CHAPTER 2

NUCLEAR CHARACTERISTICS OF
ONE~ AND TWO-REGION
HOMOGENEOUS REACTORS*

Nuclear characteristics refer to the conditions and material concentra-
tlons under which reactor systems will remain critical, the relative changes
in concentration of materials within the system as a function of reactor
operation, and the time behavior of variables in the reactor system which
occur when deviations from criticality take place. The material concen-
trations are closely connected with fuel costs in power reactors, while
reactor behavior under noneritical conditions is closely related to the safety
and control of the reactor system. These nuclear characteristics are de-
termined from the results obtained from so-called “‘reactor criticality’
and “reactor kinetic”’ calculations. In such studies, certain parameter
values pertaining to nuclei concentrations and reaction probabilities are
used ; for convenience some of these are listed in Section 2-2.

2-1. CriticaLiTY CALCULATIONS

Criticality studies are also termed ‘“‘reactor-statics studies.”” In these
studies the concentration of the various nuclei present can vary with
time, but it is assumed that the condition of criticality will be maintained.

The statics of chain reactions in aqueous-homogeneous reactors are of
interest primarily in connection with the estimation of the inventory of
fuel and fertile material, power density at the wall, the flux distribution
inside the reactor, and the rate of production of fissionable isotopes. These
enter into economic calculations pertaining to fuel costs in power reactors
and also into criteria specifying the design of the system. The most im-
portant factors determining criticality are geometry; nature, concentra-
tion, and enrichment of the fuel; nature and distribution of other com-
ponents in the reactor; and operating temperature and pressure. The
production of fissionable isotopes depends primarily on the neutron economy
of the system and will be a function of the relative competition for neu-
trons between the fertile material and the various other absorbers. The
latter include materials of construction, moderator, fuel components,
fission products, and various nonfissionable nuclei formed by parasitic
neutron capture in the fuel. In designing a reactor for the production of

*By P. R. Kasten, Oak Ridge National Laboratory.
29
30 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

fissionable isotopes, it is therefore important to choose materials (other
than fuel and fertile material) which have low neutron-capture cross
sections. This, in turn, leads to the selection of D20 as the moderator in
nearly all cases.

Although criticality i1s assumed at all times, the concentration of fuel
isotopes can change appreciably with time owing to the relative competi-
tion between isotope formation, decay, and neutron-absorption processes.
In many studies it has been assumed that the reactor has operated for
such times that the steady-state conditions apply with regard to the nuclei
concentrations. This simplifies the isotope equations but may not always
give an adequate picture of the actual concentrations which would be
present in an operating reactor.

2-1.1 Calculation methods. Since both light and heavy water are ex-
cellent moderators, the energy of fission neutrons is rapidly degraded, with
the result that most of the fissions are produced by thermal neutrons.
Under these conditions either the modified one-group or the two-group
diffusion equations are usually applicable for criticality calculations. For
simplicity, this discussion will be limited to spherical reactors.

I'or a bare, spherical reactor the criticality condition (assuming Fermi
age theory) is given by

2} prne” Bren o Zy(')

I=viser ooz ), B

 

[p(u)e~ B"’T(“')]du']’ (2-1)

where B2 = (w/R)?,

Dy, = thermal diffusion coefficient,

D(u) = diffusion coefficient as a function of lethargy,

P = resonance escape probability to thermal energy,

p(u) = resonance escape probability to lethargy wu,

R =radius of reactor plus extrapolation distance,

u = lethargy of neutrons,

w, = u evaluated at thermal energy,

v = neutrons emitted per fission,

£ = average lethargy increment per neutron collision,

2 = macroscopic cross section; superseript th refers to thermal

value; subseripts f, a, and ¢ refer to fission, absorption,
and total cross sections, respectively; Z(u) refers to Z
evaluated as a function of lethargy,

T+ = Fermi age to thermal energy,

“D(') du’_
EZ(u')

7(u) = Fermi age to lethargy u=
0
2-1] CRITICALITY CALCULATIONS 31

By introducing €, the “fast fission factor,” where

 

p(u e~ BTy

E}hpthGfiBzm‘ futh Zi(u")
0

 

c— total fissions  Z% 4+ Dy, B2 2 (u')
~ thermal fissions =P e Borth ’
2+ Dy B?
(2-2)
Eq. (2-1) becomes Joe— BoTth
'=1Trp (2-3)
where &k = n€pwmfin = infinite multiplication constant,
n = neutrons emitted per neutron absorption in fuel,

fw = fraction of thermal neutrons absorbed in fuel,

_ Dy thermal diffusion coefficient
2% T macroscopic thermal absorption cross section

 

2
Lig,

Replacing the exponential term in Eq. (2-3) by (1/1+ B?ry), the two-
group criticality condition is obtained as

k= (14 B2ra)(1 -+ B2L2). (2-4)

Although Egs. (2-1), (2-3), and (2-4) imply that resonance fissions in
the fuel are considered, in usual practice Eqgs. (2-3) and (2-4) are used on
the basis that (€p)se is equal to unity. Using this assumption, the values
of € and p to be used in evaluating & are those associated with the fertile
material. In the subsequent results and discussion, calculations based on
Egs. (2-3) and (2-4) consider that (ep)sue is equal to unity, while calcula-
tions based on Eq. (2-1) explicitly consider resonance absorptions and
fissions in fuel based on a 1/F resonance-energy flux distribution.

The breeding ratio (BR) is defined as the number of fuel atoms formed
per fuel atom destroyed for the reactor system. For a bare reactor, assum-
ing that the resonance flux is independent of lethargy and that absorptions
in fertile material produce new fuel, the BR 1s given by

i [1 —v [ R du] oz [ Zuote 6~ 77
ertile
0 0

 

BR= £t £
- “n 20" BT p du o oo (fuel) pe B du’
Tk (fuel [1—1}] th ~L———:| VE“’I el =
MueDlI=v), g, [T, £,
(2-5)
where 22 .. = thermal absorption cross section of fertile material,
Zieile = absorption cross section of fertile material at lethargy u,

2.(fuel) = fuel absorption cross section at lethargy u.
32 ONE- AND TWO-REGION HOMOGENEOUS REACTORS fcoap. 2

If resonance absorptions in fuel are neglected, the conventional two-group
formula is obtained as

th
_ Ziertile (1 — Prertite)

7
BR=Su tuel) T 1 ¥ B2rg

 

(2-6)

In the subsequent discussion, reference to one-region, two-group calcula-
tions implies use of Eqs. (2-4) and (2-6) to calculate critical mass and
breeding ratio, respectively.

The two-group diffusion equations were used for two-region reactor
calculations. The effect of a thin shell between the two regions upon
reactor criticality and breeding ratio was taken into account by considering
the shell absorptions by means of a boundary condition; the effect of a
pressure-vessel wall was taken into account by using an “‘effective’” extra-
polation distance in specifying the radius at which the flux was assumed
to be zero. The two-group equations were written as:

Dy Vs — Zyetpse + g—j 2sotpse =0, (2-7)
DoeV2se — ZscPse + PeZrePre = 0, (2-8)
DpVPém — Zpdn + %: 2 =0, (2-9)
DaVida — Zads + pe2pdm = 0. (2-10)

The subscripts f, s, ¢, and b refer, respectively, to the fast flux, slow flux,
core region, and blanket region; D 1s the diffusion coefficient; ¢ is the
neutron flux; 2, refers to the effective cross section for removing neutrons
from the fast group; and Z, refers to the thermal absorption cross section.
Other symbols have the same meaning given previously, with %k calculated
on the basis that (€p)sea = 1.

The boundary conditions used assume that the fast flux has the same
value on the core side of the core-tank wall as on the blanket side; the
same 18 also true of the slow flux. It is also specified that the fast flux and
slow flux become zero at some extrapolated reactor radius. At the core-
tank wall, the net fast-neutron current on the blanket side is assumed equal
to that on the core side, while the net slow-neutron current on the core
side is assumed equal to the flux rate of neutron absorptions in the core
tank plus the net slow-neutron current on the blanket side.

A multigroup formulation can be obtained by adding neutron groups
with energies intermediate between the fast and slow groups specified in
Eqgs. (2-7) through (2-10). These intermediate groups would be essentially
of the form

Dwip,— Zipi+ pic12Zi1di-1=0, (2-11)

where ¢ represents the ¢th group of neutrons, and 4 increases with decreas-
2-1] CRITICALITY CALCULATIONS 33

ing neutron energy. Boundary conditions analogous to those specified
above would apply. Such a formulation assumes that neutrons in the sth
group are always slowed down into the ¢+ 1 group, corresponding to a
relatively small number of fast groups. Using about 30 groups or more,
multigroup methods [1] used in homogeneous reactor calculations consider
that fast neutrons are born in the various fast groups in accordance with
the fraction of fission neutrons generated in the particular group; that for
all materials but hydrogen, neutrons are “‘slowed down’ from one energy
group to the group immediately below; and that with hydrogen, the pos-
sibility exists for a neutron to pass from one group to any group below as a
result of one scattering collision. Other multigroup methods [2] of calcu-
lation have been devised which consider that a scattering collision degrades
a neutron into the groups below in accordance with the probability for
degradation into a particular group.

2-1.2 Results obtained for one-region reactors. The majority of critical
caleulations for large one-region reactors have been based on IEqgs. (2-3)
and (2-4), in which all the fissions are effectively assumed to take place in
the thermal group [3]. However, if the effective value of n(fuel) for the
resonance region is less than the value for the thermal region, the above
models may not be adequate. Some multigroup calculations [4] have
been done for uranium-water systems at an average temperature of 260°C;
in I'ig. 2-1 are shown critical-mass requirements for light water-uranium sys-

1000

 

500 |-

200
100

50 |

U235, kg

 

 

 

 

, L L L] |
10 20 50 100 200

Radius of Bare Sphere, cm

Fie. 2-1. U285 mass and critical size of uranium light-water mixtures at 260°C.
Assumed densities: U235 = 18.5 g/cm?®, U238 = 18.9 g/cm?, H20 = 0.8 g/cm?.
34 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

 

          
  
 

 

 

 

 

3 T
[ 17 i [ T T T 7177 l [T T TTI
2 — _
MolesHZO
(o) O
Moles (H2O + DZO)
100 --(1/40)
ol
8 =
7 L2
61— 5
51 B
=
41— o
o
> 3 °
x 5 g
m‘ ‘é O?
2 2= %’, |
> =
O
o™~
T
10 — @ ]
9 |- 5 —
8 = —
7 _
6 — ]
5 +— ]
4 — —
3 — |
2 — _
Tl 3 L L | [
78910 2 3 4 5 6789100 2 3 4 5 46 78¢9

Radius of Bare Sphere, cm

Fig. 2-2. U235 mass and critical size of bare spherical reactors moderated by
H0—D30 mixtures at 260°C. Assumed densities: U235 =18.5g/cm3, H,0 = 0.8
g/em?3, D20 = 0.89 g/cm?,

tems, while Fig. 2-2 gives the ecritical-mass requirements for U235-D>0-
H:0 systems, Initial conversion ratios for light-water systems are given
in Ref. [4].

Nuclear calculations have also been performed [5] using Egs. (2-1)
and (2-5), with the 1/E component of the flux starting at energies of 6 £7.
The calculations were for single-region reactors containing only ThOg,
U23303, and D20 at 300°C with the value of 9?3 in the resonance region
considered to be a parameter. Critical concentrations thus caleulated are
given in Fig. 2-3 for zero neutron leakage. The value for 23 in the thermal
energy region was taken as 2.25. The value for 9?3 in the resonance region
has not been firmly established; based on available data, 723,92 lies
between 0.9 and 1.

When the neutron leakage is not negligible and 723 /723 is less than 1, a
finite thorium concentration exists for which the breeding ratio is a maxi-
mum. This is indicated in Fig. 24, in which the initial breeding ratio is
2-1] CRITICALITY CALCULATIONS 39

20

 

Concentration of U233, g/liter
tn
I

 

 

 

| | | | | | } \
] . L L L L
0 200 400 600 800 1000

 

Concentration of Thorium, g/liter

Fiq. 2-3. Fuel concentration as a function of thorium concentration and value
of M,es/Men fOr an infinite reactor.

 

 

 

 

 

1.3 T T
1.2 —
1.1 —
.0
T 1.0|—
o=
o
&
T 09
o
o
—6 ’ e
£ 08— — —— = ———=— Reactor Diameter Infinite —]
= ~——o——  Reactor Diameter = 14
o7l  TTmmmmssss= Reactor Diameter = 10 ft ]
0.6 l | | |
0 200 400 600 800 1000

Thorium Concentration, g/liter

23

Fia. 2-4. Breeding ratio as a function of thorium concentration and nffs/'r] th 10

a one-region reactor. Reactor temperature = 300°C, "qzt‘?, = 2.25.
36 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

given as a function of thorium concentration, reactor diameter, and rela-
tive value of #23. The breeding ratio goes through a maximum owing to
the increase in resonance absorption in fuel as the thorium concentration
1s increased.

If the above reactors were fueled initially with U233, the initial breeding
ratio would have a maximum value of 1.08 rather than 1.25; however, the
curves would have about the same shape as those presented in Fig. 24,
and the value for 72 /92 would be between 0.8 and 0.9.

