Stability of the analytical perturbation for nonlinear
coupled kinetics equations
Ahmed E. Aboanber1,2, Abdallah A. Nahla1, Faisal A. Al-Malki3
1
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt.
2
Department of Mathematics, Faculty of Science and Arts, Qassim University, QassimBuraidah, P.B 1300, Buraidah 51431, Saudi Arabia.
3
Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi
Arabia.
E-mails: aboanber@tu.edu.eg, nahla@tu.edu.eg, faisal_malki1@hotmail.com
Abstract:
The critical problem in nuclear-reactor kinetics is to predict the evolution in time of
the neutron population in a multiplying medium. Point kinetics allows the study of
the global behavior of the neutron population from the average properties of the
medium. In this paper, an efficiency improved technique based on the small
parameter approximation is developed for the analytical solution of stiff system
kinetics equations for linear and/or nonlinear coupled differential equations. The
variations of neutron density, precursors, fuel temperature and reactivity are
obtained in terms of time. Furthermore, the stability criterion of the perturbed
method is investigated and discussed. The procedure is tested by using a variety of
reactivity functions, including step reactivity insertion, ramp input and oscillatory
reactivity changes. The solution of the developed method is compared to other
analytical and numerical solutions of the point reactor kinetics equations; the results
proved that the approach is both efficient and accurate to several significant figures.
The proposed method can be used in real time forecasting for power reactors in
order to prevent reactivity accidents.
Keywords: Point kinetics , Neutron population, Stiff system kinetics equations,
Reactivity, Stability
1
1. Introduction
The sub-critical reactor kinetics analysis is important to determine the neutron density as
function reactivity insertion rate with respect to the initial sub-criticality. The analytical
and numerical solutions of the point kinetics equations in the presence of temperature
feedback reactivity are useful to estimate the transient behavior of the reactor power and
other parameters of the reactor cores.
There are many analytical and numerical solutions available to study the reactor
kinetics in the presence the adiabatic feedback model. Chen (1990) studied a nonlinear
time dependent point power reactor problem with delayed temperature feedback on
reactivity. Dam (1996) analyzed the point reactor kinetics equations in combination with
linear temperature feedback for the reactivity and an adiabatic heating of the core after
loss of cooling. Damen and Kloosterman (2001) used a simple reactor model with one
delayed neutron group and first order fuel and temperature feedback mechanisms to
calculate the linear transfer function from reactivity to reactor power that was
subsequently used in a root-locus analysis. Aboanber and Nahla (2002) introduced a
solution of the point kinetics equations in the presence of Newtonian feedback model
using different cases of Padé approximations via analytical inversion method. Aboanber
and Hamada (2003) reduced the point kinetics equations in the presence of Newtonian
feedback to a differential equation in a simple matrix form convenient for explicitly
power series solution involving no approximation beyond the usual space-independent
assumption. Aboanber (2006) studied stability and efficiency improved class of
generalized Runge-Kutta methods for several sample problems of the stiff system point
kinetics equations with reactivity feedback. Chen et al. (2006 and 2007) presented a new
analysis for the prompt supercritical process of nuclear reactor with temperature feedback
and initial power while inserting large step reactivity  0   and small step reactivity
 0   . Nahla and Zayed (2010) developed the analytical approximation and numerical
solution of the point nuclear reactor kinetics equations with average one-group of delayed
neutron and the adiabatic feedback model. Sathiyasheela (2009) presented power series
solution method for solving point kinetics equations with lumped model temperature and
2
feedback. Tashakor et al. (2010) presented a numerical solution of the point kinetics
equations with fuel burn-up and temperature feedback using an implicit time method.
Kinard and Allen (2004) described and investigated an efficient numerical solution
for the point kinetics equations in nuclear reactor dynamics. Quintero-Leyva (2008)
developed a numerical algorithm (CORE) to solve the point kinetics equations of nuclear
reactors. Sathiyasheela (2010a) solved analytically the sub-critical point reactor kinetics
equations with one group of delayed neutrons using incomplete gamma functions and
binomial expansions. Sathiyasheela (2010b) solved analytically inhomogeneous point
kinetics equations for step and linear reactivities using the prompt jump approximation.
Theler and Bonetto (2010) studied the stability of the point kinetics equations. Espinosa
et al. (2011) derived and analyzed the fractional point-neutron kinetics model for the
dynamic behavior in a nuclear reactor
In this work, the analytical perturbation for the point reactor kinetics equations with
one-group delayed neutrons and the adiabatic feedback model is presented. This
analytical perturbation method based on the expansion of the neutrons density in powers
of the small parameter. The time dependent neutron density, and the average density of
delayed neutron precursors as functions of reactivity are derived. The relations of
reactivity, neutron density and temperature with time are calculated. The stability of the
analytical perturbation method is investigated. Finally, comparison between the results of
the method and the conventional methods is introduced.
2. Mathematical Model (Analytical Perturbation)
The point kinetics equations with one group delayed neutrons in the presence of
Newtonian temperature feedback are a stiff nonlinear coupled ordinary differential
equations (Ash, 1979; Glasstone and Sesonske, 1981; Hetrick, 1993; Stacey, 2001) which
given by:
dn(t )  (t )  
=
n(t )  C (t )
dt
l
(1)
dC(t ) 
= n(t )  C (t )
dt
l
(2)
3
 (t ) = 0   [T (t )  T0 ]
(3)
dT (t )
= K c n(t )
dt
(4)
where, n(t ) is the neutron density,  (t ) is the reactivity as function of time t,  is the
total fraction of the delayed neutrons, l is the prompt neutrons generation time,  is the
decay constant of delayed neutron precursors, C (t ) is the precursor concentrations of
delayed neutron, T (t ) is the temperature of the reactor, T0 is the initial temperature of the
reactor,  is the temperature coefficient of reactivity, and K c is the reciprocal of the
thermal capacity of reactor.
From Eqs. (1), (2), (3) and (4) we have:
d (t )
= K c n(t )
dt
(5)
d 2 n(t )
dn(t )  d (t )

