(45)Kim, 0.K.. Little, R. C.. Patterson, R. L.. Ting, R. Y., Nature (London), Phys. Sci., 250, 408 (1974). (46)Kline, S. J., Reynolds, W. C., Schraub. F. A.. Rumstadler, P. W., J. fluid Mech., 30, 741 (1967). (47)Kramer, M. O.,"Jahrbuch 1969 der Deutschen Gesellschraft fur Luftund Ramfahrt," p 1, 1969. (48)Landahl. M. T., presented at the 13th ICTAM. Moscow, 1972. (49)Leger. A. E.,Hyde. J. C., Sheffer, H., Can. J. Chem., 36, 1584 (1958). (50)Little, R. C.,J. Appl. Polym. Sci., 15, 3117 (1971). (51)Little, R. C.. J. Colloidlnterface Sci., 21, 266 (1966). (52) Little, R. C..J. Colloidlnterface Sci., 37, 811 (1971). (53)Little, R. C.. lnd. Eng. Chem., fundam., 8, 520 (1969). (54)Little, R. C..hd. Eng. Chem., fundam., 8, 557 (1969). (55) Little, R. C.. Nature (London), Phys. Sci., 242, 79 (1973). (56) Little, R. C.. Nature (London), Phys. Sci., 245, 141 (1973). (57)Little, R. C.,US. Naval Research Laboratory, Washington, D.C., NRL Report 6542, May 31, 1967. (58) Little, R. C.. Patterson, R. L., J. Appl. Polym. Sci., 16, 1529 (1974). (59)Little, R. C..Singieterry, C. R., J. Phys. Chem., 68, 3453 (1964). (60)Little, R. C.. Wiegard. M.. J. Appl. Polym. Sci., 14, 409 (1970). (61)Little, R. C., Wiegard, M., J. Appl. Polym. Sci., 15, 1515 (1971). (62) Lumley, J. L., Macromol. Rev., 7, 263 (1973). (63) Metzner, A'. B., Metzner, A. P., Rheol. Acta, 9, 174 (1970). (64)Miller, M. L.. "The Structure of Polymers," Chapter 12, Reinhold. New York, N.Y.. 1966. (65) Mino, G., Kaiserman, S., Rasmussen, E.,J. Polym. Sci., 38, 393 (1959). (66) Nychas, S. G., Hershey, H. C., Brodkey, R. S., J. fluid Mech., 61, 513 (1973). (67) Mysels, K. J., Chem. Eng. Progr. Symp. Ser. (Drag Reduction), 67, 45 (1971). (68) Oldroyd, J. G., Proc. Roy. SOC.London, Ser. A, 20, 523 (1950). (69)Parker, C. A., Hedley, A. H., Nature (London) Phys. Sci., 236,(Mar 29, 1972). (70)Paterson. R. W., Ph.D. Thesis, Harvard University, 1969. (77) Peyser, P., Little, R. C., J. Appl. Polym. Sci., 15, 2623 (1971).
(72) Peyser, P.. Little, R. C.. Singleterry, C. R., NRL Report 7227, Jan 27, 1971. (73)Peyser, P., Little, R. C., J. Appl. Polym. Sci., 18, 1261 (1974). (74)Pfansteil, R.. Her, R. K.. J. Am. Chem. Soc., 74, 6059 (1952). (75)Ritter, H., Porteous, J. S.. "Water Tunnel Measurements of Skin Friction on a Compliant Coating," Admiralty Research Laboratory Report A.R.L./ N3/G/HY/9/7 (1965). (76)Rouse, P. E., Sittel, K., J. Appl. Phys., 24, 690 (1953). (77) Ruchenstein, E.,Chem. Eng. Sci,, 26, 1075 (1971). (78)Sakamoto. R., Imahori. K., Kogyo KagakuZasshi, 83, 389 (1962). (79)Sheffer. H., Can. J. Res., 26, Sec B, 481 (1948). (80)Shin, H., Sc.D. Thesis, Massachusetts institute of Technology, 1965. (87) Sholton, W., Makromol. Chem., 14, 169 (1954). (82)Stevenson, J. F., Bird, R. B., Trans. SOC.Rheol., 15, 135 (1971). (83) Takahashi, A,. Kamei, T., Kagawa, I., J. Chem. SOC.Japan, Pure Chem. Sec., 83, 14 (1962). (84).Ting, R. Y., Bull. Am. Phys. SOC., 18, 1469 (1973). (85) Ting, R. Y., Kim, 0. K., "Water Soluble Polymers," p 51, N. M. Bikales, Ed., Plenum Press, New York, N.Y., 1973. (86) Ting, R. Y., Little, R. C.. J. Appl. Polym. Sci., 17, 3345 (1973). (87) Ting, R. Y., Little, R. C., Nature(London) Phys. Sci., 241, 42 (1973). (88) Toms, B. A,, "Proceedings of the International Congress on Rheology, Holland (1948)," Part 11, pp 135-141, North Holland, Amsterdam, 1949. (89)Ultman, J. S.,Denn, M. M., Trans. Soc. Rheol., 14, 307 (1970). (90)Van Wazer. J. R., "Phosphorus and its Compounds," Vol. 1. Chapter 12, Interscience, New York, N.Y., 1966. (91) Virk, P. S.,et al.. J. Fluid Mech., 30, 305 (1967). (92)White, A.. "Some Observations on the Flow Characteristics of Certain Dilute Macromolecular Solutions," in "Viscous Drag Reduction," C. S. Wells, Ed., p 297, Plenum Press, New York, N.Y., 1969. (93)Zimm, B. H., J. Chem. Phys., 24, 269 (1956).
Received for reuiew December 16,1974 Accepted August 4,1975
Membrane Potentials and Transport Parameters for Alkali and Alkaline Earth Chlorides across Cellulose Acetate Membranes King Wai Choi and Douglas N. Bennion" Energy and Kinetics Department, School of Engineeringand Applied Science, University of California, Los Angeles, California 90024
The relative transport rates and associated membrane potentials across modified cellulose acetate membranes cured at 85OC have been measured for alkali and alkaline earth chloride solutions during reverse osmosis experiments. An effective cation mobility is defined as the product of the salt diffusion parameter, Le, times the transference number, t+m.The magnitudes of the mobilities can be ordered as follows: K+ > Rb+ > Cs+ > Na+ > Li' and Be2+ Sr2+ Ca2+ > Ba2+ > Mg2+. The mobility of the 3e2+ ion was exceptionally large, approximately the same as for the alkali ions. All other doubly charged ion mobilities were significantly smaller than for singly charged ions. The observed potential differences across the membranes are explained as a sum of contributions from a diffusion potential across the thin rejecting layer, a streaming potential across the porous sublayer, and an asymmetric double layer on both sides of the membrane. The theoretical interpretation indicates that for BeC12 solutions Be2+ ions are preferentially adsorbed on the membrane, but for all the other salts studied CI- ions were preferentially adsorbed.
