K. Kobayashi and S. Tsumura

since the neglected terms are small.
We assume that the absorption by xenon and control rods are only relevant in the thermal
group G, and the change of iodine and xenon concentrations is expressed by the following
equations as usual,
dI(r,t)
dt = γIΣfG(r)φG(r,t) − λII(r,t), (13)
dX(r,t)
= γ
dt
XΣfG(r)φG(r,t)+λII(r,t) − λXX(r,t) − σXGX(r,t)φG(r,t), (14)
where I(r,t) and ΣfG are the iodine density and the fission cross section in thermal group,
γI and γX are the fractions of yield per fission, and λI and λX are the decay constants for
iodine and xenon, respectively.
Using the assumption that the absorption by xenon and control rods is only relevant in the
thermal group, the coupling coefficients of Eqs.(10) and (11) become
∆k X mm (t) =
∆k C mm (t) =
Vm drGGm(r)σXGX(r,t)φG(r,t)
Vm drs(r,t)
, (15)
Vm drGGm(r)δΣ C G(r,t)φG(r,t)
Vm drs(r,t)
. (16)
We assume that the neutron flux, neutron production rate, xenon and the absorption cross
section of control rod are expressed as the sum of the steady state values with superscript 0
and the deviation from them in the following form
φg(r,t)=φ 0
g(r)+δφg(r,t), δφg(r,t)=f f gm(r)δφgm(t), (17)
s(r,t)=s 0 (r)+δs(r,t), δs(r,t)=f s m (r)δsm(t), (18)
X(r,t)=X 0 (r)+δX(r,t), δX(r,t)=f X m (r)δXm(t), (19)
δΣ C
G (r,t)=f C m (r)δΣCGm
(t), (20)
where the shape functions f c m (r) are normalized as
1
Vm
Vm
f c m (r)dr =1, c = f, s, X, or C. (21)
Using Eqs.(17) to (21), we define the following integral quantities in a node;
φgm(t) =φ 0
gm + δφgm(t), φ 0
1
gm = φ
Vm Vm
0
g (r)dr, δφgm(t) = 1
δφg(r,t)dr, (22)
Vm Vm
sm(t) =s 0 m + δsm(t), s 0
1
m = s
Vm Vm
0 (r)dr, δsm(t) = 1
δs(r,t)dr, (23)
Vm Vm
Xm(t) =X 0 m + δXm(t), X 0 m = 1
X 0 (r)dr, δXm(t) = 1
δX(r,t)dr. (24)
Vm
Vm
4
Vm
Vm