0 . 5 0 0 0 0 . 3 9 8 6 1 2 6 7 0 . 3 9 8 6 1 2 6 716 11 2 7 3 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 4 9 8 2 3 1 9 0 . 6 6 4 6 7 7 4 221 11 2 1 4 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 6 2 1 6 7 6 0 0 . 6 8 4 9 6 2 6 22 6 11 2 0 5 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 6 9 0 6 1 1 6 0 . 6 9 4 6 3 0 1 031 11 17 6 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 7 3 6 5 8 0 7 0 . 7 0 0 6 3 5 5 43 6 11 1 5 7 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 7 7 0 0 9 5 9 0 . 7 0 4 b 3 5 0 14 1 11 15 8 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 7 9 5 9 9 3 2 0 . 7 0 7 9 8 9 3 54 6 11 13 9 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 8 1 6 7 9 0 3 0 . 7 1 0 4 7 6 2 65 1 11 13 1 0 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 8 3 3 9 7 2 1 0 . 7 1 2 4 9 6 3 35 6 11 13 1 1 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 8 4 8 4 8 2 0 0 . 7 1 4 1 8 1 4 86 1 11 13 1 2 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 8 6 0 9 4 6 1 0 . 7 1 5 6 1 5 3 26 6 11 12 1 3 0 . 0 0 0 0 0 . 5 0 0 0 0 . 4 8 7 1 8 0 7 3 0 . 7 1 6 8 5 4 6 6

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o o o o o o o o o o o o• * • • * • • * • • • *J ' ■p -p -P OJ OJ OJ OJ O J OJ OJ fS)IS) A— fl— o 43 43 CO - 4 'Ol -p rs j oA - O ' A— -p -4 o A— o 00 OJ -p O '43 - J f—■ QC a : o o O ' O J o - o O 'A - O ' A— ST oo ISJ O ' CO 43 " 4 03 03O ' 43 -p 03 O ' ~ 4 p o ISJ ISJ S JO ' A— OJ 00 -P o — p -0 O l -P O 'o ISJ oJ o OJ vC O ' ISJ c on t— * ■—

o o o o o o o o o o o o» * ■ • • * • • « « * •O ' O ' O ' O ' O ' O ' O ' ry* O ' 41 s jl O l-4 O ' O ' on -P ■P OJ rs j A— 4C O ' SJo O ' A— O ' o ISJ o OJ o OJ 40 ‘41C3 o r o A- Cr u 43 O ' O ' O' c . A -o no OJ ISJ -p O1 C3 (SJ -4 S ) o -4rsj A- " J CC OJ O-' ~4 cr. —I c rs; S 'rs j -p —J 43 f l- > —J SJ SJ OJ —J OJ oISJ O ' o c OJ ■OI y \ f l- A— - 4

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O ' O l O l on on O l on 41 41 on 41 41A— A- fl— fl— fl— fl— fl— fl—> fl— A- fl— A—

-PO' o O' O' O ' O ' O ' O' O' O ' O '

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v_r ro Co o X—4 o 0 - o -4

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sn on U1 on on on on U1 on 41 onfl— I—1 fl— fl— f l - fl— f l— f l—

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O J OJ OJ OJ OJ O J OJ O J OJ OJ OJ O J OJ OJ OJ OJ OJ OJ OJ OJ o O J OJ OJ OJ OJ OJ 0 J OJ OJ OJ OJ O J OJ OJ4 4 • • 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o e o o o o o o o o o o o cr o o o C- o o o c? o o o o o o o c o oo o o o o o o o o o o o o o c o o o c o o o o o o o o o o o o c o o oo o o o o o o o o o o o o o o o o o o o c o o o o o o o o o o o o o o

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0 0 CO CD 0 3 CD CD CD CD 0 3 0 0 c c -s j —4 - 4 's j - o - s j O'* o o o* o o n \T \ o n o i 4 4* O J O J ■^J r oCD CD - 4 O' o n ■Ol U J ro I—* o sO 0 0 - 4 O J p— o CD o ro o 0 0 cr O J f— o o o i r o CD 4* 0 0

