The GS plant for which the most detailed information has been published is the Savannah River plant of the U. S. AEC. This section summarizes the design and operating characteristics of this plant, which has been in operation since 1955. Section 11.8 describes improvements that du Pont personnel suggested for future GS plants, some of which presumably have been adopted in the newer Canadian plants.

Figure 13.30 is a flow sheet showing the main process equipment of the Savannah River GS plant and the principal process conditions^ as given by Bebbington and Thayer [В7]. The plant consists of 24 units of type shown, operated in parallel. Not shown in the figure are the feed-water deaerator, the tower to recover H2S from purge gas, and pumps for liquid.

Natural water feed is deaerated, brought to around 32°C, and pumped to the top of the cold tower CT-1, at 292 psia. It dissolves H2S and becomes saturated after flowing down through the first few plates, and it becomes enriched in deuterium by isotopic exchange as it flows through the entire tower. Liquid leaving the bottom of CT-1, enriched to 0.085 percent deuterium, is split into two streams. About one-fourth is pumped to the top of cold tower CT-2A; three-fourths is bypassed around CT-2A, heated to 125°C by exchange against outgoing hot waste water, and pumped to the top of hot tower HT-1. Liquid flowing down through cold towers CT-2A and 2B in series is enriched to about 15 percent deuterium by further isotopic exchange. Liquid leaving the bottom of CT-2B is heated to 120°C by exchange against outgoing hot waste water and is pumped to the top of hot tower HT-2A. In HT-2A the deuterium content of the water is reduced by exchange at the higher temperature. The deuterium content of the water is further reduced, to around 0.012 percent, by exchange in the top of hot tower HT-1; this water is drawn off above the eleventh plate above the bottom of HT-1.

Before this depleted water can be discharged from the plant, it is necessary to strip it of H2S, down to less than 2 ppm. To do this, the water is heated to 200°C in heat exchanger

^For consistency with du Pont literature, pressures are given in pounds per square inch absolute (psia) and flow quantities in pound-moles. Conversion factors are 1 psia = 0.068046 atm = 6895 Pa; 1 lb-mol = 0.4536 kg-mol.

Figure 13.30 Flow diagram for unit of Savannah River GS plant. Basis, 1 h. Plant consists of 24 units.

SX-1 and fed to the top of the 12-plate H2S stripper S-l. Heat in water leaving the bottom of this tower is recovered by heat exchange against colder water in exchangers SX-1, LH-1, and LH-2. This water, cooled, depleted in deuterium, and stripped of H2S, leaves the plant as waste.

Depleted hydrogen sulfide at 32°C from the top of cold tower CT-1 is compressed 33 psi in gas blower GB-1 and fed to the bottom of hot tower HT-1. The bottom 11 plates of this tower are used to heat and humidify the hydrogen sulfide through direct countercurrent contacting with downflowing hot water charged to the eleventh plate. As the hot humid hydrogen sulfide from the eleventh plate of HT-1 and from the stripper S-l flows up through the top 59 plates of HT-1, it is partially enriched by exchange of deuterium from the downflowing water. Gas leaving the top of HT-1 is split into two streams. About one-fourth goes to the second-stage hot tower HT-2B; three-fourths is bypassed around the second stage. As gas flows up through hot towers HT-2B and HT-2A in series, its enrichment in deuterium is completed. Gas leaving the top of HT-2A is dehumidified and cooled to 75°C in primary condenser PC-2 by closed heat exchange against cold water from the bottom of the humidifier section of HT-1. The gas is cooled further to 40°C in secondary condenser SC-2 by closed heat exchange against cooling water. The hot gas bypassed around the second stage is similarly dehumidified and cooled by closed heat exchange in PC-1 and SC-1. As gas flows up through cold towers HT-2B, HT-2A, and HT-1, its deuterium is transferred to cold water flowing down in these towers.

