Pure gases. A typical diffusion barrier consists of a thin sheet of material perforated by a very large number of small holes of nearly uniform diameter. If the diameter of the holes and the thickness of the sheet are smaller than the mean free path of UF6 at the pressure upstream of the barrier, individual molecules of UF6 will flow through the holes without colliding with other molecules in what is known as molecular flow. The rate of molecular flow through a circular capillary is given by Knudsen’s law [КЗ]:

8 Hp"-p’)

mo1 ЗІуДітЇЇТ

where G = molar velocity, kg-mol/(m2 — s) r = capillary radius, m / = capillary length, m

m = molecular weight, 349 for 235UF6 and 352 for 238UF6 R = gas constant, 8314 (Pa• m3)/(kg-mol• K)^

T = absolute temperature, К p” = upstream pressure, Pa p’ — downstream pressure, Pa

The fact that G is different for 23SUF6 than for 238UF6 is what makes separation by gaseous diffusion possible.

If the pressure is sufficiently high or the holes sufficiently large to cause the gas molecules to collide with each other a number of times during flow through the barrier, laminar or viscous flow obtains. The rate of viscous flow through a circular capillary is given by Poiseuille’s law:

r*(p"2 — p’2)
16 IpRT

where p is the viscosity. For UF6 [D5],

p = 1.67(1 + 0.0026Г) X 10"5 kg/(m-s)
t = temperature, °С

The principal differences from molecular flow are as follows:

1. The flow law is the same for 235UF6 as for ^UFs, so no separation takes place during viscous flow.

2. The flow rate is inversely proportional to the viscosity instead of to the square root of the molecular weight.

3. The flow rate is proportional to the difference in the square of the pressures instead of the first power.

11 Pa = 0.007500 Torr = 0.000750 cmHg = 9.87 X 10‘6 atm.

The openings in a diffusion barrier are neither circular, straight, nor of uniform diameter, but its flow characteristics approach molecular flow at low pressures, in the form

(14.5)

and approaches viscous flow at high pressure, in the form

(14.6)

In the intermediate-pressure region, in which flow has some features of both molecular and viscous flow, experiments reported by Present and de Bethune [P3] have shown that the flow may be expressed as a linear combination of Eqs. (14.5) and (14.6):

a(p"-p’) [ b(p"2 — p’2) /m p

where a and b are properties of the barrier.

For several different models of barrier structure, the constants a and b in Eq. (14.7) can be related to dimensions of holes in the barrier. For straight circular holes of uniform radius r occupying e fraction of a barrier of uniform thickness /, Present and de Bethune assign to the constant a the value it would have for molecular flow and to b the value it would have for viscous flow, so that for this “mixed flow” model,

For a barrier consisting of two sizes of straight circular holes, with emol fraction occupied by small holes of radius rmol through which pure molecular flow takes place and evis fraction occupied by larger holes of radius rvis through which viscous flow takes place, the molar velocity for this “viscous leak” model would be

Gfri-cmi’ leak) — 8Гто1^ —о — г*ь(Р 1~P 2^vis (uo)

Crf. viscous leak) гі^Шт ШрЯТ { )

Real barriers contain crooked, noncircular holes distributed in size about a mean radius in the range of 0.005 to 0.03 pm. Molar velocity through most barrier materials is found experimentally to depend on pressures as in Eq. (14.10):

Here Г is known as the permeability, Г0 is interpreted as the permeability for molecular flow, and S is sometimes called the “slope factor.” Comparison of Eq. (14.10) with (14.8) and (14.9) shows that a physical interpretation can be given to the parameters Г0 and S in terms of pore radius and void fraction for the mixed flow and viscous leak models:

n_______ 8 er_______ 8ето|Гтоі

1 0 ” Зіу/ШКТ ~ 3ls/2mnRT (Mixed flow) (Viscous leak)

It is desirable to have a high value of Г0, to reduce the barrier area needed for a given gas flow, and a low value of J, to reduce the fraction of flow that is nonseparating.

Another parameter used to characterize flow through a barrier is the specific permeability 7, defined as the ratio of the actual flow through unit barrier area to the flow by molecular effusion alone through a hole of unit area. Because the latter is

Pfr»-Р’)

4ЛГ

where 0 is the mean molecular speed,

(14.13)

The limiting value of 7 as the pressures p" and p’ approach zero has simple physical significance. In the mixed flow model,

_ 8re

7o = bm 7 =

p, p -*■ 0

and in the viscous leak model,

7o = lim 7 = 8Гт°‘Єто1 (14.16)

p".p’-+ 0 il

Gas mixtures. Nomenclature to be used in describing the flow of a binary gas mixture through a diffusion barrier is shown in Fig. 14.4. The problem is to determine how the molar velocities of light and heavy components, G, and G2, respectively, depend on upstream and downstream

Upstream Ш Downstream face r/j face

Molar velocity Light component

Heavy component

Mole fraction Light component

Heavy component 1 — x”—(1 — v)» 1 — y’

1

Figure 14.4 Flow of binary mixture through diffusion barrier.

pressures and compositions. We shall also be interested in the composition of the net flow through the barrier, expressed as mole fraction of light component v, defined by

G.

