72
Ind. Eng. Chem. Fundam. 1903, 22, 72-70
Diffusional Boundary Layer Resistance for Membrane Transport' M. P. Bohrer Bell Laboratories, Murray Hill, New Jersey 07974
Diffusional fluxes across well-characterized membranes were measured for a variety of solutes in a stirred diffusion cell. The results were analyzed in terms of a diffusional boundary layer model based on transport through a membrane containing a uniform array of circular pores. Diffusional boundary layer resistances (Rb) were calculated from the measured fluxes and compared to values obtained using a polarographic technique and to values determined indirectly from the stirring speed dependence of the overall mass transfer resistance. The measured values of Rb were found to be strongly influenced by the membrane porosity, Rb for the lowest porosity (1.7%) membrane being twofold greater than Rb for membranes with porosities of 6.2 or 19.9%. Values of R , measured polarographically compare well with values determined directly from flux measurements as long as the membrane porosity is high (>6%). Values of Rb determined from the stirring speed dependence of the total mass transfer resistance were found to give erratic results.
Introduction Membranes are increasingly being used for a wide variety of separation processes including ultrafiltration, reverse osmosis, dialysis, and ion exchange. One aspect of membrane transport which has been recognized as a potential factor in controlling the transmembrane diffusion of solutes is the influence of a diffusional boundary layer resistance at the membrane-solution interface. In many cases, the magnitude of this diffusional boundary layer resistance may be comparable to or greater than the intrinsic resistance of the membrane itself. When this occurs, the selectivity and overall filtration characteristics of the separation process will no longer be due to just the properties of the membrane. This is expressed in eq 1, where the total resistance to transport, R, is written as the sum of intrinsic membrane resistance, R,, and boundary layer resistance, Rb, on each side of the membrane. R = R, 4- 2Rb (1) For a stirred diffusion cell, as used in this study, R is defined as (2) R = (C, - C,)/J where C1and C2 are the high-side and low-side concentrations of the solute in the chambers on either side of the membrane and J is the solute flux. Only when R, >> Rb will measurements of solute transport across membranes yield information on the intrinsic membrane resistance alone. A quantitative understanding of the diffusional boundary layer resistance is therefore required to interpret the results of membrane transport studies and to permit quantitative design of membrane processes. A convenient device for testing membranes is the stirred diffusion cell for which a variety of experimental methods have been devised to measure the diffusional boundary layer resistance. One such method involves measuring R at various stirring rates ( w ) and then plotting R vs. w Y where y is a positive number chosen to give the best straight line fit to the data. Since the intercept of such a plot corresponds to infinite stirring rate, Rb is assumed to be zero and the intercept is therefore equal to R,. The diffusional boundary layer resistance can then be calculated for lower stirring rates through the use of eq 1. Kaufmann and Leonard (1968) used this method to measure Rb for several sugars diffusing through a cellophane 'This work was presented in part at the AIChE 1982 Annual Meeting. 0196-4313/83/1022-0072$01.50/0
membrane. They examined the effects of stirring rate, solute size (diffusivity), and temperature on the measured values of Rb A second approach to measuring Rbhas been to measure mass transfer rates to or from a homogeneous surface which has replaced the membrane in a stirred diffusion cell. In one such study, Colton and Smith (1972) measured dissolution rates of benzoic acid from the base of a diffusion cell as a function of stirring rate. They examined the dependence of the local mass transfer coefficient on radial position and developed an empirical correlation for the average mass transfer coefficient (= l/Rb). Another experimental technique involving a homogeneous surface is the electrochemical measurement of mass transfer rates which are made by replacing the membrane with an electrode. Scattergood and Lightfoot (1968) measured mass transfer rates of Ag+ ions to a silver cathode in a stirred chamber. Holmes et al. (1963) performed similar studies using a copper electrode and copper sulfate solutions. Beck and Schultz (1972) measured the diffusion rate of potassium ferrocyanide to a platinum electrode in order to estimate mass transfer resistances in their flow-through diffusion cell. A final approach to measuring Rb has relied upon recently produced "track-etched" membranes which have a uniform, well-defined pore structure. Examples