Condensate from SC-1 and PC-1 is returned to the top of hot tower HT-1, and part of the condensate from SC-2 and PC-2 is returned to the top of HT-2A. The rest of the water condensed in PC-2 and SC-2 containing around 15 percent deuterium is withdrawn as plant product. Use of this stream for product instead of water from the bottom of cold tower CT-2B, which has about the same enrichment, is preferred because the condensate is cleaner.

All heat requirements for the process are provided in the form of open steam at 400 psia. Some is used at the bottom of S-l to strip H2S and the rest is fed to the twelfth plate in HT-1 to control the temperature of the hot towers and to compensate for heat losses and heat exchanger inefficiencies. Steam consumption is 1778/0.28 = 6400 mol/mol of D20 produced. This is much less than the 200,000 mol/mol D20 needed in water distillation. Additional energy in the amount of 680 kWh/kg D20 is used to circulate gas and pump liquid. This, however, is much less than is used in electrolysis or hydrogen distillation (Table 13.7). The low energy consumption of the GS process is due in large measure to the efficient heat recovery obtainable in the flow sheet Fig. 13.30, which follows Spevack’s patent [S7].

Sweep diffusion is a form of mass diffusion column in which the screen separating the counterflowing light and heavy streams is not present. Cichelli et al. [C4] developed the theory of such a column and used it to separate hydrogen from natural gas and to enrich air. Table

14.22 gives examples of partial separation of isotopes by this process.

Because it has no screen, the sweep diffusion column is simpler to construct and has a lower transfer-unit height than the mass diffusion column. A disadvantage is the greater difficulty of maintaining undisturbed counterflow over a long column.

Both the sweep and mass diffusion columns have two advantages over the stage process. The columns are more efficient because it is possible to maintain optimum conditions over the entire screen area. In the stage, this is not possible because separating agent concentrations change from point to point. The other main advantage of the column is its ability to achieve the equivalent of many stages of separation in a single piece of equipment.

One disadvantage of the column type is its more complex construction compared with the stage, which makes scale-up more difficult. The column type has high mixing losses when the gas mixture to be separated is appreciably soluble in condensed separating agent. This is not a problem when separating isotopes of permanent gases such as nitrogen or argon, but it practically precludes use of the column type for UF6, because UF6 is readily soluble in all known separating agents that do not react with it chemically.

In a recycle cascade such as Fig. 12.13, feed, product, and tails quantities and compositions (the external variables) must satisfy the material-balance relations

F = P+W (12.52)

and Fzp = Pyp + Wxw (12.53)

Because there are two equations and six variables, it is possible to specify four external variables independently. For example, these might be product rate and product, feed, and tails compositions. In such a case the other two variables would be given by

A relation for the difference in composition between heads from one stage (y,) and tails from the next higher stage (x/+1) may be obtained from (12.58):

УР-Уі „ л ^

yi~Xi*l=N~JP (12-62)

Thus, x/+1 is less than у,- by an amount that decreases as the reflux ratio Nj+1/P increases. At total reflux (Ni+xfP-*•*). *i+i “d У і are equal

Equations for interstage flow rates and compositions in the enriching section are obtained by applying to the section of the cascade from the product end through stage і + 1 shown in Fig. 12.28 a development similar to the one used earlier for the complete cascade.

The 235 U/238 U ratios tj,• and £,+ , for this matched-ratio cascade may be related by

Figure 12.28 Enriching section of matched MSU/238U cascade for 23SU, “U, 233 U separation.

