Nanopore Diffusion Rates for Adsorption Determined by Pressure

Ind. Eng. Chem. Res. 2008, 47, 3121-3128

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Nanopore Diffusion Rates for Adsorption Determined by Pressure-Swing and Concentration-Swing Frequency Response and Comparison with Darken’s Equation Yu Wang and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt UniVersity, NashVille, Tennessee 37235

Two flow-through frequency response methods of concentration-swing frequency response (CSFR) and pressure-swing frequency response (PSFR) are applied to study the adsorption of chloroethane on BPL activated carbon over a wide range of concentrations. Chloroethane at low concentrations in helium, 500 ppmV and 10 mol %, are studied using CSFR. Two models which treat an adsorption bed with axial dispersion or a simplified well-mixed concentration have been developed for both isothermal and nonisothermal conditions. The results show that heat effects can be negligible for the CSFR method. Pure chloroethane at relatively high concentrations is studied using PSFR. The experimental data are described well by a nonisothermal nanopore diffusion model. The diffusivities obtained from these two distinct methods are identical for the same concentration. The modeling of the concentration behavior depends quite sensitively on the isotherm model and obeys Darken’s equation for the D-R isotherm. The results show that the frequency response method can distinguish the importance of heat effects and the relative importance of the mass-transfer mechanisms. Introduction Activated carbon is one of the most versatile adsorbents with high removal efficiency, low cost, and reusability. It is widely used in potable water and wastewater treatment and in filters for air purification and solvent recovery that range from large air pollution control devices to small respiratory filters.1 Release of chlorinated volatile organic compounds (VOCs), which are often suspected carcinogens, is an environmental issue. VOCs can be removed from air by adsorption on activated carbon. The design or selection of an adsorbent bed requires knowledge of the adsorption capacity and the associated breakthrough time. The frequency response (FR) method has been applied extensively to measure pure gas diffusion in solids.2-5 The major advantage of the technique is a unique capability of discriminating among different rate-limiting mechanisms because of the high sensitivity of the frequency response to the nature of governing transport resistances.4 The traditional approach to FR involves periodically varying the system volume and observing changes in system pressure within a closed system. Due to a very small perturbation introduced in the frequency response methods, most of studies have assumed isothermal conditions in the mathematical models. However, Sun et al.3,4,6 have found that heat effects cannot be neglected in the closed system. Flow-through frequency response methods have been explored to provide more practical advantages, such as easily implementing a wide range of frequencies, having the possibility of using large relative amplitudes, and having reduced heat effects.7-9 Boniface and Ruthven10 first developed a flowthrough system using chromatography based on a sinusoidally varying input 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. Harkness * To whom correspondence should be addressed. Tel.: (615) 3222441 Fax: (615) 343-7951. E-mail: [email protected].

et al.11 applied a similar approach but used a short tubular reactor, instead of a chromatographic column, and a mass spectrometer to analyze the effluent concentration. On the basis of both apparatuses, we have further developed a flow-through concentration-swing frequency response (CSFR) method and first applied it to study a mixed gas-solid system.12 A small, shallow adsorption bed is used and fed with a periodically modulated inlet concentration at a constant total flow rate. The system is different from the previous concentration variation systems and is mainly designed for measuring mixture diffusion. However, it can be conveniently applied for single gas measurement when one of the adsorbed gases is replaced with an inert gas such as helium. In this study, we show how to apply this flow-through FR method to measure the single gas mass transfer rate of chloroethane on BPL carbon. Other flow-through FR methods involve an inlet flow rate perturbation7-9 or pressure perturbation13-15 around the equilibrium state. The pressure-swing frequency response (PSFR) method has been successfully applied to study pure light gases, including N2, O2, CO2, and CH4, and their mixtures adsorbed on carbon molecular sieve systems using an isothermal model13-15 and CO2 on BPL activated carbon using a nonisothermal model.16 This method provides reliable and simultaneous measurement for both equilibrium and kinetics. The heat effect is shown to be system dependent and can complicate model analysis even for an open system. The aim of the present study is three-fold. The main purpose is to validate the applicability of the two different flow-through frequency response methods, CSFR and PSFR, by comparing mass transfer measurements at the same conditions. Second, we investigate the heat effects in both systems, especially for the CSFR method, which has only been treated isothermally before. A model that considers simultaneously diffusion mechanisms and heat effects is solved analytically to give the relative importance of different mass transfer resistances. Heat effects in both systems are examined to determine their importance. Last, we consider chloroethane (ethyl chloride), which is a representative VOC, adsorbed on activated carbon to study

10.1021/ie070673r CCC: $40.75 © 2008 American Chemical Society Published on Web 04/10/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008

dy1,out,V1 dt

Figure 1. Concentration-swing FR apparatus.

