Kinetics of Sorption in Biporous Molecular Sieves

35 Kinetics of Sorption in Biporous Molecular Sieves L. K. L E E , H . YUCEL, and D. M. RUTHVEN

Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0040.ch035

Department of Chemical Engineering, University of New Brunswick, Fredericton, N.B., Canada

ABSTRACT A mathematical model is developed to describe kinetics of sorption in a "bi-porous adsorbent pellet for systems which exhibit a highly favourable (rectangular) adsorption isotherm. The theoretical uptake curves differ significantly from the predictions of a linear dual resistance model. The theory is used to analyze experimental data for sorption of cis-2-butene in a 5A molecular sieve pellet at 273°K. Introduction Commercial molecular s i e v e adsorbents c o n s i s t o f small microporous z e o l i t e c r y s t a l s formed i n t o a macroporous p e l l e t , sometimes with the a i d o f a c l a y b i n d e r . The k i n e t i c s o f s o r p t i o n are theref o r e determined by t h e combined e f f e c t s o f two d i s t i n c t d i f f u s i o n a l r e s i s t a n c e s : macropore and micropore. The r e l a t i v e importance o f these r e s i s t a n c e s v a r i e s g r e a t l y depending on the p a r t i c u l a r system and the c o n d i t i o n s . The s o r p t i o n behaviour can be p r o p e r l y des c r i b e d by a simple d i f f u s i o n model only when one or other o f the r e s i s t a n c e s i s n e g l i g i b l e (the extreme cases o f micropore or macropore c o n t r o l ) . For many systems, under p r a c t i c a l l y important c o n d i t i o n s , both r e s i s t a n c e s are s i g n i f i c a n t and t o provide a r e a l i s t i c k i n e t i c model f o r such systems r e q u i r e s s o l u t i o n o f the coupled d i f f u s i o n equations. H i t h e r t o such s o l u t i o n s have been obtained only f o r systems with a l i n e a r e q u i l i b r i u m isotherm. A numerical s o l u t i o n was given by Sargent and W h i t f o r d ^ ) and a more elegant a n a l y t i c a l s o l u t i o n was obtained by Ruckenstein et a l . ( 2 ) . T h i s was used by Ma and Ho(_3) t o i n t e r p r e t k i n e t i c data f o r s o r p t i o n o f aliène and methyl acetylene i n Linde 13X s i e v e . The l i n e a r model i s appropriate f o r t h e exchange o f i s o t o p i c a l l y tagged species studied by Sargent and W h i t f o r d ^ ) but f o r most systems, i n c l u d i n g those s t u d i e d by Ma and Ho (3) the assumption o f l i n e a r i t y i s a severe approximation. The e q u i l i b r i u m isotherms f o r many systems of p r a c t i c a l importance are h i g h l y n o n - l i n e a r and i n order to 9

417 In Molecular Sieves—II; Katzer, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

418

MOLECULAB

SIEVES—Π

increase our understanding of the behaviour of such systems we consider here a system i n which the isotherm i s r e c t a n g u l a r (irreversible). This i s the extreme l i m i t of h i g h l y favourable type I isotherm f o r which the curve approaches a step f u n c t i o n : q* = o, c = o; q* = q^, c > ο

(l)


Mathematical Model As an i d e a l i z e d r e p r e s e n t a t i o n we consider a macroporous s p h e r i c a l p e l l e t composed o f an assemblage of small uniform s p h e r i c a l microporous c r y s t a l s . Transport w i t h i n both micropores and macropores i s assumed t o occur by d i f f u s i o n with the co e f f i c i e n t s D and Dp independent of c o n c e n t r a t i o n . Neglecting accumulation with the macropores which, f o r molecular sieve adsorbents i s g e n e r a l l y small i n comparison with the c a p a c i t y of the z e o l i t e c r y s t a l s , the k i n e t i c s of s o r p t i o n may be d e s c r i b e d by a p a i r o f coupled d i f f e r e n t i a l equations: z

where Q(n,x) i s the dimensionless averaged over a c r y s t a l : 1 Q = ^

= 3

I J

(macropore d i f f u s i o n )

(2)

(micropore

(3)

diffusion)

adsorbed phase concentration

2

Q(n,Y,xh -d

Y

(k)

