Stage separation efficiency. Figure 14.7 illustrates the nomenclature to be used in describing flow rates, compositions, and degree of separation in a cross-flow gaseous diffusion stage, with v = y. The stage separates feed containing Xp mole fraction light component into a light fraction containing у mole fraction and a heavy fraction containing x mole fraction.

X <xF <y

The separation factor of the stage a is defined as

*0-y>

A stage separation efficiency E, analogous to the overall Murphree plate efficiency in distillation, may be defined as

where x0 is the composition of gas that, on the high-pressure side of an ideal barrier, would give low-pressure gas of composition у. From (14.23) it follows that

_ _ (a0 ~ l)y(l ~y)

y x° y + Mi-y)

In the close separation case,

, У-x

a — 1 = —r.—- г

*0 — x)

Our problem is to determine the relationship between the stage separation efficiency, given by (14.82), the barrier separation efficiency EB, defined by (14.26), and the local mixing efficiency EM, defined by (14.64). The stage separation efficiency depends on the relative direction of flow of the high-pressure and low-pressure streams and the degree of mixing of these streams.

No mixing, cross flow. In a common type of gaseous diffusion stage, high-pressure gas flows along the inside of a number of barrier tubes in parallel without significant mixing in the direction of flow, and the low-pressure gas that has passed through the barrier is removed in cross-flow paths approximately perpendicular to the barrier tubes. With cross flow on the low-pressure side, the composition of the gas at each point on the low-pressure side of the barrier, /, equals the composition of the net flow through the barrier at the point, v. This

Light

pressure N. j Ni-dN; ________ *i!!*/ + d*i

fraction

Figure 14.7 Cross-flow gaseous diffusion stage. Stage separation factor a = y( — x)/x( — y) stage efficiency E = (y — x)/(y — x0); Np, Nt, N, M = molar flow rates; xF, xf, x, y, v = mole fraction light component.

practically important condition obtains in most barrier testing experiments and is a condition for Eqs. (14.47) and (14.55) for barrier separation efficiency. When у = v, there is no mixing efficiency correction on the downstream side of the barrier.

Figure 14.7 shows the nomenclature to be used in deriving an equation relating the stage separation efficiency to the barrier efficiency EB and the local mixing efficiency Em for the above kind of cross-flow diffusion stage. A material balance on light component over the portion of the high-pressure side of the stage in which the flow rate decreases by dN{ may be expressed as

vdNi = NfCf — (Nt — dN,) (xf + dxt)

This leads to the differential equation

(14.85)

In the close-separation case, from (14.65),

from (14.27). In the close-separation case v changes so little from point to point in a diffusion stage that v in (14.87) may be replaced by x, the outlet heavy fraction composition. With this substitution, Eqs. (14.87) and (14.85) become

xF-x = (a0- l)x(l — x)EMEB In ^ The fraction diffused is the stage cut в:

(14.90)

By material balance,

From (14.82), the stage separation efficiency is

Because — [ln(l — 0)] /0 is greater than unity, the stage efficiency in cross flow exceeds the local efficiency Em^b — a similar result is found in distillation, where the plate efficiency is greater than the point efficiency when there is cross flow of liquid without mixing across the plate.

In an ideal cascade in which 0 = |,

E= 13&6EMEB (14.94)

Stage performance equations for a mixture in which a0 differs substantially from unity have been derived by Weller and Steiner [Wl].

Stage design variables. The principal independent variables involved in designing a gaseous diffusion stage to serve in an ideal cascade are as follows:

1. The product rate P and product composition yp of the cascade of which the stage is a member

2. The fraction of light component у in the stage of interest

3. The quality of the barrier selected, as measured by its characteristic pressure pc and its permeability у

4. The diameter d and length L of barrier tubes

5. The high-side pressure p" and low-side pressure p’

6. The barrier absolute temperature T

The principal stage characteristics that depend on the choice of the above independent variables are as follows: [47] [48] [49] [50] [51] [52] [53] [54] [55]

a — 1 = 1.386(ao — 1 )EMEB (14.95)

from (14.83) and (14.94).

For a barrier whose separation performance on UF6 is given by the Bosanquet equation (14.58),

1 _ ‘ j "

о -1 = 1.386(«o — l)EM J — . (14.96)

1 + ІР ~P )IPc

where Oo — 1 = 0.00429 (14.97)

and Em is given by (14.65) and (14.66).

Heads flow rate. The flow rate M of stage heads of composition у in the enriching section of a close-separation ideal cascade producing product at rate P and composition yP is

_2P(yP-y)_ M (a — l)y(l —y)

as can be seen by a development similar to Eq. (12.125).

Stage separative capacity. The separative capacity Д of a stage in a close-separation cascade with a cut of is

from Eq. (12.174).

