A systemic approach for the characterization of ion exchange in the altered layer of a non-radioactive nuclear glass

Chemical Engineering Science 57 (2002) 3427 – 3438

www.elsevier.com/locate/ces

A systemic approach for the characterization of ion exchange in the altered layer of a non-radioactive nuclear glass Vincent Bleta; ∗ , David Rudlo-a , Philippe Bernea , Patrick Jollivetb , Daniel Schweichc a CEA=DAMRI=SAR,

17 rue des Martyrs, F38054 Grenoble Cedex 9, France BP 171, F30207 Bagnols sur C'eze, France c CNRS=LGPC, CPE, 43 Bd du 11 Novembre, BP 2077, F69616 Villeurbanne Cedex, France b CEA=DRRV=SCD,

Received 5 April 2001; received in revised form 3 October 2001; accepted 3 October 2001

Abstract In order to correctly design the storage of nuclear wastes in nuclear glasses it is necessary to characterize the mechanisms of migration of silicon especially in the gel layer which forms at the glass=environment interface. Among the di-erent available techniques, the chromatography theory coupled with the systemic approach developed by Villermaux and coworkers is particularly indicated when dealing with trace level ion-exchange processes. This paper illustrates the advantages and limitations of this methodology through the study of 23 Na= 22 Na isotopic exchange and 23 Na=H ion-exchange processes. Despite the fact that the coe:cient of ionic di-usion is determined with a worse accuracy than with the classical “through di-usion” technique, the interest of such a methodology lies not only in the richness of the collected information on the underlying processes occurring in the migration of the studied species, but also in the easiness and rapidity of related experiments. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Ion exchange; Di-usion; Systemic approach; Nuclear glass; Gels; Parameter identi?cation

1. Introduction Although the containment of highly activated nuclear wastes in nuclear glasses has been proved for more than 30 years, these glasses and especially the French SON68 glass are still extensively studied due to the complexity of the mechanisms involved in the migration of radioactive ions like actinides, rare earths, etc. (Trocellier, 1997). More particularly, it has been shown that the altered layer (gel) which forms at the pristine glass=environment interface exhibits two fundamental characteristics related to the dissolution of the glass (Vernaz & Dussossoy, 1992), one being a di-usional barrier for silicon (Delage, 1992), the other consisting of the retention of some radionuclides present in the glass (Fillet, 1987). In order to correctly design nuclear waste storage, it is then essential to precisely characterize both the mechanisms of di-usion of silicon—the alteration

∗ Corresponding author. Tel.: +33-(0)438-78-31-60; fax: +33-(0)438-78-50-45. E-mail address: [email protected] (V. Blet).

rate is based on silicon concentrations coupled with di-usion (Jollivet, Nicolas, & Vernaz, 1998)—and the properties of retention (reversibility, durability) of the SON68 glass. Among the di-erent available techniques (Ricol, 1995; Gens et al., 1996), the one based on the chromatography theory should be particularly interesting since the related concepts have been developed for long and successfully applied to the characterization of gel type ion exchangers (Dodds & Tondeur, 1974; Nicolas-Simonnot, Sardin, & GrIevillot, 1995; Natarajan & Cramer, 1999) to which the SON68 gel should conform (Ricol, 1995). The systemic approach and more especially the Mixing Cells in series with Exchange (MCE) model developed by Villermaux (1981) leads to a quantitative description of the phenomenology of the interactions through macroscopic relationships. Although global, this modular approach enables distinguishing between elementary mechanisms without the complexity of a microscopic molecular modeling. The aim of this paper is to highlight the potentialities of such a method through the particular examples of the sodium=hydrogen (23 Na+ =H+ ) and isotopic (22 Na= 23 Na) exchanges in the sodium (23 Na)-saturated SON68 gel. As a key feature, this

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 0 8 - 7

3428

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

chromatographic method coupled with the MCE modeling not only enables the determination of the physical structure of the gel but also reveals mass-transfer limitations which can be associated with particular mechanisms of di-usion and reaction in the liquid or solid phase depending on the external environment of the gel. Clearly, these limitations can be strong enough to make an equilibrium approach wrong. 2. The MCE modeling of ion exchange We will describe the transient percolation of interacting solutes through a column packed with gel grains (see Fig. 1). Let us ?rst remark that isotopic or ion exchanges at trace level are linear adsorption processes. Linear system dynamics is thus the best tool to describe such experiments. In this approach, the response of the column to an inlet perturbation of composition is characterized by a transfer function, G(s), which is the ratio of the Laplace transforms of the inlet, CM in , and outlet, CM out , concentrations of the species under study. For one-dimensional Now through systems, the MCE model has been developed for a broad range of con?gurations of sorbing aggregates and accounting for various equilibrium and kinetic processes (Schweich, 1993). The ?nal result is CM out (s) G(s) = = G0 (s[1 + M (s)]); (1) CM in (s) where s is the Laplace variable. Function M (s) accounts for local interactions and is de?ned by the following equation: 1 − ext CM p  M (s) = ; (2) ext CM ext where ext denotes the external porosity of the bed, and CM ext and CM p , respectively, the Laplace transforms of the concentration in the external Nuid (mobile phase) and of the space-averaged concentration in the gel grains (stationary phase—see Fig. 2). For a given Cin (t) (postulated or measured) in the time domain, Eq. (1) allows calculating the corresponding column response, Cout (t) using Fast Fourier transform algorithm (Leclerc, Detrez, Bernard, & Schweich, 1995). Furthermore, the mathematical expression of G(s) allows one to relate the time moments of the response to the parameters of the underlying processes. The inverse Laplace transform of G is the residence time distribution (RTD). It is worth noting that according to this basic equation, the global transfer function G can be obtained only by the independent determinations of the transfer function of the external Nuid G0 , and the local interaction model M. The rigorous derivation of M (s) leads to more or less complex mathematical expressions which hide the physical meaning and implications of the underlying processes. Fortunately, in most situations, these expressions are well approximated by a ?rst-order dynamical system (Villermaux,

Fig. 1. Schematic diagram of the experimental apparatus for transient ion exchange of 22 Na and H onto Na-saturated gel grains.