are the mica membranes utilized by Malone and Anderson (1977) which have rhomboidal-shaped pores or the commercially produced polycarbonate membranes (Nuclepore Corp.) which have circular pores of nearly uniform radius. The advantage in using these membranes is that R, can be calculated for a given solute once the pore radius (r),pore length (L),and pore density (n)are known. For a membrane with cylindrical pores, R, is given by R, = L/(D,nw2) (3) where D , is the diffusion coefficient of the solute within the pore. For the case when the solute sue is small relative to the pore size, D, can be replaced by D,, the free solution diffusion coefficient. This technique was used by Malone and Anderson (1977) to measure Rb at different stirring rates for potassium chloride diffusion through well-characterized mica membranes of low porosity. Direct comparison of the results of these various experimental techniques for measuring diffusional boundary layer resistances in stirred diffusion cells have generally not been possible due to differences in cell/stirrer geometries. Presumably these differences affect the degree of 62 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 73
mixing within the cell chambers and therefore the magnitude of the diffusional boundary layer resistance. An additional concern in comparing Rb measured by these various methods is the effect of the surface homogeneity. As indicated in the analysis of porous membrane transport by Keller and Stein (1967) and more recently by Wakeham and Mason (1979), the actual mass transfer rate may be significantly different than that which would be expected, based solely on the geometrically open area of the porous membrane. Therefore, studies using homogeneous surfaces to estimate Rb may not be directly comparable to studies with porous or heterogeneous membranes. It is the goal of this study to compare several different methods for measuring Rb using the same stirred diffusion cell. In addition, membranes of well-defined pore structure and of different porosity will be used in order to further examine the effects of membrane porosity on the measured values of Rb. Experimental Procedure Membrane Characterization. "Tract-etched" polycarbonate membranes with nominal pore diameters of 0.1, 1.0, and 5.0 pm were obtained from Nuclepore Corp. (Pleasanton, CA). Since the pores of these membranes are essentially uniform circular cylinders (Liabastre and Orr, 1978), pore length is very nearly equal to membrane thickness, exceeding membrane thickness only to the extent that the pores are not aligned normal to the membrane surface. Information supplied by Nuclepore indicated that the maximum deviation from the normal is 29O. Assuming all deviations of 0 to 29" to be equally probable, average pore length would exceed membrane thickness by 6.8%. Employing this factor, pore length (L)is determined by weighing a membrane of known area. L is given by 1.068W L= (4) ( 1 - nnr2)Ap where W is the weight and p, the density of polycarbonate, is taken to be 1.19 g/cm3. The denominator includes a correction for membrane porosity (nnr2),which is significant for one of the membranes used. Scanning electron microscopy (ETEC Autoscan, ETEC Corp., Hayward, CA) was used to measure pore density ( n )for all membranes. Samples from several membranes of each size were mounted and lightly coated with gold by sputter coating (Mini-coater, Film-Vac, Inc.). Pore density was determined by counting the number of pores in scanning electron microscope (SEM) photographs obtained at 500-15OOOX magnification. The area in the photographs was calibrated with diffraction grating replicas (11340 lines/cm) or latex spheres (2.02 pm diameter, E. F. Fullam, Inc., Schenectady, NY) which were placed directly on the membrane samples. Scanning electron microscopy was also used to measure pore radii ( r ) for membranes with 5.0 pm nominal pore diameters. Water flow measurements were not used as discussed below due to the possibility of significant end effects influencing the flow at the entrance and exits of the pores. A cathetometer was used to measure dimensions (&O.OOOl cm) on photographs of pores at 2000-3OOOX magnification. A 1000-mesh gold grid (E. F. Fullam, Inc.) photographed under the same conditions was used for calibration. Pore radii were measured for the smaller pore membranes by determining water flow rates as described previously (Deen et al., 1981). Briefly, membranes were mounted in the diffusion cell (see below) and subjected to applied pressures of 5 to 20 cm H20. Flow rates measured at pressures less than 5 cm H 2 0 were found to vary
Table I. Solutes Used for Diffusion Experiments solute D,, cm*/sa source urea 1.38 x Longsworth (1954) glucose 6.73 x Longsworth (1953) sucrose 5.23 X Gosting and Morris (1949) K,Fe(CN), ficoll
dextran a
5.0 6.2 4.0
X X X
Jahn and Vielstich (1962) present study present study
10.'
