On the Driving Force of Methanol Pervaporation through a

potential gradient of the permeating component is an appropriate driving force to describe the ... chemical potential gradient is a proper driving for...
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Ind. Eng. Chem. Res. 2007, 46, 4091-4099

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On the Driving Force of Methanol Pervaporation through a Microporous Methylated Silica Membrane Frans de Bruijn, Joachim Gross, Z ˇ arko Olujic´ , and Peter Jansens Department of Process and Energy, Delft UniVersity of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands

Freek Kapteijn* Catalysis Engineering, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

The pervaporation transport of methanol through an amorphous microporous methylated silica membrane was studied experimentally and analyzed through modeling within a Maxwell-Stefan (MS) framework. The experimental conditions cover a temperature range of 60-155 °C and absolute pressures up to 16 bar. Exerting higher absolute pressures on the liquid feed-side of the membrane did not lead to enhanced fluxes, confirming that (i) the selective layer of the 56.8 cm2 membrane was a closed layer and defect-free, and (ii) the chemical potential gradient of the permeating component is an appropriate driving force to describe the transport through the microporous membrane. Both the adsorption isotherm and the heat of adsorption (-∆Hi,ads) were determined, and a two-site Langmuir isotherm adequately correlated the heterogeneous adsorption behavior of methanol in amorphous silica. Different levels of detail were adopted for modeling the diffusion transport through the selective layer, after having accounted for the resistance of the different support layers. Two models based on the MS approach described best the data: one model had a constant diffusivity and the other model had a loading-dependent diffusivity. The latter is equivalent to the classical pervaporation model, where the flux is proportional to the fugacity difference over the membrane and an exponential temperature dependency of the permeance. The presented derivation eliminates the inconsistencies of earlier interpretations given in the literature. Although there is a slight preference for the latter, easy-to-use model, no further statistical discrimination could be made based on the data. 1. Introduction The modeling of transmembrane mass transport is only valuable for the purpose of design if the model not only correlates experimental measurements, but also exhibits robustness toward extrapolation outside the conditions where model parameters were obtained. There is a broad consensus that the chemical potential gradient is a proper driving force to describe pervaporation using microporous ceramic membranes.1-3 Nevertheless, models that use the partial pressure difference instead of the chemical potential gradient as driving force are also applied. Wijmans and Baker4 proposed a well-known model for the description of pervaporation through polymeric membranes. This model was successfully applied to the description of solvent dehydration and organic-organic separations using ultrathin silica membranes,5,6 and for the transport of pure components.7 From experimental data, a pre-exponential factor for permeance and an apparent activation energy are obtained. The advantage of this modeling approach is its simplicity, compared to models based on the Maxwell-Stefan (MS) or Onsager formulations for mass transport. For the latter two approaches, detailed information regarding the material properties and the adsorption behavior of the components in the material must be known.8 One assumption underlying the use of partial pressure differences as the driving force is that the adsorption of the components from the bulk liquid into the membrane can be described by Henry adsorption.5 This also applies for pure component pervaporation through a silica membrane.7 Bettens et al.7 excluded any contribution of convective transport, because * To whom correspondence should be addressed. Tel.: +31 15 278 6725. Fax: +31 15 278 5006. E-mail address: [email protected].

their experimental data did not show any correlation with the viscosities. Bowen et al.3 assumed saturated monolayer coverage at the feed side of a zeolite membrane during pervaporation. The authors further stated that the application of modeling as proposed by Wijmans and Baker4 is accurate if the (Fickian) diffusion coefficients are independent of concentration and the isotherms are linear. Also, Marx2 assumed the complete selective layer of a zeolite membrane to be saturated with methanol in the modeling of pervaporation using Onsager equations. Under saturation conditions, however, the assumption of a linear adsorption isotherm is questionable. The adsorption of a component as a function of its partial pressure (or, as referenced hereafter, its fugacity) is usually described by a (e.g., Langmuir-type) strongly nonlinear isotherm that shows linearity at low fugacities and the approach of saturation at high fugacities. The permeate side of the selective layer is usually maintained under low-pressure conditions, resulting in a relatively low fugacity of the permeating species; therefore, it can be expected that the loading at the permeate side is much lower than that at the feed side. This implies a loading profile over the selective layer, so that both the assumptions of linear adsorption (the Henry regime) and the assumption of full saturation in the entire selective layer do not apply. Therefore, the existing literature does not really shed light on these incompatibilities between model assumptions and experimental observations in pervaporation. Furthermore, many authors have assumed that, for the driving force for transport, the saturation pressure at the (liquid) feed side of the permeating component should be used, which is equivalent to the use of the chemical potential of that component. However, the implication that the absolute feed pressure should have no role has never been tested.

