Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 184　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1970

SELECTION OF OPTIMAL ELECTRODIALYSIS PARAMETERS

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

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

To determine optimal electrodialysis regimes, it is useful to know the relationship between the efficiency coefficient of the electrodialyzer and other parameters, in particular those parameters that can be readily determined or directly measured. The present work is an attempt to give some of these relationships and concerns electrodialysis using ion-exchange membranes.

The basic scheme of the process is shown in Fig. 1. The directions of the flows are indicated by arrows; \(K\) is a cation-exchange membrane, \(A\) is an anion-exchange membrane.

Fig. 1. Electrodialysis scheme

With the arrangement of chambers shown in Fig. 1, anions move to the left, while cations move to the right. Concentration takes place in chamber II, and desalination in chambers I and III. The transfer of substance in electro-

in a dialyzer is described by Faraday’s law, which can be written as:

\[ G=\eta \cdot G_{\text{theor}}=\frac{\mathcal{E}}{F} I\eta, \tag{1} \]

where

\[ \begin{aligned} G &\text{— actual transfer of substance, g/sec;}\\ G_{\text{theor}} &\text{— theoretical transfer of substance, g/sec;}\\ \eta &\text{— efficiency coefficient;}\\ \mathcal{E} &\text{— equivalent amount of the substance converted;}\\ I &\text{— electric current, A;}\\ F &\text{— Faraday constant, A·sec/equivalent.} \end{aligned} \]

Taking into account that the current through the membranes is determined by the electric fluxes of positive ions (current \(I_+\)) and negative ions (\(I_-\)), we find the total currents through the membranes:

\[ I_a=(I_a)_-+(I_a)_+ \text{ — for anion-exchange membranes,} \]

\[ I_k=(I_k)_+ +(I_k)_- \text{ — for cation-exchange membranes.} \tag{2} \]

The useful transport of substance through the membranes is carried out by the currents:

\[ I'_a=(I_a)_--(I_a)_+ \]

\[ I'_k=(I_k)_+-(I_k)_-, \tag{3} \]

then the current efficiencies of the ion-exchange membranes are written in the form

\[ \eta_a=\frac{(I_a)_--(I_a)_+}{(I_a)_-+(I_a)_+} \text{ — for anion-exchange membranes,} \]

\[ \eta_k=\frac{(I_k)_+-(I_k)_-}{(I_k)_+ +(I_k)_-} \text{ — for cation-exchange membranes.} \tag{4} \]

Assuming that the current efficiency of the electrodialyzer is determined by the current efficiencies in the membranes and that \(I_a=I_k=I\) (i.e., there is no current leakage), one may write the following expression for the overall current efficiency:

\[ \eta=\frac{1}{2n}\sum_{j=1}^{n} \left[ \frac{(I_{aj})_--(I_{aj})_+}{(I_{aj})_-+(I_{aj})_+} + \frac{(I_{kj})_+-(I_{kj})_-}{(I_{kj})_+ +(I_{kj})_-} \right] = \frac{1}{2n}\sum_{j=1}^{n}(\eta_{aj}+\eta_{kj}), \tag{5} \]

where \(n\) is the number of pairs of chambers of the electrodialyzer.

In the case where the electric current is also carried by ions whose extraction or concentration is not required, it is necessary to take their contribution into account. In this case the expression for \(\eta\) is derived analogously.

It may be seen that formula (5) is easily transformed so that the current efficiency is determined through the values of the transport numbers of individual types of ions or through the values of the membrane selectivities.

For \(I_a \ne I_k \ne I\), formula (5) is not applicable.

Using the methods for determining current efficiency set forth in works [1] and [2], one can solve the problem for a more general case.

Let us find an expression for the degree of desalination and the efficiency coefficient of an electrodialyzer as functions of the process parameters—the magnitude of the electric current, water leakage, flow velocity, differences in hydrostatic pressures, and concentrations in the chambers.

