Coupled Ionic Migration and Diffusion During Liquid-Phase Controlled Ion Exchange Lester P. Van Brocklin*1 and Morton M. David Department of Chemical Engineering, University of Washington, Seattle, Wash. 98106
The effects of ionic migration on cation exchange were studied for the case of liquid-phase controlled mass transfer. Based upon three mass-transfer models-the film, the boundary layer, and the penetration-calculational techniques were developed and used to predict the effects of coupled ionic migration and diffusion. Predicted results are reported in terms of Ri factors, which for a given ion in an exchange process are ratios of the mass transfer rate with ionic migration at a given set of conditions to the mass transfer rate at the same set of conditions without ionic migration. Predicted variation in mass transfer rates ranges up to a high of about 13 times for the Cu2+-H+ exchange system based on the film model. Also presented are calculated concentration profiles for the region near the interface.
I n the unit operation of ion exchange, mobile cations (or alternatively anions) initially present in insoluble but permeable particles are replaced by other ions of the same charge sign from an electrolyte solution contacting the particles. The rate of exchange is governed by one or more of several mass transfer steps. For a given particle, the resistance to mass transfer within the particle, between the bulk solution and particle surface, or both, may be rate determining. Characteristically, mass transfer of ions in the absence of convection occurs by simultaneous diffusion and ionic migration. The ionic migration is present because the mass fluxes resulting from concentration differences combined with the constraint of electroneutrality induce a n electric field. I n the constitutive equation, the ionic migration term is the product of the electric field gradient, the concentration, the valence, and a constant. Equations 1 and 2 are the one-dimensional constitutive equations for nonelectrolytes and electrolytes, respectively (activity coefficients may be introduced into these equations (Helfferich, 1966)).
Jc/S =.-DI
(2)
Significant differences between ion-exchange and nonelectrolyte mass transfer rates can occur; the ratio of the two right-hand terms in eq 2 can vary from such larger than 1 to much smaller than 1. For exchange controlled by particle diffusion, the theory is well developed (Helfferich, 1960). The particular transport equations have been solved numerically (Helff erich and Plesset, 1958) and tested experimentally with reasonable success. However, for most industrial applications, which utilize packed beds and usually treat dilute solutions, the external resistance is the important rate-controlling factor. This step is the more complex one because of convective effects and the presence of mobile eo-ions, factors which are not significant in particle internal mass transport. Correspondence should be sent to this aubhor a t Shell Development Co., Emeryville, Calif. 94608.
For packed beds a further complexity is added to analysis of the liquid-resistance-controlled mass-transfer step, since usually local mass transfer in a packed bed is inherently a random process as shown by Jolls (1966). I n the typical bed of dumped particles, the flow field around a particular particle and the geometry of the active mass-transfer surface of that particle can differ considerably from those of its neighbors; therefore, thorough analysis of a particular particle in a bed does not necessarily enable the prediction of accurate mass transfer rates throughout the bed (Jolls, 1966). For nonelectrolytes, empirical equations for packed-bed mass transfer have been developed which are of the form (Lightfoot, et al., 1966) kL€
j = - (Sc)'I3 = A(Re)" U
(3)
This type of equation works reasonably well if allowances are made for different techniques of dumping particles and particle size distribution. This equation, however, makes no allowances for the effect on rates by ionic migration during ion exchange. Previous investigators have chosen one of two methods to correlate or describe the effects of ionic migration. First, diffusion coefficients, either arbitrarily chosen such as by Selke, et al. (1956), or calculated by film-model theory by Kataoka, et al. (1968), or as described by Helfferich (1966), have been used in correlations of the form of eq 3 to allow fitting of a particular set of data. Second, the absolute values of k~ have been ignored, and as Turner, et al. (1966), demonstrated, the relative values of k~ have been shown to change as boundary concentrations change while hydrodynamic conditions remain constant. The latter method has been used primarily for stirred-vessel systems, while the theory developed for both methods has assumed a stagnant uniform film surrounding the particles. I n most packed-bed ion exchange, however, where flow rates are relatively low and in the laminar range (Van Brocklin, 1968), a boundary-layer model is more appropriate. The present paper presents theory and techniques first to calculate, using various models, the relative changes in liquidphase mass-transfer coefficients as exchange proceeds and, secondly, a method is proposed for combining these results Ind. Eng. Chem. Fundarn., Vol. 1 1 , No. 1, 1972
91
with existing empirical correlations for nonelectrolyte mass transfer t o permit prediction of absolute values of coefficients.
