Separation factor. In the simplified analysis of the water-hydrogen sulfide exchange process in Sec. 113, the effects of the solubility of hydrogen sulfide in water and the vaporization of water into hydrogen sulfide were neglected. In the following they will be taken into account. The deuterium separation factor a for the hydrogen sulfide exchange process is defined as

_*(1 ~y)

а~У( 1-х)

where у and x are the atom fractions of deuterium in the vapor and liquid, respectively. In terms of the molecular species H20, HDO, D20, H2S, HDS, and D2S that make up each phase, a is given by

Table 13.20 Heavy-water production cost at Savannah River Quantity per kg D2 0 Cost, $/kg D20 Direct production cost Feed water, kg 24,000 5.07 Hydrogen sulfide 0.66 0.24 Salaries 2.89 Operating labor 4.37 Miscellaneous 1.12 13.69 Direct maintenance cost Labor 4.37 Materials 7.38 11.75 Utilities Electricity, kWh 604 13.76 Steam, kg (900 psig equiv.) 5,660 41.45 Cooling water, kg 125,000 1.48 Miscellaneous 0.84 57.53 Depreciation 24.95 Administrative and general 15.01 Total cost of production 122.93

(2*d, o + *hdo + 2*d. s + *hdsX2.}’h, o +Yhdo + 2^h. s +Thds)

* 22 a ("13 1331

(2^DjO + УНЭО + 2^DjS + ^HDsX^HjO + *HDO + 2*HaS + *hds) ’

where у refers to the mole fraction of the indicated species in the vapor and x in the liquid.

An expression will be derived for the dependence of a on the physical properties of the water-hydrogen sulfide system, temperature and pressure. The slight dependence of a on deuterium content will be neglected by considering only low deuterium abundances, at which *D, o ^ *hdo і etc. In this limiting case, the expression for a reduces to

(*hdo + *hdsX2th, o + 2thjs)

o = ;—————— . ‘ (13.134)

(Phdo + ThdsX2^h, o + 2xHjs)

The following properties of water, hydrogen sulfide, and their mixtures are used to evaluate

1. The humidity H of H2 0-H2 S vapor in equilibrium with liquid mixtures, defined as

Th2o H = ——

УН, й

2. The solubility S of H2S in liquid in equilibrium with vapor, defined as

*HsS Ss—Ї — *HaO

The dependence of H and S on temperature and pressure has been determined experi­mentally [S4] and is shown in Figs. 13.31 and 13.32.

3. The relative volatility a* of H20 to НЕЮ, defined by Eq. (13.5).

The dependence of a* on temperature has been given in Table 13.4; it is assumed to be unchanged by the presence of H2 S.

4. The relative volatility у of H2 S to HDS, defined as

JHjS*HDS

THDS^HjS

In the design of the Savannah River plant [B7] it was assumed that у equaled a*.

Roth et al. [R9] have determined у for anhydrous hydrogen sulfide and have found it to be substantially equal to unity. No data are available for values of 7 in aqueous solutions of hydrogen sulfide, but its value probably lies in the range 1.00 to 1.05.

5. The equilibrium constant k for the gas-phase deuterium exchange reaction,

H2 0(g) + HDS(g) ^ HDOfc) + H2 S(g)

defined by

T’HDOT’HjS

^HjOJ’HDS

The mole fractions Ущо, *h5s> *hds> /hdo. and xHdo occurring in Eq. (13.134) will be expressed in terms of yHjs> xh, o. and Thds by the following equations derived from those given above defining H, S, 7, к, and a*:

Th, o = HyHiS *Has = s*H2o

•THjS

*a**H30.yHDS J’H. S

The result of substituting Eqs. (13.139) through (13.143) into (13.134) is

(fca*XHao3,HDs/VHas) + (t^h^o^hdsA’h. s) tfyHjS + ^H2s _ ka* + yS # + 1

kHyHDS + ^hds *h, o + SxHio kH+ 1 1+5

(13.144)

The remaining mole fractions have cancelled out, and a has been expressed in terms of H, S, y, k, and a*.

Equation (13.144) is the exact expression for the deuterium exchange separation factor in liquid-vapor mixtures of water and hydrogen sulfide at low deuterium abundances. Values evaluated from it are customarily used without correction up to 15 percent deuterium. When the vaporization of water into H2 S is small (H< 1) and the solubility of H2 S in water is small (S < 1), Eq. (13.144) reduces to Eq. (13.77).

A number of experimental measurements and theoretical calculations have been made of the equilibrium constant к for the gas-phase reaction that have been correlated by the equation

к =АевЯ~ (13.145)

Values of A and В given by four investigators and к zX 32 and 138°C from Eq. (13.145) are listed in Table 1321.

The equilibrium constant ka* for the gas-liquid reaction has also been determined by a number of investigators. Results at several temperatures are given in Table 13.22. Data of Geib and Seuss have been computed from their equation for к given in Table 13.21 and their equation (13.146) fora*:

a* = 0.8624e6s-43/r (13.146)

30 40 50 60 70 во 90 Ю0 HO 120 130 140 150 160 —————————————————- TEMPERATURE *C———————————————————— Figure 13.32 Solubility of H2S in liquid water.

Table 13.21 Comparison of equilibrium constants for gas-phase reaction H20 + HDS ^ HDO + H2Sf Source Geib and Suess Bigeleisen Varshavskii and Vaisberg Roth et aL Reference [C31 [Bill [VI) [R91 A 1.010 1.051 1.0084 1.001 В 233 218 219.0 221.3 к at 32°C 2.167 2.147 2.067 2.067 138°C 1.780 1.786 1.718 1.715 t k = AeB/TM

There are substantial differences among the results for к and for ka* given by the various investigators. The equations of Geib and Seuss have been used by Bebbington and Thayer [B7] in the most complete published account of the Savannah River plant. Table 13.23 compares values of the separation factor a for the hot and cold towers of the Savannah River plant computed by Eq. (13.144) from the data recommended by Bebbington and Thayer with values computed from the data recommended by Roth et al. The data recommended by Bebbington and Thayer have been used in this chapter because they have been successful in interpreting the performance of the Savannah River plant.

The dependence of a on temperature and pressure, as computed from Eq. (13.144), is shown in Fig. 13.33. In the cold tower an increase in pressure decreases a because it increases the concentration of H2 S in the liquid more than it decreases the concentration of H2 О in the vapor. In the hot tower, an increase in pressure increases a because it decreases the concentration of H20 in the vapor more than it increases the concentration of H2S in the liquid.

Optimum operating conditions. Because the deuterium recovery increases with increasing ratio of a in the cold tower to a in the hot, it might be supposed that the optimum operating conditions would be the lowest possible cold tower temperature, the highest possible hot tower temperature, and low pressure. Other factors beside a must be considered, however.

An increase in pressure above atmospheric leads to lower costs, despite the reduced spread in a’s between the hot and cold tower, because of the greater mass flow rate of gas per unit area that can be taken through the towers at higher pressure. At a pressure of 300 psig, however, there is a discontinuous increase in the cost of equipment, because of the need to

Table 13.22 Equilibrium constant ka* for gas-liquid reaction H20(/) + HDS(g) ^ HDO(/) + H2S<g) Temperature, °С ka* Calculated from Geib and Seuss [C31 McClure and Herrick [М3] Haul et al. [H4] Interpolated from Roth et aL [R9] 24 2.38 2.38 2.267 25 2.37 2.35 2.259 78 2.03 2.02 1.948 141 1.79 1.82 1.729

Table 13.23 Comparison of separation factors at conditions of Savannah River plant

Tower

Cold

Temperature, °С 32 138

Pressure, psia 292 313

Humidity Я 0.0036 0.215

Solubility 5 0.027 0.0096

Source of data	DP-400 [B7]	Roth [R9]	DP-400 [B7]	Roth [R9]
к	2.167	2.067	1.780	1.715
a*	1.0686		1.0111
kot*	2.316	2.218	1.800	1.737
У	1.0686	1.000	1.0111	1.000
a	2.275	2.178	1.576	1.518

change to the heavier pipe and fittings required for use in this higher pressure range. This sets the optimum pressure around 300 psi. The same pressure is used in each tower, except for pressure drop due to flow, to keep gas-recompression costs at a minimum.

The optimum temperature of the cold tower is as low as possible without risking formation of a third phase in addition to vapor and aqueous solution. Table 13.24 gives the temperatures at which solid hydrogen sulfide hydrate or liquid hydrogen sulfide form in the system H2S-H20. At 300 psi, the minimum safe cold tower temperature is around 30°C. The rapid increase in condensation temperature above 300 psi is another reason for this being the optimum pressure. Before the first pilot plant for the GS process was operated, the possibility of hydrate formation was not recognized, and freeze-ups occurred until the cold tower temperature was raised above 30°C.