l
= (  (t )    l )

  (t ) n(t )
2
dt
dt
 dt

(6)
Starting Eq. (5) into Eq. (6) and multiplied by (  K c ), we get
d 3  (t )
d 2  (t )  d (t ) 
d (t )
l
=
(

(
t
)



l

)



(
t
)


dt
dt 3
dt 2
 dt 
2
(7)
Treating Eq. (7) with the initial conditions given by:
d
d 2
(0) = K c n0 yield:
(0) = 0 and
n(0) = n0 ,  (0) = 0 ,
2
dt
dt
l
where
d 2  (t )
d (t )  2
=
(

(
t
)



l

)
 (   2 (t ))
2
dt
2
dt
(8)
 2 =  02  2K c n0 (   l   0 )/
Beginning Eqs. (5) and (8):
dn(t )
 ( 2   2 (t ))
l
=  (t )    l n(t ) 
dt
2K c
and consequently, we get:
4
(9)
dn(t )  d (t ) 
 ( 2   2 (t ))
l

 = (  (t )    l )n(t ) 
d  dt 
2K c
(10)
Eqs. (5) into (10) gives
dn(t )
 ( 2   2 (t ))
lK c n(t )
 (  (t )    l )n(t ) = 
d
2K c
(11)
Starting from the fact that (lK c ) is very small, equal 2.5  10 10 for a pressurized- water
reactor with
U as fissile material, let us consider  = lK c and  = /K c into Eq.
235
(12), then
 n(t )
dn(t )

 (  (t )    l )n(t ) =  ( 2   2 (t ))
d
2
(12)
Assume the neutron density are perturbed so that:
n(t ) = n1 (t )  n2 (t )  
(13)
An application of Eq. (12) yields
 [n1 (t )  n2 (t )  ]
d
[n1 (t )  n2 (t )  ]
d
 (  (t )    l )[ n1 (t )  n2 (t )  ] = 

