Ind. Eng. Chem. Res. 2005, 44, 4745-4752
4745
Investigation of Mixture Diffusion in Nanoporous Adsorbents via the Pressure-Swing Frequency Response Method. 2. Oxygen and Nitrogen in a Carbon Molecular Sieve Yu Wang and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
A simple apparatus using the pressure-swing frequency response method has been extended to measure multicomponent diffusivities. Transfer phenomena are investigated for different compositions of a binary mixture of nitrogen and oxygen in a carbon molecular sieve. The paper has two objectives. First, we consider the applicability of some predictive multicomponent diffusion models, which predict the mixture diffusivities based on pure-component information. The results show that these models can provide qualitative but not quantitative descriptions. Second, we determine mixture diffusivities, including Fickian diffusivities and Maxwell-Stefan surface diffusivities, from the data using a new mathematical model. These diffusivities are concentration-dependent. A simple relationship that introduces a thermodynamic factor is used to describe the concentration dependence of the Fickian diffusivities. It reduces to Darken’s equation for pure components. This approach, based on nonconstant Fickian diffusivities, provides an excellent description for all of the experimental data. An alternative approach is also developed for obtaining the Maxwell-Stefan surface diffusivities without using the empirical Vignes relationship. Introduction A thorough understanding of the diffusion of multicomponent mixtures inside nanoporous materials is essential for the accurate simulation of many adsorption processes. A variety of models have been used to describe multicomponent diffusion including the Fickian model,1,2 the Maxwell-Stefan (MS) formulation,3-5 Onsager’s formulation6-8 derived from irreversible thermodynamics, Monte Carlo (MC) simulations, and molecular dynamics (MD). Because the MC and MD techniques are computationally expensive, most process simulations rely on the first three models.9 Fick’s law of diffusion has been used traditionally as a basis for separation design. It has the advantage of the simplest appearance, and it is often possible to derive analytical expressions for describing the masstransfer behavior; however, compared to the other two models, it needs more parameters because the number of diffusion cross-coefficients increases rapidly with the number of components.10 Krishna and Wesselingh5 have shown some shortcomings of the Fickian constitutive relationship, which fails to describe some situations such as osmotic diffusion, reverse diffusion, individual ionic diffusion, and selectivity-reversal phenomena for changing from a single component (pure adsorbate in helium) to a binary mixture (two adsorbates) in membrane transport. The MS theory can be successfully applied to these systems and others to describe diffusive transport in multicomponent mixtures.3-5,9,11-13 The MS diffusivity has the clear physical significance of an inverse drag coefficient and is more easily interpretable and predictable than the Fickian diffusivity.5 With the binary MS * To whom correspondence should be addressed. Tel.: (615) 322-2441. Fax: (615) 343-7951. E-mail:
[email protected].
diffusivities, the transport behavior can be described theoretically for any number of components. For those reasons, the use of the MS theory to describe multicomponent mass transport is rapidly increasing. Because experimental measurements of mixturediffusion coefficients are difficult, prediction from purecomponent parameters is desirable. Predictive theories for multicomponent diffusion are principally based on kinetic theory,14 the MS formulation,4 and irreversible thermodynamics.7,8,15 One of the main questions underlying predictive theory is whether to consider the adsorbate-adsorbate interaction and, if so, how to do so.12 The most commonly used and simplest case is to consider no diffusional interaction. Consequently, a diagonally constant Fickian diffusivity matrix is applied to predict mixture diffusion, as has been used in early work.16 Habgood6 presented a more complicated theory derived from irreversible thermodynamics theory. This formulation, without adsorbate-adsorbate interactions, has been used to describe the uptake of binary mixtures in zeolites,8 microporous activated carbon, and carbon molecular sieves (CMSs).4 Yang et al.15 derived an expression for predicting the cross-term phenomenological coefficients from the main-term coefficients based on irreversible thermodynamics theory. The results showed that, for both counterdiffusion and codiffusion, the multicomponent effects are most pronounced for the fastdiffusing component. Chen and Yang14 developed a kinetic approach to predict the Fickian diffusivity for nanopore diffusion in multicomponent systems. They assumed a geometricmean rule for interaction between unlike adsorbate molecules. The model requires knowledge of the concentration dependence of the pure-component diffusivities and the heats of adsorption of the pure components. A blocking parameter λ was introduced to relate the interaction energies between molecules and between
