Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 175
Ion-Exchange Rates in Bifunctional Resins Kent S. Knaebel, David D. Cobb,' Thomas T. Shih,' and Robert L. Pigford' Department of Chemical Engineering, University of Delaware, Newark, Delaware 197 11
Rates of exchange of sodium and chloride ions were measured using two resins containing both acidic and basic functional groups. The first was a composite resin, prepared by separately grinding Duolite CC-3 and A-368 commercial resin particles, mixing the powders, and cementing them together with an inert polymeric matrix. The second was Amberlite XD-2, an amphoteric resin containing both functions in a common polymeric network. The measurements were conducted batchwise and included variations in particle diameter, initial resin content of salt, and initial solution concentration. The rate data for both resins were well represented by a pore diffusion model in which mass transfer rates were assumed proportional to local concentration gradients. The diffusivities of NaCl in the pore fluid were 6.9 X IO-' cm'/s at 23.5 OC for the composite resin and 9.2 X lo-' cm2/s at 25 OC for the amphoteric resin. Amberlite XD-2 is the superior material in processes which require fast mass transfer.
Introduction Several investigators have studied the application of the Cycling Zone Adsorption (C.Z.A.) process (Pigford et al., 1970) to desalination of brackish water (Ginde and Chu, 1972; Latty, 1974; Shih and Pigford, 1977). The C.Z.A. process involves cyclic adsorption and desorption of solute from a fluid flowing steadily through a fixed bed, or zone, of solid adsorbent. The sorption is driven by changes in a thermodynamic variable, such as temperature, which changes the equilibrium distribution of solute between the fluid and adsorbent. In this case the adsorbent is a thermally regenerable ion-exchange resin with both acidic and basic functional groups. In addition, resin of this type has been used in other desalination processes including the Sirotherm process (Weiss et al., 1965, 19661, parametric pumping (Gregory and Sweed, 19721, and a moving bed process (Dabby et al., 1976). It is expected that the results obtained here will apply to these as well as other salt removal processes that employ bifunctional resins in which mass transfer rates are important. Early studies with thermally regenerable resins showed that mixed, commercial size resins of the weak-acid and weak-base types would not be satisfactory due to poor adsorption rates (Latty, 1974; Light, 1973; Bolto, 1970). Acceptable rates were possible with resin particle diameters of about m, but these were impractical due to excessive pressure drop. One approach toward obtaining high adsorption rates with low pressure drop involved binding together microparticles of the acidic and basic resins using a porous, inert matrix to form a composite resin. Another approach was to develop amphoteric resins in which the acidic and basic functional groups were chemically or physically bound to a common polymeric network (Bolto and Weiss, 1977). Both composite and amphoteric resin particles can be made sufficiently large that pressure drop through a fixed bed need not be excessive. Thus, the remaining concern was that the adsorption rate be as large as possible. In the present work, experiments were conducted to determine the mass transfer rates of aqueous NaCl in a composite resin and a commercial amphoteric resin. The experiments were designed to account for the effects of variations in resin particle size, initial resin salt content, Fluor Corporation, Irvine, Calif. 92714. Allied Chemical Corporation, P.O. Box 1021R, Morristown, N.J., 07960. 0019-7874/79/1018-0175$01 .OO/O
Table I.
Resin Properties
property
units
intraparticle void fractiona Langmuir isotherm parameters A B T A B T a
amphoteric resin
spherical spherical
physical particles form of mean diameter, dry mean diameter, wet densi tya
composite resin
cm
0.072
0.042
cm
0.085
0.057
g of dry resin/ 0.47 c m 3 of wet resin c m 3 of void/ 0.45 cm3 of wet resin
0.57
cm3/g cm3/mequiv "C cm3/g cm'/mequiv "C
189.5 173.8 25.0 23.21 27.82 95.0
92.4 61.7 23.5
____
____--_
0.56
Excluding interstitial void volume.
