Ind. Eng. Chem. Res. 2007, 46, 2141-2154
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Mixture Diffusion in Nanoporous Adsorbents: Development of Fickian Flux Relationship and Concentration-Swing Frequency Response Method Yu Wang and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt UniVersity, NashVille, Tennessee 37235
A frequency response method using concentration variation is developed theoretically and experimentally and applied to investigate mixture diffusion in nanoporous adsorbents. The method is based on periodically time-varying species feed concentrations with a constant total molar inlet flow rate. It can be used without the need of a carrier gas. A mathematical model is formulated considering nanopore diffusion, surface barrier resistance, external film resistance, and axial dispersion. The related analytical solutions for frequency response are derived. For nanopore diffusion, a theory for nonconstant mixture Fickian diffusivity with cross-terms is developed from irreversible thermodynamics and shows that mixture Fickian diffusivities can be expressed as the product of corrected diffusivities and a thermodynamic factor that accounts for concentration dependence. The number of unknown variables for Fickian diffusivities is the same as the number for Onsager coefficients or Maxwell-Stefan diffusivities. Adsorption of CO2, CH4, and their mixtures on carbon molecular sieve (CMS) is investigated systematically for equilibrium and mass transfer rates. The mass transfer mechanisms for pure CO2 and CH4 on CMS measured by a pressure-swing frequency response method are found to be differentsthe rate-controlling mechanism for CO2 is only nanopore diffusion, whereas the diffusion rate for CH4 is limited mainly by a surface barrier resistance at the pore mouth of the CMS. All of the mixture experimental data are measured by the new concentration-swing frequency response method and are described well by the nonconstant Fickian diffusivity model with the thermodynamic factor derived for a multicomponent multisite-Langmuir isotherm. Introduction The characterization of nanoporous materials is important because of their widespread application as adsorbents and catalysts. Compared to equilibrium properties, kinetic information is more complicated and difficult to obtain, especially for mixtures. Available experimental techniques for adsorption rates are limited due to difficulties in measuring the rate of uptake of each adsorbate. They include chromatography, batch adsorber, NMR, zero length column, frequency response (FR), and differential adsorption bed methods.1 Frequency response is a relaxation technique in which one system variable is perturbed around an equilibrium state and the response of one or more other system variables is used to characterize the system. It has the potential to discriminate easily among various mass transfer mechanisms by examining the response over a large frequency spectrum. In addition, it has advantages of minimizing measurement errors as it is a periodic process and allows small changes of system conditions compared to typical step change methods.2 Many theoretical and experimental aspects of FR have been considered for a batch system in which the gas pressure is changed by a forced periodic volume fluctuation.2-10 Only a few applications have been reported in flow systems involving the flow rate,11,12 concentration,13-15 or pressure variations.16-18 An advantage of a flow-through FR method is the relative ease of maintaining more isothermal conditions in comparison to a batch system.15 Other advantages are outlined by Park et al.11 for the flow FR method with a modulation of an inlet molar flow rate. Boniface and Ruthven13 first developed a flow FR method using chromatography based on a sinusoidally varying input * Author to whom correspondence should be addressed. Vanderbilt University, VU Station B 351604, 2301 Vanderbilt Place, Nashville, Tennessee 37235-1604 USA. Tel.: (615) 322-2441. Fax: (615) 3437951. E-mail:
[email protected].
concentration near equilibrium, which was achieved by adding a sinusoidally perturbed flow rate of a sample gas in a carrier gas having a constant flow rate. The parameters in their model include an equilibrium constant, an axial dispersion coefficient, a macropore diffusivity, and a nanopore diffusivity. Generally, several combinations of the parameters can yield virtually the same relationship of amplitude ratio on frequency. Thus, the simulation results were presented only for some limiting cases where only one of the diffusive resistances was rate controlling. Single-component gas systems were studied including Ar adsorbed in NaX as an example of negligible nanopore diffusional resistance, Ar or O2 adsorbed in Na mordenite as an example of negligible intracrystalline diffusional resistance, and Ar, O2, or N2 adsorbed in NaA zeolite as an example of negligible axial and macropore resistances. This method was shown to be valuable for obtaining kinetic information on a relatively fast-diffusing species compared to step uptake methods. The fastest frequency in this study was 0.1 Hz. Harkness et al.15 applied the FR method using concentration variation in a novel manner. A tubular reactor, which held about 30 mg of zeolite powder in a 20 cm length of 1/4 in. od stainless steel tubing, was used instead of a chromatographic column, and the outlet concentration was measured with a mass spectrometer. An adsorbable gas (propane) and a reference gas (argon) were mixed, and a small modulation in the frequency range of 0.02-2 Hz was produced in the flow of the combined gas. Then, the gas was injected into a much larger constant flow of a carrier gas. The resultant gas stream had a near constant volumetric flow rate but an oscillating composition. In this method, zeolite powder was used instead of biporous zeolite pellets and is thus only nanoporous. The method was shown to give the nanopore diffusivity and the Henry’s law constant simultaneously, independent of experimental conditions such as the modulation amplitude of the concentration, flow velocity, and column packing.
10.1021/ie061214d CCC: $37.00 © 2007 American Chemical Society Published on Web 03/07/2007
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Do and co-workers12,19 investigated a semibatch and a continuous flow adsorber using modulation of the inlet molar flow rate perturbed by four modes: sinusoidal, square, triangular, and sawtooth waves. A stream of pure adsorbate with a periodic flow rate was introduced into the adsorber, and at the same time, a stream flowed out of the adsorber through a leak valve. This stream was assumed to have a constant volumetric flow rate with its molar flow rate proportional to the adsorber pressure. Thus, an overflow parameter was introduced in the model to describe the outlet molar flow rate. The simulation results showed that the in-phase characteristic function does not depend on the outlet flow rate, but the out-of-phase characteristic function, amplitude ratio, and phase shift are highly dependent on the overflow parameter. This technique was applied to sorption kinetics of methane, ethane, and propane on activated carbon.12 Sward and LeVan16 and Wang et al.17 have presented a new method with modulation of pressure in the adsorption system. This new method has been used to examine adsorption behavior of carbon dioxide on BPL activated carbon described by a nonisothermal surface diffusion model16 and pure-component nitrogen or oxygen on Takeda carbon molecular sieve (CMS) described by a combined resistance model.17 As the FR technique has been shown to be very powerful in measurement of pure-component mass transfer, it is of interest to determine whether the FR method can be applied successfully for mixture diffusion measurement. Very few studies have been published in this area. Yasuda et al.20,21 first applied