Experimental Design in the Separation of Isotopic Ions by Electromigration in an Ion Exchanger–Solution System
Abstract Generated abstract
This paper applies statistical experimental design to the separation of isotopic ions by countercurrent electromigration in an ion exchanger, solution system. Using a half replicate of a four factor factorial design, the authors model the degree of separation and unit productivity as polynomial regression functions of solution concentration, solution and ion exchanger counterflow velocities, and applied voltage. Fisher criterion tests indicate that the resulting models are adequate within the studied factor ranges. The regressions suggest that applied voltage has the strongest effect on separation degree, while productivity is chiefly influenced by electrolyte concentration and voltage, providing guidance for subsequent optimization experiments.
Full Text
UDC 621.039.3
PLANNING OF EXPERIMENTS IN THE SEPARATION OF ISOTOPIC IONS BY ELECTROMIGRATION IN AN ION EXCHANGER–SOLUTION SYSTEM
I. A. Tikhomirov, L. A. Avdeev, A. P. Vergun
(Presented by the scientific seminar of the Faculty of Physics and Technology)
The application of statistical methods of experimental design to the investigation of the process of separation of isotopic ions by countercurrent electromigration in an ion exchanger–solution system is considered. The regression equations obtained make it possible to evaluate the degree of influence of the parameters on the principal characteristics of the separation process. Tables 2, references 2.
In [1], a mathematical model of the process of separation of isotopic ions during electromigration in an ion exchanger–solution system is represented by a partial differential equation.
Methods of experimental design make it possible to describe the separation process in such systems by a function represented in the form of a polynomial. This character of the model makes it possible not only to take into account the variety of factors affecting the principal indices of the process, but also to estimate the contribution of each factor to the value of the function under study. The use of design methods also makes it possible to reduce the amount of experimental work, which is very significant in view of the long duration of experiments on the separation of isotopic ions under the indicated conditions.
In constructing the model of the process of separation of isotopic ions, the response function considered was both the degree of separation \((y_1)\) and the productivity of the separation unit \((y_2)\). The factors were the following quantities: the concentration of the solution in the separation unit \(Z_1\)
\[ \left[\frac{\text{g-equiv}}{\text{cm}^3}\right], \]
the electric-field voltage \(Z_4\)
\[ [\text{V}], \]
and the counterflow velocities of the solution and of the ion exchanger—respectively \(Z_2\) and \(Z_3\)
\[ \left[\frac{\text{cm}^3}{\text{h}}\right]. \]
Taking account of prior information on the process, the basic levels were chosen; from the standpoint of reproducibility of the response function, the magnitudes of the intervals of variation were estimated. The values of the basic levels and intervals of variation are presented in Table 1. To obtain the regression equation of the process, a design of type \(2^{4-1}\) was used, i.e., a \(1/2\) replicate of the full factorial experiment with defining contrast \(I = x_1, x_2, x_3, x_4\); (where \(x_1, \ldots, x_4\) are the values of the factors in coded variables)
\[ x_i=\frac{Z_i-A}{\Delta}; \]
The design matrix for the separation process is presented in Table 2.
The regression equation for each response function has the form
\[ \begin{aligned} y_1 &= 0{,}63 + 0{,}07 x_1 + 0{,}08 x_2 + 0{,}08 x_3 - 0{,}10 x_4 - 0{,}08 x_1x_4 +{}\\ &\quad + 0{,}09 x_3x_4 - 0{,}06 x_2x_4, \end{aligned} \tag{1} \]
\[ \begin{aligned} y_2 &= 0{,}53 + 0{,}24 x_1 - 0{,}07 x_2 - 0{,}14 x_3 + 0{,}20 x_4 +{}\\ &\quad + 0{,}07 x_1x_4 + 0{,}24 x_2x_4 - 0{,}17 x_3x_4. \end{aligned} \tag{2} \]
The coefficients of these equations were calculated by the formulas of [2].
Table 1
Intervals of variation of the variables
| Levels of variation | \(Z_1\) | \(Z_2\) | \(Z_3\) | \(Z_4\) |
|---|---|---|---|---|
| Basic level \((A)\) | 0,3 | 3 | 0,5 | 145 |
| Variation interval \((\Delta)\) | 0,1 | 1 | 0,5 | 45 |
| Upper level | 0,4 | 4 | 1 | 190 |
| Lower level | 0,2 | 2 | 0 | 100 |
Matrix for planning experiments
Table 2
| No. | \(x_0\) | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) | \(x_1x_4\) | \(x_2x_4\) | \(x_3x_4\) | \(y_1\) | \(y_2\) |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | + | − | − | − | − | + | + | + | 0,50 | 0,114 |
| 2 | + | + | − | − | + | + | − | − | 0,44 | 1,758 |
| 3 | + | − | + | − | + | − | + | − | 0,47 | 0,321 |
| 4 | + | + | + | − | − | − | − | + | 0,80 | 0,481 |
| 5 | + | − | − | + | − | + | + | + | 0,46 | 0,208 |
| 6 | + | + | − | + | + | + | − | − | 0,78 | 0,325 |
| 7 | + | − | + | + | + | − | + | − | 0,82 | 0,530 |
| 8 | + | + | + | + | − | − | − | + | 0,75 | 0,521 |
The adequacy of the mathematical models obtained to the process of separation of isotopic ions under consideration was checked by Fisher’s criterion \((F)\).
The values of this criterion for equations (1) and (2), respectively, are 17.3 and 11.3. The tabulated value \(F_{0.05}\) is 19. Thus, the adequacy condition is satisfied.
As can be seen from equation (1), the degree of separation of isotopic ions is most strongly affected by the magnitude of the voltage on the installation, for a specified interval of variation of the counterflow magnitude. The productivity of the installation (2) is decisively influenced by the concentration of the electrolyte solution and the magnitude of the voltage.
The equations obtained provide the necessary information on how to set up further experiments in order to determine the optimal conditions for the process of separation of isotopic ions.
REFERENCES
- I. A. Tikhomirov, V. T. Doronin, A. P. Vergun, V. V. Larionov. “Elektrokhimiya,” 1970, 10, 1467.
- V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Moscow, “Nauka,” 1965.