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Nonisothermal Gas Sorption Kinetics SHIVAJI SIRCAR and RAVI KUMAR Air Products and Chemicals, Inc., Allentown, PA 18105
Analytical equations for adsorbate uptake and radial adsorbent temperature profiles during a differential kinetic test are derived. The model assumes that the mass transfer into the adsorbent can be described by a linear driving force model or the surface barrier model. Heat transfer by Fourier conduction inside the adsorbent mass in conjunction with external film resistance is considered. Experimental uptake data for sorption of i-octane on 13X zeolite and n-pentane on 5Å zeolite were quantitatively described by the model. The results show that internal thermal resistance of the adsorbent mass plays a significant role during the uptake for these systems even though the adsorbent temperature changes are small. The model shows that the non-isothermal uptake curve for an adsorbent mass which has low effective thermal conductivity (k ) is identical in form to that of the isothermal Fickian diffusion model for mass transport. It is shown that k can be significantly low for an assemblage of microparticles at low pressure and high temperatures. A parametric study of the effects of the equilibrium and the transport properties of the adsorption system on sorption kinetics is carried out. Complex interactions between these properties in determining the shape of the uptake curve are observed. e
e
0097-6156/83/0223-0171$07.00/0 © 1983 American Chemical Society
Whyte et al.; Industrial Gas Separations ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
172
I N D U S T R I A L GAS
SEPARATIONS
The i m p o r t a n c e o f a d s o r b e n t n o n - i s o t h e r m a l i t y d u r i n g t h e measurement o f s o r p t i o n k i n e t i c s h a s b e e n r e c o g n i z e d i n recent years. S e v e r a l m a t h e m a t i c a l models t o d e s c r i b e t h e n o n - i s o t h e r m a l s o r p t i o n k i n e t i c s have b e e n f o r m u l a t e d [ 1 - 9 ] . Of p a r t i c u l a r i n t e r e s t a r e t h e m o d e l s d e s c r i b i n g t h e u p t a k e d u r i n g a d i f f e r e n t i a l s o r p t i o n t e s t because they p r o v i d e r e l a t i v e l y simple a n a l y t i c a l s o l u t i o n s f o r data a n a l y s i s [6-9]. T h e s e m o d e l s assume t h a t mass t r a n s f e r c a n b e d e s c r i b e d by t h e F i c k i a n d i f f u s i o n model and h e a t t r a n s f e r from t h e s o l i d i s c o n t r o l l e d by a f i l m r e s i s t a n c e o u t s i d e the adsorbent p a r t i c l e . D i f f u s i o n o f adsorbed molecules i n s i d e t h e adsorbent and gas d i f f u s i o n i n t h e i n t e r p a r t i c l e v o i d s have b e e n c o n s i d e r e d a s t h e c o n t r o l l i n g mechanism f o r mass t r a n s f e r . Comparisons o f e s t i m a t e d d i f f u s i v i t y v a l u e s on z e o l i t e s f r o m s o r p t i o n u p t a k e measurements a n d t h o s e o b t a i n e d f r o m d i r e c t measurements b y n u c l e a r m a g n e t i c r e s o n a n c e f i e l d g r a d i e n t t e c h n i q u e s have i n d i c a t e d l a r g e d i s c r e p a n c i e s b e t w e e n t h e two f o r many s y s t e m s [ 1 0 ] . I n a d d i t i o n , t h e f o r m e r method h a s o f t e n r e s u l t e d i n a n a d s o r b a t e d i f f u s i v i t y d i r e c t l y p r o p o r t i o n a l t o the adsorbent c r y s t a l s i z e [11]. T h i s l e d some r e s e a r c h e r s t o b e l i e v e t h a t t h e r e s i s t a n c e t o mass t r a n s f e r may be c o n f i n e d i n a s k i n a t t h e s u r f a c e o f the adsorbent c r y s t a l o r p e l l e t ( s u r f a c e b a r r i e r ) [10,11]. The i s o t h e r m a l s u r f a c e b a r r i e r m o d e l , h o w e v e r , f a i l e d t o d e s c r i b e experimental uptake data q u a n t i t a t i v e l y [10,12]. Measurement o f r a d i a l t e m p e r a t u r e g r a d i e n t i n s i d e t h e a d s o r b e n t p a r t i c l e d u r i n g t h e s o r p t i o n t e s t showed t h a t t h e i n t e r n a l r e s i s t a n c e t o h e a t t r a n s f e r may n o t be n e g l i g i b l e i n comparison w i t h t h e e x t e r n a l f i l m [13-16]. I n v i e w o f t h e s e o b s e r v a t i o n s , we p r o p o s e a n o n i s o t h e r m a l s o r p t i o n k i n e t i c s model w i t h t h e f o l l o w i n g assumptions : (a)
(b)
S o r b a t e mass t r a n s f e r c a n be d e s c r i b e d b y a l i n e a r d r i v i n g f o r c e m o d e l (LDF) u s i n g a "lumped-up" mass transfer coefficient k [17]. F o u r i e r c o n d u c t i o n i n s i d e t h e a d s o r b e n t mass i n c o n j u n c t i o n w i t h an e x t e r n a l f i l m r e s i s t a n c e d e s c r i b e s the heat t r a n s f e r from t h e adsorbent.
