- Email: [email protected]
Sorption rate of bimodal microporous solids with an irreversible isotherm
Chemical Engineering Science, Vol. 44, No. 8, pp. 1707-1713, 1989. Printed in Great Britain.
0009-2509/89 $3.00+0.00 © 1989 Pergamon Press pie
SORPTION RATE OF BIMODAL MICROPOROUS SOLIDS WITH AN IRREVERSIBLE ISOTHERM D. D. DO Department of Chemical Engineering, University of Queensland, St. Lucia, Q. 4067, Australia
(Received 7 June 1988; acceptedfor publication 23 January 1989) Abstract--The sorption rate of gases into a microporous adsorbent having a bimodal pore size distribution is analysed. The adsorption isotherm is assumed to exhibit a rectangular shape (highly favourable isotherm) which is commonly observed in systems of zeolites and carbon molecular sieves. Degenerate models are derived when one mode of diffusion is faster than the other, and the conditions when one mode of diffusion controls the global uptake have been delineated. l. INTRODUCTION The global rate of adsorption of gases in microporous solids exhibiting a bimodal pore size distribution depends on the relative rates of diffusion in the macropore and micropore. The first work on modelling of adsorption in bimodal solids in the Western literature was probably by Ruckenstein et al. (1971). They assumed a pore diffusion mechanism in both the macropore and the micropore, and allowed for an adsorptive capacity in both pores. Since that work, other researchers in the area of zeolite and activated carbon [e.g. Sargent and Whitford (1971), Kawazoe et al. (1974) and Ma and Ho (1974)] have proposed a different bimodal diffusion model. The essential features of this model are: no adsorption in the macropore, pore diffusion in the macropore, adsorption isotherm at the pore mouth of the micropore and activated diffusion in the micropore. However, all the work above assumed a linear adsorption isotherm at the pore mouth of the micropore. This would render the mass balance equations linear and they were susceptible to linear analysis, like the Laplace transform or finite integral transform. Recently, Polte and Mersmann (1986) have assumed a Langmuir adsorption isotherm at the pore mouth of the micropore. A finite difference technique was used to solve the mass balance equations numerically. When the adsorption isotherm is rectangular, i.e. an irreversible adsorption isotherm, analyses were presented for pseudo-homogeneous solids. Suzuki and Kawazoe (1974) and Do (1986) dealt with batch reservoir problems. Film resistance was allowed for. Fixed-bed problems without film resistance were considered by Vermeulen (1958), Hall et al. (1966), Cooper (1965) and Cooper and Liberman (1970). Film resistance in a fixed bed was considered by Weber and Chakravorti (1974), Brauch and Schlunder (1975), Lee and Ruthven (1976) and Do (1985). This paper presents a theoretical analysis of the sorption rate in a bimodal solid with an irreversible adsorption isotherm, i.e. the surface concentration of the microsphere is instantly equal to the saturated value as long as the adsorbate concentration in the
macropore void is nonzero. The next section deals with a bimodal solid with a spherical microparticle. Analysis of a bimodal slab microparticle is then included because Prinz and Riekert (1986) in their work of the sorption of benzene, 3-methyl pentane and p-xylene found that the solution of diffusion into a slab fitted their experimental data reasonably well. 2. SPHERICAL MICROPARTICLE
Consider a bimodal solid particle comprised of many small microparticles. This particle is exposed to an environment of constant bulk concentration, Co. The following assumptions are made: (a) isothermal system, (b) constant diffusivities in the macropore and micropore, (c) spherical microparticle, (d) particle of arbitrary shape, (e) irreversible adsorption isotherm at microparticle surface. The mass balance equation in the microparticle is ~C, 1 c3 / --=~--/r2--/
c3C~\
(la)
subject to the following initial and boundary conditions: t=0: ru=0:
r~=R;
C~=0
(lb)
~C~,/t3r~,=O
(lc)
C =)'C, ~ (0
(ld) (le)
if C > 0 if C=.0
where C is the adsorbate concentration in the macropore. Equations (1 d) and (le) represent the irreversible adsorption isotherm. The mass balance equation in the macropore is OC
~C~
eu ~ t + ( 1 - e u ) ~ t
1 ~ { ~C'~
=(eM~p)~;~rk:~r )
(2)
where eM is the macropore porosity, ~p is the pore diffusivity based on the pore cross-sectional area, and
1707
1708
D.D. Do
s is the shape factor of the particle (s = 0, 1, and 2 for slab, cylinder and sphere, respectively). C e is the mean micropore concentration at position r, and is defined as follows: 3 ('R,
C-- e_- ~
2 I reCedre"
(3)
"~e do
Equation (2) is subject to the following initial and boundary conditions: t=O:
C=O
(4a)
r=O:
OC/~r=O
(4b)
r=R: (eu~p)OC/Or=k~(Co-C)
2(a) Comparable rates [7 n0(1)] The first case dealt with is the comparable-rates case, i.e. 7 is of the order of unity. The solution procedure is given in an appendix available from the author. The solutions for the fractional uptake are:
F= f
OCe
(5)
A = C/Co, A e = Ce/C~, x = r/R, x e = re/R e (6a) z=~t/R2'7
(1-eu)C,( R ~ 2 ~ k=R euC~o \R-~J ~ ' Bi=eu~ p
(6b)
/,)/
z+-----
fo
xL~edx
for z>zo
.. )
(x2-1 [1
1
[
5
+~_sJ~ [1
for s ¢ l
1 "~ 2
~,~+~)(X-1)
fors=l.
(11)
The time To is the time when the adsorption front reaches the center of the particle. It is obtained from eq. (11) by setting X to zero. Then one has 1
2 o~ e u.%\
1
/1
~)%zo+~-~ ~=l~-)=~s~+~).
OAe 1 6~ (x2~Ae'~ Oz - x~ Oxe \ e Oxe ]
(lOb)
~ ~-
,,
= )2(~_s)- ~
1\/
eqs (1) and (3)-(5) become
forO
where X(z) is the adsorption front at time ~ and is given by
|X21nX By defining the following nondimensional variables and parameters:
(l+s)f x~fi.edx tJx") (l+s)
(4c)
where k= is the fluid mass transfer coefficient. The left-hand side ofeq. (2) contains two terms. One is the hold-up capacity in the macropore void, and the other is the capacity in the micropore. In most practical cases the macropore void capacity is very small compared to the capacity of the micropore. Hence, eq. (2) can be replaced by
1 0 { OC'~ (1-eu)~-~=(eu~,)~r(r'~r ).
pore commonly used in chemical engineering literature (Vermeulen, 1958; Hall et al., 1966).
