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Local scale power distribution

When we want to calculate the thermal-hydraulics behavior of the reactor core we have to consider not only the large scale power distribution but also the more local distribution of power among fuel, moderator and structural materials present in the core. We will, however, not go into a detailed discussion of this subject. We assume that a fraction, R_f, of the total energy release occurs in fuel and the rest, 1-R_f, is released in the moderator. A mathematical definition of R_f is

$$R_f \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{q_{k,fuel}'}{q_k'}$$ (1.84)

where q_k,fuel' and q_k' are the linear heat generation rate in fuel and the total linear heat generation rate respectively measured in [W/m].

According to [11, p. 158], we are safe to assume that more than 97 per cent of the energy release occurs in the fuel [11, p.158]. This is supported by figures [12, p. 21] of the power distribution in a reactor design similar to that of the Swedish Oscarshamn reactor. In this case 97.1 percent of the energy is generated within the fuel.

The part of the power which is generated within the fuel is conducted and convected to the coolant through the fuel rods by means of a heat flux, q''. The heat flux at node k, q_k'', is determined by

$$q_k'' = \frac{R_f q_k'}{P_H}$$ (1.85)

where P_H is the heated perimeter in $[ \mbox{m} ]$ and q_k'' is measured in $[\mbox{W} / \mbox{m}^2]$. Since the rest part of the power, generated in the moderator, is mainly generated by neutron thermalization and $\gamma$-capture this power production is controlled solely by the number density of the moderator. In general we have a two-phase mixture of water and steam as the moderator and coolant. Since steam has a much smaller density than water the power generation in the moderator is controlled by the following two volumetric heat sources, $q_{\ell,k}'''$ and q_g,k''' in $[\mbox{W}/\mbox{m}^3]$,

$$\begin{eqnarray*} q_{\ell,k}''' &=& K_\ell \frac{\rho_\ell N_{\mbox{\protect\sc... ...mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}} \end{eqnarray*}$$

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

where $K_\ell$ and K_g are constants with a dimension of $[ \mbox{W} ]$, $N_{\mbox{\protect\scriptsize av}}$ is the Avogadro number and ${\cal M}_{{\mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}$ is the molar mass of water $[ \mbox{kg}/\mbox{mole} ]$. Since we assume that $\mbox{H}_2\mbox{O}$ molecules react identically in liquid and gas phases we state that

$$K = K_\ell = K_g$$ (1.86)

An energy conservation for the vapor-liquid mixture states that

$$\begin{eqnarray*} (1 - R_f) q_k' &=& A_c( q_{\ell,k}'''(1 - \mbox{$<\!{\alpha}\... ... - \mbox{$<\!{\alpha}\!>$}) + \rho_g \mbox{$<\!{\alpha}\!>$}) \end{eqnarray*}$$

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

where A_c is the cross-sectional flow area in [${\mbox{m}}^2$] and $\tilde{K}$ is defined by

$$\tilde{K} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scrip... ...mbox{\protect\scriptsize H}}_2{\mbox{\protect\scriptsize O}}}}$$ (1.87)

Using (1.111) we can write $q_{\ell,k}'''$ and q_g,k''' as follows

$$\begin{eqnarray*} q_{\ell,k}''' &=& C \rho_\ell \\ q_{g,k}''' &=& C \rho_g \end{eqnarray*}$$

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

where

$$C \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscripts... ...- \mbox{$<\!{\alpha}\!>$} ) + \rho_g \mbox{$<\!{\alpha}\!>$}}$$ (1.88)

The magnitude of q_g,k''' is usually much smaller than $q_{\ell,k}'''$ due to the smaller density of the former. The fraction of the volumetric power dissipated in vapor, W, is defined by

$$\begin{eqnarray*} W \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptsty... ...a}\!>$} + (1-\mbox{$<\!{\alpha}\!>$})\frac{\rho_\ell}{\rho_g}} \end{eqnarray*}$$

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

W for water at saturation is depicted in Figure 1.4 for three different pressures.

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