230
Ind. Eng. Chem. Fundam. 1984, 23, 230-234
Mass Transport of Electrolytes in Membranes. 1 Development of Mathematical Transport Model Peter N. Plntauro and Douglas N. Bennion” Chemical Engineering Department, University of California, Los Angeles, Los Angeles, California 90024
A model of transport across semipermeable membranes is proposed. The model is based on the Stefan-Maxwell equations and differs from previous models in that the transport parameters are considered to be functions of composition, pressure, and position within the membrane. Activity coefficient corrections and distribution coefficients at the membrane-bulk solution interface are considered. Two thermodynamic coefficients or distribution coefficients are defined along with six transport coefficients. With the concentration of fixed charge in a membrane, these parameters and equations completely define a membrane’s transport performance for any set of boundary conditions. Relationships between various equivalent forms of the transport equations are defined. Experimental approaches for measuring the necessary and sufficient parameters are suggested. The thermodynamic parameters can be measured by combinations of absorption and analysis, and the six transport parameters can be defined through measuring salt and water flux for dialysis, electrodialysis, and reverse osmosis experiments.
Over the past 25 years a wide variety of membranes have been synthesized in the laboratory for research purposes and manufactured on a large scale for industrial processes. Concurrent with the production of new membranes has been the development of mathematical models to predict and correlate membrane transport rates. A pore flow model has been used to describe the permeability properties of biological (Panganelli and Solomon, 1957; Side1 and Solomon, 1957; Solomon, 1968) and track etched mica membranes (Quinn et al., 1972; Bean, 1969; Beck and Schultz, 1970; Anderson and Quinn, 1974). The differences in solubility between closely similar ions and molecules in biological and cellulose acetate reverse osmosis membranes have been analyzed in terms of intermolecular and interionic forces external to and within the membrane (Eisenman, 1969; Diamond and Wright, 1969). Many of the early treatments of ion-exchange membrane transport used the Nernst-Planck equations to describe the relationship between the flows of permeating species and the forces acting on the system (Helfferich, 1962; Schloegl, 1954). The theory of irreversible thermodynamics has been used by a number of investigators to describe solvent and solute transport across membranes. This approach has been applied to binary solutions in biological membranes (Kedem and Katchalsky, 1958,1962) and dilute electrolytic solutions in ion-exchange (Spiegler, 1958; Scattergood and Lightfoot, 1968) and cellulose acetate reverse osmosis membranes (Bennion and Rhee, 1969; Osborn and Bennion, 1971; Choi and Bennion, 1975; Re and Bennion, 1973). A review of experimental and theoretical research in membrane transport has been recently published by the present authors (Bennion and Pintauro, 1981). In the present work, a mathematical model based on irreversible thermodynamics has been developed. In this work the transport theories of Kedem and Katchalsky (1958, 1962) and Spiegler (1958) have been expanded so that multicomponent transport of concentrated electrolytic solutions in either neutral or ion exchange membranes can be described. The Mathematical Transport Model The relative transport of ions and solvent in membranes is affected by the physical and chemical properties of the
ions and solvent in solution, the properties of the membrane, and the mutual interactions between solutes, solvent, and membrane. The proposed model takes into account all of these factors and is based on concentrated solution theory (Newman, 1973) and the Stefan-Maxwell transport equations (Hirschfelder et al., 1954). The fundamental equation describing isothermal mass transport in the membrane is
where C,, k,, and u, are the concentration, electrochemical potential, and velocity of species i, respectively, and K , is the friction coefficient describingthe interaction between species i and j . The left side of eq 1 is the driving force per unit volume on species i and the right side is the sum of frictional forces which oppose the driving force. The friction coefficients can be replaced with binary interaction coefficients, by using the relationship RTC,C,
all,
K,, =
___
where R and T are the gas constant and absolute temperature and CT is the total concentration of all species present in the membrane. For a simple 1:l binary saltsolvent-membrane system there are four components: J = cation (+), anion (-), solvent (0) and membrane (m). However, through the application of the Gibbs-Duhem equation (Newman, 1973), there are only three independent transport equations (i = +, -, 0). This analysis assumes that the membrane is stationary and u, = 0. For a binary saltsolvent membrane system eq 1 can be written as
* Chemical Engineering Department, Brigham Young University, Provo, U T 84602. 0196-4313/84/1023-0230$01.50/0
(2)
CTal
C 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 231
In the above transport equations there are a total of six independent binary diffusion parameters, Dij, with Dij = Dji (Onsager, 1931). To obtain velocities and molar fluxes, Ni, as a function of the gradients in electrochemical potential, eq 3-5 are inverted. The resulting flux equations are
N + = -L++C+2Vp+- L+-C+C-Vp-- L+oC+CoVpo (6) N- = -L-+C-C+Vp+- L-C-2Vp- - L-&-CoVpo