Diffusion coefficients at 25 "C.
considerably with time. Pressures greater than -50 cm H20 were found to stretch the membrane. Flow rates were determined by weighing timed samples and pore radii were calculated using the Poiseuille equation
=
[
~ P L Q1 1 ' 4
nnAAP
(5)
Solute Preparation. The solutes used for diffusion experiments and their diffusion coefficients are listed inTable I. The urea (J. T. Baker), D-glucose (Mallinckrodt), sucrose (J. T. Baker), and potassium ferrocyanide (Fisher Chemicals) were used as received. In addition to these low molecular weight solutes, two polysaccharides, dextran and ficoll (Dextran T70 and Ficoll70, Pharmacia Fine Chemicals, Piscataway, NJ) were utilized for diffusion experiments. The dextran and ficoll samples were fractionated by gel chromatography (Sephacryl300,Pharmacia Fine Chemicals) in order to obtain narrow molecular weight samples. The eluant used on the gel columns was 0.02 M ammonium acetate, pH = 7.0. For the first fractionation, 2 mL containing 0.2 g of the polymer was run on the column (column diameter = 2.6 cm, total volume z 230 mL) and 23 mL centered at the peak concentration was saved. Flow through the gel column was controlled at 1.0 mL/min with a peristaltic pump (Varioperplex 11, LKB Instruments, Rockville, MD), concentrations were measured with a differential refractive index monitor (Model R403, Waters Associates, Milford, MA), and samples were collected on an automatic fraction collector (2070 Ultrorac 11, LKB Instruments). This was repeated five times, the samples were pooled, freeze-dried (Model 10-010, Virtis Co., Gardiner, NY), and then redissolved in eluant (-0.35 g in 6.5 mL). Samples were then refractionated (2 mL at a time) collecting 23 mL centered at the peak concentration. These fractions were then pooled and freeze-dried. Weight-average molecular weights were measured by light scattering (KMX-6 Low Angle Light Scattering Photometer, Chromatix, Sunnyvale, CA) and number average molecular weights were measured by membrane osmometry (Model 231, Wescan Instruments, Santa Clara, CA). Diffusion coefficients for the dextran and ficoll fractions were measured by quasielastic light scattering. Limiting Current Measurements. A technique based on polarography (Kolthoff and Lingane, 1941) was used to determine the diffusional boundary layer resistance for potassium ferrocyanide (K,Fe(CN),). Under the influence of an applied potential, Fe(CN),4-is oxidized to Fe(CN)63at the surface of a platinum electrode immersed in an electrolyte solution. As the applied voltage is increased, a point is reached at which the concentration of Fe(CN)," is zero at the surface of the electrode and the measured limiting current is directly related to the mass transfer rate of Fe(CN)64-. For the experiments in this study, a platinum foil electrode was placed in the diffusion cell (see below) in place of the membrane and a second platinum wire electrode was placed in the cell chamber. The platinum foil electrode acted as the working anode at which
74
ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
oxidation of the ferrocyanide occurred. A calomel reference electrode was placed a t the platinum foil electrode surface by means of a small Teflon capillary tube. The cell was filled with a 1.0 M KC1,O.Ol M K,Fe(CN)6, and 0.01 M K4Fe(CN)6solution. An increasing voltage was applied across the platinum electrodes, and the limiting current measured at various stirring rates. The diffusional boundary layer resistance was determined frorn
Table 11. Dextran and Ficoll Molecular Weights 51 000 48 200
M, LGn __
Mn
43 700 44 800
1.17
1.08
Table 111. Membrane Characterization no tn pore diam, pm
110.
Values of R are obtained from the slope of a plot of in (AC,/AC) VS. t. The procedure used for the diffusion experiments was similar in each case. Initially, one chamber was filled with an aqueous solution of the solute (0.02-0.05 g/100 mL) while the other side was filled with deionized water. The only exceptions are the diffusion experiments with potassium ferrocyanide in which the ”downstream” side was filled with 1.0 M KCl and the “upstream” side contained 0.001 M K4Fe(CN), in 1.0 M KC1. Stirring was initiated and the concentration difference was recorded for 30-200 min. The slope of a plot of In (AC,/AC) vs. t was obtained by linear regression and regression coefficients usually exceeded 0.99. Experimental Results The results of the molecular weight measurements for the dextran and ficoll fractions are shown in Table 11. The value of A&/&&, prior to fractionation was 1.26 for dextran and 1.63 for ficoll. Diffusion coefficients for these two fractions are given in Table I.