1981; Schweich, 1993): M (s) =

K ; 1 + tm s

(3)

where K depends on thermodynamics only, and tm is an overall time constant which can be related to individual physical processes. In the MCE model, the axial dispersive Now is accounted for by mixing cells in series. G0 is thus determined by the mean residence time of the Nuid, T , and the number of mixing cells, J . Let us recall that J is equivalent to a PIeclet number based on column length (Villermaux, 1981) as far as J is su:ciently high (i.e. ¿ 20 (Schweich, 1993)). K is called the capacity factor: K = (int + g KA )

1 − ext 1 − ext = ; ext ext

(4)

where KA denotes the adsorption equilibrium constant, g is the apparent (bulk) density of the gel particle and int denotes the internal porosity of the bed. In our study, it is important to notice that the internal porosity is related to the total volume of the grain and not to the volume of the gel layer (see Fig. 2). The characteristic mass-transfer time, tm , accounts for several di-erent mass-transfer limitations which have been pointed out since long in most of the ion-exchange studies (Boyd, Adamson, & Myers,1947). First, the adsorbate ion has to di-use through the boundary layer around the grain (external di-usion). Then, the adsorbate di-uses through the pores of the gel layer from its surface to a sorption site (internal di-usion). Finally, it must sorb to the site

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

3429

Fig. 2. Schematic representation of gel grain.

(intrinsic adsorption process). These three processes in series are characterized by their respective time constants as de?ned by Nicoud and Schweich (1989): External di-usion : text =

re re  R; R= 3k 3Dext

Internal di-usion :   s   w = ; e = r e − ri ;   D  int       3re [we ch(we) + (w2 ri re − 1) sh(we)] H∗ = ;  (re3 − ri3 )w2 [sh(we) + wri ch(we)]       ∗   @H 15ri2 re e3 + 6ri e5 + e6    tint = − = ; @s s=0 15(re3 − ri3 )re Dint Intrinsic adsorption : ta =

 − int ; kdes

(5)

?lm. R is a factor which takes into account the electrical ?eld e-ects on ionic migration through the Nernst–Planck formalism and then depends on the gel composition (Nicoud & Schweich, 1989). The thickness  can be estimated by correlations of the form (Wakao & Funazkri, 1978) dp = a + b Ren Scm  n  m  u 0 dp % =a+b ; % Dext

Sh =

(6)

(7)

where re ; ri and e are, respectively, the external radius, the internal radius and the thickness of the gel layer coating the glass grain assumed to be spherical (see Fig. 2). k is the ?lm mass-transfer coe:cient,  the thickness of the boundary layer and Dext the coe:cient of ionic di-usion through this

(8)

where Sh is the Sherwood number, Sc the Schmidt number and Re the Reynolds number. % is the kinematic viscosity of the 23 NaCl solution (nearly identical to the viscosity of water), u0 the super?cial velocity of the solution and dp the equivalent hydraulic diameter of the gel grains (2re ). Data for very low Reynolds numbers are quite scarce. For liquid phase systems at such Reynolds numbers, the recommended correlation is a = 0; b = 1:09=ext and n = m = 1=3 (Wilson & Geankoplis, 1966). Strictly speaking, it is only valid for Reynolds numbers larger than 1:6 × 10−3 . Unfortunately, in our experiments this number can be as low as 10−4 . In that case, the brutal application of the above Wilson=Geankoplis correlation predicts a value of about 1 for the Sherwood number. However, for such very low Reynolds number we

3430

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

shall assume Sh ≈ 2. This value is indeed both the theoretical limit in the case of pure di-usion and the prediction from the Wilson=Geankoplis correlation at the limit of its validity domain. Dint is an apparent e-ective coe:cient of di-usion which depends a priori on the composition. An attractive approach for the determination of Dint has been developed by Nicoud (Nicoud and Schweich, 1989). It takes into account the thermodynamical aspects of the ion exchange using the electrical double-layer theory in a porous medium. Nevertheless, such an approach is rather complex and simpli?ed relationships (Hel-erich, 1962; Dodds & Tondeur, 1974) based on the Nernst–Planck model can be advantageously used in order to model the internal di-usion by a constant e-ective coe:cient of di-usion. It is worth noting that this coe:cient can di-er greatly from the in?nite dilution or “free solution” di-usion coe:cients of the exchanging ions. This has been lightly evoked by Jackman and Ng (1986) as a possible cause of the overestimation of the coe:cient of internal di-usion when they used a pure Fickian modeling of this mechanism. Finally, kdes is the ?rst-order “intrinsic desorption constant” which accounts for the dynamics of the local interaction between the species in the internal pore solution and the active sites of the solid matrix. The theory of linear chromatography leads eventually to

were carried out using a two-syringe pump (Pharmacia P500) coupled to a pressure sensor and placed outside the oven. The tubing between the components was made of polyethylether-ketone (PEEK). The percolating solutions have been prepared using ultrapure water ◦ Milli Q plus (18:2 mU cm at 25 C) and sodium chloride 23 ( NaCl) purchased from Aldrich Chemical (99.5% purity). The conductivity and the pH of the solution are continuously monitored at the outlet of the column using, respectively, the conductimeter (the volume of the measurement cell is 14 l) and the pH-meter (dead volume less than 40 l) both from Pharmacia. The radioactive tracers used in these experiments were tritium as tritiated water (HTO) and 22 NaCl at trace level dissolved in an aqueous solution of 23 NaCl. The respective activities of the injected solutions were about 22 kBq (HTO) and 2 kBq(22 Na). These radiotracers have been continuously monitored by a speci?c scintillation counter (Packard FloOne 500 TR). The Now rates range from 1 to 200 ml=h and the concentration solution of 23 NaCl has been varied between 4 × 10−3 and 1 mol=l. After the 23 Na= 22 Na isotopic-exchange experiments, the column has been repacked in order to limit any risks of ?ssures during the following 23 Na=H ion-exchange experiments.