10.1021/ie0610445 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/13/2006

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The objective of this paper is to discuss the interpretation of models for the pervaporation of methanol through a supported tubular methylated silica membrane, based on permeation data at different temperatures and absolute pressures. From the comparison of four different models based on different assumptions and simplifications for the driving force, a recommendation will be given for modeling pervaporation. 2. Theory Adsorption. Methylated silica is assumed to have two distinct adsorption sites for the adsorption of CO2 on silica: the silanol groups and the siloxane bridges.9 The dual-site Langmuir adsorption isotherm for this case is given by

qi )

bi,1fi bi,2fi + qsat,2 i 1 + bi,1fi 1 + bi,2fi

qsat,1 i

Ji ) -Fsilicaqsat i θi,av (1)

where the numbers in the subscripts and superscripts refer to the two distinct adsorption sites. The temperature-dependent adsorption parameter bi (expressed in units of Pa-1) is expressed as

bi ) bi,0 exp

(

)

-∆Hi,ads RT

θi )

qi,psat

)

qi

(3)

qsat i

Note that qsat i is not the sum of the saturation loadings of the correlation that is described by eq 1. Selective Layer TransportsMaxwell-Stefan Formulation (Model 1). Applying the Maxwell-Stefan (MS) approach to mass transport, the model can be described by10

-

n ci θjJi - θiJj Ji + (4) ∇T,pµi - V h i∇Pabs ) RT RT }ij }i j)1,j*1

ci

∑

The second term with the absolute pressure gradient reflects the influence of the absolute pressure on the driving force rather than viscous flow contribution. In this work, absolute pressure effects are calculated within the chemical potentials via the Poynting factor, as shown later. In the aforementioned discussion, it is assumed that, for a defect-free membrane, a viscous flow contribution that is due to an absolute pressure gradient is absent. Through replacement of the concentration ci by the product of the porosity (), density (Fsilica), and the loading of component i (θiqsat i ), one can rewrite eq 4 for pure components as

( )

Ji ) -Fsilicaqsat i θi}i

∇µi RT

(5)

In application of eq 5, one assumes that the diffusivity is independent of the loading and is dependent only on temperature, according to

}i ) }i,0 exp

( ) -Ei,dif RT

(6)

Consequently, the product of locally evaluated values of both

( )( ) }i ∆µi δ RT

(7)

where θi,av is the average of the degrees of occupancy for the feed side and the permeate side, which is dependent only on temperature and fugacity, through eqs 2 and 3, respectively. The occupancy is averaged in two different ways. The first way is a linear average, which is defined by the equation

θfi + θpi θi,av,lin ) 2

(2)

Often, the loading is expressed in dimensionless form as the degree of occupancy θi. Because, at saturation pressure (or fugacity), the loading must be maximum (this is equivalent to the liquid feed exposure), the occupancy can be represented as

qi

the occupancy θi and the gradient in the dimensionless chemical potential ∇µi/(RT) in eq 5 must be constant over the membrane thickness, providing a relationship between occupancy and driving force over the selective layer. Equation 5, in combination with the expressions for the loading as given in eqs 1-3 and the expression for the diffusivity in eq 6, is collectively referenced as model 1. Simplification: Approximation of Potential Gradient (Model 2) and Occupancy (Model 3). To circumvent a spatial resolution of properties over the selective layer, one can simplify eq 5 by assuming the chemical potential gradient to be constant over the layer thickness δ, resulting in

(8a)

The second averaged occupancy is a logarithmic average, which is given by11

(1 - θi,av,ln) )

(1 - θpi ) - (1 - θfi ) ln[(1 - θpi )/(1 - θfi )]

(8b)

Equation 7 is referenced as model 2a when eq 8a is used to describe the occupancy and as model 2b when eq 8b is applied. In both cases, the diffusivity (}) is given by eq 6. Equation 7 can further be simplified by eliminating the occupancy, by assuming that the selective layer is fully saturated, i.e. θi,av equals unity,2 so that

Ji ) Fsilicaqsat i

( ) ( )

}i,0 -Ei,dif ∆µi exp δ RT RT

(9)