The optimum hot tower temperature is around 130 to 140°C and is determined by a balance between the improvement in separation at higher temperature and the increased costs for heat and for humidifying the gas entering the hot tower at higher temperature.

Effect of hydrogen sulfide solubility and water volatility on analysis of process. The solubility of hydrogen sulfide and the volatility of water introduce changes in flow rates of gas and liquid and deuterium concentrations at the top and bottom of the hot and cold towers. Figure 13.34 illustrates the flow scheme and nomenclature to be used in working out these effects.

The flow rate of liquid into the cold tower is increased from F, in feed water, to Lc leaving the top tray of the tower, owing to formation of a saturated solution of hydrogen sulfide. Lc then remains constant throughout the cold tower. Between the cold and hot tower the liquid flow rate is changed to Lh because of withdrawal of product P, addition of condensate La, and vaporization of some gas, Ga. Lh remains constant through the exchange section of the hot tower down to the point where liquid is drawn off to the H2S stripper and vapor from the humidifying section is returned.

Vapor flows up through the cold tower at a constant rate Gc until in leaving the tower the rate is reduced to G0 owing to solution of some H2S in incoming feed water. The vapor flow rate to the hot tower is increased from G0 to Gh by hydrogen sulfide from the stripper and by the water vapor needed to saturate the hydrogen sulfide at the temperature of the hot tower. Gh remains constant in the hot tower.

It is possible to set independently three of the nine flow rates F, Lc, P, Lh, La, Gc, G0, Gh, and Ga. The other six are determined by the following material-balance equations:

Table 13.24 Equilibrium conditions for three phases in H20-H2S system^ Pressure, psia Temperature, °С Third phase 15 1.1 Hydrate 30 7.5 Hydrate 50 12.2 Hydrate 100 18.6 Hydrate 200 25.0 Hydrate 300 28.9 Hydrate 325 29.5 Hydrate + liquid H2 S 400 38.6 Liquid H2 S 500 48.3 Liquid H2 S 600 56.1 Liquid H2 S tData from Bebbington and Thayer [В7].

1+SC 1 +Hh 1+Sft 1 +HC Around the vapor coolers and condensers (DCHG, Fig. 13.34) Total flow: G* + Ga= La + Gc

In designing a plant, G0 and P might first be set. At several values of F, Eqs. (13.151) through (13.154) would then be used to evaluate Gc, Lc, Gh, and Lh. The ratios a^/G,. and GhlLf, ah would be determined; the optimum value of F that leads to the minimum number of plates is the one at which

Gh <*C^C

Lhah Gc

This is equivalent to Eq. (13.114).

With the values of the flow rates thus determined, the nine atom fractions of deuterium xct, ул, xcb, ycb, xP, xht, yht, xhb, ^ Унъ may be related to the composition of feed xF and the number of plates nc and nh in the cold and hot towers, respectively, by the nine equations (13.161) through (13.169), derived as follows.

At the top of the cold tower, a deuterium balance on the streams above and below the point of H2 S solution gives

_ (Le F)xct LcXa Fxf —

where хл/(/ка*)с is a sufficient approximation for the atom fraction of deuterium in the hydrogen sulfide transferred from gas to liquid. Because Lc « F( 1 + Sc), this may be approxi­mated by

(13.161)

A deuterium balance over the cold tower gives

Ge _ _£c

I УсЪ xcb. yct xct

r-c ljc

The Kremser-type equation (13.120) for the streams at the top and bottom of the cold tower, converted to the notation of Fig. 13.34, leads to

A deuterium balance between the hot and cold towers, on streams flowing across CDEFGH in Fig. 13.34, gives

Lcxcb+Ghyht = PxP + Lhxht + Gcycb (13.165)

Similarly, a deuterium balance over the vapor coolers and condensers, on streams flowing across CDGH in Fig. 13.34, gives

~~ f G^fa ~ ^а^-сУсЬ Gcycb ah

A deuterium balance over the hot tower, on streams flowing across FGHIJ, gives:

Gh _ Gh

xht~J^yht~ xhb ~J^yhb

The Kremser-type equation (13.120) for the streams at the top and bottom of the hot tower, converted to the notation of Fig. 13.34, leads to

(13.168)

The final equation is obtained by making a deuterium balance on the vapor stream entering the bottom of the hot tower. The hydrogen sulfide content of this stream consists of G0/(l + Hc) mol from the top of the cold tower plus LhShj{ + Sh) mol recycled by the stripper and humidifier from the liquid leaving the hot tower. The deuterium content of this latter hydrogen sulfide is approximately xhbj{ka*)h. The water content of this stream consists of G0Hc/(l + Hc) mol from the top of the cold tower plus GbHbl( 1 + Hh) — G0HCI( 1 + Hc) mol supplied by the humidifier and steam from the stripper. The deuterium content of this latter water vapor is approximately xhb. The balance equation expressing the deuterium content of the vapor entering the exchange section of the hot tower is

Abelson and Hoover [Al], working in the U. S. Naval Research Laboratory, found that thermal diffusion in UF6 at pressures above the critical (4.6 MPa) resulted in small but measurable enrichment of M5U at the hot wall. Because of the simplicity of thermal diffusion equipment compared with the advanced technology needed for the gaseous diffusion process, the Manhattan District in the United States in 1944-1945 used thermal diffusion of UF6 to raise its MSU content to 0.86 percent, to serve as partially enriched feed for the Y-12 electromagnetic plant. Energy for the S-50 thermal diffusion plant was obtained from steam which later powered the 150-MW electric generating station which drove the compressors of the K-25 gaseous diffusion plant.

The thermal diffusion plant [Al] contained 2100 columns, each with an effective height of

14.6 m. Each column consisted of three concentric tubes, the innermost being made of nickel,

»’Oak Ridge National Laboratory, U. S. AEC, Oak Ridge, Tennessee.

* Mound Laboratory, U. S. AEC, Miamisburg, Ohio.

§ Feed not of normal abundance, contained 1 percent 3He from nuclear reaction.

the middle of copper, and the outer of iron. The inner tube, about 5 cm in diameter, carried condensing steam at temperatures that could be varied from 188 to 286°C. The annular space (about 0.025-cm gap) between nickel and copper was filled with UF6 at a pressure of 6.7 MPa, well above the critical. The outer annular space between copper and iron carried cooling water at 63°C, slightly above the freezing point of UF6. The columns were operated batchwise, with periodic removal of slightly enriched UF6 from a header connected to the top of a group of columns and slightly depleted UF6 from a larger reservoir connected to the bottom. Operation of the complete plant of 2100 columns was affected by frequent leaks and freezeups, so that its performance is less representative than that of tests made in individual columns, which are summarized in Table 14.25.

Their separation performance was characterized by two parameters. Y is In yp/yp, the overall separation between top and bottom when equilibrium is attained at total reflux, ф is a parameter that was inferred from the rate at which product composition at total reflux approached equilibrium. The theory of the time-dependent separation performance of a thermal diffusion column developed by Cohen [C6] and others shows that ф is given by

(14.338)

where Ci and Cs are the parameters in the differential equation for the steady-state separation performance of a countercurrent column:

Operating conditions

Pressure, MPa	Steam, T	UF	6	Annular spacing Ar, cm	UF6 inventory, g		kW	K = CtLfCs	0 = CL/CS, g UF6 SWU1 day	capacity, ^max » kg U SWU/ yr	sep. cap., 6/Л max kW/(kg U SWU/yr)
Hot, Ґ	Cold, r"	Heat, H	Availability, Q
UF6	Steam
6.7	1.1	461	438	340	0.0273	2040	109	38	0.50	13.6	0.67	57
6.7	4.0	527	497e	341e	0.0256	1720	172	74	0.53	27.3	1.34	55
6.7	6.7	559	517	342	0.0248	1600	201	93	0.6	50	2.46	38
6.7	1.1	461	438	340	0.0253	1860	117	41	0.6	13.3	0.66	62
6.7	5.3	544	504e	341e	0.0230	1500	214	96	0.65	44.1	2.18	44
6.7	1.1	461	438	340	0.0225	1600	131	46	>0.7	6.2	0.31	148
6.7	5.0	540	500e	341e	0.0200	1320	216	96	0.8	26.4	1.30	74
6.7	6.7	559	517	342	0.025	1600	203	94	0.6	44.4	2.19	43
10	6.7	559	517	342	0.025	1700	198	92	0.77	31.5	1.55	59
20	6.7	559	517	342	0.025	1800	188	87	—	31.2	1.54	56

Temperature, К

Max. sep. Power/

tc, estimated, Q = H(1 — 300/T). L = 1460 cm. Y = In (yp/yF) at steady state at total reflux.

Д kg U SWU/yr = (0.238 kg U/352 g UF6) (365 day/yr) (0.80 C? L/4CS) (g UF6 SWU/day) = 0.0494 ф.

dy _ СіУ( — у) P{yp — у)

dz С$ Cs

An equation of the same form (14.181) was derived for the gas centrifuge treated as a countercurrent column. L is the active length of the column, 1460 cm.

