Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...
Transcript of Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...
![Page 1: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/1.jpg)
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
Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses
This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion inLSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please [email protected].
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
![Page 2: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/2.jpg)
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
![Page 3: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/3.jpg)
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
![Page 4: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/4.jpg)
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 .
![Page 5: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/5.jpg)
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
![Page 6: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/6.jpg)
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
![Page 7: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/7.jpg)
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
![Page 8: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/8.jpg)
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
![Page 9: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/9.jpg)
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
8 . 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 87
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
11 . 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 = 8 0 90
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
![Page 10: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/10.jpg)
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 ..................... ...................................................................................
![Page 11: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/11.jpg)
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
![Page 12: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/12.jpg)
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
![Page 13: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/13.jpg)
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 .
![Page 14: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/14.jpg)
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
![Page 15: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/15.jpg)
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 .
![Page 16: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/16.jpg)
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
![Page 17: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/17.jpg)
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 .
![Page 18: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/18.jpg)
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 .
![Page 19: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/19.jpg)
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
![Page 20: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/20.jpg)
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
![Page 21: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/21.jpg)
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
![Page 22: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/22.jpg)
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
![Page 23: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/23.jpg)
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
![Page 24: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/24.jpg)
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
![Page 25: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/25.jpg)
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 .
![Page 26: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/26.jpg)
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
![Page 27: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/27.jpg)
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
![Page 28: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/28.jpg)
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 ;
![Page 29: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/29.jpg)
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
![Page 30: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/30.jpg)
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
![Page 31: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/31.jpg)
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
![Page 32: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/32.jpg)
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
![Page 33: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/33.jpg)
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 )
![Page 34: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/34.jpg)
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
![Page 35: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/35.jpg)
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 )
![Page 36: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/36.jpg)
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 )
![Page 37: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/37.jpg)
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
![Page 38: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/38.jpg)
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
![Page 39: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/39.jpg)
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 .
![Page 40: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/40.jpg)
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
![Page 41: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/41.jpg)
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 ;
![Page 42: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/42.jpg)
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
![Page 43: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/43.jpg)
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
![Page 44: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/44.jpg)
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
![Page 45: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/45.jpg)
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^ .
![Page 46: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/46.jpg)
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 .
![Page 47: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/47.jpg)
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 :
![Page 48: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/48.jpg)
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
![Page 49: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/49.jpg)
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
![Page 50: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/50.jpg)
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
![Page 51: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/51.jpg)
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 )
![Page 52: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/52.jpg)
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
![Page 53: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/53.jpg)
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
![Page 54: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/54.jpg)
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.
![Page 55: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/55.jpg)
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
![Page 56: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/56.jpg)
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
![Page 57: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/57.jpg)
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
![Page 58: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/58.jpg)
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
![Page 59: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/59.jpg)
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
![Page 60: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/60.jpg)
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 .
![Page 61: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/61.jpg)
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)
![Page 62: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/62.jpg)
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
![Page 63: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/63.jpg)
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
![Page 64: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/64.jpg)
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
![Page 65: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/65.jpg)
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
![Page 66: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/66.jpg)
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
![Page 67: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/67.jpg)
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 .
![Page 68: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/68.jpg)
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
![Page 69: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/69.jpg)
56
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 ;
![Page 70: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/70.jpg)
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
![Page 71: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/71.jpg)
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 .
![Page 72: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/72.jpg)
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 .
![Page 73: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/73.jpg)
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 ;
![Page 74: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/74.jpg)
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
![Page 75: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/75.jpg)
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 ,
![Page 76: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/76.jpg)
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
![Page 77: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/77.jpg)
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
![Page 78: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/78.jpg)
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
![Page 79: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/79.jpg)
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 .
![Page 80: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/80.jpg)
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
![Page 81: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/81.jpg)
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
![Page 82: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/82.jpg)
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 .
![Page 83: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/83.jpg)
LITERATURE CITED
1 . Amundson, N e a l R . , J . P h y s . and C o l l o i d Chem. , 5 2 , 1153 ( 1 9 4 8 ) .
2 . _____________ , J . P h y s . and C o l l o i d Chem. , 5 4 , 8 1 2 - 2 0 ( 1 9 5 0 ) .
3 . B a r r e r , R. M . , J . P h y s . Chem. 5 7 , 35 ( 1 9 5 3 ) .
4 . B e r g , C ly d e , P e t . Eng. 1 8 , 1 1 5 -8 ( 1 9 4 7 ) .
5 . Carnahan, B . , H. A, L u t h e r , and J . 0 . W i l k e s , A p p l i e d N u m e r ic a lM e t h o d s , John W i l e y and S o n s , I n c . , New Y ork , 1 9 6 4 .
6 . C o n s t a b l e , F. H . , T r a n s . F a r . Soc . , 2 8 , 2 2 7 - 8 ( 1 9 3 2 ) .
7 . De V a u l t , J . Am. Chem. S o c . . 6 5 , 5 3 2 - 4 0 ( 1 9 4 3 ) .
8 . E a g l e , Sam, and J . W. S c o t t , I n d . E n g . Chem. , 4 2 , 1 2 8 7 , ( 1 9 5 0 ) .
9 . E b e r l y , P . E . , and E. H. S p e n c e r , T r a n s . F a r . S o c . , 5 7 , 2 8 9 - 3 0 0( 1 9 6 1 ) .
1 0 . E d e s k u t y , Fred J . , and N ea l R. Amundson, I n d . E n g . Chem., 4 4 ,1 6 9 8 - 1 7 0 3 ( 1 9 5 2 ) .
1 1 . F o r s y t h e , G. E . , and W. R. Wason, F i n i t e - D i f f e r e n c e Methods f o rP 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 , John W i l e y and S o n s , I n c . ,New York , 1 9 6 0 .
1 2 . F u r n a s , C. C . , T r a n s . Am. I n s t . Chem. E n g r s . , 2 4 , 142 ( 1 9 3 0 ) .
1 3 . Gamson, B. W. , G. T h od os , and 0 . A. Hougen, T r a n s . Am. I n s t . o fChem. E n g r s . , 3 9 , 1 - 3 5 ( 1 9 4 3 ) .
14 . G e s e r , J . J . , and L. M. C a n j a r , A . I . Ch. E . J o u r n a l , 8 , 4 9 4 ( 1 9 6 2 ) .
15 . G i l l i l a n d and Baddour , I n d . Eng. Chem., 145: No. 2 , 3 3 0 - 7 ( 1 9 5 3 ) .
1 6 . G l u e c k a u f , E . , J . Chem. S o c . , 1 4 9 , 1302 ( 1 9 4 7 ) .
17 . _____________ , and J . I . C o a t e s , J . Chem. S o c . , 149 , 1315 ( 1 9 4 7 ) .
1 8 . ____________ , J . Chem. S o c . , 1 4 9 , 1321 ( 1 9 4 7 ) .
70
![Page 84: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/84.jpg)
19 .
2 0 .
2 1 .
2 2 .
2 3 .
2 4 .
2 5 .
2 6 .
2 7 .
2 8 .
2 9 .
3 0 .
3 1 .
3 2 .
3 3 .
3 4 .
3 5 .
71
Hamming, R. W. , N u m e r ic a l Methods f o r S c i e n t i s t s and E n g i n e e r s , M cG raw -H il l Book C o . , New Y ork , 1 9 6 2 .
Hernad, H. S . , J . Am. Chem. S o c . , 4 2 , 3 7 2 - 9 1 ( 1 9 2 0 ) .
H i e s t e r , N. K . , S . B. R a d d in g , R. L. N e l s o n , J r . , and Theodore V e r m e u le n , A . I . Ch. E . J o u r n a l , 2* No. 3 , 4 0 4 - 1 1 ( 1 9 5 6 ) .
_____________, and Theodore V e r m e u le n , Chem. E n g . P r o g . , 4 8 , 505( 1 9 5 2 ) .
H i l d e b r a n d , F. B . , I n t r o d u c t i o n t o N u m e r ic a 1 A n a l y s i s , McGraw- H i l l Book C o . , New York, 1 9 5 0 .
H i r s c h l e r , A. E . , and T. S . M e r t e s , I n d . E n g . Chem. , 4 7 , 193 ( 1 9 5 5 ) .
Hougen, 0 . A . , and W. R. M a r s h a l l , Chem. E n g . P r o g . , 4 3 , 197 ( 1 9 4 7 ) .
H uckaba , C. E . , and B. F. D i c k e r t , A . I . Ch■ E . Symp. S e r . No. 3 7 ,5 8 , 91 ( 1 9 6 2 ) .
H y p e r s o r p t i o n P r o c e s s Flow S h e e t , P e t . R e f . , 2 9 , 9 ( 1 9 5 0 ) .
IBM 7040 and 7044 Data P r o c e s s i n g S y s t e m s . S t u d e n t T ex t ,I n t e r n a t i o n a l B u s i n e s s M ach in e s C o r p . , P o u g h k e e p s i e , N. Y . ,1 9 6 3 .
IBM S y s te m s R e f e r e n c e L i b r a r y , IBM 7 0 4 0 / 7 0 4 4 O p e r a t i n g Sys tem ( 1 6 / 3 2 K) , F o r t r a n IV L a n g u a g e , I n t e r n a t i o n a l B u s i n e s s
M a c h in e s C o r p . , F i l e No. 7 0 4 0 - 2 5 , Form C 28-632 9 - 1 , P o u g h k e e p s i e , N. Y . , 1 9 6 3 .
J o h n s o n , A d r a in E . , J r . , A p p l i c a t i o n o f N u m e r ica l Methods i nA n a l y s i s o f F i x e d - Bed A d s o r p t i o n F r a c t i o n a t i o n . P h .D . D i s s e r t a t i o n , U n i v . o f F l o r i d a , D e p t , o f Ch. E . , 1 9 5 8 .
J u r y , S. H . , and Wm. L i c h t , J r . , Chem. E n g . P r o g . , 4 8 , 102 ( 1 9 5 2 ) .
K a s t e n , P a u l R . , and Neal R. Amundson, I n d . E n g . Chem. , 4 2 , 1 3 4 1 -6 ( 1 9 5 0 ) .
L ap id u s , L eon , D i g i t a l Computat i o n f o r C hem ica1 E n g i n e e r s , McGraw- H i l l Book C o . , New York, 1 9 6 2 .
