Ind. Eng. Chem. Fundam. 1904, 2 3 , 234-243
234
S,= stoichiometric number of species i
t = time, s t! = anion transference number of the external electrolytic
solution T = absolute temperature, K u, = velocity of species i, m/s u = volume average velocity, defined by eq 35, m/s V = volume, m3 VI = partial molar volume of species i, m3/kg-mol z, = charge number of species i Greek Symbols a = parameter relating membrane pressure and water con-
centration in eq 12, kg-mol/(N m) fl = parameter relating membrane anion and water concentration in eq 1 2 p r = electrochemical potential of species i, J/kg-mol v = number of anions and cations per molecule of electrolyte v r = number of species i ions per molecule of electrolyte @ = electrical potential, V Subscripts
e = external electrolyte as a neutral species em = membrane electrolyte as a neutral species i, j = any arbitrary species 0 = solvent (water) m = membrane + = membrane cations - = membrane anions Superscripts
a = atmospheric 0 = initial or in some cases a reference state u = upstream I = compartment I I1 = compartment I1
L i t e r a t u r e Cited Anderson, J. L.; Quinn, J. A. Siophys. J . 1974, 14, 130. Bean, C . P. US. Department of Interior, Office of Saline Water, Research and Development Progress Report No. 465, Washington, DC, 1969. Beck, R. E.; Schultz, J. S. Science 1970, 170, 1302. Bennion. D. N.; Pintauro, P. N. AIChE Symp. Ser. 1981, No. 204, 190. Bennion, D. N.; Rhee, B. W. Ind. Eng. Chem. fundam. 1969, 8 , 36. Choi, K. W.; Bennion, D. N. Ind. Eng. Chem. fundam. 1975, 14, 296. Diamond, J. M.; Wright, E. M. An. Rev. Phys. 1969, 3 1 , 581. Eisenman, G. "Theory of Membrane Electrode Potentials. Ion Selective Electrodes", Durst, R. A., Ed.; National Bureau of Standards Special Publication 314, Washington, DC, 1969; pp 1-54. Helfferlch, F. "Ion Exchange"; McGraw-Hill: New York, 1962; pp 339-420. Curtiss, C. F.; Bird, R. B. "Molecular Theory of Gases and Hirschfelder, J. 0.; Liquids"; Wiley: New York, 1954; p 714. Kedem, 0.; Katchalsky, A. Biochem. Siophys. Acta 1958. 2 7 , 229. Kedem, 0.; Katchalsky, A. Siophys. J . 1962, 2 , 53s. Lightfoot, E. N. "Transport Phenomena and Living Systems": Wiley: New York, 1974; p 236-265. Newman, J. "Electrochemical Systems"; Prentice-Hall: Englewood Cliffs, NJ, 1973; pp 239-250. Newman, J. Ind. Eng. Chem. Fundam. 1988, 1 , 514. Onsager, L. Phys. Rev. 1931, 3 8 , 2265. Osborn, J. C.: Bennion, D. N. Ind. Eng. Chem. Fundam. 1971, 10, 273. Paganelli, C. V.; Solomon, A. K. J . Gen. Physiol. 1957, 4 1 , 259. Pintauro, P. N. Ph.D. Dissertation, University of California, Los Angeles, 1980. Quinn, J. A.; Anderson, J. L.; Ho, W. S.:Petzny, W. J. Biophys. J . 1972, 12, 990. Re, M. F.; Bennion, D. N. Ind. Eng. Chem. Fundam. 1973, 12, 69. Scattergood, E. M.; Lightfoot, E. N. Trans. Faraday SOC. 1968, 6 4 , 1135. Schloegl, R. Z . Phys. Chem. (Frankfurt am Main) 1954, 1 , 305. Sidel, V. W.; Solomon, A. K. J . Gen. Physiol. 1957, 4 1 , 243. Solomon, A. K. J . Gen. Physiol. Suppl. 1968, 5 1 , 335s. Spiegler, K. S. Trans. faraday Sac. 1958, 5 4 , 1408.
Received for review March 7 , 1983 Accepted December 23, 1983
This work was supported by the State of California through the University of California Statewide Water Resource Center. Publication costs have been provided in part by the U.S. Department of Energy and Brigham Young University.
Mass Transport of Electrolytes in Membranes. 2. Determination of NaCl Equilibrium and Transport Parameters for Nafion Peter N. Pintauro and Douglas N. Bennlon" Chemical Engineering Department, University of California, Los Angeles, Los Angeles, California 90024
Six transport parameters and two thermodynamic distribution coefficients along with knowledge of any fixed charge concentration in a membrane are necessary and sufficient data to describe mass transport of water and a binary salt across a membrane. These data have been obtained by absorption, dialysis, electrodialysis, and reverse osmosis experiments for NaCl and water transport across a perfluorosulfonated ion-exchange membrane (Nafion). The determinations were made at 1.0, 2.0, 3.0, 4.0, and 5.0 M concentrations yielding the parameters as functions of concentration. The resulting parameters, when used with an applicable mathematical model, are shown to predict membrane performance to within 10% accuracy. Radio tracer experiments are shown to yield results which differ significantly from results not including radioactive species due to an interaction between tagged and untagged species. This interaction is not present in the untagged type experiments.