The FR method is one of the relaxation techniques in which a system is perturbed periodically around a steady state to measure the response of state variables. It has the advantage of clearly and unambiguously indicating the controlling transfer mechanism among different rate-limiting mechanisms through the shapes of response curves, which depend on the time scales of the dynamic processes affecting the state variables relative to the time scale of the perturbation, the transfer mechanisms, and the physical characteristics of the system.17 A linear system is maintained through the use of a small sinusoidal perturbation of amplitude ∆A. The response in the periodic steady state is also a sinusoidal wave of the same frequency, but with a different amplitude ∆A′. The amplitude ratio can be obtained simply from the input and output amplitudes. Depending on the different kinds of perturbing and responsive variables chosen, the FR methods differ and thus the mathematical models vary. Here, we validate our methods using models for both the PSFR and CSFR methods under isothermal and nonisothermal conditions. Concentration-Swing Frequency Response. For the CSFR method, both total inlet flow rate and pressure in the adsorption bed are constants, but the inlet concentration is a time-varying sinusoidal wave. Gases diffuse into and out of the adsorbent particles, which in turn causes the mole fractions outside of the adsorbent to change and respond sinusoidally. We have developed a model for mixture adsorption,12 which is complicated because of consideration of the diffusion and equilibrium interference for different molecules. Here, we only consider a simple case for a pure component diluted with helium gas. The model assumes that the helium gas has no effect on the nanopore diffusion and equilibrium capacities of the pure component. The system is shown in Figure 1 and consists of the volume between the mass flow controllers and the mass spectrometer. The total system can be divided into three parts: the inlet region, the adsorption region, and the outlet region. In our experiments, the outlet volume is very small and is considered negligible. Only the inlet and the adsorption region will be considered here. For the adsorption region, we consider both conditions of spatially uniform concentration and a length-varying concentration with axial dispersion. Both isothermal and nonisothermal conditions are considered in the models. For the inlet region, the material balance for the adsorbable component (component 1) is

F (y (t) - y1,out,V1(t)) V1 1,in,V1

(1)

where V1 is the volume of the inlet region, y1,in and y1,out are the mole fractions for component 1 in the inlet stream and outlet stream for the inlet region, and F is a constant total flow rate entering the system. Helium is component 2 and is treated as nonadsorbing. For the adsorption region, two cases are considered. First, the adsorption bed is treated simply as a shallow bed with a uniform concentration. The material balance for component 1 in the adsorption region can be written

dy1,out,Vb

equilibrium and kinetics by the two FR methods, using PSFR for pure chloroethane and CSFR for a dilute concentration of chloroethane in helium. Theory

)

dt

+

Mb dn1 F ) (y1,in,Vb(t) - y1,out,Vb(t)) Vbc0 dt Vb

(2)

where Vb is the void volume in the adsorption bed, and Mb is the adsorbent amount. The gas composition which enters the adsorption region is the same as that coming out of the inlet region, giving

y1,in,Vb(t) ) y1,out,V1(t)

(3)

The adsorbed-phase concentration n1 can be related to the gasphase concentration y1 in a general way in the Laplace domain giving

nj1 ) Gn yj1

(4)

Substituting eqs 3 and 4 into eq 2 and combining it with eq 1 gives the transfer function for the total system

GT )

yj1,out,Vb yj1,in,V1

)

(F/V1)(F/Vb) (s + F/V1)[s(1 + Mb/(Vbc0)Gn) + F/Vb]

(5)

The adsorbed-phase transfer function Gn depends only on the properties of the adsorbate, which depend on the mass transfer mechanisms within the particle. We have developed different models to describe different controlling mechanisms, including nanopore diffusion, a barrier resistance characterized by a linear driving force (LDF), a combined resistance model, and a kinetic distribution model.13 For adsorption kinetics in activated carbon, the mass transfer phenomenon is mostly controlled by nanopore diffusion. For an isothermal condition and spherical geometry, the real and imaginary parts of Gn for the nanopore diffusion model are given by13

Re[Gn(jω)] ) 3K Im[Gn(jω)] ) 6K

1 sinh ν - sin ν ν cosh ν - cos ν

1 ν sinh ν + sin ν -1 ν2 2 cosh ν - cos ν

[

]

(6)

where ν ) x2ωR2/D with R being the radius of the controlling domain for nanopore diffusion, and K is the local slope of the isotherm. For a nonisothermal condition and spherical geometry, the bed is assumed to be at a uniform temperature and the heat transfer resistance is between the bed and the bed wall. An energy balance can be written16

MbCs

dn dT + Mbλ ) Finhˆ in - Fouthˆ out - hA(T(t) - Twall) (7) dt dt

Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3123

where Cs is solid heat capacity, λ is isosteric heat of adsorption, hˆ is enthalpy, h is the heat transfer coefficient between the bed and the wall, and A is the area of the bed for heat transfer. Since the frequency response method makes only small perturbations to the system, the adsorption equilibrium can be considered linear about the equilibrium point, giving

Writing the equations in terms of deviation variables, substituting eq 4 into eq 13, and taking the Laplace transform gives

n* ) n*(P0,T0) + K(P - P0) - KT(T - T0)

