History. The UCOR process, developed by the Uranium Enrichment Corporation of South Africa, has been operated on a large pilot-plant scale at Valindaba, Union of South Africa. Partial information on the process, its separation factor and specific power demand, and its projected economics was given by Roux and Grant [R3]. The ingenious Helikon cascade technique developed for this process, in which a single axial-flow compressor handles several process streams simultaneously, was described by Grant et al. [G2] and analyzed theoretically by Haarhoff [HI]. Cost estimates, prepared in 1974 and converted to dollars with the purchasing power of that year, predicted that the capital cost of the 5000 MT/year plant would be $1,350 million, and that the cost of separative work from it, using electricity priced at 6 mills/kWh, would be $74/kg SWU. This cost was close to the price then charged by the U. S. Atomic Energy Commission.

The UCOR project is a major effort. In 1975, some 1200 persons were employed, and $150 million had already been spent on development. Extensive experiments had confirmed the separation performance and power consumption of individual stages. A “prototype module” with design separative capacity of 6000 kg SWU/year had been built and tested. The design of a full-scale prototype, expected to have a capacity of 50,000 kg SWU/year, was well advanced. On February 14, 1978, S. P. Botha, South African Minister for Mines and Industry, announced [B18] that South Africa would expand the pilot enrichment plant to meet domestic needs, but had abandoned plans to build a full-scale plant.

Description of process. Because many features of the process, including details of the separating element, have not been disclosed, this description is necessarily incomplete. The following partial description has been given by Roux and Grant [R3]:

The South African-ог UCOR-process is of an aerodynamic type. It has been possible to develop a separating element which in effect is a high performance stationary-walled centrifuge using UF6 in hydrogen as process fluid. All process pressures throughout the system will be comfortably above atmospheric and depending on the type of “centrifuge” used, the maximum process pressure will be in a range of up to 600 kPa (6 bar). The UF6 partial pressure will however be sufficiently low to eliminate the need for process heating during plant operation, and the maximum temperature at the compressor delivery will not exceed 75°C.

The process is characterised by a high separation factor over the element, namely from

1.25 to 1.030 depending on economic considerations. Furthermore it has a high degree of asymmetry with respect to the UF6 flow in the enriched and depleted streams, which emerge at different pressures. The feed to enriched stream pressure ratio is typically 1.5 whereas the feed to depleted stream pressure ratio is typically only 1.12.

To deal with the small UF6 cut, a new cascade technique was developed, the so-called “helikon” technique, based on the principle that an axial flow compressor can simultaneously transmit several streams of different isotopic composition without there being significant mixing between them. The UCOR process must therefore be regarded as a combination of the separation element and this technique, which makes it possible to achieve the desired enrichment with a relatively small number of large separation units by fully utilising the high separation factor available. . . .

The theoretical lower limit to the specific energy consumption of the separation element can be shown to be about 0.30 MWh/kg USW. The minimum figure we have been able to obtain with laboratory separating elements is about 1.80 MWh/kg USW, based on adiabatic compression and ignoring all system inefficiencies. Although we do not believe that the present energy consumption can, in the short term be drastically reduced, the discrepancy between the above figures illustrates that the UCOR process still has a large development potential.

In discussion following presentation of the above information, the actual power consumption of a complete UCOR plant, allowing for pressure drops, and other process inefficiencies, was given as 3.5 MWh/kg SWU, or 0.40 kW/(kg SWU/year). This is to be com­pared with 0.50 estimated by Geppert [Gl] for a complete nozzle plant and 0.266 for the improved U. S. gaseous diffusion plants.

An additional important bit of process information, from Grant et al. [G2], is: “For the UCOR process, the cut is typically 0.045 to 0.055.” Figure 14.29 is a flow sheet for one stage of the UCOR process on which the preceding information has been represented, with a particular cut of в = 0.050. This cut requires use of a 19-up, 1-down cascade. The only important process variable not stated in published information is the UF6 content of the mixture with hydrogen fed to the stage. As will be shown in the next section, a UF6 feed composition of 0.032 mole fraction is consistent with the reported process information.

Enriched stream to stage i-i-19, pressure * (УІ5

Depleted stream from stage i*l, pressure = p/l.12

Enriched stream (Л from stage і -19, pressure * p/l.5

I Depleted stream to stage і — I, pressure * p/l. 12

Cut: 0.045 < в < 0.055 Separation factor: 1.025 < a: < 1.030 Specific power: Q/A — і.80 MWh/kg SWU Temperatures < 75°C

Theoretical analysis of UCOR process. Because the UCOR process has been characterized [R2] as a “stationary-wall centrifuge,” its performance for 23SUF6/238UF6 separation can be repre­sented by Eq. (14.276). The speed parameter A2 is related to the given pressure ratio р/р =

1.5 by

as may be seen from Eqs. (14.288) and (14.289).

In the UCOR process, unlike the separation nozzle process, the depleted stream is recompressed through a smaller pressure ratio (1.12) than the enriched stream (1.5). Hence, to evaluate the energy used in compression it is necessary to know the hydrogen cut 0H, the fraction of hydrogen fed that leaves in the enriched stream, and the composition of the enriched stream represented by the mole fraction fx of UF6 in it. A development analogous to the one that led to Eq. (14.271) for the UF6 cut results in Eq. (14.296) for the hydrogen cut:

because the molecular weight of hydrogen is mH =2. Because the fraction of flow area used by the enriched stream, c2/a2, is given by (14.272), the hydrogen cut 0H is related to the UF6 cut в by

л [6 + (1 — 6) exp (—A2)] ‘^176 — exp (—A2/176) вн= —

The mole fraction UF6 in the enriched stream 1, Fig. 14.29, is

_ Bf

Bf+e„( 1 — f) and the moles of UF6 (M) plus hydrogen (MH) in the enriched stream per mole of UF6 fed (M

+ N) is

Temperatures at points 1, 3, 6, 7, and 8 are assumed to equal T, the temperature of feed to the separating element. Then, the power input from compression is

K = (M + MH+N + NH) (T5 — T) (14.306)

because the heat capacity yR/{y — 1) = Cp is a linear function of mole fraction, Eq. (14.281). The energy input in joules per kilogram of UF6 fed, K/Z, is

the adiabatic, reversible energy input in joules per kilogram separative work is

К 2yR(TS — T)

A " 238(7 — 1)/ 6(1 -0X«-l)a

From an assumed feed temperature T = 313 K, a UF6 cut в = 0.05, and the stated expansion pressure ratios of 1.12 and 1.5 for the heavy and light fractions, a value for the mole fraction of UF6 in feed of / = 0.03225 was found by trial to lead to the value of 1.80 MWh/kg SWU given by Roux and Grant [R2] for the energy per kilogram uranium separative work.

Table 14.19 summarizes the steps in the calculation of compositions, properties, and flow rates of the numbered streams in Fig. 14.29, and from them, the energy per kilogram of uranium fed, the separation factor, and the separative work.

The following should be noted:

1. The high hydrogen cut, 0.73, coupled with the low UF6 cut, 0.05, causes the mole fraction UF6 in the enriched stream, 0.0029, to be much lower than in the feed, 0.032, and the mole fraction UF6 in the depleted stream, 0.105, to be much higher.

2. For every mole of UF6 fed, 21.9 mol of enriched stream and 9.1 mol of depleted stream are processed.

3. The maximum calculated temperature, 340.35 K, provides margin below the 75°C (348 K) maximum temperature cited by Roux and Grant, to allow for process inefficiencies.

4. The heavy fraction containing 0.105 mole fraction UF6 would start to condense at a pressure of 3.8 bar at 313 K. Hence the pressure of the heavy stream must be below this value and the feed pressure, p, must be below (1.12X3.8) = 4.3 bar. This pressure is much higher than the subatmospheric pressures reported for the nozzle process and would result in much lower volumetric flow rates in a UCOR plant than in a nozzle plant of the same separative capacity.