(7)
No = -Lo+CoC+Vp+- L+CoC-Vp- - LooCo2V~o (8) There are six independent Lij transport parameters in eq 6-8 with Lij = Lji. The Lij coefficients are complicated functions of concentration, temperature, and the six Dij diffusion parameters. The mathematical relationships between the Lij and Dij parameters is given in the Appendix. It is desirable to substitute into eq 6-8 expressions which relate the electrochemical potential of the solute and solvent to the experimentally measurable quantities of membrane anion concentration, C-,pressure, P, and electrical potential, a. The concept of electrical potential is introduced into the model by defining a reference electrode half-cell reaction involving the solute and/or the solvent in the membrane. Application of chemical thermodynamic principles to this half-cell reaction results in the following relationship between Vp;s and V O S-Vp- S+Vp+ SoVpo = -nFVO (9)
+
+
where Si are the stoichiometric coefficients of species i in the reference electrode half cell reaction, n is the number of electrons involved in the half-cell reaction, and F is Faraday's constant. Standard relationships have been used to describe the electrochemical potential of neutral electrolyte, pem,and solvent, po, in the membrane (Newman, 1973). For the particular case of a cation-exchange membrane, pemcan be written as a function of the anion concentration in the membrane pem=
uRT In a,"femC-/u-
(10)
and po can be described in terms of the membrane water content po
a,BfomCo COO
= RT In -
where u is the sum of the numbers of cations (u+) and anions (v-) in a neutral salt molecule, f,, is the molar activity coefficient of the electrolyte in the membrane, a," is a constant describing the secondary reference state of the salt cf,, 1as C- 0 ) and C,O is the membrane water concentration when the external salt concentration is zero. Cooappears in eq 11 as a consequence of the choice of the secondary reference state for membrane water (fo, 1 as Co/C$ 1). The symbol a$ represents the absolute activity of pure water. In order for the driving forces in the membrane flux equations to be VC-, V P , and VO, a relationship defining the functional dependence of the membrane water concentration on the membrane salt concentration and pressure must be substituted into eq 11. Such a membrane water "equation of state" is analogous to the relationship in liquids between solvent concentration, solute concentration, and density (which itself is a function of solute concentration and pressure). Although the functional dependence of Coon C- and P in a membrane may be quite complex, a linear relationship has been proposed
- -
-
-
where PO is atmospheric pressure and the constants CgO, 0,and a can be determined either experimentally or theoretically. Using eq 9-12 to eliminate Vp+, Vp-, and Vpo from eq 6-8 results in the following membrane flux equations
N+ = -L11VC- - L12VP - L13VO
(13)
- L23Va
(14)
No = -I31VC- - L32VP - IC33Va
(15)
N- = -L21VC-
- L22VP
It is not true that I,= Llkin eq 13-15. The nine L, parameters are compficated functions of concentration, pressure, temperature, the six L, coefficients from eq 6-8, the membrane salt and solvent activity coefficients and the constants in the water equation of state. A detailed method of deriving eq 13-15 from eq 6-12 along with the mathematical relationships between the L, and I , parameters has been recently published (Bennion and Pintauro, 1981). I t is desirable, when modeling actual membrane experiments, to replace the electrical potential driving force terms in eq 13-15 with terms involving constant applied current. This is done by combining eq 13-15 with the definition of current density, i, which, for a simple 1:l binary salt is
i = F(N+- N-) The resulting flux equations are N + = -L'llVC- - L f I 2 V P- L f I 3 i
(16) (17)
N- = -SC'21VC-- I'22VP - I'23i
(18)
No = -I',,VC- - S'32VP- L'33i
(19)
In eq 17-19 there are only six independent LfU parameters with Lfl1= L'21,LtI2= I' 2 2 , and FL'23 = 1 + FL',,. Mathematical relationships between the L', and L, parameters have been presented elsewhere (Bennion and Pintauro, 1981). The classical transport properties of diffusion coefficient ( D J ,transference number (t,") and electrical conductivity (K,) in a membrane can be related to the fundamental Q1 binary diffusion parameters through the I , or LfVparameters. The membrane diffusion coefficients of cations and anions are the I, coefficients of the VC- terms in the membrane flux equations explicit in potential (eq 13-15). The Fick's law type diffusion coefficient of solvent in the membrane is obtained by combining eq 15 with the water equation of state (eq 12). The following relationship results f-31
Do=P The electrical conductivity of a membrane is given by algebraic sum of two I i jparameters. Km
F(L13 - f - 2 3 )
(21)
The membrane cation transference number, t+m,and the membrane water transference number, tom(also known as the electroosmotic coefficient) are given as
Equations 21-23 are specific forms of the generalized transference number and conductivity relationships that
232 Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
have been previously derived in concentrated, multicomponent electrolyte theory (Newman, 1973). Flux eq 13-15 and 17-19 apply only within the membrane phase. At the membrane solution interface, the pressure and electrochemical potential of salt and solvent are assumed continuous. A partition coefficient, K , is used to describe the concentration jump at the interface.
