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Separation Performance of Gas Centrifuge
Notation. As used for isotope separation, the gas centrifuge is a cylinder of radius a and length L, rotating about a vertical axis with angular velocity w rad/s. Cylindrical polar coordinates are used, with the following notation for position and velocity components:
|
Direction |
Position |
Velocity, relative to solid cylinder rotating about axis with angular velocity ш |
|
Radial, out from axis |
r |
u |
|
Tangential |
в (angle) |
V |
|
Axial, up from midplane |
z |
w |
Properties of the light component of a binary mixture are denoted by subscript 1; heavy component by subscript 2.
Equilibrium separation. When a gas mixture in a centrifuge rotates as a solid body without motion relative to the wall of the cylinder, its pressure and composition are independent of в and г and vary with r according to the equations for equilibrium in a centrifugal field.
In a centrifuge rotating at со rad/s, gas of density p at radius r is subjected to centrifugal force of co2rp per unit volume, which equals the pressure gradient at that point.
|
2- C4 3 II |
(14.154) |
|
|
Because |
pm P~RT |
(14.155) |
|
1 dp тшгг p dr RT |
(14.156) |
This equation is analogous to the equation for the change in barometric altitude h under gravitational acceleration g:
By integration, the pressure ratio or density ratio between an interior radius r and the outer wall of the centrifuge at radius a is
where va is the speed of rotation coa at the outer wall, termed the peripheral speed.
Table 14.12 illustrates pressure ratios for UF6 gas (m = 352) at several values of r/a for peripheral speeds of 400, 500, and 700 m/s at 300 K. Most of the gas is in a thin shell near the wall.
In a binary mixture of gases of molecular weights mx and m2, an equation like (14.157) describes the partial pressure ratio of each component,
where x is the mole fraction of light component. The local separation factor a(a, r) between radii r and a, obtained by dividing (14.159) by (14.160), is
The separation factor in the gas centrifuge thus depends on the difference between molecular weights, whereas in gaseous diffusion it depends on their ratio. Table 14.13 gives local separation factors for mixtures of 23SUF6 and 238UF6 (Am = 3) for the same speeds and radial locations as Table 14.12. Because most of the gas is in a thin shell adjacent to the wall, the more significant values are those for r/a near unity. Even with this restriction, the separation factor for the centrifuge is much more favorable than a0 = 1.00429 for gaseous diffusion.
Pressure ratio p(r)/pe for speed
va of
d lnpc/(l — x)] I _ — Anwlr (14.163)
dr ( equii RTa2
■— -(f) <14l64)
Transport equations. When centrifugal equilibrium is disturbed, as by establishment of counterflow or injection of feed and removal of effluents, flow of the gas mixture and of its individual components takes place relative to the rotating centrifuge. The analysis to be given has the following restrictions.
1. All gas is rotating at angular velocity w so that there is no angular motion relative to the rotating centrifuge. In a coordinate system rotating with angular velocity u>, v = 0.
2. Analysis is to be limited to the case of no radial motion of the gas as a whole, и = 0. This condition cannot hold at the top and bottom of the centrifuge, but may be nearly correct away from the ends, in the so-called long bowl development.
3. The change of pD with temperature and pressure, and thermal diffusion effects, are neglected.
Transport of light component is to be described in terms of its mass velocity, the vector J, with component Jr in the radial direction and Jz in the axial. In the coordinate system rotating at angular velocity a>, the angular component Jg is zero.
When the radial composition gradient 9x/9r differs from the gradient at equilibrium (9x/9r)equi|, transport of light component against the composition gradient takes place with radial mass velocity
The axial mass velocity Jz is the sum of a convective term pwx and a diffusive term —Dp(dx/dz):
Differential enrichment equation. Under steady-state conditions, the differential equation for conservation of light component, in cylindrical polar coordinates, is
1 9 (rJr) dJz 1 9 2Je
— —- + — +——— —
r dr dz г dB2
With Jr from (14.165), Jz from (14.166), and/g = 0, Eq. (14.167) becomes
Cohen [C6] made the following assumptions to simplify solution:
1. x(l — x) is treated as a constant.
2. d2x/dz2 is neglected.
3. dxjdz is independent of r.
4. pw is independent of z.