>
>
Introduction The mobilities of ions and water in modified cellulose acetate membranes are affected by the physical and chemical properties of the ions in solution, the properties of the membrane, and the mutual interactions between solvent, solutes, and membrane. In order to improve descriptions and understanding of the properties which affect mobilities, the relative transport rates of various ions are being measured and interpreted (Fayes, 1970; Re and Bennion, 1973). The present work is to extend the previous data on alkali metal ion relative transport rates to include the alkaline earth ions. In addition, the electrical potential differ296
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
ences across the membranes during reverse osmosis are investigated. There have been a number of equations proposed, using various theoretical bases, to predict or a t least correlate transport rates of ions and water across cellulose acetate membranes. Comparisons of various approaches have been summarized elsewhere (Osborn and Bennion, 1971; Choi and Bennion, 1973). It has often been convenient to place the various models into two categories. The first are modifications of Fick's law or the Nernst-Planck equations, frequently with pore flow terms added to provide better correlation for the salt flux observations. Lonsdale et al. (1965), Sherwood e t al. (1967). and Govindan and Sourirajan
(1966) might be considered examples in this category. The so-called friction models, frequently using irreversible thermodynamic arguments as a basis, might be a second category. Papers by Spiegler (1958), Kedem and Katchalsky (l958), Wills and Lightfoot (1966), and Bennion and Rhee (1969) might be included here. Strict cataloging of various contributions has limitations, however. The work of Sherwood et al. (1967) is presented in terms of diffusion and pore flow. However, as shown by Bennion and Rhee (1969) and discussed by Osborn and Bennion (1969), the equations of Sherwood et al. are mathematically equivalent to the Bennion and Rhee model except for the interpretation of the coupling terms. Although Spiegler's early work (1958) was almost entirely in terms of a friction model, more recent work (Spiegler and Kedem, 1966) is a combination of the Nernst-Planck equations and friction models. Cellulose acetate membranes have been shown to be nonhomogeneous and composed of two layers (Banks and Sharples, 1966). Re and Bennion (1973) have used a twolayer model to help explain some of the phenomena they observed. The two-layer model is extended farther and used to help interpret the membrane potentials measured as part of this present work. The model of the rejection layer can be considered a friction or solution diffusion model. The porous sublayer can be viewed as a pore-flow model. However, the fundamental bases for each region are essentially the same. Different terms are retained in each region and different interpretations applied, reflecting our conceptions of the dominant or controlling physical processes in each region. The electrical potential gradient in the rejection layer is assumed to be a diffusion potential characterized by the pressure gradient, activity gradient, and a transference number measured using a Hittorf cell. Entrainment of the double layer by bulk flow in the pores of the sublayer is assumed to give rise to a streaming potential. Nonsymmetric absorption of ions at the membrane-solution interfaces is also assumed to contribute to observed potentials across the membrane. Membrane flux rates for reverse osmosis experiments have been interpreted using the following equations.
The a, and bo symbols imply a geometric average of upstream and downstream activities of electrolyte and water, respectively. Symbols are defined in the Nomenclature section. In this paper, it has proven convenient to define both a, and a. such that they are dimensionless. For the salt, a, is referred to a hypothetically ideal solution of unit molality; that is, a, is the mean molal activity coefficient, y i , times the moles of salt per kilogram of water, m e , divided by one mole of salt per kilogram of water. For the water, a0 is referred to pure water, that is, pure water has an activity of 1. The ion transport with electrical current in the Hittorf cell measurements is interpreted in terms of the following equations.
dP
uRT
RTda
(3)
copper wire spot welding junction
I
\
I
[
eipdxy
silver-silver chloride
f \
ewov plastic insulator electrode
Figure 1. The upstream silver-silver chloride electrode
The experimental measurements include water and salt flux as a function of applied pressure for alkali and alkaline earth chlorides. Potential differences were measured between silver/silver-chloride electrodes on each side of the membrane during reverse osmosis tests. Hittorf cell type transference number measurements were also made using the same membrane as used in the pressure flow tests.
-
Experimental Section Membrane Casting a n d Precompaction. The casting solution was 23 w/o Eastman 400-25 cellulose acetate, 27 w/o formamide, and 50 w/o acetone. The membrane was cast on a flat glass plate at room temperature (78OF). The air dry time was 8 sec. The immersion ice water bath was made from Los Angeles tap water. The bath was at a temperature of 0.2OC. The immersion time of the membrane in the ice water bath was 2 hr. The "curing step" was in tap water at 85OC for 20 min. The membrane was precompacted for 258 hr under an applied pressure of approximately 1000 psi using distilled water. Silver/Silver Chloride Reference Electrodes. Two silver/silver chloride reference electrodes were used to make potential measurements across the membrane during reverse osmosis experiments. The substrate for the electrodes was 0.019-in. diameter silver wire. Silver chloride was deposited on the surface of the silver wire. The downstream electrode was made by sealing a silver wire in a soft glass stem leaving 2 in. of wire, for use as the electrode. The upstream electrode was made by spot welding a silver wire to a 0.02-in. copper wire which was covered with polyethylene insulation. The insulated copper wire passed through a drilled out plug and was sealed in place with epoxy cement (Figure 1). The junction of silver wire and copper wire was also sealed with epoxy cement. The exposed silver wires of both electrodes were cleaned before anodizing by immersing the exposed wire in concentrated nitric acid for 1 hr, immersing in concentrated ammonium hydroxide for 4 hr, and finally washing with distilled water for 1 day. The electrodes were coated with silver chloride by anodizing in 0.1 M hydrochloric acid a t an applied current of approximately 3 mA/cm' for 45 min. The electrode potentials were compared in a single compartment right before and after the experiments. Differences were always less than 1 mV. Reverse Osmosis Cell. The main part of this cell was a stainless steel chamber. The inside was coated with Teflon to minimize any corrosion associated with the stainless steel. A mechanical, rotating stirrer passed through a highpressure seal and was driven by a motor outside the cell. A drawing of the cell is shown in Figure 2. The flux rate and product concentration at pressure differences of 600 and 800 psi and a feed concentration of 0.1 M NaCl are shown Ind. Eng. Chem., Fundam., Vol.
