Theoretical Prediction of Breakthrough Curves for Molecular Sieve

31

Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0121.ch031

Theoretical Prediction of Breakthrough Curves for Molecular Sieve Adsorption Columns D. R. GARG and D. M. RUTHVEN Department of Chemical Engineering, University of New Brunswick, Fredericton, Ν. B., Canada

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass -transfer rate equation with boundary conditions imposed by the differentialfluidphase mass balance. For systems which obe a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of non linear equations must be solved numerically. Solutions hav been obtained for saturation and regeneration of molecular siev adsorption columns. Predicted breakthrough curves are com pared with experimental data for sorption of ethane and ethylen on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regene ation is slower than saturation. This is a consequence of non linearities of the system and does not imply any difference in intrinsic rate constants. I^he b r e a k t h r o u g h c u r v e for a fixed-bed a d s o r p t i o n c o l u m n m a y be p r e dieted t h e o r e t i c a l l y f r o m the s o l u t i o n of t h e a p p r o p r i a t e mass-transfer r a t e e q u a t i o n subject t o t h e b o u n d a r y c o n d i t i o n s i m p o s e d b y t h e differ e n t i a l fluid phase mass b a l a n c e for a n element of t h e c o l u m n . F o r m o l e c u l a r sieve a d s o r b e n t s t h i s p r o b l e m is c o m p l i c a t e d b y t h e n o n l i n e a r i t y of t h e e q u i l i b r i u m i s o t h e r m w h i c h leads t o n o n l i n e a r i t i e s b o t h i n t h e differential equations a n d i n the b o u n d a r y c o n d i t i o n s . T h i s p a p e r s u m m a r i z e s t h e p r i n c i p a l conclusions r e a c h e d f r o m a recent n u m e r i c a l s o l u t i o n of t h i s p r o b l e m (1). T h e a p p r o x i m a t i o n s i n v o l v e d i n t h e a n a l y s i s are r e a l i s t i c for m a n y p r a c t i c a l systems, a n d t h e v a l i d i t y of t h e t h e o r y is c o n f i r m e d b y comparison w i t h experiment. 345 In Molecular Sieves; Meier, W., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

346

MOLECULAR SIEVES


Theoretical Prediction of Breakthrough Curve W e consider a p a c k e d c o l u m n , i n i t i a l l y a t e q u i l i b r i u m w i t h t h e feed s t r e a m , s u b j e c t e d at t i m e zero t o a step change i n t h e i n l e t c o n c e n t r a t i o n of adsorbable c o m p o n e n t . T h e f o l l o w i n g a p p r o x i m a t i o n s are i n t r o d u c e d t o s i m p l i f y t h e a n a l y s i s . (1) T h e feed s t r e a m is assumed t o consist of a d i l u t e m i x t u r e c o n t a i n i n g o n l y a single adsorbable c o m p o n e n t . (2) The s y s t e m is a s s u m e d t o be i s o t h e r m a l , a n d pressure d r o p t h r o u g h t h e c o l u m n is neglected. (3) T h e fluid v e l o c i t y is t a k e n t o be u n i f o r m across a n y section of t h e c o l u m n , a n d a x i a l d i s p e r s i o n is neglected ( p l u g flow a s s u m p tion). (4) T h e size of t h e adsorbent pellets is assumed t o be sufficiently s m a l l so t h a t changes i n t h e fluid phase c o n c e n t r a t i o n over a single pellet m a y be neglected. (5) E q u i l i b r i u m between fluid phase a n d a d s o r b e d phase is a s s u m e d t o o b e y a L a n g m u i r e q u a t i o n .

g

q

s

= — 1+

(1) be

A l t h o u g h t h e a s s u m p t i o n s of t h e L a n g m u i r m o d e l are g e n e r a l l y n o t f u l filled for m o l e c u l a r sieve adsorbents, t h i s e q u a t i o n has been f o u n d t o p r o v i d e a useful e m p i r i c a l r e p r e s e n t a t i o n of the isotherms (2). T h e p a r a m eters b a n d q m u s t , h o w e v e r , be r e g a r d e d s i m p l y as e m p i r i c a l constants. B

