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1918
NUMERICAL SOLUTION OF POINT KINETIC EQUATIONS USING
RK2-2ST WITH ADIABATIC DOPPLER EFFECTS CONSIDERING
COMPENSATED RAMP REACTIVITY
Daniel Suescún-Díaz1, Nathaly Roa-Motta2 and Freddy Humberto
Escobar3 1Departamento de Ciencias Naturales, Avenida Pastrana,
Universidad Surcolombiana, Neiva, Huila, Colombia
2Programa de Ing. de Petróleos, Universidad
Surcolombiana/CENIGAA, Avenida Pastrana, Neiva, Huila, Colombia
E-Mail: [email protected]
ABSTRACT
The second order second stage stochastic Runge-Kutta method
(RK2-2st) is implemented for solving stochastic point kinetic
equations with Newtonian temperature feedback effects, taking into
consideration external ramp reactivity. The feedback temperature is
included in the reactivity; it is an entry variable of point
kinetic equations. Doppler feedback in thermic reactors is mainly
due to epithermal capture resonances in non-fissionable combustible
isotopes. Different numerical experiments have been carried out in
which calculations are made of reactivity, mean values and standard
deviation of neutron density, and the concentration of delayed
neutron precursors. The results obtained are compared with other
methods reported in literature, and it is found that the method
proposed is sufficiently precise to give a solution to the
stochastic point kinetic equations of an adiabatic reactor.
Keywords: neutron density, concentration of delayed neutron
precursor density, doppler effects, reactivity, runge-kutta
stochastic method.
INTRODUCTION
Stochastic point kinetic equations [1-3] are a rigid system of
ordinary differential equations which describe neutron population
density and the concentration of delayed neutron precursors. This
rigidity represents a great challenge in solving the system of
equations and obtaining efficient and accurate results. In a
nuclear reactor the fission process is carried out which occurs due
to the high probability of interaction between the thermal neutrons
with the fissionable material [4]; this nuclear reactor is closely
coupled according to time.
Stochastic point kinetic equations present the form of
stochastic differential equations (SDE) [5], which describe the
dynamic evolution which implies randomness in neutron density
functions and the concentration of delayed neutron precursors. Not
only do SDEs have application in the field of nuclear physics, it
has also been very useful in studying different phenomena that
arise in various fields of application, such as in population
genetics, finance, communication systems among others.
There is a model that describes the dynamic of nuclear reactors
in which the effects of temperature feedback are especially
considered [6-7]. Based on these models it is possible to estimate
the transient behavior of neutron density and the concentration of
delayed neutron precursors, enabling timely control of the nuclear
reactor. Reactivity is affected by temperature feedback, and this
is one of the most important variables because it constitutes a
nonlinear term with the product of neutron density. Reactivity is
an input variable of point kinetic equations, and for this work it
is modeled as a reactivity ramp.
It has become a challenge to give a solution to the
deterministic equations of the point nuclear reactor, for which
there have been many attempts at detailed studies. Some of these
are: the Power series solutions (PWS) method [8] which is based on
the partial approximation of
reactivity and functions of sources neutrons, converged
accelerated Taylor series (CATS) method[9] based on accelerators of
non-lineal and lineal convergence, and which is the best method of
estimation for solving point kinetic equations, enhanced piecewise
constant approximation (EPCA) method [10], which is a
semi-analytical method which solves point kinetic equations with a
technique which repeatedly corrects the error in the source term,
the ITS2 method [11] is an explicit method of solution based on
Taylor’s lower order series expansions, the generalized
Adams-Bashforth-Moulton Method [12], the 8th-order
Adams-Bashforth-Moulton (ABM8) method temperature [13] which are
predictor-corrector methods with their respective modifiers which
increase precision, and finally the Euler Maruyama method, the
derivative-free Milstein method [6] and Taylors strong-order 1.5
method [14]which are methods which give solutions to point kinetic
stochastic equations with temperature feedback effects.