Comparison of the above results with those obtained using a two-group
model shows that if n.,/7 is equal to 1, the breeding ratio obtained by
the two methods is about the same; however, the critical concentration is
about 309, higher when the two-group model is used. If %% /7% < 1, the
value for the breeding ratio obtained using the two-group model will tend
to be higher than the actual value; however, if the fertile-material con-
centration is low (about 200 g/liter or less) and the reactor size large, little
resonance absorption occurs in fuel. Under these conditions the two-group
model should be adequate for obtaining the breeding ratio and conservative
with respect to estimating the critical concentration.

2-1.3 Results obtained for two-region reactors. Most two-region re-
actors have been calculated on the basis of the two-group model. Results
have also been obtained using multigroup calculations which indicate that
the two-group method is valid so long as the value of n(fuel) is independent
(or nearly so) of energy, or so long as nearly all the fissions are due to ab-
sorption of thermal neutrons.

To compare results obtained by different calculational methods, breed-
ing ratios and critical fuel concentrations were obtained [6] for some two-
region, Ds0-moderated thortum-blanket breeder reactors using a multi-
group, multiregion Univac program (“Eyewash’) [1} and a two-group,
two-region Oracle program [7]. In these calculations operation at 280°C
was assumed ; a 4-in.-thick Zircaloy—2 core tank separated the core from the
blanket; a 6-in.-thick iron pressure vessel contained the reactor; and ab-
sorptions occurred only in U233 and thorium in the core and in thorium in
the blanket. Twenty-seven fast groups, one thermal group, and four
regions (core, Zircaloy—2 core tank, blanket and pressure vessel) were
employed in the multigroup model. The two-group parameters were com-
puted from the multigroup cross sections by numerical integration [8].
In the two-group, two-region calculations a ‘‘thin-shell” approximation [9]
was used to estimate core-tank absorptions, while the effect of the pressure
vessel was simulated by adding an extrapolation distance to the blanket
thickness.

In the multigroup studies, various values for %23 in the resonance region
were considered. In one case the value of 25 was assumed to be constant
2-1] CRITICALITY CALCULATIONS 37

and equal to the thermal value.* In another the variation of 723 in the
resonance region was based on cross sections used by Roberts and Alex-
ander [10], which resulted in an 7% /52 of about 0.95; in the third case the
effective value for 7?3 in the resonance region was assumed to be essen-
tially 0.8 of the thermal value of 2.30.

The initial breeding ratios and U233 critical concentrations obtained from
the Eyewash and two-group, two-region calculations are given in Fig. 2-5
for slurry-core reactors (zero core thorium concentration also corresponds
to a solution-core reactor). With solution-core reactors the effect of the
value of 723 upon breeding ratio was less pronounced than for slurry-core
reactors, since fewer resonance absorptions take place with the lower fuel
concentrations. The blanket thorium concentration had little influence
upon the above effect for blanket thorium concentrations greater than
250 g/liter.

As indicated in Fig. 2-5, the breeding ratio is rather dependent upon
the value of #23; the loss in breeding ratio due to a reduced value of %3
in the resonance region is most marked for the slurry-core systems. Ior
these reactors, relatively more fissions take place in the resonance-energy
region as the core loading is increased, owing to the “hardening” of the
neutron spectrum. Figure 2-5 also shows that the two-group model gives
breeding ratios which are in good agreement with those obtained from
the multigroup model, so long as 723 does not deviate significantly from
723, Reported measurements [11-13] of #*? as a function of energy ndi-
cate that for the reactors considered here, the value of 5% /7% would be
about 0.95; the results given by curve “‘a’” in Fig. 2-5 are based effectively
on such a value of % /7% and indicate that two-group results are valid.

In general, for the cases studied it was found that for a heavily loaded
blanket (or core), the two-group values of total neutron leakage were
larger than the total leakages obtained from the multigroup calculation
(the multigroup model allowed for competition between fast absorptions
in fuel and fast leakage, while the two-group model assumed that fast
leakage occurred before any resonance absorption occurred). The multi-
group results were also used to calculate the fast effect, €, previously
defined in Eq. (2-2). It was found that resonance fissions accounted for
109, to 409, of the total fissions in those reactors containing from 0 to
300 g Th/liter in the core region. With no thorium in the core region,
changing from a 4-9 reactor (4-ft-diameter core and a 9-ft-diameter pres-
sure vessel) to a 6-10 reactor decreased core resonance fissions from about
149, to 109%,.

If the reactor core size is small, the two-group method does not ade-
quately treat leakage of fast neutrons; for this case two-group results may

*The thermal value for 723 was assumed to be equal to 2.30 instead of 2.25 used
in more recent calculations (the 2.25 value is believed to be more accurate).
38 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHaPp, 2
1.30 .

 

Two-Group, Two-Region

Breeding Ratio

 

1.10

 

 

o. Estimated from Averaged Cross Sections
“Eyewash” used by Roberts and Alexander (11}
| b, Estimated for 123 independent of Energy

     
      

 

 

 

E 10 +—
g Two-Group, Two-Region
™)
m .
o~
=2 , 23
° “Eyewash’” _res
o 423
- th 7/ eff
c
L 5k —
B
T
@
W
&
o
o
T
]
G
x J
0 100 200 300

Core Thorium Concentration, g/liter

Fig. 2-5. Breeding gain and critical concentration for slurry-core reactors vs.
core thorium concentration. Core diameter = 6.0 ft, pressure vessel diameter = 10.0
ft, blanket thorium concentration = 1000 g/liter, blanket U233 concentration = 3.0
g/kg of Th.

not be adequate even though 7, is equal to n,,. This is indicated by the
experimental [ 14] and caleulated results for the HRE-2 given in Fig. 7-15.
Ag illustrated, there is excellent agreement between the experimental
data and the data calculated by a multigroup method and by u “har-
monles” method, but not with the results from the two-group model.
The harmonics caleulation [15] referred to in Fig. 7-15 does not take into
account fast fissions but does treat the slowing-down of neutrons in a more
realistic manner than does the fwo-group caleulation. The multigroup
result [16] indicated that about 139, of the fissions were due to neutrons
having energies above thermal.

A comparison [13] of the U235 critical concentrations predicted by the
2-2] NUCLEAR CONSTANTS USED IN CRITICALITY CALCULATIONS 39

 

     
   
    

 

 

 

700 1 T T T T T T T T 1
800 |— —
700 (— —
. L |
3
s 600 |— Age
o L |
a8 Age-Yukawa
@“ 500 |— —
= Yukawa
@
2 —_
2 L
¥
400 — _
300 |- —
200 |— —
100 E | | | | | | | | L | |
50 60 70 80 90 100 110 120

Outer Radius of Reflector, cm

Fic. 2-6. Comparison of critical concentrations obtained for various slowing-
down kernels in D20. Core radius = 39 cm, total age =237 cm? for all kernels,
diffusion length of pure moderator, L¢? = 40,200 cm?. Fuel only in core.

use of different slowing-down kernels in D2O-moderated reactors is shown
in Fig. 2-6. In the age-Yukawa kernel [given by (e ™/*"1/(4mwr)3/2) X
(e”/\/g,ffllwrzr)], the “age’” parameters were taken to be 158 em? for 71
and 79 cm? for 72. For both the age kernel (given by e~ /47/(47w7)3/2)
and the Yukawa kernel (given by e~/ ‘/;/ 477 r), T was taken to be 237 cm?2.
The results show that in small reactors the calculated critical concentration
obtained using the two-group method (Yukawa kernel) is substantially
lower than that obtained using either an age-Yukawa or an age kernel to
represent the neutron distribution during the slowing-down process. Since
the age-Yukawa kernel is believed to be the proper one to use for D0,
and the HRT is a “small” reactor (in a nuclear sense), it is not surprising
that the two-group results given in Fig. 7-15 are appreciably different
from the experimental results.

2-2. NucLEAR ConsTanTs UskED IN CrRITICALITY CALCULATIONS

In obtaining the nuclear characteristics of reactors, it is necessary to
know the probabilities with which different events occur. These reaction
40 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHaP. 2

probabilities are usually given on an atomic basis in terms of cross sec-
tions [17]. Because of their diverse applications, it is necessary to present
reaction probabilities in this manner; however, in calculating the nuclear
characteristics of homogeneous reactors, it is convenient to combine funda-
mental data concerning atomic density and reaction probabilities so as to
facilitate critical calculations. This has been done to a limited extent in
this section. Listed here are some nuclear data and physical properties of
uranium isotopes, uranyl sulfate, heavy water, thorium oxide, and Zirca-
loy-2 used in two-group calculations for thorium breeder reactors [18].

TABLE 2-1

TaermalL Microscoric ABsorprTioN CROSS
SECTIONS AT VARIOUS TEMPERATURES

(Corrected for Maxwell-Boltzmann distribution and also non-(1/v) correction)

 

 

 

 

 

 

 

 

Element Ne‘;ggg’gi‘;zm 1 20°C 100°C 280°C
g, barns [17]
{233 588 526 460 376
234 92 82 72 59
235 689 595 515 411
236 6 5.3 4.7 3.9
17238 2.93 2.42 2.15 1.76
Pa233* [19] 60 130 130 130
Th232 7.45 6.60 5.85 4 81
Py 1025 975 905 950
Py 2a0* 250 600 700. 1000
Pu24l 1399 1240 1118 952
S 0.49 0.43 0.39 0.32
1i7 (99.989%) [20] 0.23 0.20 0.18 0.15
gy, barns [17]

U233 532 472 412 337
[z23s 582 506 438 350
Py239 748 711 660 693
Py24t 970 860 776 660

 

 

 

 

 

 

 

*Estimates of the effective cross section in typical homogeneous-reactor neutron
spectrums (except for 2200 m/sec value); these values include contributions due to
resonance absorptions. (Although these values were not used in the calculations
presented, they are believed to be more accurate than the ones employed. Values
used for Pa233 were 133, 118, and 97 barns at 20, 100, and 280°C, respectively.)
2-2] NUCLEAR CONSTANTS USLD IN CRITICALITY CALCULATIONS 41

2-2.1 Nuclear data. Table 2—1 lists thermal microscopic absorption and
fission cross sections for various elements and for various temperatures.
Table 2-2 lists thermal macroscopic absorption cross sections for HzO,
D20, and Zircaloy—2, and the density of H>0 and D20 at the various

TaABLE 2-2

TnerMar Macroscoric ABSORPTION (CROSS SECTIONS
AND DExNsITiES AT VARIOUsS TEMPERATURES [ 18]

(Corrected for Maxwell-Boltzmann distribution on basis of 1/2 cross section)

 

 

 

 

Ilement 20°C 100°C 280°C
Z(H20) 0.0196 0.0167 0.0107
2,(99.759% D20) 8.02x 1075 (G.85x 1075 4.44 x 1075
Z  (Zirealoy-2) 0.00674 0.00598 (0.00491
p(D:0) 1.105 1.062 0.828
p(H20) 1.000 0.962 0.749

 

 

 

 

 

 

temperatures. All cross sections listed under the eolumns headed by °C
have been corrected for a Maxwell-Boltzmann flux distribution.

Values of 7 and v for the various fuels, and the fast and slow diffusion
coeflicients for Zircaloy are given in Table 2-3.

TaBLE 2-3

SoME NUCLEAR CONSTANTS FOR
UraNIUM, PLUTONIUM, AND ZIRCALOY 2

Values of 7 and » for U and Pu [21]

 

 

 

Itlement ’ N 1 v
233 2.25 2.50
U235 2.08 2.46
Pu?3? 1.93 3.08
Pu??! 2.23 3.21

 

Diffusion coefficients for Zircaloy—2: [22]
Dy = Do=0.98 for all temperatures,
where Dy = fast diffusion coefhicient,
Do = slow diffusion coeflicient.

 

 

 
42 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHap. 2

Data for two-group calculations are summarized in Table 2—1 for 7, Dy,
Do, and p as functions of fertile-material concentration in mixtures of

TABLE 2-4

Two-Grour NucLeEar ConNsTaNTs® Fonr 1),0-MoODERATED
SysTEMs AT 280° C [18]

 

 

 

 

I'ertile-material concentration, T, Dy, D2,
g /liter em? ¢m em P
Th (in ThQz-D20)
() 212 1.64 1.24 1.000
100 212 1.62 1.23 0509
250 213 1.60 122 | 0.825
500 215 1.56 1.20 0.718
1000 215 1.50 116 0.554
U2 (in U02804-1:20) |
0 212 1.64 1.24 1.000
100 200 1.57 1.20 0.875
250 189 1.49 1.15 (0. 801
n00 179 1.40 1.10 0.720
1000 173 1.28 1.04 0.595
U238 (in U02504-LioS04-D20)
0 212 1.64 1.24 1.000
100 198 1.55 1.19 0.873
250 185 1.45 113 0.797
500 173 1.33 1.07 0.705
1000 165 1.18 0.99 0.525

 

 

 

 

 

 

 

*r = Fermi age; Dy = fast diffusion coefficient; D = slow diffusion coeflicient;
p = resonance escape probability.
TLi2S0 4 molar concentration equal to U080 molar concentration.

fertile material and heavy water (99.755; D20) at 280°C. Materials con-
sidered are ThOs-12.0, TO80,-D20, and U080 4-LieS0,-1:0 where
the molar concentration of LieSOy is the same as the UOoSO 4 molar con-
centration. Reference [18] gives corresponding data at other temperatures
and also gives some values for the case of 1120 ax the moderator,

The diffusion coefficients and ages were calculated by a numerieal inte-
aration procedure [8]. The fast diffusion constant, Dy, and the IFermi
age, 7, are based on a 1/ flux distribution, and the slow diffusion con-
stant, Do, 1s based on a Maxwellian flux distribution.

