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Langmuir 1999, 15, 6428-6437
Multicomponent Adsorption Kinetics of Gases in Activated Carbon: Effect of Pore Size Distribution Xijun Hu,*,† Shizhang Qiao,† and Duong D. Do‡ Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, and Department of Chemical Engineering, University of Queensland, Brisbane, Qld 4072, Australia Received October 19, 1998. In Final Form: May 18, 1999
The heterogeneous structure of activated carbon is studied by using a pore size distribution concept. The pore size is related to the adsorbate-adsorbent interaction energy by the Lennard-Jones potential. The competition of different species for a given pore is considered via their own interaction strength with the local micropore. A pore and surface diffusion model taking into account the pore size distribution is proposed to investigate the effect of surface heterogeneity on the prediction of binary sorption kinetics. The results are compared with those previously obtained using a uniform energy distribution.
Introduction Adsorption by activated carbon is an efficient tool for gas separation and in the environmental control of chemical process industries. However, the complex micropore and surface phenomena involved in carbon adsorption have not been adequately investigated, especially for multicomponent systems. It has been observed that activated carbon is heterogeneous in the sense that it has a range of micropore size distribution. The majority of research done so far in the literature assumes some form of energy distribution to describe the system heterogeneity.1-5 This method is reasonable for a singlecomponent adsorption system, since the micropore size is related to an adsorbate-adsorbent interaction energy. However, it fails to properly correlate the energies of different species in a multicomponent system, which is required to calculate the relative competence of each species. Another disadvantage associated with this energy distribution concept is that the size exclusion effect cannot be incorporated, which is important in dealing with mixtures of large and different molecular sizes, such as the organic components in air pollution control and wastewater treatment. Recently the pore size distribution (PSD) methodology has been used to study the adsorption equilibrium6 and kinetics7,8 of single-component systems, by using the * To whom all correspondence should be addressed. E-mail:
[email protected]. † Hong Kong University of Science and Technology. ‡ University of Queensland. (1) Do, D. D.; Hu, X. An Energy Distributed Model for Adsorption Kinetics in Large Heterogeneous Microporous Particles. Chem. Eng. Sci. 1993, 48 (11), 2119-2127. (2) Horas, J. A.; Saitua, H. A.; Marchese, J. Surface Diffusion of Adsorbed Gases on an Energetically Heterogeneous Porous Solid. J. Colloid Interface Sci. 1988, 126 (2), 421-431. (3) Hu, X.; Do, D. D. Effect of Surface Energetic Heterogeneity on the Kinetics of Adsorption of Gases in Microporous Activated Carbon. Langmuir 1993, 9 (10), 2530-2536. (4) Kapoor, A. and Yang, R. T., Surface Diffusion on Energetically Heterogeneous Surfaces. AIChE J. 1989, 35, 5 (10), 1735-1738. (5) Okazaki, M.; Tamon, H.; Toei, R. Interpretation of Surface Flow Phenomenon of Adsorbed Gases by Hopping Model. AIChE J. 1981, 27 (2), 262-270. (6) 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 (1), 225-237.
Lennard-Jones potential.9 By assuming the micropore size distribution is the sole source of system heterogeneity, the adsorbate-adsorbent interaction energy is determined as the depth of the Lennard-Jones potential minimum. As the pore size distribution is a property of the adsorbent, it can be determined by simultaneously fitting adsorption equilibria of multiple adsorbates at various temperatures to the isotherm model.10 A finite kinetics model allowing for pore size distribution is recently proposed to improve the predictions on the desorption kinetics.11,12 In this article the pore size distribution approach will be extended to predict multicomponent adsorption equilibrium and kinetics in heterogeneous systems. A micropore size distribution is assumed to describe the system heterogeneity. In the equilibrium aspect, all adsorbates compete in micropores except in those smaller than the molecular diameter of a specified species (exclusion phenomenon). This means that fewer components compete for smaller pores due to the size exclusion. The observed adsorbed phase concentration is the integration of the local value over the entire pore size distribution, which has to be done numerically. The optimization of equilibrium isotherm parameters involves three steps of numerical evaluations: the minimization of the LennardJones potential to obtain the adsorbate-adsorbent interaction energy, the integration of the local isotherm over the accessible pore size distribution, and the optimization of isotherm parameters. Isotherm parameters obtained from single-component data analysis will be used (7) Hu, X.; Do, D. D. Effect of Surface Heterogeneity on the Adsorption Kinetics of Gases in Activated Carbon: Pore Size Distribution vs Energy Distribution. Langmuir 1994, 10 (9) 3296-3302. (8) Do, D. D. Dynamics of adsorption on heterogeneous solids. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; pp 777-835. (9) Everett, D. H.; Powl, J. C. Adsorption in Slitlike and Cylindrical Micropores in the Henry’s Law Region. J. Chem. Soc., Faraday Trans. 1 1976, 72, 619-636. (10) Wang, K.; Do, D. D. Characterising micropore size distribution of activated carbon using equilibrium data of many adsorbates at various temperatures. Langmuir 1997, 13, 6226-6233. (11) Do, D. D.; Wang, K. A new model for the description of adsorption kinetics in heterogeneous activated carbon. Carbon 1998, 36, 15391554. (12) Do, D. D.; Wang, K. Dual diffusion and finite mass exchange model for adsorption kinetics in activated carbon. AIChE J. 1998, 44, 68-82.
