zy zyxw zyx Permeability of Pure and Mixed Gases in Silicone Rubber at Elevated Pressures S. M. JORDAN and W. J. KOROS,* The University of Texas at Austin, Department of Chemical Engineering, Austin, Texas 78712-1062 Synopsis The transport properties of silicone rubber are reported at 35°C for a series of pure gases (He, N,, CH,, CO,, and C,H,) and gas mixtures (CO,/CH, and N,/CO,) for pressures up to 60 atm. The effects of pressure and concentration on the permeability of various gases have been analyzed to consider plasticization and hydrostatic compression effects. Over an extended pressure and concentration range, both compression of free volume and eventual plasticization phenomena were observed for the various penetrants. In pure component studies, plasticization effects tended to dominate hydrostatic compression effects for the more condensible penetrants (C,H, and CO,) while the reverse was true for the low sorbing N, and He. These issues are discussed in terms of penetrant diffusion coefficients versus pressure to clarify the interplay between the opposing effects for the penetrants of markedly different solubilities. Additional insight into the somewhat complex interplay of the plasticization and hydrostatic cornpression effects are given by mixed gas permeation results. It was found that the permeability of nitrogen in a 10/90 CO,/N, and a 50/50 CO,/N, mixture was increased by the presence of CO, because the plasticizing nature of CO, is able t o overcome nitrogen’s compression effect. zyxwvut zyxw INTRODUCTION Typically the permeabilities of rubbery materials have been studied a t low pressures, and the permeabilities are usually found to be independent of pressure for most gas penetrants.’ NMR work2 indicates that the effect of sorbed molecules on the molecular motions of rubbery polymers can be quite different depending on whether the penetrant acts primarily as a “ plasticizing” or as a “pressurizing” medium. A pressurizing agent will tend to decrease the free volume of the polymer, thereby reducing its molecular motions. On the other hand, a plasticizing penetrant tends to increase free volume and molecular motions. These changes in molecular motions and free volume are expected to be reflected in the transport properties measured. To consider the competing effects between plasticization and hydrostatic compression mentioned above, the present study extends the pressure and concentration range beyond previous studies.’ In addition to pure gas permeability measurements, mixed gas permeability measurements using gaseous penetrants, of which one is primarily a “plasticizer” and the other is primarily a “compressor,” have been performed. The change in the free volume of the polymer is directly reflected by a change in the mobility of each penetrant through the polymer matrix. *To whom all correspondence should be addressed. zyx Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 795-809 (1990) CCC 0887-6266/90/060795-15$04.00 0 1990 John Wiley & Sons, Inc. zyxw zyxwv zyxw zyxwvu z zyx JORDAN AND KOROS 796 BACKGROUND Permeation through a thin, flat, dense film can be characterized in terms of a simple one-dimensional diffusion model.3 The local flux through a film is given by the following expression: N = - dC Deff(C ) dx where the diffusion coefficient DeEmay be a function of the local concentration, C and temperature. The permeability P can be defined as shown in eq. (2) in terms of the steady state permeation flux through a membrane of thickness 1. where N is the steady state flux, cc(STP)/(cm2 sec), and A p is the pressure difference between the upstream and downstream membrane faces ( ~ m H g ) . ~ Substituting the expression for the flux given in eq. (1) into eq. (2) yields the permeability as a function of the local concentration gradient: zy zyxw Equation (3) can be rearranged and integrated for the appropriate boundary conditions used in most of the experiments in this study: C = C, a t x = 0 (upstream face of the membrane) and C = 0 a t x = I (downstream face of the membrane) : s where and are the average solubility and diffusivity coefficients.6 The local diffusion coefficient can also be evaluated using eq. (5), which allows analysis of permeability and solubility data to directly determine this coefficient a t an arbitrarily selected upstream pressure p , , assuming the downstream