Because of the complexity of the GS process flow sheet, there are a number of opportunities for making improvements in the process that, taken together, should increase deuterium production, reduce the number of separate pieces of equipment, improve energy utilization, and reduce costs. U. S. work on improvements in the early 1960s was described by Proctor and Thayer [P4] and has been used in the first Canadian plants. Later improvements patented by Thayer [T3] have been considered for the newer Canadian plants.

This section will describe one flow sheet improvement patented by Babcock [Bl], which

would increase deuterium production by providing supplementary natural water feed to the hot tower. Burgess [B14] describes computer calculations of the increased production that would be possible if additional natural water were fed to the first stage hot tower of one of Savannah River GS units. This section will derive equations for the improved deuterium production obtainable by feeding natural water to one of the stages of the hot tower of the 24-stage example used in Sec. 11.3 of this chapter in the simplified analysis of the process.

The McCabe-Thiele diagram for this process, Fig. 13.28, shows that the deuterium content of the liquid phase flowing down through the hot tower drops to feed level xF between the third and second stages from the bottom of that tower. By feeding additional hot water at rate F’ to the second or first stage, it should therefore be possible to increase deuterium production P at constant H2S circulation rate G, although at the cost of increased tails assay xw, reduced fractional deuterium recovery, and higher heat requirements.

Analysis of the increased deuterium production made possible through use of supple­mentary hot water feed will be made by reference to Fig. 13.35. Here it is assumed that the flow rate of supplementary feed F’ to the top of stage number ns of the hot tower and the product rate P are so adjusted that the deuterium content of water flowing from stage ns + 1 to stage ns equals that of natural water feed xF, to prevent mixing loss at the supplementary

feed point. Primary natural water feed rate F and gas circulation rate G are assumed to remain unchanged, with their ratio at the optimum value G/F = 2.03 found for the previous case with no supplementary feed. Burgess [B14] has shown that G/F should be increased slightly to obtain maximum benefit from supplementary feed, but this refinement has been neglected to simplify the subsequent analysis somewhat.

Under these assumptions, conditions in the cold tower will remain unchanged. Equation (13.122) still relates the product liquid assay xP, natural feed liquid assay xF, and cold-tower gas effluent assay yF. With nc = 16, G/F = 2.03, xP/xF = 4.0, and = 2.32, yF/xF = 0.4558.

Equation (13.170) for the separation performance of the stripping section of the hot tower is obtained by analogy with Eq. (13.125), by substituting xF for xP and ns for nh.

An equation for the separation performance of the enriching section of the hot tower is derived by reference to Fig. 13.35. The deuterium balance for the entire plant above stage і of the hot-tower enriching section (between the dashed lines) is

PxP + (F-P)xi+1 + GyF = Fxf + Gyt (13.173)

At low deuterium content, the equilibrium relation for stage і is

Fi = XF — (13-174)

ah

By eliminating yt from (13.173) and (13.174), difference equation (13.175) for the liquid-phase deuterium atom fraction x is obtained:

(13.175)

The solution of (13.175) for the boundary conditions xt = xP at і = nh + 1 and xt = xF at ;’ = + 1 is

[P(xP — xF) + GyF — GxF/ah [G/(F — P)ah ] nh ~nS + fxF — GyF
F — G/ah

Equation (13.176) provides an implicit relation between the product/feed ratio P/F and the number of enriching stages nh—ns. For the present case, with xP/xF = 4, yF/xF = 0.4558, ah = 1.80, G/F = 2.03, and nh = 24.

To complete the analysis, it is necessary to find the amount of supplementary feed for the hot tower, F for a given number of stripping stages ns. To do this, W and xw are eliminated from (13.170) by means of (13.171) and (13.172), and the resulting equation is solved for ns + l:

‘ (P/F)(xp/xf -1) + (G/F)(yF/xF — І/a,) 1 / Г G/F

(1 +F’fF-P/F)(ahyF/xF)-( +F’/F-Pxp/Fxf) ]/ + F’/F-P/F)

(13.178)

Substitution of the given values for xP/xF = 4, G/F =2.03, )>f/xf = 0.4558, and ah = 1.80 yields

Numerical solution of Eqs. (13.177) for P/F and (13.179) for F’/F with auxiliary feed to the top of the second stage of the hot tower (ns = 2) or to the first stage (ns =1) yields the results of Table 13.25, where they are compared with the case of no supplementary feed.