10.1021/la981460x CCC: $18.00 © 1999 American Chemical Society Published on Web 07/23/1999
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to predict the multicomponent adsorption equilibria without any extra fitting parameters. For the dynamic study, diffusion of both the free species in the macropore phase and the adsorbed species in the solid phase will be considered. Because of the heterogeneous nature of the solid phase, the observed diffusion flux is the integration of the local flux at a patch of micropore over the full pore size distribution. For a given pore the driving force for the diffusion of the adsorbed species is its chemical potential gradient. Both single- and multicomponent systems will be studied. The equilibrium and dynamic experimental data of Hu et al.13 and Hu and Do14 will be used to verify the model simulation results. Theory Let us consider a flowing gas mixture stream containing NC components. At time t ) 0 a large microporous particle is exposed to this environment. The flow rate of the gas stream is so high that the gas concentrations can be considered constant. The particle is heterogeneous, and this heterogeneity can be induced by either an adsorbateadsorbent interaction energy distribution or a micropore size distribution. In the development of model formulations the system is assumed isothermal 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 by Hu and Do.14 In this section we present a model based on the micropore size distribution. Adsorption Isotherm and Micropore Distribution. The adsorbent particle is assumed to have a pore size distribution which causes an adsorbate-adsorbent interaction energy distribution. In this article the pore size distribution is assumed to take a Γ form:
F(rp) )
qν+1rpνe-qrp
(1)
Γ(ν + 1)
where rp is the half-width of the pores. An extended Langmuir equation for the adsorption of component k in the presence of components j ) 1 to NC is assumed to describe the local adsorption isotherm for a given micropore:
Cµ(k,E) ) Cµs(k)
b0(k)eE(k)/RgTCp(k)
1+
NC
If the energy distribution for species k is F*(k,E), the observed adsorption isotherm is:
Cµ(k) ) Cµs(k)
∫E
Emax(k)
b0(k)eE(k)/RgTCp(k)
min(k)
1+
b0(j)eE(j)/R TCp(j) ∑ j)1 g
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 a
{( ) ( ) ( ) (
r0(k) 4 + z 10 r0(k) 2 r0(k) 5 2rp - z 2rp - z
5 2 r0(k) up(k,z) ) us*(k) 3 3 z
10
-
)} 4
(4)
where z is the distance between the molecule and a given atom of a pore surface layer separated by the distance 2rp and r0 is the collision diameter, that is9
1 r0 ) [r0(bulk gas) + 2 (lattice spacing between graphite planes)] ) 1 g [r + 3.40 Å] 2 0 The parameter u/s is the depth of the Lennard-Jones potential minimum for a single lattice plane, and this depth occurs at the position r0. The properties of the potential were illustrated in ref 9. The depth of the potential minimum up* is obtained numerically from eq 4, and it ranges from the value for the interaction potential minimum depth with a single wall, us* to the value of 2us* for rp ) r0. The adsorbate-adsorbent interaction energy is the negative of the potential minimum, which is related to the micropore half-width rp by
E(k,rp) ) up*(k,rp)
(2)
b0(j)eE(j)/R TCp(j) ∑ j)1
F*(k,E) dE(k) (3)
NC
(5)
g
where Cµ(k,E) is the adsorbed amount of species k corresponding to the adsorbate-adsorbent interaction energies of all species at the levels of E, Cp is the gasphase concentration in the bulk, Cµs is the maximum adsorbed phase concentration, Rg is the gas constant, T is temperature, NC is the number of components, and b0 is the affinity constant at zero energy level and can be treated as temperature independent over a limited temperature interval. (13) Hu, X.; Rao, G. N.; Do, D. D. Effect of Energy Distribution on Sorption Kinetics in Bidispersed Particles. AIChE J. 1993, 39 (2), 249261. (14) Hu, X.; Do, D. D. Role of Energy Distribution in Multicomponent Sorption Kinetics in Bidispersed Solids. AIChE J. 1993, 39 (10), 16281640.
