ORNL-1368.txt

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3 4456 D360L0O3 3 2 é

ORNL 1368
Reactors-Research and Power fd./

 

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Myl .fiw-__
SOME EFFECTS OF TRANSMQT)*%TQN‘

PRODUCTS ON U233 BREEDER PILE OPERATION

     
   
    

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OAK RIDGE NATIONAL LABORATORY

OPERATED BY

CARBIDE AND CARBON CHEMICALS COMPANY

A DIVISION OF UNION CARBIDE AND CARBON CORPFPORATION

T

=

POST OFFICE BOX P
OAK RIDGE. TENNESSEE

 

 
ORNL~-1368
. This document consists of 40 pages,

Copy é? of 158 , Series A.

Contract No. W-740S5, Eng. 26

SOME EFFECTS OF TRANSMUTATION PRODUCTS
ON U233 BREEDER PILE OPERATION

CHEMISTRY DIVISION

J. Halperin and R. W. Stoughton

| DECLASS’HEU

CLASSIPICATION CRANGRD To:

- - -

s AEC 7
BN il teT .zl .-i?

BY - W iy - "‘-"_"--““
-

 

Date Issued: SEe g

OAK RIDGE NATIONAL LABORATORY
Operated by
CARBIDE AND CARBON CHEMICALS CCMPANY
A Division of Union Carbide and Carbon Corporation
Post Office Box P
Oak Ridge, Tennessee

d 4456 03L0L0O3 3

  
 
  
  
   

 
   
 
     
    
    
   
 
 

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ry

B hrology (Kaufmann)
;?;f ology (Benedict)
INTRODUCTION *
Two mutually dependent factors influencing the‘feasibility of breeding are the losses
of fuel atoms in chemical processing and the losses of neutrons due to absorption by fission
producta. If the fission products are removed by processing exceedingly frequently, the
neutron losses mentioned would ge low.but the fuel atoms lost would be exhorbitant; con-
versely if processing were conducted less and less frequently, the fuel atoms lost in pro-
cessing would diminish but the neutrons absorbed by fission products would become prohibitive.
Hence it seems desirable to minimize the sum of these two losses with respect to processing
period and to estimate the magnitude of the losses around the optimum value. It is felt
that sufficient data are available on certain processing losses and cross-section values to
glve a reasonable estimate of the probable range of these combined losses and of the optimum
processing periods.
J. A Lane, et al,(l) have considered the various factors influencing the financial
and neutron efficiencies of a UPJDJ breeder. For a particular pile configuration; they com-
puted the U233 production as a fudction of processing period.
; The purpose of the first part of the current paper was to comstruct an expression for
the fuel losses due to chemical processing plus the neutron losses due to absorption by
fission products and to investigate the influence of the various parameters on the magnitude
of the losses and on the optimum processing period.
| In these calculations the fission products have been divided into three classes: those
removed continuously as rare gases, those with relatively low cross-sections and those with
quite high cross-sections. Then the sum of the two types of losses under consideration
has been minimized with respect to processing period. Our interest here has been limited
largely to the 0233 breeder although the treatment should hold for any homogeneous thermal
reactor.
The second part of the paper deals with the build-up of heavy isotopes both in the
reactor and blanket of a U233 breeder and the effects of these species on neutron economy
and chemical processing. The build-up and the effects of U25h, U255, and U256 have been

(2)
quite thoroughly considered by S. Viener . Some of these higher isotope computations

 

(1)
(2)

~ o . __. AT - - % A a9A fA_4 T M\

ORNL-1096, Part IV (Dec. 10, 1951); also see ORNL-855, pp. 50-55 (Oct. 16, 1950).
 

have been repeéted here, however, since it was felt desirable to include effects of U237
and some still higher species.
I. Fiesion Product Poison lLosses vs Pfocessing Losses

In considering the factors determining reactor efficiency, one must optimize with re-
spect to some pertinent parameter. For converter and power reactors onme wants the cost per
unit product optimized. However, for breeder piles, until we are comvinced that breeding is
feasible, it seems more reasonable to minimize neutron plus fuel losses.

A calculation of the optimum processing period was carried out on the basis of a num-
ber of simplifying assumptioms. All the variables considered have been extended through a
reasonable range of values, and 1t is felt that the actual values to be realized in a given
reactor system should lie within the range covered.

The fission products were rather arbitrarily divided imto three groups:

 

 

Average
Fission yield Neutron capture cross-section
symbol value symbol value
G: Bemoved from reactor as rare gases --  0.385 - --
R: Highly capturing Rare Eerths ¥y 0.015 - 50,000 b
A; Remaining Yo 1.6 (, 50 b

The values of y. and.d; for the highly ebsorbing rare earths are rounded off figures
from the results of Ingraham, Hayden and Hess LEhys. Rev. 79, 271 (195017, end consist
mainly of Smlhg (y = 0.011, ¢ = 47,000) and sm1”1 (y = 0.004%, 0 = 7200). The actual value
of 6; is not very important as this group is essentially entirely removed by neutron capture,
and this condition would mot be altered significantly by rather large changes in 6;.

The yield for all the fissiom products with rare gas ancestors was estimated by Coryell,
Turkevich et al. in 1944k to be about 30%; it was estimated that this fraction of all the -
fission products could in primciple be removed as gases leaving T0% or 1.4 atoms per fission
in a fiqmogeneous reactor solution. A yield of 0.6 for the removable fiéeion products 1s
probably optimistic under amny practical conditiofiss perhaps 0.4 is more realistic. The
actual value used was 0,385 (i. e. 0.4 less 0.015) so the total yield per fission would be

exactly two. The cross-section value of 50 barnms is somewhat larger than the value of

 
~3-
(3)

E, P. Steinberg <for the average cross-sections for fission products (other than rare
earths) resulting from a Hanford slug which had beenm irradiated for 10 months and cooled
for three years; their value was 38 barns. A more pessimistic value was takem since the
average value for short-lived fission products almost certainly will be different from that
of long-lived species and‘the value may be higher. Steinberg et al. comcluded that with
the possible exception of the 275 4 Celhh (yield = 0.053), there are mo lomg-lived fission
products of unknown high cross-section.

If no fission products were removed as gases, the value of v, would become essentially
2,0 (actually 2.0 less 0.015 less 0.061) and a nev "group" of fission products would be
added, namely 19135 with a yield of 0.061 and an essentially infinite cross-sectionm.

The total lossee per fission, L, is here defined as the sum of the chemical losses of
fuel atoms plus the neutron losses due to fission product capture weighted by the relative
importance of a fuel atom and a neutron. This relative importance is here assumed to be
urity. (Actually a better figure is the ratio of fuel atoms produced to fuel atoms burned. )
Hence the total loss at any time t 1s given by

L = h (fuel atoms lost/fission) + neutroms lost to poisoms/fission

= h (U°27 atoms lost in chem. proc./cycle)/(fissions/cycle) +(n's captured by F. P.'s/

cycle)/(fissions/cycle),
vhere h may be comsidered equal to the breedirg gain; we shall let h = unaity.
Then the first term om the right is equal to 1b/f Ge T
vhere 1, = chemical losses, i. e., atoms U233 1ost/atom processed.
f = neutron flux

# = fission cross-section of the fuel atoms
T = processimg period.

It is assumed here that the chemical processing losses are directly proportiomal to the

amount of fuel processed.

