Assume that ρ is very small.
This causes the first root, s0
, of the reactivity to be also very small. In this case, s0 ≪ λi
and therefore, we can ignore the term s0
in the denominator of the inhour equation. The inhour equation
then takes form:
As can be seen, this special case results in the same period as in the case of simple point kinetics equation, which also uses the mean generation time with delayed neutrons (ld):
Thus for small reactivities—positive or negative—the reactor period is governed almost completely by the delayed neutron properties. Despite the fact the amount of delayed neutrons is only on the order of tenths of percent of the total amount, the timescale in seconds (τi) plays the extremely important role.
The assumption is s0 ≪ λi. The largest value of τi = 1/λi = 80s, therefore the this formula is valid for periods, τe, higher than 80s. On the other hand, this formula is useful and accurate enough for most purposes for reactivities up to about ρ = 0.0005 = 50pcm.
Mean generation time with delayed neutrons (ld):
ld = (1 – β).lp + ∑li . βi => ld = (1 – β).lp + ∑τi . βi
- (1 – β) is the fraction of all neutrons emitted as prompt neutrons
- lp is the prompt neutron lifetime
- τi is the mean precursor lifetime, the inverse value of the decay constant τi = 1/λi
- The weighted delayed generation time is given by τ = ∑τi . βi / β = 13.05 s
- Therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s-1
The number, 0.08 s-1, is relatively high and have a dominating effect of reactor time response, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:
ld = (1 – β).lp + ∑τi . βi = (1 – 0.0065). 2 x 10-5 + 0.085 = 0.00001987 + 0.085 ≈ 0.085
In short, the mean generation time with delayed neutrons is about ~0.1 s, rather than ~10-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.
Let us consider that the mean generation time with delayed neutrons is ~0.085 and k (k∞ – neutron multiplication factor) will be step increased by only 0.01% (i.e. 10pcm or ~1.5 cents), that is k∞=1.0000 will increase to k∞=1.0001.
It must be noted such reactivity insertion (10pcm) is very small in case of LWRs (e.g. one step by control rods). The reactivity insertions of the order of one pcm are for LWRs practically unrealizable. In this case the reactor period will be:
T = ld / (k∞-1) = 0.085 / (1.0001-1) = 850s
This is a very long period. In ~14 minutes the neutron flux (and power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.