- Email: [email protected]
The kinetics of film-diffusion-limited ion exchange
c-d
~ylrnrrrino
Rimed
in Great
srirnn.
ooo!-2509/93
VoL 48. No. 3. pp. 467-473.1993.
ss.00
+ om
o1992Pcnuwapm=plf
Britain.
THE KINETICS
OF FILM-DIFFUSION-LIMITED EXCHANGE
ION
G. KRAAIJEVELD and J. A. WESSELINGH’ University of Groningen, Department of Chemical Engineering, Nijcnborgb 16, 9747 AG Groningen, The Netherlands (Received 30 September 1991; accepted@ publication 12 May 1992) A--The NoAiffimion-limited ion exchange kinetics for the HCl-NaCl and HCl-CaC12 systems on a Lewatit SiOO ion exchanger are investigated. The ion exchange processes are modelled using the Maxwell-St&n transport equations. The model uses only one fitting parameter, the film thickness, the MaxwcllStefSn equations were found to give a good description of the ion exchange procc88, Value8 between 8 and 20 m were fotmd for the 6lm thickness.
ION EXCHANGE
P-
Ion exchange is a process used for separating ions from an electrolyte solution. The process uses two phases: the solution and the ion exchange resin. The solution is usually aqueous, containing any number of positively and negatively charged ionic species. The ion exchange resin consists of a permeable solid matrix, containing fixed electrical charges (in this paper sulphonic acid groups). These fixed charges all have the same sign. Each of these f&d charges has an associated oppositely charged ion, the “counter-ion”. If the salt concentration in the external solution is low relative to the concentration of fixed charges in the r&n, then, due to the electrochemical equilibrium, there will be an almost complete exclusion of “co-ions” (ions with the same charge sign as the fixed charges). The rate of the ion exchange process is usually determined by two maSs transfer resistances:
la I I
.
.
I
I
.,
,-
.
.
.
.
.
.
l
I’
..:
.
l
..I I. . I_0: _I
.-
.
.
00
l
J
Fig. 1. Experimental set-up.
-the resistance inside the particle and -the resistance outside the particle, also known as the “film” resistance.
MODEL
For solutions with low salt concentrations outside the particle, the film resistance usually dominates. Mass transfer under these conditions is the subject of this study. An interesting aspect of the ion exchange kinetics is that they depend on the direction of the ion exchange reaction. This has been carefully analysed by Helfferich (1962X for the situation where the resistance inside the particle is the controlling mass transfer resistance. There are also calculations by Wesselingh and Krishna (1990) which indicate that similar (but different) effects are to be expected when the film resistance is controlling. This paper provides experimental evidence for situations where the film resistance is the dominating one.
In the experiments in this paper, we have a simple stirred beaker glass (Fig. l), which contains one cation. At time zero, we then introduce the ion exchanger beads, which are completely loaded with a different cation. The change of the liquid concentration with time, is then measured using a conductivity probe. For more details, see the experimental section. As shown in Fii. 2, the model of this ion exchange process consists of three parts: -the solution, which is described by a bulk solution and a diffusion film, -the interface, on which equilibrium is a8~umed to exist and -the particles, which are homogeneous. The following
assumptions
are made:
-The liquid is well-mixed, so that the conccntrations in the bulk liquid are the same everywhere.
+Author to whom correspondence should be addressed. 467
468
G.
KRAALJEVELDand J. A. WESELINGH
OUTSIDE - planar 6Im - mixed bulk
INSIDE - homogeneous
fluid
The right-hand side of the equation represents the frictional interactions between species i and each of the other species present. These frictional interactions are proportional to the local concentrations of both species and the velocity differenoe between the two species. They are inversely proportional to the “Maxwell-Stefan diffusivity”. The velocities are related to the fluxes of the species by Ni = ci(u, - IJ’)
(2)
where v* is the reference velocity.
Interfacial equilibrium Fig. 2. Model aspects.
-Transport in the liquid is assumed to be by diffusion through a stagnant film around the particle. The film thickness, 6, is the only fitting parameter in the model. -The resin phase activity coefficients are equal to one. -At the interface, the particle and the liquid are in equilibrium. -All particles are homogeneous, spherical and of equal size. -Transport in the particle is described as diffusion through a stagnant sphere. -There are no co-ions present in the particle. -The water content of the particle is constant. -The ion exchange process itself is taken to be instantaneous; there is no “reaction rate limitation”. Based on these assumptions, we can set up a model consisting of diffusion equations, component mass balances, equilibria, reference frames and no-current conditions. These equations are developed below.
REFERENCE,FRAMES
AND ELECTRONEWI-RALITY
For an N-component mixture, the Maxwell-Stefan equations provide relations between N - 1 velocity differences (Lightfoot, 1974). So only N - 1 of these equations are independent. Hence, in order to calculate the N fluxes, a frame of reference for the species velocities (fluxes) has to be chosen. The most common reference frames are the solvent-fixed reference frame, No=0
(3)
and the volume-fixed reference frame,
1 r,N, = 0. The ionic partial molal volumes, which are required for the above equation, cannot be measured experimentally, because one cannot independently add cations to a system without adding also anions, Hence, the calculations of the ionic partial molal volumes are usually based on certain assumptions relating them to ionic radii, or to the ionic partial molal volume of a reference ion. The partial molal volumes which were used are given in Table 1. From computer simulations it was found that the difference between the volume-fixed and the solventfixed reference frame was not significant (less than 3% in the fitted film thickness). For the resin phase, the reference frame is more obvious; here the resin-fixed reference frame may be used: N,=O.
