M A S S T R A N S P O R T OF B I N A R Y E L E C T R O L Y T E S IN M E M B R A N E S Concentration Dependence for Sodium Chloride Transport in Cellulose Acetate D . N. B E N N I O N A N D B. W. R H E E Department of Engineering, University of California, Los Angeles, Calif. 90024
Equations describing the transport of water and a binary salt through semipermeable membranes are presented. Methods of experimentally determining the resulting six independent transport parameters have been tested. Measurements of these parameters on Loeb-type membranes are presented for curing temperatures of 75', 84", and 95" C. If the membrane concentration i s characterized b y the geometric mean activities of adjacent solutions, the osmosis parameters as defined are concentration-independent up to 0.1M. They are also independent of the pressure difference across the membrane up to 600 p.s.i.g. Above 600 p.s.i.g. compaction causes a decrease in the transport parameters. Coupling between the different driving forces is quantitatively measured. During reverse osmosis the water flux is not influenced b y coupling and can be described by one parameter. The salt flux shows coupling to the water flow and requires two parameters to describe its variations. The reciprocal relations are valid to within at least an order of magnitude. Since water flux coupling i s small, the uncertainty in the magnitude of the corresponding coupling parameter is large. Electrical conductivity in a membrane i s one third or one fourth of that in the adjacent solution. The transference number of the sodium ion decreases from 0.85 at 0.0035M to about 0.3 at 0.075M.
QUATIONS
describing diffusive flow of water and a binary
E salt through a semipermeable membrane are presented.
The six independent transport parameters defined by these equations have been measured for sodium chloride and water flow through Loeb-type cellulose acetate membranes (Loeb and Manjikian, 1965; Loeb and Sourirajan, 1963; Manjikian, 1967). The binary interaction coefficients or friction coefficients are calculated from the observed transport parameters using a calculation procedure not previously utilized. A concentration dependence for the transport parameters is proposed and tested. The pressure dependence is also observed for low and modest pressure regions. These observations have been made for four different membranes. Three of these are tube cast membranes, identical except for the socalled curing temperatures of 75', 84', and 95' C.-Le., a loose, medium, and tight membrane, respectively. The fourth membrane is a flat cast membrane cured at 75' C. However, its air-drying time was appreciably longer than that of the tubular membranes. The general formulation of the transport equations presented here is compared to two somewhat different approaches which have been used recently. Theoretical Development
The general concept of relating a driving force on a species to a linear sum of frictional interactions was proposed by Maxwell, often referred to as the Stefan-Maxwell equations. However, it is only recently that this concept has been applied to electrolyte transport by Klemm (1953, 1962), Laity (1959a, b), Lamm (1964), Newman et al. (1965), and others. I t appears that the first to apply this concept to membranes was Spiegler (1958) in a study of ion exchange membranes. The general approach has been applied to reverse osmosis membranes by Kedem and Katchalsky (1958, 1961, 1963). Recently a modified approach was presented for ion exchange membrane application by Wills and Lightfoot (1966). 36
I&EC FUNDAMENTALS
The friction coefficients are not directly measurable parameters. Using straightforward experimental procedures, values for diffusion coefficients, conductivities, transference numbers, and similar parameters are obtained. A calculation procedure is necessary to obtain the friction coefficients. Approximate methods have been described. Spiegler (1958) and Wills and Lightfoot (1966) both propose different methods applicable to ion exchange membranes where one ionic species is absent from the membrane. Kedem and Katchalsky (1961) proposed a method applicable to reverse osmosis-type membranes, but used approximations. The errors introduced by these assumptions have not been quantitatively evaluated. As shown below, the equations are all linear and an exact relationship is possible. The resulting expressions are lengthy, but they can be easily manipulated using a computer. The friction coefficients are defined by the relation
di =
Kij(vj
-~
t
)
(1)
j
For a binary electrolyte solution, Equation 1 represents three independent equations. Here di is the vector driving force per unit volume on species i. I t can usually be written as c i v ~ twhere pi is the electrochemical potential of species i (Bennion, 1966). The vi represent the velocities of species i . The molar flux of species i is civi. The Kij are the friction coefficients or binary interaction coefficients. The nomenclature for these parameters has been unfortunately varied. Except for minor differences in units, the Kij used here correspond to the r i j used by Klemm (1953) and the X i j used by Spiegler (1958). I t is often useful to replace the Kij's with binary diffusion coefficients, Dij, using the relation RT cicj K 2.1. -GTDDij
I n this discussion it is assumed that Kij =
Kji
Coleman and Truesdell (1960) along with many others (Denbigh, 1951; Fitts, 1962; Newman, 1965) have discussed Equation 2 or equivalent expressions. As Duncan (1965) has pointed out, Equation 1 defines nine transport parameters. With Equation 2, there are only six independent parameters. After applying Equation 2, Equation 1 can be written as follows for water plus a binary salt being transported across a membrane :
- v+) + K+4vo - v+) - K+,v+ c-Vp- = K+(v+ - V-) + K-,(v0 - K,V~oVpo = K + ~ ( v ++ K - ~ ( v - - v,) c+vp+ = K+(v-
V-)
Komvo
vO)
(la)
(lb) (IC)
brane. The necessity for this and the assumptions involved have been discussed by Kirkwood (1954). Before integrating, it proves convenient to replace the chemical potentials with activity and pressure terms. V p L s= P,VP
V p o = V o V P -I-R T V In a.
