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Ind. Eng. Chem. Res. 2005, 44, 3692-3701
Investigation of Mixture Diffusion in Nanoporous Adsorbents via the Pressure-Swing Frequency Response Method. 1. Theoretical Treatment Yu Wang and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
A new model has been developed for investigation of mixture diffusion in nanoporous adsorbents by the pressure-swing frequency response method; that is, the total flow rate out of an adsorption bed is measured, while the system with a constant feed rate is subjected to a sinusoidal perturbation of pressure. The model takes into account both nanopore diffusion and a surface barrier resistance for three kinds of geometric microparticles: slab, cylinder, and sphere. To describe accurately the mole fraction variation caused by mixture diffusion, the adsorption bed is characterized by three spatially uniform regions: inlet, adsorption section, and outlet. The model is linearized, and analytical solutions for the whole system are derived by decoupling rate equations, which are connected through cross-term Fickian diffusivities. It is shown that, for the model binary system CO2 and C2H6 in 4A zeolite, the mole fraction changes slightly for the flow system investigated. At high frequencies, the difference between the mole fractions in the adsorption and outlet regions becomes significant. The total transfer function is affected by both diffusional and equilibrium interference. Introduction Experimental data and models for mixture diffusion in nanoporous adsorbents are important in adsorption technology, including for the design of rate-based separations. Because of the difficulties in measuring the uptake of each adsorbate, experimental techniques in this area are limited. They include the microscopic NMR, differential adsorption bed,1,2 zero-length column,3 frequency response (FR),4,5 and constant-volume techniques. Recently, the FR method has been shown to be very powerful in measurements of pure-component masstransfer rates, but it has very few applications for multicomponent systems,4-6 and the related analysis has not been fully developed. Yasuda et al.4 reported the first application of the batch FR method with volume modulation in the measurement of binary mixtures of methane, helium, and krypton in zeolites and showed that a small amount of helium would affect the diffusion of methane or krypton molecules in the binary gas mixture. A Fickian model was proposed in which no equilibrium interference (i.e., coupling) occurred between the two adsorbates. Diffusional interactions were accounted for by means of a dense matrix of diffusion coefficients. A transformation was adopted that requires a relation among the four Dij’s, and consequently the analysis is mathematically restrictive. The same method and model have been applied to a binary N2-O2 mixture on 4A zeolite.6 Shen and Rees5 used a similar apparatus but with a square-wave modulation of volume to study mixtures of benzene and two xylene isomers. They adopted Yasuda’s model, and the results showed that a film resistance was observed in the diffusion of the mixtures. Sun et al.8 developed a model of the batch FR * To whom correspondence should be addressed. Tel.: (615) 322-2441. Fax: (615) 343-7951. E-mail: m.douglas.levan@ vanderbilt.edu.
system for multicomponent adsorption in monodispersed sorbent materials. This model included both equilibrium and diffusional interactions between the adsorbed species and the effects of heat dissipation and surface barriers. An analytical solution for a linear system and zero-order moments of temperature and uptake was obtained. The binary data of Chen and Yang9 for CO2 and C2H6 in 4A zeolite were investigated theoretically. The analysis revealed that the contribution of the cross-term diffusivities is significant. Compared to the FR method for a batch system with a volume modulation, the flow FR methods have the advantage of maintaining a more isothermal condition. The FR methods in continuous-flow systems involve a periodic perturbation of the inlet gas concentration,10 inlet molar flow rate,11 or pressure of the system.12,13 Park et al.14 presented a theoretical analysis of the FR behavior of a multicomponent system for a continuousflow system with a periodic modulation of the inlet flow rate. Simulation results for a binary system showed that the FR of the faster-diffusing component is strongly influenced by the slower component. The out-of-phase transfer function for the faster-diffusing component shows a maximum and a minimum, and the deviation between these two points depends on diffusional and equilibrium interference. The objective of this paper is to present a theoretical analysis for mixture diffusion in a flow FR system with periodical modulation of pressure (pressure-swing FR). In previous work, we have developed both nonisothermal12 and isothermal13 models for the pressure-swing FR method and have applied those models to the analysis of experimental data. In the model developed here for a mixture, both equilibrium and diffusional interference effects are considered. Microparticle shape, nanopore diffusion, and a surface barrier are accounted for in the intraparticle transfer mechanism. The mole fraction is assumed to be spatially uniform in different domains of the system: inlet, adsorption, and outlet
10.1021/ie048936u CCC: $30.25 © 2005 American Chemical Society Published on Web 04/09/2005
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because the mole fraction out of the adsorption bed changes. The material balances are
V1 dPT y ) Fi,in(V1) - Fi,out(V1) RT i,V1 dt
i ) 1, ..., m (3)
dyi,V2 V2 V2 dPT PT + y ) Fi,in(V2) - Fi,out(V2) RT dt RT i,V2 dt i ) 1, ..., m (4) Figure 1. Pressure-swing FR volume.