Эх _ Amai2 r2 x(l — x)____ 1_ Эх f
Г dr RT Dp dz Jo
because r(dx/dr) = 0 at r = 0.*
Integration of (14.169) requires use of boundary conditions for the net flow. In the enriching section, the net flow P is
r
P = 2n І pwrdr
Jo
pwr’dr’) Dp fzrdr
(14.173)
Because of assumption (3), this may be solved for dxjdz
+This condition and the lower integration limit of 0 are strictly correct only when a tube at the axis of the centrifuge is not present. In most centrifuges, with such a tube, the lower limit of integration should be the outer radius of the tube. However, at peripheral speeds of 400 m/s or higher, the density of gas at the central tube is so low that use of 0 for the lower limit of integration introduces no significant error, and r(dx/dr) at the lower limit is, much smaller than at the upper limit.
F(r) = 2ir
Cohen used the notation
C, = f ЯгУ dr
C2 — itDpa2
[m»г
Jo
C5 =C2 + C3
In terms of these functions, the differential enrichment equation (14.174) becomes
dx C і ч P(xp-x)
dz=rsxil-x)——————
Here the variation of x with r has been neglected, as it is small compared to its change with z in a long centrifuge.
For the same reason, it is permissible to write a similar equation for the composition у of the enriched stream:
Because the coefficients Ct and C5 are to be evaluated for the velocity distribution with zero net flow, (14.181) is as valid an approximation as (14.180). When feed is added to the enriched stream, as in a centrifuge with feed introduced by a tube at the axis, Eq. (14.181) is easier to use than (14.180).
The equation corresponding to (14.181) for the stripping section is
In an exact treatment, values of Ct and C5 in the stripping section would differ slightly from the enriching section because of the slightly different flow profile. In the present approximate treatment, the constants are to be evaluated for the total reflux case in which the flow patterns in both sections are the same. If the net flow rate is a small fraction of the circulation rate, studies by Parker [PI] and others have shown that the effect on Ci and C5 of the changed flow pattern with net flow is small.
Equation (14.181) may be compared with the corresponding differential equation for the enriching section of a two-stream, close-separation, countercurrent column like a distillation column:
Ppp — y )
N
^Physically, F(r) is the total mass upflow rate between the center and radius r.
where h is the height of a transfer unit, a is the local separation factor, and TV is the flow rate of the stream moving from the product end of the column. Comparison of (14.181) and (14.183) shows that Cs may be interpreted as
|
C5 = hN |
(14.184) |
|
|
and Ci/C5 as |
C, 0- 1 Cs ~ h |
(14.185) |
|
Thus |
й_£* h~ N |
(14.186) |
|
and |
Q 1 II |
(14.187) |
|
In the countercurrent centrifuge N is the depleted stream flow rate |
||
|
Ґ N = 2л І pwr dr |
(14.188) |
where r, is the internal radius at which the axial velocity w changes sign. In the present approximation, in which the centrifuge parameters are evaluated for the velocity profile at total reflux, the flow rates of enriched stream and depleted streams are equal and an equivalent equation is
(14.189)
Local separative capacity. The separative capacity of a gas centrifuge per unit length, dA/dz, may be derived from Eq. (14.181) for the composition gradient, dy/dz. In the enriching section of a gas centrifuge the net flow rate of light component toward the product end is Pyp and of heavy component is P(1 — yP). As these flows make their way through gas of composition у against a composition gradient dyjdz, the rate of production of separative work per unit height dA/dz is
where S is the separative work associated with ri mass of component 1 and пг mass of component 2.
From Eq. (14.118):
|
£ — +?] I+«■ — «> [- _ P(yp y) dy уЧ 1 — J’)5 * |
Using (14.181) for dy/dz,
dA ^ Сі P(yp — у) 1 Hyp — у)
dz “ Cs XI ~X C5 XI “X
The optimum value of the group of variables
is the value that maximizes dA/dz, at which
|
, _ Pb>p ~ У) ф~у( 1 — у)
|
In a centrifuge with axial flow independent of height, C1 and C5 are constant and condition (14.199) can be satisfied at only one height.
In terms of the parameters a — 1, A, and N,
Equation (14.201) is analogous to condition (12.125) for an ideal cascade, and (14.202) is the separative capacity of a stage of an ideal cascade divided by h.
