Measurements of Membrane Permeabilities Using a Rotating Batch Dialyzer Richard P. Wendtl and Rene J. Toups Loyola University, N e w Orleans, L a . 70118
J. K. Smith, Natasha Leger, and Elias Klein Gulj South Research Institute, N e w Orleans, L a . 70182
The rotating batch dialyzer, an exceptionally simple cell for measuring membrane permeabilities, has been characterized in terms of boundary layer permeabilities for six solutes diffusing through Cuprophane. An empirical expression was obtained for calculating boundary layer permeabilities from known values for the rotational speed of the cell, the density and viscosity of the solutions, and the diffusion coefficient of the solute in solution. Permeabilities determined with the rotating cell were confirmed for three solutes b y comparing results from an entirely different type of cell, the flat plate dialyzer.
T h e measurement of membrane permeability is complicated by t'he need to separate the resistance to mass transport into those components characteristic of the membrane and those attribut'able to the device in which the measurements are performed. Perhaps the simplest apparatus yet devised for such measurements is the rotating dialysis cell described by Regan, et al. (1968, 1970). Their cell consists of two hollowed-out disks which are clamped together with the membrane to be tested separat'ing the t'wo halves. The cylindrical compart'ments which result are filled only partially with solution, and while diffusion proceeds the cell is rotated about its horizontal shaft att,ached to a motor. Stirring of the solut'ions on either side of the membrane is accomplished by the vertical membrane as i t rotates past t'he liquid-air surface in each compartment. The apparent' membrane permeability is calculated from the rate of change of the solute concentrations in both compartments. The ease of construction and operation of this cell compares favorably with more complex devices reported in the literat'ure, i.e., the flat plate flow dialyzer and the stirred batch dialyzer. However, the nature of the boundary layers in the rotating cell is presently unknown, and this limits the u.sefulness of the device in precise measurements. For the stirred reactor bat'ch cell and the flat plate dialyzers, boundary layer theory has been applied with some rigor (Babb, et al., 1968; Colton, 1969; Kaufman and Leonard, 1968; Smith, et al., 1968) so that a t least in principle-if not always in practice-the t,rue permeability of the membranes under test can be calculated from the measured apparent permeability if various geometric and solution parameters are known. T o be accepted as both a n accurate and convenient method the magnitude cf boundary layers in the rotating dialysis cell must' be determined and a method found a t least for calculating the true permeability. Regan, et al. (1970), studied the effects of rotational speed on the apparent permeability of one solute (NaC1) through several membranes, but the effects were not quantified. The [Tork reported here for several solutes may be regarded as a more quantative estension of their work. Experiments were performed using a single sheet of memTo whom correspondence should be sent. 406
Ind. Eng. Chem. Fundom., Vel. 10, No. 3, 1971
brane material with six solutes over a fairly wide range of rotational speeds. This led to an empirical relation between the boundary layer permeability, the rotational speed of the cell, and the diffusion coefficient of the solute in the liquid phase. The results of these analyses were confirmed for three solutes bv measuring apparent membrane permeabilities of the same membrane sheet in a flat plate dialyzer a t varying feed rates and extrapolating the data to infinite velocity. Boundary layer Theory
The approach taken by us is similar to that used by Kaufman and Leonard (1968) in their work with the stirred batch dialyzer. A permeability coefficient P is defined by
J = -PAC
(1)
where J is the solute flux density aiid AC is a concentrat'ion difference. Small volume flows due to osmosis would not invalidate this equation, since the volume flows would be proportional t'o AC through a constant subsumed in P. When AC is the overall concentration difference between the bulk solutions on either side of the membrane, then P = P,I,,,I, the measured apparent permeability; when AC is the concentration difference between the bulk liquid coiicentration and the concentration of the liquid a t the membrane-liquid interface, then P = P,>I,the permeability of the boundary layer. Finally, when AC is the concentration difference between the two surfaces of the membrane, then P = Pu,the true permeabilit,y of the membrane. By assuming a pseudosteady state for the solute transport between the t v o liquid phases, arid by considering the boundary layers and membrane to be resistive elements in series, one finds
Here P b l is an empirical average of boundary layer permeabilities on either side of the membrane. The factor of 2 in eq 2 implies that to a perhaps first-order approsirnation the permeabilities may be equal on either side. The general form of a n expression for P b l in terms of geometric and solution parameters is suggested by Kaufmaii and Leonard's eq 1, which yields after rearrangement
Pb I
= CY D'
-'( p / o ) '-'w'
(3)