S u b j e c t t o t h e a p p r o x i m a t i o n s n o t e d a b o v e , t h e d i f f e r e n t i a l fluid phase mass b a l a n c e f o r a n element of t h e c o l u m n m a y be w r i t t e n ,?5

$5 +

+

àz

bt

lS_o

(2)

m bt

a n d t h e i n i t i a l a n d b o u n d a r y c o n d i t i o n s for a step change i n feed c o n c e n t r a t i o n a t t i m e zero are saturation

q(z, 0) = 0, c(0, t) = c , c(z, 0) = 0

(3)

regeneration

q(z, 0) = q , c(0, t) = 0, c(z, 0) = Co

(4)

0

0

T h e d e r i v a t i v e àq/àt represents t h e r a t e of mass t r a n s f e r f r o m t h e fluid phase t o t h e zeolite c r y s t a l s . T h e f o r m of t h e mass-transfer r a t e e q u a t i o n depends o n t h e n a t u r e of t h e c o n t r o l l i n g resistance. C o m m e r c i a l m o l e c u l a r sieve pellets consist of s m a l l zeolite c r y s t a l s f o r m e d i n t o a m a c r o p o r o u s pellet g e n e r a l l y w i t h t h e a i d of a n i n e r t c l a y b i n d e r . I n t h e present a n a l y s i s t h e pellets are considered t o be s p h e r i c a l , a n d each p e l l e t is a s s u m e d t o c o n t a i n a n assemblage of u n i f o r m l y sized s p h e r i c a l zeolite c r y s t a l s . T h i s i d e a l i z a t i o n m u s t be t r e a t e d w i t h c a u t i o n since zeolite c r y s t a l s are n o t s p h e r i c a l , a n d the range of c r y s t a l sizes present i n some c o m m e r c i a l m o l e c u l a r sieve pellets m a y be q u i t e large (3).

In Molecular Sieves; Meier, W., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

31.

347

Breakthrough Curves

GARG AND RUTHVEN


T h e r e are three d i s t i n c t mass-transfer resistances: (1) t h e e x t e r n a l resistance of t h e fluid film s u r r o u n d i n g t h e pellet, (2) t h e diffusional resist ance of t h e macropores of t h e pellet, a n d (3) t h e diffusional resistance of t h e zeolite c r y s t a l s . T h e e x t e r n a l mass-transfer resistance m a y be e s t i m a t e d f r o m w e l l - e s t a b l i s h e d correlations (4, 5) a n d is g e n e r a l l y negligible for m o l e c u l a r sieve adsorbers so t h a t , u n d e r p r a c t i c a l o p e r a t i n g c o n d i t i o n s , t h e r a t e of mass t r a n s f e r is c o n t r o l l e d b y either m a c r o p o r e diffusion o r zeolitic diffusion. I n t h e present a n a l y s i s w e consider o n l y systems i n w h i c h one or other of these resistances i s d o m i n a n t . I f b o t h resistances are of c o m p a r a b l e i m p o r t a n c e t h e a n a l y s i s becomes m o r e difficult. T h e d r i v i n g force f o r t r a n s p o r t w i t h i n t h e zeolite c r y s t a l s appears t o be t h e g r a d i e n t of c h e m i c a l p o t e n t i a l r a t h e r t h a n t h e c o n c e n t r a t i o n g r a d i ent, a n d , f o r systems w i t h a n o n l i n e a r e q u i l i b r i u m i s o t h e r m , t h e d i f f u s i v i t y is therefore c o n c e n t r a t i o n dependent (6-8). D D alna D dine = D* — — = D* — ο In q ο In q

(5)

D

z

T h i s e q u a t i o n , w i t h a c o n s t a n t v a l u e f o r Z)*, has b e e n s h o w n t o p r o v i d e a s a t i s f a c t o r y c o r r e l a t i o n of e x p e r i m e n t a l d i f f u s i v i t y d a t a for several zeolitic systems (6-8).