G, +G2

Ideal separation. When the upstream pressure is so low that only molecular flow takes place and when the downstream pressure is negligible (p’Ip" -*• 0), Eq. (14.7) shows that the molar velocity of each component is proportional to its partial pressure on the upstream faces and inversely proportional to /m:
W W r _ ap x s/mx	(14.18)
„ ap"(l-x") — >—————- */m%	(14.19)
The composition of the net flow through the barrier in this case is
x" /m2 lm! V G, +G2 x" /m2lmi + (1 — x")	(14.20)
и(1 —Xй) [m^ x"(l-v) ~ V mi ~ a°	(14.21)

(14.17)

Jm2mx is known as the ideal barrier separation factor a0. For 235UF6-238UF6 mixtures,

The composition of the upstream gas x0 that would give a net flow of composition u under these ideal conditions is

When a0 is as close to unity, as it is for UF6, many equations are simpler when expressed in terms of

Barrier separation efficiency. In practice, the difference in composition between gas on the upstream face of the barrier x" and gas flowing through the barrier is less than under ideal conditions for the following reasons, among others:

1. Downstream pressure p is not negligible, and some molecular flow takes place from downstream to upstream faces, partially offsetting separation achieved by flow in the forward direction.

2. Some of the flow through the barrier is of a nonseparating type, such as viscous flow of the gas acting as a continuous fluid.

Because of (14.25) this is

(14.33) through (14.35), and flow through the large holes of radius rvis is of the nonseparating, viscous type at the rate given by Poiseuille’s law (14.3).

Gl feV -РУ) + £ *Va — pn) = — J= b"(1 — X")-p'( 1 — у’)] + ^ (1 — x")(p"2 — p’2) Molecular Viscous n _ 8rmo|6mo| “ зі^ДШт r2- e ■ ^ ‘V15eVIS

The composition of the net flow is <*o(x" — qy’) + x"(p" + p’)( 1 — q)lpc

In this model, net flows for each component, from (14.7), are

Equation (14.42) suggests that the separation parameter pc for a mixture could be evaluated from measurement of the slope factor 5 obtained from the pressure dependence of the permeability for a pure gas, Eq. (14.10). For real barrier materials it is found that the separation parameter pc is appreciably smaller than would be predicted from the slope factor in Eq. (14.10).

Equation (14.40) may be solved for x":

v + v(p" + p’)(l — q)/pc + q(a0y’ — a0v — v + uy’) v + a0(l ~v) + (p"+p’)( 1 ~q)/pc

ap/O ~u)-u(l -/) (<*o — 1M1 — i>)

The barrier separation efficiency, from (14.26), is

To the first order in oto — 1 = 6 and у — v, Eq. (14.43) reduces to

1 ~q ~q(y’ — v)/Su(l — v)

B 1 + (ff" + ff’Xl — q)

where

is a dimensionless pressure and (14.46)

When the composition of the downstream gas у equals that of the net flow v,

E = 1 ~q

B 1 + (її" + тг’ХІ — Я)

Note that EB = 0.500 when p’ — 0 and p" = pc.

Barrier separation efficiency, mixed flow model. Present, Pollard, and de Bethune [P3,P4] have worked out the transport equations for each component of a two-component mixture flowing through a circular capillary of radius r and length / under conditions in which both molecular and viscous flows are taking place in the same capillary. They find that separation is impaired over what would be predicted from the slope factor by the viscous leak model because of an effect important at pressures below the pure viscous flow regime, in which occasional collisions between faster-moving lighter molecules and slower-moving heavier molecules slow down the former and speed up the’ latter and thus reduce separation. Their derivation is limited to the practically important case in which the composition of the gas downstream of the barrier equals that of the net flow through the barrier (y’ = v). They give a rather complex set of equations for the case in which o0 differs appreciably from unity, which reduce for the close-separation case of interest in uranium isotope separation to Eq. (14.48) for the barrier separation efficiency EB.

