The construction cost of the Savannah River heavy-water plant, built in 1951/1952, is summarized in Table 13.19 [B6]. The unit investment cost of this plant, capable of producing 454 Mg D20/year, then was $163,000,000/454,000 kg/year = $359/(kg/year).

The cost of Atomic Energy of Canada, Ltd.’s 800 Mg/year plant at La Prade, item 21, Table 13.2, was predicted [H5] to be $300 million in 1974, for a unit investment cost of $375/(kg/year), exclusive of escalation and interest during construction.

Heavy water from the Savannah River plant was sold by the U. S. AEC in the 1960s for a price of $61.73/kg. Demand for heavy water subsequently decreased, and two of the three original wings of the plant were shut down. In 1976, when one wing was operating at its full

Source: W. P. Bebbington and V. R. Thayer, Chem. Eng. Progr. 55(9): 70 (1959).

capacity of 177 MT/year, production costs [J5] were as summarized in Table 13.20. In 1977, when the one remaining wing was operating at reduced capacity, the price charged by U. S. ERDA for heavy water was $245/kg [FI].

Utility requirements reported for heavy-water production by the GS process are as follows:

7.1 General Description

When heat flows through a mixture initially of uniform composition, small diffusion currents are set up, with one component transported in the direction of heat flow, and the other in the opposite direction. This is known as the thermal diffusion effect. The existence of thermal diffusion was predicted theoretically in 1911 by Enskog [El, E2] from the kinetic theory of gases and confirmed experimentally by Chapman [Cl, C2] in 1916. It is not surprising that the effect was not discovered sooner, because it is very small. For example, when a mixture of 50 percent hydrogen and 50 percent nitrogen is held in a temperature gradient between 260 and 10°C, the difference in composition at steady state is only 5 percent. In isotopic mixtures the effect is even smaller.

Thermal diffusion column. Thermal diffusion remained a scientific curiosity until 1938, when Clusius and Dickel [C5] developed their thermal diffusion column, which made possible useful separations in simple equipment. In the Clusius-Dickel column the mixture to be separated is confined in a long, vertical tube, cooled externally and heated internally by a hot wire at the axis of the tube. Other workers have used the annular type of equipment shown in Fig. 14.38. In both types, the mixture to be separated is confined in a narrow space between an inner heated and an outer cooled surface. The outward flow of heat sets up a small difference in isotopic composition through the thermal diffusion effect, with the light isotope usually concentrating in the inner zone at the higher temperature. At the same time, convection currents are set up, as indicated by the arrows, with the lighter heated fluid adjacent to the inner wall moving upward and the heavier cooled fluid adjacent to the outer wall moving downward. This counterflow multiplies the small composition difference obtained from the thermal diffusion effect and makes possible substantial degrees of separation in a practical length of column. For example, in a column 36 m long, Clusius and Dickel were able to separate the isotopes of chlorine, producing HC1 containing 99.6 percent 35C1 at one end of the column and HC1 containing 99.4 percent 37C1 at the other.

For most isotopes it is preferable to work with gases rather than liquids, because the higher diffusion coefficients result in higher separative capacity. The optimum pressure is usually near atmospheric. However, when 235U was first found to be fissionable, Nier [N3] attempted to separate it by thermal diffusion of UF6 vapor at low pressure without success, so that it was necessary to work with the liquid at high pressures [Al] to obtain useful separation. The optimum spacing between hot and cold surfaces is a few millimeters for gases and fraction of a millimeter for liquids.

The degree of separation obtainable in thermal diffusion (the difference in composition between hot and cold walls) is much less than in other diffusion processes, so that use of a column to multiply the composition difference is practically essential. The stage type of thermal diffusion has been used only to measure the thermal diffusion coefficient and is never used for practical separations. In some thermal diffusion columns, htu’s are as low as 1.5 cm, and as many as 800 stages of separation have been obtained from a single column. Even with such a great increase in separation, it is often necessary to use a tapered cascade of thermal diffusion columns for isotopic mixtures, to minimize hold-up of partially enriched isotopes and to reduce equilibrium time.

Isotopes separated. Table 14.24 gives examples of some of the highest reported concentrations of separated isotopes that have been obtained by thermal diffusion. Most of these separations were on a small laboratory scale. The high purity to which scarce isotopes such as 13C, 15N, and leO have been concentrated is a notable feature of these examples of thermal diffusion. The feasibility of concentrating rare isotopes of intermediate mass, such as J1Ne and “A, by thermal diffusion is also noteworthy. These separations are facilitated by the large number of stages obtainable from a single thermal diffusion column.