In this work, an analysis of the efficiency and accuracy of the
stochastic second order two-stage Runge-Kutta method (RK2-2st) is
performed to provide solutions to stochastic point kinetic
equations with temperature feedback effects, considering external
ramp reactivity. Several numerical experiments are proposed which
are compared with different methods reported in literature, in
which the CATS method is one of the reference methods for its high
degree of accuracy. THEORETICAL CONSIDERATIONS STOCHASTIC
POINT KINETIC EQUATIONS The deterministic point kinetic
equations with
effects of temperature feedback with m group of delayed neutron
precursors are given by the following system
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VOL. 15, NO. 18, SEPTEMBER 2020 ISSN 1819-6608
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1919
( ) ( )
( ) ( )
1
mt t
t i ti t
i
dnn c Q
dt
(1)
( )
( )i t i
t i i t
dcn c
dt
(2)
0
0( ) ( ) ( )
t
t
t a t t b n t' dt' (3)
( )( )
tt
da b n
dt
(4)
Subject to the initial conditions
0 0( )t nn (5)
0 0( ) ii t cc (6)
0 0( )t (7)
Where ( )tn is the neutron density, ( )i tc is the
delayed neutron precursor density of the i-th group of
precursors, i is the fraction of delayed neutrons of the i-th
group of precursors, is the total fraction of delayed
neutrons , i is the decay constant of the i-th group of
precursors, is the fission velocity of the neutrons, time of
neutron generation , ( )tQ is the external neutron
source, ( )t is the reactivity, a is the impressed reactivity
variation, b is the shutdown coefficient.
Equations (1-3) are a system of rigidly coupled
non-linear differential equations in which function ( )t has a
dependency on neutron density ( )tn . These
parameters represent the effects of Newtonian temperature
feedback with ramp reactivity insertions. Nevertheless, in nature,
the dynamic processes of a nuclear reactor are stochastic.
[2-3]:
12
( ) ( ) ( ) ( )ˆˆ ˆ
t t t tdx Ax Q dt B dW (8)
where
( )
1( )
( ) 2( )
( )
ˆ
t
t
t t
m t
n
c
x c
c
,
( )
( )
0
ˆ 0
0
t
t
q
Q
y
1( )
2( )
3( )( )
1( )
ˆ
t
t
tt
t
W
W
WW
W
.
( )ˆ
tW are Wiener processes, A is the coefficient
matrix which is given in the following way
( )
1 2
11
22
0 0
0 0
0
0 0
t
m
m
m
A
(9)
B is the covariance matrix which presents an approximation of
pulsed neutrons [15], defined as
* * *
* * * *
* * * *
* * * *
*
1 2 1
1 1,1 1,2 1,
2 2,1 2,2 2,
,1 ,2 ,
m
m
m m m m m
a a a
a b b b
a b b bB
a b b b
(10)
where,
2*1
( )1 ( ) 2 1 ( ) ( ) ( )
m
i i
i
tt n t c t q t
* ( )1 1 ( ) ( )i i iit
a n t c t
* ,,
( )( ) ( )
i j i j i ji j
tb n t c t
,0 si
1 si i j
i j
i j
By providing solutions to the stochastic point
kinetic equations given by equation (8) the temporal dependence
of the processes involved in the temporal behavior of neutron flow
in the reactor, such as neutron density and concentration of
delayed neutron precursors can be found. When in equation (8) the
covariance matrix
B is equal to zero, replicating the system of equations of the
deterministic case of the point kinetic equations given by equation
(1) and equation (2). SECOND ORDER, TWO STAGES STOCHASTIC
RUNGE-KUTTA SCHEME The stochastic point kinetic equations given
by
equation (8) correspond to the Itô process in 0 ,t t Twhich
presents a stochastic differential equation structure (SDE) given
as follows:
, ,T T T TdX b t X dt t X dW (11)
With the initial condition 0X x . Where X is a stochastic
process which has continuous sample paths
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VOL. 15, NO. 18, SEPTEMBER 2020 ISSN 1819-6608
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1920
which take values in dR , for a number of d positive
integer, b and are functions with values of given coefficients
and W is a Brownian movement, which must
comply with the following conditions:
a) (0)0W
with probability 1.
b) For 0 s t T the random variable given by the
increase ( ) ( )t sW W
for a normal distribution with
zero mean and variance t s .
c) For 0 s t u v T the increments ( ) ( )t sW W
y ( ) ( )v uW W
are independent.
