Langmuir 1994,10, 3296-3302
3296
Effect of Surface Heterogeneity on the Adsorption Kinetics of Gases in Activated Carbon: Pore Size Distribution vs Energy Distribution Xijun Hu and Duong D. Do* Department of Chemical Engineering, University of Queensland, Brisbane, Queensland 4072, Australia Received March 24, 1994. I n Final Form: May 26, 1994@ The effect of surface heterogeneity on the adsorption dynamics of ethane and propane in activated carbon is studied in this article. Assuming the micropore size distribution, from which the adsorption energy distribution can be derived, is the sole source for system heterogeneity and the local equilibrium isotherm is described by a Langmuir equation, we have developed a heterogeneous pore and surface diffusion model and tested its model capability with experimental data of hydrocarbons obtained over a range of conditions. A gamma distribution is used to describe the micropore size distribution. Results of this model are compared with those obtained from a previously proposed model based on a uniform energy distribution.
Introduction There are two approaches in the description of surface heterogeneity. In one approach, a n energy distribution concept is utilized, and the other assumes a pore size distribution to describe the structural heterogeneity. The first method has been widely studied in the literature in the measurement of adsorption e q ~ i l i b r i u m l and - ~ in the steady-state surface diffusivity calculations.4-9 This energy distribution concept is recently applied to transient adsorption dynamic^.'^-'^ On the other hand, the pore size distribution concept is only studied for adsorption equilibriuml4-I6 and steady-state surface d i f f ~ s i 0 n .To l~ the best knowledge of the authors it has not been applied
* Author to whom all correspondence should be addressed. Abstract published in Advance ACS Abstracts, J u l y 15, 1994. (1)Roginski, S. S. Adsorption and Catalysis on Heterogeneous Surfaces, Academy of Sciences of USSR Moscow, 1948 (in Russian). (2)Ross, S.; Olivier, J. P. On Physical Adsorption. X I . The Adsorption Isotherm and the Adsorptive Energy Distribution of Solids. J . Phys. Chem. 1961, 65 (41, 608-615. (3) Sicar, S.;Myers, A. L. EquilibriumAdsorption ofGases and Liquids on HeterogeneousAdsorbents - APracticalViewpoint. Surf. Sci. 1988, 205,353-386. (4) Okazaki, M.; Tamon, H.; Toei, R. Interpretation of Surface Flow Phenomenon of Adsorbed Gases by Hopping Model. AIChE J . 1981, 27 (2), 262-270. (5) Zgrablich, G.; Pereyra, V.; Ponzi, M.; Marchese, J. Connectivity Effects for Surface Diffusion ofAdsorbed Gases. AIChEJ. 1986,32 (7), 1158-1168. (6) Horas, J. A.; Saitua, H. A.; Marchese, J. Surface Diffusion of Adsorbed Gases on an Energetically Heterogeneous Porous Solid. J. Colloid Interface Sei. 1988, 126(2), 421-431. (7) Seidel, A.; Carl, P. The Concentration Dependence of Surface Diffusion for Adsorption on Energetically Heterogeneous Adsorbents. Chem. Eng. Sci. 1989,44 (11, 189-194. (8) Kapoor, A.; Yang, R. T. Surface Diffusion on Energetically Heterogeneous Surfaces. AIChE J. 1989, 35 (lo), 1735-1738. (9) Kapoor, A.; Yang, R. T. Surface Diffusion on Energetically Heterogeneous Surfaces-An Effective Medium Approximation Approach. Chem. Eng. Scz. 1990,45 (ll), 3261-3270. (10) Do, D. D.; Hu, X. An Energy Distributed Model for Adsorption Kinetics in Large Heterogeneous Microporous Particles. Chem. Eng. Sci. 1993,48 (ll),2119-2127. (11) Hu, X.; Rao, G. N.; Do, D. D. Effect of Energy Distribution on Sorption Kinetics in Bidispersed Particles. AIChEJ. 1993,39(2),249261. (12) Hu, X.; Do, D. D. Effect of Surface Energetic Heterogeneity on the Kinetics of Adsorption of Gases in Microporous Activated Carbon. Langmuir 1993,9 (101,2530-2536. (13) Hu, X.; Do, D. D. Role ofEnergy Distribution in Multicomponent Sorption Kinetics in Bidispersed Solids. AIChEJ. 1993,39(10),16281640. (14) Dubinin, M. M. Physical Adsorption of Gases and Vapors in Micropores. Prog. Surf. Membr. Sci. 1976, 9, 1-70. @
in the transient adsorption dynamics, although the relationship between the energetic and structural heterogeneity is recently studied in the calculation of adsorption equilibrium isotherm.18J9 It is the purpose of this article to investigate the role of micropore size distribution in the adsorption kinetics of gases in activated carbon. "ry Let us consider a large particle exposed to a gas stream having a constant concentration of C,. The particle is heterogeneous and this heterogeneity can be induced by either an adsorbate-adsorbent interaction energy distribution or a micropore size distribution. In the development of model formulations the system is assumed isothemal and the particle is large enough so that the mass transfer is controlled by the intraparticle pore and surface diffusions. The model using an energy distribution concept has been stated in ref 11. In this section we present a model based on the micropore size distribution.