and at the exchanger-solution interface
+
ZlJ1*
ZJ2*
=
0
Theory
The three mass-transfer models used are the following: first, the two-dimensional, steady, asymptotic, laminar boundarylayer model for a n infinitely high Schmidt number; second, the Nernst film model; and third, the Danckwerts penetration model. The effects of local bed geometry and flow fields are canceled by the formulation of a n Ri factor. This is the ratio of electrolyte t o nonelectrolyte mass-transfer rates (of species i) for a single particle under pseudo-steady-state conditions. For a given exchange system Ri depends only upon interface and bulk concentration ratios; it is independent of film thickness, residence time distributions, or boundary layer thickness, velocity gradient, and mass transfer length. The theory and calculated results are presented only for two exchanging cations and one co-ion, but the extension to more species or anion exchange is obvious. Definition of Ri.T h e Ri factor is defined b y
The numerator, Ji’/S, represents the molar flux for species i with the Nernst-Planck equation (eq 2) as the constitutive equation and the denominator, J i / S , represents the molar flux for species i with Fick’s law (eq 1) as the constitutive equation. Reformulation in the usual manner to equivalent fraction driving forces reduces Ri to a ratio of mass transfer coefficients. This permits definition of a new j ’ factor shown in eq 5.
ji’ = j i R i
kLi’€
= - (SC)”’
U
Mass-Transfer Models. T h e assumptions appropriate for all models are the following: the system is isothermal, the bulk solution is homogeneous, and the solutions are dilute, so that activity coefficients are unity and there is negligible co-ion content in the exchanger. Further, D,, p , and P are constant. There are no chemical reactions. The exchanger has uniform suiface composition and mass transfer is pseudo steady state. The assumptions appropriate for each model are the following. Film Model. I n the film model of mass transfer, first proposed by Nernst, a n ion-exchange particle is assumed t o have a thin stagnant film of thickness 6 adhering t o the surface. All velocity components in the film are zero. The bulk liquid a t distances greater than 6 from the surface is well mixed and of uniform composition Cl0and C20.The quantity 6 cannot be calculated directly, but is a parameter which is adjusted to fit experimental data. For the film model only one dimension, y, in the direction of mass transfer is considered, and eq 6-9 reduce to
2
ZiJi (5)
Consequently, the electrolyte coefficient, h i ’ , can be calculated if j i and Ri are known. Alternatively, Ri can be calculated if j i snd j,’ are determined experimentally. Fundamental Equations. T h e fundamental equations are t h e conservation and constitutive equations plus the constraints of no electric current and electroneutrality. For dilute solution mass transfer of aqueous electrolytes these equations are t h e following (Newman, 1966)
=
0
i=l
(15)
Penetration Model. T h e penetration model was brought to prominence for packed-bed mass transfer by Danckwerts (1951). I n this model there is a well mixed bulk fluid of composition Cl0 and C20. Elements of the bulk fluid are randomly thrust to the interface, where mass transfer occurs in an unsteady-stdte manner. After a certain “residence time” the fluid element leaves the interface and is replaced by a fresh element having uniform bulk concentration. The fluid element is assumed large enough so that the concentration a t the rear does not change. For the penetration model, eq 6-9 reduce to at
3
e i
i
ZiJi
zici
=
0
(9)
Boundary Conditions. With t h e limitation t o three ionic species and reduction of eq 6-9 t o forms appropriate t o mass-transfer models stated below, the same four boundary conditions are necessary for integration of t h e set of equations corresponding to each of the models. T h e boundary conditions are t h e following: in the bulk solution Cl = C1Q (loa)
cz 92
=
o
(17)
2 = 1
=
czo
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
(lob)
Boundary Layer Model. The two-dimensional, steady, asymptotic boundary-layer model for infinitely high Schmidt number (Acrivos, 1960) is shown schematically in Figure 1 . This model requires the additional assumptions t h a t laminar boundary-layer flow prevails and t h a t the asymptotic solution for a n infinitely high Schmidt number is valid for ion-exchange processes. This can be restated as assuming that the region of concentration change is sufficiently thin so that the velocity uz changes nearly linearly with y over the mass transfer path. The first assumption is correct for the usual Reynolds numbers in ion-exchange
Table 1. Model
Transformed Equations
Transformation
Film
Y/6
Penetration
Y/ 2 d D T
Equation
Boundary layer
beds, as shown by Jolls (1966). The second is sufficiently correct for the Schmidt numbers encountered in ion exchange (which are above 1000 except for hydrogen which is as low as 300) as demonstrated by Wilson and Geankopolis (1966), who compared packed-bed results for very high Schmidt numbers with those around 1000. Equations 6-9 reduce t o
E!
