Theoretical prediction of the constants Cx and Cs of the UF6 thermal diffusion column would be very difficult because of the great difference in properties of UF6 between the liquid at the cold wall and the dense gas at the hot wall. For other gases at pressures around atmospheric, at temperature differences between hot and cold walls small enough so that separation performance can be characterized by gas properties at a mean temperature, closed expressions can be given for the separation parameters Cj and C5. Quantities involved are

Hot wall temperature Ґ Cold wall temperature T”

Density at mean temperature p Viscosity p

Thermal diffusion constant у Diffusion coefficient D

Thermal diffusion effect. When a composition gradient ду/dr and a temperature gradient ЪТ/Ъг occur together in a stationary gas mixture, the usual diffusion mass velocity —Dp Ъу/Ъг is modified by the thermal diffusion effect so that the mass velocity Jr in the r direction becomes

7 is known as the thermal diffusion constant. When у and ЪТ/Ъг have the same sign and are large enough, this lermal diffusion effect can cause transport of an isotope against a composition gradient and thus produce separative work. If a steady state has been established with zero transport, the mole fraction gradient is related to the temperature gradient by

(fr) =УУО~У)Щ^~ (14.346)

or/zero transport or

Integration of Eq. (14.346) between (уТ’) and (у", T") leads to Eq. (14.347) for the separation factor a:

1/ d fVI f

, 1(1-7) , ^

In a = In— ——- г = 7 In —yr (14.347)

/(1-У) T

Measurement of steady-state compositions у and y" after thermal diffusion equilibrium has been established between temperatures Ґ and T" is the most accurate way of determining the thermal diffusion constant. The next-to-the-last column of Table 14.26 gives values of the measured thermal diffusion constant for several binary isotopic mixtures. In all these cases, у is positive, which means that the light isotope concentrates at the higher temperature under the experimental conditions listed.

Values of the thermal diffusion constant can be calculated by the kinetic theory of gases if the intermolecular potential energy is known. Because the calculation is quite sensitive to the detailed intermolecular interaction, calculated values of the thermal diffusion constant are in less satisfactory agreement with experiment than other transport properties.

Hirschfelder et al. [H9] gives a generalized relation for the variation of the thermal diffusion constant with temperature for gases whose molecules interact with the so-called Lennard-Jones potential function, the difference between a repulsion energy inversely

Table 14.26. Thermal diffusion constants for isotopic mixtures Isotopes Log mean reciprocal temperature, К Thermal diffusion constant, 7 Reference Theory Measured 3 He-4 He 398 0.079 0.059 [M7] J0Ne-MNe 408 0.026 0.023 [M9] “A-«°A 450 0.023 0.020 [S2] *3Kr-*4Kr 449 0.0019 0.0013 [M9] 131Xe-132Xe 448 0.0011 0.00085 [M9] 1602-160l80 443 0.015 0.013 [W2] 12С160-13СІ60 431 0.0085 0.0070 [M9] 12CH4-13CH4 407 0.011 0.0073 [Dl]

proportional to the twelfth power of the intermolecular distance and an attraction energy inversely proportional to the sixth power. Shacter et al. [S3] have used this theory to predict the thermal diffusion constants in the third column of Table 14.26. The agreement with experimental values is only semiquantitative.

Figure 14.39, based on this theory, may be used to predict the magnitude and temperature dependence of the thermal diffusion constant for 15 isotopic mixtures. The quantity plotted, /rf, is the ratio of the calculated thermal diffusion constant to the thermal diffusion constant

Gas	elk, К	Gas	elk, К	Gas	elk, К	Gas	elk, К
He	10.2	CO	110	CH4	137	Xe	229
Ne	35.7	o2	113	Kr	190	S02	252
H2	38.0	NO	119	C02	190	Cl2	357
n2	91.5	A	124	n2o	220

Figure 14.39 The function к j — for calculating thermal diffusion constants of isotopic mixtures from Lennaid-Jones 6-12 potential function.

7* for rigid, spherical, nonattracting isotopic molecules, for which

. _ IQS г»г ~mi ^ 108 m2 + mi

Because the maximum theoretical value of fcj — for this intermolecular potential is 0.627, у in thermal diffusion is smaller than a0 — 1 in gaseous diffusion, Eq. (14.24). kj becomes negative at temperatures near the normal boiling point and changes back to positive at still lower temperatures.

Figure 14.40 shows the most accurate measurements of the thermal diffusion effect in UF6 vapor at low pressure, by Kirch and Schiitte [К2]. Results are plotted both as k’T, for comparison with other gases in Fig. 14.39, and as the thermal diffusion constant y. The very low values, under 0.00005, explain Nier’s [N3] inability to detect a thermal diffusion effect in UF6 vapor. The thermal diffusion coefficient у is so much smaller than the analogous parameter in gaseous diffusion, a0 — 1 = 0.0043, that vapor-phase thermal diffusion cannot compete economically with gaseous diffusion for uranium enrichment.

Equations for thermal diffusion column. Equations for the separation performance of a thermal diffusion column can be derived in somewhat similar fashion to the countercurrent gas centrifuge of Sec. 5.5. The results will be summarized for the simplest case to treat theoretically, that of an annular column in which the spacing d between the heated and cooled

tubes is much smaller than the log mean radius f, and in which the temperature difference between heated and cooled walls, AT — Ґ — T", is small enough so that the gas properties can be evaluated at the log mean temperature T. This theory was developed first by Jones and Furry [J5], but using different notation.

Thermal convection between the hot and cold walls under gravitational acceleration g induces longitudinal countercurrent mass flow at the rate

2nfgp2 d3 AT 384 p T

Because longitudinal velocity is zero at the heated and cooled walls, the logarithm of the effective separation factor is found to be that of Eq. (14.347). For у < 1 and AT/T < 1, this becomes

The differential enrichment equation for the countercurrent thermal diffusion column is of the same form as for the countercurrent gas centrifuge, Eq. (14.181), here written as

dy _ Ciyjl ~y)-P(yP ~у) dz C2 + Сз

2nfgp2d3y /дг* 720 p T )

Form (14.187),

For the annular geometry presently considered, the parameter C2, representing longitudinal back diffusion, is

C2 — 2-nfDpd

Jones and Furry’s development leads to Eq. (14.354) for C3:

_ 128 dN2 _2irfg2p*d’1 /atV 3 ~ 315 pD(2vr) ~ 9! p2 pD T /

Generalizations of these equations for larger АТ/T and for wider annuli (d « r) have been given by Mclnteer and Reisfeld [M8].