For the rotating cell the application of this equation is without theoretical justification since conditions are considerably different from those in the stirred dialyzer. However, it was found t h a t eq 3 was successful in correlating the data. All geometric parameters such as 'the volume of solution and radius of the cell have been combined to form the term a ; D is the diffusion coefficient of the solute in solution, p is the density, and q is the viscosity of the solution in the boundary layer region; w is the rotational speed of the cell, and b and c are constants to be determined empirically. For a specific solute diffusing across any given membrane, the constant c is that power to which w must be raised in order t o linearize a plot of (1/Pobsd) us. (l/w)'. This plot, called a Wilson plot, in the stirred batch dialyzer, has a n intercept a t (I/W = 0)which ~ is presumably equal to ( ~ / P M )Of . course, the value for PMobtained from such a n extrapolation depends on the premise that the nature of the boundary layer remains similar throughout the range of rotational speeds used in the experiment and a t the infinitely high speeds near the intercept. Smith, et al. (1968), in their analysis of the stirred batch dialyzer, discuss the magnitude of errors resulting from this assumption and also point out the possible dependence of CY on Par,D, q , and w for that apparatus. We have not tested the constancy of CY with Pv, but assume an insignificant change in CY for hemodialysis membranes of similar permeabilities. The validity of our Wilson plots was tested by intercomparison of PMvalues determined by entirely different methods. Once c was established, then a plot of In (Pbl) us. a n appropriate function of D , p , and q for several solutes was used to find b and CY.
4 " 5
Figure 1 .
The rotating dialysis cell, disassembled
Experimental Procedure
The Rotating Cell. T h e disassembled cell is shown in Figure 1. Each compartment was machined from 2.4-cm thick X 12.7-cm diameter Lucite disks to form cavities 0.55 cm deep and 9.3 cm in diameter. Two filling holes were drilled into each compartment and provided with threaded nylon plugs fitted with O-rings. Large O-rings (10.0-cm diameter) fit into grooves machined in the face of each disk in order to seal the membranes clamped between the two halves. The maximum available volume in each compartment is 60.0 em3, and the membrane area exposed to solution is 73.24 em2. The following methodology was used for all but the first three series of experiments; the three special series will be explained in the section to follow. T o begin the experiments, the wet membrane was clamped tightly between the two cell halves, one compartment (side A) was rinsed with several cubic centimeters of the solution and side 13 was rinsed with solvent (HZO). With the filling holes open the cell was thoroughly drained and shaken out, leaving about 0.1 cm3 of liquid adhering t o the walls of each compartment. These residues were established gravimetrically, and were included in later calculations. The shaft on the cell was then clamped in the motor chuck, and the pitch of the cell was adjusted until the shaft was horizontal to within 1'. A weighed syringe was used to introduce 35 c1n3 of solvent (water) into side B. Another weighed syringe was used to introduce 35 cm3 of solution into compartment A, and the filling plugs were quickly screwed tight; the motor was then started. The time that filling of compartment A commenced was designated tl and the time that rotation began was designated t2. The time difference was usually about 20 see. The empty syringes
were weighed to establish accurately the mass of solution and solvent introduced into the dialysis cell. The S'ariac used to regulate the motor speed had been adjusted previously to give the desired rotational speed, and every 200 sec during the experiment a lapsed time tachometer with a 6-sec period was used to measure w (in rpm) for the cell. Variations of +5% in w were observed during most runs. The value of w for the experiment mas taken to be the simple average of all tachometer readings. At the end of a preselected time, t3,the motor was stopped, the plugs were removed, and a fen- cubic centimeters of solution was withdrawn from each compartment. The time when withdrawal was complete was designated t 4 ; it was usually within 40 sec of t?. The diffusion time of the experiment was calculated from t = (t4 t 3 ) / 2 - (t? tl)/2, which corrects approximately for the time during filling and sampling when the cell was not rotating but some diffusion was occurring. -4Brice-Phoenix Model BP-2000-V differential refractometer was used to determine Ano, the refractive index difference between the solvent and the solution placed initially into the dialysis cell, and An, the refractive index difference between the solutions removed from the cell a t the end of the experiment. The ratio Ano/An can be taken (within 0.1%) as equal to the ratio of the concentration differences initially and a t time t , ACo/AC, so that the diaphragm cell equation (Gordon, 1945) which describes the diffusion process for these experiments can be modified t o read
+
+
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
407
5.41
0 Urea 0 NaCl A Glycolamide X Atonme @ Glucose
\ V v)
4.2
io3 x
I/W
(r.p.m.)