tm = tint + text + ta :

3.2. Procedures

(9)

3. Experimental 3.1. Material The gel SON68 has been prepared by lixiviation of 22 g of SON68 glass powder under a Now rate of 400 cm3 =day for a volume of leaching solution of 1 l. The size fraction of glass was included between 20 and 30 m and the glass alteration rate was about 0:05 g=m2 =day leading to a thickness of the gel layer of about 7 m (see Fig. 2). In order to avoid any risks of cracking, this gel has been preserved in a 1 M aqueous 23 NaCl solution (pH = 9). This solution has then been put into a 1-cm-diameter glass column and the gel has been packed by carefully sedimenting the solid phase (gel grains) by means of a light vacuum pumping and then removing of the liquid phase. This procedure led to a bed of 4:81 g of gel packed over a length of 4:25 cm (apparent density app = 1:44 g=cm3 ). Microporous ?lters are placed at both ends of the medium preventing the outNow of the gel grains and ensuring uniform Nuid distribution over the full cross section of the medium. As described in Fig. 1, the column has been put into a controlled temperature oven in order to perform the ◦ ◦ experiments at 50 C (±1 C) which corresponds to the temperature of the glass in the geological repository. A pre-warming loop ensures the heating of the solution upstream of the column. The experiments

Prior to all the tracings, the medium is equilibrated with the tested 23 NaCl solution by continuously injecting the 23 NaCl solution into the porous medium for 24 h. This duration ensures that both the inlet and outlet solutions are identical and that it is being characterized for example by conductivity. Starting at t = 0, a pulse of either a weakly overconcentrated (about 1% more) 23 NaCl solution or the tested 23 NaCl solution tagged with the radioactive tracer is injected into the bed. The ?rst experiment enables the physical characterization of the column and the determination of the transfer function G0 (under the form of the corresponding RTD in the time domain) as far as no interaction can occur between the tracer (23 Na+ ) and the sodium-saturated gel grains. For the whole conductimetric tracings, this function can be assimilated to a cascade of more than 20 mixing cells in series. The dispersion length can then be estimated as the ratio of the length of bed divided by 2J (Schweich, 1993) and ranges from 0.03 to 0:09 cm. These low values con?rm the homogeneous packing of the bed in regard to the limiting value of dispersion length which is 4 times the hydraulic diameter (i.e. about 0:01 cm) of the gel grain assumed to be spherical. For any experiment, the outlet curves are deconvoluted by the corresponding input functions which have been determined prior to the experiment by performing exactly the same tracing but with by-passing the column. Last, it has been observed that the transfer functions G0 determined by the conductimetric tracings are almost

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

3431

0.035

0.03

up-flow 0.025

down-flow

RTD

0.02

0.015

0.01

0.005

0 0

20

40

60

80

100

120

140

160

180

Time(min)

Fig. 3. Comparison of down- and up-Now con?gurations on RTD determination.

identical whatever the direction of the vertical Now (i.e. up-Now from the bottom to the top or inversely down-Now). As depicted in Fig. 3, no gravity-induced segregation is signi?cantly occurring in these experiments where the density di-erences between the injected and displaced liquids are the highest. In that sense, the MCE model can apply to this system. 4. Result and discussion 4.1. In9uence of the ionic strength on the structure of the gel Experiments have been performed in order to quantify the sensitivity of the gel to the ionic strength (i.e. the concentration of 23 NaCl) of the percolating solution Nowing through the bed at three di-erent super?cial velocities. For that purpose, the apparent (global) porosity, , and the permeability, B, of the bed have been determined, respectively, by conductimetric tracings and pressure-drop measurements (see Table 1). The apparent porosity is calculated as the ratio of the total volume accessible for sodium ion divided by the apparent geometrical volume. Fortunately, the accessible volume can be unambiguously determined as the product of the volumetric Now rate by the mean residence time whatever the Now pattern (Villermaux, 1985). As expected,  has been found to be e-ectively independent of the Now rate. On the other hand it has been veri?ed that the measured pressure drop across the bed is always proportional to the super?cial velocity whatever the ionic strength. Thus, the permeability B is calculated from the pressure drop per unit length (WP=L) according to the Kozeny–Carman equation widely used for packed beds (Mac Donald, El-Sayed, Mow, & Dullien, 1979): B=

3 %u0 1 ext = d2 WP=L 150 (1 − ext )2 p

(10)

Table 1 Experimental apparent porosity  and permeability B as functions of the normality N0 of the external NaCl solution N0 (M)

1014 B (m2 )

(%)

0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.030 0.07 0.1 0.3 0.7 1

1.40 1.47 1.62 2.76 2.85 2.92 2.99 3.10 3.29 3.86 4.04 3.97 3.97

45.4 47.7 48.6 46.4 49.9 52.1 56.6 62.1 64.5 70.7 72.7 72.8

Finally, the various porosities are related by  = ext + (1 − ext )int :

(11)

