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Ind. Eng. Chem. Res. 2002, 41, 1098-1106
Identification of Transport Mechanism in Adsorbent Micropores from Column Dynamics S. Farooq,* Huang Qinglin, and I. A. Karimi Department of Chemical and Environmental Engineering, The National University of Singapore, 4 Engineering Drive 4, Singapore 117576
The possibility of distinguishing between barrier and pore diffusional resistances in adsorbent micropores from dynamic column response has been explored. The modeling of transport through a barrier resistance confined at a micropore mouth is mathematically equivalent to the linear driving force (LDF) model. By comparing the pore diffusion and LDF models for dynamic column response for a linear isotherm, it is shown that the extent of agreement between the two models depends on the product of the two parameters R [) (1 - p)Kc/p] and γ () DcL/rc2vo). By an appropriate choice of operating conditions, it is indeed possible to clearly distinguish between the two mechanisms. The proposed criteria have been verified experimentally for the breakthrough of oxygen and nitrogen in a 4A zeolite and two carbon molecular sieve samples. Introduction It is commonly assumed that, unlike uptake in an adsorbent particle, the breakthrough dynamics obtained using the linear driving force (LDF) approximation with the LDF rate constant k, given by eq 1, is practically indistinguishable from that obtained using the full pore diffusion model with constant diffusivity D.
D k ) 15 2 Rp
(1)
Equation 1 is commonly referred to as the Glueckauf1 approximation. Through a moment analysis of the pulse response from a chromatographic column model that explicitly allowed for linear adsorption equilibrium and external film, macropore, and micropore resistances, Haynes and Sharma2 showed that
Rp2K rc2 1 RpK ) + + k 3kf 15pDp 15Dc
(2)
Equation 2 is actually an extension of the Glueckauf approximation for systems in which more than one mass transfer resistance is significant.3 Although eq 2 is strictly valid for a linear isothermal system, it is also known to work reasonably well for nonlinear systems with K replaced by qo/Co, where Co is the feed concentration of the adsorbate in the gas phase and qo is the corresponding equilibrium in the adsorbed phase.4 The separation of gases and vapors by adsorption can be broadly classified into two categories, namely, equilibrium-controlled and kinetically controlled. Most of the known kinetically selective processes use carbon molecular sieves (CMSs), although such processes based on zeolite adsorbents are also known. Takeda Chemical Company of Japan and Bergbau Forschung (BF) of Germany are the leading manufacturers of CMSs used all over the world. In the product literature, Takeda * Author to whom correspondence should be addressed. Tel.: (65) 874 6545. Fax: (65) 779 1936. E-mail: chesf@ nus.edu.sg.
Chemical Company calls its product molecular sieving carbon (MSC). In this paper, products from both manufacturers will be termed CMS. CMSs have a bidisperse pore structure with clearly distinguishable macropore and micropore resistances to the transport of sorbates. However, the controlling resistance for the uptake of sorbates is typically diffusion in the micropores. The design of a kinetically controlled process using CMSs is entirely dependent on the fundamental understanding of the diffusion of gases in the micropores of the adsorbent. The transport mechanism in CMS micropores is still not fully understood. Distributed pore diffusional resistance5-9 or barrier resistance confined at the pore mouth10-13 or a combination of two11,14 might be encountered. The modeling of transport through a barrier resistance confined at a micropore mouth is mathematically equivalent to the linear driving force (LDF) model.15 The wide success of the Gleuckauf approximation in column dynamics calculations has given rise to a misconception that the details of the transport mechanism in the micropores of an adsorbent in which the controlling step is uptake in the micropores are not captured in the breakthrough response from a column. Although it has been pointed out in some studies that the Glueckauf approximation breaks down for short columns or short contact times,16,17 the possibility of using this observation as an alternative route for understanding the transport mechanism in the micropores has not been explored. In this study, it is shown that the extent of agreement between the pore diffusion and barrier models depends on the product of two parameters, namely, R [) (1 - p)Kc/p] and γ () DcL/rc2vo). Through the appropriate choice of operating conditions, it is possible to distinguish between the two transport mechanisms in CMS micropores from breakthrough experiments. The application of this method seems to indicate that the transport of oxygen and nitrogen in the micropores of both BF and Takeda samples is controlled by barrier resistance confined at the pore mouth. The breakthrough of oxygen and nitrogen was also studied in a 4A zeolite sample as a reference point for unambiguous pore diffusional behavior.