Thermal diffusion is a convenient way of separating isotopes on a small scale. It is a very inefficient process for large-scale use because of its high energy consumption.

The number of stages required to separate feed into product and tails of specified composition is a minimum at total reflux, when N(+1/P-*». Under this condition we have seen that

*i+i =Уі (12.63)

Abundance ratios in these two streams are also equal:

?,+,=% (12.64)

Because of the definition of separation factor (12.15), abundance ratios on adjacent stages at total reflux are related by

t? i’+i = ar>i (12.65)

When applied to stage 1, this equation is

i?2=ar? i (12.66)

When applied to stage 2 it is, for constant a,

%=at? j=a2r?1 (12.67)

This is the familiar Underwood [Ul]-Fenske [FI] equation for total reflux. The ratio of abundance ratios appearing in (12.72) is the overall separation (П) of the recycle cascade:

■УрО — xw)

(1 — yp)x w

Equation (12.72) gives the minimum number of stages for a particular overall separation. The minimum number of stages requires that the ratio of interstage flow rate to product be infinite.

The minimum number of stages increases as the overall separation increases and as the separation factor approaches unity. Because both these conditions hold in a typical isotope separation plant, the minimum number of stages is often very large. For example, in a 235 U gaseous diffusion plant (a = 1.00429) making product containing 90 percent 233 U and tails 0.3 percent,

Table 13.2 lists all plants in the non-Communist world that have been built or are planned for production of deuterium, in the form of heavy water, at a rate of 1 t/year or more.

The following general comments may be made about these plants and processes:

0. All plants, except 16 and 18, have a different process for primary enrichment than for final concentration.

1. Those plants that for primary concentration use water distillation (WD) or the dual­temperature, water-hydrogen sulfide (GS) process are self-contained plants whose sole product is heavy water.

2. All other plants that for primary concentration use water electrolysis (WE), steam-hydrogen exchange (SH), synthesis gas distillation (SD), hydrogen distillation (HD), or ammonia — hydrogen exchange (AH) are parasitic to a synthetic ammonia plant. Heavy water is a by-product of these plants, and its production rate is limited by the amount of deuterium in the ammonia plant feed.

3. Water distillation is used for final concentration in all plants still operating, except 16 and 18

4. The relative amount of heavy water produced by each primary concentration process up to 1975 was reported [M7] to have been

90%, GS process

6%, water electrolysis and steam-hydrogen exchange 2%, hydrogen and synthesis gas distillation

+ Organizations: AECL, Atomic Energy of Canada, Ltd.; DAE, Dept, of Atomic Energy, India; GELPRA, Groupement Eau Lourde Precede Ammoniac; ONIA, Organisation Nationale Indus — trielle de l’Azote; SCC, Societe Chimique de Charbonnage; U. S. AEC, U. S. Atomic Energy Commission; U. S. DOE, U. S. Department of Energy.

* Processes: AD, ammonia distillation; AH1, monothermal ammonia-hydrogen exchange; AH2, dual-temperature ammonia-hydrogen exchange; GS, Girdler-sulfide, dual-temperature, water- hydrogen sulfide exchange; HD, hydrogen distillation; SD, ammonia synthesis gas distillation; SH, steam-hydrogen exchange; WD, water distillation; WE, water electrolysis.

§ Scheduled start-up year.

1%, ammonia-hydrogen exchange

0. 3%, water distillation

The rest of this chapter is organized according to process rather than individual plants. The simplest and most familiar process, distillation, is taken up first.

Section 3 describes the separation factors obtainable in distillation of the principal substances used in isotope separation. Section 4 describes deuterium concentration plants using distillation of hydrogen or ammonia synthesis gas. Section 5 describes use of water distillation for primary deuterium concentration, for final deuterium concentration, and for separation of oxygen isotopes.

Section 6 describes the enrichment of deuterium in electrolysis of water. Section 7 describes how steam-hydrogen exchange has been used to increase the recovery of deuterium in electrolytic hydrogen plants.

Section 8 summarizes separation factors obtainable in isotope exchange reactions and their temperature dependence. The latter is the key property in dual-temperature exchange processes. Section 9 develops equations to be used for calculating the number of theoretical stages needed in exchange separation towers.

Section 10 describes monothermal exchange processes, with principal emphasis on ammonia-hydrogen exchange.