14,
No. 4,
1975
297
caDillary PULLEY
I
TO REFILL
mogneiic stirrer
,electrode
~VALVE STEEL RING OIAPHRIGM PRESS CON1
c TO
-COOLING
PIPE
e
-
0
SEIL
~
~
ring motrix matrix
VALVE
-"O"RING - S T E E L RING
electrode
RUBBER RING GASKET REFERENCE SILVER CHLORIDE ELECTRODE
/
(mopnotic tiirrer
Figure 4. Hittorf cell.
STIINLESS STEEL CHAMBER THERMISTOR
~
STIRRER PLEXIGLAS CYLINDER
~~
COOLING PIPE .MEMBRANE NYLONCLOTH "0"RING POROUS STAINLESS STEEL -STEEL RING
~
REFERENCE SILVER CHLORIDE ELECTRODE
OIMENSIONS ARE IN INCHES
Figure 2. Reverse osmosis cell.
i
2 t
I
I
I
-
.04
CONCENTRATION
.03
Results
4
-gz
.02
AP=600 psi
G
2 I-
5
.01
LP=600 PSI
U
8
0-U 0 0
,005
I 0
I 4
I 8
I 12
I 16
20
S T I R R I N G SPEED irprn.10-21
Figure 3. Product flux rate dependence on stirring rate: --, centration; - - - -, flux.
con-
as functions of stirring rate in Figure 3. Above 928 rpm there is no dependence on stirring rate. Since it was desired to reduce concentration variation at the membrane solution interface boundary layer to a negligible value while minimizing heat generation in the seal, a rotation rate of 928 rpm was used in the experiments. The temperature in the cell was held constant a 25 f 0.1OC by pumping water from a Fisher Isotemp Bath through the cooling coils of the cell. Pressure was applied to the fluid in the cell through two diaphragms in series. The diaphragms separated oil from a water solution and the water solution from the test solution. A constant pressure was applied by a weight from an Amthor dead weight testing device using oil as the pressure transmitting fluid. The applied pressure was reproducible to f0.1%. Hittorf Cell. The cell body was made of Plexiglas (Figure 4). The two capillary tubes were 1.6 mm inside diameter and were used to indicate volume flow across the membrane during experiments and to ensure equal pressure in each half cell compartment. The electrodes were flat silver plates placed parallel to the membrane on both sides. This 298
geometry ensures uniform current density across the membrane. The cathode was a silver plate previously coated with silver chloride formed by making the silver plate anodic in a 0.1 M HC1 solution. A plain silver plate was used as the anode. The effective membrane and electrode areas were 32 cm2. The applied currents were approximately 20 mA for 1 hr. The volume of the cell on the cathode side was 65 cm3. The so-called rejection layer of the membrane faced to the cathode. The total charge passed through the membrane was counted using a Model 541 Koslow Scientific Co. coulometer. The experiments were run a t 25 f 1.2OC. Magnetic stirring bars were used to provide agitation. After the current was shut off, the catholyte solutions were analyzed using a conductivity cell in an isothermal bath. The anolyte solutions were analyzed in some cases to check the consistency of the data. Agreement was found to be within f 1%comparing the two observations.
Ind. Eng. Chem., Fundam., Vol.
14, No. 4, 1975
Observed mass flux, product concentration, and potential differences across the membrane during reverse osmosis experiments are reported in Table I. The upstream or feed concentrations were 0.100 M in each case. The reported membrane potentials, A+, are the downstream electrode minus the upstream electrode. It is interesting to note that depending on pressure and salt species the potential difference can be positive or negative suggesting that the explanation of these membrane potentials may be rather complex. The data are plotted in Figures 5-8 in the manner described in previous publications (Bennion and Rhee, 1969; Johnson and Bennion, 1968; Osborn and Bennion, 1971; and Re and Bennion, 1973). The slopes in Figures 5 and 6 are the salt diffusion parameters, Le, and the intercepts are the coupling parameters Leo. The slopes in Figures 7 and 8 are the water diffusion parameters, LO,and the intercepts are the coupling parameters Loe. These diffusion parameter results are tabulated in Table 11. The unusually large value of Le for beryllium chloride is particularly noticeable. The Le's for the alkali metal salts as a group are significantly larger than for the alkaline earth metal salts. The Le's can be ordered as follows within each group
K+ > Rb+ > Na+ 7 Cs+ > Li+ Be2+ > Sr2+> Ca2+ > Ba2+ > Mg2+ The values of LO are relatively independent of the salt species. The average value for the alkali metal salts LOis 5.18 X mol/cm2 sec, and for the alkaline earth metal salts the average Lo is 5.42 X mol/cm2 sec. The difference is significant, but not large. Ignoring the fact that the L's are average or effective parameters and not point values, the basic rules of irreversible thermodynamics suggest that Leo should equal Loe. For the alkali metal salts the Leo's do not show any obvious
Table I. Observed Data for Reverse Osmosis Experiments= AP =
AP=
200 psi
Salt
Flow, cep x g'min lo3,hl A @ , V
Flow, g/min
NaCl LiCl KC1 RbCl CsCl NaCl MgC1, CaC1, SrC1, BaC1, NaCl BeC1,
0.15728 9.63 +0.0685 0.16279 8.66 +0.0450 0.16442 10.38 +0.0540 0.16595 10.30 +0.0590 0.16621 9.56 +0.0505 0.16270 9.96 +0.0787 0.12893 6.30 -0.0980 0.13381 6.34 -0.0450 0.13515 6.40 -0.0240 0.13391 6.28 -0.0080 0.16546 10.36 +0.0880 0.13521 14.81 +0.0540
0.3809 0.39346 0.39764 0.40282 0.40191 0.39605 0.36866 0.37687 0.37488 0.37628 0.40975 0.37535
(1
400 psi
AP=
cep x
lo3,M A @ , V 5.49 5.10 5.99 6.01 5.67 5.80 3.95 4.04 4.06 4.02 6.33 8.97
+0.0770
+0.0750 +0.0865 +0.0750 +0.0765 +0.0976 -0.117 -0.0420 -0.0140 -0.0040 +0.1175 +0.0380
AP=
600 psi
Flow, cepX g/min lo3,M
Flow, :c x g/min lo3,M A @ , V
0.6010 4.23 0.6182 4.02 0.62475 4.68 0.63483 4.65 0.63228 4.49 0.62549 4.35 0.60857 3.36 0.61723 3.43 0.60935 3.51 0.62143 3.47 0.64439 5.13 0.61444 6.86
800 psi
+0.0930 +0.1010 +0.1040 +0.0940 +0.0870 +0.1134 -0.0960 -0.0360 -0.0055 +0.0260 +0.1390 +0.0270
0.81210 3.595 0.83290 3.501 0.84972 3.976 0.84959 4.026 0.85817 3.930 0.84798 3.907 0.83890 3.105 0.85338 3.145 0.84709 3.228 0.84739 3.227 0.87831 4.605 0.84932 5.720
A@, V
Date
+0.1090 9/11/72 +0.1280 9/11/72 10.1327 9/12/72 +0.1124 9/12/72 +0.0975 9/13/72 +0.1362 9/13/72 -0.0850 9/19/72 -0.0295 9/20/72 +0.0075 9/20/72 +0.0420 9/21/72 +0.1480 9/21/72 +0.0165 9/22/72
Effective membrane area equaled 40.7 cm2 The feed concentration was 0.100 M in all cases. Membrane curing temperature was 85°C.