ГФ" / , exp [(1 + xyp + (X/2)02 ] (іф J<P Ed — J^ в Ф" exp [( + Х)ф" + (ХІ2)ф"г] (14.48) Here ■о- III oouT ЛІ ЧІ5* (14.49) and у — 256 м 9її pD (14.50)

where p is the density and D is the diffusion coefficient. Ney and Armistead [N2] have found that pDlp for mixtures of 235UF6 and 23®UF6 is close to . With this value,

*(UFe)=g (14.51)

Numerical inversion of Eq. (14.48) shows that EB = 0.500 when <p’ = 0.00 and ф" =

0. 1834. To provide an equation that may be compared with (14.47), the characteristic pressure pc is defined by

Рс =	(0.1834X16м) ISRT 3 Г V 7I77I	(14.52)
and the dimensionless pressure її is	II	(14.53)
Hence	ф * 0.1834	(14.54)

In Eq. (14.48), substitution of (14.51) for X and change of variable from ф to я through (14.54) results in

Comparison of Eqs. (14.52) and (14.12) shows that for this mixed flow model,

0. 1834

Comparison with Eq. (14.42) shows that pc evaluated from the slope factor with the mixed flow model is only 9.17 percent the value of pc evaluated from the slope factor with the viscous leak model. Equation (14.56) comes closer to representing the characteristics of actual barriers.

Empirical equations for barrier separation efficiency. Even Present and de Bethune’s develop­ment does not represent accurately conditions in an actual diffusion barrier because gas flow paths are neither straight, circular, nor of uniform cross section. Consequently, a number of empirical equations have been suggested to characterize the separation performance of barriers. Bilous and Counas of the French CEA [B17] have proposed the empirical equation

=(!-<?) (l-y) (14.57)

valid for a limited range of values of я". С. H. Bosanquet [K5], of the British gaseous diffusion project, proposed Eq. (14.58), a modification of the viscous leak formula (14.47):

which brings its results closer to the Present and de Bethune formula (14.55). Table 14.5 compares the barrier separation efficiencies predicted by the Bilous and Counas Eq. (14.57), the viscous leak Eq. (14.47), Bosanquet’s Eq. (14.58), and Present and de Bethune’s Eq. (14.55). For gaseous diffusion process analysis, this text will use Bosanquet’s Eq. (14.58) because of its comparatively simple form and its fairly close correspondence with the theoretically based Eq. (14.55) of Present and de Bethune. As Table 14.5 shows, all four equations give a barrier efficiency of 0.500 at upstream condition я" = 1.00 and downstream condition я’ = 0.00.

Diffusion barrier characteristics. Because of security classification, quantitative information on barrier characteristics is scarce. The most comprehensive report in the open literature was made

Table 14.5 Comparison of equations for barrier separation efficiency H n. * = P ІРс я’ = p’lPc t, П <t = p Ip Barrier efficiency given by Bilous & Counas Eq. (14.57) Viscous leak Eq. (14.47) Bosanquet Eq. (14.58) Present & de Bethune Eq. (14.55) 1.00 0.00 0.00 0.500 0.500 0.500 0.500 0.72 0.144 0.20 0.512 0.473 0.508 0.528 0.72 0.18 0.25 0.480 0.448 0.487 0.507 0.72 0.24 0.333 0.427 0.406 0.450 0.469 0.4 0.10 0.25 0.600 0.545 0.577 0.604 0.5 0.125 0.25 0.563 0.511 0.545 0.572 0.72 0.18 0.25 0.480 0.448 0.487 0.507 1.0 0.25 0.25 0.375 0.387 0.429 0.436 1.2 0.30 0.25 0.300 0.353 0.395 0.391

by Frejacques et al. [F3] in 1958. The first two columns of Table 14.6 give properties reported by these workers for five different barrier types developed by the French CEA. These reported properties have been converted to the units given in the last two columns as follows. Frejacques et al. state that the barrier separation efficiency depends on upstream pressure p", downstream pressure p’, and mean pore radius f as

(14.59)

Bilous and Counas [B17] recommend for the parameter A a value of 3 OanvcmHg) as providing an adequate correlation between their pore size measurements and barrier separation perform­ance on UF6 at temperatures between 35 and 85°C. Hence the upstream pressure p" = pc in ton at which the barrier would have an efficiency of EB = 0.500 at a downstream pressure p = 0.00 is

(1 — 0.5)3(Mm, cmHg)10(Torr/cmHg) 15 p‘(Torr) = Цmj = fbm)

The relation between the observed permeability Г reported in units of gram-moles air per square centimeter per cmHg pressure difference per minute and the dimensionless permeability у defined earlier is

. r[g-mol air/(cm2,min*cmHg)]

Here 82.06 (cm3 •atm)/(g-mol*K) is the gas constant and 293 К is the test temperature. Because the mean speed of air molecules at 293 К is

In subsequent analysis of the gaseous diffusion process, the diffusion barrier will be assumed to have the properties of the French barrier listed second in Table 14.6, made by anodic oxidation of aluminum, with pc = 1500 Torr and у = 15.6 X 10’5. Over the range of operating conditions of economic interest, the specific permeability у will be treated as independent of pressure and temperature.