_, and N. R. Amundson, J . P h y s . Chem. 5 6 , 3 7 3 , 984 ( 1 9 5 2 ) .
, and J . B. R o s e n , Chem. Eng. Symp. on I o n -E x c h a n g e . 50;No. 1 4 . 9 7 - 1 0 2 ( 1 9 5 4 ) .
![Page 85: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/85.jpg)
3 6 .
3 7 .
3 8 .
3 9 .
4 0 .
4 1 .
4 2 .
4 3 .
4 4 .
4 5 .
4 6 .
47 .
4 8 .
4 9 .
5 0 .
5 1 .
5 2 .
5 3 .
5 4 .
5 5 .
72
M a i r , R. J . , I n d . Eng. Chem. , 4 2 , 1 2 7 9 - 8 6 ( 1 9 5 0 ) .
M a r s h a l l , W. R . , and R. L. P l g f o r d , "The A p p l i c a t i o n o f D i f f e r e n t i a l E q u a t i o n s t o C hem ica l E n g i n e e r i n g P r o b l e m s , " U n i v . o f D e la w a r e , Newak, D e law are ( 1 9 4 7 ) .
Masamune, S. and J . M. S m i t h , A . I . Ch. E . J o u r n a l . 1 0 , 246 ( 1 9 6 4 ) .
McCracken, D. D . , and W. S . Dorn , N u m e r ic a l Methods and FORTRANP rogram m in g , John W i l e y and S o n s , I n c . , New Y ork, 1 9 6 4 .
M i c k l e y , H. S . , T. K. Sherwood, and C. E. R e e d , A p p l i e d M a th e m a t ic s i n C hem ica l E n g i n e e r i n g , M cG raw -Hil l Book C o . , New York, 1957.
O r g a n i c k , E. I . , A FORTRAN P r i m e r . A d d i s o n - W e s l e y P u b l i s h i n g C o . , I n c . , R e a d i n g , M a s s . , 1963 .
R o g i n s k y , S . , N a t u r e , 134 , 935 ( 1 9 3 4 ) .
R o s e , A r t h u r , R. J . Lombardo, and T. J . W i l l i a m s , I n d . E n g . Chem. , 4 3 : No. 11 , 2 4 5 4 ( 1 9 5 1 ) .
R o s e n , J . B . , J . Chem. P h y s . , 2 0 , 387 ( 1 9 5 2 ) .
Round, G. F . , Robin Newton, and P . J . R e d b e r g e r , A. I . Ch. E .Symp. S e r . , No. 3 7 , 5 8 , 2 9 ( 1 9 6 2 ) .
S c h m e l z e r , E. R . , M. C. M o l s t a d , and P. F. H a g e r t y , A. I . Ch. E ■ Symp. S e r . , No. 2 4 . 5 , 2 0 9 - 1 9 ( 1 9 5 1 ) .
Schumann, T. E. S . , J . F r a n k l i n I n s t . , 2 0 8 , 405 ( 1 9 2 9 ) .
S e l k e , W. A . , Y. Bard , A. D. P a s t e r n a k , and S . K. A d i t y a , A. I . Ch. E. J o u r n a l , 2 : No. 4 , 4 6 8 - 7 0 ( 1 9 5 6 ) .
T a y l o r , Hugh S . , J . Am. Chem. Soc ■ , 5 3 , 5 7 8 - 9 7 ( 1 9 3 1 ) .
Thomas, H. C. Ann. N. Y. A c a d . S c L , 4 9 , 161 ( 1 9 4 8 ) .
, J . Am. Chem. S o c ■, 6 6 , 1664 ( 1 9 4 4 ) .
_____________, J . Chem. P h y s . , 19 , 1213 ( 1 9 5 1 ) .
W i l s o n , J . N . , J . Am. Chem. S o c . , 6 2 , 1 5 8 3 -9 1 ( 1 9 4 0 ) .
Oth er R e f e r e n c e s
A n z e l i u s , A. Z . , Angew , M a th . und M ech . , 6 , 291 ( 1 9 2 6 ) .
B e r t z e r , F . , A nn . P h y s i k . , 3 7 , 4 7 2 - 5 0 7 ( 1 9 1 2 ) .
![Page 86: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/86.jpg)
5 6 .
5 7 .
5 8 .
5 9 .
6 0 .
6 1 .
6 2 .
6 3 .
6 4 .
6 5 .
6 6 .
6 7 .
6 8 .
6 9 .
70 .
71 .
73
C r e s p i , M. , A n a l e . S o c . E s p a n . F i s . Quim. , 3 2 , 3 0 - 4 2 ( 1 9 3 4 ) .
____________ , and V. A l e x a n d r e , A n a l e . S o c . E s p a n . F I s . Quim, 3 3 ,3 5 0 - 9 ( 1 9 3 5 ) .
____________ , A n a l e . S o c . E s p a n . F I s . Qulm. , 3 2 , 6 3 9 - 5 7 ( 1 9 3 4 ) .
D am kohler , G . , Z. P h y s i k . Chem. , A174 , 2 2 2 - 3 8 ( 1 9 3 5 ) .
D i e t 1, A . , K o l l . Chem. B e i h e s t e , 6 , 127 ( 1 9 1 4 ) .
F r e u n d l i c h , Z, P h y s i k . Chem. , 8 5 , 6 6 0 - 8 0 ( 1 9 1 4 ) .
G u r v i c h , L. G . , Z. Chem. I n d . K o l l o i d e . 1 1 , 1219 ( 1 9 1 3 ) .
I l i n , B. V . , J . R u s s . Chem. P h y s . S o c . , ( 1 9 2 5 ) .
____________ , J . Gen. Chem. ( U . S . S . R . ) , 2 , 4 3 1 - 4 1 ( 1 9 3 2 ) .
K r u y t , H. R . , R ec . T r a n . Chem. , 4 0 , 2 4 9 - 8 0 ( 1 9 2 1 ) .
M a r t i n , J . P . , and R. L. M. S yn ge , B r o c h e n . J . , 3 5 , 1 3 5 8 -6 8 ( 1 9 4 1 ) .
N i z o v k i n , V. K . , T r a n s . VI. M e n d e l ia n E n g r s . , 2 , 2 1 8 - 3 2 ( 1 9 3 5 ) .
P o l a n y i , M . , E l e k t r o c h e m , 2 7 , 1 4 2 - 5 0 ( 1 9 2 1 ) .
R a k o v s k i i , V . , J . R u s s . P h y s . Chem. S o c . , 4 4 , 8 3 6 - 4 9 ( 1 9 1 2 ) .
S m ik in , J . K . , and A. I . Kondrashon, K o l l o i d e Z . , 5 6 , 2 9 5 - 9 ( 1 9 3 1 ) .
T o l l o s i z k o , S ta n is lo -w , C o l l . C z e c k . Chem. Comm. , 2 , 3 4 4 -6 ( 1 9 3 0 ) .
![Page 87: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/87.jpg)
APPENDIX A
NUMERICAL METHODS
![Page 88: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/88.jpg)
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
![Page 89: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/89.jpg)
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 :
![Page 90: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/90.jpg)
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
![Page 91: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/91.jpg)
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
![Page 92: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/92.jpg)
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 .
![Page 93: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/93.jpg)
APPENDIX B
RESULTS
![Page 94: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/94.jpg)
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
![Page 95: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/95.jpg)
TABLE I I I
WAVE FORMATION TIMES, BED DEPTH, AND SATURATION
ICurve f a s ^a 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
![Page 96: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/96.jpg)
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
![Page 97: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/97.jpg)
o
00
r-
O
C?
C3
![Page 98: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/98.jpg)
5
1 . 2 5
2 . 2 5
3 . 2 5
4 . 2 5
5 . 2 5
6 . 2 5
8 . 2 5
&
![Page 99: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/99.jpg)
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'
![Page 100: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/100.jpg)
Oata K 3urve (xlO^)
500 .0
FIGURE 8. METHYLCYCLOHEXANE-TOLUENE ON SILICA GEL A = 4000-vj
![Page 101: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/101.jpg)
6 f t
Oo
![Page 102: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/102.jpg)
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
![Page 103: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/103.jpg)
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
![Page 104: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/104.jpg)
91
a .o01
Ct
o
![Page 105: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/105.jpg)
9a
![Page 106: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/106.jpg)
.5
H = 0 .3 7 5
CD
o
r °o
CO
1 0 . 5
o
^ . s w
C6
![Page 107: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/107.jpg)
- nC l
1 1 . 5
1 3 . 5
1 5 . 5
rw1 7 . 5
1 9 . 5
r o
co
^ 6
![Page 108: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/108.jpg)
95
e»
C3
C3
ru
![Page 109: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/109.jpg)
9 fe
![Page 110: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/110.jpg)
91
u»
![Page 111: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/111.jpg)
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
![Page 112: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/112.jpg)
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
![Page 113: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/113.jpg)
T .