The appropriate i n i t i a l and boundary c o n d i t i o n s , assuming a step change i n adsorbed phase concentration at the p a r t i c l e surface at time zero, are: χ(η,ο) = ο ;

χ(ΐ,τ) = 1

(5)

Q(n,o) = ΰ(η,γ,ο) = ο ; Q(n,l,x) = Q*

(6)

=

(Τ)

ο

η,ο,τ For the case of l i n e a r isotherm (q* = Kc, Q* = Kc/q^) t h i s set of equations i s f o r m a l l y s i m i l a r to the model of Ruckenstein et alS^)

m

A r e c t a n g u l a r isotherm i m p l i e s that t h e e q u i l i b r i u m absorbed phase concentration approaches s a t u r a t i o n f o r any f i n i t e f l u i d

In Molecular Sieves—II; Katzer, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

35.

Sorption

LEE ET AL.

in Biporous

Molecular

Sieves

419

phase c o n c e n t r a t i o n . Under c o n d i t i o n s o f macropore d i f f u s i o n c o n t r o l s o r p t i o n t h e r e f o r e proceeds from l a y e r t o l a y e r through the p e l l e t i n a manner s i m i l a r t o the s h r i n k i n g core model o f a gas - s o l i d r e a c t i o n ^ >5). The uptake curve i s then given "by: 0 0 = 2/3 = = f - f(l-Q) - Q

% a c r o = 9 τ/3

ξ = 1 - j§-

Cos j j


+

i

+

COB-1(1

(8)

Jfl^j3

(9)

For the other extreme case o f micropore c o n t r o l the uptake curve i s given by the w e l l known s o l u t i o n o f eqn. 3 f o r a step change boundary c o n d i t i o n : M ξ

=

00

.

Ϊ Γ

=

1

Ρ π

"

^ _βχρ[-η π τ] =ι η 1

η

2

(10)

2

In order t o d e s c r i b e the coupled d i f f u s i o n problem which a r i s e s when both macropore and micropore r e s i s t a n c e s are s i g n i f i c a n t we note t h a t , f o r a r e c t a n g u l a r isotherm, the progress o f the c o n c e n t r a t i o n f r o n t p e n e t r a t i n g the p e l l e t can be represented as a time dependent step f u n c t i o n :

τ
k the uptake curves showed very l i t t l e change. f

f

f

f


f

n

T h e o r e t i c a l Uptake Curves A f a m i l y of t h e o r e t i c a l uptake curves i s shown i n f i g u r e 1. For small values o f 3 (< 0 . 1 ) the curves approach the l i m i t i n g curve f o r complete micropore c o n t r o l (eqn. 1 0 ) . As 3 i s increased the shape o f the curve changes but the l i m i t i n g form f o r macropore c o n t r o l (eqn. 9) i s approached o n l y when 3 i s very l a r g e (3 > ΙΟ* ). For intermediate values of 3 the uptake curves cannot be p r o p e r l y represented by a s i n g l e r e s i s t a n c e d i f f u s i o n model. Matching uptake curves o f t h i s form t o eqn. 9 (or eqn. 10) w i l l l e a d to apparent d i f f u s i v i t i e s which vary with f r a c t i o n a l uptake. Such trends have been r e p o r t e d i n the l i t e r a t u r e ( 8 , 9 ) and t h i s i s one p o s s i b l e explanation. As the two d i f f u s i o n a l r e s i s t a n c e s are not s t r i c t l y i n s e r i e s the assumption of l i n e a r a d d i t i v i t y , introduced by Roberts and York(10) cannot be j u s t i f i e d . Such an assumption cannot account f o r the changing shape o f the uptake curves. In f i g u r e 1 uptake curves f o r the present system (rectangular isotherm, dual r e s i s t a n c e model) are a l s o compared with curves c a l c u l a t e d from the modified l i n e a r dual r e s i s t a n c e , Ruckenstein model(ll_). For the r e c t a n g u l a r isotherm there i s a d i s t i n c t con c e n t r a t i o n f r o n t which penetrates the p e l l e t and c r y s t a l s behind the f r o n t a t t a i n immediately the f i n a l s a t u r a t i o n c o n c e n t r a t i o n at t h e i r s u r f a c e s . For a l i n e a r system the sorbate penetrates the p e l l e t more r a p i d l y although the c o n c e n t r a t i o n i n the c e n t r a l r e g i o n i s low. The i n i t i a l r a t e o f uptake i s f a s t e r f o r a l i n e a r system as more c r y s t a l s are i n contact with the sorbate. However as s o r p t i o n proceeds uptake by the r e c t a n g u l a r system becomes r e l a t i v e l y more r a p i d due t o the e f f e c t of the higher sorbate con c e n t r a t i o n a t the surface of the c r y s t a l s . The q u a l i t a t i v e d i f f e r e n c e in. the shapes o f the curves can be understood on t h i s basis. 1


