JOSEPH HALPERIN and R. W. STOUGHTON O a k Ridge National Laboratory, O a k Ridge, Tenn.
Batch Versus Continuous Processing in Fluidized Nuclear Reactors I N THE o A u - r I o N of a nuclear reactor where all or part of the reactor exists in fluidized form, it is interesting to examine the intrinsic efficiency of both batch and continuous processing in separating the transmutation products that are formed. Young ( 5 ) considered this probIem with respect to fission-product poisoning in 1946. More recently the present authors (2) and others (3) have considered various aspects of the problem.
Two special cases are considered: First, the initial material, N O ,is not replenished and therefore disappears exponentially according to
dNi dt
(1
5 i I n)
(5)
The solution is then
and second, the initial material No is maintained constant, Le., dNo/dt equals zero. In the first instance, the general solution for the n'th product where the initial material, NO, is not replenished can be written
z=1
Implicit in Equations 3, 4, and 6 is the requirement that the circulation rate of the fluidized reactor contents and the conThe build-up of transmutation prodsequent mixing be rapid compared t o ucts under the restraints of both batch any of the transmutative modes. T h e and continuous processing is compared bracket on the left side of these equations here for an idealized model which api=O carries the subscript B or C which proximates a representative reactor coni#j identifies them relative to the description figuration. However, consideration has In the second instance, the general of N,,/iVo for either batch or continuous not been given to factors relating to solution for the n'th product at time, t , processing respectively. Similarly, T , engineering or economic feasibilitywhere No is maintained constant can be and T C refer to batch and continuous e.g., no attempt is made to evaluate diswritten processing periods ( T advantages connected with shutdowns for is the time- to process batch processing. A sequence of reace-bi T B the contents of an entions is considered in which a product is bi 1 x 1 bi (bi b,) (4) tire reactor) which for formed by nuclear transformations from a comDarison will usustarting material, NO: J ally be for T B = Tc. a1 a2 a, a,+l Frequently it is desirNo + Ni . . . +Ni + able to ascertain the atom ratio, NJN,,,, where m is other than zero among the bl - a2 bi - a , + i bo - a1 separated products. This can be done Continuous processing refers here to by taking the ratio of NJNo to NJNo In Expression 1, the formation constant that method of processing in which a shown in Equation 7 for the case deof Ni is a,, whereas the destruction confraction of the fluidized reactor products scribed in Equation 6. stant is bi and includes all transmutative is continuously withdrawn, chemically modes of removing N i . I n general, processed to remove all newly formed b, = pu,(a) Xi and ai = pui-1 (c) or species (Nifor i > 1) from NO, and purified X,-l, where p is the neutron flux, No is returned to the reactor a t the same u(a) the neutron-absorption cross secrate that it is removed. A steady-state tion, u(c) the neutron-capture cross secequilibrium concentration of each species i=m+l tion, and X the radioactive decay conprevails. The differential equations in stant. p r implies the use of both an Equation 2, modified to include removal Batch ratios can be similarly obtained. energy-averaged cross section and a by chemical processing, are all set equal For simplicity in subsequent expressions, volume-averaged flux. The differential to zero and No is maintained constant. m = 0 will be used. equations describing Expression 1 are This is in contrast to batch processing, Batch Equations 3 and 4 are ,unwhere NO is not necessarily constant; wieldy and to compare the batch and continuous processing as defined here recontinuous cases, they should be simpliquires that NObe constant. Thus, fied. This can readily 6e accomplished The conditions imposed upon No are by examining the special case where specified later. _- 0,N o # 0 all bd are equal. Equation 4 thus I n batch processing, the transformabecomes : tions described in Equation 2 proceed for a period (t = T B ) , a t which time the reactor contents are chemically (b 4 ( bt)" - 1 ( N ~= constant) processed to separate all newly formed NO B - e - " ' ( l + b t + T + . . . (n l)! species (Ni for i > l), from NO,and the cycle is repeated. Thus, a t the beginning of each cycle only No is present in the reactor; all other N i equal zero. Transmutation Products
6
1 1
-
1
1
+
d2
3 1
=T[l
'
-)]
-
VOL. 48, NO. 12
DECEMBER 1956
21 15