the batch FR method to investigate binary mixtures of O2 and N2 in NaA zeolite. In their mathematical model, the diffusional interaction was accounted for but the equilibrium interference was neglected. Also, a restrictive state transformation was used that requires a relation between the four diffusivities, Dijs, to decouple the particle material balance equations. Recently, Wang and LeVan18,22,23 developed a flow-through pressure-swing FR method to measure multicomponent diffusivities. Transfer phenomena were investigated for different compositions of a binary mixture of nitrogen and oxygen in CMS. A new isothermal FR model, which considers both nanopore diffusion and a surface barrier, was proposed to describe the FR curves. Mixture diffusivities, including Fickian diffusivities and Maxwell-Stefan surface diffusivities, were extracted from the data using a new mathematical model. A simple relationship was suggested to describe the concentration dependence of mixture diffusivities. The mixture diffusivities reduce to those given by Darken’s equation for pure components. This approach, based on nonconstant Fickian diffusivities, provided an excellent description of all of the experimental data. Separation of CO2 and CH4 has attracted interest because of several potential applications. One prospect is enhanced coalbed methane (ECBM) production using CO2 injection. This is an emerging technology that has the potential to improve coalbed methane recovery through a competitive desorption mechanism and to simultaneously result in the sequestration of large volumes of CO2 in deep coal seams.24 Another important application is separation of CO2 from CH4, which is important in natural gas recovery and processing because CO2 reduces the energy content of natural gas and CO2 is acidic and corrosive in the presence of water for transportation and storage.25 Currently, these separations are accomplished predominantly by membrane permeation with CO2 as the high purity product, but pressure-swing adsorption (PSA) and vacuum-swing adsorption (VSA) have the potential to perform the separation to obtain CH4 as the desired product for the chemical industries.26 Kapoor
and Yang27 suggested that kinetic separation using CMS is possible and has some advantages such as low feed gas pressure and ambient temperature compared to membrane processes. The separation is applied mainly to landfill gas or coalseam gas which contains approximately 50% each of CO2 and CH4 and tertiary oil recovery where the effluent contains approximately 80% CO2 and 20% CH4.27 Various studies for pure CO2 and CH4 on CMS have been reported in the literature using gravimetric uptake measurement. Kapoor and Yang27 measured equilibrium isotherms and diffusivities for these gases in Bergbau Forschung (BF) CMS. They found that the Fickian diffusion model could be used to describe the diffusion process in CMS and that the diffusivity ratio for CO2/CH4 is 180 at 298 K. The results show that the strongeradsorbing CO2 is the faster-diffusing component, which is favorable for kinetic separation, especially when CH4 is the desired product. Ma et al.28 reported an experimental investigation involving N2, O2, CH4, and Ar in three different adsorbents, Takeda 5A and 3A and a BF CMS, for adsorbate pressures of 0-1.3 MPa and three temperatures, 273, 303, and 323 K. The equilibrium adsorption data for the four gases were described by the Langmuir model and the vacancy solution model. The adsorption rate of methane is the slowest of the four gases, and its diffusivity value was shown to have no dependency on pressure. Rutherford et al.29 observed that the mass transfer rate for carbon dioxide obeys the Fickian diffusion law and for methane it follows non-Fickian behavior in a commercially available Takeda 3A CMS. The non-Fickian response is attributed to transport resistance at the pore mouth for the methane molecules but not for the carbon dioxide molecules. Kinetic studies on the CO2-CH4 mixture in nanoporous materials are quite limited. Huang et al.30 used a differential adsorption bed to measure single-component, binary, and ternary integral uptakes of oxygen, nitrogen, carbon dioxide, and methane in BF and Takeda CMS. Their study shows that the multisite-Langmuir (MSL) isotherm is a reliable model for representing single-component equilibrium data and predicting mixture equilibrium in CMS for all of the gases investigated. The transfer of CO2 and CH4 in BF CMS was found to be controlled by a dual resistance. Empirical correlations successfully accounted for the strong concentration dependence of the thermodynamically corrected transport parameters and verified the single, binary, and ternary integral uptake experiments with the fitting parameters from the single-component experimental data. Recently, Kim et al.31 investigated kinetic separation of landfill gas (50% CO2, 50% CH4) on a CMS bed experimentally and theoretically via breakthrough experiments. They found that the adsorption rate depends strongly on pressure and temperature. A nonisothermal and nonadiabatic model using the linear driving force (LDF) model with concentration-dependent diffusivity was applied to predict the experimental breakthrough curves. In this paper, a new flow-through FR method using concentration variation is presented to investigate mixture diffusivities in nanoporous adsorbents. A small, shallow adsorption bed is fed with a periodically modulated inlet concentration at a constant total flow rate. The outlet concentration is analyzed using a mass spectrometer. This system is different from previous systems that used a concentration variation FR method, as follows. First, the system was developed to investigate mixtures, not pure systems. Second, the system can be operated without a carrier gas. Third, a relatively wide range of frequencies (10-4 to 3 Hz) can be applied. With optimizing the flow rate, system volume, and weight of adsorbent, the
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2143
maximum frequency should not limit the kinetic measurement for moderately short time scale systems. Fourth, the total inlet flow rate is precisely constant in this system, in comparison to previous systems in which a small time-varing quantity of the absorbable component was added to a large constant flowrate of an inert gas, with the total flow rate nevertheless assumed to be constant in the mathematical models. Finally, the model considers a series of mass transfer resistances including nanopore diffusion, surface barrier resistance, external film resistance, and axial dispersion. We apply this method to obtain kinetic information for binary mixtures of CO2 and CH4 at different compositions, which include an approximate composition of landfill gas and tertiary oil gas. A theory for nonconstant Fickian diffusivities is derived to account for concentration dependence. Pure-component equilibrium and kinetics results are also included in this paper to provide necessary information for the mixture studies. The mass transfer rates of the pure components are measured by the pressure-swing FR method.17 The experimental data are described by models that consider different adsorbed-phase mass transfer mechanisms, including nanopore diffusion, a surface barrier resistance model described by the linear driving force (LDF) model, and a combined resistance model which considers series resistances of nanopore diffusion and a surface barrier. To the best of our knowledge, there are no reports of using the concentration-swing FR method to determine mixture diffusivities in adsorbents. Also, this paper provides a clear way to study mixture diffusivities at different concentrations with the theory developed to consider the concentration dependence of Fickian mixture diffusivities. Even though this concentrationswing FR method pertains to mixture studies, it can also be used to determine adsorption rates for pure components in inert gas. The paper contains two distinct theoretical parts. First, we develop relations for multicomponent diffusion in nanoporous adsorbents. Then, we develop the theory for our concentrationswing frequency response method. Following these, we consider the CO2/CH4 system through experiments and analysis. Theory for Nonconstant Fickian Mixture Diffusivities Fickian and Onsager Formulations. In a previous study,22 we suggested a simple relationship to consider the concentration dependence for mixture diffusivities. It was empirical but was applied successfully for all of the compositions investigated. Here, we develop a relationship theoretically for Fickian diffusivities for mixtures that are dependent on concentrations for all of the adsorbates. There are three different ways to describe multicomponent diffusion in porous materials: Fick’s law, Onsager irreversible thermodynamics, and the Maxwell-Stefan formulation. Among them, the most commonly used approach is Fick’s law, which considers the flux to be proportional to the adsorbed-phase concentration gradient. For a binary mixture, it is expressed by
