Isothermal and Nonisothermal Permeation of an Organic Vapor

We report hereafter an experimental attempt to reach such a situation, as well as an interpretation of some of the early results obtained, showing the...
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Ind. Eng. Chem. Res. 1999, 38, 211-217

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SEPARATIONS Isothermal and Nonisothermal Permeation of an Organic Vapor through a Dense Polymer Membrane Anne Hillaire and Eric Favre* LCPM, CNRS UMR 7568, ENSIC, 1 rue Grandville, 54001 Nancy, France

The permeation of an organic vapor (acetone) through a dense polymer membrane (poly(dimethylsiloxane)) has been systematically investigated in a specially designed two-compartments module under isothermal and nonisothermal conditions. The influence of temperature on permeation flux under isothermal conditions for constant pressure and activity is discussed and compared to the results obtained when a temperature gradient is superimposed. A heat transfer analysis suggests that the strong enhancement observed in the latter case results from a temperature polarization effect at the upstream membrane side. The implications of this phenomenon are discussed both from a fundamental and applied point of view. Introduction The transport of chemical coumpounds through dense polymer films has already received considerable attention and finds applications in several fields of industrial relevance (e.g., gas and liquid separations, polymer devolatilization, barrier materials, ...). For the sake of simplicity, the design and analysis of dense polymer films for either packaging or separation applications (i.e., membrane operations) most often postulate isothermal conditions.1-3 Surprisingly, the influence of a controlled heat flux through a dense polymer film, consecutive to nonisothermal conditions and acting simultaneously to mass transport, remains an open question. Recently, experimental attempts to develop such a situation on real separation systems have been reported and seem to indicate a tremendous effect of heat flux on both global mass transport and separation selectivity.4 Unfortunately, the complexity of the latter experimental system (multicomponent transfer with coupled heat and mass fluxes) impedes a detailed analysis of the results on fundamental grounds. Such a goal could be best reached on simple experimental systems, with a single permeant; this strategy has been worthwhile, particularly for processes based on porous membranes such as transmembrane distillation.5,6 It is striking that the same has not been applied so far for dense membrane processes. For the latter, indeed, attention has focused almost exclusively on liquid transport, to describe the mass transfer through a dense membrane consecutive to a temperature gradient, a socalled thermoosmosis phenomenon.7 Apart from the two previous cases, summarized in Figure 1a,b, it would be worthwhile to examine another situation, dealing with vapor permeation under an imposed temperature gradient. To our knowledge, this condition, described in Figure 1c, has never been rigorously explored. We report hereafter an experimental attempt to reach such a situation, as well as an interpretation of some * Phone: 33-03-83-17-52-89. Fax: 33-03-83-37-99-77. Email: [email protected].

of the early results obtained, showing the impressive effect of a temperature gradient on the vapor permeation flux of an organic compound (acetone) through an elastomeric matrix (silicone rubber membrane). Apparatus and Experimental Procedures The lab-scale setup used for the experiments is shown in Figure 2 and consists of a permeation cell with an effective membrane area of 16 cm2. The cell, represented in Figure 3, is a cylindrical double-walled stainless steel vessel with two jackets (permeate and feed side) that can be thermostated independently by water circulation ((0.1 °C). For so-called isothermal experiments, water was recirculated in both the upper and lower compartments, to ensure isothermicity. An air gap (3 mm thick) has been interposed between the two compartments in order to minimize heat losses when nonisothermal conditions are imposed. A vapor generator is placed at the upstream side of the membrane: a reservoir containing pure acetone (analytical grade) is placed in a thermostated bath and generates its saturated vapor pressure at a temperature below that at the neighborhood of the membrane. The vapor activity can be simply calculated from the ratio of the solvent pressure in the cell to its saturated vapor pressure at the membrane compartment temperature ai ) pi/p/i (T) ) p/i (Treservoir)/p/i (Tcell). It can be adjusted by tuning the two temperatures of the feed reservoir and membrane compartment. The vapor phase produced in the liquid vessel is connected to the feed compartment of the membrane cell by tubings heated by heating tapes to prevent condensation. On the permeate side, a vacuum pump (Alcatel 1012A) provides the necessary driving force (downstream pressure maintained at 0.1 mmHg). The membrane used is a dense poly(dimethylsiloxane) (PDMS) film, 125 µm thick, of filled type (ca. 30% silica by weight) purchased from Sigma Medical (Silastic from Dow Corning). It is mounted between a stainless steel plate sintered support (purchased from Poral) at the

