Sep 5, 2014 - same time they differ with respect to the geometric parameters and rotational speed ..... D is its comparison for the centrifuges TC-12 and TC-21.
Atomic Energy, Vol. 116, No. 5, September, 2014 (Russian Original Vol. 116, No. 5, May, 2014)
COMPARISON OF THE CIRCULATION EFFICIENCY IN GAS CENTRIFUGES WITH DIFFERENT GEOMETRIC AND SPEED CHARACTERISTICS FOR URANIUM ENRICHMENT
V. D. Borisevich,1 O. N. Godisov,2 and D. V. Yatsenko2
UDC 621.039.31
The circulation efficiency of gas centrifuges for uranium enrichment is compared for domestically produced gas centrifuges, the TC-12 and TC-21 models from the European concern Urenco, the American AC-100 model, and the Iguasu model centrifuge in order to search for general physical mechanisms. It is found that for the domestic centrifuges the optimal gas pressure Pa near the wall of the rotor is a linear function of the optimal temperature difference ΔT along the rotor. It is shown that the degree of optimality of the circulation flow in a gas centrifuge has a maximum as a function of the dimensionless rotational speed of the rotor, which is associated with the difficulties of organizing the optimal circulation flow in a thin layer of viscous gas at the rotor wall at high rotational speeds.
Gas centrifuges for uranium enrichment in operation in different countries share many basic elements, but at the same time they differ with respect to the geometric parameters and rotational speed of the rotor. The main characteristic of a centrifuge is its separative power, which depends on the length, diameter and rotational speed of the rotor as well as other structural elements and process characteristics, which secure the optimal conditions for developing axial reflux flow (circulation), multiplying the radial effect of separation during operation. Since the centrifugal uranium enrichment technology is dual-use, as a rule, the theoretical dependences of the separative characteristics on the parameters of a centrifuge are discussed [1–9]. Such dependences are obtained by means of different approximations and often differ considerably. As a result, it is not possible to conclude unequivocally which centrifuge lengths – short (rotor length L to diameter D ratio ~4–10), medium (L/D ~ 10–20), or very long (L/D > 25) – are best for optimal axial circulation of the flow and, correspondingly, maximum separative power. The present article compares the efficiency of the circulation flow in gas centrifuges used at various times in domestic industrial facilities for uranium enrichment, the centrifugal machines TC-12, -21 from the European concern Urenco and the American machine AC-100 as well as a theoretical model, known as the Iguasu centrifuge, in order to look for general rules. In performing the analysis, it was assumed that the structural characteristics of the centrifuges examined secure the optimal axial circulation flow, which maximizes the separative power. The validity of this assumption is based on the fact that industrial application starts only after the completion of a computational-design and experimental optimization cycle aimed at maximizing the separative power of a centrifuge for the chosen rotational speed and geometry of the rotor.
1 2
National Nuclear Research University – Moscow Engineering-Physics Institute (NIYaU MIFI), Moscow. Tsentrotekh-SPb Company, St. Petersburg.
Translated from Atomnaya Énergiya, Vol. 116, No. 5, pp. 300–306, May, 2014. Original article submitted February 11, 2014.
1063-4258/14/11605-0363 ©2014 Springer Science+Business Media New York
363
Fig. 1. Schematic diagram of the interior arrangement of a gas centrifuge: 1, 3) extraction of the heavy and light fractions, respectively; 2) feed flow; 4) protective jacket; 5) magnetic suspension; 6) molecular pump; 7, 9) upper and lower diaphragm, respectively; 8) rotor; 10) stator; 11) needle; 12, 14) bottom and top gas extractor, respectively; 13) feed flow entry.