-ace'at
C,OAU
A V
iA
+-V FV
- --(N-)
(33)
where u is the volume average velocity through the membrane and is given by u =
CViNi
(35)
i
K is usually a function of concentration and can be determined either experimentally or theoretically (Pintauro, 1980). The membrane flux equations can now be combined with the conservation of species equation and the proper boundary and initial conditions to simulate various membrane processes.
Steady-State Reverse Osmosis For reverse osmosis transport, the current is zero and the cations and anions move through the membrane as neutral species. The flux equations for this case are N+ = N - = Ne, = -L'21VC- - L'22VP (25)
No = -Lf31VC- - L',,VP
(26)
Equations 25 and 26 are combined with the steady-state conservation of species equations
V.Ni = 0
(i = 0, e )
(27)
Upstream, the boundary conditions are that the concentration of the salt solution and the constant applied pressure are known
c _ =Ce -; P = P (28) K The downstream boundary conditions are atmospheric downstream pressure and the downstream salt concentration, which is produced entirely by the ultrafiltration process (Lightfoot, 1974). Equations 25,26, and 27 form a set of coupled, nonlinear partial differential equations with four unknowns, C-, P, N+, and N-. Transient Electrodialysis, Dialysis, and Membrane Conductivity A typical experimental electrodialysis cell consists of two chambers filled with electrolytic solution and separated by a membrane. In each chamber there is a capillary tube to measure the volumetric flow rate of material through the membrane and from working electrodes. For long hold-up times in the membrane, solvent and solute transport can be described by two flux equations and two conservation of species equations
The flux of cations can be obtained from the current density definition (eq 16) once N- has been determined. Changes in concentration with time in the compartments on either side of the membrane are described by mass balance equations
The first terms on the right sides of eq 33 and 34 represent the change in concentration due to the flux of material through the membrane. The second terms represent material entering or leaving the chamber by way of the capillary tubes. The third terms describe the effects of the electrode reactions on the bulk concentrations in each chamber. A term of the form (vel- - VAgC!)iA/F which accounts for volume changes of the silver-silver chloride electrode has been neglected. Equations 30-34 represent six coupled nonlinear partial differential equations containing six unknowns (C:, C:I, C-, P, N-, and No). The initial conditions for the transient problems are that the pressure and concentrations in each chamber and in the membrane are known
Ce' =
(C,I)O;
CeII
= (Ce")O;
co = (cop;P = Pa; C- = (C,')O/K (36)
At the two membrane-solution interfaces the pressure and anion flux are continuous. The continuous flux boundary conditions for compartments I and I1 are similar and of the form
-Lf21VC-- Lf22VP- L2&= k,(Ce
- KC-)
t-Oi
++ C,u ZP
(37) The left side of eq 37 is the expression for the membrane anion flux (eq 18); the right side is the anion flux in the solution, which consists of diffusion, migration, and convection terms. The mathematical analysis of a dialysis experiment requires the same equations as the electrodialysis simulation with the current set equal to zero. A membrane conductivity experiment can also be simulated by using the membrane flux equations. For a given applied current, the potential gradient across the membrane can be found by substituting flux eq 17 and 18 into the current density relationship (eq 16). The resulting expression is
A conductivity simulation would use flux equations explicit in current density (eq 18 and 19) to determine VC- and V P as a function of position within the membrane. Equation 38 would then be used to determine A@ by integrating stepwise across the membrane. Solutions to the one-dimensional steady-state reverse osmosis and the transient dialysis/electrodialysisequations can be obtained by using numerical techniques (Newman, 1968). The system to be analyzed is divided into a number of mesh points. The differential equations are placed in finite difference form, linearized and solved simultaneously at each mesh point. In order to solve the membrane transport equations for a 1:l binary salt-solvent-membrane system, two equilibrium parameters and six transport parameters must be specified (using theory or experiments). The equilibrium
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 233
