Research Journal of Applied Sciences, Engineering and Technology 3(3): 172-178, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Received: December 28, 2011
Accepted: January 27, 2011
Published: March 30, 2011
Analysis of the Temperature Effect on the Infinite Multiplication Factor for
HEU-UAl4 and LEU-UO2 Lattices of GHARR-1
1
E. Alhassan, 2N.A. Adoo, 2G.K. Appiah, 1A.N. Adazabra, 1E.A. Alhassan, 2S.E. Agbemava,
2
V.Y. Agbodemegbe, 2R. Della and 2C.Y. Bansah
1
University for Development Studies, P.O. Box 24, Navrongo Campus, Navrongo, Ghana
2
National Nuclear Research Institute, Ghana Atomic Energy Commission,
P.O. Box LG 80, Legon-Accra, Ghana
Abstract: The purpose of the study is to analyze the temperature effect on the infinite multiplication factor for
light water moderated High Enriched Uranium (HEU)-UAl4 and Low Enriched Uranium (LEU)-UO2 lattices
of the Ghana Research Reactor-1 (GHARR-1). To quantify the contribution of each component of the infinite
multiplication factor with respect to temperature within the 20 to 140ºC range, cell calculations were performed
for the two MNSR typical lattices: the 90.2% enriched HEU-UAl4 and 12.6% enriched LEU- UO2 proposed fuel
of the Ghana Research Reactor-1 (GHARR-1) using SCUBA, a locally developed FORTRAN 95 code for the
calculations and analysis of temperature coefficients of GHARR-1. It was observed that at the beginning of life
of the core, the temperature coefficient of the resonance escape probability and that of the thermal utilization
factor, contributed significantly to the negative temperature coefficient of the infinite multiplication factor
obtained for both fuels.
Key words: Core conversion, Ghana Research Reactor-1, infinite multiplication factor, reactor inherent safety,
temperature effect, zero Burnup
ascertain the polarity, magnitude and effects of fuel and
moderator temperatures on its physical components for
the safe operation of the reference 90.2% HEU-UAl4 and
the proposed 12.6% LEU-UO2 fuel lattices of the Ghana
Research Reactor-1 (GHARR-1) at zero burnup. The
research was done as part of the core conversion studies
currently on-going at the Department of Nuclear
Engineering and Materials Science (NEMS) in the
National Nuclear Research Institute (NNRI) of Ghana
Atomic Energy Commission (GAEC).
INTRODUCTION
During reactor operation, numerous parameters such
as temperature, pressure, power level, and flow are
continuously monitored and controlled to ensure safe and
stable operation of the reactor (DOE, 1993). The specific
effects of variations in these parameters vary greatly
depending upon reactor design. The most significant
effect of a variation in temperature when a reactor is in
operation is the temperature coefficient (DOE, 1993;
Lamarsh and Baratta, 1982). The temperature coefficient
which is the relative change of a physical property with
respect to changes in temperature is considered as one of
the most important physical quantities used to assess the
inherent safety of a reactor (DOE, 1993). Temperature
variation in a reactor changes the microscopic reaction
rate and the density of the moderator and consequently,
the infinite multiplication factor. This results in the
addition of either positive or negative reactivity
(DOE, 1993; Ott and Bezella, 1989) and hence changes in
reactor power. The temperature dependence of the infinite
multiplication factor is considerably important in
connection with the transient behavior of a reactor (Ott
and Bezella, 1989) and this have associated nuclear
criticality safety issues and must therefore be investigated
since it has a large influence on the estimation of
operational reactivity balance.
In this research, the effect of temperature on the
infinite multiplication factor has been analyzed in order to
Reactor description: GHARR-1 is a Chinese built
Miniature Neutron Source Reactor (MNSR) with a
maximum thermal neutron flux of 1 x 1012 ncmG2sG1 and
a thermal power of approximately 30 kW. The MNSR is
used for Neutron Activation Analysis (NAA), training of
Scientist and Engineers in nuclear science and technology
and small scale radioisotope production (AnimSampong et al., 2007; Akaho et al., 2003). In Fig. 1 and
2, the horizontal and vertical cross sectional views of the
GHARR-1 core are presented. A detailed description of
the reactor is presented elsewhere (Alhassan et al., 2010;
Akaho et al., 2003).