- O~ o - o —
![Page 114: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/114.jpg)
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
![Page 115: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/115.jpg)
.5
-n
O
o
o
1 0 .5
*-* 7lO
o
z o i
![Page 116: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/116.jpg)
.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\
![Page 117: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/117.jpg)
rtf u ~
• *
![Page 118: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/118.jpg)
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-
![Page 119: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/119.jpg)
VO 6
IT* .~ • o
a
![Page 120: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/120.jpg)
0-1
w
![Page 121: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/121.jpg)
\pfc
<3,
01
![Page 122: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/122.jpg)
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
![Page 123: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/123.jpg)
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
![Page 124: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/124.jpg)
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
![Page 125: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/125.jpg)
* - 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
![Page 126: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/126.jpg)
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
![Page 127: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/127.jpg)
" 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 - :
x , C oncentration o f More Adsorbable Component in E fflu en t
::Ur
S
![Page 128: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/128.jpg)
•*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
x , C oncentration o f More Adsorbable Component in E fflu en t
fin
m
S
![Page 129: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/129.jpg)
APPENDIX C
COMPUTER PROGRAM
![Page 130: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/130.jpg)
oo
o
oo
o
o o
o o
o o
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
![Page 131: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/131.jpg)
oo
oo
o
o o
o o
oo
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
![Page 132: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/132.jpg)
oo
o
oo
o
oo
o
oo
o
oo
o
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
![Page 133: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/133.jpg)
oo
o
oo
oo
o
oo
o
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
![Page 134: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/134.jpg)
oo
o
oo
oo
o
oo
o
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
![Page 135: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/135.jpg)
oo
o
oo
o
oo
o
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
![Page 136: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/136.jpg)
oo
o
oo
o
oo
o
oo
o
oo
o
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)
123
![Page 137: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/137.jpg)
oo
o
oo
o
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)
124
![Page 138: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/138.jpg)
no
r*
oo
o
oo
oo
o
oo
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
![Page 139: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/139.jpg)
oo
oo
n
on
o
oo
o
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
![Page 140: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/140.jpg)
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
![Page 141: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/141.jpg)
oo
o
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
![Page 142: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/142.jpg)
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
![Page 143: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/143.jpg)
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
![Page 144: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/144.jpg)
no
n
oo
o
■*> ro
►-
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
![Page 145: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/145.jpg)
o n
n
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
132
![Page 146: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/146.jpg)
oo
o
oo
o
oo
oo
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
133
![Page 147: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/147.jpg)
oo
o
oo
oo
o
oo
-t" N>
H-
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
![Page 148: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/148.jpg)
GO TO 98 E Q = . U 7 5 * Y / l l . - . 9 3 9 8 * Y )9 EQBZHX=EQ
RETURN END
![Page 149: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/149.jpg)
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
![Page 150: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/150.jpg)
r v ) f N > f N > f N ) N ) f N J P ' o r N > f ' ) f t — *— f t - f t — ft— »— X - I — I—‘ X - x - x - f — I—P r — ft— ► - » — p—P X -^ W W M N M M O O ' O v O Q O C n ^ - j O ' f f - J f U l X ' I ' W W M M H ' f - O O ' O ' O C B C D - s l ' J i—p O' ft— O' »—■ o X— 0 s P—* O' ft— 0 s p— O' X— 0 s ft— O' x-p O' ft— O' I—* C? p— 0 s *—p O' ft— o ft— O' ft— O' X—
x-p 1—ft XXI—* XXft—ft—p—ft ft— ft—ft— ft—XXX- p— X - ft—ft—o o o O' O' O' O' O' O' O' O' O' O' O' CD CD CO o o o o o o o o o o o o o o rsj IV) IN) IN)
X J ' X XX* I' XXXN» NX NX NX NX NX NX NX NX NX IN) IN) IN) r*oIN) IN) ro IN) IN) IN) I—1 X“ftXxft—1—ftCD -J O' Nfl x NX in i ft—o -o CD 'J O' Nfl XNX i\j ft—ft o o CD -XO ' NJ1 X' NX r\j ft—o >o CD -J O' Ol X '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 o o• • • • • • • • • * • • • • * • ■ • • • * • ft • « • ft • • • • * • • •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 o oo 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 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 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 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• • • • • • ft ft • • • ft • • ■ ft ft ft ft a ft ft ft ft ft ft ft ft ft ft ft ft ft ft ftnd Nil Nfl Nfl Nn Nn Nfl Nfl Nfl Ol VI o i Ol XI XI XI XI X X X X X X X X X X X X XI X X X X Xo 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 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 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 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 oft • ft ft ft ft ft ft ft ft ft ft ft ft ft 4 ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ftX' X*X X*X X XXXXXXXXXXXXXXXXXXXXXXXXXXXXXsO © vO ©.©©©•o <£> SO o sO o o %£> o sO o X) xosO X3NOo o o ■o>0XOo o CD CD coCD CD CD 00 CD CD -J -J -X-X -X -X -X O ' O ' O ' O ' O ' X X X X X X NX NX NX IN) IN) 1— r— o sD oo 00X X* X) IN) XX o o —J O ' X X) H- o oo O ' X IN) o -X X fN) XD O ' IN) vO X p—■ O ' O ' o X —1 X) ft—TNI X) X ) X X ) IN) © -) X o XI XO IN) X X X x ) o X rv) IN) t— >—p IN) p—' O ' ' J X -X X X vO NXin> Nx X) o X IN) X) -X X) p—* p— K- o -X r x IN) CD CD O ' X -X X o X -X NX X X o -XX -X 00 X— X X O ' -X ■Q X p—< X 00 -X -X p—' X X t— 1 X NX U) I—p X) -^J - J NX in: CD X o NX o X-X -X O ' -X CD ft— IN) o CD p—p IN) h— -X X) o X o p— IN) X 00 NX vO X O ' CD o X X X X X IN) NX X
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 oft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft- J -X -X -X -X -X -X -X -X -X -X -X -X -X •X -X -X -X -X —J -X -X -X •X -X -X •X -X -X -X -X pX -XIN) IN) rN) IN) IN) IN) fN) IN) r x IN) rN) IN) IN) INJ fN) fN) INI IN) IN) IN) INI IN) INJ r x r x IN) IN) IN) INI INI IN) rv ft— ft— P“*0 X3 vO O CD 00 00 00 CD oo 00 -X -X - J -X O ' O ' O ' O ' X X X X X NX NX IN) Pn) - - ft— o vC o r -XNX fN) ft— c No a -X X X IN) ft— nC •X X Nx ft— CD O ' NX c r —J X ^p NX vO X ftO X Or NJ X •X cc ftCCD 00 00 -X O ' X -X IN) -X )—p NX X X X fN) n0 X -X O o —«J IV) O ' O ' NX CO CD X O ' NX IN) X vO NXIN' o - X O ' ►- o e — X nC UP IN) vC X NO CD NX NX -X X NX X- o r © X vO ft-p - J ft— OS PN) CD -X sC ©X nX X NX -X IN) nX CD CD 'X O ' X X X PN) O ' IN) X ■o CD NX X 00 p—■ X O ' 00 X O ' a s IN) oo X VO NDO ' NX p—1 O ' X ■£ o NX X o NX X p—p X IN) X X S ' IN) NX X CD NX O ' ftO NX X NX O' -o —J o