Interstage flow rate of 236U is obtained from Eq. (12.334) and Eq. (12.338), derived in similar fashion to Eq. (13.323):

МіУб. і _ ЛГі+ i*6./+1 _ Py6j>

W* (S,+ i)1/3

236 тт. л г „ — о.. (l/t7c)1/3 “ОЛ? р)1/3 _ о.. 1-/з(‘-п)/3

и. Ni+ i*6,,+l — Py6J, (1/€(+l)./3_(1/t?/y/3 — *УЬ* ~pt/3 _ ! о2-339)

The total interstage downflow rate, Ni+l, is obtained as the sum of Eqs. (12.336),

(12.337) , and (12.339):

^T1 = prj bs^O — fl’-") + Уъ, р№п~‘ ~ О] + [1 — ^~n)’3] (12.340)

When y6’P is zero, this equation reduces to Eq. (12.106) for a two-component ideal cascade.

For the uranium isotope separation case in which

0-i=!<i

Eqs. (12.336), (12.337), (12.339), and (12.340) may be approximated by

A similar derivation leads to the following equations for the stripping section.

Щ _ lx5’W(em — 1) + 2W1 ~ e-W) Hr 6×6’W(e+il6 — 1) W ф

— 1)_________________

*5,w(ewp — 1) + *e. w(l — е-да) + 3*6,^//* — 1)

__________ 3xt. vy*,/e — 1)________________

*5,w(e*W — 1) + *8,w0 — e-W/*) + 3×6>*(e«/« _ !)

Figure 12.29 shows the fraction of 23SU and of 236U as a function of stage number for the three-component separation example of Table 12.13. Characteristic features of this plot are as follows: (1) The 23SU composition gradient is nearly linear on a semilog scale. This feature holds only for low-23SU and low-236 U fractions. (2) The composition gradient of 236 U is about two-thirds that of 235 U on a semilog scale. (3) The 236 U plot has a noticeable discontinuity in slope at the feed point and is noticeably curved upward in each section. (4) The 236 U content of the cascade at the feed point is substantially higher than the 236 U content of the feed.

This buildup of 236 U at the feed point and the bulge in 236 U gradient in each section are characteristic of a third component whose molecular weight is between those of the key components. It is responsible for the increase in separative capacity caused by the presence of the third component, de la Garza [D1 ] gives an extreme example of buildup of concentration of a component of intermediate molecular weight.

de la Garza has shown that it would be impossible to separate 236 U completely from 235 U in the product of a cascade designed with matched 235 U/238 U ratio, no matter how many enriching stages were used. The property that determines whether a component can be completely separated from product is the arithmetic mean of the molecular weights of the key components, which de la Garza has called the key weight. This is 236.5 when 235 U and 238 U are key components. Only those components with molecular weight greater than the key weight can be fractionated completely out of the product. An example of a cascade that would do this for 236 U would be one that matched 235 U/236 U ratios.

Uranium isotope separation plants may be fed either with natural uranium, which contains only the isotopes 234U, 235U, and 23SU in nearly fixed proportions; or uranium discharged from a nuclear’reactor, whfch contains %e above three isotopes in many possible proportions, together with 236U from neutron capture in M5U, some 233U from neutron capture in thorium present in the irradiated uranium, and traces of 232U from fast-neutron irradiation of thorium or decay of 236Pu.

Until recently it has been assumed that natural uranium from all sources had exactly the same content of 234U, M5U, and 238U. As lately as 1977, U. S. Energy Research and Development Administration (ERDA) used 0.711 w/o (weight percent) as the 235U content of all natural uranium feed supplied to U. S. ERDA plants for enrichment. However, accurate measurement of the 23SU/23*U ratios of uranium from various minerals and various locations has shown significant variations. Cowan and Adler [C9] have summarized measurements of the weight percent of asU in 90 samples of natural uranium stated by the measuring laboratories to have a relative error of 0.0003 or less at the 95 percent confidence level. Average values of the weight percent S5U in different classes of samples and the standard deviation as reported by Cowan and Adler are listed in Table 14.2.

The difference between sandstone-type minerals and high-temperature minerals is con­sidered to be significant. It is attributed probably to isotopic fractionation that occurred when uranium initially deposited at high temperatures from magmas was dissolved by water at lower temperature and reprecipitated in sandstones. The difference between non-U. S. and U. S. samples is explained in the same way, as most non-U. S. samples were of magmatic origin and most U. S. samples were of the sandstone type.

