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STUDIL3 o f t h e r e a c t i v i t y TEMPERATURE COEFFICIENT IN LIGHT WATER REACTORS
MALTE EDENIUS
DEPARTMENT OF REACTOR PHYSICS
Studies of the reactivity temperature coefficient in
light water reactors
Malte Edenius
AKADEMISK AVHANDLING
som framlagges till offentlig granskning
vid sektionen for teknisk fysik for av-
laggande av teknisk doktorsexamen i reak-
torfysik den 4 maj 1976, kl 10.00, semi-
narierummet, institutionen for reaktor-
fysik, Gibraltargatan 3.
Goteborg
Mars 1976
AE—RF—76—3160
STUDIES OF THE REACTIVITY TEMPERATURE COEFFICIENT IN
LIGHT WATER REACTORS
Malte Edenius
AB Atomenergi, Studsvik
March 1976
LIST OF CONTENTS
1. Introduction 1
2. Influence of temperature on the neutron
transport 4
2.1 Thermalization 4
2.1.1 Neutron scattering laws 5
2.1.2 Scattering models for water 7
2.2 The Doppler effect 9
2.3 Density effects 14
3. The cell code AE-BUXY 17
3.1 Nuclear data library 17
3.2 Resonance treatment 19
3.3 Micro group calculation 21
3.4 Macro group calculation 23
3.5 Fundamental mode calculation 24
4. Comparison between theoretical results
and experimental information 28
4.1 Description of the measurements 29
4.2 Description of the calcula-
tional methods 30
4.3 The reactivity worth of spacers 31
4.4 Results of comparison between
theory and experiment 33
4.4.1 Lattices with 1.35 % enriched
U 0 2 33
4.4.2 Lattices with 1.9 % enriched
U O 2 rods and PUO 2 rods 35
4.4.3 Comments to the results 39
5. The components of the temperature co
efficient 44
5.1 Partial temperature, coefficients 46
5.2 Contributions to the tempera
ture coefficient calculatedby COEFF 52
5.2.1 Description of COEFF 52
5.2.2 Results from calculations with
COEFF 55
6. The influence of approximations in the
theoretical treatment on calculated temperature coefficients
6.1 The crystalline binding in
6.2 The cylindricalization of pincells
6.3 Comparison between pin cell
calculations using isotropic and anisotropic scattering
6.4 The calculation of leakage
6.5 The number of energy groupsand space points
6.5.1 Macro groups and Gauss points in AE-BUXY
6.5.2 Energy groups and mesh points in the diffusion theory calcu
lation
6.6 The influence of thermal expansion on reactivity
6.7 The plutonium fission neutron spectrum
7. The influence of nuclear data on the
calculated temperature coefficient
7.1 Comparison of temperature co
efficients calculated by use of ENDF/B and UKNDL data
7.2 The effective resonance integral of U-238
7.3 Thermal scattering data
7-3.1 Scattering in water
7.3.2 Scattering in UC^
7.4 Thermal absorption and fission
cross sections
7.4.1 Group cross sections for a
1/v-absorber in Maxwellian spectra of various temperatures
7.4.2 Thermal data for U-235 and U-238
8. Summary
9. Acknowledgement
10. References
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1. INTRODUCTION
One important quantity in determining the operating charac
teristics and safety of nuclear reactors is the temperature
coefficient of reactivity. The isothermal temperature coef
ficient in Light Water Reactors (LWR) varies considerably
with the design, the moderator temperature and the boron con
centration in the moderator etc. Typical values for a fresh
Boiling Water Reactor (BWR) core is -5 pcm/°C (1 pcm = 10
at 20°C and -25 pcm/°C at 280°C. Typical values for a Pres
surized Water Reactor (PWR) are in the range -5 to -30 pcm/°C.
Much effort has been devoted to the development of proper
methods for the calculation of temperature coefficients. De
spite this, the methods in current use to calculate the tem
perature dependence of reactivity in light water reactors are
not altogether successful [l - 4]. As an example measured and
calculated values of dk ,-,/dT from the Swedish Oskarshamn-Ierr
BWR reactor are shown in Fig 1.1.
The calculated temperature coefficient is usually 2 - 5
pcm/°C too negative compared to the measured value. Going
from room temperature to operating temperature this means an
error in the predicted reactivity of about 1 %. Thus an in
centive exists for further studies in this field of reactor
physics. The literature lacks information about clean
(simple, well-defined) and precise high temperature experi
ments. With the extensive series of experiments which have
been performed in the high temperature facility KRITZ at
Studsvik sample data to compare with calculated ones have
been produced.
In the present paper a survey of the temperature effects in
a nuclear reactor is first given. Then follows a description
of the analysis of the KRITZ experiments and a comparison
between theoretical results and experimental information.
The theoretical methods used in the analysis are described.
Different components of the temperature coefficient are
studied in chapter 5. Uncertainties in the employed theore
tical methods are discussed in chapter 6 and the influence
of nuclear data on the temperature coefficient is investiga
ted in chapter 7. A summary is given in chapter 8.
Tem
p co
eff
( pcm
/ °C
)
Fig. 1.1 Oskarshamn I. The isothermal temperature coefficient
versus temperature [1].
2. INFLUENCE OF TEMPERATURE ON THE NEUTRON TRANSPORT
The influence of temperature on the neutron transport is
caused by the thermal movement of nuclei influencing the
scattering of thermal neutrons and the Doppler broadening of
resonances and by the thermal expansion of different mate
rials.
2.1 Thermalization
In many materials which are present in a nuclear reactor the
atoms may be considered to be free. The energy distribution
of the atoms is then the Maxwell distribution and it is poss
ible to derive an accurate expression for the scattering of
neutrons. This model can be used e.g. for scattering against
heavy nuclei which does not affect the neutron spectrum very
much.
For light nuclei, however, the treatment of scattering re
quires a consideration of the chemical binding and for a sa
tisfactory treatment of the thermalization in water it is
necessary to use scattering cross sections computed according
to a relevant model for the scattering process. A number of
approximations are involved in these scattering models. The
accuracy of the model is of great importance when predicting
the temperature dependence of the neutron spectrum and hence
for predicting the temperature effects in thermal reactors.
It may be noted that an accurate treatment of the thermali
zation is especially important in systems with only partial
thermalization, i.e. when the neutrons do not obtain a Max
wellian velocity distribution before they are absorbed. This
is the case in light water reactors.
The influence of binding on absorption is negligible and
absorption cross sections can be taken to be the same as for
free nuclei.
2 . 1.1 Neutron scattering laws
The scattering in a monoatomlc gas is treated in most books
on reactor physics and will not be discussed here. In
applications all scattering except that in the main con
stituent of the moderator is usually considered to obey
the free atom scattering law.
For neutrons undergoing scattering in a medium containing
bound atoms it has been shown [5] that the scattering function
can be written as the sum of differential coherent and in
coherent cross sections
E and ft are the energy and the unit vector in the direction
of motion before the collision. E* and ft* are the corre-
are the macroscopic bound coherent and incoherent cross
sections, e = E-E* is the energy change of the neutron and
•tfic = m ( v - v T) is the neutron momentum change vector.
The functions S( k ,c ) and S. (k ,e ) are defined by— inc — J
Z (E-*E 1 1) * Z L---- coh
(E->E *, 1) +1. (E-*E f 1)— m e ------ (2 .1)
(2 .2 )
(2.3)
spending quantities after the collision. Z , and E. r n coh im e
(2.4)
eiUr-xt/fi)
G (r,t)drdt s
(2.5)
The pair distribution functions Gs (£> t) and G^Cr,t) de
defined as the probability that a nucleus which is at origin
at time t = 0 will be present within a unit volume at posi
tion _r at time t. G^(r,t) is the probability that a nu
cleus other than that at origin at time zero will be at po
sition r at time t.
The interference effects of the scattering are contained in
G^(_r, t). These interference effects are important for elastic
scattering in many materials, but for inelastic scattering
in most liquids and polycrystalline materials they can be neg
lected, i.e. G^(r,t) can be put equal to zero. Then
which is known as the incoherent approximation. For light water
the incoherent approximation is especially good because
neutron scattering by hydrogen is almost completely inco
herent (a = 1.8 b and o. = 80 b ) . coh m e
One further approximation is employed in many scattering
models, viz. the Gaussian approximation. A function inter
mediate between G(^r,t) and S(k ,g ) is defined by
The Gaussian approximation of the intermediate scattering
function is given by
termine the dynamics of the scattering systems. Gg (r^t) is
(2 .6)
(2.7)
(2 .8)
with
f (w)e~fiu/2kT
2ui s i n h ( W 2 k T )-not ,
e -1
A is the mass of the scattering atoms, k the Boltzmann
constant, T the temperature, w is the angular oscillation
frequency and f(w) the frequency spectrum normalized so
that
ff(w)du) = 1
For simple scattering models assumes the Gaussian
form and this form is also applied in many sophisticated
models for scattering in water. In order to determine
>'(t)-y(0) the frequency spectrum, f(ui), is estimated from
physical considerations.
2^1.2 _____§cattering_models_for_water
One of the early models is the Nelkin model [6]. In this
model the hindered rotational motion is approximated by a
torsional oscillation. The incoherent approximation is used
for the scattering by hydrogen and the Gaussian approxima
tion is employed with a spectrum representing a set of
discrete oscillator frequencies. The spectrum is written
I 1f(u>) = I -±- 6(u>-w.) (2.10)i=l i L
A 1 =18 ' = 0
2.32 •Ftu>2 = 0.06 eV
A 3 =5.84
fiw3= 0.205 eV
> Jill 2.92 ■fiu>.
4= 0.481 eV
The first term in the summation represents the translational
motion of the free gas molecules and the second term the
hindered rotation which is assumed to be a torsional oscilla
tion. The remaining two terms represent vibrational modes.
It is possible in the framework of the Gaussian approximation
to reconstruct the complete S(a,|3) function from p(6) .
Egelstaff and Schofield [7] have proposed a scattering model,
which is known as the effective width model, with
where K^(x) is the Bessel function of the second kind.
The effective width, q, is a measure of the width of p(3).
The model degenerates to that of a free gas as q-K) . The
values of q to be used are determined by correlation to
integral experimental data.
Aj, is an effective mass of the i:th quantum state.
Defining
S(a,3) H kTe6/2 S(k ,e ) (2 .11)
with
2(2 .12)
6-G
(2.13)kT
the spectral density function can be introduced as
p (8) s 26 sinh(8/2) Him S(a,B)] rv-vn a Ja-’-O
(2.14)
(2.15)
The Haywood model [8] is based on experimental values of
p(B) for water.
The three scattering models for water mentioned above are
the ones most frequently used in reactor physics calcula
tions. They are also the models which have been used in the
present work.
The complete scattering cross section for the water moleeule
is obtained by the addition of oxygen as a free gas.
2 . 2 The Doppler effect
The resonance cross section is for nuclei at rest given by
the Breit-Wigner formula
E x ( E ) = N o o rx /ITE 2 2
4 (E-E ) +r o
/IT 2 rrI (E) = No V - J ---- — - ■ - -S.
E 4 (E-E ) +r2 L1
F_ 4(E-Eo ) R
+ r *+ No
(2.16)
(2.17)
The index x denotes absorption or fission. N is the atom
number density. T , T and T are, respectively, the x n
width for reaction x , the width for neutron emission and
the total width of the resonance. E is the resonanceo
energy. K is the reduced de Broglie wave length of the
neutron and R is the nuclear radius, o is the potential
scattering cross section and a denotes the peak value ofo
the total resonance cross section which is given by
? T'n(EJ. n o°o = *o 8 r—
(2.18)
The statistical factor g is expressed by
2J+1
g ~ 2(21+1)
where I is the spin of the target nucleus and J the
spin of the compound state.
When the nuclei are in thermal motion, the resonances are
broadened as a result of the Doppler effect. The Doppler
broadened cross section can be written [9 ]
S(£,e) is the scattering law function defined in Eq. (2.4)
and £ is the momentum of the neutron. In most applications
it is assumed that the absorbing nuclei have a Maxwellian
velocity distribution at the temperature of the medium. At
temperatures above the Debye temperature this approximation
is a good one, because the velocity distribution of the
nuclei is then insensitive to the chemical binding. The
Debye temperature is about 620 K [10] for UO^ and 200 K
for metallic uranium. The effects of crystalline binding on
the resonance absorption will be discussed in § 6.1.
Assuming that the nuclear velocities have a Maxwellian
distribution one obtains [11]
where the Doppler functions t|>(£,Y) and x(?>Y) are
defined by
(2 .20)
(2 .21)
Is (E) = N o q - p *(C,Y)+Noo | x(?,Y)+Nap (2 . 22)
g
2/tT
exp
/iT ■
- \ g(X-Y)2]
i+x2dX (2.24)
and
A = / H
2 1 2 X e — ( E ~ E ) : E = — m v where v is the
T r o r 2 r rrelative neutron-nucleus speed.
The Doppler width, A, represents the effect of temperature
on the shape of the resonance and is a measure of the width
of the Doppler broadened resonance. A is proportional to
the square root of the temperature, T, and the energy, E.
At low temperature A is small so that C is large and
the integral (2.23) becomes
*(C,Y) » — K (2-25)1+Y
Inserting (2.25) into (2.21) one obtains the Breit-Wigner
formula for the unbroadened resonance. At the other extreme,
i.e. at very high temperature, t, is small and
U.(?,Y) « ^ x. exp(- \ C2Y2) (2.26)
so that near the resonance peak
. . X / O / nSc(E) = Nao — V E ~ T exp
In this Gaussian expression A determines the width.
The shape of a resonance is changed markedly by the Doppler
broadening but the integral
stant
o(E)dE is approximately con-
a (E)dE = - a r i o
4*(C»Y)dY = - t a r z o (2.28)
In deriving the expressions (2.21) - (2.28) for the Doppler
broadened resonance cross sections it was assumed that the
velocity of resonance nuclei is much smaller than that of the
neutron. Except for low energy resonances in light nuclei
this approximation is justified, and it introduces negligible
errors when considering the U-238 resonance absorption.
The reaction rate within a resonance is a (E)<|>(E)dE and
although the integral (2.28) does not change with temperature
the resonance absorption is increased at higher temperature
because the absorption rate is changed as a result of less
marked dips in the flux at Doppler broadened resonances.
If <j>(E) is normalized to <f>(E) = 1/E above the resonance
0 ^(E)<j>(E)dE is called the effective resonancethe quantity
integral and is represented by RI
In the narrow resonance approximation [12] the effective
resonance integral is written
N.RI = x
£x (E)£p dE
Zt(E) E
where is the potential scattering cross section and
I (E) the total cross section. If E/E is set equal to t o
unity and the interference between resonance and potential
scattering is neglected, one obtains for a single resonance
f Na (r / r ) * (c ,Y ) £R I = — — - — - — -------------------E
x E I No !{.(£,Y)-!-!o 3 o p
Defining the function
»(C,Y)<KC.Y)+B
dY C
RI can be written x
N'RI = I — • Ji x p E
o ('■ £ c
For a series of resonances the total resonance integral is
obtained by summation of Eq (2.32) over all individual
resonances.