The second-order, second stage Runge-Kutta method (RK2-2st) [16]
is a numerical derivation method which permits the solution of
equations in the form of
equation (11) for a discrete time o 0 1 ... nt t t T , given
by
11 1 1 1 1ˆ ˆ, ,2 2 2 2 2
n n n n n nb
X X a a t b W b t W bx
(12)
where
,n na a t X ,n nb b t X
ˆ, ,n n n n n nX a t X b t X W Functions a and b are constant
for the scalar
case of the RK2-2st method. This method (RK2-2st), given in
equation (12) is equivalent to the Taylor second order simplified
scheme as given by [17]
22 2 2 21 2
22
2
1 1ˆ ˆ2 2
1 1 ˆ ˆ 2 2
n n n n
n n
b a a aX X a b W b W a b
x t x x
b b a ba b b W W
t x x x
(13)
Comparing Taylor’s second order scheme of
equation (13) with Runge-Kutta second order, second stage method
given by equation (12) it can be seen that the que RK2-2st is an
easily applied numerical method since it requires he calculation of
only one derivative.
In this study a solution will be given to the stochastic
equations which model the dynamic of a nuclear reactor with
feedback effects of Newtonian temperature described by equation (8)
and equation (3) with the RK2-2st method given by equation (12).
RESULTS
This section will discuss the efficiency and accuracy of the
proposed RK2-2st method for understanding neutron density,
concentration of delayed neutron precursors and reactivity when
temperature feedback effects exist for the case of insertions of
ramp reactivity. Different numerical experiments will be
carried
out which present the same initial conditions (in 0 0t ) for
neutron density and the concentration of delayed
neutron precursors, 3
0 1 /n n cm y (0)
(0)i
ii
nC
.
Parameters of the groups of delayed neutron precursors of
the nuclear reactor of graphite 235
U , which correspond to : the decay constant a:
10.0124, 0.0305, 0.111, 0.301, 1.13,3.0i s , fraction of delayed
neutrons
0.00021,0.00141, 0.00127,0.00255,0.00074,0.00027i , total
fraction of delayed neutrons 0.00645 , time of
neutron generation 55.0 10 s . In the programmed
numerical simulations the command ‘state’ was used to generate
the pseudorandom numbers with normal distribution and seed 1000,
and a fixed step size
310h , using 500 samples for each experiment.
Compensated Ramp Change For a compensated response with ramp
reactivity
inserts described by equation (4) the impressed reactivity
variation 10.003a s is considered, using a fixed
shutdown coefficient11 310 /b cm s . Table-1 shows the
mean values ( )E and standard deviation ( ) for neutron density,
the concentration of the sum of delayed neutron precursors and
reactivity. The mean values of neutron density and reactivity are
compared ITS2 [21]. Figure-1a shows the variation of neutron
density according to time. Figure-1b shows the variation of the sum
of the density of delayed neutron precursors according to time, and
Figure-1c shows the behaviour of reactivity through time. For the
graphics two random samples were taken of the numerical experiments
(purple and blue line) and the average of the 500 samples (red
line). It can be seen that the RK2-2st method presents values which
are very close to those reported with method ITS2.
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1921
The peak of neutron density and the respective time in which the
peak is generated are recorded in Table-2. The numerical results
obtained with RK2-2st are compared with the deterministic methods
the power series methods [19], CATS [20], ITS2 [21] andABM8 [22].
The
results of neutron density and reactivity confirm the extreme
precision of the CATS method. Comparing method RK2-2st it is shown
that this is accurate for studying compensated response for ramp
reactivity insertions.
Table-1. Mean values and standard deviation of neutron density
and the sum of precursor density, and reactivity using
compensated response for ramp reactivity with 10.003a s and 11
310 /b cm s .