J1 ) -D11Fp∇n1 - D12Fp∇n2 J2 ) -D21Fp∇n1 - D22Fp∇n2
(1)
where Dij are the Fickian diffusivities, which can depend on concentrations, Fp is particle density, and n is the adsorbedphase concentration. Onsager’s irreversible thermodynamics formulation is based on chemical potential gradients being the driving forces for diffusion. For a binary system, the fluxes are
J1 ) -L11∇µ1 - L12∇µ2 J2 ) -L21∇µ1 - L22∇µ2
(2)
where a linear relation between diffusion velocities and potential gradients for the solutes is assumed.32 Lij are the Onsager phenomenological coefficients, which satisfy reciprocal relations, i.e., Lij ) Lji. The chemical potential can be written in terms of the equilibrium pressure given by the adsorption isotherm using
µi ) µ0i + RT ln Pi
(3)
where Pi is the partial pressure of species i. Thus, the gradients of µi can be written N
∇µi ) RT∇ ln Pi ) RT
∑ j)1
∂ ln Pi ∇nj ∂nj
(4)
With the help of multicomponent equilibrium information, the Onsager phenomenological coefficients and Fickian diffusivities are related without any approximation. Chempath et al.33 have shown that [D] can be calculated as [D] ) [L][Γ′], where the elements of the square matrix [Γ′] are given by Γ′ij ) [RT/ (Fpnj)](∂ ln Pi/∂ ln nj). Their relationship can be rewritten in the form
[
RT ∂ln P1 D11 D12 L11 L12 Fp ∂n1 D21 D22 ) L21 L22 RT ∂ln P2 Fp ∂n1
[
] [
]
RT ∂ln P1 Fp ∂n2 RT ∂ln P2 Fp ∂n2
]
(5)
Onsager Cross-Term Coefficients. It remains to determine the appropriate relations for the cross-coefficients Dij. We first consider corrected cross-coefficients D°ij, which are the values of Dij approached at low coverage. These are then modified using thermodynamic factors to allow for equilibrium interference caused by nonlinear multicomponent isotherms. Molecular simulation has been used to compute the matrix of Onsager coefficients, and then, the Fickian diffusivities can be obtained using eq 5.34,36 The off-diagonal Onsager coefficients, Lij, have been predicted using
Lij ) Lji ) xLiiLjj
(6)
Arora and Sandler35 applied this relation to describe diffusion of N2 and O2 in single-walled carbon nanotubes (SWCN). Chen and Sholl36 found that this expression accurately describes data for diffusion of CH4 and H2 in SWCN, but they emphasized that the prediction deviates strongly from the off-diagonal Onsager coefficients for CH4 and CF4 mixtures diffusing in silicalite. In developing a more general relation than eq 6, we consider an important limiting case, namely that if two diffusing components are identical, the sum of their fluxes should be the same as the total flux if the components were indistinguishable. Adding eqs 2 we obtain
J1 + J2 ) -(L11 + L21)∇µ1 - (L12 + L22)∇µ2 ) -LT∇µT (7) or
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J 1 + J2 ) -
L11 + L21 L12 + L22 n1∇µ1 n2∇µ2 ) n1 n2 LT - nT∇µT (8) nT
where the T subscript symbolizes the components if indistinguishable, and nT ) n1 + n2. The Gibbs adsorption isotherm is the Gibbs-Duhem equation for a surface at constant temperature and can be written for our case as
Adπ )
∑nidµi ) nTdµT
(9)
for the identical and indistinguishable components, with π being the spreading pressure. Comparing eqs 8 and 9, it is apparent that eq 8 is satisfied in general if and only if
L11 + L21 L12 + L22 LT ) ) n1 n2 nT
(10)
With L12 ) L21, the leftmost equality in eq 10 gives
n2L11 - n1L22 L12 ) L21 ) n1 - n2
Here, A and A* represent indistinguishable molecules, which have the same mobility b. Thus, with LAA/nA ) LA*A*/nA* ) b, eq 13 reduces to LAA* ) 0. Yang et al.38 and Sundaram and Yang39 suggested a somewhat similar result to eq 6 based on previous work by Haase,40 who considered entropy production and showed L12 ) L21 e ( xL11L22 for nonequilibrium processes and nondependent fluxes. Yang et al. recommend the relationship
L12 ) L21 ) λxL11L22
(14)
for a binary system with an interaction coefficient λ bounded by -1 e λ e 1, where λ is positive for codiffusion and negative for counterdiffusion. We develop our cross coefficients for our Fickian flux relationship with eq 14 written in the form
Lij ) Lji ) RijxLiiLjj
(18)
where D°ij ) D°ji. General Fickian Mixture Diffusion. With eq 18 for the corrected cross-terms, we obtain binary Fickian diffusion coefficients by substituting eq 4 into eq 2 and comparing with eq 1. This gives
)
|
∂ ln P1 Fp∂n1
+ L12RT
n2
[ ][
|]
L11RT ∂ ln P1 n1 F pn1 ∂n1
n2
|
∂ ln P2 Fp∂n1
n2
[ ][ L12RT
+
Fpxn1n2
xn1n2
|]
∂ ln P2 ∂n1
n2
) D°11Γ111 + D°12Γ112 D12 ) L11RT
)
|
∂ ln P1 Fp∂n2
+ L12RT
n1
|]
[ ][
L11RT ∂ ln P1 n1 F pn1 ∂n2
n1
|
∂ ln P2 Fp∂n2
n1
[ ][ L12RT
+
Fpxn1n2
xn1n2
|]
∂ ln P2 ∂n2
n1
) D°11Γ121 + D°12Γ122 D21 ) L21RT
)
|
∂ ln P1 Fp∂n1
+ L22RT
n2
[ ][ L21RT
Fpxn1n2
xn1n2
|
∂ ln P2 Fp∂n1
|]
∂ ln P1 ∂n1
n2
+
n2
[ ][
|]
L22RT ∂ ln P2 n2 Fpn2 ∂n1
n2
) D°21Γ211 + D°22Γ212 D22 ) L21RT
)
|
∂ ln P1 Fp∂n2
[ ][ L21RT
Fpxn1n2
+ L22RT
n1
xn1n2
|
∂ ln P2 Fp∂n2
|]
∂ ln P1 ∂n2
n1
+
n1
[ ][
|]
L22RT ∂ ln P2 n2 Fpn2 ∂n2
) D°21Γ221 + D°22Γ222
n1
(19)
(15) where
The value of Rij will be different for different pairs of molecules and, from the discussion above, will be zero for self-diffusion, i.e., for diffusion of identical molecules. The main-term Onsager coefficients can be written as proportional to loading as in purecomponent diffusion, i.e.,
Lii ) D°iiFpni/RT
D°ij ) RijxD°iiD°jj ) LijRT/(Fpxninj)
(12)
nA*LAA(nA,nA*) - nALA*A*(nA*,nA) (13) nA - nA*
(17)
Substituting eqs 16 and 17 into eq 15 gives
(11)
Thus, since both molecules are identical in their mobility, the cross-term diffusivities are zero. Ka¨rger37 has derived an identical result via a different path for the diffusion of indistinguishable molecules, obtaining
LAA* (nA,nA*) )
Lij ) D°ijFpf(ni,nj)/RT
D11 ) L11RT
Onsager referred to the ratio L/n as mobility, which is considered to be a constant. Then, for the identical molecules, with L11/n1 ) L22/n2, we obtain
L12 ) L21 ) 0
Here, D°ii is the corrected main-term diffusivity, or diffusivity in the Henry’s law limit, which is a constant at a fixed temperature. For a pure component, this is Darken’s relation, Di ) D0i Γi, where D0i ) LiRT/(Fpni) and Γi ) nid ln Pi/dni. D0i is the corrected diffusivity, which is constant, and Γ is the thermodynamic factor. We let the Onsager cross-term be
(16)
Γijk ) xnink ∂ ln Pk/∂nj
(20)
is a thermodynamic factor and will be negative for j * k for the common case of competitive adsorption. Also, by straightforward extension, for an m-component mixture, we obtain the simple general relation
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2145 m
Dij )
∑D°ikΓijk k)1
(21)