L Z I
![Page 151: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/151.jpg)
246
11
6 4
90.
00
00
0.50
00
0.4
96
60
60
2
0.7
29
47
32
525
1 11
6
50
0.0
00
0
0.50
00
0.4
98
68
49
6
0.7
29
55
86
925
6 11
6
51
0.00
00
0.50
00
0.4
98
75
92
6
0.7
29
63
96
326
1 11
6
520
.000
0 0.
5000
0
.49
88
29
54
0
.72
97
15
R7
138
05 3> cc r n (N r - cc i n m (Nj p - fN, o •O CC m. r" 4 i n pn in (N m n*y LT O >r 4-433 o O ' n - n"l o O ' 3 33 o JC o o o i n m r~ .■4 35 P3 (NI in LT PvJ ao O ' CC
i''- ON O C5 -O O ' P - O ' CO <Nj r \i CO O' (NJ o -o PSj —« CO —« 35 m m > r m 3 \CO m ~ h ac m CO m r— in cc o n j (NI rsj t n O ' 00 o CO m (Nj O 4—1 r\j o 00 (Nl COp— -O O O ' O o -4 >NI n j 'AJ m n O' N? nQ -NJ O ' NJ o O ' —4 GO —4 ^0 nT XJ —4 -n mc r O ' O ' O ' o o O o o O O c ; O pn c r pn co pn 3 ) o (NI in 00 o m 3 mINI (Nj rsj (NJ rr\ 'O m m ’■"I m m m m n r o n - oo CO O ' O ' o O o O o w—t ■ i 44 4p - r - r - r - P— p - r - r - p - r» r - p - r— m >c sO - o 35 .0 35 r~ r— n - r - h - n - p - r -
« . « . . . . « . « • . 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 rsj O ' (N 3 r - •o 3 3 c 3" o < r o 00 i n pn (NI O ' 05 m m co r - m P -n - (V O ' cc m 33 in O ' p 4 3 O ' o i r CO C m i n O ' 00 —J c c (NJ oo nT rH (NJ p-^ m (Nlj n 00 r - 3 - 4 i n p— n j n j o sO O r— (NJ rrv c r >0 O ' o 35 i n (NJ pn pn CA r - in 00 3 INIO ' i n —H p - n j >0 o 3 r - o 4-4 m m (NJ ro cc GO i n r - 3 o O ' (NJ O O n - CM o COCO O ' o o —j - 4 (NJ (Nl (Ni n j m m m O ' rn o p - 35 O ' 3 ) —4 CNJ *-4 30 nT ao —4 (NJCO CO O ' O ' O ' O ' O ' O ' O ' o O ' O ' O ' m , , c r O CO 3 00 (NI i n ao —H CA la 35 CO O ' 44 (NJO ' O ' o O ' O ' O ' O ' O ' O ' O ' O ' o O ' m 1—1 CNJ 3 3 i n i n 35 35 35 r - r - r - P - r - GO CO3 3 3 3 3 ■J" 3 3 3 3 3 3" r n 3 "T 3 3 3 3 3 3 3 3 s3- %r 3 > r 3 3
. . • * . * • . • * « . 4 4 4 • . 4 * • • • » 4 • • • • • * •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 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 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 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 oi n i n m i n i n m i n i n i n i n i n in m o o o o o o o o □ o o o o o o o o o
* • * . * . . . . . . 4 * 4 ft • ft ft ft • . • 4 ft ft • ft • • ft *o o o o o o o o o o o o o ■~4 ^4 fH 1—I I I *-4 44 pH pH
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 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 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 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
. . . . . . * . . 4 . 4 ft • 4 ft ft ft ft ft ft ft ft ft ft • ft f ft ft fto 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 om 3 i n o p - CO O ' o 4-4 (NJ m 3 i n 3^ i n nO p“ 00 O' o (NJ m 3 i n f - GO O' o pH
i n m m m m i n i n 35 33 o 33 o *o ^ 4 pH 4—I 4-4 44 44 pH p H (NJ CM
35 35 35 35 3 3 'j - 3 3 <r 3 3 - (NJ 3 - O ' 3 o to p - - c i n i n pn CA m pn m m (NJ PM(NJ (NJ (NJ (NI 4—1 ^ H ^H
rH w~4 •—< 4—4 ft-4 * 4 F—I pH 4-4 44 44
CM (NJ (NI (NJ e g (NJ r g (NJ (NJ (NJ CM CM CM CM (NJ (NJ (NJ (NJ
3 ) 1 35 4 4 •o ,-4 >o <5 35 o <3 —4 vO i-H o 3 ) 35 NO r-H nO rH 35 4 1 353> a - p - CO CO O ' O ' o O 1-4 (NJ (NJ e g (NI m PC 3 i n m 3 ) 35 r - r - GO GO O ' O' o o( N i ( \ | { N J ( N j ( N i ( \ l < \ J ( n m ( n ( n m r n
![Page 152: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/152.jpg)
N> IS) fS) fS) IS) rs> IS) IS) IS) IS) IS) fS) IS) rs) IS) IS) IS) aa00 -J -J o O' o’ O’ p -S' o j O) rs) IS) aa aa o o S01—< o> r— o aa o aa O' aa O' aa O' t—1 O' O' l-a O'
rs> IS) IS) rs) IS) N) IS) IS) IS) IS) IS) rs> fS) IS) IS) IS) IS) IS)a—* aa aa ►—■ HJ aa aa aa1 aa 1—■a-a a—1 aa a** aa a—■I—* aa
O 'O O O 'f f 'O 'O 'O O 'O 'W f f l f f l O J O O O O
O ’ o ’ o i O ’ O ' O ’ o ’ P p p p p P P p p J ' OJO ' O ’ p O) rs> F—• o >0 00 O ' O’ J ' u ) IS) a a o sOo o o o o o o o o o o o o o o o o o• • • • • • • • • • » • • • • • * to o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o o
a-a a a a-a a-a l-a - a l-a l-a l-a ►— ►— 1—* • • • • ft • * » • • • » ft a a a ao o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o oo 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 o o• • • • • • * • • • • • ■ • a a ft a-S' p p p p p p P ' P ' p p ' p ' p p> p* p* P ' P '■o vO vO o o o o sO sO sO s0 >£) o •o >0 sO - o sO
-J ~vl a-l - J O' O ' O ' O ' O ' sn sn Ol O’ p* P ' p> lj)O ' O’ P f\) o nC -o O ’ OJ K- P) O ' p> e- a O’ IS) COsO O ’ a—» O ' sO rs) p O’ U1 U ) 1—1 - J 1—• p* O ’ sn I jj ou> O ' IS) o sO oc - J O ' o ) o H - *—• O ' O ' P3 O ' O ’ p 'p IS) o ’ o j IS) o i o p 03 f— 00 r—* rs> o oo p o rs)o> 0D sO O ' O ' p a a O ' O ' U ) CO IS) IS) 00 O ' o O ’ IS)
o o o o o o o o o o o o o o o c o o• • * * • • • * • • * • » * a a a ft
- J a j -o - J ' J - J -4 a j - J ■o - J - J - JIS) IS) IS) IS) rs> IS) N ) IS) IS) IS) ISJ IS) ■S) IS) IS) IS) fs> IS)oo 0D oo 00 00 oo OS --J - J ' J O ' O ' O ' O ' 'O’ srt
(X o p o j a- vO 0= O ' P ' rs) o P ' IS) oC o -- J O ’ fS) sO o i o p cn o »—■ h— o CO P ' o OJ s i O’1— IS) —J J ' p o or. a - o ’ cx O ' O ' p ' o c r P ' Su O '00 ■aj O ’ sO O ' o O ' -o O ’ P ' >— 0 J o 03 IS) co O '-a) o sO -o ■—K a a O’ vO O’ - J “S o j ) O ' 0D 1—*
e e i
t—• a-* a— M a—• a - a-a a—■ 1—a a-a a a a-a a a a a a a<£) 03 00 -0 -o O ' O ' O ’ O ’ P P OJ O) rs) rsj a-a a-aM O ' a—* O ' a - O ' a - O ' a—* O ' a-a O ' O ' O ' a-a
N ) rs) IS) IS) ISJ ISJ IS) IS) IS) IS) IS) IS) rsj rs) IS) IS) IS)e — a—* *-* H— a — a—1 a—i a - a — a— a a a—* a a a—* a a a a
a-> a^- p« a — a— a—* i—* a— t—• a—• a a a a a-a a-a 1—■ a a a-ao o o o O O O O C O o O o rs j IS) IS) rs j
so oo oo OJ OJ OJ OJ O ) O J IS) IS) IS) IS) IS) IS) rsj IS)CO - J O ' O’ P OJ ISJ a—i o >£ CD —J O ' O ’ p OJ ISJo o o o o o o o o o O o O O o O oa a a a a a a a a a a a a a a a ao o o o O O O O O o o o o O o o Oo o o o o o o o o o o o o O o o oo o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o
a— a“ a—■ a—< a - 1—* a — a — a - a a a a a aa a a a a a a a a a a a a a a a ao o o o o O O O o O o O o O o o oo o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o oo 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 oa a a a a a a a a a a a a a a a ap ' p ' P ' p* P p P P P P p p p P P P pP3 o vO vO aC o vO 'O CC CD 03 03 CD CD CD CD CD00 OJ IS) IS) a * a—> o o o o 03 - J O ' O ' O ’ P OJO’ a— -0 OJ 03 O) 03 O ) a — P " J O a—a OJ Oj OJoo 01 p> a — O ’ O ’ O ISJ O OJ OJ 03 o - J O 'oo o p* OJ O ' O ’ O ' O) a— vO OJ o a a p IS) jOJ O’ O ' O ' O’ o OJ ~*l IS) oo p O ' sO ISJ O ’ oo O ’ o o O) -o a-* OJ aO a—* OJ ISJ 03 O ’ OJ Cc OJ
o o o o o o o o o c o o o o o o o• a ft a ft a a a a a a a a a a • ft-0 -o - J •aj - J -J -aj -J -O -0 —J - J ->l - JIS) IS) r o )Sj ISJ IS) IS) IS) rs) f\) IS) ro a-* a-a a a a-a a-a0*1 O ’ p* p O) o0 rs) IS) rs> M o o -C 03 03 - J O 'Oj cr O ' rv Cc, p >0 O ' o P CD IS) 0*1 C C rs.' O ’p> o O’ -o -sl O’ o a— o O ’ ^1 O ’ c c O ' sO O ’ O 'a— 'J IS) on O ’ o CD O ' P OJ c o> a— C j sC op o O ) p p IS) a—1 - J p OJ IS) —J P p O 'IS> o vC O ’ -o C O ’ OJ p P P p ISI a— ISJ ——J
![Page 153: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/153.jpg)
w >— J— ft— •—> OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ rsj IS) ISJrs) ►- J - o o o sO OD 09 -J -J o o on OI -ps J* OJ OJ OI OI OI ■f" ftp- OJ OJ ISJ ISJ ft—I ft—> o o vO o 09i—* O' J - O' t—l 0 ft— O' *-> O' ft— o t—' O' ft—- o ft— o ft— cc o — o 1—* O' ft— O' J - O' ft— o O' ft— O'
OJ OJ OJ OJ OJ O J OJ O J O J OJ OJ O J O J O J O J O J O J O J O J rs j rsj i s j rs) rsj rsj ISJ is j is j rs j ISJ NJ ISJ i s j isj r oft— I— J - ft—1 I-* 1— ft—> >— ft— •—1 ft— ft— »— >—* ft— h— i—* r— ft— ft— ft— ft— ^ ft— ft— ►ft ft— ft— f t- ft— ft- »—• 1—* ft—