10.1021/ie980491k CCC: $18.00 © 1999 American Chemical Society Published on Web 12/09/1998

212 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

Figure 3. Detailed view of the two independent jackets membrane module used for the experimental work: (1) feed reservoir connection; (2) upper compartment; (3) membrane; (4) sintered support; (5) lower compartment; (6) vacuum pump connection.

can be checked by comparing the experimental pressure value to the pressure computed from the Antoine relationship at the reservoir temperature). After attainment of steady state, the permeated vapor was trapped for a known period of time and the condensate was then weighed in order to calculate the permeate flux. A minimum of three measurements were performed for each set of operating conditions and showed good reproducibility (discrepancies below 5%). Results and Discussion

Figure 1. Schematic representation of three types of nonisothermal membrane permeation: (a) transmembrane distillation (hydrophobic microporous membrane); (b) thermoosmosis; (c) nonisothermal vapor permeation.

Theoretical Background. Before studying the incidence of a thermal gradient on permeate flux, a rational understanding of temperature influence under isothermal conditions is an essential prerequisite. This is classically achieved in the literature by determining the permeate flux (J) under constant upstream and downstream conditions while the system temperature (T) is gradually changed.8-11 The relationship proposed in order to describe the results is almost invariably of Arrhenius type:

( ) -EaJ

Ji ) Jo exp

RT

(1)

Nevertheless, some authors incidentally prefer to express the influence of temperature based on the change in permeability value,12,13 leading to a so-called activation energy of permeation (EaP): Figure 2. Overall experimental setup: (1) membrane module; (2) feed reservoir; (3) cool trap; (4) vacuum pump; (5) thermostated bath; (6) upstream manometer; (7) desiccant; (8) downstream manometer.

downstream face and a EPDM (ethylene-propylenediene rubber) O-ring at the upstream side. In a first step, the overall system (upper and lower compartments) is degassed by vacuum, permeation rates being determined after at least 30 h in order to ensure stationary flux conditions. A high precision manometer (Edwards EMV 251, precision (1 mbar) allowed us to detect any leak in the upper compartment, as well as the absence of air in the feed mixture (this

( )

Pi ) Po exp

-EaJ RT

(2)

It is recalled that the permeability (P) relates to the permeation flux (J) through

Ji )

Pi(p′i - p′′i ) z

(3)

with (p′i - p′′i ) the solvent vapor pressure difference across the membrane and z the membrane thickness. This strategy, leading obviously to a different activation energy value, has given rise to controversial conclusions

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 213 Table 1. Various Experimental Activation Energy Values for Acetone in PDMS Ea (kJ‚mol-1)

constant activity

constant pressure

EaP EaJ

-9.6 21.5

-17 -17

EaJ and EaP, respectively corresponding to fluxes and permeabilities, are summarized in Table 1. The need to unambiguously specify which type of activation energy is used is self-speaking. The most outstanding example is the positive value of EaJ for constant activity conditions, indicating that the flux increases with temperature, whereas the opposite conclusion can be deduced with the variation of J at constant pressure. We develop hereafter the theoretical origin of such apparent discrepancies. Combining equations (1) and (3) leads to Figure 4. Arrhenius plot of experimental data under isothermal conditions: (b) permeation flux under constant upstream pressure; (9) permeability under constant upstream pressure; (O) permeation flux under constant upstream activity; (0) permeability under constant upstream activity. Fluxes have been converted to a 10 µm thick membrane.

for membrane applications dealing with liquid permeation, recently discussed.14 When a vapor is permeated, the situation is further complicated by the fact that the conditions to be kept constant when temperature is changed can be defined either on a pressure or on a thermodynamic activity basis. For an experimental setup where vacuum is applied at the downstream part of the membrane (such as the one used throughout this study), both activity and pressure values vanish; for the upstream side, however, working at constant vapor activity while increasing temperature obviously implies a change of the solvent pressure p′i. Consequently, different activation energy values will be obtained depending on the way experiments are performed. In summary, expressing the influence of temperature in vapor permeation can be performed either on a flux or permeability basis and experimentally achieved either under constant pressure or constant activity conditions. Thus, the experimental protocol should clearly specify which of the four possibilities described above has been selected when an activation energy value is reported; it should be stressed that this criteria is not always fulfilled, especially for vapor permeation. This situation complicates comparative studies and can lead to contradictory conclusions. In the next part of this paper, we will compare the various activation energies resulting from the four cases mentionned above and discuss their dependencies in a rigorous way when the membrane is placed under isothermal conditions. Influence of Temperature on Permeation Flux under Isothermal Conditions. A series of permeation experiments have been carried out with operating temperatures ranging between 30 and 70 °C; two sets of measurements have been performed for each temperature, corresponding to constant activity and constant pressure upstream conditions. Results are shown in Figure 4; it can be seen that the experimental data are extremely well fitted by a linear relationship between the log of flux versus the reciprocal temperature. The activation energies values

ln Ji ) ln Pi + ln(p′i - p′′i ) + cst

(4)