Figure 13.36 compares the above results for this 24-plate case, without reoptimization of the feed rate to the cold tower, with Burgess’ [B14] calculations for the Savannah River plant, in which feed to the cold tower was reoptimized for maximum production.

Determination of the economic proportion of supplementary feed to the hot tower involves balancing the advantage of increased production against the extra costs of preheating additional feed water and stripping H2S from additional waste. In the stripping section of the hot tower, larger downcomers would be needed for the increased liquid flow, and at some value, a larger tower diameter. In a new plant designed for it, some supplementary feed to the hot tower would seem to be advantageous. It would probably be neither practical nor economical to use more than 50 percent extra feed to the hot tower.

Theoretical prediction of the constants Cx and Cs of the UF6 thermal diffusion column would be very difficult because of the great difference in properties of UF6 between the liquid at the cold wall and the dense gas at the hot wall. For other gases at pressures around atmospheric, at temperature differences between hot and cold walls small enough so that separation performance can be characterized by gas properties at a mean temperature, closed expressions can be given for the separation parameters Cj and C5. Quantities involved are

Hot wall temperature Ґ Cold wall temperature T”

Density at mean temperature p Viscosity p

Thermal diffusion constant у Diffusion coefficient D

Thermal diffusion effect. When a composition gradient ду/dr and a temperature gradient ЪТ/Ъг occur together in a stationary gas mixture, the usual diffusion mass velocity —Dp Ъу/Ъг is modified by the thermal diffusion effect so that the mass velocity Jr in the r direction becomes

7 is known as the thermal diffusion constant. When у and ЪТ/Ъг have the same sign and are large enough, this lermal diffusion effect can cause transport of an isotope against a composition gradient and thus produce separative work. If a steady state has been established with zero transport, the mole fraction gradient is related to the temperature gradient by

(fr) =УУО~У)Щ^~ (14.346)

or/zero transport or

Integration of Eq. (14.346) between (уТ’) and (у", T") leads to Eq. (14.347) for the separation factor a:

1/ d fVI f

, 1(1-7) , ^

In a = In— ——- г = 7 In —yr (14.347)

/(1-У) T

Measurement of steady-state compositions у and y" after thermal diffusion equilibrium has been established between temperatures Ґ and T" is the most accurate way of determining the thermal diffusion constant. The next-to-the-last column of Table 14.26 gives values of the measured thermal diffusion constant for several binary isotopic mixtures. In all these cases, у is positive, which means that the light isotope concentrates at the higher temperature under the experimental conditions listed.

Values of the thermal diffusion constant can be calculated by the kinetic theory of gases if the intermolecular potential energy is known. Because the calculation is quite sensitive to the detailed intermolecular interaction, calculated values of the thermal diffusion constant are in less satisfactory agreement with experiment than other transport properties.

Hirschfelder et al. [H9] gives a generalized relation for the variation of the thermal diffusion constant with temperature for gases whose molecules interact with the so-called Lennard-Jones potential function, the difference between a repulsion energy inversely

proportional to the twelfth power of the intermolecular distance and an attraction energy inversely proportional to the sixth power. Shacter et al. [S3] have used this theory to predict the thermal diffusion constants in the third column of Table 14.26. The agreement with experimental values is only semiquantitative.

Figure 14.39, based on this theory, may be used to predict the magnitude and temperature dependence of the thermal diffusion constant for 15 isotopic mixtures. The quantity plotted, /rf, is the ratio of the calculated thermal diffusion constant to the thermal diffusion constant

7* for rigid, spherical, nonattracting isotopic molecules, for which

. _ IQS г»г ~mi ^ 108 m2 + mi

Because the maximum theoretical value of fcj — for this intermolecular potential is 0.627, у in thermal diffusion is smaller than a0 — 1 in gaseous diffusion, Eq. (14.24). kj becomes negative at temperatures near the normal boiling point and changes back to positive at still lower temperatures.

Figure 14.40 shows the most accurate measurements of the thermal diffusion effect in UF6 vapor at low pressure, by Kirch and Schiitte [К2]. Results are plotted both as k’T, for comparison with other gases in Fig. 14.39, and as the thermal diffusion constant y. The very low values, under 0.00005, explain Nier’s [N3] inability to detect a thermal diffusion effect in UF6 vapor. The thermal diffusion coefficient у is so much smaller than the analogous parameter in gaseous diffusion, a0 — 1 = 0.0043, that vapor-phase thermal diffusion cannot compete economically with gaseous diffusion for uranium enrichment.