1\
(12)
(7b)
Finally, "4e in the fractional uptake solution [eq. (10)] is the mean sorbate concentration in the micropore, given as
x e = 0: OAe/Oxe = 0
(7c)
6 o~ ,'te = 1 - - - - X n - 2 exp {--n2n2[z--z*(x)]} (13)
A
(7d) (7e)
z=0:
xe = l :
(7a)
Ae=0
fl e=_0
(13110(x
if A > 0 if A = 0
A)]
x/j
(8b)
x = I: OA/Ox=Bi(1-A)
(8c)
where Ae = 3
fo
x 2 A e dx e
where z*(x) is the time when the adsorption front reaches position x, and is determined from eq. (11).
(8a)
aA/Ox=O
x=0:
7r2n= 1
(9)
The parameter 7 measures the relative rates between the micropore diffusion and the macropore diffusion. When this parameter is of the order of unity, both diffusions would be expected to contribute to the global uptake. When this parameter is very small, micropore diffusion controls the uptake and the response is the typical Fickian uptake of the microparticle. However, when 7 is very large, one would expect the classical shrinking core model of the macro-
2(b) Micropore diffusion control (7 ~ 1) When 2, is very small, the dynamics start with the physical macropore void uptake. This stage is normally very fast and the uptake capacity of the macropore void is very small. Therefore, this stage is neglected. The next stage is the micropore uptake: all microparticles are exposed to the same bulk adsorbate concentration, Co. The mass balance equation (7 ~ 1) is
1 ±(x20Ae
Oz - x 2 tOXe\ ~'Ox.,] z=O: Au--0
(14a) (14b)
xe=0:
OAe/~xe=O
(14c)
xe = l :
Ae = l
(14d)
The solution is straightforward, and the mean micropore concentration is
Sorption rate of bimoda ® 1 Z ~ s e x p ( - ~ . 2z) n=l %
"4u = 1 - 6
((,=nz).
.~croporous solids with an irreversible isotherm (15)
In this case all the microparticles behave identically. The fractional uptake is just equal to the mean micropore concentration defined in eq. (15). When the micropore diffusion controls the global uptake the system dynamics are independent of the type of isotherm, whether linear, rectangular, Langmuir or Freundlich. The half-time can be readily obtained from eq. (15):
Solving eqs (17) and (18) gives the following equation for the adsorption wave front position: dX
~ = -P(X) dt X-S 1
1 --X
Bi
1 -s
for s # 1
(21a)
fors=l.
(21b)
1-s
e(X) =
I
(16)
The half-time is independent of all macropore characteristics, and is proportional to the square of the microparticle radius.
(20)
where
R2
to. 5 = 0.03055 - -u.
1709
X
l~+ln (1 ]
Bi
\XJ
The solution of the adsorption wave front is
2(c) Macropore diffusion control (7 >>1) When ~ is very large, the system dynamics are controlled by the macropore diffusion. This case is the
H(X) =
Ix
P(m)
= ["
(22)
where
H(X) =
(2~t + ~)(1- X2)+12x21n X 1
classical shrinking-core model commonly used in the adsorption and the gas-solid reaction literature. The mathematical proof for the shrinking-core model was presented by Do (1982) for gas-solid reaction using a singular perturbation method, and for adsorption with reaction by Do (1984). For this case (7 >>1), there is no gradient (or no mass transfer resistance) in the microparticle. The mass balance equations are
1~ (x ~t~A~ 0 xj=
x~ ~-x\
for X ( t ' ) < x < 1
x = X(t'): a = 0 x=l:
(lVa) (17b)
M1-X 3)
fors=l
(23b)
for s=2.
(23c)
Knowing the adsorption wave front, the fractional uptake is determined as F = 1 - X s + 1.
(24)
The adsorption wave front at 50% uptake is given by 11/~1+~ ( Xo.5=2
=l
0.5
fors=0
0.707
fors=l
0.794
for s=2.
(25)
Substitution of eq. (25) into eq. (22) yields the half-time t0. 5
(1 --F,M)CsR2
H(Xo.5)
(26)
(17cl
where X(7) is the position of the adsorption wave front, given by
i'=0:
/ 1
(23a)
gMC~']pC O
OA/Ox=Bi(1-A)
dX dF'-
2
fors=0
t~A ~x x(?~
(18a)
X-- 1
where H(Xo.5) is calculated from eq. (23) after X is replaced by Xo.5:
H(Xo.5)=
f 0.125+1/2Bi 0.03836+ 1/4Bi 0.01835 + 1/6Bi
for s = 0 for s = 1
(27)
for s=2.
(18b)
where 7is defined as
3. SLAB M I C R O P A R T I C L E
(eM~ p )
Co
_ z
t'= R2(1 --eM)C~ 7"
(19)
Although most work in the literature of adsorption assumed a microparticle with a spherical shape, Prinz and Riekert (1986) in their recent study of hydrocar-
1710
D. D. Do
bon uptake in ZSM-5 found that a theory using a slab crystal fits the data better than for other geometries. To make the analysis as general as possible, the theory of the sorption rate in a bimodal solid with a slab microparticle is presented here.
where ~, is given in eq. (31). The half-time of the global uptake is R2 to.5 =0.19674 -~ • (35) @
3(a) Comparable diffusion rates [Y"~0(1)] The mass balance equations in nondimensional form are identical to eqs (7)-(9) except that eqs (73) and (9) are replacd by
3(c) Macropore diffusion control (7 >>1) When macropore diffusion controls the uptake, the shape of the microparticle is irrelevant. This analysis was presented in Section 2(c), and the solutions are eqs (22) and (24).
dAu _ 02Au t3z
(28a)
Ox2
and "4u= Jo A~,dx~,
(28b)
respectively. The nondimensional variables and parameters are defined in eq. (6). Following the procedure of the previous section: , = , ~ 2 e x p {-~2[~-¢*(x)]}
A(x,~)=27 2 e-e~ n= 1
(29a)
IV(x)-V ( y ) ] y ~
d X(r)
x exp [ ~2¢,(y)] dy
1
(29b)
® e-a1~
4. D I S C U S S I O N
Solutions for the spherical microparticle and the slab microparticle are summarized in Table 1. When the macropore diffusion controls the global uptake, the solutions are the same irrespective of whether the shape of the microparticle is a slab or spherical. When the particle and the microparticle are spherical and the external mass transfer resistance is negligible, the solution reduces to that of Ustinov and Akulov (1982). 4(a) Spherical microparticle
When the microparticle is spherical and the uptake is controlled by the dual mode of diffusion, the solutions are given in eqs (10)-(13). The time at which the adsorption wave front arrives at the center of the particle is given implicitly in eq. (12). When y is large, this time is given explicitly as y
where V(x) is defined in eq. (A11), H(z) is defined in eq. (A15), and: ~,=(n-~)~
(31a)
2, = nm (31 b) Equation (30) defines the adsorption wave front as a function of time, and eqs (29a) and (29b) are solutions for the mean concentration in the micropore and the concentration in the macropore, respectively. In obtaining eqs (29) and (30) one uses the following relation (Appendix 2):
1 P ® ,--x=p+¢.z
1 x/P tanh x/P"
2
(32)
Let to be the time at which the adsorption wave front hits the center of the particle, i.e. X(%)=0. Then one has h.
