Choosing a grid--Neutronics

Before making any practical calculations we have to choose an appropriate computational grid, ie choose the values of the steplengths h_k for the regions 1 through (K-1) depicted in Figure 1.3. This subject is highly a matter of experience. According to [1, p. 510] it is necessary to have steplengths around $5\;\mbox{cm}$ in core regions and $1\;\mbox{cm}$ in reflector regions for thermal light water reactors. It is known from many applications in many different areas of science that an abrupt change of steplength from, say, $5\;\mbox{cm}$ to $1\;\mbox{cm}$ is a bad idea since these changes in unlucky circumstances can result in unwanted (erroneous) oscillations in the numerical solution. It is in general better to change the steplength gradually^1.27. Before we specify the form of a suitable grid we have to specify the geometry of a general reactor. We assume the reactor consists of a bottom reflector, a core plate (for carrying the weight of the fuel assemblies), the core and finally a top reflector as depicted in Figure 1.5.

$\begin{figure} % latex2html id marker 9604\rule{\textwidth}{0.2mm} \rule{0cm}{... ...sl Figure~\thefigure}\hspace{1em}The geometry of a general reactor.}\end{figure}$

We propose to use a grid which has uniform steplengths in the two reflector areas, in the core plate and in most of the core. The steplength in the core plate and in the bottom reflector is matched as closely as possible. Schematically we illustrate the variation of the steplength in Figure 1.6. Recognizing the presence of steam voids in the top reflector we will allow the steplengths in the two reflectors to differ. As seen in the figure we have to calculate two steplength transitions; one from the core plate to the core and one from the core to the top reflector.

$\begin{figure} % latex2html id marker 9624\rule{\textwidth}{0.2mm} \rule{0cm}{... ...space{1em}Schematic variation of steplengths for the proposed grid.}\end{figure}$

In order to calculate the grid we assume that the following quantities are specified

The used difference scheme (see section 1.5) demands that material interfaces lie on grid points. Consequently, we have to make sure that points $z_{{\mbox{\protect\scriptsize plate}}}$ , $z_{{\mbox{\protect\scriptsize c,bot}}}$ and $z_{{\mbox{\protect\scriptsize c,top}}}$ are aligned with grid points.

Since it is most unlikely that the steplengths for the reflectors specified above fit the geometry, we calculate some new steplengths which do. The new steplengths are calculated as follows

$\begin{displaymath} \hat{h}_{{\mbox{\protect\scriptsize refl}},1} \;\hbox{$=$\k... ...tsize plate}}}}{\hat{N}_{{\mbox{\protect\scriptsize refl}},1}} \end{displaymath}$

(1.89)

$\begin{displaymath} \hat{N}_{{\mbox{\protect\scriptsize refl}},1} \;\hbox{$=$\k... ...size plate}}}}{h_{{\mbox{\protect\scriptsize refl}},1}}\right) \end{displaymath}$

(1.90)

$\begin{displaymath} \hat{h}_{{\mbox{\protect\scriptsize refl}},2} \;\hbox{$=$\k... ...tsize c,top}}}}{\hat{N}_{{\mbox{\protect\scriptsize refl}},2}} \end{displaymath}$

(1.91)

$\begin{displaymath} \hat{N}_{{\mbox{\protect\scriptsize refl}},2} \;\hbox{$=$\k... ...size c,top}}}}{h_{{\mbox{\protect\scriptsize refl}},2}}\right) \end{displaymath}$

(1.92)

$\begin{displaymath}\mbox{ceil}(x) \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\sc... ... \mbox{$[x]+1$} &\mbox{for} & x-[x] > 0 \end{array} \right. \end{displaymath}$

(1.93)

The steplength in the core plate, $h_{{\mbox{\protect\scriptsize plate}}}$ , and the specified steplength in the bottom reflector is matched as follows

$\begin{displaymath} h_{{\mbox{\protect\scriptsize plate}}} = \frac{z_{{\mbox{\p... ...ptsize plate}}}}{\hat{N}_{{\mbox{\protect\scriptsize plate}}}} \end{displaymath}$

(1.94)

$\begin{displaymath} \hat{N}_{{\mbox{\protect\scriptsize plate}}} \;\hbox{$=$\ke... ...ize plate}}}}{h_{{\mbox{\protect\scriptsize refl}},1}} \right) \end{displaymath}$

(1.95)

The two transitions are most easily accomplished by means of a so-called growth-factor, G_r. That is, the transition from the core plate steplength $h_{{\mbox{\protect\scriptsize plate}}}$ to the core steplength $h_{\mbox{\protect\scriptsize c}}$ is obtained by choosing (see Figure 1.6)

$\begin{eqnarray*} h_{{\mbox{\protect\scriptsize c}},1} &=& G_{r,1} h_{{\mbox{\p... ... c}},i} &=& (G_{r,1})^i h_{{\mbox{\protect\scriptsize plate}}} \end{eqnarray*}$

$\textstyle \parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

where $h_{{\mbox{\protect\scriptsize c}},i}$ is the ith steplength in the core. Since we want as smooth a transition as possible we have the constraint on the last steplength, $h_{{\mbox{\protect\scriptsize c}},N_{tr}}$ , that

$\begin{displaymath} h_{{\mbox{\protect\scriptsize c}},N_{tr}} = h_{\mbox{\protect\scriptsize c}}. \end{displaymath}$

(1.96)

$\begin{eqnarray*} \lefteqn{ \quad\quad (G_{r,1})^{N_{tr}} = h_{\mbox{\protect\s... ...x{\protect\scriptsize plate}}}}}, \hspace{2em} (G_{r,1} > 1)} \end{eqnarray*}$

$\textstyle \parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

The transition length, $\ell_{tr,1}$ , ie the length used for the transition is computed from

$\begin{displaymath} \ell_{tr,1} = \sum\limits_{i=1}^{N_{tr}} (G_{r,1})^i h_{{\mbox{\protect\scriptsize plate}}} \end{displaymath}$

(1.97)

Analogously, for the second transition from the core to the top reflector we state (see Figure 1.6)

$\begin{eqnarray*} h_{{\mbox{\protect\scriptsize c}},\tilde{\imath}} &=& (G_{r,2... ...} (G_{r,2})^{\tilde{\imath}} h_{\mbox{\protect\scriptsize c}} \end{eqnarray*}$

$\textstyle \parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

Since it is unlikely that the specified $h_{\mbox{\protect\scriptsize c}}$ fits the area between the two transitions we calculate a new steplength for the core, $\hat{h_{\mbox{\protect\scriptsize c}}}$ , as follows

$\begin{displaymath} \hat{h}_{\mbox{\protect\scriptsize c}} \;\hbox{$=$\kern-0.6... ...r,1} + \ell_{tr,2})}{ \hat{N}_{\mbox{\protect\scriptsize c}}} \end{displaymath}$

(1.98)

$\begin{displaymath} \hat{N}_{\mbox{\protect\scriptsize c}} \;\hbox{$=$\kern-0.6... ...r,1} + \ell_{tr,2})}{h_{\mbox{\protect\scriptsize c}}}\right) \end{displaymath}$

(1.99)

$\begin{displaymath} \ell_{\mbox{\protect\scriptsize c}} \;\hbox{$=$\kern-0.68em... ...t\scriptsize c,top}}} - z_{{\mbox{\protect\scriptsize c,bot}}} \end{displaymath}$

(1.100)

We note that the transition calculations assumed a steplength in the core of $h_{\mbox{\protect\scriptsize c}}$ and not $\hat{h}_{\mbox{\protect\scriptsize c}}$ --this discrepancy is negligible in most cases.