factor of l/e, or to 36.870 of its original value. The mean life i s related to the half life by the expression half life/ mean life = log, 2 = 0.693. When the b T product is equal to unity, a significant transmutative change has occurredA convenient method of comparing namely, a change equivalent to a dethe amount of N , produced batchwise crease to 36.8% of the original content with that produced by continuous procwith no interim additions. Thus, for essing, is to evaluate the ratio of the those classes of reactors where bT will two cases, R ( B / C ) = (N,,/NO)B/(N,/NO)C not be greater than about unity, it for a given processing time where appears that batch processing will t = T B = Tc. A ratio less than unity produce less N,(for n > 1) than continsignifies that less component i2;, is uous processing. For example, in a produced in batch processing. The transformation involving neutron degeneral case where No is maintained struction of uranium-233 for a flux constant is expressed : (o of loL4n/sq. cm./second and an abEquation
G becomes:
/ n
\
-
\i
=1
/
L
The behavior of R i B / C ) can be evaluated more easily for the special case where all bi are equal. This condition is expressed : R(B/C) =
T h e factorial relationships can be seen more readily for the short time limit where the batch equations may be written :
- - - aiiyi-1 dNi
(neglecting destruction
dt
terms)
i#
i
sorption cross section u , ( U ~ ~of~ )585 barns, (6T) = (OUT= 1 a t T = 1.7 X 107 seconds or about 200 days. For a blanket containing thorium-232 having an effective r(ThZ39of about 10 barns, ( b T ) = 1 at T = l o 9 seconds or about 30 years. The question of the practical length of processing periods and the related degree of "burn up" for reactor fuels is difficult to assess a t present and beyond the scope of this work. However, certain considerations seem valid now. Clearly, reduced processing costs result from lengthening processing periods. I n this regard, there are certain advantages inherent in fluidized over heterogeneous reactors; fuel element damage caused by burn-up need not limit fuel life and replenishment can be carried out without removing the partially spent fuel. How-
ever, fluidized reactors also offer the possibility of more economical and hence more frequent processing, for solid fuel elements need not be dissolved nor is it necessary to have fuel assemblies fabricated. The more efficient and economical the processing, the shorter the processing cycle that can be tolerated if an appropriate advantage is to be gained. The principal advantage of processing fluidized fuels is the removal of fission products or other poisons-Le., parasitic neutron absorbers-formed during reactor operation or the removalofcertain products of value which would otherwise be lost. Neutron economy is a foremost consideration in certain classes of reactors. In the thermal "breeder,y9where more fuel is produced than consumed, neutron losses must be minimized if a positive breeding gain is to be achieved. In the breeding cycle of thorium-232 and uranium-233, the 27.4 day protactinium233 intermediate is an example of a product removable by processing to diminishits loss by neutron absorption. In addition, in those reactors whose excess reactivity is limited, as in certain natural uranium reactors, the permissible level of poisons is restricted. In all such cases where neutron economy is paramount, it appears that processing periods, T , where the bT product is less than unity, will prevail. Thus, for such conditions, the qualitative and in the limit quantitative factorial relationship brought out here, will properly describe the effects of batch and continuous processing. In other cases such as burners (reactors which burn fuel for power only where neutron economy is not important), bT values greater than unity may well obtain; then the relative advantages
T h s , the ratio
( n u 0 T" 1 _. n!
where
( t = T B = T c ) and ( T + 0 )
A plot of Equation 11 is shown in Figure 1. For finite n, R ( B / C ) approaches 1/n! as b T approaches zero, goes through a maximum greater than unity, and then approaches unity from above as bT approaches infinity. The point where R ( B / C ) becomes greater than 1 lies at b T values in excess of about lforn>l. The transmutative rate constant, b, is the i-eciprocal of the mean life of the species-Le., mean life = l / b . The mean life of a substance, diminishing exponentially, is the time to decrease a
2 1 16
1
1 4
0
-
'0'2
3
L
10
10.'
L
10
102
X = bt
Figure 1. Comparison of N, produced with butch and continuous processing where N O i s maintained constant during the batch period
INDUSTRIAL AND ENGINEERING CHEMISTRY
of batch and continuous processing must be evaluated by using the actual values of aiand bi in the general formulas. In batch processing where No is not replenished and where all bi are set equal, the following equations describe the N , content.
4o.ot
, ,
1
n=.l
I
I
1 I IIIII
////
i.0
///
d
I n this case, ratio, R ( B / C ) , in Equation 14 is exact at all times, whereas it is only the short time limit in those cases where NOis maintained constant.