regions. For each domain, the mole fraction may change because of effects of the pressure modulation on adsorption. This is the first theoretical treatment of the pressure-swing FR method for mixture diffusion in adsorbent particles. Mathematical Model We consider a mixed gas of m components with constant mole fractions flowing into an adsorption bed, which is subjected to a small sinusoidal pressure perturbation of frequency ω and amplitude ∆P. The pressure 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 total flow rate responds in a periodic sinusoidal manner with a different amplitude ∆F depending on the input concentration and mass-transfer rates of the mixture. The system is shown in Figure 1. The pressurecontrolled domain includes the total volume from the mass flow controllers to the mass flowmeter. This volume can be divided into three parts: the inlet volume V1, the adsorption bed volume Vb, and the outlet volume V2. Gas is assumed to have a spatially uniform mole fraction in each region, and the adsorbent has a single particle size. The system is justifiably assumed to be isothermal.13 Perturbations in pressure are small ( 0, the diffusion rate for each component will increase compared to that of the pure component, so the total amplitude ratio curve will shift to the right, showing a faster diffusion rate for the total system; otherwise, for Dij < 0, the diffusion rate for each component decreases compared to the pure component, and the total amplitude ratio curve will shift to the left, yielding a slower diffusion rate compared to the system with zero cross-term diffusivities. If the system has one positive and one negative diffusivity, the total amplitude ratio curve may shift either way depending on which component is the more dominant. The behavior of the transfer functions for mole fractions, Gy, in the adsorption and outlet regions is shown in Figure 4. At low frequencies, the amplitude ratio curve in the adsorption region is similar to that in the outlet region. With increasing frequency, the deviation of the curves in the two regions increases significantly. If the pressure perturbation is 5% of the initial pressure of 15 kPa, the mole fraction change at 1 Hz is 0.008 in the adsorption region while that in the outlet region is only 1.6 × 10-5, i.e., 500 times less. Thus, the mole fraction variation in the adsorption region is attenuated in the outlet volume, giving a small change in the mole fraction at the outlet. The variation of the mole fractions in the two regions can be predicted theoretically using an expression obtained from eqs 30 and 31 that gives the relationship of Gy in two regions as
1 Gy1,V2 ) G 1 + P0V2/(F0RT)s y1,Vb From this equation, it is clear how the ratio of Gy is affected by the system variables and the frequency. For large frequency ω, outlet volume V2, initial pressure P0, or small initial inlet flow rate F0, the magnitude difference of the Gy’s will be great. If we allow the mole fraction in the outlet region to have a gradient (i.e., as described by a series of wellmixed cells) instead of being spatially uniform, the transfer function for the mole fraction in the outlet
Figure 6. Amplitude ratio and phase angle curves of the partial and total flow rates for the reference case. Short dashed curve: CO2. Long dashed curve: C2H6. Solid curve: total.