where D°ik and Γijk are given by eqs 18 and 20, respectively. From these expressions, the Fickian diffusivities are interpretable and predictable like the Maxwell-Stefan diffusivities. They can be considered as combinations of mobility effects and thermodynamic effects. Mobilities or corrected diffusivities are assumed to be constants and can be obtained from singlecomponent and binary experiments. They are easily applied to multicomponent systems to predict Fickian diffusivities for systems with more than two components. In this way, the Fickian description has the same advantages as the MaxwellStefan formulation in simple physical interpretations and easy prediction of the elements of [D]. It is notable that the Fickian diffusivity matrix [D] does not include n2 unknown variables as mentioned in the literature. Because there exists a relationship between Fickian diffusivities and Onsager coefficients, models based on Onsager coefficients and Fickian diffusivities have the same number of unknown variables. For a binary system, the number of unknown Fickian diffusivities is not four but three, which is the same as the number of Onsager coefficients or Maxwell-Stefan diffusivities. Equation 19 demonstrates that the cross-term Fickian diffusivities D12 and D21 are not equal even though the cross-term Onsager coefficient L12 and L21 are identical, and the corrected diffusivities D012 and D021 are equal. For example, if the crossterm Onsager coefficients are negligible, then for an ideal Langmuir isotherm, the above equation will reduce to a simple result which has been used frequently to predict diffusion in multicomponent systems.41,42
[D] )
[
D1 0 0 D2
[
]
1 - θ2 θ1 1 - θ1 - θ 2 1 - θ 1 - θ 2 θ2 1 - θ1 1 - θ1 - θ 2 1 - θ 1 - θ 2
]
(22)
Theory for Concentration-Swing Frequency Response We consider a gas mixture flowing through an adsorption bed. The mole fractions in the feed are subjected to a small sinusoidal perturbation of frequency ω and amplitude ∆yi,in. Both total inlet flow rate and pressure in the adsorption bed are constant. The concentration variation causes the gases to diffuse into or out of the adsorbent particles, where they adsorb and desorb, which in turn causes the mole fractions outside of the adsorbent and the flow rate out of the adsorption bed to change. The mole fractions in the effluent of the adsorption bed respond in a periodic sinusoidal manner with an amplitude ∆yi,out that depends on the feed concentration and mass transfer rates of the mixture. The amplitude ratio ∆yi,out/∆yi,in is used to extract mass transfer rates from the mathematical model. We consider a continuous flow system near an equilibrium state. The system is shown in Figure 1 and consists of the volume between the mass flow controllers and the mass spectrometer. The total system volume, shown in Figure 2, can be divided into three parts: the inlet volume V1, the adsorption bed void volume Vb, and the outlet volume V2. In our experiments, the outlet volume was very small and could be neglected. Only the inlet and the adsorption bed void volumes will be considered here. Gas is assumed to have spatially uniform mole fractions in the inlet region but to have a concentration distribution along the length of the adsorption bed because of mass transfer and axial dispersion. The system
behaves isothermally. Since the pressure is kept constant in the shallow bed, the total molar concentration is constant. Perturbations in composition are small (≈10%) to ensure that the system is appropriately linearized. Material Balance. For the inlet region, the material balance is simple as no adsorption takes place. The total molar flow rate of the mixture that enters the system is constant and the composition of each component is perturbed in a sinusoidal wave. Because of mixing, the composition out of the inlet region is generally different from the inlet composition and depends on the perturbing frequency. The total flow rate out of the inlet region is the same as the feed rate for this continuous system. The material balance for component i in the inlet region is
V1
dci,out,V1 dt
) Fin,V1ci,in,V1(t) - Fout,V1ci,out,V1(t)
(23)
where V1 is the volume of the inlet region, Fin,V1 is the constant total flow rate coming into the system, and Fout,V1 () Fin,V1) is the total flow rate out of the inlet region. The term ci,in,V1 is the molar concentration of component i coming into the inlet region, and ci,out,V1 is the molar concentration of component i leaving the inlet region. The concentrations can be expressed as a function of the mole fraction in each region by
ci,in )
P0 y ) c0yi,in RT i,in
ci,out )
P0 y ) c0yi,out RT i,out
(24)
where yi,in and yi,out are the mole fractions of component i in the inlet stream and outlet stream for a particular region. Introducing the parameter R1 ) Fin,V1/V1, the material balance can be written
dyi,out,V1 dt
) R1(yi,in,V1(t) - yi,out,V1(t))
(25)
For the adsorption region, the material balance for component i is
Fb
∂ci ∂(Vci) ∂ni ∂2ci + ′ + ) Dz 2 ∂t ∂t ∂z ∂z
(26)
where ′ is the total bed voidage (inside and outside particles), is the void fraction of packing, Fb is the bulk density of packing which equals (1 - )Fp, ci is the concentration of component i, Dz is a Fickian axial dispersion coefficient, and V is the interstitial fluid velocity. It should be noted that V is not constant but rather varies because components of the mixture can be adsorbed or desorbed as they pass through the adsorption region. Introducing eq 24 into eq 26 and simplifying, we obtain
∂ 2y i Fb ∂ni ′ ∂yi ∂(Vyi) + + ) Dz 2 c0 ∂t ∂t ∂z ∂z
(27)
For each region, the compositions satisfy ∑yi ) 1. Rate Equations. We adopt a general form of Fick’s law to represent the mixture adsorption rate processes. Mass transfer inside a particle is described by
∂ni ∂t
m
)
∇ ‚ (Dij∇nj) ∑ j)1
(28)
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Figure 1. Concentration-swing frequency response apparatus.
Otherwise, an external film resistance needs to be considered as given by
Fb
dni,ave ) kfa(ci - cs,i) dt
(30)
where kf is the external mass transfer coefficient, a is the specific surface area, and ci and cs,i are adsorbate concentrations in the bulk phase and at the fluid-pellet interface. Equilibrium. An adsorption equilibrium relation of the general form Figure 2. Concentration-swing frequency response volume.
n/i ) n/i (P1,P2,...,Pm)
where Dii is a main-term diffusivity and Dij (i * j) is a crossterm diffusivity. The diffusivities are functions of concentration, but they can be assumed to be constant for a small concentration variation. Since the FR method is a differential technique with a small amplitude perturbation near equilibrium, we have20
Dij(c1, ‚‚‚, cm) ≈ Dij(ce1, ‚‚‚, cem) CMS has a bidisperse pore structure with clearly distinguishable resistances for macropore and nanopore transport of adsorbates. The mass transfer is represented by a dual resistance modelsnanopore diffusion and a surface barrier.30 This combined resistance model was described in our previous paper on pure-component adsorption17 and applied successfully to a binary mixture on CMS using the pressure-swing frequency response method.18,22 It is of the form
∂ni ∂t m
m
)
∑ j)1
Dij ∂ rσ ∂r
( ) σ
∂r
∂nj
/ Dij - ni) ) kBi(ns,i ∑ ∂r j)1
∂ni ) 0 at r ) 0 ∂r
can be linearized for small variations of concentration and written in the form m
n/i ) ni,0 +
∑ j)1
m
Kij(Pj - P0j ) ) ni,0 +
K′ij(cj - c0j ) ∑ j)1
(32)
where K′ij ) ∂ni/∂cj, which for a pure component is the slope of the isotherm. With a constant total concentration, it is written in the form m
n/i ) ni,0 +
K′′ij(yj - y0j ) ∑ j)1
(33)
where K′′ij ) P∂ni/∂Pj. Analytical Solution of the Mathematical Model
∂nj
r
(31)
at r ) 1
(29)
where σ is the geometric factor for the particle (0 for slab, 1 for cylinder, and 2 for sphere), and kBi is the surface barrier resistance coefficient for component i. If the external mass-transfer resistance is negligible compared to the pore diffusion resistance, the adsorbate concentration in the bulk gas and at the interface can be assumed to be identical.