Figure 2. Wilson plots of l/Pobsd vs. I / w for five solutes in the rotating cell with 35-cm3 solution volumes
Here A is the measured area of the membrane, and TiA and TiB are the volumes of solvent and solution injected into each compartment. Volumes were calculated from densities and the measured mass; each volume was corrected by 0.1 om3 t o include the residual liquid volume from the rinse step. Our assumption t h a t VA and V B in eq 4 are constant during the experiment may not be valid for all systems. Changes of ca. 20/, in V were observed for sucrose and glucose, with no changes for other solutes. Kaufman and Leonard (1968) corrected for these small osmotic effects in their work, but we did not. Values of PMfor sucrose and glucose given here have a n additional inaccuracy of ca. 2y0 due to this effect. The membrane used in these experiments came from one sheet of Bemberg PT-150 Cuprophane, having a wet thickness of 21.6 X 10-4 cm. Six solutes were used: NaCl, urea, glycolamide, DL-a-alanine, D-glucose, and sucrose. Diffusion cot o 0.510 X efficients of these solutes range from 1.477 X om2 sec-1, respectively. Reagent grade crystals and deionized conductance water were used to prepare the dilute (ca. 2-3%) stock solutions. Each experiment lasted between 800 and 1600 sec, as measured by an electric stopwatch. The experiments were carried out in a laboratory with regulated air temperatures near 24' which fluctuated somewhat from day t o day, but never more than 0.1' during a run. Solution temperatures before and after a run did not differ by more than 0.loC, indicating no significant heating effect attributable t o rotation of the cell. T o determine the effect of solution volume on apparent permeability, a series of runs was carried out using sucrose as the solute with solution volumes of 25, 35, and 45 cm3. These early experiments utilized a different filling technique than that described previously. Instead of adding the entire solvent volume t o one compartment and then adding the solution volume to the other compartment, both compartments were loaded with solvent only, using a buret. Sufficient concentrated solution was then added to side A t o achieve the desired initial concentration. I n this way it was expected that distention of the membrane could be avoided. Subsequent experiments indicated this was a n unnecessary precaution. Using the five other solutes and the simplified procedure, experiments were carried out with 35 cm3 of solution in each 408 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
compartment, at several rotational speeds ranging from 100 t o 200 rpm. T h e Flat Plate Dialysis Cell. T h e permeabilities of three solutes-NaC1, urea, and sucrose-through the same membrane sheet were examined also in a flat plate dialysis cell. This cell was provided through t h e Artificial Kidney Program, National Institutes of Health. T h e experimental arrangement has been given in a previous publication (Klein, et al., 1969); modifications since that time have led t o enclosure of the entire test apparatus in a n air thermostat for more precise temperature control and the incorporation of a differential manometer t o minimize transmembrane pressure gradients. For this work, the feed flows through the cell were varied from 30 to 540 cm3/min for the sucrose runs and from 150 to 540 cm3/min for the urea and NaCl rum. This is a steady-state measurement, the upstream concentration remaining essentially constant during the experiment. The downstream side, which was initially pure solvent, was monitored for small concentration changes with a Waters R-4 flow-through differential refractometer. The Cuprophane PT-150 membrane used was cut from the same sheet as the samples used in the rotating dialysis cell, so t h a t comparisons between the techniques are only confounded b y within-sheet variances of permeability. Results
I n Figures 2 and 3 are shown plots of l / P o b s d vs. 1 / for ~ the rotating cell. For each solute the standard deviation of the data from the least-squares fit to a straight line, using eq 2 and 3 with c = 1, is about 1% of the measured values of I/Pobsd. I n Table I are given the intercept 1/Par and its probable error, the slope m and its probable error, and the standard deviation, U, of the data from the straight line 1
1 f rn, P M w
-
Pobsd
where
m = -2w= ( y )Dp
0-'
(5)
(2)
pbl
The approximately 1% deviaticn of the data from linearity is slightly less than the estimated probable errors in the raw data used t o calculate the P o b s d values from eq 4. A 20-sec uncertainty in time contributes a 2% error, the error in Ano is O.l%, and the error in An at time tq is approximately 1.0%; errors in VA and VB aside from those discussed previously can amount to 0.1/35 = 0.2870. The estimated absolute error in A , the membrane area, is 2y0,due t o the 9.6
9.4-
0 0
A
- 25 - 35 - 45
CC.