Hence the minimum and maximum adsorption energies in eq 3 are Emin(k) ) us*(k), and Emax(k) ) 2us*(k). Let the micropore size distribution be F(rp); then observed adsorption isotherm can be written as a function of micropore half-width:
Cµ(k) ) Cµs(k)
∫r
b0(k)eE(k,rp)/RgTCp(k)
∞ min(k)
1+
NC
F(rp) drp (6)
b0(j)eE(j,r )/R TCp(j) ∑ j)1 p
g
where rmin is the minimum pore size accessible for the gas molecule and is assumed to be r0,6 which corresponds to the maximum adsorption energy. Another criterion of rmin
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Hu et al.
) 0.858r0 corresponding to a zero potential energy is also used,15 and it was concluded that the effect of minimum accessible pore size is negligible in the adsorption energy distribution calculations. By substituting the pore size distribution in eq 1 into eq 6, the final expression for the adsorbed concentration could be obtained. Therefore, the observed adsorption equilibrium isotherm in terms of pore size is
Cµ(k) ) Cµs(k) ×
∫r
b0(k)e
∞ min(k)
1+
E(k,rp)/RgT
Cp(k)
NC
b0(j)eE(j,r )/R TCp(j) ∑ j)1 p
g
q
ν+1
Γ(ν + 1)
∂Cp(k) )0 ∂r
r ) 0;
∫r∞
min(k)
Jµ(k,rp) F(rp) drp )
km(k)[Cp(k) - Cb(k)] (13)
drp (7)
where R is the radius of the particle and Cb is the adsorbate concentration in the bulk phase.
Local Surface Diffusion Flux. The driving force for the diffusion of adsorbed species is assumed to be the chemical potential gradient, hence the local surface diffusion flux of species k, Jµ(k,rp), can be written as
The initial conditions of the model equations are
t ) 0; Cp(k) ) Cpi(k); Cµ(k) ) Cµs(k)
Cµ(k,rp) ∂Cp(k) Jµ(k,rp) ) -Dµ(k,rp) ∂r Cp(k)
∫r∞ (k) × min
b0(k)eE(k,rp)/RgTCpi(k)
(8)
qν+1rpνe-qrp Γ(ν + 1)
NC
where r is the coordinate in the particle and Dµ is the zero coverage surface diffusivity and is related to the micropore half-width by
(
)
a(k) E(k,rp) Dµ(k,rp) ) Dµ0(k) exp RgT
(9)
where a is the ratio of surface activation energy to the adsorption energy, Dµ0 is the zero coverage surface diffusivity at the zero energy level, and the adsorption energy E is calculated from the micropore half-width rp. Mass Balance Equations. Since the particle is large, the mass transfer can be assumed to be under pore and surface diffusion control, so the mass balance equation in the particle is
∂Cp(k) ∂ ∞ + (1 - M) C (k,rp) F(rp) drp ) ∂t ∂t rmin(k) µ ∞ 1 ∂ 1 ∂ -M s (rsJp(k)) - (1 - M) s (rs r Jµ(k,rp) × min(k) ∂r ∂r r r F(rp) drp) (10)
M
∫
∫
where M is the macropore porosity, s is the particle geometric factor having a value of 0, 1, or 2 for a slab, a cylinder, or a sphere, respectively, and Jp is the macropore diffusion flux:
Jp(k) ) -Dp(k)
(12)
Another boundary condition is at the particle exterior surface:
r ) R; MJp(k) + (1 - M)
ν -qrp
rp e
One of the boundary conditions of eq 9 is zero flux at the particle center:
∂Cp(k) ∂r
(11)
with Dp being the pore diffusivity. (15) Jagiello, J.; Schwarz, J. A. Relationship between Energetic and Structural Heterogeneity of Microporous Carbons Determined on the basis of Adsorption Potentials in Model Micropores. Langmuir 1993, 9 (10), 2513-2517.