It is now convenient to combine all material constants in eq 9 as one activated parameter, F, as

( )

Ji ) Fi

( )

app ∆µi -Ei,dif ∆µi ) Fi0 exp RT RT RT

(10)

In eq 10, which is referenced as model 3, the activation energy for diffusion is an apparent one, because the effect of the temperature on the adsorption is not considered separately. The benefit of this approach is that no material properties must be app known and that the parameters Fi,0 and Ei,dif can be adjusted to 4 the experimental data. Simplification: Linear Fugacity Difference (Model 4). Because ∆µi/(RT) ) ln(f fi /f pi ), for small fugacity differences over the selective layer, the expression for the dimensionless chemical potential difference can be approximated by the fugacity difference ∆fi via a Taylor expansion. Alternatively, the assumption of linear adsorption (the Henry regime) also allows the driving force to be expressed in terms of larger fugacity differences,5,12 which leads to similar expressions, and is referenced as model 4:

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() () (

Ji ) Hi ≈

}i ∆fi δ

)

( )

-∆Hi,ads -Ei,dif 1 H exp }i,0 exp ∆fi δ i,0 RT RT

≡ Qi,0 exp

( )

app -Ei,per ∆fi RT

) Qi∆fi

(11)

with

Hi ≡ Fsilicaqsat i bi ) Hi,0 exp

(

)

-∆Hi,ads RT

(12)

app where Ei,per denotes the apparent activation energy for permeance, the sum of the enthalpy of adsorption, and the activation energy for diffusion. Although, in eq 11, the driving force is expressed as the fugacity difference and not as the partial pressure difference, we still consider model 4 to be equivalent to the model proposed by Wijmans and Baker,4 because the fugacity coefficient that directly relates the partial pressure to the fugacity was equal to unity for all experimental cases described in this work. It is emphasized that this model also follows from the MS approach if the assumption of a loading-independent diffusivity is released and is assumed to be proportional to the fraction of unoccupied adsorption sites.13 In that case, this Fickian-type relation (eq 11) results,1 with the same interpretation of the parameters and the diffusivity referring to zero loading. This “classical model” was applied to the description of dehydration of alcohols,5,6 and for the description of pure component transport.7,14 In Table 1, an overview of the four models used is given. Support Layer Correction and Driving Force from Bulk Fluid Properties. Figure 1 shows a schematic presentation of a supported membrane. It has been shown that, for pervaporation and vapor permeation, the fugacity at the permeate side of the selective layer should be corrected for the fugacity difference over the support layers,15 which is analogous to the corrections during gas separation with silicalite-1 membranes.12,16 For a single component, the flux through porous alumina layers can be described by the attributions of Knudsen diffusion and viscous flow. A strong adsorption behavior of methanol in the γ-Al2O3 layer has been reported earlier,17 which suggests that a surface diffusion mechanism contributes to the overall transport. The mass transport through the γ-Al2O3 layer is assumed to be governed by surface diffusion at a rate that is not limiting, compared to the diffusivity through the selective layer, so that

Figure 1. Schematic representation of the selective and support layers and a profile of the driving forces over the membrane layers.

the fugacity difference over this layer is assumed to be negligible. Capillary condensation in this layer does not have a role as could be verified by the Kelvin equation. In addition to the feed-side fugacity, the fugacity at the interface between the selective and the support layer determines the driving force through the selective layer. This interfacial fugacity can be determined by successive backward calculations, starting from the determined flux and the pressure in the vapor phase at the permeate side of the membrane. For a pure component, the chemical potential gradient as the driving force can, in difference form, be written as18

() {

}

sat f fi φsat h Li (P fabs - psat ∆µi i pi exp[V i )/(RT)] ) ln p ) ln p p RT fi φ i P abs

(13)

Note that the absolute pressure effects are taken into account in the chemical potential difference via the Poynting factor. As in eq 13, the permeate fugacity appears in the denominator; thus, small differences in this fugacity that are due to support resistance might lead to relatively large differences in the dimensionless chemical potential difference. This differs from models where the driving force is expressed in terms of the fugacity difference. There, the numerical value of the fugacity difference is dominated by the feed-side fugacity and the permeate-side value often can be neglected completely, so

∆fi ) f fi - f pi = f fi

(14)

3. Experimental Procedure Bulk Silica: Adsorption Isotherm, Skeletal Density, and Porosity. The adsorption of methanol into bulk methylated silica