<1
2-3] STEADY-STATE FUEL CONCENTRATIONS 43

2-2.2 Resonance integrals. Formulas used in calculating resonance in-
tegrals (RI) are given below.

 

 

 

 

For U238;
3, \0-471 s, \

In RI = 5.64 — (_2'/11\?2% 7578 > 1% 107, (2-13)
RI{> )= 280 barns. (2-14)

For Th232:;

3, \0-253 >, _ L

RI=8.33 ( NO2> ;0= = 4500, (2-15)
RI = 70 barns, J\?OSQ > 4500, (2-16)
RI(>}= 70 barns. (2-17)

2-3. I'veEL CONCENTRATIONS AND BREEDING RATios UNDER INITIAL
AND STEADY-STATE CONDITIONS

The relationships between breeding ratio and reactor-system inventory
determine the fuel costs in homogeneous reactors. The breeding ratio
depends on neutron leakage as well as relative absorptions in fuel fertile
material and other materials present, while material inventory is a funetion
of reactor size and fuel and fertile-material concentrations; thus a range
of parameter values must be considered to aid in understanding the above
relationships. DBased on results given in Section 1-1.3, it appears that the
two-group method gives satisfactory results for critical concentration and
breeding ratio for most of the aqueous-homogeneous systems of interest.
This permits survey-type calculations to be performed in a relatively short
time interval., The results given below are based on the conventional two-
group model.

In steady-state operation, the concentration of the various nuclides
within the reactor system does not change with time. During the initial
period of reactor operation this situation is not true, but 18 approached
after some time interval if ncutron poisons are removed by fuel processing.
Under steady-state operation 1t is necessary to consider the equilibrium
isotope relationships. In thorium breeder reactors this involves rate ma-
terial balances on Th, Pa?33 U233 234 1235 1236 fission-product polsons,
and corrosion products. (The uranium isotope chain is normally cut off at
U236 gince this Is a low-cross-section isotope, and neutrons lost to the suc-
cessors of U?36 would tend to be compensated for by fission neutrons gen-
44 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

erated by some succeeding members of the chain.) In uranium-plutonium
reactors, steady-state rate material balances were made on U235 17236
U238 Pu23® Pu?4 and Pu®*'; all other higher isotopes were assumed either
to be removed 1n the fuel-processing step or to have a negligible effect
upon the nuelear characteristics of the reactor.

Although equilibrium results give the isotope ratios which would be
approached in a reactor system, much of the desired nuclear information
can be obtained by considering “‘clean” reactors, i.e., reactors in which
zero poisons exist, corresponding to initial conditions, or to criticality con-
ditions at reactor startup. This is a result of the rather simple relationships
which exist between breeding ratio, critical concentration, and fraction
poisons, and the ability to represent the higher isotopes by their fraction-
polson equivalent.

2-3.1 Two-region reactors. In order to estimate the minimum fuel costs
in a two-region thorium breeder reactor, it i1s important to determine the
relation between breeding ratio and the concentrations of fuel (U233) and
fertile material (Th?32) in the core and blanket. Similar considerations
apply to uranium-phitonium converter reactors. The breeding or con-
version ratio will depend on neutron leakage as well as relative neutron
absorptions in fuel, fertile material, and the core-tank wall; therefore a
range of core and pressure-vessel sizes must be considered.

Since fabrication problems and the associated cost of pressure vessels
capable of operating at 2000 psi increase rapidly for diameters above 12 ft,
and since the effect of larger diameters on the nuclear characteristics of the
two-region reactors is relatively small, 12 ft has been taken as the limiting
diameter value. Actually, in most of the calculations discussed here, the
inside diameter of the pressure vessel has been held at 10 ft and the core
diameter allowed to vary over the range of 3 to 9 ft.

In addition to the limitation on the maximum diameter of the pressure
vessel, there will also be a limitation, for a given total power output, on
the minimum diameter of the core vessel. This minimum diameter is de-
termined by the power density at the core wall, since high power densities
at the wall will lead to intolerable corrosion of the wall material (Zircaloy—2).
In order to take this factor into consideration, the power densities, as well
as critical concentrations and breeding ratios, were calculated for the
various reactors.

2-3.2 Two-region thorium breeder reactors evaluated under initial con-
ditions. The results given here are for reactors at startup; although the
trends indicated apply to reactors in steady-state operation, the values
given here for the breeding ratio and fuel concentration would be some-
what different than those for steady-state conditions.
2-3] STEADY-STATE FUEL CONCENTRATIONS 45

Calculations of breeding ratio, the power density at the inside core
wall, and the maximum power density were carried out for some spherical
reactors with 200 g Th/liter in the core. The blanket materials considered
were heavy water (99.759%; D20), beryllium, and ThOs-heavy water sus-
pensions. The inside diameter of the pressure vessel was fixed at 10 ft for
one set of calculations and at 12 ft for a second set; core diameters ranged
from 6 to 9 ft in the first set and from 6 to 11 ft in the second set. The
average temperature of all systems was taken as 280°C, and for the purpose
of caleulating power densities at the core wall, the total thermal power
was taken as 100 Mw. A i-in-thick Zircaloy—2 core tank was assumed to
separate the core and blanket in all reactors, and the value of %23 was
taken as 2.32. (A more accurate value of 923 is presently considered to be
n=2.25.) No account was taken of fission-product-poison buildup, pro-
tactinium losses, or fuel buildup in the blanket. The results obtained [23]
indicate that the breeding ratio increases for any core diameter by re-
placing either a D20 or Be blanket with one containing ThOg; no sig-
nificant increase in breeding ratio is obtained by increasing the blanket
thorium concentration above 2 kg Th/liter; for reactors with fertile ma-
terial in the blanket, the breeding ratio and wall power density increase
with decreasing core diameter.

 

1.3 ' T T
Thorium in Blanket, g/liter ‘

° ., 2000 6 ft Dic. Core
— - |
= ol
2 Dia® Coare
o
£ Dia' Core
<
3
@
@

 

 

 

 

 

 

 

 

30N [ ’ 500
= b ft Diat
- \\\ ~
10 P \\\\"‘--///
G \\\\ == 500 |
~

él" \\\\\ \\___'_’_/__Q_(&_]fl Dia. Core
- ™ N NN !
s 5|\ O~z
3 ~ 2ooo<
3.. ~— 8 ft Did Core
=
z ‘ ‘ L

0 100 200 300

Thorium in Core, g/liter

Fig. 2-7. Effect of core thorium concentration on breeding ratio and wall power
density of two-region slurry reactors. n=2.25, total reactor power =100 Mw
(heat), pressure vessel = 10 ft 1D, U233 in blanket = 3 g/liter, poison fraction = 0,
temperature = 280°C.
46 ONE- AND TWO-REGION HOMOGENEQOUS REACTORS [cHAP, 2

Additional results [24] obtained on the same bases given above except
that 723 was considered to be 2.25, and that the U233 concentration in the
blanket region was varied, are given i Iigs. 2-7 through 2-9. Results
were obtained for 10-ft-diameter pressure vessels and for core diameters
of 6, 7, and 8 ft; the blanket thorium concentration was 500, 1000, or
2000 g/liter, while the core thorium concentration was 0, 100, 200, or
300 g/liter. Generally, the results in Figs. 2-7 through 2-9 are complete
for the 7-ft-diameter-core reactors, while for the 6- and 8-ft-diameter-core
reactors results are shown only for the parameter-value extremes., The
variation of results with parameter value is practically the same for all
three core diameters, and so all the essential results are presented in the
figures.

In all cases slurries of D2O-ThO.-U23303 are assumed in both the core
and blanket regions, and all power densities are based on the assumption
that the total reactor power 1s 100 thermal Mw. Whenever the power
density on the blanket side of the core-tank wall was greater than that on
the core side (owing to a fuel concentration which was higher in the blanket
than in the core), the greater value was plotted. This situation is indicated
by the dashed lines in Fig. 2-7.

Typical information obtained for these shlurry reactors is given in
Table 2-5 for two of the cases considered. The values of breeding ratio for

TABLE 2-5

SLURRY-REACTOR CHARACTERISTICS

 

Pressure-vessel inside diameter, ft 10 10
Core inside diameter, ft 7 7
Core thorium concentration, g/liter 100 200
Blanket thorium concentration, g/liter 1000 2000
Blanket U233 concentration, g/liter 3 5
Total power, Mw 100 100
Blanket power, Mw 8.5 4.2
Critical core concentration, g U233 /liter 3.0 6.0
Breeding ratio 1.18 1.21
Power density at core center, kw/liter 43 45
Power density at core wall, kw/liter 5.4 49
Flux at core center X 10714 5.5 2.9
Neutron absorptions in core wall, % 0.7 0.3

 

 

 

 

 

the two-region reactors given may be compared with those in Table 2-6
for a one-region reactor having the same size and same diameter pressure
vessel. These results indicate that for a given size, two-region reactors
have significantly higher breeding ratios than do one-region reactors.
2-3] STEADY-STATE FUEL CONCENTRATIONS 47

TABLE 2-6

BreEEDING RaATIO IN 10-FT-DIAMETER
One-REGcion REAcrors (Zrro PoisoNs)

 

 

 

Thorium conecentration, . }
. Breeding ratio
g/liter
100 0.875
200 0.993
300 1.037

 

 

 

For all reactors having thorium concentrations of at least 100 g/liter
in the core and 500 g/liter in the blanket, the neutron absorption by the
core-tank wall was less than 19, of the total absorptions. The variations
of breeding ratio and wall power density with core size, thorium concentra-
tion, and blanket U233 concentration are plotted in Figs. 2-7 and 2-8 for
reactors of 10-ft over-all diameter. The variations of eritical concentration,
fraction of total power generated in the blanket region, and the ratio of
fuel to thorium required in the core for criticality are given in Fig. 2-9 for
different thorium econcentrations and blanket fuel concentrations. The
curves for the ratio of U233/Th versus core thorium concentration are of
value in determining the reactivity which would occur if there were a rapid
change in core thorium concentration. If the reactor operating conditions

 

 

 

 

 

 

o 1.3 ] \ ‘ ! Th in Core,
< Thorium in Blanket, g/liter 5404 g/liter
o 2N 300
£ 1.2 = |
'§ 1000 200
& 500 )
1.1 100
|
1.0 |
,-;: 10
& 300
e 5 |
5= 200
S
S 100
o
3
2

 

0 1 2 3 4 5
U233 jn Blanket, g/liter

Fic. 2-8. Effect of blanket U233 concentration on breeding ratio and wall power
density of two-region slurry reactors. 5= 2.25, total reactor power = 100 Mw
(heat), pressure vessel=10 ft ID, core diameter=7 ft, poison fraction =70,
temperature = 280°C.
48 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

were such that the flat region of the appropriate curve applied, small
uniform changes in core thorium concentration would have a negligible
effect upon reactor eriticality. The location of the minimum in these
curves did not vary appreciably with changes in blanket thickness, blanket
thorium concentration, and blunket fuel concentration. The results given
in Iigs. 2-7 through 2-9 indicate that for large spherical reactors the
breeding ratio increases when the core size decreases, when the blanket 17233
concentration decreases, and when the thorium concentration is inereased
in cither the core or blanket region. The core-wall power densily decreases
when the thorium concentration is increased in the blanket region and
when the core size increases, but is relatively insensitive to changes in the
blanket fuel concentration for thorium concentrations greater than 100 and
500 g/liter in the core and blanket, respectively. The eritical concentration
of U?33 in the core decreases with decreusing core thorium concentration
and with increasing core diameter, and varies only slightly with changes
in blanket thickness, blanket U233 concentration, and blanket thorium

 

 

 

 

 

 

 
  

i5 r
? [ [ f [ \ ?
= Therium in Bianket, g/liter 8 ft Core Dia.
S ';': 2000 17 ft Core Dia.—
Y 6 ft Core Dia
oo™ .
53
5 .
o
|
0.044
| | | | E \ !
6 ft Core Dia.
@
o 036 — \_/ B
: \ 7 ft Core Dia.
T 00 -
&= 10 \_’,/ 8 ft Core Dia.
o |- 028 — ( _/ —
=
0.4
W ] l ! | I
031 —
g -
HE
o[22 0.2)~ ]
xflv‘ _—
c .g 500
‘_g - 500 6 ft Core Dia.
0.1 ) ]
1000 ' 7 ft Core Dia.
2000 i
8 {t Core Dig.

 

 

 

 

0 100 200 300

Therium in Core, g/liter

F1e. 2-9. Effect of core thorium concentration on U233 critical concentration,
ratio of U233 to Th required for criticality, and fraction of total power generated in
blanket for some two-region slurry reactors. 72* = 2.25, pressure vessel = 10 ft
ID, poison fraction =0, U233 in blanket = 3 g/liter, temperature = 280° C,
2-3] STEADY-STATE FUEL CONCENTRATIONS 49

concentration. The fraction of tolal power generated in the blanket increases
nearly linearly with increasing blanket U?33 concentration, increases with
decreasing core diameter, and also increases with decreasing core and
blanket thorium concentrations. Finally, the UZ33-fo-thorium ratio re-
quired in the core for criticality passes through a minimum when the core
thorium concentration is permitted to vary. These variations show that
desirable features are always accompanied by some undesirable ones. For
example, increasing either the core or blanket thorium concentrations
results in an inerease in breeding ratio, but there is also an accompanying
increase in inventory requirements; decreasing the core radius increases
breeding ratio and possibly decreases inventory requirements but increases
wall power density. These types of variations illustrate that minimum
fuel costs will result only by compromise between various reactor features.

