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Two-site equilibrium model for ion exchange between multi-valent cations and zeolite-A in a molten salt
Microporous and Mesoporous Materials 84 (2005) 366–372 www.elsevier.com/locate/micromeso
Two-site equilibrium model for ion exchange between multi-valent cations and zeolite-A in a molten salt Mary Lou Dunzik-Gougar a
a,*
, Michael F. Simpson b, Barry E. Scheetz
c
Idaho State University/Idaho National Laboratory, 1776 Science Center Dr., Idaho Falls, ID 83402, USA b Idaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415, USA c The Pennsylvania State University, Materials Research Laboratory, University Park, PA 16802, USA Received 3 November 2004; received in revised form 26 March 2005; accepted 28 April 2005 Available online 5 August 2005
Abstract A model has been developed and tested for its ability to describe the ion exchange of fission product species between zeolite and a molten salt bath used for pyroprocessing of spent nuclear fuel. The model is based on a system at equilibrium and assumes species interacting with zeolite are chemically independent of each other. This assumption was tested via a set of salt–zeolite contact experiments. Zeolite 4A was batch contacted with Cs ternary salt (CsCl–LiCl–KCl), Nd ternary salt (NdCl3–LiCl–KCl) or Cs–Nd quaternary salt (CsCl–NdCl3–LiCl–KCl) until ion exchange steady state was reached. Results from ternary salt and zeolite chemical analyses were used to determine model parameters for all species present. These parameters, in turn, were applied to compute estimated compositions of zeolite contacted with quaternary salt. The estimated compositions compare favorably with results from chemical analyses of the zeolite–quaternary salt experiment samples. Based on the data collected, the model is modified for predictive capability for a range of salt compositions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Fission products; Ion exchange; Zeolite; Model; Molten salt
1. Introduction The electrometallurgical treatment of spent nuclear fuel (SNF) has been developed at Argonne National Laboratory (ANL) and has been demonstrated through processing the sodium-bonded SNF from the Experimental Breeder Reactor-II in Idaho. In this process, components of the SNF, including U and species more chemically active than U, are oxidized into a bath of lithium–potassium chloride (LiCl–KCl) eutectic molten salt. Uranium is subsequently removed from solution
*
Corresponding author. Tel.: +1 208 282 7809; fax: +1 208 282 7735. E-mail address: [email protected] (M.L. Dunzik-Gougar).
1387-1811/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.micromeso.2005.04.014
by electrochemical reduction. The noble metal fission products and inactive fuel/clad components are not oxidized into the salt solution but are removed in solid form and subsequently melted into a metal waste form. The residual salt solution contains most of the fission products and transuranic elements from the SNF. One technique that has been identified for removing these fission products and extending the usable life of the molten salt is ion exchange with zeolite-A. Zeolite batch contact experiments have been performed using radioactive salt from the treatment of EBR-II fuel and non-radioactive salt of various compositions. From these experiments, salt and zeolite composition data have been collected to support development of a multi-component equilibrium model for the exchange of fission products between molten chloride salt
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372
and zeolite-A. Such a model may be used to support the development of a salt recycling process for the future electrometallurgical treatment of various SNF types.
2. Model development In a previous paper [1], early research on the application of ion exchange with zeolite-A to removal of fission products from LiCl–KCl was reviewed. Based on this work, Simpson and Gougar [1] proposed a model for a salt–zeolite system containing purely monovalent ion components and found that it exhibited an excellent fit to the data. The monovalent model and the new, multi-valent model will be compared in this section. 2.1. The multi-valent model The goal in modeling the equilibrium behavior of this system is to develop an ability to predict the distribution of multi-valent cations between the zeolite and the salt given a specific salt/zeolite ratio and starting compositions. To this end, the behavior of each different type of cation must be examined to learn how it interacts with the zeolite and its relative affinity for association with the zeolite. Ion exchange between a cation species, A, in the salt, and cation species B, in the zeolite, is represented in Eq. (1), A A B B nB Aþn þ nA Bþn $ nB Aþn þ nA Bþn : s z z s
ð1Þ
Subscripts s and z denote salt and zeolite phases, respectively. The ionic charge of species i is ni. In the multi-valent model, forward and reverse ion exchange reaction rates for this system (Eq. (1)) are formulated as follows: n
rf;A ¼ k f;A C A;s ð1 C A;z Þ A ; A rr;A ¼ k r;A C nA;z ð1 C A;s Þ.
ð2Þ ð3Þ
In the forward (2) and reverse (3) reaction rate equations, CA,s represents the mole equivalent fraction of A with charge nA in the salt phase, and CA,z is the mole equivalent fraction of zeolite sites occupied by A. The rate at which A is taken in to the zeolite structure is proportional to its concentration in the salt and to the fraction of sites in zeolite occupied by all potential reaction partners. Clearly, all potential reaction partners, or counter cations, will not have the same affinity for ion exchange with species A; however, a very effective simplifying assumption is made when considering the counter ions as a whole. In turn, the rate of species A leaving the zeolite is proportional to its concentration in the zeolite and the fraction of all other (potential replacements for A) species in the salt phase. It is noted here that, because the experiments were run under equilibrium conditions, the authors did not consider whether ion
367
exchange kinetics or diffusion of the salt species would be more rate limiting. The model equations (2) and (3) were formulated based on preliminary experimental data and, as described in this paper, were verified via additional experimentation. The value of the model is further increased through consideration of the physical system being described. While the model was not derived from first principles, a perspective on the relationship between equation and ion may be gained by drawing an analogy to probability. Consider salt cation species x, y and z with charges +1, +2, and +3, respectively. As the salt phase species enter the zeolite, they will either occupy a site(s) in the zeolite or continue flowing through the pore system to exit the zeolite macrostructure. Each site, in this analogy, is considered an equivalent site or an area in the zeolite carrying a negative 1 charge. If it is assumed that no species is preferred over any other, the probability of one equivalent of species x (+1) occupying a specific site in the zeolite is the same as that of one equivalent of y (+2) occupying that same site, although y will also satisfy an additional site due to its charge. Likewise, the probability of a site being occupied by one equivalent of y is the same as that of one z (+3) equivalent. Because Ci,s and Ci,z in the model equations are fractions, the term Ci,z is an expression of the probability that species i is in a particular site in the zeolite. If species i has a +2 charge, we must consider the probability that the ion occupies two specific sites. In a system with effectively infinite zeolite sites, each of the two sites individually has the same probability of being occupied by the cation. Then the combined probability for a +2 ion to occupy two sites is proportional to the product of the individual probabilities.
probability of species i1+ occupying one equivalent site in zeolite / Ci, probability of species i2+ occupying one equivalent site in zeolite / Ci Æ Ci, probability of species i3+ occupying one equivalent site in zeolite / Ci Æ Ci Æ Ci. Conversely, the term (1 Ci,z) is proportional to the probability that a given site in the zeolite is associated with some species other than i. probability of species i1+ not occupying one equivalent site in zeolite / (1 Ci), probability of species i2+ not occupying one equivalent site in zeolite / (1 Ci) Æ (1 Ci), probability of species i3+ not occupying one equivalent site in zeolite / (1 Ci) Æ (1 Ci) Æ (1 Ci).
