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2013 International Nuclear Atlantic Conference - INAC 2013
Recife, PE, Brazil, November 24-29, 2013 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN
ISBN: 978-85-99141-05-2
ESTABILIZATION TIME FOR SUBCRITICAL SYSTEM WITH
DIFFERENT EXTERNAL NEUTRON SOURCE
Brayan S. Fonseca1, Fernando C. Silva
1 and Zelmo R. Lima
2
1 Programa de Engenharia Nuclear - COPPE
Universidade Federal do Rio de Janeiro
Av. Horácio Macedo, 2030
21941-972 Rio de Janeiro, RJ
brayanuff@hotmail.com; fernando@con.ufrj.br; zelmorod@gmail.com
2 Instituto de Engenharia Nuclear (IEN/CNEN – RJ)
Rua Hélio de Almeida,75
21941-906 Cidade Universitária RJ
zelmolima@yahoo.com.br
ABSTRACT
This work aims at studying one dimensional spatial kinetics equations for ADS (Accelerator
Driven System) type reactors, through computational simulations. Where the neutrons
diffusion equations are discretized espatially by the finite differences cell centered mesh
scheme and time discretized using the Crank-Nicolson method. Discretized equations used
with fixed source problem are programmed in FORTRAN and used to simulate the system
behavior, with different neutron source types. In this way, we can analyse the instant of time
when the subcritical system becomes stationary, or the time interval corresponding to the
stationary flux, for different external sources. The result shows that, it is always possible to
reach system stability where no oscillation is detected, even for pulsed sources.
1. INTRODUCTION
The ADS (OECD, 2002) is an innovated system that uses a fast neutrons source through a
reaction entitled “spallation”. During spallation, neutrons are ejected from a heavy nucleus,
and then target a high energy particle; in the case accelerated protons that are derationed to a
material target in solid or liquid state (OECD, 2002).
The ADS reactor is a subcritical system that offers an intrinsic security in the potential of a
critical accident because the subcritical reactor estate avoids the fission chain reaction
without control; and so this system naturally turns off the reactor. In actual reactors, the
process turning off the device needs to be done through control bars. Thus, in accident cases,
this can not occurs and consequently, a without control fission chain reaction.
For the stability subcritical studying, we simulate through computer reenactment, a reactor
core Benchmark (NAGAYA, 1995) using the spatial kinetics equations (DUDARSTADT e
HAMILTON, 1976) discretized spatially by the finite differences cell centered mesh scheme
(Alvim, 2007) and time discretized using the Crank-Nicolson method (NAKAMURA, 1977).
INAC 2013, Recife, PE, Brazil.
In this way, we analysed the time interval corresponding to the stationary flux, for different
external sources.
This paper is structured in the following way: In section 2, we show the one dimensional
spatial kinetics equations discretization for two energy groups by the finite differences cell
centered mesh scheme and the time discretization using the Crank-Nicolson method. In
section 3, we demonstrate the reactor core used for simulation and its nuclear dates. In
section 4, we display the results and further discuss for the simulations. In section 5, we
present the conclusions.
2. SPATIAL KINETICS EQUATIONS DISCRETIZATION
In this section we show the spatial kinetics equations and its spatial discretizations by the
finite differences cell centered mesh scheme (Alvim, 2007) and the time discretization using
the Crank-Nicolson method (NAKAMURA, 1977).
The one dimensional spatial kinetics equations and two energy groups can be represented in
the following way:
2
1
6
1
2
1
,,1,
,,,,v
1
g´ l
lgllg´fg´g
gg`g`
g`gg`
gRggg
g
txCχλt)(xxν χ t)(xx
t)S(xtxxtxJx
t) (x
t
(1)
and
2
1
),(),(),(g
llgfgll txCtxxtxCt
. (2)
Where txS , represents a external source term. Here, the nuclear parameters are time
independent and there are six precursors groups. In accordance with Fick’s law, we can write
the current density as:
txx
xDtxJ ggg ,,
. (3)
The nomenclature of these equations and their respective units are:
txg , = Neutrons flux in group g , in position x and time t (number of neutrons/cm2.s);
txCl , = Delayed neutrons precursors concentrations for group l , position x and time t
(nuclei/cm3);
INAC 2013, Recife, PE, Brazil.
l = Fraction of all fission neutrons emitted per fission that appears from l th precursor
group;
l = Decay constant of l th precursor group (s-1
);
gv = Neutron speed for group g (cm/s);
xDg Diffusion coefficient for group g in position x (cm);
xRg = Removal macroscopic cross section for group g in position x (cm-1
);
xgg` = Group-transfer macroscopic cross section from a group 'g to the group g in
position x (cm-1
);
xfg = Average number of neutrons released per fission multiplied by the fission
macroscopic cross sections in position x (cm-1
);
g = Neutrons fission spectrum for group g ;
gl = Delayed neutrons fission spectrum for energy group g and precursor group l .