Introduction The mathematical model describing the transport of water and binary salt across a membrane contains two equilibrium and six transport parameters (Pintauro and Bennion, 1984). These parameters must be determined either experimentally or theoretically for a given mem-
* Chemical Engineering Department, Brigham Young University, Provo, U T 84602.
rane-solute-solvent system if the model is to be used to predict solute and solvent fluxes through a membrane. An experimental method of calculating all of the membrane parameters is presented in this paper. The two equilibrium parameters, the salt partition coefficient and the water equation of state, were experimentally measured by standard laboratory techniques (Kamo et al., 1971). The six transport parameters were obtained by performing dialysis, electrodialysis, and reverse osmosis experiments. In each experiment, a concentration change across the
0196-4313/84/1023-0234$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 235
membrane and a volumetric flow rate of fluid through the membrane were measured. The six transport parameters were found by matching the experimental data with the corresponding predicted quantities determined from the dialysis, electrodialysis, and reverse osmosis computer simulation programs (Pintauro and Bennion, 1984; Pintauro, 1980). The transport parameters in the computer simulations were varied until an objective function defined as
was minimized. In eq 1, I'YP is a measured experimental concentration change or flow rate and Fith is the corresponding value of ri obtained theoretically by solving the proper differential equations with a set of transport parameters. In the present analysis, the six Bijparameters were determined. A computer optimization program using the direct search techniques of Hooke and Jeeves (1961) was used systematically to find the set of Bijparameters yielding the minimum value of the objective function.
Experimental Section Experiments were performed on unreinforced Nafion 110 perfluorosulfonic cation-exchange membranes (manufactured by E.I. du Pont de Nemours & Co., Inc.). Ndion 110 has a theoretical ion-exchange capacity of 0.909 mequiv/g of dry membrane. Transport and equilibrium parameters were determined as a function of concentration by performing the experiments in 1,2,3,4, and 5 M NaCl solutions. Reagent grade chemicals and doubly distilled water were used to prepare the salt solutions. Temperatures were maintained constant at 25 f 0.1 "C for the transport experiments and 25 f 1.5 "C for the equilibrium experiments. All membranes were pretreated in 100 "C water for 30 min. Equilibrium Experiments. In order to determine the salt and water uptake by the membrane, four experiments were performed: wet membrane density, membrane fixed ion concentration, membrane anion concentration, and membrane water content. Measurement of the Wet Membrane Density. Nafion membrane disks, with a known dry membrane diameter and weight, were immersed in an NaCl solution and allowed to equilibrate over a 24-h period. After 24 h, the membranes were removed from the solution and excess electrolyte was wiped off the surface with filter paper. Quickly, an outline of the membrane disk was made on a sheet of paper; then the membrane was weighed and its thickness was measured at 5 locations on the disk with a micrometer. A planimeter was used to determine the area of the wet membrane outline. Measurement of the Membrane Fixed Ion Concentration. The membrane was immersed in 1.0 M HC1 for 24 h (to ensure that all of the membrane sites were in the acid form) and then leached for 24 h with distilled water. A known weight of wet membrane was placed in either a 2.0 or 3.0 M NaCl solution while stirring for 2.5 h. The NaCl solution was replaced repeatedly until no extra H+ was detected in the NaCl rinsing solutions. All of the solutions used in rinsing the membrane were collected and the H+ concentration was determined by titrating with a standardized 0.446 M NaOH solution with phenolpthalein as the indicator. Measurement of the Membrane Anion Concentration. A known weight of wet membrane was immersed in 300 mL of an NaCl electrolytic solution at a given concentration and allowed to stand about 3 h while stirring.
Over a 24-h period the NaCl solution was periodically replaced with a new solution of the same concentration. After 24 h the membrane was removed from the salt solution, wiped with filter paper to remove surface electrolyte, and then immersed in 125 mL of distilled water for 2 h while stirring. The membrane water soak was repeated 5 times, until no C1- ions were detected in the external water. The C1- ions contained in the combined washings were analyzed by potentiometrically titrating with AgNO,. Measurement of the Membrane Water Concentration. A membrane disk was immersed in an NaCl solution and allowed to equilibrate. The membrane was then removed from the electrolytic solution, wiped with filter paper to remove excess salt solution, and weighed. The membrane was dried in a vacuum oven at 37 "C for 48 h (until its weight became constant) and then reweighed. The difference in weights represented the membrane water content. Transport Experiments. Dialysis and Electrodialysis. The apparatus used in the dialysis and electrodialysis experiments is shown in Figure 1. The cell consisted of two chambers each having a volume of 50 cm3. The compartments were separated by a Nafion membrane. The area of membrane in contact with salt solution was 20 cm2. In each chamber there were reference electrode ports extending down to the membrane surface. Bulk flow of fluid across the membrane was measured for each compartment with horizontal 2-mm glass capillary tubes. The solutions were mixed by magnetic stirring bars and the entire cell was immersed in a 25 "C constant-temperature water bath. For the electrodialysis experiments, porous Ag/AgCl working electrodes were placed at the far ends of each chamber and parallel to the membrane. In the dialysis experiments, an initial concentration difference across the membrane was allowed to decay with time. The concentration difference was the same for all experiments and bracketed 1, 2, 3, 4, and 5 M NaCl by f0.2 M. The lower compartment of the dialysis cell was always filled with the less concentrated NaCl solution. Each experiment began when the upper chamber was completely filled with electrolyte. The position of fluid in both capillary tubes was recorded every 15 min. After 6 h the upper and lower compartments were carefully emptied and the final concentrations of the NaCl solutions were determined by potentiometric titration with AgN03. The electrodialysis experiments were similar to standard Hittorf transference number experiments (Robinson and Stokes, 1959; Sinha and Bennion, 1978). Initially, both compartments were filled with the same NaCl solution. A constant current (= 0.11 A) (Pintauro and Bennion, 1984) was applied for a specified time (== 85 min) during which the constant flow in the capillary tubes was recorded. A coulometer counted the total charge passed during an experiment. Bulk concentration changes in both compartments were determined by pipetting 25 mL of each final solution into clean 250-mL beakers, followed by drying and weighing the salt. Reverse Osmosis. The apparatus used in the reverse osmosis (RO) experiments is shown in Figure 2. The main component of the RO cell was a 10-L Teflon-coated stainless steel chamber. A Nafion membrane, supported by a porous stainless steel disk, was placed at the base of the chamber. Mixing was provided by a mechanical rotating stirrer. A high rotational speed (928 rpm) was used to ensure that boundary layer effects were negligible (Choi and Bennion, 1975). A constant pressure was applied by a dead weight testing device and the temperature was maintained constant by pumping water at 25 "C through
236 Ind. Eng. Chem. Fundam., Vol. 23. No. 2. 1984
I IDWN
Comparment
Figure 1. Dialysis/electrodialysis cell.