and initial solution concentration in an isothermal batch system. The experimental data were analyzed using a solution of the continuity equation in which the rate of mass transfer within the resin particles was assumed proportional to the instantaneous local concentration gradient, the diffusion coefficient being assumed constant in any experiment. Properties of Resins The composite ion-exchange resin used in this study was developed by Light (1973) following procedures suggested by Weiss et al. (1972). Light studied several pairs of monofunctional, weak electrolyte resins, and, from equilibrium data, found an optimum mixture of Duolite CC-3 (acidic) and Duolite A-368 (basic) resins. These are products of the Diamond Shamrock Chemical Co., Redwood City, Calif. The optimum was based on maximum exchange capacity of the composite resin between 23 and 80 "C, and was found to be 59 mol % of the acidic resin and 41 mol 70 of the basic resin. The composite resin was made by grinding the commercial size resins separately until the particle size was about m. Then the mixture was dispersed in a solution of polyvinyl alcohol and glutaraldehyde, and spherical composite particles were formed by suspension polymerization. When the reaction ceased, the particles were washed until fully regenerated, dried, 0 1979 American Chemical Society
176
Ind. Eng. Chem. Fundam., Vol. 18,No. 2, 1979 I .oo
~
AMPHOTERIC RESIN Hlo
. 0.80 >
L
V
0.60 W
COMPOSITE RESIN
3 I
8
0.40
4
5 AMPHOTERIC RESIN
0.20
0
0
0005
0.015
0010
0020
SOLUTION CONCENTRATION
0.025
0.030
5 cm3
Figure 1. Equilibrium data and Langmuir isotherms for the composite resin and amphoteric resin.
and then dry-screened using standard sieves. The rootmean-square diameter was determined using a microscope. The properties of the composite resin are listed in Table
I. The amphoteric ion-exchange resin used in this study was Amberlite XD-2, a product of the Rohm and Ham Co., Philadelphia, Pa. This resin is manufactured by partially filling the voids of a macroreticular, spherical, “host” copolymer with a “guest” copolymer. The copolymer hybrid particles are then treated by successive steps of chloromethylation, amination, and hydrolysis to produce the amphoteric functional groups (Barrett and Clemens, 1976). Such particles were regenerated using about 1000 bed-volumes of 95 “C distilled water, then wet-screened using standard sieves. The root-mean-square diameter was later determined using a microscope. The properties of the amphoteric resin are also listed in Table I. The absolute capacities of the resin were measured in separate batch equilibrium experiments. The data for both resins were well represented by Langmuir-type isotherms having the form q* = AC (1) 1 BC where q* is the resin absolute NaCl capacity, mequiv/g; C is the solution NaCl concentration, mequiv/cm3, and A and B are temperature-dependent isotherm parameters, cm3/g and cm3/mequiv, respectively (see Table I). The data are shown with superimposed isotherms in Figure 1. Although the extent of regeneration, pH, and other variables may affect the resin equilibrium properties, these were not taken into account during this comparison. As previously mentioned, these isotherms were obtained following prolonged regeneration of the resin with distilled water, which may have partially hydrolyzed bonds between pairs of acidic and basic functional groups. Such hydrolysis, and its reverse in saline water, occurs more slowly than the transfer of NaCl in and out of the particles, suggesting that the number of adsorption sites available for ion exchange may approach a somewhat lower level during prolonged exposure to saline water. R a t e Measurements A. Composite Resin Experiments. A known mass of fully regenerated, dry resin particles was soaked overnight in a large beaker of distilled water, allowing the resin to swell. To initiate each rate experiment, sodium chloride solution was quickly added to the beaker to obtain the desired solution volume and initial concentration. During ~
+
Table 11. Experimental Conditions init soln wet part. init resin expt concn, mean loading, no. mequiv/cm3 diam, c m mequiv/g Composite Resin 1 0.016 0.079 0.00 0.00 2 0.013 0.079 0.00 3 0.016 0.046 4 0.006 0.079 1.01 Amphoteric Resin 1 0.018 0.090 0.01 2 0.009 0.090 0.01 3 0.018 0.045 0.01 4 0.001 0.090 0.78 5 0.001 0.090 0.78 6 0.019 0.090 0.48
:1‘
EXPERIMENT
0.0I 5
temp, ‘C 23.5 23.5 23.5 23.5 25 25 25 25 25 95
SYMBOL
i 0
t
;
0010
Y 8 z
e
0.005
5L 2
0 0
5
IO
I5 20 TIME, min
25
30
Figure 2. Concentration vs. time for composite resin adsorption experiments at 23.5 “C (qo = 0.0 mequiv/g): 0, Co = 0.016 mequiv/cm3, d, = 0.079 cm; A, Co = 0.013 mequiv/cm3, d, = 0.079 cm; 0,Co = 0.016 mequiv/cm3, d, = 0.046 cm.