pressure is negligible. Equation (5) is derived by applying the Liebnitz rule for differentiation under an integral sign to eq. (4).6 The sorption of low-molecular-weight penetrants in rubbery materials is typically described by Henry’s law for cases where the sorbed concentrations are low, viz., C = k,p (6) K, is the Henry’s law coefficient, cc(STP)/(cc polymer atm), and p is the zyxwv zyxwv zyxw zyxwvu zyxwvu PERMEABILITY OF PURE AND MIXED GASES 797 penetrant p r e s s ~ r e .In ~ this case dC -= dP k, (7) a t all pressures. EXPERIMENTAL Materials The silicone rubber [poly(dimethyl siloxane)] used in this study was supplied by General Electric Company, Silicone Products Division. The 30 mil thick sample had been postcured for 4 min a t 400°F prior to receipt. The sample density measured to be 1.100 gm/cc using an aqueous density gradient column a t 30°C and agrees well with the reported value of 1.1 &- 0.03 g m / ~ c . ~ Fleming6 determined that the effective molar crosslink density of silicone rubber was vJV, = 1.24 X gmol/cc. No fillers were added to the sample, and birefringence measurements indicate no molecular orientation preference for the sample, as expected. Equipment and Procedures The permeabilities of various pure gases in silicone rubber were measured using standard techniques discussed by numerous author^.^^^^^^ The film was masked only on the upstream face with a piece of aluminum foil in order to define an accurate diffusion area of 9.54 cm2. The bottom side was not masked due to the adhesive nature of the silicone rubber sample. Instead, it was placed directly on the bottom half of the cell with a piece of filter paper over the sintered metal support disk. Due to its slightly adhesive nature, the silicone rubber formed a seal with the metal and additional adhesives were not required. The mixed gas permeation technique used in this study eliminates the need to rely upon a sweep gas a t the downstream face to transport the permeated species to the gas chromatograph for a n a l y ~ i s . ~The - ~ ~gas sample in the downstream receiving volume was allowed to expand into a previously evacuated line where it was then analyzed by gas chromatography. The specific details of this technique are given in a paper by O’Brien et al.14 RESULTS AND DISCUSSION Permeation Results-Pure Gases Figure l a shows the permeation data for helium and nitrogen in silicone rubber a t 35°C for pressures up to 900 psia; Figure l b shows the permeation data for methane, carbon dioxide, and ethylene. These two figures depict different types of permeation behavior in silicone rubber. For the two gases with low critical temperatures, the permeability decreases with increasing pressure. On the other hand, the permeability increases with increasing pressure for CO, and CH,. The reason for this difference in the permeability slopes will be explained below. The permeabilities of CO, and C,H, were zyxw zyx JORDAN AND KOROS 798 zyxwv 1 i 4 Carbon Dioxide * zyxwvu zyxwv zyxw 1 Methane lo000 200 400 Rcssurr (psia) (b) 600 800 zyx Fig. 1. (a) Pure gas permeability of helium and nitrogen in silicone rubber at 35°C. (b) Comparison of the pure gas permeabilities of various gases in silicone rubber at 35°C. measured only up to approximately 700 psia, because above this pressure, the downstream reservoir volume in our system reaches a pressure above the limit of the pressure transducer (10 mmHg) so quickly that the flux measurements could not be made accurately. The permeability values shown in Figure l a and b agree reasonably well with the low pressure data reported by Stern et al.' The slight difference between their data and the data of this study can be attributed to the fact zy zyxwvu zyxwvut PERMEABILITY OF PURE AND MIXED GASES 799 zyxwvuts zyxw r m I I I I 1 Carbon Dioxide X h U x U v t l m m m Methane P 0.5 t' 0 Ethylene I 200 I 1 400 600 Fugacity (psia) I I 800 1000 Fig. 2. The local effective diffusion coefficient [using eq. (5)] of various penetrants in silicone rubber at 35°C. that their silicone rubber sample contained 4.9 vol % silica. The permeability data for He and N, of Stern et a1.l show a similar decrease in permeability with increasing pressure; however, the current data are more extensive, thereby allowing more complex characterization of the effect. Figure 2 shows the effective local diffusion coefficient of several