When the optimization of a step cascade is carried out directly on the basis of an exact formula, the calculus of minimization of a multi-variable function is essential. The numerical calculation is very complex and a brief estimation becomes impossible as the number of steps increases, because the number of variables in the calculus amounts to (2ₘ₋1) in all for an m-steps cascade. In this paper an approximate formula of good feasibility and sufficient accuracy is proposed for the determination of the number of stages and interstage flow rate required for the plant of uranium enrichment by gaseous diffusion process. By using an approximate formula, the minimization of multi-variable function in the optimization of step cascades is able to be avoided. The applicability of the proposed method is verified by the calculation of a typical plant for uranium enrichment

DOI, Hideo

HIGASHI, Kunio

SAITO, Toru

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Optimum Design of Step Cascadesfor Uranium Enrichment byGaseous Diffusion ProcessHIGASHI, Kunio; SAITO, Toru; DOI, HideoHIGASHI, Kunio ...[et al]. Optimum Design of Step Cascades for Uranium Enrichment byGaseous Diffusion Process. Memoirs of the Faculty of Engineering, Kyoto University 1971,33(3): 163-1731971-09-30http://hdl.handle.net/2433/280852Optimum Design of Step Cascades for Uranium Enrichment by Gaseous Diffusion Process By Kunio HIGASHI*, Toru SAITO* and Hideo D01* (Received March 8, 1971 J When the optimization of a step cascade is carried out directly on the basis of an exact formula, the calculus of minimization of a multi-variable function is essential. The numerical calculation is very complex and a brief estimation becomes impossible as the number of steps increases, because the number of variables in the calculus amounts to (2m-l) in all for an m-steps cascade. In this paper an approximate formula of good feasibility_ and sufficient accuracy is proposed for the determination of the number of stages and interstage flow rate re-quired for the plant of uranium enrichment by gaseous diffusion process. By using an approximate formula, the minimization of multi-variable function in the optimization of step cascades is able to be avoided. The applicabiiity of the proposed method is verified by the calculation of a typical plant for uranium enrichment. 1. Introduction 163 At present one is able to analyse the characteristics of various kinds of cas-cades. The ideal cascade and more general tapered cascade have been discussed by Cohen1>, Higashi2> and others3 - 5 >. Square and step cascades have been inves-tigated by Cohen1>, Benedict6 >, Oliveri7> and others. The ideal cascade realizes the smallest size of cascade achieving desired enrichment, but this cascade is not practical in its construction and control because the number of separating elements in each size has to be varied continuously as the isotope ratio of uranium changes. The most practical cascade might be a step cascade, including a square cascade as a possible variation. When the optimization of a step cascade is carried out directly on the basis of the exact formula by Cohen1>, the calculus of minimization of a multi-variable function is essential. The number of variables in it amounts to (2m- l) in all for an m-steps cascade. Hence, the numerical calculation is very complex and a brief estimation becomes impossible as the number of steps increases. • Department of Nuclear Engineering 164 Kunio HIGASHI, Toru SAITO and Hideo Dm Oliveri 7> has developed an interesting approximation method for the opti-mization of step cascade. By means of this method the optimum compositions characterizing each step are· rather accurately, de.termined,, but it· i~ difficult to discuss the total flow rate .in "'r ,caswde which is e.quivale~~ to the .si~f .of a plant. It is often required to analyze how the size of cascade decreases with the number of steps. For instance, it is desired in the programming of a computer code for an economic evaluation of uranium enrichment by gaseous diffusion process to de-velop such a method as ca~ estimate. the size of a st~p cascade briefly with suffi-cient accuracy. In this paper an optimization method for square and step cascades will be proposed and discussed. 