.41\v __
dextran ficoll
.
where ne is the number of equivalents per mole of reactant, F is Faraday’s constant, A is the electrode area, CB is the bulk concentration of reactant, and iL is the limiting current. Diffusion Cell Experiments. The stirred diffusion cell was constructed from Lucite and is similar to that described by Deen et al. The membrane is held between two annular disks which are clamped between two identical cell chambers. The volume of each chamber is 39.1 cm3 and the exposed membrane area is 11.4 cm2. Teflon-coated magnetic stirring disks (2.2 cm diameter) with one side smooth and the other containing ridges is positioned with the smooth side 0.6 cm from the membrane surface. Each stirring disk is epoxied to a stainless steel shaft which is allowed to rotate in a Teflon bearing held inside each chamber. An external magnet connected to a motor with an optical tachometer and feedback control (Series H Motor Controller, G. K. Heller Co., Floral Park, NY) is used to rotate the stirring disks at a constant rate f 1rpm. The diffusion cell is jacketed for temperature control (25.0 f 0.1 “C) and each cell contains an inlet and outlet port for sampling. The difference in solute concentration between each chamber is continuously recorded by pumping (Model P-3 peristaltic pump, Pharmacia Fine Chemicals) the solution from each chamber through separate cells of a differential refractive index monitor (Model R403, Waters Associates) and then back to the chambers. To ensure that the concentrations measured by the refractometer were representative of the concentrations in the cell chambers, the fluid was pumped at a sufficiently high rate relative to the membrane diffusion rate. Conservation of mass for the diffusion cell experiment indicates that the solute concentration difference between the two chambers is given by
-
I
solute
--
~
1
2 3
~
~.
~
... .
r,pm
0.1 1.0
0.075 0.581
5.0
2.23
~
~
rt,
cm-2
L , pM
nnP
__.l_______.____l_ ~_
3.56 X 10’ 1.88 x 10’ 1.11 X 10’
6.68
11.08 12.00
0.062 0.199 0.017
The results of the characterization for the three membranes used in this study are shown in Table 111. A minimum of 4 water flow measurements (standard deviation = f3% of the mean) were used to calculate the pore radii for the 0.1 and 1.0 pm nominal pore diameter membranes. The pore radius for the 5.0 pm nominal pore diameter membrane was determined from 19 SEM photographs containing 40 pores. Significant differences were noted between the measured pore radii and the nominal pore radii reported by the manufacturer (0.075 vs. 0.05 pm, 0.581 vs. 0.5 pm, and 2.23 vs. 2.5 pm). Pore densities were determined using samples of 2-3 membranes of each pore size. No differences in pore densities were noted between different membranes taken from the same lot. The membrane porosity ( n w 2 )varied over an order of magnitude from 0.017 to 0.199. The highest porosity membrane (1.0 pm nominal pore diameter) was observed in SEM photographs to contain a number of pores which appeared to overlap on the membrane surface. In 16 photographs containing a total of 1625 pores, 86% appeared to be single pores on the surface, 10% were doublets (two pores intersecting), and 3% were triplets. It is likely, however, that only a fraction of the pores overlapping on the surface continue to overlap entirely through the membrane, since they are not all aligned normal to the surface. This pore overlap would be expected to cause the measured water flow rate to be high, which would result in an overestimation of the pore radius. Due to uncertainty in the exact amount of pore overlap and lack of a suitable correction factor, however, no modification was made in the measured values of r. Based on the uncertainty in the measured values of Q, W , A€‘, and dimensions in SEM photographs, the maximum error in the determination of I-, n, and L is estimated to be *2%, &5%,and *l%,respectively. The uncertainty in the membrane porosity and the membrane resistance (eq 3) are therefore calculated to be approximately &lo%. Values of the total mass transfer resistance, R, measured in 54 diffusion cell experiments are listed in Table IV. As expected, R can be seen to decrease with increasing stirring speed for