The behaviour of J(t,f3) versus 8 is shown in Fig 2.1
(reproduced from [13]). For large values of 6=Sp/NaQ the
flux depression by the resonance is small and J(?,8) is
insensitive to £ , i.e. to the temperature. J(^,6) is
independent of ? also when 8 is small. This is due to
the very strong flux dip at the resonance center causing
most of the absorption to occur at some distance from the
resonance center, where the shape of the cross section is
independent of the temperature.
• 31)
.32)
The Doppler broadening is most important when J(C,8)
varies significantly with £ at a given 3 • From Fig 2.1
this is seen to be the case for 7<j<16 , i.e. in the
range 0.001<8<1 . For U-238 this roughly corresponds to
1 00<a <1 05 .P
2. 3_________ Density effects
The effects of moderator density on reactivity are important
for the temperature coefficient in water moderated systems.
The x^ater density decreases with higher temperature causing
the leakage of neutrons and the number of fast fissions to
increase, the. resonance escape probability and the absorption
in the water to decrease and the thermal spectrum to become
harder. These effects are studied in chapter 5.
The effect of a change in water temperature is pronounced
aL high temperature due to the variation of the water d0
density derivative, — , with temperature (Fig 2.2).
A less important component of the temperature coefficient
is due to the thermal expansion of the fuel, the canning
and construction material. These effects have been investi
gated for some lattices, the results being given in § 6.6.
J(E.P)
j (where (3 =2^ x 10"5)
Fig.2.1 The Doppler broadening function J ( £ . P ) versus p for various £ .
3. THE CELL CODE AE-BUXY
AE-BUXY is the AE-version of the code BUXY (Burnup in xy-
geometry). It is a spectrum code for pin cells as well as
for BWR and PWR fuel assemblies. In the present work, we
have used only the pin cell option and the description be
low will be limited to this option. A flow chart is shown
in Fig 3.1.
3.1_________ Nuclear data library
The data library contains microscopic cross sections in 69
energy groups (Table 3.1) divided into 14 groups in the fast
region above 9 keV, 13 groups in the resonance region between
4 eV and 9 keV and 42 thermal groups below 4 eV. Most of the
calculations performed for this work have been made with a
library based on the UK Nuclear Data File [14]. A library
based on ENDF/B III has been used for some calculations de
scribed in chapter 7.
Maxwellian spectra with 1/E-tails have been used for weight
ing thermal group cross sections. Group averaged cross sec
tions in the resonance region have been obtained assuming a
1/E-specrum and a spectrum typical for a water system has been
used in the fast region. Effective resonance integrals for
U-235, U-236, U-238 and Pu-239 have been provided through
detailed slowing down calculations for homogeneous mixtures
of the absorber and hydrogen. Resonance integrals are tabu
lated as function of potential scattering cross section and
temperature.
For the principal moderators scattering matrices are avail
able based upon alternative theoretical models. We have used
the Nelkin model for water when not otherwise stated. Scat
tering matrices are tabulated for a representative range of
temperatures.
Table 3.1 Energy boundaries i
Group Energy
Mev
1 10 .0 6 .0655
2 6 .0655 -3 .679
3 3 .679 -2 .231
4 2 .231 -1 .353
5 1.353 -0 .821
6 0 .821 -0 .500
7 0 .500 -0 .3025
8 0 .3025 -0 .183
9 0 .813 -0 .1110
10 0 .1110 -0 .06734
11 0 .06734- 0 .04085
12 0 .04085- 0 .02478
13 0 .02478- 0 .01503
14 0 .01503- 0 .009118
eV
15 9118.0 5530.0
16 5530.0 3519.1
17 3519.1 2239.45
18 2239.45 -1425.1
19 1425.1 - 906.898
20 906.898 - 367.262
21 367.262 - L48.728
22 148.728 - 75.5014
23 75.5014- 48.052
24 48.052 - 27.700
25 27.700 - 15.968
26 15.968 - 9.877
27 9.877 - 4.00
28 4.00 - 3.30
29 3.30 - 2.60
30 2.60 - 2.10
31 2.10 - 1.50
32 1.50 - 1.30
33 1.30 - 1.15
34 1.15 - 1.123
35 1.123 _ 1.097
the 69 group library
Group Energy
eV
36 1.097-1.071
37 1.071-1.045
38 1.045-1.020
39 1.020-0.996
40 0.996-0.972
41 0.972-0.950
42 0.950-0.910
43 0.910-0.850
44 0.850-0.780
45 0.780-0.625
46 0.625-0.500
47 0.500-0.400
48 0.400-0.350
49 0.350-0.320
50 0.320-0.300
51 0.300-0.280
52 0.280-0.250
53 0.250-0.220
54 0.220-0.180
55 0.180-0.140
56 0.140-0.100
57 0.100-0.080
58 0.080-0.067
59 0.067-0.058
60 0.058-0.050
61 0.050-0.042
62 0.042-0.035
63 0.035-0.030
64 0.030-0.025
65 0.025-0.020
66 0.020-0.015
67 0.015-0.010
68 0.010-0.005
69 0.005-0
3. 2_________ Resonance treatment:
A special calculation is performed to determine the effective
resonance integrals in the energy region between 4 eV and
9118 eV. Resonance absorption above 9118 eV is regarded as
being unshielded. The 1.0 eV resonance in Pu-240 and the
G.3 eV resonance in Pu-239 are adequately covered by the
concentration of thermal groups around these resonances and
are consequently excluded from the special resonance treat
ment. Four nuclides viz. U-235, U-236, U-238 and Pu-239 are
treated as resonance absorbers.
The subroutine for calculation of effective cross sections
in the resonance region is based on an equivalence theorem
[15] which relates the tabulated resonance integrals to the
particular heterogeneous problem. The equivalence theorem is
derived using a suitable rational approximation for the fuel
to fuel collision probability.
The resonance integral in the fuel is in the narrow resonanct
approximation [16]
RI =- 0 (E)
a (E) *'(1-Pff-)ot + ap P ff-* E~
where is the absorption cross section, o the total
cross section, j the potential scattering cross section
of the fuel with admixed moderator, and P ^ the self col
lision probability in the fuel region.
Using a rational expression for P ^
x b . ap ff = y ——- ; x = — and J S. = 1 (3ff r x+u. o lt 1
l i e i
equation (3.1) can be expressed as a sum of homogeneous
resonance integrals.
ef fa = c +a.a
P p i e
and
a = 1/(NX,) . e
Z is Che average chord length in the fuel and N the num
ber density of the absorbing nuclide.
To obtain an expression of the form (3.2) for an infinite
uniform pin cell lattice we write
V . P f b P b f O MP ff = Pff + l-pKK (3*4)
D D
p ^ is the fuel-to-fuel collision probability for an isola
ted rod, p ^ is the probability for neutrons entering the
cell isotropically through the cell boundary to suffer their
first collision in the fuel, p^£ is the probability for
neutrons born uniformly and isotropically in the fuel to
reach the cell boundary uncollided and p., is the probabil-bb
ity for neutrons entering the cell isotropically through the
cell boundary to traverse the cell without collision.
The probabilities p,, , p. , and p., are calculatedlb br bb
using formulae given by Bonalumi [17] and the final expres
sion for can be written in the form of Eq (3.2) with
two terms if p ^ is given by the Wigner approximation [18]
Pff = x+a (3'5)
a is the so-called Bell factor and is calculated by a poly
nomial fit [19]. The final expression for the resonance in
tegral is then
R I = R I, (a +a_o ) + (1--&)RI, (a +aa ) h p £ e n p e
where is a parameter depending only on the materials between
the fuel regions and
nT / \ f P a dERI, (o )= j~ --- =—h P J o t E
is the resonance integral of a homogeneous medium.
In its original form Eq (3.6) gives the narrow resonance
approximation of the resonance integral. By modifying cr ,
the intermediate resonance approximation [20] is obtained.
The effective cross section in energy group g is given by
[21]
RI
° 8 --------- f e - « - 7>T “g oeff
P>g
t is the lethargy width of group g.8
The resonance integral is then corrected for the overlap ef
fect as described in [21]. The influence of this correction
on the Doppler coefficient is very small and it will not be
discussed here.
3.3_________ Micro group calculation
The micro group calculation is made by use of collision pro
babilities for the three-region (fuel, canning and modera
tor) cylindricalized pin cell. It provides 69 group spectra
which are used for energy condensation to broad group cross
sections. These are then used in the macro group and funda
mental mode calculations.
I. V. <j>. = I P. . V. (I I. , <t>. ,+ S. \ (3.8)i,g x ri,g ^ l-s-j.g 3 j.g^-g' j,g' j,g)
S. is the fission source in the fuel region, g is the J »g
energy group index and i and j the spatial region index
referring to fuel, canning and moderator.
Accurate collision probabilities for the cylindricalized pin
cell with flat source in each of the three regions are cal
culated using the FLURIG method developed by Carlvik [22].
In evaluating the collision probabilities we start by eli
minating the z coordinate. With notations according to Fig
3.2
TI
P(x) da = i da j exp(- ■^r^)sin0d8
0
is the probability that neutrons isotropically emitted from
a line source will travel the optical length x within the
azimuthal angle da without colliding.
Defining the Bickley functions
Ki»(x) =
we get
-x cosh u----------- du
, n cosh u
P ( t ) = Ki2(T) (3.9)
From (3.9) the collision probabilities in cylindrical geo
metry are derived
E.V.P. . = •=- | da J J 2 IT
dy[Ki ( t ..)-Ki ( t . . + t . ) 3 i ] 3 i j j
- Ki ( t .,+t.)+Ki ( t . . + t . + t . ) ] 3 i j l 3 ij l j
where t . , t . . and t . are optical lengths within andi ij J
between the regions under consideration. The numerical
evaluation of P is an integration over the system in the
spatial variable y and in the angular variable a.
Let Pj«-j '3e t'le first flight collision probability within
the three-region pin cell, p, . is the probability for a
neutron born in region j to reach the boundary of the pin
cell and the probability for neutrons isotropically
reflected at the boundary to have their next collision in
region i . p is the probability for the isotropically bb
reflected neutrons to pass the pin cell without collision.
The final expression for t
in an infinite lattice is
The final expression for the collision probability
P. . = p. . ,i-J i-J l"Pbb
+ ! i ± V i ( j . U )
3.4_________ Macro group calculation
Using the spectra from the micro group calculation macro
scopic cross sections are condensed to a maximum of 25 energy
groups to be used in the macro group calculation. The condensa
tion is made by flux weighting of the cross sections.
The macro group calculation is made in the same annular
geometry as the micro group calculation. The DIT method [23],
i.e. a point formalism is used to solve the integral
transport equation
X is the eigenvalue and x is the fission spectrum. The
transport matrices T give the flux in energy group> g
g at the point k due to a unit source at £ . The formal
volumes are given by
V2 =
are Gauss weights and r^ is the radius of a point
situated in the interval bounded by and ■
3.5_________ Fundamental mode calculation
The aim of the fundamental mode calculation is twofold.
First, the leakage and the migration area are determined
assuming a fundamental buckling mode. Secondly, the infinite
lattice results obtained from the transport calculation are
modified to include the effects of leakage in predicted
group constants.
The fundamental mode calculation is carried out by use of
the B^-leakage method. P^-scattering matrices are available
in the library for the principal moderators and the P^~
scattering terms are explicitly represented in the B^-
equations.
The fundamental mode equation solved by AE-BUXY is in
matrix form
a-E°)<j> + B2d(f> = J T* (3.13)S K
k = the eigenvalue
T , = x , where x is the fission spectrum88 8 * > 8 8
I = diag(I ) (the total cross section)t > 8
£° = £ , (the zeroth moment of the scattering matrix)® 8 8
d = (3a Z-E1)-1s'
. (the first moment of the scattering matrix)s g 8
f I f L . ) 2 . 3 \ l J
ar(B \
arctgj
. <
3 \z J A -1 x % o»8
; B > 0
2 2 ; K = -B > 0
£ I+k /ZA = - l i n ______&
o , 8 2 k 1 - tc /Es
B2 = -K2
All quantities are cell integrated values.
Three fundamental mode calculations are made, namely:
. 2k is obtained with B = 0
00
ke^ is obtained for a given geometrical buckling
2 2 B is varied so that k ,, = 1. B is then
eff 2equal to the material buckling B^ .
Equation (3.13) is solved in the macro group structure.
Fig.3.1 Flow chart of the pin cell calculation in AE-BUXY.
Fig.3.2 Elimination of the z co-ordinate in cylindrical geometry.
4. COMPARISON BETWEEN THEORETICAL RESULTS AND
EXPERIMENTAL INFORMATION
In order to compare the theoretical model for calculation
of the temperature coefficient with experimental informa
tion, it is desirable to use measurements on simple geomet
ries, i.e. uniform pin cell lattices. There are not many
papers published about reactor physics experiments on light
water moderated systems at high temperature. We have looked
at some early exponential experiments but found that the ac
curacy is not good enough for detailed comparisons with theo
ry, one of the main problems being the nonexistence of an
asymptotic region due to the small size of the subcritical
lattices.
Measurements on light water systems in critical facilities
at temperatures up to 90°C have been performed at several
laboratories, but at temperatures above 200°C we know only
of the experiments made in the KRITZ facility [24] at Studs-
vik and Soviet measurements [25] on highly enriched (m 80 %)
rods in H^O. Power reactor measurements provide valuable in
formation on the adequacy of the calculational methods, but
the conditions are usually very complex including a great
number of different absorber and fuel rods in different as
semblies, which requires two- or three-dimensional core cal
culations. We have therefore chosen to analyse some KRITZ-
experiments in detail. It is very valuable that these mea
surements have been made at temperatures up to 245°C. Com
parisons have been made on both uniform pin cell lattices
and cores containing BWR or PWR fuel assemblies. Here we
will, however, refer only to the uniform lattices since thest
are the most appropriate ones for the purpose of revealing
fundamental discrepancies between theory and experiment in
determining temperature effects in reactors.
4.1 D e s c r ip t io n o f th e m easurem ents
The critical facility KRITZ is used for reactor physics mea
surements on water-moderated cores at temperatures up to
245°C. KRITZ consists of a pressure tank with an insert ves
sel of which the outer wall is a circular cylinder loosely
fitting into the pressure tank, whereas the inner wall con
sists of a square cylinder. Vertical and horizontal cross
sections of the reactor tank and insert vessel are shown in
^igs 4.1 and 4.2.
Coarse reactivity control is achieved by poisoning the water
with boric acid and fine reactivity control is made by ad
justing the water level. No control rods are involved. It
means that uniform lattice arrangements can be loaded with
out any interference with heterogeneous devices except for
spacers within the core volume.
Critical water levels and flux distributions in the vertical
and horizontal directions are measured and experimental
values of the material buckling are evaluated. The flux
distributions are obtained from activation of copper wires.
Critical measurements have been performed on a large number
of uniform lattices of UC^ and PuC^ rods as well as on more
complex BWR and PWR geometries [26]. The experiments used
in the present study are all on uniform lattices containing
UO 2 enriched to 1.35 % or 1.9 % or mixed oxide (1.5 % PuO^ -
depl UC^)• The moderator to fuel ratio, boron contents and
core size are varied.