1+
b0(j)eE(j,r )/R TCpi(j) ∑ j)1 p
g
drp (14)
Solution Methodology. Let xp ) rp/r0(i) and ω(k) ) r0(k)/r0(i), where xp is the dimensionless pore half-width scaled with respect to the collision diameter of the smallest species i and ω is the relative molecular size, also scaled with the same component. Then we obtain
Cµ(k) ) Cµs(k)
∫ω(k) ∞
b0(k)eE(k,xp)/RgTCp(k)
1+
NC
×
b0(j)eE(j,x )/R TCp(j) ∑ j)1 p
g
[qr0(i)]ν+1xpνe-[qr0(i)]xp Γ(ν + 1)
dxp (15)
In eq 15 we note that the two parameters q and r0(i) are always grouped together in the description of the energy distribution. They can be separated only if we know one of them from other information rather than the adsorption isotherm, for example, the molecular size of gases. The gas-solid potential by using the notation of dimensionless pore half-width is
{( ) ( ) ) ( )}
ω(k) 4 5 2 ω(k) 10 + up(k,z) ) us*(k) 3 5 zx zx ω(k) 4 2 ω(k) 10 ; zx ) z/r0(i) (16) 5 2xp - zx 2xp - zx
(
The integrals over the required micropore size distribution are evaluated by the orthogonal collocation technique, and the adsorption energy is found from the pore size distribution by the univariate minimization routine DUMING in the IMSL Library.
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Langmuir, Vol. 15, No. 19, 1999 6431
Since the model equations are coupled partial differential equations, they are solved numerically by using a combination of the orthogonal collocation technique16 and an ODE integrator.17 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
σ(k)
∂Yp(k) ∂ ∞ + σµ(k) Y [k,e(k)] f(xp) dxp ) ∂τ ∂τ ω(k) µ 1 ∂ s ∂Yp(k) 1 ∂ η(k) s × x + δ(k) s ∂x ∂x x x ∂x Yµ[k,e(k)] ∂Yp(k) ∞ xs H[k,e(k)] f(xp) dxp (17) ω(k) ∂x Yp(k)
(
(
Yµ(k) )
∫
)
)
∫
Cµs(k) Cµ0(k)
∫ω(k) ∞
b0(k)eE(k,xp)/RgTCp(k)
1+
×
NC
b0(j)e ∑ j)1
E(j,xp)/RgT
Cp(j)
Figure 1. Micropore size distribution of Ajax-activated carbon.
f(xp) dxp (18) x ) 0;
∂Yp(k) )0 ∂x
(19)
x ) 1; ∂Yp(k) ∞ ∂Yp(k) Yµ[k,e(k)] + δ(k) H[k,e(k)] × ω(k) ∂x ∂x Yp(k) f(xp) dxp ) Bi(k)[Yb(k) - Yp(k)] (20)
η(k)
∫
τ ) 0; Yp(k) ) Ypi(k); Yµ(k) ) Yµi(k)
(21)
Results and Discussion The experimental adsorption data of ethane and propane in Ajax-activated carbon13,14 were used to validate the proposed model. Figure 1 shows the pore size distribution in terms of pore half-width, the collision length (rp) subtracted by the carbon lattice half-space (1.7 Å). The experimental micropore size distribution data, measured by a Micromeretics ASAP 2000 using the Horvath and Kawazoe equation, are presented as symbols. A Γ function is assumed to describe the pore size distribution. The fitting result using a nonlinear regression based on the least-squares technique is plotted as a solid line. It is seen that the model fits the experimental data well. The pore accessibility cutoff of ethane and propane in Ajax-activated carbon is also shown in Figure 1. Only a small portion of pores are excluded for ethane and propane molecules. The experimental isotherm data of ethane and propane in Ajax-activated carbon are shown as symbols in Figure 2 for three temperatures (10, 30, and 60 °C). The pore size distribution is converted to an adsorbate-adsorbent energy distribution by way of the Lennard-Jones potential and then used in the isotherm fitting. The isotherm parameters are extracted by using a nonlinear regression fitting based on the least-squares technique and tabulated in Tables 2 and 3. The result of isotherm model fitting (16) Villadsen, J.; Michelsen, M. L. Solution of Partial Differential Equation Models by Polynomial Approximation; Prentice-Hall: Englewood Cliffs, NJ, 1978. (17) Petzold, L. R. A Description of DASSL: A Differential/Algebraic Equation System Solver. Sandia Technical Report SAND 82-8637; National Laboratory: Livermore, CA, 1982.