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To create an applicable form of the multi-valent model, a state of ion exchange equilibrium is assumed for the salt–zeolite system. In a system at equilibrium, Eqs. (2) and (3) are equal and a resulting generic expression for the amount of species i in zeolite (Ci,z) is C i;z ¼
i K i C 1=n i;s
1=n
K i C i;s i þ ð1 C i;s Þ
1=ni
where
;
It has been well established that salt species are incorporated in the zeolite 4A structure in two ways, or in two types of sites [2–10]. For one, cations from the salt may exchange with sodium cations in the zeolite framework structure. These sodium ions are present initially to balance a negative charge in the framework. Secondly, salt cations may associate with chloride anions according to their charge. In this case the species are immobilized within zeolites interconnecting pore system as whole chloride molecules. Using Eq. (4) as a basis, an expression was developed to represent the two-site nature of zeolite-A, 1=ni
S 1 K IX i C i;s 1=n
i K IX i C i;s þ ð1 C i;s Þ
1=ni
1=ni
þ
S 2 K occ i Ci 1=ni K occ i C i;s
þ ð1 C i;s Þ
1=ni
:
2.2. The monovalent model In their earlier monovalent model, Simpson and Gougar [1] represent the forward and reverse reaction rates as
k f;i Ki ¼ . k r;i ð4Þ
hi ¼
where hT is the total mole equivalents immobilized by both ion exchange and occlusion per unit cell zeolite.
ð5Þ
B A rf;A ¼ k f;A C nA;s C nB;z ;
ð7Þ
B A k r;A C nA;z C nB;s ;
ð8Þ
rr;A ¼
where all terms are as previously defined. Because the species are monovalent (nA = nB = 1), the equilibrium zeolite concentration of a particular species may be solved for explicitly in terms of measurable salt phase concentrations. C A;z ¼
K A S 1 C A;s . K A C A;s þ K B C B;s
ð9Þ
If there are more than two monovalent species competing for zeolite sites, the monovalent model equation (Eq. (10)) resembles the Langmuir model equation, with the exception that no ion exchange sites may remain unfilled. The Langmuir model for sorption does accommodate empty sites. k i;1 S 1 C i;s C i;z;1 ¼ P . k j;1 C j;s
ð10Þ
j
Here hi is the mole equivalents of a species i per pseudounit cell zeolite. The first of the two quotient terms in Eq. (5) represents the ion-exchanged mole equivalents of i, which contribute to the total S1 mole equivalents necessary to balance the charge of one zeolite-A pseudo-unit cell. A pseudo-unit cell of zeolite-A, henceforth referred to simply as ‘‘unit cell’’, is defined as Mþ 12 ðAlO2 Þ12 ðSiO2 Þ12
Therefore, S1 = 12. The aluminosilicate framework of zeolite-A carries a negative charge due to trivalent aluminum atoms taking the places of tetrahedrally coordinated silicon atoms. Various alkali and alkaline earth cation species, M, commonly fill zeolite-A framework sites to balance the charge. In zeolite 4A, these sites are occupied by sodium ions. In this application for fission product waste ion removal, various monovalent, divalent and trivalent cations may fill these framework sites. The occluded mole equivalents of species i are represented in the second quotient term of Eq. (5). While it is known that there are 12 ion exchange mole equivalent sites in the 4A unit cell, the number of mole equivalents occluded in the cage structure is a variable quantity depending on the composition of the salt. Therefore, the number of occluded mole equivalents, S2, will be expressed as S 2 ¼ hT 12;
ð6Þ
The key differences between the multi-valent and monovalent models stem from different starting assumptions about the reacting cation species. The multi-valent model assumes that the extent to which a species, A, enters or leaves the zeolite is chemically independent of the nature of all potential reaction partners. So, a species, A, will enter zeolite according to its own concentration in the salt and to the number of sites not occupied by A in the zeolite. Likewise, A leaves the zeolite structure (entering the salt solution) according to its own concentration in the zeolite and the concentration of all non-A cation species (potential reaction partners) in the salt. By grouping all non-A species together, the multi-valent model drastically simplifies the mathematics of testing and verification for multi-component, multi-valent ion systems. In contrast, the monovalent model describes the forward reaction rate for A entering zeolite to be dependent on its own concentration in the salt and the number of sites occupied by some defined reaction partner, B. A reference reaction partner must be arbitrarily chosen for each type of site in this model such that all other species are described with respect to that one. In other words, the assumption in the monovalent model is that the rate of A reacting with zeolite is independent of all species except the arbitrary references. This model is also being tested for its applicability to multi-valent systems.