Q~
1.40
1.50 0
1.0
2.0
3.0
4.0
5.0
C. lkg mole/m'l
Figure 3. Nafion wet membrane density VS. the external NaCl run 1; (A) run 2; ( 0 )run 3. concentrstion: (0)
capillary tube and then diluting the samples with distilled water and analyzing for Na using a Perkin-Elmer Model 303 atomic adsorption spectrophotometer.
0,*1*.,0*.."~
I" I " C * E l
Figure 2. The reverse osmosis cell.
cooling coils'in the cell. A 2-mm glass capillary tube was connected to the exit port of the cell to measure the rate of fluid flow through the membrane. The upstream pressure for all experiments was maintained constant at 71 psig. This low pressure driving force was used to ensure that there was no compaction of the membrane during an experiment. Steady-state conditions were obtained when the flow rate of fluid in the capillary tube had stabilized. This occurred after approximately 10 h. For the next 30 h flow rate data were periodically taken. After steady state was attained, downstream salt concentrations were determined by pipetting 5-pL samples of solution from the
Results a n d Discussion Wet Membrane Density. Nafion wet membrane densities are plotted as a function of the external NaCl concentration in Figure 3. A least-squares fit of the density data, represented by the straight line in Figure 3, produced the following equation pm =
1.45 + 1.30 X 10-2[Ce]
(2)
Nomenclature and units are listed at the end of the paper. Membrane Fixed Ion Concentration. Three separate measurements of the Nafion fixed ion concentration were performed. The resulting ion-exchange capacities (kg-mol /kg of dry membrane) are 0.857 x W ,0.852 X IO9, and 0.854 X The membrane fixed ion concentration, x (kg mol/m3 of wet membrane), was obtained my combining the ion-exchange capacities with the results of the wet membrane density experiments. A plot of x vs. the ex-
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
237
2.0
1.8
"\E
0
1.6
P
5
1.4
0 '
1.2
1.03
L 1.0
0
2.0
1.0
4.0
3.0
5.0
C, (kg m o l e / m 3 )
2.0
3.0
5.0
4.0
C, (kg m o l e / m 3 )
Figure 4. Ndion 110 fixed ion concentration vs. the external NaCl concentration.
Figure 6. Membrane cation concentration vs. the external NaCl concentration: (-O-) Donnan theory (eq 7); (-) experimental; (0) run 1; (A)run 2; (0)run 3. 18.0
I
I
I
I
I
1.0
2.0
3.0
4.0
5.0
1
0.8
"E
\
0
0.6
P
5
0.4
0'
o.2
t /.'
o0v
I
2.0
1.0
I
I
3.0
4.0
I
-.- 0
5.0
C , (kg mole/m')
Figure 5. Membrane anion concentration vs. the external NaCl concentration: (0) run 1; (A)run 2; ( 0 )run 3.
ternal salt concentration is given in Figure 4. A leastsquares straight line fit of the data produced the following equation
x = 1.02 + 2.19 X
10-2[Ce]
(3)
The fixed ion concentration increases at higher salt concentrations because the membrane swells less in concentrated electrolyte, and the smaller membrane volume results in a higher fixed site concentration. Membrane Anion Concentration, A plot of C-versus the external NaCl concentration, C,, is shown in Figure 5 . A linear least-squares fitting equation of these data is c- = 0.179[Ce] - 0.011 (4) The membrane cation concentration was obtained by combining the anion and fixed ion concentration data with the electroneutrality equation. For a cation-exchange membrane the electroneutrality equation is
c,
=
c- + x
(5)
A plot of C, versus the external NaCl concentration is given in Figure 6. The cation data are represented by the following linear least-squares equation c, = 1.01 + 0.2Ol[Ce] (6) Also shown in Figure 6, for comparison, is the membrane cation concentration based on the Donnan equilibrium theory (Helfferich, 1962) assuming the activity coefficients inside and outside the membrane are equal. The Donnan
C , (kg m o l e / m 3 )
Figure 7. Nafion water concentration vs. the external NaCl concentration: (0) run 1; (A)run 2; (0)run 3.
theory equation for the membrane cation concentration in the units kg-mol/m3 wet membrane is
c+= 2 +
[ (;)
112
+ (€Ce)2]
(7)
where E is the membrane porosity, calculated from experimental data using the equation E = AV/(l + Av) (8)
AV is the volume increase of the membrane upon adsorption of an aqueous NaCl solution per unit volume of dry membrane. It is related to the experimentally measured dry and wet membrane weights (Wd and W,, respectively) by (9) A v = Pd(Ww - W d ) / P e W d where Pd and pe are the densities of the dry membrane and NaCl solution, respectively. The dry membrane density, Pd, is 1.64 x io3 kg/m3. As would be expected, Figure 6 shows that the discrepancies between the experimentally determined cation concentration and those calculated from eq 7 become larger as the external salt concentration increases. Membrane Water Concentration. Figure 7 is a plot of the membrane water concentrtion vs. the external NaCl concentration. A least-squares curve fit equation for the Co data is Co = 16.4 - 1.06[Ce] (10)
238 Ind. Eng. Chem. Fundam., Vol. 23, No. 2 , 1984
7 1.10 -
-
5.0
\
E w E
1 .oo fern
? ~
4.0
W
3.0
3
2.0
t< K
.