Compressor volumetric capacity. The volumetric capacity V of the compressor for the heads stream at pressure p and absolute temperature T flowing at molar rate M is

v_ MRT
P

The ratio of compressor capacity to separative capacity is

V = ART = ART [1 +(p"-P’)/Pc]2

Д (a-l)V (1.386)2(a0 — 1)2£m j p’(l-p’lp")2

This ratio is independent of isotopic composition. Because EM is only slightly dependent on p’ and p", the pressures p’ and p" that would result in minimum compressor capacity for a given separative capacity are close to those that minimize the term in braces in (14.101). These are found to be

p" = 2pc and p’ = pc (14.102)

at which the term in braces in (14.101) has the value 16lpc, so that the minimum compressor capacity is

Fmin _ 33.3RT

A (a0 — l)2Elfpc

Barrier area. The barrier area A required for a heads stream flowing at molar rate M between pressures of p" and p is

M j2mnRf M A G 7ІР”-РГ)

from (14.14). Hence the ratio of barrier area to separative capacity is

A ^ M _ 4V2nmRT___________ Ay/2mnRT [1 + 0" ~p’)/pc]2 / ri4 1ПЧ7

Д СД (a-l)W’-p-) (1.386)2(a0-l)2^7 )(p"-p’Xl-p7p")2i 1*

The pressures p and p" that would result in minimum barrier area for a given separative capacity are close to those that minimize the term in braces in (14.105). These are

p" = pc and p = 0 (14.106)

at which the term in braces in (14.105) has the value 4lpc, so that the minimum barrier area is

Amin 8.33 j2mnRT Д (<*o — 1)2£mPc7

This indicates the desirability of having a high value of the characteristic pressure pc (hence small pores) and a high value of у (hence many pores per unit area).

Power. Flow of gas through the barrier at rate M is accompanied by loss of availability at rate

Q = MRT0 In

where T0 is the temperature of heat rejection to the environment. This represents the minimum net power needed to recompress the gas from p to p" when the heat of compression at temperatures above T0 is converted to work in a reversible heat engine. Hence the ratio of this net power to separative capacity is

Q 4RT0]n(p"lp’) _ 4RT0 j[l +(p"-p’)/pc]2 (p_

Д (a-D2 (1.386)2(a0-l)2f& ( (l-p’/p")2 U

Minimum power results when both p and p" approach zero, with their ratio q = p’/p" selected to make [ln(l/p)]/(l — qf a minimum. This occurs at q = 0.285, at which the term in braces in (14.109) has the value 2.455. EM at zero pressure equals 1.0. Hence the minimum power per unit separative capacity is

5A1RT0 («о — l)2

This important result is independent of the type of barrier and isotopes being separated. For ^UFe and 23®UF6. with T0 = 300 K,

___________ (5,11) [8314 J/(kg-mol-K)] (300 K)_________

mi„ “ (0.00429)2(3.6 X 106 J/kWhX8760 h/yrX238 kg U/kg-mol)

= 0.0923 kW/(kg SWU/yr)

The pressure conditions that minimize compressor capacity, barrier area, and power consumption are listed in part 1 of Table 14.8. Part 2 of Table 14.8 gives for a diffusion plant with a separative capacity of 1 kg uranium/year, using anodized aluminum barrier tubes 0.014 m in diameter and 4 m long, the compressor capacity, barrier area, and power for the conditions that minimize these three plant requirements. Since these conditions are different, no one design can minimize simultaneously compressor capacity, barrier area and power.

The appropriate criterion to optimize the design of a gaseous diffusion stage is that the

Minimum Minimum

compressor barrier Minimum

capacity area power

^Diameter, 0.014 m; length, 4 m; permeability y, 15.6 X 10’5; pc, 1.974 atm; T, 358 K.

unit cost of separative work produced by the stage be a minimum. To illustrate how Eqs. (14.101), (14.105), and (14.109) may be used to select optimum values of the stage pressures p’ and p" that minimize the unit cost of separative work, specific assumptions will be made about the unit cost of the principal stage characteristics on which the cost of separative work depends. The unit costs assumed for this purpose are listed below:

Direct capital costs

Compressors and piping Converters and barrier Electrical equipment and cooling system Indirect capital costs Capital charge rate Electric power

Ratio of actual power to power for isothermal compression at T0

Other costs that make small additional contributions to the cost of separative work, which are to be disregarded in this example, include costs of operation, maintenance, and supervision; fixed stage costs for such components as instruments; and the cost of enriched UF6 inventory. With the above assumptions, the unit cost of separative work eg is

Table 14.9 gives the characteristics of a diffusion stage using these optimum conditions of p" = 0.55 atm and p’lp" = 0.32 for anodized aluminum barrier tubes 0.014 m in diameter and 4 m in length, with pc — 1.974 atm and 7 = 15.6 X 10"s.

The unit cost of $110/kg SWU is not far from the value of $100 anticipated for 1980 delivery. The power consumption of (2X0.16776) = 0.336 kW/(kg SWU/уеаг) may be com­pared with the power of 6,060,000 kWe consumed by U. S. ERDA’s diffusion plants when operating at their full capacity of 17,230,000 kg SWU/year [U1 ]: 6,060,000/17,230,000 = 0.352 kW/(kg SWU/year).

After the cascade improvement and cascade operating programs planned by U. S. DOE have been completed, their power consumption will be increased to 7,380,000 kW and their separative capacity to 27,700,000 kg SWU/year, equivalent to a specific power consumption of 0.266 kW/(kg SWU/year).