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Evaluating the nuclear cross sections

In the neutronics code we need to calculate the average values of the cross sections within a region k as shown in (1.17)-(1.20). As mentioned previously the nuclear cross sections depend on both the void fraction $\alpha^*$ and the average fuel temperature $\overline{T}{}_f$. If we consider the kth region in the computational grid of the neutronics code (see Figure 1.2 p. ) we define the following averages

$$\overline{\alpha}{}_k^* \;\hbox{$=$\kern-0.68em\raise1.1ex ... ...ac{1}{z_{k+1}-z_k} \int\limits_{z_k}^{z_{k+1}} \alpha^*(z) dz$$ (13.20)

and

$$\overline{T}{}_{f,k} \;\hbox{$=$\kern-0.68em\raise1.1ex \h... ...{k+1}-z_k} \int\limits_{z_k}^{z_{k+1}} \overline{T}{}_f(z) dz$$ (13.21)

With these averages at hand we define the average nuclear cross sections^13.5 as

$$\overline{D}{}_k^g \;\hbox{$=$\kern-0.68em\raise1.1ex \hbo... ...iangle$}}\;D_k^g(\overline{\alpha}{}_k^*,\overline{T}{}_{f,k})$$ (13.22)

$$\overline{\Sigma}{}_{a,k}^g \;\hbox{$=$\kern-0.68em\raise1.... ...e$}}\;\Sigma_a^g(\overline{\alpha}{}_k^*,\overline{T}{}_{f,k})$$ (13.23)

$$\overline{\Sigma}{}_{s,k}^{g \rightarrow g'} \;\hbox{$=$\ke... ... \rightarrow g'}(\overline{\alpha}{}_k^*,\overline{T}{}_{f,k})$$ (13.24)

$$\overline{\Sigma}{}_{f,k}^g \;\hbox{$=$\kern-0.68em\raise1.... ... \Sigma_{f,k}^g(\overline{\alpha}{}_k^*,\overline{T}{}_{f,k})$$ (13.25)

In practice, we substitute the integrals in (13.23) and (13.24) with suitable approximations. As a result we obtain the following second order accurate approximations

$$\overline{\alpha}{}_k^* \simeq \frac{1}{2} [ \alpha^*(z_{k+1}) + \alpha^*({z_k}) ]$$ (13.26)

and

$$\overline{T}{}_{f,k} \simeq \frac{1}{2} [ \overline{T}{}_f(z_{k+1}) + \overline{T}{}_f(z_k) ]$$ (13.27)

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