3 LL
1 .o
L 0.2
0.4
0.8
0.6
Figure 8. Salt activity coefficient in the membrane vs. the membrane anion concentration.
1 I
t
1.04
i
I
I
I
1.0
2.0
3.0
4.0
5.0
6.0
UPPER CHAMBER Ct (kg rnole/rn’)
Figure 10. Experimental data from the dialysis experiment. Volumetric flow rates after 3 h vs. the initial upper chamber concentration.
i
1
-3
1
\
1 -
1.00
0
1.0
C- (kg rnole/rn3)
1.16
0
\
-
-
i
I
I
I
I
0
0
1.0
2.0
3.0
4.0
5.0
0.0
UPPER CHAMBER C i (kg mole/m3)
Figure 11. Dialysis concentration data. The percent change in the upper chamber concentration after 6 h vs. the initial upper chamber concentration.
Membrane Activity Coefficients. Membrane salt and solvent activity coefficients were calculated using the equilibrium data and the standard definitions for the electrochemical potentials of salt and water in the external electrolytic solution (Newman, 1973). The activity coefficients were used in the determination of the six Bij membrane transport parameters. To calculate the membrane activity coefficients, the electrochemical potentials of salt and solvent inside and outside the membrane were set equal to one another. Equations for the electrochemical potentials of salt and water in the membrane phase were given in the previous paper (Pintauro and Bennion, 1984). The secondary reference states for solute and solvent inside and outside the membrane were chosen to be the same. The resulting mathematical relationships are
where f* and fo are the molar activity coefficients of salt and water in the external electrolyte, respectively, a, is the water activity in the external electrolyte, and Coois the membrane water concentration in the membrane when the external salt concentration is zero. Values off* and a, were taken from Robinson and Stokes (1959). In eq 12, the superscripts s refer to concentrations in the external solution. Consistent concentration units in eq 11 were obtained by using the membrane porosity, t. Plots of fern
vs. the membrane anion concentration and f o m vs. the membrane water concentration are shown in Figures 8 and 9, respectively. Dialysis Experiments. The information obtained from the dialysis experiments was a volumetric flow rate through the membrane 3 h into an experiment and the final salt concentration after 6 h. During an experiment the volume changes in the upper and lower chamber capillary tubes differed by about 10%. These differences were attributed to very small leaks in the cell. To obtain a single capillary volume change, the upper and lower tube measurements were averaged together. Instantaneous volumetric flow rates were obtained by drawing an equal area curve through the change in capillary volume divided by the time-change vs. time plots. The instantaneous volumetric flow rates at 3 h are shown as a function of the initial upper chamber NaCl concentration (C:) in Figure 10. The final salt concentrtion in the upper chamber of the dialysis cell (CeF)is plotted in Figure 11in terms of the percent change in the salt concentration. Due to mass balance considerations, the decrease in the concentration of the upper chamber must be matched by a corresponding increase in the lower chamber concentration (assuming negligible volume changes in the capillary tubes). The final concentrations in the lower chamber, although not plotted in Figure 11,were found to be within 2% of the values needed to balance the salt losses in the upper compartment. Thus, the lower chamber concentrations were used as an experimental check of the upper chamber concentration changes. Electrodialysis Experiments. The initial conditions and the final results of the electrodialysis experiments are
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
Table I.
239
Electrodialysis Experimental Results4
~
final concn, kg-mol/m3
init concn, kg-mol/m3 1.0 2.0 3.0 4.0 5.0 a
capillary flow rate, m3/s x 1 O l o
top
bottom
top
bottom
1.1211 1.8822 3.1056 4.0550
0.8838 2.1065 2.8821 3.9474 5.1146
2.44 2.00 1.51 0.894 1.03
2.48 2.25 1.66 0.957 1.04
T = 2 5 "C; membrane area = 1 . 7 8 X
1.0 2.0 3.0 4.0 5.0
0.927 0.870 0.794 0.777
10.04 7.79 6.36 4.93 4.30
=
IVoC,O - VFCZlF
Q
(13)
where VO and VF are the initial and final chamber volumes, C,O and C/ are the initial and final bulk solution concentrations, and Q is the total charge passed during an experiment. Electrode and capillary tube volume changes were taken into account when calculating VO and VF. The apparent membrane water transference number (also known as the electroosmotic coefficient) is given by (Sinha, 1977) tom
=
[capillary flow rate1F IVO
5046 5942 6634 4965 6113
665.0 824.6 928.4 511.6 820.3
upstream concn, ka-molim3
volum flow rate, m3is x 10"
downstream concn, kg-mol/m3
1.0 2.0 3.0 4.0 5.0
9.40 8.07 4.53 3.23 1.90
1.05 2.05 3.26 4.32 5.23
T = 25 "C; A P = 7 1 psig; membrane area
listed in Table I. As in the dialysis experiments, the concentration changes in the two chambers should be equal and opposite in direction for the initial concentrations. Discrepancies in the concentration changes were attributed to errors in the NaCl analysis technique. A single concentration change data point was obtained for each experiment by averaging the concentration changes from the upper and lower chambers. The capillary tube flow rates in Table I were obtained by averaging approximately ten volume change/time interval measurements. Flow rates fluctuated by, at most, f5Y0 from the average values listed. The flow rates were also corrected to take into account volume changes in the Ag/AgCl working electrodes (Sinha and Bennion, 1978). A single capillary tube flow rate for each experiment was obtained by averaging the upper and lower chamber capillary flows. The experimental data in Table I were initially interpreted in terms of standard Hittorf transference number calculation procedures. These calculations assume that there is negligible back-diffusion and no pressure generated fluxes; hence, the migration terms in the membrane flux equations (Pintauro and Bennion, 1984) dominate. Apparent cation transferase numbers in the membrane (t+") were determined using the following relationship (Sinha and Bennion, 1978) t+m
0.002 0.002 0.002 0.002 0.002
charge passed, C
Table 111. Steady-State Volumetric Flow Rates and Downstream NaCl Concentrations from the Reverse Osmosis Experiments4
upper lower chamber chamber 9.87 7.63 5.74 4.59 4.32
0.884 0.817 0.840 0.766 0.729
0.132 t 0.140 t 0.139 i 0.103 t 0.135 i
exP time, s
m2.
