Ind. Eng. Chem. Res. 1997, 36, 5377-5383
5377
Cs+ Ion Exchange Kinetics in Complex Electrolyte Solutions Using Hydrous Crystalline Silicotitanates Ding Gu, Luan Nguyen, C. V. Philip, M. E. Huckman, and Rayford G. Anthony* Kinetics, Catalysis and Reaction Engineering Laboratory, Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
James E. Miller and Daniel E. Trudell Sandia National Laboratories, Albuquerque, New Mexico 87185
TAM-5 is a hydrous crystalline sodium silicotitanate inorganic ion exchanger with a high selectivity for Cs+. The kinetics of Cs+-Na+ ion exchange using TAM-5 in multicomponent electrolyte solutions were determined using batch experiments. For the powder, which is composed of crystals, a single-phase, homogeneous model fit the data best. For the granules, which were prepared from the powder, a two-phase, heterogeneous model resulted in an excellent fit of the data. Macropore and crystal diffusivities were determined by fitting the model to experimental data collected on the powder and the granules. Intracrystalline diffusivities were concentration dependent and were on the order of 10-19 m2/s. Macropore diffusivities were on the order of 10-10 m2/s. Resistance to diffusion in the macropores was not significant for granules with diameters less than 15 µm. A two-phase, homogeneous model, where liquid within the pores is in equilibrium with the solid, was also evaluated for the granules. Surprisingly, for the granules, an excellent fit of the data was obtained; however, the effective macropore diffusivity was 1.1 × 10-11 m2/s, an order of magnitude smaller than the macropore diffusivity found using the two-phase, heterogeneous model. Introduction Cesium removal from aqueous solutions is a major priority of radioactive waste treatment because radioisotopes of cesium emit gamma radiation, have long half-lives, and have a high mobility in the biosphere (Moore, 1993). Cesium can be removed from aqueous supernate by ion-exchange materials in an ion-exchange column, with the resulting solid waste vitrified into glass logs and ultimately stored in a high-level waste, federal geologic repository (Babad and Deichman, 1993; Gray and Becker, 1993; Illman, 1993). A large number of synthetic ion exchangers have been prepared during the past forty years (Amphlett, 1964; De and Sen, 1978; Kepak, 1981; Komarneni and Roy, 1982; Clearfield, 1982; Marsh et al., 1993). However, few ion exchangers have the ability to effectively remove cesium from solutions with high sodium concentrations. Anthony et al. (1993) synthesized a new hydrous, crystalline silicotitanate (CST) which not only has a very high cesium selectivity in solutions with a broad range of sodium and alkaline concentrations but also is stable under high radiation (Anthony et al., 1993; Marsh et al., 1993). Optimal design of an industrial ion-exchange process requires knowledge of the ion-exchange kinetics. The ion-exchange process is composed of the following three basic steps: (1) transport of ions to the ion-exchange sites, (2) counterions replacing ions that occupy the ionexchange sites, and (3) transport of the exchanged ion away from the exchange site. Transport in ion exchange is governed by gradients of concentration, electric potential, and temperature as well as by hydrodynamic interactions and internal relaxation (Turq et al., 1992). Ion exchange is also subject to electroneutrality con* Author to whom correspondence should be addressed. Phone: 409-845-3370/3361. Fax: 409-845-6446. S0888-5885(96)00338-7 CCC: $14.00
servation in addition to conservation of mass and energy (Huckman et al., 1996). In most cases, the rate-controlling step of ion exchange is either pore (intraparticle) or film (interparticle) diffusion (Weber and Smith, 1987; Crittenden and Weber, 1978; Merk et al., 1980; Crittenden et al., 1986; Kapoor and Yang, 1988; Robinson et al., 1994). In an intermediate range of conditions, both mechanisms may affect the rate. A quantitative criterion has been derived by Helfferich (1962) with which the ratecontrolling step can be predicted. The experimental method to distinguish between particle and film diffusion is to evaluate the effect of particle size and mixing speed on the rate of ion exchange. A number of models have been used to simulate the kinetics of ion exchange. Among these, the singlephase, homogeneous as well as the two-phase, homogeneous and the two-phase, heterogeneous diffusion models are most widely used (Rosen, 