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372
3. Experimental methods and results
369
Table 1 Model parameters optimized with ternary experimental data Species
KI
K0
Cs Na Li K Nd
0.10 3.2 3.0 0.076 0.61
4.0 0 0.75 0.91 1.5
6 5 4 3 2 model 1
experiment
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Equivalent fraction Cs in salt
Fig. 1. Comparison of measured and model values for Cs loading in zeolite from Cs ternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Cs and Al.) To view this figure in color, see the web version of this paper. 14
Equivalents Nd per unit cell zeolite
To support the validity of this multi-valent model as well as to determine parameters for a number of key species, a set of salt–zeolite batch contact experiments was designed and executed. A monovalent (CsCl) or trivalent (NdCl3) salt was mixed with the LiCl–KCl solvent salt and heated to 500 °C. Non-radioactive Cs and Nd chloride salts, radioactive ‘‘surrogates’’, were used to chemically represent the fission product isotopes of those elements. Zeolite 4A (Na12(AlO2)12(SiO2)12) pellets, consisting of 10–20 wt% clay binder, were contacted with the molten salt mixtures. When ion exchange equilibrium in the contact system was reached, salt and zeolite samples were collected for chemical composition analysis. From salt composition data the mole equivalent fraction of each cation species in the mixture was determined. Zeolite composition data were converted to species mole equivalents per unit cell zeolite. Conversion was accomplished by normalization of the measured species concentrations with respect to aluminum, which is present at a uniform concentration of 12 mol equivalents per unit cell. Data from each salt–zeolite contact experiment were combined in a single data set and used to fit the model. Values for model parameters KIX and Kocc (Eq. (5)) were obtained for each species using a constrained optimization routine that minimizes the Euclidean norm of the error vector. Table 1 provides values of the model parameters obtained for each species in the ternary (three-component) contact systems. Figs. 1 and 2 show results of contact experiments in ternary (CsCl–LiCl–KCl and NdCl3–LiCl–KCl) salt systems. Concentrations of surrogate fission product species (Cs and Nd) in zeolite are plotted as a function of the equilibrium salt concentrations of that same species. Typical exchange behavior is demonstrated in these salt– zeolite systems. At lower concentrations (in this case, less than about 0.1 mol equivalent fraction of species in the salt), the Cs and Nd zeolite loading increases significantly with the increased salt concentrations compared to the changes in loading seen above 0.1 salt mole equivalent fraction. As the species concentration in the salt increases beyond 0.1, the zeolite concentration of that species shows small increases. Indeed, if considering the measurement uncertainty, it could be stated that there is no significant change in the concentration
Equivalents Cs per unit cell zeolite
7
12 10 8 6 experiment 4
model
2 0 0
0.1
0.2
0.3
0.4
0.5
Equivalent fraction Nd in salt
Fig. 2. Comparison of measured and model values for Nd loading in zeolite from Nd ternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Nd and Al.) To view this figure in color, see the web version of this paper.
of Cs or Nd in zeolite past 0.2 salt equivalent fraction. This may indicate the capacity of zeolite for these species. The Cs loading reached almost 6 equivalents, or 6 Cs ions, per unit cell, while the Nd ion loading was almost 12 equivalents (4 ions) per unit cell zeolite. Also plotted in Figs. 1 and 2 are model values for zeolite concentration calculated using the empirical parameters for each species (Table 1) and empirical values for the total number of mole equivalents immobilized per unit cell zeolite, hT. Because hT is a measured quantity dependent on salt composition, the value varies with each experiment and therefore for each point plotted. This phenomenon leads to a non-smooth model curve if the calculated points are connected. To test the assumption of species chemical independence upon which the multi-valent model was based, the model parameters determined with ternary salt system data were used to estimate final zeolite concentrations for a quaternary salt system
370
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372 30.0
Equivalents per unit cell zeolite
Equivalents per unit cell zeolite
12 10 8 6 4
Nd expt.
Nd model
2
Cs expt.
Cs model
25.0 20.0 15.0 Cs, Rb ternary Nd, U, Sr ternary
10.0
Linear (Nd, U, Sr ternary) 5.0
Linear (Cs, Rb ternary)
0.0
0 0
0.1
0.2
0.3
0.4
0
0.5
(CsCl–NdCl3–LiCl–KCl) contact experiment. Experimental results, of the same type presented for the ternary system contacts, are plotted in Fig. 3 along with the model estimates. The most significant result illustrated in the (Figs. 1–3) is the reasonably good fit between values measured in both ternary and quaternary salt systems and values calculated with the same model parameter constants. This qualitative assessment of fit is supported by a quantitative evaluation of the model fit to data as described next. 3.1. Model verification and modification The model in the form of Eq. (5) allowed preliminary verification of fit to experimental data. It is limited, however, by the fact that hT is a measured value. In other words, it is not possible, to calculate the composition of a zeolite sample without first having analyzed the sample and measured the total mole equivalents of species immobilized in the zeolite structure. Clearly, it would be more useful if zeolite composition could be estimated a priori, with the model equation using equilibrium salt concentration measurements. To this end, an empirical relationship was sought between the salt concentrations of species at equilibrium (Ci) and the total mole equivalents of all species in the zeolite after contacting (hT). Fig. 4 reveals such a relationship. In Fig. 4, data from ternary salt +1 species (Cs and Rb) experiments were plotted with identical symbols. The same was done for ternary multi-valent (Nd3+, U3+ and Sr2+) species data. Rubidium, uranium and strontium data came from a paper on related work by Lexa [11,12]. The data indicate a linear relationship between hT and the total solvent equivalents. The positive slope of the multi-valent species line (y = 3.1x + 24.3) appears
0.2
0.3
0.4
0.5
0.6
Total non-solvent equivalents in the salt
Equivalent fraction of species in salt
Fig. 3. Comparison of measured and model values for Cs and Nd loading in zeolite from Cs–Nd quaternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Cs, Nd and Al.) To view this figure in color, see the web version of this paper.
0.1
Fig. 4. Correlation between total zeolite loading (hT) and total surrogate fission product/TRU-species salt concentrations. (Data for Rb, U, Sr and some for Cs are from references [11,12].) To view this figure in color, see the web version of this paper.
to indicate that there is a gradual increase in the total mole equivalents occluded in zeolite as the concentration of fission products and actinides in the salt increases. The negative slope of the +1 species line (y = 8.4x + 22.3) in Fig. 4 is more pronounced and also consistent with a space-filling model for loading the zeolite unit cell. The solvent cations (Li1+ and K1+) share the same charge as fission product species Cs and Rb, but differ significantly in size (see Table 2). It is clear that more Li and K ions would fit in a given unit cell than Cs and Rb ions due to the larger radii of the latter two. As the concentration of Cs or Rb increases in the salt solution, the number of ions of these species per unit cell zeolite also increases. However, while the number of Cs and Rb mole equivalents per unit cell increases, the total number of mole equivalents in the unit cell decreases. This may be explained by the larger ions taking up more of the available space than the solvent ions while carrying the same +1 charge. The linear equations associated with Fig. 4 allow hT to be estimated from salt measurement, thereby removing any dependence on zeolite analysis (see Eq. (11)). Note the reference to single valent species concentrations, sv, and multi-valent, mv, species concentrations. Table 2 Effective ionic radii for ionic species in the experimental salt–zeolite contact system [13] Cation
Effective ionic radius (nm)
Li1+ Nd3+ Ca2+ Na1+ U3+ Sr2+ K1+ Rb1+ Cs1+
0.076 0.0983 0.100 0.102 0.103 0.118 0.138 0.152 0.167
hT ¼ bððC sv =ðC sv þ C mv ÞÞð8.37C sv þ 22.32ÞÞ þ ððC mv =ðC sv þ C mv ÞÞð3.10C mv þ 24.29ÞÞc.