-
N
6
P
9 ,
I
I
I
I
I
I
I
1
I
I
I
I
v LlCI 0
2
I
I
I
I
I
I
3
4
5
6
7
8
NaCl RbCl
9 0
CrCl
x 102
Figure 5. Salt flux, plot for determining Le (slope) and Leo (intercept) of alkali metal salts. Figure 7. Water flux, plot for determining LO (slope) and L h (intercept) of alkali metal salts. 30
20
I
I
'
-
/@'-
.
/'
&ClZ
10 -
4
/**
I
i C 1 P ae
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10
0
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0
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LCIZ
I
I
I
v IB.ClZ
20
30
40
50
-
60
Figure 6. Salt flux, plot for determining Le (slope) and Leo (intercept) of alkaline earth metal salts.
trends. The average Leo for the alkali metal salts is 3.11 X mol/cm2 sec, and the average Loe is 2.88 X 10+ mol/ cm2 sec. The difference is probably not significant. However, for the alkaline earth chlorides the values of the Leo's appear significantly larger than the L0,'s. The value of Leo for BeC12 appears significantly larger than Loe for the other alkaline earth salts. Once again the unusually high mobility of the beryllium ion is evident. The results of the Hittorf transference measurements are presented in Table 111. The same membrane was used in
Figure 8. Water flux, plot for determining LO (slope) and Loe (intercept) of alkaline earth metal salts.
both the Hittorf measurement and the reverse osmosis tests. The ordering of the magnitudes of the transference numbers is exactly the same as for the Le's. However, although t + O for beryllium is larger than for the other alkaline earth metal ions, it is substantially smaller than for the alkali ions. Since Le for BeC12 is much larger than for the alkali chlorides, it appears that the beryllium ion not only has a high mobility itself, but it has some sort of cooperaInd. Eng. Chern., Fundam., Vol. 14, No. 4, 1975
299
Table 11. Transport Coefficients for Alkali and Alkaline Earth Chloride Salts and Water in a Cellulose Acetate Membrane
Run o r d e r
L , x 1 0 ' . L o x 104. Le,]x l o ' ,
L,, x 106,
mol ' cm'-sec
mol cm'-sec
mol cm'-sec
mol cm'-sec
4.83 5 -43 5.53 5.02 6.17
5.05 4.98 5.20 5.40 5.18 5.17 5.25 5.48 5.43 5.47 5.38 5.40
2.88 2.68 3.05 3.85 3 .OO 3.07 3.22 5.88 4.73 4.75 4.72 4.82
4.87 2.35 2.25 0.817 2 -78 4.55 2.52 -0.867 -1.32 -0.147 1.02 1.35
____.
(2) LlCl (1) NaCi (6) NaCl (11) NaCl (3) KC1 (4) RbCl (5) CSCl (12) BeC1, (7) MgC1, ( 8 ) CaC1-
(9) SrC1, (10) BaC1,
6.10
5.23 7.68 1.95 2.13 2.23 2.03
tive effect which enhances the effective mobility of the chloride ion. Discussion Control of Variables a n d Accuracy. Activity coefficients were calculated using formulas and constants recommended by Robinson and Stokes (1959) and Harned and Owen (1958). The formulas and constants used have been tabulated by Re (1970) and Choi (1973). Accuracy is f0.1%, except for BeC12 for which accurate free energy of solution data has not been located. Pressure was measured to an estimated accuracy of 0.1% by calibration of pressure equipment with a dead-weight test unit and standard weights. The brine concentration was adjusted to 0.1 f 0.0001 A4 using gravimetric procedures to prepare the solutions and a combination of conductivity and atomic absorbtion measurements to check the concentrations. The temperature in the reverse osmosis cell was reproducible to 25 f 0.1OC. However. the absolute calibration of the thermometers was done only with an ice bath and was probably accurate to only fO.3OC. Considering only the accuracy of the measurments, the measured Le and Lo are estimated to be accurate to f0.4%. The Leo and Loe come from relatively small terms in the flux equations. Stated another way, very small errors in the experimental observations can cause significant shifts in the intercepts of Figures 5 , 6, 7, and 8. The mean deviation of Leo for the alkali metal salts is 11%, which appears to be a reasonable estimate for the accuracy of the coupling parameters. The largest source of uncertainty in comparing the parameters for the different salts is probably not the accuracy of the observations but drift in the membrane properties with time. The membrane tends to change its structure as a result of the pressure differential across it, causing creep or flow, and hydrolysis (Lonsdale, 1966). Sodium chloride was run a t the beginning, middle, and end of the runs as a control. The value of L o for the sodium chloride varied 8%, monotonically increasing, over a period of 10 days, see Table I and 11. This implies a drift of 0.8% per day in membrane properties. The alkali salt data was taken over a 3-day period and the alkaline earth salt data over a 4-day period. Thus, comparisons within each series should be meaningful to within about 3%. Since the LiCl measurements were made 10 days before the BeC12 measurements, drift in membrane properties could account for approximately an 8% variation. The Hittorf cell transference number measurements 300
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
Table 111. Cation Transference Numbers for Alkali and Alkaline Earth Metal Chlorides A1kali c at i on
Li' N a' K+
Rb+ CS'
Hittort transference number
Alkaline earth met a1 cation
Hittorf transference number
0.324 0.393 0.422 0.407 0.403
Be2*
0.306 0.220 0.237 0.2 53 0.244
M?p
Ca2* Sr2+
Ba2'
were temperature controlled to 25 f 1.2OC. Since transference numbers are relatively insensitive to temperature, this implies an error of only f0.2%. In the analysis of the Hittorf cell data it was assumed that diffusion across the membrane due to small concentration differences across the membrane was negligible. IJsing the L parameters determined several weeks earlier, errors due to this effect were estimated to be 0.7%. The so-called water transport number was assumed to be zero. The maximum error from neglecting water transport is estimated to be 0.15%. Relative Permeation Rates. Neither ionic mobilities nor relative permeation rates were measured directly. However, the products of t + m and L e should yield a value, L+, similar to an ionic mobility. The values of the L+'s can be ordered as follows.