Figure 2. Adsorption equilibrium isotherm of ethane and propane in Ajax-activated carbon: (s) Γ pore size distribution; (- - -) uniform energy distribution.
using the theory proposed in this paper is presented as solid lines. It is seen that the PSD model fits the experimental data very well. For comparison the isotherm fitting using a uniform energy distribution13 is also plotted in Figure 2 as dashed lines, which are also in good agreement with the experimental data. Since an extended Langmuir equation will be used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency of the multicomponent adsorption equilibrium. Because the
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Hu et al.
Table 1. Definitions of Nondimensional Variables and Parameters
e(k) )
E(k,xp) ; RgT
ej(k) )
∫
∞
ω(k)
[qr0(i)]ν+1xpνe-(qr0(i))xp ; Γ(ν + 1)
f(xp) )
ω(k) )
r0(k) ; r0(i)
x)
r R
H[k,e(k)] ) exp[a(k)[ej(k) - e(k)]]
e(k)f(xp) dxp;
C0(k) ) max{Cp(k), Cb(k)} Yp(k) )
Cp(k) ; C0(k)
Yb(k) )
Cb(k) ;
NC
∑
M
Cµ(k) Cµ0(k)
;
Yµ[k,e(k)] )
Cµ[k,E(k,xp)] Cµ0(k)
NC
∑D
Dp(j) C0(j) + (1 - M)
j)1
τ)
Yµ(k) )
C0(k)
µ0(j)
exp[-a(j) ej(j)]Cµ0
j)1
NC
t
NC
∑C (j) + (1 - )∑C
2
R [M
0
µ0(j)]
M
j)1
j)1
MC0(k)
σ(k) )
NC
∑
M
NC
∑C
C0(j) + (1 - M)
j)1
µ0(j)
j)1
(1 - M)Cµ0(k)
σµ(k) )
NC
∑
M
NC
C0(j) + (1 - M)
j)1
∑C
µ0(j)
j)1
MDp(k) C0(k)
η(k) )
NC
∑
M
NC
Dp(j) C0(j) + (1 - M)
j)1
∑D
µ0(j)
exp[-a(j) ej(j)] Cµ0(j)
j)1
(1 - M)Dµ0(k) exp[-a(k) ej(k)] Cµ0(k)
δ(k) )
NC
NC
∑D (j) C (j) + (1 - )∑D
M
p
0
µ0(j)
M
j)1
exp[-a(j) ej(j)] Cµ0(j)
j)1
km(k) RC0(k)
Bi(k) )
NC
NC
∑D (j) C (j) + (1 - )∑D
M
p
0
M
j)1
µ0(j)
exp[-a(j) ej(j)] Cµ0(j)
j)1
Table 2. Isotherm Parameters and Pore Diffusivities for Ethane in Ajax-Activated Carbon T (°C)
Cµs (kmol/ m3)
b0 (kPa-1)
r0 (Å)
q (Å-1)
ν (dimensionless)
µs* (kJ/ mol)
Dp (×10-6 m2/s)
10 30 60
6.2792 5.8980 5.4949
2.40 × 10-5
3.922
3.732
20.53
14.68
1.51 1.68 1.96
Table 3. Isotherm Parameters and Pore Diffusivities for Propane in Ajax Activated Carbon T (°C)
Cµs (kmol/m3)
b0 (kPa-1)
r0 (Å)
q (Å-1)
ν (dimensionless)
µs* (kJ/mol)
Dp ( ×10-6 m2/s)
10 30 60
6.2792 5.8980 5.4949
8.80 × 10-6
4.259
3.732
20.53
20.20
1.20 1.30 1.56
micropore size distribution of Ajax-activated carbon is independent of the adsorbates, all experimental data of two species at three temperatures were simultaneously fitted to the isotherm equations to extract the model parameters. We can see from Tables 2 and 3 that the parameters b0 and us* were set to be temperature independent but species dependent. The saturation parameter, Cµs was set to be temperature dependent but species independent. In general the saturation parameter should be different for species of different molecular sizes. However, since an extended Langmuir equation is used to describe the local multicomponent adsorption isotherm, the saturation parameter has to be the same for different
species to satisfy the thermodynamic consistency. This is the major drawback of the extended multicomponent Langmuir equation. The converted energy distribution from the above Γ pore size distribution is shown in Figure 3 as solid lines for ethane and propane in Ajax-activated carbon. Also plotted in Figure 3 as dashed lines are the uniform energy distributions for the same adsorption systems.13 It is seen that the uniform energy distribution has a lower mean energy compared to that derived from the pore size distribution for both ethane and propane. The single-component dynamic data were used to extract the kinetic parameters of ethane and propane adsorption
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Langmuir, Vol. 15, No. 19, 1999 6433
Figure 3. Adsorption energy distribution of ethane and propane in Ajax-activated carbon. (s) Γ pore size distribution; (- - -) uniform energy distribution.