The fuel rods with bottom extensions were standing on stain
less steel beams and their radial positions were determined
by spacers in form of horizontal straight wires placed in
different planes. The wires in each plane were turned per
pendicular to those in the neighbouring planes. The distance
between the spacer layers was in the cores with the 1.35 %
UC> 2 rods 36.0 cm which means that they affect the reacti
vity. In the cores with 1.9 % UO 2 or Pu ( > 2 rods the spacers
were placed above the critical water level and below the
active length of the rods.
4. 2_________ Description of the calculational methods
The cell code AE-BUXY was used to calculate the material
buckling and for a geometrical buckling given in input.
When the input geometrical buckling is put equal to the
experimental material buckling, the deviation of from
unity gives a measure of the discrepancy between AE-BUXY
and experiment. This comparison between theory and experi-2
ment is relying on the experimental material buckling, B ^ e x p )
obtained from copper activation. Alternatively, cal
culations may be carried out on the whole core using material
composition, geometry and critical water level as input. An
accurate calculational model requires a three-dimensional
code. Three-dimensional calculations are, however, expensive
and have been avoided in this work. We have instead simulated
the experimental configuration with a two-dimensional diffusion
theory code in xy-geometry, DIXY [271, using the measured
axial buckling to account for the axial leakage. The largest2
uncertainty in the experimental B^ is due to the deter
mination of the extrapolation length from the flux mapping.
The measured extrapolation lengths have been compared with
calculated ones. These calculations were carried out in
one dimension by use of the S code DTF-4 [28]. DTF-4 wasn
also used to calculate the reactivity worth of the spacers.
Few-group constants for DIXY and DTF-4 were generated by
AE-BUXY.
Cross sections for the radial water reflector were also
generated by use of AE-BUXY. The whole core was cylindricalized
and a transport calculation was carried out on the cluster
(containing 1500-2000 fuel rods) surrounded by a thick
water region. Few group reflector cross sections are pro
duced at different distances from the fuel region. It was
found that the 4 and 8 group cross sections do not vary
much with the distance from the fuel region. Average reflec
tor cross sections were therefore used in the core calcula
tions .
All pin cell calculations were carried out with 17 energy
groups in the macro group calculation of AE-BUXY and with
3 Gauss points in the fuel, 2 in the canning and 3 in the
cylindricalized water region. The same 17 groups were used
in the fundamental mode calculation. The choice of the above
parameters is discussed in chapter 6.
The DIXY calculations were made in 4 energy groups (10 MeV-
- 0.5 MeV - 9118 eV - 0.625 eV - 0) in the 1.35 % U02
lattices and with 6 groups (10 MeV.- 0.5 MeV - 9118 eV -
- 4 eV - 0.625 eV - 0.140 eV - 0) in the 1.9 % U02 and Pu02
lattices. The number of mesh points was chosen from a few
calculations where the distribution and total number of mesh
points were varied (see chapter 6).
Axial calculations with DTF-4 were made in the S^-approximation
with 4 energy groups and including anisotropic scattering
explicitly.
4• 3_________ The reactivity worth of spaces
The reactivity worth of spacers need be taken into account
only in the cores with the 1.35 % U02 fuel. In these lattices
horizontal straight wires of stainless steel, 0.40 cm in
diameter, placed in different planes were used to determine
the radial positions of the fuel rods. The spacer plane
separation was 36.0 cm.
The reactivity worth of the spacers and its temperature
dependence was calculated by use of DTF-4 and AE-BUXY. An
example of the geometry used in DTF-4 is shown in Fig 4.3.
The axial direction of the core and bottom- and top-reflec-
tors are considered. The radial leakage is taken into accouut
by use of a transverse buckling equal to the measured radial
buckling B2 = B2 + B2 .r x y
Homogenized macroscopic cross sections were calculated with
AE-BUXY. The calculation of cross sections for the pin cell
region (material 4 in Fig 4.3) and the region above the water
level (where the s.pace between the rods consists of H^O-
steam) (material 6) is straight forward. To be able to
handle the spacer region (material 5) the SS-wire has to be
homogenized with the water.
Some results from calculations of the spacer reactivity
worth are given in Table 4.1.
Table 4.1: The influence on reactivity from spacers in 1.35 % U( > 2 lattices calculated by DTF-4
Boron
conc
(ppm)
Temperature
C°c)
Akeff
(pcm)
AB2m
(nf2)
O 90 - 500 - 1.3210 - 560 - 1.3
175 90 - 410 - 1 .0
210 - 480 - 1.1
Experimental buckling effects of spacers were determined
only at 20 °C and 0 ppm boron. The experimental result is2 -2
[24] AB = -1.3+0.2 in . This is in good agreement with m —
the calculated value.
^ 4 ^ 1 ___Lat tices_wi th_l_135_%_enr iched_UQ2
A great number of measurements has been performed with the
fuel rods containing 1.35 % enriched UC^ [24]. Two cores
have been analysed in detail. The first one contains 39x39
pins and is without boron in the water. The seconc core con
tains 46x46 pins and the boron concentration is 175 ppm.
The moderator to fuel volume ratio, V A ' , is 1.4 and them o
measurements cover the temperature range 20-210 C.
Results of the analysis are collected in Tables 4.2-4.4.
Table 4.2: Calculated k and k for the 1.35 % U0„ lattices---------- co eff 2
Core size Boron cone. (ppm)
Temp.
(°C)
AE-BUXY DIXY
k « effk03 k
eff
20 1.16294 0.99402 0.99381
39x39 090 1.15566 0.99091 0.99144
160 1.14677 0.98773 0.98876
210 1.13872 0.98587 0.98673
20 1.12172 0.99636 0.99569
46x46 17590 1.11621 0.99365 0.99383
160 1.11003 0.99186 0.99198
210 1.10448 0.99100 0.99102
It is seen in Table 4.2 that the reactivity is slightly
underpredicted. The deviation of ke££ from unity increases
with temperature, i.e. the predicted temperature coefficient
is too negative.
Ak _j-/AT in Table 4.3 is defined as the difference between eff
kg££ at two temperatures divided by the temperature differ
ence. Thus A k ^ ^ / A T gives the discrepancy between the cal
culated and measured temperature coefficient.
Table 4.3: Ak _C/AT derived from Table 4.2 ---------- eff
Core size Boronconc.(ppm)
Temp.interval
(°C)
Akg^^/AT (pcm/°C)
AE-BUXY DIXY
20- 90 -4.4 -3.4
39x39 0 90-160 -4.6 -3.8
160-210 -3.6 -4.2
20- 90 -3.9 -2.7
46x46 175 90-160 -2.6 -2.7
160-210 -1.8 -2.0
The calculated temperature coefficient is (2-4) pcm/°C
more negative than the experimental one. There are only
small discrepancies between AE-BUXY and DIXY.
k ^ £ was calculated using both DIXY and DTF-4 at 90 °C and
210 °C. In DIXY the experimental axial buckling was used,
and in DTF-4 calculations were carried out using the
experimental radial buckling to give the transverse
leakage. As seen in Table 4.4 the agreement between the
different calculations is very good. This means that mea
sured and calculated axial and radial bucklings are consistent.
Table 4.4: Comparison between calculated by use of
AE-BUXY, DIXY and DTF-4
Bcronconc.(ppm)
Code keff S k« f f /ST
(pcm/°C)90 °C 210 °C
AE-BUXY 0.9909 0.9859 -4.2
0 DIXY 0.9914 0.9867 -4.0
DTF-4 0.9898 0.9845 ■ -4.4 '
The cores containing 1.9 % enriched IK^ rods and PuO^ rods
are regular square lattices with either only UC^ or PuC^
rods or with a central square zone of PuC> 2 rods surrounded
by a UO 2 zone. Three different volume ratios were covered
by the experiments [26] with temperatures from 20 °C up to
245 °C. Lattice data are given in Table 4.5.
In the AE-BUXY calculations the plutonium was assumed to be
homogeneously distributed in the fuel. In reality the plu
tonium fuel used in these experiments contains Pu0 2 _particles
whose average size is about 25 pm. This non-homogeneity of
U0 2 ~Pu0 2 needs to be considered because of the self shielding
in the Pu02~partides. As a result of the shielding the
Pu-239 fission and capture reaction rates will be reduced.
Because of the larger proportionate increase in the shielding
of the 0.3 eV resonance the a-value of Pu-239 is reduced.
The change of the fission rate results in a reduction in
reactivity and the change of the a-value results in an
increase in reactivity. Further, the reactivity is increased
because of the shielding of the 1 eV resonance of Pu-240.
The net reactivity change is in all our cases negative.
The self shielding in Pu-239 was estimated assuming that
the incurrent into the Pu0 2 _particles has the same spectrum
as the average flux in the fuel. The corrections for
self shielding in Pu-239 are in our cases -(300-600) pcm
and for self shielding in Pu-240 about +100 pcm. Our correc
tions were found to be in good agreement with calculations
for room temperature presented by Liikala et al. [29].
Finally the statistical weight of the plutonium zone was
Table 4 .5 . L a t t ic e data and corrections for thermal expansion and p a r t ic le s ize plus ca lcu la ted
1Fuel
m rCores ize
Pu zone
i
Boron
conc
(ppm)
Temp
(°C)
Correctionsef f
Ak ,,/AT e ff
(pcm/°C/
thermalexpan*(pcm)
p a r t ic les ize(pcm)
1.9% U0o 1.2 34x34 0 22.3 0 _ 0.99547 _c. 1! - 0 90.0 + 65 - 0.99466 -1.2
44x44 _ 0 204.9 + 175 - 0.99320 -1.3n — 0 245.2 +214 — 0.99228 -2.3
1.7 27x27 — 0 20.0 0 — 0.99430 -i i — 0 88.8 +12 — 0.99309 -1.8
1.7 36x36 — 300 23.0 0 — 0.9955140x40 - 200 208.3 +30 - 0.99193 -1.9n — 200 247.2 +38 — 0.99147 -1.2
PuC2- 1.8 40x40 24x24 300 18.1 0 -280 0.99662 __
1.9% U0o r t I f 300 90.2 +33 -260 0.99670 +0.1z i i 1! 0 211.3 +83 -140 0.99481 -1.6
i : I I 0 245.5 + 104 -130 0.99568 +2.5
2.5 34x34 24x24 350 21.8 0 -410 0.99791 —i i i i 350 90.5 + 17 -380 0.99810 +0.3i » u 250 247.0 +54 -300 0.99563 -1.6
Pu0o 3.3 26x26 26x26 50 23.0 0 -500 0.99691 iiiJL m i i 50 93.4 -16 -460 0.99726 +0.5 !M i t 5.0 208.4 -43 -400 0.99541 -1.6I I i f 50 239.2 -51 -380 0.99571
<+ 1.0
* cf § 6.6
used to find the corrections in cores containing both PuC^
and UO 2 pins. The corrections are shown in Table 4.5. The
temperature coefficient is increased by about 0.5 pcm/°C
due to the particle size reactivity effect.
Table 4.6 shows the results from AE-BUXY.
Table 4.6: Results from AE-BUXY for the 1.9 % UO 2 and
the PUO 2 lattices
Fuel V vfm 1Boron konc. (ppm)
Temp.
(°C)
k00
kef £
Ak /AT eff
(pcm/°C)
1.9 % U02 1.2 0 22.390.0
204.9245.2
1.235141.224901.198841.18559
0.996570.993090.992050.99123
-5.2-0.9-2.1
1.7 0 20.088.8
1.263281.25663
0.999380.99575
-5.3
1.7 300200200
23.0216.2247.2
1.191201.196931.19125
0.997460.990350.99000
-3.7-1.1
Pu0„ 1.8 300 18.1 1.22217 _
z300 90.2 1.22039 -
0 211.3 1.25546 -
0 245.5 1.24654 -
2.5 350 21.8 1.20305 -
350 90.5 1.20867 -250 247.0 1.23400 -
3.3 50505050
23.093.4
208.4239.2
1.258221.265451.279121.28120
0.997950.995940.992540.99224
-2.9-3.0-1.0
Comparing the results in Tables 4.5 and 4.6 one finds that
the calculated k^^-values at room temperature given by
AE-BUXY are higher than those given by DIXY and at high
temperature the AE-BUXY values are lower. Thus, the experi- 2
mental B corresponds to a less negative temperature m
coefficient than what is obtained from the diffusion theory
calculations on the whole core.
At room temperature DIXY gives « 0.995 for the clean
UO^ lattices and k ^ ^ « 0.997 for lattices containing PuO^.
Corresponding reactivities at high temperature are 0.992
and 0.996, respectively. The spread in k ^ ^ is small and the
trend in k ^ ^ versus temperature is smaller than for the
1.35 % UO^ lattices.
There are several sources to experimental uncertainties.
The uncertainty in the measured boron concentration is 1 %
when there is more than 100 ppm boron in the water. 1 ppm
boron corresponds to about 20 pcm in reactivity. The error 2 .
in the measured B is expected to correspond to less than z
100 pcm in the absolute value of k ^ ^ . Further, there is
the uncertainty in material composition and geometry. The
experimental uncertainty in the difference between two cal
culated k^^-values for the same core (but at different
temperatures) is expected to be about 50 pcm.
The last column of Tables 4.5 and 4.6 gives the discrepancy
between the measured and calculated temperature coefficient.
When looking at A k ^ ^ / A T it should be remembered that the
uncertainty in A k ^ ^ / A T due to experimental errors may be
1 pcm/°C or more. The uncertainty becomes less if one cal
culates Akgj^/AT for the whole temperature range (Table 4.7).
Table 4.7: A k ^ ^ / A T for the temperature intervall 20-245 °C
Fuel V Vfm tAkeff/AT (pcm/°C)
AE-BUXY with exp B2 m
DIXY
1.9 % U0o 1.2 -2.4 -1.41.7 -3.3 -1.8
Pu02-U02 1.8 - -0.42.5 -1.0
Pu02 3.3 -2.6 -0.6
The present calculations have been performed neglecting
the difference between the uranium and plutonium fission
neutron spectra. The latter one has a higher average
energy. This gives an increased leakage (which causes a
negative correction to anc increased fast fission
(positive correction). The importance of the plutonium
fission neutron spectrum is discussed in § 6.7. The influ
ence on the temperature coefficient is negligible in all
lattices studied in the present report.
4^4^3___Comments to_the_resuits
The spread in predicted values is small. The average
value is 0.995 for cold UO^ lattices and 0.997 for PuO^
lattices. versus temperature is shown in Fig 4.4.
AE-BUXY predicts a temperature coefficient which is
(1-4) pcm/°C too negative. This discrepancy is somewhat
smaller than what has been observed in power reactor applica
tions t1]. The magnitude of the temperature coefficient is
(10-20) pcm/°C at room temperature and about 40 pcm/°C at
high temperature. The discrepancy between measured and2
calculated values is slightly larger when measured are
used to calculate k than what is obtained from two- eff
dimensional diffusion theory. Concentrating on the DIXY
results it is seen that Ak ri./AT varies little with temper-etr
ature.