region will be given by
Gy1,∆V )
y1,∆V 1 G ) PT 1 + P0∆V/(F0RT)s y1,Vb
Gy1,(i+1)∆V )
y1,(i+1)∆V PT
)
∆V ) V2/k
1 G 1 + P0∆V/(F0RT)s y1,i∆V i ) 1, ..., k
The transfer function related to the mole fraction leaving the system would be
Gy1,V2 )
[
]
1 1 ‚‚‚ ‚‚‚ Gy1,Vb 1 + P0∆V/(F0RT)s 1 + P0∆V/(F0RT)s
)
1 Gy ,V [1 + P0∆V/(F0RT)s]k 1 b
≈
1 G 1 + P0V2/(F0RT)s y1,Vb
This shows that the mole fraction leaving the system is almost the same no matter whether the mole fraction in the outlet region is treated as uniform or not. Figure 6 shows the response curves of both the partial and total transfer functions for the reference parameters
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diffusion of adsorbates. Some similar observations were found in previous work;8 the FR of CO2 exhibits an additional maximum of smaller magnitude at lower frequencies. Conclusions
Figure 7. Amplitude ratio and phase angle curves of the partial and total flow rates with zero cross-term diffusivities.
and an inlet flow rate F0 of 1 × 10-4 mol/s. It shows that the amplitude curves of the slower-diffusing component decrease monotonically as the total amplitude ratio curve decreases. However, the amplitude curve of the fast-diffusing component behaves abnormally; the curve overshoots the value of the equilibrium state. This “roll-up” phenomenon has been shown previously by Sun et al.8 and Park et al.14 in their simulation results. They used normalized in-phase and out-of-phase functions to describe the system and suggested that the “roll up” (inphase greater than unity and out-of-phase less than zero) of the CO2 partial-pressure FR curves is caused by diffusional interference because the overshoot behavior of the faster-diffusing component disappears when the cross-term diffusivities are eliminated (D12 ) D21 ) 0), which is also shown in Figure 7. This “rollup” phenomenon does not occur for all input flow rates for our simulated flow system. The effects of the equilibrium interference can be seen by comparing Figure 6 with Figure 5, which was obtained by assuming K12 ) K21 ) 0, i.e., uncoupled adsorption equilibrium. It is noticeable that the overshoot behavior of the fast-diffusing component occurs in this figure. This further confirms that the “roll-up” phenomenon is caused by diffusional interference and not by equilibrium interference. The amplitude ratio curves in Figure 5 are much higher than those in Figure 6 for the frequency range where adsorption can occur. Especially for the faster-diffusing component at very low frequencies, the value of the amplitude ratio curve approaches 0.5, which is almost triple the value in Figure 6 where equilibrium interference was considered. This can be explained by competitive adsorption for the mixture system. The isotherm of one adsorbate is affected by the presence of another adsorbate. For our system, it can be described by
nCO2 ) K11PCO2 + K12PC2H6 When equilibrium interference (K12 ) 0) is neglected, the adsorbed amount change of nCO2 should be larger than that calculated with typical equilibrium interference (K12 < 0). This leads to a large flow rate change. Accordingly, the partial amplitude ratio curve, ∆FCO2/ ∆PCO2, has a larger value at low frequencies compared with the case in which equilibrium interference is considered. A similar behavior occurs for the other component, C2H6. This indicates that equilibrium interference is a very important consideration in mixture