the addition and throughout the experiment, a magnetic stirrer was used to mix the resin particles and the solution. Temperature and concentration were monitored continuously with a thermocouple and an electrical conductivity cell which were immersed in the solution. The conductivity cell was wrapped with cheesecloth to prevent the resin particles from reaching the electrodes. Solution concentrations were measured using a circuit suggested by Khang and Fitzgerald (1975) and were recorded on a chart recorder for further analysis. Three adsorption experiments were conducted a t 23.5 “ C to determine the influence of two key parameters. Considering experiment 1 to be the base study, the initial solution concentration was reduced in experiment 2; then a smaller particle size was used in experiment 3. Experiment 4 involved desorption a t 23.5 “C, following the same general procedures as the adsorption experiments. The exceptions were that the resin was equilibrated for several days with solution having high NaCl concentration, and the experiment was started by suddenly diluting the solution to the desired initial concentration. Details of these experiments are given by Cobb (1975) and Shih (1975). The experimental conditions are summarized in Table 11,and plots of the concentration histories are shown in Figure 2 and 3. B. Amphoteric Resin Experiments. The procedures described above were modified in three respects to improve the reliability of the measurements. First, to prevent resin particle break-up caused by impact with the stirring bar at high speed, a specially designed basket stirrer was used. Second, to reduce the likelihood of data logging errors, a microcomputer system (Sutarwalla, 1978) was used to record data. Third, to reduce initial transients due to mixing, the resin was inserted into the stirred solution to initiate each experiment. In addition, an open conductivity cell was used to monitor concentration, and a thermometer
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 Thermometer
Conductivity
;
m
I
O
0
, 5
,
IO
, I5 TIME, min
, 20
, 25
177
] 30
Figure 3. Concentration vs. time for composite resin desorption experiment at 23.5 "C (Co = 0.006 mequiv/cm3, d, = 0.079 cm, qo = 1.01 mequiv/g.
I
\
Baffle-'
-Basket stirrer hot plate surface
i-Slirrmg
Figure 5. Apparatus used in rate measurements with the amphoteric resin. 10 09
I
na
5 05
80 mesh stainless
,
,
"1
EXPERIMENT
SYMBOL
A I Preliminary) B I Preliminary 1 I
0
steel cl& basket 5 7 cm dia x 4 0 c m high
0
. head netic stir bar
'
Figure 4. Basket stirrer used in rate measurements with the amphoteric resin.
was used to monitor temperature. Concentration measurements were made using a circuit suggested by Rosenthal (1977) in conjunction with a frequency meter that had digital output; the digital data were fed to the microcomputer. A sketch of the basket stirrer is shown in Figure 4, and a sketch of the vessel is shown in Figure 5. In preparation for an experiment, a known mass of fully regenerated, moist resin particles (i.e., with interstitial water removed) was placed in the basket stirrer. The dry mass of the resin was determined later by fdly drying, then weighing a sample of the same resin. The basket stirrer was held above sodium chloride solution of known concentration and volume, until the microcomputer data logging program was started. After the initial concentration was recorded, the basket stirrer was lowered quickly into the solution, and the magnetic drive was started, inducing rotation of the basket. This rotation caused circulation of the external solution through the basket stirrer. During the experiment, concentration measurements were recorded every 10 s. Three preliminary adsorption experiments were conducted in order to estimate the effect of film diffusion resistance as a function of rotating speed. The speeds selected were 44, 125, and 175 rpm. These represent the limits of steady rotation of the magnetic stirrer and an intermediate value. Figure 6 shows a semi-log plot of one minus the fractional uptake of salt by the resin (1- F) vs. time for these experiments. The large increase in the adsorption rate observed from 44 to 125 rpm is contrasted by a relatively small increase from 125 to 175 rpm. Furthermore, Huang and Li (1973) have shown that a plot of log (1 - F) vs. time is expected to be linear with an intercept of F = 0 when film diffusion is controlling, given
O i l 0
5
IO
15 TIME, m i n
1
20
25
Y
30
Figure 6. One minus fractional uptake of NaCl by the resin vs. time for preliminary amphoteric resin adsorption experiments at 25 "C (C, = 0.018 mequiv/cm3, qo = 0.01 mequiv/g, d, = 0.090 cm), speed = 44 rpm; 0 , speed = 125 rpm; 0,speed = 175 rpm.