Figure 2 illustrates Equation 14 for Figure 2. Cornpariseveral value5 of n. son of production of The conclusion that in the short time N, with batch and limit, W A C > = (N,/No)B/(N,/No)c continuous processing approaches l / n ! expresses one of the where N O is not reprincipal distinctions between the two I I 1 IIIIII I I I IIIILI plenished during the 0.01 0.1 1.0 10 processing schemes considered here. batch period bT Similarly in the limit, T -+ 0, ratio, R ( B I C ) = (Nn/Nm)B/(Nn/Nm)c, for two taminant such as zirconium-95 in the maximize the production of some bytransmutative products approaches m!/n! production of neptunium-237 from uraproduct, then, of course, the inverse (Equation l l a ) . Thus, as the order, n, nium-235. Since R = rn!/n! will vary conclusion as to choice of processing in the transmutative sequence is inmethod would apply-Le., where batch creased, the disparity of ratio, Nn/N,,,, from G to 2 during the time of interest, a substantial advantage accrues to the processing is advantageous for cases produced via the two processing methcontinuous processing method in this considered here; it would, of course, be ods is greatest. case. disadvantageous, should it be desirable "Order" as used here is unambiguous N , and N,,, have been treated as to maximize by-product production. .when applied in the short time limit originating from the same source, N O ; Several cases of practical interest have where it can be taken as the sequential this, of course, is not necessary. In been examined which illustrate the difnumber of the product. At all times, making the comparison, only the orders ferences that have been developed beorder refers to the value of the exponenof the two products need be considered, tween batch and continuous processing. tial dependence upon t, where Ni is set provided the limitations on the NO'S, Uranium-234 a n d -237 Produced i n equal to kt" and where n may vary with magnitude of n and m, and on the processa Thorium Blanket. The fraction time. (A log-log plot of N,versus time ing periods are observed. of uranium-234 in -233 produced in a will yield the order a t any point, t , as the I t has been assumed that in the prothorium blanket is of interest in breeder slope of the curve a t that point.) I n duction of certain products, by-products reactors. The ratio, U234/U233,will be general, the order will take on integral or contaminants are undesirable and to m!/ri! in the short time limit. The seor nearly integral values a t times when be minimized. Should it be desired to quence of reactions to be considered are : the rate of production of each member of the sequence either greatly exceeds or essentially equals its rate of removal. 1.40t 1 That is, each member of the sequence is either far from saturation or completely saturated. The order diminishes by one for each member that saturates. In the intermediate region, when one member of a sequence passes from the state far short of saturation to saturation, the order diminishes continuously from n to n 1, thus taking on fractional values. Where the by-product is of higher order than the product, Equation l l a indicates an advantage of the batch processing method when n > m and n > 1. However, the inverse is true when m > n and m > 1. Likewise, continuous processing is preferred when the '*Or order of the product, m, exceeds the order of the by-product, n, and is greater than unity. An example where this t ,s e c . inverse conclusion holds and where continuous processing is advantageous, Figure 3. Comparison of U234/0233 ratios produced via batch and continuous may be a long-lived fission product conprocessing
E
/
-
VOL. 48, NO. 12
DECEMBER 1956
2 1 17
II /
uf = 5 0 0 b
/
\?
i
/
t
1 I
i
/
20
40
Figure 4. Comparison of losses caused by N , with batch and continuous processing
Thus for very short times, long before thorium-233 has saturated with respect to its 23-minute beta decay, uranium-234 is produced as a fourth order producti.e., its production varies as t 4 . This condition prevails as long as the production of each member of the sequence
Table 1.
Dependence of
Figure 5.
Period Days 4
u233
+
U232/(U233 Pa233) Ratios on Processing Period and on Batch vs. Continuous Processing
+
[C'Z38T;;la~a28S]6: [ + c u2sz
pa233
Continuous Processing Blanket Fast blanket core neutrons neutrons
+
Batch Processing Fast Blanket blanket core neutrons neutron
+
c233
Fast blanket neutrons
Pa2331
Blanket core neutrons
+
0.0051
0.11
0.0002
0.05
0.046
0.48
20
0.48
0.98
0.030
0.29
0.062
0.30
40
2.6
3.6
0.21
0.74
0.080
0.20
1.45
0.099
0.177
60
6.7
8.2
0.67
80
12.6
14.6
1.45
2.5
0.115
0.171
120
28.6
31.5
4.2
5.8
0.142
0.184
0.21
0.23
1.0
1.0
200 m
2 1 18
80
IO0
120
140
Losses as a function of chemical loss, I,
substantially exceeds its destruction, When the concentration of a member approaches saturation it is no longer considered to affect the rate of growth of,Vn; thus after thorium-233 has saturated, thorium-232 can be considered to transmute directly to protactinium-233.