2.1. Spatial Discretization
The reactor core is made by M regions, as illustrated in figure 1, in which the total length
is a .
Figure 1: Core with dimension a and with M regions
We can discretize this core in figure 1, subdividing each one of the regions in discrete cell
mesh. The figure 2 illustrates a core reactor with m generic region discretization. The
distance between cell meshes have a length mx , for any cell mesh in region m, and it’s
calculated in the following way:
1, ; 1
,,1
,,
,,
mfminn
mimf
mimf
m nnnxxnn
xxx . (4)
Figure 2: Cell mesh representation in a m generic region
f,mnnn-i,m x x x x x 11
min , 1n n mfn ,
mx
1 2 Mm
2
a
2
a
INAC 2013, Recife, PE, Brazil.
Now defining:
dxtxx
tn
n
x
x
g
m
n
g
1
,1
, (5)
dxtxCx
tCn
n
x
x
l
m
n
l
1
,1
(6)
and
dxtxSx
tSn
n
x
x
l
m
n
g
1
,1
. (7)
From equations 1, 2 e 3, we obtain:
6
1
2
'1'
'''
2
1'
11
g
1
v
l
n
lml
ggg
n
gm
m
gg
n
g
g
m
m
fgm
n
g
n
g
n
g
n
g
n
g
n
g
n
g
n
gm
tCxxtxxtS
tctbtatdt
dx
(8)
and
tCxtxtCdt
dx n
lml
g
n
gm
m
fgl
n
lm
2
1
. (9)
In which the constants n
g
n
g
n
g cba e , are so determined as:
a) For 1 and 1 mn ,
1
1
1
1
1
1
2
2
x
D
x
Dxb
g
g
g
Rg
n
g
(10)
and
1
1
1
x
Dc
g
g
. (11)
INAC 2013, Recife, PE, Brazil.
b) For 1 and , mnn mi ,
1
1
12
m
m
gm
m
g
m
g
m
gn
gxDxD
DDa , (12)
)( n
g
n
gm
m
Rg
n
g caxb (13)
and
m
m
gn
gx
Dc
. (14)
c) for Mmnn mf and , ,
m
m
gn
gx
Da
, (15)
)( n
g
n
gm
m
Rg
n
g caxb (16)
and
m
m
gm
m
g
m
g
m
gn
gxDxD
DDc
1
1
12. (17)
d) for mfmi nnn ,, and Mm1 ,
m
m
gn
gx
Da
, (18)
)( n
g
n
gm
m
Rg
n
g caxb (19)
and
m
m
gn
gx
Dc
. (20)
INAC 2013, Recife, PE, Brazil.
e) For MmNn and ,
M
M
gN
gx
Da
, (21)
and
M
M
g
Mg
M
g
M
M
Rg
N
gx
D
x
Dxb
2
2. (22)
2.2. Time Discretization
For the time discretization in equations 8 and 9, we used the Crank-Nicolson method
(NAKAMURA, 1977), obtaining:
i
n
i
n
nni
n
nni
n
nn tStAtAVtA~
1
1
~1,1
~,
1
1
1
~1,
. (23)
In which the matrixes 1,,1,
1 , ,
nnnnnn AeAAV are the tri-diagonal block elements and they are
defined as:
21
1
v
x
v
xDiagV mm
n , (24)
tatadiagA nn
nn 211,2
1
2
1, (25)
tctcdiagA nn
nn 211,2
1
2
1 (26)
and
.