Table 1. Overview of the Considered Models for Describing Methanol Transport model number

model

1

Maxwell-Stefan (MS)

2

simple MS (average occupancy)

3

4

equation number(s)

equation

( ) ( ) ( )

Ji ) -Fsilicaqsat i θi}i,0 exp Ji )

-Fsilicaqsat i θi,av

Ji )

Fsilicaqsat i

Ji )

Q0i

simplified MS (saturated, θi ) 1)

“classical” model

-Ei,dif ∇µi RT RT

5, 6 7, 6

}i,0 -Ei,dif ∆µi exp δ RT RT

( ) ( ) ( )

( )

app app }i,0 -Ei,dif ∆µi -Ei,dif ∆µi exp ) Fi0 exp δ RT RT RT RT

app -Ei,per exp ∆fi RT

9, 10

11

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Table 2. Characteristics of the Selective and Support Layers of the Membrane Studieda R-Al2O3 parameter

substrate

first

second

γ-Al2O3

δ  τ dp

2.0 mm 0.43 3 3.4 µm

40 µm 0.35 3 0.16 µm

40 µm 0.35 3 0.16 µm

1.75 µm 0.575 3 4.5 nm

a

selective layer 100 nm

From ref 22.

material, prepared in a similar way as the membrane coating procedure but with a combustable support, was studied using calorimetry at a temperature of 65 °C and a pressure of 98.3 kPa. Compared to zeolite membranes,19 the amorphous structure of methylated silica complicates the comparison of adsorption in bulk material and thin layer films, as shown for the adsorption of CO2 on silica.20 In this work, however, this difference is not taken into account. The skeletal density of the bulk material was determined as Fsilica ) 1620 kg/m3 by helium pycnometry and 1670 kg/m3 by water pycnometry. These values are low, compared to literature values, which are in the range of 1850-2200 kg/m3.21 Throughout all calculations, the value of 1670 kg/m3 was applied. The porosity of the selective material was estimated to be 0.25.22 Membranes. The tubular membrane used in this study, which was developed by the Energy Research Center of The Netherlands (ECN), had outer and inner diameter dimensions of 14 and 10 mm, respectively, and a permeating surface area of 56.8 cm2. The membrane is composed of four support layers with a methylated silica coating at the outside, facing the feed. The inner support or substrate layer (layer 1) ensures the mechanical stability. Atop this layer, two R-Al2O3 layers (layers 2 and 3) and a γ-Al2O3 layer (layer 4) were deposited, followed by the

methylated silica layer. Characteristics of the support layers are given in Table 2, and a schematic representation of the membrane is shown in Figure 1. The general chemical structure and synthesis methods of methylated silicahave been described by De Vos et al.23 Pervaporation Setup. The experimental setup used for study pervaporation was purchased from Sulzer Chemtech (Neuenkirchen, Germany). The maximum design operating pressure and temperature are 16 barg and 200 °C, respectively. A schematic presentation of the setup is given in Figure 2. Liquid feed is fed to the vessel labeled as B1, which is heated by an external thermostatic bath. After the system has been closed, the fluid circulates over the membrane through the feed pump via lines 1, 2, 4, 5, and 6. The permeate side of the membrane is kept at low pressure by the vacuum pump. The permeate pressure can be regulated by a needle valve in line 10. Permeate is withdrawn from the module via line 9 and collected in two parallel cold traps that are filled with liquid nitrogen. Makeup from vessel B2 can be introduced in the system via a dosage pump to operate at constant concentration in the case of mixture separation. Increasing the absolute system pressure can be achieved by applying an overpressure of N2 to the system via line 11. Conditions and Procedures. After startup, the membrane was operated for ∼1.5 h at the operating temperature to attain steady-state conditions. The influence of the absolute pressure during pervaporation was determined at temperatures of 61, 84, and 108 °C by increasing the system pressure with N2. The first point was measured without N2 overpressure, so for the series at 84 °C and 108 °C at the bubble point. Sampling was performed at least half an hour after the pressure was changed. The permeate pressure was maintained in the range of 9-12 mbar.

Figure 2. Setup for pervaporation that allows above-atmospheric saturation conditions.

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Figure 3. (b) Adsorbed volume and (4) heat of adsorption of methanol on bulk methylated silica material at a temperature of 65 °C.

