Page 265 - Dynamics and Control of Nuclear Reactors

266 APPENDIX C Basic reactor physics

The above formula is referred to as the four-factor formula. This does not take into
account the probability of neutron leakage from finite size cores. In order to include
this effect, two more factors are added to the above calculation [1–4].

6. Fast non-leakage probability, P Fnl : The fast non-leakage factor is the number
of fast neutrons do not leak out of the core during slowing down to thermal
neutron per fast neutron produced. A typical value for P Fnl0.97 is around 0.97 for a
U-235-fueled thermal reactor.
7. Thermal non-leakage probability, P Tnl : The thermal non-leakage factor is the
number thermal neutrons do not leak out of the core per thermal neutron
produced. A typical value for P Tnl is around 0.99 for a U-235-fueled
thermal reactor.
8. Effective multiplication factor, k eff : With definition of non-leakage
probabilities, we can now calculate the effective multiplication factor as

k eff ¼ k ∞ P Fnl P Tnl ¼ ηfp ε P Fnl P Tnl (C.15)
The formula in Eq. (C.15) is generally referred to as the six-factor formula. The
reader can find more details in Ref. [1, 2].

C.10 Neutron transport and diffusion
The most complete description of the spatial distribution of neutrons in a reactor is
given by neutron transport theory. Transport theory defines a reactor in terms of
seven independent variables: three position coordinates, two direction vectors,
energy and time. The transport theory equation is called the Boltzmann equation.
Computer codes have been developed for transport theory analysis, but they suffer
from complexity, long computing time, and difficulty in determining detailed fine-
mesh parameters needed for implementation.
Most reactor studies treat neutron motion as a diffusion process – that is, neutrons
tend to diffuse from regions of high neutron density to regions of low neutron den-
sity. Diffusion theory ignores the direction dependence of the neutrons.

Exercises

13
2
C.1. The neutron flux in a certain reactor is 2 10 neutrons/(cm -sec). If the neu-
trons have a mean velocity of 3100m/s, calculate the neutron density. Indicate
the units.
13
C.2. The neutron flux in a commercial pressurized water reactor (PWR) is 2 10 /
2 2 3
(cm -s). The macroscopic cross section for fission is 30cm /cm . Calculate
3
the rate of fission reactions per cm . Indicate the units. Simplify your answer.