a linear isotherm. The portion of the isotherm that was relevant for the current experiments was nearly linear for values of F less than 0.7. Thus, according to the data shown in Figure 6, the mass transfer rate of 44 rpm was controlled by film diffusion, but the mass transfer rates at 125 and 175 rpm were not. Since the experiment a t 175 rpm was apparently least affected by film diffusion, that speed was chosen for all subsequent experiments. The experiment at 175 rpm was called experiment 1, and was the basis of comparison for the remaining experiments. As with the composite resin, the initial solution concentration was reduced in experiment 2, then a smaller particle size was used in experiment 3. In experiment 4, the initial resin salt content was increased and the initial solution concentration was reduced to study the desorption rate a t 25 "C. The conditions of experiment 4 were duplicated in experiment 5 to test the reproducibility of the experimental procedure. Finally, in experiment 6 the desorption rate a t 95 "C was measured to test the behavior of the resin during regeneration. Full details of these experiments are given by Knaebel (1978). The experimental conditions for these experiments are summarized in Table 11, and plots of the concentration histories are shown in Figures 7, 8, and 9. Theory A simple structure is chosen to represent both the composite and amphoteric resin particles. Each particle is viewed as a sphere having homogeneous distributions of uniformly sized pores and acidic and basic exchange sites. Furthermore, it is assumed that swelling due to ion exchange is negligible since both types of particles are
178 Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 !
I
I
I
EXPERIMENT
i 0
I
SYMBOL
0.01~
0
A
5
5
0.017
u
0.016
0
3
2 3
0.015
d
Y
20
0.014 0
5
IO
25
Figure 7. Concentration vs. time for amphoteric resin adsorption experiments at 25 "C (C, = 0.018 mequiv/cm3, qo = 0.01 mequiv/g: 0, d, = 0.090 cm; A, d, = 0.045 cm. I
I
I
I
:/
ei
1
1
I
EXPERIMENT
SYMBOL
0.008
K
z
ez s 5
F
(3)
30
15 T I M E , min
o n IO.Ol0
NaCl concentrations at local equilibrium at a given radial position, D is the diffusivity of NaCl in the pores, e, is the intraparticle void fraction in the resin, and p is the resin particle density. Note that the pore-fluid Jiffusivity is based on the portion of the spherical surface which is occupied by pores. Assuming that the isotherm is linear over the concentration range in a particular experiment, the continuity equation can be reduced to the form
0006
c
-I
0.004
where D* = @/(ep + ppaq*/ac), the effective diffusivity of NaCl in the resin. Neglecting accumulation in the boundary layer outside a particle, the flux of salt through the boundary layer is equal to the flux across the interface between the solution and resin phases. Thus, assuming that spherical symmetry exists, the boundary conditions at the particle surface and center are B.C. 1 ac € p D J r = R = k(C - c(m) (4) B.C. 2
3
ac
0.002
;IF0
2 m
-~ 0
15
IO
5
20
30
25
TIME, min
Figure 8. Concentration vs. time for amphoteric resin adsorption and desorption experiments at 25 "C (d = 0.090 cm): A, Co = 0.009 mequiv/cm3, qo = 0.01 mequiv/g; and' 0 , c0= 0.001 mequiv/cms, qo = 0.78 mequiv/g.