ð11Þ
To create a model curve, values for C were chosen at regular intervals between 0 and 1. The resulting h-values form the basis for a curve to which the experimental values may be compared (plotted in Figs. 5–7). 3.2. Evaluation of model fit To quantify the quality of the fit, the extent of correlation between variations in measured hi-values and variations in the model-generated values of hi was determined. The Pearson product–moment correlation coefficient (Eq. (6)) is applied to estimate the extent of linear
Equivalents Cs per unit cell zeolite
7
371
12 10 8 Nd expt. 6
Cs expt. Nd model
4
Cs model 2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Equivalent fraction Cs or Nd in salt
Fig. 7. Comparison of model prediction for Cs and Nd loading in zeolite to experimental data from the Cs–Nd quaternary system. (hT computed from salt concentrations. Error bars reflect 2-sigma analytical uncertainty in Cs, Nd and Al.) To view this figure in color, see the web version of this paper.
6
association between measured (hi) and modeled (^ hi ) zeolite concentrations [14,15].
5
P n ^ ^ ðhi hi Þ ðhi hi Þ r ¼ P i¼1 : n ^ ^ 2 2 i¼1 ðhi hi Þ ðhi hi Þ
4 3
ð12Þ
model
2
experiment
1 0 0
0.1
0.2
0.3
0.4
0.5
Equivalent fraction Cs in salt
Fig. 5. Comparison of model prediction for Cs loading in zeolite to experimental data from the Cs ternary system. (hT is computed from equilibrium salt composition data. Error bars reflect 2-sigma analytical uncertainty in Cs and Al.) To view this figure in color, see the web version of this paper.
14 Equivalents Nd per unit cell
Equivalents Cs or Nd per unit cell zeolite
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372
12
1 Cbðm þ 1Þ=2c ðm2=2Þ P ðr; mÞ ¼ pffiffiffi ð1 r2 Þ ; Cðm=2Þ p where m ¼ n 2 [16].
ð13Þ
The fit of the experimental model was quantified with calculations of r- and P-values, which are presented in Table 3. The r-values in Table 3 are close to 1 (0.9–0.99) and the probabilities of the measured and model values of hi being linearly uncorrelated are quite low
10 8 6 experiment
4
Values for r range from 1 to +1, with no correlation at 0 and complete correlation at ±1. However, the degree of correlation is not directly indicated by r. A common test of the r-value is to compare it to the probability distribution of the parent population that is completely uncorrelated (i.e. r = 0). To that end, P(r, m), as defined in Eq. (7), is the probability that a random sample of uncorrelated data points would result in a correlation coefficient equal to r,
model
2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
Equivalent fraction Nd in salt
Fig. 6. Comparison of model curve for Nd loading in zeolite to experimental data from the Nd ternary system. (hT is computed from salt composition data. Error bars reflect 2-sigma analytical uncertainty in Nd and Al.) To view this figure in color, see the web version of this paper.
Table 3 Model fitness parameters for ternary and quaternary salt–zeolite contact experiments Experiment
r
r2
P(r, m)
Cs ternary Nd ternary Cs–Nd Quaternary
0.99 0.90 r(Cs) = 0.97 r(Nd) = 0.93
0.98 0.81 0.94 0.86
3.1E13 0.00012 P(Cs) = 8.8E9 P(Nd) = 1.7E7
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(P 6 0.00012). These results clearly indicate a good model fit. Additional verification of this fit is provided with interpretation of the r2-values also presented in Table 3. The sample coefficient of determination, r2, expresses the proportion of the total variation in the measured values that can be accounted for by a linear relationship with the values computed using a model equation [13]. So, a correlation (r) of 0.99 for the Cs ternary system data means that 98% (r2) of the total variation in measured hI-values is accounted for by a linear relationship with the model values, and so on for the other experiments.
4. Conclusions A model describing fission product ion exchange between molten salt and zeolite has been developed and tested. Preliminary verification of the model and its assumptions was successfully accomplished through: (1) salt–zeolite batch contact experiments with nonradioactive isotopes representing fission product species Cs and Nd, and (2) thorough statistical analysis of the model fit to data. With the conclusion of this preliminary study, a research path forward is easily seen. Nominally twelve additional salt bath species must be investigated and model parameters determined for each. Further model modification will also be necessary to allow prediction of equilibrium salt and zeolite phase compositions based on initial salt compositions.
Acknowledgements The authors would like to acknowledge Dr. Dusan Lexas experimental work in this area and the use of his data in early modeling efforts. Support of this research was by the US Department of Energy (Contract W-31-109-ENG-38). The work was performed under the EBR-II Spent Fuel Treatment Program, which is part of the Advanced Fuel Cycle Initiative of the Department of Energys Office of Nuclear Science and Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M. Simpson, M.L. Gougar, Ind. Eng. Chem. Res. 42 (2003) 4208. M. Liquornik, Y. Marcus, Israel J. Chem. 6 (1968) 115. M. Liquornik, Y. Marcus, J. Phys. Chem. 72 (1968) 2885. M. Liquornik, Y. Marcus, J. Phys. Chem. 72 (1968) 4704. W. Platek, J. Marinsky, J. Am. Chem. Soc. 65 (1961) 2113. M. Susic, N.A. Petranovic, D.A. Mioc, J. Inorg. Nucl. Chem. 33 (1971) 2667. C. Callahan, M.A. Kay, J. Inorg. Nucl. Chem. 28 (1966) 233. C. Callahan, J. Inorg. Nucl. Chem. 28 (1966) 2743. M. Liquornik, J. Inorg. Nucl. Chem. 33 (1971) 283. M. Liquornik, Y. Marcus, J. Phys. Chem. 76 (1971) 2523. D. Lexa, I. Johnson, Metall. Mater. Trans. B 32B (2001) 1. D. Lexa, Metall. Mater. Trans. B: Process Metall. Mater. Process. Sci. 34B (2003) 201. R.D. Shannon, Acta Cystallogr. A 32 (1976) 751. R.E Walpole, R.H. Myers, S.L. Myers (Eds.), Probab. and Stat. for Eng. and Sci., sixth ed., Prentice Hall, NJ, 1998. T.P. Ryan, Nonlinear regression, Mod. Regression Methods, John Wiley and Sons, Inc., NY, 1997, p. 416. P.R. Bevington, D.K. Robinson, Data Reduct. and Error Anal. for the Phys. Sci., vol. 50, WCB/McGraw-Hill, Burr Ridge, IL, 1992, p. 196.