DIFFUSION
THEORY
A general description of diffusion is given by the “generalized Maxwell-Stefan equations” (Lighffoot, 1974). For the ion exchange process these reduce to
The left-hand side of this equation represents the product of the concentration of species i and the “driving force” on species i. This driving force consists of two terms: -The first is the chemical potential gradient of i, which often may be reduced to the concentration gradient. -The second is the electrical gradient. This gradient is caused by the different mobilities of the ionic species.
(5)
In systems where an electrical field exists, an additional equation is provided by the electroneutrality equation:
Table 1. Partial molal volumes
Component Na+ $+ clHz0
V, (cm3m01- ‘)+ 4.4 5.6 27.8 12.2 18.0
‘Based on data from Jenkins and Pritchett (1983) and Glueckauf (1965).
The kinetics of film-diffusion-limited
As was shown by Newman (1973) deviations from electroneutrality are extremely small. If there is no accumulation of charge in the system, the electroneutrality equation is equivalent to the no-current condition: c ZiNi = 0.
I MASS BALANCES
(7)
AND SYSTEM OF DIFFERENTIAL EQUATIONS
As mentioned before, the liquid phase consists of a mixed bulk solution and a diffusion film. The component mass balances for the bulk solution have the following form: dci
Vdt=
-ANi.
For both the diffusion film and the ion exchange resin, the component mass balances are given by
469
ion exchange
The system of differential equations was solved using central finite differences. For both the diffusion film and the resin, no ‘more than five grid points were needed to obtain the required numerical accuracy, RESIN-SOLUTION
INTERFACE
As mentioned before, it is assumed that there are no co-ions present in the resin, and that the water content of the particle is constant. The remaining two concentrations which need to be calculated are those of the cations. These can be calculated from the ion exchange capacity and the Donnan equilibrium. The latter is defined by (Lightfoot, 1974)
No activity coefficient model was used in the resin; it was found that for the Ca2 +-H+ system the equilibrium could be described by
(11)
Kij=(~)"(~)*
Except for the equations describing the interfacial equilibria (see the next section), we have now developed the complete system of differential equations. For the diffusion film in the liquid phase these consist of:
The equilibrium data, for the two systems studied in this paper, Ca2+-H+ and Na+-H+. are given in Fig. 3.
-the Maxwell-Stefan equations for the three ionic species present, -the component mass balances of the four components Ceq. (911. -either the volume-fixed or the solvent-fixed reference frame and -the no-current condition.
The ionic activity coefficients in the liquid phase, were calculated using the equation suggested by Bromley (1973):
For the resin, the set of equations consists of: -the Maxwell-Stefan equations for water and the two cations present, -the component mass balances of the four components Ceq- (911, -the resin-fixed reference frame and -the no-current condition. Note that there are no co-ions present in the ion exchange resin, and that the resin itself is also treated as a component. The boundary conditions used for the system of differential equations are: -the initial concentration profiles in the resin and the solution, -the four bulk concentrations, which are described by eq. (8), -the four fluxes which are zero at the particle contre and -the electrical potential, which is arbitrarily assigned to be xero in the bulk solution (since the absoiute values are not important, only the gradient is important).
ACTIVITY
h3,o
-
COEFFICIENTS
0.512: fi
Yi =
f+J7
+ F,(B).
(12)
This equation requires only one interaction parameter, B, for each salt present, and can be used for concentrations up to 3 moljl. The parameters used were 0.0574 for NaCl, 0.1433 for HCl and 0.0948 for CaCl, . Because this article is primarily concerned with Bim-diffusion-limited ion exchange kinetics, no activity coefficient model was used for the resin phase. DIFFUSION
COEFFICIENTS
The only remaining aspect of the model in this paper is to obtain diffusion coefficients in both the liquid and the resin. Newman (1973) has shown that the required diffusivities for a single salt in solution can be calculated from the transference numbers, the conductivity and the Fick diffusivity. At low concentrations, these binary diffusivities form good estimates of the required ternary diffusivities. The diffusion coefficient for the interactions between the two cations present can be estimated from the limiting tracer diffusivities:
&r =
x_+xo+x, D+-
D+o
1. -1
D+*
(13)
Table 2 gives the “free-solution diffusivities”, which are based on the data from Chapman (1967) and Mills
G. KRAAIJEVELD
410
1.0.
.
.
t
andJ.
A. WESSELINGH
vealed that the variations in resin diffusivities started to affect the fitted film thickness only at the highest external salt concentration used (0.1 mol l- ‘).
.v-
/+
COMPUTER SIMULATIONS Using the model developed above, a computer gram was developed to simulate the experiments.
$a
:;f i
0.0. 0.0 l.O*
:
_
1.0
=Ch _
_
_
:
.
.
*
8.
ZN.
o.o,f 0.0
-
-
:
-
=Nm
Fig. 3. Equilibrium data for the Na+-H+
Ca**-H+ n = 0.8589; K = 10.80, (I)
1.0 and the
systems. Upper part. (0) 0.001 M: K = 38.96, (I) 0.01 M: K = 24.88, n = 0.7505; (A) 0.1 M: n = 0.6421. Lower part. (01 0.001 M: K = 1.25; 0.01 M: K = 1.38; i.) 0.1 M: K = 1.71.
Table 2. Free-solution Components
H+. H,O
Na’, &O Ca2+, H,O
cl-.