The parameters C, Cs, and N, and from them A, a — 1, and the separative capacity, depend on the radial distribution of mass velocity pw(r). These parameters will be evaluated for two velocity distributions:
1. Mass velocity independent of radius
2. Berman-Olander distribution
Mass velocity independent of radius. The optimum radial distribution of longitudinal mass velocity wp(r) that leads to the highest possible separative capacity per unit length when condition (14.199) is satisfied is a distribution in which the mass velocity in one direction is independent of radius r up to the outer radius a, with all countercurrent flow in the opposite direction occurring in a cylindrical shell of infinitesimal thickness at the outer radius a. Under
|
the total reflux conditions to be used in evaluating Ci and C5,
|
Cohen [C6] has shown that the first factor represents the maximum possible separative capacity per unit length for a centrifuge operating at peripheral speed va, in the absence of axial back diffusion. The second factor, termed the circulation efficiency Ec, takes into account reduction in separative capacity caused by axial back diffusion. It approaches unity as the circulation rate N increases or the radius a decreases. Values of the first factor for separating 23SU from 238U (Am = 3) at 300 K, using the value of Dp = 2.161 X 10*4 g UF6/(cm*s) recommended by May [M6] 7 are
va, m/s 400 500 700
First factor, kg UF6 SWU/(yr • m) 9.912 24.198 92.959
kg U SWU/(yr • m) 6.702 16.361 62.853
+This value, 7 percent lower than 2.32 X 10"4g/(cm*s) given by Dp = 4д/3, with p from Eq. (14.4), is used so that results will be consistent with May’s.
Berman-Olander velocity distribution. Even if the optimum radial distribution of mass velocity (14.203) could be established at one elevation in a countercurrent centrifuge, it could not persist over any distance because of the great shear force at the velocity discontinuity between the counterflowing streams. Determination of a more stable countercurrent radial velocity distribution, which would persist over a substantial length of centrifuge, requires solution of the hydrodynamic equations for motion of a compressible fluid in a centrifugal field. Even for the simplified “long-bowl” case considered here, in which the radial velocity и is zero and the longitudinal velocity w is a function only of r, the solution procedure is difficult and the equations complex. During the past 20 years, long-bowl solutions of progressively increasing rigor have been given by Parker and Mayo [PI], Soubbaramayer [S7], Berman [B16], and others.^ Olander [01] showed that Berman’s solution could be approximated for the large values of
met in centrifuges of practical importance by
(14.214)
w0 is an adjustable parameter proportional to the circulation rate N. The mass velocity at radius r is the product of Eqs. (14.214) and (14.158):
wP(r) .
w0p(e)
(14.215)
Equations (14.214) and (14.215) approach zero as r -*■ a and thus properly represent the condition of no slip at the outer wall. They fail to represent exactly conditions as r -*■ 0 because w(r) should be finite for a centrifuge without a central tube or should be zero as r -* r0 when the centrifuge has a central tube of radius r0. However, for a practical centrifuge with peripheral speed over 400 m/s, for which A2 for UF6 > 11.3, wpir)/w0p(d) from (14.215) at rja = 0.1 is less than 1.2 X 10"s times its maximum value, so that little error is made in replacing wp from (14.215) by zero at r/a < 0.1.
In Fig. 14.17 the curve marked “mass velocity” is a plot of Eq. (14.215) for A = 11.3, corresponding to peripheral speed va = 400 m/s. Most of the flow occurs in the outer half of the cross-sectional area, with flow reversal taking place at r2 la2 = 0.88, so that all heavy-fraction flow occurs in the outer 12 percent of the area. At 700 m/s this area shrinks to only 3.6 percent of the total cross section.
The curve marked “flow function,” obtained from
(14.216)
Theoretical analyses of flow and separation in a gas centrifuge published too late for inclusion in this text may be found in [S2a] and [Via],
is proportional to the total mass flow in the direction of light fraction through the area between the center of the centrifuge and radius r.