2. Approximate Formula Let us consider an m-step cascade as shown in Fig. 1. P, F and Ware the rates of product, feed and waste, respectively. Xp, xF and xw denote the mole fractions of lighter isotope in product, feed and waste, respectively. X; denotes the mole p I J;X1m(=X;} ,...__ ------r-- ------Fig. 1. Enriching Section of a squared-off cascade. fraction at the top stage in the i-th step. When the separation factor of each stage, a, is close to unity, the total tails flow rate in the i-th step, ]/', can be expressed by the following exact formulae by Cohen, where ]_,_·" = ___ 1__ 1n_l_+_a_; P (a-l) 2b;c; 1-a; - (x;_ 1+x;)(l+c;)-2x;_1X;-2c;Xp' = \/1+2c;(l-2xp)+c:, p { :: C; -----L/'(a-1) ( 1 ) ( 2) ( 3) ( 4) Optimum Design of Step Cascades for Ura11ium Enrichment by Gaseous Diffusion Process 165 L;" in Eq. (4) is the talis flow rtae in the i-th step. When both xi-i and xi are negligibly small compared with 1 or very close to 1, the above equations are reduced to the following expressions. Case· I. When xi_1 ~ 1, xi~ 1, Where 1 A - Xi u--, xi-1 ).Ii= 1, gu = x,_1 {i + (a-1)).,;L/'}. Xp p Case II. When (l-xi_1) ~ 1, (1-xi) ~ 1 , where ' 1 Ahi = l-xi-1 l-xi ).hi= -1 ghi= l-xi {i + (a-1)).hiL/'} l-xp P ( 5) ( 6) ( 7) ( 8) ( 9) (10) (11) (12) Comparing Eqs. (5)-(8) with Eqs. (9)-(12) and noticing the corresponding relations in them, the following approximate formula may be proposed for J/', where A- = xi(l -xi-i) ' Xi_1(1-xi) B- = xp(l-xp) • Xi-1(1-xi) ).i = ____ B~i_l ___ _ xp/xi_ 1-(l-xp)/(l -xi_1) gi = ~,{i+(a-lt).iL/'} (13) (14) (15) (16) (17) 166 Kunio HIGASHI, Toru SAITO and Hideo Dm A; is the over-all separation factor in the i-th step. A; approaches unity when X;_u X; and Xp tend to zero, and approaches to - I when x;_ 1 and X; tend to unity. Therefore, Eq. (13) is reduced to Eq. (5) at low concentrations, and to Eq. (9) at high concentrations. In order that Eq. (13) may converge to the formula of ideal cascade with increasing m, A; should be expressed as Eq. (16). In a special case, g; is in accord with g defined by Benedict and Pigford6 >. By making the g; of Eq. (17) equal to unity one obtains the wellknown condition for minimum reflux ratio in square and step cascades, ( L/') _ 1 ( Xp P g;=i a-l x,_1 (18) 3. The Condition of Minimum Square and Step Cascades Now, let us consider the i-th step in an m-steps cascade. Differentiating Eq. (13) with respect tog; and putting the derivative equal to zero, we obtain ln A,g;-1 _ (A;-I)g;(B;g,-l) g;-1 (A;g;-I)(g1- I)(B1g1-I) (19) or (20) where D- = (A;g;-l)(g,-1) ln A,g,-I · ' (A,-l)g, g,-1 Substituting Eq; (19) into Eq. (13) we get the minimum value of (]/'JP), (J/'/P)min, for a given set of x/s, (]/') P min (21) Eq. (21) should be used in relation to Eq. (19). The relation between A;, B1 and g, given by Eq. (19) is shown in Fig. 2. In this figure two branches can be seen for a single value of A1• One branch starting from the point (B,=l, g1=/l) cor-responds mainly to the enriching section, and the other in the region of O-i:B,< I I corresponds mainly to the analysis of stripping section. It should be hoticed that Fig. 2 or Eq. (19) is available independently of xp, x/s, m and a. Ope may I thus determine (]/' /P)min by the following successive procedure. \ First, A,, B1 and i, are calculated from Eqs. (14)-(16). Then, g1 for the values of Ai and Bi is obtained from Eq. (19) or Fig. 2. Finally, substituting Optimum Design ef Step Cascades for Uranium Enrichment ~Y Gaseous Diffusion Process 167 2.----.----,--....---.-----.--,-----.---...-~---~----.-~----.-~-IA,=l.k i...-C:::: i----•~ ::.1.e ,.u. !4. ' -I .___. _ _.__,___--'---'-...--.1.-.....L--1...;...;.~-1....;......11-..;.1.' --!--...L.....J._.L.._-L----1-.L.._...J 0 2 3 , .. 