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• t * 4 4 4 4 4 4 4 4 4 4 4 • 4 4 4 4 4 4

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4 4 4 4 4 4 4 4 4 4 ■ 4 4 4 4 4 4 4 4 4 4

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l>c >o sO *■■4 sO sO sO o o sG 4—4 sC *—4

o o p J •—t NJ rs j rN rN X X LN UP sO o N - r - c o 00 CP CP om ro m ro m rO m rN rN m m ro m m rN rN ro m m X

GENERAL BLOCK DIAGRAM FOR SOLUTION OF ADSORPTION MODEL149

START

ewlnd1 , 3

Pause

Read Data

P r e l i m i n a r y C a l c u l a t i o n s : M, A H

W r i te I d e n t i - \ f i c a t i o n on Output U n i t s

S e t INSW = 1 to e n t e r EQ

2I n i t i a l i z eProblemP a r a m e t e r s :

KK = 110 = 1JO = 1

ITTOT = 1KULTX = 1

KULT = 1KQUIT = JMAX

PTSW = .FALSE.ISW = .FALSE .

I = 1J = 1

ITPT = 0

D ef i n e :YOLD(K) , YNEW( K ) ,(K = 1, N + 1)Def i n e :X(K) ,(K = 1, JMAX)

W rite Output f o r P t . (1

©

G r id P t . (J , 1 ) : S e t YEST a t s u r f a c e =

YOLD(N + 1)

C a l c . x from F u n c t i o n EQ

C a l lCK

XfESTflffNEWa t s u r f a c e

A d j u s t YEST by Eq ( 5 - 4 ) f o r a n o t h e r i t e r a t i o n

©

150

G r id P t . ( J f 2>;R e - d e f i n e :

YNEW, k

G r id P t . ( 2 , 2 ) :

YOLD' C a l l OUT

P t . ( 1 , 1 )----- ®Guess x.

I n c r e a s eR e - d e f i n e ;YOLDk = YNEWk(k = l , . . . , t * - l ) For P t . ( J + 1 , 1 )

C a l c u l a t e x from EQ ^Ênd o f

Column 2

C a lc XNEW By Q u a d r a t i c F i tI n c r e a s e

Ind o Wave

C a l lCK

s End o f Column 1 Is

S e t XBj=XBj

XSB *=XSB

I sEnd o f Wave XB XNEW

Guess y n + 1 :

Y e s t = YOLDp+i

S e t XB XNEV Guess xS e t x XB

C a l l OUT

P t . ( 2 , 2

151

. / This an x U l t i m a t e Wave

C a lc XBj from F u n c t i o n EQ

Is

C a lc XNEW By Q u a d r a t i c F i t

A d j u s t y e s t By Eq. ( 5 - 4 ) T e s t S e n s e

S w i t c h 4 f o r I n t e r r u p t

C a l lCK

C a l l OUT

P t . ( J , 2 )ONIs

R e - d e f i n e y f o r n e x t t im e s t e p :

YNEW.YOLDewind

XB * = XNEW

C a l l ' ICKPT to

Dump

I n c r e a s eC a l l

TD Pause

End x o f ProblemC a l l

CK G rid P t . ( 1 , 1 ) : S e t J - I

KULT = 0 E s t i m a t e X2 2 (XNEW) By ’ Q u a d r a t i c F i t

I s> KQUIT

C a lc XNEW from F u n c t i o n EQ

152

?C a lc xnew2 by Cubic F i t

R e - c a l c . XNEW* by F u n c t i o n EQ

S e t XNEW =XNEW2

R e-d e f in e y a t Boundary:YOLDk = XNEW2

(k = 1 , . . . , n f l )

C a l l OUT

P t . ( I . I )

c - T lR e s e t KK“ 1

■ Q

R e-d e f in e terms for next time l e v e l :

x 4 = “ lx* = XB*XBj = J»EW2

XB* = aNEW*

I n c r e a s e J by 1

Column

End ofWave

G r id P t . ( J , I ) ( T y p i c a l P o i n t ) : Guess :

y e 8 t - Y“ » lGuess x . . (XNEW)

J »1

C a lc XNEW from F u n c t i o n EQ

C a lc XNEW2 by C ubic F i t

C a l lCK

InWave

XNEW ~ XNEW2

S e t XNEWXNEW

C a l lTD

C a l lCK

153

YNEW

A d j u s t y e s t by Eq. (5 -4 )

I

C a l l OUT

vp t -

R e - d e f i n e y f o r n e x t t im e s t e p : YOLDk = YNEWk (k = 1 , . . . , n + l )

XNEW2 C YTOL ?