The maximum separative capacity, and the power consumed per unit separative

capacity, G/Дпих, given in the last two columns of Table 14.25 have been calculated from Abelson’s parameters Y and ф to permit comparison with the other processes for enriching uranium treated in this chapter. Because the thermal diffusion column operates with constant reflux ratio, its steady-state separation performance as an enricher is given by Eq. (14.237), expressed here in the form

УР=_____________ (P/Ci) + 1______________

yF CP/С,) + exp {- [(P/CO + 1] (CiL/Cs)}

Its separative capacity A, for у < 1, is

A = — P fin —— — + A (14.343)

У yF УР )

With ур/ур from (14.340) and the separative capacity A from (14.343), the maximum value of A at CtL/Cs around 0.6 is found to be

(14.344)

At P/Сі around 1.8, 0.80 is the maximum value of the ideality efficiency Ej for this thermal diffusion column considered as a square enriching cascade.

The next-to-the-last column of Table 14.25 gives maximum values of the separative capacity of this thermal diffusion column if operated at the optimum product rate for each set of the operating conditions given in the first six columns. The last column gives the ratio of the power loss from heat input to separative capacity. The optimum set of operating conditions are those in the third row of Table 14.25, with a UF6 pressure of 6.7 MPa, a steam temperature of 559 K, and an annular spacing of 0.0248 cm. At these conditions this column would have a separative capacity of 2.46 kg uranium SWU/year and would consume heat equivalent to a power loss of 38 kW/(kg uranium SWU/year). The separative capacity of 2.46 kg uranium SWU/year of this thermal diffusion column 1460 cm high may be compared with the centrifuge of Tables 14.15 and 14.16, which had a higher separative capacity of 10 kg uranium SWU/year in a lower height of 335.3 cm. The specific power consumption of 38 kW/(kg uranium SWU/year) may be compared with 0.266 kW/(kg uranium SWU/year) for the U. S. gaseous diffusion plants. The much greater specific power of thermal diffusion was the principal reason that the Manhattan District’s thermal diffusion plant was shut down as soon as the K-25 gaseous diffusion plant began operation.

Although its very poor power utilization compared with gaseous diffusion and the gas centrifuge precludes use of thermal diffusion for large-scale uranium isotope separation, the simplicity of the equipment, the absence of moving parts, and the large separation attainable in
a convenient height have led to its use for small-scale separation of many isotopes, as suggested by Table 14.24.

At total reflux, the difference in composition between corresponding streams on adjacent stages is a maximum. As the reflux ratio is decreased, the difference in composition decreases, and reaches zero at minimum reflux. A condition for minimum reflux thus is

37+i =Уі

From (12.20),

_ (a — 1)у,<1 -Уд

Уї *1+1 у. + a(! — у.)

(Ni+i _ <УР ~Уі)Уі+ eQ -37)] V P /min (<*“IMl-37) In terms of the tails composition x,+I, this is (N>*i УРІСШІ*! + 1 — xi+1)-axi+1 P / min (a l)*l+l0 */+l) Several special forms of Eq. (12.78) will be useful. When У( < 1, (NuA v УР — У і a P ) min 37 a f

But this difference in composition is already given by the material-balance equation (12.62), so that

This equation is applicable to the portions of a heavy-water separation plant or 235 U plant near the feed point. In a close-separation cascade, in which о — 1 < 1,

These equations all show that the minimum reflux ratio increases as the composition departs more from product or tails composition. In isotope separation cascades in which q is close to unity, the minimum reflux ratio is enormous. For example, at the feed point of a plant
(12.82)

Yet, as the product end of this cascade is approached, the minimum reflux ratio approaches zero.

2.1 Terminology

In analyzing processes for separating isotopes by distillation, it is desirable to select as components those species whose proportions can be varied independently. When each molecule of the mixture being processed contains only one atom of the element whose isotopes are being separated, such as H216 О and H2l80, it is immaterial whether the components be selected as the pair (H2160, H2180) or (160, 180), as the mole fraction of H2180 in (H2160, H2lsO)is identical with the atom fraction of 180 in (160, 180). However, when the molecules of the mixture being processed contain two or more atoms of the element whose isotopes are being separated, such as hydrogen containing H2, HD, and D2, or water containing H20, HDO, and D20, it is necessary to choose as components those species whose proportions can be varied independently. In distilling a mixture of H2, HD, and D2, the amount of any one of the three components can be varied independently of the other two; the mixture is therefore treated as containing the three components H2, HD, and D2, and compositions are expressed as mole fractions of H2, HD, and D2. However, in distilling a mixture of H20, HDO, and D20, equilibrium is continuously maintained in the disproportionation reaction

2HDO ** H3 О + D2 О

so that the amount of only two of the three components can be varied independently. In this case, separation performance equations are simplest if compositions are expressed as atom fractions of deuterium or hydrogen.

All of the processes for separating isotopes of hydrogen or other light elements dealt with in this chapter involve distribution between a liquid and a vapor phase. To remain consistent with standard chemical engineering usage, component fractions in the vapor phase are denoted by у and the liquid phase by x. For a two-component mixture, the symbol у от x will denote the fraction of desired component (e. g., atom fraction deuterium in a mixture of H20, HDO, and DjO) in the vapor or liquid phase. For a mixture containing three or more components, a subscript will be used to designate the component. For example, ^hd denotes mole fraction HD in a vapor mixture of H2, HD, and D2. However, in mixtures of H2, HD, and D2 whose deuterium content is so low that the fraction of D2 can be neglected, the mole fraction of HD will be denoted by у or x without subscript.

In a two-component mixture, the separation factor a is defined as the fraction of desired component in the phase in which it concentrates divided by the fraction of desired component in the other phase. Deuterium, the isotope principally discussed in this chapter, almost always concentrates in the liquid phase. For such deuterium separation processes, the deuterium separation factor a is given by

,*/( 1 — x) УІІ1 — y)

This is the reciprocal of the equation used to define the separation factor in Chap. 12, Eq. (12.1). This change in notation for Chap. 13 is regrettable, but is hard to avoid.

Laser-based processes. Laser-based processes, which use intense, narrow-frequency radiation to cause atoms or molecules containing 235U to undergo selectively a different physical or chemical process than those containing 238U, are under intensive development in many countries, but have not yet advanced to industrial use.

The principal U. S. projects of this type are research at U. S. DOE’s Los Alamos Laboratory, which uses UF6 vapor, and work by U. S. DOE’s Livermore Laboratory and a joint venture of Avco Everett Research Laboratory, Inc., and Exxon Nuclear Company, which use uranium metal vapor. The two groups [J2, T3] using uranium metal vapor reported production of milligram quantities of partially enriched uranium in 1975. Avco and Exxon applied for a license to build a pilot plant to demonstrate their process in the mid-1980s.

Improved electromagnetic processes. Developments in plasma physics and magnet design in the 30 years since the Y-12 plant was taken off uranium isotope separation have caused many groups to reexamine electromagnetic processes for separating uranium isotopes, some of which reported at the London Conference on Uranium Isotope Separation [B20]. In the United States

Table 14.4 Gas centrifuge projects Owner Location Capacity, million separative work units per year Scheduled operation 1. Now operating Urenco-Centec (United King­dom, Holland, Germany) Capenhurst, England; Almelo, Holland 0.120 Since 1975 2. Under construction Urenco-Centec Capenhurst, Almelo 0.4-2.0 1977-1985 3. To be built U. S. DOE Portsmouth, Ohio 2.2-8.8 1986-1988 4. Under consideration Urenco-Centec Capenhurst; W. Germany Add 8 Late 1980s Japan 6 1985

a company, Phrasor, Inc., has been formed to continue development of an improved process of this general type. Dawson and associates [D3] have given a partial description of a process using ion-cyclotron resonance to ionize selectively and separate K-40. This process is being investigated for 235U with funding by U. S. DOE and TRW Defense and Space Systems.

Solvent extraction. At the 1977 International Atomic Energy Agency (IAEA) Conference on Atomic Energy at Salzburg, Austria, Commissioner Giraud of the French CEA announced development of a new process for producing uranium enriched sufficiently for reactor fuel, but impractical for producing more highly enriched weapons-grade material because it has too high a specific inventory. At the same conference, Dr. Frejacques and colleagues of the CEA [F4] said that “a new process using crown compounds of uranium is currently under study.” Such a process could involve complexing and fractional solvent extraction of 235U from an aqueous solution with a crown ether dissolved in an immiscible organic solvent.