Hz0
H+, ClNa+, ClCal+ ClH+, &a+ Hf, Ca’+
diffusivities
Diffusivity (lO-qm~s-‘)
9.308
1.333 0.792 2.033 0 69C”.*3 o’13c0.63 o:07c~.~0 l.loc0.G9 n.8.
and Lobo (1989). Unfortunately, no limiting tracer _. data were available for the Cazf -H * interactions; these are probably not important and were, therefore, neglected. Only very rough estimates of the diffusivities inside the particle were available. The general rule of thumb is that these diffusivities are 10-100 times smaller than the free-solution diffusivities. Sensitivity studies re-
pro-
Besides determining the “best-fit” film thickness, the program also produced interesting concentration profiles in the particle and the diffusion film. A typical set of concentration profiles is given in Fig. 4. In this figure the profiles of the electrical potential, the H+ and the Cl- concentrations are shown both in the particle and in the diffusion film. In most of the cases three profiles are shown to illustrate the development of each profile with time. The left half of the figure shows the profiles for the ion exchange process, where the ion exchange resin is initially in the sodium form. While the right half shows the profiles for the case where the resin is initially in the hydrogen form. From Fig. 4, two important conclusions may be drawn. First, the magnitudes of the gradients in both concentration and electrical potential, which are a measure of the driving forces, are much larger in the diffusion film. (It should be noted here that the film thickness is much smaller than the particle radius.) This means that the ion exchange process is filmdiffusion-controlled. It can be shown that the ion exchange process is film-diffusion-controlled for all the experiments presented in this paper, with particle diffusion resistance occurring only to a small extent, in the experiment with the highest concentration for the HCl-CaCl, system. And second, an important role is played by the electrical potential. Firstly, the potential jump across the interface is relatively large, which indicates that the equilibrium plays an important role in the ion exchange process. Secondly, in the diffusion film, an electrical gradient is generated due to the difference in mobility of the two cations. In the case where the ion exchange resin is initially in the sodium form, this electrical gradient enhances the mass transfer, whereas for the case where the resin is initially in the hydrogen form, the electrical gradient opposes the mass transfer. Thus, the ion exchange process is asymmetrical, due to the difference in mobility of the cations. EXPERIMENTAL
The experimental set-up is shown in Fig. 1. It consisted of a 500ml Pyrex beaker glass, which was stirred magnetically (using a 4 cm magnet), at 900 rpm. The concentration was measured using a standard immersion conductivity probe. Both the response time of this probe and the mixing time for solution and resin were found to be less than 1 s. At the salt concentrations used, the conductivity was found to be a linear function of the fraction H+ in solution at a constant external salt concentration. Due to the large differences in mobilities of the cations
The kinetics of film-diffusion-limitedion exchange
0 R
0
R+
0
R
R+6
R+6
0
R
R+6
0.2
0
5 00
5 20 00
-0.2-, 0
R
Fig. 4. H* and Cl- concentration profiles and the electrical potential profile in the particle and the diffusion flIm for the Na+-H* system, for the ion exchange process in both directions (R = OSmm, Cbult- 0.1 mall-‘, d = 20pm). The label associated with each curve indicates the time in seconds.
of the ions present could be determined with 1% accuracy from the conductivity measurement. Ion exchange material was prevented from entering the conductivity probe, by an inert plastic gauze. It was found that ion exchange material did affect the conductivity measurements when it entered the probe. Lewatit SlOO ion exchange particles were used for all the experiments. The size fraction of 0.9-1.0 mm particles was used. The capacity of the resin was found to be 4.754 meq/(gram dry resin). The water content was found to be 41,42 and 45%, respectively, for the Ca*+, Na* and HC form of the resin. The average density of the resin was 1.28 g ml-‘. Before each experiment, the ion exchanger resin was regenerated into the required form, in a fluidized bed. Usually, three steps were required using a l-2 mall- ’ solution of the chloride salt of the required cation. These were followed by three steps using the same salt, but now at the salt concentration used in the actual experiment (this to ensure that sorption effects would not spoil the experiment). The forward and reverse ion exchange processes were carried out in the batch system described above, for the systems Na+-H+-Cland Ca’+-H’-Cl-, at external salt concentrations of 0.001, 0.01 and 0.1 mol/l. All experiments were carried out in duplo at room temperature (22°C) and during the experiments no temperature change was observed. The following procedure was followed for each of the experiments. used, the concentrations
At the start of the experiment, the batch system contained 400 ml of solution containing only one kind of cation. At time t = 0, 20 g of ion exchange material (which was completely loaded with the other cation) was introduced. With the aid of the conductivity probe, the conductivity was measured continuously. Furthermore, samples of the initial and final solutions were analysed using AAS, NaOH and AgNOs titrations, so that the concentrations of all the ionic species could be determined. The mass balances on the solution were found to be correct to within 2%. From the initial and final concentrations, the equilibrium coefficients could be calculated (see Fig. 3).
RESULTS AND DISCUSSION
Figure 5 gives a typical set of measured conversion-time profiles, for the forward and reverse exchange processes for the two systems. The calculated profiles can also he seen in this figure. Only one fit-parameter was used to describe these experiments, the film thickness. In both cases it is clearly demonstrated that the ion exchange process is faster when the hydrogen ion moves from the solution to the ion exchanger. The same behaviour was observed for all the experiments. However, it should be noted here that the rates are measured in terms of conversion (i.e. the ratio of the amount of cations exchanged and the total amount of cations which will be exchanged when equilibrium is
G. KRAAIJEVELDand J. A. WESSELINGH Table
3. Film thickness for the various Concentration ~(moll-‘)
System Na+-H+
Film thickness (pm) Forward
exchange’
Reverse exchange’
0.001 0.010 0.100
12.0 14.4 12.0
15.4 18.6 13.8
0.001
14.5 11.8 14.3
18.9 8.8 7.9
0.010
Ca2+-H*
ion exchange processes
0.100
‘Forward exchage is chosen to be the exchange where the ion exchange particles are initially in the hydrogen form. t Reverse exchange is the exchange where hydrogen is initially in the solution.