10.1021/ie0104621 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/04/2001
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1099 Table 1. Equations Describing the Two Models distributed pore diffusional resistance
barrier resistance at the pore mouth
fluid-phase material balance and boundary conditions
|
∂C hp h ∂C h h ∂C 1 ∂2C ) - 3ψβ ∂τ Pe ∂χ2 ∂χ ∂η
|
∂C h ∂χ
χ)0
(
)
|
∂C hp ∂2C h p 2 ∂C hp ∂Y )β + - 3Rγ ∂τ η ∂η ∂ξ ∂η2
|
∂C hp ∂η
η)0
) 0;
|
∂C hp ∂η
η)1
|
∂C h )0 ∂χ χ)1
) -Pe(C h |χ)0- - C|χ)0+);
macropore material balance and boundary conditions
ξ)1
) ∆(C h -C h p|η)1)
η)1
macropore material balance and boundary conditions
(
)
∂C hp ∂2C h p 2 ∂C hp )β + - 15Rγ(Y h* - Y h) ∂τ η ∂η ∂η2
|
∂C hp ∂η
η)0
) 0;
|
∂C hp ∂η
η)1
) ∆(C h -C h p|η)1)
micropore material balance and boundary conditions
micropore material balance
∂Y ∂2Y 2 ∂Y )γ + ∂τ ∂ξ2 ξ ∂ξ
∂Y h ) 15γ(Y h* - Y h) ∂τ
(
|
∂Y ∂ξ
ξ)0
) 0;
)
Y|ξ)1 ) C hp
Model for Predicting Column Dynamics Assumptions. The mathematical model for the breakthrough of an adsorbate in a bed that is packed with a bidisperse adsorbent and subjected to a change (step or pulse) in the feed concentration of the adsorbate has three main components: (i) adsorbate material balance for the fluid phase, (ii) adsorbate material balance in the macropores, and (iii) adsorbate material balance in the micropores. Transport in the macropores is represented by molecular-diffusion-controlled pore diffusion, where the macropore diffusivity is given by the bulk molecular diffusivity corrected for pore tortuosity. Both possibilities of barrier resistance confined at the pore mouth and pore diffusional resistance distributed in the micropore interior are considered for micropore transport in this comparative study. The steady-state assumption at the fluid-solid interface forms the link between the fluidand solid-phase balance equations. In other words, the diffusive flux of the sorbate across the external gas film around the adsorbent particle is assumed to be equal to that at the macropore surface. In most gas adsorption studies, intraparticle transport of the adsorbate is the slower step, and it is reasonable to assume negligible gas-side resistance. Nevertheless, in this study, the external film resistance is included in the model for completeness. However, it is assumed that the microparticle surface is in equilibrium with the macropore gas. The analysis is restricted to a linear isotherm. It is further assumed that a single adsorbate is fed to the bed in a stream of inert carrier. Moreover, the stream is dilute in adsorbate concentration so that the system is isothermal and the bulk velocity remains practically unchanged along the column length. An axially dispersed plug-flow pattern is assumed for the fluid phase. Frictional pressure drop is neglected, and the column is assumed to operate at constant pressure. Model Equations. Subject to the assumptions discussed in the preceding section, the dimensionless forms of the model equations and the boundary conditions are
summarized in Table 1. The equations contain the following dimensionless variables
C h )
Cp C q q j z , C hp ) , Y ) , Y h ) , χ ) z/L, Co Co qo qo L r R η ) , and ξ ) Rp rc
where C is the adsorbate concentration in the fluid phase, Cp is the adsorbate concentration in the macropore, Co is the adsorbate concentration in the feed, q is the adsorbate concentration in the micropore, q j is the average adsorbate concentration in the micropore, qo () KcCo) is the adsorbate concentration in the micropore in equilibrium with Co, z is the distance along the bed length, R is the distance along the macroparticle radius, r is the distance along the microparticle radius, L is the bed length, Rp is the macroparticle radius, and rc is the microparticle radius. In addition, the equations also contain the following dimensionless groups
R)
1 - p Dp L Dc L kb L Kc, β ) 2 , γ ) 2 or , p R vo r vo 15vo p
c
voL k fR p 1- , ψ) ∆) p, and Pe ) pDp DL In the above dimensionless groups, is the bed voidage, p is the particle voidage, Kc is the dimensionless Henry’s constant based on the micropore volume, vo is the interstitial fluid velocity in the bed, kf is the external film mass transfer coefficient, kb is the barrier mass transfer coefficient, Dc is the micropore diffusivity, DL is the axial dispersion coefficient, and Dp is the macropore diffusivity. According to the assumption made about macropore diffusion, Dp is the bulk molecular diffusivity (Dm) corrected for particle tortuosity (τp). L, Rp, and rc have already been introduced. Numerical Solution. The model equations were solved numerically by the method of orthogonal collo-
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
Figure 1. Comparison between the breakthrough responses of the two transport models. The shaded area represents I defined by eq 3. R ) 100, β ) 250, γ ) 0.0005, Pe ) 250, Ψ ) 0.5, ∆ ) 10.
