Page 57 - Dynamics and Control of Nuclear Reactors

48 CHAPTER 4 Solutions of the point reactor kinetics equations

5 s loop time
Magnitude (% change in power to per cent change in reactivity) 10 0
7.5 s loop time
1
10

10 −1 −2 −1 0 1 2 3
10 10 10 10 10 10
(A) Frequency (rad/s)
0
5 s loop time
7.5 s loop time
−20
Phase (deg) −40

−60

−80

−100
10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4
(B) Frequency (rad/s)
FIG. 4.14
(A) Fluid fuel reactor frequency response magnitude for two different loop residence times of
5 s and 7.5 s. (B) Fluid fuel reactor frequency response phase angle for two different loop
residence times of 5 s and 7.5 s.

β i
sC i C i0 ¼ n λ i C i (4.15)
Λ
Solving for n(s) gives
!
6
6
ρ β X λ i β =Λ X λ i C i0
i
s n ¼ n 0 + (4.16)
Λ
i¼1 s + λ i i¼1 s + λ i
The seven roots of the bracketed term on the left are the “characteristic values”, s i ,of
the equation. The solution has the form
7
X s i t
ntðÞ ¼ A i e (4.17)
i¼1
where, A i ¼ coefficients derived from the initial conditions.