Page 58 - Dynamics and Control of Nuclear Reactors

4.8 The inhour equation 49

The multiplier on the left-hand side of Eq. (4.16) is the characteristic polyno-
mial in the variable s. Equating this to zero and rearranging gives the following
equation:

!
6
X β i
ρ ¼ s Λ + (4.18)
s + λ i
i¼1
The characteristic roots of Eq. (4.18) may be obtained graphically. That is, plot the
right-hand side and the constant value of ρ on the same graph. The values of s at the
intersections are the roots. This graph serves to illustrate the form of the solution as is
the purpose of this exercise.
Graphs of the right-hand side of Eq. (4.18) appear in many references [1]. These
graphs are usually qualitative sketches since the roots span over three decades and
include both positive and negative values. Here we illustrate the form of the solution
of Eq. (4.18) quantitatively using two graphs. The first is a semi-logarithmic plot for
the roots that are always negative. This solves the problem of covering three decades.
The ordinate chosen for this plot is the period, the time T, required for the terms to
decrease by a factor of e (T i ¼ 1/s i ). The second is a plot that spans the positive and
negative values around the origin. Fig. 4.15 illustrates the graphs for a U-235 fueled
reactor with a generation time of 10 5 s. The roots are the values of the ordinate at
which a plot of the right-hand side (a line representing a constant value of ρ) inter-
sects the plots for the left-hand side.
These graphs reveal the following:

• All roots are negative when ρ < 0
• Six roots are negative and one is positive when ρ > 0
• The root with the least negative value has a period of around 80 s.

The least negative root limits the rate of power decrease following a negative reac-
tivity insertion. However, this does not mean that the power decrease will necessarily
change with a simple exponential drop with a period of 80 s. The more negative roots
will still contribute and their contribution will depend on continuing precursor pro-
duction during the power decrease as well as contributions from precursors present at
the start of the transient.
There is a practical aspect to the behavior described above. The inhour equation
is often cited as a relation for evaluating reactivity changes (for example in control
rod calibrations). This assumption leads to a form of the inhour equation with
s ¼ 1/T, where T is the reactor period observed after initial transients have died
away.

!
6
1 X β i
ρ ¼ Λ + (4.19)
T 1=T + λ i
i¼1