0
5
10
15
20
25
Time 63) I
1.001 .
z
Ca”+ 0.75 -
at the particle-solution interface constitutes a large part of the total difference (similar to what can be seen in Fig. 4 for the Na+-H+ system). A list of “fitted” film thicknesses is given in Table 3. These film thicknesses were fitted to the experimental concentration-time profiles, by minimizing the squares of the differences between the calculated and the experimental concentrations. The values of the film thicknesses all lie between 8 and 20 pm. These values are similar to those obtained using the existing mass transfer correlations, such as that of Calderbank and Moo-Young (1961). The film thicknesses of duplo experiments all agreed within 3-15% with each other. The dependence of the film thickness on the concentration does not show any clear trend. Sensitivity analyses reveal that the value of the equilibrium coefficient has a large effect on the value of the fitted film thickness. Hence, inaccuracies in this equilibrium coefficient may to a large extent explain the variation in the film thickness. At the highest concentrations, inaccuracies in the resin phase diffusion coefficientmay also result in an error of approximately 10% in the fitted film thicknesses. COMPARISON WITH OTHER DIFFUSION THEORIES
0
5
10 Time
15 (8)
20
25
Fig. 5. Conversion vs time, for the forward and reverse exchange, for both systems, at an external sait concentration of 0.01 mall-‘: (a) initially hydrogen is in the solution; (b) initially hydrogen is in the ion exchange resin.
The Nernst-Planck equation (14) is actually a special case of the Maxwell-Stefan equation (1). It can be obtained by neglecting all frictional interactions except those with the solvent. And, by assuming that the solvent has a velocity (flux) equal to zero, (14)
reached). In terms of “absolute rates” (i.e. the amount of cations removed per unit time), the two systems do not show the same behaviour. In this case the Na+-H+ system still shows the same asymmetry, but for the Caz+-H+ system the asymmetry is the other way around (i.e. the ion exchange process where the hydrogen ion moves from the ion exchanger to the solution is faster). This phenomena can be explained by the fact that the ion exchange equilibrium strongly favaurs the uptake of calcium ions. A further indication of this effect is the fact that the Donnan potential
For dilute solutions the MaxwellStefan and the Nernst-Planck equations give almost identical results. For the experiments presented in this paper, a difference in the fitted film thickness of less than 3% at the highest concentration was found between these two equations. The Maxwell-Stefan equations introduce rather complex calculations. To avoid these, a difference approximation may be used (Wesselingh and Krishna, 1990):
The kinetics of film-diffusion-limitedion exchange Note that in this case the bars indicate average values. This difference approximation works well for most engineering calculations (Wesselingh and Krishna, 1990), and can also be used for concentrated solutions. For the experiments presented in this paper an agreement to within 10% (in fitted film thickness) with the “exact numerical solution” was found. If one makes even more simplifying assumptions, Fick’s law of diffusion may be obtained as a special case of the Maxwell-Stefan equation. A comparison between Fick’s law and the Maxwell-Stefan equation is, of course, not sensible in the context of the present paper. Since Fick’s law does not take into account the electrical driving force, it cannot possibly predict asymmetric ion exchange rates.
CONCLUSION
This paper has clearly demonstrated that at low concentrations the forward and reverse ion exchange rates are different and, hence, that the ion exchange between ions of different mobility is asymmetric. It has demonstrated this by showing that the ion exchange process for the Nat-H+ system is significantly faster when hydrogen is moving from the solution into the ion exchange particle. For the Ca2’-H+ system, the absolute ion exchange rates do not exhibit the same behaviour, due to the fact that the ion exchange equilibrium strongly favours the absorption of calcium. The relative rates (measured in terms of conversion) do, however, show the same behaviour. Finally, it has been shown that the Maxwell-Stefan transport equations can be applied successfully to ion exchange processes. NOTATION
A B
f, F
I k
total area for mass transfer, m Bromley interaction parameter, kg mol- 1 concentration, mol mm3 diffusion coefficient rn’ s- i Faraday constant, 96,485 C molt i ionic strength on a molality basis, mol kg- ’ mass transfer coefficient, m s-i
K N R T 8 V v x z
473
equilibrium coefllclent flux, mol mm2 s-i gas constant, 8.3144Jmol-1K-1 temperature, K average species velocity, m s-r batch volume, m3 partial molal volume, m3mol-i mole fraction charge number
Greek letters activity coefficient chemical potential electrical potential, V : Y
Subscripts species i, j ii membrane solvent ;r total tot tr tracer cation + anion REFERENCES
Bromley, L. A., 1973, Thermodynamic propertics of strong electrolytesin aqueous solutiona A.I.Ch.E. J. 19, 313-320. Cslderbank, P. H. and Moo-Young, M. B., 1961. The con-
tinuous phase heat and mass-transfer properties of dispersions. Chem. Engng Sci. 16, 39-54. Chapman, T. W., 1967, The transport properties of concentrated electrolytic solutions. Ph.D. thesis, University of
California, California Glueckauf, E., 1965, Molar volumes of ions. mans. Faraday
Sot. 61,914-921. Helfferich, F., 1962, Ion-exchange kinetics. III. ExperimentaI test of the theory of particle-diffusion controlled ion exchange. J. phys. Chem. 66,39-44. Jenkins, H. D. B. and Pritchett, M. S. F.. 1983, Absolute ionic partial molal hydration ,volumes in water at 298 K. Philosophical Magazine B48,493-503. Lightfoot, E. N.. 1974, Transport Phenomena and Living Systems. Wiley, New York. Mills, R. and Lobo, V. M. M., 1989, Self-diffusion in Ebctrolyre Solutions. Elsevier, Amsterdam. Newman, J., 1973, Electrochemical Systems. Prentice-Hall, Englewood Cliffs, NJ. Wesselingh, J. A. and Krishna, R., 1990, Mass nan.$er. Ellis Horwood, London.