4 '.5 \e __ , __ .I_: 8 9 10 B· i , I ' Fig. 2. The relation between A;, B; and g;, which is obtain~d from Eq. (20). these values into Eq: (21), (J/'/ P)miu is determined. The interstage flow rate of each step corresponding to (J /' / P) min• L/' / P, and the total number of stages, n, are also easily calculated as follows: L/' _ B,g,-1 P - (a-1),l,' from Eq. (17), and "' "' n = ~ n, = ~ (J/'/P)/(L/'/ P) i:::::al i~l from Eqs. (21) and (22). When we define (J"/P)min as (22) (23) (24) this gives the minimum total tails flow rate of an m-steps cascade with a given set of Xp, Xp and intermediate compositions Xu x 2 , ···, xm-i· Therefore, (]"/ P)mm depends on x 2, x 2, •··, xm-i· In order to optimize a step cascade one should find a set of x/s which makes (J" /P)min minimum. For the discussions below, let us 168 Kunio HIGASHI, Toru SAITO and Hideo Doi define(]"/ P)opt and nopt as the minimum value of(]"/ P)min and the number of stages corresponding to it. 4. Application to Square Cascade Square casc,ade is a kind of step cascade with m=l. Then'. Eq. (21) can be made applicable to square cascade by putting x0 =x F and x1 =Xp. As an example, the calculations below were carried out on the enrichment of natural uranium. The results are shown in Fig. 3. Curve ( 1) represents the requited size of square ca$cade calculated. by Eq. (21). Curve (2) is obtained by me~ns of the trial-and-1rrror ~ethod from Cohen's exact formula Eq. (1). Curve (3) shows 5 5 108 I , /, / l // I A / ,, v ✓ -~ / 11;~ /~ .f /l" ,/.," .... '-"' ' / ,, / ·" / , ·' I , ,,, Vl // J/ 2 3 5 7 10 Xp x 100 (%) 20 30 50 70 Fig. 3. The minimum size of a square cascade. Curves (1) and (2) are calculated by means of the approximate formula (21) and the exact solution (1), respectively. Curve (3) shows the size of the ideal cascade. (a= 1.0028, XF=0.00714) the size of the ideal cascade. From Fig. 3 we may conclude that the approximate formula Eq. (13) is precisely applicable within a rather wide range of Xp and xp. A slight deviation of curve (1) from curve (2) for xp>O.l may be of little impor-tance practically, because it is unusual to adopt a square cascade up to xp>O.l in natural uranium enrichment. A single step in actual step cascade has a rather small overall separation Optimum Design of Step Cascades for Uranium Enrichment by Gaseous Diffusion Process 169 factor. Therefore, it may be expected that Eq. (21) will also provide a suffi-ciently accurate estimation for a step cascade. 5. Application to Step Cascade We shall now consider a 3-steps cascade. (]" / P)min is calculated directly from the exact formula Eq. (I) for various sets of (x1> x2) under the same value of x F and xp, as shown in Fig. 4. In this example (]" / P)opt appears at point Q (x1 =0.0124, x2 =0.0236). The parameter in Fig. 4 denotes the value of (]"/ P)min divided by (]"/ P)opt at point Q. The loops drawn in this figure are contourlines corresponding to the.same value of the parameter. As shown in Fig. 4 the value of (J"/P)min changes very slowly around the optimum point Q. For instance, let us compare the three cascades Q, R and S in Fig. 5 with one another. These cascades correspond t~ points Q, R and S in Fig. 4, respectively. Though they are much different in shlipe, as seen in Fig. 5, s 0.028 0.037 0.026 1400 0.025 1200 Xz 1000 0.023 = .a ~ 800 -0.022 .. "' E 600 "' 0.021 400 -0.020 200 0.0111 0 0.011 0.012 0.013 0.014 0.015 X, Fig. 4. The minimum size of 3-steps squared-off cascade expressed as a function of (x1 , x2). The optimum cascade is realized at point Qin this example. The parameter denotes the value of (Jn/P)min/(J8/P) 0 pt• (a= J.0028, XF=0.00714, Xp=0.05) 2 1.5 0.5 0 0.5 1.5 I I I' ! I 2 Fig. 5. The shapes of 3-steps squared-off cascades corresponding to the points Q, R, and Sin Fig. 4. (a=I.0028, xF=O, 00714, Xp=0.05). 