PROBE OF •WATER LEVEL METER
ADJUSTABI WATER LEVEL
TO COVER GAS SUPPLY
NON-RETURNVALVE
IN-LET FOR WATER CIRCULATION
U02 REGION
SQUARE-FORMED SPACE FOR EXPERIMENTS
DUMP SPACE (4 SEGMENTS)
POSITION OF NEUTRON SOURCE (SB-BE)
SPRING-LOADED SAFETY SHUTTER (7 TOTALLY)
DRAINAGE OF DUMP SF*CE
Fig 4.1 Vertical cross section of reactor tank with insert vessel
Pressure tank
Dump space (four communicating segments)
Safety shutter (seven totally)
Fig 4.2 Horizontal cross section of the insert vessel.
The lines of dots and dashes give the boundaries of the
two core sizes investigated for the 1.35 % U02 rods.
(The dash circles indicate the size and position of
sealed openings in the tank lid. The shaded circular
areas show the position of the neutron detectors.)
Fig.L 3. Axial representation of
Axial representation in DTF -4
Mat. 7 (SS ♦ Steam)
Mat. 6 (U02 + Zr ♦ Steam)
Mat. 4
Mat. 5
Mat. U-Mat. 5
Mat. U
Mat. 5 ( H,0 + U0, + Zr+:
Mat. 4 (H20 ♦ U02 + Zr)
Mat. 3 (H20 + SS+Zr)
Mat. 2 (H20 +SS)
Mat. 1 (H20 )
and reflectors in DTF-4.
keff
Fig. 4. A keff versus temperature calcu lated by DIXY.
5. The components of the temperature coefficient
Many phenomena in a reactor system contribute to the temper
ature coefficient of reactivity. Lack of agreement between
theoretical and experimental values of the temperature coeffi
cient, dk/dT, may therefore be due to errors in the calcula
tion of one or several of the different components. Further
more, agreement obtained in a limited study does not neces
sarily mean that the calculations are correct in all respects.
Compensating errors may appear and care must be exercized in
drawing conclusions.
Neglecting the effect of thermal expansion, except the water
density effect, dk/dT may be written
T^, T£ and T^ = fuel, canning and moderator temperature,
respectively.
0 = moderator density.
Theoretical values of the terms on the RHS of Eq (5.1) may
be obtained by using a suitable advanced code such as
AE-BUXY. A corresponding experimental determination is
difficult to achieve. Attempts have been made to measure
the last term of (5.1) using flashing experiments in KRITZ.
The accuracy of these measurements, especially the determina
tion of the void content and void distribution, is, however,
not high enough to allow a detailed comparison with the
theoretical results.
Other experimental methods to separate the total temper
ature coefficient into temperature and density effects have
been used elsewhere, e.g. simulating void by use of aluminium
dk(T ,T ,T ,0)3k3T
3k d0 36 dTdT
(5.1)
tubes in the moderator. The introduction of aluminium
causes, however, other problems, e.g. the not negligible
fast scattering in A1 and streaming effects in the tubes.
Another way to obtain experimental information of the
separate terms in (5.1) is to perform critical measure
ments within a wide temperature range. The factor d0/dT
in the last term is much more sensitive to the temperature
(cf Fig 2.2) than 8k/3T and 3k/30 , so the last term in
(5.1) gives a significantly higher contribution to dk/dT
at high temperature than at low temperature. The KRITZ
measurements of dk/dT which cover a wide range of temper
atures therefore provide important information, which may
help to reveal whether the discrepancy between theory and
experiment is due to pure temperature effects, water
density effects or both.
Theoretical values of the terms in Eq. (5.1) are given
in § 5.1.
In order to gain some insight into the physical significance
of the calculated coefficients, it is helpful to express them
in terms of the derivatives
dn df dp dL_ & & — i. and — £.dT dT dT dT
n is defined as \>Z,/£ in the fuel, f is the ratio g f a g
between fuel absorptions and cell absorptions, p is the2 . .®
resonance escape probability and L the diffusion length.
Contributions to dk/dT from each of the derivatives (di
vided into four energy groups, g, have been calculated using
a special code, COEFF, with four-group cross sections from
AE-BUXY as input. Results from these calculations are dis
cussed in § 5.2.
Looking at Eq. (5.1) we note that the term 9k/8Tc in a light
water moderated system with zircaloy canning can be neglected
compared to the other terms. In the 1.35 % UO 2 lattice for
example, AE-BUXY gave the very small contribution
and this term will be neglected in the following discussion.
Tables 5.1-5.3 show the terms of Eq. (5.1) as predicted by
AE-BUXY. Results are given for the temperature coefficient
of the infinite multiplication constant, Ak^/AT , the
reactivity for a critical lattice, Ap/AT , the material2 2
buckling, ABm /AT , and of the migration area, AM /AT .
The temperature intervals used in the calculations are 20 °C
to 90 °C and 160 °C to 210 °C. In this temperature range
experimental values of the total temperature coefficients
are available.
The fuel temperature coefficient is in all lattices less
negative in the temperature interval 160-210 °C than in the
interval 20-90 °C. This should be expected due to the- 1/2 . .
approximate T dependence of the Doppler coefficient.
8p/3T^ is built up of two components. The dominating part
is caused by the Doppler broadening of resonances and a
minor contribution is given by the thermal scattering in the
oxygen. In the 1.35 % UO^ fuel these two components were
determined separately and we obtained
3k^
Y f ~ = “0*02 pcm/°Cc
(I£_\ \ 3 T j
f'thermal= -0.3 pcm/°C
for the temperature interval 20-90 °C and
Boroncontent(ppm)
9X/£Tf 3X/3Tm
3X/36 *d0/dT dX/dT
Temp in te rv a l (°C) 20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210
X=k00 0 -4.8 -4.3 -3.8 -3.6 -1.8 -8.2 -10.4 -16.1
(pcm/°C) 175 -4.7 -4.1 -3.3 -3.3 +0.1 -3.7 -7.9 -11.1
X=p 0 -4.1 -3.6 -5.1 -4.7 -10.6 -29.0 -19.8 -37.3
(pcm/°C) 175 -4.1 -3.7 -4.3 -4.0 -6.9 -20.2 -15.3 -27.9
X=B2 0 -.0114 -.0088 -.0137 -.0108 -.0261 -.0598 -.0512 -.0794
( n fV c . ) 175 -.0113 -.0086 -.0113 -.0092 -.01.64 -.0406 -.0390 -.0584
X=M2 0 -.0004 -.0004 .0049 .0048 .0253 .0732 .0298 .0776
(cm2/°C) 175 -.0004 -.0004 .0054 .0046 .0251 .0730 .0301 .0772
Vm/Vfm tBoron
conc(ppm)
3X/3Tf 3X/3Tm
3X/30‘ d0/dT dX/dT
Temp in te rv a l (° C) 20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210
X=k00 1 2 200 -5.8 -5.1 -2.9 -2.8 -4.0 -13.8 -12.6 -21.7
(pcm/°C) 1 7 200 -4.6 -4.1 -2.6 -2.9 +0.7 -2.4 -6.5 -9.4
X=p 1 2 200 -4.8 -4.2 -3.7 -3.5 -13.2 -34.3 -21.7 -42.0
(pcm/°C) 1 7 200 -3.8 -3.4 -4.1 -4.1 -10.9 -30.3 -18.9 -37.8
X=B2m
1 2 200 -.0124 -.0096 -.0095 -.0080 -.0322 -.0724 -.0542 -.0900
(m-2/°C) 1 7 200 -.0110 -.0086 -.0117 -.0100 -.0291 -.0690 -.0518 -.0876
X=M2 1 2 200 -.0007 -,0008 +.0032 +.0034 +.0255 +.0728 +.0281 +.0754
(cm2/°C) 1 7 200 -.0004 -.0006 +.0042 +.0042 +.0261 +.0768 +.0300 +.0804
V Vfi m rijti
Boron
conc(ppm)
3X/8Tf 3X/3Ttn
dX/ZO‘ dQ/dT dX/dT
Temp in te rv a l (°C) 20-90 160-210 20-90 160-210 20-90
i
160-210 20-90 160-210
X=k00
(pcm/°C)
it
1.8
2.5
3.3
300
250
0
-5.1
-4.1
-3.5
-4.6
-3.7
-3.1
+4.0
+7.6
+7.7
+ 5.0
+8-7
+9.0
-1.3
+ 2.7
+3.7
-9.5
+0.1
+2.5
-2.4
+ 6.2
+8.0
“9.1
+5.1
+8.4
X=p
(pcm/°C)
1.8
2.5
3.3
300
250
0
-4.1
■3.3
-2.7
“3.7
-3.0
-2.5
+2.5
+5.1
+4.1
+3.8
+6.7
+6.2
-13.7
-11.4
-14.4
-39.5
“36.5
-44.3
-15.3
-9.6
-13.0
-39.4
-32.8
-40.5
X=B2m
(m 2/°C)
1.8
2.5
3.3
300
250
0
-.0124
-.0107
-.0092
-.0098
.-0082
-.0072
+.0075 |+.0100
+.0164 1+.0186
+.0140 !+.01821
-.0394
-.0351
-.0467
-.0972
-.0952
-.1210
i
-.0443 J -.0970
-.0294 ! -.0848i
-.0420 | -.1100i
X=M2
(cm2/°C)
1.8
2.5
3.3
300
250
0
-.0004
-.0002
-.0002
-,0006
-.0002
-.0002
+.0018
+.0025
+.0034
+.0012
+.0016
+.0020
r i+.0258 I +.0756
i
+.0262 | +.0784 1
+.0263 j +.0812— . ___ i__ _____
+.0272
+.0285
+.0300
+.0762
+.0798
+.0830
3 : 1 , = - 3 -3 ^ ° cv f7 Doppler
= -0.3 pcm/°Cj. thermal
for the interval 160-210 °C.
The moderator temperature coefficient is negative in the
UO 2 lattices and positive in the PUO 2 lattices. The different
signs of 9p/3Tm for the two types of fuel are due to the
0.3 eV resonance in Pu-239 and the non-l/v-dependence of
the thermal U-235 cross section. Thus, an analysis of both
UO 2 and PUO 2 systems constitutes a severe test of the
ability to predict the thermal neutron spectrum. An error
in the calculated temperature dependence of the spectrum will
give different errors in the predicted temperature coefficient
for UO 2 and PUO 2 lattices. Our analysis of the KRITZ experi
ments has, however, given approximately the same discrepancy
between calculated and measured temperature coefficients in
both UO 2 and PUO 2 systems. This strongly indicates that the
change of the thermal neutron spectrum with temperature has
been correctly calculated.
The large temperature dependence of the water density
coefficient is a consequence of the variation of d0/dT with
temperature. The water density, 0, and d0/dT versus
temperature are shown in Fig. 2.2. The average value of
d0/dT is -0.470*10"3 g/cm3*°C and -1.096-10_3 g/cm3*°C
in the temperature intervals 20-90 °C and 160-210 °C,
respectively. 3Bm /30*d0/dT is roughly proportional to
d0/dT , whereas the changes of 3p/30»d3/dT and
3M2/30'd0/dT are larger than they would be if they
were proportional to d0/dT . This is due to spectrum
effects. 3^/ 3 0 - d0/dT has a more complex variation
with temperature because Bk^/ST is built up from several
positive and negative contributions (cf § 5.2).
GO
The discrepancy between the theoretical and the experimental
reactivity can be described by a function f(T,0) defined
by
k.xp(T'9> ’ ‘W r y (T-e) + £« ’6> <5-«
We found in chapter 4 that f(T,0) , within reasonable
error limits, is independent of temperature, i.e.
3f , 3 f d0 _ ,3T 36 dT " const* (5.3)
There are no physical reasons why the two terms of the LHS
should be correlated so we can expect both terms to be con
stant. This means that
3 f / d e \ 1— = c o n s t ^— j (5.4)
There is also no reason to believe that 3f/30 is inversely
proportional to d0/dT so (5.4) leads to
(If 3f/33 f 0 we would have 3f/30-*<» at a small extrapola
tion in the temperature to T = 4 °C where d0/dT = 0.)
Thus we find that if (5.3) is true, which is approximately
the case in the investigated lattices, then the discrepancy
between calculated and measured temperature coefficients
must be expected to be a pure temperature effect and not a
secondary effect due to the temperature dependence of the
water density.
5.2______Contributions to the temperature coefficient _________ calculated by COEFF_____________________________
5.2.1_ _Descrigtion_of COEFF
The program COEFF was written as a tool for studying the
contributions from various parameters (n, f, P and L^) to
the temperature coefficient. Input data consist of few-group
parameters produced by AE-BUXY. These data refer to the
criticality spectrum obtained from.the fundamental mode cal-2 2
culation with B = Bm
The following nomenclature will be used to describe COEFF.
G number of energy groups
g group index
Xgfission neutron spectrum
Zag
cell averaged absorption cross section
vEfg
cell averaged v-fission cross section
g *-gcell averaged scattering cross section (from g to g')
Dg
cell averaged diffusion constant
‘"gcell integrated neutron flux
B2m
material buckling
B2 geometric buckling
<j> and g
y are normalized such that g
E xg = g s
1
?(I“s+ D B2)<|> = 1 g g
Define
R = f" Y (E .. ./d> , — E , „ d> „)8 g = l g H_g+ i g ' ^ ' V g g
R = 0o
r g =o
vE
nf =fg
ag
(5.7)
P =
8 Rg“l + Xg
(5.8)
PG = 0
The neutron balance for group g can then be written
(E + D B ) <J) = x + R i “ Rag g m *g Ag g-1 g
(5.9)
Noting that R = P (R + x ) we get g g g g
(E + D B ) <j> = (1 - P„) (x + R ,) =ag g m' *g g ' VAg g-1'
D: = (i+ b 2) L(i - p ) ) x .38 8 ag m 8 g^= l 8 g "=g ' 8
g - 1fl P„u (5.10)
where
g-1n P „ = 1 for g' = g
g ^ g ' 8
Write
k = ------ S------ T------ = I v Z ( <f> = 7’nf z <J> (5.11)
+ O 8 8 8 8 8 a§ 8g ag g "> g
k = E d - B2 ) " 1 n f (1 - P ) f X , V P „ (5.12)g ag 8 8 g'=l 8 g"=g’ 8
This expression may be written
g 1 + L 2g B 2m g ’-lV 8 g"=g
8 - 1n p „„ (5.13)
with
D(1 - P )
L2 = ^ — --------- (5.14)
8 e 21 + -=2- P B
£ g mag
P 1 = (1 + L2 B2) P (5.15)g g m g
Pg is the slowing down probability in an infinite lattice.
In the special case with P 1 = ---- — (£ = V £ , )g £ + £ rg V jl g •‘-gag rg g Fg
we get
, 2 ______
8 Eag + Erg
2 2i.e. the conventional definition of L . Replace B with the
2 m geometrical buckling, B , in (5.13). We obtain
k = T 5k (5.16)
8 S
nf (1 - P ) g g-1 P enSk = — §— y x . n ---- &=— =• (5.17)
8 1 + L2 B2 g ^ l 8 g'-g' 1 + L „ Bo o
(5.16) now gives the effective multiplication constant, k,2 . 2
at a given geometrical buckling, B , (k^k^ if B =0) and (5,
gives the contributions to k from each energy group.