2
(14)
( 2   2 (t ))
Therefore,
   2   2 (t ) 

n1 (t ) = 
2    l   (t ) 
n2 (t ) =
(15)
 2   2   2 (t )   2  2  (t )(   l )   2 (t ) 



4    l   (t )  
(   l   (t )) 3

(16)
And consequentely, the neutron density in terms of reactivity can be obtained as follow:
n(t ) =
   2   2 (t )      2  2(   l )  (t )   2 (t ) 

 1 


2    l   (t )  
2 
(   l   (t )) 3

(17)
The rate of change for reactivity in terms of neutron density is obtained by combining
Eqs. (13) and (5), neglecting the somallest term with the coefficient K c so that:
d (t )
= K c n1 (t )
dt
(18)
5
Combining with Eq. (15) yields
t=
1    l      0     l      0 
  1 

1 
 ln 
 ln 
 
      (t )  
      (t ) 
(19)
3. Stability discussion
3.1. Bounded of solution:
The bounded values for the variation of reactivity with time can be estimated by taking
the limit of Eq. (19) when the reactivity tends to   :
  l


2
2
1     0  (   0 )(   (t ))   


ln  2
=  (20)
lim t = lim
    2 (t )  (   (t ))(   0 ) 

 ( t )  
 ( t )   




which can be rewite as: lim  (t ) =  , That is lim n(t )  lim n(t )
t 
t 
 ( t )
Furthermore, the limit of Eq. (19) when the reactivity tend to   :
   2   2 (t )      2  2  (t )(   l )   2 (t ) 

 1 

 = 0 (21)
lim
3
2
 ( t )   2    l   (t ) 
(


l



(
t
))



Eqs. (20) and (21) show that: lim n(t ) = 0 , in this particular instance the neutron flux
t 
given by Eq. (19) is bounded function.
3.2. Stability of solution
The stability of the solution n(t ) obtained by Eq. (17), which satisfy the nonlinear
ordinary differential equation (12) is considered. Let us add a small function  (t ) to the
solution n(t ) such that n* (t ) = n(t )   (t ) which assure Eq. (12) also,
 [n(t )   (t )]
d

[n(t )   (t )]  (   l   (t ))[n(t )   (t )] =  ( 2   2 (t )) (22)
d
2
applying into Eq. (12), neglecting the term  (t )
d (t )
and dividing by  (t )n(t ) yields
d
1 d (t )
1 dn(t )    l   (t ) 


= 
 (t ) d
n(t ) d
n(t )


Combining Eqs. (17) and (23):
6
(23)
1 d (t )
 (t ) d
1 dn(t ) 2   l   (t )


n(t ) d
    (t )   (t )
2
=

(  l )
2
2


(24)
2
 2  (t )(   l )   (t )
    (t )   (t )  2  2  (t )(   l )   2 (t )
4
2

The solution of Eq. (23) taking the following form:
 2 0   (t ) 4 0   (t ) 



 2


 (t ) =  (0) exp 
 (  l  )2 (   l   )(   l   ) 2 4(   l )(   l   ) 


1

    0   
2
2

   (t ) 


 (  l  )2 (   l   )(   l   ) 2 4(   l )(   l   ) 


1

   (t )  
2
 2






0 

 (  l )2  2    l  (   l ) 2   2  


 

   l    (   l  ) 2   2  
2

0



2
2 
   l   (t )  (   l )   
 (  l )2  2    l  (   l ) 2   2  


 

2
   l    (   l ) 2   2  


0



2
2 
   l   (t )  (   l )   
  l   (t )
4n0 

3

  2  l   (t )    2  2  (t )  l    2 (t )
4



















 (25)










(4)



