Page 48 - Dynamics and Control of Nuclear Reactors

4.4 Analytical solutions 39

1.35

1.3
1.25

1.2
P/P(0)
1.15

1.1

1.05
1
0 20 40 60 80 100 120
Time (s)
FIG. 4.4
Fractional power response to a ramp reactivity input, followed by a step change to zero
reactivity.

4.4 Analytical solutions
Analytical solutions for the six-delay neutron precursor group zero power model and
for more detailed power reactor models are too tedious for practical implementation.
However, an analytical solution to the point kinetics equation with one delay neutron
group is easily performed for a step change in reactivity. This solution is useful in
understanding the nature of reactor transients. Laplace transforms are a convenient
tool for solving linear, constant coefficient differential equations. Appendix D pro-
vides a brief summary of Laplace transform theory. Appendix H shows the derivation
of the analytical solution of the point reactor kinetics equations for a step change in
reactivity. The parameters used in the simulation are as follows
ρ ¼ +10 cents ¼ 0.1*0.0067 ¼ 0.00067
β ¼ 0.0067
Λ ¼ 0.00001 s.
-1
λ ¼ 0.08 s
The Laplace transform of the fractional power is given by
β
Λ λ + + s
Ps ðÞ Λ
¼ (4.9)
2
P 0ðÞ Λs + β ρ + λΛð 0 Þ s λρ 0
The roots of the denominator polynomial are used to solve for P(t)/P(0). The two
roots of this polynomial are given by.