Section 11 describes the principle of dual-temperature exchange processes with particular reference to the water-hydrogen sulfide exchange reaction and gives more detailed engineering information about plants using this, the GS process, the process of greatest commercial significance.

Dual-temperature exchange processes using ammonia and hydrogen, methylamine and hydrogen, and water and hydrogen are described in Secs. 12, 13, and 14, respectively, and are compared with the GS process in Sec. 14.

Section 15 gives a brief description of exchange processes for separating lithium isotopes, and Sec. 16 gives a limited account of exchange processes for separating isotopes of carbon, nitrogen, oxygen, and sulfur.

Gaseous diffusion. Table 14.3 lists gaseous diffusion plants in operation in 1977 and those then under construction, planned, or under consideration. Part 1 of Table 14.3 lists plants in operation at that time. The three large plants of the U. S. Department of Energy (DOE) had a capacity of over 17 million kg separative work units (SWU) per year when supplied with the maximum amount of electric power, 6100 MW, they could then utilize.

The U. S.S. R. plant is rumored to have an annual capacity of from 7 to 10 million units, of which 3 million are thought available for export. The existing plants of the French Commissariat a l’Energie Atomique (CEA) and British Nuclear Fuels, Ltd. (BNFL) are too small to be a major source. Little is known about the Chinese plant.

Table 14.3 Gaseous diffusion projects

Capacity, million kg separative work

Owner Location units per year

Part 2 of Table 14.3 lists additional separative capacity by gaseous diffusion under construction. U. S. DOE is improving the barrier in its three existing plants and increasing the power input to the stage compressors to increase capacity by 10.5 million units per year. The Eurodif combination of French, Belgian, Italian, Spanish, and Iranian interests is building a 10.8 million unit per year plant in France, using French-developed technology, to start operation in

1978.

Parts 3 and 4 list additional gaseous diffusion enrichment projects likely to be built. The Coredif project uses French diffusion technology, and appears to be committed to construction of 5.4 million units of additional diffusion capacity at a European site still to be selected. Possible expansion of capacity of this plant by another 5.4 million units per year is under consideration.

Gas centrifuge projects. Table 14.4 lists gas centrifuge projects. The Urenco-Centec Organization, a combination of British, Dutch, and German interests, has been operating three pilot units at Capenhurst, England, and Almelo, Holland, since 1972. By 1982 these plants will have been expanded to an annual capacity of 2 million units. This group is seeking additional orders with intention of increasing capacity to 10 million units by 1985 if orders materialize.

President Carter announced on April 20, 1977, that the United States would expand its uranium enrichment facilities and would shortly reopen its order book for sale of additional units of separative work. After the cascade uprating and cascade improvement programs have been completed, all new separative capacity would be provided by the gas centrifuge, whose much lower energy demand and greater flexibility were perceived as decisive advantages. U. S. DOE is building a centrifuge enrichment plant with capacity of 2.2 million kg SWU/year at Portsmouth, Ohio, for operation in the late 1980s. Expansion of 8.8 million kg SWU/year is possible.

Japan is building a 7000-machine centrifuge pilot plant to operate in 1979 and is considering a 6 million SWU/year production plant to start operation in 1985.

Aerodynamic processes. Two projects have developed to industrial-scale processes for sepa­rating uranium isotopes by causing a mixture of UF6 and hydrogen to flow at high speed in a sharply curved path and thus experience centrifugal acceleration large enough to effect partial separation of a5UF6 and 238UF6. The separation nozzle process developed by Becker and his associates at the Karlsruhe Nuclear Research Center in Germany and adapted for industrial use by Steag, A. G., and Gesellschaft fur Kemforschung is being used in a plant with a capacity of 180,000 SWU/year being built in Brazil for operation in 1982. The UCOR process, developed by Roux, Grant, and their associates of the Uranium Enrichment Corporation of South Africa, has been demonstrated in a 6000 SWU/year pilot plant at Valindaba, South Africa; in 1978 a decision was to be made whether to build a commercial plant based on this process. These processes will be described in Sec. 6.

When a power company or other customer wishes to obtain Ep kg of uranium enriched to yp weight fraction, the usual arrangement is for the customer to purchase [(yp ~xw)/(zF ~xw)]Fp kg of natural uranium with zp = 0.00711, deliver it to a uranium enrichment plant providing toll enrichment services, and pay for an amount of separative work S calculated from

The U. S. DOE sets the tails assay хц> in transactions with its customers; in 1977 this tails assay was хцг = 0.002. In the future, it is likely that customers will be given some latitude in the choice of xw so as to minimize the sum of the costs of separative work and natural uranium feed.