The mathematical model pertains to an m-component system. To obtain an analytical solution, the rate equations need to be solved first. Equation 29 contains m2 unknown diffusion coefficients that are coupled to one another by adsorbed-phase loadings. They can be decoupled into m individual equations and then solved as in the case of pure-component systems. We have provided the details in a previous paper.18 An eigenvalue method developed by Toor43 is adopted to reduce the linearized equations of diffusion in a multicomponent system to a set of equivalent single component diffusion equations. The linearized theory is exact for small changes in concentration. All of the equations for the two regions can be solved in the Laplace domain. Because the rate solutions for mixtures are complicated, especially for more than two components, we show a detailed solution only for a binary system.
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2147
xs/ζ coth(xs/ζ) - 1 σ ) 2 fk ) 3βk (41) (s/ζ)[xs/ζ coth xs/ζ + βk - 1]
We introduce the following deviation variables to simplify the model equations
y′i,in ) yi,in - yi,0
where ζ ) λk/R2 and βk ) RkBk/ζ. For a binary system, ψ/i is
y′i,out ) yi,out - yi,0 n′i ) ni - ni,0
n
ψ/k )
(Vyi)′ ) V0y′i + V′yi,0
(34) where
Then, taking the Laplace transforms of the deviation variables in the material balances for the inlet region, we obtain
syj1,out,V1 ) R1(yj1,in,V1 - yj1,out,V1) syj2,out,V1 ) R1(yj2,in,V1 - yj2,out,V1)
(35)
Thus, the mole fraction for component i coming into the adsorption bed is
[
[
]
1
d11 d21 d12 d22 ) λ2 - D22 D12
dikn/i ∑ i)1
λ1 - D11 D21 1
(42)
][ )
1
D12 λ1 - D22
D21 λ2 - D11
1
(43)
1 λ1,2 ) [D11 + D22 ( x(D11 - D22)2 + 4D12D21] 2
yji,out,V1 ) yji,in,V1/(s + R1) The material balances for the fixed-bed adsorber in the Laplace domain are
(44)
where the plus and minus signs correspond to the two eigenvalues λ1 and λ2. Equation 32 for a binary system can be written / nj s,1 ) K′11cs,1 + K′12cs,2 ) K′′11ys,1 + K′′12ys,2
∂yj1 ∂2yj1 Fb ′ ∂Vj snj1 + syj1 + V0 + y1,0 ) Dz 2 c0 ∂z ∂z ∂z Fb ∂yj2 ∂ yj2 ∂Vj ′ + y2,0 ) Dz 2 snj2 + syj2 + V0 c0 ∂z ∂z ∂z
]
/ njs,2 ) K′21cs,1 + K′22cs,2 ) K′′21ys,1 + K′′22ys,2
(45)
2
(36)
Finally, the adsorbed-phase transfer function can be obtained by substituting eqs 38, 42, and 45 into eq 37 to give
The adsorbed-phase concentration ni is solved for with the help of a quasi-adsorbed phase concentration ψk by Toor’s transformation.18 The relationship is expressed as
n1 ) Gψ1,n1ψ1 + Gψ2,n1ψ2 )
(
) (
)(
)
D11 - λ2 D11 - λ2 D11 - λ1 ψ1 + ψ2 λ 1 - λ2 λ 1 - λ2 D21
n2 ) Gψ1,n2ψ1 + Gψ2,n2ψ2 )
(
)(
) (
n1 ) G11ys,1 + G12ys,2 n2 ) G21ys,1 + G22ys,2 where
G11 ) C1f1 + C2f2 G12 ) C3f1 + C4f2
)
D22 - λ2 D11 - λ2 D11 - λ2 ψ1 + ψ (37) D12 λ1 - λ2 λ1 - λ2 2
G21 ) C5f1 + C6f2
where λk are the eigenvalues associated with the matrix of elements Dij, which satisfy |Dij - λiδij| ) 0. For the three different particle geometries, ψk is solved to obtain
ψk ) ψ /k fk(βk,λk)
(38)
tanh(xs/ζ) σ ) 0 f k ) βk xs/ζ[xs/ζ tanh(xs/ζ) + βk]
(39)
σ ) 1 fk ) 2βk
[
I1(xs/ζ)
xs/ζI0(xs/ζ) xs/ζ tanh
( ) ] I1(xs/ζ)
I0(xs/ζ)
G22 ) C7f1 + C8f2
(
(
C2 ) K′′11
) (
)(
)
D11 - λ2 D11 - λ2 λ1 - D11 + K′′21 λ 1 - λ2 λ1 - λ2 D21
C1 ) K′′11
)(
)(
)
D11 - λ2 D11 - λ1 λ2 - D22 + λ1 - λ2 D12 D21 D11 - λ2 D11 - λ1 K′′21 λ1 - λ2 D21
+ βk
(40)
(46)
(
C3 ) K′′12
) (
(
)(
)(
)
)
D11 - λ2 D11 - λ2 λ1 - D11 + K′′22 λ 1 - λ2 λ1 - λ2 D21
2148
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(
C4 ) K′′12
)(
(
C5 ) K′′11
)
)(
(
)
)(
[ [
(
(
)(
(
)(
) (
)(
)
)
D11 - λ2 λ2 - D22 D11 - λ2 + K′′21 λ1 - λ2 D12 λ1 - λ2
)(
)
C8 ) K′′12
(
)(
)(
) (
)(
)
)
D11 - λ2 λ2 - D22 D11 - λ2 + K′′22 λ1 - λ2 D12 λ1 - λ2
kfac0 (y1 - ys,1) Fb
kfac0 (y2 - ys,2) sn2 ) Fb
∂Vj/∂z is obtained by adding the above two equations to give
[
Dz
∂2yj1 2
- V0
∂yj1 ∂z
{
(
)
V0 + xV02 + 4ADz z + 2Dz
G′12 ) G′21 ) G′22 )
)
where A ) s{Fb/(c0)[y20(G11 - G12) + y10(G22 - G21)] + ′/}. The terms R and β are determined by the boundary conditions, which are
Vy1,out,V1 ) Vy1 - Dz(∂y1/∂z) at z ) 0
(53)
dy1 ) 0 at z ) L dz
(54)
These give,
R ) -Bβ 2 yj 1 - B + (1 + B)δ 1,out,V1
(55)
where
2
k′f G12 k′f + (G11 + G22)k′fs + (-G12G21 + G11G22)s
(
V0 - xV02 + 4ADz z (52) β exp 2Dz
β)
n2 ) G′21y1 + G′22y2
}
Solving this differential equation for yj1, we obtain
n1 ) G′11y1 + G′12y2
2
(50)
∂z Fb ′ [y (G - G12) + y10(G22 - G21)] + yj ) 0 (51) s c0 20 11 1
(47)
We have obtained the rate expression for nji in terms of the surface mole fractions yjs,i by eq 46. Substituting this equation into eq 47 and simplifying gives
G′11 )
]
Fb ∂Vj s(G11 - G12 + G21 - G22) yj1 ) ∂z c0
yj1 ) R exp
If no external mass transfer resistance exists, the adsorbate mole fractions in the bulk gas and at the surface of the adsorbent particles are the same. Otherwise, eq 30 is needed for the transfer rate between the bulk gas and the interface. In the Laplace domain it is
sn1 )
Fb ∂yj2 ∂2yj2 ′ ∂Vj s(G22 - G21) + s yj2 + V0 + y2,0 ) Dz 2 c0 ∂z ∂z ∂z (49)
Substituting eq 50 into eq 49 and simplifying gives
D11 - λ2 D22 - λ2 + λ1 - λ2 D12 D11 - λ2 λ1 - D11 D22 - λ2 K′′22 λ1 - λ2 D21 D12
(
] ]
Fb ∂yj1 ∂2yj1 ′ ∂Vj s(G11 - G12) + s yj1 + V0 + y1,0 ) Dz 2 c0 ∂z ∂z ∂z
)
D11 - λ2 D22 - λ2 + λ1 - λ2 D12 D11 - λ2 λ1 - D11 D22 - λ2 K′′21 λ1 - λ2 D21 D12
C6 ) K′′12 C7 ) K′′12
)(
D11 - λ2 D11 - λ1 λ2 - D22 + λ1 - λ2 D12 D21 D11 - λ2 D11 - λ1 K′′22 λ 1 - λ2 D21
B ) (2 - δ) exp[Pe(1 - δ)]/δ 2
δ ) x1 + 4ADz/V02, and Pe ) VL/Dz
k′f(-G12G21s + G11(k′f + G22s)) k′f2 + (G11 + G22)k′fs + (-G12G21 + G11G22)s2
The mole fraction of component 1 leaving the adsorption bed is
k′f2G21 k′f2 + (G11 + G22)k′fs + (-G12G21 + G11G22)s2 k′f(-G12G21s + G22(k′f + G11s)) k′f + (G11 + G22)k′fs + (-G12G21 + G11G22)s2 (48) 2
where k′f ) kfac0/Fb. Thus, eq 46 or eq 48 is the solution of the rate equation model described by eqs 29-32. Substituting this solution into eq 36 and combining with yj1 + yj2 ) 0 gives
yj1(L) )
4δ exp[Pe(1 - δ)/2] (1 + δ)2 - (1 - δ)2 exp(-Peδ)
yj1,out,V1 (56)
