ON THE PROBLEM OF ISOTOPE SEPARATION IN ELECTRODIALYSIS USING ION-EXCHANGE MEMBRANES

I. A. TIKHOMIROV, V. T. DYURONIN, A. P. VERGUN,
G. S. TIKHONOV, V. V. LARIONOV

(Presented by the scientific seminar of the Faculty of Physics and Technology)

The possibility of using ion-exchange membranes for isotope separation has been shown in a number of works [1—10]. It has been proved [9] that the solvation of isotopic ions in the resin of an ion-exchange membrane is considerably less than in solution. This means that the process of separation of isotopic ions in the “ion-exchange membrane—solution” system proceeds more efficiently than in solution.

The studies on the subject were carried out using a countercurrent multichamber electrodialyzer, the equivalent scheme of which is shown in Fig. 1. Between the cathode and the anode a potential difference is applied that is necessary to satisfy the separation conditions. The flux of the light isotope through the membrane \((I)\) is composed of the flux caused by the electric field in the ion exchanger \((I''_{E})\) and in the solution \((I'_{E})\), the diffusion flux through the ion exchanger \((I''_{D})\) and through the solution \((I'_{D})\), and the counterflow of electrolyte \((I_{P})\). We assume that separation in electrodialysis occurs mainly during the electromigration of isotopic ions through the membrane, while in the gaps between the membranes separation is practically absent owing to complete mixing of the solution in them. It is additionally assumed that polarization may be neglected, since the separation process is considered at low electric-current densities. In this case the concentration profile and the profile of the electric-field strength over the conventionally assembled membrane stack may be considered continuous. In accordance with the scheme under consideration, a differential equation has been compiled for the transport of the light isotope, describing the process of isotope separation in a multichamber electrodialyzer operating in column mode. The equation of the electrodialyzer-column under consideration, as shown in [10], reduces to an equation of the following type:

Fig. 1. Scheme of the separation process

\[ \frac{dc}{dt}=\frac{d}{dx}\left[\frac{dc}{dx}+c(\kappa-c)\right], \tag{1} \]

where \(c\) is the relative concentration of the light isotope in the solution phase; \(t\) is time, \(x\) is the distance of the section of the electrodialyzer under consideration from the feed point, and \(\kappa\) is a constant determined from the parameters of the investigated “ion exchanger—solution” system. Equation (1) differs from the analogous equations given in [1], [2] by the introduction of the constant \(\kappa\).

Let us write the boundary conditions:

\[ \left. \frac{dc}{dx}+c(\kappa-c) \right|_{\substack{x=0\\ x=b}}=0, \tag{2} \]

where \(b=L/l\) is the reduced length of the separation device, taking into account the transition to new variables in setting up equation (1); \(L\) is the length of the membrane stack, and \(l\) is a constant in the transition to new variables. The steady state of the system under consideration, according to (1) and (2), is described by the equation

\[ \frac{dc}{dx}+c(\kappa-c)=0, \tag{3} \]

whence we obtain an expression for the steady-state distribution of the relative concentration of the light isotope along the length of the electrodialyzer

\[ c=\frac{\kappa}{2}\left[1-\operatorname{th}\frac{\kappa(x-x_0)}{2}\right], \tag{4} \]

where the constant \(x_0\) is determined from the condition of conservation of the number of particles in the system

\[ \int_0^b C\,dx=c_0b. \tag{5} \]

\(c_0\) is the initial concentration of the light isotope. Substituting (4) into (5) and integrating, we obtain an expression for finding the value \(x_0\).

\[ e^{\kappa x_0}=\frac{e^{c_0 b}-1}{1-e^{-(\kappa-c_0)b}}. \tag{6} \]

The nonstationary state is described by equation (1). Following the conclusions obtained in [2], we represent the solution of equation (1) in the following form:

\[ c=\frac{\kappa}{2} \left\{ 1- \frac{ \operatorname{sh}\frac{\kappa(x-x_0)}{2} +\frac{2}{\kappa}\operatorname{ch}\frac{\kappa x_0}{2} e^{-\frac{\kappa^2 t}{4}} \sum_{n=1}^{\infty} A_n\lambda_n\cos\lambda_n x\,e^{-\lambda_n^2 t} }{ \operatorname{ch}\frac{\kappa(x-x_0)}{2} +\operatorname{ch}\frac{\kappa x_0}{2} e^{-\frac{\kappa^2 t}{4}} \sum_{n=1}^{\infty} A_n\sin\lambda_n x\,e^{-\lambda_n^2 t} } \right\}, \tag{7} \]

where

\[ A_n=\frac{2\lambda_n}{b}\, \frac{\left[1-e^{y_0b}(-1)^n\right]\left(\frac{\kappa^2}{4}-y_0^2\right)} {\left(\lambda_n^2+y_0^2\right)\left(\lambda_n^2+\frac{\kappa^2}{4}\right)}. \tag{8} \]

\[ \lambda_n=\frac{\pi n}{b}; \qquad (n=1,2,3\ldots), \tag{9} \]

\[ y_0=\frac{k}{2}-c. \tag{10} \]

These theoretical conclusions agree satisfactorily with the experimental results.