For physical interpretation, it is instructive to cast these equations into a form that contains the internal circulation rate N. N is given by
(14.217)
where rt is the radius at which w changes sign and
(14.218)
From Eq. (14.200) the maximum value of the separative capacity per unit length, for a given value of N and velocity profile, is
|
Л’2 f[56] [f(r/a)]2 d(r/a)/(r/a) Jo_________________ 2irDp [/(r,/a)] 2 |
‘dA _ TiDp(bm)2 Vg 4/1 ________________ 1_____________
dz)^’ S(RT)2 I3 1 + mnDpaf/N2} (/(r1/a)]2/4/3
Equation (14.226) has been written in this form to facilitate comparison with Eq. (14.212) for the maximum separative capacity obtainable from the optimum, constant mass velocity profile. The first factor in these two equations is the same and is the maximum separative capacity per unit length obtainable for a given va. The second factor,
is called the flow pattern efficiency and represents the reduction in separative capacity caused by departure of the mass velocity profile in an actual centrifuge from the optimum, constant mass velocity profile.
The third factor,
is the circulation efficiency, which takes longitudinal back diffusion into account. It approaches unity as the radius a decreases or the circulation rate N increases. It is written in this form to facilitate comparison with Eq. (14.212) for the optimum, constant mass velocity profile. The two expressions differ by the factor [firja)]214I3 in the denominator of (14.228).
Table 14.14 gives the results of calculation of these centrifuge parameters for the Berman-Olander velocity profile (14.214) for UF6 at 300 К and peripheral speeds of 400, 500,
|
Table 14.14. Functions of Berman-Olander velocity distribution (14.214) for UF6 at 300 К
|
NS=N° exp p A* ^2 + (14.229)
where L = length of centrifuge
N% = circulation rate at bottom
A, is the decay constant for the circulation rate, whose dependence on the peripheral speed va and centrifuge radius a can be estimated from the aerodynamics of the centrifuge. Qualitatively, A; is higher the larger va and the smaller a.
End cap thermal drive. When flow is induced by heating the top and cooling the bottom of the centrifuge, as in Groth’s machine (Fig. 14.14), and the lateral wall is isothermal, the circulation rate decays exponentially from both ends and can be represented qualitatively by
Ne = №e cosh A*z (14.230)
Here Ng is the circulation rate at the midplane (2 = 0) and Xg is the decay constant, which is higher the larger va and the smaller a.
Wall thermal drive. When a linear temperature gradient is imposed on the lateral wall of the centrifuge, Durivault and Louvet [D7] have shown that the circulation direction at the wall is in the direction of increasing wall temperature. The rate is highest at the midplane and decreases to zero at the top and bottom. Hence the circulation rate for this type of drive can be modeled approximately by
0 cosh (AwL/2) — cosh Awz w w cosh (XWLI2) — 1
Here is the circulation rate at the midplane and Aw is the decay constant.
Combination of drives. In an actual centrifuge driven by a motor at the bottom, motor inefficiency introduces heat at the bottom end cap. The longitudinal variation of circulation rate then depends on where this heat is removed and whether other heat sources are present. Examples of centrifuge separation performance will be given for two cases:
1. Constant circulation rate, independent of 2
2. Optimized circulation rate, varied for maximum separative capacity at every elevation
Centrifuge considered. The centrifuge example whose separation performance is to be evaluated has the dimensions of the centrifuge tested by the Standard Oil Development Company in 1944 and described by Beams et al. [B3]:
Length, L = 335 cm Radius, a = 9.15 cm
which was tested at a peripheral speed of 206 m/s. May [M6] has evaluated separation parameters for a centrifuge of the foregoing dimensions for peripheral speeds of 400, 500, and 700 m/s possibly obtainable with more modem materials. Dimensionless integrals used in separation performance equations have been given in Table 14.14 for these speeds.