Table 14.19 Steps in calculating separation performance of UCOR process Variable Symbol Equation Value Mole fraction UF6 in feed f Assumed 0.03225 Temperatures to stage T і and Tз Assumed 313K Molecular weight feed m (14.283) 13.3036 R/Cp of feed (Г-П/7 (14.281 & 2) 0.259363 Speed parameter A2 (14.295) 11.31261 UF6 cut в Given 0.05 Hydrogen cut Єн (14.298) 0.728919 Mole fraction UF6 in enriched stream /і (14.299) 0.0022807 R/Cp of enriched stream (Ті — D/Ті (14.281 & 2) 0.286761 Mole fraction UF6 in depleted stream /з (14.301) 0.104573 R/Cp of depleted stream (Тз — D/Тз (14.281 & 2) 0.210766 Moles enriched stream^ Ш + МН)/(М + Ю (14.300) 21.9232 Moles depleted stream ^ (N + МН)/(М + Ю (14.302) 9.0845 Compression ratio, heads compressor Рі/Рі Given 1.5/1.12 Temperature from heads compressor т2 (14.303) 340.3507 К Temperature to feed compressor т* (14.304) 330.4904 К Compression ratio, feed compressor Ps/Pa Given 1.12 Temperature from feed compressor Ts (14.305) 340.3488 К Energy, MWh/kg U fed К/3.6 X 10* z (14.307) 3.1728E-5 Separation factor a (14.276) 1.027239 kg U separative work/kg U fed д/z (14.308) 1.7622E-5 MWh/kg SWU *73.6 X 109 A (14.309) 1.8005 ^Per mole UF6 fed.

5. The calculated separation factor of 1.0272 is in the range 1.025 to 1.03 cited by Roux and Grant and is higher than optimum in the nozzle process.

6. The value of A2 = 11.31 calculated for wheel flow is sufficiently high that even if the effective gas speed were well below that corresponding to the stated expansion ratio of 1.5, the separation factor would not be much below the calculated 1.027 value.

7. The specific power of 1.80 MWh/kg SWU, with no allowance for process inefficiencies, is equivalent to 0.205 kW/(kg SWU/year). This may be compared with 0.168 for gaseous diffusion (Table 14.9), and 0.31 for the nozzle process (Fig. 14.23). The higher value for the nozzle process may be due to its expanding the heavy stream through the full pressure ratio.

UCOR process equipment. The low cut, в = 0.045 to 0.055, selected for the UCOR process requires use of more stages than the gaseous diffusion or nozzle process, despite the higher UCOR separation factor. To reduce the number of independent items of process equipment, the UCOR process uses the ingenious Hilikon technique to consolidate as many as 20 stages in a single independently operable unit. Figures 14.30, 14.31, and 14.32, adapted from UCOR publications [G2, HI], provide a partial description of the Helikon principle and the process equipment used in it.

Each Helikon module uses two axial-flow compressors, one for the enriched streams (point 1, Fig. 14.29) and a second for the feed streams (point 4). The nature of flow through this type of compressor is such that there is rather little mixing of material fed into the barrel at one angular position with material of another composition fed at another angular position. Such streams of different composition flow through the compressor in helical paths and leave the compressor still relatively unmixed.

Figure 14.30 shows how the inlet end of the compressor would be divided into sectors to handle the streams fed to three stages with 23SU fractions increasing in the order zx < z2 < z3. Each feed stream is divided into two halves which are introduced symmetrically about plane AA through the axis into sectors formed by radial partitions. In this way, composition differences between adjacent streams are minimized. The partitions stop at the inlet rotor blades and begin again after the outlet blades. To deal with possible helical displacement during compression, the

Figure 14.30 Introduction of three streams of different 23SU content Zi <z2 <z3 into axial-flow compressor.

Figure 14.31 Schematic representation of flow through stage і of p-up, one-down Helikon module. (Reproduced with permission of the copyright holder, American Institute of Chemical Engineers, and Dr. W. L. Grant.)

outlet partitions may be displaced through an appropriate angle. In the UCOR plant with a cut of 55, 38 (2 X 19) sectors would be used.

The flow path through one sector of a Helikon module, containing all equipment of stage і except the light-stream compressor, is shown in Fig. 14.31. Depleted stream from stage і + 1 and enriched stream from stage і — p, both at intermediate pressure, are mixed and fed into one sector at the compressor inlet. At the compressor discharge the compressed feed is

Figure 14.32 Flow between modules of three-up, one-down Helikon cascade. (Reproduced with permission of the copyright holder, American Institute of Chemical Engineers, and Dr. W. L. Grant.)

collected in the appropriate sector, passed first through a stage cooler, and then through the separating element where it is divided into the low-pressure enriched stream and the intermediate-pressure depleted stream.

The enriched stream from each sector is transported to the enriched stream compressor for stage і + p in the module handling the next higher enrichment. The depleted stream is rotated by deflecting plates into the feed stream of stage / — 1 of the same module, or if from the least enriched stage, is sent to the highest stage of the module handling the next lower enrichment.

To illustrate the Helikon principle, flow between two adjacent modules of a three-up, one-down Helikon cascade is shown schematically in Fig. 14.32. The upper half shows the flow of depleted streams from one stage to the next lower stage; the lower half shows the flow of enriched streams from a sector of one module to the corresponding sector of the module of next higher enrichment. Because the figures are symmetric about the plane AA, the other half of the flow paths are not shown.

To permit construction of a complete plant with one size, or at most a few sizes, of compressor, while providing the variation in stage throughput desirable in an ideal cascade, it is proposed that the number of sectors in a module be varied to provide a smaller number of large sectors near the feed point and a larger number of small sectors toward the product and tails ends of the cascade.

Experiments reported by Grant et al. [G2] have shown that mixing of streams of different composition in an axial flow compressor can be kept acceptably low.

The number of stages needed for a given overall enrichment is inversely proportional to 6(a — 1). Because of its low cut the UCOR process needs more stages than the separation nozzle or gaseous diffusion process, despite its higher separation factor. This potential disad­vantage is dealt with by the Helikon technique, which combines a number of stages into a single module. Table 14.20 compares the gaseous diffusion process design of Table 14.9, the improved separation nozzle process of Table 14.18, and the UCOR process of Table 14.19 with respect to cut, separation factor, number of stages in an ideal cascade producing product containing 3 percent 235U and tails containing 0.25 percent 23SU, and the number of modules for such a UCOR plant cited by Grant et al. [G2].

In some stage processes, the heads and tails streams are separated in such a way that all portions of each stream have uniform composition. This occurs, for example, in a well-mixed electrolytic cell operated with steady flow of feed water and steady withdrawal of partially electrolyzed water. In other stage processes, the heads or tails stream may be withdrawn in such a way that the other stream changes progressively in composition during the separation process. This occurs, for example, when water flows through an electrolytic cell without mixing, and becomes progressively richer in deuterium, or when water is electrolyzed batchwise and becomes richer in deuterium as time goes on. These are examples of differential stage separation, in which successive small portions of one stream are removed from a second without mixing the second stream or giving the first stream further opportunity to exchange material with the second.

Two types of differential stage separation are illustrated in Fig. 12.10. In type A the stream being removed in small portions is depleted in the desired component, while the remaining stream becomes progressively enriched in this component; the concentration of

deuterium in batch electrolysis of water is an example of this type of differential stage separation.

In type В the stream being removed in small portions is enriched in the desired component, while the remaining stream becomes progressively depleted in this component. The flow of a mixture of 235 UF6 and 238 UF6 along the barrier of a gaseous diffusion stage is an example of this type of process. The small portions of gas that pass through the barrier are enriched in the desired component, U235 F6, and the remaining gas flowing along the upstream side of the barrier becomes progressively depleted in 235 UF6.

Equations relating the flow rates and compositions of feed and product streams in differential separation processes, first derived by Lord Rayleigh [Rl] for batch distillation, are often called the Rayleigh distillation equation. We shall derive some of these relationships for type В differential stage separation, using the nomenclature shown in Fig. 12.11.