Equations for thermal diffusion column. Equations for the separation performance of a thermal diffusion column can be derived in somewhat similar fashion to the countercurrent gas centrifuge of Sec. 5.5. The results will be summarized for the simplest case to treat theoretically, that of an annular column in which the spacing d between the heated and cooled

tubes is much smaller than the log mean radius f, and in which the temperature difference between heated and cooled walls, AT — Ґ — T", is small enough so that the gas properties can be evaluated at the log mean temperature T. This theory was developed first by Jones and Furry [J5], but using different notation.

Thermal convection between the hot and cold walls under gravitational acceleration g induces longitudinal countercurrent mass flow at the rate

2nfgp2 d3 AT 384 p T

Because longitudinal velocity is zero at the heated and cooled walls, the logarithm of the effective separation factor is found to be that of Eq. (14.347). For у < 1 and AT/T < 1, this becomes

The differential enrichment equation for the countercurrent thermal diffusion column is of the same form as for the countercurrent gas centrifuge, Eq. (14.181), here written as

dy _ Ciyjl ~y)-P(yP ~у) dz C2 + Сз

Form (14.187),

For the annular geometry presently considered, the parameter C2, representing longitudinal back diffusion, is

C2 — 2-nfDpd

Jones and Furry’s development leads to Eq. (14.354) for C3:

_ 128 dN2 _2irfg2p*d’1 /atV 3 ~ 315 pD(2vr) ~ 9! p2 pD T /

Generalizations of these equations for larger АТ/T and for wider annuli (d « r) have been given by Mclnteer and Reisfeld [M8].

In any practical separation plant, the preferred reflux ratio will clearly be greater than the minimum, which would lead to an infinite number of stages, and less than the infinite reflux ratio needed for the minimum number of stages. In most nonisotopic separation plants it is customary to select a reflux ratio somewhat greater than the minimum at the feed point and to use the same value throughout the entire enriching or stripping section, even though a smaller value would suffice toward the product or waste end of the plant. In distillation this is done because the reflux ratio in an adiabatic column remains nearly constant, and it is cheaper to add or remove heat only at the ends of the column than at a number of intermediate points. In many isotope separation plants, however, so much can be saved in the way of reduced equipment size and material holdup by reducing the reflux ratio at intervals between the feed point and the product ends of a cascade that this is usually done. Investigation of the properties of such a “tapered” cascade is therefore important in isotope separation, and of interest in other separation problems because it indicates how equipment size and holdup could be reduced in cases where the increased complexity of a “tapered” plant is justified.

Properties of a cascade with constant reflux ratio over a substantial composition interval are considered in Sec. 13.

When only two isotopic compounds are present in the mixture being separated, such as a mixture of CH4 and CH3D or a mixture of H2160 and H2I80, the separation factor in distillation may be estimated with sufficient accuracy for survey purposes from the ratio of the vapor pressures 7г of the two compounds,

aAB = (13.2)

ffB

where A is the compound with higher vapor pressure. Measurements of the separation factor in liquid-vapor equilibrium of many isotopic mixtures have shown that In a (measured) is within 10 percent of In a [calculated from (13.2)] except for 3He-4He or H2-HD-D2 mixtures. With the same exceptions, measured lna’s vary less than 10 percent with isotopic composition at constant temperature or pressure.

For Eq. (13.2) to be strictly true, it is sufficient that the liquid and vapor phases form ideal solutions, which is usually very nearly the case for isotopic mixtures at pressures up to 1 atm.

When more than two isotopic compounds are present in the mixture being separated, such as H2, HD, and D2, or H2 0, HDO, and D2, the relation between separation factor and vapor pressures becomes more involved. The situation is complicated further when the vapor pressure of a mixed isotopic compound cannot be measured, because it cannot be isolated in pure form. HDO is such a compound, because it remains in equilibrium with H20 and D20:

2HD0=*H20 + D20

The approximate relation between separation factor for hydrogen from deuterium and the measurable vapor pressures of H20 and D2Ois

= I"*’0

ffD,0

The general rule is that in a mixture of isotopic compounds

ЛГАиДА^ВДА^Вї……………….. ХЪп

the separation factor for isotopes A and В may be approximated by

2.2 Separation Factors

Table 13 3 lists for a number of isotopic mixtures the separation factor computed from vapor pressures by this general formula. This table gives separation factors at the normal boiling point and at the triple point, the lowest temperature at which distillation is possible. As this table shows, the separation factor is greatest for compounds of elements of low atomic weight and increases as the temperature is reduced.

