Construction and optimization of a separation ...

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CASCADE IN THE GAS-DYNAMIC METHOD OF SEPARATION ... effect of separation is achieved in specially organized gas flows - flows in nozzles, jets,.
CONSTRUCTION AND OPTIMIZATION OF A SEPARATION CASCADE IN T H E GAS-DYNAMIC METHOD OF SEPARATION A. V. Bulgakov

UDC 621.039.3

Among the different methods for separating gases and isotopes, great interst has been shown in recent years in the so-called gas-dynamic methods [1-7], in which the elementary effect of separation is achieved in specially organized gas flows - flows in nozzles, jets, interacting jets, etc. The separation process can occur as a result of both pressure diffusion or curvature of the stream lines in a quasiequilibrium flow [1-3, 5] and individual interaction of molecules under conditions of translational nonequilibrium [i, 4-7]. Some gas-dynamic separation schemes are already employed for the separation of uranium isotopes on a commercial scale [3, 4]. Gas-dynamic methods have some general properties: the carrier gas must be pumped through; the concentration of the components being separated in the flow field is spatially nonuniform; the separation process, leading to a marked increase in the separation factor as the relative fraction of the extracted flow decreases, is substantially asymmetric. This leads to the fact that the problem of effective connection of the separate steps with one another becomes complicated. In the available theories of symmetric [8] and asymmetric [9] cascades the specific characteristics of the gas-dynamic methods are not taken into account, and they must be generalized. In this work the questions of the construction of ideal and nonideal gas-dynamic separating cascades and the question of obtaining optimal schemes for connecting the steps neglecting the waste part of the cascade are examined. Asymmetric Cascade Consisting of Steps with One Extraction. Let a stream L of a twocomponent mixture with the required component having a concentration c be fed into the gas-dynamic separating element. Two streams exit from the element: an enriched stream eL with a concentration c+ (extract) and a depleted stream (i - e)L with a concentration c-. Let the separation of the isotopic mixture be realized in such a manner that the enrichment factor s+ = (c + - c)/[c(l - c+)] 62 > 63. The concentration in the enriched stream B equals c- = c - 6-. Using the expression

(i) and the mass balance equations

it is easy to show that

61 = 4c (t --c) 8~ in l/O,020a; 82 = 4c (1 -- c) 81 (03 In 020203 -- In OlOJ/(I - - Oa); 6a :- 4c (t --c) ~/(0~In 0,02 --In 02)/(t --02); 8------4c (l --c) etOt In 1/02/(t --0,).

The work of separation of the element is defined in this case as the sum of sequential separation acts : 0102 8U=8L~}(iO~lo, ln~Ol+i--L-~2

lnO~

9 020203

O~-~-i--~-3 ln~03+...).

(6)

The results of calculations based on the formula (6) are shown in Fig. 3. In two extractions in the element the maximum work of separation is realized for 81 = 0.37, ~2 = 0.2 (curve 5) and is 23% higher than the maximum possible value of 6U in an element with one extraction (81 = 0.2; curve i). The employment of an element with three extractions increases the gain in the work of separation of the element up to 33% (curve 6). We shall now examine a cascade consisting of steps with several extractions. In this case, the number of possible connection schemes is larger. To simplify the analysis we shall confine our attention to steps with two extract streams. Let the stream D n = 01e2L n be fed into the input of the (n + k)-st step, while the stream E n = 81(I - 82)L n is fed into the input of the (n + s)-st step, and in addition k > s. The enriched stream B n = (i - 81)" L n is fed into the input of the preceding (n - l)-st step, since the step operates most effectively for 61 < 0.5 (see Fig. 3). The equations of balance of the streams in this case can be written in the form l DB+Dn-I+'"+Dn-A+*+ En + E n - I + ' " T E n - s + I - - B n + * = P , (7)

Dn (cn@6,. n)+Dn-i (cn-2+61 ..... I)@ ..--~-Dn-I,+I(cn-1r +82. n-h+J@En (cn@82. n) @ + 62, n-l) + . . 9@ En-s+l (cn-.~+, ~-82, n-s+ 1) - - B•+I (cn~*

+ En-i (r

We assume,

as usual

-

-

87l+l) --- Pc p.

(8)

[9, 12], that O n ~ Dn_ * ~ ... := Dn_h+ 1 >> P;

6,, n = 61, n-* . . . .

