On the derivation of the equation for the process of separation of isotopic ions during ion-exchange extraction in the liquid ion exchanger–solution system

ON THE DERIVATION OF THE EQUATION FOR THE PROCESS OF SEPARATION OF ISOTOPIC IONS DURING ION-EXCHANGE EXTRACTION IN THE LIQUID ION EXCHANGER—SOLUTION SYSTEM

I. A. Tikhomirov, V. V. Larionov, A. P. Vergun, V. T. Doronin

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

At present there is extensive material in the literature describing the process of separation of ions with similar properties, including isotopic ions. However, most investigations of this kind have been carried out as applied to isotopic exchange. In the present work we attempt to derive a general equation for the process of separation of isotopic ions for the case of ion-exchange extraction.

Let us assume that, at the beginning of the process, an ordinary ion-exchange equilibrium is established in a two-phase liquid—liquid system. There then occurs a redistribution of the isotopic ions, and the lighter isotope passes into the phase of the liquid ion exchanger. The phases move countercurrently in the separating column. The material fluxes in the ion-exchanger phase and in the solution are schematically represented in Fig. 1. The notation in the figure is as follows:

Fig. 1

\(\Phi_1\) — material flux in the ion exchanger,
\(\Phi_2\) — material flux in the solution,
\(\gamma\) — flux of the light component from the solution into the ion exchanger,
\(C_1\) — concentration of the light component in the ion exchanger,

$C_2$ is the concentration of the light component in the solution,

$D_2$ and $D_1$ are the diffusion coefficients, respectively, in the solution and in the ion exchanger.

Let us isolate an element $dx$ and write the balance for the light component. The process is considered for $t \ne 0$. Because of its smallness, we neglect the dissolution of the liquid ion exchanger in the solution.

The balance for the light component is written as:

$$ \frac{d}{dx}\left(\Phi_1 C_1-D_1\frac{dC_1}{dx}\right)=\gamma, $$

$$ \frac{d}{dx}\left(\Phi_2 C_2+D_2\frac{dC_2}{dx}\right)=\gamma. \tag{1} $$

The stationarity condition here is not fulfilled only at the initial instant of time, when accumulation with respect to substance occurs. The change in the fluxes $\Phi_1$ and $\Phi_2$ occurs as a result of the overall transfer $\gamma_0$ of the isotope mixture from the solution into the ion exchanger. The isotopic redistribution has a competitive character

$$ \frac{d\Phi_1}{dx}=-\gamma_0, \tag{2} $$

$$ \frac{d\Phi_2}{dx}=-\gamma_0. \tag{3} $$

Subtracting equation (3) from equation (2), we obtain:

$$ \frac{d\Phi_1}{dx}-\frac{d\Phi_2}{dx}=0, \tag{4} $$

$$ \Phi_1-\Phi_2=q_k=\mathrm{const}. \tag{5} $$

If it is assumed that $D_1$ and $D_2$ are small, i.e., that mixing along the axis is absent, then equation (1) is written as:

$$ \frac{d}{dx}(\Phi_1 C_1)=\gamma,\qquad \frac{d}{dx}(\Phi_2 C_2)=\gamma. \tag{6} $$

Substituting equation (2) into one of equations (6), we obtain

$$ \Phi_1\frac{dC_1}{dx}=\gamma+\gamma_0 C_1. \tag{7} $$

For $C_1 \ll 1$, equation (7) is transformed into the form

$$ \Phi_1\frac{dC_1}{dx}=\gamma. \tag{8} $$

It can be shown that [1]

$$ \gamma=\Phi_0\left[\varepsilon C_1(1-C_1)-\iota(C_1-C_2)\right]. \tag{9} $$

Further, subtracting one equation (1) from the other and integrating, we have:

$$ \Phi_1 C_1-\Phi_2 C_2-D_1\frac{dC_1}{dx}-D_2\frac{dC_2}{dx} =q_k C_k=\mathrm{const}, \tag{10} $$

where $q_k$ is the withdrawal flux at the “rich” end of the column,

$C_k$ is the concentration of the isotope being withdrawn.

We write equation (5) as:

$$ \Phi_1=\Phi_2+q_k. \tag{11} $$

It is easy to see that, for \(q_k \ll 1\), the following expression is obtained from equation (8):

\[ \frac{dC_1}{dx} \approx \frac{dC_2}{dx}. \tag{12} \]

Then equation (10) is simplified:

\[ \Phi_1 C_1 - \Phi_2 C_2 - D_{\mathrm{e}} \frac{dC_1}{dx} = q_k C_k, \tag{13} \]

where

\[ D_{\mathrm{e}} = D_1 + D_2. \]

From this we find the expression for \(C_1 - C_2\) and substitute it into equation (9). After transformation we obtain

\[ \left(\frac{\Phi_1}{\Phi_0} + \frac{D_{\mathrm{e}}}{\Phi_2}\right)\frac{dC_1}{dx} = \varepsilon C_1(1-C_1) - \frac{q_k(C_k-C_1)}{\Phi_2}. \tag{14} \]

The expression \(\dfrac{\Phi_1}{\Phi_0} + \dfrac{D_{\mathrm{e}}}{\Phi_2}\) is the reciprocal of the height equivalent to a theoretical plate. Then

\[ dn = \frac{dx}{\left(\dfrac{\Phi_1}{\Phi_0} + \dfrac{D_{\mathrm{e}}}{\Phi_2}\right)}. \tag{15} \]

Finally, omitting the indices, we have

\[ \frac{dC}{dn} = \varepsilon C(1-C) - \frac{q_k(C_k-C)}{\Phi_2}. \tag{16} \]

Thus, the equation takes the form of the general cascade equation for isotope separation.

REFERENCES

1. A. M. Rozen, Theory of Isotope Separation in Columns. Atomizdat, Moscow, 1960.