10.1021/ie0489352 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/20/2005
4746
Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005
molecules and a vacant site. For λ ) 0, this kinetic model is equivalent to the irreversible thermodynamics model with vanishing cross-term interactions. Chen et al.17 used the DAB technique to measure diffusivities of O2 and N2 in a CMS over the pressure range 3.75.1 atm at 27 °C. Their mixture composition of O2 and N2 is similar to air. They investigated the capabilities of two predictive theories, irreversible thermodynamics and kinetic theory. For most of the codiffusion and counterdiffusion cases, predictions by using the theoretical binary diffusivities and pure-component diffusivities are in agreement with the experimental data. Krishna3,18 developed the MS formulation based on irreversible thermodynamics for multicomponent nanopore diffusion. In their model, cross-term diffusivities (}ij) were estimated using an empirical relationship given by Vignes19 to account for adsorbate interactions. For a single file diffusion mechanism, with no interaction between adsorbed species, the MS formulation yields the same expression as the irreversible thermodynamics theory with vanishing cross-term interactions.4 Van den Broeke and Krishna4 performed breakthrough experiments to test the predictive capability of three models: the diagonally constant Fickian diffusivity, the irreversible thermodynamics theory with vanishing cross-term interactions, and the MS with Vignes’ relationship. They examined methane, carbon dioxide, propane, and propene on microporous activated carbon and a CMS. The results showed that prediction of the complete MS formulation including adsorbateadsorbate interaction has only minor differences with the irreversible thermodynamics approach. The predictions of these two models are more accurate than those from the model based on a matrix of diagonally constant Fickian diffusivities. Even though irreversible thermodynamics and MS theory can describe the CH4/CO2/He mixture system well, they still could not provide satisfactory results for propane/propene on a CMS; the reason given was inadequate accuracy of the multicomponent adsorption equilibrium. Recent experimental studies on the permeation of binary hydrocarbon mixtures across a silicalite membrane have emphasized the need for including the adsorbate-adsorbate interaction terms.12 The results show that, with the empirical Vignes relationship for describing interaction, the MS model describes the flux reduction of the weakly adsorbing component quantitatively much better than the model without interactions. Kapteijn et al.9 verified the necessity of including }ij for mixtures of components with quite different diffusivity values, and if the differences are small, then the influence of cross terms is negligible, which was almost the case for the results of Van den Broeke and Krishna.4 The adsorbent used in the experiments reported here is a CMS, which is applied widely in a variety of gas separation proceses such as separation of air to recover N2 by pressure-swing adsorption. This is a modified form of activated carbon that has both a high internal surface area and molecular-sieving capability. Pores and surface barrier pore constrictions are close to molecular dimensions, which enables the separation of components on the basis of differences in diffusion rates. Several studies of pure-component diffusion and equilibrium adsorption of O2 and N2 on a CMS are available,17,20-24