ft— J - ft— ft— J - ►« ft— j - ft—> 1— ft—■ *“ > ISJ TSJ rsj IS) ISJ OJOJ OJ OJ o j o j on on on on O' —J -0 o o I—" OJ O M j ' M j ' j ' j ' j ' j ' j ' j ' j ' O C ' O o d ' O ' t s
IS» rsj ISJ rsj ISJ ft— ft— ft— 1—* ft— J - ft— J — I—1 ft— - J ' J -0 o O' o o O' o o o o o on on onOJ ISJ ft— o "Q 00 -0 O' on -p 0 J ISJ J - o sO 00 - J O' ft— ft— o o CO -0 o on X ' OJ ISJ ft— o oO CD ' J
o o o o o o o o o o o o o o o o o o o -p' 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 o o o o o o o o o o o o o o o o oo 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 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 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
J - ft— ft— ft— ft— ft— ft— ►- ft— ft— ft— ft— ft— ft— ft— 1—* 1—> 1— ft— ft—t—1 ►- t—1 ft- I—* ft— ft— ft— ft—ft— I—* ft—• • • * * • * » • ft • • » • • • • ft • • • • • ft • ft • • • • • ft • * ftVJ1 on on on on on OI on on on on on on on on on on on on o o o o o o o o o o o o o o o oo 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 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 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 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• • • * • • • • • • • * • • • • • * ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ftX ' X* J - JS JS 4> n - -p n - -p OJ 0 J -p* ■r* -t" J ' JS -p> ^* JS JS JS -p- j s JS-ol - J O ' O ' O ' O ' on on on -P> 0>J OiJ IS) o ' J 1—1 sO ■X3 sO O sO *0 -o •o sO 'C ■o ■3 sC sC sO o-f' no o CD O ' J* I—* sO on ISJ CD UJ 00 IS) J* OiJ o CO 00 CD CD CD CD 00 CO CD 05 09 CD CD CD 09 CD ' J►— j s —J —J —J 00 o sO p 00 o o 'O o on on O ' O ' O ' O ' O ' O ' on on 4 - ou Oft* ISJ ft— o VC CDoo sO ISJ o o r—' 00 ^0 -sj O ' -vj O ' -^1 ISJ 0D OJ O J CD CD " J O' 0*) o O ' ►— O' sO ft— Oft) -T- 0 J 1—1
O ' oo vO JS 1—• on c c 00 u J sO os) ' J - J t— ISJ ■o 00 O ' o vO O ' O ' *£3 o Oft) vO on OftJ Oft) n - - J09 O '—• O ' on o 0 J 'O 00 O o u 00 vO rs) IS) CO O ' sO O ' -J o O ' o on JS ISJ 09 O ' o 00 o OI 1—■ 'DOO CD - J 0*) OiJ sO o OU JS O ' ISJ OU r— cc o CD 0 J J ' OJ -o on O ' o 00 J ' 0*1 Ok) O ' S ' Oft) ■f' ft—
o o o o o o o o o o o o o o o o o o o o o o c o o o o a o o o o o o oft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft •-ftl - J -ftj - J ~J —J O' O' O' O' O' O' O' O' O' O ' O ' O' Oft) -J —J - J - J ftj - J - J - J —J —J -ft) -ft) - J ftftl -ftj ^ 1ft— o o o o o vC sO sC j : CD 09 - J -J O' on Oft) o ft- ISJ ISJ rs> ISJ rs* ISj IS) IS) rs* IS) rsj rs) IS) ISJ NJ njo 09 -ftj on OftJ 1—* o -P ft— -ftj OJ CD rsj Oft) ISJ O' on 00 'C vO sO o ftO ■o ftO ftO ftO ftO •o 'O ftO ftO oo O' ft— on a o ft— J ' Oft) O' -p Oft* o ftC sO 0- c or ftO < J ftO 09 a C? —J O' o or X- OJ rs; ft— oSJ O' -J on -J o Oft) o -p ft- 00 ISJ ft- o ISJ Oft) CD OftJ on S ' IS) ftO —ft ISJ rsj O' o OJ on —J CD 05 0Do ft— Xft ft- ou -J o O' CO ISJ CD »c S ' ftC on o O' or ISJ o S ' o OJ ft- ISJ —J 09 O' ft- ft— X- *— H— X - t - 'n o ISJ ft— vO ft— O' 09 Ok) —J vO on o NJ ^S ISJ on ISJ rsj o 0J S ' O' IS) JS on CD IS) 0J vO O ' vO on on O ' O'•J '1 o |SJ Oft) ft—ft— o J ' o - J ISJ o ■Jl rs) -ftj 09 ft— 0J J ' O ' 0J o ISJ Nj Nj -J rsj -ft) v J -ftl Oj o o
o n
![Page 154: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/154.jpg)
1 4 1
n p*- O' OJ lA 3 O L“ LA X rM O' CM O' CM X o- x> X CM X CO CO X CM O' —1 CM O' X 3 X XF“4 CM ce re r - ia x X x ^4 X X O' X X r - X X GO O' *3- CM rsj 'd- CM X X O' o 3- 3- X X X X< r x re r - r e CO 3 4ft- 3 X X X X CM H X O' -4 “4 o O' r - c - CM X CM r— X X O' O' O'cr X J \ o r4 cr CM O CO CM X O' 3 X O' 0- CM 4fl- in X O' CM 'd' lr \ X o X o X 3- X 3- CM O' X
3 LPl a x ■e CM o c- 3 3 X .-M CM 0- CM o o <r i^- —4 >• o X X X o M 3" o X O' —ftf\t in P" X X O' O o -ft —i CM CM X X ■j- M- X LP X X X X C' r - r - f'- r - r»- X—4 «—4 ^4 —-4 ~h •H —1 CM CM OJ CM OJ CM CM rM' CM CM CM PJ <NJ CM CM CM CM CM CM CM CM CM CM CM
r - f" r - c- r - r - r - r - r~ c- C- 0- r— r - r - O' O- r - r^ r - f - P- r - r - r - r - r - f - f~r h- r - r - r - r -* » • • • • • ■ » • * ft • • • * * • « * • » • • • * • ft • * * * • • •
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 o o
o *sf f - x r-~ X CM 0- o- X O' X o o- X -fl- X O' «r ^4 rP X CM CM 3- X X CM c- X CO Xr—- r - 0" X 'fl- X CO X 3 o X X X OJ i - t X o X O' X r4 X O r4 c c O' —-ft X X X CO c—< 03 r-- cm CC CM O' X X CM o -ft o O' O o X O' ^4 X CM X X rfN o X r - CO X X X o X X Xx r - o x CO r - X 3 3 O' O' ■J- X —ft X <■ o X *T X X X CM X c- r ~ X X X X X X Xx o re X r - X X X C- uo X X X •H X X r o X CM X o X —4 3- r~- o X X X o CM 3X r - CO O' o CM co 3 X X f - r - X O' O' o o f 4 CM CM X X 3- 3- 3" X X X X X X Xr— r - r - r - 0C X X X X X X X X X X CO O' O' O' O' O' O' O' CP O' O' O' O' O' O' O' O' O' O' O'3 3 'd" 3 3 3 3 3 3 3 3 3 3 3 3 •fl- nT -fl- ■fl- ->3- •fl- -3- ■c 3- 3- 3- 3- 3- 3- 3- 3 3 3 3
* • • • ft » * « • * * • • • ■ • * * • • * * • • • * • • ft ft • ft ft ft fto 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 ©
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 o oo 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 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 oIT. in IA in in in X X X X X X X X X X X X X X X X X LA X X X X X X X X X X X
■ • • • * • • • • • • • • * • • • • • 4 * • » * ■ ■ ft ft ft * ft * ft ■ fti—4 i i —4 —H l—1 ^4 r—4 r-4 r-4 4-M 4-J ^4 —ft
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 o oo 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 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 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
• * * « ft • • • • • ft • • • * « * • » • • • • • • « ft ft ft ft ft ft ft ft fto 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 oin x cc O' o — > CM CO 3 X X r - X O' o CM X sf X X r - X O' o 1—* CM X 3- X X CO O'CM CM CM CM CM ff\ CO CO X X X X X X X -fl- -fl- •fl- -fl- -fl- -fl- 3- 3- 3" 3- X X X X X X X X X X
X cm f\J CM CM CM CM CM o o o o o o o o o o o o o o o o o o o X X X X X X X Xr-ft ^4 •—4 —M —ft —ft •—4 -ft r-M •—4 r-4
H #—4 —ft H l—i 4—4 ^4 r—4 *•4 c 4 p*4 r—4 r-M r-4 —M —ft "dX re m m re x CO CO fO X X X X X X X X ro X X X ro r c \ X X X X X X X X X X X X
X ^4 sO f—4 X H o —4 o 'O SO o ^4 O X 4—4 X X pH X r—4 X —H X —4 X —ft X XCM m re 3 3 LA LA NO o r - r - CO CO CP CP o o r— CM CM X X 3 3 X X X X r - r - m X O' O'— 4 —4 i-H — 4 #4 #-4 ^4 ^ 4 *»4 — 4 p H f* 4 rsj rsj CM CM CM CM CM cm CM CM CM CM CM CM CM CM CM CM CM CM
![Page 155: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/155.jpg)
ft »—* »—■ >—. ►— o j OJ 0 J OJ OJ OJ OJ OJ OJ OJ 0 J OJ OJ OJ OJ OJ OJ OJps) ►— o O X X 00 00 - J - J O ' x X X p* p* 00 03 - o ' J x O' OI X -O' p - OJ OJ r o IO ft—* o of t- O ' (7* ft- x ft— x f t- x N- x f t- O' O ' ft— J* ft—. O' f t- O ' 1—. O' ft—. O' ft— O ' ft— O' * ’ O ' ft- O ' ft—
.p* ■p- > P* p* p “ p - p> P3 p ' J* p3 p ' p i JN OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJi—• ►— •—• ft- ft— ft—. f t- ft—ft ft— ft- ft— ft-— ft- f t- ft—* ft— ►- ft— f t- ft— ft-— ft— ft— ft— ft—* 1—. ft— ft - f t - ft— »—■ ft— ft—•
ft— ft— ft— ft— ft—ft— I*— ft—>— io N) no ro ro ro OJx x x x x O' O' -0 X o o ft—OJ OJ “O x P ' r o P ' P ' P ' P ' p ' X x x x x x x x x x x x
fO ro fo ro ro 1— ft— ft—ft- ft—ft—ft—ft— ft— “J -o ■O - j -0 -J O' O' O' O' X O' O' O' O' O'p* OJ ro i— o X 0D -0 O' o i p> Jj ro i—• o X CD O' O' o i uJ ro ft— o X 33 -o O' o i -O ■Jj ro ft- oO o o o o o o o o o o o o o o o o X o o o o o o o o o o o o o o o o o* ft • • • • • • • • • • • • • * » • • • • • • • * • * • • • • • • • •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 o oo 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 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 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
ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro 1—«ft—ft—ft—ft- < -* >— > * t-* r -* ►—« * • • ft • ft • • » • * * • • • * ft » * ft ft • # ■ • • » * • • • » » •o o o o o o o o o o o o o o o o o 03 OI OI o i o i o i o i OI 03 03 OI o i 03 03 03 OI o i OIo o o o o o o o o o o o o o o o o o o o o o o c c o o o o o o o o c oo 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 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 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 om • • » ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft • ft ft ft ft ft ft ft ft ftO ' -T' > - t ' -p" > J ' O ' O ' -C' > 0 J OJ O J ro J ' O ' -P* p i -F* ■p- - f l -> J3 > -S' -F' -p-c r 03 OI 03 -p> OJ OJ IO t—* »—1 o 03 -o C3 X) o o X) X) X x> X X X X X X X X X X Xo - o J ' t— oo J ' o O ' h - OI 00 o ►— X) OJ X5 CD l-¥1 co CO CD -o -0 -o - o ' J - o -o - J - o - o ■o O ' O '
U1 0 0 CD OI CD OJ O J *o cc' r o ►— c c ft- c o c o X X X a -o c r O ' o i OJ IO r— X 03 O 'OJ ■o OJ t o OJ o -0 o o -o OJ »—* OJ 03 o 03 03 O ' O' CO J> o 0J -0 X o o X - J OJ CD r o Xr o o o OJ o OJ >o •—i o O ' o i - J 03 J> 03 03 —J OJ o CD CO CD o X t— O ' 03 o i -p* o oJ X -E' Mv3 o OI -0 - J o 1—■ -X) ■OI *0 <3 CO o 03 OJ o 1—• <— ■> NJ h— o o co O ' o OJ OJ O ' o ro ro X X- J o o i -J ro 1—■ 00 CO ro o OJ -F' P ' O' ->l o x> ro o CD CD CD O' ro r - OI ro O ' ft—' -o CD O ' O '
o o o o o c o o o o o o o o o o o o o o o o o o o o o o o o c o c o oft ft ft ft ■ ft 4 ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft • ft ft ft ft ft ft ft ft ■ •O ' O' O' O' O' O ' O ' O' O ' O' O' X X X X X ro - J - J -o - J - J - J -o -o -o -J —J ->l -0 -o -JX X X X X CD CD X O' X X -T' X 1—- —J OC ro ro ro ro IO IO IO ro ro IO ro ro ro ro IO IO rsj ISJX X X ro o -o VjJ X X X OJ X X OJ OJ - J 00 X X X X X X X X X X X X CD X CD CD CD CDcr - J -r- X ro t— O' X ft— Cc X X ft— o o X *— X O' 03 X ■P- p- Om OJ ro ft— 1— c - X X - J or. >r o - t ' -o O ' o ft—1 -T- o ^1 X X X OJ co ft— oo X OJ o -o r o ft— X 03 o ft- no ft— o X oX Cr r o X o X -J X cr- ft— X C3 -F' ro OJ X 03 p* 0J CO -o c . -J p- X r o ft- ro O'- c no ft— c : sOX O ' f o O ' X r o X ■p* r o ro - J J j IO o t X p i o o o X X --J ft—■ OJ 0 J OJ X p* X X ft— X o o &X OJ X O ' - o O ' CftJ ro CftJ -c* ft—• X X ■Jj " o X - o ft— f t - ■o ■o - J o OJ X —J pft f t - co ft— f t - ■o sD
Z*7l
![Page 156: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/156.jpg)
143
i n r— <x g O ' O' co LO r-~ cc rg O ' Ni-» o ' cc rmm4 <r X) r - ro -O' (Nj cc X O' O ' ■g- O' (Nl CD o (Ni om o O' —< g G sO L*. r - o r - r g O C" ay a \ ' J 'J - •sT O ' u> cc r g cr r g r g cc ( 0 o o G o
o O' G CO r - XV i n nT ^ r CO o> rsj O' r— r o O ' pH O cO P - CO g iT\ o n - >r Nj <r o O ' (NJ (NJ•J" o sT i n G r - O ay m ■s* O ' CO Os,’ rg o r - sj* sO o eg •S nT pH X CO CO X ^H m r - r~ g (NJ CO (NJ
0 \ —1 g o 'O G r-~ CO CO r — o Xv m o s3- o ■o ro r— ■NJ *■» pH a \ O ' >* o ro o O ' (NJ Gph UN G cc O ' o f—4 <NJ m nT ay o r - 00 00 c r o o pH pH rg (Nl rn rr\ m sT o* in IT. X. i n G G gO O O o o pH pH —H fH ■ F j —H pH pH co eg rg r g r g r g rg rg rg r g r g r g (NJ (NJ (Nl (NJ (NJ (NJr - c - r - r - r » r - r - r - r - r - r - r - r - r - r - r - r - r - r - r - p - f— r~ r - r - r - r - r - r - r - c - r -
* • • • « • • • • « « • • * • • • • * • « • * • • • • * • « * • • • *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 o o
dN #—1 o r— c c (Nl O' O ' UN m nC <r p - X IT. r - a y o X *jy ■J- g - >o 00 o G o (Nl o r -O ' G —1 O (Ni O ' LT. O ' o sC r - ro X h - sC O ' X o r g r - c r 00 r o t r o hJ- sT r u o CD PN IN- N- (NlUN O' c LIN r - vO rn UN o m rH o in O r g m r - r - «—i i—4 sC X X cr* r - CD UN O' •u- UN G O 'CD G o O ' O (Nl * <r (NJ m X O CO m sO r g r o o r o rg r - CO o f-H m r o o CN UN '3 ‘ (NJ r - o pH3 g r— UN INN O UN o < r O ' o r—< o O ' r - \T\ r g O ' m ro CO rO r - (Nl G O u - r - vT(NJ 3 UD CO o (Nl CN UN o 00 o r g m vT sT irv O r - r - X O ' c r O o (NJ (Nl PN (N (O vT -3"G -O sC O r - r - (N- r~ r~ r - CO CO 00 CO X X X 00 co X X CO X 0 s <r O ' O ' O ' O ' O ' O ' O' O '•X •J" ■U- •X NT '3 ' ■J- 'T nT ■>r < r ■J- <r > r g - nJ- <■ <r s f s f g - sT ' j - ■3" «3" -J- NT %3- •3-
* * • • « * • * • • ft # ft » • « • • ft ♦ ft * ft ft • • • • • • • • • •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 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 o o o o o o oo 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 o c 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 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 c o
• • • • » • • • • • * « • • • * • • * * • • • • « • * * • • » • t * •(NJ (NJ (Nl (NJ (Nl (NJ (Nj (Nl (NJ (NJ (NJ (NJ (NJ (NJ (Nl (NJ (NJ (NJ (Nl (NJ (Nl (Nl (NJ (NJ (NJ (Nl (Nl (Nj (Nl (NJ (Nl (Nl (NJ (Nl (NJ
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 o oo 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 o o o o o o o O o o o o o o V—J G o o o o o o o o o o o o o o oo 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 o
• t t * « • • • • • • * • * • * * • * « • ft ft * • » « ■ * • • * * » #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 o oUN G e- oo O' o *—1 (NJ ro ■J" UN o !— cc O' o 1-1 (NJ CO <r UN - o r— 00 O' o p-H (NJ ro s t UN sO r - 00 O'(Nl (Nl (NJ (Nl (NJ (Cl r< \ ro ON (O CO CO CO co -31 -3- "3- ■3- St *t «t «t UN UN UN UN UN UN UN UN UN UN
c o r o r o <o r o r o r o rg r g (NJ (Nl r o r g rg r g o o o o o o o o o o o o o o o o o 0 8 8p*H■1 ^ *—(t-i pH —HpH 1-4■4 1-H PHP-H p—H
p H i H p H p H p H p ^ •p-H p H • 4 p H • i ^ H ■—1 — 4 • 4 P*0 p H p H p H p 'H p H p H — 4 ——J H p H
s t ■3- s t ' f >t >t ' t s t s t ■3- 3 - s t ■T sr s t > t s t 3 - S t s t 3 - s t s t 3 3 3 3 3 3 3 3 3
G G p ^ G G G #pH G G p ^ G p-g sO f —H G r - 4 G G p H G H G G * - 4 G —4 G H G(NJ co ro -3- sr UN G G G r~ O- CO CD O' O' o o ppH (Nl ru ro CO 3 - 3 in UN G G r - O- G G O' O'
^ • 4 p H p 4 r - 4 p H p H P»H (Nl ro rg (Nl (NJ (N J (NJ (NJ (NJ (NJ e g (NJ (Nl (NJ (Nl CNJ (NJ (Nl (Nl (Nl
![Page 157: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/157.jpg)
ft— (—I 1—* X" OJ OJ OJ OJ OJ o j OJ OJ OJ OJ OJ OJ OJ o J OJ OJ OJ OJ OJ OJft— ft— o o o o 00 00 -J O' O' Vn o ft0 o 00 00 ' J - J O ' O' on O ' x> X- OJ OJ IN) r o ft— o oo ft- O' *—• O' ►—• O' ft—* O' ft— O' M O' ft— ft—1 O' O' ft— O' O' ►—» O' O' ft— O' I—1 O' ft— O ' ft— O' ft—
vn on \n on o n o i o n o n o n o i o n o i j i j i ^ J> X- X' X* X- X- X' X» X' X> J ' -> X' X- X* X* X* X* X*>— h- *— *—• ft— ft— H- ft— i— i— ►— —— *—* ft— ft- 1—I ft— ►— ft— ft— M I—1 ft- ft— ft— >—* >— ft— ft—* ft— ft— ft—
ft- ft— ft- I—1ft- fo) no r o r o foj (OJ OJ OJO ' O ' ' J vD X3 Oo O J OJ OJ-0O- 0 ^ ' ^ ' ^ ' J ' J ' C ' 0 ' 0 ' 0 ' 0 ' 0 ' 0 ' 0 ' C ' 0 ' 0 ' 0 ' 0 ' 0 ' 0 ' 0 0 3 3
n o ro n o foj ft— 1—* ft— ft— ft— p— ft— ft- ft—1— 03 ' J —J —J - J -X - J -0 -v l O ' O ' O ' O ' O ' O ' O ' O ' O ' O 'OJ ro f t- o ■o 0 0 -0 O ' on X- J j N) 1—’ o o sO 03 - J O ' on X* OJ IN) o o 33 - J O ' on X*OJ no ft-— oO o 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 oft • • • * • • * • » • * * # • • • • • » * » * * • • • » * • • • • • •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 o oo 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 o o o o oo 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 o o 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
