The Optimal Dimensions of a Clusius-Dickel Isotope Separator

Abstract

A Clusius-Dickel isotope separator is treated on basis of the theory of Furry, Jones and Onsager. The dependence on the dimensions of the apparatus of $H$, $A$, $\frac{K_d}{K}$ and $P_c$ are derived without any assumption regarding the intermolecular forces in the isotope mixture. The problem treated is: How to choose the dimensions of the apparatus in order to obtain, at the smallest possible cost, a given quantity of isotope mixture, containing one isotope in a given concentration. The costs are assumed to consist of the costs of the power expended and the costs of the apparatus, the latter costs being proportional to the surface of the tubes. We need not include costs, which are independent of the dimensions of the apparatus, as, e.g., the cost of the natural isotope mixture. The question of how to choose the dimensions in order to obtain the output with the minimal power expenditure or with the minimal surface of the tubes is a special case of the general problem. The problem is solved for a single-stage apparatus, in which the concentration of the isotope in question always is small compared with unity. Formulae are given for the distance between the tubes, the mean circumference of the annular space between them and the length of the tubes, as well as for the minimum costs. Similarly, the problem is treated for an "ideal" multi-stage apparatus consisting of very many stages. In this case, the relative abundance of the isotope in question is not restricted to small values. It is shown that the costs are a minimum if the distance between the tubes has a constant value for all stages, this value being the same as for single-stage operation. The mean circumference however is different for different stages. Both in the single- and the multi-stage operation an increase in the pressure reduces the costs somewhat. The production of ${\mathrm{C}}^{13}$ is given as a numerical example.