In theory, Eqs. (10) and (11) should give ext (or int ) from B and . As depicted in Fig. 4, the evolution of B as a function of the 23 NaCl concentration of the solution reveals three distinct con?gurations of the gel bed although the apparent porosity  of the bed increases more continuously with the ionic strength of the solution. The ?rst drastic change in B occurs in the range 0.006 –0:007 M in 23 NaCl concentration. When confronted with the relative apparent porosities this change should preferentially be due to a brutal increase in dp (∼ 40%) which could occur when the gel grain is removed from its storage solution to these experimental conditions corresponding to a su:cient swelling. For larger 23 NaCl concentrations, the permeability and the apparent porosity continuously increase up to constant values both corresponding to 23 NaCl concentrations larger than 0:1 M from which the contraction of the gel could be at its

3432

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438 4.5E-14

70

Apparent porosity (%)

65

Permeability

4E-14

3.5E-14 60 55

3E-14

50

2.5E-14

45

Permeability (m2 )

Apparent porosity (%)

75

2E-14 40 1.5E-14 35 30 0.001

1E-14 0.01

0.1

1

[NaCl] (M)

Fig. 4. Evolution of the apparent porosity and permeability of the bed as functions of the normality of the

maximum. Some additional assumptions must be made to interpret these results: • If Donnan exclusion dominates for the lower ionic strength, then  = ext . If one assumes further that this external porosity is independent of ionic strength, then int is deduced from Eq. (11). Unfortunately, this interpretation leads to a constant permeability contrary to the experimental results. • If one assumes that the hydraulic diameter dp is constant and equal to the original size of the glass particles (25 m), then Eqs. (10) and (11) allow estimating ext . However, this second assumption leads to an irrelevant increase of the internal porosity from about 30% to 50% since the natural trend of the gel would be to contract when the 23 NaCl concentration of the solution increases from 0.004 to 1 M (see Fig. 5). • Finally, according to the well-known swelling of the ion exchangers (Hel-erich, 1962), changes in tortuosity and in gel grain diameter are likely to occur when the 23 NaCl concentration varies. In that case the Kozeny–Carman equation based upon the assumption of constant solid phase surface area no longer applies even if replacing the constant (1=150) by another one could compensate X these variations (Ostergren, TrXagardh, Enstad, & Mosby, 1998). When the 23 NaCl concentration is 0:004 M we shall assume that the gel grain diameter is equal to its original size of 25 m and the internal porosity is roughly a constant equal to about 30% as shown in Fig. 5. It is then interesting to notice that the skeletal density, s , of the gel given by s =

app (1 − ext )(1 − int )

(12)

with ext =22% and int =30% is equal to 2:64 g=cm3 , which is quite close to the range (2.8–3 g=cm3 ) reported in the literature.

23 NaCl

solution.

Unfortunately, due to the above-mentioned swelling of the gel, no assumption can reasonably be made on the diameter of the gel grain and then no estimation of both porosities can be given through Eqs. (10) and (11) when the 23 NaCl concentration of the solution is larger than 0:004 M. 4.2. Hydrodynamic dispersion For both ion-exchange processes (23 Na= 22 Na and Na=H), di-erent experiments have been conducted at various super?cial velocities and 23 NaCl concentrations of the external solution. For each operating condition, the corresponding (T; J ) hydrodynamic parameters were previously determined by conductimetric tracings (see Tables 2 and 3). Since the dispersion length is the sum of the ionic “free” di-usion, +free , and statistical or mechanical dispersion, +stat , lengths, the number J of mixing cells is related to the super?cial velocity by 23

J=

L L = : 2+free + 2+stat 2ext Dext =,u0 + 2+stat

(13)

Dext is 1:3 × 10−5 and 9:3 × 10−5 cm2 =s for sodium and hydrogen ions, respectively (Li & Gregory, 1974). The external porosity has been determined upon the assumption of internal porosity equal to 30% and the tortuosity factor, ,, has been taken equal to 2 (Schweich, 1993). At the lowest super?cial velocity, free ionic di-usion prevails while dispersion is mainly statistical when the super?cial velocity increases for both ions sodium or hydrogen (see Table 4). Then, Eq. (13) indicates that J should practically not depend on the super?cial velocity when higher than 1:83 × 10−3 cm=s. On the other hand, at the lowest super?cial velocity of 3:5 × 10−4 cm=s; J should be roughly inversely proportional to the external porosity and then should decrease when the ionic strength of the solution increases. Experimental evolutions of J conform with these trends in the case of the 23 Na=H ion-exchange process (see Table 3)

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

3433

0.55

0.5

Internal porosity

0.45

0.4

0.35

0.3

0.25

0.2 0.001

0.010

0.100

1.000

[NaCl] (M)

Fig. 5. Evolution of the internal porosity as a function of the normality of the NaCl solution (constant diameter of the gel grain). Table 2 Operating conditions and corresponding optimized parameters of the di-erent MCE models used to ?t the outlet functions measured in the isotopic exchange

u0 (cm=s) × 104

23 Na= 22 Na

N0 (M)

T (s)

J

tm (s)

K

18.3

0.01 0.004

1180

34.94

1700 1380

119.66 349.6

51.0

0.01

362

22.98

164

130.39

206.5

0.01

101

27.05

79.5

108.96

Table 3 Operating conditions and corresponding optimized parameters of the di-erent MCE models used to ?t the outlet functions measured in the 23 Na=H exchange

u0 (cm=s) × 104

N0 (M)

T (s)

J

tm (s)

K

3.5

0.01 0.004

6240 5340

49 67

35.7 102

0.66 0.86

18.3

0.01 0.004

1280 1090

50 68

∼0 ∼0

0.64 0.84

51.0

0.01 0.004

428 366

46 63

∼0 ∼0

0.65 0.88

206.5

0.01 0.004

107 91.8

50 68

but not in the case of the 23 Na= 22 Na one (see Table 2). In this latter case, these deviations are probably due to existing ?ssures and cracks generated by the relatively bad packing of the column (low J ) but unrevealed by conductimetric tracings. Then, we shall assume that for the 23 Na=H ion-exchange process, J is equal to 49 ± 2 and to 67 ± 2 for 23 NaCl concentrations equal to 0.01 and 0:004 M, respectively. For the 23 Na= 22 Na ion-exchange process, although no real hydrodynamic meaning can be given to a mean value of J , we will take J equal to 28 ± 5 for the 0:01 M 23 NaCl concentration.