cc. cc.
I
40
50
60
lo3
70
80
90
I1
0
x l/u,(rpm?
Figure 3. Wilson plots of l/Pobsd vs. 1 / w for sucrose in the rotating cell a t three solution volumes
Table 1. Permeability Data for Several Solutesa -
Cell type
Rotating
r,
Solute
Ct g/100 cm3
OC
cm3
NO. Of expts
sec/cm
m X IO3, sec/(cm min)
sec/cm
Urea NaCl Glycolamide Alanine Glucose Sucrose Sucrose Sucrose
1.50 1.50 1.oo 1.00 1.oo 1.75 2.40 1.30
23.4 23.4 25.2 23.4 26.5 23.5 23.5 23.5
35 35 35 35 35 35 25 45
7 6 6 5 6 12 8 7
1389 =t 30 1384 =t 19 1666 rt 61 2951 + 31 4326 + 120 7902 f 317 8083 =t 257 8188 + 212
7 6 . 2 =k 4 . 0 62.3 + 2 . 5 108.7 =t9 . 3 8 8 . 7 =t4 . 7 115.0 + 1 0 . 6 145.9 f 2 1 . 4 150.2 =t 1 2 . 8 99.3 5 7.0
23 15 36 19 53 150 89 33
VA
1 /Par,
U,
DP/V
x
1.397 1.511 1.264 0.973 0.790 0.510 ... I
Urea 0.08 25.0 ... 10 1443 + 68 60.7 5 8 . 2 152 NaCl 2.85 25.0 ... 8 1349 =t 56 61.9 & 15.5 93 Sucrose 0.08 25.0 ... 7 8197 + 222 7 2 . 7 =t 1 5 . 0 367 Note: Deviations in values of 1/Pn for glucose and sucrose include 1% and 2% error, respectively, for volume transfer.
difficulty of measuring the diameter across the flattened O-ring which circumscribes the area. This constant error would not introduce scatter into the d a t a but would affect the extrapolated value of l/f". The variation of rotational speed introduces one of the largest sources of error, estimated to be as high as 4y0. The latter can be eliminated in future work by use of better speed-controlled motors. I n Table I, in g/100 cm3 represents the mean solute concentration between compartments A and B, and T is the average temperature of the series of runs for a given solute. Small corrections for variations of about 0.1"C in T within a series of experiments were made using an assumed activation energy of 3.5 kcal/mole in a n Arrhenius equation for the temperature dependence af solute permeability through Cuprophane (Colton, et al., 1971). Agreement within experimental error between values for ~/PM a t three solution volumes for sucrose (Table I) implies the area, A , t o be used in eq 4 for calculating P o b a d should be the entire measured area of the membrane and not merely the submerged portion of the membrane, which represents only 60% of the available area for 35-cm3 solution volumes. This finding agrees with that of Regan, et al. (1970). The existence of a thin film of solution carried over by the submerged portion of the membrane as it traverses the arc above the solution level apparently permits the diffusion process to continue for the short period of time t h a t the solution film is out of the bulk layer. This behavior would explain the independence of A from VA and V g . However, the slope m for the 45-cm3 series of runs with sucrose is significantly different from the slope found a t 25 and 35 cm3, indicating a basic difference in the boundary layer a t the highest volume measured. Especially a t high rotational speeds the membrane tends to create a solution annulus when 45 cm3 is contained in each compartment; this is not observed with lower solution volumes. Because the flow pattern a t 35 cm3 appeared reproducible and the 25-cm3 solution volumes resulted in diff usion times that were inconveniently short, all subsequent experiments were conducted with 35-cm3 solution volumes. Permeabilities measured in the flat plate dialysis cell are illustrated graphically in Figure 4 ; in Table I1 are given the parameters calculated from a least-squares fit to the equation
.