For batch processimg, i. e. periodic processing of the emtire reactor fuel the nmeutron

losses may be computed as follows:

 

3) _ . S
ANL-bh49, pp. 82-5 (Oct. 1950).
%%E = rate of change of highly absorbing rare earth atoms within a processing period
= ypf NgOg - £Olp - | s

Qr Nra - "__g (l - € f a‘rt) v -

where t' = time after the end of the last period
Ng = number of fuel atoms (held constant ) v
STt = fission corss-section of fuel atams

It is assumed that the loss of atoms Np by beta decay is negligible campared to loss
by neutron capture; if this is not the case, the above differential equation should
contain an additional term; = A Nro

In the case of the remaining fission products (not removed as gases) it is

{-
¢

assumed that neutron absorption or decay results in transmutation to a species of ;

the same average capture cross-section. With this assumption these poison atoms growil. .

e e e g

in linearly with time, i.e.,

Ng = yNrf Ot

The term involving neutron loss per fission will then be

5T
l-e 7))
j (NaGa + Nr r)fdt - § ya T‘i yr [1 ( T ;" ’r J

 

fo‘l'?_i"r

Putting this term back in the original expression for total losses per fission,

with I being replaced by Ly indicating batch processing,

Ip = Lo +Ya0"7‘+ [1 L___m:)

Opf T for T

 

Tt is interesting that ’7’ always appears in the equation as the product £7 ; hence Iy

can be optimized with respect to £7 and then for any value of £ the optimum 7 is
readily obtained. To obtain the optimum period one may differentiate with respect to
f7T, equate to zero and solve for £ or one may simply plot I against 77 or £7T . The
latter method gives more information for relatively 1little more work since solving the
differential equation would be done by trial and error or by plotting anyway. Inspection
of this equation sh‘pws that for positive values of f T and for the range of variables

studied here only one minimm is possible.
-5

A simplified approximation for the optimum £7 results if f o7, is quite large.
In this case Np rapidly approaches its equilibrium value of y Ng 0‘}/ O3 and then

Ly, becomes

 

1
Lp = e + 3 y,0ufT +7 .
OefT r

On differentiation and setting the derivative equal to zero, the resulting optimum

£fT is given by

7 = 2 1o
YaCaCr

 

Perhaps a more meaningful basis than loss per fission would be loss per fuel
atom destroyed. The values given by the above equétion may be converted to losses per
J".‘uel atom destroyed by dividing Iy by (1 +0t), where A& is the neutron capture to
fission ratid for fuel atoms. (1 +& ), of course , equals 0_10'_’_;'_3 .

For continuous processing the fission producf.- concentrations fipproach constant

 

valies rather soon and then remain constant.

aN N
—£ = yrfp Oy - Npl{fop ¢ 1/T) =0

or _
N _Yrmffi'i' . Yflfc'ffr

rafs‘r-}l/r 1+ oy f7T

 

AN

— Yalp®ef - Ngfyr = 0

The neutron loss term per period becomes

 

1 (NgS°g + NpO2)ET = 50T ¢ _?:&‘_fif
NeSf T 15T

Then

 
6=

For'G;f1’ greater than unity tlhe approximate expression for the optimum value of f1

1c
TT = Yalalf

As more detailed information on fission product yields and cross-sections becomes

becomes

available, the neutron absorption effects can be broken up into several terms like the
last two in the equations for L, and I; above. The magnitudes of the cross-sections and
yields would determine the number of terms desired and the bulk of the species with smaller
crogs-sections wouid as here be lumped into a single term. Also for radioactive species
such terms should include decay constants; these 6f course go into the differential equa-
tions as additional coefficients of the Ny and N, terms, . Tfia present status of our knowledge
- concerning chemical processing losses as well as fission product yields and cross-sections
does not Jjustify a more detailed calculation at presenf. The data which now exist may be
found in the National Bureau of Standards Circular 499 and Supplements to this circular by
K. Way, L. Farro, M. R. Scott and K. Thew. Théae data have been summarized by R. P. Schuman,
KAPL-634 (August 1951).

L. and 15, are plotted against f” for various values of the variables in Figs. 1,52,
3 4 5 and a summary of the optimum values is given in Table I. 1In Table I the first
column indicates the variables in question, the second column lists the standard values of
these variables and the third column shows the values of the variables in question used in
calculating the results given on each line.

Fig. (1) showé the effect of changing the chemical processing losags lc from 6.0003 to
0.005. PFig. 1 can also be interpreted as showing fhe influence of varying the factor h
(the relative value of a U233 atom and a neutron) while keeping 1, and the other variables
constant. The 1lc = 0.0003 curves correspond to h = 0.3 and 1 = 0.001 and the 1, = 0.005
curves correspond to h = 5 and 1, = 0.001. If h is considered as the breeding gain, its
value would very likely lie between 0.90 and 1.25; however, if h is used to signify tfie
relative dollar-value of a U233 atom and a neutron it may differ frbm unity by as much as

a factor of five.
DW
l 1 ] | I I | I l 1 l 1 ] 1 I T ] T I ] l T r ¥ ] ¥ ' T
- | y, O fT 7
L = —< Jre
| CONT. O}f‘[,' +y00(-:lf1: * |+0".fT, -
200k —
-— L e L lugfrs [l_u—e"”r“)] J
BATCH™ o= ft * 2 Yo% TTT + Y, it o; =500
y 0'=80b
a a da -
| \ yr = . 0I5 i
5ol 7, =50,000 b
L(%)
{00
50

ounay
G.16045

 

 

 

 

 

 

 

 

E 16 20 30 40 50 60 T'OM 80 90 100 o 120
t (days) at f=10 ]
| ] ] | l ] 1 | L ] ] | ] i 1 | I 1 1 | I | 1 ] ] 1 ] ] ]
{ {0 20 30 40 50 60 70 80 90 {00
frx 10"19 ( nt::urri;rzonS)
FIG.1 LOSSES AS A FUNCTION OF THE CHEMICAL LOSS, I,

150
~8-

Fig. (2) shows the effects of increasing the fission cross-section of the fuel fl}
from 500 to 800 barns; this is about the difference which would occur if the fuel were
changed from 0233 to Pu239. In Fig. (3) the product of fission yleld and absorption cross-
section, ya6;, is varied from 20 to 200 barms; this essentially indicates the effect of
varying the value of(fg'frqm 12.5 to 125 barns. Also from Fig. (3) an indication of the
magnitude of the change to be expected on raising Y, from 1.6 to 2.0 may be deduced. The
effects of varying y, and 6;‘are shown in Figs. (&) and (5). Fig. (6) shows the effects
of adding another fission product of yield 0.05 end cross-section of 300 or 3000 barns to
those already considered.

