Dependence of diffusive permeation rates on upstream and downstream pressures

Journal ofMembrane Science, 2(1977)141-151 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

DEPENDENCE OF DIFFUSIVE PERMEATION AND DOWNSTREAM PRESSURES

RATES ON UPSTREAM

I. SINGLE COMPONENT PERMEANT

F.W. GREENLAW*, Department (U.S.A.)

W.D. PRINCE *t, R.A. SHELDEN and E.V.

of Chemical Engineering,

THOMPSON

University of Maine at Orono, Orono, Maine 04473

(Received November 22, 1976)

summary Permeation rates for hexane through polyethylene film were measured over the range of pressures 1 to 20 atmospheres on the upstream side of the membrane and 1 mm to 300 mm Hg on the downstream side. It was found that the rates were linear with upstream pressure when liquid was in contact with the downstream face, but were independent of upstream pressure when vapor at low pressure was in contact with the downstream face. Excellent agreement was obtained between the experimental results and the predictions of a semi-empirical model.

Introduction In reverse osmosis, there is typically a liquid solution under high pressure on one side of the membrane and liquid at approximately atmospheric pressure on the downstream side. In pervaporation, there is typically liquid at approximately atmospheric pressure on the upstream side and vapor at low pressure on the downstream side. We wished to study the general case including that of high pressure on the upstream side accompanied by low pressure on the downstream side. We were interested in the possibility that the downstream pressure could be a useful process variable for improving permeation rates or separations. Before considering the behavior of multi-component permeants, we wished to first study and understand the behavior of single component permeants. The present paper reports on the results of our studies of hexane permeating through polyethylene. Experimental The apparatus used is shown schematically in Fig.1. Pressure was applied *This work is drawn from the theses of William D. Prince and Frederick W. Greenlaw submitted in partial fulfi!lment of the Master of Science degree requirements of the University of Maine at Orono. ‘Present address: Union Carbide Corporation, South Charleston, West Virginia.

NZ TANK VACUUM

PERMEATION -W

-

CELL

____--__--_----r-,--I ; I I

P+R CONTROLLER

-

I

1 D/P CELL

ti

PNEUMATIC CONTROL VALVE

I I I I I I

j

L______________-----A

Fig. 1. Schematic diagram of the experimental

apparatus.

to t,he hexane on the upstream side of the membrane by a cylinder of compressed nitrogen through a pressure reducing valve. The pressure downstream was regulated by a pneumatic control system consisting of a differential pressure sensing cell, proportional plus reset controller, and a pneumatic control valve, operating in conjunction with a vacuum pump. The downstream pressure was maintained constant within It1 mm Hg. Permeation rates were measured by observing the movement of the liquid miniscus in a calibrated sightglass. Hexane could be fed from the pressurized storage tank to reposition the miniscus without disturbing the membrane. All connecting lines between the nitrogen cylinder and the permeation cell were 114” stainless steel tubing. The membrane was mounted in a high pressure permeation cell (Abcor Corporation, Cambridge, Mass.). The cell is shown m Fig.2. The membrane was supported by a porous sintered stainless steel disc. The permeation area was 25.7 cm’. The membrane was polyethylene film (Exxon Chemical Co., Pottsville, Pennsylvania), 0.00254 cm in thickness, with a density of 0.920 g/ cm3. It was used as received, without pretreatment. The hexane was n-hexane (Fisher Scientific Co., Fairlawn, New Jersey), certified 99 mole % purity. The cell was mounted in a thermostated bath and maintained at 30 f 0.1%. Results The experimental results are summarized in Figs.3 and 4. In Fig.3 the permeation rate versus downstream pressure is plotted for various fixed upstream pressures. In each case, when the downstream pressure is below the saturation pressure of hexane (188 mm Hg), i.e., when the downstream face of the membrane is in contact with vapor, the permeation rate decreases in a non-linear fashion with increasing downstream pressure. When the downstream pressure

Fig.2. Schematic diagram of the permeation

0.01 0

I

I

I

50

100

150

DOWNSTREAM

PRESSURE

-,-_

200 (mm

cell.