penetrants in silicone rubber at 35°C [calculated using eq. (5)]. The diffusion coefficients for CO, and CH, at zero concentration agree well with the data reported by Stern et al.' a t 35°C. The diffusivity of nitrogen decreases sharply with increasing pressure until around 300 psia, where it levels off and slowly begins to rise with increasing pressure; however, even a t 900 psia the diffusivity of nitrogen is 23% lower than that measured at low pressures. These results suggest that the fractional free volume of the polymer is reduced because of nitrogen's hydrostatic pressure. A t higher nitrogen pressures, however, the solubility level is presumably sufficient to begin to counteract the compressive effect and cause the diffusivity to increase with pressure. The methane and carbon dioxide diffusivity data initially decrease with increasing pressure but eventually reapproach their original values a t higher pressures. Presumably, the solubilities of these gases are sufficient to dominate in the competition between compression and dilation a t elevated pressures. NMR relaxation work by Assink' supports the concept that a penetrant can act on a polymer in two different fashions-either as a dilator or as a compressor of free volume. Assink concludes that, within the experimental limitations of measuring NMR relaxation times, helium gas applied to silicone rubber acts only as a pressuring medium in the sense that it decreases the free volume of the polymer and thus decreases its molecular motion. The negative permeability slopes for the gases in Figure l a and the positive permeability slopes for the gases in Figure l b can be rationalized by examining the actual volume changes of silicone rubber caused by exposure to He pressures up to 1000 psia, as shown in Figure 3. The data indicate that the sample volume actually undergoes compression in the presence of helium, even 800 zyxwvutsr zyx JORDAN AND KOROS zyxwv zyxwvutsrq 0 200 400 800 600 1000 P(PW Fig. 3. Volume change of silicone rubber caused by exposure to high-pressure helium, confirming the compression of silicone rubber under high helium pressure? zyxwvutsr zyxwv zyxw zyx zyxwv though a small but measurable amount of sorption was observed under these conditions. The net result of the two opposing tendencies of dilation and compression of free volume can be quantitatively described by eq. (8), which represents the fractional volume of the polymer as a function of the penetrant pressure and volume fractionl5916*24: uf = vg + Aa( T - Tg) - Ab( p - p,) + yu (8) where vfs is the fractional free volume of the pure polymer a t the reference temperature Tgand p , = 1 atm; v is the volume fraction of penetrant sorbed by the polymer; and Acy and A/3 are the differences in the thermal expansion coefficient and compressibility above and below Tg,respectively. The parameter y is the concentration coefficient which describes the plasticizing tendency of the penetrant for a given polymer. According to Fujita,17 the self-diffusion coefficient depends upon the fractional free volume, as given by eq. (9): 0: = RTA,exp( -Bd/uf) = A,RT (9) where the fractional free volume, uf, is given by eq. (8), and A, and Bd are characteristic parameters related to the size and shape of the penetrant molecule. Both A , and B d are ideally considered to be independent of temperature and penetrant concentration.18 In general, the change in free volume will be reflected by a change in the mobility of the penetrant in the polymer. The mutual diffusion coefficient obtained from permeability data is related to the mobility and mass fraction ( w ) of the penetrant, viz.,19 where dl is so-called “mobility” of the penetrant and is the inverse of the so-called “resistance coefficient” of the medium, which is equal to the product of the effective viscosity of the medium (7)and the effective diameter (a,) of zy zyxwv zyxwvu zyxwvu zyxwvu zyxwvuts PERMEABILITY OF PURE AND MIXED GASES 801 TABLE I Free Volume Parameters for Various Gases in Silicone Rubber at 35°C Penetrant Ad X lo7 (cm2 gmol/J s) Bd Y Nitrogen Methane Carbon dioxide 1.797 1.527 1.955 0.4074 0.4296 0.3957 0.0718 0.226 0.144 the penetrant.,O For cases where Henry’s law applies, as is true in this study, the term ( d In a,/d In wl) is equal to unity. Fleming8 showed that all penetrants can be represented reasonably well by Henry’s law