Schematic diagram of the boundary layer model
formation and Di multiplies the diffusion term. Table I shows the similarity transformations used and the resulting equations for electrolytes. For nonelectrolytes the ionic migration terms disappear. Calculation of R , Factors. F r o m the definition of R t and eq 1 and 2 there results
3 ZtJl =
Figure 1 .
0
i= 1
Since the concentration change occurs only within the linear part of the velocity profile, the velocity components may be represented by uz = Y m ) (19) (20)
where a(z) is the velocity derivative at the wall (y = 0) and the prime denotes the derivative with respect to 2 . Equation 20 is a consequence of the continuity equation.
With this approximation the concentration boundary layer equations become yP(z)
dCi
-
bx
-
1 ~
2
y*P’(x)
ac, = -
bY
34)
RT a y
(i
=
1, 2)
(22)
plus eq 14 and 17. Calculation Techniques
Similarity Transformations. I n order t o integrate the differential equations above, similarity transformations were used to change the equations t o ordinary differential equations. Although the technique was not necessary for the film model, it is shown for this model t o permit comparison with the transformed equations for the other models. Since only one diffusion coefficient could be used in each of the transformations the coefficients had to be redefined as
D , = DXD,
Since the factor (da/dy),=o cancels, the effects of film thickness, average eddy residence time, or the liquid flow field length and particle geometry disappear from the Ri calculation (if sufficient info:mation is available to calculate (as/ by),,^ the ionic flux could be determined directly). For a given system and boundary conditions, [bC,/bq z&‘i* ( F / R T ) /(b+/bq)lV = 0 can be found by integrating equations for the particular model employed and using the results to obtain (C,),,=o, (bC,,@?),=o, and (b+/bo),=o. The same approach could be used t o evaluate (bC&), = 0 in the denominator. However, analytical expressions for C, as a function of 7 are available for certain limited nonelectrolyte cases of the thiee mass transfer models. As examples, for the film model, C, is simply a linear function of y or 7; and for laminar boundarylayer flow with mass transfer in the asymptotic case of a very high Schmidt number, (bC,/bq),=o = 1.12(C,0-Ci*) (Lightfoot, et al., 1966). Therefore for the boundary-layer model
+
(23)
where D B is a constant and normally equal to one of the diffusion coefficients. Then D N appears in the similarity trans-
R,
=
rz
9
+ z,c,R T b q - --
,,=o
(25)
1.12(C,O - C,*)
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
93
1.6
14
-
Cu++-H+ EXCHANGE
0
CALCULATED USING EQUATION (29)
A CALCULATED USING EQUATION
CALCULATED BY ANALOG COMPUTER (291
1.5
FILM MODEL
- - -
Na+- Cs+ EXCHANGE 2,
8,
-
-z3
ZZ
1
1.4
-
1.0 .Oz = 12.975 1.3
1.2
1.1
1.o 0
0.4
0.2
0.6
0.8
1.0
**
x;
ZSCS*
Figure 4.