cation. The partial differential equations were discretized in space, which converted them into a system of ordinary differential equations in the time domain. Eleven collocation points were used along the bed length, and seven points were used along the macroparticle/microparticle radii. The ordinary differential equations were integrated in the time domain using Gear’s variable-step integration algorithm as provided in the FORSIM package.18 Although an analytical solution is available for the model that considers distributed pore diffusional resistance,19 numerical evaluation of the associated slowly converging complicated functions did not appear to be advantageous over the numerical solution adopted here. In fact, it was much simpler to modify similar routines for column dynamics calculations with nonlinear isotherms that were developed earlier in this laboratory.20 Comparative Study. R, β, γ, ∆, ψ, and Pe are the six dimensionless groups that constitute the inputs for the two models. These parameters were systematically varied to study their effects on the difference in the solutions of the two models when kb ) 15Dc/rc2. The shaded area between the solutions of the two models shown in Figure 1 was normalized with respect to R and taken as a measure of the difference. The shaded area, I, is given by the integral
I)
(
)
∫0∞| CCo|χ)1 pore -
( | ) | C Co
dτ
(3)
Figure 2. (a) Plot of I vs γ showing dependence on R and negligible effects of β and Peclet number; (b) plot of E vs γ showing that the peaks are separated by the ratio of their R values; (c) plot of E vs Rγ showing that all of the results converge to a single curve.
χ)1 barrier
Note that the integrand is taken as an absolute value. The trapezoidal rule with a dimensionless time interval of 1 was used to calculate the area. Range of Investigation. R contains the equilibrium constant Kc and was varied from 10 to 100. β and Pe are measures of the significance of macropore resistance and axial dispersion, respectively. β was varied from 10 to 500, and Pe from 50 to 500 to cover the range typically encountered in a system controlled by transport in the micropores. Values greater than 500 for those two parameters are not significant in micropore-controlled systems. ∆ is a measure of the ratio of the external film transport coefficient to the macropore transport coefficient. In the limit of no flow (i.e., Sherwood number, kf2Rp/Dm ) 2), ∆ reduces to τp/p, which is typically taken as 10. This value is already too high to have any effect on the solution. Typical values of and p are 0.4 and 0.33, respectively, which makes ψ ≈ 0.5. Hence, values of ∆ and ψ were fixed at 10 and 0.5, respectively. γ is the most crucial parameter, and it was varied in this comparative study from 10-6 to 1.
Results and Discussion The results of the simulations are summarized in Figure 2a. From Figure 2a, it is clear that the difference between the two models, denoted by I according to eq 3, depends on R and γ and is practically insensitive to the parameters β and Pe. The area difference goes through a maximum with increasing γ and practically vanishes at the two ends of the range covered. For the same γ value, the area difference between the two models increases with increasing R, i.e., with increasing adsorption capacity. Figure 2a apparently carries the impression that, for comparable γ values, the difference will be more clearly identifiable for a more strongly adsorbed system. On closer examination, it was found that the increase in the integral defined by eq 3 with increasing R was due to longer breakthrough times coming from increased column capacities. It was, therefore, concluded that a more meaningful measure of the difference between the two models would be the area, I, normalized with respect to R. We denote this normalized area as E () I/R). Plots of E vs R are shown in Figure 2b.
