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The multi-group approximation

In practice when one wishes to calculate the flux distribution in a reactor core, the continuous energy diffusion equation (1.1) is not used directly. This is partly because solving (1.1) is simply too big a computational task and partly because of limitations in accuracy and completeness of the cross section libraries used for these calculations. These circumstances makes it plausible to divide the energy axis up into intervals, or groups, as shown in Figure 1.1.

If we make the assumption that the energy dependent flux density separates in space and energy, ie if we assume that the flux density in energy group g can be written as

$$\Phi(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex},E) = \phi... ...{r}{}\hspace{0.15ex})\psi^g(E), \quad\quad E \in [E_{g-1};E_g]$$ (1.1)

where

$$\int\limits_{E_{g-1}}^{E_g}\psi^g(E) dE = 1, \quad\quad g=\{1,2,\ldots,G\}$$ (1.2)

and if we integrate the continuous energy equation (1.1) on an energy interval corresponding to group g we obtain

$$\begin{eqnarray*} \lefteqn{-\nabla\cdot \left [ \int\limits_{E_{g-1}}^{E_g} D(\... ...i^{g'}(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}) \right \} \end{eqnarray*}$$

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

At this stage we define the following group constants

$$D^g(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}) \;\hbox{$... ...(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex},E) \psi^g(E) dE$$ (1.3)

$$\Sigma_a^g(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}) \;... ...a(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex},E) \psi^g(E) dE$$ (1.4)

$$\Sigma_{s}^{g' \rightarrow g} \;\hbox{$=$\kern-0.68em\raise... ...space{0.15ex},E' \rightarrow E) \psi^{g'}(E') dE' \right ] dE$$ (1.5)

$$\nu^{g'}\Sigma_f^{g'}(\hspace{0.2ex}\underline{r}{}\hspace{... ...ce{0.2ex}\underline{r}{}\hspace{0.15ex},E') \psi^{g'}(E') dE'$$ (1.6)

$$\chi^g \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\int\limits_{E_{g-1}}^{E_g} \chi(E) dE$$ (1.7)

In terms of (1.5)-(1.9) the multi-group diffusion equations may be written in the form

$$\begin{eqnarray*} \lefteqn{-\nabla\cdot [ D^g(\hspace{0.2ex}\underline{r}{}\hsp... ...{r}{}\hspace{0.15ex}) \right), \quad\quad\quad g=1,2,\ldots,G \end{eqnarray*}$$

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

The two first terms (from the left) in (1.10) closely resemble the ones in the continuous energy case, viz they describe neutron loss in group g at position r due to diffusion and absorption respectively. The next term specifies the loss of neutrons in group g through out-scattering, ie scattering from group g to group g'. We note that the term out-scattering is somewhat misleading in so far as the sum includes self-scattering, ie scattering from group g into itself. The first term on the right hand side of (1.10) describes the in-scattering of neutrons to group g including self-scattering. The last (source)term is the gain of neutrons in group g caused by fission (in the critical case, $\lambda_0 = 1$).

In regard to the multi-group diffusion equations one could say that we force the flux solution to fit the continuous energy diffusion equation not in every energy-point but in an integral sense over an energy interval $\Delta E_g$, such that the reaction rates are correctly modeled.

We impose the following continuity conditions on a potential solution of the multi-group equations

$$\begin{eqnarray*} \phi^g(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}) \quad\qu... ...ex}$\ at material interfaces (discontinuities)\footnotemark } \end{eqnarray*}$$

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

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