For the process parameters we introduce the following notation (see Fig. 1):

$v_1'$ — volumetric flow rate of the stream entering desalting chamber I, ml/sec;

$v_2'$ — volumetric flow rate of the desalted stream, ml/sec;

$v_1''$ — volumetric flow rate of the stream entering concentrating chamber II, ml/sec;

$v_2''$ — volumetric flow rate of the stream leaving the concentrating chamber, ml/sec;

$C_1'$, $C_2'$, $C_1''$, $C_2''$ — respectively, the concentrations of the solutions, g/ml;

$W$ — volumetric rate of water transfer from the desalting chamber, ml/sec;

$G$ — rate of substance transfer, g/sec.

Using the indicated notation, relations of the form

\[ v_1' = v_2' + W, \tag{6} \]

\[ v_1'' = v_2'' - W, \tag{7} \]

\[ C_1'v_1' = G + C_2'v_2', \tag{8} \]

\[ C_1''v_1'' = C_2''v_2'' - G \tag{9} \]

and relation (1) can be written as:

\[ \eta=\frac{\left[C_1'(v_2' + W)-C_2'v_2'\right]\cdot F}{\mathcal{E}I}. \tag{10} \]

We shall assume that, over broad ranges,

\[ \frac{\partial W}{\partial \eta}=0,\quad W \ll v_2' \ \text{and}\ W \ll v_2''. \]

Then, instead of expression (10), we obtain

\[ \eta=\frac{Fv_2'\Delta C'}{\mathcal{E}I}, \tag{11} \]

where it is assumed that:

\[ \Delta C' = C_1' - C_2'. \]

It is necessary to distinguish the actual transfer of substance

\[ G=\eta G_{\text{theor}}=C_1'(v_2' + W)-C_2'v_2' \tag{12} \]

from the apparent transfer

\[ G_1=C_1'v_2' - C_2'v_2' = v_2'\Delta C'. \tag{13} \]

The latter is used in practice to determine the current efficiency of an electrodialyzer, $\eta_1$.

Between the actual transfer of substance $G$ and the apparent transfer $G_1$ there is the relation:

\[ G=G_1+C_1'W, \tag{14} \]

\[ G_1=G-C_1'W=\eta G_{\text{theor}}-C_1'W. \tag{15} \]

Between the current efficiency and the current efficiency of an electrodialyzer usually measured in practice (apparent current efficiency) $\eta_1$, the relation is written in the form:

\[ \eta_1=\frac{G_1}{G_{\text{theor}}}=\eta-\frac{C_1'W}{G_{\text{theor}}}; \tag{16} \]

\[ \eta_1=\eta\left(1-\frac{C_1'W}{G}\right)= \eta\left(1-\frac{C_1'W}{v_2'\Delta C' + C_1'W}\right); \tag{17} \]

\[ \eta_1=\eta\left(1-\frac{1}{1+\dfrac{v_2'\Delta C'}{W C_1'}}\right). \tag{18} \]

From the last formula the following conclusion can be drawn: the apparent efficiency of the electrodialyzer is proportional to the current efficiency and increases with increasing desalination and flow velocity, and also with decreasing water leakage and stream concentration.

To estimate the dependence of the apparent efficiency of the electrodialyzer on the magnitude of the current and on the difference in hydrostatic pressures in the chambers, let us express the quantities \(W\), \(\Delta C'\), and \(v_2'\) entering formula (18) through the current \(I\) and the hydrostatic pressures.

According to [3] and [4], the water leakages can be written as:

\[ W=W_I+W_p, \tag{19} \]

where \(W_I\) is the volumetric rate of water transfer due to electroosmosis, ml/sec;

\(W_p\) is the volumetric rate of water transfer under the action of the hydrostatic pressure difference, ml/sec.

In turn, for currents below the limiting values, according to [4], we have:

\[ W_I=k_1 I, \tag{20} \]

where

\(k_1\) is the volume of electroosmotically transferred water in 1 sec at a current of 1 A, and

\[ W_p=k_2(P'-P''), \tag{21} \]

where

\(P'\) is the hydrostatic pressure to the left of the membrane under consideration, cm water column;

\(P''\) is the hydrostatic pressure to the right of the membrane, cm water column;

\(k_2\) is the volume of water transferred through the membrane under the action of a difference in hydrostatic pressures of one cm water column in one second, ml/sec \(\times\) cm water column.

\[ k_2=\frac{kS}{\delta\nu}, \tag{22} \]

where

\(k\) is the absolute permeability coefficient;

\(S\) is the useful flow area of the membrane, cm\(^2\);

\(\delta\) is the membrane thickness, cm;

\(\nu\) is the viscosity of water (for distilled water \(\nu=1\)).