S e t KULT = 1

I sT h is an

l t i m a t e Wave

S e t KULT = 1

U l t i m a t e Wave R o u t i n e :T e s t KULTX

KULTX

C a lc H

W r i te H on 1 a n d / o r 3

S e t KULTX « 1 C a lc KQUIT

r.W r i t e T o t a l I t e r a t i o n s on 3 (ITTOT)

ToSTART

154

SUBROUTINE TD

I n i t i a lE n try

^ i r s X , Use o f TD

a t t h i s Pt

S e t ISW=.TRUE.

S e t PTSW=.TRUE.

C a lc YEND = YOLDn+1 + WAX

( n - l ) S ;

C a lc B by E q (A l8 )

C a lc W by Eq(A20)

C a lc T^YOLDx / S i

t 2- yold2 / s 2

C a lc Tk =

PfOLD|(+ (m- R r x )

• Tk - l 1 / s k(k = 3 , . . . , n )

C a lc Tn+1YEND + 2mTn / S , ^ x

C a lc YNEW^.1 =

TN+1C alc YNEWk

<k = n , • ■ • »2)

C a lc YNEWx =+ 6m(YNEW?)

C RETURN

c SUBROUTINE CK

i t

S e t KK * 1

)

A ccu m u la te I t e r a t i o n Count (ITPT)

W r i t e o u t p u t on 1 ( f o r t r a c i n g u s e )

W r i te L o ca t i o n o f t h i s P o i n t ( I , J )

S e t KK = 2

RETURN

155

SUBROUTINE OUT

C h an ging ?

IO

I s

/ i s N J < JMAX

End o f Wave

S e t KK

S e t KK = 3

I n c r e a s e 10 by HFREQS e t JO = 10

C a lc H and T I s

W r i t e R e s u l t s on 1 a n d / o r 7

I n c r e a s e JO by TFREQ

KDIF

I < MAXOUT

I s

n S e t ITPT = 0 PTSW = .FALSER e - d e f I n e

YNEW, = x

(k -1

RETURNrW r i te y^ on 3 a n d / o r 7 (k = l , . . . , n + l )

156

O p e r a t i n g I n s t r u c t i o n s

A l th o u g h t h e program a s w r i t t e n i s i n t h e FORTRAN l a n g u a g e ,

some m o d i f i c a t i o n s may be r e q u i r e d b e f o r e i t i s u s e d on a n o t h e r com

p u t e r , p a r t i c u l a r l y t h e i n p u t / o u t p u t o p e r a t i o n s . The o p e r a t i n g p r o c e

d u re o u t l i n e d h e r e i n w i l l i n v o l v e p r i m a r i l y t h e p r e p a r a t i o n o f in p u t

d a t a , a l o n g w i t h c e r t a i n o p t i o n s t h a t a r e a v a i l a b l e f o r r e s u l t s .

D e t a i l s o f m ach in e o p e r a t i o n may be found i n t h e a p p r o p r i a t e r e f e r e n c e

manuals ( 2 8 ) .

Data Cards

Card Format Columns

E 1 0 . 0 1 -1 0 XF,

E 1 0 . 0E 1 0 . 0E 1 0 . 0E 1 0 . 06A5

1 1 - 2 02 1 - 3 03 1 - 4 04 1 - 5 05 1 - 8 0

QQ,DT,A,XNR,P1-P6

E 1 0 . 0E 1 0 . 0

1 - 1 01 1 - 2 0

YT0L, EX,

E 1 0 .0 2 1 - 3 0 EY,

15 1-5 JMAX,

15 6 - 1 0 ITMAX

15 1 1 - 1 5 HFREQ

15 1 6 - 2 0 TFREQ

15 2 1 - 3 0 RFREQ

15 3 1 - 3 5 KDIF,

V a r i a b l e

f e e d c o n c e n t r a t i o n , f r a c t i o n o f more a d s o r b a b l e component damping f a c t o rd i m e u s i o n l e s s t im e in c r e m e n t d i m e n s i o n l e s s p a r t i c l e s i z e number o f r a d i u s i n c r e m e n t s p roblem i d e n t i f i c a t i o n

