Because the atom fraction of light component in the net transport of gas through the barrier, v, is greater than the atom fraction of light component in the gas at the high-pressure face of the barrier, x", there must be a difference between the average composition of the gas flowing past the high-pressure side of the barrier, x(, and the gas at the barrier face, x", to maintain the required transport of light component to the barrier surface. The local barrier mixing efficiency is defined as

(14.64)

The purpose of this section is to show how this mixing efficiency depends on conditions on the high-pressure side of the barrier.

Figure 14.5 is a transverse section of a circular barrier tube of diameter d with high-pressure flow along the inside of the tube. With turbulent flow of gas inside the tube, molar velocity is practically uniform at a value slightly above the average, H, over most of the tube diameter, but drops to zero at the tube wall. Atom fraction light component is practically constant at a value slightly above the average, X{, over most of the tube diameter owing to turbulent mixing where the velocity is uniform, but drops to a lower value of x" adjacent to the tube wall to provide the required transport through the poorly mixed gas adjacent to the

tube wall. The molar velocity of gas flow through the barrier is G, with i> atom fraction light component.

Bilous and Counas [B17] have used an equation derived originally for wetted-wall gas-absorption towers to evaluate as a function of the molar velocities G and H. The basic assumption is that the actual gas flow pattern behaves as if there were a stagnant film of thickness t adjacent to the tube wall, through which light component is transported by molecular diffusion, with diffusion coefficient D. As will be shown later, EM in this model is given by

(14.65)

Here 352 is the molecular weight of UF6 and p is the mass density. The empirical correlation for the thickness of the stagnant film t, obtainable from standard chemical engineering texts such as [S4], is

Here pD/p has the value for UF6, and Re is the Reynolds number on the high-pressure side of the barrier:

p is the viscosity of UF6, Eq. (14.4).

Equation (14.65) for the composition gradient in mass transfer through a stagnant film of thickness t may be derived with the aid of Fig. 14.6. At a distance tj into the film, where the atom fraction of light component is Xf, the required net transport of Gv mol of light component per unit area per second is the resultant of that due to flow, Gxj, and that due to diffusion:

r r Dp dxf Gv =Gxf~ 352 dFf

The solution of this equation, with boundary condition Xf — x{ at tf — 0, is

In —^- = -352Gt/Dp

V—Xf r

At the barrier surface, where // = f and Xf = x",

which is (14.65).

The molar velocity H of the gas along the high-pressure side of the barrier may be obtained from the dimensions of the barrier as follows: The total number of moles of gas flowing through a barrier tube d in diameter and L long is ■ndGL. In a well-designed diffusion stage, one-half of the gas entering the stage is diffused. The molar velocity of gas at the inlet end of each tube of the stage then is

r _ 2 ndGL 8 GL, n 77<f2/4 d

Figure 14.6 Nomenclature for de­riving Eq. (14.65) for mixing ef­ficiency.

For a rough estimate of average mixing efficiency, the value of H at midlength of the tube may be used, at which

For a barrier tube of given diameter and permeability, the mixing efficiency is higher the longer the barrier tube, because the molar velocity along the tube is proportional to the length. However, the pressure drop experienced by the gas flowing along the tube is greater the longer the tube, both because of the increased flow path and the increased molar velocity. This pressure drop is detrimental for three reasons: The barrier separation efficiency is decreased, more barrier area is needed, and more energy must be expended to restore the pressure drop. Determination of optimum tube length requires an economic balance among the gain in mixing efficiency, the loss of barrier efficiency, and the cost of increased energy input. Such detailed balance is beyond the scope of this text. Instead, calculations will be given for the mixing efficiency and pressure drop for several tube lengths, and an arbitrary choice of length will be made for subsequent design examples.

The pressure gradient in a circular tube of diameter d through which turbulent flow at molar velocity H is taking place is

dp _ 0.046m2H2 dz (Re)02 dp

Here p is the gas density:

Tube lengths of 2, 4, and 6 m will be considered. A specific high-side inlet pressure of 1 atm (101,325 Pa) and low-side pressure of 0.25 atm (25,331 Pa) will be used.

= 14.61 X 10"5 kg-mol/(m2 *s)

For the example of the French aluminum barrier with у — 15.6 X 10_s, the molar velocity G of UF6 through the barrier at 358 K, an upstream pressure of p" = 1.0 atm (101,325 Pa), and a downstream pressure of p’ = 0.25 atm (25,331 Pa), from (14.14), is

For this barrier at 358 K, in general, when pressures p" and p are expressed in atmospheres,

G = 14.61 X 10’s = 19.48 X 10’5 (p" — p) kg-mol/(m2-s) (14.77)

Table 14.7 gives the mixing efficiency and pressure gradient at the inlet, midlength, and outlet of barrier tubes of these three lengths, and the overall pressure drop in the direction of flow down the tube. The overall pressure drop is approximated by multiplying the average of the pressure gradient at the three calculated points by the length of the tube.

Table 14.7 shows that increasing the tube length from 2 to 4 m increases the mixing efficiency at midlength by 9 percent with an increase in pressure drop under 1 percent. Further increase in tube length to 6 m increases mixing efficiency by less than 4 percent, with an increase in pressure drop of 2 percent. Determination of the optimum tube length would require an economic balance that is beyond the scope of this text. A length of 4 m will be used

Table 14.7 Variation of local mixing efficiency and pressure drop with length of barrier tube’*’ Tube length L, m 2 4 6 Molar velocity along tube, kg-mol/(m2 — s) Inlet 0.1669 0.3339 0.5008 Midlength 0.1252 0.2504 0.3756 Outlet 0.0834 0.1669 0.2504 Mixing efficiency Inlet 0.823 0.896 0.925 Midlength 0.781 0.870 0.906 Outlet 0.708 0.823 0.870 Pressure gradient, Pa/m Inlet 114 396 821 Midlength 68 236 489 Outlet 33 114 236 Overall pressure drop Pa 143 995 3092 Fraction of inlet 0.0014 0.0098 0.0305

^Tube diameter d, 0.014 m; barrier specific permeability 7, 15.6 X 10”s; temperature T, 358 K; high-side pressure p”, 101,325 Pa; low-side pressure p’, 25,331 Pa.

in Sec. 4.7 in examining the effect of various combinations of high-side and low-side pressures on stage design and plant requirements.

Table 14.7 also shows that the mixing efficiency at midlength is close to the average value over the tube length. To simplify the calculations to be made in Sec. 4.7, the mixing efficiency will be treated as if constant at its value at midlength, and the pressure drop along the tube will be neglected. In accurate design calculations, point-by-point calculations along the barrier tube should be made of pressure drop, flow through the barrier, flow along the tube, and mixing efficiency, refinements that are neglected in the remaining treatment of gaseous diffusion.