Another possible cause of lower 23SU content more dramatic than isotopic fractionation during mineral deposition is possible occurrence of a critical fission chain reaction in a uranium deposit subsequent to primary mineralization, which would deplete 235U relative to 238U. One such deposit has been found and extensively studied at Oklo in the Republic of Gabon, West Africa. One uranium sample from this mine contained only 0.3 percent 235U [Nl], and much of the ore contains substantially less than 0.711 w/o 235U. Extensive nuclear chemical research reported in the proceedings containing [Nl] has found higher than normal concentrations of fission-product nuclides such as 143Nd and 145Nd in regions where the 23SU content of uranium is lower than normal. The evidence is conclusive that a fission chain reaction operated for many years in this deposit about 2 billion years ago. At that time the 235U content of natural uranium would have been around 3 percent, compared with today’s 0.711 percent, because of the shorter half-life of 23SU compared with 238U. In portions of the Oklo deposit where the uranium content was high and neutron absorbers were scarce, water made its way into the ore in sufficient concentration to establish a low-power fission chain reaction that persisted for thousands of years and used up a substantial fraction of the 23SU present at the start of the reaction. Cowan [C8] has summarized salient findings about this dramatic natural event and has given reasons for anticipating future discoveries of other one-time natural uranium reactors where the present ^U/23^ ratio would also be less than normal.

Because of the possibility of natural depletion of MSU and because of the availability of tails from isotope separation plants that might become mixed with natural uranium, it is important that natural uranium feed for an isotope separation plant be analyzed for its 235U content.

The 2,4U/238U ratio of natural uranium is generally assumed to be the same as the ratio of the half-lives of these elements, 2.47 X 10s years/4.51 X 109 years = 0.000055.

Source: G. A. Cowan and H. H. Adler, Geochim. et Cosmochim. Acta 40:1487 (1976).

3 URANIUM ENRICHMENT PROJECTS

AND SEPARATION POTENTIAL

9.1 Definitions

The second factor appearing in Eqs. (12.122) and (12.139) for the total flow rate in an ideal cascade is known as the separative capacity, or separative power [C3], D. For a plant with a single tails, product, and feed stream, it is given by

D = W(2xw — 1) In + Pilyp — 1) In J£—FQzF — 1) In (12.141)

The separative capacity has the same dimensions as used for the flow rates. It is a measure of the rate at which a cascade is performing separation.

The separative capacity concept may be generalized to a plant with any number of external streams of composition xk and molar flow rate Xk (positive when a product, negative when a feed). The total internal flow rate in such a plant, J + K, is^

The function ф defined by (12.144) is called the separation potential, or the elementary value function [C3]. It is a function only of composition and is dimensionless. It is plotted in Fig. 12.18. It is symmetrical about x = 0.5, at which value it vanishes. It is positive for all other x and increases without limit as x approaches zero or unity. This expresses the fact that a plant of infinite size is required to produce a pure isotope. The curve of ф versus x is convex downward, because

is positive.

t This holds for a close-separation, ideal cascade. When /3 — 1 is not small relative to unity, the more general equation is

(0 + 1 )D

J + К —————-

(0-l)ln0

Figure 12.18 Separation potential.

Because ф is convex downward, D is always positive.

The importance of the separative capacity in isotope separation lies in the fact that it is a good measure of the magnitude of an isotope separation job. Many of the characteristics of the plant that make important contributions to its cost are proportional to the separative capacity. For example, in a gaseous diffusion plant built as an ideal cascade of stages operated at the same conditions, the total flow rate, the total pump capacity, the total power demand, and the total barrier area are all proportional to the separative capacity. In a distillation plant, the total column volume and total rate of loss of availability are proportional to the separative capacity.