Figure 4. Comparison of single-site (light dashed line) and dual-site (soild line) Langmuir adsorption isotherms of methanol on bulk methylated silica at a temperature of 65 °C.

To test the durability of the membrane, we performed two series of measurements, varying the temperature, with a period of four weeks in between tests. During the first series, the temperature was varied from 60 °C to 155 °C; during the second series, the temperature was varied from 84 °C to 149 °C. Permeate pressure was maintained in the range of 9-12 mbar. Samples for determining the permeate composition were taken in duplicate. When the error (which was defined as 2 times the standard deviation divided by the average) was larger than 5%, a third sample was taken. The permeation flux was determined by collecting permeate in a cold trap for a known period of time (at least 2 min). After this time, the glass vessel in which permeate had been collected was disconnected from the vacuum system, without interrupting the permeate pressure, and the frozen permeate was carefully brought to room temperature. The vessel with permeate was weighed on a balance that was accurate to three digits and the difference between the empty and filled vessel was divided by the sampling time and membrane area to determine the permeate flux.

Figure 5. Flux of methanol versus absolute system pressure by varying N2 overpressure at three temperatures: (O) 108 °C, (0) 84 °C, and (4) 61 °C. Ppermeate ) 15 mbar.

experimental data (see Figure 4) and was further used in the permeation modeling. Because of the fact that, under liquid feed conditions, saturation must occur, the calculated loading at the corresponding saturation fugacity was taken to be the saturation loading for the occupancy calculation. Nair et al.24 found somewhat lower values for the saturation loading of methanol in bulk silica, namely 4.1 mol/kg at 30 °C and 3.3 mol/kg at 50 °C, but this evaluation is quite sampledependent. The density of our sample was lower, which implies a larger pore volume. Pervaporation at an Elevated Absolute Feed Pressure. In Figure 5, the methanol flux is plotted versus the absolute system pressure at three temperatures. Increasing the feed pressure caused a slight drift in temperature. Theoretically, the absolute pressure affects the fugacity of liquid methanol via the Poynting factor. However, for the pressure condition studied here (up to 16 bar), this effect on the fugacity is calculated to be 0.8, the volume of adsorbed methanol increases sharply, whereas the heat of adsorption decreases. The difference between the average heat of adsorption (42.1 kJ/mol) and the lowest absolute value measured in this region (6.1 kJ/mol) amounts to the heat of condensation of methanol, indicating that the sample was saturated and pore and interparticle condensation occurred at p/p0 > 0.8. Both the single-site as well as the dual-site Langmuir adsorption model was used to correlate the experimental data, up to a relative saturation pressure of 0.8 (see Table 3). The dual-site model resulted in a very satisfying correlation of the

Table 3. Estimated Parameter Values for Single- and Dual-Site Adsorption Isothermsa dual-site model single site model bi,0 qisat a

10-11 Pa-1

3.88 × 4.87 mol kg-1

Heat of adsorption: -∆Hads ) 42.1 kJ/mol.

site 1 bi,0,1 qisat,1

site 2 10-12 Pa-1

6.78 × 4.97 mol kg-1

bi,0,2 qisat,2

3.00 × 10-10 Pa-1 1.69 mol kg-1

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Figure 6. Flux of methanol versus temperature for two experimental series: (4) Ppermeate ) 12 mbar for series 1 and (O) Ppermeate ) 8 mbar for series 2.

Figure 7. Flux of methanol versus fugacity at the permeate side at 97.6 °C, (9) with and (]) without correction for the fugacity difference over the support layers.

observed. Therefore, contributions due to viscous flow can be excluded7 and the membrane is free of defects. This is direct proof for the absence of convective transport, whereas Bettens et al.7 formulated this conclusion on the basis of circumstantial evidence. Pervaporation as a Function of Temperature. In Figure 6, the methanol flux is plotted versus temperature. This figure includes results of a second temperature run that was performed four weeks after the first one, which indicates the reproducible dependency of the flux on temperature. The activation energy for flux was determined at 21.7 kJ/ mol. The exponential increase of the saturation vapor pressure with temperature is proportional to the heat of vaporization.7 Our data confirm that the difference between the apparent activation energies for flux (21.7 kJ/mol) and for permeance (-15.0 kJ/mol) is indeed equal to the heat of vaporization of methanol (35.1 kJ/mol at the atmospheric bubble point of 65 °C). Pervaporation as a Function of Permeate Pressure. Figure 7 depicts the methanol flux for different permeate pressures at 98.3 °C, with and without correction for the fugacity difference over the support layer. A slight decrease in flux with increasing permeate pressure can be observed. The support correction increases the permeate-side pressure of the selective layer by 2-3 kPa. Support Correction: Comparison of Models. Figure 8a displays the pervaporation fluxes versus the fugacity difference measured at various temperatures and permeate pressures (using