Deuterium. The first part of Table 13.3 lists vapor-pressure ratio data for four compounds of hydrogen that are handled in large enough volumes to be possible feed materials for a plant to concentrate deuterium by distillation.

H2 + HD is the only mixture of compounds of hydrogen that has a separation factor as favorable as in conventional industrial distillation. In this case, however, the true separation factor is less favorable than here calculated from the vapor-pressure ratio, because of nonidealities in gaseous and liquid mixtures of hydrogen and HD. Moreover, it is desirable to operate above atmospheric pressure, to preclude in-leakage of air. Under practical conditions, at 1.6 atm, the relative volatility obtainable is around 1.6 [Nl]. This is the most favorable relative volatility for separation of deuterium by distillation.

Although water has a slightly less favorable relative volatility than ammonia, water makes the better working substance because it is available in unlimited quantities, whereas the amount of deuterium that could be extracted from ammonia is limited to the amount present in ammonia produced industrially.

Methane cannot be used as working substance in a distillation process because its relative volatility is so close to unity. This is regrettable in view of the large amount of natural gas that might be used as a source of deuterium.

Concentration of deuterium by distillation of hydrogen will be discussed in Sec. 4 and water in Sec. 5.

Noble gases. The second part of Table 13.3 lists vapor-pressure ratios for isotopes of the noble gases helium, neon, argon, and xenon. The vapor-pressure ratio is very high for helium, much smaller for neon, scarcely different from unity for argon, and precisely 1 for xenon. This illustrates the general rule that distillation is a possible separation method for isotopes of the lightest elements, but becomes useless at atomic weights much over 20. Distillation is the preferred method for separating helium isotopes.

Carbon, oxygen, and nitrogen. The only other compounds listed in Table 13.3 whose isotopic species have been concentrated to a significant degree by distillation are CO, NO, and H2 О (for oxygen isotope separation). Distillation becomes unattractive as a method for separating an isotope of low natural abundance when the vapor-pressure ratio is below 1.01, because the plant required for a given output becomes very large and the time required to bring the plant into steady production becomes very great. This is a consequence of the high holdup per unit separation capacity in this method in which the process fluid is liquid. Gas-phase separation processes such as gaseous diffusion are less subject to this difficulty.

Derivation of Eq. (13.3). The following derivation of Eq. (13.3) relating the deuterium separation factor in the distillation of water to the vapor pressure rr of H20 and D20 is similar to that given by Urey [Ul]. It is assumed that:

1. Liquid and vapor phases form ideal solutions.

2. The vapor pressure of НЕЮ is the geometric mean of the vapor pressures of H20 and D20.

3. Equilibrium in the reaction

H20 +D20 s* 2HDO

is maintained in the liquid phase.

4. The distribution of deuterium and hydrogen atoms among the three species of water is random, so that the equilibrium constant for this reaction has the value of 4.0. These assumptions are plausible, but are not subject to complete experimental confirmation because liquid НЕЮ cannot be isolated, because it disproportionates into H20 and D20. Values for the equilibrium constant calculated by statistical mechanics are around 3.8.

Water contains the three molecular species H20, НЕЮ, and D20. In concentrating heavy water by distillation, the deuterium separation factor is defined as the ratio of the atomic ratio of deuterium to hydrogen in the liquid to the corresponding ratio in the vapor. In terms of the mole fractions of individual compounds in the liquid дг and vapor y, the separation factor a* is

