Page 31 - Dynamics and Control of Nuclear Reactors

22 CHAPTER 3 The point reactor kinetics equations

6
dn X
¼ 1 βÞP Τ L a + L + λ i C i (3.4)
ð
dt
i¼1
dC i
¼ β P Τ λ i C i i ¼ 1,2,…,6 (3.5)
dt i
Eq. (3.4) may be written as.
6
dn L a + L X
¼ P T ð 1 βÞ + λ i C i (3.6)
dt P T
i¼1
Note that P T is simply the ratio of total neutron productions (by fission) to total neu-
L a+L
tron losses (by absorption and leakage) and is equal to the effective multiplication
factor, k eff . So Eq. (3.6) becomes.
6
dn 1 X
¼ P T ð 1 βÞ + λ i C i (3.7)
dt k eff
i¼1
or
6
dn k eff 1 X
¼ P T β + λ i C i (3.8)
dt k eff
i¼1
or
6
dn X
¼ P T ρ βð Þ + λ i C i (3.9)
dt
i¼1
where
k eff 1
ρ ¼ (3.10)
k eff
The total neutron production, P T , is given by.
P T ¼ νΣ f Φ
or

P T ¼ νΣ f nv (3.11)
where
ν¼average number of neutrons produced per fission
Σ f ¼macroscopic fission cross section
Φ¼neutron flux
v¼neutron velocity

As shown in elementary reactor physics books [2, 3], the reciprocal of (νΣ f v) is the
time between production of a neutron and the production of new neutrons by fission,
and is called the generation time, Λ. Therefore, the final form of the point kinetics
equations is