Two-site equilibrium model for ion exchange between multi-valent cations and zeolite-A in a molten salt Mary Lou Dunzik-Gougar a
a,*
, Michael F. Simpson b, Barry E. Scheetz
c
Idaho State University/Idaho National Laboratory, 1776 Science Center Dr., Idaho Falls, ID 83402, USA b Idaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415, USA c The Pennsylvania State University, Materials Research Laboratory, University Park, PA 16802, USA Received 3 November 2004; received in revised form 26 March 2005; accepted 28 April 2005 Available online 5 August 2005
Abstract A model has been developed and tested for its ability to describe the ion exchange of fission product species between zeolite and a molten salt bath used for pyroprocessing of spent nuclear fuel. The model is based on a system at equilibrium and assumes species interacting with zeolite are chemically independent of each other. This assumption was tested via a set of salt–zeolite contact experiments. Zeolite 4A was batch contacted with Cs ternary salt (CsCl–LiCl–KCl), Nd ternary salt (NdCl3–LiCl–KCl) or Cs–Nd quaternary salt (CsCl–NdCl3–LiCl–KCl) until ion exchange steady state was reached. Results from ternary salt and zeolite chemical analyses were used to determine model parameters for all species present. These parameters, in turn, were applied to compute estimated compositions of zeolite contacted with quaternary salt. The estimated compositions compare favorably with results from chemical analyses of the zeolite–quaternary salt experiment samples. Based on the data collected, the model is modified for predictive capability for a range of salt compositions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Fission products; Ion exchange; Zeolite; Model; Molten salt
1. Introduction The electrometallurgical treatment of spent nuclear fuel (SNF) has been developed at Argonne National Laboratory (ANL) and has been demonstrated through processing the sodium-bonded SNF from the Experimental Breeder Reactor-II in Idaho. In this process, components of the SNF, including U and species more chemically active than U, are oxidized into a bath of lithium–potassium chloride (LiCl–KCl) eutectic molten salt. Uranium is subsequently removed from solution
*
Corresponding author. Tel.: +1 208 282 7809; fax: +1 208 282 7735. E-mail address: [email protected] (M.L. Dunzik-Gougar).
1387-1811/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.micromeso.2005.04.014
by electrochemical reduction. The noble metal fission products and inactive fuel/clad components are not oxidized into the salt solution but are removed in solid form and subsequently melted into a metal waste form. The residual salt solution contains most of the fission products and transuranic elements from the SNF. One technique that has been identified for removing these fission products and extending the usable life of the molten salt is ion exchange with zeolite-A. Zeolite batch contact experiments have been performed using radioactive salt from the treatment of EBR-II fuel and non-radioactive salt of various compositions. From these experiments, salt and zeolite composition data have been collected to support development of a multi-component equilibrium model for the exchange of fission products between molten chloride salt
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372
and zeolite-A. Such a model may be used to support the development of a salt recycling process for the future electrometallurgical treatment of various SNF types.
2. Model development In a previous paper [1], early research on the application of ion exchange with zeolite-A to removal of fission products from LiCl–KCl was reviewed. Based on this work, Simpson and Gougar [1] proposed a model for a salt–zeolite system containing purely monovalent ion components and found that it exhibited an excellent fit to the data. The monovalent model and the new, multi-valent model will be compared in this section. 2.1. The multi-valent model The goal in modeling the equilibrium behavior of this system is to develop an ability to predict the distribution of multi-valent cations between the zeolite and the salt given a specific salt/zeolite ratio and starting compositions. To this end, the behavior of each different type of cation must be examined to learn how it interacts with the zeolite and its relative affinity for association with the zeolite. Ion exchange between a cation species, A, in the salt, and cation species B, in the zeolite, is represented in Eq. (1), A A B B nB Aþn þ nA Bþn $ nB Aþn þ nA Bþn : s z z s
ð1Þ
Subscripts s and z denote salt and zeolite phases, respectively. The ionic charge of species i is ni. In the multi-valent model, forward and reverse ion exchange reaction rates for this system (Eq. (1)) are formulated as follows: n
rf;A ¼ k f;A C A;s ð1 C A;z Þ A ; A rr;A ¼ k r;A C nA;z ð1 C A;s Þ.
ð2Þ ð3Þ
In the forward (2) and reverse (3) reaction rate equations, CA,s represents the mole equivalent fraction of A with charge nA in the salt phase, and CA,z is the mole equivalent fraction of zeolite sites occupied by A. The rate at which A is taken in to the zeolite structure is proportional to its concentration in the salt and to the fraction of sites in zeolite occupied by all potential reaction partners. Clearly, all potential reaction partners, or counter cations, will not have the same affinity for ion exchange with species A; however, a very effective simplifying assumption is made when considering the counter ions as a whole. In turn, the rate of species A leaving the zeolite is proportional to its concentration in the zeolite and the fraction of all other (potential replacements for A) species in the salt phase. It is noted here that, because the experiments were run under equilibrium conditions, the authors did not consider whether ion
367
exchange kinetics or diffusion of the salt species would be more rate limiting. The model equations (2) and (3) were formulated based on preliminary experimental data and, as described in this paper, were verified via additional experimentation. The value of the model is further increased through consideration of the physical system being described. While the model was not derived from first principles, a perspective on the relationship between equation and ion may be gained by drawing an analogy to probability. Consider salt cation species x, y and z with charges +1, +2, and +3, respectively. As the salt phase species enter the zeolite, they will either occupy a site(s) in the zeolite or continue flowing through the pore system to exit the zeolite macrostructure. Each site, in this analogy, is considered an equivalent site or an area in the zeolite carrying a negative 1 charge. If it is assumed that no species is preferred over any other, the probability of one equivalent of species x (+1) occupying a specific site in the zeolite is the same as that of one equivalent of y (+2) occupying that same site, although y will also satisfy an additional site due to its charge. Likewise, the probability of a site being occupied by one equivalent of y is the same as that of one z (+3) equivalent. Because Ci,s and Ci,z in the model equations are fractions, the term Ci,z is an expression of the probability that species i is in a particular site in the zeolite. If species i has a +2 charge, we must consider the probability that the ion occupies two specific sites. In a system with effectively infinite zeolite sites, each of the two sites individually has the same probability of being occupied by the cation. Then the combined probability for a +2 ion to occupy two sites is proportional to the product of the individual probabilities.
probability of species i1+ occupying one equivalent site in zeolite / Ci, probability of species i2+ occupying one equivalent site in zeolite / Ci Æ Ci, probability of species i3+ occupying one equivalent site in zeolite / Ci Æ Ci Æ Ci. Conversely, the term (1 Ci,z) is proportional to the probability that a given site in the zeolite is associated with some species other than i. probability of species i1+ not occupying one equivalent site in zeolite / (1 Ci), probability of species i2+ not occupying one equivalent site in zeolite / (1 Ci) Æ (1 Ci), probability of species i3+ not occupying one equivalent site in zeolite / (1 Ci) Æ (1 Ci) Æ (1 Ci).