+ v-p-
or v p ,
= v+vp+
+ v-vp-
=
Fz+c+v+
+ Fz-c-v-
Y-
(3)
2, RT
Lo=-(4)
(5)
K
k F -
6
6 is the effective membrane thickness across which the integration is performed. When the electric current density, i, is zero, the salt flux, N,, can be defined as -Ye = N-/v- = N+/v+. By making these substitutions and assuming effective values for the activity of salt and water in the membrane, a, and d o , the flux equations can be integrated to yield the following when the electric current is zero.
Ne = -L,Aa,
Most of the symbols have conventional definitions. I t follows from the algebraic manipulation that the transport parameters C,,sea, So,, Lo,t+, t-, and to can be written as functions of K, K OK+,, , K-,,,, and K O , (Bennion, 1966). The relationships are given in the Appendix. Since t+ is equal to (1 - t-), and as a result of Equation 2 , it follows that C,, equals so,, there are only five independent parameters defined by Equations 5, 6, and 7. The sixth parameter is the electrical conductivity. The method for introducing the current equation is basically the same as described by Kedem and Katchalsky (1963) and presented in somewhat more general form by Newman et al. (1965). In this approach, potentials are defined in terms of reference electrodes, for which the generalized reaction is
s+w+z+
+ S - V ~ + sovp0 = - n F W
de
- Le, -Y Aao -
When the driving forces Aae, Aao, and AP are all zero, the flux equations reduce to
N +
t+i -z+F
No =
t,i
i = -kA@
+ s-w-2- + sowo= ne-
From thermodynamics s+vp+
6
a,
Equations la, l b , IC, 3, and 4 are five linear equations containing eight variables. By algebraic manipulation they can be rearranged so that any three variables appear as the independent variables. For experimental work it is convenient to rewrite the transport equations as follows.
(8)
Substitution into Equation 8 from Equations 1, 3, 4, 5, 6, and 7 yields
Here K is defined in terms of the K’s as shown in the Appendix. These flux equations must be integrated across the mem-
(11)
C, RT a, 6
z+ L e-= - y - -
The definition of the electrical current density is
i
(10)
In addition, as predicted by classical electrochemical transport theory (Newman et al., 1965), C , and $,, are highly concentration-dependent. If these parameters are redefined as follows, they are nearly concentration-independent
A fourth equation might be written, in which the driving force on the membrane is on the left-hand side of the equation. However, as discussed by Hirschfelder et al. (1954), the sum of the driving forces is zero and the fourth equation would not be independent. The following is simply a standard thermodynamic relationship defining the chemical potential of the electrolyte (Denbigh, 1957). p s = v+p+
+ v R T V In a ,
(22)
The thermodynamic variables a,,, a,, and P are continuous across the membrane boundary. Thus Aa,, Aao, and A P can be determined by simply obtaining the difference of the values in the bulk solutions on either side, adjacent to the membrane. However, in general, the potential Q is not continuous across the membrane solution interface. Direct current measurements using reference electrodes in the bulk solutions yield k only after complex corrections. Simple alternating current measurements as described by Wills and Lightfoot (1966) yield k directly if the concentration is the same on both sides of the membrane and a suitable frequency range is employed. There VOL. 8
NO. 1
FEBRUARY 1969
37
are only three independent transport parameters in Equations 17 and 18, since L o , equals L a , as a direct consequence of Equation 2. I t is proposed to use the geometric mean of the activities in the bulk solutions on either side of the membrane as the effective or characteristic membrane activities, 6, and d o . The experimental procedures described allow independent measurement of groups ( L e o a s / ~and ) (Loeao). If L,, equals Lo,, it is taken as evidence supporting the use of the geometric mean for the effective membrane activity. Often the coupling term or terms from which L e , and L o , are determined are small. Thus, small measurement errors may cause significant variations in L e , or Lee. Therefore, the test is not highly sensitive and variations of 100% or less are about the best that can be expected with present experimental techniques. It has been shown conclusively (Banks and Sharples, 1966; Riley et al., 1964) that the Loeb-type membrane is not homogeneous but has a thin, tight skin on one surface where the actual salt rejection takes place. As done earlier by Reid and Breton (1959), there may be advantages in principle to studying the normal cellulose acetate membranes which are thought to be homogeneous. This approach has been discussed (Lonsdale, 1966; Lonsdale et al., 1965). However, the high flux Loeb-type membranes are of more practical interest. I t has been assumed in the analysis and interpretation of experimental data presented here that most of the resistance to the various fluxes occurs in the thin dense layer mentioned above. The parameters reported are to be viewed as effective or average parameters applicable to the membrane as a whole. A more detailed analysis of the series resistance of this double structure-type membrane has been given by Jagur-Grodzinski and Kedem (1966). A number of other formulations of the flux equations are in current use. The basic principles behind these approaches have been discussed by Merten (1966). I t appears useful to compare briefly the equations presented here with two other popular approaches. The following equations have proved useful for engineering applications as presented by Sherwood et al. (1967).