The last two columns of Table 12.9 give the unit cost of product, in dollars per kilogram of uranium and dollars per gram of 235U, calculated from Eq. (12.152), with zp = 0.00711, the assumed transaction tails assay of хц/ = 0.003, and the unit costs employed in Sec. 5 of Chap. 3, cp = $89.11/kg uranium and eg = S100/SWU, which lead to the equation:

(2yP — 1) In + 219.5666yp — 6.4300

(12.157)

Because costs change frequently, this equation must be considered an example rather than a permanent relation.

Figure 12.19 has been calculated from Eq. (12.157). It shows the dependence of the unit cost of enriched uranium, in dollars per gram of 235 U, on the weight percent 235 U and shows the contributions to this cost from natural uranium feed and enrichment.

The deuterium exchange reaction between water and hydrogen discussed in Sec. 7 is one of a group of deuterium exchange reactions that have been extensively studied and are the basis for most of the world’s heavy-water production. Table 13.17 lists deuterium separation factors between liquid water and gaseous compounds of hydrogen for temperatures in the range 0 to 200°C. The ratio of the separation factor at 25°C to that at 125°C, a2s/ai2s> is also given. The higher this ratio is, the greater is the fractional recovery of deuterium and the smaller is the number of stages needed in the dual-temperature exchange process to be described in Sec. 11.

^A flow sheet like Fig. 13.21 would concentrate deuterium even if electrolysis produced no separation at all.

The reactions of Table 13.17 have been listed in order of increasing values for this ratio. Because water is one of the components of each pair in Table 13.17, processes based on these reactions could use liquid water as feed and thus would not be limited in output by limited feed availability.

Table 13.18 lists deuterium separation factors between gaseous hydrogen and liquid ammonia or methylamine, two compounds of hydrogen proposed for deuterium separation processes. The ratios of separation factors between the temperatures marked by a dagger, which have been proposed for dual-temperature processes based on these reactions, are also given. Both the separation factors and the separation factor ratios of the reactions involving hydrogen are greater than those involving water in Table 13.17. These higher values are what give the reactions of Table 13.18 their practical importance. A disadvantage of the reactions of Table

13.18 is that their deuterium production is limited to the amount present in commercially available hydrogen.

In all systems deuterium tends to concentrate in the phase normally liquid except ammonia-water at high temperature. Separation factors in chemical exchange are much higher than separation factors in distillation for the corresponding materials (cf. Table 13.3) except for ammonia-water. The high value of these separation factors and their strong dependence on temperature are what give the chemical exchange process its importance for separation of deuterium and isotopes of other light elements.

The deuterium exchange reaction between water and ammonia, water and hydrogen sulfide, or water and the hydrogen halides proceeds rapidly in the liquid phase without catalysis, because of ionic dissociation. In the case of a mixture of water and hydrogen sulfide, for example, the ionic equilibria

H20=*H+ + 0H h2s=*h+ + sh-

HDO-H+ + OD — HDS-H+ + SD-

HDO^D+ + OH’ HDS^D+ + SH-

permit rapid exchange of H+ and D+ between the two materials. Deuterium exchange between water and phosphine, water and hydrogen, ammonia and hydrogen, or methylamine and hydrogen does not proceed without catalysis. The water-phosphine reaction can be catalyzed by

Table 13.18 Separation factors in liquid-vapor deuterium exchange reactions involving hydrogen Reactants NH3 + HD CH3NH2 + HD Products NH2D + H2 CH3NHD + H2 а/К Separation factor a at 2 3 1 -50 6.6 7.90+ -25 5.19t 6.04 0 4.25 4.85 25 3.62 — 40 3.32 3.6+ 50 3.15 _ 60 2.99+ — 100 2.55 — 125 2.34 — Ratio at t 1.74 2.19 Reference [PI], [R4] averaged [R7]

strong acids [W4], the water-hydrogen reaction by nickel or platinum-metal catalysts (see Sec. 7), the ammonia-hydrogen reaction by potassium amide dissolved in liquid ammonia [C2], and the methylamine-hydrogen reaction by potassium methylamide.

Solutions used in the ammonia-water, water-hydrogen, and ammonia-hydrogen processes are relatively noncorrosive and may be handled in ordinary steel equipment. Solutions used in all of the other processes are relatively corrosive, and require use of stainless steel or other expensive construction materials.

The constant-boiling mixtures formed by water and the hydrogen halides make it difficult to use these systems in a practical exchange process.

Of the reactions listed, the water-hydrogen sulfide case has the greatest practical impor­tance because it needs no catalysis and has a fairly large change of separation factor with temperature. This case is discussed in detail in Sec. 11. The water-hydrogen reactions discussed in Sec. 7 and the ammonia-hydrogen and methylamine-hydrogen reactions, with their large separation factors and large change of separation factor with temperature, are also of practical importance.

In some cases the separation factors given in these tables have been determined experi­mentally from equilibrium constants К for gas-liquid reactions such as

H2 0(0 + HDS(?) * HDO(I) + H2 S(?)

In other cases, they have been derived from experimental measurements of the equilibrium constants к for gas-phase reactions such as

H2 0(g) + HDSQr) * HDO(g) + H2 S(?)

In still other cases gas-phase equilibrium constants have been computed by statistical mechanics from molecular properties. Procedures for calculating к have been described by Bigeleisen and Mayer [B12]. Varshavskii and Vaisberg [VI] have given a very extensive tabulation of values of к calculated for many deuterium exchange equilibria.

(*hdo+ ^DjoM^HjO +*hdo)

O’HDS + 2yD, s)/(2yHsS +3’hds) When the deuterium content of liquid and vapor is low, under a few percent, Xd3o ^*hdo> ^hdo^^HjO > etc., so that the above equation reduces to *hdo/xh, o

Expressions for the relation between К, к, and the chemical exchange separation factor a will now be derived. Let us consider first the exchange reaction between liquid water and gaseous hydrogen sulfide. As in distillation, the deuterium separation factor in the chemical exchange reaction is defined as the ratio of the abundance ratio of deuterium to light hydrogen in the liquid to the corresponding ratio in the vapor. In terms of the mole fractions of individual compounds in the liquid and vapor, the separation factor is’*’

HDO(I) + D2S(?)^ D2 0(0 + HDS(?)

must be also taken into account. These do not greatly affect the value of a, however. The equilibrium constant к for the gas-phase reaction is defined as

t In this equation the solubility of hydrogen sulfide in the liquid and the vaporization of water in the vapor have been neglected. These effects are treated in Sec. 11.

Because liquid-vapor exchange reaction is the resultant of vapor-phase exchange reaction and the vaporization equilibrium reaction

H, O(0 + HDCfe) ^ HDO(0 + H2OC?)

for which the equilibrium constant is the relative volatility a.*, defined by ^ *hdo/*h3o Унхю/Ун2о (13.75) it follows that K = ka* (13.76) so that a = ka* (13.77)

In the more general exchange reaction

MHm (0 + ZH2 ., Dfr) — MHm., D© + ZHrfr)

in which the liquid compound MHm and the gaseous compound ZH2 contain different numbers of hydrogen atoms, the separation factor is related to the equilibrium constant by

Values of ajK have been listed in Tables 13.17 and 13.18.

Notation. As used for isotope separation, the gas centrifuge is a cylinder of radius a and length L, rotating about a vertical axis with angular velocity w rad/s. Cylindrical polar coordinates are used, with the following notation for position and velocity components:

Direction	Position	Velocity, relative to solid cylinder rotating about axis with angular velocity ш
Radial, out from axis	r	u
Tangential	в (angle)	V
Axial, up from midplane	z	w

Properties of the light component of a binary mixture are denoted by subscript 1; heavy component by subscript 2.

Equilibrium separation. When a gas mixture in a centrifuge rotates as a solid body without motion relative to the wall of the cylinder, its pressure and composition are independent of в and г and vary with r according to the equations for equilibrium in a centrifugal field.

In a centrifuge rotating at со rad/s, gas of density p at radius r is subjected to centrifugal force of co2rp per unit volume, which equals the pressure gradient at that point.

	2- C4 3 II	(14.154)
Because	pm P~RT	(14.155)
	1 dp тшгг p dr RT	(14.156)

This equation is analogous to the equation for the change in barometric altitude h under gravitational acceleration g:

By integration, the pressure ratio or density ratio between an interior radius r and the outer wall of the centrifuge at radius a is

where va is the speed of rotation coa at the outer wall, termed the peripheral speed.

Table 14.12 illustrates pressure ratios for UF6 gas (m = 352) at several values of r/a for peripheral speeds of 400, 500, and 700 m/s at 300 K. Most of the gas is in a thin shell near the wall.