1952; Helfferich, 1966; Sherry, 1971; Garg and Ruthven, 1974; Saunders et al., 1989; Weber and Smith, 1987; Kapoor and Yang, 1988; Sun and Meunier, 1991; Komiyama and Smith, 1974; Nagel et al., 1987; Robinson et al., 1994). The homogeneous models assume the particle has a homogeneous pore structure, and the heterogeneous model assumes there are two different diffusion coefficientssan effective diffusivity for the macropores in between the crystals and an intracrystalline diffusivity used to describe Cs+ diffusion through the crystals. These diffusion mechanisms occur in series (Weber and Smith, 1987; Kapoor and Yang, 1988; Sun and Meunier, 1991; Robinson et al., 1994) or in parallel (Komiyama and Smith, 1974; Nagel et al., 1987; Robinson et al., 1994). In this paper, we address the intraparticle mass transfer of Cs+ in TAM-5 using complex electrolyte solutions and batch reactor experiments. We use simple binary exchange models and account for multicompo© 1997 American Chemical Society
5378 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 Table 1. Composition of Solution 3 (DSSF5 Simulant) component
concn (mol/L)
component
concn (mol/L)
NaNO3 KNO3 KOH Na2SO4 Na2HPO4‚7H2O Cs+
0.83 0.14 0.54 5.49 × 10-3 0.01 varies
NaOH Na2CO3 NaCl NaNO2 Al(NO3)3‚9H2O
2.78 0.11 0.07 1.08 0.51
nent effects by introducing a concentration-dependent intracrystalline diffusivity for Cs+. The film masstransfer resistance is insignificant at high agitation speeds (Nguyen, 1994). Intraparticle diffusivities of Cs+ in TAM-5 are estimated by fitting homogeneous or heterogeneous diffusion models to the batch experimental data. These diffusivities can be used in a general column model to predict the dynamic behavior of an ionexchange column. Experimental Procedure A batch technique is used to conduct all of the kinetic experiments. A 500 mL multicomponent solution is placed in a 600 mL plastic beaker with a magnetic stirrer. A measured amount (usually 5 g) of TAM-5 powder or granules is added to the solution, which is stirred at a constant rate. At selected time intervals, a sample of approximately 3 mL of slurry is collected in a syringe and quickly pressed through a 0.2 µm syringe filter to get a clear liquid sample. The liquid is analyzed for cesium content using a Varian atomic absorption spectrometer. Previous work in our laboratory indicates that this spectrometer is accurate to within 5% for this range of Cs+ concentrations. Six TAM-5 samples were used in this study. Sample 1 was from a batch of UOP IONSIV IE-910 powder (batch no. 993794040002) with particle diameters between 0.25 and 0.45 µm. Sample 2 was from a laboratory batch of powder with particle diameters between 0.2 and 0.4 µm. Samples 3 and 4 were 75-150 and 250-420 µm diameter granules, respectively, made from UOP IONSIV IE-910 powder without the use of any binder. Samples 5 and 6 were UOP IONSIV IE-911 granules (batches 8671-08 and 07398-38B, respectively) and have size distributions such that for sample 5, d < 297 µm, negligible; 297 µm < d < 420 µm, 3.7%; 420 µm < d < 590 µm, 44.0%; 590 µm < d < 840 µm, 52.3%; and for sample 6, d < 297 µm, 11.6%; 297 µm < d < 420 µm, 34.7%; 420 µm < d < 590 µm, 53.7%. Three electrolyte solutions were used to investigate the effect of composition on the rate of mass transfer. Solution 1 contains 5.1 M NaNO3, 0.6 M NaOH, and 50-150 ppm (mg/L) Cs+. Solution 2 contains 5.1 M NaNO3, 0.6 M NaOH, 0.67 M KNO3, and 100 ppm Cs+. Solution 3 has the compositions listed in Table 1; solution 3 is also called a DSSF5 simulant.
of uniform size, and the granules are considered to be an assembly of these crystalline particles. Homogeneous Diffusion Models The homogeneous diffusion model considers the powder to be a single, uniform phase, and there is one intracrystalline phase diffusivity for each species. The granules are considered to be two-phase porous particles, and there is a single effective diffusivity for each species. In the granules, the model assumes that diffusion occurs through the liquid-filled pores with local equilibrium between the solid and the pore liquid. Single-Phase, Homogeneous Model for the Powder. The mass balance for the powder is
∂q 1 ∂ ∂q ) 2 Dcr2 ∂t r ∂r ∂r
(
(1)
and the boundary and initial conditions are
q(r,0) ) 0, 0 < r < r0
(2)
∂q (0,t) ) 0 ∂r
(3)
A linear and a Langmuir isotherm are used to relate the Cs+ concentration on the surface of the powder particle to the Cs+ concentration in the bulk liquid solution.