Expression (19), taking (20) and (21) into account, is written as

\[ W=k_1 I+k_2(P'-P''). \tag{23} \]

Moreover, the actual transfer of substance can be written in the form

\[ G=k_3 I+k_4(P'-P''), \tag{24} \]

where \(k_3\) and \(k_4\) are constants.

The change in concentration \(\Delta C'\) is written as:

\[ \Delta C'=\frac{(k_3-k_1 C_1')I+(k_4-k_2 C_1')(P'-P'')}{v_1'-k_1 I-k_2(P'-P'')}. \tag{25} \]

The volumetric velocity of the desalinated stream is determined by the expression:

\[ v_2'=v_1'-k_1 I-k_2(P'-P'') \tag{26} \]

and formula (18) takes the form:

\[ \frac{\eta_1}{\eta} = 1- \frac{1}{ 1+ \dfrac{(k_3-k_1C_1')I+(k_4-k_2C_1')(P'-P'')} {[k_1I+k_2(P'-P'')]C_1'} }. \tag{27} \]

All the parameters entering formula (27) are readily determined in practice.

It follows from expression (27) that, in the absence of water leakage due to the difference in hydrostatic pressures in the chambers of the electrodialyzer, the ratio of the apparent current efficiency of the electrodialyzer to the current efficiency does not depend on the electric current strength.

If the expression

\[ \eta_1=\frac{v_2'\Delta C'}{G_{\text{theor}}} =\frac{v_2'\Delta C'}{\mathcal{E}I}F \tag{28} \]

is transformed to the form

\[ \Delta C' = \eta_1\frac{\mathcal{E}I}{Fv_2'} = \frac{1}{v_2'} \left\{ \frac{\mathcal{E}I\eta}{F} - [k_1I+k_2(P'-P'')]C_1' \right\}, \tag{29} \]

then, for the case when \(k_2(P'-P'')=0\), we obtain

\[ \Delta C' = \frac{I}{v_2'} \left( \frac{\mathcal{E}\eta}{F} - k_1C_1' \right). \tag{30} \]

From the last expression it is easy to see that, in this case, at constant values of \(\eta\) and \(C_1'\), the degree of desalination is determined not by the individual values of the current strength and the flow rate of the desalinated solution, but by the ratio \(I/v_2'\). The degree of desalination increases in direct proportion to \(I/v_2'\). However, the value of \(I/v_2'\) can be increased only to a limited extent, since at a critical value of this quantity concentration polarization of the membranes sets in, the current efficiency decreases, and the increase in the degree of desalination slows down.

All the foregoing applies to a pair of electrodialyzer chambers with any scheme of connection (with respect to flow) of the desalination and concentration chambers. By introducing the quantity \(n\) (the number of pairs of electrodialyzer chambers) into the principal relations derived, the chamber connection scheme is taken into account.

Conclusions

1. A relation has been found between the apparent efficiency coefficient of an electrodialyzer and the current efficiency, depending on the electric current strength, the difference in hydrostatic pressures in the electrodialyzer chambers, and the concentration of the solution entering the desalination chamber.

2. A relation is given for the magnitude of solution desalination as a function of the concentration of the solution entering the desalination chamber and of the parameters indicated in item 1.

It is shown that, in the absence of water leakage due to the difference in hydrostatic pressures in the electrodialyzer chambers, the degree of desalination is determined by the ratio of the electric current strength to the flow rate of the desalinated solution.

REFERENCES

1. D. R. Uilson. Demineralization by Electrodialysis, ed. by B. N. Laskorin and F. N. Rauzen, Gosatomizdat, Moscow, 1963.

2. Kalman I., Siros I. and Mikes I. A. Chem. Techn., 18, No. 7, 411—413, 1966.

3. B. N. Laskorin, N. M. Smirnova, M. N. Gantman. Ion-Exchange Membranes and Their Application, Gosatomizdat, Moscow, 1961.

4. Oda, Jawataja. Bull. Chem. Soc. Japan, 30, 3, 1957.