__+_

250 Hg)

Fig.3. Permeation rate vs. downstream pressure for fixed values of upstream pressure: 0,ll.g atm; 0, 5.4 atm;f, 1.0 atm.

is above the saturation pressure, i.e., when the downstream membrane face is in contact with liquid, downstream pressure has relatively little effect on permeation rates compared to the effect with vapor at the downstream face. From the data in Fig.3 it is difficult to determine the effect of variation in upstream pressure on rate for a fixed downstream pressure, because the same

144

ii

___P------

~___~_p__~~__--“-------

“E Y 5I ::

0.0-

F i zi i= I.z

*.-

_--

__r-

_.--;

,‘/

r

, /’

0.4

-

/z

2 it

,

H ,#’ O.O_/” 0

Permeation

I IO PRESSURE rate

/’

/’

I /

,c’

/’

/NY

/

/I

I 5 UPSTREAM

Fig.4.

z /’

/y/F

/XX.

CH rd

I 15

1 20

1 25

(atm)

vs. upstream

pressure

for fixed values of downstream

pressure:

o,1.5mmHg;*,150mmHg;,b,300mmHg.

membrane was not used for each study and small differences in rate from membrane to membrane tend to obscure the effect of upstream pressure variation. Nevertheless, Fig.3 does suggest that varying the upstream pressure has a strong influence on rate when the downstream pressure is above the saturation pressure (liquid in contact with the downstream face), but little effect when the downstream pressure is well below this pressure (vapor at low pressure in contact with the downstream face). These observations are confirmed by the data shown in Fig.4. In these experiments, for a given membrane, upstream pressure was varied while downstream pressure was held constant. When the downstream pressure is greater than saturation (300 mm Hg) the permeation rate depends strongly on the upstream pressure and is proportional to the pressure difference across the membrane. With low pressures downstream (1.5 mm Hg), variation in the upstream pressure had virtually no measurable effect, in spite of the fact that the pressure difference across the membrane was varied by more than 20-fold. The behavior of the system at 150 mm Hg downstream (vapor near the saturation pressure) is intermediate between that at 300 mm and 1.5 mm downstream. Theory We have compared the experimental results with the predictions of a semiempirical model characterized by the following [l---3] o (1) Permeation is the result of diffusion across the membrane through the amorphous regions of the polymer.

145

(2) Fick’s

law is applicable

but with a concentration

dependent

diffusivity,

i.e. j = -D

dc/dx

(1)

where D is the diffusivity in cm/se?, j is the permeation rate in cm3 hexane/ cm2 polymer-sec., c is the concentration of permeant in the polymer in cm3 hexane/cm3 polymer, and x is the position in the membrane measured from and normal to the upstream face in cm. (3) At each membrane face, the pure permeant in contact with the membrane is in equilibrium with the permeant dissolved in the polymer. (4) The pressure within the membrane is constant at a value equal to the upstream pressure. For this model, it is possible to calculate the permeation rates as a function of upstream and downstream pressures if (1) the relationship between diffusivity and concentration of permeant is known, and (2) the equilibrium relationship between permeant concentration in the polymer and permeant pressure is known. Upon examining the data of Rogers et al. [4] on the sorption and diffusion of hexane vapor through polyethylene of the same density and at the same temperature as in our work, we found t.hat a good fit to their data could be obtained with the following empirical equations: for diffusivity, where D,,

D=D,(l

+ac”)

(Y, and n are constants,

for equilibrium

sorption,

(2)

and

c = S@/P,)

(3)

+ TOJIP,I~

where S, T, and m are constants. Here p is the pressure and p. is the saturation pressure of hexane in mm Hg. If hexane is assumed to be a perfect gas, the ratio p/p0 is equal to the activity, a, of the hexane dissolved in the polymer, and we may rewrite eqn. (3) as

c=Sa+Tam

(4)

If D from eqn. (2) is substituted into eqn. (1) and the result integrated the thickness of the membrane L (in cm), we obtain :

across

x=L s

j dx

x=0

= Jc’

Do(l

+crcn)dc

Here c1 and c2 refer to the concentrations polymer at the upstream and downstream the indicated integration we obtain

jL =D,(c,

(5)

C2

- c,) +D,

-Fn+l

(,:+I - cy+‘)

of permeant dissolved in the faces, respectively. Carrying out

146

To relate the concentrations of permeant in the polymer at the membrane faces to the pressures upstream and downstream we make use of the equality of chemical potential between dissolved permeant and contacting fluid which follows from the assumption of thermodynamic equilibrium. The relevant equations for chemical potential are the following: (1) For the pure liquid upstream at pressure pl, assuming constant molar volume u, (7)

ccl=~“+NPl-Po)

where p” is the chemical potential of the pure liquid at its saturation pressure. (2) For the dissolved liquid at the upstream face, assuming the effect of pressure on activity coefficient (but not chemical potential) is negligible, and assuming the partial molar volume of hexane equal to the molar volume, ~-(~=j.~~+v@,-p~)+RTlna,