Substitution of these values into (12.153) and using (12.144) for ф yields

c yp

rr — (ERDA, 1977) = (2xp -1) In t-^— +258.0964 yp-6.7039 (12.154)

£p 1 —yp

This equation has been used in the U. S. “Standard Table of Enriching Services” [U2]. This text has used хц/ = 0.003 as a more probable value of the tails assay in enrichment transactions after 1977. With zp = 0.00711 and xw = 0.003, Eq. (12.153) becomes

щ; (this text) = (2xP — 1) In + 219.5666 yP — 6.4300 (12.155)

The second column of Table 12.9 gives, for different values of the product weight fraction 235U, yp, the kilograms of natural uranium feed required to produce 1 kg of product, Ep/EP,

from the material-balance relation

Ep yp—0.003

£> (this teXt> = 0.00711 — 0.003 (12.156)

The third column gives the number of separative work units required to produce 1 kg of product, S/Ер, from Eq. (12.155). The units of S are kilograms of uranium, but are conventionally referred to as SWUs (for separative work units).

The high cost of recovering heavy water from electrolytic hydrogen by exchange with steam is due largely to the cost of making and condensing steam and to the large mass of catalyst needed for this vapor-phase reaction at low pressure. These difficulties would be avoided if the reaction could be carried out at an acceptable rate in the presence of liquid water. Becker [BIO] developed a colloidal platinum-on-charcoal catalyst, suspended in liquid water, which was circulated in countercurrent flow to gaseous hydrogen through a conventional sieve-plate column. This catalytic exchange system was tested on a semicommercial scale at Dortmund,

Figure 13.19 Flow sheet for one ex­change tower and electrolytic cell group.

Germany [Wl], in the early 1960s, using a dual-temperature flow sheet. Even with this finely divided catalyst and a pressure of 200 bar, the plate efficiency at 30°C was only around 1 percent. The resulting large column and catalyst volume made the process appear to be only marginally economical.

The low plate efficiency is due to the low solubility of gaseous hydrogen in liquid water, which results in a low mass-transfer coefficient for hydrogen to and from the catalyst surface, which is wetted by liquid water. Stevens [S8], in Canada, has recently developed a catalyst for the deuterium exchange reaction that is not wetted by liquid water and is much more active. The catalyst consists of nickel or platinum deposited on a conventional support such as silica gel and then coated with a thin layer of a water-repellent resin through which hydrogen can rapidly diffuse. Liquid water and gaseous hydrogen flow countercurrent through a column packed with particles of such a catalyst. The gas-phase deuterium exchange reaction between water and hydrogen takes place on the catalyst surface, while H20 and HDO are transferred simultaneously between the gas and liquid phases. From experiments described in Stevens’ patents, transfer-unit heights of around 1.5 m are predicted [H2] at a superficial gas velocity of 3 m/s evaluated at standard conditions of 0°C and 1 atm for actual conditions of 60°C and pressures in the range of 14 to 40 bar.

Figure 13.20 shows how exchange towers packed with such a catalyst permitting counterflow of hydrogen and water might be used to increase the recovery of deuterium from

the cascade of electrolytic cells shown in Fig. 13.16. Exchange towers are used to reduce the deuterium content of hydrogen leaving electrolytic stages 2, 3, and 4 to 0.0563 a/о, the same value as in hydrogen leaving the first electrolytic stage. In this flow sheet it has been assumed that the exchange towers are so designed and operated that water and hydrogen leaving them have the same deuterium content as the corresponding streams leaving the electrolytic stages with which they are mixed. This ideal cascade condition minimizes exchange tower volume.

Comparison of Fig. 13.20 with Fig. 13.16 shows that the use of exchange towers would increase heavy-water production from 0.3553 to 0.930 mol and deuterium recovery from 23.8 to 62.2 percent.