In a binary mixture of gases of molecular weights mx and m2, an equation like (14.157) describes the partial pressure ratio of each component,

where x is the mole fraction of light component. The local separation factor a(a, r) between radii r and a, obtained by dividing (14.159) by (14.160), is

The separation factor in the gas centrifuge thus depends on the difference between molecular weights, whereas in gaseous diffusion it depends on their ratio. Table 14.13 gives local separation factors for mixtures of 23SUF6 and 238UF6 (Am = 3) for the same speeds and radial locations as Table 14.12. Because most of the gas is in a thin shell adjacent to the wall, the more significant values are those for r/a near unity. Even with this restriction, the separation factor for the centrifuge is much more favorable than a0 = 1.00429 for gaseous diffusion.

Pressure ratio p(r)/pe for speed
va of

d lnpc/(l — x)] I _ — Anwlr (14.163)

dr ( equii RTa2

■— -(f) <14l64)

Transport equations. When centrifugal equilibrium is disturbed, as by establishment of counterflow or injection of feed and removal of effluents, flow of the gas mixture and of its individual components takes place relative to the rotating centrifuge. The analysis to be given has the following restrictions.

1. All gas is rotating at angular velocity w so that there is no angular motion relative to the rotating centrifuge. In a coordinate system rotating with angular velocity u>, v = 0.

2. Analysis is to be limited to the case of no radial motion of the gas as a whole, и = 0. This condition cannot hold at the top and bottom of the centrifuge, but may be nearly correct away from the ends, in the so-called long bowl development.

3. The change of pD with temperature and pressure, and thermal diffusion effects, are neglected.

Transport of light component is to be described in terms of its mass velocity, the vector J, with component Jr in the radial direction and Jz in the axial. In the coordinate system rotating at angular velocity a>, the angular component Jg is zero.

When the radial composition gradient 9x/9r differs from the gradient at equilibrium (9x/9r)equi|, transport of light component against the composition gradient takes place with radial mass velocity

The axial mass velocity Jz is the sum of a convective term pwx and a diffusive term —Dp(dx/dz):

Differential enrichment equation. Under steady-state conditions, the differential equation for conservation of light component, in cylindrical polar coordinates, is

1 9 (rJr) dJz 1 9 2Je

— —- + — +——— —

r dr dz г dB2

With Jr from (14.165), Jz from (14.166), and/g = 0, Eq. (14.167) becomes

Cohen [C6] made the following assumptions to simplify solution:

1. x(l — x) is treated as a constant.

2. d2x/dz2 is neglected.

3. dxjdz is independent of r.

4. pw is independent of z.

Эх _ Amai2 r2 x(l — x)____ 1_ Эх f

Г dr RT Dp dz Jo

because r(dx/dr) = 0 at r = 0.*

Integration of (14.169) requires use of boundary conditions for the net flow. In the enriching section, the net flow P is

r

P = 2n І pwrdr

Jo

pwr’dr’) Dp fzrdr

(14.173)

Because of assumption (3), this may be solved for dxjdz

+This condition and the lower integration limit of 0 are strictly correct only when a tube at the axis of the centrifuge is not present. In most centrifuges, with such a tube, the lower limit of integration should be the outer radius of the tube. However, at peripheral speeds of 400 m/s or higher, the density of gas at the central tube is so low that use of 0 for the lower limit of integra­tion introduces no significant error, and r(dx/dr) at the lower limit is, much smaller than at the upper limit.

F(r) = 2ir

Cohen used the notation

C, = f ЯгУ dr

Jo

C2 — itDpa2

[m»г

Jo

C5 =C2 + C3

In terms of these functions, the differential enrichment equation (14.174) becomes

dx C і ч P(xp-x)

dz=rsxil-x)——————

Here the variation of x with r has been neglected, as it is small compared to its change with z in a long centrifuge.

For the same reason, it is permissible to write a similar equation for the composition у of the enriched stream:

Because the coefficients Ct and C5 are to be evaluated for the velocity distribution with zero net flow, (14.181) is as valid an approximation as (14.180). When feed is added to the enriched stream, as in a centrifuge with feed introduced by a tube at the axis, Eq. (14.181) is easier to use than (14.180).

The equation corresponding to (14.181) for the stripping section is

In an exact treatment, values of Ct and C5 in the stripping section would differ slightly from the enriching section because of the slightly different flow profile. In the present approximate treatment, the constants are to be evaluated for the total reflux case in which the flow patterns in both sections are the same. If the net flow rate is a small fraction of the circulation rate, studies by Parker [PI] and others have shown that the effect on Ci and C5 of the changed flow pattern with net flow is small.

Equation (14.181) may be compared with the corresponding differential equation for the enriching section of a two-stream, close-separation, countercurrent column like a distillation column:

Ppp — y )

N

^Physically, F(r) is the total mass upflow rate between the center and radius r.

where h is the height of a transfer unit, a is the local separation factor, and TV is the flow rate of the stream moving from the product end of the column. Comparison of (14.181) and (14.183) shows that Cs may be interpreted as

	C5 = hN	(14.184)
and Ci/C5 as	C, 0- 1 Cs ~ h	(14.185)
Thus	й_£* h~ N	(14.186)
and	Q 1 II	(14.187)
In the countercurrent centrifuge N is the depleted stream flow rate
	Ґ N = 2л І pwr dr	(14.188)

where r, is the internal radius at which the axial velocity w changes sign. In the present approximation, in which the centrifuge parameters are evaluated for the velocity profile at total reflux, the flow rates of enriched stream and depleted streams are equal and an equivalent equation is

(14.189)

Local separative capacity. The separative capacity of a gas centrifuge per unit length, dA/dz, may be derived from Eq. (14.181) for the composition gradient, dy/dz. In the enriching section of a gas centrifuge the net flow rate of light component toward the product end is Pyp and of heavy component is P(1 — yP). As these flows make their way through gas of composition у against a composition gradient dyjdz, the rate of production of separative work per unit height dA/dz is

where S is the separative work associated with ri mass of component 1 and пг mass of component 2.

From Eq. (14.118):

£ — +?] I+«■ — «> [- _ P(yp y) dy уЧ 1 — J’)5 *

Using (14.181) for dy/dz,

dA ^ Сі P(yp — у) 1 Hyp — у)

dz “ Cs XI ~X C5 XI “X

The optimum value of the group of variables

is the value that maximizes dA/dz, at which

, _ Pb>p ~ У) ф~у( 1 — у) О II (14.197) Because d (d£ Ci 20 d<j> dz J Cs C5 (14.198) II a (14.199) and (dA C[ W2 / шах 2Cs 4Cs 4Cs (14.200)

In a centrifuge with axial flow independent of height, C1 and C5 are constant and condition (14.199) can be satisfied at only one height.

In terms of the parameters a — 1, A, and N,

Equation (14.201) is analogous to condition (12.125) for an ideal cascade, and (14.202) is the separative capacity of a stage of an ideal cascade divided by h.

The parameters C, Cs, and N, and from them A, a — 1, and the separative capacity, depend on the radial distribution of mass velocity pw(r). These parameters will be evaluated for two velocity distributions:

1. Mass velocity independent of radius

2. Berman-Olander distribution

Mass velocity independent of radius. The optimum radial distribution of longitudinal mass velocity wp(r) that leads to the highest possible separative capacity per unit length when condition (14.199) is satisfied is a distribution in which the mass velocity in one direction is independent of radius r up to the outer radius a, with all countercurrent flow in the opposite direction occurring in a cylindrical shell of infinitesimal thickness at the outer radius a. Under

the total reflux conditions to be used in evaluating Ci and C5, pvv = ^7 — NS(a) 1Ш (14.203) where 8(a) is the delta function in cylindrical coordinates defined by 8(r) = 0 (г Ф a) (14.204) and 2rr J r8(r) dr = 1 (14.205) From (14.175), the flow function F(r) for this case is Mr1 F(r) = — (г Фа) a (14.206) F(a) = 0 (14.207) From (14.176), ^ AmNa>2a2 AmNv Cl “ 4RT ~ 4RT (14.208) From (14.178), C * 3 8 nDp (14.209) Hence Amvl a 1 “ 4RT (14.210) from (14.187), and N 7Фра1 П ~ 8 nDp N (14.211) from (14.186). The maximum separative capacity per unit length, from (14.200), (14.208), (14.177), and (14.179), is (14.209), ( fdA TrDp(Am)2Va 1 ydz J mM _ 8(RT)2 1 + 8(nDpaflFf2 (14.212)

Cohen [C6] has shown that the first factor represents the maximum possible separative capacity per unit length for a centrifuge operating at peripheral speed va, in the absence of axial back diffusion. The second factor, termed the circulation efficiency Ec, takes into account reduction in separative capacity caused by axial back diffusion. It approaches unity as the circulation rate N increases or the radius a decreases. Values of the first factor for separating 23SU from 238U (Am = 3) at 300 K, using the value of Dp = 2.161 X 10*4 g UF6/(cm*s) recommended by May [M6] 7 are

va, m/s 400 500 700

First factor, kg UF6 SWU/(yr • m) 9.912 24.198 92.959

kg U SWU/(yr • m) 6.702 16.361 62.853

+This value, 7 percent lower than 2.32 X 10"4g/(cm*s) given by Dp = 4д/3, with p from Eq. (14.4), is used so that results will be consistent with May’s.