F o r a s y s t e m w h i c h obeys t h e L a n g m u i r i s o t h e r m , E q u a t i o n

5 becomes

D,

(6)

=

1 -

q/q*

T h e a p p r o p r i a t e f o r m of t h e diffusion e q u a t i o n , w h e n zeolitic diffusion is the c o n t r o l l i n g resistance, is t h u s Γ

aq _ D* a at

=

r

2

àrll

r

agi

2

-

q/q,àr]

w i t h the initial and boundary conditions given b y q(r 0) = 0 (saturation) or q (regeneration) }

0

q(r , z

t -

ζ/υ)

= bq c(l s

aq/àr(0, t -

ζ/υ)

+ 6c)"

1

= 0

T h e average solid-phase c o n c e n t r a t i o n , w h i c h is r e l a t e d t o t h e

(8) (9) (10)

fluid-phase

c o n c e n t r a t i o n b y E q u a t i o n 1, is g i v e n b y 3

r

qrWr


(11)


348

MOLECULAR SIEVES

T h e s o l u t i o n of E q u a t i o n s 2 - 4 a n d 7-11 gives t h e t h e o r e t i c a l b r e a k t h r o u g h c u r v e for t h e case of z e o l i t i c diffusion c o n t r o l . If zeolitic diffusion is sufficiently r a p i d so t h a t t h e sorbate c o n c e n t r a t i o n t h r o u g h a n y p a r t i c u l a r c r y s t a l is essentially c o n s t a n t a n d i n e q u i l i b r i u m w i t h t h e m a c r o p o r e fluid j u s t outside t h e c r y s t a l , t h e r a t e of mass t r a n s f e r w i l l be c o n t r o l l e d b y t r a n s p o r t t h r o u g h t h e macropores of t h e pellet. T r a n s p o r t t h r o u g h the macropores m a y be assumed t o occur b y a diffusional process c h a r a c t e r i z e d b y a c o n s t a n t pore diffusion coefficient D p . T h e r e l e v a n t f o r m of t h e diffusion e q u a t i o n , n e g l e c t i n g a c c u m u l a t i o n i n t h e fluid phase w i t h i n t h e macropores w h i c h is generally s m a l l i n c o m p a r i s o n w i t h a c c u m u l a t i o n w i t h i n t h e zeolite c r y s t a l s , is

gg

D p Α Γ ρ * Ί

p

=

at

-

w(l

e ) R

aR

a

Γ

2

p

(i2)

L aR J

a n d for a L a n g m u i r i s o t h e r m aq àt

eD

=

p

w(l

-

p

R

e )bqM àRl(l p

agi

2

-

2

q/q*) àRj 2

w i t h the appropriate initial and boundary conditions given b y g(R, 0) = 0 (saturation) or q (regeneration)

(14)

q(R , t - ζ/υ) = 6g c(l + 6c)"

(15)

0

p

B

(0, < - Φ)

1

= 0

(16)

T h e average adsorbed phase c o n c e n t r a t i o n at a n y p o i n t i n t h e c o l u m n is given b y

qRHR

(17)

" £p J o 3

T h e b r e a k t h r o u g h c u r v e for the case of m a c r o p o r e diffusion c o n t r o l m a y t h u s be o b t a i n e d f r o m t h e s o l u t i o n of E q u a t i o n s 2 - 4 a n d 13-17. T h e equations were t r a n s f o r m e d i n t o dimensionless f o r m a n d s o l v e d b y numerical methods. S o l u t i o n s of t h e diffusion equations (7 or 13) were o b t a i n e d b y t h e C r a n k - N i c h o l s o n m e t h o d (9) w h i l e E q u a t i o n 2 was s o l v e d b y a f o r w a r d finite difference scheme. T h e t h e o r e t i c a l b r e a k t h r o u g h curves were o b t a i n e d i n t e r m s of t h e f o l l o w i n g dimensionless v a r i a b l e s K =

QO/QS

= &Co(l + fcco)" ; 1

φ

= c/c ; 0

X

=

bq z/mvr; B

Τ =

(t —

z/v)/r


31.