Adsorption Isotherm and Micrpore Distribution. The local adsorption isotherm for a given micropore is assumed to follow the Langmuir equation
where C,(E) is the adsorbed amount a t an adsorbateadsorbent interaction energy level of E, C,, is the maximum adsorbed phase concentration, R, is the gas (15)Dubinin, M. M. Generalization of the Theory ofvolume Filling of Micropores to Nonhomogeneous Microporous Structures. Carbon 1985,23,373-380. (16) Jaroniec, M.; Choma, J.; Lu, X. An Improved Method for Evaluating the Micropore-size Distribution from Adsorption Isotherm. Chem. Eng.Sci. 1991,46 (12), 3299-3301. (17) Do, D. D.; Do, H. Effect of Micropore Size Distribution on the Surface Diffusivity in Microporous Solids. Chem. Eng. Sei. 1993,48 (141, 2625-2642. (18) Jagiello, J.; Schwarz, J. A. Energetic and Structural Heterogeneity of Activated Carbons Determined Using Dubinin Isotherms and an Adsorption Potential in Model Micropores. J . Colloid Interface Sci. 1992, 154 (l), 225-237. (19) Jagiello, J.; Schwarz, J. A. Relationship between Energetic and Structural Heterogeneity of Micropous Carbons Determined on the basis of Adsorption Potentials in Model Micropores. Langmuir 1993,9 (101, 2513-2517.
0743-746319412410-3296$04.50/00 1994 American Chemical Society
Langmuir, Vol. 10,No. 9,1994 3297
Adsorption Kinetics of Gases in Activated Carbon constant, T is temperature, bo is the affinity constant at zero energy level and can be treated as temperature independent over a limited temperature interval. If the energy distribution is F*(E), the observed adsorption isotherm is
In this article the energy distribution is considered to be induced by the variation of the micropore size. Assuming that the micropore is slit-shaped, the gas-solid potential of a molecule confined in two parallel lattice planes, up, is given by Everett and Pow120
(AT} 2rp - z
(3)
where z is the distance between the molecule and one of the pore surface walls separated by a distance of 2rp and ro is the molecular size. The parameter u.* is the depth of the Lennard-Jones potential minimum for a single lattice place and this depth occurs at the position ro. The properties of the potential were illustrated in ref 20. The depth of the potential minimum up*is obtained numerically from eq 3 and it ranges from the value for the interaction potential minimum depth with a single wall, us*, to the value of 2us* for r p = ro. The adsorbateadsorbent interaction energy is the negative of the potential minimum, which is related to the micropore halfwidth, rp,by
E = up*( r p )
(4)
Hence the minimum and maximum adsorption energies in eq 2 are: Ed,, = us*,and E,, = 2us*. Let the micropore size distribution be F(rp),the observed adsorption isotherm can be written as a function of micropore half-width:
where rminis the minimum pore size accessible for the gas molecule and is assumed to be rO,l*which corresponds to the maximum adsorption energy. Another criterion of rmh= 0.858ro corresponding to a zero potential energy is also used by Jagiello and Schwarz,lgand they concluded that the effect of minimum accessible pore size is negligible in the adsorption energy distribution calculations. In this paper a gamma distribution is assumed to describe the micropore size distribution: 4 F(rp)
v+l
Y
rp e
= r(Y
rium isotherm in terms of pore size is
where xp is dimensionless pore half width scaled with respect to the molecular size. In eq 7 we note that the two parameters, q and ro, are always grouped together in the description of the energy distribution and hence cannot be separated unless we know one of them from other information rather than adsorption isotherm. The gas-solid potential by using the notation of dimensionless pore half width is
up (2)= u,*
52110 -{-(-) 35 2,
1 ’( 5 23tp-z,
--
)lo -
(y +
(ky};
z, = z/ro (8)
The Local Surface Diffision Flux. The driving force for the diffusion of adsorbed species is assumed to be the chemical potential gradient; hence the local surface diffusion flux (JJ can be written as
where r is the coordinate in the particle and D, is the zero coverage surface diffusivity and related to micropore half width by
( j);u:
D,(rp) = D,, exp - where a is the ratio of surface activation energy to the adsorption energy, D,o is the zero coverage surface diffusivity a t zero energy level, and the adsorption energy E is calculated from the nondimensional micropore half width xp. Mass Balance Equations. By assuming pore and surface diffusion control, the mass balance equation in the particle is
-qr,
+ 1)
(6)
Substituting this form of pore size distribution into eq 5 , we obtained the final expression for the adsorbed concentration. Therefore, the observed adsorption equilib-
where EM is the macropore porosity, s is the particle geometric factor having a value of 0, 1, or 2 for slab, cylinder, or sphere, respectively, and Jpis the macropore diffusion flux (12)
(20) Everett, D. H.; Powl, J. C. Adsorptionin Slit-likeand Cylindrical Micropores in the Henry’s Law Region. J . Chem. SOC.,Faraday Trans. I 1976,72,619-636.