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1101 Table 2. Summary of the Experimental Runs
run
operating pressure, P (bar)
feed composition
column length, L (cm)
interstitial velocity, v0 (cm/s)
operating temperature, T (°C)
R
γ
Rγ
Pe
β
ψ
∆
20.4 17.5 13.3 68 41.9 31.5
0.04 0.055 0.14 0.0013 0.0038 0.0067
0.82 0.96 1.86 0.09 0.16 0.21
731.5 716.1 485.4 221.0 208.8 202.6
261.4 275.8 600.9 2531.1 2712.0 2812.9
0.65
10
1 2 3 4 5 6
1.4 1.4 1.4 1 1 1
4.2% O2 4.2% O2 3.9% O2 4.8% N2 4.8% N2 4.8% N2
60 60 60 60 60 60
9.4 10.1 5.1 1.8 1.9 2
RS-10 zeolitea -10 10 26 -10 10 26
7 8 9 10 11 12
1.4 1.4 1.4 1.4 1.4 1.4
2.7% O2 2.8% O2 3.1% O2 2.9% N2 3.0% N2 4.7% N2
40 40 40 40 40 40
8.3 4.7 5.1 0.9 1 1.2
BF CMSb -30 1 29 1 29 70
82.6 40.4 19.3 34.7 17.9 14
0.0039 0.015 0.039 0.004 0.014 0.039
0.32 0.61 0.75 0.14 0.25 0.55
246.8 201.0 195.3 85.6 82.3 80.3
25.8 55.5 60.2 271.6 286.6 296.1
0.8
10
13 14 15 16 17 18
1.4 1.2 1.2 1.4 1.4 1
2.7% O2 3.0% O2 3.0% O2 2.8% N2 3.1% N2 4.7% N2
40 40 40 40 40 80
8.3 4.7 5.1 1.9 2 1.8
Takeda CMSb -30 1 29 1 29 70
98.1 48.7 25.4 45.3 22.4 13.5
0.0044 0.021 0.05 0.0022 0.0066 0.064
0.43 1.02 1.27 0.10 0.15 0.86
246.8 190.0 184.1 138.0 129.9 169.0
25.8 75.6 82.0 128.6 143.3 773.9
0.61
10
a
For RS-10, Rp ) 0.06 cm, p ) 0.29 (Farooq22). b For CMSs, Rp ) 0.15 cm, p ) 0.33 (assumed).
Figure 3. Comparison of experimental and theoretical breakthrough of oxygen in RS-10 zeolite. See Table 2 for experimental conditions and values of various parameters.
The normalized plots in Figure 2b revealed an interesting feature in that they were identical in shape but shifted from one another by the ratio of their R values. This clue paved the way for further improvement. As can be seen from Figure 2c, all of the results converged to one curve when E was plotted as a function of the product of R and γ. Figure 2c provides a measure of the expected difference in the dynamic column response due
to the two possible transport mechanisms in the adsorbent micropores. The level of difference that is visually distinguishable will be discussed in the next section, where the experimental results are analyzed. At this point, it is important to recall that the comparison made in this study is based on kb ) 15Dc/ rc2. Perhaps more important is the fact that, when the two models differ, one cannot be made to fit the other
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
even if the micropore parameter is varied. In other words, an unambiguous distinction between the two transport mechanisms is possible throuh the appropriate choice of operating conditions. Significance of rγ. The concentration wave velocity is given by
ωc )
vo 1- Kp 1+
where Kp ) p + (1 - p)Kc ) p(1 + R). The following alternative expression can be obtained for γ by substituting the interstitial fluid velocity in the column, vo, with an equivalent expression in terms of concentration wave velocity, ωc
γ)
Dc L r 2 ωc c
[
]
[
]
Figure 4. Comparison of experimental and theoretical breakthrough of nitrogen in RS-10 zeolite showing the fit of the pore model. For experimental conditions and values of various parameters, see Table 2.
1 1- 1+ (1 + R) p
or
Rγ )
Dc L R 2 ω 1 rc c 1 + (1 + R) p
The term in the square brackets on the right-hand side of the above equation varies from 1.55 to 1.96. Hence, it is not unreasonable to assume that this quantity is approximately constant. One can then take Rγ ∝ (Dc/ rc2)(L/ωc), which can be viewed as the dimensionless residence time of an adsorbing molecule in the bed. Experimental Verification. Several breakthrough experiments of oxygen and nitrogen in one zeolite (modified 4A called RS-10) and two CMS (BF and Takeda) samples were conducted to verify the theoretical results presented in Figure 2c. These experiments are summarized in Table 2. In all runs, the column pressure was nearly atmospheric, and the concentration of adsorbate in the feed was kept low (