MATERIALS AND METHODS
The study was carried out in the year 2009 at the
Graduate School of Nuclear and Allied Sciences of the
University of Ghana, Ghana, West Africa.
Corresponding Author: E. Alhassan, University for Development Studies, P.O. Box 24, Navrongo Campus, Navrongo, Ghana
172
Res. J. Appl. Sci. Eng. Technol., 3(3): 172-178, 2011
Fig. 1: A horizontal cross section view of GHARR-1 (Anim-Sampong et al., 2007)
Fig. 2: A vertical cross section view through the GHARR-1 (Anim-Sampong et al., 2007)
A locally developed computer code named SCUBA,
written in FORTRAN-95 programming language was
used in the computation and analysis of the effect of the
fuel and moderator temperature changes on the infinite
multiplication factor based on Eq. (1) to (16). A detailed
structure and development of the code has been presented
elsewhere (Alhassan et al., 2010).
Theoretical consideration: For the maintenance of a
self-sustaining chain reaction, it is necessary that every
neutron produced in fission initiate another fission
(DOE, 1993). The minimum requirement is for each
nucleus undergoing fission to produce on the average, at
least one neutron that will cause the fission of another
nucleus. This condition is conveniently expressed in
173
Res. J. Appl. Sci. Eng. Technol., 3(3): 172-178, 2011
terms of the multiplication factor, which is given by
(Gladstone and Sesonke, 1981):
k∞ = εpfη
Temperature effect on the fast fission factor: For the
calculation of the fast fission factor, it was assumed that,
thermal fission occur only in U-235 whiles fast fission
and epithermal absorptions occur in U-238 only. For a
heterogeneous reactor, is given by (Bennet, 1972):
(1)
where, g is the fast fission factor, p is the resonance
escape probability, f is the thermal utilization factor and
0 is the thermal fission factor.
The Reactivity Temperature Coefficient ("T) which
is defined as the change in reactivity with respect to
temperature is expressed as (Ott and Bezella, 1989):
αT =
dρ
dT
⎛
1 + 0.690⎜⎜
⎝
ε=
⎛
1 + 0.563⎜⎜
⎝
where, N 28 N H O is the ratio of the number of atoms
2
keff − 1
keff
of U-238 to that of water. To analyze the temperature
effect on the fast fission factor, the moderator densities
were varied with temperature and the corresponding
number densities computed. The temperature coefficients
for the various temperatures were calculated using Eq. (8)
by applying polynomial of best fit and then differentiating
and the results are plotted in Fig. 3.
(3)
Differentiating equation (3) with respect to
temperature (T) gives (Lamarsh and Baratta, 1982):
αT =
Temperature effect on the thermal utilization factor:
The thermal utilization factor (f) by definition is given by
(Akaho, 2007):
dρ
d ⎛ k eff − 1⎞
⎜
⎟=
=
dT dT ⎜⎝ k eff ⎟⎠
1 dk eff
1 dk eff
≈
2
dT
k eff dT
k eff
f =
(4)
1 dkeff
1 dk∞
1 dPNP
=
+
keff dT
k∞ dT
PNL dT
∑
(6)
c
c c
(9)
a
m
a
=
103 ρm N A m
σa
Am
(10)
⎛ N 28 ⎞ 28
⎟σ
⎝ N 25 ⎠ a
(11)
f =
⎛N ⎞
⎛ NH 2O ⎞ H 2 O ⎛ NAl ⎞ Al
σ a25 + ⎜ 28 ⎟ σ a28 + ⎜
⎟ σa + ⎜
⎟ σa
⎝ N 25 ⎠
⎝ N 25 ⎠
⎝ N 25 ⎠
σ a25 + ⎜
where, the first term is the temperature coefficient of the
infinite multiplication factor and the second term
represent the leakage effect. By using the four factor
formula, "k4 can be expressed as follows:
α k∞ =
f f f
a
m m m
a
where, σ am is the microscopic cross section for the
moderator,Am is the atomic weight of the moderator, Dm is
the density of the moderator and NA is the Avogadro’s
number. After substituting Eq. (10) into (9) and dividing
through by N25, f becomes:
(5)
Equation (5) can be written as:
αT = α k∞ + α L
∑a φ f v f
f
∑φ v
+∑ φ v +∑ φ v
where, the subscripts f, m, c represents fuel, moderator
and cladding respectively. The moderator macroscopic
cross section is given by:
where, keff is the effective multiplication number.