Using the concept of an equivalent electrolyte for the system under consideration, we obtained an expression for determining the separation coefficient in electrodialysis in the “ion-exchange membrane—solution” system:

\[ \alpha=1+\frac{\Delta c\,V_k}{[(1-c)S\Delta t W-\Delta c V_k]c}, \tag{11} \]

where \(\Delta c\) is the change in the relative concentration of the light isotope in the cathode chamber with volume \(V_k\) over the time \(\Delta t\), \(W\) is the linear counterflow velocity, and \(S\) is the cross section of the electrodialyzer. Relation (11), unlike the relation derived in [13] for the process of separation during electrolysis in solution, has a different structure and determines the value of \(\alpha\) for the “ion-exchange membrane—solution” system during electrodialysis.

Figure 2 shows a diagram of the apparatus used for continuous isotope separation by the electrodialysis method. The electrolyte solution, which serves to create counterflow in the multichamber electrodialyzer, flows from vessel 1 into vessel 2 with an adjustable level. From vessel 2 the electrolyte, through filter 3, thermostating coil 4, and flowmeter 5, continuously enters the cathode chamber of the electrodialyzer. After passing through all the chambers (the direction of counterflow through the electrodialyzer is shown by the arrow), the solution flows through a branch pipe into vessel 6. Similarly, through elements of the circuit 7–11, a solution with the isotopes to be separated is fed into the large-volume anode chamber. Under the action of the voltage applied to the electrodes, the isotopic ions migrate to the cathode, enriching the solution of the near-cathode chambers in the most mobile ion. Separation of isotopic ions is carried out under the condition \(v_1>W>v_2\), where

Fig. 2. Diagram of the apparatus used in enriching a mixture in the more mobile ion. Designations are in the text.

$c_1$ and $c_2$ are the velocities of the ions being separated under the action of the electric field, and $W$ is the counterflow velocity. High voltage stabilization is ensured by introducing AC and DC voltage stabilizers into the power-supply system. The filter-press-type electrodialyzer was assembled from Plexiglas plates, between which ion-exchange membranes were clamped. Such a design makes it possible to change the number of chambers rapidly, to disassemble and assemble the electrodialyzer easily, and also to use it for separating ionic mixtures of very diverse composition. The main experiments were carried out on a 59-chamber electrodialyzer. MK-40-S and MA-40-S type membranes were used as ion-exchange membranes. The distance between the electrodes of the anode and cathode chambers was 533 mm, the volume of one electrodialyzer chamber was 21.16 cm$^3$, and the temperature in the chambers was maintained at $(10 \pm 0.1)^\circ$C. The duration of the experiments was brought up to 900 hours. The process was carried out in a non-withdrawal regime.

Figure 3 shows the distribution of the relative concentration of the most mobile isotopic ions along the length of the separation device. The form of the experimental dependence obtained agrees with the theoretical conclusions. Figure 4 shows the dependence of the relative concentration of the light isotope in the cathode chamber on the duration of the separation process. Processing this dependence makes it possible, using relation (11), to determine the single-stage coefficient

Fig. 3. Distribution of the relative concentration of the most mobile isotopic ions along the length of the separation device

Fig. 4. Dependence of the relative concentration of the light isotope in the cathode chamber on the duration of the process

of isotope separation in electrodialysis. Knowing the single-stage separation coefficient and the total separation effect, one can determine the number of theoretical plates and the HETP. For the electrodialyzer used by us, the number of equivalent theoretical plates is equal to 800, i.e., 1500 plates per 1 meter of length of the separation device.

It should also be noted that, when separating isotopic cations in electrodialyzers with anion-exchange membranes, along with an increase in the total separation, a decrease in the transport of the light isotope through the electrodialyzer is observed. It is much more economical to use electrodialyzers with cation-exchange membranes, where a sufficiently high separation effect is combined with a large amount of transport (up to 90%).

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