.
fl'~l"
1
oo e ;'~.~.\
tw=/-/t+o+--2 Y+~ + - /
(33) \7]\ 3 ,=, 2, / where H(zo) is defined in eq. (A15). The fractional uptake is determined in eq. (10). 3(b) Micropore diffusion control (7 ~, 1) When micropore diffusion controls the uptake, the fractional uptake is given by ® 1 F = "tu = 1 -- 2 ,~--1~2 exp ( -- ~2 z)
(34)
f 1
1"~
1
(36)
15 Table 2 tabulates this nondimensional time as a function of 7 and Bi. The time at which the adsorption wave front arrives at an arbitrary position x (x~[0, 1]) is denoted as z*(x), and is given implicitly in eq. (11). When the particle is spherical, these adsorption wavefront times are tabulated in Table 3 for Bi --* oo. The results are also shown graphically in Fig. 1. The continuous lines are for Bi ~ c~ and the dashed lines for Bi = 10. The line 0.03055 [half-time for the micropore diffusion, from eq. (16)] is also shown in Fig. 1. If the z*(x) curves lie well below the 0.03055 line, the system is micropore diffusion controlled. This is physically expected because the time for the adsorption wave front to penetrate the particle is much shorter than the halftime for diffusion into the microparticle. This means that the macropore void is initially equilibrated with the adsorbate. After this is achieved (which is very fast) micropore diffusion starts to occur. Thus, from Fig. 1 one can conclude that, when y<0.1, micropore diffusion controls the uptake. It is also clear in Fig. 2(a) that the profile of .4, vs x is practically uniform over the entire practical time range. When y lies between 0.1 and 10, the system is controlled by both micropore and macropore diffusion (Fig. 1). This is further illustrated in Fig. 2(b)-(d) ,as plots of .4~ vs x with z as the parameter. Figure 2(e) shows A, plotted vs x for y = 50. It shows that the absorbed concentration .4u behaves like the
1711
Sorption rate of bimodal microporous solids with an irreversible isotherm Table l. Solutions of the fractional uptake 741 Micropore diffusion control
y~ 0 ( 1 )
7"~1 Macropore diffusion control
Equation (15) Equation (34)
Equations (10)-(13) Equations (29) and (30)
Equations (22) and (23)
Spherical microparticle Slab microparticle
Table 2. Time zo to reach the core of a spherical particle (s = 2) comprised of spherical microparticlest Bi y
oo
0.05 0.07 0.1 0.2 0.3 0.5 0.7 1 2 3 5 7 10
0.000 0.000 0.001 0.006 0.014 0.034 0.060 0.104 0.266 0.433 0.766 1.100 1.600
100 472 911 818 762 193 599 121 061 82 34 67 00 00
2 5 7 9
0.000 0.000 0.001 0.007 0.014 0.035 0.062 0.107 0.27.3 0.443 0.783 1.123 1.633
50 490 9 947 3 889 4 016 2 706 772 046 15 47 34 33 33 33
0.000 0.000 0.001 0.007 0.015 0.036 0.063 0.110 0.280 0.453 0.800 1.146 1.666
20 509 9 983 8 961 3 273 3 226 958 986 25 12 34 00 67 67
0.000 0.001 0.002 0.008 0.016 0.040 0.069 0.119 0.300 0.483 0.850 1.216 1.766
10 569 2 097 1 184 3 067 9 828 587 892 63 08 34 00 67 67
0.000 0.001 0.002 0.009 0.019 0.046 0.079 0.135 0.333 0.533 0.933 1.333 1.933
674 8 298 8 580 3 467 4 629 857 994 48 37 33 33 33 33
tWhen 7 is large (7 > 10)
r~(s÷l)\2
~,/
15"
Table 3. Adsorption wave front time z*(x) for spherical particle comprised of spherical microparticles (Bi = oo) ~*(x) x
7=0.05
y=O.1
7=0.5
7= 1
7= 5
7 = 10
1
0
0
0
0
0
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
3.844x10 -7 5.288× 10 -6 2.271x 10 - s 5.999x 10- 5 1.203x 10 -4 2.010x 10 -a 2.927× 10 -4 3.806x10-4 4.466x 10 -4 4.722 × 10 -4
1.536×10 6 2.107 × 10- s 9.008 x 10 -5 2.367 x 10 -4 4.722× 10 -4 7.842 × 10 -4 1.136 × 10 - 3 1.471×10 -3 1.722 x 10 -3 1.819 × 10 3
3.807x10-5 5.099 × 10 4 2.109 x 10 3 5.328 x 10-3 1.020× 10 -2 1.629 x 10 -2 2.279 × 10 -2 2.871×10 2 3.298 x 10 2 3.460 x 10- 2
1.506x10 4 1.961 x 10-3
3.461x10 3 3.696 x 10-2 1.165 x 10-1 2.270 × 10-1 3.500× 10 -1 4.733 x 10-1 5.867 x 10-1 6.800×10 1 0.743×10-1 0.767 × 10-1
1.253x10-2 1.102 × 10-1 2.934 x 10-1 5.200 × 10-1 7.667 x 10 -1 1.013 1.240 1.427 1.553 1.600
traditional shrinking-core model, that is the spatial d o m a i n is practically divided into two. In the interior the particle is free from any adsorbate and, in the remaining domain, the particle is saturated with adsorbate. Thus, when ~ > 50, the global uptake is controlled by m a c r o p o r e diffusion, and the solution is given in eqs (22) and (24). The fractional uptakes vs z when Bi = oo are s h o w n in Fig. 3 for 7 =0.1, 1, 10 and 50. The continuous lines are obtained from the general solution [eqs (10)-(13)]. The dashed line is obtained from eq. (15), i.e. when micropore diffusion controls the uptake. It is clear from the previous discussion that, when 7 < 0.1, microCES 44:8-H
7.800×10-3
1.887 x 10-2 3.460× 10 -2 5.311 × 10 -2 7.189 × 10 -2 8.828x10-2 9.977 x 10-2 1.041 x 10-1
pore diffusion controls the uptake. The d o t - d a s h e d curve is obtained from eqs (22) and (24), i.e. when m a c r o p o r e diffusion controls the uptake. It is clear that m a c r o p o r e diffusion controls uptake when 7 > 50. The same conclusion was obtained when one investigated the behaviour of/T~ [Fig. 2(e)]. In conclusion, the following criteria are summarized for the d o m i n a t i n g mechanism of global uptake: 7<0.i:
micropore diffusion control
0.1<7<50:
m i c r o p o r e - m a c r o p o r e diffusion control
7>50:
m a c r o p o r e diffusion control.
1712
D. D. Do > 500:
macropore diffusion control.