At a point in the stage where a small amount of heads stream having flow rate dM’ and composition у is separated, the flow rate of the remaining depleted stream is changed by amount dN’ and its composition is changed by dx’. The material balance equation on total flow is

dM’=—dN’ (12.22)

and the material balance equation on flow of desired component is

у dM’ = —dix’N*) (12.23)

The result of elimination dM’ is

(12.24) (12*25)

-y’dN’ = — d(x’N’)

dN’ dx’

N’ y’ — x’

Heads

The result of integrating this equation from the feed end of the stage at which the flow rate is Z and composition z to the tails end where the flow rate is N and the composition x is

(12.26)

This is the general form of the Rayleigh equation. When the relationship between у and x is known, the equation may be integrated graphically or numerically.

For a two-component mixture, the relationship between y’ and x may be expressed in terms of a local separation factor a’, defined as

,_//(! -/) “ *70 -*’)

in analogous fashion to the stage separation factor defined by (12.15). The result of using this equation to eliminate у from (12.26) is

7d-*) =

(12.28)

When a is constant throughout the stage, this equation may be integrated to give . N ot, 1 — z . 1 . x Ы7= ‘ , Ы 1 _v-+ — , Ы 7 L а—і 1 х а—і z (12.29) Because (12.30) this may be transformed to

(1 — Є)(ах + 1 -*)

A relation between the stage separation factor a and the local separation factor a may be obtained from Eq. (12.31) by using (12.12) to replace z by у and (12.15) to eliminate y:

When a’ — 1 < 1, as in separating uranium isotopes by gaseous diffusion, this equation reduces to

In this form it can be seen that a is greater than a’, and becomes much greater as 8 approaches unity. Thus, differential stage separation may be used to enhance the difference in composition attainable in simple stage separation.

For type A differential stage separation, a similar derivation leads to

(12.34)

Because of Eq. (12.13) defining r and (12.16) defining /3,

_ 1

ol/(a’-l)

The equation corresponding to (12.33), applicable when a — 1 < 1, is

(a’-l)lnfl

This type of cascade may have practical application in a Becker nozzle plant or centrifuge plant for producing low-enriched uranium, with individual stages operated at a cut of around Figure 12.26 is a schematic diagram of stage connections showing the nomenclature to be used in solving the enrichment equations for such a cascade. Olander [01] has solved the enrichment equations for such a cascade.

The cascade receives feed of fraction zF at flow rate F and produces an upper product of fraction yP at flow rate P, a lower product of fraction yg at flow rate Q, and tails of fraction Хцг at flow rate W. For this two-up, one-down cascade, p = 2, <7=1, the heads separation factor (3 is

(3 = ctpl(p+q) = a2/3 (12.263)

and the tails separation factor 7 is

7 = aql(p+q) = a113 (12.264)

By counting the number of stages ns in the stripping section of Fig. 12.26, it is seen that

(1+є)5 («"у) -«(«-у) +*[(2 + 5)е(б-^) — о * *>» (‘ — т)]| ■ ТТТГГ, [-‘ + f ■ ~ 8,‘ + 0U61)

ZF _ У ПК— 1 — yHg +1 xw

— zF —y„s—x 1 — xw

Similarly, the total number of stages n satisfies

Ур _ n+l xw — Ур 1 — xw

For this low-enrichment case in which 1 — x <* Г,

_ j. , _ HzfIxw ) ”s + 1——— і——- In 7 _ + о _ fa(yp/xw) In 7 „+ 1 = lnQg/**) In 7

The separative capacity D of the two-up, one-down cascade is D = Щур — 1) In -2*— + Q(2ye — 1) In 1 —yp v 1-у + W(2xw — 1) In

Enriching section. Material balances above line BB in Fig. 12.26 are М,+Мі-і = P + Q + JV/+! and МіУі + Л/,_! у і — , = РуР + QyQ +Ni+l хі+ j but Уі = X*+l =уЙ+2

{[—(7 "*■!)] n~i — 7n—*)

f = H7 + 1)]

7(7 + Q/P)

(7 — 1)(27 + 1)

This may be confirmed by substitution of (12.287) into (12.286). For the three top stages, Eq. (12.187) gives

w—4 II ^ck	(12.288)
m„- 1 _ Q p p	(12.289)
— = 1 + 7 + 73	(12.290)

which are also obtainable by inspection of Fig. 12.26.

External flow rates. One relation among the external flow rates W, Q, and P is obtained by equating the heads flow rate from the top stage of the stripping section Mj evaluated from Eq. (12.278) with / = ns to the heads flow rate into the bottom stage of the enriching section Mj evaluated from Eq. (12.287) with і = ns. From Eqs. (12.278) and (12.280),

where r, s, and t are functions of n, ns, and 7. A second relation between W, P, and Q in terms of these variables may be obtained from the material-balance relations

F = W + P + Q (12.297)

and Fzf = Wxw + Pyp + Q}>q (12.298)

Inspection of Fig. 12.26 shows that

zF=XjyynS+1 (12.299)

yP=xwyn+2 (12.300)

and y<2=*w 7n+1 (12.301)

The result of eliminating F, zF, Xw, yP, and yg from Eqs. (12.297) through (12.301) is

(7"s+1 ~ 1)Н’ = (7И+2 — тЛ5+1)7>+ (7n + 1 “7ns+1)Q (12.302)

Equations (12.296) and (12.302) make it possible to determine the flow ratios Q/P and W/P as functions of n, ns, and y:

Q _ Kt”*2 — і)

P tiys+i — l)-r(y»+1 — yns+i)

W _ f(r"+2 -T,,S+»)-s(y> + 1 — ys+1) P t(ynS+1 — 1) — r(yn + i — ynS+1)

Design example. The foregoing equations will be applied to the two-up, one-down ideal cascade considered by Olander [01] having three stripping stages (ns = 3), seven total stages (n = 7), and a tails separation factor (7) of 1.3027. Values of r, s, and t then are

	Equation	Value
r	(12.292)	0.592674
s	(12.294)	15.99783
t	(12.295)	-2.99973

Table 12.12 gives compositions and flow rates relative to top product calculated from the preceding equations for feed containing 0.71 percent 235 U.

Section 7.4 described the development in Canada [S8] of a catalyst for the deuterium exchange reaction between hydrogen and liquid water that is not inactivated when submerged in water.

	Fig. 13.37 Ammonia process	Fig. 13.40 Methylamine process
Deuterium content relative to feed
First-stage product	100	95
Stripped synthesis gas	0.15	0.2
Number of stages
Cold	32.9	9.4
Hot	40.0	18.6
Molal flow rates, relative to product D2 0
Hydrogen feed	8,898	10,237
Hydrogen, cold tower, stripping	25,119	10,237
Liquid	3,200	2,160

Availability of this catalyst has led to interest in its possible use in dual-temperature water-hydrogen exchange processes. With liquid-water feed and recirculated hydrogen gas, this catalyst could be used in a dual-temperature process similar in principal to the GS process, with a schematic flow sheet like Fig. 1325. With ammonia synthesis-gas feed and recirculated water, this catalyst could be used in a dual-temperature process similar to the ammonia-hydrogen process flow scheme of Fig. 13.37, provided that impurities in synthesis-gas feed that would poison the catalyst can be recovered sufficiently completely.

Miller and Rae [M7] have suggested process conditions for a dual-temperature process using this catalyst at 69 atm pressure and temperatures of 50°C for the cold tower and 170°C for the hot. These conditions have been used to estimate optimum flow rates and numbers of theoretical stages for dual-temperature water-hydrogen processes using these two flow schemes. The results are tabulated in Table 13.28 and compared with similar data for the other dual-temperature processes discussed previously.

With water feed, the water-hydrogen exchange process has the advantages of lower gas and liquid flow rates and fewer stages than the water-hydrogen sulfide process. Utility requirements would also be smaller. Disadvantages of the hydrogen process are the higher pressure and the need to use large volumes of an expensive catalyst. If the catalyst were sufficiently active and not too expensive, the hydrogen process might be economically attractive.