q(r0,t) ) FpKdCb; linear isotherm q(r0,t) )
QTKCb
(4)
; Langmuir isotherm
(1 + KCb)
(5)
The bulk material balance is
Vsolution
∂Cb ∂q ) N4πr02Dc , at r ) r0 dt ∂r
(6)
where N is the number of particles. An analytical solution to this problem using a linear isotherm and constant Dc is given by Crank (1957),
F)1-
∑
[
6a(1 + a)
exp 9 + 9a + hn2a2
]
-hn2Dct r02
(7)
where F is the fractional attainment of equilibrium, and
a)
Vsolution VsolidKdFp
(8)
and hn are the non-zero roots of
Mathematical Model All the models assume that diffusion within the powder or granules is the rate-controlling step and follows Fick’s law. The batch reactor is assumed to be isothermal (Nguyen, 1994), and the bulk liquid concentration is uniform throughout. Local equilibrium is assumed between the solid and the pore liquid inside the particles; both linear and Langmuir isotherms are used to describe this equilibrium. The powder is considered to be small, spherical, crystalline particles
)
tan hn )
3hn
(9)
(3 + ahn)
An analytical solution for the Langmuir isotherm does not exist. Two-Phase, Homogeneous Model for the Granules. The mass balance for the granules is
(
)
∂C ∂q ∂2C 2 ∂C + (1 - ) ) De + ∂t ∂t ∂R2 R ∂R
(10)
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5379
where C is the Cs+ concentration in the pore liquid and q is the solid-phase Cs+ concentration. The solid-liquid equilibrium can be described by either a linear or Langmuir isotherm. For a linear isotherm and constant De, we can write
(
∂C ∂2C 2 ∂C ) Da + ∂t ∂R2 R ∂R
)
(11)
where Da is
Da )
De 1- 1+ FpKd
(
(12)
)
C(R0,t) ) Cb
(13)
∂C (0,t) ) 0 ∂r
(14)
Figure 1. Experimental data and single-phase, homogeneous model fits for TAM-5 powder in DSSF5 using a constant diffusivity. Exp 1: Initial Cs+ ) 10 ppm, L/S ) 100 mL/g, Dec ) 3.17 × 10-19 m2/s. Exp 2: Initial Cs+ ) 60 ppm, L/S ) 100 mL/g, Dec ) 1.25 × 10-19 m2/s. Exp 3: Initial Cs+ ) 100 ppm, L/S ) 50 mL/g, Dec ) 1.19 × 10-19 m2/s. Exp 4: Initial Cs+ ) 100 ppm, L/S ) 100 mL/g, Dec ) 0.71 × 10-19 m2/s.
C(R,0) ) 0, 0 < R < R0
(15)
The initial and boundary conditions are
The boundary and initial conditions are
The bulk phase material balance is
Vsolution
Vsolution a) 1- 1+ FpKd Vsolid
( (
(17)
) )
There is no analytical solution for the Langmuir isotherm.
∂q ) 0 at r ) 0, t > 0 ∂r
(23)
This model is used only for the granules. It assumes that diffusion occurs in the macroporous spaces between the crystals and then into the crystals. The former is described with a macropore diffusivity and the latter with an intracrystalline diffusivity. The mass balance around the crystals is
∂q ∂q 1 ∂ ) D r2 ∂t r2 ∂r c ∂r
(
q)
QTKC (1 + KC)
at r ) r0, t > 0
(24)
C ) 0 at t ) 0, 0 < R < R0
(25)
∂C ) 0 at R ) 0, t > 0 ∂R
(26)
C ) Cb at R ) R0, t > 0
(27)
Equations 18-27 are solved numerically.