 

 

 

 

 

 

70
w= 1\ 1 | ¥

60 —W 3 —
@ R = Core Tank Radius
S
o _ R=2H ]
‘5.“1:,50——W—5 _____ R=13H
-
2_540_ -
a=
-3 30— —
3
n-: W=
w% 2 20—W = —
= W=

10— —

0

1.25

l | I

2 n
T
o el
o
£
-
@ ]
g
o

 

 

 

 

o 10 20 30 40 50
Reactor Length, ft

Fic. 2-10. Gross breeding ratio and maximum power density at core wall for
two-region cylindrical reactors. W =g of U2?3/kg of Th?*? in blanket, blanket
radius = 5 ft, power = 450 Mw (heat).

The breeding ratio, the power density at the core wall for a given total
power, and the required fuel concentration have also been evaluated for
eylindrical reactors [25]. The results, plotted in Fig. 2-10, are based on
two-group calculations for eylindrical reactors; the diameter of the pressure
vessel was assumed to be 10 {t, the total reactor power 450 Mw, the reactor
50 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

temperature 300°C, and the core diameter either 4 or 6 ft. The Zircaloy-2
core tank was assumed to be 3-in. thick when the core diameter was 4 ft,
and § in. when the core diameter was 6 ft. The reactors were assumed to
contain D20, U, and 69, fraction poisons in the core region and D0,
U233, and 1000 g Th/liter in the blanket region. I'igure 2-10 gives the
breeding ratio and the maximum power density at the core wall (core side)
as a function of reactor length for different core radii and blanket fuel con-
centrations. The highest breeding ratios are associated with small core
radii, thick blankets, and long reactors; however, these reactors also have
relatively high power densities at the core wall. Increasing the reactor
length increases the breeding ratio and decreases the wall power density
and the critical fuel concentration but appreciably increases the inventory
of material.

Im other studies [26] of eylindrical reactors, results were obtained which
indicated that breeding ratios for eylindrical reactors of interest were about
the same as those obtained for spherical reactors. The required fuel con-
centrations were higher, as expected, so that the average flux was lower
for the cylindrical geometry. Although the maximum core-wall power
density decreased with inercasing eylinder dianmeter, it was always higher
than the wall power density obtained for the spherieal reactors of equal
volume. The results of these caleulations are given in Table 2-7. These
reactors were assumed to be at 280°C with 79, core poisons, 3 g of 1233
per liter and 1000 g of Th per liter (as ThO2) in heavy water in the blanket,
and operated at a total power of 60 Mw. A eylindrical core was assumed
to he positioned within a cylindrical pressure vessel such that a 2-ft blanket
thickness surrounded the core.

The results given in Table 2-7 show that increasing the reactor height
had only a slight effect on breeding ratio; also, although the critical con-
centration declined with increasing height, the corresponding total fucl
mventory increased. While not shown, the ratio of blanket power to core
power did not vary significantly with reactor height. Inecreasing the core
volume caused 2 pronounced decrcase in core-wall power density and an
increase in fuel inventory. Thus, for a 3-ft-diameter core, Increasing the
core length from 4.8 to 8 ft decreased the power density by 369;. However,
the blanket and core fuel nventory increased by about 409%.

2-3.3 Nuclear characteristics of two-region thorium breeder reactors
under equilibrium conditions. Results [27] of some nuclear computations
associated with the conceptual design of HRT-3 are given below for
spherical two-region reactors i which the following conditions were
specified:

(1) The reactor system is at equilibrium with regard to nuclei concen-
trations.
TABLE 2-7

Breepinc Ratios, CorE-WALL Power Dexsity, AND CriTicar U233
CONCENTRATION TOR CYLINDRICAL Rreacrtors or Varrous HrrcuTs

 

Core diameter, ft

 

 

 

 

 

 

 

 

Height of 2% 3 33
core, ft , | L
U233, Breeding Power U2ss, Breeding Power U233 Breeding Power .
g/liter ratio dcn&t; o g/liter ratio dengztj\ o o/liter ratio density,
' kw/liter ’ kw,/liter = kw, liter
3.5 4.1 1.10 25
4.8 4.6 1.12 29
6.0 4.2 1.12 24 3.0 1.12 16
6.8 6.5 1.13 36
8 6.2 1.13 31 3.9 1.13 18 2.7 12 12
10 6.0 1.13 26 3.7 1.13 15 2.6 2 10
12 5.8 1.13 21 3.6 1.13 13 2.5 2 8

 

 

 

 

 

 

 

 

 

 

 

 

*Core-wall power density, based on total reactor power of 60 Mw (heat).

R

ALVIS-AAVILY

TaNA

SNOILVUINIOINOD
52

OXNE- AND TWO-REGION HOMOGENEQOUS REACTORS

[cHap. 2

(2) Hydroclone separation of poisons from the core system is emploved

in addition to Thorex processing.

(3) The core Thorex cycle time is a dependent function of the specified
total poison fraction; the blanket cyele time is a function of the blanket

U233 concentration.

CHARACTERISTICS OF INTERMEDIATE-ScarLkt (HRE-3)

TasLe 2-8

Two-RecioNn Reactors (EquiLisrivM CoONDITIONS)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Core diameter, ft 4 4 4 4 5
Pressure vessel 1D, ft 8 8 9 9 9
Blanket thorium, g/liter 500 1000 a00 1000 . | 1000
Blanket U232 o /kg Th 3.0 3.0 3.0 3.0 3.0
Core poison fraction 0.07 0.07 0.07 0.07 0.07
Concentration of 1233
g/liter (core) 3.68 4 .04 3.63 4.02 219
Concentration of U233,
g/liter (core) 0.41 0.39 0.37 0.37 0. 21
Breeding ratio 1.041 1.0047 1.086 1.123 1.089
Core-wall power density
(inside), kw/liter 27 23 27 23 10
Core cyele time, days 833 901 817 893 616
Blanket cycle time, days 220 371 288 504 486
Core power, Mw (heat) 51.5 51.9 51.0 51.7 51.2
Blanket power, Mw (heat) 10.0 9.6 10.5 9.9 10.3
Neutron absorptions and leakages per 100 absorptions in fuel
Absorptions in core by
7233 4.7 76.4 4.6 76.4 75.3
(7234 9 2 8.2 S.4 7.8 8.2
235 9.1 8.1 8.3 7.6 8.0
[J236 0.7 0.4 0.4 0.3 0.4
Poisong 5.2 5.3 n.2 5.3 0.2
Heavy water 0.9 0.8 0.9 0.8 1.6
Absorptions in core tank 2.1 1.6 2.1 1.6 2.4
Absorptions in blanket by :
233 16.2 15.5 17.1 16.0 16.6
T234 (7235 7236 0.1 0.1 0.1 0.1 0.1
Th 95.8 101.7 [100.9 104 .8 101.0
Pa233 09 0.3 0.7 0.4 0.4
Poisons 0.4 0.4 0.4 0.4 0.4
Heavy water 0.5 0.2 0.5 0.2 0.2
Fast leakage 4.3 3.6 2.2 1.6 3.0
Slow leakage 3.4 0.9 1.7 0.4 0.8

 
2-3] STEADY-STATE FUEL CONCENTRATIONS 53

(4) The poison fraction due to samarium is 0.80;; that due to xenon
is 197, [poizon fraction is the ratio of Z,(poison)/Z s(fuel)].

(5) The external core system has 1.0 liter of volume for every 20 kw of
core power; the blanket external system has 1.0 liter for every 14 kw of
blanket power.

(6) The total reactor power is 61.5 thermal M,

(7) The average core and blanket temperatures are 280°C.

The breeding ratio for the reactor variables considered is plotted in IFig. 2-11
as a function of pressure-vessel size for 4- and 5-ft-diameter cores with
several blanket thorium concentrations. More extensive results, including
neutron balances, are given in Table 2-8 for selected reactors. The neutron

114 l |

 

  
   

Thorium in Blanket,
g/liter

1.12 ]
// —

//
110 — —
-

—
— _]

.08
~

1.06

1.04 / — A-ft -Diameter Core ]

— e 5.1t -Diameter Core

Breeding Ratio

 

 

 

100 | | l
8 9 10 1

Inside Diameter of Pressure Vessel, ft

Fig. 2-11. Breeding ratio as function of pressure vessel size for various core
diameters and blanket thorium concentrations, Core poison fraction= 0.07, cor-
rosion products =0, copper concentration =0, blanket U2 =3.0 g/kg of Th,

9

n%3 = 2.25, mean temperature = 280°C, equilibrium isotope concentrations.

balances are normalized to 100 absorptions in U233 and U232; therefore the
numerical values represent approximately the percentage effects of the
arious items on the breeding ratio (however, the effect of Pa=34 losses on
hreeding ratio woukd be obtained by doubling the values given).

Some of the materiils which act as neutron poisons can be altered by
reactor-system design; these include fission-product poisons, core-tank
material, contaminants such as 11,0, and additives such as the cupric
ion. The effect of these on breeding ratio is discussed below.

Fission-product poisons. The effect of total core poison fraction, fpe
(ratio of absorption cross section of poisons to fission cross section of fuel),
o4 ONE- AND TWO-REGION HOMOGENEQUS REACTORS [cHAP. 2

.14

 

1.12

1.10

o
=

o
o

 

BR, Breeding Ratio

1.04 |— —

 

 

o0 | | [ | | | J | |
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Q.10 0N 0.12

 

ch, Total Core Poison Fraction

I1G. 2-12. Effeet of core poison fraction on breeding ratio. Core diameter = 4 ft,
blanket U#33 = 3.0 g kg of Th, 2% = 2.25, mean temperature = 280°C, equililrium
Isotope concentrations.
on breedimg ratio is shown in Ilig. 2-12. The results indicate that for these
reactors the change i breeding ratio with change in poison fraction can
be estimated from the relation

ABR =—0.75 Af,.. (2-18)

Achievement of a xenon poison fraction of 0.01, as postulated in the
computations for Iig. 2-11 aud Table 2-8, requires the removal of most of
the xenon or its lodine precursor before neutron capture oceurs. 1If there
15 no fast-cycle syvstem for iodine or xenon removal, the poison fraction
resulting from xenon will be about 0.05.  According to Ilq. (2-18), the
breeding ratio would be reduced by about 0.03 below the values given in
g, 2-11 if all xenon were retained in the core system.

Core-tank absorptions. The thickness of the Zirealoy core tank was taken
as 0.42 m. {or the 5-ft core and 0.33 . for the 4-t vessel. As shown in
Table 2-8, neutron captures in the core tank reduce the breeding ratio
about 0.02 in the 4-ft core and 0.03 in the 5-ft core. 1f the core-tank thick-
ness were iltered, the losses would be changed proportionately,

Absorptions in copper. Copper can be added to act as a recombination
catalyst for decomposed water. No allowance for neutron absorptions in
the copper recombination catalyst was made in these computations. The
poison fraction attributable to copper in various concentrations and the
effects on breeding ratio (ABR) are estimuted in Table 2-9.  or other
copper concentrations the poisoning effects would be proportionate to the
values in Table 2-9.
2-3] STEADY-STATE FUEL CONCENTRATIONS 55

TABLE 2-9

Errect oF CorPeER ADDITION ON BREEDING RATIO

 

 

 

l
Core ‘ Copper Poison
diameter, | concentration, fraction ABR
ft 1 g-mole/liter
0.004 —0.003
(0.008 | — 0.006

 

 

 

The copper concentration required for 1009}, recombination in a 4-ft
core at 61.5 Mw has been estimated to be 0.018 g-mole/liter. For this core
size and copper concentration, the loss of neutrons to copper would reduce
the breeding ratio by 0.005.

Ho0 contamination. Any 11,0 contained in the heavy-water moderator
will act as a poison and reduce the breeding ratio. The above results are
based on the use of heavy water containing 0.259, H20O. Neutron captures
in the moderator in a 4-ft-core reactor (see TLLble 2-8) were found to be
about 0.009 per absorption in fuel, of which about 6095 were in Ha. Thus
0.25%, 120, which is 9 liters in a 3600-liter system, reduced the breeding
ratio by 0.005. Other values are given in Table 2-10. Different concentra-
tions of Hao0Q would cause changes in breeding ratio proportionate to the
values in Table 2-10.

TasLE 2-10

Errect oF HoO CONCENTRATION ON
Brreping Ratio

 

 

 

 

f
Core diameter, ft H-0 concentration ABR
4 067 (36 hiters) —0.02
5 09, (44 Titers) —0.04

 

 

 

A specified volume of I120 added to the blanket has much less effect on
breeding ratio than the same amount added to the core. This is a result of
both the lower flux in the blanket region and the larger volume of the
system.

Corrosion products.  Assuming the surface area of stainless steel in the
core high-pressure system to be 6000 ft2, corrosion to an average depth of
0.001 in. would remove 250 1h of metal. If this were distributed uniformly
throughout the fuel solution, the poison fraction resulting from it would be
o ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cuaPp. 2

about 0.18 in a 4-ft core and 0.36 in a 5-ft core. However, iron and chro-
mium would precipitate and be removed by hydroclones. If the hydro-
clones were operated on a fast cycle time (several days), neutron capture
in iron and chromium would be unimportant. Nevertheless, the absorption
cross section of the nickel and manganese (which probably remain in solu-
tion) amount to about 159, and 99, respectively, of the total absorption
cross section of type—347 stainless steel. Thus, from 0.001 in. of corrosion,
the nickel and manganese would yield a poison fraction of about 0.04
(ABR = 0.03) in a 4-ft core, and 0.08 (ABR = 0.06) in a 5-t core.

