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Interpolation of q', $\mbox{$<\!{\alpha}\!>$}$ and $\overline{T}{}_f$

The interpolation of q' and $\overline{T}{}_f$ is straightforward if we just remember that the neutronics code assumes that the z-coordinates are given in [cm] and not [m] (for historical reasons).

Interpolation of $\mbox{$<\!{\alpha}\!>$}$ is a bit more involved since as we will see shortly we have to perform a transformation of the $\mbox{$<\!{\alpha}\!>$}$ values calculated by the hydraulics code before we can perform the interpolation. The necessity of this transformation arises from the fact that the tables of nuclear cross sections are based on an assumption of the liquid and vapor densities. For further reference we denote the density assumptions by

$$(\rho_g^*,\rho_\ell^*)$$ (13.14)

where $\rho_g^*$ [kg/${\mbox{m}}^3$] and $\rho_\ell^*$ [kg/${\mbox{m}}^3$] are densities of vapor and liquid respectively. In order to account for the actual densities encountered in the fuel channel we have to correct the $<\!{\alpha}\!>$ values calculated by the hydraulics code such that we end up with the correct number density of ${\mbox{H}}_2{\mbox{O}}$ molecules, $n_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}$ [molecules/${\mbox{cm}}^3$], in the moderator.

The number density, n [molecules/${\mbox{cm}}^3$], is defined by

$$n \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscripts... ...ngle$}}\;\frac{\rho N_{\mbox{\protect\scriptsize av}}}{\cal M}$$ (13.15)

where $\rho$ is the density of the medium [g/${\mbox{cm}}^3$], $N_{\mbox{\protect\scriptsize av}}$ is the Avogadro number and ${\cal M}$ is the molar mass of the molecules $[ \mbox{g}/\mbox{mole} ]$.

Equation (13.18) reveals that the number of ${\mbox{H}}_2{\mbox{O}}$ molecules per unit volume in the moderator with void fraction $<\!{\alpha}\!>$ is given by

$$n_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsi... ...mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}}$$ (13.16)

where the densities $\rho_g$ and $\rho_\ell$ correspond to the local state (p,T) in the fuel channel.

In the table of cross sections the number density of ${\mbox{H}}_2{\mbox{O}}$ molecules, $n_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}^*$ [molecules/${\mbox{cm}}^3$], is given by

$$n_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsi... ...mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}}$$ (13.17)

Since it is mainly the number density which determines the moderator neutronics behavior^13.4 we will demand that

$$n_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsi... ...\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}$$ (13.18)

which after some manipulation gives

$$\alpha^* = \frac{\rho_g - \rho_\ell}{\rho_g^* - \rho_\ell^*... ...\!>$}+ \frac{\rho_\ell - \rho_\ell^*}{\rho_g^* - \rho_\ell^*}$$ (13.19)

The interpolation of $<\!{\alpha}\!>$ is therefore carried out after we have transformed $\mbox{$<\!{\alpha}\!>$}\rightarrow \alpha^*$. Note that his transformation can result in negative void fractions (ie $\alpha^* < 0$) which means that we have to extrapolate in the table of cross sections.

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