R1 factors
Calculated
0 0
0.2
0.4
0.8
0.6
x:
2q
exiting ions, these quantities are necessarily related. The relations, as developed by Van Brocklin, are the following
'3'3
Figure 2.
Calculated R1 factors
Ri
Dz
1. film model: R 2 = Di Cs+- Na' EXCHANGE CALCULATED USlNO EOUATION (29) FILM MODEL
2,
=
ZZ
-
-2,
2. penetration model:
-
Rz
(27)
= 1
3. boundary layer model:
-
R2
=
(28)
Integration. T h e film model equations for two exchanging cations were solved analytically. The general solution for the binary exchange of cations 1 and 2, with 1 entering the exchanger, is given in eq 29, where
a
C Y =
0.5
0
0.2
Figure 3.
0.4
Calculated
0.6
R1
0.8
factors
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
22Dz 22D2 - ziDi
1.o
Similar forms are obtained for the film and penetration models; these and more details of the analysis are given by Van Brocklin (1968). Since R c factors can be calculated for either entering or
94
=
M
=
(1 -
CY
xi =
(1- 2))
Z*Ct z3c3
and ion 3 is the co-ion. The boundary-layer and penetration model equations had
to be solved numerically. An Electronics Associates TR-48 analog computer was used for this purpose. These equations plus the constraint equations (eq 14 and 17) were integrated over 7 for a given ionic system and a given set of boundary conditions corresponding to eq 10 and 11. Usually a trial and error approach was necessary since the analog computer solves initial value problems and these were boundary value problems.
and the third vertex is given by computer calculation a t 210 = 1.0 and z1* = 0. 1.8 r
-
Results and Discussion
Because of the large amounts of calculation time which would be required, particularly for the trial-and-error solutions of the boundary-layer and penetration problems, a comprehensive parametric study of the three models was not attempted. However, sufficient calculations were made to permit a general understanding of the behavior of the R.I factor and its potential uses and of some important effects of the electiic field on the ion-exchange process. Figures 2-12 are examples of the results obtained. Figures 2-6 show part of a parametric study of Rt factors as a function of equivalent fractions of species 1 a t the interface and in the bulk. As defined above, species l is the entering cation and 3 is the co-ion. Figures 7-12 show concentration profiles in the liquid phase. R i Factor Characteristics. Figures 2 and 3 show plots of R1 us. 210 and XI* based on the film model, calculated using eq 29. The two cases shown, Cs+-Na+ and Cu2+-H+ exchange, with a monovalent anion, represent typical cases of important industrial separation processes. Many cation-exchange processes are designed for recovery of either univalent or divalent ions. The regenerating species is usually N a + or H+. Most important divalent ions have a diffusivity near that of copper ion. The diffusivity of cesium ion is typical for common univalent ions except hydrogen which has a uniquely high diffusivity. The results shown are valid for any monovalent anion, since the anion diffusivity is eliminated in deriving eq 29. Several important characteristics of the plots can be noted. First, if one ion is present as a very small fraction of the total concentration, the mass-transfer rate of t h a t ion is independent of ionic migration. For example, in Figure 2 for CuZf-H+ exchange, R1 = 1.0 if XI* a a 0. This fact also permits calculating R1 a t 210 = 1, z1* = 1, where xz0 = x2* = 0 and R2 = 1. Then from eq 26 R1 = D2/D1 X RZ = 12.97. Second, R1values increase monotonically in Figure 2 (slow ion entering) and decrease monotonically in Figure 3 (fast ion entering) as z10 and zl*increase. Third, R2factors for ions exiting can be calculated from eq 26 for the film model. Fourth, the Rt factor plots have roughly the shape of a distorted triangle. The location of two of the vertices is found as immediately above. The location of the third vertex a t no= 1.0 and $I* = 0 can be determined exactly only by calculation; however, it always (in the calculations made) lies much closer to 1.0 than to the other limiting value of R1. Figures 4, 5, and 6 show R1 factor plots at 210 = 1.0 for several exchanging systems, calculated from all three masstransfer models. Results for values of 22 # 1.0 are given by Van Rrocklin (1968) and Au Yong (1969). Complete R1 factor plots for the boundary-layer and penetration models are similar in appearance to the R1 plots for the film model. At the limits of concentration the dilute ion controls mass transfer. This means that for the boundary layer model
Ri
=
1.0
e)
(21' a
xi*
c
0)
RI =
(Z1O x
and for the penetration model
L2 2,
EXCHANGE
= -2, = 1 = 2
-
1.7
x; = 1.0
I
1.6
0
4,=1 a , = 1.87 a s= 2.79
c
1.4
1 . 3 t
/
$ 0 7
0 CALCULATED BY ANALOG COMPUTER A CALCULATED USING EQUATION (29) 1.o
1
0
I
0.4
Figure 5.