GARG AND RUTHVEN

349

Breakthrough Curves

where t h e t i m e c o n s t a n t τ = r /D* for zeolitic diffusion c o n t r o l , a n d τ = Rj> bq w(l — e ) / D p € p for m a c r o p o r e diffusion c o n t r o l . T h e c o m p u t a t i o n s are b u l k y , a n d t o generate t h e solutions for p r a c t i c a l c o l u m n lengths r e q u i r e d considerable c o m p u t e r t i m e . F o r the l i n e a r case (bc 0, λ = 0) t h e b r e a k t h r o u g h curves c a l c u l a t e d b y the n u m e r i c a l m e t h o d agreed w e l l w i t h the a n a l y t i c s o l u t i o n of R o s e n (10, 11) t h u s c o n f i r m i n g , a t least for a linear s y s t e m , t h e v a l i d i t y of the c o m p u t a t i o n a l a l g o r i t h m . 2

2

2

s

p


0

T y p i c a l t h e o r e t i c a l b r e a k t h r o u g h curves are s h o w n i n F i g u r e s 1 a n d 2 for a range of λ v a l u e s a t one p a r t i c u l a r X v a l u e . A m o r e extensive s u m m a r y of t h e n u m e r i c a l s o l u t i o n , for t h e case of z e o l i t i c diffusion c o n t r o l , has been g i v e n b y G a r g (1). F o r t h e l i n e a r s y s t e m t h e b r e a k t h r o u g h curves are s y m m e t r i c , a n d t h e curves for s a t u r a t i o n a n d regeneration of t h e c o l u m n are m i r r o r images. T h i s s y m m e t r y disappears a t t h e higher v a l u e s of λ, a n d t h e s a t u r a t i o n b r e a k t h r o u g h c u r v e approaches a step f u n c t i o n w h i l e t h e r e g e n e r a t i o n c u r v e develops a p r o n o u n c e d t a i l . T h u s i t is clear t h a t for n o n l i n e a r systems t h e regeneration process w i l l be m u c h slower t h a n column saturation under comparable conditions. A t sufficiently large values of X t h e s a t u r a t i o n curves a p p r o a c h a " c o n s t a n t p a t t e r n " f o r m , a n d thereafter t h e c o n c e n t r a t i o n f r o n t progress t h r o u g h t h e c o l u m r i a t a steady v e l o c i t y , g o v e r n e d b y t h e c a p a c i t y of t h e adsorbent a n d t h e feed c o n c e n t r a t i o n , w i t h n o f u r t h e r change i n t h e shape of t h e c u r v e . S u c h b e h a v i o r is c h a r a c t e r i s t i c of systems w i t h a f a v o r a b l e e q u i l i b r i u m i s o t h e r m (12). T h e c o n s t a n t p a t t e r n l i m i t is reached w h e n the dimensionless c o n c e n t r a t i o n profile i n fluid phase a n d adsorbed phase become p r a c t i c a l l y coincident, a n d t h e a s y m p t o t i c f o r m of t h e b r e a k l.0i

τ Figure 1.

Theoretical breakthrough curves for zeolitic diffusion control at X = 1.0: saturation ( ), regeneration ()



350

Figure 2.

MOLECULAR SIEVES

Theoretical breakthrough curves for macropore diffusion control at X 1.0: saturation ( ), regeneration ( )

Figure 3. Comparison of asymptotic constant pattern saturation breakthrough curves for λ = 0.445: (1) zeolitic diffusion control with D independent of con centration, (2) zeolitic diffusion control, (3) macropore diffusion control z

t h r o u g h c u r v e m a y be d e r i v e d s i m p l y f r o m t h e s o l u t i o n of t h e differential r a t e e q u a t i o n ( E q u a t i o n 7 or 13) subject t o the c o n d i t i o n Φο

=

g/go


(18)

=

31.