.
... ...
Flat (NBS)
4
103
...
e
1/Pobsd
=
1/PM
+ m/v
(7)
where v is the feed flow velocity in cma/min. The scatter from linearity is about equal to the observed reproducibility of 5y0 in the runs.
i! -
Pox
30
I O '
-
2I0
IO
310
'
40
610
5 I0
70 I
80 I
I
io3 x IN, min/cc Figure 4. Wilson plots of 1/Pobsd vs. 1/v for three solutes in the flat plate dialyzer
The intercepts ~ / P M obtained with the rotating cell for urea, NaC1, and sucrose would be about 3y0 lower if corrected to 25"C, using a 3.5 kcal/mole activation energy, and would differ by an average of 4.5y0 from corresponding values obtained with the flat plate dialysis cell. The average of expected percentage differences calculated from probable errors in Table I is 4.37,. The variability of membrane
Table II. Observed and Calculated Boundary layer Permeabilities for the Rotating Cell at 150 rpm X I O 3 from Wilson plot (eq 5 and 2 )
Pbi
Solute
Urea NaCl Glycolamide Alanine Glucose Sucrose
3 4 2 3
937 815 760 382 2 609 2 056
Pbl
X I O 3 from
a 878 080 635 068 680 019
=q
3 4 3 3 2 2
% difference
~
Av
=
1 15 31 9 2 1 10
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
5 3 7 3 7 8 4
409
12 0 ,
/
/
I
1111s
I
/ :ose
116
11.31-
0
Glycoiomide
/
1
1I.lt
In (Dplq) Figure 5.
Plot of In m vs. In ( D p / r ] ) for the rotating cell
samples cut from the same sheet of Cuprophane membrane would probably introduce another few per cent difference between measured values for ~ / P M (Smith, et al., 1968). Thus it is fair to conclude that the rotating cell and the flat plate dialysis cell yield equivalent results for the three solutes studied. The flow characteristics are quite different for these two methods, and it would seem reasonable to accept the Wilson plot intercepts, ~ / P I Idetermined , with either method as true values for the membrane. The accuracy and linearity of Wilson plots for the rotating cell appear t o be well verified. The method of least squares mas used to find the slope and intercept of a graph of In m us. In (LIP/?) for the solutes studied; only the 35-cma data were used. Values of D , p, and 7 were obtained from the literature for the solutes in question. (See Longsworth (1957)) for a compilation of diffusion data.) Rather than list all sources of data, the following are recommended for reliable data a t 2 5 O , for density, viscosity, and diffusion coefficients a t several solution concentration levels: urea (Gosting aiid Akeley, 1952) ; glycolamide (Dunlop and Gosting, 1953); alanine (Gucker and Allen, 1942; Gutter and Kegeles, 1953); and sucrose (Gosting aiid Morris, 1949). All literature values were corrected t o the experimental temperature by assuming t h a t viscosities and densities varied for the dilute solutions as they did for pure water and by using an Arrhenius-type activation energy for diffusion in liquids of 5 kcal/mole (Longsworth, 1955). I n Figure 5 is shown a plot of In m us. In ( D p / q ) . The graph is linear within the probable error of I n m. From the intercept, In (2/a), the value for CY was found to be (1.83 =t1.57) x the slope is ( b - 1) = -0.648 i. 0.124 (eq 6). Discussion
According to the preceding data analysis, the boundary layer permeability for a given solute in the rotating cell can be predicted for experiments with 35 em3 in each compartment by the equation Pbl =
1.83
x ioT3 X
w
x
(Dp,/17)”,648
(8)
Here, w is expressed in rpm and all other quantities are expressed iii cgs units. Differences between Pbl calculated from 410 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