U232
lo'
60
+ T x 4 0 - ' 9 (;z)
bT
73.6
78.2
15.0
18.0
2050.0
2070.0
2050.0
2070.0
INDUSTRIAL AND ENGINEERING CHEMISTRY
Similarly after this saturates, thorium232 is considered to convert directly into uranium-233. In estimating the orders, rn and n, for evaluating the ratio, m ! / n ! , only those members far from saturation are counted. Thus, the formula is valid in the short time limit and approximately valid where saturation of some components has occurred while others are still far from saturating. The order is then decreased by one for each member that has attained saturation. n may be considered a reasonable approximation of the order for production of N , until the elapsed time exceeds about one half of a half life for any member, Ni (o 5 i 5 n ) , of the sequence. This approximation improves as n increases. For times greater than about one half of a half life and until saturation of a particular member, iv,, takes place, the instantaneous rate of production of N, will change from order n to n - 1, if only one member of the sequence saturates during this time. Using a similar approximation, a member may be considered saturated after times equal to several half lives have elapsed. Here, as n increases, greater times will be required to achieve saturation. Thus, the order for uranium-234 production is considered to be about 4 up to about 10 minutes, whereupon it will gradually reduce to about 3 in an hour: The order is considered to be.
if the uranium-237 in -233 were to cause
about 3 up to approximately two weeks and then it will gradually reduce to order 2 in several months. This or-der will then prevail for months or yearsuntiluranium233 and then -234 saturate, depending upon the exact flux of the reactor. The following equation illustrates some ratios, R ( B / C ) , that are thus derived and shows the lower content of uranium-234 in -233 for the batch processing case.
After Th233 saturates After Paza3saturates
a delay in its use to permit cooling, a time savings of more than a month could result when batch rather than continuous processing is used. Uranium-232 Produced in a Thorium Blanket. Seven short-lived members of the 4 N decay chain-Le., thorium series of naturally radioactive nuclidesgrow with a 1.9 year half life into equilib-
- 2! - 1 -
3!
(16)
-5
1 = -1 2
E -
2!
The ratio, R ( B / C ) is plotted as a function of time in Figure 3 for thermal neutron fluxes of 1012, 1013, and 1014 n/sq.cm./second. The limits indicated in Equation 16 appear as plateaus in the curve for (p = 101%n/sq. cm./second. However, the plateau a t R = ‘/z is decreasingly apparent in the (o = 1013 and (o = 1014 n/sq. cm./second curves. This is caused by the difference in magnitude of the destruction constants, b,, of uranium-233 and protactinium233, which at p = 1012 n/sq. cm./second, differ by a factor of about 500 and a t p = 1014 by a factor of about 5 . It appears that a factor in excess of 100 between destruction constants is required in order that distinct plateaus be observed. Plutonium-240 in -239 is similar to uranium-234 in -233 except that the decay constants and cross sections differ somewhat. Uranium-237 (6.7-day@and y emitter) production involves more sequential reactions than does uranium-234 production so that the contrast between batch and continuous processing is greater. I n the region of interest, R ( B / C ) = m!/n! will vary from about 2!/6! = 1/360 to 1/4! = 1/24 (Expression 15). After the uranium-237 product saturates, it is in equilibrium with uranium-236. The comparison given in this case by R ( B / C ) is between the reactor contents of the batch and continuously processed reactors at the end of the processing period, T. R ( B / C ) explicitly gives the ratio where the byproduct does not decay. Since uranium237, does decay and since the batch product is necessarily held up for a longer time than in the continuously processed reactor, a direct comparison of the processed products in this case cannot be made without a detailed schedule. However, if the specific beta and gamma activity associated with the uranium-233 product limits the subsequent ability to handle it, and if the “cooling” time is measured from the time the uranium-233 is removed from the reactor, then the above value of R ( B / C ) is also valid. In that case,
rium with uranium-232. Because of their associated high-energy alpha, beta, and gamma radiations, uranium-232 may be an undesirable contaminant of uranium-233. I n its production in a thorium blanket the following sequence of reactions must be considered : Th*Bz(n,2n)ThZ3’8-+ 25 .G hr.