Theoretical Prerequisites and Computational Relations. As a rule, the radial or primary effect of separation is considerably multiplied along the axis of the rotor owing to the development of a reflux flow, which makes it possible to greatly reduce the number of separation steps in industrial cascades. The circulation plays another important role: the fractions separated in the axial direction are much easier to remove from a centrifuge by means of stationary gas extractors placed near the top and bottom ends of the rotor. The circulation of the working gas in a rotor is produced by different methods [10]. The internal reflux parameters are determined by the combined effect of two factors: 1) mechanical excitation by the stationary gas extractor of the heavy fraction, which can be installed at the top and bottom of the rotor, thereby determining the direction of circulation; 2) thermal excitation, for which the circulation arises as a result of a temperature gradient along the rotor. To decrease the braking effect, the stationary, heavy-fraction, gas-extractor can be screened by a diaphragm rotating together with the rotor. In this case, the diaphragm is equipped with two systems of annular openings: at the wall of the rotor (outer system) and at a distance from it (inner system). The heavy-fraction extractor slows the rotating gas, decreasing its angular speed. In the process, a stream flow through the system of inner openings is realized, forming a single circulation flow at a distance of several diameters of an opening (Fig. 1). The pressure distribution of the working gas along the radius of the rotor in accordance with Boltzmann’s law has the form P(r) = P0exp(Δµω2r2/2RT), where P(r) and P0 are, respectively, the gas pressure at the radius r and the axis of the rotor; μ is the molecular mass of the working gas; Δμ is the molecular mass difference between the uranium isotopes 238U and 235U; ω is the angular speed of the rotor; R is the universal gas constant; and T is the temperature of the gas. Near the stationary gas extractor, the gas distribution along the radius becomes more sloping. In addition, at a shorter radius the pressure of the working gas above the diaphragm becomes higher than at the same radius in the working chamber of the rotor. In addition, the stationary gas extractor not only slows down the working gas but it also brings about a significant 364
release of heat in the region of its ‘nose’. The pressure and temperature differences give rise to a counter flow (from the heavyfraction extractor toward the light-fraction extractor and, correspondingly, a return flow along the wall of the rotor). Theoretical studies are continuing in order to determine the effect of the periodic action of a shock wave, arising at the nose of the gas extractor, on the outflow of the working gas from each opening of the inner annular system of diaphragms [11]. The studies have not been completed. The circulation flow owing to the slowing of the gas is greatest near the gas extractor and rapidly diminishes away from it. It has been shown theoretically [6] that the distance l at which the mechanical circulation decreases as a result of viscous friction is a function of the fifth power of the linear speed of the rotor l ~ Pa /Va5, where Pa = exp(A2) is the pressure of the working gas near the wall of the rotor; A = (Δµω2Ra2/2RT)1/2 is a constant characterizing the ratio of the linear speed of the rotor to the speed of sound in UF6, which we shall call the compressibility of the gas or simply the compressibility; Va = ωRa is the linear speed; and Ra is the radius of the rotor. Hence it follows that the mechanical circulation should decrease with increasing length of the rotor and correspondingly the thermal circulation should increase. In contrast to the mechanical component of the circulation, for which it is difficult to find a numerical characteristic since it is determined by internal processes, the thermal component is uniquely related with the axial gradient of the temperature on the outer surface of the rotor. Another fundamental parameter on which optimal circulation depends is the gas-filling or gas content of the working gas M in the rotor, which is directly proportional to the pressure Pa at the wall: M=
2 Pa πL ω2
[1 − exp ( − A2 )].
Theoretical studies show that the optimal circulation is a complicated function of the rotational speed and geometric dimensions of the rotor and the temperature gradient on its surface as well as the gas-filling. In the present work, an attempt is made to find a relation between the pressure on the wall of the rotor and the temperature gradient giving the thermal component of the circulating flow. In separation theory, an estimate is used for the maximum possible capacity of a single centrifuge, which is proportional to the fourth power of the linear speed of the rotor and directly proportional to its length and is determined by the expression [12] 2 ⎛ ΔμV 2 ⎞ πL a ⎟ δUmax = ρD⎜ , (1) ⎜ 2 RT ⎟ 2 ⎠ ⎝ where ρ is the density of the working gas, and D is the self-diffusion coefficient of the uranium isotopes. This relation was derived by Dirac without any assumptions about the type of centrifuge so that it can be used to evaluate the degree of optimality of the circulating flow in the rotor by means of the efficiency of the centrifuge: E = δU/δUmax,
(2)
where δU is the separative power obtained by experimental or computational means. In the analysis of the characteristics of flows of the working gas, it was shown that relation (1) holds only for an evaporative centrifuge and for concurrent and countercurrent machines the factor 0.81 must be introduced. For this reason, the maximum possible efficiency E does not exceed 81%. Experimental and computational studies have shown that the dependence of the separative power of a centrifuge with axial circulation on the linear speed of the rotor differs significantly from the fourth power. The separative power is proportional to Va3 up to rotational speeds of the rotor ~400 m/sec and ~Va2 at higher speeds [13]. This is because the intensity and radial and axial profiles of the circulation flow affect the separative power of the centrifuge. To determine the separative power taking these factors into account, it was proposed that dimensionless corrections be introduced into the Dirac relation [12]: δU = δUmaxeFeCeI.