1."
0.5
I
I
I
0.2
I
0.6
xy
=
I
I
0.8
1
m 2,c;
Calculated
r
Rl factors
Na'-Cut'
EXCHANGE
5 0.6
0
2/3
Cut*-Nat
0.2
0.8
0.4
zl*a 1.0)
z:
r,ct ZJCI;
Figure 6.
Calculated
R1
factors
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
95
BOUNDARY LAYER MODEL Nat- HC EXCHANGE
x; = 1.0
6, I = 0.144
x: = 0
4, = 1.0 A , = 0.205
VN
Figure
7. Calculated concentration profiles
"."
"'"I
*-----------
-
& [ = 1.0 * 0.144 Q, = 0.205
x; = 1.0 xt 0
0.2
-
Ht- Na' EXCHANGE BOUNDARY LAYER MODEL
I 0
0.1
0.2
0.3
0.4
I 0.5
I 0.6
I
0.7
I (
8
'7N
Figure 8.
Calculated concentration profiles
I n all calculations the deviations of R1from unity were the largest for the film model. The boundary layer results for a given zIo and zl* always showed significantly less effect of ionic migration than the film model results, and the penetration results less effect than the boundary layer results. The general shapes of the R , plots for various models were similar, but no simple relation between film, boundary layer, and penetration results could be found for a given exchange system. Equation 29 shows that the co-ion diffusivity should have no effect 011 the RI values foi the film model, and a limited numbei 96
Ind. Eng. Chem. Fundom., Vol. 1 1,
No. 1 , 1972
of calculations for the boundary-layer and penetration (Au Yong, 1969; Van Brocklin, 1968) mddels also indicate that the co-ion diffusivity is not significant. Equation 29 indicates moderate effects of eo-ion valence on R i values a t large diffusivity ratios of the exchanging ions, for small values of a* (see also Kataoka, et al., 1968). However, for larger values of zl* and/or nominal ratios of diffusivities for the counter ions, both eq 29 and a small number of calculations for the boundary-layer model show the co-ion valence does not influence Rt in a n important manneI. Similarly, the limited available results indicate that counterion valences do not in general
x', = 1.0 x: = 0
= 1.0 1.0 a, = 1.0
2J
a, =
N' 1
Figure
9. Calculated concentration profiles
BOUNDARY MODEL Na+- C r' EXCHANGE
x", = 1.0 x: = 0
a1 = 1.0 .D,
= 1.56
D 3 = 1.505
RN
Figure 1 0.