any given solute/membrane pair. For a given solute and stirring speed, R decreases with increasing membrane porosity. These relationships are further demonstrated in Figure 1, where the overall mass transfer coefficient, K (K = l / R ) ,is plotted vs. stirring rate (a)for the diffusion of sucrose in each of the three membranes. As can be seen, increasing the stirring speed increases the mass transfer coefficient, the effect being greatest for the highest porosity membrane. In fact, the lowest porosity membrane shows almost no dependence on stirring, indicating that the boundary layer resistance is a small fraction of the total mass transfer resistance for this membrane. Diffusional boundary layer resistances were calculated from the membrane characterization data and R values
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 75
Table IV. Total Mass Transfer Resistance Data membrane 2
membrane 1 solute
w,
R , s/cm
rev/min
urea
glucose
sucrose
solute
w , revlmin
50 100 150 200 250 300 350 50 100
3630 2445 2026 1793 1626 1508 1368 6289 4369
urea
50 100 150 200 250 300 325
glucose
50 100 150 200 250 300 350
7174 53 24 4478 3773 3619 3295 3203
sucrose
75 100 150 200 50 100 150 200 250 300 325
ficoll
dextran
50 100 150 200 250 300 100
membrane 3
R , s/cm
solute
3 356 2 229 1840 1623 1448 1329 1278 4 404 3 863 3 243 2 812 6 519 4 716 3 940 3 553 3 179 2 991 2 858 33 640 244 454 22 752 18 976 1 7 674 14 807 34 960
urea
50 100 150 200 250 300
11024 9 572 8 982 8 717 8 468 7 821
glucose
100
19 699
sucrose
50 100 150 200 250
27 178 24 815 22 438 22 285 21 608 20 493
W,
rev/min
300
R , s/cm
Membrane numbers refer to Table 111. 4600
40
I
I A
V
3 p:
I
5 22oot A
\\
\\
SUCROSE GLUCOSE UREA
I
i
, i L MEMBRANE 3. nar2=0017
0
50
100
150
200
250
300
350
STIRRING SPEED IREVIMINI
Figure 1. Effect of stirring rate (w) on the overall mass transfer coefficient (K).
using eq 1 and 3. The results of these calculations are shown in the third column for each membrane in Table V. In general, Rb decreases with increasing stirring speed, and for a given stirring speed, Rb decreases as the size of the solute increases (diffusivity decreases). In the above calculations, it is assumed that the solute flux occurs in only one direction, i.e., normal to the membrane surface. For membranes with discrete pores, however, solute diffusion near the pore openings is no longer one-dimensional. Keller and Stein (1967) have analyzed this entrancefexit effect for the case of diffusion through a membrane containing a uniform array of parallel, circular pores. In their analysis the diffusional boundary layer resistance is modeled by an unstirred layer of thickness 6 on each side of the membrane, where 6 is related to Rb by 6 = D,Rb (8) Keller and Stein’s results can be expressed as
Figure 2. Effect of membrane porosity on the measured boundary layer resistance. Values were obtained at a stirring speed of 100 rev/min.
where the function is plotted in their Figure 4 and given analytically in terms of a Bessel function expansion. The second term on the right-hand side of eq 9 contains both the Rb and entrance-and-end-effect contributions to R. Using the measured values of R and R,, this equation was solved iteratively for 6. The results, expressed in terms of corrected Rb values, are given in the last column of Table V. The end effect correction has the greatest impact for the membrane with lowest porosity (1.7%), reducing the uncorrected Rb values 39% on average. This correction amounts to only 10% and 11% for the membranes with porosities of 6.2 and 19.9%, respectively. In spite of this correction, values of Rb for the lowest porosity membrane (membrane 3 in Table V) are -2-3-fold greater than corresponding values for either of the other two membranes. This is demonstrated in Figure 2 where corrected Rb values are plotted vs. membrane porosity for sucrose, glucose, and urea diffusion at a stirring rate of 100 rev/min. Rb values for the membranes with porosities of 6.2 and 19.9% are within the estimated experimental error ( 10%) of each other as shown. Several investigators have used Keller and Stein’s analysis in the study of transmembrane diffusion. Malone
76
Ind. Eng. Chem. Fundam., Vol. 22,
Table V.