(3) 365
This approach makes it possible to investigate the effect of the circulation flow in the centrifuge, dividing it into individual components. The factor eF, called the utilization factor of the circulation flow, describes the difference of the actual radial profile of the circulation flow from the ‘ideal’ profile. The factor eC characterizes the effect of the circulation flow on the separative power of the centrifuge, and the factor eI makes it possible to determine the difference of the distribution of the axial circulation flow in a centrifuge and an ideal cascade and for this reason it is called the efficiency of the form of the centrifuge. The corrections in relation (3) can be combined into a single coefficient that can be determined experimentally and, therefore, is unique for each centrifuge. It what follows, we shall call it the experimental realizability factor ee of the circulation efficiency. Then the dependence under discussion can be represented in the form [14] δU =
Va2 L e , 33000 e
(4)
where the linear speed of the rotor is expressed in m/sec and the rotor length in m. The constant 33000 was obtained under the following conditions: • a linear temperature gradient on the side surface of the rotor; • the coefficient eC = m2/(1 + m2) = 0.9 (m = 3); • the average temperature of uranium hexafluoride 310 K; and • the radial distribution of the circulation flow along the radius of the rotor (circulation flow utilization efficiency) is given in the form eF = 14.4RT/μVa2 (so-called Martin profile). It is precisely the last assumption that leads to the quadratic dependence of the separative power δU on the linear speed of the rotor. The parameter ee can be interpreted as the ratio of the separative power of a real centrifuge to an optimized model with circulation flow distribution called the Martin profile. Evidently, a circulation profile giving separative power higher than the Martin profile can be realized in a real centrifuge, just as the coefficient eC can be greater than 0.9. Other dependences of δU on the linear speed, length and diameter of a rotor are known, for example, semiempirical. As a rule, they make it possible to obtain a satisfactory estimate of the main parameter of a centrifuge in a comparatively small range of these parameters. One is presented in [15]: 2
⎛V ⎞ ⎛ D ⎞ δU = 12 L⎜ a ⎟ ⎜ ⎟ ⎝ 700 ⎠ ⎝ 0.12 ⎠
0.4
,
(5)
where the linear speed of the rotor is expressed in m/sec and the length and diameter in m; the dimensions of the separative power are obtained in kg·SWU/yr. Relation (5) was used to propose a relation, which, being independent of the linear speed, length, and diameter of the rotor, characterizes in its general form the degree of optimality of the circulation flow multiplying the radial effect of separation [9]: 2 0.4 ⎛ Va ⎞ ⎛ D ⎞ (6) k = δU / L⎜ ⎟ ⎜ ⎟ . ⎝ 700 ⎠ ⎝ 0.12 ⎠ It is difficult to judge the degree of optimality of the characteristics of the circulation flow of any particular centrifuge on the basis of the absolute magnitude of the dimensionless constant k. However, when comparing different centrifuges the following conclusion can be drawn. Actually, constant 12 in relation (5) is a measure of the optimization of the circulation for centrifuges whose parameters were used to derive this approximate relation. According to the data in [9], which were obtained using relation (6), the degree of optimality of the circulation in a centrifuge for numerical modeling of flow and diffusion with the rotor length increasing from 0.5 to 5 m at first decreases and then changes very little in the interval 3–5 m. The data for the serial centrifuges TC-12, -21 [16] confirm the behavior found for rotor length 3–5 m. In the present work, the efficiency of a centrifuge, the experimental realization factor of the circulation efficiency ee, and the circulation flow utilization factor k were calculated on the basis of experimental data using relations (2), (4), and (6) 366
Fig. 2. Separation efficiency E (B), experimental realization factor of the circulation efficiency ee (B) and the degree of optimality of the circulation flow k (b) of the domestic centrifuges (1–9), the Urenco centrifuges TC-12 (10) and TC-21 (11), the American centrifuge AC-100 (12), and the Iguasu model centrifuge (13).