The actual value of the poison fraction from corrosion products would
depend on the corrosion rate and the chemical processing rate. If the cor-
rosion products are assumed to change to isotopes of the same cross sec-
tion upon neutron capture, the following relations are obtained under
equilibrium conditions:

fo=0.01tX R X (1./365)  (Core ID = 4 ft), (2-19)
fo=008 X R X (T./365)  (Core ID =5 ft), (2-20)

where f,, 1s the equilibrium core poison fraction from corrosion products, B
the mean corrosion rate in mils/yr, and T, the core cyele time in days.

An additional point of concern resulting from corrosion of stainless steel
is the adverse effecet of high corrosion-product concentrations on the
stability of fuel solution. The concentration of nickel resulting from
0.001 in. of corrosion would be 0.052 g-mole/liter, and that of manganese
would be 0.011 g-mole/liter. Unless adjustments were made to the acid
concentration, the fuel solution would probably form a second phase
before the above concentration of nickel was attained.

The corrosion products from the Zircaloy—2 core vessel would not appre-
ciably affect the breeding ratio, even if they remained in suspension.
Owing to the dilution effect associated with the large external volume, cor-
rosion of the core tank would result in a slight increase in the breeding ratio.

2-3.4 Equilibrium results for two-region uranium-plutonium reactors.
Initial reactor-fuel materials which have been considered [28] in uranium-
plutonium systems are UQ2504-120, UO2(NO3)2~D20, and TO3z-D20.
Of these, the system which gives the highest conversion ratio is the one
containing UO3-D20. IHowever, because of the relatively low values for
n(U235) and n(PPu¥?), it is presently considered that the attainment of a
conversion ratio as great as unity under equilibrium conditions is imprac-
tical beecause of the high fuel-processing rates and the large reactor sizes
that would be required. However, many uranium-plutonium reactor
systems which will operate on either natural-uranium feed or on fuel of
lower enrichment than natural uranium appear feasible. A two-region
reactor can be operated by feeding natural uranium (or uranium of lower
2-3] STEADY-STATE FUEL CONCENTRATIONS o7

enrichment in U235) into the blanket region, and plutonium (obtained by
processing the blanket) into the core region.

The reactor system considered here is one containing UQy, PuO», and
D-0; steady-state concentrations of U233 U U238 Pu? Pu> and
Pu2!!t are considered. Iruel is removed and processed at a rate required to
maintain a specified poison level. The reactor consists of a core region in
which plutonium is burned and of a blanket region containing uranium
and plutonium, Under equilibrium conditions the net rate of production
of plutonium in the blanket is equal to the plutonium consumption in the
core. In Table 2-11 are given [29] some of the nuclear characteristies for

TaprLy 2-11

Darta rorR Two-Reciox, UO3—Pu0s—120 REAcTORS OPERATING AT 250° C,
Having 4 Core Diaverer oF 6 FT, A BLANKET THICKNESS OF 3 FT, AND
VARIABLE BLANKET-I't1iL INRICHMENT

 

U235/U238 1n blanket 0.0026 0.0035 0.0040
Blanket U cone., g/liter 500 500 500
Pu239,/1U2%8 in blanket 0.0010 0.0018 0.0022
Pu240 /1238 in blanket 0.00013 0.00041 0.00060
Pu2# /U238 in blanket 0. 00002 0.00007 0.00011
TFeed enrichment, 1235/T 0.0031 0.0047 0.0058
Blanket power, Mw 247 411 519 ‘
(ore power, Mw 320 320 320

I Core Pu cone., g/liter 1.66 1.48 [.40
Pu2/Pu?t? in core 0.99 0.99 .99
Pu?*', Pu*# in core 0.35 0.35 (.35
Fraction of fissions in U233 0.25 0.28 0.30
Fraction of U consumed 0.017 0.016 0.015
Total power, Mw (heat) 67 731 339

 

 

 

 

 

 

two-region, UOy~Pu0.-D20 reactors having o core diameter of 6 {t and an
over-all diameter of 12 ft, and having various U#33 "U”** ratios in the blanket
region,

2-3.5 One-region reactors. Single-region reactors have simpler designs
than two-region reactors by virtue of having only a single fuel region; alxo,
fuel processing costs for one-region reactors are generally lower than for
two-region systems.  Iowever, to attain brecding or conversion ratios
comparable to those in a 10-ft-diameter two-region reactor, the diumcter
of a one-region reactor has to be about 15 ft or greater. The construction
58 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

of pressure vessels of such diameters is difficult, and relatively little ex-
perience on such construction has been obtained to date.
In the succeeding sections some equilibrium results are given for the
nuclear characteristics of one-region breeder and converter reactors.
1.2

 

 

 

 

| | r ! | {
Diemeter
(Ft}
1.7 - 20 —
18
16
14
° 1.0 — 12 —
T
e 10
o
£
209 —
o 8
©
Zz
0.8 —
0.7 |— -
ok | | | | 1 |
50 100 150 200 250 300 350 400

Thorium Concentration (g/liter)

Fi1c. 2-13. Breeding ratio vs. thorium concentration for one-region reactors of
various diameters. Poison fraction = 0.08, % = 2.25.

2-3.6 Equilibrium results for one-region thorium breeder reactors. Re-
sults have been obtained [30] for one-region thorium breeder reactors
operating under equilibrium conditions.  Critical concentrations and
breeding ratios were obtained by means of Ilgs. (2-4) and (2-6). Iig-
ure 2-13 gives the breeding ratio as a function of thorium concentration
and reactor diameter. Comparison of these results with those obtained for
two-region reactors illustrates that reactor diameter influences breeding
ratio to a greater extent in one-region systems than in two-region systems.
Also, increasing the reactor diameter increases the breeding ratio signif-
icantly even for 14-ft-diameter reactors. Although breeding ratio can be
inereased by increasing the thorium concentration, there is an accompany-
mg increase in fuel inventory. To keep inventory charges at a reasonably
low level and yet permit a breeding ratio of unity to be attained requires
thorium concentrations between 200 and 300 g/liter and reactor diameters
of about 14 ft.

The equilibrium isotope concentrations as a function of thorium concen-
tration for a l4-ft-diameter reactor are given in Iig. 2-14, This diameter
2-4] UNSTEADY-STATE FUEL CONCENTRATIONS 59

 

 

 

 

 

 

3
=
o
c
2
z
€
g
€
S
0 t } 1' ; [
50 100 150 200 250 300 350

Thorium Concentration (g/liter)

Fig. 2-14. Uranium isotope concentrations under equilibrium conditions vs.
thorium concentration in a one-region reactor of 14-ft diameter.

aalue has been chosen because it represents the diameter which gives
minimum fuel costs, although it is realized that construction of the cor-
responding vessel may be hevond present technology. The fuel-processing
eyele time which minimized fuel cost was that which corresponded to a
poison fraction (due to fission products) of about 0.08, and so this is the
ralue used for poison {raction in the results given here.  Absorptions in
higher isotopes contributed an additional poison fraction of about 0.03.

2-3.7 Equilibrium results for one-region uranium-plutonium reactors.
The fertile-material concentrations and reactor diameters have essentially
the saume cffects on conversion ratio and fuel inventory for one-region
uranium-plutonium systems as they do for thorium breeder systems;
however, since the 7’s for U235 and Pu®* are lower than for U#%3, it is more
difficult to attain a conversion ratio of unity in U-Pu gystems than it is in
U233-Th systems. It is still possible, though, for UO3-PuO2-D20 systems
to operate on natural-uranium feed, as evidenced by the results for two-
region reactors. Minimum fuel costs (based on n*' = 1.9; a4 more accurate
value 1= now believed to be 2.2) for one-region reactors, however, occur
when the uranium feed is slightly enriched in U232 [32]. Table 2-12 gives
results [32] of some nuclear calculations for these one-region systems
operating under equilibrium conditions. The reactor diameter was taken
to be 15 {t; the fuel-processing rate was such as to maintain a poison frac-
tion of 707 in the reactor core. Iuel feed was considered to be obtained
from an isotope-enrichment diffusion plant. The results indicate that the
uranium feed for these reactors would have to contain between 1 and
1.59, U239,

91 UNSTEADY-STATE FUEL CONCENTRATIONS AND BRrREEDING RaTIOS

2-4.1 Two-region reactors. During the period following reactor startup,
there is a buildup of fission-product poisons and higher isotopes with time,
which results in varying nuclear characteristics. This section presents some
calculations relative to the HRE-3 conceptual design for the initial period
of reactor operation.
 

60 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cmar, 2
TasLe 2-12
ReacTtor CHARACTERISTICS FOR SOoME ONE-REGION,
UO3-PuO2-D20 ReacTors* OPERATING UNDER
Equmisrium CoNDITIONS

Reactor temperature, °C 250 250 250 300
U cone., g/liter 334 253 170 183
1235 conc., g/liter 2.65 1.47 0.83 1.35
U235 cone,, g/liter 0.39 0.22 0.12 0.20
Pu?* cone., g/liter 3.43 1.74 0.84 1.08
Pu2%0 conc., g/liter 3.39 1.72 0.83 1.17
Pu®# conc., g/liter 1.21 0.61 0.30 0.41
Initial enrichment (no Pu),

U239/ (total) 0.0116 0.0106 0.0098 0.0140
Steady-state enrichment,

U235/T (total) 0.0081 0.0059 0.0050 0.0073
Steady-state feed enrichment,

U235/ (total) 0.0153 0.0113 0.0096 0.0136
Fraction of fissions in U235 0.24 0.25 0.29 0.31
I'raction of U consumed 0.018 0.017 0.015 0.014

 

 

 

 

 

 

 

*Reactor diameter = 15 ft; poison fraction = 79,; reactor fuel returned to diffu-
sion plant for re-enrichment; tails from diffusion plant are assumed to have U235
content of U?35/U = 0.0025; processing losses are neglected.

Computations have been performed for several spherical reactors using
an Oracle code [31] for two-region, time-dependent, thorium breeder
systems. The variation with time of the breeding ratio and the concentra-
tions of U233, 234 U235 1236 Pa233 and fission-product poisons were
obtained. Calculations were first confined [32] to solution-core reactors
initially containing either U233 or U235 and generating a core power of
50 My, Core and blanket Thorex processing was considered only when the
initial fuel was U233, The use of centrifugal separation (hydroclones) for
core-solution processing was assumed in all cages. The time dependence of
the concentrations of xenon, the samarium group of poisons, and the
poisons removable by hydroclone processing were neglected, since the time
required for these poisons to reach near-equilibrium conditions is relatively
short. Account hag been taken of their presence by the use of fixed poison
fractions. The effect of copper added for internal gas recombination was
also ineluded in this way. Core diameters of 4 and 5 {t, pressure-vessel
diameters of 8 and 9 ft and thorium blanket concentrations of 500 and
1000 g/liter were considered. Iixed core poisons in terms of percentage of
2-4] UNSTEADY-STATE FUEL CONCENTRATIONS 61

core fission eross section were: samarium group, 0.89;; xenon group, 19%;
copper, 0.89; poisons removable by hydroclones, 19;. [Iixed blanket
poisons were: samarium group, 0.8%; xenon, 19,. It was assumed that
the core solution was processed both by hydroclones and by the Thorex
process described in Chapter 6; the blanket slurry of thorium oxide in
heavy water was processed by Thorex only, and fuel produced in the
blanket was drawn off from the Thorex plant and returned to the core at a
rate sufficient to maintain criticality.

The system was assumed to start “clean,”” except for the poisons men-
tioned, with either U233 or U235 in the core. Muakeup fuel (same as initial
fuel) was fed as needed while the concentration of fuel in the blanket was
increasing.  When the blanket fuel concentration reached a predetermined
level, blanket processing was initiated at the rate which would be required
if the reactor were at equilibrium; however, for the U235-fueled reactors,
calculations were performed only up to the time at which processing
would start. At the start of processing, the fuel feed for the core was
assumed to come from the processed blanket stream. When the core
poison level built up to a predetermined point (87 for the U233 reactors),
processing of the core solution was started. The processed core-fuel stream
was considered mixed with the processed blanket stream; part of the mix-
ture was used as core feed while the excess was drawn off as excess fuel.
The calculations were continued until most of the concentrations ap-
proached equilibrium values. Time lags due to chemical processing holdups
were neglected. The chemical processing rates employed were those cal-
culated earlier for equilibrium reactors [27].

The curves in I'ig. 2-15 show results for some representative U%33-fueled
reactors. As shown in Iig. 2-15(a), with 1000 g Th/liter in the blanket
region, the breeding ratio falls steadily for about 900 days until core proc-
essing starts. Although blanket processing, begun at 490 days, arrests the
growth of U233 in the blanket rather suddenly, the slope of the breeding-
ratio curve does not change markedly because the buildup of core poisons
is controlling the breeding ratio. When core processing interrupts the
growth of core poisons, the breeding ratio levels off sharply. The variation
of relative leakage and poison losses with time and the variation of blanket
power with time are illustrated in I'ig. 2-15(b) for the case of a blanket
thorium concentration of 500 g/liter.