eq 8 and P b l observed are about 7% of the true boundary layer values. We again emphasize the empirical nature of this analysis and do not suggest any theoretical implications of this equation. For the fast-moving solutes with Dp/q = 1.5 X (such as NaC1) diffusing through Cuprophane, the calculated value of ~ / P ax t a rotational speed of 150 rpm would have a probable error of 2% if P b l were calculated directly from eq 8 and 2. If a Wilson plot were made of measured values of 1/w and P o b s d for a n y membrane, and a value of 2W/Pb1 obtained as the slope, then for any other membrane tested with the same solute the boundary layer correction could be made a n even more negligible error. If CY depends on Pnl as Smith, et al. (1968), h a r e suggested, then additional errors may appear when values for P b l obtained with one membrane are used t o correct 1/Pobed obtained on other membranes of considerably different permeabilities. A method for avoiding part of the tedious calibration procedure for the stirred batch dialyzer was suggested by Colton (1969) and also Smith, et al. (1968): P b l was directly determined by measuring the dissolution rate of a disk of solid benzoic acid clamped in place of the membrane. From known values of D , p , and q , for benzoic acid in water, the geometric coefficient corresponding t o OL in eq 3 was determined accuately (the other parameters b and c were predicted theoretically and the “membrane” was considered to be infinitely permeable). Attempts t o apply this procedure t o confirm the values of the parameters in eq 8 for the rotating cell, using a solid disk of benzoic acid and measuring the apparent “permeability” of the solute into 35 ema of solution, were so irreproducible that the experiments were abandoned. The scatter in these data was attributed to two causes. First, the small solution volume and the relatively large Pbl value for beiizoic acid resulted in times of only 200-300 sec for complete equilibration; since the filling time and withdrawal time errors approached 20 see, the error expected in P b l was about 10%. The second and probably more important source of error was the observed behavior of the solutions as the cell rotated: instead of a continuous film being carried over as was the case with the Cuprophane membranes, the solution carried over on the face of the benzoic acid disk quickly coalesced into droplets or irregular patches before the area was again submerged into the liquid layer. Apparently the nonwetting nature of the benzoic acid with respect to aqueous solutions resulted in an irreproducible film formation on the unsubmerged portion of the disk. As a result the disk area A was not constant, and the value used in a modified form of eq 4 to calculate Pblwas not correct. From these observations we judge that the rotating cell method may not be reliable for testing membranes which are not wetted by the solutions in the cell compartments. However, with the precautions indicated, the rotating cell provides a relatively simple and inexpensive method for the measurement of membrane permeabilities. It is possible to calculate the contribution of boundary layer resistance to such observations by the use of the techniques outlined in this work. .Is a result, the technique can be recommended for rapid membrane screening. d comparison of the boundary layer resistance between the rotating cell and the flat plate dialysis cell indicates that the flat cell can be operated readily under conditions which reduce the boundary resistance t o 10% of the total, whereas the rotating cell a t 150 rpm may retain boundary resistances of nearly 25YG of the total for fast membranes such as Cuprophane. Thus it is most important that corrections for boundary layer contributions
be made when this type of cell is used and t h a t the basis of the Correction be reported. Acknowledgment
Work with the rotating cell ,was carried out a t Loyola University and supported by a grant from Schlieder Educational Foundation, S e w Orleans, La. Work at Gulf South Research Institute was supported in part by Contract KO. PH 43-68-709 from the Kational Institute of Arthritis and Metabolic Diseases, NIH, USPHS, DHEK. literature Cited
Babb, A. L., RIaurer, C. L., Fry, D. L., Popovich, 11. P ., McKee, R. E., Eng. Prog?. Synip. Ser. 64 (84), 59 (1968). Colton, C. K., Ph.D. Thesis, Massachusetts Institute of Technology, 1969, 1971. Colton, C. K., Smith, K. A., hlerrill, E. W., Farrell, P. C., J . Biomed. Mater. Res., in press, 1971. Dunlop, P. J., Gosting, L. J., J . Amer. Chem. Sac. 75, 5073 (1953).
s:ifc;, j!: ~ ; i & ~ y ~ f i y + . f ,$;28ij(l.9;f,)i058 ~ ~ ~ ~ ~ : Gosting, L. J., Morris, hl. S., J . 71, 1998 (1952).