in Ajax carbon. A slab particle of 4.4 mm full length has been shown to be under macropore and surface diffusion control.18 In this paper, a tortuosity of eight7 was used to calculate the pore diffusivities of ethane and propane from the combined molecular and Knudsen diffusivities. The ratio of the surface activation energy to the heat of adsorption a is set to be 0.5 for the adsorption of ethane and propane in Ajax carbon. Therefore, the only extracted parameter is the zero coverage surface diffusivity at the zero energy level Dµ0, which is independent of concentration and temperature. Figure 4b shows the adsorption dynamics of ethane in a 4.4 mm full length slab of Ajax-activated carbon at 30 °C and 1 atm. The results are shown as the fractional uptake versus time. The fractional uptake is defined as the uptake
{
Rs+1 S+1
∫0 [MCp + (1 - M)Cµ] dr} R
at any time divided by its value at final equilibrium (time ) ∞). Three concentrations (5, 10, 20%) of ethane are studied. The fittings using the PSD model (solid lines) with an extracted zero coverage surface diffusivity at the zero energy level, Dµ0, of 1.473 × 10-7 m2/s are in good agreement with the experimental uptake data (symbols). The dynamic parameters of ethane adsorption are tabulated in Table 4. For comparison, the fittings of the heterogeneous macropore and surface model using an uniform energy distribution (UED)13 are also plotted in Figure 4 as dashed lines. After the zero coverage surface diffusivity at the zero energy level is obtained, the PSD model is used to predict the desorption kinetics of 10% ethane in a 4.4 mm full length slab of Ajax-activated carbon at 30 °C for 1 atm, (18) Hu, X. Fundamental Studies of Multicomponent Adsorption, Desorption and Displacement Kinetics of Light Hydrocarbons in Activated Carbon. Ph.D. Thesis, University of Queensland, 1992.
Figure 4. Adsorption and desorption kinetics of ethane in Ajax-activated carbon of 4.4 mm full length slab at 1 atm. (a) 10 °C; (b) 30 °C; (c) 60 °C; (s) Γ pore size distribution; (- - -) uniform energy distribution. Table 4. Surface Diffusivities for Ethane in Ajax Activated Carbon distribution
Dµ0 (×10-7 m2/s)
a
uniform energy Γ pore size
7.02 1.473
0.5 0.5
which is plotted in Figure 4b as a solid line. The prediction result using the energy distribution concept is also plotted in Figure 4b as a dashed line. It is seen that both models can reasonably predict the experimental data, but the UED model is in better agreement with the experimental data before 500 s while the PSD theory gives a slightly better result especially when the desorption process has progressed to a significant extent. This may be explained by the spread of the energy distribution and the mean energy. As seen from Figure 3a, the energy distribution of ethane varies from 23 to 60 kJ/mol for the PSD model (width of 37 kJ/mol), which is higher and wider than that for the UED model of 6.26-30.75 kJ/mol (width 24.5 kJ/ mol). The desorption process would start first from the low-energy “patches” (flat surface and large pores), and intuitively, toward the end of the desorption process, it is the high-energy “patches” (very small pores) which would undergo desorption. If the energy distribution is more spread, the desorption rate will also vary more for molecules in the low- and high-energy ends. If the mean
6434 Langmuir, Vol. 15, No. 19, 1999
Hu et al.