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To create an applicable form of the multi-valent model, a state of ion exchange equilibrium is assumed for the salt–zeolite system. In a system at equilibrium, Eqs. (2) and (3) are equal and a resulting generic expression for the amount of species i in zeolite (Ci,z) is C i;z ¼
i K i C 1=n i;s
1=n
K i C i;s i þ ð1 C i;s Þ
1=ni
where
;
It has been well established that salt species are incorporated in the zeolite 4A structure in two ways, or in two types of sites [2–10]. For one, cations from the salt may exchange with sodium cations in the zeolite framework structure. These sodium ions are present initially to balance a negative charge in the framework. Secondly, salt cations may associate with chloride anions according to their charge. In this case the species are immobilized within zeolites interconnecting pore system as whole chloride molecules. Using Eq. (4) as a basis, an expression was developed to represent the two-site nature of zeolite-A, 1=ni
S 1 K IX i C i;s 1=n
i K IX i C i;s þ ð1 C i;s Þ
1=ni
1=ni
þ
S 2 K occ i Ci 1=ni K occ i C i;s
þ ð1 C i;s Þ
1=ni
:
2.2. The monovalent model In their earlier monovalent model, Simpson and Gougar [1] represent the forward and reverse reaction rates as
k f;i Ki ¼ . k r;i ð4Þ
hi ¼
where hT is the total mole equivalents immobilized by both ion exchange and occlusion per unit cell zeolite.
ð5Þ
B A rf;A ¼ k f;A C nA;s C nB;z ;
ð7Þ
B A k r;A C nA;z C nB;s ;
ð8Þ
rr;A ¼
where all terms are as previously defined. Because the species are monovalent (nA = nB = 1), the equilibrium zeolite concentration of a particular species may be solved for explicitly in terms of measurable salt phase concentrations. C A;z ¼
K A S 1 C A;s . K A C A;s þ K B C B;s
ð9Þ
If there are more than two monovalent species competing for zeolite sites, the monovalent model equation (Eq. (10)) resembles the Langmuir model equation, with the exception that no ion exchange sites may remain unfilled. The Langmuir model for sorption does accommodate empty sites. k i;1 S 1 C i;s C i;z;1 ¼ P . k j;1 C j;s
ð10Þ
j
Here hi is the mole equivalents of a species i per pseudounit cell zeolite. The first of the two quotient terms in Eq. (5) represents the ion-exchanged mole equivalents of i, which contribute to the total S1 mole equivalents necessary to balance the charge of one zeolite-A pseudo-unit cell. A pseudo-unit cell of zeolite-A, henceforth referred to simply as ‘‘unit cell’’, is defined as Mþ 12 ðAlO2 Þ12 ðSiO2 Þ12
Therefore, S1 = 12. The aluminosilicate framework of zeolite-A carries a negative charge due to trivalent aluminum atoms taking the places of tetrahedrally coordinated silicon atoms. Various alkali and alkaline earth cation species, M, commonly fill zeolite-A framework sites to balance the charge. In zeolite 4A, these sites are occupied by sodium ions. In this application for fission product waste ion removal, various monovalent, divalent and trivalent cations may fill these framework sites. The occluded mole equivalents of species i are represented in the second quotient term of Eq. (5). While it is known that there are 12 ion exchange mole equivalent sites in the 4A unit cell, the number of mole equivalents occluded in the cage structure is a variable quantity depending on the composition of the salt. Therefore, the number of occluded mole equivalents, S2, will be expressed as S 2 ¼ hT 12;
ð6Þ
The key differences between the multi-valent and monovalent models stem from different starting assumptions about the reacting cation species. The multi-valent model assumes that the extent to which a species, A, enters or leaves the zeolite is chemically independent of the nature of all potential reaction partners. So, a species, A, will enter zeolite according to its own concentration in the salt and to the number of sites not occupied by A in the zeolite. Likewise, A leaves the zeolite structure (entering the salt solution) according to its own concentration in the zeolite and the concentration of all non-A cation species (potential reaction partners) in the salt. By grouping all non-A species together, the multi-valent model drastically simplifies the mathematics of testing and verification for multi-component, multi-valent ion systems. In contrast, the monovalent model describes the forward reaction rate for A entering zeolite to be dependent on its own concentration in the salt and the number of sites occupied by some defined reaction partner, B. A reference reaction partner must be arbitrarily chosen for each type of site in this model such that all other species are described with respect to that one. In other words, the assumption in the monovalent model is that the rate of A reacting with zeolite is independent of all species except the arbitrary references. This model is also being tested for its applicability to multi-valent systems.
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3. Experimental methods and results
369
Table 1 Model parameters optimized with ternary experimental data Species
KI
K0
Cs Na Li K Nd
0.10 3.2 3.0 0.076 0.61
4.0 0 0.75 0.91 1.5
6 5 4 3 2 model 1
experiment
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Equivalent fraction Cs in salt
Fig. 1. Comparison of measured and model values for Cs loading in zeolite from Cs ternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Cs and Al.) To view this figure in color, see the web version of this paper. 14
Equivalents Nd per unit cell zeolite
To support the validity of this multi-valent model as well as to determine parameters for a number of key species, a set of salt–zeolite batch contact experiments was designed and executed. A monovalent (CsCl) or trivalent (NdCl3) salt was mixed with the LiCl–KCl solvent salt and heated to 500 °C. Non-radioactive Cs and Nd chloride salts, radioactive ‘‘surrogates’’, were used to chemically represent the fission product isotopes of those elements. Zeolite 4A (Na12(AlO2)12(SiO2)12) pellets, consisting of 10–20 wt% clay binder, were contacted with the molten salt mixtures. When ion exchange equilibrium in the contact system was reached, salt and zeolite samples were collected for chemical composition analysis. From salt composition data the mole equivalent fraction of each cation species in the mixture was determined. Zeolite composition data were converted to species mole equivalents per unit cell zeolite. Conversion was accomplished by normalization of the measured species concentrations with respect to aluminum, which is present at a uniform concentration of 12 mol equivalents per unit cell. Data from each salt–zeolite contact experiment were combined in a single data set and used to fit the model. Values for model parameters KIX and Kocc (Eq. (5)) were obtained for each species using a constrained optimization routine that minimizes the Euclidean norm of the error vector. Table 1 provides values of the model parameters obtained for each species in the ternary (three-component) contact systems. Figs. 1 and 2 show results of contact experiments in ternary (CsCl–LiCl–KCl and NdCl3–LiCl–KCl) salt systems. Concentrations of surrogate fission product species (Cs and Nd) in zeolite are plotted as a function of the equilibrium salt concentrations of that same species. Typical exchange behavior is demonstrated in these salt– zeolite systems. At lower concentrations (in this case, less than about 0.1 mol equivalent fraction of species in the salt), the Cs and Nd zeolite loading increases significantly with the increased salt concentrations compared to the changes in loading seen above 0.1 salt mole equivalent fraction. As the species concentration in the salt increases beyond 0.1, the zeolite concentration of that species shows small increases. Indeed, if considering the measurement uncertainty, it could be stated that there is no significant change in the concentration
Equivalents Cs per unit cell zeolite
7
12 10 8 6 experiment 4
model
2 0 0
0.1
0.2
0.3
0.4
0.5
Equivalent fraction Nd in salt
Fig. 2. Comparison of measured and model values for Nd loading in zeolite from Nd ternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Nd and Al.) To view this figure in color, see the web version of this paper.