From these curves it can be concluded that in all cases batch processing affords (a)
smaller losses, (b) minimum Iosses at a larger value of f4, and (c) a flatter minimum, than
the continuous processing. Tt may further be concluded that for any reasonable value of
the variables in a particular case, the losses due to processing and neutron absorption by
fission products are expected to be in the range of 2.5 to 6.0%. The best estimates at
present seem to be about 3.0% for batch processing and about 3.5% for continuous processing,
the percentages here being given on the basis of neutron losses per fissionable atom de-
stroyed (by neutron absorption). Incidentally conclusion (a) holds for any conceivable
combination of half lives and cross-sections among the fission species; see Appendix A.

In spite of the advantages of batch processing mentioned here, any isolated reactor
system would undoubtedly be processed on a continuous basis because of the large hold-up
of fisfiionable material which would be required for batch processing. At least twice the
capacity of the reactor would have to be on hand if it were desirable to keep the pile
operating while processing the removed fuel; intermittent pile operation and processing, to
avoid such hold-up of material, would seem to be at leéat equally undesirable. If on the
other hand, several reactors were located at one installation then only one additional
reactor-full of held-up material should be required if all pilea were processed batchwise
in series. The percentage of material held-up and not in pile operation would be much

smaller--perhaps as small as in the case of continuous processing. Under these circumstances

~
 

L
DWG. 16046
_ I I I l 1 l 1 ‘ 1 ‘ 1 I 1 ] 1 I l 1

 

 

L
0.0 J _“- LCONT‘ |
| BATCH |
i le =.004 B
- Yq0, = 80D .
_ y, = 015 |
o = 50,000 b
150+ _
'1 —
L (%) 5 .
10.0H

 

50

 

56 60 7Q 80 90 100 10
II-(doys) at f =10

i | 1 1 1 | ] | 1 | |

{ {0 20 30 40 50 60 70 80 90 {100

-{9 ;, neutrons
frx10 oz )

10 20 30 40

 

 

 

FIG. 2 LOSSES AS A FUNCTION OF FISSION CROSS-SECTION, o
L (%)

 

S
DWG. 16040

 

20.0 J-

15.01

{0.0H

 

 

 

=l

 

[ | I 1 I 1 I 1 ! 1 | | | 1 I ' ‘ 1
l'CONT. 4
—=== LpgatcH 1
oc~200 b -
o * 500b Yaa
. =.001 ]
y, =.015 ]
o, = 50,000 b
Y, = TOTAL FISSION YIELD OF GROUP A ]
FISSION PRODUCTS . _
oy = AVERAGE CROSS-SECTION FOR
GROUP A FISSION PRODUCTS
yo0o,= 200b ]
-~
-
,/’ ]
,/
/’, —60b -
pad 60’
,’/, \10 3
///,’
_ .
,//’ GOb ]
o Yol =
/”/ ’—--‘—'—-—-_-
\::—— T Ya%a -:2_0_b___ —
i i ! 1 1 1 ] 1 1 1 ! ]
10 20 30 40 50 60 7014 80 90 {00 10 i
t (days) at f =10
| 1 | 1 I 1 l } I ] l i l 1 l 1 I 1

 

 

10 20 30 40 50 60 70 80 90 100

-49, neutrons
ftx10 oz )

FIG. 3 LOSSES AS A FUNGCTION OF THE PRODUCT y,o,
 

20.0

15.0

L (%)

10.0

5.0

 

 

 

 

 

 

FISSION PRODUCTS

DWG. 16043
¥ I 1 l I I 1 I I I 1 l 1 I 1 l 1 l I
1 Lcont —
. T LaatcH 4
| o, =500 b ]
- . = .00{ -
_ y,c,=80Db i
g, = 50,000 b
" ‘ <
| \h' —
. . 0\5 -
No©
= . 006 -
Yoo
_ o
¥ 2 0% -~
= \\ ‘-_,—--_-—--— -1
i — - 5 _‘—"-__
NN - yr :9-‘*5" —
e — -—___,-—" _ ’—___
- \\\ ‘____,—-"' y(._.Q.Q——"‘" B
\ \ ___.—"'— _____-—’
\ \\-.,_ —__—_____.--— ’-—___..—-
- \ - - __——“-’ -
\\\- —-"'—‘-—--—‘-
L i L-—---l_—-- ! § 1 1 ! ! 1 1 ]
1 10 20 30 40 50 60 70 80 90 {00 {10
|- 14 -
t (days) at f =410
| | ] | { | 1 ] ! l 1 | I ] 1 | I | L
{ 10 20 30 40 50 60 70 80 90 100
, -{9 t
fe x 10 (neéjmraonS)
FIG. 4 LOSSES AS A FUNCTION OF FISSION YIELD OF GROUP R
 

 

willanye
DWG. 16044
T T

 

  

  
 
  

 

 

 

 

=T I T ‘ T I T | T [ | T I T | ' ]
20.0H LconT —
- === Lgarch .
i . =.001 |
s = 500 b
) Yooq =80 b il
150 y, = .015 —
L (%) | -
10.0— —
! o, =150,000 b _
sol o, = 50,000 b ez
o, =10,000b ____—===%

N T -S;—:jt:—f"in 10,000 b g
| o, =150,00057 ' °r =50,000b ]

i 1 1 1 1 1 1 1 1 i 1 1

{ 10 20 30 40 50 60 794 80 90 {00 110
- r(days) at f =10 7]

ol L o 1 1 1)y 4

{ 10 20 30 40 50 60 70 80 90 {00
frx IO—19(niL:!t1|;ons

FIG. 5 LOSSES AS A FUNCTION OF THE AVERAGE CROSS-SECTION
OF GROUP R FISSION PRODUCTS, o,
 

DWG. 16042

 

 

 

 

 

 

1 I 1 l v I 1 l T I 1 I ) I T I 1 I L
| -
20.0H I“CONT. —
| —=== LparcH i
- le =.001 |
= O':f = 500 b -
y, o, = 80D
y, =.015
15.01— o, = 50,0000b ]
i Y, = .05 i
©
N ’3000 ]
i 6\; -
L (%) - o >
0
10.0 z 30 —
Oy
- - 0 "
Oo_~o==%
| :-3 00 ® i
e)
i ’,/, »
,,__.-‘ . 300 b -
i =~ ’ —?““—’” -—';
; - o —— g’
- ’.—"‘— b _
R N I
L 1 ] I ! ] ] I t ] 1 ]
{ 10 20 30 40 50 60 4 70 80 90 {00 1O
- r(days) at f = 10 )
0 | | : | 1 | I | I | ! | i | I | I | 1
{ 10 20 30 40 50 60 70 80 S0 {00
fr x 10-19( necu;]rzons)

FIG. 6 LOSSES AS A FUNCTION OF THE AVERAGE CROSS—SECTION

OF GROUP B FISSION PRODUCTS, a
<1k

the advantages of batch processing may well outweigh the disadvantages. Tt has been pointed
out that the assumption that the processing losses will be proportional to the fuel atoms
processed may not be valid for all mefhods of processing. For example, with an ion exchange
method, fuel solution could be poured through an absorption column until the radiation had
destroyed the usefulness of the resin or until the column was loaded with fission products
and the urenium losses on the column might be essentlally 1ndependent_of the rate of
throughput. This might be true and in such a case the treatment given here would not nec-
essarily be expected to hold for ion-exchange processing; it is8 not entirely clear, however,
exactly how an ion-exchange continuous process would be carried out. It is felt thaf the
calculations made in this report would be pertinent to a solvent extraction process vwhether

conducted in 1light water or directly in heavy water.
-15-

TABLE 1
Minimum Losses in % Due to Chemical Proceasing Plus Fission Product Neutron Absorption Per

Fuel Atom Destroyed. (Fuel Atom Lost Assumed Equivalent to Neutron Lost).