1
.
.
.
.
,
.
,
0.0200
where k is the film mass-transfer coefficient, C is the bulk solution NaCl concentration, r is radial position, and R is the resin sphere radius. The initial conditions require uniform concentration in the resin pores and external solution a t t = 0 I.C. 1 c(r,O) = co (6) I.C. 2
C(0) =
0.0196
5
0.0194
0.0190 0
1
co
(7)
Finally, the flux of salt from the external solution into the resin is assumed proportional to the concentration difference between the bulk solution and the solution in the pores a t the particle surface aC M - = ha-(C - c(R)) at V
t /
8
(5)
=0
, I
'I
, 2 TIME, min
I
1
3
I
4
Figure 9. Concentration vs. time for amphoteric resin desorption experiment at 95 "C (Co = 0.019 mequiv/cm3,d, = 0.090 cm, qo = 0.48 mequiv/g).
highly crosslinked. Measured particle sizes showed that the particles in the samples tested were nearly but not perfectly uniform. The surface-average sizes were used in the rate computations. The experimental system is an isothermal batch in which resin is exposed to a sudden change of solution concentration in a finite medium. Although the mass transfer rate of sodium chloride from solution possibly involves several transport steps as well as the exchange reaction, the rate-controlling step is assumed to be diffusion through the pores. The pore diffusion process is described by Fick's equation in which the diffusion coefficient is taken as constant. Thus the continuity equation is written
where c and q* represent the pore-fluid and resin-phase
where a = 3 / p & , the resin interfacial area per unit mass, M is the mass of resin, and V is the volume of solution in the experiment. This set of equations is solved by Laplace transformation, assuming that the film mass transfer coefficient is infinite so that C = c(R), and inversion by the method of residues. The following expression for external solution concentration is obtained (Cobb, 1975)
c = (C, - co)
+-c6a9exp(-D*txi2/R2) + 9a +
L
1+
CY
i=l
CY%?
(9) where P,VD* a=--
@fD
PPV
+ Ppaq*/aC)
(10)
and the x i represent the nonzero roots of the transcendental equation f(xi)
3x1
= tan x i - --
3 + ani
-0
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 I
-
I
I
I
I
-
0.030
0.025
179
-
-
-
0.30
EXPERIMENT 3
D" t 0.20 -
-
R*
Dm t R2
-
0015-
0.10-
0
EXPERIMENT I
5
0
IO
20
15
25
O0
-
Figure 10. Data and fitted lines for typical experiments that were used to determine the effective diffusivity of the composite resin at 23.5 'C.
Table 111. Effective Diffusivities expt no.
0.25
1 2 3
0 20 EXPERIMENT 3 (dp=0045cm)
4
0 15
5 6
D't Rz.
0 IO
composite resin D * , l o 7 cm'/s
amphoteric resin D*, l o 7 cm'/s
0.120 0.129 0.124 0.433
2.30 1.82 1.69 2.83 2.76 29.7
-_-_
____
Table IV. Pore-Fluid Diffusivities
0 05
expt no.
-
EXPERIMENT I 0 090cm)
( dp
0 0
5
10
I5
T I M E , min
Figure 11. Data and fitted lines for typical experiments that were used to determine the effective diffusivity of the amphoteric resin at 25 "C.
This equation may be solved by Newton's method. Rearrangement of eq 9 using material balance equations gives
co c 6 a ( l + a ) exp(-D*tx,2/R2) =1--x co - c, i=l 9 + 9CY + CY2xi2 -
4
Figure 12. Data and fitted line from the desorption experiment at 95 "C that was used to determine the effective diffusivity of the amphoteric resin.