To show the practical implication of the theory, the following practical range of parameters encountered in the zeolite literature is used: eM=0.33 R u = l x 10-4 cm R=0.01, 0.02, 0.05, 0.1 cm Cs = 0.0025 mol/cm 3
Co = 10-1°-10- 7 mol/cm 3 ~p=0.1, 0.5 cm2/s = 10- t t, 10- lo cm2/s b
10--~)
'
'
I
I
/
I
I
I
~t
1 X
Fig. 1. Adsorption wave front time, z*(x), as function ofx for spherical particle comprised of spherical microparticles: ( ) Bi= ~, (. . . . ) Bi = 10. (a) 4
Acknowledgements This work is a by-product of a project funded by University of Queensland Foundation Limited and a CSIRO/University of Queensland Grant. Support from the ARC is also gratefully acknowledged.
(b)
1
Figure 4 shows a plot of ~ vs the bulk adsorbate concentration, Co. If the microparticle and particle are spherical, the parametric domain of validity of dual micropore-macropore diffusion control is 0.1 < ~ < 50. This domain is indicated in Fig. 4. Thus the dual micropore-macropore diffusion control mechanism is easily the prevailing mechanism in zeolite adsorption.
NOTATION
X
I
1
X
1
(c)
ml,
Bi C C~ Co Cs
x
1
(d)
~p F
(e)
n[xh)] k= P P(X) r
0
x
1
0
x
1
Fig. 2. Mean micropore concentration, .~u, vs x for spherical particle comprised of spherical microparticles: (a) 7=0.1, (b) ~=0.5, (c) 7=1, (d) 7=10, (e) 7=50.
ru
R R~ $
t
? 4(b) Slab microparticle The solutions for slab geometry are summarized in Table 1. The same behaviour is observed for a slab microparticle as for a spherical microparticle. The following criteria are derived for a slab microparticle < 1: 1 < ~,< 500:
micropore diffusion control micropore-macropore diffusion control
X X~
xh) V
nondimensional macropore concentration nondimensional micropore concentration Biot number for mass transfer, eq. (6b) macropore concentration micropore concentration bulk adsorbate concentration saturated micropore concentration micropore diffusivity macropore diffusivity based on pore area fractional uptake function defined in eq. (24) mass transfer coefficient Laplace transform variable function defined in eq. (21) radial coordinate microparticle radial coordinate particle radius microparticle radius shape factor time time, defined in eq. (19) nondimensional radial coordinate nondimensional microparticle radial coordinate adsorption wave front position at time z function, defined in eq. ( )
Greek letters e~t
parameter, defined in eq. (6b) macropore porosity
Sorption rate of bimodal microporous solids with an irreversible isotherm
•0001
.001
.01
C
.1
1
1713
10
Fig. 3. Fractional uptake vs z for spherical particle comprised of spherical microparticles (Bi= oe).
104
103
102
I0
0.1
10-10
1159
1[~8
1(~7
co Fig. 4. Parameter y vs the bulk adsorbate concentration, Co, for ~ = 1 x 10-lo cm2/s: ( ) ~p =0.1 cm2/s, (- - - -)
~p=0.5 cm2/s.
~n 2,
~*(X)
eigenvalue, defined in eigenvalue, defined in eigenvalue, defined in eigenvalue, defined in nondimensional time, adsorption wave front tion
eq. (15) eq. (A22) eq. (31b) eq. (31a) defined in eq. (6b) time to reach posi-
REFERENCES Brauch, V. and Schlunder, E. U., 1975, The scale-up of activated carbon columns for water purification, based on results from batch tests--II. Theoretical and experimental determination of breakthrough curves in activated carbon columns. Chem. EnonO Sci. 30, 539-548. Cooper, R. S., 1965, Slow particle diffusion in ion-exchange columns, lnd. Enon9 Chem. Fundam. 4, 308-313.
Cooper, R. S. and Liberman, D. A., 1970, Fixed bed adsorption kinetics with pore diffusion control. Ind. Engn9 Chem. Fundam. 9, 620-623. Do, D. D., 1982, On the validity of the shrinking core model in non-catalytic gas solid reaction. Chem. Engn9 Sci. 37, 1477-1481. Do, D. D., 1984, Determination of boundary conditions for zero-order chemical reaction inside a single catalyst pellet under transient condition. Chem. Engn9 Commun. 25, 251-265. Do, D. D., 1985, Discrete cell model of fixed-bed adsorbers with rectangular adsorption isotherms. A.I.Ch.E. 3. 31, 1329 1337. Do, D. D., 1986, Analysis of a batch adsorber with rectangular adsorption isotherms. Ind. Engn9 Chem. Fundam. 25, 321-325. Hall, K. R., Eagleton, L. C., Acrivos, A. and Vermeulen, T., 1966, Pore- and solid-diffusion kinetics in fixed-bed adsorption under constant-pattern conditions. Ind. Enon9 Chem. Fundam. 5, 212-223. Kawazoe. K., Suzuki, M. and Chihara, K., 1974, Chromatographic study of diffusion in molecular sieving carbon. 3. chem. Enyn 9 Japan 7, 151-157. Lee, L. K. and Ruthven, D. M., 1976, Kinetics of adsorption in pore diffusion controlled systems. Chem. EnonO Sci. 31, 851-85Z Ma, Y. H. and Ho, S. Y, 1970, Diffusion in synthetic Faujasite powder and pellets. A.I.Ch.E. 3. 20, 279-284. Polte, W. and Mersmann, A., 1986, Ad- and desorption in bidispersed adsorbents, in Fundamentals of Adsorption (Edited by A. I. Liapis). New York. Prinz, D. and Riekert, L., 1986, Observation of rates of sorption and diffusion in zeolite crystals at constant temperature and pressure. Ber. Bunsenges. phys. Chem. 90, 413417. Ruckenstein, E., Vaidyanathan, A. S. and Youngquist, G. R., 1971, Sorption by solids with bidisperse pore structures. Chem. Engn9 Sci. 26, 1305 1313. Sargent, R. W. H. and Whitford, C. J,, 1971, Diffusion of carbon dioxide in type 5A molecular sieve, in Molecular Sieve II, Advances in Chemistry Series 102, American Chemical Society, Washington, DC. Suzuki, M. and Kawazoe, K., 1974, Batch measurement of adsorption rate in an agitated tank-pore diffusion kinetics with irreversible isotherm. J. chem. Engn9 Japan 7, 346-350. Ustinov, E. A. and Akulov, A. K., 1982, Adsorption kinetics for systems with strongly convex iostherm. I. Model of diffusional mass transport in transport pores and microporous formations. Russ. 3. phys. Chem. 56, 1028-1030. Vermeulen, T, 1953, Theory for irreversible and constant pattern solid diffusion. Ind. Engn 9 Chem. 45, 1664-1670. Vermeulen, T., 1958, Separation by adsorption methods. Adv. chem. Enon O 2, 147-208. Weber, T. W. and Chakravorti, R. K., 1974, Pore and solid diffusion models for fixed bed adsorbers. A.1.Ch.E.J. 20, 228-238.
