ABSTRACT
In this section, we consider the classical integer order neutron point kinetic equations for m-delayed groups as follows [24,53]:
dn t dt
t l n t c S ti ii
m( ) ( ) ( ) ( )= −� ��
�
�� + +
= ∑ρ β λ 1
(4.1)
dc t dt l n t c t i m i i
i i ( ) ( ) ( ), , , ,= − =β λ 1 2 … (4.2)
where: n(t) is the time-dependent neutron density c ti( ) is the ith precursor density ρ(t) is the time-dependent reactivity function βi is the ith delayed fraction β β=
l is the neutron generation time λi is the ith group decay constant S(t) is the neutron source function
The classical neutron point kinetic equation is considered in matrix form as follows [60]:
dx t dt Ax t B t x t S t ( ) ( ) ( ) ( ) ( )= + + (4.3)
where:
x t
n t c t c t
c tm
( )
( ) ( ) ( )
( )
=
•
–
³ ³ ³ ³ ³ ³ ³ ³
—
˜
µ µ µ µ µ µ µ µ
with initial condition x x( )0 0=
Here, we define A as
A
l
l
l
l
=
−
−
−
−
�
�
� � � � � � � � � � �
�β λ λ λ
β λ
β λ
β λ
0 0
0 0
0 0
�
� � � � � � � � � � �
B(t) can be expressed as
B t
t l
( )
( )
=
�
�
� � � � � � �
�
�
� � � � � � �
ρ 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1
and S t( ) is defined as
S t
q t
( )
( )
=
•
–
³ ³ ³ ³ ³ ³ ³ ³
—
˜
µ µ µ µ µ µ µ µ
0 0
where: q(t) is the time-dependent neutron source term
In this section, we will apply the MDTM to obtain the solution for classical neutron point kinetic equation (Equation 4.3).