t o l e r a n c e f o r end o f wave t o l e r a n c e f o r f i t t i n g p o l y n o m i a l s i n x by t r i a l and e r r o r t o l e r a n c e f o r f i t t i n g s u r f a c e c o n c e n t r a t i o n

number o f g r i d p o i n t s i n J - , o r T - d i r e c t i o nmaximum number o f i t e r a t i o n sa l l o w e d p e r p o i n tf r e q u e n c y o f o u t p u t a l o n g J -a x i s , a t g i v e n Hf r e q u e n c y o f o u t p u t a l o n g I -a x i s , a t g i v e n Tf r e q u e n c y o f o u t p u t o f i n t e r n a ld i f f u s i o n c u r v e ( o n l y e f f e c t i v ewhen KDIF * 1)code f o r o u t p u t o f d i f f u s i o n c u r v e s : z e r o f o r no o u t p u t ; one f o r o u t p u t on l o g i c a l u n i t one a n d / o r t h r e e

157

Card Format Columns V a r i a b l e

3 15 3 6 - 4 0 IPCHX, code f o r o u t p u t o f b u l k - p h a s e( c o n t ) s o l u t i o n s : one f o r b o t h p r i n t e d

o u t p u t on l o g i c a l u n i t one and punched o u t p u t on l o g i c a l u n i t s e v e n ; z e r o f o r p r i n t e d o u t p u t o n l y

15 4 1 - 4 5 IPCHY, code f o r o u t p u t o f d i f f u s i o nc u r v e s : one f o r punched o u tp u t on l o g i c a l u n i t 7 , and p r i n t e d o u t p u t on l o g i c a l u n i t t h r e e ; z e r o f o r p r i n t e d o u t p u t (only e f f e c t i v e f o r KDIF ■ 1)

15 4 6 - 5 0 MAXOUT, column number up t o w h ich o u t p u tf o r e v e r y g r i d p o i n t i s g i v e n o v e r r i d i n g HFREQ and TFREQ. Same l o g i c a l u n i t s a s f o r IPCHX and IPCHY

E q u i l i b r i u m F u n c t i o n

B e f o r e t h e program can be u s e d w i t h any s y s t e m o t h e r than

t h o s e i n c l u d e d i n t h i s w ork , an a p p r o p r i a t e s u b r o u t i n e f o r e q u i l i b r i u m

r e l a t i o n s h i p s must be p r e p a r e d . The r e q u i r e m e n t s o f t h i s s u b r o u t i n e

a r e a s f o l l o w s . The name o f t h e s u b r o u t i n e must be g i v e n by t h e FORTRAN

s t a t e m e n t , FUNCTION EQ (a r g u m e n t ) . The FORTRAN COMMON s t a t e m e n t s

l i s t e d i n t h e main program must be r e p r o d u c e d and i n c l u d e d w i t h th e

s u b r o u t i n e . In th e f i r s t u s e o f t h e s u b r o u t i n e , t h e argument i s t h e

•jff e e d c o n c e n t r a t i o n , f o r w hich y (YS i n t h e program) must be computed.

W hatever i n s t r u c t i o n s a r e n e c e s s a r y , i n c l u d i n g in p u t or o u t p u t , may be

JL

i n c l u d e d t o o b t a i n y . There must be a s t a t e m e n t to t e s t a p aram eter

INSW, w h ic h i n i t i a l l y i s s e t t o 1 by t h e main program, t h e n t o z e r o

a f t e r y i s computed . T h e r e a f t e r t h e c a l c u l a t i o n f o r y w i l l be s k ip p e d

and t h e s u b s e q u e n t u s e o f EQ w i l l be t o compute x i n term s o f t h e g i v e n

158

argument y . Any a p p r o p r i a t e i n s t r u c t i o n s may be g i v e n , t h e l a s t e x e c u t e d

s t a t e m e n t c a l l i n g f o r a r e t u r n t o t h e m ain program, A f l o w d iagram

i l l u s t r a t i n g t h e g e n e r a l r e q u i r e m e n t s f o r t h i s s u b r o u t i n e i s g i v e n b e lo w .

