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Next: Grid points influenced by Up: Grid generation Previous: Grid generation Contents Index

Grid points not influenced by a spacer

The first requirement is strongly related to the linear heat generation rate which is displayed in Figure 8.7. The user supplies the desired mean steplength $h_{\mbox{\protect\scriptsize av}}$ which corresponds to the average linear heat generation rate, $q_{\mbox{\protect\scriptsize av}}'$ , where $q_{\mbox{\protect\scriptsize av}}'$ is defined by

$\begin{displaymath} q_{\mbox{\protect\scriptsize av}}' \;\hbox{$=$\kern-0.68em\... ...e\triangle$}}\;\frac{1}{\ell_c} \int\limits_0^{\ell_c} q'(z)dz \end{displaymath}$

(8.16)

Now the steplengths in the grid (excluding the steplengths in proximity to the spacers) are calculated such that

$\begin{displaymath} \int\limits_{z_k}^{z_{k+1}} q'(z)dz = q_{\mbox{\protect\scriptsize av}}' h_{\mbox{\protect\scriptsize av}} \end{displaymath}$

(8.17)

ie one could say that the grid is constructed such that the (mixture) enthalpy increase per steplength is fixed.

$\begin{figure} % latex2html id marker 31074\rule{\textwidth}{0.2mm} \rule{0cm}... ...scriptsize av}}'$\ denotes the average linear heat generation rate.}\end{figure}$

Since we do not want to spend too much computing time on the grid generation we approximate the integral in (8.17) by the 2nd order accurate expression

$\begin{displaymath} \int\limits_{z_k}^{z_{k+1}} q'(z)dz \approx (z_{k+1}-z_k)\frac{q'(z_k)+q'(z_{k+1})}{2} \end{displaymath}$

(8.18)

which inserted into (8.17) yields

$\begin{displaymath} (z_{k+1}-z_k)\frac{q'(z_k)+q'(z_{k+1})}{2} = q_{\mbox{\protect\scriptsize av}}'h_{\mbox{\protect\scriptsize av}} \end{displaymath}$

(8.19)

For every steplength in the grid (excluding the spacer grid points) we have to solve (8.19) for z_k+1. In practice it is sufficient to solve (8.19) by the fixed point iteration scheme shown below

$\begin{displaymath} z_{k+1}^{(j+1)} = \frac{2q_{\mbox{\protect\scriptsize av}}'... ...box{\protect\scriptsize av}}}{q_k' + q'(z_{k+1}^{(j)})} + z_k \end{displaymath}$

(8.20)

where the superscripts indicate the iteration number. A suitable start guess is (simply putting q_{z_k+1}' = q_{z_k}')

$\begin{displaymath} z_{k+1}^{(0)} = \frac{q_{\mbox{\protect\scriptsize av}}'}{q_k'}h_{\mbox{\protect\scriptsize av}} + z_k \end{displaymath}$

(8.21)

Next: Grid points influenced by Up: Grid generation Previous: Grid generation Contents Index

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