Owing to its high sensitivity to recognize the masstransfer mechanism, the FR method offers the potential to determine adsorption rates not only for pure components but also for mixtures. We have developed a new mathematical model for mixture diffusion measurements for an isothermal pressure-swing flow-through system. Both diffusional and equilibrium interferences are included in this model. Compared with other models proposed for the FR method, this model considered both nanopore diffusion and a surface barrier for the intrapartical transport mechanism for different geometric nanoporous regions: slab, cylinder, and sphere. Furthermore, the flow-through system is characterized by three spatially uniform regions: inlet, adsorption, and outlet. Each region allows individual mole fractions to change relative to other regions. For mixture sorption rates, there can exist significant kinetic interferences between components during their simultaneous transport. Therefore, it is necessary to account for the cross-term diffusivities, and it is relatively complicated to evaluate the resulting dense diffusivity matrix. In this work, an eigenvalue method developed by Toor15 has been adopted to decouple the rate equations. Analytical solutions have been derived for a binary system. The overall method provides fast and accurate determination of the diffusivity matrix. A model system has been tested, and the simulation verified some previous results. The fast-diffusing component is strongly influenced by the slower-diffusing component and overshoots general trends in some situations. This roll-up phenomenon for the fast-diffusing component is caused by diffusional interference. Moreover, the simulation results have shown that the mole fraction changes only slightly for the flow system with a small perturbation. At low frequencies, the mole fraction changes are almost the same for different regions of the system, but at higher frequencies, the mole fraction shows more of a distribution among the system components. This mole fraction variation also depends on the system variables of the initial pressure, initial flow rate, and outlet volume. The total transfer function shifts toward lower frequency when the crossterm diffusivities change from positive to negative. Equilibrium interference affects all but the highest frequencies and is especially significant at low frequencies, where the system is close to equilibrium. Acknowledgment We are grateful to the U.S. Army ECBC for the support of this research. Appendix The method of solution involves the Laplace transform method, which converts the set of partial differential equations into the set of ordinary differential equations presented in the main body of the paper. We can write this linear system in matrix form as [A][x] ) [b]. The solutions can be obtained by matrix inversion:
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[x] ) [A]-1[b] The solution to eqs 24-26 in the Laplace domain is
Inlet region V1 F1,V1 ) - y10P hT RT V1 F2,V1 ) -s y20P hT RT V1 h FT,V1 ) -s P RT T Adsorption region y1,Vb ) [MbRTs(-G12y202 + y10(-G11 + G22 + (G11 + G21 - G22)y10))]/[F0RT + hT P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10))]P y2,Vb ) [MbRTs(G12y202 - y10(-G11 + G22 + (G11 + G21 - G22)y10))]/[F0RT + hT P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10))]P F1,Vb ) P h T[s(F0RT(-G12MbRTy20 - (G11MbRT + V1 + Vb)y10) - P0sy10((G11MbRT + Vb)(G22MbRT + V1 + Vb) - (G11 + G21 - G22)MbRTV1y10 - G12MbRT (G21MbRT + V1y20)))]/(RT(F0RT + P0s(Vb + MbRT (G11y20 - G12 + (G12 - G21 + G22)y10)))) F2,Vb ) P h T[s(-F0RT(G22MbRT + V1 + Vb) + F0RT((-G21 + G22)MbRT + V1 + Vb)y10 P0sy20((G11MbRT + Vb)(G22MbRT + V1 + Vb) (G11 + G21 - G22)MbRTV1y10 G12MbRT(G21MbRT + V1y20)))]/(RT(F0RT + P0s(Vb + MbRT(G11y20 - G12 + (G12 - G21 + G22)y10)))) P1,Vb ) [-G12MbP0RTsy20 + (F0RT + P0s(G22MbRT + Vb))y10]/[F0RT + P0s(Vb + MbRT(G11 - G12 - G11y10 + hT (G12 - G21 + G22)y10))]P P2,Vb ) [F0(RT - RTy10) + P0s(G11MbRT + Vb ((G11 + G21)MbRT + Vb)y10)]/[F0RT + P0s(Vb + MbRT(G11 - G12 - G11y10 + (G12 - G21 + G22)y10))]P hT n j 1 ) [F0RT(G12y20 + G11y10) + P0s((G11G22 G12G21)MbRTy10 + Vb(G11y10 + G12y20))]/[F0RT + P0s(Vb + MbRT(G11 - G12 - G11y10 + (G12 - G21 + G22)y10))]P hT n j 2 ) [F0RT(G22y20 + G21y10) + P0s((G11G22 G12G21)MbRTy20 + Vb(G22y20 + G21y10))]/[F0RT + P0s(Vb + MbRT(G11y20 - G12 + (G12 - G21 + G22)y10))]P hT