in the lower pressure regions ( < 600 psia). For silicone rubber, one can use eqs. (9) and (10) to relate the mobility of the penetrant to the free volume of the polymer as follows: RTA, = - Bd RTA,exp Vfs + Aa(T - T,) - P,( P -1 - PLP + Y % zyxwv zyxw In the analysis for silicone rubber, the following approximations were used: P g ( p - 1 atm) = 0 for the glass, Aa = Aa for uncrosslinked silicone rubber, since the crosslinking should affect both the rubber and glass expansivity, ha = 7.35 x K-lZo; vfS = 0.025 based on the WFL treatment of the free volume in terms of mechanical properties22;and PL = 0.933 x atm-l.8 PL and Pg are the compressibility coefficients above and below Tg. The mobility data for the pure gases were calculated using eq. (10) for the Henry’s law limits of the data in Figure 2 and were analyzed as a multivariable nonlinear least squares problem in terms of eq. (11)in order to determine A,, Bd, and y. The parameter values for the various gases determined from such an analysis are tabulated in Table I. The values reported for CO, by Kulkarni et a1.l’ for polyethylene a t 35°C are similar in magnitude to those reported in Table I. Since the ( d In a / d In w ) and 1/(1 - wl)terms in eq. (10) are essentially unity over most of the pressure range studied, changes in D,, determined from eq. (7) are reflected directly by changes in Al. The mobility of N2, therefore, decreases with increasing pressure by 12%,while the mobility of CO, increases by 8.5% over the pressure range, thus verifying the notion that N, acts predominantly as a pressurizing agent while CO, acts primarily as a plasticizing agent. Permeation Results-Mixed Gas Since the permeability of helium and nitrogen decrease with increasing pressure due to the reduction in free volume caused by the hydrostatic pressure of the gas, introduction of a mixed gas feed stream containing He or N, should show interesting effects. It is expected that under these conditions, He or N, would act to lower the permeability of all components unless a JORDAN AND KOROS 802 zyx pure zyxwvu zyxwvu zyxwvuts zyxwvut Mixed-pressure 4000 0 20 40 60 80 100 120 140 Pressure or Fugacity(psia) Fig. 4. Permeability of pure CO, (0,pressure; m, fugacity) and CO, in a 10 : 90 mixture of CO, and N, in silicone rubber at 35°C. component in the feed stream has sufficient solubility to overcome this effect. To test this hypothesis, a silicone rubber film was exposed to a 10 : 90 mol% mixture of CO,/N,, and a 50 :50 mixture of CO,/N,, and a 50 :50 mixture of CO,/CH,. The mixture containing methane was also included, since pure methane (Fig. lb) exhibited both types of behavior: a slight decrease in permeability at lower pressures and a gradual increase in permeability a t higher pressures. zyxwvut Carbon Dioxide / Nitrogen Mixtures The CO, permeabilities as a pure component and in a 10 :90 mixture of CO, and N, are shown in Figure 4. The abscissa is based on either pressure or fugacity, and the permeability is then calculated using either the corresponding pressure or fugacity. The fugacity curves are included since gas phase nonidealities for mixtures can be quite important and reduce the mixed gas permeability values a t higher pressures"; hence, only the fugacity-based curves should be compared when rigorously discussing the behavior observed. The upper two curves in Figure 4 represent the permeability of pure CO,; the lower two curves show the CO, permeability in a 10 :90 mixture of CO, and N,. At low CO, pressures the pure and mixed gas curves converge to a CO, permeability value of 5600 Barrers. There is approximately a 3% offset between these two curves, but this is well within the experimental uncertainty of 3-4%. The permeability of CO, is depressed due to the presence of the nitrogen and its permeability decreases with increasing pressure. The permeability value for CO, a t 100 psia reflects the hydrostatic nitrogen pressure of 900 psia which depresses the permeability of GO, and overcomes the plasticizing nature of GO, observed in the pure gas case (Fig. lb). The presence of another penetrant in the feed stream can be accounted for by the addition of a term in the expression for free volume eq. (18) that accounts for the plasticizing contribution of that penetrant 25*26: zy zyx zyxwvutsr zyxw PERMEABILITY OF PURE AND MIXED GASES 0.16 h P PI E v - x .B P 803 zyxwv I I I zyxwvutsrqpon 0.15 0.14 - - . \. 