422

MOLECULAR SIEVES—II

isotherms were e s s e n t i a l l y the same (Dofc* 2 . 5 x l 0 cm .sec" as compared with βχΙΟ" ^ cm .sec" from e x t r a p o l a t i o n of high temper ature data obtained with the l a r g e r Linde 5A c r y s t a l s The t h e o r e t i c a l l i n e s i n f i g u r e k are back c a l c u l a t e d from eqn. 23 and i t i s evident that the d i f f e r e n c e s i n the d i f f e r e n t i a l c o e f f i c i e n t s (D ) are due mainly t o the d i f f e r e n c e s i n the isotherms. The values o f D determined from the i n t e g r a l experiments (h.2,5.0 and 6 . 0 χ 1 0 " cm .sec" ) are somewhat higher than the l i m i t i n g values o f D . This i s t o be expected since t h e i n t e g r a l d i f f u s i v i t y should correspond t o an average value o f Dz c a l c u l a t e d over the range ο -* qm: *η· qm _ 1 1 +

1

2

2

1

1

z

z

l i +

2

1


Q

D -d z

%η

Z

(2k)

q

J

P e l l e t diameter 0 . 3 8 7 cm. C r y s t a l diameter 0 . 7 micron. Experi mental p e l l e t d e n s i t y 1 . 1 2 gm.cm"^. P o r o s i t y , c a l c u l a t e d assuming s o l i d density 1 . 5 7 gm.cm"3 ε = 0 . 2 9 . Mean macropore radius ^ U50Â. Values o f Κ from slopes o f experimental isotherm. I n t e g r a t i o n o f the curve o f f i g u r e k y i e l d s values o f D close t o the experimental values and showing the observed i n c r e a s i n g trend with i n c r e a s i n g pressure. The importance o f considering both macropore and micropore r e s i s t a n c e s may be seen by examining t h e s o r p t i o n curves f o r t h e d i f f e r e n t i a l runs which, i n the i n i t i a l r e g i o n , are almost l i n e a r i n / T " The time constants c a l c u l a t e d from the i n i t i a l slopes o f these curves, assuming only a s i n g l e d i f f u s i o n a l r e s i s t a n c e ( e i t h e r micropore o r macropore as i n eqn. 9 or 1 0 ) , are compared i n t a b l e I I with the values derived from the l i n e a r dual r e s i s tance model. I t i s evident t h a t , under these c o n d i t i o n s , t h e assumption o f a s i n g l e r e s i s t a n c e w i l l l e a d t o l a r g e e r r o r s i n the calculated d i f f u s i v i t i e s . Since a bed o f z e o l i t e c r y s t a l s can act l i k e a macroporous p e l l e t the p o s s i b l e i n t r u s i o n o f secondary d i f f u s i o n a l r e s i s t a n c e i s a f a c t o r which should always be considered i n the a n a l y s i s o f t r a n s i e n t s o r p t i o n curves. z

Conclusions In the a n a l y s i s o f the s i n g l e r e s i s t a n c e d i f f u s i o n problem i t has been shown t h a t , f o r systems i n which the d i f f u s i v i t y increases with sorbate concentration, t h e form o f the uptake curve d i f f e r s TABLE I I - Comparison o f D i f f u s i o n Time Constants C a l c u l a t e d from S i n g l e Resistance and Dual Resistance Models (sec" !" 1

Run

C a l c u l a t e d From Linear Dual Resistance Model ( D / r ) χ 10 * ( D / R ) 2

z

2

1

z

p

p

P

C a l c u l a t e d from S i n g l e s i s tance Models Eqn. 8 Eqn. 9 ( D / r ) χ 10U (Dp/Rp ) 2

z

D-l D-2 D-3

2.1 2.9 k.l

0.20 2.21 0.063

2

z

0.5 0.57 0.U6

0.06l 0.055 0.031


35.