~ylrnrrrino
Rimed
in Great
srirnn.
ooo!-2509/93
VoL 48. No. 3. pp. 467-473.1993.
ss.00
+ om
o1992Pcnuwapm=plf
Britain.
THE KINETICS
OF FILM-DIFFUSION-LIMITED EXCHANGE
ION
G. KRAAIJEVELD and J. A. WESSELINGH’ University of Groningen, Department of Chemical Engineering, Nijcnborgb 16, 9747 AG Groningen, The Netherlands (Received 30 September 1991; accepted@ publication 12 May 1992) A--The NoAiffimion-limited ion exchange kinetics for the HCl-NaCl and HCl-CaC12 systems on a Lewatit SiOO ion exchanger are investigated. The ion exchange processes are modelled using the Maxwell-St&n transport equations. The model uses only one fitting parameter, the film thickness, the MaxwcllStefSn equations were found to give a good description of the ion exchange procc88, Value8 between 8 and 20 m were fotmd for the 6lm thickness.
ION EXCHANGE
P-
Ion exchange is a process used for separating ions from an electrolyte solution. The process uses two phases: the solution and the ion exchange resin. The solution is usually aqueous, containing any number of positively and negatively charged ionic species. The ion exchange resin consists of a permeable solid matrix, containing fixed electrical charges (in this paper sulphonic acid groups). These fixed charges all have the same sign. Each of these f&d charges has an associated oppositely charged ion, the “counter-ion”. If the salt concentration in the external solution is low relative to the concentration of fixed charges in the r&n, then, due to the electrochemical equilibrium, there will be an almost complete exclusion of “co-ions” (ions with the same charge sign as the fixed charges). The rate of the ion exchange process is usually determined by two maSs transfer resistances:
la I I
.
.
I
I
.,
,-
.
.
.
.
.
.
l
I’
..:
.
l
..I I. . I_0: _I
.-
.
.
00
l
J
Fig. 1. Experimental set-up.
-the resistance inside the particle and -the resistance outside the particle, also known as the “film” resistance.
MODEL
For solutions with low salt concentrations outside the particle, the film resistance usually dominates. Mass transfer under these conditions is the subject of this study. An interesting aspect of the ion exchange kinetics is that they depend on the direction of the ion exchange reaction. This has been carefully analysed by Helfferich (1962X for the situation where the resistance inside the particle is the controlling mass transfer resistance. There are also calculations by Wesselingh and Krishna (1990) which indicate that similar (but different) effects are to be expected when the film resistance is controlling. This paper provides experimental evidence for situations where the film resistance is the dominating one.
In the experiments in this paper, we have a simple stirred beaker glass (Fig. l), which contains one cation. At time zero, we then introduce the ion exchanger beads, which are completely loaded with a different cation. The change of the liquid concentration with time, is then measured using a conductivity probe. For more details, see the experimental section. As shown in Fii. 2, the model of this ion exchange process consists of three parts: -the solution, which is described by a bulk solution and a diffusion film, -the interface, on which equilibrium is a8~umed to exist and -the particles, which are homogeneous. The following
assumptions
are made:
-The liquid is well-mixed, so that the conccntrations in the bulk liquid are the same everywhere.
+Author to whom correspondence should be addressed. 467
468
G.
KRAALJEVELDand J. A. WESELINGH
OUTSIDE - planar 6Im - mixed bulk
INSIDE - homogeneous
fluid
The right-hand side of the equation represents the frictional interactions between species i and each of the other species present. These frictional interactions are proportional to the local concentrations of both species and the velocity differenoe between the two species. They are inversely proportional to the “Maxwell-Stefan diffusivity”. The velocities are related to the fluxes of the species by Ni = ci(u, - IJ’)
(2)
where v* is the reference velocity.
Interfacial equilibrium Fig. 2. Model aspects.
-Transport in the liquid is assumed to be by diffusion through a stagnant film around the particle. The film thickness, 6, is the only fitting parameter in the model. -The resin phase activity coefficients are equal to one. -At the interface, the particle and the liquid are in equilibrium. -All particles are homogeneous, spherical and of equal size. -Transport in the particle is described as diffusion through a stagnant sphere. -There are no co-ions present in the particle. -The water content of the particle is constant. -The ion exchange process itself is taken to be instantaneous; there is no “reaction rate limitation”. Based on these assumptions, we can set up a model consisting of diffusion equations, component mass balances, equilibria, reference frames and no-current conditions. These equations are developed below.
REFERENCE,FRAMES
AND ELECTRONEWI-RALITY
For an N-component mixture, the Maxwell-Stefan equations provide relations between N - 1 velocity differences (Lightfoot, 1974). So only N - 1 of these equations are independent. Hence, in order to calculate the N fluxes, a frame of reference for the species velocities (fluxes) has to be chosen. The most common reference frames are the solvent-fixed reference frame, No=0
(3)
and the volume-fixed reference frame,
1 r,N, = 0. The ionic partial molal volumes, which are required for the above equation, cannot be measured experimentally, because one cannot independently add cations to a system without adding also anions, Hence, the calculations of the ionic partial molal volumes are usually based on certain assumptions relating them to ionic radii, or to the ionic partial molal volume of a reference ion. The partial molal volumes which were used are given in Table 1. From computer simulations it was found that the difference between the volume-fixed and the solventfixed reference frame was not significant (less than 3% in the fitted film thickness). For the resin phase, the reference frame is more obvious; here the resin-fixed reference frame may be used: N,=O.