2
1
2
1
21
2
1
21
2
1
2
1
221
2
6
1
1
6
1
1
,
tbtx
txt
ttx
t
ttb
An
m
n
m
m
f
l l
ll
m
m
f
l l
lln
nn
(27)
While the source term i
ntS
~ in equation 8 is:
in
nni
n
nni
n
nni
n
ii
ntAtAVtAtStS
1
~1,
~,
11
~1,
~~
. (28)
INAC 2013, Recife, PE, Brazil.
In which:
tS
tCt
xttS
Sn
i
l
i
n
l
l
mln
in
i
,2
6
1
,1
~2
2
. (29)
So, to solve the equation (23), in each step of time 1it , we use the called Thomas Algorithms
(Alvim, 2007), besides a known spatial and time neutron source.
3. BENCHMARK ANL BSS-6-A2 PRESENTATION
In this section, we show the Benchmark problem used in the simulations done during the
studying of this work. This Benchmark core (NAGAYA, 1995), whose geometry can be seen
at figure 3, has three regions, with the regions 1 and 3 having the same material composition
and the same size. The central region, with 160 cm length, it’s a different type, containing,
for example, more absorbers.
Figure 3: ANL-BSS-6 Benchmark problem geometry
In this Benchmark the flux is zero at extremity as boundary condition. The nuclear dates,
velocity and delayed neutron constants are shown, respectively, in tables 1, 2 e 3:
INAC 2013, Recife, PE, Brazil.
Table 1: ANL-BSS-6 Benchmark problem nuclear parameters
Constant Region 1 e 3 Region 2
)(1 cmD 1.5 1.0
)(2 cmD 0.5 0.5
)( 1
1
cmR 0.026 0.02
)( 1
2
cmR 0.18 0.08
)( 1
12
cms 0.015 0.01
)( 1
1
cmf
0.01 0.005
)( 1
2
cmf
0.2 0.099
Table 2: ANL-BSS-6 Benchmark problem neutron velocity
g )/( scmvg 1 7100.1
2 5100.3
Table 3: ANL-BSS-6 Benchmark problem delayed neutron constants
l l )( 1sl
1 0.00025 0.0124
2 0.00164 0.0305
3 0.00147 0.1110
4 0.00296 0.3010
5 0.00086 1.1400
6 0.00032 3.0100
In order to obtain different subcritical systems, from Benchmark, removing macroscopic
cross section values were altered, through maintaining invariable the others nuclear
parameters. The table 4 shows the removing macroscopic cross section values in each region,
for the two energy groups, and the multiplication factor associated.
Table 4: Removing macroscopic cross section ( m
Rg ) and the multiplication factor ( effk )
associated
11
1
cmR 11
2
cmR 12
1
cmR 12
2
cmR effk
0.025 0.176 0.019 0.079 0.951142
0.0245 0.176 0.019 0.079 0.962169
0.0245 0.176 0.0185 0.079 0.971960
0.0245 0.176 0.0185 0.077 0.980073
0.0239 0.176 0.0185 0.077 0.991892
INAC 2013, Recife, PE, Brazil.
4. RESULTS ANALYZED
We show, in this section, the results obtained with the computational simulation. For all the
cases, the distance between each cell mesh ( mx ) was 1 cm, in any region. This totalizes 240
cell meshes.
The neutron external source was adjusted to have a neutron flux in order 1015
nêutrons/cm2.s,
existing for the fast energy group and being zero for the termical group, because we are
simulating subcritical systems like ADS reactor type. In all cases considered, the time
interval between steps of time were s510 , with flux always equal to zero at initial instant.
The results shown are as follows; the flux stabilization time , for the different subcritical
systems and external neutron source. Some different cases were considered for this finality: i)
punctual source localized in 0x , emitting scmneutrons ./10 214 ; ii) Source centered in
reactor core, occupying 10 cm length, emitting scmneutrons ./10 313 ; iii) distributed source
through the entire reactor core, emitting scmneutrons ./10 312 .
In the next subsections are shown, separately, the results for the cases with external source
with constant and varying time intensity, respectively.
4.1. External Source with Constant Time Intensity
In the tables 5, 6 e 7 are shown the neutron flux stabilization time for the cases i, ii e iii,
respectively, with constant time source, considering the different subcritical systems.
Table 5: Time stabilization for source in central reactor core
effk (s)
0.951142 246
0.962169 200
0.971960 231
0.980073 253
0.991892 259
Table 6: Time stabilization for source centered in reactor core with length 10 cm
effk
(s)
0.951142 274
0.962169 221
0.971960 201
0.980073 288
0.991892 303
INAC 2013, Recife, PE, Brazil.