To see how a change in n . f , p1 and L effects k we diffe-g g g g
rentiate (5.16)
Ak = T An f X+5"Af n X +y'AP1 Y + ^ A L 2 Z (5.18)“ g g g “ g g g ^ g g g g g
with the coefficients X , Y and Z given by Table 5.4.g g g
Table 5.4: Coefficients in equation (5.18)
Xg
1 i r- 8 1 Pg"2 2(1- y I v „n , 2 2
1 + LT B 8 g^=l 8 g"=g' 1 + l/„ B© . O
Yg
1 ^ 8~^ PE»2 2 11 2 ? +
1 + i/ B 8 g'=l 8 g"=g' 1 + l/„ B •8 8
! G g g"'-l Pg"
+ i L 2 2 I ^ c 1 n 2 2g "’=g+l 1+L‘„BZ 8 8 gV =l 8 g"=g' l+l/„B
8 8 8
Zg
b 2 i 8 8-1 p i "2 2 2 nfg (1_Pg) I V " *
(l+LgB ) 8 8 gV=l 8 g"=g' 1+L2 Bo o
B2 ° 1 ~i 8 2 2 L 2 2 Po"') C.
1+l V g"'-g+l l+L2,, ,B 8 8 gV =l 8o o
g’"-l pi,n —
g"=g' l+lr,,Bo
G]T which appears in the second term of Y and Z is put
g - 'o g + i § 8
equal to zero for g = G.
5^2^2___Results_from_calculations_with_COEFF
The coefficients An /AT , Af /AT , Ap /AT and AL2/AT g g g g
and their influence on the partial and total temperature
coefficients have been calculated using four energy groups
with the boundaries 10 MeV - 0.5 MeV - 9118 aV - 0.625 eV -
The results have been collected in Figs 5.1-5.12. There is
no experimental information available which corresponds to
these theoretical results. The value of this kind of calcula
tions is that they give a detailed picture of the temperature
dependence of the neutron balance and its influence on the
reactivity coefficient.
The fuel temperature coefficient is mainly caused by changes
in n and p in the resonance group (Figs 5.1, 5.5 and
5.9). A small contribution to 3p/3T^ comes from 3n/3T^
and 3f/3Tf in the thermal group due to the thermal scatter
ing in the oxygen. The thermal contribution to 3p/3T^ is
0.2-0.3 pcm/°C. The increased resonance absorption gives
rise to a very small positive contribution due to decreased
leakage. This effect is seen in 31,2/3T^. The numerical
value is about 0.1 pcm/°C. It is also seen in the diagrams
that the Doppler effect is less in the high temperature
interval than in the low one.
Figs 5.2, 5.6 and 5.10 show the contributions to the modera
tor temperature coefficient. In the IK^ lattices there is a
large negative effect caused by 3n^/3Tm and a smaller
positive effect caused by 3f./3T > so that the total thermal4 m
temperature because the water density is lower. In the Pu02
lattices there is also a negative contribution to 3p/3Tm
due to 3ri,/3T , but the positive influence of 3f,/3T is4 m '* m
larger than the negative contribution from Sn^/ST , so
that the coefficient, Sp/ST^, is positive. The thermal leakage
increases with higher temperature due to the harder spectrum.
This leakage effect is nearly proportional to B and ism
(1-2)pcm/°C.
The water density influence on the temperature coefficient
is shown in Figs 5.3, 5.7 and 5.11. This effect is, as we
already have seen in § 5.1,larger at high temperature than
at low one due to the temperature dependence of d0/dT .
The density change contributes to the reactivity coeffi
cient in all energy groups, the largest contributions being
from 9n^/36'd0/dT (positive due to less absorption in 1^0
and boron), 3p^/30-d0/dT (positive due to increased fast
fission), 3p../30• d0/dT and 3p„/30'd0/dT (negative due to
decreased resonance escape probability) and from 3L /30-d0/dT
(in all energy groups negative due to the larger leakage at
low water density). The negative contribution to 3p/30-d0/dT
from the change of is much smaller than the corresponding
contribution to 3p/3T .m
Figs 5.4, 5.8 and 5.12 show the contributions to the total
temperature coefficient, i.e. the sum of the contributions
in the other diagrams.
We have seen that the temperature coefficient is built up
from a number of negative and positive contributions, each
of which has to be correctly predicted. Small errors in
several of the calculated contributions can give a signifi
cant error in the total calculated coefficient or compensate
each other, so that the total coefficient is in better agree
ment with experiments than should be expected from the
accuracy of the theory and the nuclear data. Since we do
not have detailed experimental information about the sepa
rate contributions, we must rely on careful theoretical
analyses of the approximations made in the calculations
and on estimated uncertainties in the nuclear data in order
to determine the accuracy in the theoretical methods.
-5
AT!
g=1 2 3 i*
.0.0 0 0
S 2 S S' M I No i o I
° i o oto ' <JD CQ i=- OQc CD £ CD
Q- e Q-eQ- a a a - a ioa
Af
2 3 A
Ap
2 3
AL
2 3
to
Fig. 5.1 Contributions to the fuel temp, coeff. 1.35% UO2 lattices
Total
AT)
g=1 2 3 U Hi
Ap
1 2 3
A ^
1 2 3 /
-5
1 2 3 A
Af
o <_>oo • • • •o O O o
o oo oCN ID CN ID
0 3 m m m£ E Q. Q.Q . Q . CL CLa. cun in o
Total
5jo d0_ 69 dT
pcm/*C
15
10
-5
-10
-15
o u u u « • • • o o o o cn cn *-
CN cn , i i io o o o CN to CM UD
' c o mCD CD £ £
c r Q-Q- E £ CL CLCL Q . _ _
=0=0 g=1 2 3 I*
All
2 3 U
Af
Ap
2 3
AL
1 2 3 A
Total
rat
Fig. 5.3 Contributions to the water density coeff.
1.35% U02 lattices
20
15
10
=OUDL
Ap
2 3
AL
1 2 3 A
Total
-10 -
-15
o ooo o O or n i— CD
(Nl CM
o o o o(N ID CN ID
CD CQCQ CD
l l l l in inO O
2 3 U
Af
9=1
At|
2 3
Af
2 3 AUM I
Ap
1 2 3
AL2 [Total
1 2 3 A
-5
o ooo
CN (NI i I I
O O O Q CM tO ( N U 5
CN (n W i" ; '
II II H >1
Fig.5.5 Contributions to the fuel temp, coeff.
1.9% U02 lattices.
At]
g=1 2 3 4 JlJ
Ap
1 2 3
AL
1 2 3 A
-5
y y y [
2 3
Af
oCN
Ocn
O o • •o o, cn r;
o o'CN CD
CN CN
"h h :> :>
’m ’n
> >
Total
- b i -
5p d8
50' dT
pcm/ G
Total
Fig. 5.7 Contributions to the water density coeff.
1.9% UO2 lattices
- 64 -
+10
At)
g=1 2 3 4=D=cr
-5
-10
-15
-20
-25
W \ 1 2 3
Af
O o O O
CN CMCT)I I ' *O O O O CN tO CN lO
II II II II
> > > > w w
>£>E>e>E
Ap
1 2 3
m
AL2
1 2 3
Fig.5.8 Contributions to the total temp, coeff.
1.9% UO2 lattices.
|Total
I*
Fig. 5.9 Contributions to the fuel temp, coeff. 1.5% Pu02 lattices
To
tal
+10
g=i
Ail
2 3 4
Ap | ALZ
1 2 3 M 2 3
-5
3 4 TF Total
Af o 0 0 . 0O O O o
I I * 1
c3 § R §
oo oo co oo ^ ^ ro n ii n n )■ > > > >
Fig. 5.10 Contributions to the moderator temp, coeff.
1.5% U02 lattices
Total
Fig. 5.12 Contributions to the total temp- coeff. 1.5% Pu02 lattices.
6. THE INFLUENCE OF APPROXIMATIONS IN THE THEORETICALTREATMENT ON CALCULATED TEMPERATURE COEFFICIENTS
Practical reactor physics calculations have to be performed
in a limited number of energy groups and spatial regions.
We will, show that in our calculations the energy groups and
the spatial representation were chosen so that the calculated
temperature coefficient is insensitive to our specific
choice. Other effects that for practical purposes are often
neglected in the calculations will also be discussed here.
The influence of uncertainties in nuclear data will be dealt
with in the next chapter.
6. L___________The crystalline binding in UP,,
Due to crystalline binding the velocity distribution of
the fuel atoms is not Maxwellian as it was assumed when
calculating the Doppler broadened line shape of the reso
nance cross section in Eqs (2.21) and (2.22). The uranium
atoms in U02 vibrate with an average kinetic energy larger
than that in a free gas state. L a m b [30] has shown that for
weak binding (or the Short Compound Nucleus Lifetime
approximation, SCNL) the atoms behave like a free gas with
an effective temperature which is higher than the temper
ature of the medium.
The SCNL approximation is applicable when
r + A >> 2 k e D ( 6 - D
where
9^ is the Debye temperature
/ AkTEA = / — - is the Doppler width
A
r is the total width of the resonance at half maximum.
The effective temperature, Te££» is i-n the SCNL approxima
tion given in terms of the average kinetic energy per mode
of oscillation as [31].
where v is the phonon frequency and g(v) the phonon
frequency distribution.
It is convenient to describe the crystalline effects in
terms of the Debye temperature of the medium. This descrip
tion is approximate, because in general the crystal does
not have a Debye phonon distribution. However, an effective
Debye temperature, which varies with the temperature of the
medium, can be defined so that it produces the correct
crystalline effects in the medium. Dolling [32] has deter
mined the effective 0Q versus T for U02 . For temperatures
above 300 K, 0p is fairly constant (e^ ^ 620 K ) .
Shenter [33] has used the phonon distribution measured by
Dolling [32] to calculate T ... He has also given resultseff
for a Debye distribution
CO
O
g(v) = 3 ^ 3
v_
2for \> < v
D
g ( v ) = 0 for v > vD
DkO
D
with Q_ = 620 K. T obtained with the Dolling distribu- D eff
tion and the Debye distribution agree very well for temper
atures above about 300 K.
The SCNL approximation does not hold for the low energy
resonances of U-238. F is 0.025-0.1 eV for resonances in
U-238 below 200eV. The Doppler width, A, is for the 6.68
eV resonance 0.05 eV at 300 K and 0.07 eV at 600 K and for
the 190 eV resonance the values of A are 0.3 eV and 0.4 eV
at 300 K and 600 K, respectively. 2k©D is equal to 0.1 eV
in U O 2 so relation (6.1) is fulfilled for the 190 eV re
sonance and resonances at higher energies but not for the
6.68 eV resonance.
The chemical binding effects on resonances have been studied
by Adkins [31]. He used several models for the phonon fre
quency distribution, among others the one measured by
Dolling, and reported results for the 6.68 eV and 190 eV
resonances. In Table 6.1 the resonance integral change be
tween 300 K and 500 K is shown as obtained by Adkins using
a detailed crystalline model, CRYS, the SCNL approximation
with the Dolling frequency spectrum and the free gas model.
Regarding the CRYS model as giving the correct result his
results show that the SCNL approximation reduces the error
in ARI/AT by a factor 3 compared with the free gas model.
The small overestimation of the Doppler effect in low energy
resonances that remains can in most applications be accepted,
because the dominating contribution to the Doppler coeffi
cient in normal LWR lattices comes from the energy range
above 100 eV for which energies A > 2k0^.
Table 6 . 1 : Calculated U-238 resonance integral change
between 300 K and 500 K [31] using various
crystal models
Model 6.68 eV, 0 =17.52 bp
190 eV, 0 =40 b P
ARI/RI
(%)
Error
(%)
ARI/RI
(%)
Error
(%)
CRYS .2534 0 .7791 0
SCNL Dolling .2739 8 .8152 5
Free gas .3092 22 .8981 15
Thus, for calculating the Doppler effect in U-238 it may be
recommended to use an effective Doppler temperature above
the physical temperature of the fuel. This is a very simple
way to account for the crystal binding effects without
having to consider complicated calculations of the reso
nance cross section line shape. Well established codes
using the ip and x formalism can be used without any
modification. In cell programs which use tabulations of the
effective resonance integral as function of the temperature
Te^ should then be used instead of T when interpolating
in the tables. This procedure has been built into the
A.E-BUXY code with Tg££ given by Eq (6.2) with the Debye
frequency distribution and
1
T
eD = 620 K
eff ! » »(6.3)
AT = T . -T as function of T is plotted in Fig 6.1. eff
The influence of the effective temperature on calculated
fuel temperature coefficients in three of the lattices
which were analysed in chapter 4 is shown in Table 6.2.
Table 6.2: Comparison between fuel temperature coefficients
calculated using T ,, = T and T defined efr eff
by Eq (6.3)
Case Ok /3T,to (pcm/°C)
Temp.interval(°C) 20-90 160-210
1.35 % U02
Te£f-TT ef£-E q (6.3)
-4.7 -4.1V /V =1.4 in f175 ppm B
-3.8 -3.5
1.9 % U02T =T -4.6 -4.1
V /V =1.7 m f
200 ppm BTeff=Eq(6.3) -3.8 -3.5
Pu02
V /V =2.5 m f
1eff=T
Teff= E q '6,3)
-4.1
-3.5
-3.7
-3.2250 ppm B
The cylindrrealization of pin cells
The lattice cell is in AE-BUXY replaced by a circular
Wigner-Seitz cell with white boundary conditions. In order
to investigate the validity of this approximation we
carried out calculations on the actual square cell in two
lattices and compared calculated values of and its
temperature dependence with results obtained in the corre
sponding circular cell.
Similar investigations have been done previously by Honeck
[34, 35], Fukai [36, 37], Sauer [38], Carlvik [39], Dudley
[40], Newmarch [41] and Weiss [42] among others. These
investigations are, however, limited to the study of the
flux distribution in one-group calculations with a flat
source in the moderator. We found it therefore worthwhile
to carry out multi-group calculations where the spatial
variation of the source in each group is taken into account.
The two square lattices contain 1.9 % IK^ rods and the
moderator to fuel ratios are 1 arid 2. To simplify the cal
culations the fuel has no canning. Calculations were done
for 20 °C and 245 °C.
The multi-group transport codes BOCOP [43] and FLUCAL [44]
were used for the calculations. BOCOP is a two-dimensional
collision probability code in xy-geometry and the circular
fuel region was approximated by a polygon. The polygon was
chosen so that the volume of the rod is preserved. The outer
square boundary of the cell is correctly represented in
BOCOP with reflecting boundary conditions. FLUCAL solves
the transport problem in annular geometry by use of the
DIT-method. The cell boundary was cylindricalized preserving
the cell volume and white boundary conditions were applied.
All calculations were made with 8 energy groups with the
group boundaries 10 MeV - 9118 eV - 4 eV - 0.625 eV -
-0.14 eV - 0.058 eV - 0.030 eV - 0. Group cross sections
were generated by AE-BUXY.