with D, being the pore diffusivity.
Hu and Do
3298 Langmuir, Vol. 10, No. 9, 1994 Table 1. Definitions of Nondimensional Variables and Parameters
5 4 h
3
M
z
h2
E l
v
1Jo
$
0
20
40
60
80
100 120
80
100 120
L
0 " " " " ' " " '
0
20
40
60
Pressure ( k P a )
Bi =
Figure 1. Adsorption equilibrium isotherms of ethane and propane in Ajax activated carbon: (a)ethane, (b)propane; (-1 gamma pore size distribution, (- - -) uniform energy distribution.
k&CO
cMD,Co+ (1 - EM)D,O exp(-aS)C,,
One of the boundary conditions of eq 9 is the zero flux a t the particle center
r = 0;
ac
-p-O
ar
(13)
x = 0;
-aY_p - O ar
Another boundary condition is a t the particle exterior surface
where R is the radius of the particle and Cb is the adsorbate concentration in the bulk phase.
Solution Methodology Since the model equations are coupled partial differential equations, they are solved numerically by using a combination of the orthogonal collocation techniquez1 and a n ODE integrator.22 To facilitate the analysis, the model equations are cast into nondimensional form by using the nondimensional variables and parameters defined in Table 1. The resulting nondimensional model equations are
(21) Villadsen, J.; Michelsen, M. L. Solution of Partial Differential Equation Models by Polynomial Approximation; Prentice-Hall: Englewood Cliffs, NJ, 1978. (22) Petzold, L. R. A Description of DASSL: A DifferentiaUAlgebraic Equation System Solver. Sandia Technical Report; SAND 82-8637, Livermore, CA, 1982.
The integrals over the required micropore size distribution are evaluated by a n adaptive integration routine DQDAGI and the adsorption energy is found from the pore size distribution by a univariate minimization routine DUMING.~~
Results and Discussions The single-component experimental data of adsorption equilibrium and kinetics of ethane and propane in Ajax activated carbon collected in our laboratory" are used in this article to study the role of micropore distribution in the adsorption processes. Figure 1shows the adsorption equilibrium of ethane and propane in Ajax activated carbon for three temperatures (10, 30, and 60 "C). The experimental result is presented as symbols and the isotherm model fitting using the theory proposed in this paper as solid lines. It is seen that the model fits the experimental data well. Isotherm parameters are extracted by using a nonlinear regression fitting based on the least-squares technique, and they are tabulated in Tables 2 and 3. It is noted that parameters q and ro are grouped together. For comparison the isotherm fitting using a uniform energy distributionlz is also plotted in Figure 1a s dashed lines, which is also in good agreement with the experimental data, although slightly worse than the model using pore size distribution. (23) IMSL Library, version 1.1. 1989.