Berman-Olander velocity distribution. Even if the optimum radial distribution of mass velocity (14.203) could be established at one elevation in a countercurrent centrifuge, it could not persist over any distance because of the great shear force at the velocity discontinuity between the counterflowing streams. Determination of a more stable countercurrent radial velocity distribution, which would persist over a substantial length of centrifuge, requires solution of the hydrodynamic equations for motion of a compressible fluid in a centrifugal field. Even for the simplified “long-bowl” case considered here, in which the radial velocity и is zero and the longitudinal velocity w is a function only of r, the solution procedure is difficult and the equations complex. During the past 20 years, long-bowl solutions of progressively increasing rigor have been given by Parker and Mayo [PI], Soubbaramayer [S7], Berman [B16], and others.^ Olander [01] showed that Berman’s solution could be approximated for the large values of

met in centrifuges of practical importance by

(14.214)

w0 is an adjustable parameter proportional to the circulation rate N. The mass velocity at radius r is the product of Eqs. (14.214) and (14.158):

wP(r) .

w0p(e)

(14.215)

Equations (14.214) and (14.215) approach zero as r -*■ a and thus properly represent the condition of no slip at the outer wall. They fail to represent exactly conditions as r -*■ 0 because w(r) should be finite for a centrifuge without a central tube or should be zero as r -* r0 when the centrifuge has a central tube of radius r0. However, for a practical centrifuge with peripheral speed over 400 m/s, for which A2 for UF6 > 11.3, wpir)/w0p(d) from (14.215) at rja = 0.1 is less than 1.2 X 10"s times its maximum value, so that little error is made in replacing wp from (14.215) by zero at r/a < 0.1.

In Fig. 14.17 the curve marked “mass velocity” is a plot of Eq. (14.215) for A = 11.3, corresponding to peripheral speed va = 400 m/s. Most of the flow occurs in the outer half of the cross-sectional area, with flow reversal taking place at r2 la2 = 0.88, so that all heavy-fraction flow occurs in the outer 12 percent of the area. At 700 m/s this area shrinks to only 3.6 percent of the total cross section.

The curve marked “flow function,” obtained from

(14.216)

Theoretical analyses of flow and separation in a gas centrifuge published too late for inclu­sion in this text may be found in [S2a] and [Via],
is proportional to the total mass flow in the direction of light fraction through the area between the center of the centrifuge and radius r.

For physical interpretation, it is instructive to cast these equations into a form that contains the internal circulation rate N. N is given by

(14.217)

where rt is the radius at which w changes sign and

(14.218)

From Eq. (14.200) the maximum value of the separative capacity per unit length, for a given value of N and velocity profile, is

Л’2 f[56] [f(r/a)]2 d(r/a)/(r/a) Jo_________________ 2irDp [/(r,/a)] 2

‘dA _ TiDp(bm)2 Vg 4/1 ________________ 1_____________

dz)^’ S(RT)2 I3 1 + mnDpaf/N2} (/(r1/a)]2/4/3

Equation (14.226) has been written in this form to facilitate comparison with Eq. (14.212) for the maximum separative capacity obtainable from the optimum, constant mass velocity profile. The first factor in these two equations is the same and is the maximum separative capacity per unit length obtainable for a given va. The second factor,

is called the flow pattern efficiency and represents the reduction in separative capacity caused by departure of the mass velocity profile in an actual centrifuge from the optimum, constant mass velocity profile.

The third factor,

is the circulation efficiency, which takes longitudinal back diffusion into account. It approaches unity as the radius a decreases or the circulation rate N increases. It is written in this form to facilitate comparison with Eq. (14.212) for the optimum, constant mass velocity profile. The two expressions differ by the factor [firja)]214I3 in the denominator of (14.228).

Table 14.14 gives the results of calculation of these centrifuge parameters for the Berman-Olander velocity profile (14.214) for UF6 at 300 К and peripheral speeds of 400, 500,

Table 14.14. Functions of Berman-Olander velocity distribution (14.214) for UF6 at 300 К Peripheral speed = wa, m/s 400 500 700 , /352 v[57] 1 [58] [59]a a’-**rt ) 11.3 17.6 34.6 — , at which w — 0 a 0.938 0.963 0.982 Eqs. (14.215) and (14.218) 0.02253 0.01638 0.00861 — і’лт 2.4945E-3 1.1841E-3 0.2904E-3 2 d(r/a) r/a 44.24E-6 14.60E-6 1.736E-6 4 Flow pattern efficiency, Ep = -7— h 0.5626 0.3841 0.1943 [/(»Т/«)]* 4/3 Amvl /. 2.868 4.594 10.68 Separation factor, a 1— . RT /(r,/e) 0.02131 0.02174 0.01988

NS=N° exp p A* ^2 + (14.229)

where L = length of centrifuge

N% = circulation rate at bottom

A, is the decay constant for the circulation rate, whose dependence on the peripheral speed va and centrifuge radius a can be estimated from the aerodynamics of the centrifuge. Qualitatively, A; is higher the larger va and the smaller a.

End cap thermal drive. When flow is induced by heating the top and cooling the bottom of the centrifuge, as in Groth’s machine (Fig. 14.14), and the lateral wall is isothermal, the circulation rate decays exponentially from both ends and can be represented qualitatively by

Ne = №e cosh A*z (14.230)

Here Ng is the circulation rate at the midplane (2 = 0) and Xg is the decay constant, which is higher the larger va and the smaller a.

Wall thermal drive. When a linear temperature gradient is imposed on the lateral wall of the centrifuge, Durivault and Louvet [D7] have shown that the circulation direction at the wall is in the direction of increasing wall temperature. The rate is highest at the midplane and decreases to zero at the top and bottom. Hence the circulation rate for this type of drive can be modeled approximately by

0 cosh (AwL/2) — cosh Awz w w cosh (XWLI2) — 1

Here is the circulation rate at the midplane and Aw is the decay constant.

Combination of drives. In an actual centrifuge driven by a motor at the bottom, motor inefficiency introduces heat at the bottom end cap. The longitudinal variation of circulation rate then depends on where this heat is removed and whether other heat sources are present. Examples of centrifuge separation performance will be given for two cases:

1. Constant circulation rate, independent of 2

2. Optimized circulation rate, varied for maximum separative capacity at every elevation

Centrifuge considered. The centrifuge example whose separation performance is to be evaluated has the dimensions of the centrifuge tested by the Standard Oil Development Company in 1944 and described by Beams et al. [B3]:

Length, L = 335 cm Radius, a = 9.15 cm

which was tested at a peripheral speed of 206 m/s. May [M6] has evaluated separation parameters for a centrifuge of the foregoing dimensions for peripheral speeds of 400, 500, and 700 m/s possibly obtainable with more modem materials. Dimensionless integrals used in separation performance equations have been given in Table 14.14 for these speeds.

Circulation rate independent of height. The case of circulation rate independent of height is analogous to distillation at constant reflux ratio. For this case, explicit equations can be given for overall separation performance of the centrifuge. Conceptually, a constant circulation rate might be realized by a proper combination of end cap thermal drive, Eq. (14.230), and wall thermal drive, Eq. (14.231). Specifically, if

= X

¥-■)

the circulation rate has the constant value

N = Ne + Nw = №e cosh — V" (14.234)

When N and the radial velocity profile are constant, the separation parameters Ci and Cs in the differential equations (14.181) for the enriching section and (14.182) for the stripping section are independent of position z. For the low-enrichment case (y < 1) to be treated here, the equations may be linearized to

Enriching: C5 ^ = (C, + P)y — Pyp (14.235)

Stripping: Cs ^ = (Ci — W)y + Wyw (14.236)

The integral of (14.235) between у = yp at the feed point, z = 0, and у = Ур at the top, z =

Le, is

Ур_ P+Ct

У§ ~ p + c, exp [- (P + C,)WCs] (14.237)

The integral of (14.236) between у = y-ц, at the bottom, z = —ід, and у = yp at z = 0 is

W-C:

Л W-C, exp [-(W-COLsICs]

The material-balance equation on light component at the feed point where feed rateP+W joins enriched stream at rate N — W is

(TV — + (P + W)yF = (N + P)yf (14.239)

Overall material balance on light component is

Wyw + Pyp = (W + P)yF (14.240)

For given values of yP, LE> P, W, and TV, these equations are sufficient to determine the product composition yP, tails composition уц>, and the heads compositions yp leaving the stripping section and yp entering the enriching section.