 

 

T (Days) T (Days)
std.

ariable | value | value | Ly |trx 10719 | at £ = o | 1, |erx107 |at £ = 10%Y
o= - std. | 2.9 21 24 3.5 15 18
1, 0.001 | 0.0003 | 2.0 10 12 2.h 10 12
1 .01 | .005 | 5.0 50 58 6.5 35 b1
67 |500 800 2.5 16 19 3.0 12 14

Yy 015 | .005 | 2.0 22 26 2.6 15 17.5

Yy 015 | .03 b1 19 22 n,7 14 16
6 150,000 | 10,000 2.4 18 21 3.1 14 16
@ |50,000 [150,000 3.0 20 23 3.6 16 19
yafa | 80 20| 2.1 4o 47 2.k 30 35
yafa | 80 200 | 3.8 13 15 %8 | 10 | 12
6o 0 300 | 3.0 19 22 3,7 m 16
6o 0 3000 | 3.8 14 16 b7 10 12

 

 

 

 

 

 

 

 

 

 

 
-16-

II. The Effects of Bulld-Up of Heavy Isotopes

In a U233 thermal breeder the U233 concentration in the core will refiain essentially
constant by addition of new material as the fuel is burmed, and the isotopes U234, y235
and U236 will 8lovly grow in and attaln concentrations of roughly the same order of magni-
tude as that of the U233, Other species, e. g. U237, Np237, Wp238, Pu238, pu239, y231,
U232, etc., will also grow in in smaller amounts and the methods and schedule of processing
the fuel will determine the maximum levels of the Np and Pu isotopes. The following seche-
matic diagram indicates moet of the pertinent reactions which will occur in the core.
Neutron fission reactions are omitted although U231, 0932, 0255, U235, U237 and Pu?39 are

known or expected to undergo fission with thermal neutrons.

Pu38(n,y)Pu239

.0d 2.3d
s‘r B-’T ’

239
p>37(n,7) ¥p238(n,7)Np

6.
U231(n,2n) U232(n,2n) U233(n,y) U234(n,y) U235(n,y) UR36(n,y) UR37(n,y) U238(n,r) U39

The nuclides U231, 0238, U239, Np239 and Pu239 will not be discussed subsequently

 

 

 

 

g8ince they will exist in rather small concentrations and ainfie calculations concerning
their build-up would be very unreliable.

The effects of the uranium isotopes consiet largely of 1ncreasing the total uranium
concentration and specific alpha activity. The total uranium concentration at equilibrium
becomes about 2.18 times that at the start-up of the pile. The alpha activity change will
depend largely on the U232/UR33 ratio as discussed below. In addition, the U237 growing
in will cause even the "decontaminated" fuel to contain apprecisble quantities of beta and

gamma radioactivitlies. The effect on neutron economy is small,
-17-

Considering first the major heavy isotopes, the differential equations for growth are

dNok

dt

Noz£6c(23) - Nouflc(2h)

25
at

Nouflc(2h) - Nostla(25)

dN26

————

dat

Nos£(c(25) - Nagede(26)

Where f represents the neutron flux, 0? and 0’;3. indicate respectively cross-sections
for neutron capture and for meutrom absorption (1. e. fission plus capture). The N's indi-
cate the concentrations of the various species; the subscript and parenthetical numbers are
the usual code symbols for the heavy isotopes, e. g. 23 represents elément 92, masse 232.‘
The fission cross-sections for both Uejh and U’256 are negligibly small.

The final equilibrium values of the relative concentrations of U233, U234, ¥235, and
U236 are obtained by equating these differential equations to zero and solving for the vari-
ous isotopic ratios. The following ratios are obtained using the cross-sections given in

Table II.

Mol 6c(23) 50

 

 

et =7 ‘= 0.71L
Moz Gc(oh) 70
Nas _ €c(23) _ 50 = 0.078
Nos  (a(25) 6ho
N 25) 06

26 _ (c(25)0e(23) _ 100 x 50  _ 0.391

 

 

Nps  Go(26)0a(es) 20 x 6h0

From these values one sees that the final equilibrium number of uranium atoms per atom

of UPDD is 2.18, 1. e. the uranium concentration increases by this factor.
-18-

TABLE II

Thermal Neutron Cross-Sections of Heavy Isotopes Used in this Report. (Values Given in

Nuclide
Th232

The33

Pal3l

Pa233

uR32

ye33

yash

@35

1236
237
Np237

pu238

(.
7.0
1350

150

50

50

50

70

100

20

180

460

T.0
1350

100

250

TO

640

20

- 8ho

480

Barns).

Remarks
Value from HE. S. Pomerance, ORNL-51, p. 16 (19.48).

Hyde, et al. ANL-4165, 6-25-48 reports 0, = 1350 §
100 barns. G, assumed equal to (.

A better value is probably (; = 290 + 20k, reported
by R. E. Eleon and P. Sellere, ANL-4112, p. 27 (1947).

L. I. Katzin and F, Hagemann, CC-3699 (1946),
report 37 + 14 for O'C(lj) , but there is evidence from
Hanford irradiatione of Th that their figure is too Jlow

A. Van Winkle, R. Olson, W. C Bentley and A. Ghiorso,
CF-3795 (1947), obtained U ¢(22) = The value of
(g (22) used here is purely a guess

G. Haines and K. Way, ORNL-86, report as a consistent
set of values, 0o = Th, Og = 564, N = 2.35.

(r (24) = 88 was reported by H. Pomerance, Reactor Sci-
ence and Technology 2, Fo. 1, p. 83 (April 1952); this
is probably the best value.

G. Haines and X. Way, ORNL-86, report as a consistent
set of values, ¢, = 98, (O = 644, n = 2.12.

P. R. Fields and G L. Pyle ANL-4490, p. 5 (1950)
give Jc = 23.5.H. Pomerance, ibid., gives 5.8 .

A guess, giving 3(27) + 227 /f 2 x 10°°L at £ =
1015, T1/2(27) = 6.9 4.

Value quoted for pile neutrons by P. R. Fields and
G. L. Pyle, ibid.