of Cs or Nd in zeolite past 0.2 salt equivalent fraction. This may indicate the capacity of zeolite for these species. The Cs loading reached almost 6 equivalents, or 6 Cs ions, per unit cell, while the Nd ion loading was almost 12 equivalents (4 ions) per unit cell zeolite. Also plotted in Figs. 1 and 2 are model values for zeolite concentration calculated using the empirical parameters for each species (Table 1) and empirical values for the total number of mole equivalents immobilized per unit cell zeolite, hT. Because hT is a measured quantity dependent on salt composition, the value varies with each experiment and therefore for each point plotted. This phenomenon leads to a non-smooth model curve if the calculated points are connected. To test the assumption of species chemical independence upon which the multi-valent model was based, the model parameters determined with ternary salt system data were used to estimate final zeolite concentrations for a quaternary salt system
370
M.L. Dunzik-Gougar et al. / Microporous and Mesoporous Materials 84 (2005) 366–372 30.0
Equivalents per unit cell zeolite
Equivalents per unit cell zeolite
12 10 8 6 4
Nd expt.
Nd model
2
Cs expt.
Cs model
25.0 20.0 15.0 Cs, Rb ternary Nd, U, Sr ternary
10.0
Linear (Nd, U, Sr ternary) 5.0
Linear (Cs, Rb ternary)
0.0
0 0
0.1
0.2
0.3
0.4
0
0.5
(CsCl–NdCl3–LiCl–KCl) contact experiment. Experimental results, of the same type presented for the ternary system contacts, are plotted in Fig. 3 along with the model estimates. The most significant result illustrated in the (Figs. 1–3) is the reasonably good fit between values measured in both ternary and quaternary salt systems and values calculated with the same model parameter constants. This qualitative assessment of fit is supported by a quantitative evaluation of the model fit to data as described next. 3.1. Model verification and modification The model in the form of Eq. (5) allowed preliminary verification of fit to experimental data. It is limited, however, by the fact that hT is a measured value. In other words, it is not possible, to calculate the composition of a zeolite sample without first having analyzed the sample and measured the total mole equivalents of species immobilized in the zeolite structure. Clearly, it would be more useful if zeolite composition could be estimated a priori, with the model equation using equilibrium salt concentration measurements. To this end, an empirical relationship was sought between the salt concentrations of species at equilibrium (Ci) and the total mole equivalents of all species in the zeolite after contacting (hT). Fig. 4 reveals such a relationship. In Fig. 4, data from ternary salt +1 species (Cs and Rb) experiments were plotted with identical symbols. The same was done for ternary multi-valent (Nd3+, U3+ and Sr2+) species data. Rubidium, uranium and strontium data came from a paper on related work by Lexa [11,12]. The data indicate a linear relationship between hT and the total solvent equivalents. The positive slope of the multi-valent species line (y = 3.1x + 24.3) appears
0.2
0.3
0.4
0.5
0.6
Total non-solvent equivalents in the salt
Equivalent fraction of species in salt
Fig. 3. Comparison of measured and model values for Cs and Nd loading in zeolite from Cs–Nd quaternary contact experiments. (The measured experimental value of hT was used for each model calculation. Error bars reflect 2-sigma analytical uncertainty in Cs, Nd and Al.) To view this figure in color, see the web version of this paper.
0.1
Fig. 4. Correlation between total zeolite loading (hT) and total surrogate fission product/TRU-species salt concentrations. (Data for Rb, U, Sr and some for Cs are from references [11,12].) To view this figure in color, see the web version of this paper.
to indicate that there is a gradual increase in the total mole equivalents occluded in zeolite as the concentration of fission products and actinides in the salt increases. The negative slope of the +1 species line (y = 8.4x + 22.3) in Fig. 4 is more pronounced and also consistent with a space-filling model for loading the zeolite unit cell. The solvent cations (Li1+ and K1+) share the same charge as fission product species Cs and Rb, but differ significantly in size (see Table 2). It is clear that more Li and K ions would fit in a given unit cell than Cs and Rb ions due to the larger radii of the latter two. As the concentration of Cs or Rb increases in the salt solution, the number of ions of these species per unit cell zeolite also increases. However, while the number of Cs and Rb mole equivalents per unit cell increases, the total number of mole equivalents in the unit cell decreases. This may be explained by the larger ions taking up more of the available space than the solvent ions while carrying the same +1 charge. The linear equations associated with Fig. 4 allow hT to be estimated from salt measurement, thereby removing any dependence on zeolite analysis (see Eq. (11)). Note the reference to single valent species concentrations, sv, and multi-valent, mv, species concentrations. Table 2 Effective ionic radii for ionic species in the experimental salt–zeolite contact system [13] Cation
Effective ionic radius (nm)
Li1+ Nd3+ Ca2+ Na1+ U3+ Sr2+ K1+ Rb1+ Cs1+
0.076 0.0983 0.100 0.102 0.103 0.118 0.138 0.152 0.167
hT ¼ bððC sv =ðC sv þ C mv ÞÞð8.37C sv þ 22.32ÞÞ þ ððC mv =ðC sv þ C mv ÞÞð3.10C mv þ 24.29ÞÞc.