LEE E T AL.

Sorption

in Biporous

Molecular

Sieves

423

Comparison of Theory and Experiment Experimental isotherms f o r cis-2-butene i n Davison 5A s i e v e ( c r y s t a l s and b i n d e r l e s s p e l l e t at 2T3°K are shown i n f i g u r e 2. The isotherms are h i g h l y favourable and at t h i s temperature sorp t i o n i s slow and approximately i s o t h e r m a l . In an i n t e g r a l adsorp t i o n experiment the requirements o f the r e c t a n g u l a r iostherm bipore model are t h e r e f o r e approximately f u l f i l l e d whereas i n a d i f f e r e n t i a l experiment the system w i l l be approximately l i n e a r . Under d i f f e r e n t i a l c o n d i t i o n s adsorption and d e s o r p t i o n curves are i d e n t i c a l but d e s o r p t i o n measurements a r e experimentally e a s i e r . The experimental values o f 3 and D / r (Table I) were determined by matching the experimental uptake curves t o the appropriate f a m i l y of dimensionless t h e o r e t i c a l curves (rectangular or l i n e a r model). The d i f f e r e n c e between d i f f e r e n t i a l and i n t e g r a l curves i s evident i n f i g u r e 3. Deviations between t h e o r e t i c a l and experimental curves are more pronounced f o r the i n t e g r a l curves but t h i s i s t o be expected s i n c e the assumption o f a constant d i f f u s i v i t y i s a more severe approximation i n the i n t e g r a l case. Furthermore, the lower value of 3 i m p l i e s greater i n f l u e n c e of micropore r e s i s t a n c e and under these c o n d i t i o n s d e v i a t i o n s i n the t a i l o f the curve are to be expected due t o c r y s t a l s i z e d i s t r i b u t i o n ( l 2 ) . A l s o given i n Table I are the values o f 3 c a l c u l a t e d a p r i o r i assuming a t o r t u o s i t y f a c t o r of Knudsen d i f f u s i o n i n the macro-


2

z

z

pores i s dominant but at the higher pressures molecular d i f f u s i o n i s a l s o s i g n i f i c a n t . Although the t h e o r e t i c a l l y estimated values of 3 are smaller than the experimental values there i s order o f magnitude agreement. Micropore d i f f u s i v i t i e s c a l c u l a t e d from the d i f f e r e n t i a l runs are shown i n f i g u r e k which i n c l u d e s a l s o the values obtained from d i f f e r e n t i a l experiments with unaggregated c r y s t a l s (3*o). The d i f f e r e n t i a l d i f f u s i v i t i e s show the u s u a l strong c o n c e n t r a t i o n de pendence a r i s i n g from the n o n - l i n e a r i t y o f the e q u i l i b r i u m i s o t

h

e

m

:

D

z

= D (dlnc/dlnq)

(23)

0

For both s i z e s o f c r y s t a l and f o r the p e l l e t the values of D c u l a t e d from eqn. 23 using the values o f (dlnc/dlnq) from the

cal

Q

TABLE I - D e t a i l s of Experiments and C a l c u l a t e d Parameters From. Curve Matching 10^(D /r 2) 3

Ρ •> Έ>2 Differential D-l D-2 D-3 Integral 1-1 1-2 1-3

(torr)

z

z

Κ

3.31 3.U6 3.55

2

1

(sec" )

- 1

(sec )

2.1 U . 2 - 2 . 9 505 7 . 5 - ^ . 2 390 2.9 k.l 8 . 6 - 7 . 5 210 qm(mmoles.cm"3) 0-51 0-99 0-199

Estimated Values Dp/Rp 3

0.3b O.Ul 0.52

k k

8 1 1 1

2 2.1 1.6 0.75 0.5 O.h


0.1* O.h O.k

0.38 0.35 0.3


424

MOLECULAR SIEVES—Π

10'

10"

3

10'

2

1

1

10

ίο

2

Figure 1. Comparison of theoretical uptake curves calculated from linear and rectan gular isotherm dual resistance models. The line for macropore control (equation 9) is plotted for β = 10* on the micropore time scale.