With synthesis-gas feed, the water-synthesis-gas exchange process appears to be at a dis­advantage relative to the ammonia and methylamine exchange processes because the water pro­cess has the highest flow rates and the largest number of stages.

Absorption spectrum of uranium metal vapor. The absorption spectrum of uranium metal vapor is very complex, with over 300,000 lines at visible wavelengths. However, many of these absorption lines are very sharp, with sufficient displacement between a 23aU absorption line and the MSU absorption line for the corresponding transition, and without overlap of the 23 line with the «’U line for a different transition, to permit selective excitation of the 235U atoms. However, choice of the wavelength most suitable for a practical process is made difficult by the large number of possibilities. Janes et al. [J2] discuss some of the alternatives.

History. In the United States, laser isotope separation (LIS) with uranium metal vapor has been investigated experimentally by the Lawrence Livermore Laboratory (LLL) of the U. S. DOE at Livermore, California, and by Jersey-Nuclear-Avco Isotopes, Inc. (JNAI), a joint venture of Exxon Nuclear Company of Bellevue, Washington, and Avco-Everett (Massachusetts) Research Laboratory, which holds a number of patents on this method, of which the most significant are those of Levy and Janes [Jl, L2].

Workers at LLL [T3, D2] have reported production of milligram quantities of uranium enriched to 2.5 percent 235U by this method. In the LLL work, the source of uranium metal vapor was a uranium-rhenium alloy, chosen to reduce attack by the hot metal on the containing crucible. Deflection of 235U ions was by an electric field.

In the JNAI work, solid uranium metal is vaporized by an electron beam impinging on its surface, and deflection of 23SU ions is by either space charge expansion, a magnetic field, an electric field, or a combination. The JNAI process, as described in patents [Jl, L2] and a 1977 article [J2], has evolved through several stages. The next section describing the uranium metal LIS process follows the 1977 article.

Process description. Figure 14.41 is a schematic assembly drawing of one module of the JNAI uranium metal laser isotope separator. Figure 14.42 is a transverse section of this module. In Fig. 14.41, separation takes place inside a vacuum chamber about 1 m long. The uranium vapor source at the bottom consists of a charge of uranium metal, held in a water-cooled crucible, whose top surface is heated to 3000 К by a sheet of high-energy electrons curved and focused in a line on the uranium by a magnetic field of 100 to 200 gauss. Uranium vapor atoms diverge radially upward from the heated line source and travel in straight lines because of the absence of collisions in the high vacuum. These atoms flow upward between longitudinal, cooled,

Figure 14.41 Schematic diagram illustrating basic elements of the JNAI atomic LIS process. (Re­produced with permission of the copyright holder, American Institute of Chemical Engineers.)

product collector plates, oriented so that the atoms move parallel to them and do not impinge. The space between the plates is illuminated by light from a system of lasers, to be described later, which ionize most of the 235U atoms selectively, while leaving most of the 238U atoms un-ionized. The 235U ions, being electrically charged, can be deflected from the outward flowing uranium vapor and caused to impinge on and adhere to the product collector plates. Three possible methods for deflecting the 23SU ions are (1) expansion with energetic electrons released when the uranium is ionized, (2) motion in circular orbits around longitudinal magnetic field lines, or (3) deflecting by electric fields produced by giving adjacent collector plates alternative positive and negative charges. Un-ionized 238U atoms move outward beyond the product collector plates and condense on the upper cooled tails collection surface.

For maximum capacity, the uranium vapor density should be as high as possible. An upper limit is around 1013 atoms/cm3, because at higher density collisions between 235U ions and 238U atoms, or charge exchange between them, would occur too frequently, resulting in too high deflection of 238U. Assuming a plate height of around 4 cm, a flow area 4 cm wide by 100 cm long, and a uranium vapor thermal velocity of 40,000 cm/s, the uranium feed rate per module would be

— °-ш 8 a«*>

which represents a maximum daily 235U production rate of

(0.063 g uranium/sX0.00711 g 235U/g uranium)(86,400 s/day) = 39 g 235U/day (14.356) per module 1 m long.

The lasers used to ionize the 235U should be pulsed sufficiently often to irradiate all 23SU atoms passing between the plates. With a plate height of 4 cm and a vapor velocity of 40,000
cm/s, this requires a pulse repetition rate of 10,000 Hz. This, and other requirements to be described below, require development of lasers more advanced than any now available.

The light path through the module is limited to around 1 m to prevent the uranium metal vapor from itself becoming a laser, with consequent loss of selectivity. This length limitation prevents full utilization of laser photons in a single module and makes desirable connecting several physically separate modules optically in series as suggested by the second chamber shown in Fig. 14.41.

Laser requirements. In order to utilize photons efficiently, absorption by M5U should be selective. A 235U absorption line should be found that (1) occurs at a frequency at which 238U does not absorb, and (2) has a high absorption cross section, to reduce the light path needed for efficient use of photons. Because the isotope shift between 23SU and 238U absorption frequencies is of the order of 1 in 50,000, the first requirement calls for use of a very narrow 235U absorption line. Because the absorption lines for transitions in which uranium is ionized are very broad, it is necessary to ionize the 235U atoms in two or more steps, in which the first step is selective excitation of 235U to an energy level below the uranium ionization potential of

6.18 eV. This would be followed by less selective absorption of one or more additional photons of sufficient energy to ionize the excited 23SU atoms but of too little energy to ionize the unexcited 238U atoms. One of the JNAI patents [L2] suggests use of a narrow-frequency laser supplying visible light at 502.74 nm to excite 23SU, followed by ultraviolet light at 262.5 nm to carry the excited atoms over the 6.18 eV ionization level. At 502.74 nm, the 235U absorption line, of half-width around 0.001 nm, is displaced 0.01 nm from the 23®U absorption line, so that the required selectivity is obtained.

The energy E imparted to the 23SU atom by absorption of a photon of wavelength X is evaluated from Planck’s law,

(14.357)

h is Planck’s constant, 6.62559 X 10"M (J • s). c is the velocity of light, 2.997925 X 108 m/s. The energy in electron volts V is

where e is the electron charge, 1.60210 X 10 19 C, so that

he (6.62559 X Q-UX2.99192S X 108) _ 1,23981 X 10~6 eX ~ 1.60210 X КГ19 X (m) “ X (m)

Hence the energy given the 23SU atom by successive absorption of photons of wavelength 502.74 and 262.5 nm is

X (nm) E (eV)

502.74 2.466

262.5 4.723

7.189

Since the total 7.189 eV absorbed by 235U exceeds its ionization potential of 6.18 eV, this two-step photon absorption process imparts sufficient energy to ionize 235U. But since 238U absorbs only the 262.5-nm photon, 238U receives only 4.723 eV and is not ionized. Other possible combinations of two or more photons are described by Janes et al. [J2].

Even though the foregoing photon absorption process selectively ionizes 235U, charge exchange between 23SU ions and neutral 238U atoms and atomic collisions deflect enough 238U atoms to the collector plates to limit the heads enrichment factor to around 10. For example, a product content of 6 percent 23SU is the highest value that has been obtained from natural uranium. At the same time, however, very complete stripping of 23SU from tails is claimed. Three advantages cited for this kind of separation performance are as follows:

1. A single stage of separation suffices to produce uranium of high enough enrichment for light-water reactors.

2. More complete stripping is achieved than is economical in gaseous diffusion or the gas centrifuge.

3. This LIS process can be used to produce uranium containing 2 to 3 percent 235U from tails from these other processes.

The lasers for the process just described have three requirements more exacting than any yet developed:

1. They must deliver pulses with a frequency of 10,000 Hz.

2. To be economical they must last for a year or more to deliver over З X 10n pulses before replacement.

3. They must deliver far more energy per pulse than any high-frequency lasers now available.

Development problems. Despite the promise apparent in this laser enrichment process, it has a number of development problems. As just stated, lasers with higher repetition rate, longer life, and higher energy must be developed. Optical windows that do not lose transparency or mechanical integrity from deposition of uranium or intense illumination must be developed. Materials problems associated with handling corrosive uranium metal at high temperatures must be solved. Perhaps most important of all, convenient means must be developed for charging uranium to the high-vacuum, high-temperature system and for collecting and removing the separated uranium product and tails fractions. This LIS process appears to have one of the
disadvantages of the Y-I2 electromagnetic process, of having feed material deposit all over the vacuum chamber, necessitating troublesome interruptions for disassembly and clean-out.