DIFFUSION
THEORY
A general description of diffusion is given by the “generalized Maxwell-Stefan equations” (Lighffoot, 1974). For the ion exchange process these reduce to
The left-hand side of this equation represents the product of the concentration of species i and the “driving force” on species i. This driving force consists of two terms: -The first is the chemical potential gradient of i, which often may be reduced to the concentration gradient. -The second is the electrical gradient. This gradient is caused by the different mobilities of the ionic species.
(5)
In systems where an electrical field exists, an additional equation is provided by the electroneutrality equation:
Table 1. Partial molal volumes
Component Na+ $+ clHz0
V, (cm3m01- ‘)+ 4.4 5.6 27.8 12.2 18.0
‘Based on data from Jenkins and Pritchett (1983) and Glueckauf (1965).
The kinetics of film-diffusion-limited
As was shown by Newman (1973) deviations from electroneutrality are extremely small. If there is no accumulation of charge in the system, the electroneutrality equation is equivalent to the no-current condition: c ZiNi = 0.
I MASS BALANCES
(7)
AND SYSTEM OF DIFFERENTIAL EQUATIONS
As mentioned before, the liquid phase consists of a mixed bulk solution and a diffusion film. The component mass balances for the bulk solution have the following form: dci
Vdt=
-ANi.
For both the diffusion film and the ion exchange resin, the component mass balances are given by
469
ion exchange
The system of differential equations was solved using central finite differences. For both the diffusion film and the resin, no ‘more than five grid points were needed to obtain the required numerical accuracy, RESIN-SOLUTION
INTERFACE
As mentioned before, it is assumed that there are no co-ions present in the resin, and that the water content of the particle is constant. The remaining two concentrations which need to be calculated are those of the cations. These can be calculated from the ion exchange capacity and the Donnan equilibrium. The latter is defined by (Lightfoot, 1974)
No activity coefficient model was used in the resin; it was found that for the Ca2 +-H+ system the equilibrium could be described by
(11)
Kij=(~)"(~)*
Except for the equations describing the interfacial equilibria (see the next section), we have now developed the complete system of differential equations. For the diffusion film in the liquid phase these consist of:
The equilibrium data, for the two systems studied in this paper, Ca2+-H+ and Na+-H+. are given in Fig. 3.
-the Maxwell-Stefan equations for the three ionic species present, -the component mass balances of the four components Ceq. (911. -either the volume-fixed or the solvent-fixed reference frame and -the no-current condition.
The ionic activity coefficients in the liquid phase, were calculated using the equation suggested by Bromley (1973):
For the resin, the set of equations consists of: -the Maxwell-Stefan equations for water and the two cations present, -the component mass balances of the four components Ceq- (911, -the resin-fixed reference frame and -the no-current condition. Note that there are no co-ions present in the ion exchange resin, and that the resin itself is also treated as a component. The boundary conditions used for the system of differential equations are: -the initial concentration profiles in the resin and the solution, -the four bulk concentrations, which are described by eq. (8), -the four fluxes which are zero at the particle contre and -the electrical potential, which is arbitrarily assigned to be xero in the bulk solution (since the absoiute values are not important, only the gradient is important).
ACTIVITY
h3,o
-
COEFFICIENTS
0.512: fi
Yi =
f+J7
+ F,(B).
(12)
This equation requires only one interaction parameter, B, for each salt present, and can be used for concentrations up to 3 moljl. The parameters used were 0.0574 for NaCl, 0.1433 for HCl and 0.0948 for CaCl, . Because this article is primarily concerned with Bim-diffusion-limited ion exchange kinetics, no activity coefficient model was used for the resin phase. DIFFUSION
COEFFICIENTS
The only remaining aspect of the model in this paper is to obtain diffusion coefficients in both the liquid and the resin. Newman (1973) has shown that the required diffusivities for a single salt in solution can be calculated from the transference numbers, the conductivity and the Fick diffusivity. At low concentrations, these binary diffusivities form good estimates of the required ternary diffusivities. The diffusion coefficient for the interactions between the two cations present can be estimated from the limiting tracer diffusivities:
&r =
x_+xo+x, D+-
D+o
1. -1
D+*
(13)
Table 2 gives the “free-solution diffusivities”, which are based on the data from Chapman (1967) and Mills
G. KRAAIJEVELD
410
1.0.
.
.
t
andJ.
A. WESSELINGH
vealed that the variations in resin diffusivities started to affect the fitted film thickness only at the highest external salt concentration used (0.1 mol l- ‘).
.v-
/+
COMPUTER SIMULATIONS Using the model developed above, a computer gram was developed to simulate the experiments.
$a
:;f i
0.0. 0.0 l.O*
:
_
1.0
=Ch _
_
_
:
.
.
*
8.
ZN.
o.o,f 0.0
-
-
:
-
=Nm
Fig. 3. Equilibrium data for the Na+-H+
Ca**-H+ n = 0.8589; K = 10.80, (I)
1.0 and the
systems. Upper part. (0) 0.001 M: K = 38.96, (I) 0.01 M: K = 24.88, n = 0.7505; (A) 0.1 M: n = 0.6421. Lower part. (01 0.001 M: K = 1.25; 0.01 M: K = 1.38; i.) 0.1 M: K = 1.71.
Table 2. Free-solution Components
H+. H,O
Na’, &O Ca2+, H,O
cl-.