The principal path for forming uranium232 is that shown in Equation 17. However, since the threshold for the (n, 2n) reaction on thorium-232 is -6 m.e.v., only the tail of the high-energy neutrons created in fission can effect the (n, 2n) reaction. There are two sources of these fast neutrons: fissioning of uranium-233 produced in the blanket, and leakage into the blanket from the core. As previously stated, uranium-233 growth and, therefore, the fast neutron flux dependent upon it will for times of interest grow with an order of 1 to 2, depending upon whether protactinium-233 has saturated. The fast neutron cQntribution from the core is considered constant. Thus, since the order for uranium-232 production from thorium-232 is 2 after several days, the ratio for fast neutrons originating in the blanket is
up to about 1 month and approaches 1 1 R(B/C) = - = until uranium-233 3!
begins to saturate. However, the ratio for fast neutrons originating in the 2! 1 core varies from R ( B / C ) = to 5.
a
Frequently of greater interest than the ratio, U”*/U*33, is Us‘/(U233 Pa233). For the latter ratio, R = ‘ / 6 for blanketoriginated neutrons and R = ‘/z for core-originated neutrons are appropriate in the range from several days until uranium-233 begins to saturate. Protactinium-233 does not contribute to the order here. The relative contribution of fast neutrons from the core and blanket has been examined (7, 4) for a representative, large homogeneous reactor (Table I). An optimum ratio, R ( B / C ) 0.17, is to be found for a processing period of about 80 days. Furthermore, R(B/C) is relatively insensitive to the processing period in this range so that from about 40 to 150 days R ( B / C ) is less than 0.20. Thus, uranium-232 in -233 can be substantially reduced by producing the latter in a batch processed rather than a continuously processed blanket.
+
Neutron losses from Production of Tran sm u fa tion Products
Closely related to the production of transmutation products in a fluidized reactor are the resulting neutron losses in the reactor produced by their formation. The equations for these losses are similar to those for formation of these products. Consider losses involving neutron capture by the n’th product in as equence of transmutation products. If batch processing is employed and No is maintained constant, the neutron loss can be expressed by Equation 19 where k is proportional to the rate of neutron absorption by the species under consideration. I t is convenient to express the neutron loss due to the n’th product, L,, in terms of the number of fissions or absorptions taking place in the fuel or neutron absorptions in the fertile material. Thus, k = cpa,No/N~cpu~T where A is most conveniently taken as the fissionable fuel when core processing is being, considered and as the fertile thorium-232 or uranium-238 where blanket processing is being considered. uA correspondingly refers to either the fission or absorption cross section of the fuel or the absorption c r o s section of the fertilem aterial in the blanket.
r I’
i = l i
#i
For the special case in which all 6 ; are equal, Equation 19 reduces to
-
( ~ T B n) f i = l
n - 1 e-bTB
i = O
1 VOL. 48, NO. 12
(all b, are equal)
DECEMBER 1956
(20)
2 1 19
The analogous losses suffered in continuous processing are shown by
amined in which the application of these equations are useful. Neutron Loss Caused by Xenon-I35 Growth in Reactor Blanket of Thorium-232. In operating a thorium oxide slurry blanket under conditions where the particle size is large enough to retain virtually all fission products within the slurry particles, the gaseous fission products could not be purged from the system, as they could if the blanket were a soluble solution. Xenon135, because of its high yield in fission and high absorption cross section, is potentially one of the larger sources of neutron loss in the reactor system. The pertinent sequential reactions to be considered are given in Expression 27.