For constant pressure conditions (p′i ) cst) and when downstream pressure is negligibly small (p′′i ≈ 0) compared to the upstream pressure (p′i), it follows

ln Ji ) ln Pi + cst

(5)

With the difference between flux and permeability logarithm being constant and independent of temperature, the same activation energy will be obtained in terms of flux (EaJ) or permeability (EaP) when constant pressure conditions prevail. This conclusion is experimentally verified (Table 1, last column). On the other hand, if the upstream activity is kept constant while permeate pressure remains negligibly small, eq 4 has to be written

ln Ji ) ln Pi + ln ai + ln p* i (T) + cst

(6)

The effect of temperature on the saturated vapor pressure, p*i can be approximated by the well-known expression derived from the Clausius-Clapeyron equation, relating the first derivative of vapor pressure with respect to temperature and the molar heat of vaporization:

(

p* i ) A exp

)

-∆Hv RT

(7)

thus, at constant activity, the two activation energies EaJ and EaP are not equal and should be related by

EaJ ) EaP + ∆Hv

(8)

This relationship is experimentally verified, since we obtain a difference between EaJ and EaP very close to the tabulated acetone heat of vaporization value (31 kJ/ mol for a theoretical ∆Hv value of 31.2 kJ/mol). It should be recalled that expression 8, already proposed by Gee15 but seldom employed, is valid only for a downstream pressure value close to zero. The vacuum imposed throughout the experiments (p′′ ) 0.1 mmHg) seems to be low enough for this condition to be fulfilled. It is interesting at this stage to examine the implications of the above conclusions when a more detailed interpretation of activation energy values is attempted. Based on the solution-diffusion model, the permeability can be written as the product of the diffusivity Di and the sorption coefficient Si: Pi ) DiSi. The temperature

214 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

Figure 5. Schematic thermodynamic path representation of the various energies involved in sorption.

dependence of D and S is again classically expressed by an Arrhenius relationship:3

( )

D ) Do exp

-EaD RT

and

( )

S ) So exp

-∆Hs RT

(9)

where EaD is the activation energy of diffusion and ∆Hs is the enthalpy of sorption of a vapor in the polymer. The activation energy of permeation can then be related to the previous energies by

EaP ) EaD + ∆Hs

(10)

Diffusion coefficients increase with temperature (positive value of EaD), while sorption is usually exothermic (negative value of ∆Hs). For organic vapors, the sorption coefficient dominates, the resulting activation energy EaP is then negative, corresponding to a decrease of permeability with temperature. Concerning the activation energy EaJ, it follows

EaJ ) EaD + ∆Hv + ∆Hs

(11)

The highly positive value of ∆Hv dominates; consequently, a positive value of EaJ occurs when activity remains constant. The activation energy EaJ includes two contributions, one of the diffusion type and the other equivalent to the enthalpy of sorption (∆H ) ∆Hv + ∆Hs). As represented in Figure 5, ∆H corresponds to the enthalpy of sorption of a liquid in a polymer, whereas ∆Hs corresponds to the enthalpy of dissolution of a vapor. Again, values reported in the literature can be controversial; for instance, heat of sorption derived from swelling in liquid studies corresponds to ∆H, while values obtained by chromatographic measurements are equivalent to ∆Hs. For acetone in PDMS, an experimental value of -21.8 kJ‚mol-1 has been reported for the former,16 compared to 9.8 kJ‚mol-1 for the second case.17 The corresponding heat of vaporization value computed by a combination of these two results (31.6 kJ‚mol-1), according to the thermodynamic path sketched in Figure 5, compares favorably well with the theoretical one (31 kJ‚mol-1). In conclusion, we developed at this stage a rational understanding of the influence of temperature on vapor permeation, based on various means of expression or measurement; the apparent discrepancies between the four types of activation energy values have been resolved according to simple relationships. From a fundamental point of view, the activation energy of permeation EaP should be preferentially used to express the effect of temperature, since it takes implicitly into account the fact that the permeated compound is a vapor. Nevertheless, the flux activation energy EaJ for constant pressure conditions is more convenient for studies dealing with engineering aspects (i.e., design or analysis of a vapor separation unit). In any case,