The LDF m o d e l i s a r e a l i s t i c r e p r e s e n t a t i o n o f t h e system w i t h a s u r f a c e b a r r i e r . O t h e r w i s e , k c a n be t r e a t e d as a n a p p a r e n t mass t r a n s f e r c o e f f i c i e n t i r r e s p e c t i v e o f t h e t r u e t r a n s p o r t mechanism w h i c h c a n b e d i r e c t l y u s e d i n t h e d e s i g n and o p t i m i z a t i o n o f a d s o r b e r s . T h i s concept has been s u c c e s s f u l l y used t o a n a l y z e column b r e a k t h r o u g h data f o r p r a c t i c a l n o n - i s o t h e r m a l systems [18-20]. I t substantially
Whyte et al.; Industrial Gas Separations ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
9.
SIRCAR
Νonisothermal
AND KUMAR
Gas
Sorption
Kinetics
s i m p l i f i e s the mathematical modelling of the behavior of a d s o r b e n t columns i n c o m p a r i s o n w i t h t h e u s e o f F i c k i a n d i f f u s i o n models [22,23]. N o n - i s o t h e r m a l LDF M o d e l We assume t h a t t h e a d s o r b e n t mass u s e d i n t h e k i n e t i c t e s t c o n s i s t s o f a s p h e r e o f r a d i u s R. I t may be composed of s e v e r a l m i c r o s i z e p a r t i c l e s ( s u c h as z e o l i t e c r y s t a l s ) bonded t o g e t h e r as i n a c o m m e r c i a l z e o l i t e b e a d o r s i m p l y a n assemblage o f t h e m i c r o p a r t i c l e s . I t may a l s o be composed o f a n o n c r y s t a l l i n e m a t e r i a l s u c h as g e l s o r a l u m i n a s o r a c t i v a t e d c a r b o n s . The r e s i s t a n c e t o mass t r a n s f e r may occur a t t h e s u r f a c e o f t h e sphere o r a t t h e s u r f a c e o f each microparticle. The h e a t t r a n s f e r i n s i d e t h e a d s o r b e n t mass i s c o n t r o l l e d by i t s e f f e c t i v e t h e r m a l c o n d u c t i v i t y . Each m i c r o p a r t i c l e i s a t a u n i f o r m t e m p e r a t u r e d e p e n d e n t on t i m e and i t s p o s i t i o n i n t h e s p h e r e . We assume t h a t t h e a d s o r b e n t mass i s i n i t i a l l y i n e q u i l i b r i u m w i t h t h e a d s o r b a t e a t p r e s s u r e Ρ and t e m p e r a t u r e Τ , and a d i f f e r e n t i a l s t e p change i n t h e g a s p h a s e p r e s s u r e t o P^ i s a p p l i e d a t t i m e t = 0. The p r e s s u r e i n s i d e t h e a d s o r b e n t mass ( o r t h e m i c r o p a r t i c l e s ) , P ( t ) , i n c r e a s e s w i t h t i m e , which i s g i v e n by:
^ k
g
=
k
s
(D
[P. - Pit)]
i s t h e "lumped-up" mass t r a n s f e r
coefficient.