Table 11. Nafion Cation Transference Numbers and Electroosomotic Coefficients external NaCl 4" concn, upper lower kg-mol/ m3 chamber chamber
current, A
(14)
where I is the applied current and Vo is the partial molar volume of water. Corrections were not made to take into account the volume of the salt when determining the capillary tube flow rate. Table I1 lists experimentally determined values of t,"' and tomfor external NaCl concentrations ranging from 1.0 to 5.0 M. The agreement be-
=
4.07
X
m2.
tween transference numbers determined from the upper and lower chamber data is good. The high values of t+m are consistent with the idea that the cations are the major current carriers in cation-exchange membranes. These apparent transference and electroosmotic numbers are not corrected for salt and water diffusion across the membrane due to small concentration differences or pressure gradients. Reverse Osmosis Experiments. The data obtained from the reverse osmosis experiments were the steady-state volumetric flow rate through the membrane and the steady-state downstream salt concentration. The steadystate flow rates were obtained by averaging together approximately ten experimental flow rate measurements. The steady-state downstream bulk solution salt concentrations and flow rates are listed in Table 111. The highest and lowest flow rates during a single experiment varied by at most f10% and usually by no more than f 5 % from the avexage flow rates in Table 111. Each concentration in Table I11 is the average value of three micropipet samples taken over a 20-h period. The measured downstream concentrations are all greater than the upstream concentrations. In reverse osmosis experiments with cellulose acetate membranes it is known that salt rejection decreases as the AP driving force decreases (Lonsdale, 1966). Also, low-pressure reverse osmosis experiments with ionic membranes were found to reject only 0-5% of the upstream salt (Michaels, 1965). The experimental errors associated with the sampling and analysis technique in the present work were estimated to be between 5 and 7%. The high downstream salt concnetrations were, therefore, attributed to the low salt rejection character of the Nafion membranes in these low-pressure RO experiments coupled with experimental error. Since downstream salt concentration data were needed from this experiment, it was assumed that there was no salt rejection by the Nafion membranes and the upstream and downstream concentrations were equal. Boundary Layer Corrections. An experiment was performed to calculate the mass transfer coefficient, k,, at the membrane-solution interface in the upper and lower chambers of the electrodialysis cell. This quantity appears in one of the boundary conditions of the dialysis and electrodialysis computer simulation models (Pintauro and
240
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
Table IV. Optimized
Di;Parameters for Nation-NaCl-Water System (2'
= 25 " C )
external NaCl concn, kg-mol/m3
D3,,, m2/s ID,. x l o i o ID x 1 0 ' " ID x 10"' x 1O'O B+m x 1 O ' O D,,
x
1.0
2.0
3.0
4.0
5.0
1.80 7.51 5.53 4.50 2.62 5.27
2.37 13.8 3 95 3.23 3.44 3.32
1.08 66.3 3.39 2.69 2.66 2.14
3.43 53.5 3.05 2.49 1.64 0.976
0.211 103.0 1.76 1.92 0.756 1.66
Bennion, 1984). A limiting current method was used to calculate k , (Scattergood and Lightfoot, 1968) in which the membrane in the electrodialysis cell was replaced by a flat copper sheet, the Ag/AgCl working electrodes were replaced by flat copper sheets, and the NaCl electrolyte was replaced by a 0.05 M CuS04-0.5 M H2S04solution. Cathodic limiting currents (13were measured at the center Cu electrode for a given stirrer speed and k , was found from the following relationship
3.2 3.0
2.8
2.6
2.4
IL
k, = nFC,
2.2
where n is the number of electrons participating in the Cu2+/Cuelectrode reaction. The resulting k, values are 1.22 and 1.66 X m/s for the upper and lower chamber membrane-solution interfaces, respectively. Transport Parameter Calculations and Evaluation. A set of six Dij binary interaction coefficients at each NaCl concentration were determined from the equilibrium and transport experiments. The computational technique of determining the Dij parameters assumes that the set of optimized parameters is constant throughout the membrane at that particular external NaCl concentration. However, due to the nature of the transport experiments, the salt content on either side of the membrane differed. Each set of optimized parameters is valid over a small concentration range (approximatelyf0.2 M external NaC1, based on the initial concentrationdifferences in the dialysis experiments). Table IV lists the optimized Dij parameters as a function of the external NaCl concentration. These values of ai,, when inserted into the reverse osmosis, dialysis, and electrodialysis computer simulation programs, were able to predict the experimental results to within f3%. Each optimized Dij parameter has an interval of uncertainty of &lo%, meaning that the true minimum in the objective function (eq 1) is somewhere within *lo% of those values listed in Table IV. The choice of *10% was a reasonable compromise between computer costs and precision. To check for multiple roots of the objective function, the optimization computer program was started using different initial guess values for the Dij parameters. All of the optimization searches resulted in the same set of ai, parameters (within the f10% interval of uncertainty). Mathematical relationships between the D , parameters and the membrane anion concentration were found by combining the results in Table IV with eq 4 and then applying a linear least-squares or Lagrange polynomial regression technique. The resulting equations are
a+-= (-9.95[C-I2 + 8.334[C-] + 0.683) X lo-''
(16)
+ 3.64[C-]) X lo-''
(17)
lo-''
(18)
a+,= exp(l.61 - 1.06[C-]) X a+,- (6.86[C-]3- 22.9[C-] + 0.668) X
(19)
= exp(1.34
= exp(1.92 - 1.38[C-]) x
Bo, = exp(2.07 -2.38[C-])
X
low9
(20) (21)
2.0
-
0.0
1.0
2.0
3.0
4.0
5.0
6.0
TIME (hour)
Figure 12. Upper chamber concentration decay curves from the 1.0-3.0 M dialysis experiment: (0) experimental data; (-) theoretical decay curve.