ο C-521 Bead

25 -

+ 035AJ Crystal χ 0-70AI Crystal

2.0

15

j^a^-o—

rz

χ

+

-x—*—^~

Ό

* "°

l

+•

p

*

"

x-

*

1 70

1 80

1.0h

05 h 1 10

I

20

1 30

1 40

J 50

1 60

1 90

100

Ρ(Torr) Figure 2.

Experimental isotherms for sorption of cis-2-butene at 273°Κ in Davison 5A sieve. 0.7 micron crystals, X; 0.35 micron crystals, +; pellet, O .


35.

LEE E TAL.

Sorption in Biporous

Molecular

Sieves

425


ο ο

140 Jl

(sec*)

Figure 3. Comparison of theoretical and experimental sorption curves. Theoretical curves for run D-l from linear dual resistance model and for run 1-1 from rectangular dual resistance model Parameters are given inTable 1.

100 90

δ

C-521 Bead

ο

0.70AJ C r y s t a l

χ

035AJ Crystal

80

70

60

50

0.5

1.0

1.5

20

2-5

Figure 4. Variation of micropore dif fusivity with sorbate concentration. Pellet, A; 0.7 micron crystab, O; 0.35 micron crystals, χ. Theoretical curves are from equation 23 with D = 2.5 X 10~ cm - sec' . 0

q_ ( millimoles / g m adsorbent )

14

2


1


426

MOLECULAR SIEVES—II

only s l i g h t l y from the constant d i f f u s i v i t y case (15). The as sump- ' t i o n of a constant e f f e c t i v e micropore d i f f u s i v i t y may t h e r e f o r e be an acceptable approximation i n the present model although i n a r i g o r o u s a n a l y s i s the concentration dependence would have t o be considered. The t h e o r e t i c a l uptake curves f o r an i r r e v e r s i b l y adsorbed species i n a bi-porous molecular sieve p e l l e t d i f f e r s s i g n i f i c a n t l y from the equivalent curves f o r a l i n e a r system. The proposed model provides a s a t i s f a c t o r y i n t e r p r e t a t i o n of experimental i n t e g r a l uptake curves f o r cis-2-butene i n Davison 5A sieve at 2T3°K since f o r t h i s system, under the experimental c o n d i t i o n s , the assumption o f a r e c t a n g u l a r isotherm i s a reasonable approximation. By c o n t r a s t , i n d i f f e r e n t i a l experiments, the system can be con s i d e r e d as l i n e a r and the s o r p t i o n curves conform to the p r e d i c t i o n s o f the l i n e a r dual r e s i s t a n c e model. For more complete confirmation of the v a l i d i t y of the t h e o r e t i c a l a n a l y s i s f u r t h e r experiments with p e l l e t s of d i f f e r e n t s i z e are d e s i r a b l e . Notation c c D Dp q qm q* q t r r R Rp t 0

z

z

f l u i d phase c o n c e n t r a t i o n of adsorbate f l u i d phase concentration at e x t e r n a l surface of p e l l e t d i f f u s i v i t y of sorbate i n z e o l i t e c r y s t a l macropore d i f f u s i v i t y (based on pore area) adsorbed phase c o n c e n t r a t i o n (moles/unit s o l i d volume) s a t u r a t i o n c a p a c i t y of adsorbed phase e q u i l i b r i u m adsorbed phase c o n c e n t r a t i o n adsorbed phase concentration averaged over a c r y s t a l adsorbed phase concentration averaged over the p e l l e t r a d i a l coordinate f o r c r y s t a l equivalent r a d i u s of z e o l i t e c r y s t a l r a d i a l coordinate f o r p e l l e t equivalent r a d i u s o f p e l l e t time

Dimensionless V a r i a b l e s Κ mt/m^ Mt/M^ Q Q

dimensionless e q u i l i b r i u m constant f o r l i n e a r system f r a c t i o n a l uptake by a c r y s t a l f r a c t i o n a l uptake by p e l l e t as d e f i n e d by eqn. 20 q/qm |/Qm

Q* η nf γ χ

Kinetics of Sorption in Biporous Molecular Sieves

Recommend Documents