Figure 5. Adsorption and desorption kinetics of propane in Ajax-activated carbon of 4.4 mm full length slab at 1 atm. (a) 10 °C; (b) 30 °C; (c) 60 °C; (s) Γ pore size distribution; (- - -) uniform energy distribution. Table 5. Surface Diffusivities for Propane in Ajax-Activated Carbon distribution
Dµ0 (×10-7 m2/s)
a
uniform energy Γ pore size
13.4 1.941
0.5 0.5
energy is higher, the desorption process will be slower. Since the energy distribution for the PSD model has a wider range and a higher mean energy, the desorption process will become slower when it progresses to small pores. To further test its potential, the PSD model is used to predict the dynamics of ethane in the same carbon at two other temperatures, 10 and 60 °C, without any extra fitting parameter. This is shown in Figure 4a and c. Like the uniform energy distribution model (dashed line), the PSD model well predicts the adsorption kinetics at these two temperatures. Figure 5b shows the adsorption kinetics of propane in Ajax-activated carbon of 4.4 mm full length slab at 30 °C and, 1 atm. The extracted dynamic parameters of propane adsorption in Ajax-activated carbon are tabulated in Table 5. Again the two models fit the experimental data well. The models are further used to predict the adsorption kinetics of propane in a 4.4 mm full length slab of Ajaxactivated carbon at two other temperatures, 10 and 60 °C, and the desorption kinetics of 10% propane in Ajax carbon at 30 °C and 1 atm, using the above extracted surface diffusivities. The prediction results on the adsorption kinetics at 10 and 60 °C are shown in Figure 5a and c, respectively. It can seen that both the PSD model and
Figure 6. Binary adsorption dynamics of ethane and propane onto an Ajax-activated carbon of 4.4 mm full length slabs at 10 °C, 1 atm: (a) 5% ethane and 5% propane; (b) 10% ethane and 5% propane; (c) 20% ethane and 5% propane; (s) pore size distribution; (- - -) uniform energy distribution.
the uniform energy distribution model predict the adsorption uptakes quite well, implying that the temperature dependency of the adsorption kinetics is correctly represented by the model. The desorption prediction at 30 °C is plotted in Figure 5b. Unlike the kinetics of ethane, the predictions of the two models on the desorption kinetics of propane are nearly superimposed on each other. This is so because the effect of the more spread uniform energy distribution (from 0 to 42.75 kJ/mol) is offset by the higher mean energy of the PSD model (from 23 to 56 kJ/mol). Having obtained the adsorption equilibrium and masstransfer parameters of single-component systems, we can now use these parameters to predict the binary adsorption dynamics. Figure 6 shows the simultaneous adsorption kinetics of 5% propane together with different concentrations of ethane (5, 10, 20%) onto 4.4 mm slab particles of Ajax-activated carbon at 10 °C and 1 atm. The predictions of the PSD model (solid lines) are in very good agreement with the experimental data (symbols). The predictions of the energy distribution model are plotted in Figure 6 as
Adsorption Kinetics of Gases in Activated Carbon
Langmuir, Vol. 15, No. 19, 1999 6435
Figure 7. Binary adsorption dynamics of ethane and propane onto an Ajax-activated carbon of 4.4 mm full length slabs at 30 °C and 1 atm: (a) 10% ethane and 10% propane; (b) 5% ethane and 10% propane; (c) 5% ethane and 20% propane; (s) pore size distribution; (- - -) uniform energy distribution.
Figure 8. Binary adsorption dynamics of ethane and propane onto an Ajax-activated carbon of 4.4 mm full length slabs at 30 °C and 1 atm: (a) 5% ethane and 5% propane; (b) 10% ethane and 5% propane; (c) 20% ethane and 5% propane; (s) pore size distribution; (- - -) uniform energy distribution.