The transfer function for the adsorption region is
GT,Vb )
jy1(L) 4δ exp[Pe(1 - δ)/2] ) yj1,out,V1 [(1 + δ)2 - (1 - δ)2 exp(-Peδ)]
(57)
The transfer function for the total system including both the inlet and adsorption regions is
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2149
GT )
jy1(L) yj2(L) ) ) yj1,in,V1 yj2,in,V1 4F/V1δ exp[Pe(1 - δ)/2] (F/V1 + s)[(1 + δ)2 - (1 - δ)2 exp(-Peδ)]
(58)
Experiments The adsorbent used in this study was Takeda MSC-3K type 161. It is in the form of cylindrical pellets of particle size 1.182.80 mm (also reported as 7 × 14 mesh). The CMS has a mean pore size of 0.4 nm and is intended for pressure-swing and vacuum-swing adsorption separations of small molecules. CO2 and CH4 (99.6%) were obtained from Air Liquide. Pure-component equilibria for carbon dioxide and methane on Takeda CMS were measured gravimetrically using a Cahn D-200 microbalance. The adsorbent was placed in a sample pan suspended from the microbalance and was regenerated in situ at 393 K under vacuum using a heated nichrome wire coil in the glassware surrounding the sample. Additionally, a constant temperature was maintained during adsorption measurements using a temperature bath with circulating water. Adsorption equilibrium data were measured at 298 K. Pure-component kinetics were measured by the pressureswing FR method, which has been described in our previous paper.17 The adsorption bed was filled with 5.51 g of CMS, regenerated at 150 °C for 24 h with helium flowing. The pressure was perturbed in a sinusoidal wave over the range of 5 × 10-5-0.2 Hz around an equilibrium pressure of 1 atm. Small amplitude (< (5%) pressure variations were applied to satisfy the assumption of linearity. Measurements were carried out at 298 K with a constant flow rate of 1.5 sccm. Mixture kinetics were measured using the concentrationswing FR method. The schematic of the concentration-swing FR apparatus was shown in Figure 1, which is suitable for measuring mass transfer rates for mixtures from vacuum to high pressure. The mixture was generated by mixing two pure adsorbates; their flow rates change in a sinusoidal wave with the same amplitude but reversed in phase. The small modulation of flow rates was produced by mass flow controllers (MKS 1179). A carrier gas is optional and was not used. The resulting mixture had a desired constant total flow rate with a sinusoidal composition. After blending, the mixed gas passed through the adsorption bed. The effluent from the adsorption bed was sampled by a quadrupole mass spectrometer (MS) HP 5972 via a bypass pumped stainless capillary column with the remainder of the flow passing through a pressure controller (MKS 690) and then exiting to vacuum. The mixture experiments were conducted with different compositions of CO2 and CH4: 80% CO2 + 20% CH4, 50% CO2 + 50% CH4, and 20% CO2 + 80% CH4. The CMS sample (0.156 g) was regenerated at 150 °C for 24 h with helium flowing. The regenerated adsorbent was then equilibrated with the gas of the desired composition flowing at 1 atm. The system compositions were perturbed in a sinusoidal wave over the range from 1 × 10-4-0.5 Hz. Measurements were carried out at 298 K and 1 atm with a constant total flow rate of 1 sccm. System control and data acquisition were accomplished using a National Instruments 6052E multifunction board and a PCI 6711 analog output board installed in a Dell computer. Programs were written in LabVIEW 6i to automate operation of the equipment and analyze the raw data. For the MS, data acquisition and analysis were achieved using HP Chemstation G1701 AA software. The mole fraction profiles were obtained
Figure 3. Measured equilibrium isotherms of pure CO2 and CH4 on Takeda CMS and fits of the Langmuir and multisite-Langmuir models.
from the mass abundance data measured, and they were analyzed subsequently to extract amplitude information. Results and Discussion Equilibrium. Pure-component adsorption isotherms for CO2 and CH4 at 25 °C are shown in Figure 3 along with different model fits. First, we apply the Langmuir model in the form
ni )
nsibiPi 1 + biPi
(59)
where nsi is the saturation loading and b is the Langmuir constant. The fitting results, which are represented by dashed curves, demonstrate that the Langmuir model describes the isotherm of CH4 well but that is not the case for CO2. A variation of the Langmuir model has been developed for pure and mixed-gas components by Nitta et al.55 for localized monolayer adsorption. For a simple case of no adsorbateadsorbate interaction on a homogeneous surface, this multisiteLangmuir (MSL) isotherm is given by
biPi )
ni/nsi (1 - ni/nsi)ai
(60)
where ai is the number of sites occupied by one adsorbate molecule and nsi is the saturation concentration, which is related to the active site concentration n0 by nsi ) n0/ai. It reduces to the pure-component Langmuir isotherm when ai ) 1. Because the MSL model works well with the adsorption of hydrocarbons and carbon dioxide on activated carbon or CMS with ai ranging from 2 to 6,1 we adopted this MSL model to describe pure-component adsorption data. The isotherm parameters were obtained from the pure-component data by a nonlinear regression with the constraint that the values of ainsi were set equal to one another for both gases. The fitting results are represented as solid curves in Figure 3. It is clear that the MSL model shows an improvement over the Langmuir model for the description of CO2 adsorption, and for CH4, both model results are indistinguishable. Both the Langmuir and multisite-Langmuir model can be extended to describe mixture isotherms from pure-components parameters. For the Langmuir model, thermodynamic consistency is only satisfied when the monolayer capacities are the same for all of the components. Because the monolayer capacity
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Figure 4. Predicted binary isotherms for CO2 and CH4 on Takeda CMS using the multicomponent multisite-Langmuir model.