0009-2509/89 $3.00+0.00 © 1989 Pergamon Press pie
SORPTION RATE OF BIMODAL MICROPOROUS SOLIDS WITH AN IRREVERSIBLE ISOTHERM D. D. DO Department of Chemical Engineering, University of Queensland, St. Lucia, Q. 4067, Australia
(Received 7 June 1988; acceptedfor publication 23 January 1989) Abstract--The sorption rate of gases into a microporous adsorbent having a bimodal pore size distribution is analysed. The adsorption isotherm is assumed to exhibit a rectangular shape (highly favourable isotherm) which is commonly observed in systems of zeolites and carbon molecular sieves. Degenerate models are derived when one mode of diffusion is faster than the other, and the conditions when one mode of diffusion controls the global uptake have been delineated. l. INTRODUCTION The global rate of adsorption of gases in microporous solids exhibiting a bimodal pore size distribution depends on the relative rates of diffusion in the macropore and micropore. The first work on modelling of adsorption in bimodal solids in the Western literature was probably by Ruckenstein et al. (1971). They assumed a pore diffusion mechanism in both the macropore and the micropore, and allowed for an adsorptive capacity in both pores. Since that work, other researchers in the area of zeolite and activated carbon [e.g. Sargent and Whitford (1971), Kawazoe et al. (1974) and Ma and Ho (1974)] have proposed a different bimodal diffusion model. The essential features of this model are: no adsorption in the macropore, pore diffusion in the macropore, adsorption isotherm at the pore mouth of the micropore and activated diffusion in the micropore. However, all the work above assumed a linear adsorption isotherm at the pore mouth of the micropore. This would render the mass balance equations linear and they were susceptible to linear analysis, like the Laplace transform or finite integral transform. Recently, Polte and Mersmann (1986) have assumed a Langmuir adsorption isotherm at the pore mouth of the micropore. A finite difference technique was used to solve the mass balance equations numerically. When the adsorption isotherm is rectangular, i.e. an irreversible adsorption isotherm, analyses were presented for pseudo-homogeneous solids. Suzuki and Kawazoe (1974) and Do (1986) dealt with batch reservoir problems. Film resistance was allowed for. Fixed-bed problems without film resistance were considered by Vermeulen (1958), Hall et al. (1966), Cooper (1965) and Cooper and Liberman (1970). Film resistance in a fixed bed was considered by Weber and Chakravorti (1974), Brauch and Schlunder (1975), Lee and Ruthven (1976) and Do (1985). This paper presents a theoretical analysis of the sorption rate in a bimodal solid with an irreversible adsorption isotherm, i.e. the surface concentration of the microsphere is instantly equal to the saturated value as long as the adsorbate concentration in the
macropore void is nonzero. The next section deals with a bimodal solid with a spherical microparticle. Analysis of a bimodal slab microparticle is then included because Prinz and Riekert (1986) in their work of the sorption of benzene, 3-methyl pentane and p-xylene found that the solution of diffusion into a slab fitted their experimental data reasonably well. 2. SPHERICAL MICROPARTICLE
Consider a bimodal solid particle comprised of many small microparticles. This particle is exposed to an environment of constant bulk concentration, Co. The following assumptions are made: (a) isothermal system, (b) constant diffusivities in the macropore and micropore, (c) spherical microparticle, (d) particle of arbitrary shape, (e) irreversible adsorption isotherm at microparticle surface. The mass balance equation in the microparticle is ~C, 1 c3 / --=~--/r2--/
c3C~\
(la)
subject to the following initial and boundary conditions: t=0: ru=0:
r~=R;
C~=0
(lb)
~C~,/t3r~,=O
(lc)
C =)'C, ~ (0
(ld) (le)
if C > 0 if C=.0
where C is the adsorbate concentration in the macropore. Equations (1 d) and (le) represent the irreversible adsorption isotherm. The mass balance equation in the macropore is OC
~C~
eu ~ t + ( 1 - e u ) ~ t
1 ~ { ~C'~
=(eM~p)~;~rk:~r )
(2)
where eM is the macropore porosity, ~p is the pore diffusivity based on the pore cross-sectional area, and
1707
1708
D.D. Do
s is the shape factor of the particle (s = 0, 1, and 2 for slab, cylinder and sphere, respectively). C e is the mean micropore concentration at position r, and is defined as follows: 3 ('R,
C-- e_- ~
2 I reCedre"
(3)
"~e do
Equation (2) is subject to the following initial and boundary conditions: t=O:
C=O
(4a)
r=O:
OC/~r=O
(4b)
r=R: (eu~p)OC/Or=k~(Co-C)
2(a) Comparable rates [7 n0(1)] The first case dealt with is the comparable-rates case, i.e. 7 is of the order of unity. The solution procedure is given in an appendix available from the author. The solutions for the fractional uptake are:
F= f
OCe
(5)
A = C/Co, A e = Ce/C~, x = r/R, x e = re/R e (6a) z=~t/R2'7
(1-eu)C,( R ~ 2 ~ k=R euC~o \R-~J ~ ' Bi=eu~ p
(6b)
/,)/
z+-----
fo
xL~edx
for z>zo
.. )
(x2-1 [1
1
[
5
+~_sJ~ [1
for s ¢ l
1 "~ 2
~,~+~)(X-1)
fors=l.
(11)
The time To is the time when the adsorption front reaches the center of the particle. It is obtained from eq. (11) by setting X to zero. Then one has 1
2 o~ e u.%\
1
/1
~)%zo+~-~ ~=l~-)=~s~+~).
OAe 1 6~ (x2~Ae'~ Oz - x~ Oxe \ e Oxe ]
(lOb)
~ ~-
,,
= )2(~_s)- ~
1\/
eqs (1) and (3)-(5) become
forO
where X(z) is the adsorption front at time ~ and is given by
|X21nX By defining the following nondimensional variables and parameters:
(l+s)f x~fi.edx tJx") (l+s)
(4c)
where k= is the fluid mass transfer coefficient. The left-hand side ofeq. (2) contains two terms. One is the hold-up capacity in the macropore void, and the other is the capacity in the micropore. In most practical cases the macropore void capacity is very small compared to the capacity of the micropore. Hence, eq. (2) can be replaced by
1 0 { OC'~ (1-eu)~-~=(eu~,)~r(r'~r ).
pore commonly used in chemical engineering literature (Vermeulen, 1958; Hall et al., 1966).
1\
(12)
(7b)
Finally, "4e in the fractional uptake solution [eq. (10)] is the mean sorbate concentration in the micropore, given as
x e = 0: OAe/Oxe = 0
(7c)
6 o~ ,'te = 1 - - - - X n - 2 exp {--n2n2[z--z*(x)]} (13)
A
(7d) (7e)
z=0:
xe = l :
(7a)
Ae=0
fl e=_0
(13110(x
if A > 0 if A = 0
A)]
x/j
(8b)
x = I: OA/Ox=Bi(1-A)
(8c)
where Ae = 3
fo
x 2 A e dx e
where z*(x) is the time when the adsorption front reaches position x, and is determined from eq. (11).