Langmuir, Vol. 10, No. 9, 1994 3299
Adsorption Kinetics of Gases in Activated Carbon 0.15
1
- ethane
pore
3 -
0.10
h
0
L
a '
2
-
v
0.05
LL h
b
1 -
Y
1
I
\
\ 4
I
0
06
08
10
1 2
14
16
18
+O
Figure 2. Pore size distributionof ethane and propane in Ajax activated carbon in terms of dimensionlessmicropore half width.
il
Table 2. Isotherm Parameters and Pore Diffusivities for Ethane in Ajax Activated Carbon
e,,
T ("C) (kmoUm3) bo (Wa-l) qro 10 30 60
8.00 8.22 8.87
3.78 x
US*
v
79.4 100.7
(kJ/mol) 19.22
DP m2/s)
1.51 1.68 1.96
Table 3. Isotherm Parameters and Pore Diffusivities for Propane in Ajax Activated Carbon
c#
US*
8
T ("C) (kmoVm3) bo (kPa-') 10 30 60
7.33 7.52 8.06
4.57 x
lo-*
qro
Y
68.4 77.7
(kJ/mol) 23.01
DP m2/s) 1.20 1.30 1.56
The pore size distribution in terms of dimensionless pore half width is plotted in Figure 2. It is seen that this distribution has the same shape for ethane and propane, with the distribution for propane shifted to the left from that of ethane. As a result, a larger portion of micropores becomes inaccessible to propane than ethane. This is physically expected as a propane molecule is larger and hence has a higher value ofro than ethane. The similarity of the dimensionless pore distribution functions for ethane and propane confirms that the pore size distribution should be independent of the adsorbate. To have a clearer picture about the difference betweeen the two models: one is based on the pore size distribution and the other on energy distribution, the energy distributions for ethane and propane in Ajax activated carbon are shown in Figure 3. The energy distribution is calculated from the pore size distribution by the following equation:
where the derivative dEldx, is numerically computed. It is seen in Figure 3 that the uniform energy distribution has a lower mean energy (18.5 kJ/mol for ethane and 21.4kJ/mol for propane) compared with that derived from pore size distribution (25.5 kJ/mol for ethane and 31.5 kJ/mol for propane). The ratio of the mean energy of the uniform distribution to that based on the gamma pore size distribution is 0.725 for ethane and 0.679 for propane. One point worthwhile noting is that the energy distribution derived from the micropore size distribution is more spread and does not reduce to zero in the high energy range. The reason for this is that the dependence function of adsorption energy on the micropore size is a n asymmetrical one. The potential energy changes more rapidly for small pores than for large pores; hence the energy distribution is broadened toward higher energies.18
0.05 0.00
1 p1 ....................
0
,
,
10
20
.............
30
40
50
Energy ( k J / m o l ) Figure 3. Adsorption energy distribution of ethane and propane in Ajax activated carbon: (a) ethane, (b) propane; (-) gamma pore size distribution, (- - -) uniform energy distribution, (. *) polynomial fit.
Since the calculation of the energy from the pore size requires the minimization of the potential function, the computing process is very slow. In the optimization of the isotherm parameters from the experimental data, the model fitting takes more than 14 h on a n Intel 486DX33 personal computer. The adsorption equilibrium routine is frequently called during the adsorption dynamics computation and will increase the computing time drastically. To simplify the kinetics calculations, we use a polynomial function to regress the energy distribution derived from the micropore size distribution. Such regressed polynomials are shown in Figure 3 as dotted lines, with degrees of 9 for ethane and 7 for propane. It should be noted that a t least 9 digits of the polynomial parameters should be kept in order to give adequate results, because of the high degree. It is seen that the polynomials are quite similar to the original energy distributions with very small deviations. To ensure that the model using a polynomial energy distribution can adequately describe the system, isotherm model calculation is carried out using the fitted polynomial parameters. As plotted in Figure 4, although the results are not as good as that based on the original energy derived from pore size (Figure 11, the model using the polynomial energy distribution is considered adequate. Therefore, it is quite reasonable for us to use the simplified polynomial function in the dynamic calculations. Having solved the isotherm equilibrium calculations, we now study the adsorption kinetics. Since a polynomial energy distribution is used we can simplify the dynamic computation by integrating the required functions over the finite energy range instead of the inifinite pore size distribution. Therefore, a Gaussian quadrature by Villadsen and MichelsenZ1is used in the kinetics simulation to replace the adaptive integration routine as used in the equilibrium isotherm parameter optimizations. The adsorption processes of ethane and propane in a n Ajax activated carbon slab having a full length of 4 mm are found to satisfy the macropore and surface diffusion
3300 Langmuir, Vol. 10,No.9,1994
Hu and Do 10 0 8
0 6 0 4 02
E
r
m 0 2
E
4
n
3 o 0 1 a 0 1.0
"
"
4
'
'
"
"
500
g
"
' 1500
"
1000
._
;
4
0.8
G
i
i
0.6
1 0
10°C
0
30°C
v
60°C
0.4
0.2
20
40
60
80
and propane in Ajax activatedcarbonusing a polynomial energy distribution: (a) ethane, (b) propane.