Taking the limit,
lim  (t ) = lim  (t ) = 0
t 
(26)
 ( t )  
That is, the perturbed solution of Eq. (17) is stable.
3.3. Liapunov stability:
The stability of the solution of the autonomous system is also analyzed according to
Jordan and Smith, 2007, as follows:
7
Assume that n (t ) is an equivalent solution as n(t ) given by Eq. (17) so that:
   2   2 (t )      2  2(   l )  (t )   2 (t ) 
 1 


n (t ) = 
2    l   (t )  
2 
(   l   (t )) 3


(27)
where  2 =  02  2K c n0 (   l   0 )/
From Eqs. (17) and (27), for which |  02   02 |<  , the inequality is:

 2 ( 2   2  2(   l )  (t ))
| n(t )  n (t ) |<

2(   l   (t ))
4(   l   (t )) 4

(28)
Taking the limit of Eq. (28) when time tend to  which means the reactivity tend to
   which
  = min { ,  } .

2 ( 2   2  2  (  l ))

<
lim | n(t )  n (t ) |<
t 
2(   l    )
4(   l    ) 4

(5)
Then, the perturbation solution n(t ) of Eq. (19) is Liapunov stable.
4. Results and discussions
Consider the prompt supercritical process in a pressurized-water reactor with
235
U as
fissile material. It is assumed that  = 0.0065, l = 0.0001 s,  = 0.07741s1 , K c = 0.05
K/MW  s, and  = 5 10 5 K 1 . The supercritical process will take place when three
initial reactivities are inserted into the reactor. This reactor is operating in critical state
with initial power n0 = 10.0MW . The comparison between conventional numerical and
analytical solution for the neutron density is shown in Fig (1).
Variation of neutron density n(t ) as a function of time t at the initial reactivity
0 = 0.25 , 0.5  and 0.75  is drawn in Fig (1). It is found that variation of neutron
density with time for different methods, developed analytical perturbation, Chen (1990)
approximation and numerical method, is corresponding. Furthermore, the relation
between reactivity and time at the initial reactivity 0 = 0.25 , 0.5  and 0.75  is
shown in Fig (2).
Finally, the relation between time, reactivity, neutron density and temperature using
8
the analytical perturbation, Chen approximation and numerical methods is presented in
Tables (1), (2) and (3) with initial reactivity 0 = 0.25 , 0.5  and 0.75  respectively.
The numerical method based on Taylor's series method (Nahla and Zayed, 2010). It is
found that the difference between the developed analytical perturbation results, Chen
approximation and numerical reaults are small less than 10 3 .
5. Conclusions
The developed analytical perturbation method for nonlinear couple reactor kinetics
equations with one group of delayed neutrons in presence of temperature feedback have
been derived. The analytical perturbation technique based on the small term, l  Kc which
equal 2.5  10 10 for a pressurized-water reactor with
235
U as fissile material. The stability
of the analytical perturbation technique are studied with different method. The analytical
results of the supercritical process in a pressurized-water reactor with
235
U as fissile
material are drawn and tabulated. It is found that the neutron density using the analytical
perturbation method are identical with the neutron density using Chen approximation and
numerical method which based on Taylor's series.
References
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coupled differential equations. Journal Physics A: Mathematics and General 30,
1859-1876.
[2] Aboanber, A.E., Hamada, Y.M., 2003. Power series solution (PWS) of nuclear reactor
dynamics with newtonian temperature feedback. Annals of Nuclear Energy 30, 11111122.
[3] Aboanber, A.E., Nahla, A.A., 2002. Solution of the point kinetics equations in the