In all cases the time required to reach the 89 core poison level was of
the order of two years. The average breeding ratio during this period was
about 0.02 higher than the equilibrium value. The effect of poisoning due
to buildup of corrosion products is not included in this estimate.

While the above statements concerning breeding ratio are characteristic
of U233-fueled reactors, U235 reactors show quite a different variation of
breeding ratio with time, due to changes in the effective 5 for the system.

)
62 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

| l | 5 8 T T T
: oz : | Core Poison
:g B -é E 06— : : Fraction -
o —=Qa | Neutron
£ | Lleakage
g’ — ;9_ & .04 : 2 —
3 , 8% '
g ! L o
-7 | — £ 9 .02 | Blcmkzlet Poison Fraction —
1 2 <
‘5 | |
ol N | | s oLl i 1|
5 t+—Core Processing Starts 15 fe—Blanket Proc. Starts
S — l | T T T ]
— F—Blclnkei Processing Starts I }*—Core Proc. Starts
4l . _] < 19l— -]
@ e Core U233 32 |
£ ! ore U 2 Blank
| anket Power at — — — —
2 41 Blanket U233 5 o . ]
g / 1 3 Equilibrium
£ ; ' < ) f
g 20— | — % 6+ ]
e | X |
S ' | £ | |
I | = I
1 ! ' ] ] | : ]
|
P by 233
b . L , ra J I
] T 1 [ I 1 l 1 i I {
0 800 1600 2400 3200 4000 0 500 1000 1500 2000 2500
Time (days) Time {days)
{a) Breeding Ratio, Blanket Conc. of u233 {b) Neutron lLeakage, Poison
and Pa233 and Core Conc. of U233 Fraction and Blanket
{Blanket Conc.=1000g Th/I}. Power (Blanket Conc.=
500g Th/ ).

Fi1a. 2-15. Nuclear characteristics of a 60-Mw (heat) two-region U233 breeder
during initial operating period. Core diameter = 4 ft, pressure vessel diameter
= 0 ft, 280°C, solution core.

Such reactors have appreciably lower breeding ratios than do U233-fueled
systems. This is primarily due to the relatively low value of n?5 compuared
to 9?* (n** = 2.25 while n?°= 2.08). As U233 builds up in the blanket of
a U2*5-fueled reactor, the average value of % for the reactor as a whole
increases. This helps compensate for the increase in core poison fraction
and causes the breeding ratio, which initially drops from 1.008 to 1.002
during the first 200 days of operation, to risc again. A maximum of 1.005
in the breeding ratio is reached at about 600 days, after which a slow
decrcase follows.

Three types of isotope growth were obtained in this study. First, the
concentration of the main isotope, U233 or U??3, remained relatively con-
stant despite sizeable changes in other concentrations. Second, the pro-
tactinium concentration-time curve was found to “knee-over’” even before
processing started. This behavior was due to the radioactive decay rate
being several times larger than the chemical processing rates employed.
Third, the concentration of the heavier isotopes of uranium built up slowly
with time for the assumed power level; even after 10 years’ operation their
concentrations were much less than the equilibrium values. The tolal core
concentration of uranium was thus significantly less than the equilibrium
value. The isotope concentrations after various operating times are shown
in Table 2-13, along with equilibrium values.

 
2-4] UNSTEADY-STATE FUEL CONCENTRATIONS 63

TasLe 2-13

Core CoNCENTRATION oF Uranium IsoTOPES
As A Yuxcrion oF TimMeE For A U23.FueLeEp REACTOR
AND FOR A U23-FurLEp REACTOR

(Core diameter = 4 ft; pressure-vesse! diameter =9 ft; ThO: concentration in
blanket = 1000 g/liter; solution core, total power = 60 Mw of heat)

 

 

 

 

 

 

 

 

 

 

 

 

Time, days U233 234 U235 } 1236 l U total

Concentrations of isotopes for U233 fuel, g/liter

0 3.75 — —_ — 3.75

200 3.77 0.33 0.02. — 4.12

500 3.74 0.74 0.07 0.01 4.56

920 3.91 1.29 0.14 0.04 5.38

2000 3.98 2.06 0.26 0.18 6.48

3000 4.02 2.40 0.31 0.34 7.07

Equilibrium 4.02 2.74 0.37 1.57 8.07
Concentrations of isotopes for U23 fuel, g/liter

0 -— — 4. 21 — 4.21

200 — — 4.19 0.57 4.76

500 — — 4.12 1.43 5.55

900 - — 4.03 2.54 6.57

 

 

 

 

While chemical processing sharply discontinues the growth of U233 in
the blanket, the power level in the blanket may markedly overshoot the
equilibrium level, as shown in Fig. 2-15(b). An overshoot is obtained when
the blanket-U?33 concentration reaches its maximum value before core
processing starts. Table 2-14 shows the peak values of blanket power
computed for the reactors studied, along with the equilibrium values.

Results similar to those given above have also been obtained [33] for
two-region breeders having various concentrations of thorium in the core.
The core diameter was set at 4 ft, the pressure-vessel diameter at 9 {t, and
the blanket thorium concentration at 1000 g Th/liter. The core thorium
concentrations studied were 100, 150, and 200 g Th/liter. As before, the
moderator was heavy water in both core and blanket volumes, and both
regions operated at a mean temperature of 280°C; the Zircaloy—2 core tank
was 0.33 in. thick. Calculations were performed at a constant {ofal power
of 60 thermal Mw. The same chemical processing conditions were assumed
O+

ONE-

AND TWO-REGION

TaBLK 2—-

14

HOMOGLENEOUS REACTORS

[cHAP. 2

Prax anp EquirisritM Varnturs or BLANKET POwERr
CoMpPUTED FOR UZB-FurLkp REACTORS

{Reactor power = 60 thermal Mw)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

('ore | Reactor | Blanket Equilibrium Peak value of | Time at which
dia, dia., Cone., value of blanket | blanket power peak value
ft ft o Th/liter power, Mw M occurs, dayvs
4 9 1000 9.6 10.8 520
4 9 500 10.4 12.5 350
4 8 1000 0.3 10.8 380
D 9 1000 10.0 11.5 490
3
=
o
c
2
S
€
g
c
Q
o
1.22 T T T T T T T 0.08
[ !
0~ g
01.20 0 | J0.06 =
= 0?0 | o
S ) | I g
o | t Core Processing Starts =
218 | l -10.04 8
- ! | ©
@ .
e | i Blanket Poison Fraction e
o0 | L
1.16 ! i Jo.02
Blanket Processing SM Breeding Ratio
1.14 | i | i :l | [ ! 1 0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (days)

Fig. 2-16. Variation with time of U233 T?234 and Pa?33 in the core, U233 and
Pa233 in the blanket, breeding ratio, and core and blanket poison fractions as func-
tions of time for a two-region 60-Mw thorium breeder reactor. Core thorium con-
centration = 150 g Th/liter, blanket thorium concentration = 1000 g Th/liter,
temperature = 280°C.
-

2-4] UNSTEADY-STATE FUEL CONCENTRATIONS 6D

as before, with the important exception that no hydroclone processing was
employed for the slurry-core cases.

Results given in Iig. 2-16 are for a core thorium concentration of
150 g Th/liter and are typical of all cases studied in their important features.
As with solution-core reactors, there is a small but relatively rapid initial
drop in breeding ratio during the first 100 days following reactor startup.
This 1s due to the neutron captures in protactinium as its concentration
rises and approaches equilibrium conditions. The buildup of protactinium
is quite similar to the behavior seen in the solution-core cases. IEqui-
librium protactinium levels are reached in both core and blanket long
before chemical processing is started. This period is followed by a more
or less linear fall in breeding ratio with time, due to core poison-fraction
buildup. Initiation of blanket processing produces a relatively minor
change in slope of the breeding ratio-time curve, in contrast to the sharp
break caused by the start of core processing. Linear buildups with time
of the poison fractions and the higher isotopes are also observed, with
U2 in the core being the most important higher isotope. Other isotope
concentrations are comparatively low even at 1500 days.

Figure 2-17 lists some results for the various cases considered (except
for the case already considered in Fig. 2-16). Comparison of the results
shows that the time at which core processing is initiated increases with
increasing core thorium concentration (except for the solution-core case);
this is due to the higher U233 critical concentration associated with higher
thorium concentration, resulting in a longer period to build up to a specified
core poison fraction. For the solution-core case, hydroclone processing was
assumed, so that 759, of the poisons were removed by means other than
by Thorex processing of the core (the time specified for core processing to
begin in Fig, 2-17 corresponds to the initiation of Thorex processing).
It is interesting to note that with slurry cores, despite the absence of hydro-
clone separation, the core poison fraction still takes two or more years to
reach a 79 level.

In these calculations there was no allowance for the poisoning effect of
accumulated corrosion products. Absence of hydroclone processing would
result in retention of all corrosion products in the core system, and for
the same corrosion rate the cross section of corrosion products is about
four times as great as in a solution-core system using hydroclones. How-
ever, the higher critical concentrations in slurry-core systems compensate
for this, and the poison fraction in the core with 200 g Th/liter could be
about the same as in a solution-core reactor.

The critical concentration of U233 in the core displays a small but steady
rise; this contrasts with solution-core reactors, where the critical core
concentration is found to be strikingly constant. Solution-core reactors,
because of their longer thermal diffusion length, are more sensitive to the
 

 
   
 

 

 

 

   

 

  

 

 

 

    
   

 

 

 

66 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP, 2
16 R R 1 | é
(e} Solution Core Reactor {with Hydroclones) B
| |
14 J‘ 4
7‘ : U233 1,
Blanket Proce;sing Starts BER
A2 ! :Core Processing Starts 12
1 1o | | o4 { | 1
18 17~
] {b) Slurry-Core Reactor— [ | I 5
o =
BR 100g of Th/liter in Core as ThGC 233 — 10 =
. 2 U o
; Z
ot16 | B s 3
;é i+;~Core Processing Starts g :E:
@
£ ' 1 Blanket Processing Starts S
I - |7 S
& | | | | | 1 6 @
20 . 20 S
L N L o
(c} Slurry Core Reactor—200g of Th/liter in Core 19 &
| -
| |
18 — ! 18
17
16— ! 16
l ! =115
li Core Processing Starts |
14— Blanket Processing Starts | : — 14
| | ! L j { L1 13
0 200 400 600 800 1000 1200 1400 1600

Operating Time at Reactor Power of 600 MwiHeuat! (days)

Fra. 2-17. Comparison of breeding ratios and U233 core concentrations for two-
region breeder reactors with various core-thorium concentrations. Core diameter
=4 ft, reactor diameter =9 ft, 280°C, blanket thorium concentration = 1000
g/liter, blanket processing started when U233 blanket concentration reached
3 g/liter, core processing started when core poison fraction reached 79, (89, for
solution-core case).

growth of fuel in the blanket than slurry-core machines. The latter have
less thermal leakage from core to blanket, and core-poison and higher-
isotope growth are less well compensated by the rise of fuel level in the
blanket.

2-4.2 One-region reactors. The critical cquation for one-region reactors
18 not so involved as that for two-region systems, and so the mathematical
system analogous to the one used above for two-region systems is com-
paratively simple. However, relatively few nuclear results for one-region,
time-dependent systems have been obtained specifically. Some results
are given in Reference [29] for a 15-ft-diameter reactor containing 200 g of
uranium per liter and operating at a constant power of 1350 Mw (heat)
at 250°C. Concentrations of U235 Pu23® Pu?t0 and Pu?%! and the fraction
of fissions due to U235 were obtained as functions of time.
2-5] HOMOGENEQOUS REACTORS AND REACTIVITY ADDITIONS 67

Under certain conditions the mathematical system involving nuclide
concentrations, criticality, and fuel processing can be solved analytically
[341: the solutions obtained have been used to calculate fuel costs directly,
but they could also be used to calculate breeding ratio and critical fuel
concentrations explicitly.

2-5. SAFETY AND STABILITY OF HOMOGENEOUS REACTORS
FoLrLowiNg REACTIVITY ADDITIONS

Reactor kinetic studies are usually broken down into investigations of
reactor safety and reactor stability. Both safety and stability are deter-
mined by the generalized equations of motion involved. The two categories
are considered, however, because the time scales in safety studies are
usually much shorter than those involved in stability studies; hence, the
generalized equations of motion can be simplified in accordance with the
specified study. As used here, safety refers to the events which happen as
a result of the initial power excursion following a reactivity addition;
stability refers to the events which occur as a result of subsequent power
surges. By these definitions, it is possible that a reactor can safely with-
stand the first power surge following a reactivity addition, but still not
be stable; under these conditions, the reactor would be safe with respect
to the first power surge but would not be safe to subsequent power surges.
On the other hand, a reactor system can be stable and still not be safe;
i.c., permissible changes in reactor variables can be exceeded as a result
of the first power surge, even though the system would have reached a new
equilibrium condition had 1t been able to withstand the first power surge.

The majority of work on the kinetics of homogeneous reactors has con-
cerned circulating, pressurized-type systems, and the discussion below
pertains primarily to those systems. There has been relatively little effort
devoted to boiling homogeneous reactors; for these reactors the control
and safety problems may be more difficult than for the nonhoiling reactors
[35]. In boiling systems, the most important parameters are those asso-
ciated with vapor formation and bubble growth, vapor removal from the
system, and the control system. With regard to control, boiling reactors
have the disadvantage of producing more power when less power is de-
manded unless 2 control system is used in which an increase In reactor
pressure leads to ejection of fuel from the core region. Alternatively, power-
demand control can be obtained by permitting the boiling process to
generate only a fraction of the total reactor power, the remainder being
removed by fuel fluid which is circulated through a heat exchanger.