Circulation rate independent of height. The case of circulation rate independent of height is analogous to distillation at constant reflux ratio. For this case, explicit equations can be given for overall separation performance of the centrifuge. Conceptually, a constant circulation rate might be realized by a proper combination of end cap thermal drive, Eq. (14.230), and wall thermal drive, Eq. (14.231). Specifically, if
= X
¥-■)
the circulation rate has the constant value
N = Ne + Nw = №e cosh — V" (14.234)
When N and the radial velocity profile are constant, the separation parameters Ci and Cs in the differential equations (14.181) for the enriching section and (14.182) for the stripping section are independent of position z. For the low-enrichment case (y < 1) to be treated here, the equations may be linearized to
Enriching: C5 ^ = (C, + P)y — Pyp (14.235)
Stripping: Cs ^ = (Ci — W)y + Wyw (14.236)
The integral of (14.235) between у = yp at the feed point, z = 0, and у = Ур at the top, z =
Le, is
Ур_ P+Ct
У§ ~ p + c, exp [- (P + C,)WCs] (14.237)
The integral of (14.236) between у = y-ц, at the bottom, z = —ід, and у = yp at z = 0 is
W-C:
Л W-C, exp [-(W-COLsICs]
The material-balance equation on light component at the feed point where feed rateP+W joins enriched stream at rate N — W is
(TV — + (P + W)yF = (N + P)yf (14.239)
Overall material balance on light component is
Wyw + Pyp = (W + P)yF (14.240)
For given values of yP, LE> P, W, and TV, these equations are sufficient to determine the product composition yP, tails composition уц>, and the heads compositions yp leaving the stripping section and yp entering the enriching section.
To avoid mixing losses at the feed point, the solution for which
yjr=yF (14.241)
is desired. When this is true, yp also equals yF, from (14.239). When LE, ід, yF, N, and the feed rate F = P + W are given, it is necessary to find by trial the value of P (or W) at which the preceding five equations are satisfied. The condition for this is obtained by substitution into the overall material-balance equation (14.240) for yplyp from (14.237) and Ур/yw from
(14.238):
“ W /ур
-j— + P [Zy=W + F) (14.242)
Ур/yw Ур/
The separative capacity Д for this low-enrichment case, from Eq. (12.141), is
Д —P In yP — W In yw + F In yp
(14.243)
When the no-mixing-loss condition (14.241) is satisfied, the separative capacity, from (14.243), (14.237), and (14.238), is
(14.244)
This separative capacity is lower than the product of the maximum separative capacity per unit length (dA/dz)max, given by Eq. (14.200), and the length L = LE + L$. The ratio
is termed the ideality efficiency Ej. In terms of the three efficiencies: ideality efficiency £>, Eq.
(14.243) , circulation efficiency Ec, Eq. (14.228), and flow pattern efficiency Ep, Eq. (14.227), the overall separative capacity is
Centrifuge example. Tables 14.15 and 14.16 summarize calculations of the separative capacity of a centrifuge 335.3 cm long, 18.29 cm in diameter, run at a peripheral speed of 400 m/s at 300 K, with circulation rate independent of height. Some of these were given by May [М6]. In Table 14.15, the circulation rate for all cases is 0.1884 g UF6/s, and the feed rate is varied. The separative capacity has a maximum of 10.05 kg uranium SWU/year at a feed rate of 0.038052 g UF6/s (1200 kg UF6/year. Д remains close to 10 with a variation in feed rate of ± 20 percent, but decreases considerably at feed rates outside of this range. The axial separation factor is 1.67 at the lowest feed rate, decreases steadily with increasing feed rate, and equals 1.37 at optimum. The height of a transfer unit is 12.39 cm. The maximum ideality efficiency, at the optimum feed rate, from (14.245), is 0.8147. The circulation efficiency, from (14.228), is Ec = 0.9757. The overall efficiency E = EpEcEj has a maximum value of 0.4472.
In Table 14.16, the feed rate is held constant at 0.03171 g UF6/s (1000 kg UF6/year) and the circulation rate is varied. The separative capacity has a maximum of 10.03 kg uranium SWU/year at an optimum circulation rate N = 0.1884 g UF6/s and decreases rather rapidly with changes from this rate. The axial separation factor has a maximum of 1.41 at the optimum circulation rate. The height of a transfer unit increases almost proportionally with circulation rate. The circulation efficiency increases from 0.9095 at the lowest circulation rate of 0.0942 g/s to practically unity at the highest, showing the decreasing influence of axial back diffusion as circulation rate increases.