--. E
w
2 W
c a K
3
s Y
TIME (hour)
Figure 13. Capillary tube volumetric flow rates from the 3.6-4.8 M dialysis experiment: (0) experimental data; (-) theoretical curve.
Applications of Results to Additional Measurements. To test the applicability of using eq 16-21 over wide concentration ranges, two additional dialysis experiments were performed. In one experiment a Nafion membrane separated 1.0 and 3.0 M NaCl solutions. In the other experiment 3.6 and 4.8 M NaCl solutions were used. During an experiment the volumetric flow rates in the capillary tubes were periodically recorded and micropipet samples were periodically withdrawn from the top compartment of the dialysis cell and analyzed for Na by atomic adsorption spectrophotometry. Figure 12 compares the experimental and theoretical concentration decay curves for the upper chamber of the 1.0-3.0 M dialysis experiment. The theoretical solid line was obtained by inserting eq 16-21 into the dialysis computer simulation model (Pintauro and Bennion, 1984). Theoretical and experimental volumetric flow rates in the upper chamber capillary tube of the 3.6-4.8 M dialysis experiment are shown in Figure 13. Initially, the upper chamber was filled with 4.8 M NaCl electrolyte and the volumetric flow was into the upper chamber. Discrepancies between the theory and experiments at very short times can be attributed to small movements of the membrane in the direction opposite the
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 241 3.0
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I
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Figure 15. Nafion membrane conductivity vs. the external NaCl concentration: (0) run 1; (0) run 2; (-) theoretical.
1 .o
0.0
Y 16
Figure 14. Anion concentration profiles from the 1.0-3.0 M dialysis computer simulation program.
osmotic flow. These membrane movements have been observed by other investigators (Kressman et al., 1963) and result in low initial experimental flow rates. At longer times agreement between theory and experiment is within 10%. The 1.0-3.0 M dialysis computer simulation program with the correlated transport parameters was also used to compute the anion concentration and pressure profiles within the membrane. Figure 14 shows the concentration profiles in a Nafion membrane of thickness 6 at three times. Initially the membrane was presoaked in a 1.0 M NaCl solution; hence the concentrations at t = 0 are constant throughout the membrane. As time progresses, the bulk concentration on the right (initially 3.0 M) decreases and the bulk concentration on the left increases. The theoretical pressure profiles for the 1.0-3.0 M dialysis simulation showed a pressure bulge of =22 psi within the membrane. One possible explanation for this pressure fluctuation is that more water is carried into the membrane in the hydration sheaths of the Na+ ions than leaves the membrane. The buildup of water inside the membrane results in the increase in membrane pressure. The magnitude of the theoretical pressure maximum was found to be sensitive to the coefficient of the pressure term in the water equation of state (Pintauro and Bennion, 1984) and a more in depth analysis of this term is needed to obtain more accurate predictions of the membrane pressure excursions. As a final test for the optimized transport parameters, theoretical and experimental membrane conductivities were compared. Nafion membrane conductivities were experimentally measured by using the dialysis cell (Figure 1)with flat plate Ag/AgCl working electrodes. For a given applied dc current, the initial potential drop across the membrane was measured by using two saturated calomel reference electrodes. Since small currents were passed through the cell (between 0.067 and 0.136 A) (Pintauro and Bennion, 1984) for short periods of time, concentration changes in the solutions adjacent to the membrane were negligible and Ohm’s law was used to calculate membrane conductivities, i.e.