0.36 0.42

0.62 0.87

4.3. Hydrodynamics and mass-transfer kinetics Once (T; J ) have been determined and ?xed, the local interaction parameters (K; tm ) have been optimized by a least-squares ?tting procedure in order to get the best agreement between experimental RTD and the MCE model. This is illustrated in Figs. 6 and 7 corresponding to typical 23 Na= 22 Na and 23 Na=H ion-exchange processes, respectively. The MCE model parameters K and tm are recapitulated in Tables 2 and 3 corresponding to both ion-exchange processes.

3434

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

Table 4 Contribution of the free ionic dispersion regime (+free =+stat ) as a function of super?cial velocity and 23 NaCl concentration in both ion-exchange processes u0 × 104 (cm=s)

[NaCl] (M)

3.5 3.5 18.3 18.3 51.0 51.0 206.5 206.5

0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.01

22 Na= 23 Na

23 Na=H

+free =+stat (%)

+free =+stat (%) 92.2 96.7 17.9 18.8 5.9 6.2 1.6 1.7

2.0 0.5 0.1

When the mobile phase Now hydrodynamics characterized by J varies, so does the mass-transfer kinetics since tm and J are both related to the variance, -2 , which is a measure of the width of the response curve. According to the MCE model, 2

2

T (1 + K) + 2Ktm T: -2 = J

(14)

For a given experiment -2 is ?xed. This means that uncertainties on J are reNected in tm . Thus, di-erentiating Eq. (14) with respect to J at constant -2 yields @-2 =@J T (1 + K)2 dtm =− 2 = : dJ @- =@tm 2KJ 2

(15)

Values of uncertainty on tm ; dtm , resulting from uncertainty on J; dJ , are given in Table 5 for both ion-exchange processes. The relative deviation dtm =tm ranges from 27% to 85% for the 23 Na= 22 Na isotopic exchange and is equal to 5% and 31% for the 23 Na=H ion-exchange process. Although

relatively high, these values indicate that the obtained values of tm are signi?cant thus enabling to interpret these measurements. 4.4. Thermodynamics of the isotopic exchange 23 Na= 22 Na As an element of con?dence in the MCE modeling, one can observe from Table 2 that K does not depend on the super?cial velocity of the (0:01 M) 23 NaCl solution but rather on its ionic strength. The mean value of K for the 0:01 M 23 NaCl concentration is about 120 and is equal to 350 for the 0:004 M 23 NaCl concentration. From a theoretical point of view, it is convenient to note that whatever the isotopic nature of sodium ions the concentration (qNa ) sorbed in the gel grain is proportional to the concentration ([Na]) in the bulk external solution via the ratio of the cationic exchange capacity (CEC) divided by the normality of the external solution (N0 ) (i.e. the total NaCl concentration): qNa =

CEC [Na] = KA [Na]: N0

(16)

This relationship has been established upon the assumption that the total anionic concentration in the gel grain (Donnan uptake) due to the electrolyte sorption is negligible compared to the exchanging sites concentration (CEC) and that both isotopes have identical adsorption constant. Due to the high values of the K parameter (see Table 2) the contribution of the internal porosity can be neglected in Eq. (4). Although unusual, qNa and CEC can then be expressed in equivalent per unit volume of interstitial (external) solution. This system of units avoids knowing the external porosity and the gel density and leads to CEC = KN0 ∼ = 1:3 mol=l of external solution whatever the ionic strength. It is worth noting that when expressed in unit mass of gel grain, CEC is equal to KA N0 = KN0 ext =app ∼ = 2 × 10−4 mol=g of gel

0.0025

Measured RTD 0.002

MCE Model

RTD

0.0015

0.001

0.0005

0 0

200

400

600

800

1000

1200

1400

Time (min)

Fig. 6. Typical experimental and calculated RTDs for the

23 Na= 22 Na

exchange.

1600

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

3435

0.3 input function 0.25

Measured outlet function

MCE model convoluted by the input function

RTD

0.2

0.15

0.1

0.05

0 0

5

10

15

20

25

30

Time (min)

Fig. 7. Typical experimental and calculated RTDs for the

23 Na=H

exchange.

Table 5 Variation of dtm in the global mass-transfer characteristic time tm as a function of super?cial velocity and processes

u0 × 104

[NaCl]

22 Na= 23 Na

(cm=s)

(M)

tm (s)

3.5 3.5 18.3 18.3 51.0 51.0 206.5 206.5

0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.01

4.5. Thermodynamics of the ion-exchange

23

164

140

79

39

Analysis of Table 3 indicates that the partition parameter K is well independent of the super?cial velocity and that the ion-exchange process is instantaneously equilibrated when this super?cial velocity is higher than 3:5 × 10−4 cm=s. Provided that the separation factor (/Na=H ) is a constant, one can deduce the theoretical sorption isotherm of the Na=H ion-exchange process:

qH =

(CEC + A)[H] +



dtm (s)

460

Na=H

tm (s)

dtm (s)