Table 6.3 shows the comparison of and temperature
coefficients, Ak /AT, calculated in the square and theOO
circular cell.
Table 6.3: Comparison of k and Ak /AT calculated■ " ■ ■ — ■ ■■■■ - ■ ■ * OO oo
in square and circular cell
V /V, m f
Temp.
(°C).
kOO Ak^/AT (pcm/°C)
Square Circ. Square Circ.
1.0
2.0
20
245
20
245
1.23366
1.17817
1.27605
1.25800
1.23427
1.17851
1.27654
1.25834
-24.66
- 8.02
-24.78
- 8.09
The difference between the results obtained in the two
geometries is small, k^ is overestimated by about 50 pcm
when the square cell is cylindricalized. Ak^/AT is about
0.1 pcm/°C more negative in the circular geometry. This
small discrepancy may, however, be due to numerical uncer
tainties in the calculations which are about 0.1 pcm/°C.
We conclude that the error introduced by the cylindricaliza-
tion of the pin cells is negligible. This conclusion is in
agreement with the results of [34-42], where the authors
in most cases found that the flux distribution obtained
from a one-group calculation in a Wigner-Seitz cell with
white boundary conditions is a good approximation for the
solution in the exact geometry.
Comparison between pin ceil calculations using isotropic and anisotropic scattering___________
In cell calculations transport corrected cross sections
are usually used to account for anisotropy in scattering.
The transport cross section used in AE-BUXY is derived by
expanding the transport equation in Legendre polynomials.
P^-row sum and weighted P^-column sum corrections [21] are
used in the thermal and epithermal regions, respectively.
In order to check the adequacy of the use of a transport
corrected total cross section, pin cell calculations were
performed using S -theory. These calculations were madeO
with DTF-4 in 6 energy groups with the boundaries 10 MeV -
- 0.5 MeV - 9118 eV - 0.625 eV - 0.140 eV - 0.058 e V - 0.
Cross sections were generated by AE-BUXY and DTF-4 was run
on a 1.35 % U0£ pin cell at 20 °C and 40 °C with either P^-
scattering and transport corrected diagonaT’elenents or P^-
scattering. The results are shown in Table 6.4. It is seen
that the use of transport corrected isotropic scattering is
a very good approximation for the P^-scattering. The very
small discrepancy between the P^- and P^-results is not
significant and we conclude that transport corrected cross
sections give the same k^ as calculations using explicit
P^-scattering.
Table 6.4: Comparison of DTF-4 calculations using P -
and P -scattering. 1.35 % U0„, V / V c = 1 . 4 I I m f
k00 Ak /AT00
20 °C oo
o (pcm/°C)
Pg-scattering 1.16024 1.15834 -9.50
P^-scattering 1.16028 1.15837 -9.55
The prediction of leakage in small cores is a crucial part
of the reactor physics calculations. The calculated leakage
is sensitive to nuclear data in the fast energy region
(fission neutron spectrum, inelastic scattering cross
sections for U-238, hydrogen and oxygen scattering data),
and a proper treatment of the anisotropic scattering is re
quired. One must therefore expect that in most cases there
may be an uncertainty of several per cent in the calculated
leakage. The error in the predicted leakage due to erroneous
fast data or an inadequate treatment of the anisotropic
scattering is, however, insensitive to the temperature and
the influence on calculated temperature coefficients is less
significant than the influence on the k^^-value.
We have seen in chapter 4 that the fundamental mode calcula-2
tion using the measured B and the two-dimensional diffusionm
theory calculation give approximately the same reactivity,
i.e. the predicted leakage is the same. DIXY-calculations
have also been compared with SQ-calculations using DTF-4 inO
one-dimensional cylindrical geometry [45]. The agreement in
calculated temperature coefficients is good. The axial2
leakage obtained using the measured agrees well with that
calculated by DTF-4 (Table 4.6). Thus, we do not expect that
the observed discrepancies between experimental and theoret
ical temperature coefficients can be explained by errors
in the leakage calculation.
This conclusion is supported by the observation that the
error in predicted temperature coefficients is the same at
low and high temperature although the influence of the
leakage on the coefficient is much larger at elevated
temperatures. Further, we note that the leakage in the
cores of chapter 4 varies from 10 % to 25 but no trend
is seen in the discrepancy, Ak ^ / A T , as function of the
1cakage.
6.5^1_____ Macro grougs_and Gauss H 2 iD£®_iD_^E~BUXY
AE-BUXY calculations were made on the 1.35 % UC> 2 lattice
with 175 ppm boron in the water using 17 macro groups and
8 Gauss points (3 in UO^, 2 in canning, 3 in H^O) or 25
groups and 16 Gauss points (6 in UC^, 4 in canning, 6 in
1^0). Condensation vectors for the group structures are
(cf Table 3.1).
17 groups 2, 4, 6, 14, 21, 25, 26, 27, 35, 38,
45, 48, 51, 55, 59, 63, 69.
25 groups 2, 4, 6, 14, 21, 23, 25, 26, 27, 32,
35, 38, 41, 45, 48, 51, 54, 55, 56,
57, 59, 61, 63, 66, 69.
The two group-structures differ mainly in the thermal region.
Results from calculations at two temperatures are given in
Table 6.5. The discrepancies between the 17 and 25 group
results are negligible.
Table 6.5: Comparison between AE-BUXY calculations with
17 macro groups, 8 Gauss points and 25 macro
groups and 16 Gauss points. 1.35 % 175 ppm B
Temp
(°C)
Numberofmacrogroups
NumberofGausspoints
k o k c . eft
B 2m
(m'2)
2M
( ^ (cm )
20 17 8 1.12172 0.99640 29.91 40.7
25 16 1.12171 0.99630 29.88 40.7 '
210 17 8 1.10448 0.99060 20.77 50.3
25 16 1.10459 0.99060 20.77 50.3
To investigate how the leakage depends on the number of
fast groups, calculations were made on the 1.35 % UO^
lattice without boron in the water using 17 and 23 groups
in the fundamental mode calculation. The 17 group structure
is the same as given above. 6 groups were added in the fast
region to obtain the 23 group structure which is
23 groups 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 21,
25, 26, 27, 35, 38, 45, 48, 51, 55,
59, 63, 69.
The 17 and 23 group structures are identical below 9118 eV
(gr 14). Results are given in the tables below.
Table 6.6: AE-BUXY calculations with 17 and 23 macro groups.
1.35 % U02, 0.8 ppm B
Temp
(°C)
Mumberofmacrogroups
k00 kef f
B2m
(m “)
r2 ;
M
(cm )
90 17 1.15566 0.99585 36.01 43.2
23 1.15567 0.99415 35.61 43.7
21.0 1.7 L. 1.3872 0. 99147 27.28 50.9
23 1.13874 0.98983 26.97 51.4
Table 6.7: Temperature coefficients derived from Table 6.6.
Number of macro groups
Alc^/AT(pcm/°C.)
Ak ff/AT (cpm/°C)
17 -14. 1. -3.65
23 -14.1 -3.60
It is seen from Table 6.6 that the leakage is underestimated
by about 1 % in the 17 group structure compared to the 23
group structure. This corresponds to about 0.2 « in keff
The temperature coefficients were changed very little when
the number of groups was changed (Table 6.7).
The results in Tables 6.5-6.7 are in agreement with previous
experience and justify the use of 17 macro groups and 8 Gauss
points in AE-BUXY.
6.5.2 Energy groups and mesh points in the diffusion theory calculation
The number of energy groups and mesh points was varied in
diffusion theory calculations on the 1.35 % U02 lattices
with surrounding reflectors. Results are given below.
Two group-structures were compared with the condensation
vectors
4 groups 6, 14, 45, 69.
8 groups 2, 6, 14, 45, 55, 59, 63, 69.
Table 6.8: k rr from DIXY. 1.35 % U0„ lattice with 46x46 ---------- err 2
fuel pins, 175 ppm B
Number of groups
Number of mesh points
keffAkeff/AT
(pcm/°C)20°C 210°C
4 28x28 0.99691 0.99301 -2.05
4 39x39 0.99730 0.99349 -2.01
8 39x39 0.99730 0.99350 -2.01
It is seen from Table 6.8 that k __ and Ak AT areeff eff
insensitive to the number of energy groups and mesh points
in the diffusion theory calculation. The exact agreement
between the 4 and 8 group calculations is accidental. Similar
comparisons made for other purposes have given discrepancies
of about 100 pcm between 4 and 8 group calculations.
6.6_________ The influence of thermal expansion on reactivity
For practical reasons the thermal expansion of fuel, canning
and construction material is often neglected in calculations,
because this effect in most cases gives only a small correc
tion.
The influence of thermal expansion on the reactivity was
estimated separately for all the KRITZ lattices analysed
in chapter 4, and the corrections have been applied in the
results given in Tables 4.2-4.7.
The following coefficients of thermal expansion were used
The variation of the lattice pitch with temperature is due
to expansion of stainless steel in which the spacers are
fastened.
In Table 6.9 the correction for thermal expansion is split
up into contributions from the change of fuel radius, can
radius, cell radius, fuel density and can density. It is
seen that the effects on dk /dT are very small except for(X> *
the contribution due to l\r , - . The dominating effects oncell
dk rp/dT are due to changes in the leakage in the fast
SS
i i - i o ' V c
7•10_6/°C
18‘10~6/°C
energy region. The individual contributions compensate
each other to some extent. In the 1.35 % UO^ lattice with
0 ppm boron in the water the total correction to the
temperature coefficient is only -0.05 pcm/°C.
Table 6.9: Individual contributions to the temperature
coefficient due to thermal expansion.
1.35 % ITC>2 lattice with 0 ppm boron in the water
dk /dTOO
dk /dT eff
dB2/dTm
dM2/dT
(pcm/°C) (pcm/°C) (m"2/°C) (cm2 /°C)
“uo2 -.04 + .18 + 5-10_A-4
- 5-10
SrZr-2-.03 -.27 - 7 + 5
Ar ., cell
+ .20 + .38 + 10 - 5
AeU02-.04 -.34 - 9 + 10
A0Zr-2+ .01 0 0 0
Total + .10 -.05 - 1 + 5
The expansion of the spacers increases the lattice pitch
at high temperatures. This affects not only the cell cal
culations 'but also the core calculation since the cor- size
is changed. For the core with 46x46 pins we get
AB2/AT = -6*10~4 m _2/°C . AB2/AT is -0.05 m ~ 2/°C so the g m
correction is in this lattice about 1 % of the total temper
ature coefficient which corresponds to 0.2-0.4 pcm/°C.
Table 6.10 shows corrections for thermal expansion cal
culated using AE-BUXY.
Fuelm f
Boron conc. (ppm)
dk /dTOO dk _./dT eff
dB2/dTm
dM2/dT
(pcm/°C) (pcm/°C) (m'2/°C) (cm2/°C)
1.35 % U02 1.40
175+ .10 + .02
-.05-.14
- 1-10"4 - 3
-4+ 5-10 + 10
1 9 0 + 1.05 + .65 +13 + 9
1.9 % U0o1 ♦ z.
300 +1.01 + .65 +13 + 9z
1.7 300 + .13 -.09 - 2 + 7
1.8 300 + .82 + .50 +10 + 6Pu0o 2.5 300 + .10 -.14 - 3 + 5
3.3 j - .10 -.29 - 8 + 5
6.7_________The plutonium fission neutron spectrum
The fission neutron spectrum used in the analysis is a
Maxwellian with ~ 1*33 MeV. The average energy of
fission neutrons from fissions in plutonium is higher than
that given by this spectrum. The use of a plutonium fis-2
SLon neutron spectrum increases the predicted and M ,
but has a negligible influence on the temperature coeffi
cient. The table below shows results from AE-BUXY calcula
tions with two different fission neutron spectra represen
tative for fissions in uranium and plutonium. The Maxwellian
temperatures are 1.33 MeV and 1.385 MeV, respectively. The
change in k is due to an increased number of fast . 2
fissions m U-238. M is 1.4 % larger with the plutonium
spectrum.
and plutonium fission spectra.
PuO- lattice with V /V, = 2.52 m f
T ’mp.
( u c )
Fission neutron
spectrum
koo k
effM 2
(cm )
U 1.22588 1.00000 36.60
20 Pu 1.22685 .99781 37.11
diff +97 pcm -219 pcm + .51
U 1.23829 1.00000 46.31
210 Pu 1.23938 .99775 46.96
diff +109 pcm -225 pcm + .65
Fig. 6.1 Te f f as function of T
7. THE INFLUENCE OF NUCLEAR DATA ON THE CALCULATED
T E MP E RATI IRE COE F F1CIE NT
7.L Comparison of: Lemper;i lure coefficients calculated
__ ___________by use of ENDP/B and UKNI) 1^..lata______________________
The standard library in AE-BUXY is based on UKNDL but a data
library in the BUXY format has also been generated from
ENDF/B-III. Epithermal cross sections and effective reso
nance integrals were processed by Che SPENG and DORIX
codes [46]. Thermal group cross sections were generated
by use of FLANGE-2 [47] which was modified to calculate
group scattering matrices instead of point-to-point matrices.
Three of the lattices studied in chapter 5 were recalculated
using the data library generated from ENDF/B-III. Partial
and total temperature coefficients calculated with the two
libraries are compared in Table 7.1.
The agreement between the temperature coefficients obtained
with the two different libraries is remarkable (the levels2
of calculated k and M differ, however, more than the
coefficients in Table 7.1). The two AE-BUXY libraries are
entirely independent. Both the basic nuclear data and the
processing codes used for generating group cross sections
differ.
Scattering matrices for water are based on the Nelkin
model in the "UK.\'DL"-library and on the Haywood model in the
"ENDF/B" library (cf § 7.3).
The good agreement in the moderator temperature coefficients
shows that the thermal, scattering matrices and thermal
reaction cross sections are in agreement. The water density
coefficients obtained with the two data libraries are
practically the same.
f ......