Solving this equation with the boundary conditions given in eq 14 gives

(8)

where n* is the amount adsorbed at equilibrium at pressure P and temperature T, P0, and T0 are the equilibrium pressure and temperature about which the pressure is varied, and KT is the negative of the slope of the isobar. The energy balance can be further linearized and simplified in the Laplace domain giving

GE )

Mbλs T h )nj MbCss + R

(9)

where R ) FinCp + hˆ A. The adsorbed-phase transfer function Gn has been derived previously and is given as16

6K Π+M + Π-N ν2 M2 + N2 6K Π+N - Π-M Im[Gn] )- 2 ν M2 + N2

Dz

For a more realistic case, we consider an adsorption bed with axial dispersion. The material balance for the adsorbate is

(12)

where ′ is the total bed voidage (inside and outside particles),  is the void fraction of packing, Fb is the bulk density of packing, c1 is the concentration of component 1, Dz is the Fickian axial dispersion coefficient, and V is the interstitial fluid velocity. As the concentrations of the adsorbable component in the system are dilute, the amount adsorbed or desorbed will not change total flow rates appreciably, and thus V can be assumed constant for all of the experiments. Thus, the values of Fin,Vb and Fout,Vb are assumed to be equal, being dominated by the constant flow rate of helium. As the total concentration c0 is keep constant, using the mole fraction instead of the concentration and simplifying eq 12, we obtain

∂y1 Fb ∂n1 ∂2y1 ′ ∂y1 +V + ) Dz 2  ∂t ∂z c0 ∂t ∂z

(13)

The boundary conditions are

Vy1,out,V1 ) Vy1 - Dz(∂y1/∂z) at ∂y1/∂z ) 0

at z ) L

z)0

[(

)]

dyj1 ′ FGn yj ) 0 - s + dz  c0 1

4δ exp[Pe(1 - δ)/2] (1 + δ)2 - (1 - δ)2 exp(-Peδ)

(16)

yj1out,V1 (17)

GT )

jy1(L) ) yj1,in,V1 (F/V1 + s)[(1 + δ)2 - (1 - δ)2 exp(-Peδ)]

(18)

Pressure-Swing Frequency Response. For the pressureswing FR method, pressure in the adsorption bed is perturbed sinusoidally around an equilibrium state, and the outlet flow rate responds with a sinusoidal wave. The apparatus and mathematical model have been described in previous papers.13,16 The transfer function is given by

Π- ) MbCsωΨ- - RΨ+ D N ) MbCsω - β 2 Ψ- (11) R

∂(Vc1) ∂c1 ∂n1 ∂2c1 + + Fb ) Dz 2 ∂t ∂z ∂t ∂z

-V

where δ ) x1+4ADz/V2, A ) s[Fb/(c0)Gn + ′/], and Pe ) VL/Dz. Gn is given by eq 4. We have shown previously that the transfer function for the system can be obtained by combining eqs 1, 4, and 17 to give12

(10)

ν sinh ν + sin ν ν sinh ν - sin ν - 1 Ψ- ) 2 cosh ν - cos ν 2 cosh ν - cos ν

′

2

4F/V1δ exp[Pe(1 - δ)/2]

where β ) 3KT M bλ and

Π+ ) MbCsωΨ+ + RΨD M ) R - β 2 Ψ+ R

dz

yj1(L) )

Re[Gn] )

Ψ+ )

d2yj1

(14) (15)

GT )

F h ) ω x(MbRe[Gn(jω)] + V/RT)2 + (MbIm[Gn(jω)])2 P h (19)

in which V is the total system volume, and Gn is the adsorbedphase transfer function defined by nj ) GnP h . For the nanopore diffusion mechanism controlling, Gn has the same form shown in eq 6 or 10 depending on whether the system is described by isothermal or nonisothermal behavior. Experiments The adsorbent used in this study was BPL activated carbon (lot no. 4814-5) in 6 × 16 mesh form, supplied by Calgon Carbon Corp. It is predominantly nanoporous with a macropore structure. It has a BET specific surface area of ∼1200 m2/g and a mean pore width of about 1 nm.18 The bottled gases used to prepare feeds were chloroethane of 99.7+% purity (SigmaAldrich), 1000 ppmV chloroethane in helium (Air Liquide), and research grade helium (Air Liquide). The apparatus for PSFR and operating procedures have been discussed in a previous paper.13 Pure chloroethane entered into the system at a constant flow rate controlled by a mass flow controller (MKS 1179), and the system pressure was perturbed sinusoidally using a pressure controller (MKS 640) over the range of 2 × 10-5-0.2 Hz at different equilibrium pressures of 0.05, 0.1, 0.5, and 1 atm. The flow rate leaving the adsorption bed responded accordingly with a sinusoidal wave, which was measured by a mass flow meter (MKS 179). Small-amplitude pressure variations (