From the basic relationship between keff and k4,keff
PNL , where k4 is the infinite multiplication factor and PNL
is the non-leakage probability; we obtain by
differentiating Eq. (4):
αT =
(8)
N 28 ⎞
⎟
N H2O ⎟⎠
(2)
where, reactivity (D) which is the fractional departure of
a system from criticality is expressed as (Lamarsh and
Baratta, 1982):
ρ=
N 28 ⎞
⎟
N H2O ⎟⎠
1 dk∞ 1 dε 1 dp 1 df
1 dη
(7)
=
+
+
+
ε dT p dT f dT η dT
k∞ dT
where, N25 and N28 are the number densities for U-235 and
U-238, NAl and NH2O are the number densities for Al and
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Res. J. Appl. Sci. Eng. Technol., 3(3): 172-178, 2011
H2O, σ a25 and σ a28 are the absorption microscopic cross
Temperature effect on the resonance escape
probability: With the assumption that all the resonances
are resolved and do not overlap, the resonance escape
probability, P for a heterogeneous reactor is given
mathematically by (Zweitfel, 1985):
sections for U-235 and U-238 respectively and, σ aAl and
σ aH2O are the absorption microscopic cross sections for
Al and H2O, respectively. To calculate the thermal
utilization factor, the density of the light water moderator
was varied with temperature and the corresponding cross
sections and number densities computed. The temperature
coefficients for the various temperatures were calculated
using Eq. (11) by applying polynomial of best fit and then
differentiating and the results are plotted in Fig. 4.
⎡ N V AI
eff
28
p = exp − ⎢
⎢ ξ
p
⎣
∑
with,
ξ∑ p =
Temperature effect on the thermal fission factor: The
thermal fission factor which is the number of fast neutrons
produced per thermal neutron absorbed in the fuel is given
by (Akaho, 2007):
η=
g (T )σ 0( 28) (T0 ) ⎛ T0 ⎞
⎟⎟
⎜⎜
1128
.
⎝ T ⎠
+VC
ξ ∑ p is the average slowing down power in the
moderator, N28 is the density of atoms/nuclei of U-238,
VA, VM and VC are the volume of fuel region, moderator
and clad respectively and Ieff is the effective Resonance
Integral from the Strawbridge correlation.
The temperature effect on the resonance escape
probability was calculated by varying the fuel temperature
and computing the corresponding cross sections. The
temperature coefficient for the various fuel temperatures
were calculated using Eq. (15) by differentiating the
polynomial of best fit and the results are plotted in Fig. 5.
RESULTS
2
(13)
The results of the factors of the temperature
coefficient of the infinite multiplication factor calculated
and averaged over the temperature range of 20 to 140ºC
for the LEU and HEU fuels of GHARR-1 and
experimental results are compared in Table 1.
The factors of the temperature coefficient of the
infinite multiplication factor plotted against temperature
(of 20 to 140ºC) for the LEU and HEU fuels of GHARR1 are presented in Fig. 3, 4, 5, 6 and 7.
where, F0(28) and Fa28(T) are the microscopic cross section
of U-238 at temperature T0 and T respectively, T0 = 293 K
and T = (2+293) K with 2 in ºC.
Whiles for U-235, the absorption cross section was
given by (Lamarsh and Baratta, 1982):
g (T )σ 0( 25) (T0 ) ⎛ T0 ⎞
⎟⎟
⎜⎜
σ a 25 (T ) =
1128
.
⎝ T⎠
M
region A, >m is the log energy decrement of the moderator,
∑ sM is the scattering macroscopic cross section of the
microscopic cross sections for U-235, σ a25 and σ a28 are
the absorption microscopic cross sections for U-235 and
U-238 and,N25 and N28 are the number densities of U-235
and U-238, respectively.