To avoid mixing losses at the feed point, the solution for which

yjr=yF (14.241)

is desired. When this is true, yp also equals yF, from (14.239). When LE, ід, yF, N, and the feed rate F = P + W are given, it is necessary to find by trial the value of P (or W) at which the preceding five equations are satisfied. The condition for this is obtained by substitution into the overall material-balance equation (14.240) for yplyp from (14.237) and Ур/yw from
(14.238):

“ W /ур

-j— + P [Zy=W + F) (14.242)

Ур/yw Ур/

The separative capacity Д for this low-enrichment case, from Eq. (12.141), is

Д —P In yP — W In yw + F In yp

(14.243)

When the no-mixing-loss condition (14.241) is satisfied, the separative capacity, from (14.243), (14.237), and (14.238), is

(14.244)

This separative capacity is lower than the product of the maximum separative capacity per unit length (dA/dz)max, given by Eq. (14.200), and the length L = LE + L$. The ratio

is termed the ideality efficiency Ej. In terms of the three efficiencies: ideality efficiency £>, Eq.

(14.243) , circulation efficiency Ec, Eq. (14.228), and flow pattern efficiency Ep, Eq. (14.227), the overall separative capacity is

Centrifuge example. Tables 14.15 and 14.16 summarize calculations of the separative capacity of a centrifuge 335.3 cm long, 18.29 cm in diameter, run at a peripheral speed of 400 m/s at 300 K, with circulation rate independent of height. Some of these were given by May [М6]. In Table 14.15, the circulation rate for all cases is 0.1884 g UF6/s, and the feed rate is varied. The separative capacity has a maximum of 10.05 kg uranium SWU/year at a feed rate of 0.038052 g UF6/s (1200 kg UF6/year. Д remains close to 10 with a variation in feed rate of ± 20 percent, but decreases considerably at feed rates outside of this range. The axial separation factor is 1.67 at the lowest feed rate, decreases steadily with increasing feed rate, and equals 1.37 at optimum. The height of a transfer unit is 12.39 cm. The maximum ideality efficiency, at the optimum feed rate, from (14.245), is 0.8147. The circulation efficiency, from (14.228), is Ec = 0.9757. The overall efficiency E = EpEcEj has a maximum value of 0.4472.

In Table 14.16, the feed rate is held constant at 0.03171 g UF6/s (1000 kg UF6/year) and the circulation rate is varied. The separative capacity has a maximum of 10.03 kg uranium SWU/year at an optimum circulation rate N = 0.1884 g UF6/s and decreases rather rapidly with changes from this rate. The axial separation factor has a maximum of 1.41 at the optimum circulation rate. The height of a transfer unit increases almost proportionally with circulation rate. The circulation efficiency increases from 0.9095 at the lowest circulation rate of 0.0942 g/s to practically unity at the highest, showing the decreasing influence of axial back diffusion as circulation rate increases.

In Tables 14.15 and 14.16, the cut (ratio of product flow rate to feed flow rate) at conditions that lead to maximum separative capacity is 0.45. Because the centrifuge is

Table 14.15 Effect of feed rate on separation performance of gas centrifuge

Length: stripping, L$ = 167.65 cm; enriching, LE = 167.65 cm Radius: a = 9.145 cm Temperature: 300 К

Peripheral speed: va = 40,000 cm/s;A2 = 11.3 Circulation rate: N = 0.1884 g UF6/s Centrifuge parameters: Сг = 0.00402 g UF6/s

Cs = 2.3347 (gUF6-cm)/s Radial enrichment factor: a — 1 = 0.02131 Height transfer unit: h = 12.39 cm

Efficiencies: flow pattern, Ep — 0.5626; circulation, Ec = 0.9757

UF6flow rate, g/s Feed 0.006342 0.019026 0.031710 0.038052+ 0.044394 0.06342 Product 0.002749 0.008409 0.014239 0.017200f 0.020200 0.02928 Tails 0.003593 0.010617 0.017471 0.020852+ 0.024194 0.03414 Axial separation factors Heads, (3 1.29638 1.23601 1.19165 1.17394 1.15851 1.12318 Tails, 7 1.29314 1.22993 1.18510 1.16752 1.15246 1.11812 Overall, /З7 1.67640 1.52020 1.41222 1.37060 1.33514 1.25585 Cut, в 0.433 0.442 0.449 0.452 0.455 0.462 Separative capacity, kg U SWU/yr 4.484 8.859 10.03 10.05 9.830 8.749 Efficiencies Ideality, Ej 0.3635 0.7182 0.8132 0.8147 0.7970 0.7093 Overall, E = EfEcEj 0.1996 0.3943 0.4464 0.4472 0.4375 0.3894 ^Optimum.

connected in a cascade, cascade conditions may require the centrifuge to operate at a somewhat different cut. This would violate the no-mixing-loss condition at the feed point (14.241) (unless the feed location can be changed) and reduce the separative capacity.

May [M6] has calculated the effect of varying the circulation rate N on the separative capacity of this centrifuge example operated at peripheral speeds of 400, 500, and 700 m/s, at a feed rate of 0.03171 g UF6/s, using the parameters of Table 14.14, with results shown in Fig. 14.18. The optimum heavy-stream flow rate N increases with increasing speed. The separative capacity at optimum N increases as u2’02.

Optimum distribution of circulation rate. When the circulation rate N is independent of height, the parameters Ct and C5 are constant. It is thus possible to satisfy condition (14.199) for the composition variable and (14.200) for the maximum separative capacity gradient at only one elevation and one value of у in each of the enriching and stripping sections. This is what causes the ideality efficiency for the constant N condition to be less than unity. We shall now give an example of a centrifuge in which the heavy-fraction flow rate N is varied so as to have its optimum value at every height in the centrifuge, and will find by how much the overall height of a centrifuge of a given capacity could be reduced compared with one with uniform N.

The composition gradient dy/dz in the enriching section may be expressed as a function of the composition у and the circulation rate N by Eqs. (14.181) and (14.179):

dy N(CxjN)y{ — у) —Р(ур-у) „ „

— =————————————- (14.247)

dz N* (C3/N2) + C2

Table 14.16 Effect of circulation rate on separation performance of gas centrifuge

Length: stripping, Ls = 167.65 cm; enriching, Lp = 167.65 cm Radius: a — 9.145 cm Temperature: 300 К

Peripheral speed: i>„ = 40,000 cm/s; A1 = 11.3 Radial enrichment factor: a — 1 =0.02131 Feed rate: F = 0.03171 g UF6/s (1000 kg UF6/yr)

~ о * —»-1 ОІ Circulation rate, N	0.0942	0.1884+	0.2200	0.3768	0.5652
Product	0.01431	0.014239f	0.014345	0.014796	0.01511
Tails	0.01740	0.017471 +	0.017365	0.016914	0.01660
Centrifuge parameters
Ci, g/s	0.00201	0.00402	0.00469	0.00804	0.01205
Cs, (g • cm)/s	0.6262	2.3347	3.1630	9.169	20.56
Height transfer unit h, cm	6.65	12.39	14.38	24.3	36.4
Axial separation factor
Enriching, p	1.1384	1.19165′	1.1856	1.1366	1.0966
Stripping, у	1.1285	1.18510+	1.1810	1.1357	1.0964
Overall, Py	1.2847	1.41222+	1.4002	1.2908	1.2023
Cut, в	0.451	0.449t	0.452	0.467	0.477
Separative capacity,
kg U SWU/yr	5.29	10.03T	9.53	5.50	2.87
Efficiencies
Flow pattern, Ep	0.5626	0.5626	0.5626	0.5626	0.5626
Circulation, Eq	0.9095	0.9757	0.9821	0.9938	0.9972
Ideality, Ej	0.4601	0.8132	0.7676	0.4378	0.2277
Overall, E = EpEqEj	0.2354	0.4464	0.4241	0.2448	0.1277

UF6 flow rate, g/s

^ Optimum.

For a given speed va, Ci/N is a constant,

£■

N independent of N and y, as shown by Eq. (14.224). Similarly, C3/N2 is a constant,

B3-N*

independent of N and y, as shown by Eq. (14.225). In these terms,

dy _ 1УДіу(1 — у) ~ P{yP ~ у) dz ~ І^В3 + С2

The optimum value of N for a given у is the value at which

Э In (dy/dz) Віу(1 —у) IN В з _

bN NBiy( 1 — у)— Р(.Ур — у) І’РВз + Сг

or — ІЇВзВзУІ 1 — у) + ‘ШРВзІУр — У) + ВіС2у{ — у) = 0

In a centrifuge with the optimum value of N at enrichment dy from (14.250) is

QTfBj + Cj = 2*3 ^op, WDBx-PYe В, E

The last expression results from using (14.253) to eliminate Ye — In the low-enrichment case,

f dz _ 2*з d hi уJ min B

(14.257)

ftE 2е(Уе) = <*zmin =

The minimum length of enriching section Ze necessary to increase у from the feed value Ур to a higher value yE is

Figure 14.18 Effect of heavy — stream flow rate on separative ca­pacity of centrifuge at peripheral speeds of 400, 500, and 700 m/s. Length 335.3 cm; diameter 18.29 cm; feed rate 1000 kg UF6/year. (From May [M6J.)