In Tables 14.15 and 14.16, the cut (ratio of product flow rate to feed flow rate) at conditions that lead to maximum separative capacity is 0.45. Because the centrifuge is
Table 14.15 Effect of feed rate on separation performance of gas centrifuge
Length: stripping, L$ = 167.65 cm; enriching, LE = 167.65 cm Radius: a = 9.145 cm Temperature: 300 К
Peripheral speed: va = 40,000 cm/s;A2 = 11.3 Circulation rate: N = 0.1884 g UF6/s Centrifuge parameters: Сг = 0.00402 g UF6/s
Cs = 2.3347 (gUF6-cm)/s Radial enrichment factor: a — 1 = 0.02131 Height transfer unit: h = 12.39 cm
Efficiencies: flow pattern, Ep — 0.5626; circulation, Ec = 0.9757
|
UF6flow rate, g/s
^Optimum. |
connected in a cascade, cascade conditions may require the centrifuge to operate at a somewhat different cut. This would violate the no-mixing-loss condition at the feed point (14.241) (unless the feed location can be changed) and reduce the separative capacity.
May [M6] has calculated the effect of varying the circulation rate N on the separative capacity of this centrifuge example operated at peripheral speeds of 400, 500, and 700 m/s, at a feed rate of 0.03171 g UF6/s, using the parameters of Table 14.14, with results shown in Fig. 14.18. The optimum heavy-stream flow rate N increases with increasing speed. The separative capacity at optimum N increases as u2’02.
Optimum distribution of circulation rate. When the circulation rate N is independent of height, the parameters Ct and C5 are constant. It is thus possible to satisfy condition (14.199) for the composition variable and (14.200) for the maximum separative capacity gradient at only one elevation and one value of у in each of the enriching and stripping sections. This is what causes the ideality efficiency for the constant N condition to be less than unity. We shall now give an example of a centrifuge in which the heavy-fraction flow rate N is varied so as to have its optimum value at every height in the centrifuge, and will find by how much the overall height of a centrifuge of a given capacity could be reduced compared with one with uniform N.
The composition gradient dy/dz in the enriching section may be expressed as a function of the composition у and the circulation rate N by Eqs. (14.181) and (14.179):
dy N(CxjN)y{ — у) —Р(ур-у) „ „
— =————————————- (14.247)
dz N* (C3/N2) + C2
Table 14.16 Effect of circulation rate on separation performance of gas centrifuge
Length: stripping, Ls = 167.65 cm; enriching, Lp = 167.65 cm Radius: a — 9.145 cm Temperature: 300 К
Peripheral speed: i>„ = 40,000 cm/s; A1 = 11.3 Radial enrichment factor: a — 1 =0.02131 Feed rate: F = 0.03171 g UF6/s (1000 kg UF6/yr)
|
~ о * —»-1 ОІ Circulation rate, N |
0.0942 |
0.1884+ |
0.2200 |
0.3768 |
0.5652 |
|
Product |
0.01431 |
0.014239f |
0.014345 |
0.014796 |
0.01511 |
|
Tails |
0.01740 |
0.017471 + |
0.017365 |
0.016914 |
0.01660 |
|
Centrifuge parameters |
|||||
|
Ci, g/s |
0.00201 |
0.00402 |
0.00469 |
0.00804 |
0.01205 |
|
Cs, (g • cm)/s |
0.6262 |
2.3347 |
3.1630 |
9.169 |
20.56 |
|
Height transfer unit h, cm |
6.65 |
12.39 |
14.38 |
24.3 |
36.4 |
|
Axial separation factor |
|||||
|
Enriching, p |
1.1384 |
1.19165′ |
1.1856 |
1.1366 |
1.0966 |
|
Stripping, у |
1.1285 |
1.18510+ |
1.1810 |
1.1357 |
1.0964 |
|
Overall, Py |
1.2847 |
1.41222+ |
1.4002 |
1.2908 |
1.2023 |
|
Cut, в |
0.451 |
0.449t |
0.452 |
0.467 |
0.477 |
|
Separative capacity, |
|||||
|
kg U SWU/yr |
5.29 |
10.03T |
9.53 |
5.50 |
2.87 |
|
Efficiencies |
|||||
|
Flow pattern, Ep |
0.5626 |
0.5626 |
0.5626 |
0.5626 |
0.5626 |
|
Circulation, Eq |
0.9095 |
0.9757 |
0.9821 |
0.9938 |
0.9972 |
|
Ideality, Ej |
0.4601 |
0.8132 |
0.7676 |
0.4378 |
0.2277 |
|
Overall, E = EpEqEj |
0.2354 |
0.4464 |
0.4241 |
0.2448 |
0.1277 |
|
UF6 flow rate, g/s |
|
^ Optimum. |
For a given speed va, Ci/N is a constant,
N independent of N and y, as shown by Eq. (14.224). Similarly, C3/N2 is a constant,
B3-N*
independent of N and y, as shown by Eq. (14.225). In these terms,
dy _ 1УДіу(1 — у) ~ P{yP ~ у) dz ~ І^В3 + С2