Economic estimates. Despite these problems, JNAI was sufficiently optimistic about the ultimate economics to go ahead with pilot-plant construction. At this stage of development, however, economics are very uncertain. This is illustrated by Table 14.27, which compares estimates of process characteristics and costs made by JNAI and a Japanese group. The specific power estimate of Janes et al. is around that predicted for the gas centrifuge. The estimate of Ozaki et al., although 10 times higher, is lower than that of gaseous diffusion (Sec. 4.7). The unit investment costs predicted by both groups, although very different, are much lower than for gaseous diffusion or the gas centrifuge and are the principal reason for the interest being shown in this process.

Two features that make separative work cost estimates very uncertain are uncertainty about laser energy efficiencies and ignorance of operating and maintenance costs, which can be obtained only by completing the development and making life tests on plant equipment.

The number of stages in an ideal cascade may be evaluated by a procedure similar to that used in deriving Eq. (12.72) for the minimum number of stages at total reflux. The result is

In [yf(l ~xw)l( 1 — yp)xw] In ypQ. — xw)l( 1 —J’pfrK’]

In |5 In a

Thus the number of stages required for a given separation in an ideal cascade is just twice the minimum number needed at total reflux minus 1.

By a similar procedure, the number of stages in the stripping section is found to be

and in the enriching section

A relation between composition and stage number may be derived by a procedure similar to that which led to (12.68):

(12.96)

Уі = *l+i

Because q„ (12.97)

*/ = */-! =У/-2

The corresponding equations in the stripping section are

8.2 Reflux Ratio

The reflux ratio required to bring about condition (12.83) defining an ideal cascade may be found as follows. From (12.62),

Nt*i ^ УР ~Уі P Уі-Хі+і

But уі = r,+1 in an ideal cascade, and z,-+i is given in terms of xI+i by (12.19) with a = 0s, so that

Мы yptfxj+i + 1 -*f+i)-flXf+r _ 1 Гyp № — зуЛ

P (fi— l)*r+i(l “■*<♦!) 0-l[xi+l 1 —ЛГ/+1 J

This equation is the same as for minimum reflux (12.79), except that 0 replaces a.

Figure 12.16 is a McCabe-Thiele diagram for an ideal cascade. The equilibrium line, relating

Уі to Xf, is represented by the solid curved line, with the equation

Уі

1 — уі

The operating line, relating у,- to jci+], is represented by the dashed curved line, with the equation

у і _ y/uxj+1 1 — уі 1 — хі+1

The graphic construction shows that with these two lines x,-+1 = уui, as required for an ideal cascade. The straight line connecting the product point (yp, yp) with the point on the operating line (yi, x,+i) has a slope (ур—Уі)І(ур—Хі+1), which equals Ni+l /(jV,+1 +F), the ratio of tails to heads flow at this point in the cascade. Ni+1/P is given by Eqs. (12.101) and (12.102).

In the stripping section, the equation corresponding to (12.102) is

Щ 1 (l-xw folA

IV 0-ll-уі Уі)

An equation for the reflux ratio in the enriching section as a function of stage number may be obtained by substituting x,+1 from (12.99) into (12.102):

^ET =^Tbp(l — U’-") + (1 — ypW1-*- 1)] (12.106)

Similarly, in the stripping section, Eqs. (12.105) and (12.100) lead to

jbi l*wf& -1) + (1 -*wXl — Ґ)]

Although water distillation is no longer used for primary concentration of deuterium because of its high energy consumption, the principal features of water distillation plants for this purpose will be described briefly because they illustrate isotope separation principles so well.

Process requirements. Distillation of water for deuterium separation differs from all other industrial distillation processes in the extremely small difference in normal boiling point between the key components, 0.7°C between H2 О and HDO. This, coupled with the very low natural abundance of deuterium, leads to an extraordinarily high reboil vapor ratio, so that the heat consumption per unit of D20 product is enormous.

In fa<l — Хр)ІХр( 1 — Xp)] In o*

A rough idea of the requirements of the water distillation process may be derived from a representative separation factor of 1.05. The minimum number of theoretical plates (итіп) needed to enrich deuterium from the natural concentration of xp = 0.000149 atom fraction to product concentration of Xp = 0.998 is

The optimum number will be somewhat more than twice this, or around 700 plates.

The minimum consumption of steam per mole of heavy water produced is secured when an infinite number of plates is used, so that the outgoing steam depleted in deuterium may be in equilibrium with incoming feed.

From Eq. (12.80), the minimum molar ratio of steam flow rate G to product P is

For a practical plant, with a finite number of stages, around 200,000 mol of steam must be provided per mole of heavy water produced. Because of the small difference in boiling point between the two products, this large amount of heat flows through a relatively small temperature difference; in fact, the principal temperature differences are due to pressure drop across the column and temperature difference across reboiler and condenser heat exchange surface, rather than differences between the boiling points of the components. Economical operation requires that the large heat demand be supplied as nearly reversibly as possible, with the minimum practicable loss in availability. Reboil heat should be supplied with good thermodynamic efficiency, and column pressure drop should be minimized.

History of process. Despite these severe requirements, the water distillation process has been of interest because of its simple, conventional equipment. For primary concentration of deuterium from natural water, it received attention in Germany, where pilot-plant work was done by I. G. Farben during World War II [C3], and in the United States [M8], where most of the heavy water used by the Manhattan District was produced in this way.

Manhattan District plants. The water distillation plants of the Manhattan District were built to provide a simple and certain way of producing heavy water, although not necessarily at minimum cost. Because speed was more important than economy, it was not possible to explore fully developments that might have permitted more economical production. These plants are described briefly in this section; more detailed information has been given by Maloney and Ray [M8] and by Selak and Finke [S3].

Plants. Three water distillation plants were designed and built for the Manhattan District by

E. I. du Pont de Nemours and Company, Inc. These plants were located at Morgantown, West Virginia, Childersburg, Alabama, and Dana, Indiana. Parts of the plants were started up in June 1943, and concentrations reached steady-state values in June 1944. About 90 days were needed to reach steady state. The plants were shut down in October 1945 because of reduced demand and because of the high cost of their heavy water.

These distillation plants concentrated deuterium from 0.0143 a/о (atom percent) to 87 to 91 a/о. Further concentration to 99.8 percent was effected by electrolysis. The average recovery of D20 from the steam fed was only 1.94 percent; 360,000 mol of steam were, fed per mole of DjO produced.

The combined capacity of the three plants was 1.15 MT/month. The total production of

99.8 percent D2 was 20.7 MT.

The total cost of the plants was $14.5 million. The unit investment cost was therefore

$14,500,000

(1.15X12X1000)

The operating costs were as follows: Per month Per kg D2 0 Steam $295,000 $271 Other 127,000 117 Total $422,000 $388

Process. A simplified flow sheet of the process used at the Morgantown plant, the smallest and most efficient of the three, is shown in Fig. 13.3. This plant produced 254kg D20/month, with a deuterium recovery of 2.8 percent. The plant consists of an eight-stage cascade of distillation towers, with associated reboilers, condensers, and pumps. Summary data on the towers of each stage are given in Table 13.9.

The first stage consists of five parallel groups of two series-connected towers, of which one group, 1A and IB, are shown in Fig. 13.3. Feed for each 1A tower consists of condensate from the reboiler of the associated IB tower. Feed is introduced at the top of the 1A tower. Stripped vapor from the top plate is condensed in a barometric condenser, vented to a steam ejector that maintains a pressure of from 50 to 90 Torr at the top of the tower.