170 Kunio HIGASHI, Toru SAITO and Hideo Doi all of them achieve the same job of enrichment and both the cascades R and S are larger in size by only 0.5% than the cascade Q. One should remark the rather broad flexibility existing in the design of a real plant. Generally, we may conclude from this fact that a rough estimation without any complicated procedures will suffice in determining the set of intermediate compositions which optimizes a step cascade . . Then, we may simply determine x/s with little error in the size of optimum cascade by putting all the over-all separation factors in each step equal to a constant value, that is hence A; =A= {xp(l-xp)}t/m (i = 1, 2, ·•· m-1), xp(l-xp) (25) X; = XpA' /(1 + XpAi). (26) 1-xF 1-xF Point A in Fig. 4 was plotted according to Eq. (26). It is remarkable that (J" /P)min at poitt A is only 0.3% larger than at point Q. Step cascade should approach to ideal cascade as m increases. In the extreme case where m is equal to the number of stages of ideal cascade, nideal• A defined by Eq. (25) is in accordance with the so-called heads separation factor of each stage in ideal cascade, f:J( =va), and X; obtained from Eq. (26) gives the composition at the i-th stage of ideal cascade. Thus, it is expected that a gradually more accurate value of optimum composition will be calculated by Eq. (26) with the increase of m. This is also one reason we adopt Eq. (26) to determine x/s. When A is equal to fl and very close to unity, we get ln A,g,-1 ~ (f:J-l)g, g;-1 g;-1 and then 1 g, ~ 2-- (when A=f:J~l), B; from Eq. (19). Substituting Eq. (28) into Eqs. (21) and (23), we obtain (]·") (L·") 2 ( x -J,- opt = Y opt~ a-1 X~1 l-xp) (whenA=f:J~l) 1-x,_1 (27) (28) (29) This is twice the minimum reflux ratio (Eq. (18)) and equal to the tails flow rate in the i-th stage of ideal cascade, (L/'/P)ideaJ• In this extreme case the number of stages in each step, n;=]/'/L;'', is reduced to unity as seen in Eq. (29). There-Optimum Design of Step Cascades/or Uranium Enrichment by Gaseous Diffusion Process 171 fore, n is equal to m, that is, (30) This is in accord with the expression for nideai· Hence, it is obvious that (]" /P)opt calculated by using Eqs. (19), (21) and (26) approaches (J"/P)ideal with the increase of m. It is important in a design of an uranium enrichment plant to investigate how the size of a step cascade approaches that of an ideal cascade. (]" /P)opt/(J"/P)ideal and n0 pt/ nideal are calculated by the proposed method and shown in Figs. 6 and 7 as a function of m. Fig. 6 shows the value of (J" /P)opt decreases rapidly with the increase of m. The larger the value of m is, the more slowly a cascade decreases in size and the less simple it becomes. Therefore, we may conclude that in most cases the number of steps fewer than 5 will be sufficient in an enriching section as well as in a stripp-ing section. So, serious technical problems, we think, will not be arised in squaring-off a cascade into several steps. The separative work ratio10> of a step cascade with 2.5 I 2 V I/ / / 1.5 vv m-1 / / V vi;' ,,,/ ~- ..,,.,,.,,,. - -----........ 3 -. I 5 ...-fo-I 2 3 5 7 10 20 30 50 70 XpX 100 (%) Fig. 6. The convergence of the size of a squared-off cascade to that of an ideal cascade with the increase of m. The parameter m denotes the number of steps of a squared-off cascade. (a= 1.0028, XF=0.00714) 0.9 ;; 0.8 . ... i= :a-,! 0.7 0.6 0.5 :---- -- :::::.5..--- --- t--. ~- r-3......_ ~ ~L', '"-- -- 1---.. " ~ ........... ~2 ...... I......_ t--,~~ --..... r--..... t---1', " t'--m=I r---... r---......_ ,.... I 2 3 5 7 10 20 30 50 70 Xp xtOO (%) Fig. 7. Comparison between the total number of the stages of the optimum m-steps squared-off cascade and that of the corresponding ideal cascade. Parameter m denotes the number of steps of the squared-off cascade. (a= 1.0028, XF= 0.00714) 172 Kunio HIGASHI, Toru SAITO and Hideo Dot several steps to a corresponding ideal cascade is close to l, as seen in Fig. 6. Thus it may be more realistic in the economics <?f large scale separation of uranium-235 to evaluate enriched uranium on the basili of an ideal cascade. The optimum waste· composition may be determined in like manner. The point T in Fig. 4 is determined by the approximatation developed by Oliveri. This point T is rather