(8a)
aA/Ox=O
x=0:
7r2n= 1
(9)
The parameter 7 measures the relative rates between the micropore diffusion and the macropore diffusion. When this parameter is of the order of unity, both diffusions would be expected to contribute to the global uptake. When this parameter is very small, micropore diffusion controls the uptake and the response is the typical Fickian uptake of the microparticle. However, when 7 is very large, one would expect the classical shrinking core model of the macro-
2(b) Micropore diffusion control (7 ~ 1) When 2, is very small, the dynamics start with the physical macropore void uptake. This stage is normally very fast and the uptake capacity of the macropore void is very small. Therefore, this stage is neglected. The next stage is the micropore uptake: all microparticles are exposed to the same bulk adsorbate concentration, Co. The mass balance equation (7 ~ 1) is
1 ±(x20Ae
Oz - x 2 tOXe\ ~'Ox.,] z=O: Au--0
(14a) (14b)
xe=0:
OAe/~xe=O
(14c)
xe = l :
Ae = l
(14d)
The solution is straightforward, and the mean micropore concentration is
Sorption rate of bimoda ® 1 Z ~ s e x p ( - ~ . 2z) n=l %
"4u = 1 - 6
((,=nz).
.~croporous solids with an irreversible isotherm (15)
In this case all the microparticles behave identically. The fractional uptake is just equal to the mean micropore concentration defined in eq. (15). When the micropore diffusion controls the global uptake the system dynamics are independent of the type of isotherm, whether linear, rectangular, Langmuir or Freundlich. The half-time can be readily obtained from eq. (15):
Solving eqs (17) and (18) gives the following equation for the adsorption wave front position: dX
~ = -P(X) dt X-S 1
1 --X
Bi
1 -s
for s # 1
(21a)
fors=l.
(21b)
1-s
e(X) =
I
(16)
The half-time is independent of all macropore characteristics, and is proportional to the square of the microparticle radius.
(20)
where
R2
to. 5 = 0.03055 - -u.
1709
X
l~+ln (1 ]
Bi
\XJ
The solution of the adsorption wave front is
2(c) Macropore diffusion control (7 >>1) When ~ is very large, the system dynamics are controlled by the macropore diffusion. This case is the
H(X) =
Ix
P(m)
= ["
(22)
where
H(X) =
(2~t + ~)(1- X2)+12x21n X 1
classical shrinking-core model commonly used in the adsorption and the gas-solid reaction literature. The mathematical proof for the shrinking-core model was presented by Do (1982) for gas-solid reaction using a singular perturbation method, and for adsorption with reaction by Do (1984). For this case (7 >>1), there is no gradient (or no mass transfer resistance) in the microparticle. The mass balance equations are
1~ (x ~t~A~ 0 xj=
x~ ~-x\
for X ( t ' ) < x < 1
x = X(t'): a = 0 x=l:
(lVa) (17b)
M1-X 3)
fors=l
(23b)
for s=2.
(23c)
Knowing the adsorption wave front, the fractional uptake is determined as F = 1 - X s + 1.
(24)
The adsorption wave front at 50% uptake is given by 11/~1+~ ( Xo.5=2
=l
0.5
fors=0
0.707
fors=l
0.794
for s=2.
(25)
Substitution of eq. (25) into eq. (22) yields the half-time t0. 5
(1 --F,M)CsR2
H(Xo.5)
(26)
(17cl
where X(7) is the position of the adsorption wave front, given by
i'=0:
/ 1
(23a)
gMC~']pC O
OA/Ox=Bi(1-A)
dX dF'-
2
fors=0
t~A ~x x(?~
(18a)
X-- 1
where H(Xo.5) is calculated from eq. (23) after X is replaced by Xo.5:
H(Xo.5)=
f 0.125+1/2Bi 0.03836+ 1/4Bi 0.01835 + 1/6Bi
for s = 0 for s = 1
(27)
for s=2.
(18b)
where 7is defined as
3. SLAB M I C R O P A R T I C L E
(eM~ p )
Co
_ z
t'= R2(1 --eM)C~ 7"
(19)
Although most work in the literature of adsorption assumed a microparticle with a spherical shape, Prinz and Riekert (1986) in their recent study of hydrocar-
1710
D. D. Do
bon uptake in ZSM-5 found that a theory using a slab crystal fits the data better than for other geometries. To make the analysis as general as possible, the theory of the sorption rate in a bimodal solid with a slab microparticle is presented here.
where ~, is given in eq. (31). The half-time of the global uptake is R2 to.5 =0.19674 -~ • (35) @
3(a) Comparable diffusion rates [Y"~0(1)] The mass balance equations in nondimensional form are identical to eqs (7)-(9) except that eqs (73) and (9) are replacd by
3(c) Macropore diffusion control (7 >>1) When macropore diffusion controls the uptake, the shape of the microparticle is irrelevant. This analysis was presented in Section 2(c), and the solutions are eqs (22) and (24).
dAu _ 02Au t3z
(28a)
Ox2
and "4u= Jo A~,dx~,
(28b)
respectively. The nondimensional variables and parameters are defined in eq. (6). Following the procedure of the previous section: , = , ~ 2 e x p {-~2[~-¢*(x)]}
A(x,~)=27 2 e-e~ n= 1
(29a)
IV(x)-V ( y ) ] y ~
d X(r)
x exp [ ~2¢,(y)] dy
1
(29b)
® e-a1~
4. D I S C U S S I O N
Solutions for the spherical microparticle and the slab microparticle are summarized in Table 1. When the macropore diffusion controls the global uptake, the solutions are the same irrespective of whether the shape of the microparticle is a slab or spherical. When the particle and the microparticle are spherical and the external mass transfer resistance is negligible, the solution reduces to that of Ustinov and Akulov (1982). 4(a) Spherical microparticle
When the microparticle is spherical and the uptake is controlled by the dual mode of diffusion, the solutions are given in eqs (10)-(13). The time at which the adsorption wave front arrives at the center of the particle is given implicitly in eq. (12). When y is large, this time is given explicitly as y
where V(x) is defined in eq. (A11), H(z) is defined in eq. (A15), and: ~,=(n-~)~
(31a)
2, = nm (31 b) Equation (30) defines the adsorption wave front as a function of time, and eqs (29a) and (29b) are solutions for the mean concentration in the micropore and the concentration in the macropore, respectively. In obtaining eqs (29) and (30) one uses the following relation (Appendix 2):
1 P ® ,--x=p+¢.z
1 x/P tanh x/P"
2
(32)
Let to be the time at which the adsorption wave front hits the center of the particle, i.e. X(%)=0. Then one has h.