Vapor-pressure ratio

V^HjO/^DjO Separation factor

3.1 Principle

The gaseous diffusion process makes use of the phenomenon of molecular effusion to effect separation. In a vessel containing a mixture of two gases, molecules of the gas of lower molecular weight have higher speeds and strike the walls of the vessel more frequently, relative to their concentration, than do the molecules of the gas with higher molecular weight. If the walls of the vessel have holes just large enough to allow passage of molecules one by one without permitting flow of the gas as a continuous fluid, more of the lighter molecules flow through the wall, relative to their concentration, than the heavier molecules. The flow of individual molecules through minute holes is known as molecular effusion. The possibility of separating gases by effusion through porous media was discovered experimentally by Graham over a hundred years ago. Maxwell showed that this separation was due to the fact that the relative frequency with which molecules of different species enter a small hole is inversely proportional to the square root of their molecular weights. For a mixture of 235UF6 and 238UF6 this ratio, the ideal separation factor for gaseous diffusion a0, is

(14.1)

Because this value is so close to unity, to obtain a useful degree of separation the process must be repeated many times in a countercurrent cascade of gaseous diffusion stages, such as was shown in Fig. 12.2.

3.2 History

The first use of gaseous diffusion for isotope separation was by Aston [A4], who in 1920 effected a slight separation of the isotopes of neon in a single stage of gaseous diffusion through a porous clay tube. Hertz [H2, H5, H6] greatly increased the separation obtainable by this method by using a countercurrent recycle cascade of from 24 to 50 stages of the type shown in Fig. 12.2. This apparatus effected practically complete separation of the neon isotopes of mass 20 and 22 and completely separated hydrogen and deuterium. With a 34-stage cascade, Wooldridge, Jenkins, and Smythe [W3, W4] enriched 13CH4 from 1 to 16 percent.

When World War II created a demand for 235U, the proved ability of gaseous diffusion to effect isotope separation and the existence of a stable, volatile compound of uranium, UF6, led

to intensive development of this process in England and the United States. Because of greater security against attack and more abundant energy supplies, the two governments decided that the first gaseous diffusion uranium enrichment plant would be built in the United States. The Manhattan Project, under the leadership of General Leslie ft.. Groves, built the first gaseous diffusion plant, the K-25 plant, at Oak Ridge, Tennessee, which began operation in 1945. Partial descriptions of this plant and the demanding development effort that led to its successful operation have been given by Smyth [S6], Keith [Kl], Hogerton [H10], Groves [G5], and Groueff [G4], and the official U. S. history by Hewlett and Anderson [Н7]. The development effort in England and the construction of the British gaseous diffusion plant at Capenhurst in the 1950s has been described by Jay [J3]. The independent development of the gaseous diffusion process in France in the 1950s and the construction of the first French plant at Pierrelatte in 1964-1967 has been described by CEA [С7].

In the future, it is probable that the supplier of enrichment services will permit a customer to specify the assay (®*U content) of the tails to which feed is to be stripped so as to minimize the combined cost to the customer of natural UF6 feed and separative work. Figure 12.20 shows qualitatively the effect of tails composition on the contributions to product cost arising from costs for feed and for separative work in stripping and enriching sections. The amount of separative work required in the enriching section is independent of tails composition. But the cost of separative work required in the stripping sections varies from zero when хц/ = Zp (no stripping) to infinity when хц/ = 0. Conversely, the cost of feed varies from infinity when хц/ — zp to a minimum at хц/ = 0, as may be seen from Eq. (12.152). There is therefore an optimum tails assay x0 between Хц/ = 0 and хц> = zp, at which the sum of the cost of separative work and the cost of natural uranium feed is a minimum.

An equation for evaluating the optimum tails composition is derived by substituting explicitly into Eq. (12.152) for the unit cost of product cp the separation potentials фр, фц/, and фр expressed in terms of the corresponding weight fractions xp, хц>, and xp by Eq.

(12.144) :

0 zF Figure 12.20 Effect of tails composi-

Tails composition, xw tion on cost of product.

yp-xw

zp-xw

Optimum tails composition occurs when

When cp from Eq. (12.158) is substituted into Eq. (12.159) and optimum tails composition x0 is substituted for xw, the result is

— УР ZF (Izp — 1) In j~~— і (12.160)

(zp-x o)2 1~zf

This may be simplified to

Figure 12.21 shows the dependence of optimum tails composition on the feed-to-separative work cost ratio.

An interesting interpretation may be given Eq. (12.161). Hie optimum tails composition is

0 12 0.20 0 30 0.40 Figure 12.21 Optimum tails com-

Optimum % U-235 in tails position.

the composition of material from which natural uranium can be produced in an ideal cascade without stripping section for the same cost as natural uranium from an external source. This may be seen by comparing the right side of Eq. (12.161) with the term in brackets of Eq. (12.137) for the total flow rate in the enriching section of an ideal cascade. Further discussion of these equations is given by Hollister and Burlington [H3].