The separative capacity is analogous to the heat duty of an evaporator or other process equipment. The separation potential is analogous to the enthalpy per mole of the streams entering or leaving an evaporator. Calculations of material balances and separative capacity in an isotope separation plant are made in similar fashion to conventional material and heat balances. A form for such calculations is illustrated in Table 12.8, which illustrates the calculation of the separative capacity of an isotope separation cascade producing 1 mol/day of 235 U at 0.80 mole

+Given: Pxp = 1.0 mol/day desired isotope in product; xp = 0.800 mole fraction desired isotope in product; xp = 0.0072 mole fraction desired isotope in feed; xy/ = 0.0036 mole fraction desired isotope in tails. Required: net separative capacity of cascade.

fraction of 233 U, from normal uranium feed, with cascade waste containing 0.0036 mole fraction 235 U.

It is also useful to have a measure of the amount of separation performed by a cascade in making Ep moles of product and Ещ moles of waste from Ep moles of feed. This measure is provided by the separative work S, defined in similar fashion to the separative capacity.

S = Ew(2xw — 1) In + Erflyr — 1) In — Ep{2zF — 1) In

(12.146)

Separative work S’ has the same dimensions as used for the amounts of material E. Each term of the form Еф in Eq. (12.146) represents the separative work content associated with amount of material E in the corresponding stream.

Generalization to more than three streams is treated in Sec. 11.

Practical applications of Eq. (12.141) for separative capacity and Eq. (12.146) for separative work are usually expressed in terms of kilograms of uranium and weight fractions rather than moles and mole fractions. When atomic weights of the components are as close together as 235 U and 238 U, the equations on a weight basis still provide a valid measure of the magnitude of a job of separation.

9.2 Applications of Separative Capacity and Separative Work

If the rate of production of an ideal cascade at one set of feed, product, and tails compositions is known, so that its separative capacity can be evaluated, the best possible performance of the cascade for another set of compositions can be calculated by treating its separative capacity as a constant property of the cascade. This will be true if under the changed conditions the number of separating units in series and parallel are so rearranged that mixing of streams of different compositions is avoided.

The following may be cited as examples of the kinds of problems that may be solved by this means:

1. The effect of change in product rate on product purity

2. The effect of change in feed rate on product rate at constant product purity

3. The effect of providing supplementary feed of a different composition on product rate

4. The effect of withdrawing partially enriched product on product rate

Problem 12.3 illustrates how problems of this kind can be solved.

Most large isotope separation plants have so much flexibility that their separative capacity can be kept very nearly constant under moderately changed conditions.

6.3 Principle of Process

In the electrolytic cascade shown in Fig. 13.16, 76.2 percent of the deuterium in the water fed leaves with the hydrogen product at too low a concentration for economical recovery by electrolysis, even though over half of the deuterium in the hydrogen product is at or above the natural abundance. Some of the deuterium in this hydrogen may be recovered economically by the steam-hydrogen exchange process. The principle of this process may be seen through an example. Consider the effect of mixing hydrogen from stage 3 of Fig. 13.16 containing 0.0394 percent deuterium with an equal volume of steam containing the natural abundance of deuterium, 0.0149 percent, and passing the mixture over a catalyst at 80°C in which the exchange equilibrium

HD + H2O^H2 + HDO

is established. Because the equilibrium constant for this reaction is 2.8 at this temperature, the deuterium content of hydrogen and steam will be changed as follows:

By cooling the mixture to condense the steam and separating the hydrogen and water, it is possible to transfer deuterium from hydrogen gas to water without burning the hydrogen, and thus to increase the heavy-water output from an electrolytic plant without having to sacrifice hydrogen production. This principle would be applicable not only to hydrogen slightly enriched in deuterium, as from stage 3 of Fig. 13.16, but to any hydrogen containing more than 1/2.8 of the natural abundance of deuterium, because deuterium would be transferred to natural water from such hydrogen.