Figure 8. Flux of methanol versus (a) the fugacity difference over the membrane and (b) the dimensionless chemical potential difference over the membrane. Data were measured as a function of both temperature (Figure 6) and permeate pressure (Figure 7). Key: (2) over the entire membrane, not corrected for support, and (O) over the selective layer, corrected for support.

data from Figures 6 and 7, respectively). The flux measured at 60.1 °C, which is below the bubble point, is presented separately. The fugacity difference is calculated for two different cases: (i) over the entire membrane from the bulk-phase conditions at the feed and those at the permeate side of the membrane; and (ii) over the selective layer by correcting for the fugacity difference over the support layer. Because of a low contribution of the support resistance, in the present study, the difference between the two approaches is small to negligible, and the approximation of the fugacity difference using the feed-side fugacity, as proposed in eq 14, seems to be allowed. The situation is different if the chemical potential difference is conceived as the driving force for mass transport, as illustrated in Figure 8b for the temperature data series. The chemical potential difference is calculated both with and without supportlayer correction, and a marked difference between both series is observed. Despite the small fugacity difference over the support layers, the difference in chemical potential over the selective layer is significantly influenced, because of the dependency on the ratio of the fugacities at both sides of the membrane, rather than on their difference. Modeling Adsorption Loading and Chemical Potential. The one-component MS formulation for permeation from eq 5 was solved numerically using the dual-site adsorption model. The feed-side fugacity was determined from bulk properties, and the permeate-side fugacity was determined from a correction for the fugacity difference over the support layers. For experimental data acquired at 60, 106, and 155 °C, the profiles over the selective layer of the occupancy θi and dimensionless

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Figure 10. Maxwell-Stefan diffusivity } (expressed in units of m2/s) in model 1 versus temperature.

Figure 9. (a) Degree of occupancy θi and (b) dimensionless chemical potential (µi - µ0i )/(RT) versus the dimensionless position in the selective layer modeled for three experimental conditions using the Maxwell-Stefan model 1. The value of zero (0) on the x-axis represents the feed side, whereas the value of one (1) represents the permeate side.

chemical potential (µi - µ0i )/(RT) were calculated by introducing a spatial resolution of 30 grid points and applying an iterative prediction-correction procedure; results are shown in Figures 9a and 9b, respectively. The occupancy profiles in Figure 9a are not completely linear, although a linear approximation would not deviate to a great extent. As a consequence of the decaying occupancy θi toward the permeate side and the condition that θi∇µi is constant (eq 5), the chemical potential shows a nonlinear profile (Figure 9b). It is apparent that an approximation of a linear driving force, in terms of the chemical potential in combination with a constant, average occupancy, does not do full justice to the actual model. In Table 4, the values for the parameters found from modeling the data using the models as presented in Table 1 are presented. After the profiles of the occupancy and the dimensionless chemical potential gradient are known, the MS diffusivity can easily be determined from eq 5. The resulting diffusivities of the full MS model (model 1) are plotted versus the temperature in Figure 10. The diffusivities of the series where the absolute pressure was varied (data denoted by the rectangular symbols

in Figure 5) follow a different exponential trend, compared to the diffusivities determined from experimental series with varying temperature (data denoted by the diamond-shaped symbols in Figure 6) and permeate pressure (data denoted by the diamond-shaped symbols in Figure 7). The exponential increase of the diffusivity with temperature of all series is in agreement with the expected activated nature of micropore diffusion (see eq 6). Because there was a time gap of two months between these two experimental series, this difference is attributed to stabilization effects of the fresh membrane.25 It has been reported that methanol can react with the silica, resulting in methoxy groups and thereby reducing the permeability.26 To compare the different models, only the data measured as a function of temperature (Figure 6) and permeate pressure (Figure 7) are used further. The pre-exponential factors for either diffusivity or permeance and the activation energies were determined from Arrhenius plots that incorporate all data (see Table 4). The residuals, which are defined as the difference between the modeled and experimental fluxes, are presented in Figures 11a and 11b. From these residuals, the sum of squared residuals (SSR) is calculated for all series, and the corresponding values are given in Table 4. From the SSR values, it follows that the classical pervaporation model (model 4) and the full MS model (model 1) give the best fit. Also, the simplified MS model, assuming a completely saturated selective layer (model 3), describes the results fairly well. The SSR values are one order of magnitude larger when the occupancies are averaged using a dimensionless chemical potential difference (model 2). A linear average of the occupancies gives a slightly better accuracy, compared to that obtained using logarithmic average occupancies. The residual distribution of the three best models (Figure 11a) seems fairly random, which indicates their adequacy, unlike the systematic deviating trends in the distributions for the two other cases of model 2, as is clearly visible in Figure 11b. The use of a linear dimensionless chemical potential difference over the selective layer (model 3) makes modeling for applications such as design purposes much easier, compared to