parameters are the salt partition coefficient (eq 24) and the water equation of state (eq 12). Standard experimental techniques (Pintauro, 1980) can be used to determine these two parameters. The concentration of fixed ion-exchange sites in the membrane must also be known. The six transport parameters appear in different forms in the development of the membrane flux equations, Le., the six Dij parameters in eq 3-5, the Lij L 'ij parameters in eq 6-8 and the six I';parameters in eq 17-19. These six transport parameters cannot be obtained directly from experimental measurements. It would be highly desirable to be able to measure each transport parameter one a t a time. For example, if VP, i, N+, and No could be held at zero, then N- and VC- could be measured in an experiment and I'21 would be simply -N-/VC-, as follows from eq 18. The problem faced in multicomponent transport in membranes is that, so far as appears known a t this time, it is impossible to have VC- finite and maintain V P in the membrane at zero or to keep N - finite and N o zero. One possible solution to this coupling phenomenon is to make several nearly "orthogonal" experiments. The word orthogonal is used here to describe experiments which are coupled at right angles in a multidimensional space uniquely relating one force to one flux. Since a truly orthogonal set of experiments has yet to be formulated, nearly orthogonal experiments can be performed and the six transport parameters fitted to the experimental data simultaneously. Electrodialysis, dialysis, and reverse osmosis are three nearly orthogonal experiments which can be used. Each can yield two independent experimental observations, a concentration change and a volume or weight change in a compartment of known volume. The resulting six independent experimental observations can be used to determine the six transport parameters. In the electrodialysis experiments the coefficients of the current terms in the flux equations dominate. In the dialysis experiments the VC- terms dominate while the pressure terms dominate in the reverse osmosis experiments. By selecting a set of six Dij parameters, six Lij parameters or six & I i j parameters and solving the proper differential equations, each of the three membrane experiments can be simulated on a computer and theoretical predictions of the experimental results can be made. The sum of the squares of the differences between observed and calculated results defines an error. Values of the six transport parameters are selected so that, by a directed trial and test procedure, the optimum set of parameters is found which yields the minimum error. In the following paper, a detailed description of the experimental procedure and apparatus for determining the two equilibrium and six membrane transport parameters is given. For a Nafion cation exchange membraneNaC1-water system, the six Dij parameters are determined by performing electrodialysis, dialysis, and reverse osmosis experiments, as described above.
Appendix Interconversion of the D i , and Lij Membrane Transport Parameters. The determination of the six Lijcoefficients as functions of the six Dijparameters begins by expanding eq 1in terms of the Kij friction coefficients. For the binary salt-membrane-solvent system in question (with urn = 0), eq 1 becomes
where
RTC;C; and KG = K,? To simplify the mathematics, eq A1-A3 are rearranged to give the following C+Vp+= M++u+ M+-u- M+ouo (A41
+
+
C-Vp- = M-+u++ M--u- + M + u ~ CoVpo = M ~ + u + + M@U-+ MWUO
(A5) (A61
where
M..I K... 115 M ll. , = M..* i # j
(A71
Mi; = -E Ki;
(AN
and j
Equations A4-A6 are mathematically inverted to give three velocity equations (u+, u-, and u,) in terms of the three Vpi driving forces. The velocity equations are then multiplied by the corresponding concentration (C+, C-, or C,) to give the three molar flux equations (eq 6-8). The relationships between the Lij and Mij parameters are given as follows
L+- =
-[M+,-M,
- M+-.M,]
M -[M+-*M, - M--*M+o] L+o = M L-- =
-W++*M, M
- M+O21
-[M+-.M+o - M++44_oI M -[M++.M-- - M+?] L, = M M = M++.M--.M, + BM+o.M_o.M+o - M--.M+; M_02- M,*M+-' L_o =
(A10) (All) (-412) (A13) (A141 - M++.
(A15)
Nomenclature as = property expressing the secondary reference state, m3/ kg-mol A = cross-section membrane area, m2 Ci = concentration of species i, kg-mol/m3 CT = total concentration, kg-mol/m3 Di = diffusion coefficient of species i, m2/s Do = water diffusion coefficient in eq 20, m2/s ai,.= diffusion parameter describing the interaction of species L and j , m2/s e- = symbol representing an electron fi = membrane molar activity coefficient of species i F = Faraday's constant, 9.6487 X lo7 C/kg-equiv i = electric current density, A/m2 k , = mass transfer coefficient at the membrane-solution interface, m/s K = salt artition coefficient, (kg-mol/m3of solution)/(kgmol/m! of wet membrane) Kij.= friction coefficient describing the interaction of species 1 and j , kg/[s m3) L, = membrane transport parameters appearing in eq 6-8, m 4 / b N) I.. = membrane transport appearing in eq 13-15 L Yij = membrane transport parameters appearing in eq 17-19 n = number of electrons involved in an electrode reaction N i = molar flux of species i, kg-mol/(m2 s) P = pressure, N/m2 R = universal gas constant, 8.314 x lo3 J/(kg-mol K)
Ind. Eng. Chem. Fundam. 1904, 2 3 ,
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
234-243
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. U S . 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. Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. "Molecular Theory of Gases and Liquids"; Wiley: New York, 1954; p 714. Kedem, 0.; Katchalsky, A. Biochem. Siophys. Acta 1958. 2 7 , 229. Katchalsky, A. Siophys. J . 1962, 2 , 53s. Kedem, 0.; 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