E n t e r F u n c t i o n E Q ( z ) , where V z ■ x f ) or y

s e t INSW R eturn

. / 1 8 INSW

Q = °

I n c r e a s e ITTOT by i

compute

1 - J L C t f

Each t im e th e f u n c t i o n i s u sed one i s t o be added t o a

v a r i a b l e , ITTOT, to r e c o r d th e t o t a l number o f i t e r a t i o n s e x e c u t e d .

G e n e r a l Comments on Use o f th e Program

I f a l l o p t i o n s a r e c a l l e d f o r , a t o t a l o f f o u r o u t p u t d e v i c e s

a r e n e c e s s a r y , e x c l u s i v e o f w h a t e v e r d e v i c e s a r e r e q u i r e d by th e computer

s y s t e m i t s e l f . The normal mode o f o p e r a t i o n i s t o o b t a i n b o th p r i n t e d

and punched o u t p u t f o r o n l y t h e b u lk p h a se c o n c e n t r a t i o n , a s i t i s t h i s

o u t p u t w h ich i s compared w i t h e x p e r i m e n t a l d a t a t o d e t e r m i n e th e mass

t r a n s f e r c o e f f i c i e n t .

The i n p u t d a t a , e x c e p t f o r f e e d c o n c e n t r a t i o n and i d e n t i f i c a

t i o n , i s t o some e x t e n t a m a t t e r o f e x p e r i m e n t a t i o n . Once a f e a s i b l e

o p e r a t i n g s e t o f p a r a m e t e r s i s f o u n d , s l i g h t v a r i a t i o n s may be made t o

o p t i m i z e computer u s a g e . For i n s t a n c e t h e damping f a c t o r , QQ, must be

d e t e r m i n e d , a s t h e r e i s u s u a l l y some v a l u e above w h ich d i v e r g e n c e w i l l

159

o c c u r . I f QQ i s made t o o low, t h e number o f i t e r a t i o n s i n c r e a s e s . The

optimum v a l u e may be o b t a i n e d by making a few f a l s e s t a r t s a t v a r i o u s

QQ v a l u e s t o g e t th e number o f i t e r a t i o n s made a t some s e l e c t e d p o i n t ,

t h e n p l o t t i n g QQ v e r s u s number o f i t e r a t i o n s . S i m i l a r t e s t s may be

run t o g e t optimum v a l u e s o f t h e e r r o r t o l e r a n c e s , w h i c h i f t o o h i g h ,

c a u s e i n a c c u r a t e r e s u l t s , or i f t o o low i n c r e a s e t h e number o f i t e r a

t i o n s , w i t h a w a s t e o f a c c u r a c y .

For e a c h s y s t e m ch eck ed i n t h e p r e s e n t s t u d y , a c o n v e r g e n t

p r o c e s s , no m a t t e r how t im e - c o n s u m i n g , c o u l d a l w a y s be f o u n d . There

i s no g u a r a n t e e , h o w e v e r , t h a t any s y s t e m w i l l b eh a v e s o w e l l . One

must e x p e c t t o spend c o n s i d e r a b l e e f f o r t b e f o r e o b t a i n i n g t h e f i r s t

s u c c e s s f u l r u n .