GARG AND RUTHVEN

351

Breakthrough Curves


C u r v e s c a l c u l a t e d i n t h i s w a y for m a c r o p o r e c o n t r o l a n d z e o l i t i c diffusion c o n t r o l are c o m p a r e d i n F i g u r e 3 for one p a r t i c u l a r v a l u e of λ. A l s o s h o w n i n t h i s figure is t h e t h e o r e t i c a l c u r v e for z e o l i t i c diffusion c o n t r o l w i t h a c o n s t a n t d i f f u s i v i t y . Differences between t h e shapes of these c u r v e s are not large a l t h o u g h t h e case of zeolitic diffusion c o n t r o l w i t h a c o n s t a n t d i f f u s i v i t y leads t o s u b s t a n t i a l l y greater t a i l i n g .

t -minutes

Figure 4- Comparison of experimental and theoretical breakthrough curves for CiHv-He in SA molecular sieve: (0 and X) experimental results for saturation and regeneration, respectively; ( ) Rosen linear solution; ( ) present nonlinear solution for λ = 0.05 and D*/r = 2.20 X 10~ sec' 2

t

z

1

Comparison of Experimental and Theoretical Breakthrough Curves E x p e r i m e n t a l b r e a k t h r o u g h curves for t h e s a t u r a t i o n a n d regeneration of a c o l u m n p a c k e d w i t h L i n d e 5 A m o l e c u l a r sieve, w i t h a feed s t r e a m c o n t a i n i n g a s m a l l c o n c e n t r a t i o n of ethane i n h e l i u m , are s h o w n i n F i g u r e 4. E x p e r i m e n t a l details are s u m m a r i z e d i n T a b l e I . F o r t h i s s y s t e m t h e Table I.

Details of Experimental Conditions and Diffusional Time Constants» V,

Sorbate C H 2

6

C H

4

2

α

SOT-

CO,

bent

%

Linde 2.14 5A Linde 1.99 4A

D€/ R bq X P

Temp, °C

cm/ sec

cm

m

50

11.2

91.5

1.08

24

22.0

5.2

1.16

z,

D*/r , 2

2

z

bq

s

313

sec"

p

w(l — e )

1

2.20 Χ 10"

P

e

p

8

24,692 1.19 Χ 10"

4

19.50 Χ 10"

8

6.46 X 10"*

Operating pressure was atmospheric in all cases.


352

MOLECULAR SIEVES

d e v i a t i o n of t h e i s o t h e r m s f r o m l i n e a r i t y is s m a l l (λ = 0.05), b u t t h e effect o n t h e b r e a k t h r o u g h c u r v e is s t i l l significant. T h e slope of t h e s a t u r a t i o n c u r v e is a p p r e c i a b l y greater t h a n t h a t of t h e regeneration c u r v e , a n d t h e m e a n of t h e t w o curves lies close t o t h e t h e o r e t i c a l c u r v e for a l i n e a r s y s t e m c a l c u l a t e d f r o m R o s e n ' s a n a l y s i s (10, 11). T h e t h e o r e t i c a l curves, c a l c u l a t e d f r o m t h e dimensionless s o l u t i o n of t h e n o n l i n e a r p r o b l e m u s i n g t h e same v a l u e s of bq a n d D * / r for b o t h s a t u r a t i o n a n d regeneration give a g o o d r e p r e s e n t a t i o n of t h e e x p e r i m e n t a l d a t a . F u r t h e r m o r e , t h e v a l u e s of D * ( c a l c u l a t e d a s s u m i n g r = 1.8 μ) (3) agree w e l l w i t h d i f f u s i v i t y d a t a o b t a i n e d b y a n i n d e p e n d e n t g r a v i m e t r i c s t u d y of t h e s o r p t i o n of ethane i n 5 A zeolite c r y s t a l s , t h u s p r o v i d i n g a v e r y s a t i s f a c t o r y c o n f i r m a t i o n of t h e v a l i d i t y of b o t h t h e e x p e r i m e n t a l t e c h n i q u e a n d t h e t h e o r e t i c a l a n a l y s i s s