close to the optimum point Q, and it shows the superiority of Oliveri's method in the determination of the set of x/s. The value of(]" /P)opt obtained, however, increases with m and does not approach the value of(J"/P)ideaJ• Though the above discussions are concerned with the enriching section of uranium enrichment plant, one may be able to extend the analysis for the optimiz-ation of the stripping section. It is briefly shown below how to calculate the minimum value of J /' in the stripping section. Steps in the stripping section of the cascade are numbered consecutively from l at the waste end of the plant to the feed step. A, and B, of a given step are obtained from Eqs. ( 14) and ( 15). The corresponding g, is determined by the relation of Eq. (19) or Fig. 2. When the uranium is enriched from natural ura-nium (xF=0.0072), the value of J, in stripping section may be approximated to I. g, in stripping section should be defined as follows, g •. __ Bl- {1 _(a-Wl)L/'} stripping section . I (31) Then W/L/' can be obtained by using Eq. (31). The absolute value ot the right hand side of Eq. (21) substituting A,, B,, and g, is equal to (J/'/W)min in a step in the stripping section. The number of stages of the· step can be obtained by multiplying of (J/'/W) and (W/L/'). The relation of W = Xp---'-XF p XF-XW is available to convert (J"/W)min to (J"/P)min• The preceding results may be summarized as follows: (32) I. The approximate formula Eq. ( 13) was proposed as against the exact solution Eq. (1). 2. It was pointed out that there exists in practice a rather brnad flexibility in the determination of intermediate compositions x/ s. On the basis of this fact Eq. (26) was proposed for x,. 3. The size of the optimum m-steps cascade, (J"/P) 0 pt, is obtained by the following successive procedures: Optimum Design of Step Cascades for Uranium Enrichment by Gaseous Diffusion Process 173 a) x/s are calculated from Eq. (26). b) A, B; and g; are obtained from Eqs. (14), (15) and (19) (or Fig. 2), respectively. c) (J"/P) 0 pt is determined by means ofEqs. (21) and (22). 4. The behavior of (J" /P)opt and that of nopt are investigated for the variation of m and Xp by using the method described above. The results are shown in Figs. 6 and 7. 5. We have concluded. from Fig. 6 that the number of steps fewer than 5 may be sufficient in moi.t cases. 6. It is shown briefly how to optimize the stripping section of a plant. Most of the calculctions required in this report were carried out by using the Kyoto University Digital Computer II. References l) K. Cohen: The Theory of Isotope Separation, McGrawHill Book Co., New York (1952) 2) K. Higashi, A. Oya, J. Oishi: Characteristics of a Tapered Cascade for IsoLpe Separation by Gaseous Diffusion, Nucl. Sci. Engng., 32, 159 (1968) 3) A. de la Garza, G.A. Garret,J.E. Murphy: Multicomponent Isotope·Separation in Cascades, Chem. Engng• Sci., 15, 188 (1961) 4) A. de la Garza: A Generalization of the Matched Abundance Ratio Cascade for Multicom-ponent Isotope Separation, K-1527 (1962) 5) A. Apelblat and Y. Hamed-Lehrer: The Theory of a Real Isotope Enriching Cascade, Israel A.E.C. IA759 (1962) 6) M. Benedict and T.H. Pigford: Nuclear Chemical Engineering, chapt. 11, .McGraw-Hill Book Co., New York (1957) 7) E. Oliveri: Squaring Criteria for a Diffusion Cascade,Energia Nucleare, 8,453 (1961) 8) C. Frejacques and .R. Galley: Deductions Based on Studies of Uranium Isotope Separation and French Achievements in this Field, Proceedings of the 'fhird International Copference on the Peaceful Uses of Atomic Energy, vol. 12, P. 329 (1965) · 9) AEC Gaseous Diffusion Plant Operations, OR0--658 (1968) 10) M. Martensson: Symposium on Economics of Nuclear Fuels, Gottwaldov, CSSR, 27-31 May 1968 

Optimum Design of Step Cascades for Uranium Enrichment by Gaseous Diffusion Process

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Optimum Design of Step Cascades for Uranium Enrichment by Gaseous Diffusion Process

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