(8)

(3) For the dissolved liquid at the downstream face (recall that the pressure in the membrane is assumed constant at p 1), p2=go

+v(p,-po)+RTlna,

(9)

(4) For pure liquid in contact with the downstream face,

Pz=PO +v(P,-Po) (5) For pure

gas behavior,

(10)

vapor in contact with the downstream face, assuming perfect

1-12 = PO + RT In (p2/po) Equating eqns. (7)

(11)

and (S), we find that the upstream activity must be unity:

al = 1

(12)

Equating eqns. (9) and (10) we obtain the following relation for the downstream activity, a2, applicable when we have liquid at, the downstream face, a, = exp

- ZT CPI- PA] [

(13)

Equating eqns. (9) and (11) we obtain the following relation for a2, applicable when we have vapor at the downstream face a?=-

Pz

PO

exp

[ ;Tol,-Po)]

(14)

Eqn. (4) together with eqns. (12) through (14) permits the calculation of values of cl and c2 as a function of upstream and downstream pressures, given values of the empirical constants S, T, and m. Substitution of these values of cl and c2 into eqn. (6) permits the calculation of permeation rates as a func-

147

tion of upstream and downstream pressures given the values of the empirical constants D,, 01,and II. Figs.5 and 6 show the calculated permeation rates using the values: D, = 24.2 X lo-’

cm/sec2;

I*

cm3 hexane

T = 0.0712

cm3 hexane -n

LY= 262

cm3 polymer

cm3 polymer )

m = 3.4;

’

1

S = 0.018

n = 1.33; - cm3 hexane I-cm3 polymer 3

all of which were obtained by fitting the data of Rogers et al. to eqns. (2) and (3). The agreement between the experimental results and those calculated for the semi-empirical model can be seen by comparing Fig.3 with Fig.5 and Fig.4 with Fig.6. The shapes of the curves are very similar, and the values of rate predicted by the model are within a factor of two of those found experimentally (the experimental rates are higher). Considering the numerous assumptions and approximations of the model, and the fact that the empirical constants were fit to data obtained on different samples of polyethylene than those used in our work, we feel the agreement is excellent. It should be noted, for purposes of comparison, that the ordinate scales in Figs.5 and 6 differ from those in Figs.3 and 4 by a factor of two. The applicability of the model described above is somewhat limited by the fact that there are six experimental constants to be determined. If the relationships in eqns. (2) and (4) are simply approximated by straight lines through the origin, that is,

0

I

I

1

1

I

I

50 100 150 DOWNSTREAM PRESSURE

I

200 (mm Hg)

I

2%

Fig.5. Permeation rate vs. downstream pressure for the six-parameter 11.9 atm; middle curve, 5.4 atm; lower curve, 1 0 atm.

model: upper curve,

148

UPSTREAM

PRESSURE

(atm)

Fig.6. Permeation rate vs. upstream pressure for the six-parameter 1.5 mm Hg; middle curve, 150 mm Hg; lower curve, 300 mm Hg

model: upper curve,

(15)

D=k,c

and c = k,a

(16)

where k 1 and kz are constants, a considerable simplification results. Substituting eqn. (15) into eq. (1) and integrating we obtain jL =_“2’ (c,” - czz)

(17)

Now substituting eqn. (16) into eqn. (17) yields

hk22

jL=------

2

a1

2

(

-

a2”)

(18)

Noting that al = 1 in our system (eqn. (12)) and combming the constants we obtain, jL = K(1 - az2)

(1%

where K = k, k22/2

It will be recalled that a2 can be calculated from the system pressures using eqns. (13) and (14).