Table 4. (Apparent) Activation Energies, Diffusivities, and Pre-exponential Factors for Several Models model number

model

1 2a 2b 3 4

Maxwell-Stefan (MS) simplified MS θi,av,lin simplified MS θi,av,ln simplified MS (saturated, θi ) 1) “classical” model

pre-exponential factor

1.89 mol m-2 s-1 8.12.10-10 mol m-2 s-1 Pa-1

Di,0 (× 10-10 m2/s)

Edif or Eapp act (kJ/mol)

sum of squared residuals, SSR

6.84 3.61 4.65 0.678

21.2 19.7 20.2 16.6 -15.0

0.33 × 103 2.1 × 103 2.3 × 103 0.63 × 103 0.21 × 103

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independent diffusivity or one proportional to the fraction of empty sites. Because a loading dependency is generally observed for microporous zeolite membranes,27 there is a preference for the “classical” model (model 4), although, statistically, no real discrimination can be made on the basis of the fitting results.28 The generally proposed interpretation of the linear adsorption isotherm for the linear dependency on the fugacity difference over the membrane is incorrect. In fact, the model directly follows from a chemical potential gradient of the permeating component over the membrane as the driving force, by assuming a loading-dependent diffusivity and a nonlinear isotherm. For completeness, the pervaporation modeling approach of Asaeda and co-workers is mentioned.29,30 They assumed that the rate-determining process is not the transport through a selective layer, but the evaporation of the permeating component into and transport through the support layer. Because of capillary forces in the support, a liquid/vapor interface exists, whose location in the support moves with increasing flux from smaller to larger pores, corresponding to increasing saturated vapor pressures. This model especially applies to very high permeation fluxes,30 which is a situation that does not exist in this work. 5. Conclusions

Figure 11. Residuals as a function of the experimental flux (a) for the full Maxwell-Stefan model 1 ((0) the simplified Maxwell-Stefan model with θ ) 1 (([) model 3) and (4) the classical pervaporation model 4) and (b) for the simplified Maxwell-Stefan equations for a linear averaged value of θi (((0) model 2a) and the logarithmic averaged value of θi ((b) model 2b), and the model with θ ) 1 ((]) model 3)).

the application of the full MS equation (model 1), as Marx2 recommended, assuming a fully saturated selective layer. From the modeling results of model 1 (Figure 9a), it is apparent that the assumption of full saturation does not hold and this may indicate the limitations of this “short-cut calculation” model for other conditions or for a description of the permeation of mixtures. However, the classical model (model 4) performs best and is easiest to apply. It can be reconciled as an MS model in which the loading-dependent diffusivity compensates for the loading effect of the thermodynamic factor in the flux equation.12 From eq 5, it can be derived that the group }(qi/fi) should be constant. For a Langmuir adsorption isotherm, this implies a loading dependency of the diffusivity of

}i )

}i,θ)0 1 - θi

(15)

This implies an increase of the MS diffusivity with loading, which is a phenomenon that has been well-described for diffusion in zeolites.27 The apparent activation energy represents the sum of the activation energy for diffusion at zero loading and the adsorption enthalpy. Because the latter is larger in negative magnitude than the former, the apparent activation energy for the permeance is negative.5 Evaluation of these results indicates that the two model approaches perform well to describe the pervaporation of methanol. Their basic difference is the assumption of a loading-