z

2


z

(18). A l s o s h o w n i n T a b l e I are t h e e s t i m a t e d v a l u e s of t h e t i m e c o n s t a n t for m a c r o p o r e diffusion based o n e s t i m a t e d m a c r o p o r e diffusivities. F r o m t h e r a t i o of t h e t i m e constants for m a c r o p o r e diffusion a n d z e o l i t i c diffusion, i t is clear t h a t t h e a s s u m p t i o n of z e o l i t i c diffusion c o n t r o l is a v a l i d a p p r o x i m a t i o n f o r these systems. F i g u r e 5 shows t h e e x p e r i m e n t a l b r e a k t h r o u g h curves o b t a i n e d b y S h e t h (14) for s a t u r a t i o n a n d regeneration of a 4 A m o l e c u l a r sieve c o l u m n w i t h a feed s t r e a m c o n t a i n i n g a s m a l l c o n c e n t r a t i o n of ethylene i n h e l i u m . T h e e q u i l i b r i u m i s o t h e r m for t h i s s y s t e m is h i g h l y n o n l i n e a r , a n d , as a r e s u l t of t h i s , t h e s a t u r a t i o n a n d regeneration curves h a v e q u i t e different shapes. H o w e v e r , t h e t h e o r e t i c a l curves c a l c u l a t e d f r o m t h e n o n l i n e a r a n a l y s i s u s i n g t h e same v a l u e s of t h e p a r a m e t e r s bq a n d D*/r for b o t h 2

s

g

80

Figure 5. Comparison of experimental and theoretical breakthrough curves for C H -He system in 4A molecular sieve: (O and X) experimental results for saturation and regeneration, respectively; ( ) present nonlinear solution for λ = 0.667 and D*/r* = 1.19 X W~* sec' 2

i

1


31.

GARG AND RUTHVEN

Breakthrough Curves

353

s a t u r a t i o n a n d regeneration give a good r e p r e s e n t a t i o n of t h e observed b e h a v i o r . A n e q u a l l y good fit was o b t a i n e d for t h e e x p e r i m e n t a l curves measured at other t e m p e r a t u r e s a n d ethylene concentrations. F o r this s y s t e m the v a l i d i t y of t h e a s s u m p t i o n of zeolitic diffusion c o n t r o l was v e r i fied d i r e c t l y b y v a r y i n g t h e size of t h e m o l e c u l a r sieve pellets (14), a n d t h i s is confirmed b y t h e r e l a t i v e m a g n i t u d e s of t h e diffusional t i m e c o n stants g i v e n i n T a b l e I .