To reduce the number of electrolytic stages to three, Hammerli and co-workers have suggested [H2] the flow sheet of Fig. 13.21. The big advantage of this flow sheet is the great simplification of interstage connections compared with Fig. 13.20. The principal disadvantage of Fig. 13.21 is its much greater catalyst volume. Hammerli [HI] estimates, however, that the cost of catalyst and catalyst towers for a flow sheet like Fig. 13.21 is only 15 percent of the cost of the electrolytic cells, so that it is cost-effective to simplify the flow sheet at the expense of increased catalyst volume. The dimensions of the catalyst towers of Fig. 13.21 for a superficial hydrogen gas velocity of 3 m/s at standard conditions are

In comparing Figs. 13.20 and 13.21, the following should be noted. Hydrogen product rates are substantially equal, 10,000 kg-mol/h. The total amounts of water electrolyzed are about the same. Use of an exchange tower on hydrogen from the first cell coupled with the closer approach to exchange equilibrium at the top of the towers of Fig. 13.21 permits reduction in deuterium content of depleted hydrogen from 0.0563 to 0.050 percent and

Natural water
Feed

10,000 9 8569 0.0149%

increases heavy-water production from 0.930 to 0.982 kg-mol/h. More separative work is performed in the exchange towers of Fig. 13.21 than in those of Fig. 13.20, primarily to compensate for the loss of separative work in Fig. 13.21 where the water recycled from each burner is mixed with water of quite a different composition from an exchange tower. Other factors increasing the separative work demand on the towers in Fig. 13.21 are the lower electrolytic separation factor of 6^ used in that figure compared with 7 in Fig. 13.20 and the lower deuterium content of hydrogen product.

One possible difficulty with Fig. 13.21 is the much higher average deuterium content of water in the electrolytic cells compared with Fig. 13.20. This requires that cell leak rates and water holdup be kept small.

As will be shown in Sec. 5.5, the separative capacity of a countercurrent gas centrifuge is proportional to its length L and increases rapidly as the peripheral speed va increases. Hence it is advantageous to run at the highest practical speed and to use centrifuges of the greatest practical length. An absolute limit to the speed is reached when tangential stresses caused by centrifugal forces equal the tensile strength of the rotor material. Limitations on practical values of the length are set by the need to avoid combinations of length, radius, and speed at which
the rotor experiences resonant vibrations. These two factors limiting centrifuge mechanical performance, which have the greatest effect on separation performance, will be discussed in this section. Many other relevant mechanical topics, such as design of bearings, motor drives, and damping mechanisms, are beyond the scope of this text.

Maximum peripheral speed. Consider a cylindrical shell of radius r and thickness dr, made of material of density p and rotating at angular velocity « rad/s. Figure 14.16 represents a volume element of the shell of height dz subtending an angle dd. The mass of the element

dm = prdrdzdd (14.145)

experiences a centrifugal force

ш1 rdm = pw1 r* drdzdd (14.146)

in the outward r direction. This must be balanced by the components in the opposite direction, Off sin(d6/2), of the tangential stresses oe acting on the two surfaces drdz offset by angle dd.

Figure 14.15 The Zippe centrifuge. (Adapted from Shacter et al. [S3.])

where umax is the maximum tangential speed, at which the tangential stress reaches the tensile strength a of the material.

Table 14.11 gives the density, tensile strength, and modulus of elasticity E of six possible high-tensile materials for centrifuge rotors. These properties are given in metric units and SI units.

The maximum tangential speed ranges from 400 m/s for aluminum alloy to 720 m/s for a carbon fiber-resin composite.

Conditions for resonant vibrations. Certain angular velocities со,- cause a thin, hollow cylinder to go into resonant longitudinal vibrations. If a centrifuge rotor is driven for any length of time at or near one of these angular velocities, rotational energy is used to increase the amplitude of longitudinal vibrations until the rotor or its bearings may be wrecked. Consequently, it is important to avoid tangential speeds at which a rotor of given length and radius will be in resonance. Texts on mechanical vibrations such as [D4] show that the longitudinal vibration frequency of a thin hollow cylinder of radius a, modulus of elasticity E, and length L, unrestrained at the ends, in the ith mode is

where the eigenvalues X,- are

і 1 2 3 4 5

X, 22.0 61.7 121.0 200.0 298.2 і is the number of loops into which the profile of the cylinder is displaced. Because

Vі СО/Г

the length-to-radius ratio at which rotors of each of the materials run at maximum tangential speed umax would be in resonant vibration is

The last part of Table 14.11 gives values of L/a for the first five resonances in cylinders of the five materials operated at the maximum speed, at which tangential stress equals the tensile strength of the material. At lower speeds u, the critical L/a ratio is obtained by multiplying the values of Table 14.11 by /vmaxlv.