.
fl'~l"
1
oo e ;'~.~.\
tw=/-/t+o+--2 Y+~ + - /
(33) \7]\ 3 ,=, 2, / where H(zo) is defined in eq. (A15). The fractional uptake is determined in eq. (10). 3(b) Micropore diffusion control (7 ~, 1) When micropore diffusion controls the uptake, the fractional uptake is given by ® 1 F = "tu = 1 -- 2 ,~--1~2 exp ( -- ~2 z)
(34)
f 1
1"~
1
(36)
15 Table 2 tabulates this nondimensional time as a function of 7 and Bi. The time at which the adsorption wave front arrives at an arbitrary position x (x~[0, 1]) is denoted as z*(x), and is given implicitly in eq. (11). When the particle is spherical, these adsorption wavefront times are tabulated in Table 3 for Bi --* oo. The results are also shown graphically in Fig. 1. The continuous lines are for Bi ~ c~ and the dashed lines for Bi = 10. The line 0.03055 [half-time for the micropore diffusion, from eq. (16)] is also shown in Fig. 1. If the z*(x) curves lie well below the 0.03055 line, the system is micropore diffusion controlled. This is physically expected because the time for the adsorption wave front to penetrate the particle is much shorter than the halftime for diffusion into the microparticle. This means that the macropore void is initially equilibrated with the adsorbate. After this is achieved (which is very fast) micropore diffusion starts to occur. Thus, from Fig. 1 one can conclude that, when y<0.1, micropore diffusion controls the uptake. It is also clear in Fig. 2(a) that the profile of .4, vs x is practically uniform over the entire practical time range. When y lies between 0.1 and 10, the system is controlled by both micropore and macropore diffusion (Fig. 1). This is further illustrated in Fig. 2(b)-(d) ,as plots of .4~ vs x with z as the parameter. Figure 2(e) shows A, plotted vs x for y = 50. It shows that the absorbed concentration .4u behaves like the
1711
Sorption rate of bimodal microporous solids with an irreversible isotherm Table l. Solutions of the fractional uptake 741 Micropore diffusion control
y~ 0 ( 1 )
7"~1 Macropore diffusion control
Equation (15) Equation (34)
Equations (10)-(13) Equations (29) and (30)
Equations (22) and (23)
Spherical microparticle Slab microparticle
Table 2. Time zo to reach the core of a spherical particle (s = 2) comprised of spherical microparticlest Bi y
oo
0.05 0.07 0.1 0.2 0.3 0.5 0.7 1 2 3 5 7 10
0.000 0.000 0.001 0.006 0.014 0.034 0.060 0.104 0.266 0.433 0.766 1.100 1.600
100 472 911 818 762 193 599 121 061 82 34 67 00 00
2 5 7 9
0.000 0.000 0.001 0.007 0.014 0.035 0.062 0.107 0.27.3 0.443 0.783 1.123 1.633
50 490 9 947 3 889 4 016 2 706 772 046 15 47 34 33 33 33
0.000 0.000 0.001 0.007 0.015 0.036 0.063 0.110 0.280 0.453 0.800 1.146 1.666
20 509 9 983 8 961 3 273 3 226 958 986 25 12 34 00 67 67
0.000 0.001 0.002 0.008 0.016 0.040 0.069 0.119 0.300 0.483 0.850 1.216 1.766
10 569 2 097 1 184 3 067 9 828 587 892 63 08 34 00 67 67
0.000 0.001 0.002 0.009 0.019 0.046 0.079 0.135 0.333 0.533 0.933 1.333 1.933
674 8 298 8 580 3 467 4 629 857 994 48 37 33 33 33 33
tWhen 7 is large (7 > 10)
r~(s÷l)\2
~,/
15"
Table 3. Adsorption wave front time z*(x) for spherical particle comprised of spherical microparticles (Bi = oo) ~*(x) x
7=0.05
y=O.1
7=0.5
7= 1
7= 5
7 = 10
1
0
0
0
0
0
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
3.844x10 -7 5.288× 10 -6 2.271x 10 - s 5.999x 10- 5 1.203x 10 -4 2.010x 10 -a 2.927× 10 -4 3.806x10-4 4.466x 10 -4 4.722 × 10 -4
1.536×10 6 2.107 × 10- s 9.008 x 10 -5 2.367 x 10 -4 4.722× 10 -4 7.842 × 10 -4 1.136 × 10 - 3 1.471×10 -3 1.722 x 10 -3 1.819 × 10 3
3.807x10-5 5.099 × 10 4 2.109 x 10 3 5.328 x 10-3 1.020× 10 -2 1.629 x 10 -2 2.279 × 10 -2 2.871×10 2 3.298 x 10 2 3.460 x 10- 2
1.506x10 4 1.961 x 10-3
3.461x10 3 3.696 x 10-2 1.165 x 10-1 2.270 × 10-1 3.500× 10 -1 4.733 x 10-1 5.867 x 10-1 6.800×10 1 0.743×10-1 0.767 × 10-1
1.253x10-2 1.102 × 10-1 2.934 x 10-1 5.200 × 10-1 7.667 x 10 -1 1.013 1.240 1.427 1.553 1.600
traditional shrinking-core model, that is the spatial d o m a i n is practically divided into two. In the interior the particle is free from any adsorbate and, in the remaining domain, the particle is saturated with adsorbate. Thus, when ~ > 50, the global uptake is controlled by m a c r o p o r e diffusion, and the solution is given in eqs (22) and (24). The fractional uptakes vs z when Bi = oo are s h o w n in Fig. 3 for 7 =0.1, 1, 10 and 50. The continuous lines are obtained from the general solution [eqs (10)-(13)]. The dashed line is obtained from eq. (15), i.e. when micropore diffusion controls the uptake. It is clear from the previous discussion that, when 7 < 0.1, microCES 44:8-H
7.800×10-3
1.887 x 10-2 3.460× 10 -2 5.311 × 10 -2 7.189 × 10 -2 8.828x10-2 9.977 x 10-2 1.041 x 10-1
pore diffusion controls the uptake. The d o t - d a s h e d curve is obtained from eqs (22) and (24), i.e. when m a c r o p o r e diffusion controls the uptake. It is clear that m a c r o p o r e diffusion controls uptake when 7 > 50. The same conclusion was obtained when one investigated the behaviour of/T~ [Fig. 2(e)]. In conclusion, the following criteria are summarized for the d o m i n a t i n g mechanism of global uptake: 7<0.i:
micropore diffusion control
0.1<7<50:
m i c r o p o r e - m a c r o p o r e diffusion control
7>50:
m a c r o p o r e diffusion control.
1712
D. D. Do > 500:
macropore diffusion control.