The variation of this equilibrium constant with temperature is given in Table 13.16.

The exchange reaction proceeds at a negligible rate unless catalyzed, and the only catalysts available until recently lost activity in the presence of liquid water. It was therefore necessary to use a gas-phase catalytic reactor, as described in the previous example.

The recovery of deuterium from hydrogen by exchange with water could be increased over the single-stage example just cited by using a multistage countercurrent cascade. The simplest arrangement, consisting of a tower packed with catalyst through which liquid water flows down and gaseous hydrogen flows up, was not practical, because of the inactivation of catalyst mentioned above. One possible arrangement of gas-phase exchange reactors is shown in Fig. 13.17. In such a cascade each exchange reactor, with its associated evaporator, condenser, and separator, acts like a single plate of a distillation column. If the condenser condenses all the steam leaving a stage, the separation factor is simply the equilibrium constant к for the above exchange reaction, with a value around 2.8. In such a cascade, the electrolytic cell acts like a reboiler to provide hydrogen recycle for the exchange cascade. In fact, the additional enrichment of deuterium occurring in the electrolytic cell is not essential for the operation of the process, because any desired degree of enrichment could be obtained by using a sufficient number of exchange stages.

The maximum recovery of deuterium possible with such a cascade is achieved by increasing the number of stages indefinitely and reducing the reflux ratio of depleted hydrogen to product heavy water until the depleted hydrogen is in deuterium exchange equilibrium with feed. The deuterium recovery r is given by

Figure 13.17 Cascade of exchange reactors.

where yw and xp are the atom fraction of deuterium in depleted hydrogen and feed, respectively, and W and F are the corresponding flow rates. At the low atom fractions in natural water and depleted hydrogen, W/F is very nearly unity, and y-wjxp may be approximated by

xP~ к

Therefore, the maximum recovery of deuterium obtainable by steam-hydrogen exchange at 80°C is

/•тах «1-^8 = 0.64 (13.71)

This is more than twice the maximum recovery attainable economically by electrolysis alone. The deuterium content of depleted hydrogen in equilibrium with natural steam is 0.0053 a/o.

6.4 History

Use of this vapor-phase deuterium exchange reaction between steam and hydrogen was proposed independently in 1941 by Harteck and co-workers in Germany and by Urey and co-workers in the United States as a means for recovering deuterium from electrolytic hydrogen. Harteck and Seuss [C3, G3] developed a supported nickel catalyst that caused the
reaction to take place at an acceptable rate below 100°C, where the high value of the equilibrium constant favors high recovery. In 1942, a set of exchange reactors containing this catalyst was installed to treat hydrogen from the sixth stage of the Norsk Hydro electrolytic plant at Rjukan, Norway [S10], and additional reactors were planned for the fourth and fifth stages, but operation was interrupted by the war before this could be completed. After the war, catalytic reactors were installed at Norsk Hydro plants at Rjukan and Glomfjord, Norway, bringing their combined heavy-water production at one time to around 20 МТ/уеаг [B9].

In the United States, a similar vapor-phase, steam-hydrogen, deuterium exchange process was the first one selected for large-scale production by the Manhattan District [M8], and a heavy-water plant using this process was built at the synthetic ammonia plant of the Consolidated Mining and Smelting Company at Trail, British Columbia. Urey and co-workers at Columbia University developed a nickel-on-chromia catalyst, and Taylor and co-workers at Princeton University developed a platinum-on-charcoal catalyst, both of which were used in this plant. The exchange tower system used in the Trail plant was devised by Barr and co-workers [B5] of the Standard Oil Company of New Jersey, which was responsible for the basic design of the plant.

