Page 35 - Dynamics and Control of Nuclear Reactors

26 CHAPTER 3 The point reactor kinetics equations

P ¼ P0ðÞ + δP

C ¼ C0 ðÞ + δC

ρ ¼ ρ 0ðÞ + δρ
where

P(0)¼initial steady state value of P
δP¼deviation of P from its initial steady state value
C(0)¼initial steady state value of C
δC¼deviation of C from its initial steady state value
ρ(0)¼initial steady state value of ρ (¼ zero because the initial state is a critical
reactor)
δρ¼deviation of ρ.

Substituting these definitions into the point kinetics equations gives
6
dP 0ðÞ dδP δρ β X
+ ¼ ð P 0ðÞ + δPÞ + λ i C i 0ðÞ + δC i Þ (3.26)
dt dt Λ
i¼1
dC i 0ðÞ dδC i β i
+ ¼ ð P 0ðÞ + δPÞ λ i C i 0ðÞ + δC iÞ (3.27)
dt dt Λ
Note that (at steady state)
dP 0ðÞ
¼ 0
dt

dC 0ðÞ
¼ 0
dt
ρ 0ðÞ ¼ 0

β i
Λ P 0 ðÞ λ i C i ¼ 0
This leads to the following equations:
6
dδP δρ β δρ X
¼ P 0ðÞ δP + δP + λ i δC i (3.28)
dt Λ Λ Λ
i¼1
β
dC i i
¼ δP λ i δC i (3.29)
dt Λ
In general, one may also arrive at Eqs. (3.28) and (3.29) by subtracting the point reac-
tor kinetics equations at a specified nominal condition from the two Eqs. (3.26) and
(3.27).