(all bi are equal) ( 2 2 ) The ratios of losses suffered in batch to those in continuous processing are shown in Equations 23 and 24 for T B = TC. n
L
(all bg are equal)
(24)
Time
Until
1135
& Xe135saturate
Until Pa233 saturates
no
+ N1--+
No
+
Equation 24 is illustrated in Figure 4 for several values of n. The loss ratios, 1 R'(B/C), approach ap(n I ) ! as proaches zero, and unity from below as T approaches infinity. Thus, for all values of T , batch processing provides lower losses. I n batch processing where h ' o is not dni, maintained constant-Le., dt -6ONo; No = h';e-*o'-and where all bi are equal, the following equation expresses the neutron losses. ~
+
-
1
The ratio of losses for the batch and continuous processing where TB = T , is
Nc
Nz
Thus, at very short times before thorium-233 has saturated, Xenon-135 is produced as a fifth order product, and thereafter until about one third of a day, a fourth order product, then until protactinium-233 saturates, a second order product. Since the ratio of losses in batch to continuous processing 1 is approximated by R'(B/C) = -' (n I)! where n is chosen as previously described for different processing periods, the ratio 1 1 1 will vary from - to - to - before pro6! 5! 3! tactinium-233 saturates. The neutron loss caused by Xenon-I35 per fission of fuel atoms in the blanket will equal the fission yield (-6%), since Xenon-I35 will be almost completely removed by neutron capture due to its high cross section. The neutron loss caused by Xenon-I35 per neutron absorption in the thorium blanket will then depend
Several cases of interest have been ex-
2 1 20
N--+
Ng -+ h73
INDUSTRIAL AND ENGINEERING CHEMISTRY
upon the uranium-233 level in the blanket. A continuously processed blanket containing 1000 grams of thorium and 3 grams of uranium-233 per liter would suffer a neutron loss of about l~2yo per neutron absorption in the thorium. This loss would be at least sixfold less in the batch processed blanket for sufficiently short processing times. Fission Product Losses in the Core of a Homogeneous Reactor. The importance of neutron losses caused by fission products formed in the core, makes a more detailed analysis of interest. For this article, the fission products may be divided into three groups (2): those that can be removed as gases or because of rare gas precursors in a solution constantly purged with gas, the highly absorbing rare earths, and the remaining fission products. The fission yield of the first group is taken as 0.385; of the second group as y, = 0.015; and of the third group as ya = 1.60. The average cross section, ur, of the highly absorbing rare earths is rounded off at 50,000 barns (the actual value of u, is not important since the group is removed almost entirely by neutron capture) and the remainder are taken conservatively to average 50 barns. Since the first group of fission products is swept out, no neutron losses arise from it. The following equation describes the behavior of the r group, the highly absorbing rare earth fission products, in batch processing.
+
N , and u, again refer to the number of atoms and microscopic cross section o f the fissionable material in the core, averaged over the entire volume of the core system. -VT, ur, and y, similarly refer to number of atoms, average cross section, and total fission yield of the r group. It is implicitly assumed in Equation 28 that neutron capture provides the principal mode of removal of" A',, since the r group consists mainly of stable samarium-I49 and 73-year samarium-l 51. The a group of fission products may be described similarly except that the capture of a neutron results in the transmutation to another species of the same average cross section. ddt N a = yahrjujp or N , = y-Njujpt
(29)
The sum of the losses of the a and the r groups of fission products can then be expressed as (La)B (Lr)B, the neutron losses per fission as in the following equation :
+
Although in the previous examples, the chemical losses-Le., loss of N f in sepa-
The neutron losses per fission per period become
+ Nra,)] = y a u o ~ T+c 1YrurQTc + QMTC rating from N , and N,.-were assumed negligible, it will be of interest in this case to consider this parameter too. Since the core fuel may be processed a number of times before it is burned up, the chemical loss is a more significant factor in the processing of the core than in the processing of the blanket, since the product in the latter is not recycled. The chemical loss per fuel atom fissioning is ICh/Njaf(oT, where / o h is the chemical loss per atom processed. The total loss of both neutron and fissionable material per fission for the batch period, T , is L , as given in Equation 31. More exactly, the neutron loss and the fuel atom loss may be weighted in accordance with their relative importance. This weighting factor, taken here equal to unity, appears to be approximately that in the seactor, since each fuel atom upon abrorbing one neutron yields 7 neutrons. 9(UZ39- 1 1.30, 9(U236)- 1 = 1.08. I)
Since shortening the processing period (processing more frequently) increases the chemical loss per unit time but decreases the neutron loss terms, the losses may be optimized (minimized) with respect to the processing time, T. I n Equation 31, the terms p and T always appear together as the product, p T ; thus the optimization may be carried out with respect to pT. A simplified approximation for the optimum, pT, results when pa, is large, which will be true for most practical cases. In that event, N, rapidly approaches its equilibrium value and the last term in Equation 31 reduces to y?. The optimum pT that follows is
For continuous processing, the fission product concentrations are at their equilibrium values. The concentration of N, and N , are given in Equations 33 and 34, respectively.
By substituting the optimum values of p T into Equation 39, the minimum batch loss is found to be about 70% of the minimum continuous loss.
(35)
If the chemical losses are completely negligible, the loss ratio is
. For all values of cpT where T = T B = Tc, the ratio of losses for the a group,
The total losses of neutrons and fuel atoms for continuous processing is
LC = ~-l c h
OB equals (La)c
TC + .Y.u~'PTc+ 1 + aTvTc Y~QM
uiv T c
(36)
The ratio of the optimum p T for batch and continuous processing can be readily obtained; for a given p level, the optimum T B occurs at a time about 1.4 times longer than does Tc.