Outlet region y1,V2 ) [-F0Mb(RT) 2s(G12y202 - y10(-G11 + G22 + h T]/[(F0RT + P0sV2)(F0RT + (G11 + G21 - G22)y10))P P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10)))] y2,V2 ) [F0Mb(RT) 2s(G12y202 - y10(-G11 + G22 + (G11 + G21 - G22)y10))P h T]/[(F0RT + P0sV2)(F0RT + P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10)))] F1,V2 ) (s(-F02(G12Mb(RT) 3 - (F02(RT) 2((G11 G12)MbRT + V1 + V2 + Vb) + P02 s2V2(-G12MbRT (G21MbRT + V1 + V2) + (G11MbRT + Vb)(G22MbRT + V1 + V2 + Vb)) + F0P0RTs(-G12MbRT(G21MbRT + V1) + (G11MbRT + V2 + Vb)(G22MbRT + V1 + V2 + Vb)))y10 + (G11 - G12 + G21 - G22)MbP0RTs h T/(RT(F0RT + (F0RTV1 + P0sV2(V1 + V2))y102))P P0sV2)(F0RT + P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10))))
F2,V2 ) (s(-F02(RT) 2(G22MbRT + V1 + V2 + Vb) + F02(RT) 2((-G21 + G22)MbRT + V1 + V2 + Vb)y10 P02s2V2y20(-G12MbRT(G21MbRT + V1 + V2) + (G11MbRT + Vb)(G22MbRT + V1 + V2 + Vb) + G12MbRT(V1 + V2)y10 - (G11 + G21 - G22)MbRT (V1 + V2)y10) - F0P0RTsy20((G11MbRT + V2 + Vb) (G22MbRT + V1 + V2 + Vb) - (G11 + G21 h T/ G22)MbRTV1y10 - G12MbRT(G21MbRT + V1y20))))P (RT(F0RT + P0sV2)(F0RT + P0s(Vb + MbRT((G11 G12)y20 + (G22 - G21)y10)))) FT,V2 ) -(s(P0s(-G12MbRT(G21MbRT + V1 + V2) + (G11MbRT + Vb)(G22MbRT + V1 + V2 + Vb) + G12MbRT(V1 + V2)y10 - (G11 + G21 - G22)MbRT (V1 + V2)y10) + F0RT(V1 + V2 + Vb + MbRT(G12 + h T/(RT(F0RT + G22 + (G11 - G12 + G21 - G22)y10))))P P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10)))) P1,V2 ) (F02(RT) 2y10 + F0P0RTs(-G12MbRTy20 + (G22MbRT + V2 + Vb)y10) + P02s2V2y10(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10)))P h T/((F0RT + P0sV2)(F0RT + P0s(Vb + MbRT((G11 - G12)y20 + (G22 - G21)y10)))) P2,V2 ) (F02(RT) 2y20 + F0P0RTs(G11MbRT + V2 + Vb - ((G11 + G21)MbRT + V2 + Vb)y10) - P02s2V2y20 (-Vb + MbRT(G12 - G11y20 - (G12 - G21 + h T/((F0RT + P0sV2)(F0RT + P0s(Vb + G22)y10)))P MbRT((G11 - G12)y20 + (G22 - G21)y10)))) Notation b ) Langmuir constant, bar-1 Dii ) main-term diffusivity of i, m2/s
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3701 Dij ) cross-term diffusivity of i, m2/s Fi,in ) gas mass flow rate of i into the system region, mol/s Fi,out ) gas mass flow rate of i out of the system region, mol/s GT ) transfer function for the entire system Gn ) transfer function for the adsorbed phase kB ) barrier resistance coefficient, m/s K ) local slope of isotherm, mol/(g bar) Mb ) amount of the adsorbent, g n ) adsorbate concentration, mol/kg P ) pressure, bar R h µ ) microparticle half-width, m R ) ideal gas constant, J/(mol K) T ) temperature, K Ve ) volume of the chamber excluding the adsorbent, m3 Vs ) bulk volume occupied by the adsorbent, m3 Greek Letters h µ2) β)R h µkB/(D/R ) void fraction of the adsorption bed or heat of adsorption γ ) λk/R h µ2 χ ) porosity of the adsorbent λ ) eigenvalue or blocking diameter θ ) fractional coverage of adsorption sites σ ) geometric factor for the particle ω ) frequency, Hz Superscripts ′ ) deviation variable * ) equilibrium state - ) Laplace domain Subscripts 0 ) initial state or mean value s ) saturation loading
Literature Cited (1) Yang, R. T.; Chen, Y. D.; Yeh, Y. T. Prediction of cross-term coefficients in binary diffusion: diffusion in zeolite. Chem. Eng. Sci. 1991, 46, 3089-3099. (2) Chen, Y. D.; Yang, R. T.; Uawithya, P. Diffusion of oxygen, nitrogen and their mixtures in carbon molecular sieve. AIChE J. 1994, 40, 577-585. (3) Brandani, S.; Jama, M.; Ruthven, D. Self-diffusion and counter-diffusion of benzene and p-xylene in silicalite. Microporous Mesoporous Mater. 2000, 35-36, 283-300.