Table 7: Time stabilization for source distributed through the entire reactor core
effk
(s)
0.951142 205
0.962169 217
0.971960 253
0.980073 255
0.991892 255
4.2. External Source with Varying Time Intensity
For this case, the time shape, adopted for the source, is descripted below:
oo S
T
tt
1 . (30)
In which 0S is the intensity of the source, in accordance with the cases i, ii e iii and T is the
period in seconds. The graphic representation of this source, for a period T equal to 0,01s and
oS from case i, is shown in figure 4.
0,0 0,5 1,0 1,5 2,0
0,00E+000
2,00E+009
4,00E+009
6,00E+009
8,00E+009
1,00E+010
Inte
nsid
ade
(nêu
tron
s/cm
3.s
)
Tempo (s)
T=0,5 s
Figure 4 – Source intensity in time function
The neutron flux stabilization time, considering the different subcritical systems, for oS from
the cases i, ii and iii, respectively, they are shown in tables 8, 9 e 10.
INAC 2013, Recife, PE, Brazil.
Table 8: Time stabilization for source in central reactor core
(s)
effk
T=0,01s T=0,001s
0.951142 195 214
0.962169 224 247
0.971960 197 230
0.980073 272 269
0.991892 274 309
Table 9: Time stabilization for source centered in reactor core with length 10 cm
(s)
effk
T=0,01s T=0,001s
0.951142 200 208
0.962169 191 285
0.971960 265 222
0.980073 194 258
0.991892 261 271
Table 10: Time stabilization for source distributed through the entire reactor core
(s)
effk
T=0,01s T=0,001s
0.951142 287 201
0.962169 216 255
0.971960 197 198
0.980073 271 202
0.991892 237 228
We can see, with the results shown in tables 8, 9 and 10, that, the flux time stabilization
doesn’t vary so much between the positions and intensity of the varying time external source.
Although, for the cases with constant time external sources, we see that the flux stabilization
time tends to increase at the criticality proximity.
INAC 2013, Recife, PE, Brazil.
5. CONCLUSIONS
In accordance with the results obtained by the spatial kinetics program (1-D/2-G), we
conclude that constant time external sources, in all the cases examined, the flux stabilization
time increases with multiplication factor effk , or the stabilization increases at the criticality
proximities.
For the varying time external source we see that the system reached the stabilization for all
the periods or frequency of neutron pulsed emitting and for different position or intensity of
the neutron source. None oscillation was detected in these results, even with pulsed source.
We can conclude also, that according to the methodology work, we were able to obtain,
neutron flux stabilization time for subcritical systems with different external neutron sources.
In spite of future studying perspectives, we can extend this numerical computing to more than
one dimension; use others external neutron sources types or shapes, use algorithms and
different discretization techniques for the process of neutrons diffusion equations.
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(2007).
2. J.J. Dudestadt and L.J. Hamilton, Nuclear Reactor Analysis, New York-USA, John Wiley
& Sons (1976).
3. F. Inanc, “A Coarse Mesh Nodal Method For One-Dimensional Spatial Kinetics
Calculations”, Ann. NucL Energy, V. 24, No. 4, pp. 257-265 (1997).
4. Y. Nagaya and K. Kobayashi, “Solution of 1-D Mult-Group Time-Dependent Diffusion
Equations Using The Coupled Reactor Theory”, Ann. NucL Energy, V.22, N.7, pp.421-
140 (1995).
5. S. Nakamura, Computational Method in Engineering and Science, John Wiley & Sons,
New York-USA (1977).
6. “OECD-Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear
Fuel Cycles”, https://www.oecd-nea.org/ndd/reports/2002/nea3109-ads.pdf (2002).
7. T. M. Sutton and B. N. Aviles, “Diffusion Theory Method For Spatial Kinetics
Calculations”, Progress in Nuclear Energy, V.30, N.2, pp.119-182 (1996).
8. R. L. Zelmo, “Aplicação do Método dos Pseudo-Harmônicos à Cinética
Multidimensional”, Tese de D.Sc., COPPE/UFRJ, Rio de Janeiro-Brasil (2005).
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