Rotors that are shorter than the first critical length are said to be subcritical. Such rotors do not need special means to avoid resonant speeds. Rotors that are longer than the first critical length are called supercritical. They must be operated at speeds away from resonance

and must be provided with drives of sufficient power to accelerate them quickly through resonant speeds and brakes of sufficient power dissipation to decelerate them quickly. All of Groth’s rotors listed in Table 14.10 have length-to-diameter ratios below the first critical at the listed peripheral speeds. However, if ZG7 had been made of titanium and operated at its maximum peripheral speed of 442 m/s, it would have run between the first and second resonance.

Power consumption. Because the separation performed by the gas centrifuge is a thermodynamically reversible process, the minimum energy input necessary to separate an isotopic mixture is merely the small difference in free energy between the separation products and the feed. The actual energy input is thousands of times greater because it is dominated by the work necessary to overcome mechanical friction in bearings, aerodynamic drag, and pressure drops in gas circulation. These energy inputs are specific to details of centrifuge and plant design and cannot be estimated from principles of the separation process, as was possible for gaseous diffusion. The U. S. DOE stated [U3] that the power consumption of a centrifuge plant per unit separative capacity would be around 4 percent of the power consumption of a gaseous diffusion plant.

The comparatively low power consumption of a gas centrifuge plant is its greatest advantage over competing processes. The relatively low separative capacity of a single centrifuge is its greatest disadvantage.

Means for estimating the separative capacity of a centrifuge will be developed in Sec. 5.5.

3.1 Separating Unit, Stage, and Cascade

The smallest element of an isotope separation plant that effects some separation of the process material is called a separating unit. Examples of a single separating unit are one stage of a mixer-settler, one plate of a distillation column, one gas centrifuge, one calutron, or one electrolytic cell.

A group of parallel-connected separating units, all fed with material of the same composition and producing partially separated product streams of the same composition, is known as a stage. Often a single unit serves as a stage, like a plate of bubble-plate column. However, in some separation methods whose units have low capacity, such as an electrolytic cell or centrifuge, it is necessary to use many units in parallel.

When the degree of separation effected by a single stage is less than the degree of separation desired between product and waste, it is necessary to connect stages in series. Such a series-connected group of stages is known as a cascade. Examples of a cascade are a complete distillation column or a battery of solvent extraction mixer-settlers.

The relation between unit, stage, and cascade is illustrated by Fig. 12.8. Each unit of this cascade might represent, for example, an electrolytic cell. The group of parallel-connected cells, each of which separates feed of composition zx into a partially enriched stream of composition у і and a partially depleted stream of composition, constitutes the first stage of this cascade. The cascade is the entire group of series — and parallel-connected cells.

A cascade that has the same number of units (i. e., the same capacity) in all stages of a group is known as a “squared-off cascade. A cascade in which the number of units, or the capacity, in each stage decreases as the produce and waste ends of the cascade are approached is called a tapered cascade. A single multiplate distillation column is an example of a squared-off cascade; a gaseous diffusion plant for uranium separation is an example of a tapered cascade.

The engineering analysis of separation processes frequently employs the concept of an ideal, or equilibrium stage. In such a stage, the feed streams, which may be one or two in number, are acted upon to produce two product streams that are in equilibrium. The use of such a concept can be employed in the design and analysis of both stagewise and continuous contacting equipment. Determination of the number of stages in a cascade required to achieve a given separation involves the determination of the number of such ideal stages followed by application of a stage efficiency, which expresses the fraction of ideal transfer achieved in the actual stages employed.

In some isotope separation plants, notably those using distillation or exchange processes, it is more economic to use a constant interstage flow rate over a considerable composition interval rather than a flow rate that decreases steadily from the feed point to the product ends, as is characteristic of an ideal cascade. Cohen [C3] has called such cascades “squared-off’ cascades and has derived equations for their separation performance. This section summarizes the derivation for a close-separation, squared-off cascade.

In the enriching section of a cascade with constant tails flow rate N, the change in composition x with stage number і is given by differential equation (12.128). The number of
enriching stages nn needed to span the composition range between Xi and x2 is then obtained by integration of

di_ _ _____________ 1_____________

dx (a — 1)*(1 — x) — (P/N)(yp — x)

If a constant value of N is used for the entire enriching section spanning the composition range from zF to yp,

________ Ыур ~ zF)__________

yp + zp — 2yP zF-c(yp — Zp)

In the stripping section similar equations hold, with substitution of — W for P and xw for yp in Eqs. (12.224) through (12.227). Equation (12.228) for a square stripping section, with constant value of N on all stages, becomes

__________ b(zF ~ xw)__________

zF + xw — 2zF xw — c(xw — zF)