Case Data l ib 3X/3Tf 3X/3Tm
3X/36‘dQ/dT dX/dT
Temp
in te rv a l(°C)
20-90 160-210 20-90 160-210 20-90 160-210 20-90 160-210
1.35% UO9 X=km UKNDL -4.7 -4.1 -3.3 -3.3 +0.1 -3.7 -7.9 -11.1
Vm/Vf-1.4 ( n r m ^VDF/R -4.8 -4.2 -3.3 -3.3 0 -4. 1 -8.3 -11.6
175 ppm B X=p UKNDL -4.1 -_J. 1 “ *4 . 3 -4.U “ D . 9 -20.2 -15.3 -27.9
(pcm/°C) I'NDF/R -4.2 -3.8 -4.3 -4. 3 -6.9 -20.2 -15.4 -28.3
- .U113 -.0086 -.0113 * .0092 -.0164 -.0406 -.o3yo -.0584
(m-2/°C) ENDF/B -.0121 -.0094 -.0120 -.0100 -.0164 -.0406 -.0405 -.0600
X=M2 UKNDL -.1)004 -.0004 .0046 .0046 .0251 .0730 .0293 .0772
(cm2/°C) ENDF/3 -.0004 -.0004 .0046 .0048 .0239 .0692 .0281 .0736
1-9% U09 X=k01 UKNDL -4.6 -4.1 -2.6 -2.9 +0.7 -2.4 -6.5 -9.4
Vm/Vf =l77 (pcm/°C) ENDF/B -4.7 -4.2 -2.7 -3.0 +0.8 -2.2 -6. 6 -9.3
200 ppm B X=p UKNDL -3.8 -3.4 -4.1 -4.1 -10.9 -30.3 -18.9 -37.8
(pcm/°C) ENDF/B -3.8 -3.4 -4.2 -4.3 -11.0 -30.5 -19.1 -38.2
X=BI UKNDL -.0110 -.0086 -.0117 - .0100 -.0291 -.0690 -.0518 -.0876(m**2/oc) ENDF/B -.0116 -.0090 -.0129 -.0112 -.0300 -.0706 -.0544 -.0908
X-M2 UKNDL -.0004 -.0006 +.0042 +.0042 +.0261 +.0768 +.0300 +.0804
(cm~/°C) ENDF/B -.0003 -.0004 +.0043 +.0044 +.0250 +.0734 +.0290 +.0774
Pu09 X=kOT UKNDL -4.1 -3.7 + 7.6 +8.7 + 2.7 +0.1 +6.2 +5.1
V V 2*5 .(pcm/°C) ENDF/B -4.1 -3.8 +7.5 +8.4 +2.7 +0.2 +6.1
00t
+
250 ppm B |X*o UKNDL -3.3 -3.0 +5.1 +6.7 -11.4 -36.5 -9.6 -32.8
(pcm/°C) ENDF/B -3.3 -3.0 + 5.0 +6.4 -11.5 -36.5 -9.8 -33.1■X=B& UKNDL -.0107 -.0082 +.0164 +.0186 -.0351 -.0952 -.0294 -.0848i (m~2/oc) ENDF/B - .0110 -.0086 +.0160 +.0176 -.0354 -.0956 -.0304 -.0866
X=M2 UKNDL -.0002 -.0002 +.0025 +.0016 +.0262 +.0784 +.0285 +.07981
i(cm2/°C) ENDF/B
!-.0003 -.0004 +.0027 +.0018 +.0257 +.0766 +.0281
1+.0780
7 • 2 _ The effective-' resonance i»tegral of U-238
Shielded resonance integrals for U-238 processed from UKNDL
of ENDF/B-IT.1 give an overestimation of the resonance
absorption in cell calculations [48-50]. Tt is there
fore necessary to reduce the resonance integral and corre
late it to integral experiments.
In the data library based on UKNDL which was used in our
analysis of experiments, the U-238 absorption cross section
was reduced uniformly by 0 . 2 0 barn in the energy interval
4 eV - 9118 eV. This correction was originally suggested
by Askew [48]. The effective resonance cross section for
energy group g is given by Eq (3.7)
RI (a ,T)
■ W w ” ■ — L J ; ; »t )x - __ 5__P.? .5___
p»g
where RI (o ,T) is the tabulated effective resonance g P»g
integral as function of the background cross section, ,
and the temperature, T . t is the lethargy width ofg
group g . Differentiating Eq (7.1) one obtains
ARI„(o„ „,T) RI„(o „ -T) 2Ao£ ■ .P»g_____ [ 1 _____________ g.P.1« . ] (7.2)
a , g t o8 8 P>g
which with Ao = -0.20 b for all e is the correction useda »g
in the tables of the library.
In the last column of Table 7.3 resonance integrals cal
culated by AE-BUXY using the Wigner approximation for the
fuel self collision probability
Et 1 Pff = ——- ; x = — and E = — (7.3)f f a+x Z e 2r v ;
e
and the above correction are compared with resonance inte
grals measured by Hellstrand [51] in isolated fuel pins con
taining UO^ or U-metal. The comparison is done for UO^ rods
with two different radii. The smaller radius is typical for
a BWR rod and the larger one gives background cross sections
corresponding to a Dancoff factor of about 0.5, which is
representative for a BWR assembly with average void contents
in the coolant. Resonance integrals for U-metal rods are
also compared at two radii. The experimental values are
given in Table 7.2. The tables also show the temperature
dependence of the resonance integrals. The calculated reso
nance integrals are here compared with experimental results
from references [51, but a comparison with others experi
ments would give essentially the same results. See e.g. the
review by Hellstrand in [53]. The experimental uncertainties
are about 4 % in the room temperature integrals and about
10 % in the Doppler coefficients.
It is seen that AE-BUXY, ' ■ Lth the resonance integral correc
tion which was used in the analysis in chapter 4, overesti
mates the Doppler effect by about 10-15 %, corresponding to
0.4-0 . 8 pcm/°C in the fuel temperature coefficient. In order
to see how the correction in itself affects the temperature
coefficient, three different corrections were tested, viz.
Correction la : .'a = -0.20 barna
11 _ __2a : ;'o = -0. 20[ i *-0.007 (/T-/300) ] barn
" 3a : .MU = -0.06 RI
A rational approximation Cor tin? fuel self collision pro
bability suggested by C.ai ivik [54]
was used instead of the Wigner approximation because the
Carlvik approximation is more accurate.
Using uncorrected data the calculated resonance integral
is overestimated by about 6 % in UC^ rods and 5 % in
U-metal rods (Table 7.3). The temperature dependence is
predicted within about 5 %. The correction la is somewhat
too large giving a slightly underestimated resonance integral
in the UO^ rods. For the U-metal rods, which have a smaller
resonance integral, this correction is about twice the
required one. The Doppler coefficient is increased by 9 %
due to the correction. In order to preserve the temperature
dependence of the uncorrected resonance integral correction 2a
was tested. This correction is identical with la at 300 K.
but does not change the /F-dependence of the resonance inte
gral. As seen in the table ARI/AT remains the same as
with the uncorrected data. Correction 3a reduces the reso
nance integral by & % at all temperatures and o^-values. This
correction also leaves the Doppier coefficient unchanged.
Table 7.4 shows the corresponding comparison for the
library based on ENDF/B-III. The results are similar to
those for the UKNDL-library. The uncorrected resonance inte
gral is about 10 % too large, compared to 6 % for the UKNDL
data, and the following corrections were tested
Correction lb : Ao -0.27 barna
2b : Aa = -0.27[l+0.007(/T-/300)]barna
" 3b : ARI = -0.10 RI
The uniform reduction of a is larger than in the UKNDL
library and the discrepancy in ARI/AT is consequently
larger. Corrections 2b and 3b reproduce the measured Doppler
coefficient better.
The corrections la and lb have the advantage that the un
shielded resonance integral is reduced by 1.5 b and 2.1 b,
respectively, which is within the experimental uncertainties.
They do, however, change the temperature dependence of the
resonance integral. Corrections 3a and 3b do not affect the
Doppler coefficient but give a value of the unshielded reso
nance integral which lies outside the experimental uncer
tainty. Corrections 2a and 2b combine the advantages of
corrections 1 and 3. A drawback is, however, that although
ARI/AT over the whole resonance energy region becomes
correct, ARI /AT in individual energy groups will be §
erroneous. For example, in a group without any resonance
where ARI /AT « 0 one obtains ARI /AT < 0 . g g
The measured temperature dependence of the resonance inte
gral was fitted to the expression [52].
RI(T)-<5 = [RI(T )-5 ] -11+3 (v T'-v/T—)] (7.5)o o
where 6 is the 1/v-part of the resonance integral and
the coefficient 6 varies slightly with the surface tp mass
ratio, S/M, but is assumed to be independent of T . The
measurements cover the temperature range up to 1000 K.
The theoretical values of the Doppler coefficients in
Tables 7.3 and 7.4 were determined using the real temperature,
T, of the fuel. When the crystalline binding effects are taken
into account using an effective Doppler temperature, Tej£>
defined by Eq (6.3) one finds that the calculated Doppler
coefficient in the UO^ rods is reduced by about 9 % over
the temperature interval 300 - 1000 K. The influence of
crystalline effects in the U-metal rods can be expected to
be small.
Table 7.2: Measured U-238 resonance integrals [51] and
Doppler coefficients [52] for isolated fuel rods
Fuel Radius
(cm.)
RIexp
300 °K
6 '*10“
u°? 0.52 21.6 0.77
u o2 1.04 16.8 0.67
u 0.50 16.1 0.73
u 1.00 12.6 0.62
cf Eq (7.5)
Table 7.3: Comparison between calculated U-238 resonance integrals
and Hellstrand's measurements. UKNDL data
Fuel Radius
(cm)
Carlvik approximation Wigner
Corr laNo corr Corr la Corr 2a Corr 3a
RIRI - 1 <% >
exp
RI (ARI/AT)exp th 1
RI., (ARI/AT) th exp
(%)
U°2
U02
U-metal
U-metal
u o2
u o 2
U-metal
U-metal
0.52
1.04
0.50
1.00
0.52
1.04
0.50
1.00
+5.7
+6.4
+4.9
+4.4
+6
-4
+1
- 2
- 0.9
- 1.7
- 3.7
- 5.9
+15
+ 5
+11
+ 9
-0.9
-1.7
-3.7
-5.9
+6
-4
0
-5
0.0
+1.1
-0.4
-0.5
+6
-4
i~ L
-3
- 1.2
- 1.8
- 4.0
- 6.0
+14
+11
+15
+11
Table 7.4: Comparison between calculated U-238 resonance integrals
and Hellstrand's measurements. ENDF/B-III data
Fuel Radius
(cm)
Carlvik approximation Wigner ;
Corr laNo corr Corr la Corr 2a Corr 3a
U°2 0.52 + 9.2 + 0.4 +0.4 -0.5 0.0
RI UCL 1.04 +10.1 - 0.7 -0.7 +0.8 0.0
RI - 1 « >2
U-metal 0.50 + 10.8 - 0.6 -0.6 +1.4 - 0.2exp
U-metal 1.00 +12.1 - 1.3 -1.3 +3.1 - 1.0
U0„ 0.52 + 6 +6 +4 + 16RI (ARI/AT) 2
exp th iU0„ 1.04 - 1 +12 0 -3 +15
RI (ARI/AT) 2tn exp
U-metal 0.50 + 5 + 18 +3 +3 + 21
(%) U-metal 1.00 - 2 +14 -4 -4 + 11
The analysis in chapter 4 was performed using the Wigner
approximation and correction la to the U-238 resonance inte
gral. Table 7.3 shows that using this option the Doppler
coefficient is overpredicted by about 10-15 %. This over-
estimation is reduced to 0-5 % when the crystalline binding
effects are accounted for. It should, however, be remembered
that the experimental uncertainty is of the magnitude 10 %
and one can therefore not draw too detailed conclusions
about the ability of the theory from these comparisons.
When the crystalline binding effects are neglected the shape
of the calculated RI versus temperature can be fitted to
expression (7.5) with a 0-value which is different from the
experimental one. Using the effective Doppler temperature,
T^j;, the calculated Doppler coefficient is reduced more
at low temperature than at high, thus giving a slightly
different shape of RI(T). The precision in Doppler coefficient
measurements has usually not been hi°h enough to detect any
temperature dependence in g . There are, however, some
experimental results indicating a deviation from the in
dependence, e.g. [55].
7._3I_1_____ Scatter ing_i.n_water
The Nelkin scattering model was used for water in the
analysis in chapter 4. Below, temperature coefficients
calculated by use of the Nelkin, the effective width and
the Haywood model are compared. Scattering matrices derived
from the effective width model were tabulated at 293 K w i t h q =
4.3, 450 K with q = 2.8 and at 600 K with q = 2.1 (cf eq 2.15).
Scattering cross sections at intermediate temperatures wer'i
obtained by linear interpolation. The Haywood scattering
cross sections are those obtained from ENDF/B-III by use
of FLANGE-2.
Moderator temperature coefficients for three lattices using
the three different scattering models are listed in Table 7.5.
The agreement in calculated coefficients is very good con
sidering that three entirely different models have been
used.
Table 7.6 shows calculated reaction rate ratios in the fuel
region using the different scattering models. The reaction
rates obtained with the Nelkin and the effective width model
are equal, whereas the Haywood model gives a somewhat harder
spectrum at both low and high temperature.
7 ^ 3 ^ 2 __________§ £ 2 t t e r i n g _ i n _ U 0 2
Calculations using the IK^ scattering model in ENDF/B-III
have been compared with calculations using the free gas model
for both uranium and oxygen in • Although the scattering
cross sections differ considerably in the two cases, the
influence on calculated reactivities and coefficients is
negligible. This is due to the small importance of the
thermal scattering in the fuel in light water reactors. (It
was shown in chapter 5 that the thermal contribution to the
fuel temperature coefficient is only about -0.3 pcm/°C).
Table 7 ,5 : Ca lcu la ted moderator temperature c o e f f ic ie n ts using d i f fe r e n t
sca tte r ing models fo r water
Case Sca tte r ing model dk /9T 00 m
3p/3Tm
3B2/3T m m
3M2/3Tm
Temp in te rv a l (°C) 20-90 160-210 20-90 160-210' 20-90 160-210 20-90 160-210
1.35 % U09 Nelkin -3.3 -3.3 -4.2 -3.9 -.0113 -.0092 +.0046 +.0046
V /V *1.4 m f
E ff .w id th -3.3 -3.2 -4.5 -4.1 -.0124 -.0096 +.0060 +.0058
175 ppm B Haywood -3.2 -3.2
r-H |
•1
i |
-3.8 -.0114 -.0094 +.0046 +.0044
1.9 % U09 Nelk in -2.6 -2.9 -4.0 -3.9 -.0117 -.0100 +.0042 +.0042
V /V _= l.7 m r
E ff .w id th -2.6 -2.8 -4.6 -4.3 -.0136 -.0110 +.0057 +.0054
200 ppm B Haywood i •
1 U1
-2.8 -3.9 -3.9 -.0119 -.0104 +.0041 +.0042
Pu09 Nelk in +7.6 +8.7 +5.0 +6.3 +.0163 +.0184 +.0025 +.0016
V /V =2.5 m f
E f f .w id th +8.9 +9.2 +5.5 +6.4 +.0179 +.0182 +.0037 +.0024
250 ppm B Haywood +7.4 +8.6 +4.7 +6.2 +.0158 +.0186 +.0026 +.0016
Table 7 .6 : Ca lcu la ted re ac t ion ra tes fo r U-235, Pu-239, Pu-241 and Dy-164
re la t iv e to the re ac t ion ra te in a 1/v-absorber
Case Sca tte r ing modal ( 2 3 5 / l / v ) ^(239/1 /v )f is s
(241/1/v) .f 1 S S
(D y / i/v )abs
Temp# (°C) 2 0 2 1 0 2 0 2 1 0 2 0 2 1 0 2 0 2 1 0
1.35 % U02 . Nelk in .993 .968 1.288 1.498 1.109 1.216 .926 .902
V /V =1.4 m f
E f f .w id th .994 .968 1.286 1.500 1 . 1 1 0 1.218 .926 .902
175 ppm B Haywood .990 .966 1.307 1.514. 1.116 1 . 2 2 0 .924 ,901_
1.9 % JJO Nelk in .993 .969 1.305 1.515 1.116 1 . 2 2 2 .920 .897
V /V =1.7 tn f
^ f f .w id th .994 .969 1.304 1.517 ' 1.117 1.224 .920 .897
200 ppm B Haywood .990 .967 1.326 1.531 1.123 1.226 .918 .895
Pu02 Nelk in .997 .973 1.185 1.357 1.081 1.174 .926 .894
V /V =2.5 m i
E f f .w id th .997 .974 1.184 1.357 1.082 1.175 .927 .895
250 ppm B Haywood .994 .971 1.197 1.365 1.085 1.175 .924 .893
Thermal absorption and fission cross sections
7.4.1 Group cross sections for a 1/v-absorber in____________ Maxwel_Han_sgectra_of_various_temperatures
The AE-BUXY data library contains group cross sections which
have been generated by energy condensation of the cross
sections using typical weighting spectra. The energy groups
have been chosen so that group constants may be considered
to be insensitive to the weighting spectra used to generate
them. In the thermal region Maxwellian spectra have been
used for weighting. Scattering matrices are tabulated for
a representative range of temperatures, whereas absorption
and fission cross sections for most nuclides have been
assumed to be temperature independent.