This section derives a general equation for the dependence of stream compositions in an exchange column on stream flow rates and number of equilibrium stages. In Fig. 13.22, the vapor flow rate is V kg-mol of exchangeable element in unit time, and the liquid flow rate is L in the same units. In the present simplified derivation these flow rates are treated as constant throughout the column. Vapor compositions у and liquid compositions x are expressed as atom fraction of desired isotope of exchangeable element. To keep the derivation simple, atom fractions are to be restricted to values below 0.05, as are found in the large stages of plants to concentrate deuterium, 13 C, 15 N, or 18 О from the natural element.

The equilibrium relation between vapor and liquid leaving stage і is

*, = <W — (13.79)

where a is the separation factor. This convention is used to make a greater than unity for exchange separation of deuterium, the example of greatest practical importance.

The material-balance equation for the section between the top of the column and the top of stage і is

Hence

y, xQ Atom fraction

V L Molal flow rote

yn+i *„ Atom fraction

V L Molal flow rote Figure 13.22 Flow rates and compositions in exchange

Vapor Liquid column.

From (13.82),

This is the general equation for the number of theoretical stages needed in exchange columns. It is a form of the Kremser [K5, S5] equation, derived originally for gas absorption.

6.1 Introduction

Processes in which isotopic composition changes are produced when a flowing gas mixture experiences large linear or centrifugal acceleration are termed aerodynamic processes. Of the many aerodynamic processes that have been proposed or investigated experimentally, only two have been carried through large-scale pilot-plant experiments to intended commercial deployment. These are the separation nozzle process, developed by Becker and his associates of the Nuclear Research Center at Karlsruhe, West Germany, and the UCOR process, developed by the Uranium Enrichment Corporation of South Africa. The separation nozzle process has passed through a number of development stages, which have been described in detail by Becker and his associates [B5-B12, G1 ]. These will be summarized in Sec. 6.2. The South African process is subject to considerable industrial secrecy; a brief summary of three published articles on this process [G2, HI, R3] will be given in Sec. 6.3.

Numerous other schemes for separating isotopes in flowing gas streams have been conceived and subjected to small-scale test, but none has appeared sufficiently promising to enlist the major development support given the nozzle and UCOR processes. Summary descriptions of other aerodynamic processes are in references [T2] and [М2].

4.1 Terminology

The simplest type of separating unit or stage is one that receives one feed stream and produces one heads stream enriched in the desired component and one tails stream depleted in the desired component. Figure 12.9 shows such a stage, which is fed at rate Z with z fraction desired component and which produces a heads stream at rate M with у fraction desired component and a tails stream at rate N with x fraction desired component. Flow rate and composition should be on the same basis, e. g., weight, mole, or atom.

Because of overall material balance,

The fraction of a component appearing in the hepds stream is known as the recovery r of that component. The recovery of the desired component, for example, is

Weight, mole or atom fractionof Weight, desired mole or alom Flowrate isotope ratio

M у n

Figure 12.9 Flow rates, composi­tions, and separation factors.

The separative capacity Д of a stage receiving feed of atom fraction z at rate Z and producing heads of atom fraction у at rate M and tails of atom fraction x at rate N is

Д = M(2y — 1) In + N(2x — 1) In jZ— — Z(2z — 1) In (12.253)

Substitution of Eq. (12.10) for M, (12.11) for N, (12.18) for у in terms of z, and (12.20) for x in terms of z into Eq. (12.253) and simplification leads to

Л = ^TZ-f W — 1) In 7 ~ (7 — 1) In (3 + z [((З + 1X7 — 1) In/3 — (7 + 1 № — 1) In 7] )

(12.254)

In this general expression, the ratio of total flow to a stage to separative capacity of the stage, Z/Д, is a function of stage feed composition z. Hence, for this general case, the total flow to all stages cannot be obtained simply as DfZ/A), as was done in Sec. 11. That is, the concept of separative capacity does not provide a simple, accurate way of evaluating the total flow in general for a p-up, q-down cascade. There are, however, a number of practically important special cases for which the term in braces of Eq. (12.254) is substantially or completely independent of z, in which the separative capacity may still be used.