dashed lines which are also in good agreement with experimental data. The proposed model is further tested with the binary adsorption dynamics of ethane and propane in Ajax carbon of 4.4 mm slabs at 30 °C under different concentration combinations, which are shown in Figures 7 and 8. Because ethane is a fast-diffusing/less-strongly-adsorbed species in this system, ethane will penetrate into the carbon first, behaving as a pseudo single component, and then be displaced by the later-coming propane. Therefore, an overshoot is observed in the fractional uptake of ethane. Although the predictions of both models are in good agreement with the experimental data here, the predictions on the overshoot degree of ethane kinetics are quite different for the two models. This may be attributed to the different energy matching schemes adopted by the two models. The model comparisons are further carried out with the binary adsorption dynamics of ethane and propane in Ajax carbon of 4.4 mm slab at 60 °C under different concentration combinations, which are shown in Figure
9. It can be seen that the PSD model predicts the degree of overshoot of the faster diffusing/less-strongly-adsorbed species better than the energy distribution model. Finally, we study the binary desorption dynamics of 10% ethane and 10% propane in 4.4 mm full length slabs of Ajax-activated carbon at 30 °C and 1 atm. The experimental data (symbols) are shown in Figure 10 together with the corresponding predictions of the PSD model (solid lines) and the energy distribution model (dashed lines). Although the results of both models are close to each other for the ethane uptake rate, the PSD model provides better predictions for propane the desorption uptake rate at the later stage (>1000 s). Conclusions Using a pore size distribution to describe the system heterogeneity of adsorption in microporous activated carbon can better describe the competition of different species. The pore size distribution can be converted to an
6436 Langmuir, Vol. 15, No. 19, 1999
Hu et al.
Figure 10. Binary desorption kinetics of 10% ethane and 10% propane in Ajax-activated carbon of 4.4 mm full length slab at 30 °C and 1 atm. (s) pore size distribution; (- - -) uniform energy distribution.
found to be able to accurately predict the multicomponent sorption dynamics and to adequately represent the concentration and temperature dependency of the adsorption rate process. The agreement between the theory and the experimental data is excellent. Acknowledgment. Financial support from the Croucher Foundation, the Research Grants Council of Hong Kong, and the Australian Research Council is greatly acknowledged. Glossary a Bi Cp Cb C0 Figure 9. Binary adsorption dynamics of ethane and propane onto an Ajax-activated carbon of 4.4 mm full length slabs at 60 °C and 1 atm: (a) 5% ethane and 5% propane; (b) 10% ethane and 5% propane; (c) 20% ethane and 5% propane; (s) pore size distribution; (- - -) uniform energy distribution.
energy distribution by the Lennard-Jones potential relationship. The PSD model can fundamentally address the local interaction energy (or matching energy) between various adsorbate molecules via their own interaction strength with the local micropores. This matching scheme, which is termed the adsorbate-pore interaction scheme, introduces the physical condition for determining the matching energies between different species on heterogeneous surface. In comparison, when an energy distribution is used, the traditional accumulative energymatching scheme is arbitrary and lacks the fundamental ground, since the matching does not bring out the feature of competition of adsorbate residing in a pore. In this paper a multicomponent sorption dynamics model using a pore size distribution concept has been successfully developed to study the adsorption and desorption kinetics of gases in a large activated carbon. By using information of singlecomponent equilibrium and mass transfer, this model is
Cµ Cµ0 Cµs Dp Dµ Dµ0 E e F F* f H Jp Jµ km PSD q r R
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/m3) characteristic concentration for the adsorbed concentration (kmol/m3) saturation adsorbed concentration (kmol/m3) macropore diffusivity (m2/s) surface diffusivity (m2/s) surface diffusivity at the 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 (kmol/(m2 s)) flux through the solid (kmol/(m2 s)) film mass transfer coefficient (m/s) pore size distribution gamma distribution parameter (1/m) particle radial position (m) particle radius (m)
Adsorption Kinetics of Gases in Activated Carbon r0 Rg rmin rp s T t UED up up* u s* x xmin xp Yp
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 ()0, 1, 2 for slab, cylinder and sphere respectively) temperature (K) time (s) uniform energy distribution gas-solid potential in slitlike parallel walls (kJ/ kmol) 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
Langmuir, Vol. 15, No. 19, 1999 6437 Yb Yµ z zx
nondimensional adsorbate concentration in the bulk nondimensional adsorbed concentration in the particle distance between the molecule and the pore wall (m) dimensionless distance between the molecule and the pore wall
Greek Symbols δ M η Γ ν σ σµ τ
model parameter defined in Table 1 particle macropore porosity 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
LA981460X