G. Reed and W. Bentley, cc-3780(19h7) report (@ =
300 - 800.
-19-

On solving the differential equations, the isotopic ratios as a function of

time become

 

 

 

 

 

ok Go(3) | T@ITy  ogsm (1 - e Setehiee,
No3 Oc(2h)
v  Oe(2)  Geen)  Teemre ge(z) oY) _ -Ga(es)se
N,  0aes)  Ta(@5)-le(2h) Oa(25)/07a(25)-0c(24)] |
- Oc(ak)ft fa(25)ft
- 0.078125 - 0.0877192982 e + 0.0095942982ke |
Mo _ 6e(23)0c(25) (1;e'°2(26)fl5+0:’(23)0:(25) [;-63(21&)&_;06(26):?1:]
N,s Ca(25)0c(26) [Ba(25)-Ge(24)] [Tclan)-Tc(26)] -
_0c(23) 0c(24) Gc(25) -0a(25)ft -Jc(26)ft,
[33125).63(26)7[33(25)-62(2&)]63(25) e - )
~fc(eh)tt _Ga(25)ft Fec(26) £t

= 0.390625 + 0.1754385965 e -0.001547467458e -0.02052785923ke

Values of méh/u23; H25/H23 and N26/né3 as a function of £t are given in Teble III. At

short times, 1. e. up to ft = 1021 neutrons/cm?, the approximation

2

26

———— ——
—

5

Go(23)0c(24)0c(25) £t

mfl
N\

may be used with a maximum error of 20% at the highest ft.
-20-

TABLE III

Relative Concentrations of U25317023h, U235 and 0256 as a Function of Flux times Time, ft

ft x 10-19

1
3

10

30

100
300
1000
3000
5000
7000
10,000
15,000
20,000

30,000

Nok/Nps
¥.998 x 10~
1.498 x 1073
1.983% x 1077
1.484 x 1072
4.829 x 1072
0.1353

.3596

.6268

.6927

. 7090

L7143

.T143

7143

.T1h43

Nos/Nos
1.746 x 107"

 

1.564 x 1076

1.709 x 1072

1.468 x 10-%
1.395 x 1072
8.428 x 107
0.03458
06738
.0T548
LOTTHT
.07812
.07812
.07812

.07812

N26 /Mo
5,845 x 10"11

1.567 x 1077
5.728 x 10-5
1.492 x 10"6
h.89% x 1077
9.646 x 107
1.556 x 1072

.10230

.1882

.2527

314k

.3625

.3803

.3892

.391
~21-

In view of the beta and gamma activity associated with 0237 ag well as its possible
fissionability, the concentration of this isotope as a function of time was also determined
ag follows:

Symbolic forms of the equations for Ny Ny; and Naq/w,,

- oe(24)ft - Ga(25)ft - Ge(26)1¢,
1. e Egé =8+Dbe e(24) +ce ' +de
N , -Cc(ak)rt -0a(25)ft -O0c(26)ft - q ft
and 2T - ai + bt e +c’2 +d' e + gl e
NQ}

were put into the differential equation,

N .

where q = 6_:E;,(Q"{) + AET/f’ 227 being the radioactive decay constant of U237.

The values of a', b', c', d' and g' were obtained in terms of g, the various cross-
sectiona, and the values of a, b, ¢, and 4; the latter numerical values are those in the

last form of the equation for N,./N»-z presented previously.
| 26/ 723

at =~ 80¢(26) ; b' =bCe(26) ; c' =cOc(26) ; d' =a0e(26) ; g = -a'-b'-c'-d'.
qa q - Oc(24) q -0a(25) q -G c(26)

Then at £ = 1015,

N | - Cc(2k)ft -0a(25)rt
27 = 0.00390625 + 0.00181801654395 e - 0.000022756874378 e +
Koz -G c(26)ft -q ft

- 0.0057021831206 e + 0.000000673451984 e

The value of q used for f = 101 was 2 x 10721 om”.

While these calculations are quite elaborate, the methods used here are'conaidered
better, if a desk calculator is available, then using sufficiently precise approximate
expressions.

Values of H2.T/N23 and curies of U~ per gram of U7 are shown in Table IV. The
subsequent approximate expresaions were used to obtain the values of Né7/flé3 at the

shortest times and to check the values at t x 107 = 1, 3 and 10 and at the longest times.
-22-

The first involves substituting the approximate expression for H26/H23, i.e.,
the expression proportional to 33, into the differential equation, for H27 growth
glven previously, solving the resulting equation, expanding the exponential term

in the solution and dropping off higher terms.

aN
'&'%I = Npgf 05(26) - Npp q f

- 2 G(23) 05(21) 03(25) a5(26)2"7 - Mpq q £
Ny =1 Gi(23) 0G24 GG(25) G(e6)e T - LT

The second approximation, which actually can be as accurate as one wishes with
enough works providing N26/N23 is known as a function of time, utilizes the assumption
that N2§ can be considered constant over small increments of ft at the higher values

of the latter. On this assumption the differential equation becomes
dNg7/at = Wog £0(26) - Nop q £,

where Nog is the average value of N26 over the time increment in question At = t-t!.
If now No7 is the concentration of U237 at time t, N£7 the value at t' and ANp7 =

No7 - N£7, the solution may be expressed alternatively

Ne7 | N2 Tc(26) [1 _ (1 _ 1 N§7/N23 ) o=d A(ft)J

N2j Noj q 0-c(26)-1\T;6-/N25
! _
or ANz . {E&E_‘i@fl - N_?z} {1 ) e-qA(ft)}
N23 N23 q N23

The equilibrium value of Np7/N,j, unlike the ratios of the lower uranium

isotopes, is flux dependent.

N2y _ N26 o c(26) - N2g fa-e(26)
Ne3 Nej q Na3 (g7 + £0(27))

 

0.00391 at a flux of 1015
Table IV

 

U237 /4233 Ratios as a Function of Irradiation Time at Flux of 1015

t(sec x 10-5)*
Oel

o3
1.0

10
30
100
300
1,000
3,000

10,000

N26/No3

 

5.83 x 10-11
1.57 x 10=7
5,73 x 1078
1.49 x 1076
L.89 x 10-5
9.65 x 10~k
1.56 x 10~2
0.102

314

0389

2391

391

3
One day is 0.86L x 10° sec.

sz/N23
2,91 x 10715

2.36 x 10713

2.8 x 10-11
2,01 x 10-9
1,77 x 10=7
6.6 x 10-6
1.40 x 10~k
9.99 x 10-4
j.lh x 10-3
3.89 x 10-3
3,90 x 10-3

3.91 x 1073

Curies y237/g y233

 

2,35 x 10-10
1.91 x 10-8
2.27 x 106
1.63 x 10-b
1.43 x 10-2
5.22 x 10~1
11.3
80.8

25k.

31L.

315.

316.
—2l-

The concentration of Np237 will depend on the processing method, 1.e. whether
or not neptunium is removed from the fuel during processing. If it is not removed

it will continue to build up with time and its relative concentration will be given

by
t
N7 . A27N27dt
No3 N3

°
until its destruction rate by neutron absorption becomes significant. The accurate
expression for N27/N23 ‘given previously may be put into this equation and integrated,
thereby giving an accurate expression for N37/N23 for shorter times. At longer times
the concentration would be cbtained by integrating the equapion

N3

T - ?\27 N27 - N37 £ 06(37)

If the neptunium isrpartly removed in chemical processing the differential equation

becomes

dN
—E‘gl = )\27 No7 = N3, (£ 08(37) + a/7)

Where 7 is the processing period and a is the fraction of Np removed from the fuel
in processing. (Complete removal in processing corresponds to as 1), The maximum
possible relative concentration of Np237 at equilibrium, assuming a = O and assuming

the values of cross-sections given in Table II, would then be

N3z 25
‘N7 qf%c(37)

or Ny . Nyg Moy _ Nps DBy oo(26)
23 Np7 Np3  Np3  q°f2Cc(37)

= 0.037 at £ = 1015, g = 2 x 10721
~25.