The temperature dependence of the thermal fission
factor (0) is contained in the microscopic cross sections
of the various isotopes in the fuel. Since U-238 and U-235
are both non-1/v nuclides, the Westcott correction factor
g(T) was used for the correction of the absorption cross
sections.
For U-238 the absorption cross section is given as
(Lamarsh and Baratta, 1982):
σ a 28 (T ) =
V +V
A
the moderator, ∑ ApM is the potential cross section of
where, g is the enrichment of the fuel, σ f 25 is the fission
1
)
V A ξ F ∑ Fp + ξmA ∑ ApM + V C ∑ Cp + V M ∑ M
p
non-resonant region, ∑ M
p is the potential cross section of
(12)
N 25σ a 25 + N 28σ a 28
(
(15)
(16)
where,
vN 25σ f 25
⎤
⎥
⎥
⎦
12
DISCUSSION
(14)
In Table 1, the temperature coefficient of the infinite
multiplication factor averaged over the 20 to 140ºC
temperature range have been presented. The results
obtained from the code showed the values of the
temperature coefficient of the thermal utilization factor
and that of the resonance escape probability to be -0.070
where, F0(25) and Fa25(T) are the microscopic cross section
of U-235 at temperature T0 and T, respectively. SCUBA
calculates using Eq. (12) to (14) for both the 12.6% UO2
and 90.2% UAl4 enrichments for temperatures between 20
and 140ºC.
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Res. J. Appl. Sci. Eng. Technol., 3(3): 172-178, 2011
1/ε(dε/dT)in ∆k/k/°C
Table 1: Temperature effect of k4 in a UAl4 and UO2 lattice of GHARR-1
HEU-UAl4This work
HEU-UAl4 Experimental
Fuel lattice
(20 to 45 ºC)/mk
(20 to 140 ºC)/mk
1/p (dP/dT)
Negligible
- 0.00056
1/f (df/dT)
- 0.092
- 0.070
1/0 (d0/dT)
+0.00056
1/e (dg/dT)
+0.00084
1/k4 (dk4/dT)
- 0.092
- 0.073
0.0000045
90.2% HEU-UAI4 fuel
0.000004
12.6% LEU-UO2 fuel
4.00E-06
1/η(dη/dT)in ∆k/k/°C
90.2% HEU-UAl4 fuel
3.00E-06
0.000003
2.50E-06
0.0000025
0.000002
2.00E-06
0.0000015
1.50E-06
0.000001
1.00E-06
0.0000005
5.00E-06
0
0
20
40
60
Temperature in °C
80
100
0.00E-06
0
0
100
50
100
150
Fig. 6: Effect of temperature on the thermal fission factor for
HEU and LEU cores of GHARR-1
0.0001
0
50
Temperature in °C
Fig. 3: Effect of temperature on the fast fission factor for HEU
and LEU cores of GHARR-1
0.00E+00
150
0
50
100
150
-0.0001
-1.00E+04
-0.0002
-0.0003
1/k4(dk4/dT)in ∆k/k/°C
1/f(df/dT)in ∆k/k/°C
12.6% LEU-UO2 fuel
(*0.1)
3.50E-06
0.0000035
90.2% HEU-UAl4 fuel
12.6% LEU-UO2 fuel
-0.0004
-0.0005
Temperature in °C
-0.0006
-2.00E+04
-3.00E+04
-4.00E+04
-5.00E+04
Fig. 4: Effect of temperature on the thermal utilization factor
for HEU and LEU cores of GHARR-1
-6.4E-06
1/p(dp/dT)in ∆k/k/°C
LEU-UO2This work
(20 to 140 ºC)/mk
- 0.0058
- 0.160
+0.028
+0.0029
- 0.131
-6.00E+04
-5.8E-06
-5.6E-06
-5.4E-06
-5.2E-06
-0.000005
0
100
50
Temperature in °C
Temperature in °C
and -0.00056 mk for the HEU and -0.160 and -0.0058 mk
for the LEU respectively whereas the results from
literature [5] is -0.092 mk for the temperature coefficient
of the thermal utilization factor and that of the
temperature coefficient of resonance escape probability
was found to be negligible for the 90.2% HEU fuel. These
results are therefore in reasonable agreement. The
contribution of the temperature coefficient of the fast
fission factor and the thermal fission factor were found to
be positive but less dominant.