-- 61;

de Cn-l--Cn-2=cr~--cn-l=~

Neglecting

infinitesimals

of

higher

order,

from

etc.

the

formulas

(7)

kO,02 + sO1 (1 - - O J + 0 1 = t; de [Ok ( k + t ) + Es (s+ 1)] d--~= 2 [sE (~2 + 8 - ) @ k h

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(81-~8-) - - p (cp--c)].

and

(8)

we o b t a i n

(9) (lO)

Z :J

ZL

~,,

J

2

Ii

~,~ :,7 /.5

5/

11

0,3 Fig.

150

"100

0,,: 8~

Fig.

3

20a

?,iO iV

4

Fig. 3. Work on separation of the gas-dynamic separating element: i) 02 = 83 = i; 2) 02 = 0.8, 03 = 1; 3) 0 2 = 0 . 5 , 03 = 1; 4) 82 = 0 . 1 , 03 = 1; 5 ) 02 = 0 . 2 , 0 s = 1; 6) 02 = 0 . 4 , 03 = 0 . 2 . Fig. 4. The total flux in a real cascade consisting of steps with two extractions versus the number of steps: i) 01 = 5/12, 82 = 0.4, k = 2, s = i; 2) 01 = 5/12, 02 = 0.5, k = 3, s = 2; 3) 01 = 0.5, 02 = 1/3, k = 3, s = 0; 4) 81 = 0.4, 02 = i/6, k = 4, s = i; 5) 01 = 82 = 1/3, k = 4, s = I; 6) 01 = i/3, 02 = 0.25, k = 5, s = 1. Equation (9) imposes restrictions on the extraction factors - an arbitrary set of values of 01 and 02 cannot be realized in the cascade, but rather the set must satisfy the relation (9) for fixed integers k and s. The concentration distribution in the cascade follows from Eq. (I0). It is easy to show that for a cascade consisting of identical steps (L = const) the form of the solution of Eq. (i0) is identical to the expression (5), where (s+ 1) In (UO1)+(k--s) Oe In [1/(0102)] e n': 88:01 k (k - - 1) 0~02 + s (s - - I) 01 (1 - - 0~) q- 2 (I - - 01) ; Ln

L [k ( k - - ~ ) O l 0 2 + s ( s - - t ) O

1(1-02)+2(t__01)

].

The results of the calculation of such a cascade for separation of uranium hexafluoride are shown in Fig. 4 for several sets of 91 and @2The total flux here is referred to the same value of (ZL) I as in Fig. I. Comparison of Figs. 1 and 4 shows that the use of steps with two extractions gives a substantial advantage both in ZL and in the number of steps. The value of XL in a cascade with a minimum total stream (k = 4, s = i, curve 4) is 40% lower than in a cascade consisting of steps with one extraction with N = ii0, while for N = i00, it is 54% lower. The difference between the total fluxes of the cascade with k = 4, s = i and simpler cascades with k = 3, s = i (curve 3) and with k = 3, s = 1 (curve 2), the last of which is the most advantageous of all cascades studied, is small. For N = II0 this difference is ~50. Even in the simplest scheme k = 2, s = 1 (curve I) ZL and N are lower than in any scheme consisting of steps with one extraction. We shall now examine the possibility of constructing steps with two extractions. In this case the conditions c,~_h~-5~-'"cn_~+52

an ideal cascade

consisting

of

c~+1-5-~c,

or for k + s > L [10], where Ls and L are the fluxes of the carrier gas and of the mixture being separated. In the general case

~+=A~ ~)~ In(I/O), where x = Ls

~(x) + 8 as

x + 0; @(x) + 1 as

x + =.

We define overall total flux in cascade as the sum (EL)o = EL + ELs To obtain the optimal ratio of the fluxes Ls and L, corresponding to the mlnimum value of (ZL) o in the ideal cascade, the condition

(i+qx)/~(~=min, where

(16)

q is the cost factor for pumping the carrier gas, must hold.

As an example we shall make use of the results of [13] on the separation of uranium hexafluoride in hydrogen in a "separating nozzle," The experimental data from this work on the effect of the hydrogen concentration on g+ are approximated well by the expression @(x) = x/(20 + x). Setting q = i, we find that the condition (16) is satisfied by x = 22, which corresponds to a uranium hexafluoride concentration of 4.3%. It is interesting that precisely this concentration (molar fraction of UF 6 equals 4-5% in the mixture UF6-H 2) was chosen in the practical realization of this method [2]. We note, however, that the total flux is no longer the only characteristic of the efficiency of the cascade, since it determines only the energy consumed by the cascade, while the number of elements, as before, is proportional to the total flux of the mixture to be separated EL. This analysis of the costs of pumping the carrier gas can be easily extended to a real cascade. The author thanks Yu. S. Kusner for useful discussions and critical remarks. LITERATURE CITED i. 2. 3. 4.

