Page 30 - Dynamics and Control of Nuclear Reactors

3.3 Development of the point reactor kinetics equations 21

fraction of naturally-occurring Deuterium. Deuterium is also produced in LWRs by
neutron absorption in Hydrogen. Consequently, they have a similar, but smaller
photoneutron effect than CANDU reactors [1].
Molten salt reactors have three components (Li, Be, and C) that are susceptible to
photoneutron production with fission gamma rays.

3.3 Development of the point reactor kinetics equations
The purpose here is to arrive at reactor kinetics equations using the simplest and most
intuitive approach. We want to develop a set of ordinary differential equations
describing the time evolution of neutron density or reactor power. Many develop-
ments have been published that derive the equations from first principles of reactor
physics. But all approaches give the same final results and all have the details swept
into simple quantities such as k eff and reactivity. Simulations involve the use of
specified values of k eff or reactivity as input disturbances or as feedback terms in
power reactor simulations where k eff or reactivity feedback is proportional to various
process variables such as temperature or pressure.
The basic idea in formulating dynamic equations is as follows:
ð
ð
ð Rate of change of some quantityÞ ¼ Rate of productionÞ Rate of lossesÞ (3.1)
For the point kinetics model, the equations are presented below.
dn
¼ F p P Τ + P d L a + L (3.2)
dt

dC i
¼ F d P Τ L decay i ¼ 1,2,…,6 (3.3)
dt
where

n¼neutron density (neutrons per unit volume)
F p ¼fraction of neutrons that are released promptly
P T ¼total rate of neutron production by fission
P d ¼rate of release of delayed neutrons
L a+L ¼rate of neutron losses by absorption and leakage
F d ¼fraction of neutrons that are released after a delay
C i ¼concentration of ith delayed neutron precursor
L decay ¼rate of decay of delayed neutron precursors (equal to P d )

Let β be defined as the total fraction of neutron production that is delayed and β i be
the fraction of fissions that result in production of delayed neutron precursor, C i . The
loss of precursor, C i , is given by λ i C i , where λ i is the radioactive decay constant, and
is the contribution of the ith precursor to the delayed production term, P d . Since there
are multiple precursor species, typically represented by six groups, the equations
become