K+ > Rb+ > Cs+ > Na+ > Li+ Sr'+
> Ca2+ > Ba2+ > Mg'+
The value of L+ for K+ is the largest for the alkali ions, and L+ for Sr2+is largest for the alkaline earth ions. The value of L for Be2+ is larger than for Sr2+,but most theories on alkaline earth ion behavior exclude Be2+ as part of the series. The relative mobilities of the alkali metals correspond to series four predicted by Eisenman (1969) and further discussed by Diamond and Wright (1969). Eisenman's theory has a thermochemical basis and does not really include mobilities, although procedures for including mobility effects are discussed. The basic concept of the theory is that ions which are absorbed onto sites in the membrane are passed more rapidly than ions less strongly adsorbed. Whether or not ions are absorbed depends on the site-ion energy minus water-ion energy. If water-ion interactions are much stronger than site-ion interactions, the water-ion energies will be controlling, and the ions with the weakest water-ion interactions will pass through the membrane preferentially suggesting the sequence Cs+ > Rb+ > K + > Na+ > I i + . If site-ion interactions are much stronger than water-ion interactions, the site-ion energies will be controlling, and the ions with the strongest site-ion interactions will pass through the membrane preferentially suggesting the sequence Li+ > Na+ > K+ > Rb+ > Cs+. Our observed sequence for the alkali ions is intermediate and implies that the site-ion interactions are larger than the water-ion interactions for K+, Rb+, and Cs+ ions since their mobilities are in the order of the smallest crystallographic radius ion being most mobile. Similarly, the Na+ and Li+ water-ion interactions are presumed larger than the site-ion interactions since their mobility order is the ion with the smallest hydrated radius being most mobile. As discussed by Diamond and Wright (1969), Sherry (1968) has predicted seven possible mobility sequences for the alkaline earth ions. The sequence observed in our study does not correspond to any of Sherry's seven sequences. However, Sherry based his calculations on closely spaced
sites. In addition, for closely spaced sites he predicts higher transport rates for alkaline earth (divalent) cations than for alkali ions. But for widely spaced sites, higher monovalent ion transport rates are predicted. Our results, excluding Be", show the monovalent alkali cations to pass approximately five times faster than the divalent alkaline earth cations, implying widely spaced sites. Thus, correspondence with Sherry's sequences is not to be expected since his calculations were done only for closely spaced sites. The simpler form of the Eisenman (1969) theory applies to charged ion-exchange sites within a membrane. The cellulose acetate membranes which we have studied are believed to contain hydroxyl and acetate cites. The complete theory would require including effects of coordination number (Eisenman's q . N ) and effective charge spacing within the site (Eisenman's rp and r,,). Including these effects appears necessary for quantitative comparisons, but the simple form of the theory seems adequate to explain qualitatively the ordering of the alkali ion transport rates. The Eisenman (1969) theory only includes cation-water and cation-membrane interactions in a thermodynamic or solubility based model, the more soluble ions being the most readily passed through the membrane. Anion-site and anion-water interactions are discussed, but only in the context of relative transport rates of anions. A complete theory should include both the thermodynamic or solubility and the rate effects and all six binary interactions a t once: cation-anion, cation-water, cation-membrane, anionwater, anion-membrafie, and water-membrane. Spiegler (1958) showed how binary friction type interactions can be used to calculate transport parameters for ion-exchange membrane coefficients. Newman et al. (1965), using procedures similar to those presented by Hirshfelder et al. (1954), presented a general procedure for relating the binary diffusion coefficients, Q,, to commonly measured transport parameters. Beniiion (1966) and Bennion and Rhee (1969) showed explictly how to relate D+-, D+o, D+,, 3-0, .D-m, 3 o r n to the more readily measured parameters L e , Lo, Leo (or Lo,), t+"', torn, and K (values of Le, Lo, Leo, and t+"' are reported on in this paper). It seems that a combination of Eisenman's thermodynamic theory and some rate theory such as Eyring's (1954) membrane transport theory could be used to predict the ~,,'s, and Bennion's inversion theory used to predict the L's, t,'s and K . Such a theory might take the form D,, = A,, exp(E,,;/kT)
The El,: might be calculated based on relative interactions of the i and j species. Following Eisenman (1969), coulombic type interaction energies might be used to estimate EL,:. The preexponential factor might be a product of a frequency factor, kT/h, and the square of a characteristic length, l,,' The length l,, might be the distance of movement of species i relative to species j for the occurrence of a single movement event or the equivalent. In the inversion process of converting the D,,'s to the L's, the concentration of the various species within the membrane enter in a prominent manner. The concentration of sites on the membrane can be determined from knowledge of the chemical structure of the membrane. The concentration of permeating species can be estimated by calculating equilibrium constants following Eisenman's (1969) theory. It is beyond the scope of the present paper actually to calculate the L parameters using the above procedure. The theory will no doubt need more precise definitions of the energy and distance parameters making numerical work a t this time premature. However, it appears that some ex-
tended form of the Eisenman theory, such as indicated above, will be able to explain transport rates in cellulose acetate, reverse osmosis membranes as well as other membrane transport processes. Membrane Potentials. If it is assumed that the measured membrane potentials tabulated in Table I are simply diffusion potentials, they should be correlated by the following equation.
The complete derivation of eq 1 is given by Choi (1973). It follows closely the procedures outlined by Newman (1973a). W 'is the potential of the silver in the reference electrode a t the downstream or low-pressure side of the membrane. am is the potential of the silver in the reference electrode a t the upstream or high pressure side of the membrane. The superscripts 6 and 6 imply solution in the highpressure and low-pressure sides of the membrane, respectively. Other symbols have customary meaning and are defined in the Nomenclature section a t the end of the paper. The first term on the right-hand side of eq 5 simply corrects for the pressure on the emf of the reference electrode. It is usually very small and often neglected. The second term on the right-hand side is the dominant term. Some authors (Minning, 1973) subdivide i t into two subterms calling the first the streaming potential and the second the diffusion potential. In this paper the total - is defined as the diffusion potential and the term streaming potential is used to describe a potential contribution in the porous sublayer to be discussed later. The third term on the right-hand side of eq 5 is the contribution associated with water diffusion. If W'- amwere sufficiently accurate, it might be possible to determine tom using eq 5 since all ofher terms in eq 5 can be evaluated from the data in Table 1.