Hz0
H+, ClNa+, ClCal+ ClH+, &a+ Hf, Ca’+
diffusivities
Diffusivity (lO-qm~s-‘)
9.308
1.333 0.792 2.033 0 69C”.*3 o’13c0.63 o:07c~.~0 l.loc0.G9 n.8.
and Lobo (1989). Unfortunately, no limiting tracer _. data were available for the Cazf -H * interactions; these are probably not important and were, therefore, neglected. Only very rough estimates of the diffusivities inside the particle were available. The general rule of thumb is that these diffusivities are 10-100 times smaller than the free-solution diffusivities. Sensitivity studies re-
pro-
Besides determining the “best-fit” film thickness, the program also produced interesting concentration profiles in the particle and the diffusion film. A typical set of concentration profiles is given in Fig. 4. In this figure the profiles of the electrical potential, the H+ and the Cl- concentrations are shown both in the particle and in the diffusion film. In most of the cases three profiles are shown to illustrate the development of each profile with time. The left half of the figure shows the profiles for the ion exchange process, where the ion exchange resin is initially in the sodium form. While the right half shows the profiles for the case where the resin is initially in the hydrogen form. From Fig. 4, two important conclusions may be drawn. First, the magnitudes of the gradients in both concentration and electrical potential, which are a measure of the driving forces, are much larger in the diffusion film. (It should be noted here that the film thickness is much smaller than the particle radius.) This means that the ion exchange process is filmdiffusion-controlled. It can be shown that the ion exchange process is film-diffusion-controlled for all the experiments presented in this paper, with particle diffusion resistance occurring only to a small extent, in the experiment with the highest concentration for the HCl-CaCl, system. And second, an important role is played by the electrical potential. Firstly, the potential jump across the interface is relatively large, which indicates that the equilibrium plays an important role in the ion exchange process. Secondly, in the diffusion film, an electrical gradient is generated due to the difference in mobility of the two cations. In the case where the ion exchange resin is initially in the sodium form, this electrical gradient enhances the mass transfer, whereas for the case where the resin is initially in the hydrogen form, the electrical gradient opposes the mass transfer. Thus, the ion exchange process is asymmetrical, due to the difference in mobility of the cations. EXPERIMENTAL
The experimental set-up is shown in Fig. 1. It consisted of a 500ml Pyrex beaker glass, which was stirred magnetically (using a 4 cm magnet), at 900 rpm. The concentration was measured using a standard immersion conductivity probe. Both the response time of this probe and the mixing time for solution and resin were found to be less than 1 s. At the salt concentrations used, the conductivity was found to be a linear function of the fraction H+ in solution at a constant external salt concentration. Due to the large differences in mobilities of the cations
The kinetics of film-diffusion-limitedion exchange
0 R
0
R+
0
R
R+6
R+6
0
R
R+6
0.2
0
5 00
5 20 00
-0.2-, 0
R
Fig. 4. H* and Cl- concentration profiles and the electrical potential profile in the particle and the diffusion flIm for the Na+-H* system, for the ion exchange process in both directions (R = OSmm, Cbult- 0.1 mall-‘, d = 20pm). The label associated with each curve indicates the time in seconds.
of the ions present could be determined with 1% accuracy from the conductivity measurement. Ion exchange material was prevented from entering the conductivity probe, by an inert plastic gauze. It was found that ion exchange material did affect the conductivity measurements when it entered the probe. Lewatit SlOO ion exchange particles were used for all the experiments. The size fraction of 0.9-1.0 mm particles was used. The capacity of the resin was found to be 4.754 meq/(gram dry resin). The water content was found to be 41,42 and 45%, respectively, for the Ca*+, Na* and HC form of the resin. The average density of the resin was 1.28 g ml-‘. Before each experiment, the ion exchanger resin was regenerated into the required form, in a fluidized bed. Usually, three steps were required using a l-2 mall- ’ solution of the chloride salt of the required cation. These were followed by three steps using the same salt, but now at the salt concentration used in the actual experiment (this to ensure that sorption effects would not spoil the experiment). The forward and reverse ion exchange processes were carried out in the batch system described above, for the systems Na+-H+-Cland Ca’+-H’-Cl-, at external salt concentrations of 0.001, 0.01 and 0.1 mol/l. All experiments were carried out in duplo at room temperature (22°C) and during the experiments no temperature change was observed. The following procedure was followed for each of the experiments. used, the concentrations
At the start of the experiment, the batch system contained 400 ml of solution containing only one kind of cation. At time t = 0, 20 g of ion exchange material (which was completely loaded with the other cation) was introduced. With the aid of the conductivity probe, the conductivity was measured continuously. Furthermore, samples of the initial and final solutions were analysed using AAS, NaOH and AgNOs titrations, so that the concentrations of all the ionic species could be determined. The mass balances on the solution were found to be correct to within 2%. From the initial and final concentrations, the equilibrium coefficients could be calculated (see Fig. 3).
RESULTS AND DISCUSSION
Figure 5 gives a typical set of measured conversion-time profiles, for the forward and reverse exchange processes for the two systems. The calculated profiles can also he seen in this figure. Only one fit-parameter was used to describe these experiments, the film thickness. In both cases it is clearly demonstrated that the ion exchange process is faster when the hydrogen ion moves from the solution to the ion exchanger. The same behaviour was observed for all the experiments. However, it should be noted here that the rates are measured in terms of conversion (i.e. the ratio of the amount of cations exchanged and the total amount of cations which will be exchanged when equilibrium is
G. KRAAIJEVELDand J. A. WESSELINGH Table
3. Film thickness for the various Concentration ~(moll-‘)
System Na+-H+
Film thickness (pm) Forward
exchange’
Reverse exchange’
0.001 0.010 0.100
12.0 14.4 12.0
15.4 18.6 13.8
0.001
14.5 11.8 14.3
18.9 8.8 7.9
0.010
Ca2+-H*
ion exchange processes
0.100
‘Forward exchage is chosen to be the exchange where the ion exchange particles are initially in the hydrogen form. t Reverse exchange is the exchange where hydrogen is initially in the solution.