but studies of the mixture diffusivities are very limited.17,25-27 Recently, the frequency response (FR) method has been applied to investigate the pure-component diffusivities of N2 and O2 in a CMS.28,29 Because this technique has been shown to be very powerful in the measurement of pure-component mass transfer, it is of interest to determine whether the FR method can be applied successfully in mixture-diffusion measurement. Yasuda et al.25,30 applied the batch FR method to investigate O2/N2 on 4A zeolite at 273 K and pressures near 0.013 bar. Using a restrictive state transformation, which requires a relationship between the four Dij’s to decouple the micropartical mass balance equations, the results showed that the main-term O2 diffusivity (D11) decreases with an increase in the concentration of N2 and approaches that of pure N2, whereas the main-term N2 diffusivity (D22) is almost unaltered with an increase in the concentration of O2. The cross-term diffusivities are of the same sign, and D21 (relating the flux of N2 to the gradient of O2) is always larger than D12. In their mathematical model, the diffusional interaction was accounted for but the equilibrium interference was neglected. In this paper, we apply pressure-swing FR to measure transport coefficients of binary mixtures of N2 and O2 in a CMS at different compositions. Although we have developed nonisothermal models for the pressure-swing FR method for adsorption of pure components,31 we have shown that N2 and O2 adsorbed on a CMS can be justifiably treated using an isothermal model under the conditions of our experiments.28 A new isothermal FR model that considers both nanopore diffusion and a surface barrier, proposed in the companion paper,32 is applied to describe the FR curves. With the new binary experimental data, the proposed model has been used to check the capability of three predictive mixture diffusivity models: the constant Fickian diffusivity model, the irreversible thermodynamics theory with vanishing cross-term interactions, and the MS with Vignes’ relationship. Moreover, to confirm the utility of the FR method, the experimental data have been analyzed to extract mixture diffusivities for Fick’s law and MS theory. Both the constant Fickian diffusivity and the nonconstant Fickian diffusivity have been examined. Theory The mathematical model was presented in the companion paper.32 The total transfer function GT, dependent on the adsorbed-phase transfer function for each component, is derived there for a binary system. The intraparticle transport mechanism is considered to be controlled by a dual-resistance mechanism, which is determined by the mixture diffusivities (D11, D12, D21, and D22) and parameter β, which is the ratio of the nanopore diffusion time scale to the surface barrier time scale.33 The FR curves can be predicted by using different models to obtain the mixture diffusivities and then compared with the experimental data to test the capability of the predictive models. Alternatively, the mixture diffusivities can be extracted from the experimental data. The model presented in the companion paper32 is based on Fick’s law. Krishna and Wesselingh5 elaborated on the relationship of the Fickian diffusivities and the MS diffusivities in their review paper. With this
[
]
Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005 4747
relationship, we can predict the Fickian diffusivities from the MS diffusivities. Alternatively, we can obtain the MS diffusivities directly from the experimental data. Krishna3,18,34 applied Maxwell and Stefan’s idea to describe the nanopore diffusion of adsorbed molecules in an n-component mixture starting from
θi
n
∑
njNi - niNj
The first term on the right-hand side reflects the friction exerted by adsorbate j on the surface motion of species i, and the second term reflects the friction between species i and the surface. }ij and }i represent the corresponding MS surface diffusivities. The }i ’s are the diffusivities that reflect interactions between species i and the adsorbent surface, and the }ij ’s are the diffusivities that reflect sorbate-sorbate interactions.34 The gradient of the thermodynamic potential can be expressed in terms of the thermodynamic factors Γ3
RT
∂ ln fi Γij ≡ θi , i, j ) 1, 2, ..., n ∂θj (2)
n
∇µi )
Γij∇θj ∑ j)1
For example, for the Langmuir isotherm for an ideal gas mixture (fi ≡ Pi), we have
θi )
ni n
biPi
)
sat
n
1+
(3)
biP i ∑ j)1
and the elements Γij are found to be
Γij ) δij +
θi n
(4)
θi ∑ j)1
1-
(8)
Taking the inverse of matrix B gives34
Ni
-F ∇µi ) + sat sat RT j)1,j*i n n } nsat i j ij i }i i ) 1, 2, ..., n (1)
θi
θ2 θ1 1 + } } }12 B ) 1 θ 12 θ1 2 1 + }12 }2 }12
B-1 )
1 }2 }1 1 + θ1 + θ2 }12 }12
[
}1}2 }1 }2 }1 + θ1 θ1 }12 }12 }1}2 }1}2 θ2 }2 + θ2 }12 }12
]
(9)
Thus, for the Langmuir isotherm, the Fickian diffusivities for a binary mixture can be predicted from the MS diffusivities from eqs 4, 7, and 9 combined to give34
[ [
}1 + θ1
D)
}1}2 }1}2 θ1 }12 }12
]
1 }1 }2 }1 }2 }2 }1 θ2 } 2 + θ2 1 + θ1 + θ2 }12 }12 }12 }12 1 - θ2 θ1 1 - θ1 - θ2 1 - θ1 - θ2 (10) θ2 1 - θ1 1 - θ1 - θ2 1 - θ1 - θ2