ITS2 RK2-2st
t(s) ( )tE n ( )t ( )tE n ( )t ( )tn ( )i tE c ( )i tc
0 1 0 1 0 0 1.6765,E+03 -
0.5 1.3247E+00 2.3256,E+08 1.0831,E+00 2.3256E-01 3.4040,E+00
1.6858,E+03 3.9518,E+01
1 2.0532E+00 4.6512,E+08 2.1512,E+00 4.6512E-01 2.5525,E+00
1.7221,E+03 5.7236,E+01
1.5 4.3472E+00 6.9767,E+08 4.2369,E+00 6.9767E-01 2.1003,E+00
1.8343,E+03 7.9087,E+01
2 2.3921E+01 9.3023,E+08 2.3687,E+01 9.3023E-01 3.7551,E+00
2.3551,E+03 1.5290,E+02
2.5 1.4390E+04 1.1628,E+07 1.4232,E+04 1.1628E+00 2.2736,E+03
8.9758,E+04 1.3962,E+04
2.7 4.6419E+06 1.2556,E+07 4.5610,E+06 1.2555E+00 7.2888,E+05
1.7813,E+07 2.8475,E+06
2.8 1.9053E+08 1.2946,E+07 1.8666,E+08 1.2921E+00 2.9139,E+07
6.2342,E+08 9.8483,E+07
2.9 4.8596E+09 1.0595,E+07 4.9235,E+09 1.0087E+00 1.6824,E+08
2.3661,E+10 2.2486,E+09
3 8.0571E+08 6.4673,E+08 7.9858,E+08 6.3481E-01 5.4592,E+07
6.1090,E+10 1.8893,E+07
4 3.4133E+08 5.0440,E+08 3.3862,E+08 5.0236E-01 2.8983,E+05
8.9115,E+10 1.0660,E+08
5 3.2933E+08 4.5283,E+08 3.2801,E+08 4.5279E-01 3.0982,E+05
1.0941,E+11 4.6246,E+07
10 3.1459E+08 2.7934,E+08 3.1465,E+08 2.7954E-01 4.6935,E+03
1.8361,E+11 5.5692,E+06
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1922
0 2 4 6 8 100,00E+000
1,00E+009
2,00E+009
3,00E+009
4,00E+009
5,00E+009
6,00E+009
Ne
utr
on
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
1a) Neutron density
0 2 4 6 8 100,00E+000
5,00E+010
1,00E+011
1,50E+011
2,00E+011
Su
m o
f p
rec
urs
or
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
1b) Sum of precursor density
0 2 4 6 8 100,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
Re
ac
tiv
ity
($
)
Time (s)
1c) Reactivity
Figure-1. Variation of neutron density sum of precursor density
and reactivity according to time with
10.003a s and 11 310 /b cm s .
Table-2. Peak of neutron density and time to peak in compensated
ramp with
10.003a s and 11 310 /b cm s .
Method Peak neutron Time to peak
Power series 5.0537E+09 2.84999
CATS 5.1141566E+09 2.910581
ITS2 5.11415988E+09 2.910582
ABM8 (h=0.001s) 5.11375187E+9 2.911
RK2-2st 5.2394E+09 2.912
Tables 3-4 and Figures 2-3 show numerical
results for different values of a , 10.001s and 10.003s ,
for the fixed shutdown coefficient 13 310 /b cm s .
With the proposed RK2-2st method, numerical calculations are
made of neuron density, the sum of
delayed neutron precursor concentration and reactivity using
changes in reactivity with compensated ramp, which are presented in
Table-3. Reactivity values are expressed in $ (dollar).
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1923
Table-3. Mean values and standard deviation of neutron density
and the sum of precursor density
and reactivity for10.01a s and 10.003a s with 13 310 /b cm s
.
t(s) ( )tE n ( )tn ( )i tE c ( )i tc ( )t
a) 10.01a s
0.1 1.0004E+00 3.4831E+00 1.6767E+03 1.6599E+01 1.5504E-01
0.5 4.1507E+00 2.1830E+00 1.7323E+03 4.1404E+01 7.7519E-01
5.0 1.0340E+11 7.9123E+05 3.8545E+13 6.3865E+08 4.3645E-01
7.5 1.0195E+11 1.3594E+05 5.1182E+13 3.0308E+08 3.3638E-01
10 1.0124E+11 9.3704E+04 6.1137E+13 1.8360E+08 2.7590E-01
b) 10.003a s
0.1 0.8820E+00 3.7150E+00 1.6760E+03 1.6598E+01 4.6512E-02
0.5 1.0831E+00 3.4040E+00 1.6858E+03 3.9518E+01 2.3256E-01
5.0 3.1959E+10 2.9551E+07 1.1081E+13 5.4386E+09 4.3287E-01
7.5 3.2090E+10 7.1755E+06 1.5131E+13 3.8991E+08 3.5024E-01
10 3.1464E+10 1.6650E+05 1.8345E+13 6.1148E+08 2.8053E-01
0 2 4 6 8 100,00E+000
5,00E+011
1,00E+012
1,50E+012
2,00E+012
2,50E+012
3,00E+012
Ne
utr
on
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
2a) Neutron density
0 2 4 6 8 100,00E+000
1,00E+013
2,00E+013
3,00E+013
4,00E+013
5,00E+013
6,00E+013
7,00E+013
Su
m o
d p
rec
urs
or
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
2b) Sum of precursor density
0 2 4 6 8 100,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
Re
ac
tiv
ity
($
)
Time (s)
2c) Reactivity
Figure-2. Variation of neutron density, sum of precursor density
and reactivity according to time with
10.001a s and 13 310 /b cm s .
