Order test

In this section we will perform an order test of the finite difference scheme. Such an order test is important since an implementation may produce correct result even though the scheme is implemented wrongly, for instance if one or more occurrences of $\rho_{\ell,i+1}$ were replaced by $\rho_{\ell,i}$ . In such a case the results seem correct even though the desired accuracy of the scheme is destroyed, ie the implementation produce results which are more inaccurate than expected.

Similar to the work done in section 4.4.3 which treated an order investigation of the neutronics finite difference scheme we calculate an error estimate for successive grid reductions. The error definition we will use in this context is given by

$\begin{displaymath} E_\xi \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscr... ...e{0.2ex}\underline{\xi}{}\hspace{0.15ex}^{(r+1)} \Vert _\infty \end{displaymath}$

(9.18)

To prevent the calculation of the point of void departure from disturbing the results we firstly run the program with a very fine grid in order to calculate the "correct" $h_{\ell,d}$ . The value of $h_{\ell,d}$ obtained in this fashion is incorporated into the source code such that the value of $h_{\ell,d}$ is fixed at this value for all grid reductions performed! With this problem fixed we calculate the solution for a coarse grid, ie with an average steplength of $0.1 {\mbox{ m}}$ , 5 uniform steplengths for every spacer and with a length of the spacer pressure disturbance of $0.05 {\mbox{ m}}$ . The linear heat generation profile was obtained from a reactor physics calculation (in which the void fraction, $<\!{\alpha}\!>$ , and the fuel temperature, T_f, increased linearly with z) and the mass flux was $\mbox{$<\!{G}\!>$} = 750 {\mbox{ kg}}/({\mbox{s}}\cdot{\mbox{m}}^2)$ . For successive grid reductions we calculated the solution and calculated the error given by (9.19) for the three coupled dependent variables $<\!{\alpha}\!>$ , $<\!{x}\!>$ and p. The results are given in Table 9.1. In section 7.2 we concluded that the difference scheme employed had order 2--this is confirmed by the experiments given in the table.

$\begin{table} % latex2html id marker 33182 \rule{\textwidth}{0.8mm} \refstepco... ...07\cdot 10^{-3}$& $2.00$\ & $2.00$\ & $1.96$\\ \hline \end{tabular*}\end{table}$

Really sceptic readers could argue that the 1.96 for the order on the pressure error, E_p, is not close enough to 2.00. In order to rule out any doubt we have carried the calculation for the same case but now starting with an average steplength $h_{\mbox{\protect\scriptsize av}} = 0.05 {\mbox{ m}}$ instead of $0.1 {\mbox{ m}}$ . The results are displayed in Table 9.2.

$\begin{table} % latex2html id marker 33258 \rule{\textwidth}{0.8mm} \refstepco... ...6860\cdot 10^{-2}$& $2.00$& $1.99$\ & $1.99$\\ \hline \end{tabular*}\end{table}$

One final investigation we will make is to look at the difference between the coarse grid (r=0) and fine grid (r=7) solutions. This difference is a good estimate for the error induced by the numerical method on the coarse grid solution. In Figures 9.7, 9.8 and 9.9 we have depicted the difference for $<\!{\alpha}\!>$ , $<\!{x}\!>$ and p. If we consider that relative to the total core pressure drop the maximum pressure derivation is only 0.17 %^9.7 we can conclude that even the very coarse grid (r=0) is adequate for practical calculations!

We note that the discretization error in the pressure distribution (see Figure 9.9) seems to be dominated by the local truncation error in the neigbourhood of the spacers. We conclude therefore that the discretization error can be reduced by choosing a higher number of uniform steplengths, N, but as noted previously the error on the pressure is acceptably low.

$\begin{figure} % latex2html id marker 33324\rule{\textwidth}{0.2mm} \rule{0cm}... ...m}}$) core flow quality results calculated by the C implementation.}\end{figure}$

$\begin{figure} % latex2html id marker 33375\rule{\textwidth}{0.2mm} \rule{0cm}... ...}}$) core void fraction results calculated by the C implementation.}\end{figure}$

$\begin{figure} % latex2html id marker 33426\rule{\textwidth}{0.2mm} \rule{0cm}... ...e pressure distribution results calculated by the C implementation.}\end{figure}$