No. 1, 1983
Diffusional Boundary Layer Resistance Data membrane 1a
membrane 2
’
Rb, S/Cm solute w , rev/min uncor cor urea 50 1427 1291 100 835 7 54 150 363 625 200 457 509 250 382 425 300 366 328 350 265 296 glucose 50 2350 2123 100 1390 1252
_ _ _ I
sucrose
50 100 150 200 250 300 350
2564 1639 1216 8 64 787 625 579
solute urea
glucose
2315 1476 1092 772 702 555 514
sucrose
ficoll
dextran a Membrane numbers refer
to Table 111.
membrane 3
Rb, s/cm W , rev/min uncor cor 50 1477 1333 100 91 3 822 150 719 645 200 610 547 250 523 468 463 41 4 300 390 325 438 1789 1610 75 100 1364 1518 150 1208 1083 200 993 887 50 2 728 2 458 100 1826 1640 150 1438 1288 1112 1245 200 250 943 1058 300 964 8 57 325 897 797 50 1 2 336 11 046 100 7 743 6 878 6 892 6 106 150 5 004 200 4 393 4 353 2 50 3 802 2 920 300 2 501 100 9 326 1 0 530
26
uncor
cor
urea
50 100 150 200 250 300
3005 2279 1984 1851 1727 1403
2171 1512 1245 1124 1011 718
glucose
100
4708
3133
sucrose
50 100 150 200 250 300
6973 5792 4603 4527 4188 3631
4861 3790 2711 2642 2335 1829
W ,
rev/min
40
,
I
ar/2
where the first term on the right-hand side is the intrinsic membrane resistance, the second term is the boundary layer resistance, and the final term is due to the entrance and exit effects. They report that this approximation is valid for nar2 < 0.5 and 8 < 6 / r < 500. Using eq 10 to calculate Rb yields values which range from 2-10% greater than those obtained using eq 9. A closer examination indicates that for this approximation to be within 5% of Keller and Stein’s solution, the following condition must also be satisfied (nar2)(6/r) < 10
(11)
Wakeham and Mason (1979) have also introduced an approximation for determining Rb based upon Keller and Stein’s analysis. Their results can be expressed as
R = R,
solute
’ See text for explanation of corrected and uncorrected Rb values.
and Anderson (1977) have devised an approximation based upon Keller and Stein’s results which yields
L
Rb,S/cm
+ D, E( L,
where the term, A/AeffOrif,is given in their eq 18. Values of Rb calculated with eq 12 were found to be 10% higher than values calculated using Keller and Stein’s solution for all cases. The stirring speed dependence of the measured R values was used to obtain a second estimate of the intrinsic membrane resistance, R,, as described by Kaufmann and Leonard (1968). Plotting R vs. UT gives a reasonably straight line where the intercept (correspondingto infinite stirring) is assumed equal to R,. Figure 3 shows an example of such a plot where y was chosen to give the best straight line fit by maximizing the linear regression coefficient of determination. Table VI lists the values of y
0
10
20
30 w y110
40
’ MIN/REV)
50
60
70
Figure 3. Plot of overall mass transfer resistance, R, vs. the inverse of the stirring speed. The value of y for this plot is 0.71. The intercept of this plot represents the intrinsic membrane resistance, Rm.
Table VI. Comparison of Methods for Determining the Intrinsic Membrane Resistance R,,
membrane a 1
2 3
solute urea sucrose urea sucrose urea sucrose
y
0.71 0.51 0.73 0.54 0.25 0.14
s/cm
intercept of R vs. W - Y L / ( D ” r r 2 ) 666 750 590 801 2793 -2130
775 2 046 403 1063 5 014 1 3 231
Membrane numbers refer t o Table 111.
which were calculated for urea and sucrose diffusion in each of the three membranes. Also shown in Table VI is a comparison of the values of R, obtained from plots of R vs. UT and the values of R, obtained from the measured membrane parameters using eq 3. As can be seen, the values of R, obtained by these two methods are not in
Ind. Eng. Chem. Fundam., Vol. 22,
Table VII. Diffusional Boundary Layer Resistance for K,Fe( CN), Rb, sicm diff in memb
W,
a
rev/ min
electrochem
50 100 150 200 250 300 350
1515 1124 87 6 732 627 562 51 5
la
memb 2
memb 3
1330
1259
3640
791 47 3
Membrane rnumbers refer to Table 111.