for some types of centrifuges having different speed and geometric dimensions as well as foreign centrifuges and the Iguasu model centrifuge. Results and Discussion. It can be supposed that the large difference of the estimates of the experimental separative power is most likely due to the difference of the changes in the optimal parameters of the circulation flow in the rotor as a function of the angular speed and geometric dimensions of the rotor from the assumptions adopted to derive relation (5). Analyzing the data, it can be concluded that the dependence of the separative power for A2 > 30 on the linear speed and length of a rotor is the same and can be expressing in the form: δU = A0(Va2L)n,
(7) where A0 is a constant. For relation (5), the parameter n = 1, but this does not mean that it will always be the same for all types of centrifuges. As follows from the data presented in Fig. 2, for some types of centrifuges the realization of the experimental circulation efficiency ee exceeds 100% (AC-100), whence it follows that the exponent n in relation (7) must be greater than 1. Taking account of the experimental and theoretical proofs showing the dependence of δU on L to be close to linear, it can be concluded that the speed dependence of the separative power can be greater than quadratic. The length L to diameter D ratio of the rotor is usually used to compare the geometric parameters of the centrifuges being studied. This ratio lies in the range 4–6 for the domestic centrifuges (see Fig. 2, 1–9), close to 20 (10, 12) for TC-12 and AC-100, and is greater than 30 (11) for TC-21. The dependence of the parameter k on the degree of compression A2 or the dimensionless angular speed of the rotor was constructed using relation (6) for all types of centrifuges (Fig. 3). The dependence obtained has a maximum attesting to the difficulty of creating the optimal circulation with increasing linear speed of the rotor above a certain value. This difficulty is due to an increase in the zone of the ‘vacuum’ nucleus near the rotational axis of the rotor and correspondingly a reduction of the thickness of the zone of viscous gas flow on its side wall. For most types of gas centrifuges, the parameter k for degree of compression 20–45 lies in the range 11–13 (see Fig. 3). However, the value is only 8.6 for the high-speed AC-100 centrifuge, which could attest to the difficulty of optimizing a centrifuge with such a long rotor as well as to the need to refine the exponent 0.4 for the dependence of the separative power on the diameter of the rotor. For example, in the case of AC-100, we have k ~ 12 for the weaker dependence of δU on D with exponent 0.2. This is supported by calculations for the model Darmstadt centrifuge with rotor diameter 0.5 m [9], according to which k = 9 but for the exponent in the dependence from D = 0.2 the parameter k increases to 12. 367
Fig. 3. The parameter k versus the degree of compression A2 of the gas for domestic centrifuges 1–9 (E), the centrifuges TC-12, TC-21 (10, 11), and AC-100 (12) (B) and the Iguasu model centrifuge (13, C).
Another example confirming the possibility of a different dependence of the separative power on the rotor diameter D is its comparison for the centrifuges TC-12 and TC-21. It is well known that both centrifuges have the same diameter but different rotor length (~3.3 and ~5.1 m, respectively) and linear speed of the rotor. In addition, the parameter k is almost the same for both machines. This shows that in both centrifuges it was possible to attain almost the same degree of optimality of the circulation flow even though the rotor length is different. In another case, as Fig. 3 shows, an increase of the diameter of the rotor of the AC-100 centrifuge resulted in an appreciable reduction of the quality of the circulation. Optimal Parameters of the Circulation Flow. The character of the change in the parameters of the optimal circulation flow can be determined in principle but a difficulty arises because the solution of the gas-dynamic problem is not unique. Therefore, in order to pick a solution that is realized in practice and to refine the understanding of the physics of the processes occurring in a real centrifuge the results of an analysis of the experimental data must be used. The following relation for determining the optimal pressure of the working gas at the wall of the rotor was proposed in [6] for the example of the computational optimization of the Isuagu centrifuge:
*
Pw* = µ(8RT)1/2H*L/RaA5 /(1 + µR/2A2)1/2,
(8)
where H = 10 [6]. In accordance with relation (8), the optimal pressure at the wall of the rotor increases as the ~4th power of the linear speed of the rotor and linearly as a function of the length of the rotor and deceases inversely as the radius of the rotor. The equations of the fitting functions for the optimal pressure at the wall of the rotor versus the gas compression A2 have the following form: Pa = –0.013A4 + 3.258A2 – 1.335 (R2 = 0.950); Pw* = 0.0306A4 – 5.785A2 + 51.878 (R2 = 0.916), where R2 is the coefficient of determination, which determines the reliability of the approximation. According to Fig. 4, relation (8) describes the experimental values quite accurately for low compression (A2 < 25). However, as the linear speed of the rotor increases, the computed optimal gas-filling Pw* is found to be 1.5–2 times higher than the experimental value. This result makes it possible to judge the limits of applicability of the pancake approximation used in [6], where relation (8) was obtained. The experimental Pa and computed Pw* pressures of the working gas near the wall of the rotor satisfactorily agree with one another to within the compression of the gas A2 < 25. The discrepancy increases for higher values of the linear speed of the rotor. 368
Fig. 4. Computed from relation (8) (C) and experimental (B) for domestic centrifuges 1–9 gas pressure near the rotor versus the compression of the working gas.