In order to verify this assumption, group cross sections for
a 1/v-absorber were calculated for E <_ 0.625 eV using Max
wellian spectra with various temperatures for weighting.
An analytical expression for the group cross sections of
a 1/v-absorber is easily derived
(7.6)
M(E,T)dE
E£g
where
The evaluation of the integrals in (7.6) gives
a g (T) = (7.7)
g+ and g- are the upper and the lower energy boundaries of
group g .
Calculated a (T) in the AE-BUXY group structure are
listed in Table 7.7 for the temperatures 300, 600, 900 and
1200 K and for E 0.625 eV. CT2200 ecIua to uHity- It
is seen that the group cross sections are insensitive to the
temperature of the Maxwellian spectrum used for weighting.
This conclusion may be expected to hold for all smooth cross
sections. Thus, the error in predicted reactivities and
reactivity coefficients caused by the use of a limited
number of thermal energy groups is negligible as far as
reaction cross sections are concerned (if all materials
are containing only nuclides with smoothly varying cross
sections).
Z •--- _Thermal_data_for_U-235_and_U-238
Figures 7.1-7.4 show the thermal U-235 data, a , a ,sl r
a = o /o, - 1 and n = vo,/o in the libraries generated a f r a
from UKNDL and ENDF/B-III. The agreement between the two
libraries is very good and in the scale used in Figs 7.1
and 7.2 the cross sections coincide. There are, however,
other evaluations giving different data and it has been
pointed out, e.g. in [56]> that the spread in measured data
for U-235 is relatively large.
In order to study the effect of modifications in U-235 data
on calculated reactivities and temperature coefficients a
was changed keeping a unchanged. The change in the shapecl
of a versus energy is shown in Fig 7.3. This modification
lies within the scatter of experimental points and is the
same one as has been considered by Askew L563• Our choice
for this modification of a is arbitrary. It represents
one among several modifications which may be considered.
The influence on calculated reactivities and temperature
coefficients is shown in Table 7.9.
Table 7.7: Group cross sections for a L/v-absorber in the .
AE-BUXY group structure using Maxwellian weighting
spectra of various temperatures.
Group Q 1
00
300 K 600 K 900 K 1200 K
46 2. 21447E-01 2.16559E-01 2.15074E-01 2.14321E-01
47 2.45213E-01 2.41125E-01 2.39788E-01 2.39105E-01
48 2.62920E-01 2.61079E-01 2.60619E-01 2.60392E-01
49 2.76325E-01 2.75378E-01 2.75185E-01 2.75088E-01
50 2.86520E-01 2.85957E-01 2.85864E-01 2.85817E-01
51 2.96194E-01 2.95673E-01 2.95571E-01 2.95517E-01
52 3.10725E-01 3.09782E-01 3.09507E-01 3.09367E-01
53 3.30109E-01 3.20967E-01 3.28735E-01 3.28566E-01
54 3.60025E-01 3.57808E-01 3.57049E-01 3.56666E-01
55 4.03698E-01 4.00596E-01 3.99528E-01 3.98991E-01
56 4.68427E-01 4.63581E-01 4.61926E-01 4.61100E-01
57 5.33721E-01 5.31824E-01 5.31190E-01 5.30874E-01
58 5.88684E-01 5.87 594E-01 5.87233E-01 5.87053E-01
59 6.37430E-01 6.36764E-01 6.36544E-01 6.36435E-01
60 6.85634E-01 6.8497 9E-01 6.84764E-01 6.84655E-01
61 7.43046E-01 7.42215E-01 7.41941E-01 7.41802E-01
62 8.12023E-01 8.11195E-01 8.10919E-01 8.10779E-01
63 8.83177E-01 8.82635E-01 8.82452E-01 8.82358E-01
64 9.60238E-01 9.59539E-01 9.59302E-01 9.59182E-01
65 1.06175E+00 1.06080E+00 1.06048E+00 1.06032E+00
66 1.20412E+00 1.20273E+00 1.20226E+00 1.20203E+00
67 1.42486E+00 1.42256E+00 1.42179E+00 1.4214IE+00
68 1.83773E+00 1.83282E+00 1.83120E+00 1.83040E+00
69 3.03882E+00 3.0J 881E+00 3.01227E+00 3.00901E+00
The absorption cross section of U-238 is generally assumed
to obey the 1/v-law at thermal energies. This is the case
also in UKNDL and ENDF/B-III. Table 7.8 shows the lowest
positive energy resonances of U-238. There are no significant
discrepancies between data from different references.
Table 7.8: U-238 resonance parameters in BNL-325
E
(eV)
Srn(meV)
ry(meV)
r
(meV)
I
4 . 4 1 . 0 0 0 1 1 1 1
6 . 6 7 1 . 5 2 26 2 7 . 5 0
1 0 . 2 5 . 0 0 1 5 6 1
1 1 .3 2 . 0 0 0 3 5 1
1 6 .3 .0 0 0 0 5 1
1 9 . 5 0 .0 0 1 4 1
2 0 . 9 0 8 . 7 25 34 0
3 6 . 8 0 3 2 . 0 25 5 7 . 0 0
6 6 . 1 5 2 6 . 0 22 48 0
Extrapolating from the positive energy resonances one
finds that the first negative energy resonance may be
expected at approximately -10 eV, which means that its
contribution to at thermal energy is of the 1/v-shape.
The contribution from this negative energy resonance is
of importance when deriving the thermal . At
E = 0.0253 eV the first 22 positive levels contribute
2.38 barns [57] and with °2200 = the remaining 0.35 b
usually is attributed to a single negative energy resonance.
When generating U-238 data for ENDF/B-III a single
negative energy resonance was placed at -15 eV [57].
The possibility of a resonance close to zero energy can,
however, not be excluded. If such a resonance exists, its
contribution to the thermal absorption cross section will
be such that the gradient of a versus energy will be
more negative than for a 1/v-cross section.
In order to investigate how a non-l/v cross section affects
the predicted temperature coefficient a change of the absorp
tion cross section was made below 0.3 eV. The modification
was made so that the reaction rate in a 20 °C Maxwellian
spectrum was unchanged. Modified and unmodified cross
sections are shown in Fig 7.5 and results from the calcula
tions with the modified data are collected in Table 7.9
Table 7.9: k and temperature coefficients calculated by AE-BUXY using modified
thermal U-235 and U-238 data
Lattice Modification
kOO
20°C 210°C
Ak /ATOO
(pcm/°C)
k f ef20°C
f210°C
Ak ,,/AT ett
(pcm/°C)
1.35% U02
V /Vf=1.4 m r0 ppm B
Reference 1.16299 1.13875 -12.76 1.00000
i
1.00000 I 0 1
A°f235
A°a238
+483
+500
+741
+ 1172
+ 1.36
+3.54
1
+412 1 +645
+380 i +966i
+ 1.23
+3.08
1.9% U02
V /V,=1.7 m r200 ppm
Reference 1.21010 1.19573 -7.56 1.00000 1.00000 0
A°f235
A0a238
+519
+392
+785
+909
+1.40
+ 2.72
+426
+338
+650
+765
+ 1.18
+2.25
In the 1.35 % U02 lattice the temperature coefficient was
predicted about 4 pcm/°C too negative (cf Table 4.3) and in
the 1.9 % UO^ lattice 2 pcm/°C too negative (cf Table 4.7).
The modification of U-238 gives the correction +3 pcm/°C
and +2 pcm/°C in the 1.35 % U02 and 1.9 % U02 lattices,
respectively. The U-235 modification gives about +1 pcm/°C
in both lattices.
Data for some lattices which were analysed in chapter 4 are
collected in Table 7.10. The thermal absorption in U-235
and U-238 as calculated by AE-BUXY is given in percent of
the total cell absorption plus leakage. Assume that the
change in calculated Ak^^/AT due to modified U-235 and
U-238 cross sections is proportional to the relative absorp
tion in the nuclide. Then corrections to Ak ,.r/AT areeff
obtained by combining the results of Table 7.9 with the
relative absorptions listed in Table 7.10. These estimated
corrections are given in the last two columns and are com
pared with the discrepancies obtained in the comparisons
between theory and experiment.
It is seen that for all these lattices a close agreement
between theory and experiment can be obtained by modifying
the shape of the thermal U-238 absorption cross section as
function of energy.
The influence of a modification in the shape of the thermal a
for U-235 on the temperature coefficient has previously been
studied elsewhere, e.g. in [56, 58]. Askew et al. [4]
conclude that nuclear data uncertainties indicate that
extreme changes within the uncertainties of nuclear data
in the energy dependence of r) for U-235 could contribute
about +1 pcm/°C to the temperature coefficient. Our results
are in agreement with his conclusion.
Basiuk et al. [59] have made calculations on a PWR lattice
assuming a p-wave resonance in U-238 at 0.1 eV. This did
not change the calculated temperature coefficient and they
conclude that a p-wave resonance at a lower energy would
not influence the results.
Table 7.10: Estimated corrections to Ak £C/AT due to modified a anu ------------ eff a238 f235
for some KRITZ lattices.
Fuel
V / V f and m f
boron conc
Temp
°C
Therma
U235
abs (%)
U238
ikef£/4T
Discrepancyth-exp
(pcm/°C
Correct modifie
°f 235
ion due to i
°a238
1.35% U0 9 V /V,=1.4 m f
0 ppm B
20
210
41.1
40.6
12.5
12.7-4.4 + 1.2 +3.1
V /Vf=1.4 m f
175 ppm B
20
210
40.9
40.1
12.4
12.5-3.3 + 1.2 +3.1
1.9% U0 2 V /Vf=1.2 m i
200 ppm B
20
210
39.1
38.0
8.6
8.6-1.4 + 1.1 +2.1
V /V,=l.7 m f
200 ppm B
20
210
41.2
40.4
9.0
9.1-1.8 + 1.2 +2.3
1.5% Pu02 V /V,=l.8 m 1
300 ppm B
20
210
2.5
1.8
5.7
4.9-0.4 +0.1 +1.3
V /V,=2.5 m t
250 ppm B
20
210
2.3
1.9
6.1
5.2-1.0 +0.1 + 1.4
V /Vf=3.3 m r
0 ppm B
20
210
2.5
2.3
6.4
5.5-0.6 +0.1 + 1.5
Fig. 7.1 oQ for U-235 in the AE-BUXY library-
Fig.7.2 of for U-235 in the AE-BUXY library
106
Fig. 73. a for U -23 5 in the A E - B U X Y library and in E N D F / B m .
s.
^ M o d fied shape of cfa in the present study
....
>vS s\
............. ....v v
0.001 0.01 0.1 eV E
Fig. 7.5 orQ for U-238 in the AE-BUXY library
Our comparison between calculated and measured temperature
coefficients shows that the employed theory predicts a too
negative coefficient, the discrepancy being 1-4 pcm/°C.
The partial temperature coefficients - the fuel temperature
coefficient, the moderator temperature coefficient and the
water density coefficient - were theoretically determined.
The fuel temperature coefficient is approximately proportional - 1/2
to T . The dominating component is caused by the Doppler
broadening of the U-238 resonances and an accurate calcula
tion of the Doppler coefficient requires that the crystalline
binding in the U0„ is taken into account./This effect is^ V
largest at low temperature. It can be accounted for by using
an effective Doppler temperature which is higher than the
true temperature of the medium. The correction of the
Doppler coefficient due to the crystalline binding is
about 15 % at room temperature and 3 % at 1000 K.N
The moderator temperature coefficient consists mainly of three
components 3p/3rr3n/3T , 3p/3f-3f/3T and 3p/3L2 -3L2 /3Tm m m
in the thermal energy group. 3p/3f•3f/3Tm is in all. lattices. . /.
positive and the other two components are negative./The mag
nitude of 3p/3f'3f/3Tm is much larger in Pu02 lattices
than in UO 2 lattices. The discrepancy between calculated
and measured temperature coefficients is approximately the
same for all lattices, however, and one may therefore expect
that the employed model for the thermal scattering in water
predicts the temperature dependence of the spectrum correctly.
This conclusion is supported by the comparison between re
sults using the Nelkin, the effective width and the Haywood
models for scattering in water. All three models give the
same temperature coefficient although the Haywood model pre
dicts a harder spectrum than the other two models.)
The water density coefficient varies considerably with
temperaturefwhereas the discrepancy between theoretical
and experimental temperature coefficients is nearly the same
at low and high temperature. This observation indicates that
the inconsistency between theory and experiments is not due
to effects caused by the water density variation with tem
perature. '
Various approximations in the theory have been validated. We
have found that the cylindricalization of pin cells, the
utilization of transport corrected cross sections to account
for anisotropic scattering in the cell calculation and the
use of a limited number of energy groups and spatial meshes
in the calculations introduce very small errors in the pre
dicted temperature coefficients.
The comparison between results using the UKNDL and the
ENDF/B III data shows that these data sets provide the same
temperature coefficient.!
The temperature coefficient is sensitive to the energy
dependence of the thermal U-235 and U-238 cross sections.
An error of % 1 pcm/°C is possible due to the uncertainty
in the capture to fission ratio as function of energy for
U-235. The thermal U-238 absorption cross section is usu
ally assumed to obey the 1/v-law. However, if a negative
energy resonance exists close to zero energy its influence
on a will be such that a decreases with higher energy & a
faster than the 1/v-law prescribes. Such a shape of aa
versus energy would produce a less negative temperature
coefficient. It was found that the calculated temperature
coefficients for all investigated cores can be brought in
agreement with experimental values by a modification of the
thermal U-238 absorption cross section shape. This modifi
cation would also give calculated values of closer to
unity and with a smaller spread.
The work presented here was carried out at the section
for Reactor Physics of AB Atomenergi, Studsvik. Part of
the work was sponsored by the Swedish Board for Technical
Development (STU).
I would like to thank my colleagues for many fruitful dis
cussions. My thanks are particularly due to Drs E Hellstrand,
H Haggblom and R Persson at a B Atomenergi and Prof N G
Sjostrand at Chalmers University of Technology.
- Ill -
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