agreement (differing by as much as a factor of -3) and there is no obvious trend of a relationship between them. It is unlikely that this discrepancy is due to errors in estimating R, from the measured membrane parameters, since as described above, the error in calculating R, from eq 3 is f10% or less. In addition, the assumption of cylindrical pores oriented perpendicular to the membrane surface, upon which eq 3 is based, has been well documented for these membranes (Deen et al., 1981; Liabastre and Orr, 1978; Schultz et al., 1979; and Van Bruggen et al., 1974). One likely reason for the large discrepancies noted is simply the error incurred by extrapolating the data to = 0. The value of the intercept is very sensitive to slight changes in the values of R, especially since a rather limited range of stirring speeds was examined. It is estimated that the values of R measured in the diffusion cell experiments are accurate to -10%. Other possible explanations for this discrepancy will be discussed in the next section. The final results to be presented are the polarographically determined values of Rb for potassium ferrocyanide diffusing to a platinum electrode. Table VI1 lists these values along with Rb values determined by measuring K,Fe(CN), diffusion through the porous membranes. The values of Rb obtained with the highest porosity membranes (membranes 1and 2 in Table VII) are essentially the same as those obtained electrochemically, suggesting that these membranes behave similar to a completely homogeneous surface. Decreasing the porosity, however, can be seen to cause Rb to increase, the value for the lowest porosity membrane being -3-fold greater than that determined electrochemically. The values of Rb listed in Table VI1 have been corrected for entrancefexit effects as described above.
Discussion Interfacial mass transfer data are often expressed in a correlation of the form NSh a N S c b N b c
(13)
where NShis the Sherwood number, Ns, is the Schmidt number, and NRe is the Reynolds number. Using a multiple regression analysis, the data in Table V for the 1.0 pm nominal pore diameter membrane yield NSh 0: Nsc0.33NR$66.The Schmidt number dependence agrees very well with the theoretical prediction of I f 3 based on boundary layer theory (Levich, 1962) as well as with previous experimental findings (Kaufmann and Leonard, 1968). The exponent for the Reynolds number is higher than the 0.58 found by Malone and Anderson (1977) or the 0.57 found by Colton and Smith (1972). The most likely reason for this difference is the fact that the cellfstirrer geometries reported for these studies are different from that used here.
No. 1, 1983 77
Three experimental methods for determining diffusional boundary layer resistances at membrane surfaces were examined in the present study. In one method, Rb was determined directly from the measured overall mass transfer resistance, the membrane resistance having been calculated independently using known values of r, n, and L. The other two methods for determining Rb, are compared to this "direct" method. First, extrapolation of measured R values to infinite stirring speed in order to determine R, is found to give highly erratic results (see Table VI). As mentioned in the above section, one reason for this is probably the limited range of stirring speeds examined, resulting in large uncertainties in locating the intercept of a plot of R vs. w?. A second possible cause for these highly erratic results is suggested by the findings of Smith and Colton (1972),who analyzed the problem of mass transfer between a fluid undergoing solid body rotation and a coaxial circular disk in an otherwise impermeable, infinite surface. Their analysis indicates a laminar-to-turbulent type transition in the diffusional mass transfer resistance at a Reynolds number of 3 X lo4. Smith and Colton confirmed the existence of this transition experimentally by measuring dissolution rates of a disk of benzoic acid embedded in the base of a stirred diffusion cell at various Nb. Since the data in the present study were obtained at relatively low Reynolds numbers (NReI 2.4 X lo3),extrapolation to high stirring rates (beyond the transition point) would not be expected to give accurate results. Kaufmann and Leonard (1968), who used this approach to estimate Rb values for transport across a cellophane membrane, measured diffusion rates at Reynolds numbers from 2.4 X lo4to 4.4 X lo5. Their data appear to give consistent results; however, uncertainties in the intrinsic structure of their membranes preclude any direct comparison with the present results. The second method for measuring R b which has been examined relies upon polarographically determined mass transfer rates of electroactive species to a platinum electrode. Comparison of Rb measured in this manner to values measured from diffusion experiments (see Table VII) indicates excellent agreement between these two methods when the membrane porosity is relatively high (>6%). As the porosity of the membrane decreases, Rb is found to increase relative to the value obtained electrochemically. Beck and Schultz (1972) used this method to measure Rb in a flow-through diffusion cell. They found good agreement between values of Rb measured electrochemically and values determined directly from diffusion through a well-characterized membrane with a porosity of 10%. As indicated by the above results, as well as from a comparison of the data presented in Table V, membrane porosity can be seen to have a significant effect on the measured value of Rb. In fact, Rb values for the lowest porosity membrane ( n d = 0.017) are 2-3-fold greater than values for the other two membranes examined. The small difference between Rb values for the membranes with porosities of 6.2 and 19.9% and the polarographic data (Table VII) suggests that membranes with porosities of -6% or greater behave essentially as homogeneous surfaces. Further increases in porosity would not be expected to affect Rb. Similar results were found by Malone and Anderson (1977) in their study of potassium chloride diffusion through well-characterized mica membranes with porosities of 0.3-5.7%. They noted that their measured values of Rb were -3 times greater than those obtained by Colton and Smith (1972), who measured dissolution rates from a homogeneous surface of benzoic acid. In
78
Ind. Eng.
