Page 49 - Dynamics and Control of Nuclear Reactors

40 CHAPTER 4 Solutions of the point reactor kinetics equations

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2
s 1 ,s 2 ¼ β ρ + λΛÞ ð β ρ + λΛÞ +4λρΛ
ð
2Λ
s 1 t
s 2 t
The general solution has the form PtðÞ ¼ Ae + Be .
P 0ðÞ
The solution for the example case is given by
PtðÞ 0:0088t 603:09t
¼ 1:111e 0:111e (4.10)
P 0ðÞ
Note that the solution for reactor power contains two exponential terms, one of the
exponents is positive and the other is negative. The term with the negative exponent
goes to zero rapidly, leaving the term with the positive exponent to define the con-
tinuing transient. The reactor period for this example is given by T ¼ 1/
0.0088 ¼ 113.6 s.
Now consider the response to a negative step change in reactivity. The following
equation is the solution for the same reactor as above for a 10¢ reactivity step.
PtðÞ
¼ 0:909e 0:0073t +0:0909e 737:07t (4.11)
P 0ðÞ
Note that for the positive reactivity step change, one of the exponents is positive and
the other is negative; and for the negative reactivity step change, both the exponents
are negative. As shown in Section 4.8, for a six-delayed neutron group model, there is
one positive exponential and six negative exponentials for a positive step and seven
negative exponentials for a negative step.
The behavior described above can be deduced by examining the equation for neu-
tron density.
6
dn ð ρ βÞ X
¼ n + λ i C i (4.12)
dt Λ
i¼1
Note that the first term on the right is negative for ρ < β and the second term is
always positive. Thus, delayed neutrons are essential to maintain an increasing neu-
tron density if ρ < β.If ρ > β, the rate of increase is positive even if there were no
delayed neutron contribution.

4.5 Solutions for small perturbations

The small perturbation form of the zero-power reactor equations is sometimes
used. But the reader must remember that it is applicable only for small perturbations
with respect to a nominal. Fig. 4.5 shows simulations performed with the full, six-
delay group model and the small perturbation model (also with six delay groups).
The discrepancies in the results obtained with the small perturbation model are
apparent.