2-5.1 Homogeneous reactor safety. The nuclear safety of homogeneous
reactors will be a function of the maximum permissible reactivity addition
68 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

and the possible reactivity additions. Despite the inherent safety associaiad
with a large negative temperature coefficient of reactivity, it cannot be
stated a prior: that homogeneous reactors will be safe under all operating
conditions. The limiting feature with respect to reactivity addition is the
permissible pressure rise within the reactor system. This is usually either
the maximum permissible core pressure rise (two-region reactor) or the
maximum permissible pressure rise within the pressure vessel (one-region
reactor).

With a given limit on the maximum permissible pressure rise, it is possi-
ble to specify the maximum permissible reactivity addition which can be
added to the reactor. Reactor operations can then be restricted by design,
s0 that reactivity additions associated with physical events will not exceed
the maximum permissible reactivity addition.

The potential reactivity available in homogeneous systems is inherently
large, because o high operating temperature is coupled with a high negative
temperature coefficient of reactivity. However, all reactivity additions in-
volve a time element. Since 1t appears desirable to allow continuity of
physical operations, the safety design criteria should be applicable to con-
tinuous, linear rates of reactivity addition. Specifically, the maximum
permissible linear rate of reactivity addition should specify what restrictions
are necessary on the physical system so that this rate is not exceeded.

Equations of motion. The neutron density is the fundamental variable
in homogeneous reactor safety and is influenced primarily by the tempera-
ture and density of the moderator and by the operational changes which
effect a reactivity change. So long as the reactor is not far above prompt
critical, the neutron density i1s given by the conventional equations of
motion. These may also be used when larger reactivity additions are
considered, 1f the prompt-neutron lifetiine is assumed to be that associated
with the region m which the neutron density is rising most rapidly with
time. Under this condition, the over-all rate of increzse in neutron density
is overestimated, so that a safety factor will exist in reactor designs based
upon these equations.

Reactivity additions which involve homogenecous reactor safety are
considerably in excess of that required for prompt eriticality, and for these
cases the reactor power reaches a maximum value in times of the order of
tenths of seconds. Such time intervals are short compared with the average
half-life of the delayed-neutron precursors, and so only a small fraction
of the precursors formed during the power rise decay during that time in-
terval. The delayed neutrons from these precursors therefore contribute
little to the reactor power while the power is rising; rather, they are formed
followmg the time of peak power and exert a powerful damping influence
on the power oscillation, leading to a single, damped power surge. For
these safety calculations, the delayed-neutron density can be considered
2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 69

constant, and so the neutron-density equations can be combined into one
equation. In terms of reactor power, this equation 1s

@_I:ke(l_fi)_l
dt [

]P+§Pm (2-21)

where k., = effective multiplication constant,

P =reactor power,

Po= P evaluated under initial conditions,

t = time,

B = effective fraction of fission neutrons which are delayed,
l = average lifetime of prompt neutrons.

The appropriate value for 8 is determined on the basis of the time spent
inside and outside the reactor core vessel by a fluid particle.

To complete the mathematical system, the relation between £, and P
is required, which requires intermediate relations. For aqueous systems
operating above 200°C, k, 1s influenced primarily by fluid-density effects,
insofar as inherent reactivity changes are concerned. Since reactivity can
also be added by physical operations, k, can be considered to be given by

 

 

ok, ,
ke= 14 A+ bt + Jp (p— po), (2-22)
where b = linear rate of reactivity addition to reactor, Ak, ‘sec,
%;;g = density coeflicient of reactivity,

A = instantaneous reactivity addition,
p — po=deviation of the average density of fuel fluid from its in-
itial value.

The core fluid density is determined from the hydrodynamic equations
of continuity and motion, in conjunction with the equation of state for
the fluid. In most studies to date, a one-dimensional flow model is as-
sumed, gas effects are neglected, the core tank is considered to be rigid,
and the core inlet fluid velocity 1s considered to be constant. The equation
for the change in flmd density is then

d A i . o
P == po(U = L), (2-23)

where A = cross-sectional area of relief pipe,
V.= volume of core region,
U — Uy = deviation of the fluid velocity in core-relief piping from
its initial value.
70 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

Assuming that the resistance to fluid flow between the core and pres-
surizer is proportional to the square of the average fluid velocity, and that
the fluid is incompressible, the one-dimensional hydrodynamic equation
of motion is given by

 

 

[F’“A A‘i’;)} G (U—=Uo)=p—p,— %@ (T! U, (229
where ny = resistance coefficient associated with velocity U,
A (z) = cross-sectional area of relief pipe,
g. = dimensional constant,
p. = core pressure,
D, = pressurizer pressure,
U = average velocity of fluid in relief piping of cross-sectional
area A,
L =length of relief piping from core to pressurizer.

Use of the absolute value of U in Eq. (2-24) ensures that fluid friction
will always oppose fluid flow.

The relation between pressure, temperature, and density of the fluid is
given by the equation of state:

p="12) = 1o~ 78)+ o= ol (2-25)

where p = pressure rise in core,
T — Ty = deviation of the average core-fluid temperature from its
nitial value.

Relations are still needed between T and PP and between p, and p.
Assuming isentropic compression within the pressurizer, p, is approxi-
mated by

 

- Vc .
Py — Po=npo LB 22, (2-26)
po V,

where V, = volume of pressurizing fluid,
pp, — po = deviation of the pressurizer pressure from its initial value,
n = ratio of heat capacity at constant pressure to heat capac-
ity at constant volume for pressurizing fluid.

The relation between T and P is obtained from an energy rate balance on
the core fluid. Since the rate of energy transport associated with fluid
2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 71

flow and thermal diffusion is small during times of interest, the energy
rate balance Is approximated by
s L _p_p, (2-27)
dt
where S, = volume heat capacity of the core fluid.

The mathematical system is now complete, and consists of Eqs. (2-21)
through (2-27). Since an analytic solution to the above system has not
been obtained, numerical integration of the above equations is necessary
in general.

Calculation of reactor pressure rise. The variable of interest is the maxi-
mum value of core pressure for a given set of parameter values. Although
not exact, an analytic expression for the maximum pressure rise, Pmax, can
be derived [36] for the case of an instantaneous reactivity addition. This
is given by

 

_ mgp m2F Com B
Pons = 5. [ o+ 1+ 2 )]+ 2, (2
-2

measure of effect of pressurizer volume upon core pressure rise, dimen-
sionless,

]

=1 %[Cz + Gy & mim —{(;%m)m] ,

= A — 8 = instantaneous prompt reactivity addition divided
o by I, sec™ 1,

Pmax = Maximum pressure rise, Ib-force/ft?,

 

 

o nlUo normaiized friction coefficient in core relief line,
Yr 4 L g d.L sec L
f Ax)
dp)(ake) normalized conversion factor between net core
T2 op density change and core pressure rise, {t?/sec-1b
force,
_ Uodpo (0. s o _ g.(dp/dp) —2
vy o= 7T \7p)’ sec 2, wi = Loy ST
“Jo A(x)

Results obtained from Eq. (2-28) have compared favorably with numerical
integration of Eqs. (2-21) through (2-27) [36].
72 ONE- AND TWO-REGION HOMOGENEQUS REACTORS [cHAP. 2

Although Eq. (2-28) is based on an instantaneous reactivity addition,
it can be applied to rate additions of reactivity in eertain circumstances.
For rate additions normally encountered in homogeneous systems, it ap-
pears that the most dangerous amounts of reactivity can be added when
the initial reactor power is low. In these circumstances, reactivity is added
during an interval which is much greater than the time interval in which
the reactivity is decreased as a result of fluid density changes. A simplified
mathematical model can then be used [36] to relate a linear rate of re-
activity addition, b, to its equivalent instantaneous reactivity addition,
Akey. This relationship is given by the following equation:

 

2 /6, 2
£ Me/20Wnp
w2 In(m2/2w2 ) (2-29)
wnp n(me/ wn'p
where w2, = wiry, sec™ 2,
w2 1 ak‘e P() qe(}mz
= | — —=} —1 8
" l 0T/ S, ’
Tpe = rcactor power at prompt critical relative to initial power
level,
Eeq —
m., = —Efll—fi’ sec”

Aleq = equivalent instantaneous Ak, associated with rate of reac-
tivity addition,

£ =b/l sec™2

The value for z,, is obtained from the equation

 

Xpe = M2 | Vi \J/;;E erf (\L/[Z%) ’ (2-30)

where M, =

l
k.(0) = initial value of ..
Fquations (2-29) and (2-30) are plotted in Figs. 2-18 and 2-19 and give

the particular combinations of initial power level and rate of reactivity
addition corresponding to a given equivalent prompt reactivity addition.
2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 73

70

 

I | | | I | | 1 i

_\qup (sec'l)
{

Mg =

 

 

 

| | | | | | I | !
0 10 20 30 40 50 60 70 80 90 100
i l?—(setz'z)
"l
FiG. 2-18. Relation between equivalent prompt reactivity and rate of reactivity
addition. Ak, o = equivalent prompt reactivity, [ = mean lifetime of prompt neu-
trons, b= rate of reactivity addition, anp = modified nuclear frequency.

 

W00 ==—T—7TTTIIT T T TTJmr T T 1T T 17

LLLLLA

T

)

ol &

X -

s S

+

s 100 — hmn
ol| & — =
= — — —
| = — ]
L

4

——

o e, o

EN

[ TTTTT
[ LTI

|

 

 

 

 

. pur e e b

0.1 10 100 1000
MS

 

V2

I'1g. 2-19. Neutron power at prompt critical relative to initial power for differ-
ent rates of reactivity addition.
74 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cuHAP. 2

Having determined the equivalent prompt reactivity addition, the
maximum pressure rise can be calculated by using Iiq. (2-28) with m
replaced by m.. To check the validity of the derived relations, values of
Pmax Obtalned analytically were compared [36] with those obtained by
numerical integration of Eqs. (2-21) through (2-27). The results obtained
by numerical integration were used to establish the relationship between
rate of reactivity addition and equivalent instantaneous rexctivity addition
on the basis of equal pressure rise. Thus, for a given rate of reactivity
addition and initial reactor power level, a particular maximum pressure
was obtained. The amount of prompt reactivity added instantancously
which gave the same maximum pressure was termed the equivalent prompt
reactivity addition corresponding to a given rate addition and initial
power level. Iror a given rate of reactivity addition, good agreement was
obtained between the equivalent prompt reactivity addition obtained by
numerical integration and that obtained by use of Eqgs. (2-29) and (2-30).

Reactivity additions. Homogeneous reactors usually have relatively
large, negative temperature coeflicients of reactivity. A high negative
value for the temperature coefficient is usually associated with a high degree
of reactor safety; however, this is true only if reactivity is introduced by
means other than the temperature cocflicient itseli. If reactivity is added
by adding cold fuel to the reactor core, e.g., by means of a heat-exchanger
accident, then the smaller the value of the temperature coeflicient the
smaller the reactivity addition. I'rom this viewpoint, a small temperature
coefficient 1s desirable if the most hazardous reactivity addition oceurs as a
result of introduction of cold fluid into the reactor. On the other hand, a
large negative temperature coefficient is desirable to compensate for
reactivity added by means other than that associated with decreasing the
core fluid temperature.

Reactivity changes can also oceur as a result of changes in fuel con-
centration or distribution. In shirry reactors, large changes in reactivity
can result from settling of the fertile material. The results [37] of two-
group, two-dimensional, two-region calculations are shown in Table 2-15
for a cylindrical reactor containing highly enriched UTO02804 in a 5-ft-
diameter core region, and ThOg in a 23-ft-thick blanket region. The aver-
age temperature was assumed to be 280°C, and the height was assumed to
be equal to the diameter for hoth core and pressure vessel.

Although the calculated reactivity additions associated with slurry
settling were quite large, such additions would not occur instantaneously;
rather, they would be a function of time, the rate of reactivity addition
being controlled by the rate at which the slurry settled. Scttling data for
slurries containing 1000 or 500 g Th/liter correspond to rates of reactivity
addition of less than 0.029; Ak./sec. These rates are well within permissible
rates of reactivity addition; however, if settling took place over a period
2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 75

 

 

 

TaBLE 2-15
ReacTivitTy CHANGE AS A FUNCTION OF SLURRY SETTLING
Initial thorium econcentration Percentagfa of Calculated reactivity

in blanket, g/lit reactor height change,
1 blanket, g/ater slurry has settled 9 Ak,

1000 30 +2.8

1000 50 10.2

500 30 2.1

500 50 8.7

 

 

 

 

 

 

1.00

0.98

   
    
 

12 f

2009 Th/liter
initially

10% Poisons

o
0
o
!

o
o
-
I

12 ft

200g Th/liter
initially

No Poisons

S
<
N
|

12 ft
3009 Th/liter
initially

Ratio of Actual Mass to Critical Mass

 

 

 

0.90 I— 10% Poisons
10 ft
0.88 |— 3004g Th/liter -
initially
10% Poisons
0.86 L [ I | |
0 8 16 24 32 40 48

Setiled Distance (in.)

Fra. 2-20. Effect of settling on eritical mass requirements in one-region reactors.

of time, the reactor temperature and pressure would eventually reach the
point where the fuel system would have to be diluted or dumped.

