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Theory

In order to obtain a reasonable simple semi-analytical^9.1 solution we have to make a number of simplifying assumptions

.

Constant properties: This assumption enables us to calculate the dependent variables, $<\!{x}\!>$, $<\!{\alpha}\!>$ and p separately along the flow path.

.

Sinusidal heat addition profile: Enables us to obtain an analytical expression for the mixture enthalpy, $<\!{h}\!>$.

.

Flat phase distributions: Makes the void fraction calculation simpler. This assumption corresponds to

$$C_0 = 1 \qquad,\qquad u_{gj} = 0$$ (9.1)

Since we have assumed a sinusidal heat addition profile we may write the linear heat generation rate for one fuel element^9.2

$$q'(z) = q_0 \sin\left(\frac{\pi z}{\ell_c}\right)$$ (9.2)

where q₀ is the maximum linear heat generation rate [W/m] and $\ell_c$ is the length of the core [m].

With q' at hand we calculate the mixture enthalpy, $<\!{h}\!>$, along the channel. According to (6.103) we have

$$\mbox{$<\!{h}\!>$}= h_{\ell,0} + \frac{q_0\ell_c}{\pi A_c \... ...}{\ell_c}\right)\right) \qquad{\mbox{for }} 0 \le z \le \ell_c$$ (9.3)

where $h_{\ell,0}$ is the core inlet enthalpy [J/kg]. In order to calculate the flow quality, $<\!{x}\!>$, we firstly have to calculate the point of void departure, ie $h_{\ell,d}$, by utilizing the Saha-Zuber correlation (see (6.93)). If we insert the expression for the wall heat flux, q'', given by

$$q''(z) = \frac{q'}{P_H}$$ (9.4)

where P_H [m] is the heated perimeter for the whole fuel element into the Saha-Zuber correlation we end up with the equations

$$\begin{eqnarray*} h_f - h_{\ell,0} + \frac{q_0 \ell_c}{\pi A_c \mbox{$<\!{G}\!... ...(\frac{\pi z}{\ell_c}) \qquad{\mbox{for }}\mbox{\bf Pe}>70000 \end{eqnarray*}$$

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

These equation has to be solved iteratively for z_d and when completed the specific enthalpy at the point of void departure, $h_{\ell,d}$, is obtained by

$$h_{\ell,d} \equiv \mbox{$<\!{h}\!>$}\vert _{z=z_d}$$ (9.5)

ie by inserting z_d into (9.3). Note that since the properties are fixed $h_{\ell,d}$ can be calculated once and for all at the start.

Inserting $h_{\ell,d}$ (from (9.5)) and $<\!{h}\!>$ (from (9.3) into Levy's profile fit (see (6.97)) gives us $<\!{x}\!>$. With $<\!{x}\!>$ known we can calculate the void fraction $<\!{\alpha}\!>$ from the Zuber-Findlay void fraction model (see (6.73)) with C₀ = 1 and u_gj = 0 in the following way

$$\mbox{$<\!{\alpha}\!>$}= \frac{\mbox{$<\!{x}\!>$}}{\mbox{$<\!{x}\!>$}+ \frac{\rho_g}{\rho_\ell}(1-\mbox{$<\!{x}\!>$})}$$ (9.6)

which actually corresponds to an assumption of homogeneous flow (no slip between the phases).

Since we at this point know both the flow quality and the void fraction we can calculate the total pressure drop along the channel, $\Delta p_{\mbox{\protect\scriptsize tot}}$ [Pa], from the equation (momentum equation)

$$\Delta p_{\mbox{\protect\scriptsize tot}} = \Delta p_{\mbox... ...ct\scriptsize el}} + \Delta p_{\mbox{\protect\scriptsize loc}}$$ (9.7)

where

>

is the friction pressure drop [Pa].

>

is the acceleration pressure drop [Pa].

>

is the elevation pressure drop [Pa].

>

is the local pressure drop due to the grid spacers [Pa].

Note that the first assumption (see p. ) enables us to calculate the pressure drops separately.

The frictional pressure drop, $\Delta p_{\mbox{\protect\scriptsize fric}}$, may be calculated by

$$\Delta p_{\mbox{\protect\scriptsize fric}} = -\frac{\mbox{$... ...{2 \rho_\ell D_e} \int\limits_{0}^{\ell_c} \phi_{\ell 0}^2 dz$$ (9.8)

where $f_{\ell 0}$ is the single-phase Darcy-Weisbach friction factor given by the Colebrook interpolation formula (see (6.81)) and $\phi_{\ell 0}^2$ is given by the Jones expression (see (6.85)). To simplify things and since the pressure dependence is very small we put p=p₀ in the Jones formula for $\Psi(\mbox{$<\!{G}\!>$},p)$ eq. (6.86) (see p. ) and count the $\Psi$-function independent of p such that $\phi_{\ell 0}^2 = {\mbox{fcn}}(\mbox{$<\!{x}\!>$})$ only. In mathematical terms we make the following approximation

$$\Psi(\mbox{$<\!{x}\!>$},p) \approx \Psi(\mbox{$<\!{x}\!>$},p_0)$$ (9.9)

where p₀ is the pressure at the inlet of the core [Pa]. Without this assumption the solution of the problem would require a step wise solution.

The acceleration pressure drop, $\Delta p_{\mbox{\protect\scriptsize acc}}$, can be expressed as

$$\Delta p_{\mbox{\protect\scriptsize acc}} = -\mbox{$<\!{G}\... ...lpha}\!>$}} \right]_{z=\ell_c} - \frac{1}{\rho_\ell} \right\}$$ (9.10)

The elevation pressure drop, $\Delta p_{\mbox{\protect\scriptsize el}}$, is given by

$$\Delta p_{\mbox{\protect\scriptsize el}} = -g \left\{ \rho_... ...l)\int\limits_{0}^{\ell_c} \mbox{$<\!{\alpha}\!>$}dz \right\}$$ (9.11)

The integral of $<\!{\alpha}\!>$ has a very complicated structure^9.3 which necessitates numerical evaluation. This is the sacrifice we have to make when we require to include the subcooled void in the calculations. If subcooled void was neglected we could in fact derive an analytical expression for $\int \mbox{$<\!{\alpha}\!>$}dz$.

The total pressure drop due to the spacers, $\Delta p_{\mbox{\protect\scriptsize loc}}$, is calculated by the expression

$$\Delta p_{\mbox{\protect\scriptsize loc}} = \frac{\mbox{$<\... ...!{x}\!>$}\vert _{z_{{\mbox{\protect\scriptsize sp}},j}}\right)$$ (9.12)

where $K_{{\mbox{\protect\scriptsize sp}},j}$ is the loss coefficient of the jth spacer [--] and $z_{{\mbox{\protect\scriptsize sp}},j}$ is the position of the jth spacer [m].

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