If the poisoning effect of r, as well as the rare-gas fission product group, is kept small because of low solubility (as in an aqueous sulfate system), only soluble members of the a group will contribute to fission product losses. Then the ratio of batch losses to continuous losses, LB/&, will a t short processing times equal unity, but will approach one half at longer times. This is apparent in the following equation when T B = Tc. Here y ; is the yield of the soluble a fission products. loh
Table
II.
U
~
Y:U~VTB B 2
lch + yduavTc
(39)
UfvTc
UIt
0.001 0.0003 0.005 0.001 0.001 0.001 0.001 0.001 0.001 0.001
~
( L r ) ~ (LJC
varies
from '/2 for small values of cpT to unity for large values. The contribution to the total loss by the r group predominates at small flux times, and the effect of the a group predominates at large flux times. The results of this combinaL,, is that for short tion of losses, La flux times the loss ratio, LB/Lc, approaches l / 2 . At greater flux times L B / L c passes through a maximum (for conditions of the first case in Table 11-at about 0.8 for a flux time of about 1020) and then approaches '/z again at large flux times. When comparing the loss equations for batch and continuous processing (Equations 31 and 36) term by term for T g = T c , chemical processing losses are identical but the neutron losses for both a and r fission product groups are less in batch processing. Figure 5 illustrates these equations for some representative values of the parameters. The effect of varying loh from 0.5 to 0.03Oj, chemical loss is shown. Batch processing provides lower losses at all times where T B = Tc. The minimum losses are not only lower for batch processing but occur a t greater values of p T and provide a flatter minimum than in continuous processing. Each of the parameters, L h , uf,y r , rr, and youo
+
Minimum Losses Caused by Chemical Processing Plus Fission-Product Neutron Absorption per Fuel Atom Fissioning
barns 500 500 500 800 500 500 500 500 500 500
UT,
UT
0.015 0.015 0.015 0.015 0.005 0.03 0.015 0.015 0.015 0.015
5 5 5 5 5 5
Lc
LB
Assumed Conditions loh
However, the ratio
of losses for the r group,
I n a manner analogous to batch processing for u,pT>>l, the optimum (oT for continuous processing is
L-B-Lc
l/~.
barns X 104 X IO4 X lo' x 104 X 104 x 104
104 1 . 5 X 106 5 X 104 5 X 104
Mini- (Minimum) Mini- (Minimum) mum Occurs at Occurs at Yaua, mum barns LB, % QT x 10-19 L C t % PT x 1 0 - 1 9 15 80 3.2 21 3.8 10 2.6 80 2.2 10 7.1 80 5.5 50 35 12 3.3 80 2.7 16 2.9 22 15 80 2.2 14 80 4.5 19 5.2 14 80 2.6 18 3.4 16 80 3.3 20 4.0 30 40 20 2.3 2.6 13 10 200 4.2 5.3
VOL. 48, NO. 12
DECEMBER 1956
2121
have been varied through a representative range and the results are shown in Table 11. With the above assumptions, losses can be expected to range between 2.0 and 6.0y0per fission, and a reasonably optimized estimate is 3.070 for batch processed and 3.5% for continuously processed fuel. Summary
In operating a fluidized nuclear reactor to produce a useful product, it is interesting to consider the extent to which higher order by-products are formed during irradiation under batch and continuous processing. ,4s long as the order of the by-product, ATn,is greater than that of the product, N,.and greater than unityi.e., n > m, n > 1-batch processing will produce material of lower byproduct content for a wide range of processing periods. This advantage, extending from the short time limit, persists for times comparable to the transmutative mean life of the species. At sufficiently long times, both methods of processing yield material of about equal by-product content with continuous processing producing a slightly lesser amount. I t appears that practical processing periods for at least certain classes of reactors are of a magnitude that falls in the earlier region, before the leveling off takes place and where a significant difference exists between the two alternative processes. Generally, if in the formation of N, and N,, they are produced by the different orders, m and n, respectively, a difference in the ratios,
s 2,will ;Zr,
the individual species considered in the sequence-Le., concentration be far short of saturation. The differential equations that describe the growth of transmutation products and neutron losses due to these products are similar. The neutron-loss and growth equations are identical in form except that the last differential equation for neutron losses contains only a production and no destruction term (compare Equations 19 and 2). The effect of this difference is that the batch to continuous ratio, R ' ( B / C ) = ( L n ) B ! (Ln)c.derived from the integrated equations for neution losses, in contrast to R ( B / C ) = (&Vn/Nm) B 1 (Nn/A',) c derived from the integrated growth equations, has no finite maximum and approaches unity asymptotically. Thus R' is less than unity for all processing times, whereas R (for n > m. rb > 1) exceeds unity for processing times comparable to the mean life of the species. Therefore, batch processing yields lower neutron losses than does continuous processing under all conditions and in the short time I)!. limit R' = l / f n Thus, under a range of conditions where higher order by-products are formed in a fluidized reactor, batch processing yields lower by-product contents and always gives rise tolower neutron losses caused by formation of these byproducts, than does continuous processing.