Conclusion T h e f o r m of the b r e a k t h r o u g h curves for m o l e c u l a r sieve a d s o r p t i o n c o l u m n s is v e r y sensitive t o the n o n l i n e a r i t y of t h e e q u i l i b r i u m i s o t h e r m s , b u t , for c o n d i t i o n s of zeolitic diffusion c o n t r o l , t h e t h e o r e t i c a l a n a l y s i s has been s h o w n to p r o v i d e a s a t i s f a c t o r y p r e d i c t i o n of t h e observed b e h a v i o r . W h e n t h e d e v i a t i o n of t h e i s o t h e r m f r o m l i n e a r i t y is s m a l l , R o s e n ' s l i n e a r s o l u t i o n gives a s a t i s f a c t o r y r e p r e s e n t a t i o n of t h e m e a n of t h e s a t u r a t i o n a n d regeneration b r e a k t h r o u g h curves. T h i s p r o v i d e s a v e r y useful m e t h o d of c a l c u l a t i n g t h e d i f f u s i v i t y f r o m e x p e r i m e n t a l b r e a k t h r o u g h curves, b u t , for t h e a p p l i c a t i o n of t h i s m e t h o d , b o t h t h e s a t u r a t i o n a n d regeneration curves m u s t be a v a i l a b l e . M a t c h i n g t h e i n d i v i d u a l b r e a k t h r o u g h curves t o t h e l i n e a r s o l u t i o n m a y l e a d t o v e r y significant errors i n t h e c a l c u l a t e d d i f f u s i v i t y v a l u e s even w h e n the d e v i a t i o n f r o m l i n e a r i t y is q u i t e s m a l l . W h e n t h e i s o t h e r m is h i g h l y n o n l i n e a r , t h e r e g e n e r a t i o n b r e a k t h r o u g h c u r v e becomes v e r y s l o w c o m p a r e d w i t h t h e s a t u r a t i o n c u r v e , b u t t h i s difference m a y be q u a n t i t a t i v e l y a c c o u n t e d for b y t h e n o n l i n e a r a n a l y s i s u s i n g t h e same diffusional t i m e constants for a d s o r p t i o n a n d d e s o r p t i o n . T h e o b s e r v e d difference between s a t u r a t i o n a n d regeneration rates i n m o l e c u l a r sieve c o l u m n s a n d t h e c o r r e s p o n d i n g difference between a d s o r p t i o n a n d d e s o r p t i o n rates i n " s i n g l e p a r t i c l e " g r a v i m e t r i c e x p e r i m e n t s has l e d several p r e v i o u s i n v e s t i g a t o r s (14-16) t o conclude t h a t t h e diffusivities for a d s o r p t i o n a n d d e s o r p t i o n are different. T h e present a n a l y s i s a n d o u r earlier a n a l y s i s (17) of t h e " s i n g l e p a r t i c l e " p r o b l e m s u g gest t h a t these differences arise s i m p l y f r o m t h e n o n l i n e a r i t i e s of t h e s y s t e m r a t h e r t h a n f r o m a n y f u n d a m e n t a l difference i n the a c t u a l r a t e constants. I n t h e e x p e r i m e n t a l systems considered here, t h e c o n t r o l l i n g resistance was i n each case z e o l i t i c diffusion, b u t systems i n w h i c h m a c r o p o r e r e s i s t ance is d o m i n a n t are e q u a l l y c o m m o n . A s examples one m a y cite t h e s o r p t i o n of l i g h t h y d r o c a r b o n s i n t h e D a v i s o n 5 A m o l e c u l a r sieves w h i c h c o n t a i n m u c h s m a l l e r zeolite c r y s t a l s a n d c o r r e s p o n d i n g l y s m a l l e r m a c r o pores t h a n t h e e q u i v a l e n t L i n d e p r o d u c t s (18). P r o b a b l y t h e m o s t serious a p p r o x i m a t i o n i n t h e present a n a l y s i s is t h e a s s u m p t i o n t h a t t h e s y s t e m is i s o t h e r m a l since t h i s l i m i t s t h e a p p l i c a b i l i t y of t h e t h e o r y t o v e r y l o w c o n c e n t r a t i o n systems. N e v e r t h e l e s s , t h e t h e o r y


354

MOLECULAR SIEVES

p r o v i d e s i n s i g h t i n t o t h e b e h a v i o r of m o l e c u l a r sieve c o l u m n s w h i c h m a y be of v a l u e even w h e n t h e a p p r o x i m a t i o n s of t h e m o d e l are n o t e x a c t l y f u l filled. Nomenclature a c t i v i t y of s o r b a t e

a b

L a n g m u i r equilibrium constant

c c

sorbate c o n c e n t r a t i o n i n b u l k phase sorbate c o n c e n t r a t i o n a t c o l u m n i n l e t