Heterogeneous Diffusion Model
)
(18)
The mass balance around the granules is
∂C ∂q ∂ ∂C + (1 - ) ) D R2 ∂t ∂t R2 ∂R ep ∂R
(
Numerical Methods The spatial derivatives are approximated using secondorder, centered, finite difference approximations. The resulting set of ordinary differential equations was solved using Dow Chemical’s SimuSolv modeling and simulation software. Adams-Moulton and RungeKutta-Fehlkberg methods are used as numerical integration methods. The program is run on a VAX computer system. Results and Discussion
)
(19)
where q j is the average concentration of exchanged ion in the crystals
∂q 3 ∂q ) D , at r ) r0 ∂t r0 c ∂r
(20)
The bulk material balance is
Vsolution
(22)
∂Cb ∂C ) N4πR02De , at R ) R0 (16) ∂t ∂R
where N is the number of particles. The solution to this problem, for the linear isotherm, is again given by eq 7, where Da replaces Dc and
q ) 0 at t ) 0, 0 < r < r0
∂Cb ∂C ) -Np4πR02Dep , at R ) R0 ∂t ∂R
(21)
Micropore Diffusivity in DSSF5 Simulant. The intracrystalline diffusivity was estimated by fitting the single-phase, homogeneous diffusion model to experimental data obtained using powder. A Langmuir isotherm was used, and Dc was assumed to be constant. As shown in Figure 1, the model accurately predicts the experimental data. The resulting diffusivities are listed in Figure 1, and they range from 0.71 × 10-19 to 3.17 × 10-19 m2/s. Not yet published work from our laboratory on similar systems indicates that intracrystalline diffusion coefficients estimated using this technique typically have a two standard deviation range of about (25%. The variation found in this study indicates that the intracrystalline diffusivity is a function of concen-
5380 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 2. Experimental data and single-phase, homogeneous model fits for TAM-5 powder in DSSF5 using the Nernst-Hartley equation for Cs diffusivity, where DT0 ) 3.24 × 10-19 m2/s, and A ) -4.41 × 10-6 m3/g. Exp 1: Initial Cs+ ) 10 ppm, L/S ) 100 mL/g. Exp 2: Initial Cs+ ) 60 ppm, L/S ) 100 mL/g. Exp 3: Initial Cs+ ) 100 ppm, L/S ) 50 mL/g. Exp 4: Initial Cs+ ) 100 ppm, L/S ) 100 mL/g.
Figure 3. Experimental data and single-phase, homogeneous model fits for TAM-5 powder in solution 1 at various Cs+ concentrations. All experiments at 23 °C. D0 ) 5.66 × 10-9 m2/s, A ) 4.69 × 10-8 m3/g, E ) 13 208 cal/mol.
Table 2. Experimental Conditions Using Solution 1 experiment no.
Cs concn (ppm)
temp (°C)
1 2 3 4 5
50 100 150 100 100
23 23 23 34 44
tration. These numbers are the same order of magnitude as solid-phase diffusivities reported for other molecular sieve ion-exchange systems, such as basic cancrinite, analcite, and ultramarine (Amphlett, 1964). The numerical solutions obtained using a Langmuir isotherm were compared to the analytic solutions obtained using a linear isotherm, and the solutions were found to be nearly identical. Thus, even though previous work (Zheng, 1995) suggests that this system follows a Langmuir isotherm, a linear approximation is appropriate for these experimental conditions. There are several equations relating diffusivity with concentration (Horvath, 1985). We used the NernstHartley equation
Dec ) DT0(1 + Aq) A)
∂ ln γ( ∂q
(28) (29)
Equation 28 was incorporated into eqs 1-6, and the resulting equations were solved numerically. The parameters DT0 and A were estimated by simultaneously fitting the experimental data of different initial concentration and solution/solid ratios. DT0 is the intracrystalline diffusivity at zero Cs+ loading and is a function of temperature. For DT0 ) 3.24 × 10-19 m2/s and A ) -4.41 × 10-6 m3/g, an excellent agreement between the predicted curves and the experimental data, especially at lower initial concentrations, is obtained (see Figure 2). Temperature-Dependent Crystal Diffusivity in Simpler Solutions The dependencies of Cs+ intracrystalline diffusivity on solution composition and ion-exchange temperature in solution 1, a simple electrolyte solution, were also determined using batch experiments. Table 2 gives the
Figure 4. Experimental data and single-phase, homogeneous model fits for TAM-5 powder in solution 1 at various temperatures. Initial Cs+ is 100 ppm for all experiments. D0 ) 5.66 × 10-9 m2/s, A ) 4.69 × 10-6 m3/g, E ) 13 208 cal/mol.