Considering the growth of Np237 for intermediate and long times inder the con-
dition where none of it is removed by chemical processing, it may be assumed that
the production of Np237 is equal to the neutron capture by U236 1ess the neutron
absorption by v237 and Np237, i.e.,

dN37

~ ot
- T, O Ll |
-4 - Nog £ c(26) q £ N37f 0 e(37)

where Nog is the average concemtration of 0236 over any interval At or A(ft).

On integration the increase of Np237 in any interval becomes

E?l {N% 0_6(26).' ;{27 i NB'Y} {1 ) e-G'c(BT)A(ft)J

No3 (N23 ©Oe(37) af N3

where Né-? is the concentration of Istz37 at the beginning of the interval A (£t).
This expression should be quite accurate for longer times of reactor operation

providing the Np237 is not removed by chemical processing. If the Np is removed
by processing then A N37 may be taken as the amount of Np237 produced since the
end of the last processing period if the processing is conducted batchwise. In
the latter case, N;7/N23 = 0 and A (ft) becomes £7 at the engl of a processing
period; with these substitutions the above equation gives N37/N23 at the end of

a period (providing the period is long compared to the 7 day half life of U237):

N37/N23 ={N_23 oe(20) 327} {1 - e'°_°(37)f7} (1)

 

N23 Oc(37) af

If continuous processing is employed and Np is removed with high chgmi’cal

-ef ficiency, an approximate expression for N37/N23 is

Ny Aer 1
= 56 £ Tc(26) — N37 {f c(37) +T}' 0
~26-

or N37 _ m A37 Ge(26) £ 7 (2)
" No3 N23 aqf 1.40¢(37)f7T

 

where 7T 1s the processing period.
In view of the uncertainty in the factors which determine the Np237 concen-
238

tration, it was not felt meaningful to calculate the Pu concentrations except
under certain extreme conditions. It seems pointless to carry out more calcu-
lations on Pu until a given reactor system is designed. As an example, however,

the N),g in a reactor where equation (2) holds and where Pu is also removed Ly

continuous chemical processing may be obtained in a manner similar to the N37:

WS - N3y £0(37) - Myg(£oa(ll) 4 =) = o
at i

Mg | Ny oONET | Wag A2 (we(26):T)  op(3DET
N23 Np3 14ca(U8)ET Moz qf (L4oe(3NET) (L +0,(L8)E T)

 

(3)

Table V shows the relative concentrations of Np237 and Pu238 compared to ye33
as a function of ft (flux times time of pile operation) for two different values of
fT for continuous processing. Column five shows corresponding N37/N23 ratios for
batch processing at £T = 1022,

The neutron loss due to the absorption by U23h, U235 and U236 less the re-
production of neutrons by fission of 1 is given by equation (L).

L, = neutron loss - N2), - 6 e(2h) + (1

Npga(25) . N2 Go(26)
U233 atoms destroyed N23 G a(23)

No3oa(23) N23zoa(23)

 

= 25)

-Co(2L) £t -0 ,(25)ft

0,0032954545) 4+ 0.02979266347 e - 0,01256025877 e

0.02052785923) e~ Tc(20)ft (L)

'Here,szs = neutrons emitted per neutron absorbed by U235 and is here assumed equal
to 2.12. A plot of L, vs. ft is shown in Fig. 7. The losses are seen to increase

to a maximum value of 0.0063 (i.e. 0.63%) at about ft = 3 x 1021, then decrease to
 

 

 

 

 

 

o
g { ] IR | { F T T T I { F T yTd [ | T TT1T1] I IDV\I,G'I“IGCI)AIH
O
az
j—
0
o +.006 + 006
s
O
—
Njf + 004} +004
N
aJ
>
E:u + 002 +.002
Qa
w
Ll
@ 0 0
O
-
5
S -002f -002
—
-
=
Ll
£
._IC 19 1 IItIIlll20 1 IIIlIIII2| 1 lIIIIlIl2 I lllIlllI ] L1 11t
2 2
10 10 10 10 0 0=

FLUX-TIME (ft)

FIGURE 7. NET NEUTRON LOSSES DUE To UZ>* (239

AND U%® BUILD-UP

“LZ—
Table V

Concentrations of Np237 [Equation (2)] and Pu230 E‘L’quation (3)_] as a

Function of ft for Continuous Processing Where Both Np and Pu Are Chemically Removed by Processing.

(t is time after reactor first starts up. )

 

 

 

3t
Ny7/N23 N37/No3 Ny,g/N5
£t x 10-21 No6/No3 £7 =100 £7 = 1022 £T =102  £T = 1020 £7 = 1022
% 1.9 x 105 5.7 x 108 2,0 x 1070 2.6 x 10~ 9.7 x 10~10 6.3 x 10~7
3 9.6 x 104 1.1 x10® 4.0 x 1075 5.2 x 10-5 1.9 x 108 1.2 x 10-5
10 1.6 x 1072 1.9 x 107 6.6 x 1074 8.6 x 10-L 3.2 x 10°7 2.0 x 10~k
30 0,102 1.2 x 104 1.2 x 10-3 5.5 x 10-3 2.0 x 10~ 1.3 x 10~3
100 .31 3.6 x 1074 1.3 x 1072 1.7 x 1072 6.1 x 100 4.0 x 10-3
300 .39 LS x 1007% 1.6 x 1072 2,1 x 1072 7.7 x 10~6 5.0 x 10™3
O .39 L5 x 1074 1.6 x 107 2.1 x 1072 7.7 x 10°° 5.0 x 107>

*Batch processing, calculated by Equation (1).

$#nity here would be about 11.6 days at £ = 1015 or 116 days at f = 101k,

-Q2-
~29 -

a minimum of -0.00L43 (i.e. a net neutron gain) at about 3 x 1022 and then increase

again and level off at an equilibrium value of 0.0033 above 3 x 1023, The equili-

brium value alone, of course, can be obtained from the above equation in its first

form by insertion of the equilibrium values of the relative concentrations N2h/N23 s

N25/N23 and N26/N23. It is interesting that while the total uranium atoms per atom

of Y233 approaches 2,18, the maximum neutron loss is only about 0.6% and the equili-

brium value is only about 0.3%.