0

l o c a l sorbate c o n c e n t r a t i o n i n m a c r o p o r e


c D Z>* Dp m q q q q* z

q

s

z e o l i t i c d i f f u s i v i t y (based o n s o l i d area) l i m i t i n g z e o l i t i c d i f f u s i v i t y a t zero sorbate c o n c e n t r a t i o n m a c r o p o r e d i f f u s i v i t y (based o n pore s e c t i o n a l area) r a t i o of b e d v o i d space t o zeolite c r y s t a l v o l u m e = e / ( l — e') l o c a l sorbate c o n c e n t r a t i o n i n a zeolite c r y s t a l average sorbate c o n c e n t r a t i o n for a c r y s t a l sorbate c o n c e n t r a t i o n a v e r a g e d o v e r a p e l l e t sorbate c o n c e n t r a t i o n i n e q u i l i b r i u m w i t h l o c a l sorbate c o n c e n t r a t i o n i n fluid phase s a t u r a t i o n sorbate c o n c e n t r a t i o n i n L a n g m u i r e q u a t i o n

q

i n i t i a l (or final) u n i f o r m sorbate c o n c e n t r a t i o n i n zeolite c r y s t a l i n e q u i l i b r i u m w i t h fluid phase c o n c e n t r a t i o n c

r r R

r a d i a l c o o r d i n a t e for zeolite c r y s t a l r a d i u s of zeolite c r y s t a l r a d i a l c o o r d i n a t e for p e l l e t

0

0

z

R t Τ ν w X ζ

p

pellet radius time dimensionless t i m e = (t — ζ/ν)/τ l i n e a r fluid v e l o c i t y v o l u m e f r a c t i o n of zeolite c r y s t a l s t o t o t a l s o l i d m a t e r i a l i n a pellet dimensionless distance bq z/mvr distance measured from bed inlet a

e

v o i d f r a c t i o n of b e d

€ 1 — e'

v o i d f r a c t i o n of p e l l e t r a t i o of zeolite c r y s t a l v o l u m e t o t o t a l b e d v o l u m e = (1 — e)

p

(1

-

€ )W P

φ

c/co

λ

qo/q = bco(l + feco)"

r

r /D*

1

s

2

z

f o r z e o l i t i c diffusion c o n t r o l o r R

2

p

bq w(l — e ) / D e s

P

P

for m a c r o p o r e diffusion c o n t r o l

Literature Cited 1. Garg, D. R., Ph.D. Thesis, University of New Brunswick, 1972. 2. Ruthven, D. M., Loughlin, K. F., J.C.S. Faraday Trans. I (1972) 68, 696.


P

31. 3.

GARG AND RUTHVEN

Breakthrough Curves

Loughlin, K. F., Derrah, R. I., Ruthven, D. M . , Can. J. Chem. Eng.

355 (1971)

66.

Petrovic, L. J., Thodos, G., Ind. Eng. Chem., Fund. (1968) 7, 274. Wilson, E. J., Geankoplis, C. J., Ind. Eng. Chem., Fund. (1966) 5, 9. Barrer, R. M., Fender, B. E. F., J. Phys. Chem. Solids (1963) 21, 12. 7. Barrer, R. M., Davies, J. Α., Proc. Roy. Soc., Ser. A (1971) 322, 1. 8. Ruthven, D. M . , Loughlin, K. F., Trans. Faraday Soc. (1971) 67, 1661. 9. Crank, J., "The Mathematics of Diffusion," Clarendon Press, Oxford, 1958. 10. Rosen, J. B., J. Chem. Phys. (1952) 20, 387. 11. Rosen, J. B., Ind. Eng. Chem. (1954) 46, 1590. 12. Vermeulen, T., Advan. Chem. Eng. (1958) 2, 147. 13. Ruthven, D. M., Loughlin, K. F., Derrah, R. I., ADVAN. CHEM. SER. (1973) 121, 330. 14. Sheth, A. C., M.Sc. Thesis, Northwestern University, 1969. 15. Eberley, P. E., Ind. Eng. Chem., Prod. Res. Develop. (1969) 8, 140. 16. Satterfield, C. N., Frabetti, A. J., A.I.Ch.E. J. (1967) 13, 731. 17. Garg, D. R., Ruthven, D. M., Chem. Eng. Sci. (1972) 27, 417. 18. Ruthven, D. M., Loughlin, K. F., Can. J. Chem. Eng. (1972) 50, 550.


4. 5. 6.

RECEIVED October 25, 1972.


49,

Theoretical Prediction of Breakthrough Curves for Molecular Sieve

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