+ kzM,APc, ATo= k i ( A P - AT) + kzM,APco
Ne
=
k&,(c,
- c),
Slight modifications have been made in the notation to conform more closely to the general notation of the previous equations in this report. Here c t is the initial concentration or the concentration on the high pressure side of the membrane and c, is the product concentration or concentration on the low pressure side of the membrane. I n presenting these equations, Sherwood et al. stated that coupling was neglected and that the last term in each equation represents “pore flow.” Coupling may be explained by various detailed mechanisms. As pointed out by Merten (1966), the transport of salt due to water flow is a form of coupling. Thus, these so-called pore flow terms are coupling terms. The k $ s in each equation are the coupling coefficients. The assumption that they are equal is the reciprocal relationship analogous to Equation 2 of this paper. T o include any other terms would be to overspecify the problem. This point is discussed in greater detail by Bennion (1966). If one assumes Aae proportional to Aa,, and replaces activities by concentrations, Equations 17 and 18 can be put in the same form as these two equations of Sherwood et al. By equating coefficients of matching driving forces there result four equations for the three parameters. The equations are not consistent. Thus the coupling terms are not equivalent in the two formulations. 38
l&EC FUNDAMENTALS
As pointed out by Banks and Sharples (1966), dye molecules will not pass through these modified cellulose acetate membranes. T o imagine macroscopic, hydrodynamic flow as a meaningful concept in pores less than a molecular diameter in size seems less reasonable than Equation 2 as the basis for flow coupling. I t is further felt that pressure and concentration dependence of the parameters will prove simpler using the equations developed here. The experimental results presented below are offered as partial justification for this claim. In addition, Vignes (1966) and Cullinan (1966) have sIio\.cn that the use of activities as driving forces reduces substantially the concentration dependence of the associated transport parameters. It thus appears that, in principle, Equations 17 and 18 are to be preferred. Another approach has recently been summarized by Spiegler and Kedem (1 966), whose fundamental equations are the same as Equations 3, 4, and 5 presented here. The most obvious difference is the use of a volumetric flux in place of a ivater, molar flux, and a different grouping of driving forces. There is another difference xchich in practice may be minor but in principle is significant. I n Equation 10 of Spiegler and Kedem (1966) the so-called osmosis driving force, d r / c 8 , which is similar to our Aa,, is equated to the salt activity differential. The justification for this is based on an application of the Gibbs-Duhem equation using only the salt and water chemical potentials. However, variation in membrane activity affects the salt and water activity. This effect was, unfortunately, not considered. The differences in activity of salt and water across a membrane are related, but, as shown by Bennion (1966), the relationship is in general nonlinear. In practice the nonlinearity may be mild. However, as has been sholvn, it is not necessary to eliminate ha,or A&. Equations 17 and 18 can be used directly. Other than these probably minor differences, the equations used here and those of Spiegler and Kedem (1966) differ only in interpretation and method of application. Experimental Equipment
Four separate types of experimental measurements were made in this work. Osmotic Cell, I n these measurements water and salt flux are observed for various concentration differences across the membrane in the absence of any pressure difference. Wills and Lightfoot (1966) refer to this as free diffusion, Shenvood et al. (1967) call it forward osmosis, and Jagur-Grodzinski and Kedem (1966) call it simply osmosis. The experimental equipment and interpretations vary, but the basic physical phenomenon being observed is the same for all of these experiments. A sketch of the osmosis cell used in this work is shown in Figure 1. The left-hand compartment is initially distilled water. The concentration in this compartment, c1, increases with time. This variation is monitored with a conductivity probe. The right compartment is initially some predetermined NaCl solution whose concentration, cp, decreases with time. This latter variation is monitored as a consistency check and to help estimate an upper error limit. This was established as between l and 2%. The water level in each compartment is maintained at the same height by a side connection or small tube around the membrane barrier. Volumetric flux across the membrane is measured by observing the movement, dlldt, of a liquid marker in the connecting tube. The marker fluid is a mixture of density one of carbon tetrachloride and benzene. The initial concentrations, clo and c p , in the tube on either side of the dye marker are the same as are initially in the compartments
RUBBER STOPPER-
-CONSTANT CONSTANT TEMP BATH
WATER
MAGNETIC -MAGNETIC STIRRING BAR SIDE TUBE
MAGNETIC -MAGNETIC STIRRER
\