The optimum value of N for a given у is the value at which
Э In (dy/dz) Віу(1 —у) IN В з _
bN NBiy( 1 — у)— Р(.Ур — у) І’РВз + Сг
or — ІЇВзВзУІ 1 — у) + ‘ШРВзІУр — У) + ВіС2у{ — у) = 0
In a centrifuge with the optimum value of N at enrichment dy from (14.250) is
QTfBj + Cj = 2*3 ^op, WDBx-PYe В, E
The last expression results from using (14.253) to eliminate Ye — In the low-enrichment case,
|
f dz _ 2*з d hi уJ min B |
(14.257)
|
ftE 2е(Уе) = <*zmin = |
The minimum length of enriching section Ze necessary to increase у from the feed value Ур to a higher value yE is
Figure 14.18 Effect of heavy — stream flow rate on separative capacity of centrifuge at peripheral speeds of 400, 500, and 700 m/s. Length 335.3 cm; diameter 18.29 cm; feed rate 1000 kg UF6/year. (From May [M6J.)
|
A similar development for the stripping section leads to
|
for the low-enrichment case. The minimum length of stripping section zs necessary to decrease у from yp to a lower value is
f 0 2B3 FF ,
Zs(ys) = dzmin = Nr dny (14.262)
To give an example of the reduction in centrifuge height for a given separative capacity that could be obtained if it were possible to use an optimized, variable heavy-stream flow rate instead of the uniform flow rate employed in Table 14.15, a centrifuge with optimized, variable heavy-stream flow rate was designed for the conditions of Table 14.15 marked with a dagger. These led to maximum separative capacity in a centrifuge operated with uniform heavy-stream flow rate. Centrifuge characteristics for the optimum flow-rate distribution are shown in Fig.
14.19 and compared with the uniform-flow-rate case in Table 14.17.
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0.2 -0.1 0 0.1 0.2 0.3 Figure 14.19 Variation of optimum heavy-stream flow rate and enrichment with height in centrifuge at 300 K, 400 m/s, and 1000 kg UF6/year feed rate. |
Conditions common to both cases
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Table 14.17 Comparison of centrifuges with uniform and optimized variable heavy-stream flow rates UF6 flow rate, g/s
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In Fig. 14.19, height above feed point is plotted vertically to correspond with orientation of an operating centrifuge. Optimum heavy-stream flow rate has a maximum of 0.2837 g UF6/s at the feed location (z = 0) and decreases to 0.0297 at the top and bottom. These are to be compared with 0.1884 g/s in the uniform-flow-rate case. This decrease in flow rate from feed location to withdrawal ends of the centrifuge is qualitatively similar to that of the ideal cascade discussed in Chap. 12. However, the tails flow rate at the top, product end of the centrifuge cannot drop to zero, as it would in an ideal cascade, because dyjdz would become zero aX N = 0, as can be seen from Eq. (14.250).
In Fig. 14.19 composition is plotted horizontally as In y/yp, to bring out another difference from an ideal cascade. A plot of distance versus In у in an ideal cascade with constant height of a transfer unit (htu) would be a straight line. In this centrifuge with variable circulation rate, the htu from Eq. (14.186) varies from 18.4 cm at the feed elevation to 3.8 cm at the top and bottom. This causes In у to change more rapidly with z at the top and bottom than at the feed elevation.
As Table 14.17 shows, optimization of flow distribution permits reduction in centrifuge length for the stated separation performance from 335.3 to 284.43 cm. The ratio of these lengths, 0.8483, is somewhat greater than the ideality efficiency of the uniform-flow case, 0.8132 from Table 14.16. The reason for this may be seen by comparing the circulation efficiencies for the two cases. With variable flow rate, the circulation efficiency ranges from 0.9891 at the feed location to 0.5000 at top and bottom, compared with a constant efficiency of 0.9757 for the uniform-flow-rate case. Thus, the average circulation efficiency with variable flow rate is lower than with constant, a disadvantage that partially cancels the use of optimum flow rate at every height.