149

The resulting simplified model is characterized hy a single empirical constant. Its value can be obtained experimentally for example by a pervaporation experiment in which a very low pressure is maintained downstream so that a2 is essentially zero. For this case the product of permeation rate and membrane thickness will equal K. Eqn. (19) together with eqns. (13) and (14) then permits the calculation of permeation rate for all other pressures upstream and downstream. In Figs. 7 and 8 are plotted the variation of permeation rates with pressure predicted using eqn. (19) for the value K = 1.02 X low7 (cm3-cm/ cm2-set). By comparing Figs. 7 and 8 with Figs. 3 and 4 it can be seen that the one-parameter model is in good general agreement with the experimental results. The shape of the experimental curves in Fig.3 is, however, better matched by the six-parameter model than by the one-parameter model. Further it should be noted that the single parameter of the latter model is fit directly to our experimental permeation rates while the six parameters of the former model are fit to the data of Rogers et al. Conclusions The good general agreement between the experimental results and the predictions of both the one-parameter and six-parameter models lends support to their validity. These models suggest that permeation rates for pervaporation with a high vacuum downstream should be the same as those obtained in reverse osmosis only with an infinite pressure upstream [ 51. This is shown in Fig.9 for the six-parameter model. At other than infinite upstream pres-

DOWNSTREAM

PRESSURE

(mm Hgl

Fig.7. Permeation rate vs. downstream pressure for the one-parameter 11.9 atm; middle curve, 5.4 atm; lower curve, 1.0 atm.

model: upper curve,

I

1

I

I

5

IO

15

20

UPSTREAM

PRESSURE

1

25

(atm)

Fig.8. Permeation rate vs. upstream pressure for the one-parameter model: upper curve, 1.5 mm Hg; middle curve, 150 mm Hg; lower curve, 300 mm Hg.

c ’

0.6

X

0.0 0

too PRESSURE

200 DROP ACROSS

300 MEMBRANE

400 (ctmb

Fig.9. Permeation rate vs. pressure drop across the membrane for the six-parameter model upper curve, vacuum downstream;

lower curve, 1 .O atm downstream.

sure, the permeation rates in pervaporation with a high vacuum downstream will be greater than those obtained in reverse osmosis. This will be true even though the pressure difference across the membrane may be far greater in reverse osmosis than in pervaporation. This was confirmed experimentally in

157

our study. We found, for example, higher permeation rates with 1 atmosphere upstream and 1 mm vacuum downstream than with 20 atmospheres upstream and 300 mm downstream. The models also suggest that for pervaporation with a high vacuum downstream, increasing the upstream pressure will have no effect on permeation rate. This was confirmed experimentally in our work, and was also reported by Binning et al. [ 61 and Aptel et al. [ 71 e It should be noted that this independence of permeation rate on upstream pressure is expected only for upstream pressures greater than the saturation pressure. For upstream pressures below the saturation pressure, the permeation rate should fall toward zero with decreasing upstream pressure. This is not discernible in Fig.9 because of the scale of the plot. The one-parameter model predicts an essentially linear dependence of permeation rate on pressure difference in reverse osmosis for low upstream pressures, but a decreasing dependence of rate on pressure difference at high pressures. This is shown in Fig.9. In practice, a departure from linearity in plots of permeation rate versus pressure difference in reverse osmosis may result from membrane compaction, but the model suggests that departure from linearity can occur at sufficiently high pressures even in the absence of compaction. The fact that for high vacuum downstream, the permeation rate is essentially independent of upstream pressure for a diffusive mechanism suggests that the observed effect of upstream pressure on permeation rate can be used to distinguish between porous and non-porous membranes. The latter would be expected to show a dependence of permeation rate on upstream pressure. Finally, the six-parameter model may facilitate the modeling of more complex multi-component systems because it sharply reduces the number of empirical constants necessary to describe the system’s behavior.

References 1 R.B. Long, Abstracts, Division of Petroleum Chemistry, Chicago Meeting of the American Chemical Society, August 30-September 4, 1964. 2 D.R. Paul and O.M. Ebra-Lima, J. Appl. Polym. Sci., 14 (1970) 2201. 3 S. Rosenbaum, Polymer Letters, 6 (1968) 30’7. 4 C.E. Rogers, V. Stannett, and M. Szwarz, J. Polym. Sci., 45 (1960) 61. 5 D.R. Paul and O.M. Ebra-Lima, J. Appl. Polym. Sci., 15 (1971) 2199. 6 P. Aptel, J. Cuny, J. Tozefonvicz, G. Hovel and J. Neel, J. Appl, Polym. Sci., 18 (1974) 351. 7 R.C. Binning, R.J. Lee, J.F. Jennings and E.G. Martin, Ind. Eng, Chem., 53 (1961) 45.