30
T I M E , min
F=-
3
W2 T I M E , min
m
(12)
where Cmis the bulk solution concentration a t equilibrium with the resin, and F is the fractional uptake of salt by the resin. An expression that is almost identical with eq 12 is given by Crank (1975) for fractional uptake by a single homogeneous sphere. R e s u l t s a n d Discussion The rate equations discussed above provide two related measures of ion-exchange resin performance. First, the effective diffusivity, D*, is useful in comparing the speeds of response for different resins, viz., a criterion for selecting a resin for an adsorption process. Second, the pore-fluid diffusivity, D , is useful for drawing conclusions about the intraparticle environment in which diffusion occurs. For example, a large value of D , Le., approaching the bulk solution value, could be attributed to the existence of pores which are fairly large compared to the dimension of the diffusing species, and easily accessible exchange sites. A much smaller value, however, could indicate the presence of less accessible exchange sites, perhaps surrounded by so-called polymeric islands (Mikes, 1970), or possibly a bidisperse pore structure. In the latter case, kinetic analysis by the method of Ma and Lee (1976) could provide additional insight about the diffusional process and structure of the resin.
1 2 3 4 5 6
composite resin D, l o 7 cm2/s
amphoteric resin D , l o s cm'/s
6.64 7.14 6.85 6.92 _-__
0.83 1.34 0.64 0.87 0.85 2.09
____
The value of D* is determined from rate data by evaluating several values of D*t/R2 by the method of successive substitution from eq 12. This requires knowledge of the constants xi and CY,and F as a function of time. Plots of experimental values of D*t/R2 vs. time are given in Figures 10,11, and 12 from typical experiments for both resins. Superimposed on the data are lines through the origin obtained by a least-squares scheme. Using the slopes through the origin for each experiment, along with the average particle radius, the value of the effective diffusivity of NaCl in the resin was calculated. The close fit of the data by the lines seems to confirm the validity of the model for these experiments. Subsequently, the pore-fluid diffusivity of NaCl was determined from
D = (1
+ :g)D*
where the slope of the isotherm was evaluated a t the arithmetic mean of the pore-fluid concentrations a t the beginning and end of an experiment. The values of the effective and pore-fluid diffusivities from both sets of experiments are listed in Tables I11 and IV. The activation energy for diffusion of NaCl in the amphoteric resin was estimated from an Arrhenius-type equation D = A exp(-E,/RT) Substitution of the values from experiment 6 a t 95 "C and the mean value from experiments 1 through 4 a t 25 "C
180
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
gave activation energies of 8.2 kcal/mol and 2.6 kcal/mol for the effective and pore-fluid diffusivities, respectively. These quantities are considered to be approximate because they are based on limited rate data and the assumption that the resin density and intraparticle void fraction did not vary with temperature. The theoretical treatment used here appears to have several advantages. First, the accuracy in fitting the rate data leaves little room for improvement. For example, there is little evidence that two diffusion coefficients representing diffusion in both macropores and micropores (Ma and Lee, 1976) would improve the quality of fit. Second, the results should be reliable over a broad range of processing conditions. Mass transfer coefficients determined from a linear or quadratic driving force rate expression (Helfferich, 1966) based on averaged phase concentrations would yield unreliable predictions for operations with a short cycling period with respect to the time constant for diffusion in the resin. Finally, since the rate expression indicates the effects of resin properties on the effective diffusivity of the solute in the resin, it could be used in conjunction with an experimental program to develop improved resins. Conclusions Since the effective diffusivity of NaCl in the amphoteric resin is an order of magnitude higher than in the composite resin, the former is an