]
For the general multicomponent mixture, it is difficult to predict all Fickian diffusivities because not all elements of D have a physical interpretation. The MS approach has the potential to predict all of the elements of D. Furthermore, it has the advantage of decoupling the drag effects (B) from the thermodynamic effects (Γ). The main difficulty in this approach is determining the cross-term diffusivities }ij . One estimation of }ij has been suggested by Krishna and Wesselingh5 based on the empirical relationship of Vignes19
where δij is the Kronecker delta. The above equations can be cast into an n-dimensional matrix notation of the form
}ij ) [}i]θi/(θi+θj)[}j]θj/(θj+θj)
-FnsatΓ(∇θ) ) B(N)
The cross-term diffusivities satisfy the Onsager reciprocal relation3
(5)
}ij ) }ji
where the matrix B has elements given by
Bii )
1 }i
n
+
θj
∑ j)1,j*i}
Bij ) -
θi }ij
ij
(6)
With a matrix of Fickian diffusivities D defined by N ) -FnsatD(∇θ), the following explicit expression for D is obtained:
D ) B-1Γ
(12)
For a special case with no interactions between the adsorbed species i and j (i.e., }ij f ∞ numerically), eq 10 can be further simplified for this single-file diffusion mechanism to give4
i ) 1, 2, ..., n
i ) 1, 2, ..., n
(11)
(7)
For a binary mixture (n ) 2), the matrix B has the form
D)
[
}1 0 0 }2
[
]
1 - θ2 θ1 1 - θ1 - θ2 1 - θ1 - θ2 θ2 1 - θ1 1 - θ1 - θ2 1 - θ1 - θ2
]
(13)
Equation 13 has the same form as the expression derived from irreversible thermodynamics.6 The fluxes for a binary diffusing pair are given by the phenomenological equations
4748
Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005
Figure 1. Flow-through apparatus for a binary pressure-swing FR. Table 1. Diffusional Time Constants and Barrier Effects for Pure O2 and N2 in a CMS component N2 O2
Figure 2. Prediction results for a binary mixture N2/O2 on a CMS.
where Lij’s are the Onsager phenomenological coef-
N1 ) -L11∇µ1 - L12∇µ2 N2 ) -L21∇µ1 - L22∇µ2
(14)
ficients. These Lij’s are strong functions of concentration, especially in a dilute solution.35 They satisfy the reciprocal relationship Lij ) Lji. The system given by eq 14 can be simplified by introducing an assumption that cross-term coefficients Lij are zero. Combined with a Langmuir isotherm, the Fickian diffusivities can be predicted in the same way as that for no cross-term interaction in the MS formulation (eq 13). In this paper, we will also refer to this approach as the irreversible thermodynamics model with vanishing interaction because it was first derived from irreversible thermodynamics theory.8 Using the different approaches given by eqs 10 and 13 for predicting the Fickian diffusivities from purecomponent diffusivities, we can obtain the adsorbedphase concentration n from eq 22 of the companion paper.32 Then, the transfer function GT of the total system, given by eq 28 of the companion paper,32 can be solved to get the FR curves. These can be compared to the experimental data to test the capabilities of the predictive models. Alternatively, we can substitute eq 1 into GT and correlate the MS surface diffusivities. Experiments Mass-transfer rates were measured using a flowthrough apparatus, which is shown in Figure 1. Two components were introduced through gas driers and mass flow controllers, which were used to set the desired composition. After blending, the mixed gas passed
D/R2 (s-1) 10-5
9.4 × 1.3 × 10-3
β 3.1 24.9
through the adsorption bed and then exited to a vacuum. The system pressure was modulated using a pressure controller. The response of the total mass flow rate was recorded. The system was controlled using LabVIEW 6i, and the amplitude and phase information were extracted directly from the data using LabVIEW. The adsorbent used in this study is MSC-3K type 161. It is in the form of cylindrical pellets of 1.18-2.80 mm diameter (also reported as 7 × 14 mesh), supplied by Takeda Chemical Industries, Ltd. Experiments were conducted with N2 (99.5%), O2 (99.6%), and their mixtures at different compositions: 100% N2, 80% N2 + 20 O2, 50% N2 + 50% O2, 20% N2 + 80% O2, and 100% O2. The CMS sample (6 g) was regenerated at 150 °C for 24 h with helium flowing at 200 mL/min. The regenerated adsorbent was then equilibrated with gas of the desired composition flowing at 1 atm. The volume of the adsorption region was about half of the bulk volume of the CMS, which was 9.17 cm3. The volumes of V1 and V2 were 34.5 and 32.0 cm3, respectively. The system pressure was perturbed in a sinusoidal wave over the range of 0.00005-0.2 Hz. Small-amplitude (