S i n c e some p rob lem s r e q u i r e much t im e on t h e com p uter a b u i l t -

i n i n t e r r u p t p r o c e d u r e has b e e n i n c l u d e d . I f , i n t h e m i d d l e o f a run,

i t i s d e s i r e d t o su sp en d c a l c u l a t i o n s t e m p o r a r i l y , a c o n s o l e s w i t c h

may be d e p r e s s e d t o c a u s e o u t p u t o f a l l i n t e r m e d i a t e r e s u l t s . T h is

o u t p u t may be u sed to r e s t a r t from t h e p o i n t o f i n t e r r u p t i o n . The

p r e s e n t program, a s u s e d on t h e 7 0 4 0 , p e r m i t s a dump o f computer

s t o r a g e , and r e g i s t e r s on a m a g n e t i c t a p e ( B - l ) by d e p r e s s i n g c o n s o l e

s w i t c h 4; l a t e r r e s t a r t from t h i s p o i n t may be a c c o m p l i s h e d by m ou n t ing

t h e dumped o u t p u t on t h e same t a p e u n i t from w hich t h e dump was t a k e n ,

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

The program a l s o p r o v i d e s a means f o r t r a c i n g t h e p r o g r e s s o f

i n t e r m e d i a t e c a l c u l a t i o n s by u s e o f c o n s o l e s w i t c h number 2 . I f t r o u b l e

i s e n c o u n t e r e d more d e t a i l may be o b t a i n e d by d e p r e s s i n g t h i s s w i t c h ,

w i t h t h e r e s u l t s o f e a c h i t e r a t i o n b e i n g p r i n t e d on l o g i c a l u n i t 1.

160

T h i s s w i t c h s h o u l d n o t be u s e d e x c e p t f o r d e b u g g i n g p u r p o s e s due t o t h e

h i g h volume o f o u t p u t .

A b u i l t - i n c h e c k f o r e x c e s s i v e i t e r a t i o n s i s p r o v i d e d . I f th e

maximum number ( c a l l e d ITMAX i n th e program) i s e x c e e d e d a m e s s a g e

w i l l be w r i t t e n on t h e main o u t p u t u n i t ( l o g i c a l u n i t 1 ) , a l o n g w i t h

t h e g r i d p o i n t l o c a t i o n . A l s o p r o v i d e d a r e c h e c k s t o p r e v e n t com put ing

g r i d p o i n t s o u t s i d e o f t h e a d s o r p t i o n w ave . I f t h e d i f f e r e n c e b e tw een

y and y i s l e s s th a n t h e v a r i a b l e YTOL, th e n c a l c u l a t i o n s a r e s to p p e d

f o r t h a t p a r t i c u l a r bed l e v e l . A l l r e m a i n i n g p o i n t s a r e th en s e t o

th e e q u i l i b r i u m v a l u e s c o r r e s p o n d i n g t o th e upper end o f t h e wave.

The maximum mesh s i z e p e r m i t t e d i n t h e p r e s e n t v e r s i o n o f t h e

program a l l o w s one th o u sa n d i n t e r v a l s i n b o th t h e H- and T- d i r e c t i o n s .

T h i s number may be a l t e r e d a c c o r d i n g t o s t o r a g e l i m i t a t i o n s o f the

co m p u ter . A p p r o x i m a t e l y 2 0 , 0 0 0 words o f s t o r a g e a r e r e q u i r e d w i t h th e

p r e s e n t program and m a ch in e c o n f i g u r a t i o n .

Machine C o n f i g u r a t i o n Used

Computer: IBM 7040 Data P r o c e s s i n g Sys temData channels A: 1622 Card Read Punch (BCD on ly )

B: 4 magnetic Cape u n i t s ( 7 3 3 0 )C: 4 m a g n e t i c t a p e u n i t s ( 7 3 3 0 )

Memory S i z e ; 3 2 , 7 6 8 b i n a r y words E xtend ed I n s t r u c t i o n s e t

M o n i to r Sys tem : IBM 7 0 4 0 / 7 0 4 4 O p e r a t i n g S y s te m (16/32K.)V e r s i o n 6

NOMENCLATURE

E f f e c t i v e p a r t i c l e r a d i u s , f t

D i m e n s i o n l e s s p a r t i c l e r a d i u s , d e f i n e d by E q . ( 4 - 7 )

S u r f a c e a r e a a c r o s s w h ich d i f f u s i o n o c c u r s on t h e s u r f a c e o f t h e s p h e r i c a l p a r t i c l e s , f t ^ / f t ^ bed