Figure 5. FR curves for CO2 and CH4 on CMS at 1 atm for three models.
obtained for CO2 is almost twice as much as that for CH4, the extended Langmuir method is not suitable to describe the binary equilibrium of CO2 and CH4 in CMS. For multicomponent equilibrium with all of the interaction energies between adsorbed molecules assumed to be zero, the MSL model is given by
ni/nsi
biPi )
(61)
n
(1 -
ni/nsi)a ∑ i)1
i
As for the pure-component case, for all adsorbates, the quantities ainsi are required to be equal to satisfy a balance on surface area or pore volume within the adsorbent. This model has the same form as the extended Langmuir model when ai ) 1. Furthermore, it has been proven to be thermodynamically consistent.56 Thus, it is an alternative for multicomponent equilibrium models even when the monolayer capacities of the components are different. We applied the multisite-Langmuir to predict the binary isotherms from eq 61 with the correlated pure-component adsorption parameters. The mixture isotherms are shown in Figure 4. An adsorption equilibrium relation in the general form of eq 31 can be linearized for small variations of concentration. For a binary system, the expression of K′′ij in eq 33 is determined from the related isotherm model. For the multisite-Langmuir isotherm used in this work, the K′′ijs have the form
K′′11 ) K′′21 ) K′′12 ) K′′22 )
n1,sθ1(1 - θ1 - θ2 + a2θ2) y10(1 - θ1 - θ2 + a1θ1 + a2θ2) - n2,sa2θ1θ2 y10(1 - θ1 - θ2 + a1θ1 + a2θ2) - n1,sa1θ1θ2 y20(1 - θ1 - θ2 + a1θ1 + a2θ2) n2,sθ2(1 - θ1 - θ2 + a1θ1) y20(1 - θ1 - θ2 + a1θ1 + a2θ2)
(62)
Pure-Component Kinetics. Mass transfer rates for pure components were investigated using the pressure-swing FR method, which has been described previously.17 The adsorption
Figure 6. Comparison of isotherm slopes for experimental data and two isotherm models.
bed in the system is subjected to a sinusoidal pressure perturbation of frequency ω and amplitude ∆P, and the flow rate out of the adsorption bed responds with a sinusoidal wave having the same frequency but a different amplitude ∆F. The amplitude ratio (∆F/∆P) is derived to extract mass transfer rates from models. The amplitude-ratio curves for pure CH4 and CO2 are shown in Figure 5. For CO2, the full spectrum of the kinetics is clearly shown. At the very low frequencies, the amplitude ratio data approaches the value [MbK + V/RT] asymptotically, which means that the system attains the full equilibrium capacity of CMS. With an increase of frequencies, the amplitude ratio data decrease, indicating that the amounts adsorbed and desorbed decrease because of the limited time for mass transfer. When the frequency becomes fast enough, the amplitude ratio data approach the inert gas response curve, corresponding to no adsorption. The capacity term K′′, i.e. the slope of the isotherm, can be calculated from the very low frequency data. We extracted K′′ from the experiments at three different pressures and compared them with the values predicted from the isotherm models. The comparison results are shown in Figure 6. The experimental values are close to the values calculated from the MSL model. This verifies the capability of measurement of isotherms using this pressure-swing FR method because we can construct an isotherm from the values of capacity terms at different pressures. This approach has also been demonstrated for nitrogen and
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2151 Table 1. Parameters for Three Different Models for Pure Components CO2 and CH4 in CMS MD model
LDF model
combined resistance model
adsorbate
D/R2 (1/s)
K
β
D/R2 (1/s)
CO2 CH4
4.5 × 10-4 1.4 × 10-6
8.7 × 10-3 2.7 × 10-5
795 3.2
4.8 × 10-4 5.5 × 10-6
oxygen in CMS in our previous paper.17 Moreover, the agreement between the experimental data and prediction from the MSL model indicates that the MSL model is more capable of describing the CO2 isotherm. The response curve for CH4 does not show that the system approaches equilibrium at very low frequencies. The lowest frequency measured was 1 × 10-5 Hz, i.e., 27.7 h for one cycle. The mass transfer rate of CH4 on CMS is very slow. We predicted the equilibrium asymptote calculated from K′′ using the MSL model in Figure 5. The frequency at which the system closely approaches equilibrium is estimated to be about 1 × 10-6 Hz whereas that for CO2 is about 1 × 10-4 Hz. The difference is almost 100 times. Thus, the mass transfer rate of CO2 is much faster than that of CH4. We applied three different models to investigate the mass transfer mechanisms for adsorbates on CMS. The correlation results are shown in Figure 5, and the extracted parameters from different models are listed in Table 1. The figure shows that the LDF model provides a much better description than the MD model for CH4, but for CO2, the MD model describes the response curves much better than the LDF model. This indicates that the primary transfer mechanisms for these two adsorbates are different. Furthermore, the results show that the combined resistance model provides an excellent description for both systems, which is reasonable because the LDF and MD models are the two limiting cases for the combined resistance model. The relative contribution of the MD resistance and the surface barrier resistance is given by the parameter β in the combined resistance model. For a value of β less than 1, the combined resistance model essentially reduces to the LDF model, and for β greater than 100, the combined resistance model very closely approaches the MD model.50 Our results show that the fitted β for CH4 is 3.2, indicating that the controlling mechanism for CH4 is mainly the surface barrier. In contrast, the fitted β for CO2 is 795, indicating that the controlling mechanism for CO2 is nanopore diffusion. If the diffusivities for both components obey Darken’s equation, Di ) D0i Γi, the corrected diffusivities (D0i /R2) can be obtained as 1.64 × 10-4 s-1 and 2.91 × 10-6 s-1 for CO2 and CH4, respectively. Additionally, the thermodynamic factors (Γ) are calculated using the multisite-Langmuir isotherm. Our results agree well with observations from Rutherford et al.29 They reported that CO2 uptake exhibits Fickian behavior and that CO2 transport in Takeda 3A CMS is governed by nanopore diffusion with diffusive constants D/R2 of 4.8 × 10-4 s-1 at 293 K. However, CH4 transport exhibited non-Fickian kinetics but obeyed a LDF model with a value of k of 3.2 × 10-5 s-1 at 323 K. Huang et al.30 showed that their differential uptake curves of CO2 and CH4 on CMS are best fitted by the dual resistance model, but not by either the pore model or the barrier model. Binary Mixture Kinetics. The binary mixture kinetics were measured by the concentration-swing FR method, where the feed composition was perturbed sinusoidally by 10% and the outlet compositions were measured using the MS. The FR curves for CO2 and CH4 mixtures at three different compositions are shown in Figure 7. CO2 (labeled 1) is the faster-diffusing component, whereas CH4 (labeled 2) is the slower-diffusing one.