10/90 coz/N2 I 0.11 0 200 1 I 400 600 Fugacity (psia) 800 zyxw Fig. 5. The mobility of CO, in silicone rubber, predicted using eqs. (11) and (12), for the 10 : 90 COJN, system. For the 10 :90 CO,/N, system, the mobility of CO,, predicted using eqs. (10) and (12), is shown in Figure 5 versus fugacity. The pure-gas mobility values are included as points of comparison. The mobility of CO, is depressed due to the significant compressive nature of N, on silicone rubber. Figure 6 shows the permeability of CO, in a 50 :50 mixture of CO,/N,. The low pressure offset between the pure and mixed gas curves is roughly 3.5% and within experimental error. The CO, permeability in the mixed gas case is still depressed below the pure CO, values (compared on a fugacity basis), but the permeability no longer displays a decreasing tendency with pressure as was observed for the 10 :90 CO,/N, mixture. Clearly, a t the lower nitrogen partial pressures in the 50 :50 mixture, the impact of hydrostatic pressure effect can be reduced by the swelling effect of CO,. The swelling effect of CO, cannot zyxwvu Pressure or Fugacity (psia) Fig. 6. Permeability of CO, in silicone rubber a t 35°C for a 50 :50 COJN, feed stream 804 zyxwvuts zyxw JORDAN AND KOROS 100 0 zyxwvu 200 300 Fugacity (psia) *, 400 500 Fig. 7. Mixed gas permeability of CO, [O, actual; eq. (13)] and N, [ O , actual; for a 50 :50 COJN, feed stream in silicone rubber a t 35°C. +, eq. (13)] completely overcome the hydrostatic pressure effect of nitrogen, but it minimizes its influence. The mobility of CO,, in the 50 :50 mixture, predicted using eq. (la), is shown in Figure 5. Once again, the trends observed for permeability are observed for mobility. Figure 5 shows that the mobility of CO, is lower than in the pure gas case but increases with increasing pressure as was observed for the permeability. The flux of a specific component in a mixed gas feed stream in a membrane can be described using Fick's first lawZ6: zyxwv zyxwvu zyx N' = RTAaexp + where up = vfs Aa(T - T,)- A& p - ps). Since the relationship between ua and ub was not known a priori, a trial-and-error method was employed to determine the mutually consistent profile of each across the membrane. The steady-state permeability of a specific component in the mixture was calculated using a forth-order RungeKutta technique similar to that used by Stern et aLZ5(and described in the Appendix). The permeability of CO, and N, predicted using eq. (13) for a 50 : 50 CO,/N, mixed gas feed stream is shown in Figure 7 along with the actual mixed gas data from Figures 6 and 9. The agreement between the predicted values using eq. (13) and the experimental values is 20% or better and similar in magnitude to results reported by Stern et a1.26 The diffusivity of CO, can be calculated for mixed gas systems using eqs. (10) and (12), as shown in Figure 8. The diffusivities of CO, determined from actual mixed-gas data are also shown. There is fairly good agreement between the free volume prediction and the actual data. The diffusivity of CO, in a 50 : 50 CO,/N, mixed determined from permeability data is slightly lower (9.0%) a t higher CO, pressures. Therefore the free volume expressions allow reasonable predictions of the diffusivity and permeability of components in a mixed gas system. z zy zyx PERMEABILITY OF PURE AND MIXED GASES t 1 805 zyxwv zyx zyxwvut ~ L.U 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED 100 200 300 400 F’ressure (psia) Fig. 8. Diffusivity of CO, for mixed gas systems calculated using free volume analysis [eqs. (10) and (12)] and from mixed gas data. (0,10 :90 CO,/N, permeability based; 0, 10 :90 CO,/N, free volume analysis.) (M, 50 : 50 CO,/N, from permeability data; 0, 50 :50 CO,/N, free volume analysis). zy The permeability of nitrogen in silicone rubber is shown in Figure 9. The lower curve represents pure nitrogen, the curve directly above it shows the permeability of nitrogen in a 10 :90 CO,/N, mixture, and the top curves show the nitrogen permeability in a 50 : 50 CO,,”, mixture. The 4% difference between the pure gas and mixed gas curves a t low N, pressures is well within the experimental uncertainties. The permeability of nitrogen in the 10 :90 CO,,”, mixture is slightly increased by the presence of CO,. Presumably this occurs because of the plasticizing