Cs+ concentrations and temperatures for these experiments. The liquid-to-solid ratio was held constant at 100 mL/g. To include temperature effects, eq 28 was modified to
Dec ) D0e-E/RT(1 + Aq)
(30)
Parameters D0, E, and A were estimated by matching the experimental results with eqs 1-5 and eq 30. The parameter estimates were D0 ) 5.66 × 10-9 m2/s, A ) 4.69 × 10-8 m3/g, E ) 13 208 cal/mol. As shown in Figures 3 and 4, the up-take curves for different initial Cs+ concentrations and ion-exchange temperatures can be predicted with a single set of parameters. The Cs+ intracrystalline diffusivity is greater in solution 1 than in DSSF5. Also, in contrast to DSSF5 simulant, the Cs+ intracrystalline diffusivity increased with Cs+ concentration. To further study the effect of solution composition on intracrystalline diffusivity, an experiment was conducted in solution 2. Solution 2 can be viewed as solution 1 plus 0.67 M KNO3. The amount of K+ in solution 2 is the same as that in the DSSF5 simulant. Equilibrium experiments show that the Cs+ selectivity drops by about 50% in this solution. The estimated Cs+ intracrystalline diffusivity in solutions 1, 2, and 3 are 2.24 × 10-19, 1.94 × 10-19, and 0.71 × 10-19 m2/s, respectively. It seems that while the effect of K+ is a slightly decreasing, concentration-dependent Cs+ intracrystalline diffusivity, anions and other cations in DSSF5 cause a more significant decrease. The ways in
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5381
Figure 5. Experimental data and two-phase, heterogeneous model fits for sample 3 (75-150 µm) and sample 4 (250-420 µm) in DSSF5 simultant. Initial Cs+ is 100 ppm for both experiments.
Figure 6. Experimental data and two-phase, heterogeneous model fits for sample 5 (07398-38B) and sample 6 (8671-08) in DSSF5 simultant. Initial Cs+ is 100 ppm for both experiments.
which competing and noncompeting ions affect the Cs+ intracrystalline diffusivity are expected to vary case by case. This variation is due, in part, to the binary simplification used in the models. Macropore Diffusivity of Granules Heterogeneous Models. A two-phase, heterogeneous model (eqs 18-27) was used to describe ionexchange kinetics in TAM-5 granules. The concentrationdependent diffusivity obtained from the powder data was used as the intracrystalline diffusivity in the heterogeneous model. The effective macropore diffusivity was estimated by matching the model predictions to experimental data from samples 3-6 in DSSF5. Figures 5 and 6 show there was good agreement between the calculated curves and the experimental data. For samples 3 and 4, the estimated macropore diffusivity is 2.3 × 10-10 m2/s, and as expected, both experiments can be fit using a single macropore diffusivity. The ionic diffusivity of Cs+, DCs, is 2.2 × 10-9 m2/s (Robinson and Stokes, 1970), and the particle porosity, , is 0.4. Therefore, the calculated tortuosity is 3.8. This indicates that surface diffusion is not significant in samples 3 and 4 (Ruthven, 1984). For samples 5 and 6, Dp/R2 ) 0.0067 and 0.01 1/s, respectively. Using a weight-averaged radius, Dp ) 2.95 × 10-10 and 9.27 × 10-10 m2/s for samples 5 and 6, respectively. With ) 0.4 and DCs ) 2.2 × 10-9 m2/s, we calculate τ ) 3.0 and 0.95 for samples 5 and 6, respectively. The tortuosity less than unity indicates surface diffusion is significant in sample 6 (Ruthven, 1984).
Figure 7. Concentration profiles in the micropores. Concentration is plotted as a function of dimensionless crystal radius.
Figure 8. Experimental results and two-phase, homogeneous model fits for sample 3 (75-150 µm) and sample 4 (250-420 µm) in DSSF5 simultant. The effective diffusivities are considered neither macropore nor micropore diffusivities.