Slightly enriched water from the bottom of tower 1A is pumped to the top of tower IB, and vapor from the top of IB flows back to the bottom of 1A.

Most of the water at the bottom of IB, now enriched to 0.117 a/о deuterium, is converted to vapor in the reboiler and returned to IB, but around 12 percent is pumped ahead to the top of 2A. Vapor from the top of 2A is condensed in a condenser refrigerated with ammonia, to prevent loss of the now valuable water. This condenser is also vented to a steam ejector, which maintains a pressure of 130 Ton.

The second stage consists of two towers, 2A and 2B, connected in series, like each 1A and IB pair. The third and higher stages consist of single towers, of progressively decreasing diameter. Arrangements for reboiling water at the bottom of each tower and condensing and returning vapor at the top of the next stage are the same as at the bottom of 2B and the top of 3. The progressive decrease in tower diameter from the feed point to the product end is characteristic of an isotope separation plant.

As the water flows through the stages of the plant, it is enriched progressively in deuterium, until it reaches 89 a/о in the bottom product of the eighth and last stage.

Most of the steam for the plants at Morgantown and elsewhere was generated at 165 psia’’’ and throttled to 55 psia, the pressure at which it was used in the reboilers, even though steam at 22 psia would have sufficed to reboil the tower bottoms, where were at subatmospheric pressure. Because low-pressure, by-product steam was not available in the required amounts, it was necessary to generate steam solely for the water distillation plant. This was inefficient and added to the operating cost in these plants.

Towers. ToweTS of these plants over 18 in in diameter were of the plate type, with plates on 1-ft spaces. All the large towers used bubble caps, except 1A, which had tunnel caps. Towers 18 in in diameter and smaller were packed with |- by |-in ceramic rings.

Possible improvements. The designers of the Manhattan District plants recognized that two major improvements could be made in a future water distillation plant designed for economy rather than speed of construction. These were as follows:

1. More efficient utilization of heat than generating 150-psig steam solely for the distillation plant

2. The use of tower internals with greater capacity per unit volume than tunnel — or bubble-cap plates, to increase plant capacity for the same capital investment

More efficient utilization of heat. In the first-stage towers of the Manhattan District plants, where most of the heat was consumed, heat flowed from the tower bottom temperature of

^1 psia (pound force per square inch absolute) = 51.7 Torr = 0.06895 bar = 6895 Pa.

Waste condensate 0.0143% D 11,670 kg/h Figure 13.3 Morgantown water distillation plant.

Table 13.9 Towers of Morgantown water distillation plant Tower Number in parallel Diameter[44] No. of plates kg vapor/h Pressure, Torr a/o deuterium, bottom Top Bottom 1A 5 15 ft 80 (80,400) 67 238 IB 5 12 ft 90 80,400 238 536 0.117 2A 1 10.5 ft 72 (9,620) 129 340 2B 1 8 ft 83 9,620 340 645 1.04 3 1 3.3 ft 72 1,380 124 343 3.8 4 1 1.5 ft 72* 330 127 440 10.0 5 1 10 in 72t 85 127 340 11.5 6 1 10 in 72* 85 124 328 21.2 7 1 10 in 72* 90 124 333 56.4 8 1 10 in 72* 90 127 308 89 Total 18 757 92,070 tl ft= 12 in = 30.48 cm. * Number of theoretical plates in packed column.

195°F (10.5 psia)* to the tower top temperature of 111°F (1.3 psia). To transfer this heat through the reboilers, steam at 233°F (22 psia) was required. Because this heat is needed only at relatively low temperatures, it is very inefficient to obtain it by burning fuel under boilers, without making use of the heat at higher temperatures first. Two possible ways of providing low-temperature heat more efficiently are these:

1. Using 22-psia exhaust steam from the turbines of a power plant.

2. Using a vapor-recompression system

Examples of these two schemes are shown in Figs. 13.4 and 13.5. The turbine-exhaust scheme of Fig. 13.4 has two advantages over the vapor-recompression scheme of Fig. 13.5.

1. The cost of the condenser and steam jet ejector is less than that of the vapor compressor and feed-water preheater.

2. The power lost in the steam turbine plant is less than the power consumed by the vapor compressor. Although the theoretical power W lost by the turbine exhausting at 22 psia instead of at 1.3 psia is exactly the same as the power consumed by a compressor taking the same amount of steam from 1.3 to 22 psia, the actual turbine efficiency of E drops the turbine power lost to WE and the compressor efficiency of E? raises the power consumed by the compressor to W/E1.

The turbine exhaust scheme has the disadvantage of making the production rate of heavy water dependent on the production rate of power from the steam turbines.

Packed towers. After these plants were built, several improved types of tower internals were developed that have higher capacity per unit volume and lower pressure drop per theoretical

Figure 13.4 Water distillation tower reboiled by steam turbine exhaust.

plate than bubble-cap plates and are claimed to be more economical. The British Atomic Energy Research Establishment has developed a tower packing known as Spraypak [M4] for use especially in the water distillation process.

Distillation-cascade design principles. Some of the principles involved in designing an isotope separation plant for minimum cost will be illustrated by roughing out optimum conditions for a water distillation plant incorporating the two improvements noted above.

Design variables. The principal design variables in a water distillation plant are

1. The type of tower internals

2. The pressure p, Ton

3. The vapor velocity v, cm/s

4. The ratio of reboil vapor to product, G/P

Figure 13.S Water distillation tower reboiled by vapor recompression.

The best inemals and the optimum values of pressure, vapor velocity, and reboil vapor ratio are those that permit production of heavy water at minimum cost. The initial cost of the plant depends on a number of factors including the total number of towers, the total amount of reboiler and condenser surface, and the total volume of tower internals. The principal operating cost is for power, which is proportional to total loss in availability of steam as it flows through the towers. A complete minimum-cost analysis requires knowledge of the unit cost of all the important cost components and is beyond the scope of this book. Design for minimum volume of tower internals or minimum loss in availability due to tower pressure drop and for minimum cost of these two important contributors to total cost can be carried out without complete unit-cost data and will be discussed. Because the same choice of reboil vapor ratio minimizes the number of towers, their volume, and the loss of availability within them, this reboil vapor ratio is close to that which leads to minimum production cost. An equation for this optimum reboil vapor ratio will now be derived, and expressions will be developed for the total volume of towers and the total loss in availability in towers designed for the optimum ratio.

Enrichment equation. The differential equation for the increase in deuterium content x with distance z from the top of the tower is

А^=(а*-1М1-*)-^,-х) (13.12)

This equation is derived in a manner similar to (12.128); h is the height of a transfer unit, h dxjdz replaces dx/di, and G, the molar flow rate of steam, plays the role of the tails flow rate N.

Tower volume. At a point in the tower where the vapor velocity is v and the absolute pressure is p, the area A needed to accommodate a steam flow rate of G mol/s is

A=— (13.13)

where R is the gas constant and T the absolute temperature. The volume of tower dV needed to increase the deuterium content of the liquid by an amount dx is

dV A____________________ hRT/pv_____________

dx ~ dxjdz ~ [(a* — 1 )x(l — x)/G] — (P/G2XxP — x)

The steam flow rate that leads to minimum tower volume is that which makes this expression a minimum at every x, or

_ Щхр-х)
bopt (a*-iyx(i-x)

This is the tails flow rate for an ideal cascade. The details are the same as in deriving (12.132). At this optimum steam rate,

/dV = AhRT Pjxp-x)

V&/min pv(a* — 1)J X2(l — x)2

pv(a* — l)2

pv(a* — 1)’

In a tower in which h, T, p, v, and a* are held constant,

where Dpp, the separative capacity for the enriching section of an ideal cascade per unit

product rate, is given by

The factor 4hRT/pv(a* — l)2] gives the tower volume required for a plant performing 1 mol of separative work per second; it is a measure of the relative volume needed for different types of packing as a function of vapor velocity and pressure. When the design objective is to minimize tower volume, the type of packing and the velocity and pressure that minimize this factor should be selected.