Figure 6. Acetone permeation flux versus dowstream compartment temperature for nonisothermal experiments. Filled symbols correspond to the isothermal case. Double dashes indicate the feed reservoir temperature. Oscillating behavior occurs for a permeate temperature below the one pointed by the arrow (condensation effect). Fluxes values have been converted to a 10 µm thick membrane. Upper compartment temperature: (O) 50 °C; (4) 60 °C; (0) 70 °C.

attention should be paid to the fact that, whereas permeability decreases with temperature, it is not obvious that flux varies in the same way. Incidence of a Temperature Gradient on Permeation Flux. For nonisothermal experiments, the upper compartment has been kept at a constant temperature (ranging between 50 and 70 °C), while the downstream jacket temperature was gradually changed from 20 to 80 °C. The results obtained are reported in Figure 6, where isothermal conditions have been precised in order to highlight the incidence of the thermal flux. It can be seen that the systematic application of a temperature gradient in a direction similar to the concentration gradient (i.e., a permeate temperature lower than the upstream temperature) induces a tremendous increase of the permeate flux; indeed, the maximal flux experimentally observed can be about 4 times superior to that of the isothermal case. The reverse is also true, to a lower extent: a temperature gradient opposite to the concentration gradient leads to a slight decrease of the permeate flux. For all the data shown in Figure 6, pressure remained unchanged in the upstream compartment at a value corresponding to the acetone vapor pressure at the reservoir temperature. Nevertheless, it is obvious that the downstream temperature could not be decreased indefinitely since acetone condensation is likely to occur at the membrane surface below a certain critical level, indicated by an arrow in Figure 6. This phenomenon leads experimentally to oscillating flux and pressure behaviors, impeding measurement exploitation. It should be noticed that the temperature for which condensation is experimentally evidenced is by far lower than the reservoir temperature, which is indicated in Figure 6 by a double dash. The unexpected results described above confirm to some extent the already observed influence of a temperature gradient on permeation flux,4 and call for a rational analysis in order to achieve a sound understanding of the phenomena involved. A knowledge of the membrane interfacial conditions is an essential starting point to such an analysis. This can be achieved by a careful comparison of the data shown in Figure 6

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 215

Figure 8. Schematic representation of the hypothetical thermal gradient postulated, based on data interpretation, for the nonisothermal experiments reported in this paper. Figure 7. Graphical comparison of the nonisothermal and isothermal results. Fluxes values have been converted to a 10 µm thick membrane. (b) Nonisothermal, X-axis: downstream temperature, Tupstream ) 50 °C. Isothermal case, X-axis: cell temperature, (O) constant upstream pressure, p′ ) 330 mbar, (0) constant upstream activity, a ) 0.4.

with those obtained under isothermal conditions; in the next part, we applied this strategy to one set of data (upper compartment temperature 50 °C), with either constant activity (a ) 0.4) or constant upstream pressure (p )330 mbars) conditions for a temperature identical to that of the permeate side. The results are shown in Figure 7. For a constant activity hypothesis, not only can the graphs not be superposed but also they do not vary in the same way. This result indicates that since vapor activity (ai ) p/i (Treservoir)/p/i (Tmembrane)) cannot be considered as constant, membrane upstream temperature should differ from feed temperature. As a consequence, an upstream temperature gradient can already be postulated at this stage. On the contrary, a comparison based on a constant pressure basis shows that the two graphs are comparable within the limits of experimental errors (Figure 7). This information suggests that, for nonisothermal experiments, the upstream membrane temperature probably corresponds to the temperature imposed at the downstream side. In other words, membrane temperature would equal permeate temperature, no temperature gradient taking place within the polymer layer. It should be stressed, however, that, for a downstream temperature below the vapor condensation temperature, a real transmembrane gradient is likely to occur. The following analysis excludes such situations and the corresponding hypothetical temperature profile is illustrated in Figure 8: the fact that the temperature drop remains located in the vapor boundary layer is a typical indication of a strong temperature polarization effect. The validity of the latter hypothesis can be theoretically checked by a thermal resistance analysis, pointing out the domination of the upper vapor layer compared to the other potential resistances. Such an eventuality is examined hereafter. For the sintered stainless steel support, as well as the dense membrane, purely conductive heat transfer has been assumed, while natural convection conditions prevail for the vapor boundary layer. The three different

