Preliminary tests--eigenvalue spectrum

The first thing one can do when the matrices, B and C in (1.91), belonging to the eigenvalue problem are calculated is to calculate all eigenvalues and eigenvectors for the system. This is in practice a very difficult operation but in Matlab we have a function, eig, which, in fact, does this.

For these tests we have chosen a nearly uniform computational grid of $10 \mbox{cm}$ . The only steplengths which are not $10 \mbox{cm}$ (but $< 10 \mbox{cm}$ ) lie next to the material interfaces {fuel and absorber $\rightarrow$ fuel} and {fuel and absorber $\rightarrow$ top reflector}.

In both tests described below we use the absorber tip position, $z_{\mbox{\protect\scriptsize abs}}$ , given by

$\begin{displaymath} z_{\mbox{\protect\scriptsize abs}} = 253.265 \mbox{cm} \end{displaymath}$

(4.1)

The first test involves a sub-critical reactor assembly with the data given in section 4.2 except for the fast absorption cross section of the two fuel slabs which we increase from $0.00086 \mbox{$\mbox{cm}^{-1}$}$ to a value of

$\begin{displaymath} \overline{\Sigma}{}_a^1 = 0.0012 \mbox{$\mbox{cm}^{-1}$} \end{displaymath}$

(4.2)

In Figure 4.2 we have depicted the real part of the eigenvalue spectrum--the imaginary part of the eigenvalues are within machine precision. We note that the second and most important theorem, Theorem 1.2 (see p.

), given by Wachspress is confirmed, since we have a unique real positive eigenvalue greater in modulo than all other eigenvalues. Moreover, since the distance between the largest and next largest eigenvalues is fairly large the prospects in regard to solving the eigenvalue problem by the power method are good. By observing the signs of the elements in the eigenvectors (not shown) we note that only one eigenvector has purely non-negative (real) elements and this eigenvector corresponds to the largest eigenvalue--all of this is also stated by Theorem 1.2. This eigenvector--the normalized flux solution--is plotted in Figure 4.3. When looking at the figure we may note the large truncation errors near the top-reflector interface due to the coarse grid. We can also see the flux suppression in the bottom half of the reactor core which is poisoned by the absorber.

$\begin{figure} % latex2html id marker 16596\rule{\textwidth}{0.2mm} \rule{0cm}... ... reactor assembly and with an uniform steplength of $10 \mbox{cm}$.}\end{figure}$

$\begin{figure} % latex2html id marker 16616\rule{\textwidth}{0.2mm} \rule{0cm}... ... 1) flux and the three vertical lines indicate material interfaces.}\end{figure}$

Repeating the test procedure for a super-critical reactor assembly with data corresponding to those given in section 4.2 and by (4.3) we obtain the results illustrated in Figure 4.4 and Figure 4.5. The same comments stated in the sub-critical case also apply here.

$\begin{figure} % latex2html id marker 16636\rule{\textwidth}{0.2mm} \rule{0cm}... ... reactor assembly and with an uniform steplength of $10 \mbox{cm}$.}\end{figure}$

$\begin{figure} % latex2html id marker 16656\rule{\textwidth}{0.2mm} \rule{0cm}... ... 1) flux and the three vertical lines indicate material interfaces.}\end{figure}$

For later reference we list the seven largest eigenvalues for the two reactor assemblies in Table 4.1.

$\begin{table} % latex2html id marker 16702 \rule{\textwidth}{0.8mm} \refstepco... ...11 \\ 0.4847 & 0.4574 \\ 0.3896 & 0.3704\\ \hline \end{tabular*}\end{table}$