Rate of loss of availability. In the scheme of Fig. 13.4 for reboiling a tower with low-pressure exhaust steam from a turbine, factors that reduce the power output of the turbine are (l)the temperature difference across the reboiler, which causes the turbine exhaust pressure to be higher than the tower bottom pressure, and (2) the steam pressure drop through the tower, which causes the tower bottom pressure to be higher than the tower top. We shall focus attention on the second of these inefficiencies and shall derive an expression for the reduction in turbine power caused by steam pressure drop through the tower.

If this were the only thermodynamic inefficiency, the loss in turbine power would equal the rate of loss of availability in the tower Q, given by

Q=T0 f (13.18)

where dS/dt is the rate of production of entropy in the tower and T0 is the temperature at which heat is rejected to cooling water. When liquid and vapor have the same temperature and when liquid-phase pressure changes are neglected, the rate of entropy production is simply that due to steam-pressure changes,

(13.19)

where Z is the height of the tower, and s is the entropy per mole of steam. If steam is treated as a perfect gas,

(13.20)

so that

The rate of availability loss per unit height is the integrand

dQ RTqG dp dz p dz

and the rate of availability loss per unit increase in deuterium content is

dQ dQ/dz_____________ (hRT0lp)(dp/dz)________

dx ~ dx/dz ~ [(a* — 1 )x(I — x)!G] -(P/G2 YxP-x)

The optimum steam rate, which makes this a minimum, is again given by (13.14), so that

(dQ 4hRT0 dp P(xp-x)

dx )min ~ (a* _ і fp dz xi(i-xf

4hRT0 dp (a*-l fp dzPDp’F

In a tower in which A, p, and a* are held constant, the minimum rate of loss of availability is obtained in the same way as the minimum volume (13.16) and is

The factor

4hRT0 dp («*-1 fp dz

gives the loss of turbine power in a plant performing 1 mol of separative work per second; it is a measure of the relative power consumption with different types of packing as a function of vapor velocity and pressure.

Costs for tower volume and power. The contribution of tower volume and availability loss to the cost of heavy water produced by the distillation of water in an ideal cascade may be evaluated when values are assigned to

/, the fractional charge against investment per year су, the unit cost of tower volume

cq, the unit cost of turbine work lost owing to tower pressure drop

. 4hRT ]c у————- 7 pita*-1)2

4hRT0 dp (a* — 1 Ур dz

The annual charge A for tower volume and power, then, is

where the numerical factor is the number of seconds per year. The contribution of tower volume and power to the unit cost of heavy water, in dollars per mole, is obtained by dividing the annual cost by the number of moles of heavy water produced per year, 3.15 X 107Acp:

jcv[4hRT/pv(a* — l)2] 4hRT0 dp) D^p_

3.15 X 107 CQ(a*-l fpdzxP

DPF may be obtained from (13.17). With xp = 0.000149 and xP = 0.998,

Dp, F

xp

Packing characteristics. We have shown that the optimum steam rate that leads to minimum tower volume and minimum power is that of the ideal cascade (13.14). The optimum type of packing, optimum pressure, and optimum vapor velocity is that which makes the expression in braces (1327) a minimum. We shall not attempt to evaluate a number of types of packing, but shall use Spraypak no. 37 packing as an example of the selection of optimum vapor velocity and pressure. This is the type of packing recommended by McWilliams and co-workers [M4] for a water distillation plant.

Figure 13.6 is a plot of the height of a transfer unit in feet, A, and the pressure drop per unit transfer unit in ton, A dp/dz, versus percent of flooding velocity, obtained from the data of McWilliams and co-workers [M4]. These data are for the system Нг O-HDO at total reflux and pressures of 420, 760, or 1245 Torr. Flooding velocities Vf reported by McWilliams et al. at

Figure 13.6 Characteristics of Spraypak no. 37 for H20-l DO, total reflux.

these three pressures are given in Table 13.10. For pressures below 420 Torr it will be assumed that the product v}p is constant at its value of 128.5 for 420 Torr and that h and h dpjdz have the same values as for 420 Torr shown in Fig. 13.6.

The following values will be assigned the parameters of the cost equation (13.27):

/ = 0.20/yr су = $0.002/cm3 cQ = $0.015/kWh

Table 13.10 Flooding velocities for Spraypak no. 37, system H20-D20, total reflux

Pressure p, Torr 420 760 1245

Temperature T, К 357.4 373.2 387.6

Vapor density p, g/cm3 (pM/RT) 0.000339 0.000588 0.000928

Flooding velocity, g/(cm2-s) [M4] 0.208 0.275 0.338

Vf, cm/s (above/p) 614 468 364

v}p, g/(cm-s2) 127.8 128.8 123.0

R = 62360 (Torr*cm3)/(ginol, K) (first term)

R = 0.000002310 kWh/(g-mol-K) (second term)

T = absolute temperature, K, corresponding to p, from Table 13.4 T0 = 310 К

a* — 1, from Table 13.4 h, cm, from Fig. 13.6 h dp/dz, Ton, from Fig. 13.6

v, vapor velocity, cm/s, and p, pressure, Ton, are independent variables

0.0213 hT 0.289 h^P (a*-l f Р» (a* —1 fpdz

With these values, Eq. (13.27) becomes

The first term gives the contribution of tower volume to the cost of heavy water; the second, the contribution of power.

Figure 13.7 represents these parts of the cost of heavy water from (13.29) as a function of vapor velocity, for pressures of 200, 420, 760, and 1245 Ton. Conditions that lead to minimum cost are listed in Table 13.11.

The above loss in availability is equivalent to 8.5 kWh/g D20. Because this takes account only of tower pressure drop, and does not include the loss in availability associated with temperature drop across heat exchange surface in reboilers and condensers, it is apparent that power consumption in water distillation is appreciably higher than in hydrogen distillation. The cost of $193/kg DjO covers only the cost of tower packing and power loss associated with tower pressure drop. If account were taken of the cost of tower shells and foundations,

reboilers, condensers, pumps, and other process equipment and other sources of power loss, the cost of producing heavy water by water distillation would be much greater than $215/kg, U. S. Energy Research and Development Administration (ERDA)’s charge in 1977 for heavy water made by the GS process.

Holdup and start-up time. McWilliams and co-workers [M4] have found that the holdup of Spraypak no. 37 under these conditions is about 4 lb water/ft3, or 0.064 g/cm3. Consequently, the total water holdup of the columns of a plant producing 1 g-mol of D20/s would be (0.064X1.02 X 1011 )/18 = 3.63 X 10® g-mol water. From Eqs. (13.17), (12.199), and (12.203), the average deuterium content of the water inventory of this plant is

. In [xP(l — xF)/xp( 1 — X/.)] — [{xP — Xp)/Xp( 1 — *F)]

Xg = ——————————————————————————————

Dp, fIxp

In [(0.998X0.999851)/0.000149X0.002)] — [(0.998 — 0.000149)/(0.998X0.999851)]

“ 6729

= 0.00209 atom fraction

The increase in D20 inventory during start-up of this plant would be

IE(xE ~xF) = 3.63 X 10® (0.00209 — 0.000149)

= 7.05 X 10s g-mol D20 (13.31)

The start-up time for this water distillation plant, evaluated from approximate Eq. (12.197), is

or 8.17 days.

The low holdup and low start-up time is another advantage of Spraypak compared with bubble-plate columns.

Squared-off cascade. The preceding treatment of a water distillation plant as an ideal cascade operated at uniform vapor velocity has required that the steam flow rate be varied continuously as its deuterium content changes and that the number of towers in parallel, or the tower area,
be changed continuously. On the other hand, a practical water distillation plant, like the Morgantown plant, will consist of a number of multiplate towers in parallel in the first group at the feed point, a smaller number in parallel in the second group at a higher deuterium content, a smaller number still or a smaller tower in the third group, and so on until at the product end a very small tower will suffice. A practical plant like this is characterized by uniform heads and tails flow over a large number of stages. Cohen [C9] has called such a plant a “squared-off’ cascade and has developed general equations for it.

