Page 44 - Dynamics and Control of Nuclear Reactors

4.2 Numerical analysis 35

Note that ρ is a constant and most of the elements of the matrix are zero for a sim-
ulation of a step change in reactivity. Matrices with most elements equal to zero are
called sparse matrices. Note also that all of the diagonal elements are negative if
ρ < β, and all but one is positive if ρ > β. Matrices with all negative diagonal ele-
ments are more likely to indicate stability of the response than matrices with one
or more positive diagonal elements. But the negativity of diagonal elements does
not ensure stability. The lesson here is that a modeler should recheck the formulation
if positive diagonal elements appear in a matrix representing a system that is thought
to represent a stable system.
The solution vector (state variables) and the initial condition vector are defined as
follows:
1
2 3
6 7
6 7
6 7
3 6 7
2
P 6 β 7
1
6 7
6 7 6
λ 1 Λ 7
6 P 0 7 6 7
6 7 6 7
2
6 C 1 7 6 β 7
6 7 6 7
6 7 6 λ 2 Λ 7
6 C 2 7 6 7
6 7 6 7
3
X ¼ 6 7 X 0ðÞ ¼ 6 β 7 (4.5b)
6 C 3 7 6 7
6 7 6 λ 3 Λ7
6 7 6 7
C 4
6 7 6 7
4
6 7 6 β 7
7
6 7 6
C 5
6 7 6 λ 4 Λ7
4 5 6 7
β 7
6
C 6 6 5 7
6 7
6 λ 5 Λ7
β
6 7
4 6 5
λ 6 Λ
Note that the vector, f, is null (or zero) in this case. The model is an initial value
problem. It is noted that a matrix formulation of a perturbation model has zero initial
conditions, but a non-zero forcing vector. Formulating a state variable perturbation
model is left as an exercise for the reader.
The solution for a linear system is a sum of exponentials. The exponential coef-
ficients may be positive or negative, real or complex. (Complex coefficients indi-
cate an oscillatory response). The coefficients are called eigenvalues and are the
same as poles of the system transfer function. It is important to note that the
above linear system is absolutely stable if all the eigenvalues (or poles of system
transfer function) are negative or have negative real parts (for the case of complex
eigenvalues).
The point kinetics model for a constant step change in reactivity is a 7-th order
linear dynamic system. Therefore, its solution is a sum of seven exponential terms.
As a transient unfolds, the most positive (or least negative term) eventually domi-
nates the response of interest, n(t). That is.
st
ntðÞ constant e , for larget: (4.6)