Calculated concentration profiles
significantly affect Ri,and that the ratio of the diffusivities of the counter ions is the most important factor. As noted pre* I, viously the maximum (or minimum) value of Ri,a t z ~ = depends only on the diffusivity ratio of the exchanging ions. The preceding discussion of Ri factor characteristics indicates a method for estimating binary-exchange R, values. Equation 29 can be used to calculate film-model results for a new system, Rt values a t zl*= 1 can be calculated readily for the other models, and other R , values for these models can be roughly estimated because of the similar appearance of the
plots, as indicated in Figures 4-6 for zl0 = 1. The same procedure can be used for either cation or anion exchange. Liquid-Phase Concentration Profiles. Concentration profiles for the liquid phase adjacent t o the exchanger were calculated for a small number of cases. Figures 7-10 are profiles for the boundary-layer model. Boundary conditions are zl0 = 1.0 and zl*= 0. Both the abscissa and ordinate are normalized. Probably the most obvious characteristic of the boundarylayer concentration profiles is the expected (Helff erich, 1960, 1966) normality increases if the fast ion is entering and deInd. Eng. Chem. Fundam., Vol. l l , No. l , 1972
97
1.4, H+- Na+ EXCHANGE
\
a,
O r
0
-
FILM MODEL
\
6.92
'
0.2
0.4
Y I ~
Figure 12. Calculated concentration profiles
creases if the fast ion is exiting. I n addition, there is a minimum point of total normality at some point between the bulk and the interface if there is a maximum at the interface and vice versa. Figures 7 and 8 show this clearly. Figure 9 shows concentration profiles for a system with equal diffusion coefficients for comparison. Profiles for the penetration model appear quite similar to those for the boundary layer model (Van Brocklin, 1968) and are not shown here. Figures 11 and 12 show calculated profiles for the film model. The nonexchanging ion has a linear profile but the counterion profiles show some curvature. Maximum or minimum normalities occur a t the interface, although not a t inter98
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
mediate points. The change in potential across the boundary layer is of the order of up to 25 mV for the boundary layer model. Comments on t h e R I Method and M a s s Transfer Models. The theory developed presents a self-consistent relation between ion exchange and nonelectrolyte liquidphase-controlled mass transfer while using a minimal number of plausible assumptions. Further, it provides an approach for calculating mass transfer coefficients for ionexchange processes. Although developed with packed beds in mind, the approach is valid for other types of operations, such as stirred vessels, provided the model most appropriate for the particular type of contacting is used. Calculation of the R I factor itself requires a minimum amount of data on boundary conditions, self-diffusion coefficients]and valences. Calculation of actual mass-transfer coefficients further requires knowledge or correlation of nonelectrolyte mass-transfer coefficients as a function of hydrodynamic conditions. However, a complete mathematical description of the hydrodynamics is not necessary and is not presently available for packed-bed flow. If such a description were available, of course, the mass-transfer rates could be calculated directly. Two assumptions used in all models deserve further discussion. One of these is that uniform bulk-liquid and surface compositions exist for a given ion-exchange particle] as stated in eq 10 and 11. I n a packed bed such an assumption is undoubtedly somewhat inaccurate, because concentration boundary layers are present near all exchanging particles. Therefore] the liquid impinging on a given particle may not be of uniform composition. Another assumption, implicit in the derivations but previously not stated explicitly, is that a*,which is in equilibrium with the particle surface composition, is not affected for a given exchanger composition by the change in normality a t the exchanger-solution interface. For most ion-exchange systems] the equilibrium is a function of total solution normality] but this effect should be negligible over the normality change between the bulk solution and the interface. If this assumption were not made, the proper boundary condition for the interface would be the surface exchanger composition, and an equilibrium relationship would have to be included in the RI calculations. As indicated by Figures 4 through 6, the three models give results which differ significantly, so the question of which model is preferable is posed. I n view of the known presence of steady, laminar boundary layers under conditions usually encountered in ion-exchange and the