Fig. 5. Experimental optimal gas pressure Pa near the wall of the rotor for domestic centrifuges 1–9 versus the optimal temperature difference ΔT on its outer surface.
Analysis of the experimental data for the centrifuges 1–9 (see Fig. 2), differing by the geometric dimensions and the rotational speed of the rotor, revealed a correlation between the pressure of the working gas near the wall of the rotor Pa and the optimal temperature difference ΔT at the wall of the rotor (Fig. 5). The dependence obtained by a fit to the experimental data has the following form: Pa = 17.431ΔT – 31.292 (R2 = 0.995). For the experimental pressure and temperature difference shown in Fig. 5, the length and linear speed of the rotor varied by a factor 1.5–2 and the diameter of the rotor by 20–30%. The dependence obtained is explained by the variation in 369
the compression A2 of the gas in the range 25–35 as a function of the pressure and temperature. It is known that for high linear speeds of the rotor (A2 > 20–25) almost all of the gas is concentrated in a thin layer on the side surface of the rotor. The thickness of this layer on the side wall of the rotor, where a circulation flow is established, depends as the powers 1/3 and –1/3 of the radius and angular speed of the rotor, respectively. For this reason, a change in the compression of the gas by a factor of 1.5 owing to a change in the radius of the rotor or its angular speed by approximately 20% results in a 6% variation of the characteristic thickness of the layer. At the same time, the change in the thickness of the layer on the wall of the rotor is directly proportional to the gas pressure Pa near the wall of the rotor. In other words, the effect of this parameter on the optimal circulation is a factor of 10–15 times higher and it is this parameter that is decisive for the established relation with the optimal temperature difference. In summary, the relation obtained can be used to determine the optimal pressure near the wall of the rotor and the gas content in the centrifuge according to the optimal temperature difference along the length of the rotor. Conclusion. Analysis of the separation efficiency, experimental realization factor of the circulation efficiency and degree of optimality of the circulation flow in a gas centrifuge with different speeds and geometric parameters suggests that the separative power of a single centrifuge with high angular speeds of the rotor (A2 > 30) can be described by the general relation δU = A0(Va2L)n, where A0 is a constant. A special study is required to determine accurately when and why n can differ from 1. The dependence of the degree of optimality of circulation flow in a centrifuge on the dimensionless rotational speed of the rotor was constructed on the basis of the experimental data. The presence of a maximum in the relation obtained is probably due to the difficulties in organizing the optimal circulation flow in a thin layer of viscous gas at the wall of the rotor for high rotational speeds (compression of the gas A2 > 35). It was found that in domestic centrifuges the optimal gas pressure Pa near the wall of the rotor is a linear function of the optimal temperature difference ΔT along its length. This information could be helpful in calculations and design work for optimizing new-generation centrifuges. The investigation of the experimental dependence of the optimal gas pressure near the wall of the rotor for machines with different speeds and geometric dimensions on the compression of the working gas made it possible to determine the limits of applicability of the pancake approximation, which gives satisfactory agreement with the experimental results on the compression of the gas A2 ≤ 25.
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