Page 66 - Dynamics and Control of Nuclear Reactors

58 CHAPTER 6 Fission product poisoning

Xe-135 concentration, σ aX is the Xe-135 absorption cross section and Φ is the
neutron flux. The absorptions in Xe-135 consume neutrons that could otherwise
be available for fissions in the fuel, thereby reducing reactivity.

6.2.3 Equations for Xe-135 behavior
The differential equations for I-135 and Xe-135 are as follows:
dI X
¼ γ I Φ λ I I (6.1)
dt f
dX
¼ γ Σ f Φ + λ I I Xσ aX Φ λ X X (6.2)
dt X
where
3
I¼I-135 concentration (number of atoms/cm )
γ I ¼ I-135 fission yield (0.063 I-135 atoms per fission)
2
Φ¼neutron flux (number of neutrons/(cm -sec))
s )
λ I ¼ I-135 decay constant (2.87 10 5 1
3
X¼Xe-135 concentration (number of atoms/cm )
γ X ¼ Xe-135 fission yield (0.003 Xe-135 atoms per fission)
λ X ¼ Xe-135 decay constant (2.09 10 5 1
s )
6
σ aX ¼ Xe-135 absorption cross section (3.5 10 b at 0.0253eV)
To solve these equations, it is necessary to know the cross sections and the neutron
flux or the reactor specific power (kW/kg of fuel), which uniquely defines the
neutron flux.
It should be noted that reported cross sections are for mono-energetic neutrons at
an energy of 0.0253eV or a speed of 2200m/s. This energy corresponds to a temper-
ature of 20°C or 293K. In a reactor, the absorption and fission cross sections must be
corrected for the actual temperature of the moderator. In a thermal reactor it is usu-
ally assumed that the Maxwell distribution applies for neutrons in equilibrium
with moderator atoms and that the cross section varies as the inverse of the neutron
velocity. As shown in elementary reactor physics books [1], the “effective” cross sec-
tion for a material at temperature, T (Kelvin), is
!
r ffiffiffiffiffiffiffiffi
1 293
σ TðÞ ¼ σ 0:0253eVÞ
ð
1:128 T
For example, the fission cross section for U-235 is 649 b for neutrons at 2200m/s, but
the effective neutron fission cross section is 411 b at 300°C (a typical moderator tem-
perature in a pressurized water reactor).
It is possible to reformulate the equations for Xe-135 transients with various def-
initions for the quantities in the equations. One such reformulation is as follows.
dI=N f
¼ γ σ f Φ λ I I=N f (6.3)
I
dt