From Fig. 3, it can be observed that, the temperature
coefficient of the fast fission factor increases linearly with
90.2% HEU-UAI4 fuel
-0.000006
12.6% LEU-UO2 fuel
Fig. 7: Effect of temperature on the infinite multiplication
factor for HEU and LEU cores of GHARR-1
12.6% LEU-UO2 fuel
(*0.1)
-6.2E-06
90.2% HEU-UAl4 fuel
150
Fig. 5: Effect of temperature on the resonance escape
probability for HEU and LEU cores of GHARR-1
176
Res. J. Appl. Sci. Eng. Technol., 3(3): 172-178, 2011
temperature increase and its contribution to the
temperature coefficient of the multiplication factor was
found to be positive for both fuels but more significant for
the LEU fuel. This can be attributed to the moderator
density effect; an increase in temperature results in a
reduction of the moderation ratio and hence leads to an
increase in the fast to thermal ratio. Since fast fission can
only occur in U-238 nuclei whiles fission in U-235 nuclei
is limited to only thermal neutrons, LEU fuel with a
higher amount of U-238 is fissioned by some of the fast
neutrons and thereby making available more neutrons in
the reactor core.
The temperature effect of the thermal utilization
factor was found to be negative for both fuels and
decreases linearly for both fuels as shown in Fig. 4. The
negative contribution of the temperature coefficient of the
thermal utilization factor is related to the water density
effect; an increase in moderator temperature reduces the
number of neutron collisions per unit volume and so
causing the neutron spectrum to harden (Alhassan, 2009;
Mahmood et al., 2008). This alters the balance between
the fission and absorption rates and reduces the thermal
fission factor and hence the temperature coefficient of the
thermal utilization factor. The hardened spectrum results
in an increase in resonance absorption for the low-lying
resonances of U-238 which are more dominant in LEUUO2 than the HEU-UAl4 fuel hence that of the LEU fuel
was seen to drop more sharply.
The linear nature of the plots in Fig. 5 illustrates the
fact that, the temperature of the fuel reacts immediately to
changes in reactor power for both fuels. With increasing
temperature, U-238 resonances broaden thereby
increasing the absorption of neutrons (Doppler
broadening). Because of the strong U-238 capture cross
section in the LEU fuel, the temperature effect resonance
escape probability is more dominant in the 12.6%
enriched LEU fuel than the 90.2% HEU fuel.
As can be observed from Fig. 6 that, the temperature
coefficient of the thermal fission factor was positive and
decreases almost linearly for LEU fuel whiles increasing
gradually for the HEU fuel for the whole temperature
range. This temperature effect is mainly due to the
spectral shift effect which is more dominant for the LEU
fuel because the thermal fission factor is enrichment
dependent. Also, the thermal energy spectrum changes
and broadens with temperature and hence alters the
balance between the fission and absorption rates in the
core.
From Fig. 7, it is observed that the temperature
coefficient of the infinite multiplication factor decreases
for the entire temperature range (20-140oC). The negative
contribution comes from the temperature coefficient of
the thermal utilization factor and resonance escape
probability whiles the temperature coefficient of the fast
fission factor and the thermal fission factor both
contributed positively.
CONCLUSION
The analyses of the temperature effect on the infinite
multiplication factor for the light water moderated 90.2%
High Enriched Uranium (HEU)-UAl4 and the proposed
12.6% Low Enriched Uranium (LEU)-UO2 lattices of the
Miniature Neutron Source Reactor (MNSR) for a fresh
core replacement has been carried out using SCUBA a
locally developed FORTRAN 95 code. The results
showed the predominance of the contribution of the
temperature coefficients of the resonance escape
probability and the thermal utilization factor to the
negative temperature coefficient of the infinite
multiplication factor. The contribution of the temperature
coefficient of the resonance escape probability which was
negligible with the HEU fuel will contribute significantly
to the inherent safety ability of the proposed LEU fuel for
GHARR-1.
ACKNOWLEDGMENT
Authors want to thank friends and colleagues at the
Ghana Research Reactor-1 Centre, National Nuclear
Research Institute, Ghana Atomic Energy Commission for
their suggestions and encouragement.
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