The h e a t b a l a n c e f o r t h e a d s o r b e n t i n t h e r e g i o n 0 ^ r < R may be w r i t t e n a s : p
C
8T 8t
=
r
1 3 r, 2 3T 2 · 3r e * 8? I k
r
r
-j 1
/ t
p
q
3n 8t
( 0
,
( 2 )
r
Where Τ i s t h e t e m p e r a t u r e o f t h e a d s o r b e n t mass a t r a d i u s r and t i m e t . η i s t h e a d s o r b a t e l o a d i n g p e r u n i t w e i g h t o f t h e a d s o r b e n t a t r a d i u s r and t i m e t . q i s t h e i s o s t e r i c heat o f a d s o r p t i o n , p, c and k a r e , r e s p e c t i v e l y , t h e d e n s i t y , t h e h e a t c a p a c i t y and t h e e f f e c t i v e t h e r m a l c o n d u c t i v i t y o f t h e a d s o r b e n t mass. The a d s o r b e n t i s a t e q u i l i b r i u m u n d e r l o c a l c o n d i t i o n s o f Ρ and T. Thus f o r a d i f f e r e n t i a l t e s t where t h e c h a n g e s i n t h e a d s o r b e n t t e m p e r a t u r e and t h e a d s o r b a t e l o a d i n g a r e s m a l l , one may w r i t e : [n ( t , r ) - n j = a [P - P j + b [ T ( t , r ) - T j
Whyte et al.; Industrial Gas Separations ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
(3)
174
I N D U S T R I A L GAS
=
SEPARATIONS
3n*
3
9P
at Τ , P ο' »
(4) /«-λ
« _ 8 η* 8Τ a t Τ , Ρ
^
;
The a s t e r i c k i n d i c a t e s t h a t t h e q u a n t i t i e s a r e e v a l u a t e d under e q u i l i b r i u m c o n d i t i o n s , η and are, respectively, the i n i t i a l and t h e f i n a l e q u i l i S r i u m a d s o r b a t e l o a d i n g s during the test. The b o u n d a r y c o n d i t i o n s f o r e q u a t i o n s ( 1 ) a n d ( 2 ) a r e : P(o)
=
P
Ρ (oo)
=
ρ
00
T(r,o) = T(r,oo) = 8T 8r 8T Br
o
Τ T°
=
0
= 0 h r = R
"
k
[
T
- V
The l a s t b o u n d a r y c o n d i t i o n a c c o u n t s f o r t h e e x t e r n a l h e a t t r a n s f e r f r o m t h e a d s o r b e n t mass, h i s the effective e x t e r n a l heat t r a n s f e r c o e f f i c i e n t , a, b , c, p, q, k , k and h c a n b e assumed t o be c o n s t a n t s f o r a d i f f e r e n t i a l test. The a v e r a g e a d s o r b a t e l o a d i n g i n t h e a d s o r b e n t mass, n ( t ) , c a n be o b t a i n e d b y : e
n(t)
=
2
— | — J R ο
η (r,t).r .dr
g
(6)
J
E q u a t i o n s ( 1 ) - ( 3 ) a n d ( 6 ) were s i m u l t a n e o u s l y s o l v e d u s i n g t h e above b o u n d a r y c o n d i t i o n s b y L a p l a c e transformation and i n v e r s i o n b y t h e method o f r e s i d u e s t o o b t a i n t h e f o l l o w i n g a n a l y t i c a l equations:
F = 1 - e
6§ ι-β
_
Π n-P
s 2
2
Ρ {P
2
2
-s(i-s)}
(e"V
nT
- e" )
(7)
Whyte et al.; Industrial Gas Separations ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
9.
SIRCAR A N D K U M A R
Nonisothermal
Sin ρ χ *η
6 = 2
Gas
η
(χ) S i n p
2
η - P
n
Sorption
2 {p
n
n
Kinetics
s -s(l-s)}
-ρ τ
-ητ
2
(e
where
χ
=
n
-
e
)
(8)
r/R „2
c
s
=
α
=
h R k e k ?
(9)
pcCl-β)
n
=
F
=
2 k R s α
n -n ο n -n
00
o T(x,t)-T θ
=
ο T* -
T ο
1
(T*-T
)
o'
n
( oo -
n 0
>
c (1 - β)
p 's a r e t h e p o s i t i v e r o o t s o f t h e t r a n s c e d e n t a l _ .n equation: r
P cot p n
n
- 1
= - s
n=
1,2,3,
(10)
I t c a n be shown f r o m a d s o r p t i o n t h e r m o d y n a m i c s t h a t :
Whyte et al.; Industrial Gas Separations ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
I N D U S T R I A L GAS
176
*