36 102 0 0 0 0 0 0

11 5 2 1 1 0 0 0

where A denotes the chloride ion (Cl− ) concentration sorbed in the gel grain (Donnan uptake), qH the concentration of hydrogen ion sorbed in the gel grain and [H] the concentration of hydrogen ion in the bulk external solution. Ke is the dissociation constant of water (about 10−14 mol2 =l2 ). In this equation, /Na=H is reputed to be larger than 1 due to the largest a:nity of the gel for the sodium than for the hydrogen cations (Hel-erich, 1962). Since [H] (about 10−9 mol=l) is much lower than N0 , Eq. (17) reduces to the linear relationship (CEC + A)[H] qH ∼ = KA [H]: = /Na=H N0

(18)

The separation factor can then be estimated by introducing Eq. (4) into Eq. (18), /Na=H =

app (1 − int )(CEC + A) ; N0 {K − int [1 −  + K]}

[H]2 (CEC + A)2 − 4Ke [H]({N0 − [H]}/Na=H + [H]) 2({N0 − [H]}/Na=H + [H])

concentration for both ion-exchange 23 Na=H

1701

grain. As mentioned above (see Section 4.1) this value only applies at the lowest 23 NaCl concentration for which an estimate of ext (22%) can be given. This is in agreement with the theoretical linear sorption isotherm (Eq. (16)) and constitutes thus an experimental evidence of the relevance of the above MCE modeling of this particular isotopic ion exchange.

23 NaCl

;

(17)

(19)

3436

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

upon the implicit assumption that the denominator of the above fraction is strictly positive so that int ¡

K : 1−+K

(20)

Eq. (20) enforces the internal porosity to be lower than 27.8% when using the values of  and K corresponding to the 0:004 M 23 NaCl concentration. On the other hand, Eq. (19) enables determining /Na=H assuming that A is negligible compared to CEC. As an illustration, using the above values of  and K leads to the fact that /Na=H has to be larger than 455 for ensuring int ¿ 0 and is about 1:2×104 for int =27%, indicating the very favorable character of this ion-exchange process. Assuming that /Na=H does not depend on the 23 NaCl concentration and A is still again negligible compared to CEC, inversion of Eq. (19) points out that K has to increase when both A or int increases. Then the observed decrease in K when the ionic strength of the solution increases should result only in the corresponding decrease in int since a decrease in the Donnan uptake, A, would be irrelevant. Using /Na=H = 1:2 × 104 this latter conclusion leads to the fact that int should be about 26% when the 23 NaCl concentration is 0:01 M for accounting for the variations in K. It is worth mentioning that this low relative variation in the internal porosity (about 4%) is in fair agreement with the former assumption of constant int within the range 0.004 –0:01 M (see Fig. 5). 4.6. Mass-transfer limitations in the isotopic exchange 23 Na= 22 Na It has been previously mentioned that the obtained values of tm account for signi?cant mass-transfer limitations in this isotopic exchange (see Table 5). It is then of practical interest to notice that the higher the super?cial velocity, the weaker the external mass-transfer resistance. Conversely, the internal di-usion and intrinsic adsorption processes are independent of Nuid velocity. At the highest super?cial velocity of 0:021 cm=s, the Reynolds, Schmidt and Sherwood numbers (Eq. (8)) are 0.00575, 702 and 5.4, respectively. For these determinations, the viscosity, %, has been taken equal to 0:00913 cm2 =s (Perry & Green, 1997). An upper estimate of external porosity (ext = 32%) is given by the assumption of internal porosity equal to 30% and with an apparent porosity equal to the largest value of 52% corresponding to the 0:01 M 23 NaCl concentration (see Table 1). The characteristic time of external di-usion can then be estimated according to Eq. (5): text =

Kd2p ext r R= 6 1:6 s: 3k 1 − ext 3 Sh Dext

(21)

In Eq. (21) the R factor introduced in Eq. (5) has been taken equal to 1 since no electrical ?eld e-ects should occur in

this isotopic exchange. Assuming that the isotopic desorption rate is instantaneous, comparison between text and tm (see Table 2) leads to the fact that at the highest super?cial velocity the isotopic exchange is mostly controlled by internal di-usion. Then equalizing tint and tm in Eq. (6) leads to an e-ective coe:cient of internal di-usion, Dint =

15ri2 re e3 + 6ri e5 + e6 = 0:9910−9 cm2 =s; 15(re3 − ri3 )re tint

(22)

whose variation ranges from 0.66 to 1:95 × 10−9 cm2 =s. ∗ , must be The actual coe:cient of internal di-usion, Dint corrected for the thermodynamics, ∗ Dint = KDint ;

and then can be directly linked to the coe:cient of self-di-usion of the ion, D0 , via the tortuosity factor, ,: ∗ Dint = (int =,)D0 :

(23)

As reported by Dodds and Tondeur, D0 has been found to be equal to 9:44 × 10−7 cm2 =s from sodium isotopic-exchange studies (Dodds & Tondeur, 1974). Thus, provided that the internal porosity is 30%, Eq. (23) enables determining the tortuosity factor ranging from 1.2 to 3.6 and whose mean value (2.4) is very close to the value of 2 recommended by Schweich (Schweich, 1993). 4.7. Kinetics of the interaction

23

Na=H

Analysis of Table 3 reveals that mass-transfer limitation occurs during the 23 Na=H ion exchange only for the lowest super?cial velocity of 3:5×10−4 cm=s. An estimation of the upper bound of the characteristic time of internal di-usion of hydrogen ion tint (H) can be given by assuming that the corresponding tint (Na) is the one determined in the previous paragraph: tint (H) = tint (Na) 6 0:57

K(H)D0 (Na) K(Na)D0 (H)