presence of Newtonian temperature feedback by Padé approximations via the
analytical inversion method. Journal Physics A: Mathematics and General 35, 96099627.
[4] Ash, M., 1979. Nuclear reactor kinetics. McGraw-Hill, Inc., USA.
[5] Chen, G.S., 1990. A nonlinear study in a reactor with delayed temperature feedback.
9
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11
Table1: Reactivity, Neutron density and Temperature as functions of time at initial
reactivity 0.25  .
Time
Reactivity
Neutron Density
(s)
($)
Chen App.
Anal. Pert.
Num. Meth.
(K )
0.0
0.25
10.0
10.0
10.0
300.0
10.0
0.207752
11.920335
11.916985
11.916994
305.492219
20.0
0.158941
13.368069
13.365992
13.366050
311.837653
30.0
0.105789
14.155938
14.155193
14.155254
318.747375
40.0
0.050948
14.249263
14.249636
14.249701
325.876708
50.0
-0.003053
13.742208
13.743366
13.743446
332.896942
60.0
-0.054208
12.796830
12.798441
12.798521
339.547050
70.0
-0.101164
11.585716
11.587518
11.587613
345.651356
80.0
-0.143195
10.257042
10.258853
10.258986
351.115344
90.0
-0.180073
8.921237
8.922945
8.923002
355.909445
100.0
-0.211921
7.651384
7.652932
7.653634
360.049753
110.0
-0.239085
6.489769
6.491133
6.491833
363.581103
115.0
-0.251051
5.956452
5.957721
5.958417
365.136628
12
Temperature
Table2: Reactivity, Neutron density and Temperature as functions of time at initial
reactivity 0.5  .
Time
Reactivity
Neutron Density
(s)
($)
Chen App.
Anal. Pert.
Num. Meth.
(K )
0.0
0.5
10.0
10.0
10.0
300.0
10.0
0.446517
18.220153
18.195046
18.195158
306.952849
20.0
0.359572
26.743477
26.725939
26.726191
318.255655
30.0
0.245386
31.900591
31.894813
31.895273
333.099883
40.0
0.120072
32.594130
32.596224
32.597210
349.390656
50.0
-0.001244
30.096715
30.102094
30.103104
365.161749
60.0
-0.109559
26.074489
26.080507
26.081541
379.242696
70.0
-0.201433
21.691699
21.697222
21.698225
391.186282
80.0
-0.276824
17.572978
17.577644
17.578548
400.987074
90.0
-0.337342
13.984105
13.987887
13.988706
408.854429
100.0
-0.385198
10.991478
10.994473
10.995258
415.075780
110.0
-0.422648
8.564157
8.566497
8.567273
419.944209
120.0
-0.451736
6.631089
6.632902
6.633668
423.725630
130.0
-0.474206
5.110976
5.112373
5.113089
426.646742
140.0
-0.491496
3.926056
3.927128
3.927845
428.894540
147.0
-0.501146
3.259237
3.260127
3.260785
430.148962
13
Temperature
Table3: Reactivity, Neutron density and Temperature as functions of time at initial
reactivity 0.75  .
Time
Reactivity
Neutron Density
(s)
($)
Chen App.
Anal. Pert.
Num. Meth.
(K )
0.0
0.75
10.0
10.0
10.0
300.0
10.0
0.647415
47.877804
47.691231
47.691120
313.336018
20.0
0.405342
71.467363
71.455316
71.456192
344.805479
30.0
0.135590
66.159691
66.179999
66.181404
379.873244
40.0
-0.094510
53.134130
53.153026
53.154675
409.786287
50.0
-0.273940
40.439861
40.453927
40.455536
433.112263
60.0
-0.408680
30.010027
30.020056
30.022448
450.628466
70.0
-0.507961
21.967440
21.974528
21.976612
463.534940
80.0
-0.580334
15.950425
15.955435
15.957500
472.943463
90.0
-0.632747
11.522864
11.526410
11.528276
479.757136
100.0
-0.670554
8.296181
8.298697
8.300844
484.672014
110.0
-0.697738
5.960236
5.962025
5.963708
488.205908
120.0
-0.717256
4.275238
4.276512
4.278079
490.743231
130.0
-0.731262
3.062159
3.063066
3.065639
492.564088
140.0
-0.741275
2.193012
2.193659
2.195231
493.865788
150.0
-0.748468
1.567664
1.568125
1.571112
494.800892
153.0
-0.750188
1.418068
1.418485
1.421002
495.024410
14
Temperature
Fig.1: Variation of neutron density for different initial reactivity.
15
Fig.2: Variation of reactivity for different initial reactivity.
16