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1924
0 2 4 6 8 100,00E+000
1,00E+011
2,00E+011
3,00E+011
4,00E+011
5,00E+011
6,00E+011
7,00E+011
Ne
utr
on
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
3a) Neutron density
0 2 4 6 8 100,00E+000
5,00E+012
1,00E+013
1,50E+013
2,00E+013
Su
m o
f p
rec
urs
or
de
ns
ity
(n
/cm
3)
Time (s)
Sample 1
Sample 2
Mean
3b) Sum of precursor density
0 2 4 6 8 100,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
Re
ac
tiv
ity
($
)
Time (s)
3c) Reactivity
Figure-3. Variation of neutron density, sum of precursor density
and reactivity according to time with
10.003a s and 13 310 /b cm s .
The density of neutrons obtained with the RK2-2st method using
changes in compensated ramp reactivity are compared with the
deterministic methods CATS [9], EPCA [10], the ITS2 method [11] and
ABM method [13],
reported in Table-4. It was confirmed that the RK2-2st is an
effective method and sufficiently precise for insertions of ramp
reactivity with temperature feedback.
Table-4. Neutron density for different values of a with 13 310
/b cm s .
t(s) EPCA CATS ITS2 ABM8 (h=0.001) RK2-2st
a) 10.01a s
0.1 1.167210838E+00 1.1672108379E+00 1.1672108379E+00
1.1672108379E+00 1.0004E+00
0.5 4.269952865E+00 4.2699528644E+00 4.2699528644E+00
4.2699528644E+00 4.1507E+00
5.0 1.033889665E+11 1.0338896655E+11 1.0338896655E+11
1.0338896655E+11 1.0340E+11
7.5 1.019499913E+11 1.0194999125E+11 1.0194999125E+11
1.0194999125E+11 1.0195E+11
10 1.012434883E+11 1.0124348832E+11 1.0124348832E+11
1.0124348832E+11 1.0124E+11
b) 10.003a s
0.1 1.045371667E+00 1.0453716665E+00 1.0453716665E+00
1.0453716665E+00 0.8820E+00
0.5 1.324661986E+00 1.3246619862E+00 1.3246619862E+00
1.3246619862E+00 1.0831E+00
5.0 3.215676113E+10 3.2156761131E+10 3.2156761131E+10
3.2156761131E+10 3.1959E+10
7.5 3.210205182E+10 3.2102051821E+10 3.2102051821E+10
3.2102051821E+10 3.2090E+10
10 3.145614687E+10 3.1456146867E+10 3.1456146867E+10
3.1456146867E+10 3.1464E+10
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1925
CONCLUSIONS Based on numerical experiments carried out to
study stochastic point kinetic equations with a nuclear reactor
with adiabatic Doppler effects considering external ramp
reactivity, it was found that the proposed stochastic second order
second stage Runge-Kutta (RK2-2st) method is the numerical
derivation method that provides a solution to the system of
equations and generates values in neutron density and reactivity
which are very similar to those reported by one of the most exact
methods (deterministic) such as CATS. The RK2-2st method is
equivalent to the second order Taylor scheme, and it is easy to
implement since its scheme implies only one derivative of covariant
matrix regarding to random variables. RK2-2st is an alternative and
efficient method to study the adiabatic nuclear reactor.
ACKNOWLEDGEMENTS
The authors thank the research seed of Computational Physics,
the research group in Applied Physics FIASUR, and the academic and
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