D0 (Na) 0:6 s: D0 (H)

(24)

Eq. (24) proves that tint (H) is negligible compared to tm whatever the K value between 0.66 or 0.86. Then the mass-transfer resistance should be either located in the ?lm or due to a ?nite desorption rate that is in accordance with the observed variations of tm as a function of the concentration of the external solution. At the lowest super?cial velocity corresponding to a Reynolds number of about 10−4 , it is assumed that whatever the 23 NaCl concentration, the Sherwood number is roughly equal to its limiting value of 2 as proposed in Section 2. Thus, the thickness of the viscous boundary layer is about 12:5 m in our operating conditions. The derivations by Nicoud (Nicoud & Schweich, 1989) enable one to estimate text as a function of the ionic fraction of hydrogen ion in

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

the solution, XH : text (H+ ) = =

r 3Dext

 4

1 + 4:5(KA + 1)XH (1 + 9KA XH )2

4 Kext r 2 ≈ 0:005 s: 3 1 − ext Dext

 →

XH →0

4r 3Dext (25)

Here again, the characteristic time of external di-usion is far lower than the observed ones (ranging from 36 to 102 s). Thus, the observed mass-transfer limitation should be generated by the intrinsic adsorption reaction with a desorption constant equal to 0.016 and 0:006 s−1 when the concentration of the external solution is, respectively, equal to 0.004 and 0:01 M. This evolution could be compatible with the following desorption rate rdes : rdes

6 × 10−5 ≈ [H]: N0

It is worth noting that these developments implicitly assume that the structure of the gel bed keeps practically constant during the exchange 23 Na=H. Although local variations in internal and external porosities as well as in tortuosity should occur it is expected here that they are not too important.

5. Conclusion The methodology proposed in this paper for characterizing ion exchangers has been developed since long by the pioneering work made by Villermaux and coworkers. In particular, the MCE modeling of percolation through a packed bed enables one to determine the mass-transfer limitations which occur in most ion exchangers. The aim of this paper was to highlight the potentialities of such an approach through the particular study of migration of sodium and hydrogen ions at trace level in the SON68 gel. Its main limitation occurs in the determination of the coe:cient of ionic di-usion whose sensitivity requires not only the number of mixing cells in cascade, J , to be independent of the super?cial velocity but also the characteristic time of internal di-usion to be su:ciently high compared to the mean residence time, T , of a non-reacting species. As an example, the relatively low accuracy on the determination of the coe:cient of di-usion of sodium ion could be improved by using the classical “through-di-usion” technique (Melkior, 2000) while the coe:cient of di-usion of hydrogen ion cannot even be estimated. On the other hand, this limitation is counterbalanced by the richness of the collected information: variations in porosity and thermodynamics, identi?cation of main kinetics limitations of sodium isotopic exchange and sodium=hydrogen exchange. It is worth noting that all of this information has been easily obtained by simple and fast (compared to the “through-di-usion” technique) experiments. More particularly, it has been shown that the 23 Na=H exchange seems to be practically instantaneous

3437

although an intrinsic adsorption limitation may arise at very low super?cial velocity. This last feature sheds doubt on the validity of the isotopic tracing of aqueous solution using tritiated water as a non-exchanging tracer. These results show that this methodology could be used to estimate the silicon di-usion coe:cient, whose knowledge is the key point for all the glass alteration models. In that sense, the “macropore column” concept of Young and Ball (1998) appears to be promising provided it can apply to the material in question. Notation A B C CM CM p  CEC dp D D∗ Dext D0 e G G0 H∗ J k kdes K KA Ke L M N0 WP q Q re ri rdes R s

chloride anion concentration in solid phase per unit mass of gel particle, mol=kg intrinsic permeability of the gel bed, m2 concentration, mol=m3 Laplace transform of the concentration C Laplace transform of the space-averaged concentration in the gel grain cationic exchange capacity, mol=kg or mol=m3 particle hydraulic diameter, m e-ective di-usion coe:cient, m2 =s actual di-usion coe:cient, m2 =s free solution (in?nite dilution) di-usion coe:cient, m2 =s self-di-usion coe:cient, m2 =s thickness of the gel layer, m transfer function of the MCE model in Laplace domain (global) transfer function of the MCE model in Laplace domain (mobile phase) transfer function in the Laplace domain accounting for the internal di-usion process number of mixing cells used in G0 ?lm mass transfer coe:cient, m=s desorption constant of the linear desorption rate, s−1 capacity factor, m3 =m3 adsorption constant, m3 =kg dissociation constant of water, (mol=l)2 length of the bed, m transfer function of the MCE model in Laplace domain (local interactions) normality or concentration of the solution, mol=m3 pressure drop across the bed, Pa concentration in solid phase (per unit mass of gel particle), mol=kg volumetric Now rate, m3 =s external hydraulic radius of the particle, m internal hydraulic radius of the particle, m linear desorption rate, mol=s correction factor used in the de?nition of characteristic time of external di-usion Laplace variable

3438

Sc Sh t T u u0

V. Blet et al. / Chemical Engineering Science 57 (2002) 3427–3438

Schmidt number (=%=D) Sherwood number (==dp ) time, s mean residence time of the mobile phase, s interstitial velocity, m=s super?cial velocity, m=s

Greek letters    ext int /Na=H % +free +stat app g s  ,

equilibrium partition factor (=int + g KA ) thickness of the viscous boundary layer, m apparent packing porosity, m3 =m3 external packing porosity (volume fraction of mobile phase), m3 =m3 internal packing porosity, m3 =m3 separation factor de?ned for sodium=hydrogen exchange kinematic viscosity of the solution, Pa=s ionic “free” dispersion length, m statistical dispersion length, m apparent (bulk) density of the gel bed, kg=m3 bulk density of the gel particle, kg=m3 (=app =(1 − ext )) skeletal density of the gel, kg=m3 (=app = (1 − ext )(1 − int )) density of the solution, kg=m3 tortuosity factor, m=m