Breeder Blanket

 

In the blanket the following scheme was considered:

 

he3h A7 X
A 2L4.1 4 7

(n,7)

233 B

Th \
A 23.5 m

(n,‘)’)

Th232

(n,2n)

¥

 

the3l  B-

25.6 h

 

 

The primary reaction sequence, of course, is

Th232(n, )Th233 e pa®33 — pe33

Pa23,4 /5’ U23h ___fi_.____>
flk T.1 m; 6.7 h 2.3 x 105 y
(n:7) (n,%)
pa233 y233 oL
27ah d i (n,fiSSo)
(n,2n) l (ny2n)
v
pa32 § y232 oL
3
P.:-1231 | oL

and the other reactions may be considered according to their effects on the production

of U233. Neutron capture by the members of the 233-chain involves a double loss, i.e.

a neutron is lost and an actual or a potential U233 atom is converted into a non-

fissionable U23h atom. Higher isotopes may be built up by neutron capture if the

blanket is not processed very frequently.

Fission of U233 need not involve a net
-30-

loss since most of the resulting fission neutrons should be abscrbed in the
blanket, especially since most of the fissions will occur toward the inner edge
of the blanket (i.e. the edge nearer the reactor). It may be reasonable to
assume then that absorption by U233 results in neither 'a neutron‘loss or gain.
Should some fast neutrons come in contact with the blanket, the (n,2n)
reactions will occur to a small extent. These will involve a small neutron

gain (a negative loss). The principle (n,2n) reaction may be expected to be

Th232(n,2n) Th?3L _B7  pa23l

in view of the high relative concentration of Th2320 The net gain would obviously

- be equal to the number of (n,2n) reactions by thorium less the number of neutrons
absorbed by the Pa231. |

The principle effect of the U232 will be to increase the specific alpha
activity of the product. U232 decays to the 1.9 y ThZ28, all the daughters
of which are much shorter than 1.9 y. Hence‘the activity resulting from any
U232 win grow with a2 1.9 y half life amd finally attain a disintegration rate
six times that of the parent U232 (i.e. five additional alphas from the
daughters). The effect of U23h, in addition to the losses mentioned above, is
merely one of diluting the mroduct ?ith a non-fissioning isctope.

All the calculations madé here assume constant flux, i.e. invariant in
time and independent of position. While this assumption is reasonably good
for the reactor, it is much poorer for the blanket. Should the fuel and blanket
be intimately mixed in a one core reactor, the results here would‘be somewhat
better for the blanket reactions and somewhat poorer for the reactions con-
cerning the fuel. In the cases of the (n,2n) reactions the calculations are
based upon cross-sections for (n,2n) reaction per unit thermal flux; obviously
any particular value for such a cross-section can hold only for a particular

geometrical configuration. Hence the (n,2n) calculations are particularly poor
-31-

unless an appreciable amount of fissioning of U233 occurs in the blanket or unless
a one region breeder reactor is being considered.

The losses due to capture by merbers of the 233-chain may be estimated as
follows:

For any reasonable irradiation time, the Th233 concentration will be at its

steady state value because of the short half-life of this species (23.5 m).

~ Hence
i‘g.s."z = Ngpf 5(02) =Aqalg3 = O
or NO3 = w
A03

The resulting loss of Th233 atoms per U233 atom produced will then be approximately

No3f 0(03)t _ £0,(03)

Noof ca(02)t 03
and will remain independent of irradiation time as long as the U233 production
is directly proportional to irradiation time,
As long as the fraction of Pa?33 and 1233 atoms absorbing neutrons is small,

the concentrations of these two species may be simply expressed:

dN]_3
dt

neutrons absorbed by Th less decay of P3233

Noof Ta(02) - A3 Ni3

=
N
W

1

Noof G.(02)t [1 - -5\—;; (l-e'h3til
=32=

For long irradiation times or very high fluxes, more accurate expressions
will be required which take into account the loss of Pa233 and U233 atoms by
neutron absorption. For the present purposes, however, this is not thought
necessary especially in view of the question concerning the value of the
capture cross-section of Pa¢33, 1In any case, the methods employed here will
be satisfactory for checking more accurate calculations; also in combination
with successive approximations for the neutron absorptions mentioned the
methods used here can be made essentially as accurate as one wishes.

The total loss (of neutrons and neutron-equivalence of 233-chain atoms)

per U233 atom produced is then

t

 

t
"

| o
(1 4 h) [£076(03)/X ¢y + IR
0

2,2 [fo-c(03)//\03 $ -—‘:9—;%(-23 (1~ —tpe(l =€ A13"'):|
13 A13 -

where h is the breeding gain (i.é; the U233 atoms produced per 1233 atom des-
troyed) and may be assumed equal to about 1.2. The absorption of neutrons by
U233 while producing little or no net neutron losses will cause a time loss
in that for each U233 atom destroyed a Thé33 atom is assumed to be produced
and this atom must deczy through Pa233 to U233,

The U23h formed is produced by capture by Th233hand Pa233 and U233, the

U23h‘to U233 ratio being given approximately by the equation

N t ’
“EE = —_— ' fdt N 23)fdt
o fc-c(03)/>\03 1 oy o(0) % / N3O o(13)8dt ¢ [ Np3G(23)

0 0

= fO—e(O3)/AOB ¥ f—%:%](-?-)- [1 - -Xi-’gg (1 - e‘=‘7\13t-{] %G‘c(23)fl" [-]é'- = 7%375

1 _=A13t
+ —T—” " (1-e 3 ):l
=33=-

If now Ug(13) = 0g(23) = G3
N2),
— I £fTL(03)/A,. +1 ot

Thus within the accuracy of the above assumptions the U234 produced by
Th233 capture per 233 produced depends on flux only while the y23k produced
from the other two members of the 233-chain per U233 atom produced depends
directly on the product ft. The ye3L/ye33 at ény given irradiation time is, of
course, directly proportional to the neutron flux. |

Some figures on y233 roduction, U23,4/U233 ratios and losses due to U23h
production are given in Table VI.

The production of Pa23l and 0232 will now be considered. In these calcu-
lations some of the cross-sections are either unknown or poorly known, and hence
the results may be considered as very rough estimates or guesses. Since the
half life of Th@3l is short compared to the probable irradiation tixfie, THh3?
may be considered to be transformed directly to Pa231 by (n,2n) reaction.

dN11

—= = Nop Sh,2n(02)f - N3G (11)E

or Ny _ dn,2n(02) - e=Q‘c(ll)ft)
No, Qe(11)

1.0 x 107U (1 - e Te(11)ft,

The symbol*crh,gn(02) stands fa a number which when multiplied by the thermal
neutron flux gives the specific rate of transformation of Th232 4o Th23l,
Actually this {n,2n) reaction can occur only with neutrons of kinetic energy
greater than the binding energy of one neutron in Th232, and hence the value
of the cross-section can vary greatly at constant thermal neutron flux. In a

given pile assembly, however, the value of (Sa’zn(02) will be constant in time,
“fiafleVI

4233 and U234 Production in U233 Breeder Blanket

Losses Resulting from Neutron Capture by Th?33 and Pa®33

(A1l figures are for flux of 104 and are directly proportional to flux)

 

No),/N°>% Resulting Fr

om

 

 