As the granule radius becomes smaller, macropore diffusivity becomes less important. Our calculations show that in granules with a radius less than 15 µm, the concentration gradient in the macropore is negligible. It is worthwhile to emphasize that Dp/R2, not Dp, represents the diffusional resistance. Larger values of Dp/R2 indicate smaller resistances to diffusion. Figure 7 shows an intracrystalline concentration profile at different times. The curves have parabolic shapes. Therefore, the coupled partial differential equations can be simplified to ordinary differential equations by using the parabolic profile assumption suggested by Do and Rice, 1986; Liaw et al., 1979; Cen and Yang, 1986. Two-Phase Homogeneous Model. The kinetic data for the granules were also fit by using the twophase, homogeneous model to determine the effective diffusivity. Equations 10-17 were solved both numerically and analytically. Figure 8 shows that there is a good match between experimental data and predicted curves and numerical results are the same as the analytical results. The estimated diffusivities for samples 3-6 were in the range of 1.0 × 10-11-3.0 × 10-11 m2/s. As the Cs+ diffusivity in aqueous electrolyte solutions is on the order of 10-9 m2/s, these values do not fall between the extremes D[/(2 - )]2 to D(/2) suggested by Helfferich (1962); thus this coefficient is considered to be neither a macropore nor an intracrystalline diffusivity. Conclusion Cs+ macropore and intracrystalline diffusivities were determined for a binary diffusion model from batch
5382 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
kinetic experiments using different particle sizes, initial concentrations, and solution/solid ratios. The intracrystalline diffusivities were on the order of 10-19 m2/s, and this was the most significant mass transfer resistance. The macropore diffusivities were on the order of 10-10 m2/s. When the granule radius is less than 15 µm, macropore resistance to diffusion can be neglected. The Cs+ intracrystalline diffusivity was dependent on the Cs+ concentration and the ion-exchange temperature, and these dependencies were quantified. The Cs+ intracrystalline diffusivity was also found to be a function of solution composition. Both competing and noncompeting ions affected the concentration-dependent intracrystalline diffusivity. This study shows that the single-phase, binary, homogeneous model with a concentration-dependent diffusivity works well for the powder, and the two-phase, heterogeneous model works well for the granules. Although the two-phase, homogeneous model can give a good fit of experimental data for the granules, the diffusivities obtained are neither macropore nor intracrystalline diffusivities and could be misleading. The two-phase, heterogeneous model, which assumes series macropore and intracrystalline diffusion, is physically realistic and fits the experimental data well. Using the diffusivity estimated for the powder as the intracrystalline diffusivity, the macropore diffusivities obtained are 1 order of magnitude smaller than the ionic diffusivity. The tortuosities of most of the granules were larger than 1, except for one sample, which indicates that surface diffusion is not significant in most of the samples. The intraparticle diffusion models and the diffusivities obtained in this study provide essential information for modeling of ion-exchange columns. Notations C ) intraparticle liquid phase concentration (g/m3) Cb ) bulk liquid phase concentration (g/m3) Dc ) intracrystalline diffusivity (m2/s) De ) effective diffusivity for two-phase, homogeneous model (m2/s) Dep ) effective macropore diffusivity in two-phase, heterogeneous model (m2/s) Dc0 ) intracrystalline diffusivity at zero loading (m2/s) E ) activation energy (cal/mol) F ) fractional attainment of equilibrium k ) pre-exponential factor (m2/s) Kd ) distribution coefficient (mg/L) K ) parameter of Langmuir isotherm q ) solid phase concentration (g/m3) qj ) average solid phase concentration over the crystal (g/ m 3) QT ) ion-exchange capacity of the ion exchanger r ) radius (m) r0 ) radius of crystal (m) R ) radius (m) R0 ) radius of granule (m) Vsolid ) total volume of ion exchangers Vsolution ) volume of ion-exchange solution γ( ) ionic activity coefficient ) void fraction Fp ) particle density