ð11Þ
To create a model curve, values for C were chosen at regular intervals between 0 and 1. The resulting h-values form the basis for a curve to which the experimental values may be compared (plotted in Figs. 5–7). 3.2. Evaluation of model fit To quantify the quality of the fit, the extent of correlation between variations in measured hi-values and variations in the model-generated values of hi was determined. The Pearson product–moment correlation coefficient (Eq. (6)) is applied to estimate the extent of linear
Equivalents Cs per unit cell zeolite
7
371
12 10 8 Nd expt. 6
Cs expt. Nd model
4
Cs model 2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Equivalent fraction Cs or Nd in salt
Fig. 7. Comparison of model prediction for Cs and Nd loading in zeolite to experimental data from the Cs–Nd quaternary system. (hT computed from salt concentrations. Error bars reflect 2-sigma analytical uncertainty in Cs, Nd and Al.) To view this figure in color, see the web version of this paper.
6
association between measured (hi) and modeled (^ hi ) zeolite concentrations [14,15].
5
P n ^ ^ ðhi hi Þ ðhi hi Þ r ¼ P i¼1 : n ^ ^ 2 2 i¼1 ðhi hi Þ ðhi hi Þ
4 3
ð12Þ
model
2
experiment
1 0 0
0.1
0.2
0.3
0.4
0.5
Equivalent fraction Cs in salt
Fig. 5. Comparison of model prediction for Cs loading in zeolite to experimental data from the Cs ternary system. (hT is computed from equilibrium salt composition data. Error bars reflect 2-sigma analytical uncertainty in Cs and Al.) To view this figure in color, see the web version of this paper.
14 Equivalents Nd per unit cell
Equivalents Cs or Nd per unit cell zeolite
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12
1 Cbðm þ 1Þ=2c ðm2=2Þ P ðr; mÞ ¼ pffiffiffi ð1 r2 Þ ; Cðm=2Þ p where m ¼ n 2 [16].
ð13Þ
The fit of the experimental model was quantified with calculations of r- and P-values, which are presented in Table 3. The r-values in Table 3 are close to 1 (0.9–0.99) and the probabilities of the measured and model values of hi being linearly uncorrelated are quite low
10 8 6 experiment
4
Values for r range from 1 to +1, with no correlation at 0 and complete correlation at ±1. However, the degree of correlation is not directly indicated by r. A common test of the r-value is to compare it to the probability distribution of the parent population that is completely uncorrelated (i.e. r = 0). To that end, P(r, m), as defined in Eq. (7), is the probability that a random sample of uncorrelated data points would result in a correlation coefficient equal to r,
model
2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
Equivalent fraction Nd in salt
Fig. 6. Comparison of model curve for Nd loading in zeolite to experimental data from the Nd ternary system. (hT is computed from salt composition data. Error bars reflect 2-sigma analytical uncertainty in Nd and Al.) To view this figure in color, see the web version of this paper.
Table 3 Model fitness parameters for ternary and quaternary salt–zeolite contact experiments Experiment
r
r2
P(r, m)
Cs ternary Nd ternary Cs–Nd Quaternary
0.99 0.90 r(Cs) = 0.97 r(Nd) = 0.93
0.98 0.81 0.94 0.86
3.1E13 0.00012 P(Cs) = 8.8E9 P(Nd) = 1.7E7
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(P 6 0.00012). These results clearly indicate a good model fit. Additional verification of this fit is provided with interpretation of the r2-values also presented in Table 3. The sample coefficient of determination, r2, expresses the proportion of the total variation in the measured values that can be accounted for by a linear relationship with the values computed using a model equation [13]. So, a correlation (r) of 0.99 for the Cs ternary system data means that 98% (r2) of the total variation in measured hI-values is accounted for by a linear relationship with the model values, and so on for the other experiments.
4. Conclusions A model describing fission product ion exchange between molten salt and zeolite has been developed and tested. Preliminary verification of the model and its assumptions was successfully accomplished through: (1) salt–zeolite batch contact experiments with nonradioactive isotopes representing fission product species Cs and Nd, and (2) thorough statistical analysis of the model fit to data. With the conclusion of this preliminary study, a research path forward is easily seen. Nominally twelve additional salt bath species must be investigated and model parameters determined for each. Further model modification will also be necessary to allow prediction of equilibrium salt and zeolite phase compositions based on initial salt compositions.
Acknowledgements The authors would like to acknowledge Dr. Dusan Lexas experimental work in this area and the use of his data in early modeling efforts. Support of this research was by the US Department of Energy (Contract W-31-109-ENG-38). The work was performed under the EBR-II Spent Fuel Treatment Program, which is part of the Advanced Fuel Cycle Initiative of the Department of Energys Office of Nuclear Science and Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M. Simpson, M.L. Gougar, Ind. Eng. Chem. Res. 42 (2003) 4208. M. Liquornik, Y. Marcus, Israel J. Chem. 6 (1968) 115. M. Liquornik, Y. Marcus, J. Phys. Chem. 72 (1968) 2885. M. Liquornik, Y. Marcus, J. Phys. Chem. 72 (1968) 4704. W. Platek, J. Marinsky, J. Am. Chem. Soc. 65 (1961) 2113. M. Susic, N.A. Petranovic, D.A. Mioc, J. Inorg. Nucl. Chem. 33 (1971) 2667. C. Callahan, M.A. Kay, J. Inorg. Nucl. Chem. 28 (1966) 233. C. Callahan, J. Inorg. Nucl. Chem. 28 (1966) 2743. M. Liquornik, J. Inorg. Nucl. Chem. 33 (1971) 283. M. Liquornik, Y. Marcus, J. Phys. Chem. 76 (1971) 2523. D. Lexa, I. Johnson, Metall. Mater. Trans. B 32B (2001) 1. D. Lexa, Metall. Mater. Trans. B: Process Metall. Mater. Process. Sci. 34B (2003) 201. R.D. Shannon, Acta Cystallogr. A 32 (1976) 751. R.E Walpole, R.H. Myers, S.L. Myers (Eds.), Probab. and Stat. for Eng. and Sci., sixth ed., Prentice Hall, NJ, 1998. T.P. Ryan, Nonlinear regression, Mod. Regression Methods, John Wiley and Sons, Inc., NY, 1997, p. 416. P.R. Bevington, D.K. Robinson, Data Reduct. and Error Anal. for the Phys. Sci., vol. 50, WCB/McGraw-Hill, Burr Ridge, IL, 1992, p. 196.