OE f f e c t i v e i n t r a p a r t i c l e d i f f u s i v i t y , f t / h r

F r a c t i o n o f s a t u r a t i o n , or d e g r e e o f a d s o r p t i o n i n a column o f l e n g t h , z

F r a c t i o n o f s a t u r a t i o n i n a l e n g t h o f column e q u i v a l e n t t o th e l e n g t h o f a wave

F r a c t i o n a l s a t u r a t i o n o f column a t b e g i n n i n g o f s t e a d y s t a t e

F r a c t i o n a l s a t u r a t i o n o f a d s o r p t i o n zone a t b e g i n n i n g o f s t e a d y s t a t e

F r a c t i o n a l s a t u r a t i o n o f a d s o r p t i o n zone a t any p o i n t w i t h i n t h e column

F r a c t i o n v o i d s w i t h i n a d s o r b e n t bed

D i m e n s i o n l e s s bed l e v e l p a r a m e t e r , d e f i n e d by E q . ( 4 - 8 )

H e ig h t o f t r a n s f e r u n i t b a s e d on l i q u i d f i l m c o e f f i c i e n t , K,

S u b s c r i p t s d e n o t i n g column number o f t r i a n g u l a r g r i d

Mass t r a n s f e r c o e f f i c i e n t f o r e x t e r n a l s u r f a c e f i l m , f t / s e c

R a t i o o f AT t o <^R)^ i n f i n i t e d i f f e r e n c e e q u a t i o n f o r i n t e r n a l d i f f u s i o n .

Number o f i n c r e m e n t s u s e d i n f i n i t e d i f f e r e n c e e q u a t i o n f o r i n t e r n a l d i f f u s i o n

Number o f t r a n s f e r u n i t s , b a s e d on l i q u i d - f i l m c o e f f i c i e n t ,

T o t a l l i q u i d f e e d f l o w r a t e , f t ^ / h r

P a r t i c l e r a d i u s v a r i a b l e , f t

D i m e n s i o n l e s s p a r t i c l e r a d i u s v a r i a b l e , d e f i n e d by E q . ( 4 - 7 )

C r o s s - s e c t i o n a l a r e a o f a d s o r b e n t co lumn, f t

D i m e n s i o n l e s s t im e v a r i a b l e , d e f i n e d by Eq. ( 4 - 9 )

D i m e n s i o n l e s s t im e c o r r e s p o n d i n g t o b e g i n n i n g o f s t e a d y s t a t e

T o t a l q u a n t i t y o f more a d s o r b a b l e component i n e f f l u e . i t s t r e a m , ft-*

Maximum q u a n t i t y o f more a d s o r b a b l e component t h a t c o u ld be c o n t a i n e d i n a l e n g t h o f bed e q u a l t o t h e l e n g t h o f a wave, f t ^

Pore vo lu m e, f t ^ / l b a d s o r b e n t

T o t a l volume o f l i q u i d i n e f f l u e n t l i q u i d , f t

R ate o f t r a n s f e r o f more a d s o r b a b l e component from b u lk phase t o adsorbed phase i n d i f f e r e n t i a l s e c t i o n , d z , f t ^ / h r

V o lu m e t r i c c o n c e n t r a t i o n o f more a d s o r b a b l e component i n non--J •} rad sorb ed phase f t J / f t J

I n s t a n t a n e o u s v o l u m e t r i c c o n c e n t r a t i o n o f more a d s o r b a b l e component a t wave f r o n t , f t - V f t ^

V o lu m e tr ic f e e d c o n c e n t r a t i o n o f more a d s o r b a o l e com p on en t , f t - V f t ^

V o lu m e t r i c c o n c e n t r a t i o n o f more a d s o r b a b l e component on b u l k - p h a s e s i d e o f i n t e r f a c e i n e q u i l i b r i u m w i t h th e ad sorbed phase i n t e r f a c i a l c o n c e n t r a t i o n , f t 3 / f t 3

V o lu m e tr ic c o n c e n t r a t i o n o f more a d s o r b a b l e component i n a d sorbed p h a s e , f t ^ / f t ^

V o lu m e tr ic c o n c e n t r a t i o n o f more a d s o r b a b l e component i n a d sorbed phase i n e q u i l i b r i u m w i t h f e e d , f t ^ / f t ^