(4) Yasuda, Y.; Yamada, Y.; Matsuua, I. New developments in zeolite science and technology; Elsevier: New York, 1986. (5) Shen, D.; Rees, L. V. C. Frequency response study of mixture diffusion of benzene and xylene isomers in silicalites1. In Proceedings of the International Zeolite Symposium; Bonneviot, L., Kaliaguine, S., Eds.; Elsevier Science: Quebec, Canada, 1995; pp 235-242. (6) Yasuda, Y.; Matsumoto, K. Straight and cross term diffusion coefficients of a two component mixture in micropores of zeolites by frequency response method. J. Phys. Chem. 1989, 93, 31953200. (7) Jordi, R. G. Batch frequency response techniques in gasphase adsorption applications. Thesis, Department of Chemical Engineering, University of Queensland, Queensland, Australia, 2001. (8) Sun, L. M.; Zhong, G. M.; Gray, P. G.; Meunier, F. Frequency response analysis for multicomponent diffusion is adsorbents. J. Chem. Soc., Faraday Trans. 1994, 90, 369-376. (9) Chen, Y. D.; Yang, R. T. Predicting binary fickian diffusivities from pure-component fickian diffusivities for surface diffusion. Chem. Eng. Sci. 1992, 47, 3895-3905. (10) Boniface, H. A.; Ruthven, D. M. Chromatographic adsorption with sinusoidal input. Chem. Eng. Sci. 1985, 40, 2053-2061. (11) Park, I. S.; Petkovska, M.; Do, D. D. Frequency response of an adsorber with modulation of the inlet molar flow-ratesii. a continuous flow adsorber. Chem. Eng. Sci. 1998, 53, 833-843. (12) Sward, B. K.; LeVan, M. D. Frequency response method for measuring mass transfer rates in adsorbents via pressure modulation. Adsorption 2003, 9, 37-54. (13) Wang, Y.; Sward, B. K.; LeVan, M. D. New frequency response method for measuring adsorption rates via pressure modulation: Application to oxygen and nitrogen in a carbon molecular sieve. Ind. Eng. Chem. Res. 2003, 42, 4213-4222. (14) Park, I. S.; Kwak, C.; Hwang, Y. G. Frequency response of continuous-flow adsorber for multicomponent system. Korean J. Chem. Eng. 2000, 17, 704-711. (15) Toor, H. L. Solution of the linearized equations of multicomponent mass transfer: I. AIChE J. 1964, 10, 448-460. (16) Gerla, P. E.; Rubiolo, A. C. A model for determination of multicomponent diffusion coefficients in foods. J. Food Eng. 2003, 56, 401-410. (17) Marutovsky, R. M.; Blow, M. Calculation of the straight and cross diffusion coefficients of a two-component mixture in a microporous zeolitic structure. Zeolites 1987, 7, 111-114.
Received for review November 2, 2004 Accepted February 28, 2004 IE048936U