Settling of fuel and fertile material in large single-region reactors may
cause the reactor to become subcritical. The results [38] of two-group
caleulations for D2O-ThO2-U?3303 reactors 10 and 12 ft in diameter and
containing 200 and 300 g Th/liter are given in Fig. 2-20. Cylindrical
76 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHaP. 2

geometry was assumed, with height/diameter ratio equal to one; the slurry
was assumed to be homogeneously distributed in the bottom portion of the
reactor, with a D,0 reflector on top. In IFig. 2-20 the amount of fuel
required for criticality (after settling) is compared with the amount of fuel
initially required (before settling). DBased on these two-group results,
it appears that large one-region reactors become suberitical following
slurry settling.

In the previous analytical presentation, the effect of decomposition
gases upon reactor safety has been tacitly neglected. However, the genera-
tion of hydrogen and oxygen from the decomposition of water represents a
possible explosion hazard. If the fuel contains no catalyst for recombina-
tion, gas bubbles will form and contain an explosive mixture of hydrogen,
oxygen, and water vapor. Explosions might occeur in the core, gas separator,
recombiner system, or connecting piping. If an explosion occurs outside
the reactor core, reactivity can be added to the reactor, resulting in an
undesirably high power surge. Under these conditions, not only must the
gas explosion be considered but also the possible reactivity addition asso-
ciated with the explosion. If the gas explosion oceurs in the core, however,
no reactivity addition should be involved, and only the pressure associated
with the explosion need be considered. If no explosion occurs, the presence
of gases will still influence the compressibility of the fuel fluid, and this
effect can be considered in the evaluation of fluid compressibility. Neglect
of gas formation In evaluating reactor safety is justified if complete recom-
bination of the decomposed gases i1s achieved within the reactor, or if the
reactor is initially operating at a power so low that decomposition gases
are not formed (corresponding to absorption of the decomposition gases by
the fuel solution). If the reactor is at low power and reactivity is added to
the system, then neglect of gas formation is conservative with respect to
safety evaluation, since in this case the formation of gases helps in de-
creasing the reactivity of the system. If the initial power is so high that
undissolved gases are initially present within the core region, then the
effect of these gases must be concidered in evaluating fluid compressibility.
The presence of undissolved gases would tend to lower the permissible
reactivity addition; however, an increase in gas volume would imply an
increase in initial reactor power, which aids reactor safety if reactivity can
be added as a rate function only.

The number of fission neutrons which are delayed usually is considered
as an Important factor in reactor safety. However, if reactivity is added
as a linear rate function at low initial reactor power, the delayed neutrons
have little influence upon the reactivity addition above prompt critical.
They will influence the stability and steady-state operational behavior of
the reactor, though, and are necessary to damp the power oscillation follow-
ing a reactivity addition.
o

2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 77

Reactor safety also mmvolves events which are not directly associated
with reactivity additions; e.g., if the circulating pump failed or if there
were sudden eessation of heat removal in the heat exchanger, after-heat
effects may raise the temperature (and pressure) of the system to unde-
sirably high values. The possibility of an instantaneous rupture of a high
pressure steam line, however, is remote, based on experience in conven-
tional plants.*

2-5.2 Homogeneous reactor stability. The purpose of stability studies
1s to determine whether the reactor power will return to a stable equilibrium
condition following a system disturbance. Although inherently connected
with safety, stability studies usually treat small reactivity additions and
concern time intervals long in comparison with those involved in safety
studies. The general stability problem can be broken into simpler problems
by eliminating those parts of the physical system which have only a small
influence upon the time behavior of the variable of interest.

The most familiar of stability studies concerns the reactor core-pressurizer
system, and will be referred to as “nuclear stability” studies. These
consider the high-pressure system alone and assume that the power demand
is constant. This is a valid assumption, since changes in power demand
and changes in variables resulting from fluid flowing between the low- and
high-pressure systems initiate only low-frequency nuclear power oscilla-
tions in comparison with those sustained by the pressurizer-core system.

Reactor instabilities can also arise due to interactions between the
reactor primary system, heat-removal system, and fuel-storage system.
These are associated with the method of reactor operation, and are
discussed under the heading of Operational Stability.

Under certain conditions it may be desirable to operate reactors with
fission gases retained within the system; in these circumstances the buildup
of Xe!?5 during periods of low-power operation may influence subsequent
reactor operation. Although no stability problem is involved, a long time
scale is associated with controlling the reactor in these circumstances.
Reactivity effects and required fuel-concentration changes associated with
Xe!?? buildup and burnup are discussed below under the appropriate
heading,.

Nuclear stability. In studying nuclear stability, the equations of motion
for a single-region reactor system are normally used. These should be
adequate 1f the reactor behavior is controlled by a single region; however,
it should be remembered that the mean lifetime of prompt neutrons should
be that for the reactor us a whole. The general mathematical system is too
complicated to handle analytically, so that it is necessary either to resort

*Conversations with utility people indicate there is no accident on record in
which the high-pressure steam line failed instantly.,
78 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

to numerical integration of specific cases or to linearize the equations so
that they can be treated analytically. The linearized approach is valid for
very small power oscillations; since the nonlinear effects appear fo intro-
duce damping of the power surges, the reactor system should be stable if
the stability eriteria for the linearized equations are satisfied. Not meeting
the linearized stability criterin may not necessarily result in the buildup
of pressure oscillations to proportions where reactor safety is concerned;
satisfying the criteria should aid in eliminating the small pressure surges
which may physically weaken the system if they occur in a repetitive
maner,

The conventional equations of motion given in Lqgs. (2-21) through
(2-27) are used, except that the delayed-neutron precursors are assumed
to decay In accordance with a single effective decay constunt, and fluid-
flow effects are considered in which the average fuel-fluid temperature is
the linear average of the core inlet and outlet temperatures. Heat-
exchanger behavior will also affect the temperature of the fuel fluid and
is to be considered. The resulting linearized equations [36] lead to an
equation defining the criteria for stability of the reactor system. If only
the reactor core and pressurizer system are considered, corresponding to a
relatively large residence time for fluid in the core, the stability criteria
reduces to

 

8 ot oyt Bl 1B By o ;
viBlarat e+ B+ 5)| > (vt ) > 0 ey
Since all quantities are positive, the last inequality is always satisfied.
By replacing the other inequality with an equal sign and fixing all param-
eters but one, the value of the remaining parameter (which barely fails to
satisfy the inequality) can be obtained. By this procedure, the results
given in Fig. 2-21 were obtained for a specific value of ;. Stable operating
conditions are those lylng above the appropriate curve.

The above treatment neglects any effects that fluid fransport and heat-
exchanger behavior might have upon the temperature of the fluid entering
the reactor. Results of studies [36] in which these effects were considered
indicate that the resulting stability criteric ave less stringent than those
for systems in which fluid transport and heat-cxchanger behavior are
neglected. On this basis, the results given iu iig. 2-21 form reasonable
bases for safe design.

A one-region reactor will operate at a power density lower than that at
which the corresponding two-region reactor is operated, if the physical
size of the one-region reactor is large. Based on the stability criteria given
in Fig. 2-21, about the same degree of stability will result if the product of
w? w? remains constant in going from a one- to a two-region reactor. On
the basis of fuel-cost studies, the optimum two-region and one-region
2-3] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 79

 

 

 

 

 

5
10
T T
— —
5»_ —
21— _|
&
$ 10— —
= — ]
™ — —]
o | —
+ 5 —i
& ]
; ) -
24 ]
103 —
5 [ 11 ]
10 2 5 102 9 5 108 2
2
o -2
Trcy e

Fra. 2-21. Stability criteria for homogeneous circulating reactors. ;= normal-
ized friction coefficient = 0.5.

reactors do have w} w® which are roughly equal, with the one-region

reactor being somewhat more stable, However, there appears to be no
distinet advantage of one type over the other as regards nuclear stability,
since both normally will operate well within the stable region.

Operational Stability., Other types of stability which might be considered
concern the response of the reactor to changes i load demand and to
changes in rates of flow between the low- and high-pressure systems. These
usually involve power oscillations whose frequencies are much lower than
those sustained by the core-pressurizer system. In these circumstances the
general equations of motion can be simplified; however, load-demand
studies will concern the particular characteristics of the heat exchanger
under consideration,

Changes in fluid flow rates between the high- and low-pressure systems
m reactors can also influence reactor stability. During eperation of the
HIRI+1, 1t was noticed that under certain conditions the temperature and
power of the reactor would rise with time. This effect was termed the
“walkaway” phenomenon [39]. Tt was associated with a slow rate of
reactivity addition produced by an Increase in core fuel concentration,
leading to an increase in operating power and temperature. This increase
in fuel concentration resulted when water was removed from the core
(as a result of decomposition-gas formation and accompanying vaporization
of water) at a faster rate than it was returned from the low-pressure system.
This process continued until it was stopped by the operator.
 

 

 

 

TaABLE 2-16
Errect or XE'3% vpoN RreacTtor ConpiTiOoNs FoLLOWING SHUTDOWN AND RETURN TO Powgr
Reactor (a) (b) (c) (d) (e) (f) (g)
4 P 1 dN dk, Ak AT tmi
1ameter, d)o P _—— s €) ’ min,
ft Po N dt dt o, o hr
sec™! sec !
4 1013 0 3.0x 106 7.4x 1077 0.86 2 10
4 1014 0.1 6.8 x 10> 1.7X 1075 9.0 20 D
4 1014 0 8.5x 10-3 2.1 x 1075 13. 30 D
4 1015 0.1 6.9x 10~¢ 1.7xX 1074 17. 60 1
4 1015 0 3.5x 104 8.6 X 1075 33. v. large 3
% 1015 0 6.9x 1074 3.6x10°¢ 52. v. large 2

 

 

 

 

 

 

 

 

 

 

(a) Initial value of thermal flux before shutdown; averaged over high-pressure system;
neutrons,/cm?-sec.

(b) Power during shutdown relative to power before and after shutdown.

(¢) Relative rate of change in U235 concentration required to maintain criticality at time of
return to power (no change in fluid temperature).

(d) Rate of reactivity addition due to xenon burnout at time of return to power.

(e) Estimate of reactivity associated with removal of Xe!3% at time of maximum xenon
concentration.

(f) Estimate of fluid temperature decrease if reactivity addition associated with xenon buildup
were compensated by fall in reactor temperature.

(g) Estimate of the time required for xenon concentration to reach a minimum value following
return to power.

08

-INO

axv

SHOLOVHY SNOUNIDOKOH XNOIDITU-OMI

7 "dVHD]
2-5] HOMOGENEOUS REACTORS AND REACTIVITY ADDITIONS 81

The above type of phenomenon is a function of fuel flow rates between
the low- and high-pressure system and can therefore be controlled. Using
the applicable linearized equations of motion, the criteria for stability
toward walkaway can be obtained. This has been done for the HRE-2
[40]. Design parameters such as operating conditions, fluid-flow rates,
vessel volumes, recombination rate of decomposition gases, and other
conditions will influence the degree of stability. Walkaway will not occur
if decomposition gases are not formed. If gas is formed, a paramount
cause of instability 1s insuflicient overpressure. Increasing the fuel feed-
pump rate or the ratio of solution volume in the high-pressure system to
that in the low-pressure system, or decreasing the reactor temperature or
power will tend to increase reactor stability against walkaway. Walkaway
can also be prevented by automatic control devices which influence the
behavior of the heat exchanger. Walkaway conditions can be instigated
by reducing overpressure or by increasing the rate of gas formation.

Ejffect of Xe'35 upon reactor behavior. If reactors are operated with com-
plete recombination of the decomposition gases and no letdown from the
high-pressure system, there will be virtually complete retention of the
fission fragments and the fission-product gases within the reactor system.
Under such conditions, partial or total reactor shutdown will lead to an
increase in Xe'? concentration, which might, in turn, lead to difficulties
in maintaining criticality or reaching criticality upon subsequent power
demand. To investigate the above effect, calculations were performed
[41] for a 4-ft-diameter spherical reactor operating at 280°C, containing
U235 heavy water, ['35 and Xe!35 and operating at initial fluxes of 10'3 to
10'3 neutrons/(em?)(sec). lor these cases the reactor power was reduced
either to zero or to one-tenth the initial value, and the fuel concentration
required for eriticality was obtained for times following the power reduc-
tion. At the time that the xenon concentration reached a maximum value,
the power was returned to its original level, and at this point the maximum
rate of reactivity addition associated with xenon burnout was obtained.
The rate of change in fuel concentration required to maintain criticality
at this point was also obtained. In addition, the time required for the xenon
concentration to reach a minimum value following return to power was
evaluated. The maximum change in xenon concentration was used to
estimate the total reactivity addition associated with xenon buildup and
was Interpreted in terms of fluid temperature changes required to maintain
eriticality if the fuel concentration were maimntained constant. Table 2-16
summarizes the results obtained and, for comparison purposes, also includes
results for a reactor of infinite diameter. As shown in Table 2-16, the
maximum rate of reactivity addition associated with xenon burnout was
less than 1073 Ak, /sce, which does not appear to constitute a dangerous
rate. However, at an operating flux of 102 the increase in fuel concentra-
82 ONE- AND TWO-REGION HOMOGENEOUS REACTORS [cHAP. 2

tion or the decrease in temperature required to maintain criticality following
partial or total shutdown is so large that provisions should be made for
elimination of xenon by external means. At 10'* average flux it appears
that xenon poisoning may be compensated by decreasing the reactor
temperature.

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84 ONE- AND TWO-REGION HOMOGENEOUS REACTORS

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