The pervaporation of methanol through an amorphous methylated silica membrane was measured for temperatures in the range of 60-155 °C and for different feed and permeate pressures. Increasing the feed-side pressure in a range of ∼15 bar, above the bubble point, was determined to have no effect on the methanol flux, (i) confirming the membrane to be defectfree, and (ii) proving that the absolute liquid feed pressure does not determine the driving force for pervaporation through microporous membranes, but that this should be best based on the chemical potential (i.e., saturation pressure at the operating temperature) of the permeating component. For the description of pure methanol pervaporation transport, the difference in accuracy between two different models based on the Maxwell-Stefan formulation is only minor. In both models, the thermodynamic potential gradient of the permeating component is used as the driving force and differ in the loading dependence of the diffusivity. The model with a loadingdependent diffusivity is identical to the classical pervaporation model with the fugacity difference as the driving force. In the literature, the assumption of a linear adsorption isotherm, which is used to explain this linear dependency on the fugacity (or partial pressure) difference over the membrane, is inconsistent. The new interpretation eliminates this inconsistency. Notations b ) adsorption constant (Pa-1) c ) concentration (mol/m3) d ) diameter (m) } ) Maxwell-Stefan diffusivity (m2/s) E ) activation energy (kJ/mol) F ) activated transport parameter (mol m-2 s-1) f ) fugacity (Pa) H ) Henry coefficient (mol/m) ∆Hads ) enthalpy of adsorption (kJ/mol) J ) flux (mol m-2 s-1) n ) number of moles p ) partial pressure (Pa) P ) pressure (Pa) q ) loading (mol kg) Q ) permeance (mol m-2 s-1 Pa-1) R ) gas constant; R ) 8.314 J mol-1 K-1

Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 4099

T ) temperature (°C or K) V h ) molar volume (m3/mol) Greek Symbols δ ) thickness (m)  ) porosity θ ) occupancy µ ) chemical potential (J/mol) τ ) tortuosity F ) density (kg/m3) φ ) fugacity coefficient Superscripts abs ) absolute app ) apparent dif ) diffusion f ) feed side p ) permeate side referring to the pore per ) permeation sat ) saturated 1, 2 ) in reference to adsorption sites 1 or 2 0 ) reference state Subscripts ads ) adsorption i or j ) referring to component i or j ij ) binary (component i and j) L ) liquid phase 1, 2 ) in reference to adsorption sites 1 or 2 0 ) reference state Acknowledgment The work presented here is supported by a grant from the Dutch Program bureau EET (Economy, Ecology, Technology), which is a joint initiative of the Dutch Ministries of Economic Affairs, Education, Culture and Sciences and of Housing, Spatial Planning and Environment (under Contract No. EETK20061). The authors wish to thank the Energy Research Centre of The Netherlands (ECN) for supplying the membrane used in this study. Marjo Mittelmeijer-Hazeleger (University of Amsterdam) is gratefully acknowledged for determination of the adsorption isotherm. Literature Cited (1) Krishna, R. A unified approach to the modelling of intraparticle diffusion in adsorption processes. Gas Sep. Purif. 1993, 7 (2), 91. (2) Marx, S. Application of pervaporation to the separation of methanol from tertiary amyl methyl ether, Ph.D. Thesis, University of Potchefstroom, Potchefstroom, South Africa, 2002. (3) Bowen, T. C.; Li, S.; Noble, R. D.; Falconer, J. L. Driving force for pervaporation through zeolite membranes. J. Membr. Sci. 2003, 225, 165. (4) Wijmans, J. G.; Baker, R. W. A simple predictive treatment of the permeation process in pervaporation. J. Membr. Sci. 1993, 79, 101. (5) Ten Elshof, J. E.; Abadal, C. R.; Sekulic´, J.; Chowdhury, S. R.; Blank, D. H. A. Transport mechanisms of water and organic solvents through microporous silica in the pervaporation of binary liquids. Microporous Mesoporous Mater. 2003, 65, 197. (6) Sommer, S.; Melin, T. Influence of operation parameters on the separation of mixtures by pervaporation and vapour permeation with inorganic membranes. Part 1: Dehydration of solvents. Chem. Eng. Sci. 2005, 60, 4509. (7) Bettens, B.; Dekeyzer, S.; Van der Bruggen, B.; Degre`ve, J.; Vandecasteele, C. Transport of pure components in pervaporation through a microporous silica membrane. J. Phys. Chem. B 2005, 109, 5216.

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ReceiVed for reView August 9, 2006 ReVised manuscript receiVed October 8, 2006 Accepted October 10, 2006 IE0610445