Bed h e i g h t v a r i a b l e , f t

Length o f a d s o r b e n t bed r e q u i r e d f o r fo r m a t io n o f u l t i m a t e w a v e , hr

Length o f an a d s o r p t i o n z o n e , f t

R ea l t ime e l a p s e d s i n c e i n i t i a l c o n t a c t o f f e e d w i t h a d s o r b e n t , hr

Real time e lap sed from i n i t i a l s ta r tu p u n t i l beginning o f s t e a d y - s t a t e , hr

Real time required for a d sor pt ion zone to pass a g iven p o in t , hr

Bed d e n s i t y , l b / f t ^

AUTOBIOGRAPHY

The a u t h o r was born J u l y 1 8 , 1 9 3 4 , i n West Palm B e a c h , F l o r i d a ,

a t t e n d e d p u b l i c s c h o o l s i n t h a t c i t y , and g r a d u a t e d from Palm Beach

High S c h o o l i n J u n e , 1951 . He a t t e n d e d t h e U n i v e r s i t y o f C h i c a g o ,

C h i c a g o , I l l i n o i s , d u r i n g 1 9 5 1 - 1 9 5 2 ; a t t e n d e d t h e G e o r g i a I n s t i t u t e

o f T e c h n o l o g y , A t l a n t a , G e o r g i a , from 1952 t o 1 9 5 5 , m a j o r i n g i n c h e m i c a l

e n g i n e e r i n g , and g r a d u a t e d w i t h t h e b a c h e l o r ' s d e g r e e i n J u n e , 1956;

worked f o r E s s o R e s e a r c h L a b o r a t o r i e s , B a ton R ouge , L o u i s i a n a , a s an

e n g i n e e r from 1955 t o 1958; e n r o l l e d i n L o u i s i a n a S t a t e U n i v e r s i t y ,

B aton Rouge , L o u i s i a n a i n 1959 , m a j o r i n g i n c h e m i c a l e n g i n e e r i n g and

m i n o r i n g i n b u s i n e s s a d m i n i s t r a t i o n , worked a s a g r a d u a t e a s s i s t a n t in

t h e D epartment o f C hem ica l E n g i n e e r i n g and t h e Computer R e s e a r c h C e n te r

and a s an i n s t r u c t o r i n the Department o f M e c h a n i c a l E n g i n e e r i n g , and

g r a d u a t e d w i t h th e m a s t e r ' s d e g r e e i n c h e m i c a l e n g i n e e r i n g i n A u g u s t ,

1960; e n r o l l e d i n L . S . U . i n 1960 on t h e d o c t o r a l program, m a j o r in g i n

c h e m i c a l e n g i n e e r i n g , and i s p r e s e n t l y a c a n d i d a t e f o r th e d e g r e e o f

d o c t o r o f p h i l o s o p h y . He i s p r e s e n t l y employed f u l l - t i m e by th e

Computer R e s e a r c h C e n te r a s A s s i s t a n t D i r e c t o r . He i s a member o f Tau

B e ta P i , P h i Lambda U p s i l o n , P h i Kappa P h i , t h e American I n s t i t u t e o f

C hem ica l E n g i n e e r s , and th e A s s o c i a t i o n f o r Computing M a c h in e r y ; and

i s a r e g i s t e r e d p r o f e s s i o n a l e n g i n e e r , S t a t e o f L o u i s i a n a . He i s

m a r r i e d t o th e former M a r i l y n J o y c e G r i f f i n o f J o n e s b o r o , A r k a n s a s .

164

EXAMINATION AND THESIS REPORT

Candidate; Elmer Lawrence M orton, J r .

Major Field: Chem ical E n g in ee r in g

Title of Thesis: A n a ly s i s o f L iq u id -P h a se A d so rp t io n F r a c t io n a t io n in F ix ed Beds

Approved:

Y c ^ J t > \ r w iMajor Professor ana ChairmanMaj

/ ; - c

Dean of the Graduate School

EXAMINING COMMITTEE:

jfcl .

f J J - '

u

Date of Examination: Ja n u a ry 1 1 , 1965

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