'
0
100 120
Pressure ( k P a ) Figure 4. Adsorption equilibrium isotherm fittings of ethane
5%
0
10%20%
v 0.0
0
1
"
"
.
0
"
200 4 0 0 600 800 T i m e (Seconds)
'
1000
Figure 6. Adsorption kinetics of ethane in Ajax activated carbon of 4.4 mm full length slab: (a) 10 "C, (b) 60 "C; (-)
gamma pore size distribution, (- - -1 uniform energy distribution.
10
Table 4. Surface Diffusivities for Ethane in Ajax Activated Carbon
2 0 8
m
distribution
4
n
~~
uniform energby
> 0 6
gamma pore size
a c
Dun
m2/s)
~~
~~
7.02 10.4
a
____~
0.5 0.5
Table 5. Surface Diffusivities for Propane in Ajax Activated Carbon distribution DLlo m2/s) a uniform energy 13.4 0.5
2 0 4 c)
0
id & 0 2
gamma pore size
0 0
0
500
1000
Time (Seconds)
Figure 5. Adsorption kinetics of ethane in Ajax activated carbon of 4.4mm full length slab at 30 "C: (-1 gamma pore
size distribution, (-
-
-)
27.0
0.5
1500
uniform energy distribution.
c o n t r 0 1 . ~Figure ~ ~ ~ ~5 shows the adsorption dynamics of ethane in a 4.4mm full length slab ofAjax activated carbon at 30 "C, 1 atm. Three concentrations (5, 10, and 20%) of ethane are studied. The pore diffusivity in Ajax activated carbon is calculated using a macropore tortuosity of eight24 and the combined molecule and Knudsen diffussivities, which has been obtained in Hu and Do.ll The ratio of the surface activation energy to the heat of adsorption is set to 0.5. Therefore, the only extracted parameter is the zero coverage surface diffusivity a t zero which is independent of concentration energy level (D,,), and temperature. The model fitting using this theory and an extracted D,o of 1.04x m2/s (solid lines) is in good agreement with the experimental uptake. Since the pore size distribution approach has a larger mean energy, the parameter D,o is higher than the value (7.02 x lo-' m2/s) obtained in Hu and Do" where a uniform energy distribution is used. It should be borne in mind that the only (24)Gray, P.G.;Do, D. D. Adsorption and Desorption Dynamics of Sulphur Dioxide on a Single Large Activated Carbon Particle. Chem. Eng. C O ~ ~ 1990,96, Z L ~ . 141-154. (25)Hu, X. Fundamental Studies of Multicomponent Adsorption, Desorption and Displacement Kinetics of Light Hydrocarbons in Activated Carbon. PhD Thesis, 1992.
difference between the theory proposed in this article and that by Hu and Doll is the energy distribution function (polynomial derived from pore size distribution or uniform). Therefore, the results here can also be viewed as the effect of energy distribution shape on the adsorption kinetics. The experimental data are also well represented by the model of Hu and Doll (dashed lines), which has a similar concentration dependency as the model presented in this article. After the zero coverage surface diffusivity a t zero energy level is obtained, the model is used to predict the desorption kinetics ofpreadsorbed 10%ethane in a 4.4mm full length slab of Ajax activated carbon a t 30 "C for 1atm, which is plotted in Figure 5. Both models can reasonably predict the experimental data, but the theory presented in this paper gives a slightly better result, especially when the desorption process has progressed to a significant extent. To further test the potential of the models, they are used to predict the dynamics of ethane in Ajax activated carbon of 4.4mm full length slab a t two other temperatures, 10 and 60 "C, without any extra fitting parameter. This is shown in Figure 6. Like the uniform energy distribution, the model in this paper well predicts the adsorption kinetics for these two temperatures. However, this study gives a stronger concentration dependency of the uptakes a t 60 "C than the corresponding uniform energy distribution model. Figure 7 shows the adsorption kinetics of propane in Ajax activated carbon of 4.4 mm full length slab a t 30 "C,
Adsorption Kinetics of Gases in Activated Carbon 1.o a,
Y 0.8 03
4
a
-a
0.6
G
,2 0.4 4
0
a
k 0.2 0.0' 0
'
'
"
'
'
"
1000
"
'
'
"
2000
'
3000
Time ( S e c o n d s )
Figure 7. Adsorption kinetics of propane in Ajax activated carbon of 4.4 mm full length slab at 30 "C: (-) gamma pore size distribution, (- - -) uniform energy distribution.