Table 2. Thermal Resistances Computations stainless steel support thermal conductivity λ (W‚m-1‚°C-1) thermal resistance Ri (m2‚°C‚W-1) a

polymer membrane

solvent vapor boundary layer

12.1

0.24

0.014

2.5 × 10-4

5.2 × 10-4

0.521a

See detailed calculation in appendix.

heat transfer resistances computed are summarized in Table 2. The stainless steel support thermal resistance appears to be the smallest compared to the two others due to its high thermal conductivity value. Comparatively, the membrane resistance is the same order of magnitude; this results from the compensation of the poor thermal conductivity of polymers by the relatively low membrane thickness value. The overhelming contribution of the vapor boundary layer to the overall heat resistance confirms the hypothesis of a temperature gradient located almost solely at the upstream membrane side; this conclusion is consistent with the temperature polarization phenomenon postulated above. Conclusion This work aimed originally to investigate the incidence of a controlled transmembrane temperature gradient on pure vapor permeation properties. It has been shown afterward that the attainment of a transmembrane temperature gradient can be problematic to achieve from the experimental point of view. In fact, similarly to studies relevant to transmembrane distillation6 or thermoosmosis,7 data interpretation suggests a strong temperature polarization phenomenon to take place in the upstream boundary layer; it should be noticed, however, that, while this effect was shown to take place on both faces of the membrane in thermoosomosis or transmembrane distillation, it remains located to one side of the membrane in the present case. The situation reported above results essentially from the fact that the downstream side shows a high thermal conductivity (metallic support); this should not be considered as a general observation for other systems where a metallic support is not present (e.g., self-supported hollow fibers or flat membrane on plastic support). Furthermore, a radial temperature gradient within the polymer membrane could also possibly be superimposed to the axial gradient suggested above and complicate

216 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

the analysis; in that event, surface temperature measurement by noninvasive techniques (e.g., pyrometric sensors) would be of value. With the applied temperature difference occurring essentially at the upstream side of the membrane, the intramembrane temperature remains essentially constant at a value close to that of the support (i.e., downstream side). The conclusions drawn from this study have several implications: (i) If a real controlled transmembrane temperature gradient is desired, attention should be paid to the upstream heat transfer resistance. A first solution would consist of adding a highly heat conducting support at the membrane upstream side, while taking care to prevent any supplementary mass transfer limitation. Forced convection (i.e., agitation) could be also applied in the upstream compartment in order to achieve the same purpose. Should one or the other of these cautions be neglected, the results obtained will essentially reflect the incidence of the upstream membrane temperature variation, while no real transmembrane temperature gradient is imposed. In any case, recognizing that temperature polarization can hardly be integrally annihilated, the upstream membrane temperature would be best assessed experimentally (e.g., by pyrometric sensors). (ii) For installations dedicated to isothermal vapor permeability determination, this study has shown that membrane temperature is essentially governed by the dowstream support temperature. As a consequence, not only the upstream compartment but also the permeate side should be precisely thermostated; otherwise, differences in permeability values obtained by different installations could result, as already reported incidentally in the literature. These discrepancies should be stronger for installations differing from the downstream heat resistance point of view; such a peculiarity has been already experimentally evidenced but remains largely unreferred.8 (iii) Vapor permeation flux being strongly affected by heat perturbations, a thermal polarization phenomenon could potentially explain the discrepancies between saturated vapor and liquid permeation rates, often reported, but largely unexplained up to now.18,19 Indeed, the strong differences in terms of thermal and mass diffusivity between a vapor and a liquid (best expressed through their respective Levis number) should lead to different upstream membrane temperatures, that is, different flux values. Thus, differences in thermal polarization intensity could possibly explain why, under a similar driving force based on bulk conditions (i.e., chemical potential difference), differences in mass transfer can appear. An analysis of heat and mass transfer rate contributions has been shown recently to give rise to strong deviations from isothermal behavior in a transitory transport regime of permeants through dense polymers;20 similar considerations applied to the stationary case could possibly lead to an explanation of the above paradox. (iv) The thermal polarization effect could be systematically exploited for vapor permeation installations; with the latter working usually under mild forced convective conditions, a lower membrane temperature would yield a higher permeate flux under similar pressure conditions. Even though this strategy is wellknown for an industrial unit,21 this study suggests achieving such an improvement by a cooling process

applied to the membrane support. A support showing good thermal conductivity should be best suited for such an application. Appendix Calculation of the Thermal Resistance of the Solvent Vapor Boundary Layer. Given the fact that no impeller is used in the membrane, natural convection has been assumed for heat transfer resistance computation in the vapor boundary layer. The properties of acetone vapor have been taken as an average at 50 °C22 and allow the calculation of the Prandlt and Grashof numbers.