If tom is calculated as suggested in the previous paragraph, values of tom equal approximately 49 are found. If tom is interpreted as the water carried by the cations minus the water carried by the anions, a maximum value for tom of 4 is expected. If one assumes tom between +4 and -4 and uses eq 1 to calculate t+m,values between 0.655 and 0.702 are found for NaC1, for example. Since the Hittorf cell measurements give t+" equal 0.393 for NaCl, the high values found using eq 5 seem physically unreasonable. Thus, it is concluded that eq 5 alone is not sufficient to explain the observed potential measurements. Since variations in tom between f4 contribute in only a minor way to - P,tom will be assumed zero in the rest of the total this discussion. A model to explain the observed potential differences is now proposed in which A@(measured) is assumed to be the sum of a diffusion potential across the rejecting layer plus a streaming potential across the porous sublayer plus a contribution from asymmetric double layers at the solutionrejection layer and rejection layer-porous sublayer interfaces.
+ + A@(asymmetricd.1.)
A@(measured) = A@(diffusion) A@(streaming)
(6)
The contribution A+ (diffusion) is assumed to arise only in the bulk region of the thin, rejecting layer of the membrane. Equation 5 is assumed to describe the contribution. Since the porous backing layer is assumed to contribute Ind. Eng.
Chem., Fundam., Vol. 14, No. 4, 1975 301
40
-20 0
I
1
I
I
10
20
30
40
1
-200
0
A V E R A G E V E L O C I T Y 105 (cm/rec)
I
I
I
I
I
10
20
30
I
40
A V E R A G E V E L O C I T Yx 105 ( c m i w )
Figure 9. Membrane potential correlation for NaC1.
Figure 11. Membrane potential correlation for alkaline earth chlo-
rides. Table IV. Correlation Constants for Asymmetric Double Layer Potential and Streaming Potential
v
V-sec/cm
(2) Licl (1) NaCl (6) NaCl (11) NaCl (3) KC1 (4) RbCl
-60.0 -90.0 -76 .O 40.5 -37.0 -76 .O -55.0 -56 .O -1 58 .O -122.0 -110.0 -1 54 .O
271.56 157.48 184.51 170.07 212.33 177.23 117.82 -118.40 171.05 63.62 108.97 240.06
0
(12) BeC1, (7) MgC1, (8) CaC1,
crc1
I
I
I
10
20
30
(9) SrC1, (10) BaC1,
40
ks >
(molarity)-l
(5) CSCl
20
Wl ,
Run order
A V E R A G E V E L O C I T Y x 105 (cmisec)
Figure 10. Membrane potential correlation for alkali chlorides.
nothing to the overall pressure drop and selective transport rates, P1 - P2 and ae6/aetare values applicable across the membrane as a whole for reverse osmosis measurements. The porous sublayer is assumed to have no semipermeable properties and to act only as a porous medium consisting of pores and pore walls. As fluid flows through the pores, part of the diffuse double layer along the walls is entrained establishing an excess of the double layer charge downstream relative to upstream positions. The phenomenon is well known. Following Newman (1973b), see his eq 63-36, the streaming potential can be written as follows. A@(streaming)= k ,
p
(7)
Zz(R0) and I l ( R 0 ) are modified Bessel functions of the first kind of order two and one, respectively. Ro is a dimensionless pore radius, rolX, where ro is an effective pore radius, The superscripts L and 0 on Ro imply values a t the backing layer-downstream solution interface and the backing layer302 Ind. Eng. Chem., Fundam.. Vol.
14, No. 4, 1975
rejecting layer, respectively. s is the variation of pore size within the porous backing layer, droldx. X is the Debye length. q2 is the charge per unit area in the double layer of the pore walls. KIeffis an effective, ionic conductivity in the pores and is more fully defined by Newman (1973b). [ is the porosity of the backing structure. I t is assumed to be a constant, but this assumption could be easily modified. It is assumed that k , is independent of concentration. This assumption can only be approximately true and will need reevaluation later. In our experiments, the feed concentration was held constant, and the product concentration varied only modestly for the range of pressure differences studied. An elementary analysis of the probable concentration dependence of the parameters in k , indicates the assumption is reasonable for Ro less than 3. For Ro greater than 8, k , is expected to decrease as one over the square root of concentration. Since the total magnitude of the streaming potential decreases rapidly for Ro greater than 3, the variable k , region is probably not important. The asymmetric double layer potential is assumed to arise from nonsymmetric shifts in the potential across the double layers on each side of the rejecting layer of the membrane. Since the double layers are formed due to ad-
Table V. Calculated Values for Potential Contributions 800 psi A@
A@
600 psi
A@
A9
A@
A9
A@
A9
(measured), (diffusion),(streaming),(double layer), (measured), (diffusion),(streaming), (double layer), mV mV mV mV mV mV mV mV (2) L i c l (1) NaCl (6) NaCl (11) NaCl (3) KC1 (4) RbCl (5) CSCl (12) BeC1, (7) MgC1, (8) CaC1, (9) SrC1, (10) BaC1,
128.0 109.0 136.2 148 .O 132.7 112.4 97.5 16.5 -a5 .o -29.5 7.5 42 .O
52.0 62.7 61.2 57.9 65.6 63.1 63.1 27.0 23.8 25.6 27.3 26.4
92.84 52.49 64.15 61.29 74.06 61.79 41.49 41.27 59 .ll 22.27 37.87 83.44
-17.35 -6.36 9.52 33.01 -8.93 -12.33 -5.62 30.76 -166.20 -77.45 -58.34 -65.79
1 0 1.o 93 .O 113.4 139.0 104 .o 94 .o 87.0 27.0 -96 .O -36.0 -5.5 26.0
400 psi A9
A9
68.91 38.92 47.34 44.96 54.52 46.17 30.56 -29.85 42.75 16.11 27.23 61.17
50.0 59.7 58.4 55.9 62.2 58.9 60.5 25.3 23.4 25.0 26.6 25.8
-16.84 -5.22 10.15 33.35 -8.53 -11.42 -5.10 31.60 -165.49 -76.95 -57.92 -65.13
200 psi 4@
A9
A@
49
A9
49
(measured), (diffusion), (streaming), (double layer), (measured),(diffusion),(streaming), (double layer), mV mV mV mV mV mV mV mV LiCl NaCl
NaCl NaCl KC 1
RbCl CSCl BeC1, MgC1, C aC 1, SrC1, BaC1,
75.0 77.0 97.6 117.5 86.5 75 .O 76.5 38.0 -117.0 4 2 .O -14 .O 4 .O