obvious choice for use in a continuous desalination process. The mean value of the diffusivity in the pore-fluid in the amphoteric resin is slightly less than the value in bulk solution a t 25 "C, 1.5 X cm2/s. Similarly, the activation energy for diffusion in the pore-fluid in the amphoteric resin is slightly less than the value in bulk solution, 3.9 kcal/mol (Fell and Hutchison, 1971). These observations indicate that the pore-fluid environment in the amphoteric resin probably resembles bulk solution, and that the exchange sites are relatively accessible. The value of the pore-fluid diffusivity in the composite resin, however, is more than an order of magnitude less than in the amphoteric resin. This may be due to a more complicated or restrictive pore structure, and to isolation of exchange sites in crosslinked polymeric islands as a result of the suspension polymerization technique used in making the composite resin. Acknowledgment The support of the Office of Saline Water (Grant 1430-2922) and the National Science Foundation (Grant ENG 77-19951) is gratefully acknowledged. The experiments on the composite resin were carried out a t the
Department of Chemical Engineering, University of California, Berkeley, Calif. Nomenclature a = resin interfacial area per unit mass, cm2/g A = Langmuir isotherm parameter, cm3/g B = Langmuir isotherm parameter, cm3/mequiv c = pore-fluid NaCl concentration, mequiv/cm3 C = bulk solution NaCl concentration, mequiv/cm3 d = average resin particle diameter, cm d = diffusivity of NaCl in pore-fluid, cm2/s D* = effective diffusivity of NaCl in resin, cmz/s E, = activation energy for diffusion, kcal/mol F = fractional uptake of NaCl by resin q* = absolute NaCl capacity of resin, mequiv/g r = radial position, cm R = average resin particle radius, cm T = temperature, "C t = time, s xi = nonzero roots of eq 11 Greek Letters = parameter reflecting resin properties and experimental conditions (see eq 10) pp = resin particle density, g/cm3 cp = resin intraparticle void fraction cy
L i t e r a t u r e Cited Barrett, J. H., Clemens, D. H., U S . Patent No. 3 9 9 1 017 (1976). Bolto, B. A,, J . Macromol. Sci. Chem., A4(5), 1039 (1970). Bolto, B. A., Weiss, D. E., "Ion Exchange and Solvent Extraction", Vol. 7, J. A. Marinsky, Ed., Marcel Dekker, New York, N.Y., 1977. Cobb. D. D., M.S. Thesis, University of California, Berkeley, Calif., 1975. Crank, J., "The Mathematics of Diffusion", 2nd ed, Ciarendon Press, Oxford, England, 1975. Dabby, S. S..et at., Paper IWC-76-20, 37th International Water Conference, Pittsburgh, Pa., 1976. Fell, C. J. D., Hutchison, H. P.. J . Chem. Eng. Data, 16, 427 (1971). Ginde, V. R., Chu, C., Desalination, 10, 309 (1972). Gregory, R. A., Sweed, N. H., Chem. Eng. J., 4, 139 (1972). Helfferich, F., "Ion Exchange and Solvent Extraction", Vol. I , J. A. Marinsky, Ed., Marcel Dekker, New York, N.Y., 1966. Huang, T.-C., Li, K.-Y., Ind. Eng. Chem. Fundam., 12, 50 (1973). Khang, S. J., Fitzgerald, T. J., Ind. Eng. Chem. Fundam., 14, 208 (1975). Knaebel, K. S., M.Ch.E. Thesis, University of Delaware, Newark, Del., 1978. Latty, J. A., Ph.D. Dissertation, University of California, Berkeley, Calif., 1974. Light, W. G., M.S. Thesis, University of California, Berkeley. Calif., 1973. Ma, Y. H., Lee, T. Y., AIChEJ., 22(1), 147 (1976). Mikes, J. A., "Pore Structure in Ion Exchange Materials", in "Ion Exchange in the Process Industries", SOC.Chem. Ind., London, England, 1970. Pigford, R. L., Baker, B., 111, Bium, D. E., U.S. Patent No. 3542525 (1970). Rosenthal, L. A.. Ind. Eng. Chem. Fundam., 16, 483 (1977). Shih, T. T., Pigford, R. L., "Removal of Salt from Water by Thermal Cycling of Ion Exchange Resins", in "Recent Developments in Separation Science", Vol. 3, N. N. Li, Ed., CRC Press, Cleveland, Ohio, 1977. Shih, T. T., Ph.D. Dissertation, University of California, Berkeley, Calif., 1975. Sutarwalla, F. T.. M.Ch.E. Thesis, University of Delaware, Newark, Del., 1978. Weiss, D. E., et al., 1st Intl. Symp. Water Desalination, 2, (1965). Weiss, D. E., et al., J . Water Pollut. ControlFed., 38, 1782 (1966). Weiss, D. E.,Bolto, B. A , , Wiilis, D., U S . Patent No. 3 645 922 (1972).
Received f o r review August 30, 1978 Accepted February 5 , 1979
