The enrichment parameters and the gas properties are given in table 2.1 and 2.2.
To determine the dependence of the calculated enrichment on the specific interaction potential model, the shape factors (c.f. eq. 2.4) are calculated for three simple but more or less realistic models, viz. the Inverse Power repulsion (IP), the Lennard-Jones 1 2 - 6 (L-J 12—6) and the Buckingham e x p . - 6 (B exp.—6). Extensive tabulations for the IP model are given by Greene et al. (1966), for the L-J 12—6 model by Mclnteer and Reisfeld (1961). The shape factors for the B exp. —6 model
were supplied by Saviron (pers. communication). The shape factors and transport coefficients have to be calculated for a range of r1 and Ty values, because the column diameter varies from 0.89 cm near V to 0.99 cm near V_, and the cooling water temperature changes by at least 8 °C from V to V_. The shape factors are given in table 2.3.
Table 2 . 1 . Parameters of the enrichment apparatus.
inner diameter columns diameter wire length of columns temperature cold wall temperature h o t w i r e number of columns in parallel
0 . 8 9 < 2 / "1 < 0.99 cm
c Boersma-Klein and De Vries (1966)
Considering the fact that the shape factors for IP and L-J 12—6 were obtained by graphical interpolation from tabulated values, the application of different models presents good agreement (within 1% with the exception of /?,p and hL_j).
The values for the L-J shape factors may be somewhat large, since a direct compu-tation of some of them by Saviron (pers. communication) yielded lower values ( ^ 4% for h and » 1 % for kc and k6).
We choose the L-J 12—6 interaction potential as our working model. Using eqs.
2.4 and tables 2.1—2.3 the transport coefficients were calculated for this potential model (table 2.4). Similar values were obtained for the other potentials. Using eq.
2.9 (qe = exp{/y/(/Cc + /Cd)}Z.) we obtain the equilibrium separation factor qe
(table 2.5). Apparently small differences in transport coefficients yield considerably
Table 2.3. Shape factors f o r different interaction potential models.
I. Inverse Power Repulsion3
TA (K)
c shape factors calculated by Saviron (pers. communication) using Simpson integration routines, a steepness parameter a = 17 (Mason and Rice, 1954) and e/k = 119 K at 2 9 8 K and 118 K at 343 K respectively
t maximum temperature of cooling water, occasionally reached; thermal protection switches off the power at this temperature
Table 2.4. Transport coefficients for the Lennard-Jones 12—6 model.
Table 2.5. Equilibrium separation factors for different interaction potential models.
Molecule
different separation factors. Considering the direct dependence of the transport coefficients on the column radius {r*, q and r~ for H, Kc and Kd respectively, eqs.
2.4) a maximum value of qe will be obtained for Kd = 2KQ. Because also the shape factors (slightly) vary with r-, the true maximum of qQ will be found near this value. The column radii r1 and r2 have therefore been chosen near the value
maximizing c7e. A larger r^ results in an increased rate of enrichment on the one hand and a lower q on the other. Therefore the optimum value of /\. is somewhat larger than the value maximizing c7e.
Similarly a maximum of qe as a function of "T, is expected near Kc = Kd. An increase of qe with 7^ is expected for Kc> /Cd and a decrease for Kc< /Cd (c.f.
table 2.4 and 2.5).
Finally we can calculate the values of q'e and tr, using eq. 2.14, 2.15 and 2.16. In eq. 2.14 in combination with the total mass in V we must use the total TD transport of 9 columns in parallel, i.e. 9 H. The q' values for standard and minimum volume and for the practical range of r- and 7"- are given in table 2.6. In fig. 2.3 and 2.4 the values of q' and f. are shown as a function of V_ for the different isotopic molecules.
Table 2.6. Equilibrium enrichment factor and relaxation time for the Lennard-Jones 12—6 potential model. the enrichment for large separation factors (<7e> 10Vt/V+). Under these conditions the enrichment set-up will operate reproducibly, insensitive to small variations in operating conditions (7^, 7"2, p). The choice of the potential model has little in-fluence on the calculated 9' and t . This is evident from a comparison of table 2.5, showing that the influence of variations of r^ and 7^ on qQ is equally large as that of the interaction potential model, with table 2.6 and fig. 2.3, showing that qe for
40-
30-
20-10
V t / V+/
343 K v 16n
308 K L u
308 K I2r 18.
343 K
•c'°o
343 K 13r 16n
308K u u
/
V. (I 100 200
Fig. 2.3. Equilibrium enrichment q'e as a function of the size of the storage volume V_ . VXIV^ — ratio of the total system volume ( V + V+ + VQ) and the enrichment volume.
80
6 0
-20
-—
-I -I
/
I . I
/ 343 K
, 343 K / V 3 4 3 K /'/ 308K
.308K
^ - 3 0 8 K
12C18Q
V
6o
13C160
12C18Q
UC1 60"
13
c
16o
-V_ ( I ) 100 200
Fig. 2.4. Relaxation time fr of enrichment as a function of the size of the storage volume V_
1 4C1 60 and 1 2C1 80 is mainly determined by V_ ( ^ Vt). Using the tabulated values of q'e and tf and eq. 2.11 and 2.12 the performance of the TD set-up can be calculated. In section 2.4.2 we give a comparison of this calculation with the enrichments observed.
Because of the similar enrichment of the C 0 and C O molecules (fig. 2.3 and 2.4) the former is considered to be a good indicator for the C enrichment obtained. The figures show that the relation between q and q will slightly depend on V a n d the temperature. Values for the coefficients a-, a« and a~ of eq.
2.19 are given in table 2.7. It should be noted that these values only apply to the molecules 14C 0 and 12C O. The experimental relation, on the contrary, will be based on a comparison of all C containing isotopic molecules, including
1 4C1 70 and 1 4C1 80 , with those of mass 30 (1 2C1 80 + 1 3C1 70 + 1 4C1 60).
for different storage volumes.
a2x 1 03
For mono-atomic molecules the theoretical models used above give satisfactory results. For poly-atomic molecules most properties of the gas can be described. The phenomenon of thermal diffusion, particularly the TD constant a, however, cannot be described accurately by these simple potential models, since it depends critically on the type of collisions. Therefore the 'theoretical' values calculated above were obtained from a theory that applies to spherically symmetric 'soft' molecules and experimental values of the thermal diffusion constant a. From the observation that
the choice of the potential model is of minor importance in the calculations given above, it follows that the main influence of the model is via a.
I t has been shown (Boersma-Klein and De Vries, 1966; Stevens and De Vries, 1968) that the use of a non-spherical interaction potential fails to describe experi-ments satisfactorily. Rather, the difference in mass-distribution w i t h i n the molecule is the determining factor. To explain the experimental results Schirdewahn et al.
(1961) considered the influence of the moment of inertia of the molecules on the thermal diffusion behaviour. Van de Ree (1967) gave a discussion in which a spherical potential located in the centre of the molecule rotates around the centre of mass.
Another effort t o account for the non-sphericity of the molecules is the description as rigid rotating ovaloids. In analogy t o the model of rigid elastic spheres this model is partially successful, but cannot predict the temperature dependence of a. Recent-ly Verlin et al. (1975a, b) calculated collision integrals for soft non-spherical molecules developed in terms of spherical £2 collision integrals. They found a satis-factory theoretical temperature dependence after the parameters had been f i t t e d to experiments for CO at one temperature.
It was discussed above that the influence of the interaction potential on the shape factors and hence on q' and f. is small. Therefore the theoretical refinements just mentioned are only needed when the thermal diffusion constant a must be obtained f r o m theory because experimental values are not available.