46.2 54.7 53.6 51.9 57.1 55.1 55.8 22.7 22.4 23.9 25.5 24.7
43.86 24.63 29.99 28.55 35.65 29.29 19.43 -18.23 25.90 9.82 16.72 37.03
-1 5.69 -3.62 12.41 34.16 -7.75 -8.18 -3.95 33.43 -163.59 -75.78 -57.03 -63.41
sorption of charged ions or dipole molecules, it is assumed that this contribution can be correlated in terms of an adsorption related theory. Following concepts presented by Davies and Rideal (1963), the surface potential can be related to the adsorption distribution parameters, Bi, as follows. A@ = -
RT (z+ - z - ) F In
where cp is the product concentration leaving the membrane and Bo and m are constants used to fit the data. Based on these considerations, we write A+(asymmetric d.1.) =
RT (z+
- z-)F
[In BO
+ In (1+ mc,)]
(11)
Since all the terms in the diffusion potential, A@(diffusion), can be calculated directly, it is convenient to define a potential A@., where A@., = A@(measured)- A@(diffusion)
37.9 43.9 43.3 42.6 44.7 44.3 45.4 18.0 18.9 20.4 21.7 21.1
18.14 10.15 12.27 11.52 14.32 12.06 8.03 -6.56 8.98 3.47 6.03 13.09
-10.96 14.52 22.74 37.35 4.74 2.62 0.84 42.60 -125.86 -68.87 -51.07 42.41
I t is possible to calculate A a C directly from the experimental measurements. Substitution of eq 7,11, and 12 into eq 6 yields A@c +
RT In (1+ mc,) F(z+ - 2-1
=
- F ( z +RT- 2-1 In Bo + k,Q
(2)
Here A@ is the potential jump across the double layer a t the interface between the membrane and bulk aqueous solution. For purposes of correlating our data, it has been assumed that
-
45.0 68.5 78.7 88.0 54 .O 59.0 50.5 54 .O -98 .O 45.0 -24 .O -8.0
(12)
(13)
The left-hand side of eq 13 is plotted as a function of Q. The parameter m is adjusted to give the least-mean square deviation to a fit of a straight line through the data. The associated calculations and curve fitting were done with an IBM 360-91 computing machine. The resulting curves are shown in Figures 9, 10, and 11. The associated values of m and k, are tabulated in Table IV. Using these parameters, the values of A@(diffusion),A@(streaming),and A@(asymmetric d.1.) have been calculated and tabulated in Table V along with values of A@(measured).Assuming values for s and using eq 8, values of q 2 can be calculated. Because of the approximate nature of the theory, such calculations are not being reported a t this time. However, it is to be noted that k, is proportional to q 2 . From Table IV it follows that q 2 is positive for all the salts studied except for BeClz for which 42 appears to be negative. Since q2 is the charge in the diffuse double layer, a positive q 2 implies adsorption of anions on the membrane because the interfacial region as a whole is electrically neutral. Thus, a negative 42 for BeC12 Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
303
implies strong preferential adsorption of Be2+ while for all other salts studied preferential adsorption of C1- is implied. This preferential adsorption of Be2+ ions may account, in part, for the unexpectedly high mobility of Be2+ ions across the membrane.
Conclusions The relative cation mobilities of alkali and alkaline earth ions are found to be K+ > Rb+ > Cs+ > Na+ > Li+ Be2+ > Sr2+> Ca2+> Ba2+ > Mg2+ Based on a theory proposed by Eisenman, the alkali metal mobility series implies that waters of hydration around K+, Rb+, and Cs+ are preferentially absorbed by absorption sites in the membrane. Waters of hydration for Na+ and Li+ are preferentially retained by the ions. The theory as applied to the alkaline earth series has not yet been extended to the point that definite conclusions as to the relative interaction energies can be given. A simple diffusion potential model is not sufficient to explain observed membrane potentials across the membrane during reverse osmosis. Using Hittorf cell measurements of transference numbers, a model is proposed which assumes that the observed potential is the sum of a diffusion potential across the thin rejecting layer, a streaming potential across the porous backing layer, and an asymmetric double layer on each side of the thin rejecting layer. The theory predicts that Be2+ ions are preferentially adsorbed from BeC12 solutions, but C1- ions are preferentially adsorbed for all other alkali and alkaline earth chloride salts studied. However, for all salts studied, the cation transference members relative to the membrane were between 0.22 for Mg2+ and 0.422 for K+.
Acknowledgments The reverse osmosis equipment used in this study was designed and built by Michel Re and Edward Selover. The assistance of Jerry Houser in taking data is also gratefully acknowledged. This work was supported by the State of California through the University of California Statewide Water Resources Center. Nomenclature = activity of solvent relative to pure water, dimensionless a, = activity of solute (Terne)with infinite dilution as the secondary reference state, dimensionless 6, = effective activity of species i in the membrane, dimensionless Aa, = activity difference of species i across the membrane U , ~ - U , ~ ,dimensionless A,, = preexponential factor, cm2/sec B, = distribution coefficient of species i relative to membrane and solution phases BO = proportional constant c p = concentration of product electrolyte, M = diffusion coefficient describing the interaction of species i and j , cm2/sec Ellt = interaction energy between species i and j , ergs h = Planck's constant, erg-sec i = current density, A/cm2 II = modified Bessel functions of first kind, first order 1 2 = modified Bessel function of first kind, second order k = Boltzmann's constant, ergsIoK k, = proportional constant in streaming potential, V-sec/ cm Kleff = effective product solution conductivity, mholcm l,, = characteristic length between the movement of species i relative to species j , cm
a0
304
Ind. Eng.
Chem., Fundam., Vol.