high Schmidt numbers for exchanging species, the boundary layer model appears most appropriate. Conversely, the film and penetration models are not consistent with the hydrodynamic situation in packed beds. The film model, however, is simpler for calculational purposes and serves to indicate the maximum effects which can result from ionic migration. Experimental data (Van Brocklin, 1968) indicate results ranging from boundary layer to film model predictions] but very few data for quantitative tests of the models are available (this topic will be included in a subsequent paper). Design Implications for Packed Beds. During the usual packed-bed operation, a n interior particle first exchanges with nearly depleted liquid and finally exchanges with liquid of entering composition. Therefore] the interface and bulk vary in some continuous fashion between zl0 = XI* = 0 and z10 = XI* = 1.0. This means that RI must vary from 1.0 to the maximum (or minimum) value during exchange. Hence, the mass-transfer coefficient varies in a similar manner. As
may be noted from Figures 2 to 7, R1 may well vary severalfold during this process. This situation casts doubt on the suitability of existing mathematical design methods for packed beds, since virtually all these methods assume a constant masstransfer coefficient in the liquid phase. For the same reason, the significance of a single-valued mass-transfer coefficient obtained from breakthrough data is open to question. I n principle, numerical methods for packed-bed design or data interpretation can accommodate varying mass-transfer coefficients. Sufficiently extensive calculations of R1 combined with appropriate data for nonelectrolyte mass transfer could provide the coefficients needed. However, the computational effort required for multi-ion exchanging systems, with significant internal resistance, appears formidable. Thus the knowledge of ion-migration effects on masstransfer rates in packed beds supplied by the theory presented does not solve the problem of packed-bed design for ion-exchange units. Rather it suggests both the need for caution in using predictive or interpretive methods employing a constant value for the liquid-phase mass-transfer coefficient and the desirability of developing breakthrough curve models which recognize properly the changes which occur in this coefficient. Nomenclature
A C D a, F I j j‘ kL kL’
J/S J’/S N R
Ri Re r*
s
sc
T
U
constant, see eq 3 concentration, moles/cma diffusion coefficient. cm2/sec normalized diffusion coefficient Faraday constant electric current, A j factor, see eq 3 j’ factor, see eq 5 liquid-phase mass transfer coefficient, cm/sec = liquid-phase mass transfer coefficient when ionic migration effects are significant, cm/sec = flux, moles/cm2-sec = flux when ionic migration effects are significant, moles/cm2-sec = convected flux, moles/cm2-sec = gas constant = Ri factor, see eq 4 = Reynolds number = ratio of equivalent concentrations of exchanging ions a t interface, see eq l l b = area, cm2 = Schmidt number = temperature = velocity, cm/sec = = = = = = = = =
2
= distance normal to interface, cm = equivalent fraction in solution; distance along ex-
z
= valence
y
change surface, cm
GREEKLETTERS p(x) = velocity gradient a t interface, I./sec 6 E
7
+ p
= = = = =
film thickness, cm void fraction similarity transform, see Table I electric potential, V density, g/cma
SUPERSCRIPTS 0 = bulk phase * = a t the interface
SUBSCRIPTS = species = liquid phase 1 = cation entering 2 = cation exiting 3 = anion N = normalized quantity
i L
literature Cited
Acrivos, A., Phys. Fluids 3,567 (1960). Seattle, Au Yone. K. S.. M.S. Thesis, Universitv of Washington, Washy 1969.‘ Danckwerts, P. V., Ind. Eng. Chem. 43,1960 (1951). Helfferich, F., “Ion Exchange,” pp 250, 10, 153, McGraw-Hill, New York, N.Y., 1960. Helfferich, F., in ‘‘Ion Exchange,” Vol. 1, J. Marinsky, Ed., p 65, Marcel Dekker, New York, N.Y., 1966. Helfferich, F., Plesset, M. S., J.Chem. Phys. 28,418 (1958). Jolls, K. R., Ph.D. Thesis, University of Illinois, Urbana, Ill., 1966. Kataoka, T., Sato, N., Ueyama, K., J. Chem. Eng. Jup. 1, 38 (1968). Lightfoot, E., Massot, C., Ivani, F., Chem. Eng. Progr. Symp. Ser. 61, No. 58, 28 (1966). Newman, J., IND. ENQ.CHEM.,FUNDAM. 2,525 (1966). Selke, W., Bard, Y., Pasternak, A., Aditya, S. K., A.Z.Ch.E. J . 4FQ flOKR\ IV (IC,“”,.
J., Church, M., Johnson, A., Snowden, C., Chem. Eng. Sci. 21, No. 4, 317 (1966). Van Brocklin, .L,. P., Ph.D. Thesis, University of Washington, Seattle, Wash., 1968. 5, 9 Wilson, E., Geankopolis, C., IND.ENG.CHEM.,FUNDAM. (1966). RECEIVED for review October 19, 1970 ACCEPTEDSeptember 27, 1971 TU;; ier,
Financial assistance was received from the National Science Foundation.
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
99