5. 6. 7.

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R. Campargue, J. Anderson, J. Fenn, et al., "On aerodynamic separation methods, ~' in: Proceedings of the European Nuclear Conference, Vol. 12, Paris (1975), pp. 5-25. E. Becket, W. Bier, W. Ehrfeld, et el., "Physics and technology of separation nozzle process," in: ibid., pp. 44-52. A. Roux and W. Grant, "Uranium enrichment in South Africa," in: ibid., pp. 39-43. K. Nguyen and R. Andres, "Free jet deceleration - a scheme for separating gas species of disparate mass," in: Proceedings of the llth International Symposium on Rarefied Gas Dynamics, Paris (1979), pp. 667-682. J. Brooc, W. Calla, E. Muntz, et al., "Jet membrane process for aerodynamic separation of mixture and isotopes," J. Energy, i, No. 5, 199-208 (1980). Yu. S. Kusner, S. S. Kutateladze, V. G. Prikhod'ko, et al., "Inertial gas-kinetic separation of gas mixtures and isotopes," Dokl. Akad. Nauk SSSR, 247, No. 4, 845-848. A. Bulgakov, Yu. Kusner, V. Prikhodko, and A. Rebrov, "Separation of gas mixture components in interacting flows," in: Progress in Astronautics and Aeronautics, Vol. 74, New York (1981), pp. 607-616.

8. 9. !0. ii.

12. 13.

K. Cohen, The Theory of Isotope Separation, New York (1951). N. A. Kolokol'tsov, "Problem of construction of ideal asymmetric separation cascades," At. Energ., 27, No. I, 9-13 (1969). Yu. S. Kusner, "Theory of gas-dynamic separation," Dokl. Akad. Nauk SSSR, 259, No. 2, 359-261 ( 1 9 8 1 ) . V. A. Chuzhinov, V. A. Kaminskii, V. A. Laguntsov, et al., "Increasing the efficiency of mass-diffusion separation cascade with the use of asymmetric step connection," At. Energ., 44, No. 3, 254-256 (1978). N. A. Kolokol'tsov, "Increasing the efficiency of cascades for separation of isotopes using steps with more than two outgoing streams," At. Energ., 37, No. i, 32-34 (1974). E. Becher, W. Bier, W. Ehrfeld et al., "Die physikalischen Grundlagen der Uran 2ss Anreicherung nach dem Trenndusenverfahren," Z. Naturforsch., 26a, 1377-1384 (1971).

OPTIMIZATION OF A RESEARCH REACTOR IN OPERATON Yu. P. Malers

UDC 621.039.5

A reactor for physical research can be considered as a production plant which supplies neutrons to a consumer [I]. Thermal-neutron flux in experimental facilities was taken in t h e p r e s e n t work as a measure of theproductivity of such a plant for the sake of simplicity

[2]. Optimization of the operation of a research reactor has a specificity. Thus, optimization at the design stage is performed with a fixed productivity whereas this approach is hardly justified in the operation. It may be more profitable to meet certain needs of the experiments by improving the performance of operating reactors rather than by constructing bigger reactors or a greater number of new ones~ Possible decisions on operating conditions of a research reactor are considered in [3]. Among other things the profitability approach was proposed. However, the model outlined has been presently unrealized and thus far the profitability criteria cannot be applied to research reactors. To formulate an optimality criterion (OC) for an oPerating research reactor, we proceed from the following assumptions: the total productivity of all reactors has been planned (this corresponds to a version of [3] with a centralized decision on the product volume and distribution); new reactors are steadily brought into use (it is possible to vary the productivity of operating reactors without disturbing the first assumption due to the planning of new reactors' productivity); the design "unit cost" in the new reactors under design is fixed (its variation with productivity is negligible when a productivity change in operating reactors is small as compared to the productivity of new reactors). The following designations are used: P and C are the productivity and design cost respectively for operating reactors; P0, Co and U0 are the productivity, reduced ocst and unit cost respectively for new reactors (U 0 = C0/P0). The OC can be formulated as follows: C + C o + min; P + P0 ~ Pt is fixed. Since Co = UoP0 = U0(P t - P), where U 0 and Pt (total productivity) are fixed, the OC can be written as C-UoP ~ min(or P-C/U0 § max). Translated from Atomnaya ~nergiya, Vol. 61, No. 6, pp. 453-454, December, 1986. Original article submitted January 20, 1986.

0038-531X/86/6106-I055512.50 9 1987 Plenum Publishing Corporation

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