(NJ ro ro ro ro (NJ (NJ IN) r o ro (NJ ro ro (NJ (NJ ro (NJ INJ (NJ (NJ NJ (NJ ro NJ NJ N J no N J ro NJ no no NJ N> N J• • ft ft ft ft ft ft ft ft ft ft ft ft • • ft ft • ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft fton OI on on o n on on on on n on on on 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 o o o o o c 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 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 o o o o o o o o o c 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 o o o o o o o o o o o o o o o o o o o o o o oft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft • ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft-> J ' -p* X* X ' x- JT oo oo oo oo oo 00 NJ JS -T' JN -F' J ' -S' X ' -T- X ' X* X* JN X ' X ' X* Xs x» X ' X ' X*OftJ oo oo NJ NJ ft- o 03 - 0 o > NJ O ' sO o o -c ~o -0 o 03 O 03 X3 sO 03 vO o J3 03 vO X3 X3<3 on o on o -r- - o vO o sO U1 - 0 ft—■ ft—* -0 - o —J -0 - i -0 - J cn O ' O ' O ' O ' O ' O ' o n o n on o n X X 'f o ft— - 0 >c o n on O ' n - Oo - 0 CD j : r o _N 00 oo oo INJ (Nj o >o CD - J o r X ' Nj f t- X3 -vj X ' NJ ftO -ftjft— 03 - 0 ft— o oo on o n o - O ' o o n o (NJ ft— o C r r o - J ft— oo on on on NJ 03 X* CD ft— (NJ ft— CD X- c o ft—r o o CO ft— N J on CD o •£> o o ’vjj ft—- —J c r o n ■S' o n vn oo 03 on o ft— vO O ' sO -J f t- o NJ - J O ' CD OftJOftJ - J o - J o o O ' ft— ’J j NJ -0 N j 00 ft— NJ on CC —J 00 -n ft- 03 o NJ on 3 vO 30 - J 33 ft— OftJ on o n ft—OftJ ftN ftO ■jj ft— O ' ft— ft— ft- ft— - o ft— - 0 on O ' o c o o o C3 OftJ o ft— o c r O ' CD NJ O ' X3 -ftj ft— ft— N j
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 o o o o o o )-N oft • • • • • • • • • • • * • » • • • • « • • • » • • • » • • • • • • •O ' o r> o o •> a* cr* O ' o v-P u s N) - o ' J - J -4 - J - J -4 —J - J -4 —J - 4 -4 - J - 4 - 4 -4 -4CD CD -Nj O' o J s J J rsj nD u s o r o r o r o ro r o Nj NJ r u N j NJ r o fsj fsj fsj ro r o f o ro ro N j N>OftJ o o rsj INJ o 00 CO w 1-^ NO o x> 'C CD CD :r> CD CJZ CD 3? oo CD X -4 - 4 "4 r s<3 a - a -Nj vC IN) a a ’ U) r v 1— *—ft c c . vT c* Cc -vj c - m X- OJ t—* c cc -o O" O-' •sC—J o oo CD o UJ o H- N j -""J -T" o O ' o Xr c r o ro SjJ s>> ro o - 4 OftJ N-ft-N- r o X* O ' on X 'o - o o u ; o p— - s ’ -Ts* - J K- h— f o O ' nC fN r*» X- u / X* O ' O' O ' O' OJ o ru 'Hftft' -4ftO o ac rsj CD OftJ r—' o S ' OI H- ro O ' co 1“ r o Nj O ' UJ •ftftH U4 CD sO -ft. OJ on o o > v lOJ- - 4 !SJ J" r-r *— o o ■NJ —J o - J a o _*N X" sP O' cc -C" o>/ vC rjs ST o r OJ ■ftft on
![Page 158: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/158.jpg)
-C *c ■s / 'l A ■XLoST —- r--* cr CO 0s I-*-,
CD o 3 sO sT -0 3 o r—9 ^O' ro >c r\j rsj CO<Mmo r-. o CO j0 o >T Or- O' rsj sr o COo rs; •*«?CO COcr C" cr cr O O OsO >C gD *c nOg: r- A- r-• • • • • • « « •o o o o o o o o o
O CO r v rss r— o r - ino t n vT rsj •H r - 4 Xcr a LT, r - !— 43 o in mc r c r m LTi CO i n 43 4CO rs) O lO O' — i (Mrs. *c* cr rsj LT. r - O ' OJ 4s r %?■ in LAin i n 43 4)
nT <T 4 4 4• • « • • • • ■ 4
o o o o o O o o O
O o O O o o o o Oo c o o o o o o oo o o o o o o o oin in in in in m in in in
• * * * « ■ • • •o i (Nl Ol Oj f\l OJ r\i OJ OJ
O o o o o o o O oo o o o o o o o oo o o o o o o o oo o o o o o o o o
• • * • 4 • 4 4 •o o o o o o o o o4 in 43 r— CC O' o rs.OJ OJ OJ 04 04 OJ ro m
in lA in in m m CA rc,w 044 fH
v-4 *—4 »■ 4 >*4* —4in in in in m in in in in
sO ■4 43 o o **4CMrsj m ro sj- <r in n sO^4 —4 ^4 •~4 4
n •sC 43 CO G nj sTin r- ro rsj O 03 Cro o- o- O o O o r-m 43 O' nT rsj CMon O' OJ n r 03 30 30in COcr O r\i mO a o o • 4 —tr*~ O' n- r- r- n- r-• 4 4 t 4 • « •o o o o o o o o
O' o ■n 43 3 o oCSi o r—CC ro m o-*c o'. OJ AJ O . . i mcr ro 43 cr ro 00 in 4"f—4 .—-s_t rn O' ro r—OsC CCO' OJ >r m 0-X 43 43 r- f—o~ r 0-
>1“ sT >r 4- <-• 4 4 4 4 4 4 4o o o o o o o o
o o o o o o o oo o o o o o o oo o o o o o o on in in in IT in IT. in• 4 4 4 4 4 4 4rsj OJ OJ OJ OJ OJ OJ Oj
O o O O O o o oO o o O o o o co o o O o o o oo o o o o o o o* 4 4 4 • 4 4 4o o o o o o o om 4 in 43 n- COO' om rororomroro-T
rr ro ro ro m OJ OJ OJ—h r—4 #«h4
4 ^ 4 *—*in in mm in in min
o 43*4o 43o r- r- COCD O' COo
ph 4 4 rsj
rsj >c Aj O' Cs o ■cnr LT cc o aj X Orr\ o O CO rO| O —4 O'r - s r r - >!" cc 00 <r sO M"n - o M" n j O ' sO - n c r ns+ in vO n- r - COcr cr O
pH fMr - r- r— r - n - r— r— r - r -
• • • • • • * • •o o O o o O o o o
LA r - 4 M m NOX p" H <x> <rcsj r - >?■ c ro o ocr in O' m o r-r r - rosO CM in m sD rr o mrg in m in sT m • ^ O'X cr O CM nr in inr—. r - X X X X X co CO
<■ M" ■*$• >1- <rt « • * * * 4 4O o o O o o o o o
o o o o o o o o oo o o o o o o o oo o c o o o o o oin in in in in in in in in4 4 4 4 4 4 4 4 4OJ OJ OJ OJ OJ OJ OJ OJ OJ
o o o o o o O o Oc o o o o o o o oo o o o o o o o oo o o o o o o o o4 4 4 4 4 4 4 4 4o o o o o o o o oOJ ro ■J- in 4) r - CO O''3‘ ■tf H- H- 4- <T •n H- ■4-
OJ OJ OJ Ol OJ O o o ow~-i ^4 —4 H
4—4«—4r—itn m in in in in in m in
43 r-4o sO 43 p 43O CM CM ro ro ■4- s fOJ CM CM CM CM OJ OJ 04 OJ
145
o sT ro in. n- ro ro 4■4 O' o <r <n CD O X<r O' r- r—OJ rr\ ro O' ■no OJ OI O' nT 43 43 ro O'—4 4) in 3 4" COOJ m4-1 OJ C 4 m ro ro 4 4OJ OJ OI OJ OJ OJ Oj OJ OJr- r— n* r- r- p- r- r-4 • 4 4 4 4 4 4 4o o o o o o o o o
O' in 4- O' m OJ 43o IT. 4> Oi m X o 43 O'r- CD o cc 4- OJ X 43o- o 43 o O' 43 o O X4) nT O r*- OJ X) 4 O' m4) r—CO00 O' O' O o pHco COco CD X X O' a O'
4- ■>n *n vT 4- 4 4 44 4 4 4 4 4 4 4 4o o o o o o o o o
o o o o o o o o oo o o o o o o o oo o o o o o o o oin in tn. tn in in in in in4 4 4 4 4 4 4 4 4
OJ OJ OJ OJ OJ OJ Ol OJ Ol
o o o o o o o o oo o o o o o o o oo o o o o o o o oo o o o o o o o o4 4 4 4 4 4 4 4 4o o o o o o o o oo OJ m vj- in 43 p- Xin in in in in in in in m
o o o o o o o o opH—* pHpH -H
pH pH pH —Hin in n, in in in in m tn
<-j 4) pH43 pH 43 rH 43in in 4) 43 n- f- X X O'OI OI OI OJ Ol OJ OJ OJ Ol
![Page 159: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/159.jpg)
fl—is j
a—a- A—
fl—o
fl-o 43 43 00 00 -4 ' J O'
i—> O ' O ' O ' A - O ' A - O' A— O'
O ' O ' O ' O ' O ' O ' O ' O ' o- O' O ' O '>—1 A— A — A - A— A - A— A- A- 1—• r—■ fl—
A - A- A- IS) M ISJ ISJ rsj fS) rsj OJ -P-O 43 43 o o OJ OJ o j O' 43 rsj (—•
rsj (SJ ISJ rsj rsj A- A- A - H t—* 1—• A—-P OJ ISJ a— o o 00 ' J O ' Ol -P LkJo o o o o o o o o o o o• t • • » • • * • • • •o o o o o o o o o o o oo o o o o o o o o o o oo o o o o o o o o o o oo o o o o o o o c o o o
OJ OJ OJ OJ OJ OJ OJ 0 J OJ OJ OJ OJfl • • * • • * # * • • ■o o o o o o o o o o o oo o o o o o o o o o o oo o o o o o o o o o o oo o o o o o o c o o CDo
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
9*71
-P O J O J O J O J OJ OJ OJ O J O J OJO ' o 43 -c 00 CO -4 -4 O ' O ' 41 onA— A— O ' fl— O ' fl— O ' O ' fl- O ' A-
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 '
( . 00 -4 -4 1-4 -4 -J -4 1 -4 —Jro o 4300—J O' 41-pOJ isj fl—oo 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 o o o o C3o o oo o o o o o o o * "Io o oo o o o o o o o o o o o
u> rs j r\> rs j ro rs j rs j rs j ro rs j N? n ;* • • • • • • • • • • •o \T \ w1 o x-n■jn ■jn■sJI 'w"o o o o o o o o o o oo o o o o o o o o o oo o o o a o c o ;j>o o o
o o o o o o o o o o o o• « • • • » • • • • • »rs j -> JS -> -c*J J sO sO o o ■c sC o o o o sCo n s o O ' Qs o c> sn onc r o s/*1 Uo' flNj •—’ o c r o o --P' r— 4N OJ o X-T c c o onO CO —J ^ s o e - CO fs j H- OJo o o o “ 4 H— CO OJ ■_oo UJ -sJ sO ■jn 'J o 'sjJ o
o o o o o o o o o o o* • • • • • • • • « tno•0 -o -J -J -J -0 -4 -4 -4 -4OJ rs j ro no (Sj ro no n j s S ) S i rs jcn JS*S XX -or* X r — X -4 -4 —4 -4C l c \J" _ s OJ rs j ►— a 4 CLo o r■t* o c r sC o rn«s O ' VjJ o -p-■sC *< rv yC o Cr r - c r C3 -4 O ' so 'oJ w o rs j c? ’NJ S ) f l- OJ —4 o
v_r ro Co o X—4 o 0 - o -4
OJ OJ OJ OJ OJ OJ OJ OJ OJ 0J rs jJS -ts OJ OJ s> s fl— fl— o o 43O ' fl— O ' fl— O ' I ’ O ' f l— O ' f l- O '
sn on U1 on on on on U1 on 41 onfl— I—1 fl— fl— f l - fl— f l— f l—
fl— fl— f l- f l— fl—O ' O ' O ' 03 03 CD o o o o o
O ' O ' O ' O ' O ' O ' O ' O ' O ' O ' 41■o 00 -4 o J1 J ' 04 rs j fl— o 43o o o o o o o o o o o• • • » • » ' • 4 » • •o o o o o o o o o o oo o o o o o o o o o oo o o o o o o o o o oo o c o o o o o o o o
S ) SJ S ) s> ro) s> s s ISJ s> S )• • ■ • • • * • • » »41 on ■oi U1 on O ' O l 41 '41 41 onc o o c o o c o o o oo o o o o o o o o o oo o o o o o o o o o o
o o o o o o o o o o o» * • » * • • • • ft ft-JS -ts Js -P -p- -p* -p- -p -P43 43 43 43 43 43 43 43 o 43 sOU1 -> -p 4S OJ OJ OJ S ) s r -c 03 O ' OJ o -4 -p o C ' s CO03 4T o OJ -p- OJ o 4> O ' O ' VjJ43 03 O ' O ' O ' 4 s o on -4 o r sCC3 o S I fl— oJ SJ C3 -p ■41 03 a-rs o’* on 03 on OJ OJ CD -4 X