14, No. 4, 1975
Li
= transport coefficient of species i, mol/cm2-sec m = empirical constant defined in eq 1O,l./mol me = molality of electrolyte, mol/kg of water N , = molar flux of species i, mol/cm2-sec P = pressure, dyn/cm2 aP = pressure difference across the membrane (Pz - PI), dyn/cm2 = surface charge density in the diffuse layer, C/cm2 = volumetric flow rate, cm3/sec = superficial average velocity, cm/sec ro = pore radius of porous layer in membrane, cm Ro = ro/X K = universal gas constant, ergs/mol°K s = gradient of pore radius in x direction t o m = water transport parameter t +m = ion transference number relative to membrane T = temperature, OK V I = partial molar volume of species i, cm3/mol x = distance from solution-membrane interface, cm zI = charge number of species i
i
Greek Letters 6 = rejection layer thickness, cm X = Debyelength,cm = porosity of membrane @ = potential of a reference electrode, V A@ = potential difference, V u = number of ions per molecule of electrolyte c = permittivity, F/m Subscripts e = electrolyte as a neutral species i = any arbitrary species j = any arbitrary species 0 = solvent 1 = upstreamside 2 = downstreamside Superscripts 6 = upstream side t = downstream side m = implies the quantity is relative to membrane N = upstream silver chloride electrode a' = downstream silver chloride electrode
Literature Cited Banks, W., Sharples. A,. J. Appl. Chem., 16, 28 (1966). Bennion, D. N., Department of Engineering, University of California, Los Angeles, Report 66-17, 1966. Bennion, D. N., Rhee. B. W., lnd. Eng. Chem., Fundam., 8.36 (1969). Choi, K. W.. Bennion, D. N.. Department of Engineering, University of California, Los Angeles, Report 7370, 1973. Davies, J. T., Rideal, E. K.. "Interfacial Phenomena," Academic Press, p 60, New York, N.Y., 1963. Diamond, J. M., Wright, E. M., Annu. Rev. Physiol., 31, 581 (1969). Eisenrnan, G:, Nat. Bur. Stand. U.S. Spec. Pub/., 314, 1-56(1969). Fayes. D. E., M.S. Thesis, School of Engineering and Applied Science, University of California, Los Angeles. Calif. 1970. Govindan, T. S., Sourirajan, S., hd. Eng. Chem.. Process Des. Dev., 5, 422 (1966). Harned. H. S., Owens, B. B., "The Physical Chemistry of Electrolyte Solutions," p 702, Reinhold. New York, N.Y.. 1958. Hirshfelder. J. 0.. Curtss. C. F., Bird, R. B., "Molecular Theory of Gases and Liquids," p 714, Wiley, New York, N.Y., 1954. Johnson, J. S.,Bennion, D. N.. Chem. Eng. Prog. Symp. Ser. 64, No. 90, 270 (1968). Kedem, O., Katchalsky. A,, Biochim. Biophys. Acta, 27, 229 (1958). Lonsdale. H. K., Merten, U.. Riley, R. L., J. Appl. Polym. Sci.'9, 1341 (1965). Lonsdale. H. K., "Desalination by Reverse Osmosis," U. Merten. Ed., pp 93-160, M.I.T. Press, Cambridge, Mass.. 1966. Minning, C. P., Ph.D. Dissertation, Department of Mechanical Engineering, University of California, Berkeley.1973. Newman, J., Bennion, D. N., Tobias, C. W., Ber. Bunseges. Phys. Chem., 69, 608 (1965). Newman, J.. "Electrochemical System," (a) pp 29-60, (b) pp 190-206, Prentice-Hall, Englewood Cliffs, N.J., 1973. Osborn, J. C., Bennion, D. N., Department of Engineering, University of California, Los Angeles, Report 69-49, 1969. Osborn. J. C., Bennion, D. N., lnd. Eng. Chem., Fundam., 10, 273 (1971). Parlin, R. B., Eyring. H.. "Ion Transport Across Membranes," H. T. Clarke, Ed., Academic Press, New York, N.Y., 1954. Re, M., M.S. Thesis, Department of Engineering, University of California. Los Angeles. Calif.. 1970.
Re, M.. Bennion, D. N., lnd. Eng. Chem., Fundam., 12,69 (1973). Robinson, R. A,, Stokes, R. H., "Electrolyte Solutions," pp 230-238 Butterworths London, 1959.
Spiegler, K. S., Trans. Faraday SOC.,54, 1408 (1958). Kedam, O.,Desalination, 1, 311 (1966). Spiegler, K. S., Wills, G.B., Lightfoot, E. N.. hd. Eng. Chem.,Fundam., 5, 115 (1966).
Sherry, H. S., "Ion Exchange," Vol. 11, Marinsky, Ed., pp 89-133, Marcel Dekker, Inc., New York, N.Y., 1968. Sherwood, T. K., Brian, P. L. T., Fisher, R. E., hd. Eng. Chem., Fundam., 6, 2 (1967).
Received for review May 8, 1974 Accepted June 5,1975
Effective Catalytic Rate Constant in Random Porous Media-Overlapping Solid Spheres and Overlapping Void Cube Surfaces Ruth A. Reck Physics Department, General Motors Research Laboratories, Warren, Michigan 48202
Gene P. Reck' Department of Chemistry, Wayne State University, Detroit, Michigan 48202
This paper presents calculations which illustrate how bounds may be obtained for the steady-state effective rate constant, k, of a surface reaction in a random porous medium of simple geometry. Upper bounds on k are obtained for a random distribution of overlapping (1) void spheres and (2) void cubes. For the cases chosen the magnitude of the bound on k is larger in the presence of surface discontinuities. This is the result of the masstransfer variational calculation and is independent of any increase in k that might occur from surface structure < 0.5, the calculated value of k is greater for convex spherical surchanges. For a void volume fraction (6) faces while the converse occurs for 4 > 0.5.
Introduction The present work extends the method of calculation of the effective rate constant, k , of a surface reaction in a random porous medium to examples of surface geometrics specified by random overlapping void spheres and random overlapping void cubes. In earlier papers (Reck and Reck, 1968a,b) a formalism was presented which calculated the steady-state rate constant when simultaneous chemical formation and mass transport occurred in the void space and disappearance of a species B* occurred a t the surface of a random porous medium. In that work application was made t o the model of overlapping void sphere surfaces only. In principle a further application of this formalism to other geometries is important in order to (1) optimize k through the proper choice of pore size and geometry and (2) separate the catalytic surface structure from the masstransfer effects within a single model. Recent work demonstrating the importance of geometric effects in the mass transfer limit of oxidation reactions with noble metal catalysts has been reported by Hegedus (1973). In that work it was shown that improved monoliths can be obtained by an optimal choice of channel shape, feed flux, and channel geometry. Even within rather limited pore models, some assessment of the role of the boundary geometry may be important (Ablow e t al., 1965). The recent work of Somorjai (1972) in the development of step-like catalytic surfaces may require a separation of the effects of surface atomic structure and the role of bulk vs. surface mass-transport for a thorough understanding of the systems.
The Variational Approach
The variational formulation has been presented previously (Reck and Reck, 1968a) and readers are referred to this earlier work for details. The present calculation is restricted to the upper bound of the simplest of the four. cases presented there, case I. Other applications follow in a similar manner. In case I a reactive species B* is formed a t a uniform rate a everywhere in the void region. The concentration of B* a t a point r is balanced (at steady state) by the infinite rate of disappearance of B* a t the void-solid surface, Z. An upper or lower bound is set on the effective rate constant k defined as the ratio of the rate of formation of B*, 0 , to the mean steady-state concentration of B*, ?
k = a/?
(1)
In the upper bound, the effective rate constant k is related to the details of the porous medium in the trial concentration function by the two-point probability function p ( t ) which is defined as the probability that a randomly selected point r, in a representative volume V (which is considered to define the void space) is a t a distance between t and t dt from the closest point on E. This p ( c ) is a special two-point function since it is stated in terms of the distance t from r to a point on the surface, 2 . Written in terms of p ( t ) the upper bound on h for case I is
+
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
305