To show the practical implication of the theory, the following practical range of parameters encountered in the zeolite literature is used: eM=0.33 R u = l x 10-4 cm R=0.01, 0.02, 0.05, 0.1 cm Cs = 0.0025 mol/cm 3
Co = 10-1°-10- 7 mol/cm 3 ~p=0.1, 0.5 cm2/s = 10- t t, 10- lo cm2/s b
10--~)
'
'
I
I
/
I
I
I
~t
1 X
Fig. 1. Adsorption wave front time, z*(x), as function ofx for spherical particle comprised of spherical microparticles: ( ) Bi= ~, (. . . . ) Bi = 10. (a) 4
Acknowledgements This work is a by-product of a project funded by University of Queensland Foundation Limited and a CSIRO/University of Queensland Grant. Support from the ARC is also gratefully acknowledged.
(b)
1
Figure 4 shows a plot of ~ vs the bulk adsorbate concentration, Co. If the microparticle and particle are spherical, the parametric domain of validity of dual micropore-macropore diffusion control is 0.1 < ~ < 50. This domain is indicated in Fig. 4. Thus the dual micropore-macropore diffusion control mechanism is easily the prevailing mechanism in zeolite adsorption.
NOTATION
X
I
1
X
1
(c)
ml,
Bi C C~ Co Cs
x
1
(d)
~p F
(e)
n[xh)] k= P P(X) r
0
x
1
0
x
1
Fig. 2. Mean micropore concentration, .~u, vs x for spherical particle comprised of spherical microparticles: (a) 7=0.1, (b) ~=0.5, (c) 7=1, (d) 7=10, (e) 7=50.
ru
R R~ $
t
? 4(b) Slab microparticle The solutions for slab geometry are summarized in Table 1. The same behaviour is observed for a slab microparticle as for a spherical microparticle. The following criteria are derived for a slab microparticle < 1: 1 < ~,< 500:
micropore diffusion control micropore-macropore diffusion control
X X~
xh) V
nondimensional macropore concentration nondimensional micropore concentration Biot number for mass transfer, eq. (6b) macropore concentration micropore concentration bulk adsorbate concentration saturated micropore concentration micropore diffusivity macropore diffusivity based on pore area fractional uptake function defined in eq. (24) mass transfer coefficient Laplace transform variable function defined in eq. (21) radial coordinate microparticle radial coordinate particle radius microparticle radius shape factor time time, defined in eq. (19) nondimensional radial coordinate nondimensional microparticle radial coordinate adsorption wave front position at time z function, defined in eq. ( )
Greek letters e~t
parameter, defined in eq. (6b) macropore porosity
Sorption rate of bimodal microporous solids with an irreversible isotherm
•0001
.001
.01
C
.1
1
1713
10
Fig. 3. Fractional uptake vs z for spherical particle comprised of spherical microparticles (Bi= oe).
104
103
102
I0
0.1
10-10
1159
1[~8
1(~7
co Fig. 4. Parameter y vs the bulk adsorbate concentration, Co, for ~ = 1 x 10-lo cm2/s: ( ) ~p =0.1 cm2/s, (- - - -)
~p=0.5 cm2/s.
~n 2,
~*(X)
eigenvalue, defined in eigenvalue, defined in eigenvalue, defined in eigenvalue, defined in nondimensional time, adsorption wave front tion
eq. (15) eq. (A22) eq. (31b) eq. (31a) defined in eq. (6b) time to reach posi-
REFERENCES Brauch, V. and Schlunder, E. U., 1975, The scale-up of activated carbon columns for water purification, based on results from batch tests--II. Theoretical and experimental determination of breakthrough curves in activated carbon columns. Chem. EnonO Sci. 30, 539-548. Cooper, R. S., 1965, Slow particle diffusion in ion-exchange columns, lnd. Enon9 Chem. Fundam. 4, 308-313.
Cooper, R. S. and Liberman, D. A., 1970, Fixed bed adsorption kinetics with pore diffusion control. Ind. Engn9 Chem. Fundam. 9, 620-623. Do, D. D., 1982, On the validity of the shrinking core model in non-catalytic gas solid reaction. Chem. Engn9 Sci. 37, 1477-1481. Do, D. D., 1984, Determination of boundary conditions for zero-order chemical reaction inside a single catalyst pellet under transient condition. Chem. Engn9 Commun. 25, 251-265. Do, D. D., 1985, Discrete cell model of fixed-bed adsorbers with rectangular adsorption isotherms. A.I.Ch.E. 3. 31, 1329 1337. Do, D. D., 1986, Analysis of a batch adsorber with rectangular adsorption isotherms. Ind. Engn9 Chem. Fundam. 25, 321-325. Hall, K. R., Eagleton, L. C., Acrivos, A. and Vermeulen, T., 1966, Pore- and solid-diffusion kinetics in fixed-bed adsorption under constant-pattern conditions. Ind. Enon9 Chem. Fundam. 5, 212-223. Kawazoe. K., Suzuki, M. and Chihara, K., 1974, Chromatographic study of diffusion in molecular sieving carbon. 3. chem. Enyn 9 Japan 7, 151-157. Lee, L. K. and Ruthven, D. M., 1976, Kinetics of adsorption in pore diffusion controlled systems. Chem. EnonO Sci. 31, 851-85Z Ma, Y. H. and Ho, S. Y, 1970, Diffusion in synthetic Faujasite powder and pellets. A.I.Ch.E. 3. 20, 279-284. Polte, W. and Mersmann, A., 1986, Ad- and desorption in bidispersed adsorbents, in Fundamentals of Adsorption (Edited by A. I. Liapis). New York. Prinz, D. and Riekert, L., 1986, Observation of rates of sorption and diffusion in zeolite crystals at constant temperature and pressure. Ber. Bunsenges. phys. Chem. 90, 413417. Ruckenstein, E., Vaidyanathan, A. S. and Youngquist, G. R., 1971, Sorption by solids with bidisperse pore structures. Chem. Engn9 Sci. 26, 1305 1313. Sargent, R. W. H. and Whitford, C. J,, 1971, Diffusion of carbon dioxide in type 5A molecular sieve, in Molecular Sieve II, Advances in Chemistry Series 102, American Chemical Society, Washington, DC. Suzuki, M. and Kawazoe, K., 1974, Batch measurement of adsorption rate in an agitated tank-pore diffusion kinetics with irreversible isotherm. J. chem. Engn9 Japan 7, 346-350. Ustinov, E. A. and Akulov, A. K., 1982, Adsorption kinetics for systems with strongly convex iostherm. I. Model of diffusional mass transport in transport pores and microporous formations. Russ. 3. phys. Chem. 56, 1028-1030. Vermeulen, T, 1953, Theory for irreversible and constant pattern solid diffusion. Ind. Engn 9 Chem. 45, 1664-1670. Vermeulen, T., 1958, Separation by adsorption methods. Adv. chem. Enon O 2, 147-208. Weber, T. W. and Chakravorti, R. K., 1974, Pore and solid diffusion models for fixed bed adsorbers. A.1.Ch.E.J. 20, 228-238.