Amer. Chem. SOC.,
11949). Gucker,’F. T., Allen, J. W., J . Amer. Chem. Sac. 64, 191 (1942). Gutter, F. J., Kegeles, G., J . Amer. Chem. Sac. 75, 3893 (1953). Kaufman, T. G., Leonard, E. F., A.I.CH.E. J . 14, 421 (1968). Klein, E., Smith, J. K., Wendt, R. P., J . Polym. Sci., Part C 28, 20 (1969). Longsworth, L. G., in “Electrochemistry in Biology and hIedicine,” T. Shedlovsky, Ed., Chapter 12, Wiley, STew York, N. Y., 1955. Longsworth, L. G., in “American Institute of Physics Handbook,” D. E. Gray, Ed., Chapter 2, LlcGraw-Hill, Sew York, N. Y . . 19.57.
Regan, Trii., Esmond, W. G., Streckfus, C., Wolbarsht, A. M., Science 162, 1028 (1968). Regan, T. M., Esm:nd, W. G., Streckfus, C., Wolbarsht, A. A I . , Backer, B., in Membranes from Cellulose and Cellulose Derivatives,” Journal of Applied Polymer Science Symposium No. 13. D 251. Interscience. Sew York. N. Y.. 1970. Smith, K.*-4.,Colton, C. K.; Merrill, E’. W., and Evans, L. B., Chem. Eng. Progr. Symp. Ser. 64 (84), 45 (1968). RECEIVED for review Kovember 18, 1970 ACCEPTED April 29, 1971
Viscoelastic Effects in the Flow of Non-Newtonian Fluids through a Porous Medium Eugene H. Wisslerl Marathon Oil Company, Littleton, Colo.
This paper contains a review of available experimental data on non-Newtonian flow through porous media, a general derivation of Darcy’s law for flow of a power-law visco-inelastic fluid, a third-order perturbation analysis of flow of a viscoelastic fluid through a converging-diverging channel, and an analysis of the flow of a power-law fluid through the same system. The perturbation analysis shows that the purely viscous force acting on a converging-diverging section must be multiplied b y a factor, 1 A(XV/r)*, to account for viscoelastic effects. A i s a constant which seems to be of order 10, X i s a characteristic relaxation time for the fluid, V i s the interstitial velocity, and r i s a measure of the minimum pore size. The proposed correction factor agrees well with experimental data reported b y Marshall and Metzner. Analysis of flow of a power-law fluid through a converging-divering channel provides a basis for experimental study of viscoelastic effects in polymer solutions. If the measured pressure drop exceeds the value predicted on the basis of viscometric data alone, viscoelastic effects are probably important and the fluid can b e expected to have reduced mobility in a porous medium.
+
S l o w f l o ~of a single r\’epitonian fluid through a porous medium is described by Darcy’s law
4
=
-(k/b)(AP/L)
(1)
in which q = superficial velocity, 72 = permeability of the medium, g = viscosity of the fluid, aiid A p / L = pressure gradient in the direction of flow. By slow flow, we mean the Reynolds number A’Re
(2)
= 6qP/bp
is much snialler than unity. Iii this expression 6
=
a charac-
Industry Fellow from The University of Texas at Austin.
teristic dimension for the porous medium, such as the mein capillaric diameter, P = porosity, and q/P = characterislk interstitial velocity. Smallness of the Reynolds iiuniker implies that inertial effects can be neglected. For a visco-inelastic, power-law fluid, Darcy’s law takes the form (3)
in which k’ is a parameter that depends on the power-l:Lw exponent n and K is the other power-law parameter. For packed beds, k’ can be estimated using arguments sinii ar to those leading to the Carniaii-Kozeny equation (Bii-d, Ind. Eng. Chem. Fundarn., Vol. 10, No. 3, 1971
41 1