nature of CO,. As the partial pressure of nitrogen is increased, however, the hydrostatic pressure effect of nitrogen begins to play a role again, causing the permeability to decrease slightly with increasing pressure. On the other hand, the permeability of nitrogen in the 50 : 50 CO,,”, mixture is significantly higher than the pure gas values and zyxwv zy Prtsglre or Fugrity (psi.) Fig. 9. CO,,”, Permeability of N, in silicone rubber a t 35°C for a 10:90 COz/N, feed system. and a 50:50 806 zyxwvutsr zyxwv JORDAN AND KOROS zyxwv Presswe or Fugacity Fig. 10. The permeability of CO, in a 50:50 mixture of CO, and CH, in silicone rubber at 35OC. increases with increasing pressure. Apparently, as was the case for permeability of CO, in this mixture, the swelling nature of CO, is able to overcome the influence of nitrogen's pressuring effect. Therefore, as the partial pressure of CO, is increased in the feed stream, the permeation behavior shifts from hydrostatic pressure control to swelling control. The mobility of N,, predicted using eq. (12), reveals that the pattern seen in the permeability data is reproduced in the mobility data. For example, the mobility of N, in the 50 :50 CO,/N, system is significantly higher than the pure gas value. zyxwvu zy zyx Carbon Dioxide / Methane Mixtures Figure 10 depicts the permeability of CO, in a 50 : 50 mixture of CO,/CH, in silicone rubber a t 35OC. Once again, the offset between the mixed gas and pure gas permeability values a t low pressures is roughly 5%.The permeability of CO, in this mixture is lower than the pure gas values a t low pressures and higher a t pressures above roughly 250 psia. The slight depression in the permeability of CO, below 250 psia is presumably due to the very slight compressive effect that methane has at low pressures. Methane's permeability decreases slightly (as shown in Fig. Ib) and then begins to increase with increasing pressure around 300 psia. Therefore, the initial hydrostatic effect caused by methane is not overcome until higher CO, pressures are reached. The permeability of CO, is very slightly increased above the pure gas value above 300 psia as shown in Figure 10. The increase in the permeability of CO, in the mixture above the gas value is roughly 3.5% a t the highest pressures and well within the limits of experimental error (3-4%). The mobility of CO, is not significantly affected by the presence of CH, based on calculations using eq. (12). PERMEABILITY OF PURE AND MIXED GASES 807 zy zyxwv zyxw zyxwv Prcsrun of Fugacity @sip) Fig. 11. Permeability of pure CH, and CH, in a 50 :50 mixture of CO,/CH, in silicone rubber at 35°C. The permeability of CH, in the 50 : 50 mixture of CO,/CH, is shown in Figure 11. The permeability of CH, is significantly enhanced by the presence of CO, in the feed stream. The pressure response of the permeability of pure methane is fairly flat with a slight decrease a t the lower pressures. Obviously, as was the case for nitrogen in the presence of CO,, CO, swelling overcomes methane’s tendency to compress the polymer, thus the permeability and mobility of CH, is increased by the presence of CO,. CONCLUSIONS This extended study of silicone rubber over a large pressure and concentration range using both pure and mixed gases has supported the hypothesis that gaseous penetrants can act as either pressurizing or plasticizing agents depending on their sorption level. The permeability of low sorbing gases, such as helium and nitrogen, in silicone rubber decreased with increasing pressure due to hydrostatic compressive effects that caused a reduction in the free volume of the polymer. This decrease in free volume results in a corresponding decrease in the penetrant’s mobility. On the other hand, the hydrostatic effect is overcome by sorption of more condensible penetrants, such as carbon dioxide and ethylene, resulting in an increase in permeability with increasing pressure. Mixed gas studies showed that these compressive effects can be overcome by the addition of a penetrant to the mixture that acts as a dilating agent. The permeability of nitrogen increased above the pure gas values when carbon dioxide was present in the mixture, in direct contrast to what was observed for pure nitrogen. In addition to the increase in nitrogen’s permeability with increasing pressure, there was also a corresponding increase in