A similar development for the stripping section leads to (14.259) where у _y-yw s Лі-у) (14.260) or 1 II £ (14.261)

for the low-enrichment case. The minimum length of stripping section zs necessary to decrease у from yp to a lower value is

f 0 2B3 FF ,

Zs(ys) = dzmin = Nr dny (14.262)

J-ZS Jys

To give an example of the reduction in centrifuge height for a given separative capacity that could be obtained if it were possible to use an optimized, variable heavy-stream flow rate instead of the uniform flow rate employed in Table 14.15, a centrifuge with optimized, variable heavy-stream flow rate was designed for the conditions of Table 14.15 marked with a dagger. These led to maximum separative capacity in a centrifuge operated with uniform heavy-stream flow rate. Centrifuge characteristics for the optimum flow-rate distribution are shown in Fig.

14.19 and compared with the uniform-flow-rate case in Table 14.17.

0.2 -0.1 0 0.1 0.2 0.3 Figure 14.19 Variation of optimum heavy-stream flow rate and enrichment with height in centri­fuge at 300 K, 400 m/s, and 1000 kg UF6/year feed rate.

Conditions common to both cases

Table 14.17 Comparison of centrifuges with uniform and optimized variable heavy-stream flow rates UF6 flow rate, g/s Feed 0.03171 Product 0.014239 Tails 0.017471 Separation factor Enriching 1.19165 Stripping 1.18510 Overall 1.41222 Separative capacity, kg SWU/yr 10.03 Conditions differing in optimized case Heavy-stream flow rate Uniform Optimized g UF6/s at feed point 0.1884 0.2837 at top 0.1884 0.0297 at bottom 0.1884 0.0297 Height, cm, enriching 167.65 140.41 stripping 167.65 142.02 total 335.3 284.43 Circulation efficiency Feed point 0.9757 0.9891 Top or bottom 0.9757 0.5000

In Fig. 14.19, height above feed point is plotted vertically to correspond with orientation of an operating centrifuge. Optimum heavy-stream flow rate has a maximum of 0.2837 g UF6/s at the feed location (z = 0) and decreases to 0.0297 at the top and bottom. These are to be compared with 0.1884 g/s in the uniform-flow-rate case. This decrease in flow rate from feed location to withdrawal ends of the centrifuge is qualitatively similar to that of the ideal cascade discussed in Chap. 12. However, the tails flow rate at the top, product end of the centrifuge cannot drop to zero, as it would in an ideal cascade, because dyjdz would become zero aX N = 0, as can be seen from Eq. (14.250).

In Fig. 14.19 composition is plotted horizontally as In y/yp, to bring out another difference from an ideal cascade. A plot of distance versus In у in an ideal cascade with constant height of a transfer unit (htu) would be a straight line. In this centrifuge with variable circulation rate, the htu from Eq. (14.186) varies from 18.4 cm at the feed elevation to 3.8 cm at the top and bottom. This causes In у to change more rapidly with z at the top and bottom than at the feed elevation.

As Table 14.17 shows, optimization of flow distribution permits reduction in centrifuge length for the stated separation performance from 335.3 to 284.43 cm. The ratio of these lengths, 0.8483, is somewhat greater than the ideality efficiency of the uniform-flow case, 0.8132 from Table 14.16. The reason for this may be seen by comparing the circulation efficiencies for the two cases. With variable flow rate, the circulation efficiency ranges from 0.9891 at the feed location to 0.5000 at top and bottom, compared with a constant efficiency of 0.9757 for the uniform-flow-rate case. Thus, the average circulation efficiency with variable flow rate is lower than with constant, a disadvantage that partially cancels the use of optimum flow rate at every height.

In Chaps. 12, 13, and 14, dealing with isotope separation, the composition of a mixture may be expressed in terms of the weight (or mass) fraction of each component, the mole fraction of

I і ♦xi Figure 12.8 Unit, stage, and cascade.

each component, or the atom fraction of each isotope. The relations among these three measures of composition may be illustrated by the example of water containing 0.79 weight fraction HjO (molecular weight 18), 0.19 weight fraction HDO (molecular weight 19), and 0.02 weight fraction Dj О (molecular weight 20). The procedure to obtain mole fractions from these weight fractions is shown below:

	Mol/g mixture	Mole fraction
H20	0.79/18 =0.0439	0.0439/0.0549 = 0.800
HDO	0.19/19 = 0.010	0.010/0.0549 =0.182
DjO	0.02/20 = 0.001	0.001/0.0549 =0.018
	0.0549	1.000

The atom fraction of deuterium is the ratio of the number of atoms of deuterium to the number of atoms of deuterium plus hydrogen in the mixture, or

(0.182 X 1) +(0.018X2)

———————— ~ = 0.109 (12.5)

The symbol z will be used to represent the fraction of a component in the feed stream to a unit, stage, or cascade; у the fraction in the enriched stream leaving a unit, stage, or cascade; and x the fraction in the depleted stream leaving a unit, stage, or cascade. The context will indicate whether weight, mole, or atom fractions are being dealt with. In the case of compounds containing a single atom of a polyisotopic element, such as UF6, atom fractions and mole fractions are identical.

For mixtures of two isotopes, the symbol z, y, or x refers to the fraction of desired isotope (for example, 235 U in the case of uranium or D in the case of hydrogen). For mixtures of three or more isotopes, the first subscript following z, y, or x indicates the specific isotope.

The location of a stream in a unit, stage, or cascade is also designated by a subscript, standing alone for a two-component system, or standing second after a comma for a multicomponent one. For example, z^p is the fraction of the ith isotope in feed.

Some relations for isotope separation plants are simpler when expressed as weight, mole, or atom ratios, defined as the ratio of the fraction of one component to the fraction of a second. These ratios are denoted by Greek letters f, £, or »j for feed, depleted, or enriched stream, corresponding to z, x, or y. In a two-component mixture, these ratios are defined as the ratio of the fraction of the desired component to that of the other component. For example, in a tails stream, the weight, mole of atom ratio for a two-component mixture is

(12-6>

For a multicomponent mixture the two components entering the ratio are designated by a double subscript, without comma, for example,

(»4 (ил)

x»

The ratio of atom fractions is frequently termed the abundance ratio. For example, the abundance ratio of 235 U to 238 U in natural uranium containing 0.007205 atom fraction 235 U and 0.99274 atom fraction 238U is 0.007205/0.99274 = 0.007258.

9.3 Separation Factor

In the ideal cascade discussed up to this point, each stage receives as feed two streams of the same composition, a tails stream from the stage next higher in the cascade and a heads stream from the stage next lower in the cascade. In such a cascade the heads separation factor Д, tails separation factor 7, and overall separation factor a are related by

P = y=aU2 (12.230)

The cut в at which condition (12.230) is satisfied is given by

1)2

fi + l

The cut thus ranges in value from l/(/3 + 1) at z = 0 to /3/03 + 1) at z = 1. Because (3 for most isotope separation processes is close to unity, в in this type of ideal cascade must be close to |.

In some isotope separation processes it is impractical to operate a stage at a cut of 5 for mechanical or hydraulic reasons, and in others the separative capacity of the stage is higher at a cut substantially different from f. In the Becker separation nozzle process described in Chap. 14, the separative capacity of a stage producing a heads stream at a given rate is substantially higher at a cut of | than at a cut of |.

To permit operation at a cut different from ; while still ensuring that the composition of heads and tails streams entering each stage be equal requires a more complex cascade

connection scheme than the one shown in Fig. 12.13. Figure 12.25 is an example of such a more complex cascade in which the cut of each stage would be approximately 5. In this cascade the heads stream leaving a stage is fed to the stage two stages up (at higher enrichment) in the cascade and the tails stream leaving a stage is fed to the stage one stage down in the cascade. Olander [01] calls this a “two-up, one-down” cascade. The condition for an ideal cascade, that the streams entering a stage have the same composition, applied to this cascade, requires that

*/+i = Уі-г

in terms of fractions, or

£i+ 1 — Vi— 2

in terms of ratios. From the definition of separation factor a,

Vi-2 ~ ab-2

From the definition of tails separation factor 7,

Іі— 1 = ?7—2 = 7І7-2

Similarly, ?/ = 7?/-і = 72£i-2

and |,+ i = 7 %i — 73£i-2

From (12.233), (12.234), and (12.237),

a = 73 7 = aW3

a =0y

7 ~ 1 = (<* , 1)g = (a — 1)6 (12.252a)

p + q

Thus, in a process like the Becker nozzle process, in which it is desirable to design stages for a cut of |, the cascade might advantageously be of the two-up, one-down type shown in Fig.

12.25 with p = 2 and q = 1.