0 8 ,
,
d
4a °0.2. 4 r ,
,
,
,
,
,
,
,
,
;5,10% 1
4
a
3
0.0
a
0
1000
2000
3000
i 116 0.4 0
0.2 0.0' 0
5%
10% v 20% "
"
"
1000
"
"
"
2000
'
'
1 I
3000
Time (Seconds)
Figure 8. Adsorption kinetics of propane in Ajax activated carbon of 4.4mm full length slab: (a) 10 "C, (b) 60 "C; (-) gamma pore size distribution, (- - -) uniform energy distribution. 1atm. Again the model from this study (solid lines) fits the experimental data well. Its prediction on the desorption kinetics is also good and marginally better than the model using a uniform energy distribution (dashed lines). The models are further compared in the predictions of adsorption kinetics of propane in a 4.4 mm full length slab of Ajax activated carbon a t two other temperatures, 10 and 60 "C,using the above extracted surface diffusivities. As shown in Figure 8, both theories predict the uptakes quite well a t these two temperatures, implying that the temperature dependency of the adsorption uptake is correctly represented by these two models.
Conclusions The surface heterogeneity of activated carbon can be viewed as being induced by either a micropore size distribution or an energy distribution. The pore size distribution is related to an energy one using the LennardJpnes potential relationship. Both approaches can describe the adsorption equilibrium of ethane and propane in activated carbon well. However, different mean and shape of adsorbate-adsorbent interaction energies are
Langmuir, Vol. 10, No. 9, 1994 3301 observed for different methods. When a pore size distribution is used, the energy distribution has a larger mean compared with a uniform energy distribution. In the description of adsorption kinetics, the model using a gamma pore size distribution can adequately represent the concentration and temperature dependency of the adsorption rate process, as a uniform energy distribution does. The prediction on the desorption kinetics of ethane in Ajax activated carbon by this study is better than that by a previous model using a uniform energy distribution. A polynomial function is found to be adequate is simulating the energy distribution derived from pore size distribution leading to a great saving in computation time when it is used in the dynamic calculation.
Glossary ratio of the surface activation energy to the heat of adsorption Biot number (defined in Table 1) adsorbate concentration in the macropore (kmol/ m3) adsorbate concentration in the bulk (kmol/m3) characteristic concentration for the fluid concentration (kmol/m3) adsorbed concentration in the particle (kmol/") characteristic concentration for the adsorbed concentration (kmol/") saturation adsorbed concentration (kmol/m3) macropore diffusivity (m2/s) surface diffusivity (m2/s) surface diffusivity at zero energy level (m2/s) adsorbate-adsorbent interaction energy (kJ/ kmol) nondimensional energy (defined in Table 1) pore distribution function energy distribution function nondimensional distribution function (defined in Table 1) function defined in Table 1 flux through the macropore (kmoU(m2 s)) flux through the solid (kmol/m2s)) film mass transfer coefficient ( m / s ) gamma distribution parameter (l/m) particle radial position (m) particle radius (m) molecular size parameter of the Lennard-Jones potential (m) gas constant (kJ/(kmol K)) minimum accessible pore size (m) micropore half-width (m) geometric factor (=O, 1 , 2 for slab, cylinder, and sphere, respectively) temperature (K) time (s) gas-solid potential in slitlike parallel walls (kJ/ mol) depth of potential minimum for slitlike walls (kJ/ kmol) depth of potential minimum for single lattice plane (kJ/kmol) nondimensional particle radial position dimensionless minimum accessible pore size dimensionless micropore half width nondimensional adsorbate concentration in the macropore
3302 Lungmuir, Vol.10,No.9,1994
yb YP z ZX
nondimensional adsorbate concentration in the bulk nondimensional adsorbed concentration in the particle distance between the molecule and the pore wall (m) dimensionlessdistance between the molecule and the pore wall
Greek symbol 6 model parameter defined in Table 1 CM particle macropore porosity
Hu and Do rl
r V
U
0,
t
model parameter defined in Table 1 gamma function gamma distribution parameter model parameter defined in Table 1 model parameter defined in Table 1 nondimensional time defined in Table 1
Acknowledgment. Financial support from the Australian Research Council and the REGS of the University of Queensland is gratefully acknowledged.