c o o o o c o o o o o• • * • * • • • • • ft-4 -4 -4 - J -4 -4 -4 -4 -J -s iSJ ISJ rs j S ) S ' s s r o ISJ s n j-4 -4 •4 O ' O ' O ' O ' 41 on ono - «— c *4 o r u . c - c o r s sc-4 43 c 43 Cc -T' o ’j j -p X- WS-4 43 cn 43 c O ' O ’ f l— fl 41 n jf l- co OJ -4 43 ■41 -4 — o -4 o■CC '■jj 04 O-l OJ rp O ' oX
![Page 160: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/160.jpg)
N } ro ro ro ro N> ro ro ro ro ro ro r j ro ro ro ro ro ro ro M t—1 l“ » t— 1—■ p- I—1 ►—* (—* t—t *— t—1 1—•vO o 00 CD -4 -J O ' O ' O l o i ■P' -P- OJ LO ro ro o O X) 'O 05 05 -0 -0 O ' O ' O l O l -C* (JJ (Jj
O ' >—* O ' • O ' »—* O ' »—» O ' *—■ O ' t— O ' i—• O ' »— O ' O ' *—* O ' I—1 O ' 1—* O ' O ' r“ O ' t— O ' M 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 ' O ' O ' O ' O '*— >—» 1—* t—■ »»•* f-* *—* t—• H—* ►— ►— M >—* I—* f—* t—• t—■ t—* *—* 1—- I—* 1—> f—* ►— t—■
*—* I—1 ►— l“ > (—* t— r - H—r-* ■—* t—> I—* h—» I— t—* I—* I— t—* * 1—* f—• r—o o o o o o o o o r o ro ro ro ro ro ro ro OJ OJ OJ OJ u J OJ OJ o j o j o i O l o i o i o i Ol cr O ' -J
Ol Ol Ol Ol Ol Ol Ol oi Ol Ol J' J' -p*J' •F*-t' J*■p' ■F*(Jj UJUJOJOJOJOJOJOJOJrororororovO 05-oO ' Ol ->OJro o •o 03-J O ' oi OJfo►—*o o 03-j O ' Ol OJro o sO 35*oO ' Olo o o o o c 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» * • • * • » • • » * • • • * • * • » ■ » • 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 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 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 o o o o o o o o o o o o o o o J—•>o o o o o o o o o o o o o o o
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
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 0 0 0 0 04 4 4 4 4 4 • * ♦ • • • • * • • • • « » ■ ft ft ft ft ft ft ft ft ft ft ft ft ft fth£> -p* -p* X ' -r* -P' -o 4> S ' 4N 4> JN S ' S ' s > S ' S - 4 " 4 - 4 * S ' S ' - o S '
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
O o oo O' vO H - u i c> c ui > N> o o N j - o 1— o n - o 'O 4 4 CD O' O J 0 0 ►— O J o^ 0 0 0
o n O J - 0 0 3 - o O l O l r>j * ro - 4 oj Ol o O J 0 0 rv o* o c —4 OJ 4" 44 -o c r S ' CO - 4 cr N Jro o o O' ro c> c n oo “*rj - o c ISJ o o 4> 1—- - 4 O J OJ cn -p* o n r o 0 Ol f—• cr sCo - J ro — ro O l o v n 0 0 ' r l Co N J »— ro CD - o 4C N> O ' - J 4> 4 s H—o O J JV - 4 0 0
oi on ■£> C r O' ro o CD o »— o No O ' CD CD cc CD o t— p— o N j N j o O'! ■Ol 44 > 0 J S cr -*4
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0ft • ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft • ft ft • ft ft *- 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 ^ 4 - J - 4 ' J - 4 - 4 - 4 - 4 - 4 - 4 - 4 -4 - 4 ~4 - 4 * 4 - 4 c r 0 c r c r /^ s c r »s 0 c rN ; N J N j N j N J N J H - ft— H - ►— »— 1—' 1— H— p—1 *—■ p— P-—• 0 0 0 0 0 0 4 0 0 40 43 4 3 CD CD CD - 4 - 4NJ N J J—- »— 0 ^■s 0 OD CO - 4 c r 0 0 4> UJ NJ H— 0 sTJ ~4 c r o i OJ NJ 0 CD c* 4> NJ 0 ■4 K— c r
N> - 4 r s . - 4 0 <rC r v O ' CD c NJ 0 . - r -N O ’ N j c : a o r h— c r 0 O-’ o ' a O ' OJ c c r c r - r cOJ -0 s > rO O l CQ 43 O I 0 c c Ov p—» OJ 0 c r c r sO NJ NJ *S>J O l c r e r - r c c ^ 4 0 Cc a c OJOJ SG N j O" 0 43 H* »—» 0 - 4 - r OJ 0 vC OJ 0 - 4 *4 c r CD - r - 4 - 4 c r OJ “4 O r r*. r> c r c r4* O l - 4 r o 0 NJ 0 *— Nj N J c r c r o J P' J sO OJ CD 0 c r »— 00 0 0 NJ S ' CD - r *— " 4 - 4 b *4 04> OJ NJ 4 on ■«£ 43 o j - 4 0 0 0 - 4 NJ IT 4 j 0 0 X" rv r*-; *4 0 c r r 1 o j ►— N^ p— * 4
m
![Page 161: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/161.jpg)
148
or* r\i Ts, r— ry\ cr. IN, rCt cO c r--4 (-0 c ro *>—i cc o —r - N c N CO p~ O' 00 sC NJ CO cc o X .—4 p - * 4 p - O cp■"Nl 10 .—4 N NJ pp X P - rO N j X •NJ o r>. IN p* x , ■*o. o _NsC sC lP X r - N- U"\ Nj Is- o NJ rO m 4 CP u- o c r - CP r-*.
-N p o O ' N! ..N rc *N r s r— cp 0> N X r s o OrN rN X vr x U"S UN LN s£ 'C -o s0 ■0 P- r - r - P - P -
n j NJ (Ni N Nj r \ , •NJ (NJ NJ NJ NJ Nl Nl N N N NJ N l Nl N Nr - n - P - r - P - P - P - P - r - P - r— p - r - r - P - p - N r - P -
• • • * • • • • * • • • ■ • • * * * • • •c o o C o o O O o o o O o o o o o O o o O
r o X o sC f f—i F—< o o CO ro. lT- f** p - r o r o LP. o lT»p - n£- NJ o c c L '' o r - p-H 4—1 X c c r c c X sC X JC cc lT .CP x X NJ p> x C 1 CC Co X r - CP O NJ X c c X CVJfN -*s —4 p - *—i fNJ o c o .—j o X rO o r - P - UP — A < , O 'NJ r o x> NJ p - —i i/s X3 NJ UP X o ■ro u o p - ?p vT COcr CP o o •r 4 »—4 Os! f\J N ) fO r o r o •X X X X X UA u '' \ I \CO c c CP CP CP O ' c - c - 0 s CP CP CP CP CP CP c r c - c -x X x X X - c •J- •T X X s t X X X X X X N?" >?■ > r < r
• • • • • « • ■ • • * • • • • ■ • • « * •o o o o o o o o O o o o o o o o o o o o Q
O o o c c O o O O o o O O O O O c O o O oo o o o o o o o O o o O O o O O c o o o oo o o o o o o o o o r—s O o o o o o o o o oo o o o o o o o o o o O o c o o o o o o c
• t * 4 4 4 4 4 4 4 4 4 4 4 • 4 4 4 4 4 4
r o O', rO r o r o r o r o r o r o r o r o ro , rO r o r o r o r o r o r o r o CO
o o CD o o o o o o o o o o O CD CD o o o o oo o o o o o c o o o o o o o O C o o o o oo o o o o o o o o o o o o CD O O o o o o oo o o o o o o o o o o o o o o o o o c o o
4 4 4 4 4 4 4 4 4 4 ■ 4 4 4 4 4 4 4 4 4 4
c o o o o o o o o o o o o c c o o o o o oc (—M OJ fO <■ o CO O ' o rs. r o sC IO sO 1 - cc O ' c<3 o sO n0 o sO -a •G «o vC r - 0 - r - r - 0 - r - r - r - 0 - r— co
c o o o o o o o o 00 co oc >D <5 vC o ■o s0 sG o sO<—. r-H ^ 4
—I 4—4 4»-4 ^-4 r—* *—1 ^ 4 4 4o •O o >o • o • o >0 sO sO sC ■sO sO o sO sC O sC sO o sC
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
![Page 162: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/162.jpg)
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
©
![Page 163: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/163.jpg)
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 ^^End 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
![Page 164: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/164.jpg)
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
![Page 165: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/165.jpg)
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
![Page 166: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/166.jpg)
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
![Page 167: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/167.jpg)
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
![Page 168: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/168.jpg)
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 )
![Page 169: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/169.jpg)
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
![Page 170: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/170.jpg)
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
![Page 171: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/171.jpg)
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
![Page 172: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/172.jpg)
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.
![Page 173: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/173.jpg)
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
![Page 174: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/174.jpg)
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
![Page 175: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/175.jpg)
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
![Page 176: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/176.jpg)
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 ^
![Page 177: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/177.jpg)
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
![Page 178: Analysis of Liquid-Phase Adsorption Fractionation in Fixed ...](https://reader031.fdocuments.us/reader031/viewer/2022012014/61591d657b793d1657335a5f/html5/thumbnails/178.jpg)
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