Subscripts a ext in int m out p

adsorption (characteristic time) external di-usion inlet concentration internal di-usion global mass transfer (characteristic time) outlet concentration particle

References Boyd, G. E., Adamson, A. W., & Myers, L. S. (1947). The exchange adsorption of ions from aqueous solutions by organic zeolites, II, kinetics. Journal of the American Chemical Society, 69, 2836–2848. Delage, F. (1992). Etude de la fonction cin@etique de dissolution d’un verre. Ph.D. thesis, Univ. Sciences et Techniques du Languedoc, Montpellier, France. Dodds, J. A., & Tondeur, D. (1974). Mesure des coe:cients de di-usion des ions Na+ et Ca2+ dans une rIesine eI changeuse d’ions cationique forte de type gel. Journal de Chimie Physique, 71, 238–242. Fillet, S. (1987). M@ecanismes de corrosion et comportement des actinides dans le verre nucl@eaire SON68. Ph.D. thesis, Univ. Sciences et Techniques du Languedoc, Montpellier, France. Gens, R., Lemmens, K., Van Iseghem, P., De Canniere, P., Put, M., & Aertsens, M. (1996). The corrosion of nuclear waste glasses in

a clay environment: Mechanisms and modelling. CEC contract N FI2W-0031. Final report 1991–1995. Hel-erich, F. (1962). Ion exchange. New York: McGraw-Hill. Jackman, A. P., & Ng, K. T. (1986). The kinetics of ion exchange on natural sediments. Water Resources Research, 22, 1664–1674. Jollivet, P., Nicolas, M., & Vernaz, E. (1998). Estimating the alteration kinetics of the French vitri?ed high-level waste package in a geologic repository. Nuclear Technology, 123, 67–81. Leclerc, J. P., Detrez, C., Bernard, A., & Schweich, D. (1995). DTS: un logiciel d’aide a[ l’Ielaboration de mod[eles d’Iecoulement dans les rIeacteurs Revue de l’IFP, 50, 641–656. Li, Y., & Gregory, S. (1974). Di-usion of ions in sea water and in deep-sea sediments. Geochimica and Cosmochimica Acta, 38, 703–714. Mac Donald, L. F., El-Sayed, M. S., Mow, K., & Dullien, F. A. L. (1979). Flow through porous media—The Ergun equation revisited. Industrial and Engineering Chemistry Fundamentals, 18, 199–208. Melkior, T. (2000). Etude m@ethodologique de la diEusion de cations interagissants dans les argiles. Application: mise en oeuvre exp@erimentale du couplage chimie-diEusion d’alcalins dans une bentonite synth@etique. Ph.D. thesis. Report CEA-R-5890, CEA=DIST Ed., Saclay, France. Natarajan, V., & Cramer, S. M. (1999). Modeling Schock layers in ion-exchange displacement chromatography. AIChE Journal, 45, 27–37. Nicolas-Simonnot, M. O., Sardin, M., & GrIevillot, G. (1995). Consequences of ion-exchange and sorption e-ects on pH waves propagating in a strong-base anion-exchange column. Journal of Liquid Chromatography, 18, 463–487. Nicoud, R. M., & Schweich, D. (1989). Solute transport in porous media with solid–liquid mass transfer limitations: Application to ion-exchange Water Resources Research, 25, 1071–1082. X Ostergren, K. C. E., TrXagardh, A. C., Enstad, G. G., & Mosby, J. (1998). Deformation of a chromatographic bed during steady-state liquid Now. AIChE Journal, 44, 2–12. Perry, R. H., & Green, D. W. (1997). Perry’s chemical engineers’ handbook (7th ed.). New York: McGraw-Hill. Ricol, S. (1995). Etude du gel d’alt@eration des verres nucl@eaires et synth'ese de gels mod'eles. Ph.D. thesis, Univ. Pierre et Marie Curie, Paris, France. Schweich, D. (1993). Transport of linearly reactive solutes in porous media. Basic models and concepts. In D. Petruzelli, & F. G. Hel-erich (Eds.), Migration and fate of pollutants in soils and subsoils, NATO series G, Vol. 32 (pp. 221–245). Berlin: Springer. Trocellier, P. (1997). Les verres nuclIeaires. Revue ScientiFque et Technique des Mus@ees de France, 6, 85–92. Vernaz, E., & Dussossoy, J. L. (1992). Current state of knowledge of nuclear waste glass corrosion mechanisms: The case of R7T7 glass Applied Geochemistry, 1, 13–22. Villermaux, J. (1981). Theory of linear chromatography. In A. E. Rodrigues & D. Tondeur (Eds.), Percolation processes, theory and applications. NATO series E, Vol. 33 (pp. 83–137). Berlin: Springer. Villermaux, J. (1985). G@enie de la r@eaction Chimique (2nd ed.). Paris: Tec & Doc (Lavoisier). Wakao, N., & Funazkri, T. (1978). E-ect of Nuid dispersion coe:cients on particle-to-Nuid mass transfer coe:cients in packed beds. Chemical Engineering Science, 33, 1375–1384. Wilson, E. J., & Geankoplis, C. J. (1966). Liquid mass transfer at very low Reynolds numbers in packed beds. Industrial and Engineering Chemistry Fundamentals, 5, 9–14. Young, D. F., & Ball, W. P. (1998). Estimating di-usion coe:cients in low-permeability porous media using a macropore column. Environmental Science and Technology, 32, 2578–2584.