Calculating the point of void departure, zd, iteratively

It is shown in this section that it is important to calculate the point of void departure accurately in order to predict the subcooled void fraction accurately. When we use a very fine computational grid it is sufficient to predict $h_{\ell,d}$ to within $\pm$ one steplength. However, for a coarse grid which is constructed by the guidelines described in section 8.5 the first steplengths are in general very coarse due to a low linear heat generation rate q'. The result is that $h_{\ell,d}$ is calculated using values of q' and the physical properties for a z-level which is off by as much as 30 cm when compared to the true z_d. Since q' is increasing rapidly in this area $h_{\ell,d}$ will be seriously affected by this calculation procedure. The error on $h_{\ell,d}$ will canalize itself to the subcooled void fraction where $\mbox{$<\!{\alpha}\!>$}$ especially in the region totally dominated by subcooled boiling may be off by more than 20 % as compared to a fine grid solution. In Figure 8.2 we have compared plots of the coarse grid void fraction calculation with a fine grid solution in which the $h_{\ell,d}$ is calculated very accurately (the accurate method of calculating $h_{\ell,d}$ is described later in this section). As we can see the error is largest in the area with a high degree of subcooling and the error decreases with decreasing subcooling according to the Levy profile fit (see (6.97)). In Figure 8.3 we can see the influence of the erroneous $h_{\ell,d}$ on the flow quality distribution. The error on $\mbox{$<\!{x}\!>$}$ is as we can see also serious--in fact the relative errors on $\mbox{$<\!{\alpha}\!>$}$ and $\mbox{$<\!{x}\!>$}$ are of the same order of magnitude. The error on the void fraction is most inconvenient since the void fraction is feed back to the neutronics calculation with an error on the power distribution as the result.

$\begin{figure} % latex2html id marker 30740\rule{\textwidth}{0.2mm} \rule{0cm}... ...he correct void fraction (942 CV's) (\mbox{---}) along the channel.}\end{figure}$

$\begin{figure} % latex2html id marker 30760\rule{\textwidth}{0.2mm} \rule{0cm}... ...the correct flow quality (942 CV's) (\mbox{---}) along the channel.}\end{figure}$

It is important to note that this 20 % error on $\mbox{$<\!{\alpha}\!>$}$ arises from an error on $h_{\ell,d}$ of only 1 %, ie $h_{\ell,d}$ is a sensitive quantity in regard to the calculation of $\mbox{$<\!{\alpha}\!>$}$ .

The remedy for the above mentioned problem is to calculate $h_{\ell,d}$ (and z_d for that matter) accurately without considering the underlying grid. In Figure 8.4 we see a close-up of the computational grid around the point of void departure. Our problem is to find the point of void departure which lies somewhere between the grid points illustrated. In our implementation we have used the properties at the previous z-level to evaluate the void departure criterion at the current z-point (see (6.93)), ie for $\mbox{\bf Pe} > 70000$ which is appropriate for our purpose we calculate the enthalpy at the point of void departure at grid point i by the expression

$\begin{displaymath} h_{\ell,d,i} = h_{f,i-1} - K q_{w,i}''' \end{displaymath}$

(8.12)

$\begin{figure} % latex2html id marker 30780\rule{\textwidth}{0.2mm} \rule{0cm}... ...around the point of void departure. VD is short for void departure.}\end{figure}$

With reference to Figure 8.4 the problem is to find the void departure point, z_d. We should be aware that a situation can occur in which the void departure point is situated a bit to the left of the point z_i. This is a result of using the old properties to calculate $h_{\ell,d}$ ^8.4. In any case we can state that our task is to solve the equation

$\begin{displaymath} \mbox{$<\!{h}\!>$}- h_{\ell,d}(z) = 0 \qquad\mbox{where}\quad z \in [z_j;z_{i+1}] \end{displaymath}$

(8.13)

Since we have to solve a non-linear equation (8.13) numerically in a limited range of z-values the best choice is to use a numerical method in the group of enclosure methods which, as stated previously, work on a range of the independent variable. We have chosen the Pegasus method described in section 7.3 in order to solve (8.13) since it is one of the fastest and more robust enclosure methods.

For comparison we have included Figures 8.5 and 8.6 which compare coarse and fine grid solutions when we employ an accurate grid independent calculation of $h_{\ell,d}$ . As we can see the coarse grid solution lies close to the fine grid solution!

$\begin{figure} % latex2html id marker 30864\rule{\textwidth}{0.2mm} \rule{0cm}... ...!>$}$, with an accurate calculation of $h_{\ell,d}$\ in both cases.}\end{figure}$

$\begin{figure} % latex2html id marker 30906\rule{\textwidth}{0.2mm} \rule{0cm}... ...!>$}$, with an accurate calculation of $h_{\ell,d}$\ in both cases.}\end{figure}$

Since we know that $h_{\ell,d}$ is one of the critical quantities it is interesting to investigate the accuracy of the Saha-Zuber correlation. As noted in section 6.7.2.4 it is the expression for $\mbox{\bf Pe}>70000$ which is of interest in BWR design. The Saha-Zuber correlation for this case may be written

$\begin{displaymath} h_f - h_{\ell,d} = \frac{q_w''}{\mbox{$<\!{G}\!>$} \mbox{\bf St}} \end{displaymath}$

(8.14)

$\begin{displaymath} \mbox{\bf St} = 0.0065 \qquad {\mbox{for}} \quad \mbox{\bf Pe}>70000 \end{displaymath}$

(8.15)

$\begin{table} % latex2html id marker 30973\rule{\textwidth}{0.8mm} \refstepcou... ...04875$\ & $1134.2$\ & $-2.2$\% \\ \hline \end{tabular*}\end{minipage}\end{table}$