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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.
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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.
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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.
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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:
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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
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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
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JORDAN AND KOROS
798
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1 i
4
Carbon Dioxide
*
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1
Methane
lo000
200
400
Rcssurr (psia)
(b)
600
800
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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
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PERMEABILITY OF PURE AND MIXED GASES
799
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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
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JORDAN AND KOROS
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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?
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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
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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 %
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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
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pure
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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.
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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:
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PERMEABILITY OF PURE AND MIXED GASES
0.16
h
P
PI
E
v
-
x
.B
P
803
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I
I
I
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0.15
0.14
-
-
. \.
10/90 coz/N2
I
0.11
0
200
1
I
400
600
Fugacity (psia)
800
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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
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Pressure or Fugacity (psia)
Fig. 6. Permeability of CO, in silicone rubber a t 35°C for a 50 :50 COJN,
feed stream
804
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JORDAN AND KOROS
100
0
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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:
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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.
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PERMEABILITY OF PURE AND MIXED GASES
t
1
805
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~
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
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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
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JORDAN AND KOROS
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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.
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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
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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.
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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
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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)
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zy
Pi,
References
S. A. Stern, V. M. Shah, and B. J. Hardy, submitted for publication.
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Received April 3, 1989
Accepted August 16, 1989