material
approaches
exist
m!
n!'
-
Thus in the limit, lower content of iVn in A', will be found in the batch material, provided n exceeds m and unity. O n the other hand, if the product is produced by a higher order reaction than the by-product-Le., m > n, m > I-the inverse holds and continuous processing is to be preferred. The advantage of continuous processing then is analogous to the advantage of batch processing where n exceeds m. Furthermore, the product and by-product need not originate from the same ancestor, for the same conclusions prevail regardless of source. The significant factors are order for production and the limitation that rates of production substantially exceed rates of destruction for
2 122
R'(B/'C) =
Ln
=
L
=
( n , ~ ) (n.J), ,
INDUSTRIAL AND ENGINEERING CHEMISTRY
number of atoms of ith nuclide = neutron flux, neutrons/sq. cm./sec. = neutron-capture cross section of the ith species for ( n ) process (barns = 1&4 sq. cm.) = neutron-absorption cross section of ith species including all processes which transmute to a new species such as h7)> (n.f), (Qn). (barns = sq. cm.) = radioactive decay constant, X = In 2/half life, set.--' = neutron, or nth member of sequence = formation constant of Ni in first order differential equation, sec,-l = destruction constant of Nt in first order differential equation. When all 6i are equal, b = b i , sec.-I = barns, sq. cm. = time, sec. = Batch processed = Continuously processed. = length of period for batch processed reactor (time to process contents of an entire reactor) = length of period for continuously processed reactor (time to process entire contents of a reactor)
used when T = T B = Tc,
by batch to that produced by continuous processing (Ln)B/(LJc. Ratio of neutron losses due to neutron absorption in Nn when batch processed, to losses in ATn when continuously processed loss from neutron absorption in the nth product, iVn, per fission in the fuel or absorption in the fertile material totaI loss due to both neutron absorption and chemical processing per fission in the fuel (n,2n) = the nuclear reaction in which the particle preceding the comma in the parenthesis is absorbed by a nucleus followed by emission of the particle(s) to the right of the comma. The example cited here illustrates neutron absorption followed by gamma radiation only, the fission process, and double neutron emission, resuectivelv I
k , .4
=
Nomen cia ture
Lim R ( B / C ) -
T+O
=
+
depending upon whether batch or continuous processing is used. In the short time limit and for equal processing periods, the ratio of batch to continuously processed
T
=
T
=
U
=
f
=
Y Y 'a
=
U
=
1,h
=
=
k =
--__ where i b T ~ ~Tu a
I
A refers
to either fuel or fertile material, dcpending upon whether the loss being considered is in the core or the blanket of a two region reactor the high cross section rareearth group of fission products all fission products other than the high cross section rare-earth group and the rare-gas group fissionable fuel fission yield fission yield of soluble products in the a group cross section chemical losses of atoms per atom processed
Literature Cited (1) Dresner, L., Oak Ridge Xational
Laboratory, ORNL-CF-55-5-140, May 23, 1955. ( 2 ) Halperin, J., Stoughton, R. W., Oak Ridge National Laboratory. ORNL-1368, Sept. 29, 1952. (3) Kasten, P. R., Oak Ridge National Laboratory, ORNL-CF-55-0-4, Sept. 1, 1955. (4) Stoughton, R. W., Halperin, J., Oak Ridge National Laboratory, ORNIP CF-55-10-104, Oct. 20, 1955. ( 5 ) Young, G., Clinton Laboratories, CLGY-1, Dec. 4, 1946. RECEIVED for review April 5, 1956 ACCEPTED September 23, 1956 Work performed for U. S. Atomic Energy Commission. Presented second annual meeting, -4merican Nuclear Society, Chicago, Ill., June 1956.