5.1 Principle

The principle of the countercurrent gas centrifuge is shown in Fig. 14.10. The device consists of a long, thin, vertical cylinder made of material with high strength-to-density ratio, rotating in an evacuated casing about its axis with high peripheral speed. The gas rotating inside the cylinder is subject to centrifugal acceleration thousands of times greater than gravity. This makes the pressure at the outer radius of the cylinder millions of times greater than at the center and causes the relative abundance of the heavier isotope to be appreciably greater at the outer radius than at the center. For UF6 at 300 K, for example, in a centrifuge rotating at a peripheral speed of 500 m/s, the abundance ratio of 238U to 235U at the outer radius is greater

than at the center by a factor of 1.162 and the pressure at the outside is greater than at the center by a factor of 46 million. By inducing countercurrent flow between the “^-depleted stream near the outer radius and the 23SU-enriched stream near the axis, the difference in composition between the top and bottom can be made much greater than between the two streams at one elevation. Three general methods have been used for inducing countercurrent flow: (1) by the system of internal scoops and baffles shown in Fig. 14.10, (2) by convection currents set up by heating one end and cooling the other or establishing a temperature gradient along the wall, or (3) by flow induced by pumps external to the machine, as shown in Fig. 14.11. The last gives greater operating flexibility, but is much more complex mechanically.

Natural oxygen contains 0.037 percent 17 О and 0.204 percent 18 O. The isotope 18 О was first concentrated by Huffman and Urey [H4] in 1937, by the distillation of water. Although the separation factor is very low (1.004 at 100° C), the method has been adapted to semi­commercial production by Dostrovsky [D3], who produced 11 g/day of 18О at an enrichment

of 95 percent. 18 О is also concentrated when water is distilled for deuterium separation, but the low separation factor for the oxygen isotopes limits the degree of enrichment.

Other methods used to concentrate 18О are the distillation of CO, referred to in Sec. 2.4, the distillation of NO, and the exchange reaction between C02 gas and water [B4], for which the separation factor is 1.02. Boyd [B3] estimated that 180 could be produced at a rate of 4 g/day at a cost of $93/g by this process.

To make use of Eq. (12.209) we need expressions for the inventory of both components /, the inventory of desired component lx and the inventory of separative work Іф in a close — separation, ideal cascade. To derive these expressions we shall assume that the stage inventory is proportional to the stage feed rate, as stated by (12.198), and that the average stage composition is that of the stage feed z,-.

Because of the first assumption, the total inventory is proportional to the total interstage flow rate, given by (12.181), so that

/=(^Ї7 I 02.210)

ф, the separation potential, may be thought of in this connection as a function for evaluating the inventory. We have proved that ф satisfies differential equation (12.172) and is given by

(12.144) .

By a development similar to that which showed the separation potential to have these properties, it can be shown that the inventory of desired component is given by

= (12.211)

and the inventory of separative work

$ = 2 V(*fc) (12.212)

The functions ф, ф, and it and their second derivatives are listed in Table 12.10.

The derivation of differential equation (12.215) for the separative work inventory function ■n is similar to the derivation of differential equation (12.172) for the separation potential ф. Equation (12.170) is valid for any function of composition that can be expressed as a Taylor series. Therefore, the feed rate to stage і may be expressed in terms of it instead of ф as

M.+Nss __8___ міФі) + ХіФі) ~ (Mi + Niivjzt)

1 ‘ (a — l)2 zf(l — zt)2[d2TT(Zi)/dz2]

Because of the assumption that the stage inventory is given by (12.198) and the assumption

that the average composition’*’ of the inventory is z,-, the inventory of separative work on the stage is

(12.218)

The separative work inventory of all stages will be given by Eq. (12.212) if and only if the second factor is independent of i, that is, if

d2it(Zj) _ (2Zj — 1) In [z,/(l — zt)] 02 2191

dz2 zf(l-z,)2 v ‘ ;

The proof is similar to that given in Sec. 11 to establish Eq. (12.182) for the total flow rate of all stages. Differential equation (12.213) for the component inventory function ip may be derived in similar fashion (see Prob. 12.9).