. 3368t
t(days) N13/N02 N23/N02 Elé_iifial Pa233 Capture 1233 Capture AfTEE:}n§£3 i?ef }g?%)
Nop
1 $.97 x 105  0.08 x 105 6,05 x 10°5 2.1 x 10k 1.82 x 10-6  5.0L x 10k 0.110
3 1.76 x 10-4  0.06 x 104 1.82 x 10k 6,35 x 1078 1.56 x 10-5  9.38 x 104 .203
1Q 5.35 x 104 0.70 x 1074 6,05 x 10-4  1.99 x 1073 1.68 x 1074 2.45 x 1073 .501
30 1.28 x 10-3  0.5L x 10-3  1.82 x 103  5.13 x 1073 1.37 x 10=3  6.79 x 1073 1.19
50 1.72 x 10-3  1.30 x 103 3.02 x 10-3  7.43 x 1073 3,37 x 103 1.11 x 102 1.70
70 2.00 x 103  2.24 x10-3  kL.2L x 103 9.15 x 1073 6.00 x 103 1.53 x 1072 2,07
100 » 21 x 103 3.8Lx 1073 6.05 x 107 1.09 x 1072 1.07 x 102 2.19 x 102 2.L6
150 2.37 x 10-3 671 x 1073 9.08 x 107 1.28 x 1072 1.96 x 1072 3,27 x 102 2.88
200 2.40 x 1073 9.7 x103 12,1 x1073 1.38 x 1072 2,93 x 1072 L.35 x 1072 3.10
300 ol x 103 15.8 x 1023 18,2 x 1073 1.51 x 1072 5,00 x 1072 6.53 x 1072 3.39

*Ratios given are for complele decay of the Pa233°

3t constant value of 2.88 x 10-l4 is included in this column for neutr

##tyeutron plus fuel atom losses due to neutron capture by T

on capture by 233,
n233 and Pa?33.
=35~

at constant power, and will vary somewhat with the space coordinateé; in an external
blanket the value will vary to a greater extent with space coordinates, decreasing
rather rapidly with increasing distance from the source of fast neutrons. The value
of Crh,2n(02) used here is 0,015 barns which is approximately the value for the

238

(n,2n) reaction on U in the Hanford piles. I. Perlman (MB-IP-62L, November 15,
1952) gives a figure of 0,007 barns for the U238 (n,2n) reaction for pile neutrons
and suggests that the Th232 (n,2n) may be a little lower. O the other hand

Perlman suggests a higher wvalue, i.e., 290 rather than 150 bams, for the cross-

section for the subsequent Pa231 (n, ¥) reaction.

 

 

 

For the U232 formed by neutron capture by Pa?3l followed by beta decay of Pa232
dN
_E%g = N11Q(11)f - Noowp(22)f
N -
sz : Un,2n駧)+ G;,zn(ozzr e S o 5n(02) Gp(1) - (222
02 U'a( Gc(ll)" a( ) 0-3.(22)[06(11)' %(22)]

105 X lO"h (1 } 2.0 e-G-c(ll)ft - 3.0 e-Fa(ZZ)ft)
At small wvalues of t

g%: ~ 1.5 x 10~k [(g—g(ll)- %6‘32(22)) £242 (%5-&2(22) - %0_03(11)) f3t3:]
The relative concentrations Nll/NOQ and'N22/N02 at various values of ft are
given in Table VII.

Tt does not seem feasible to estimate with any accuracy the amount of U232
produced by (n,2n) reaction on ye33 or by (n,2n) reaction on Pa233 followed by
beta decay of the Pa232, as thesg (n,2n) cross-sections for pile neutrons are
entirely unknown. If, however, the (n,2n) cross-sections fér Pa233 and 7233

232

are the same as for‘Th232, then the amount of U produced from these two species
~36-

will be about 1/20 that calculated in Table VII as formed from Th232 via pa®3l

up to about £t = 10%2; above this ft value the U232 formed from Pa?33 plus U233

approaches about 3£§ that shown in Table VII.

Table VII

pa23l/pn232 ang y232/7n232 Ratios Formed in Blanket Due to Reactions

ft

1019
3 x 1019

1020

3 x 1020
1021

3 x 1041
1022

3 x 1022
1023

N11/Noo

 

1.50 x 10~7
4.50 x 10-7
1.50 x 106
L.ko x 10~
1.39 x 1075
3.62 x 1075
7.77 x 10~
9.89 x 10=>
1.00 x 1074

1.00 x 10~k

Th232 (n,2n) Th231-1£i;9Pa231(n,6)Pa2327fi5::> y232

N22/N02
1.12 x 10-10
1,01 x 10-9
1.12 x 1078
0.99 x 10~7
1.04 x 1076
7.92 x 10-6
5.1k4 x 1075
1.31 x 107k
1.50 x 107b

1.50 x 10~k
-37-

Appendix A
Relative Losses for Batch and Continuous Processing

It might be argued that with some combination of half lives and cross-sections
of the shorter lived fission products it is conceivable that continuous processing
might afford lower losses than batch processing. It can easily be shown that this
is not possible,

Consider a hypothetical fission product of yleld ¥, capture cross-section
of T3, decay constant A, and concentration Ny; and define A= Ay + fay-

Then in the batch case, at time t after the last processing Ni becomes

Ny =M (1 - e=NY),

and the contribution of this species to the overall loss per fission
1 T y1O1f A
Lo! = frogt f o - [ —= )

O
In the continuous case the steady state concentration Ny and the contribution to

 

the overall losses L, become

NeOpf T . NS, f ) £
=.Z;_£_;£___ and Lcw - —;—_;h:E - Z}EZE_:E
14+ NT | NeOpf T L4AT

Now consider the total losses Ly and L; with the contribution from the

species Nj:

 

 

h 1 1 1 o fTy |, Y1S 1 | }
Iy = ¢ +lyag by |1t (- O )4 1 L (16T
e = Ple 4y oers T 4T

Opf 7 1L+GSET 14 AT
~38-

Comparing these equations term by term at T = optimum processing period for

continuous processing, it is readily seen that the two first terms are equal.

 

Each of the other terms is smaller in the batch case than the corresponding
term in the continuous case; this is obvious for the two second terms. The
third terms are very similar to the fourth ones (i.e. they become exactly
similar in the limiting case where )\i = 0); hence the argument to be made
for fourth terms will also hold for the third ones.

Take the ratio

 

It 1 4+AT
—pi- - l l (l_ e-AT) .
Le! AT NT

This ratio is equal to 1/2 at AT =0 , cor;tinues to increase as /\‘Tincreases
and finally approaches unity as /A7 gets indefinitely large. Hence it is
always lesa than unity for finite AT . Any further breaking up of fission
products into additional groups will give further terms like those already
considered, and therefore would not alter the conclusions drawn here. A
possibility not explicitly covered so far is the production of a highly
absorbing species from a moderat.eiy long-lived fission product of lower cross-
section; such a case would tend to favor batch processing even more than those
considered above.

The argument so far proves that if the optimum ‘T for continuous processing
is employed for both types of processing the batch method af fords lower losses,
Obviously if the batch case were optimized indeiaenden_tly with fespect t:o T it

would afford even lower losses.