nitrogen’s mobility in the mixed gas situation. JORDAN AND KOROS 808 The authors gratefully acknowledge support of this work by National Science Foundation (PYI) program under grant number CBT 8351932. Also support from IBMs Fellow program is acknowledge for S. M. Jordan. zy APPENDIX The flux of each component is given by: zyxwvuts zyxwvutsrqpon zyxwvuts where up = vf8 + Aa(T - Tg)- A g ( p - p s ) . Finally, using Henry's law relationship cDi = kDipi ('4.3) Differentiating again and rearranging, we get The mixed gas model can then be expressed as the following nonlinear system of differential equations: YI - ua ('4.6) zyxwv dY1 & =Y3 dY2 = Y4 (A.lO) (A.ll) Equations (A.6)-(A.11) can be solved using a fourth-order Hunge-Kutta technique to give the steady-state concentration profile across the polymer membrane corresponding to any specific zyxwv zyxwvutsr PERMEABILITY OF PURE AND MIXED GASES 809 upstream and downstream partial pressures of each component. The method of solution is as followsn: 1. Input values for the upstream partial pressure for both components and calculate the corresponding Henry’s law concentrations. 2. The downstream partial pressure for both components in this system is zero. 3. Guess values for the Henry’s Law concentration gradients at the upstream face of the membrane, dun/& and dub/&. First guess can be made assuming linear concentration profiles. 4. Integrate eqs. (A.6)-(A.11) across the membrane and check to see if the downstream compositions are both equal to zero. 5. Modify dun/& and dub/& at x = 0 if necessary and integrate across the membrane again until downstream compositions are both zero. 6. Calculate the flux and then the permeability of each component using the following definition of P,: Nil p. = (A.12) zyxwvutsrq zyxwvu zyxwvu zyxwvuts zyxwvut zy Pi, References S. A. Stern, V. M. Shah, and B. J. Hardy, submitted for publication. R. A. Assink, J. Polym. Sci. Polym. Phys. Ed., 12, 2281 (1974). J. Crank, The Mathematics of Diffclsion, 2nd ed., Clarendon Press, Oxford, 1975. R. T. Chern, W. J. Koros, H. B. Hopfenberg, and V. T. Stannett, in Materials Science of Synthetic Membranes, D. R. Lloyd, Ed., American Chemical Society, Washington, D.C., 1985, Chap. 2. 5. V. T. Stannett, W. J. Koros, D. R. Paul, H. K. Lonsdale, and R. W. Baker, Adu. Polym. Sci., 32, 71 (1979). 6. W. J. Koros, Ph.D. Thesis, The University of Texas at Austin, 1977. 7. P. Blanc, General Electric Silicone Rubber Products Division, Personal Communication. 8. G. K. Fleming, Ph.D. Dissertation, The University of Texas at Austin, 1987. 9. R. T. Chern, W. J. Koros, B. Yui, H. B. Hopfenberg, and V. T. Stannett, J . Pvlym. Sci. Polym. Phys. Ed., 22, 1061 (1984). 10. R. T. Chern, W. J. Koros, H. B. Hopgenberg, and V. T. Stannett, J. Polym. Sci. Polym. Phys. Ed., 21,753 (1983). 11. K. D. Ziegel, H. K. Frensdorff, and D. E. Blair, J. Polym. Sci. Part A-2, 7,809 (1969). 12. R. A. Pasternak, J. F. Schimscheimer, and J. Heller, J. Pvlym. Sci. Part A-2, 8, 467 1. 2. 3. 4. (1970). 13. H. Yasuda, J. Appl. Pvlym. Sci., 14.2839 (1970). 14. K. C. O’Brien, W. J. Koros, T. A. Barbari, and E. S. Sanders, J. Membr. Sci., 29, 299 (1986). 15. E. F. Castro, E. E. Gonzo, and J. C. Gottifredi, J . Membr. Sci., 31, 235 (1987). 16. S. A. Stern, S. M. Fang, and H. L. Frisch, J . Pvlym. Sci. A-2, 10, 201 (1972). 17. H. Fujita, Fortschr, Hochpolym. Forsch., 3, l(1961). 18. S. S. Kulkarni and S. A. Stern, J . Pvlym, Sci. Polym. Phys. Ed., 21, 441 (1983). 19. A. L. Hines and R. N. Maddox, Mass Transfer Fundamentals and Applicationss,PrenticeHall, Englewood Cliffs, New Jersey, 1985, p. 27. 20. R. J. Bearman, J . Phys. Chem., 65, 1961 (1961). 21. R. Simha and R. F. Boyer, J . Chem. Phys., 37, 1003 (1962). 22. C. D. Armeniades and E. Baer, in Introduction to Polymer Science and Technology: An SPE Textbvok, H. S. Kaufman, John Wiley & Sons, New York, 1977, Chap. 6. 23. S. M. Jordan, W. J. Koros, and G. K. Fleming, J . Membr. Sci., 30, 191 (1987). 24. S. A. Stern, S. S. Kulkarni, and H. L. Frisch, J . Pvlym. Sci. Pvly. Phys. Ed., 21, 467 (1983). 25. S. M. Fang, S. A. Stern, and H. L. Frisch, Chem. Eng. Sci., 30, 773 (1975). 26. S. A. Stern, G. R. Mauze, and H. L. Frisch, J . Pvlym. Sci. Pvlym. Phys. Ed., 21, 1275 (1983). 27. B. J. Story, Ph.D. Dissertation, The University of Texas at Austin, 1989. Received April 3, 1989 Accepted August 16, 1989