where A@, is the measured potential drop across the membrane, 6 is the membrane thickness, and A is the cross-sectional area of the membrane. The theoretical relationship between the membrane conductivity and the
transport parameters has been shown in the previous paper (Pintauro and Bennion, 1984). Figure 15 compares the experimental and theoretical membrane conductivities as a function of the external NaCl concentration. The circles and squares are membrane conductivities obtained from separate experiments. Except at 5.0 M NaC1, the agreement between theory and experiments is good. The 5.0 M discrepancy is attributed to poor performance of the conductivity apparatus with the high concentration salt solution. The erratic fluctuations in some of the Bzjvs. C, data in Table IV and the 10% discrepancy between theoretical and experimental flow rates in Figure 13 are attributed to the fact that a single set of Bij parameters was obtained at each concentration from experiments in which the membrane concentrations were constantly changing. In the dialysis experiments, large initial concentration differences across the membrane were required in order to produce measurable fluxes. Concentration gradients were generated in the electrodialysis experiments by the applied current. This problem of nonuniform membrane concentrations can be solved by performing experiments in which the concentration changes are small or nonexistent. Improved experimental methods of accurately measuring small concentration changes in electrolytic solutions would be needed. Also, the dialysis experiment could be replaced by a constant concentration experiment, such as a radioactive tracer diffusion coefficient measurement. Another solution would be to investigate various combinations of concentration terms and transport parameters which appear in the model in order to find a set of six concentration-independent parameters. Nonuniform concentration experiments would not affect the accuracy of optimized concentration independent parameters. The optimized 21ij parameters were used to analyze two commonly performed membrane experiments: a Hittorf membrane transference number experiment and a radiotracer membrane diffusion coefficient experiment. The determination of apparent membrane cation and water transference numbers in a Hittorf experiment assumes that migration effects dominate. True transference numbers, t+,, and electroosmotic coefficients, torn, can be calculated from the six Bij parameters. A comparison of the transference numbers calculated from the optimized ai, parameters and eq 13 and 14 indicate the relative importance of diffusional and convective effects. The comparison showed less than a 2% difference between t+, and tom calculated from eq 13 and 14 (Table 11) and those calculated from the optimized Q, parameters. Hence, the assumption of negligible diffusion and convection in a Hittorf membrane transference number experiment is valid. Comparison to Radio Tracer Measurements. In radioactive tracer diffusion coefficient experiments (Yeager et. al., 1980) a small number of either Na+, C1-, or H 2 0 species are replaced by radioactively tagged species. In a cation tracer experiment, for example, the experimental
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Ind. Eng. Chem. Fundam.,Vol. 23, No. 2, 1984
I 0
0.2
0.4
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Figure 16. Comparison of tracer and nontracer sodium diffusion coefficients: (-A-) D*Na+;(-0-) L11/1.12.
design and the similarity of cation and cation tracer species result in no net movement of anions or water through the membrane. Also, there are no pressure or potential gradient components in the cation and cation tracer flux equations because the total cation plus cation tracer concentrations are equal on either sides of the membrane. The question that arises in these experiments is whether the measured tracer diffusion coefficient in the five-component (tracer cation, cation, anion, solvent, membrane) system is the same as the nontracer cation diffusion coefficient (the mutual diffusion coefficient) in a four-component system (cation, anion, water, membrane). To examine this problem, the theoretical flux equations for a five-component, cation tracer system were derived (Pintauro, 1980; Bennion and Pintauro, 1981). From that analysis, the following relationship was obtained for the tracer diffusion coefficient of Na+ in the membrane D*Na+
The term f ll/l.12 is the diffusion coefficient of nontracer Na+ in the nontracer system. L11 is divided by 1.12 because the nontracer cation membrane flux equation was derived in terms of the gradient in membrane anion concentration (Pintauro and Bennion, 19841,whereas the cation tracer flux equation is in terms of VC+. The ratio of the ion gradients is obtained by combining eq 4 and 6. The f 1, term is the membrane transport parameter describing the interaction between tracer and nontracer species. If this term is small compared to f 11/1.12, then measured values of D*Na+ can be used in the analysis of nontracer systems. To determine the magnitude of L1., values of L11/1.12 were calculated from the optimized 2). parameters. The values of ,C11/1.12 were then comparea with values of D*Na+ found experimentally by Yeager at the University of Calgary (Yeager, 1980) using the same Nafion membrane that was used in the RO, dialysis and electrodialysis experiments explained previously. The results of this comparison, shown in Figure 16, indicate that not only is LISnonnegligible, but it actually changes sign as salt concentration increases (Le., for C- < 0.35 kg-mol/m3 f 1,* > 0 and for C- > 0.35 kg-mol/m3 L1. < 0). These findings are consistent with the theoretical analysis of tracer and mutual diffusion coefficients of proteins (Keller et al., 1971). The change in sign of Ll,* has been observed previously by investigators who analyzed a dilute NaBr-water-cation exchange membranetracer Na+ system assuming no Br- ions in the membrane (Meares et al., 1972). Conclusions Equilibrium and transport parameters have been determined as a function of electrolyte concentration for the NaC1-H20-Nafion cation-exchange membrane system.
The two equilibrium parameters (the salt and water uptake) along with fixed anion concentration in the membrane were determined from experimental absorption data, and the six transport parameters were determined by matching data from dialysis, electrodialysis, and reverse osmosis experiments with a mathematical transport model. The mathematical model with the experimentally determined parameters was used to predict, with time, the concentration changes in the bulk solutions adjacent to the membrane and the bulk flow of fluid through the membrane. The model and parameters were also used to predict membrane conductivity as a function of concentration. To reduce the labor and increase the accuracy of the transport parameter measurements, additional work is needed to find less concentration-dependent membrane parameters and to develop experiments in which the membrane concentration changes are small. This optimization technique offers the possibility of correlating membrane transport parameters as a function of a variety of variables, such as temperature, pressure, electrolyte composition, and membrane structure. Such correlations could be used to predict the optimum microscopic structure of the membrane needed to produce the desired ion and solvent permeations. Acknowledgment This work was supported by the State of California through the University of California Statewide Water Resources Center. The authors also wish to thank E. I. du Pont de Nemours & Co., Inc., for supplying the Nafion membranes, and Professor Howard Yeager at the University of Calgary, who performed the radiotracer diffusion coefficient experiments. The publication costs have been borne in part by the US. Department of Energy and Brigham Young University. Nomenclature A = cross-sectionalmembrane area, m2 C, = concentration of species i, kg-mol/m3 D, = diffusion coefficient of species i, m2/s a,,,= diffusion parameter describing the interaction of species 1 and j , mz/s f, = membrane molar activity coefficient of species i F = Faraday's constant, 9.6487 X lo7 C/kg-equiv I = current, A IL= limiting current, A k , = mass transfer coefficient at the membrane-solution interface, m/s L,, = membrane transport parameters appearing in eq 23 n = number of electrons involved in an electrode reaction P = pressure, N/m2 Q = amount of accumulated charge, C tLm= Transference number of species i in the membrane T = absolute temperature, K = volume, m3 V , = partial molar volume of species i, m3/kg-mol W = membrane weight, kg Greek Symbols 6 = membrane thickness, m t = membrane porosity K = electrical conductivity, mho/m x = membrane fixed ion concentration, kg-mol/m3 @ = electrical potential, V p = membrane density, kg/m3 Subscripts d = dry membrane e = external electrolyte as a neutral species ij = any arbitrary species 0 = solvent (water) m = membrane + = membrane cations
Ind. Eng. Chem. Fundam. 1984, 23, 243-252
- = membrane anions Superscripts 0 = initial or reference state F = final * = radiotracer species Registry No. NaC1, 7647-14-5; Nafion 110, 61261-17-4.