Figure 13.8 compares the variation of tails flow rate with stage number in a squared-off cascade with the variation in an ideal cascade performing the same job of separation in the same number of stages. Because the total flow rate in an ideal cascade is the lowest possible, the area under the stepped curve of the squared-off cascade is greater than under the smoothly tapered curve of the ideal cascade.

Consider a squared-off cascade making product containing Xp fraction deuterium at the rate P. An equation giving the number of stages n12 needed in a section of the plant that enriches the deuterium content of water from хг to x2, with a uniform steam rate G, is obtained from Eq. (12.224):

1 , 1 +a

n ————— In ——

(a* — 1

b(x2 — Xi)

where a — —- ;— гтт—■—г— —————

(x2 + Xі + c) — 2x, x2 — 2cxp

P

G(a* — 1)

Figure 13.8 Steam flow rate versus stage number in ideal and squared — off cascades.

(a* —1 )x — (P/G)xp a* — 1 g — 1

the ratio of the steam rate to the minimum steam rate at Xi.

Because the number of stages in the portion of an ideal cascade whose overall separation is

a* — 1

CO is

the value of g that leads to the same number of stages in a square section as in an ideal cascade is given by

geo — 1

g~ 1

CO + 1 CO

This is close to the optimum value of g for a square cascade section, as will now be shown. The volume of the section is given by

hRT „ hRT G, gw-1

—- nG =——— г—- In——— —

pv pv a* — 1 g — 1

The optimum value of G for a section of a squared-off cascade is one that makes nG a minimum, with n given by (13.39).

We shall evaluate this optimum value and compare the minimum size of a section of a squared-off cascade with the portion of an ideal cascade performing the same job of separation. This will give us an idea of the penalty in increased equipment size paid by using practical towers with uniform vapor flow rates instead of the constantly changing flow rate of an ideal cascade.

hRT 4P pv (a*-i)2

*p(l ~*i) *l(l — Xp)

(*i>-*iXl -2xi) ~Xi)

The volume of the portion of an ideal cascade enriching deuterium from xt to x2 is

pv (a* — l)2 *1*2

With V from (13.45), Kideal from (13.47), n from (1339), and g defined by (13.41), there results

Figure 13.9 is a plot of К/КИеа1 against g, the ratio of the steam flow rate to the minimum at Xi for several values of the overall enrichment of the section (со = x2lx). Figure 13.9 shows (l)that the optimum steam rate in a square section is less than the optimum in an ideal cascade, and (2) that the penalty in using a squared-off cascade is less than 15 percent so long as the overall enrichment of a section is under 4. At x2/x2 = 4, the optimum steam rate is 1.35 times the minimum, at point A in the figure. In practice, a somewhat lower steam rate would be used in order to reduce the size of reboilers and condensers. A point around В might be chosen, at g= 125, where the number of stages in a square section is equal to the number in the portion of an ideal cascade with the same overall enrichment [cf. Eq. (13.44)]. The tower volume and power consumption at this condition are 1.156 times those of an ideal cascade, and costs are higher by the same factor.

Figure 13.9 Volume of square sec­tion relative to ideal cascade.

Figure 13.10 Example of Sulzer CY packing for water distillation columns. Diameter, 250 mm.

These equations for a squared-off cascade and those previously given for an ideal cascade were used to work out the variation of steam flow rate with stage number shown in Fig. 13.8 for an overall enrichment per section x2/xi =4 and a steam flow rate (?) 1.25 times the minimum. The following conditions have been assigned to each cascade:

P= 1 mol/h xP = 0.998 xF = 0.000149 a* = 1.04

The portion of each cascade up to x = 0.0093 has been shown.

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.

Although the isotopes of an element have very similar chemical properties, they behave as completely different substances in nuclear reactions. Consequently, the separation of isotopes of certain elements, notably 235 U from 238 U and deuterium from hydrogen, is of great importance in nuclear technology. The fact that isotopes of an element have such similar gross physical and chemical properties, however, makes their separation unusually difficult and has necessitated the development of processes and concepts especially adapted to this purpose. Despite the novelty of some of these isotope separation techniques, they have features in common with distillation and other familiar separation methods, and study of isotope separation is helpful in under­standing more conventional separation methods.

1 USES OF STABLE ISOTOPES

Table 12.1 lists separated isotopes that are being produced on a significant industrial scale. In addition to these, separated isotopes of practically all natural elements are being produced in research quantities by the U. S. Department of Energy (DOE) and by the atomic energy agencies of England, France, the Soviet Union, and other nations.

1.1 235 U

23SU is the separated isotope of by far the greatest industrial importance, with the value of annual production throughout the world of the order of a billion dollars. Uranium enriched from the natural level of 0.7 percent to from around 1.5 to 4 percent is used as fuel in power reactors moderated by natural water or graphite.

235 U enriched to 90 percent or higher, mixed with thorium, is proposed as fuel for the high-temperature gas-cooled reactor, the light-water breeder reactor, and the thorium-fueled CANDU type of heavy-water reactor, and as an alternative fuel for light-water reactors. In these reactor systems fission of 235 U is supplemented by five times or more as many fissions from 233U produced by neutron absorption in thorium, as outlined in Chap. 3. Highly enriched 233U is used as fuel for research or testing reactors, where the highest attainable neutron flux is wanted, and in compact power reactors, where high power density is needed.

Table 12.1 Uses of separated isotopes Isotope Natural atom percent Use 235 и 0.720S Fuel for nuclear fission reactors D 0.015 1. D2 0 moderator for natural uranium reactors 2. Fuel for thermonuclear reactors 6Li 7.56 1. Source of tritium 2. Fuel for thermonuclear reactors 7Li 92.44 1. As LiOH, water conditioner for water-cooled reactors 2. As lithium metal, possible high-temperature reactor coolant 10 В 19.61 1. Neutron absorber in control rods and shielding 2. Neutron-capture medical therapy 13c 1.107) ( 1. Stable isotopic tracer in living systems 1SN 0.366 ( ) 170 0.037 ( j 2. Nuclear magnetic resonance studies of 13 0 0.204} f molecular structure

Figure 12.23 shows the nomenclature to be used in describing the operation of an isotope separation plant during the transient period in which it is approaching steady-state performance. Figure 12.24 represents qualitatively the way tails and product flow rates and compositions will change with time during this transient period. Compositions are represented by a scale linear in In [x/(l -*)].

At time zero, all stages of the plant contain material of feed composition, zP. Initially the plant is operated with no feed supply and no tails or product withdrawal. As the plant operates, the fraction of desired isotope in material at the tails end of the plant decreases and the fraction of desired isotope in material at the product end increases. At time tx material at the tails end of the plant reaches the desired steady-state level xw. At this time tails withdrawal is started at such a rate W(t) as to keep the composition at this point constant at xw. Feed is supplied at a rate equal to tails withdrawal. At first, tails withdrawal is at a rate below the steady-state value W because the compositions elsewhere in the stripping section have not yet reached steady-state values. The tails rate increases and may temporarily exceed the steady-state value for a time, until product withdrawal can be started.

The fraction of desired isotope in material at the product end of the plant continues to increase, reaching the steady-state value yP at time t2. Product withdrawal is then started at such a rate P(t) as to keep product composition constant at yP. Feed is supplied at the rate P(t) + Wt). At first, product withdrawal is at a rate below the steady-state value P because the compositions elsewhere in the enriching section have not yet reached steady-state values. As time goes on, P(f) approaches P asymptotically.

The equilibrium, or start-up, time for product withdrawal tP is defined as the number of days of equivalent production lost during the approach to steady state. In Fig. 12.24, the area of the rectangle between the vertical line at tP and the horizontal line at unity equals the area between this horizontal line and the curve for P(t)/P. Mathematically,

(12.188)

к Фе

ls4>s

Іф

(12.189)