Pr ) Gr )

Cpµ ) 0.304 λ

Fgβ(T1 - T2)D3 µ2

) 7.5 × 105

with T1 - T2 corresponding to the boundary layer temperatures (i.e., T1 ) Tfeed and T2 ) Tmembrane ) Tpermeate). The final value of the Grashof number being within the interval Gr ∈ [2 × 105, 2 × 107], laminar conditions can be assumed. Thus, the computation of the Nusselt number is possible using a relationship proposed23 for horizontal heated plates facing downward or horizontal cooled plates facing upward:

Nu ) 0.27(Gr Pr)1/4 with Nu ) hD/λ. A boundary layer heat transfer coefficient can then be deduced, h ) 1.92 W‚m-2‚°C-1, as well as the corresponding thermal resistance: R ) 1/hc ) 0.521. Notation ai:

thermodynamic activity of component i

Cp:

heat capacity (J‚kg-1‚K-1)

D:

cell diameter (m)

Ea:

activation energy (J‚mol-1)

g:

gravitational acceleration (m2‚s-1)

Gr:

Grashof number

h:

heat transfer coefficient (W‚m2‚K-1)

Ji:

flux of component i (kg‚h-1‚m-2)

Nu:

Nusselt number

p:

pressure (Pa)

p*:

saturated vapor pressure (Pa)

Pi:

permeability of component i (m3‚m‚s-1‚m-2‚Pa-1)

Pr:

Prandlt number

R:

thermal resistance (m2‚°C‚W-1)

T:

temperature (K)

Greek Letters β:

thermal expansion coefficient (K-1) ) 1/Tfilm

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 217 F:

density (kg‚m-3) (kg‚m-1‚s-1)

µ:

viscosity

λ:

thermal conductivity (W‚m-1‚K-1)

Superscripts ′:

upstream side of the membrane

′′:

permeate side of the membrane

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silicone rubber hollow-fiber membranes. Kagaku Kogaku Ronbunshu 1992, 18, 259. (12) Chung, I. J.; Lee, K.-R.; Hwang, S. T. Separation of CFC12 from air by polyimide hollow fiber membrane module. J. Membr. Sci. 1995, 105, 177. (13) Leemann, M.; Eigenberger, G.; Strathmann, H. Vapour permeation for the recovery of organic solvents from waste air streams - Separation capacities and process optimisation. J. Membr. Sci. 1996, 113, 313. (14) Feng, X.; Huang, R. Y. M. Estimation of activation energy for permeation in pervaporation processes. J. Membr. Sci. 1996, 118, 127. (15) Gee, G. Some thermodynamic properties of high polymers and their molecular interpretation. Quart. Rev. 1947, 1, 265. (16) Tian, M.; Munk, P. Characterization of polymer-solvent interactions and their temperature dependence using inverse gas chromatography. J. Chem. Eng. Data 1994, 39, 742. (17) Favre, E. Swelling of cross-linked poly(dimethylsiloxane) networks by pure solvents: influence of temperature. Eur. Polym. J. 1996, 32 (10), 1183. (18) Stannet, V.; Yasuda, H. Liquid versus vapor permeation through polymer films. Polym. Lett. 1963, 1, 289. (19) Bode, E.; Busse, M.; Ruthenberg, K. Considerations in interfaces resistances in the process of permeation of dense membrane. J. Membr. Sci. 1993, 77, 69. (20) Wang, P.; Meldon, J. H.; Sung, N. Heat effects in sorption of organic vapors in rubbery polymers. J. Appl. Polym. Sci. 1996, 59, 937. (21) Baker, R. W.; Wijmans, J. G. Membrane separation of organic vapors from gas streams. In Polymeric Gas Separation Membranes; Paul, D. R., Yampol’skii, Y. P., Eds.; CRC: Boca Raton, FL, 1994; Chapter 8. (22) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The properties of gases and liquids; McGraw-Hill Book Company: 1977. (23) McAdams, W. H. Heat Transmission; Third Edition ed: McGraw-Hill Book Company: 1954.

Received for review July 29, 1998 Revised manuscript received October 19, 1998 Accepted October 21, 1998

IE980491K