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Anthony, R. G.; Philip, C. V.; Dosch, R. G. Selective Adsorption and Ion Exchange of Metal Cations and Anions with SilicoTitanates and Layered Titanates. Waste Manage. 1993, 13, 503. Babad, H.; Cash, R. J.; Deichman, J. L. High-Priority Hanford Site Radioactive Waste Storage Tank Safety Issues: An Overview. J. Hazardos Mater. 1993, 35, 427. Boyd, G. E.; Adamson, A. W.; Myers, L. S., Jr. The Exchange Adsoprtion of Ions from Aqueous Solutions by Organic Zeolites. J. Am. Chem. Soc. 1947, 69, 2836. Cen, P. I.; Yang, R. T. Analytic Solution for Adsorption Breakthrough Curves with Disperse Sorbents (Zeolites). AIChE J. 1986, 32 (10), 1635. Clearfield, A. Inorganic Ion Exchange Materials; CRC Press: Boca Raton, 1982. Crank, J. The Mathematics of Diffusion; Oxford University Press: New York, 1957; p 85. Crittenden, J. C.; Weber, W. J., Jr. Predictive Model for Design of Fixed-Bed Adsorbers: Single-Component Model Verification. ASCE Env. Eng. Div. 1978, 104 (EE6), 1175. Crittenden, J. C.; Hutzler, N. J.; Geyer, D. G.; Gravitz, J. L.; Friedman, G. Transport of Organic Compounds with Saturated Groundwater Flow: Model Development and Parameter Sensitivity. Water Resour. Res. 1986, 22 (3), 271. De, A. K.; Sen, A. K. Synthetic Inorganic Ion-Exchangers. Sep. Sci. Technol. 1978, 13, 517. Do, D. D.; Rice, R. G. Validity of the Parabolic Profile Assumption in Adsorption Studies. AIChE J. 1986, 32 (1), 149. Garg, D. R.; Ruthven, D. M. The Performance of Molecular Sieve Adsorption Columns: Systems with Micropore Diffusion Control. Chem. Eng. Sci. 1974, 29, 571. Gray, R. H.; Becker, C. D. Environmental Cleanup: The Challenge at the Hanford Site, Washington, USA. Environ. Manage. 1993, 17, 461. Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962; p 302. Helfferich, F. Ion-Exchange Kinetics. In Ion Exchange; Marinsky, J. A., Ed.; Marcel Dekker: New York, 1966. Horvath, A. L. Handbook of Aqueous Electrolyte Solutions, Physical Properties, Estimation and Correlation Methods; John Wiley & Sons: New York, 1985. Huckman, M. E.; Gu, D.; Philip, C. V.; Anthony, R. G. Mathematical Modeling of Ion Exchange Columns with Emphasis on Hydrous Crystalline Silico-Titanates, Molecular Sieves and Non Ideal Solutions. In Emerging Separation Technologies for Metals II; Bautista, R. G., Ed.; The Minerals, Metals & Materials Society: Reno, NV, 1996. Illman, D. L. Researchers Take Up Environmental Challenge at Hanford; Chem. Eng. News 1993, June 21, p 3. Kapoor, A.; Yang, R. T. Optimization of a Pressure Swing Adsorption Cycle. DOE/C/22060-2649; State University of New York at Buffalo, February 1988. Kepa´k, F. Behaviour of Radionuclides in Aqueous Solutions and Their Sorption on Hydrated Oxides and Some Insoluble Salts. At. Energy Rev. (Suppl.) 1981, 2, 5. Komarneni, S.; Roy, R. Use of γ-Zirconium Phosphate for Cs Removal from Radioactive Waste. Nature 1982, 299, 707. Komiyama, H.; Smith, J. M. Intraparticles Mass Transport in Liquid-Filled Pores. AIChE J. 1974, 20 (4), 728. Liaw, C. H.; Wang, J. S. P.; Greenkorn, R. A.; Chad, K. C. Kinetics of Fixed Bed Adsorption: A New Solution. AIChE J. 1979, 25, 376. Marsh, S. F.; Svitra, Z. V.; Bowen, S. M. Los Alamos Report, LA12654(UC-940); October 1993. Merk, W.; Fritz, W.; Schlunder, E. U. Competitive Adsorption of Two Dissolved Organics onto Activated Carbon-III. Chem. Eng. Sci. 1980, 36, 743. Moore, M. First, Puzzlement; Then Action. Bull. At. Sci. March 1993. Nagel, G.; et al. Modeling of Non-Isothermal Multi-Component Adsorption in Non-Adiabatic Fixed Beds. Chem. Eng. Sci. 1987, 42 (1), 143. Nguyen, L. T. A Kinetic Model for Ion Exchange Between Cesium and Sodium Using Silico-Titanates. M.S. Thesis, Texas A&M University, 1994. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd Ed.; Butterworths: London, 1970. Robinson, S. M.; Arnold, W. D.; Byers, C. H. Mass-Transfer Mechanisms for Zeolite Ion Exchange in Wastewater Treatment. AIChE J. 1994, 40 (12), 2045. Rosen, J. B. Kinetics of a Fixed-Bed System for Solid Diffusion into Spherical Particles. J. Chem. Phys. 1952, 20, 387.
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Received for review June 12, 1996 Revised manuscript received September 4, 1997 Accepted September 4, 1997X IE960338V
X Abstract published in Advance ACS Abstracts, November 1, 1997.