Chem. Fundam., Vol. 22, No. 1,
1983
contrast to the present study, Malone and Anderson did not report any differences in Rb values between their lowest and highest porosity membranes. The heterogeneous nature of the membrane surface affects Rb due to the fact that when a solute molecule approaches a porous membrane is must diffuse laterally relative to the membrane surface in order to reach a pore entrance. This causes an increase in the average diffusion path length for the solute and hence a greater resistance to transmembrane diffusion. As the porosity of the membrane decreases, this average diffusion path length increases. As mentioned in the previous section, Keller and Stein (1967) have examined the effect of membrane porosity on the overall mass transfer resistance for the case of a solute diffusing through a membrane with a uniform array of circular pores. Their solution was used to obtain the corrected values of Rb listed in Table V. The difference between the corrected and uncorrected values corresponds to that fraction of R which is due to entrance and exit effects at the pore openings. As demonstrated in Figure 2 and Tables V and VII, the values of Rb for the lowest porosity membrane are significantly higher than values obtained for the membranes with porosities of 6.2 or 19.9%. These results suggest that the heterogeneity of the mass transfer surface has an effect on the boundary layer resistance in addition to the entrancelexit effect which is accounted for by Keller and Stein’s analysis. One factor which may be important is the arrangement of the pores on the membrane surface. Prager and Frisch (1975) have assessed the effect of the geometrical arrangement of pores on the surface of a thin membrane using a self-consistent field method. They found that the overall mass transfer resistance for a regular array is much less than for a random array at low porosity. Although their analysis does not permit calculation of Rb, it is evident that the arrangement of pores on the membrane surface can have an important effect. Acknowledgment The author gratefully acknowledges Dr. G. D. Patterson for his measurement of the dextran and ficoll translational diffusion coefficients and Dr. H. H. Law for his assistance in measuring the limiting currents. The assistance of Mr. L. Stillwagen in measuring the polymer molecular weights is also greatly appreciated. Nomenclature A = total membrane area, cm2 AeffORf= effective area of infinitesimally thin membrane, eq 12, cm2 a = exposed membrane radius, cm b = exponent for Nsc, eq 13, dimensionless C1 = concentration of solute in chamber 1 of diffusion cell, mol/cm3 C2 = concentration of solute in chamber 2 of diffusion cell, mol/cm3
CB = concentration of solute in bulk solution, mol/cm3 AC = C1 - C2 at time t, mol/cm3 ACo = C, - C2 at t = 0, mol/cm3 c = exponent for N R ~eq, 13, dimensionless D, = diffusion coefficient of diffusing species within the membrane, cm2/s D, = diffusion coefficient of diffusing species in free solution (continuum), cm2/s F = Faraday’s constant, C/mol i L = limiting current, C/s J = transmembrane solute flux, mol/cm2 s K = overall mass transfer coefficient, cm/s L_ = pore length, cm &fn = number-average molecular weight, g/mol M , = weight-average molecular weight, g/mol N s =~ a/R$= = Sherwood number, dimensionless N,, = p / p D , = Schmidt number, dimensionless N b = u a 2 p / p = Reynolds number, dimensionless n = pore density, cm-2 ne = number of equivalents per mole LIP = pressure difference across membrane, cm H20 Q = volumetric flow rate, cm3/s R = total mass transfer resistance, s/cm Rb = diffusional boundary layer resistance, s/cm R, = intrinsic membrane resistance, s/cm r = pore radius, cm t = time, s V = chamber volume of diffusion cell, cm3 W = membrane weight, g 6 = diffusional boundary layer thickness, cm I* = viscosity, g/cm s p = density, g/cm3 4 = function used by Keller and Stein (1967), eq 9, dimensionless w = stirring speed, rev/min
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Received for review December 17, 1981 Revised manuscript received September 3, 1982 Accepted September 17, 1982