Table 2. Parameters for Binary Mixture of CO2 and CH4 in CMS mixture corrected diffusivities, D0/R2 (1/s)
adsorbate CO2 CH4
10-4
10-5
1.7 × 1.62 × 10-5
1.62 × 2.95 × 10-6
β
Dz (m2/s)
795 1.3
2.3 × 10-5 2.3 × 10-5
The y-axis shows the amplitude modulations of the outlet composition of CO2. At low frequencies, the outlet modulations followed the inlet composition perturbation with a 10% amplitude. With an increase of frequency, the FR curves respond with less amplitude depending on the mass transfer rates of the mixture. A non-constant Fickian diffusivity model was applied here since this model has successfully described the experimental data over the entire range for N2 and O2 mixtures on CMS compared to a constant diffusivity model.18,22 The relationship for the dependence of the Fickian diffusivities and concentration has been derived in detail in the theory section. With the assumption of constant corrected mixture diffusivities, the mixture diffusivities can be simplified to
[D] )
[
D011Γ111 + D012Γ112 D011Γ121 + D012Γ122 D021Γ211 + D022Γ212 D021Γ221 + D022Γ222
]
(63)
For the multisite-Langmuir isotherm, the thermodynamic factors Γ, given by eq 20, are
Γ111 )
1 + (a1 - 1)θ1 - θ2 1 - θ1 - θ 2
x
ns2 a2 θ1θ2 x ns1 1 - θ1 - θ2
Γ112 )
ns1 a1 θ1 ns2 1 - θ1 - θ2
Γ121 ) Γ122 ) Γ211 )
( (
x x
) )
a2 ns1 1 θ1θ2 + x ns2 θ2 1 - θ 1 - θ 2 a1 ns2 1 θ1θ2 + x ns1 θ1 1 - θ 1 - θ 2 ns2 a2 θ2 ns1 1 - θ1 - θ2
Γ212 ) Γ221 )
x
Γ222 )
ns1 a1 θ1θ2 x ns2 1 - θ1 - θ2
1 + (a2 - 1)θ2 - θ1 1 - θ1 - θ 2
The model results with cross-term diffusivities are shown in Figure 7. The model provides a good description for all of the experimental data. Results are listed in Table 2. The main-term diffusivities are close to the corrected diffusivities calculated from pure-component results using Darken’s equation, as we expect. Cross-term diffusivities are similar in magnitude to the main-term diffusivities. The positive cross-term diffusivities indicates that the flux of a fast-diffusing component will be slowed down by the appearance of a slower-diffusing component in a counterdiffusion situation. The R in eq 15 is estimated by regression of D0ij and eq 18 to be 0.74 for this case, which
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Figure 7. Frequency response curves for mixture experiments and simulation results for nonconstant Fickian diffusivities at 1 atm and different compositions.
indicates that the cross-term Onsager coefficients can be described by L12 ) L21 ) 0.74xL11L22. Solely for comparison purposes, we use pure-component diffusivities to predict the mixture behavior. The main-term diffusivities are set equal to the corrected diffusivities obtained from pure-component experiments, and the cross-term diffusivities are assumed to be negligible. The prediction results are shown by the dashed lines in Figure 7. It is clear that this prediction overestimates the amount of CO2 diffusing into the adsorbent. Thus, the cross-term diffusivities are needed to accurately characterize the rate behavior, with CH4 retarding the diffusion of CO2. Conclusions A new concentration-swing FR method has been developed for performing measurements of adsorption rate parameters for mixtures. The transport behavior of CH4 and CO2 in CMS has been examined to demonstrate the method, which has the advantage of maintaining near isothermal conditions and linear behavior.19 The model considers a series of mass transfer resistances in the system, including axial dispersion, external mass transfer, a surface barrier at the pore mouth, and nanopore diffusion. Adopting an equation proposed by Yang et al.,38 a nonconstant multicomponent Fickian diffusivity model has been developed from irreversible thermodynamics. A simple general m relationship with the form Dij ) ∑k)1 D°ikΓijk, which considers the mixture Fickian diffusivities to be the product of corrected mixture diffusivities and the thermodynamic factor, has been derived to describe the concentration dependence of the diffusivities. With two further assumptions, (i) Lij ) Lji ) RijxLiiLjj and (ii) LiiRT ≈ Fpni, the corrected mixture diffusivities D°ik can be treated as constants at a fixed temperature, and the new Fickian model can be extended to predict multicomponent mixture kinetics with multicomponent equilibrium and the kinetic information of binary systems. We have shown that eq 6, Lij ) Lji ) xLiiLjj, is not robust as it cannot be applied to self-diffusion, i.e., to diffusion of identical molecules. Parameter Rij is introduced to represent the cross-interaction effect of different molecules, and thus, the value of Rij will be different for different pairs of molecules. Generally, the more dissimilar the molecules are, the larger Rij is. The term Rij will be zero for self-diffusion (i.e., for identical molecules). Because there exists
a relationship between Fickian diffusivities and Onsager coefficients, both Onsager coefficients and Fickian diffusivities have the same number of unknown variables, namely three for a binary system. Adsorption equilibrium and kinetics of pure CO2 and CH4 and their binary mixtures at three different compositions on Takeda CMS have been examined. The pure-component isotherms were measured using a gravimetric method. The multisite-Langmuir model provides a better description for CO2 adsorption on CMS than does the Langmuir model; for CH4, both models can represent experimental data well. Adsorption kinetics of pure components were measured by the pressureswing FR method. The transport mechanism of CO2 on CMS was found to be controlled by nanopore diffusion, but CH4 is mainly controlled by a surface barrier resistance. The ratio of the diffusivities of CO2 and CH4 is approximately 100 at atmospheric pressure for the Takeda CMS-3K type 161 under the experimental conditions. The new concentration-swing FR method was applied to investigate the binary mixtures for different compositions, which include the approximate compositions for landfill gas and gases encountered in tertiary oil recovery. Because the saturation loadings for pure components are quite different, the multicomponent multisite-Langmuir isotherm was used to describe the binary isotherm. The thermodynamic factor derived from the multisite-Langmuir isotherm was introduced to consider the concentration dependence of the mixture diffusivities. This nonconstant Fickian diffusivity model described mixture diffusion for all of the experimental data well. The results show that the main-term diffusivities are in very good agreement with the corrected diffusivities calculated from pure-component results using Darken’s equation. This method is capable of measuring rates for relatively fastdiffusing systems. It provides simple and fast determination of transport mechanisms by means of the theoretical analysis. The transfer behavior of mixtures can be studied independently without the need of a carrier gas. The technique can be extended to ternary or higher order multicomponent systems with no difficulties in technique because of the simultaneous detection of all components by a mass spectrometer. Acknowledgment We are grateful to the U.S. Army ECBC for the support of this research. Nomenclature b ) Langmuir constant, 1/atm c ) concentration, mol/m3 Dii ) main-term diffusivity, m2/s Dij ) cross-term diffusivity, m2/s D0 ) corrected diffusivity, m2/s Dz ) Fickian axial dispersion coefficient, m2/s Fin,Vi ) total gas mass flowrate coming into volume i, mol/s Fout,Vi ) total gas mass flowrate coming out of volume i, mol/s GT ) total transfer function for entire system Gn ) transfer function for adsorbed phase Ji ) flux of component i, mol/(m2s) kB ) barrier resistance coefficient, m/s kf ) external mass transfer coefficient, m/s K′′ ) local slope of isotherm P ∂n/∂Pi, mol/g Lij ) Onsager phenomenological coefficient n ) adsorbate concentration, mol/kg P ) pressure, bar
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ReceiVed for reView September 15, 2006 ReVised manuscript receiVed December 22, 2006 Accepted January 15, 2007 IE061214D