0
5
10
15
20
25
Time 63) I
1.001 .
z
Ca”+ 0.75 -
at the particle-solution interface constitutes a large part of the total difference (similar to what can be seen in Fig. 4 for the Na+-H+ system). A list of “fitted” film thicknesses is given in Table 3. These film thicknesses were fitted to the experimental concentration-time profiles, by minimizing the squares of the differences between the calculated and the experimental concentrations. The values of the film thicknesses all lie between 8 and 20 pm. These values are similar to those obtained using the existing mass transfer correlations, such as that of Calderbank and Moo-Young (1961). The film thicknesses of duplo experiments all agreed within 3-15% with each other. The dependence of the film thickness on the concentration does not show any clear trend. Sensitivity analyses reveal that the value of the equilibrium coefficient has a large effect on the value of the fitted film thickness. Hence, inaccuracies in this equilibrium coefficient may to a large extent explain the variation in the film thickness. At the highest concentrations, inaccuracies in the resin phase diffusion coefficientmay also result in an error of approximately 10% in the fitted film thicknesses. COMPARISON WITH OTHER DIFFUSION THEORIES
0
5
10 Time
15 (8)
20
25
Fig. 5. Conversion vs time, for the forward and reverse exchange, for both systems, at an external sait concentration of 0.01 mall-‘: (a) initially hydrogen is in the solution; (b) initially hydrogen is in the ion exchange resin.
The Nernst-Planck equation (14) is actually a special case of the Maxwell-Stefan equation (1). It can be obtained by neglecting all frictional interactions except those with the solvent. And, by assuming that the solvent has a velocity (flux) equal to zero, (14)
reached). In terms of “absolute rates” (i.e. the amount of cations removed per unit time), the two systems do not show the same behaviour. In this case the Na+-H+ system still shows the same asymmetry, but for the Caz+-H+ system the asymmetry is the other way around (i.e. the ion exchange process where the hydrogen ion moves from the ion exchanger to the solution is faster). This phenomena can be explained by the fact that the ion exchange equilibrium strongly favaurs the uptake of calcium ions. A further indication of this effect is the fact that the Donnan potential
For dilute solutions the MaxwellStefan and the Nernst-Planck equations give almost identical results. For the experiments presented in this paper, a difference in the fitted film thickness of less than 3% at the highest concentration was found between these two equations. The Maxwell-Stefan equations introduce rather complex calculations. To avoid these, a difference approximation may be used (Wesselingh and Krishna, 1990):
The kinetics of film-diffusion-limitedion exchange Note that in this case the bars indicate average values. This difference approximation works well for most engineering calculations (Wesselingh and Krishna, 1990), and can also be used for concentrated solutions. For the experiments presented in this paper an agreement to within 10% (in fitted film thickness) with the “exact numerical solution” was found. If one makes even more simplifying assumptions, Fick’s law of diffusion may be obtained as a special case of the Maxwell-Stefan equation. A comparison between Fick’s law and the Maxwell-Stefan equation is, of course, not sensible in the context of the present paper. Since Fick’s law does not take into account the electrical driving force, it cannot possibly predict asymmetric ion exchange rates.
CONCLUSION
This paper has clearly demonstrated that at low concentrations the forward and reverse ion exchange rates are different and, hence, that the ion exchange between ions of different mobility is asymmetric. It has demonstrated this by showing that the ion exchange process for the Nat-H+ system is significantly faster when hydrogen is moving from the solution into the ion exchange particle. For the Ca2’-H+ system, the absolute ion exchange rates do not exhibit the same behaviour, due to the fact that the ion exchange equilibrium strongly favours the absorption of calcium. The relative rates (measured in terms of conversion) do, however, show the same behaviour. Finally, it has been shown that the Maxwell-Stefan transport equations can be applied successfully to ion exchange processes. NOTATION
A B
f, F
I k
total area for mass transfer, m Bromley interaction parameter, kg mol- 1 concentration, mol mm3 diffusion coefficient rn’ s- i Faraday constant, 96,485 C molt i ionic strength on a molality basis, mol kg- ’ mass transfer coefficient, m s-i
K N R T 8 V v x z
473
equilibrium coefllclent flux, mol mm2 s-i gas constant, 8.3144Jmol-1K-1 temperature, K average species velocity, m s-r batch volume, m3 partial molal volume, m3mol-i mole fraction charge number
Greek letters activity coefficient chemical potential electrical potential, V : Y
Subscripts species i, j ii membrane solvent ;r total tot tr tracer cation + anion REFERENCES
Bromley, L. A., 1973, Thermodynamic propertics of strong electrolytesin aqueous solutiona A.I.Ch.E. J. 19, 313-320. Cslderbank, P. H. and Moo-Young, M. B., 1961. The con-
tinuous phase heat and mass-transfer properties of dispersions. Chem. Engng Sci. 16, 39-54. Chapman, T. W., 1967, The transport properties of concentrated electrolytic solutions. Ph.D. thesis, University of
California, California Glueckauf, E., 1965, Molar volumes of ions. mans. Faraday
Sot. 61,914-921. Helfferich, F., 1962, Ion-exchange kinetics. III. ExperimentaI test of the theory of particle-diffusion controlled ion exchange. J. phys. Chem. 66,39-44. Jenkins, H. D. B. and Pritchett, M. S. F.. 1983, Absolute ionic partial molal hydration ,volumes in water at 298 K. Philosophical Magazine B48,493-503. Lightfoot, E. N.. 1974, Transport Phenomena and Living Systems. Wiley, New York. Mills, R. and Lobo, V. M. M., 1989, Self-diffusion in Ebctrolyre Solutions. Elsevier, Amsterdam. Newman, J., 1973, Electrochemical Systems. Prentice-Hall, Englewood Cliffs, NJ. Wesselingh, J. A. and Krishna, R., 1990, Mass nan.$er. Ellis Horwood, London.