Literature Cited Bennion, D. N.; Pintauro, P.N. AICh€ Symp. Ser. 1981, (204), 190. Choi. K. W.; Bennion, D. N. Ind. Eng. Chem. Fundam. 1975, 14, 296. Helfferich, F. "Ion Exchange," McGraw-Hill: New York, 1962; pp 339-420. Hooke, R.; Jeeves, T. A. J. ASSOC. Comput. Mech. 1961, 8, 212. Kamo, N.; Toyoshima, Y.; Nozaki, H.; Kabatake, Y. KollOM-2. 2. Pdym. 1971, 248, 914. Keller, K. H.; Canales, E. R.; Yum. S.1. J. Phys. Chem. 1971, 75. 379. Kressman, T. R. E.; StanbrMge, P. A,; Tye, F. L.; Wilson, A. G. Trans. Faraday SOC. 1983, 5 9 , 2133. Lonsdale, H. K. "Desalination by Reverse Osmosis", U. Merten, Ed.; The M. I.T. Press: Cambridge, MA, 1966; pp 93-160.
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Meares, P.; Thain, J. P.; Dawson, D. G. "Membranes-A Series of Advances", G. Eisenman, Ed.; Marcel Decker, Inc.: New York, 1972; pp 55-124. Michaels, A. S.; Bixler, H. J.; Hausslein, R. W.; Fleming, S. M. US. Department of the Interior. Office of Saline Water, Research and Development Progress Report No. 149, Washington, DC, 1965. Newman, J. "Electrochemical Systems"; PrenticaHail; Englewood Cliffs, NJ, 1973; pp 239-250. Plntauro, P. N. Dlssertetion, University of California, Los Angeles, 1980. Pintauro, P.N.; Bennlon. D.N. Ind. Eng. Chem. Fundem. preceding paper in thls issue. Robinson, R. A.; Stokes, R. H. "Electrolyte Solutions"; Butterworth Scientific Publlcations: London, 1959; pp 102-104. Scattergood, E. M.; Lightfoot, E. N. Tf8nS. Faraday Sm. 1968, 6 4 , 1135. Sinha, M. M.S. Thesis, University of California, Los Angeles, 1977. Sinha, M.; Bennlon, D. N. J. Electrochem. SOC. 1978, 125, 556. Yeager, H. L. University of Calgary, Alberta, Canada, private communication, 1980. Yeager, H. L.; Klpling, B.; Dotson, R. L. J. Nectrochem. SOC. 1980, 127, 303.
Received for review March 7, 1983 Accepted December 23, 1983
EXPERIMENTAL TECHNIQUES Gradientless Reactor for Gas-Liquid Reactions Yonatan Manor Department of Chemlcal Englneerlng, Unlverslty of Illlnols, Urbana, Illinois 6 180 1
Roger A. Schmltr' Department of Chemlcal Englneerlng, Unlverslty of Notre Dame, Notre Dame, Indiana 46556
A continuous-kw gradientless reactor was designed in this research to facilitate the study of relatively fast gas-liquid reactions under Isothermal conditions and in the absence of transport limitations. The essential component of the reactor was a multi-bladed rotor which contacted the gas-liquid interface directly. The mixing characteristics were tested, and mass transfer studies yielded rates greater than 30 times those of previously employed reactors. To demonstrate the utility of a reactor of this type, kinetic studies of the oxidation of sulfite and of propionaldehyde were conducted under kinetically controlled conditions at rates not previously attainable under such conditions.
Introduction Studies of the kinetics of fast gas-liquid reactions are generally complicated because their rates are usually retarded by transport limitations, and isothermality is difficult to maintain. Most prior research has been aimed at trying to model mathematically the simultaneous processes of reaction and transport or at designing equipment for improved parameter measurement and control. Little attention has been given to the development of gas-liquid contactors which would eliminate transport interferences over the range of reaction rates of interest. In the commonly employed sparged reactor, mass transfer rates are increased by adding extra stirring through an impeller submerged in the liquid phase. The energy introduced by the impeller is transferred through the liquid bulk to the gas-liquid interface where it is utilized for mixing near the interface and for bubble re-
dispersion. Most of the energy is dissipated through internal friction and liquid turbulence. Once the boundaries of the dispersed gas phase become rigid, as in the case of small bubbles or in the presence of impurities at the interface, additional energy input by increased stirring has little or no effect. The present work was aimed at designing and testing a continuous-flow reactor in which the energy for increasing the rates of interphase transport is imparted by a multi-bladed rotor directly at the gas-liquid interface. The primary utilization of the reactor would be for studies of intrinsic chemical kinetics at the laboratory scale although it is possible that a similar design could be considered for use at the commercial scale. The results of laboratory tests reported in this paper show that significantly higher values of kL are attainable than have been reported for other types of reactors, and that the bulk fluid 0 1984 American
Chemical Society
