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Subcooled boiling model

At illustrated in Figure 6.1 (p. ) subcooled boiling can be subdivided into

.

Highly subcooled boiling or wall voidage region.

.

Slightly subcooled boiling or detached voidage region.

According to [19, p. 223, Table 6.1] the void fraction in the wall voidage region was 0.008 for $p=69 {\mbox{ bar}}$, $\mbox{$<\!{G}\!>$}=680 {\mbox{ kg}}/({\mbox{m}}^2\cdot{\mbox{s}})$, $D_e = 12.7 {\mbox{ mm}}$ and $q_w'' = 790 {\mbox{ kW}}/{\mbox{m}}^2$, ie conditions similar to normal BWR operation. Since the wall voidage void fraction is very small we will in the following neglect the wall voidage region altogether, ie we assume that

$$\mbox{$<\!{\alpha}\!>$}\simeq 0 \qquad{\mbox{ for }}\qquad z<z_d$$ (6.82)

With this assumption in effect the subcooled boiling model consists of the following items

.

Correlation for estimation of the point of void departure, z_d (see Figure 6.1, p. ).

.

Correlation(s) for estimating the true flow quality, $\mbox{$<\!{x}\!>$}$ (or alternatively the mass generation rate per unit length, $\mbox{$\dot{m}$}_{\ell g}'$).

.

Correlation for the calculation of the subcooled void fraction.

.

Correlation for the calculation of the wall shear stress during subcooled boiling.

The different parts are investigated in the subsequent.

In nearly all the literature on two-phase flow (eg [18] and [19]) the authors agree that the point of void departure, z_d, is best predicted^6.12 by the Saha-Zuber correlation^6.13 [18, p. 214] which states that

$$h_f - h_{\ell,d} = \left\{ \begin{array}{rr} 0.0022 \frac{q... ...} & {\mbox{for }} \mbox{\bf Pe} \ge 70000 \end{array} \right.$$ (6.83)

where

>

Mean liquid specific enthalpy at the point of void departure [J/kg].

>

Wall heat flux [W/${\mbox{m}}^2$].

>

Heated diameter defined by $D_H \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;4A_c/P_H$ [m].

>

The mean liquid specific heat [J/(kg$\cdot$K)].

>

The Peclet dimensionless number^6.14 defined by $\mbox{\bf Pe} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;(\mbox{$<\!{G}\!>$}D_H C_{p,\ell})/k_\ell$ [--].

>

The total mean mass flux defined by (6.20) [ ${\mbox{kg}}/({\mbox{m}}^2\cdot{\mbox{s}})$].

It is not clear how the properties should be evaluated. We will, however, evaluate the properties at the mean liquid temperature which we could call the mixing cup temperature.

Alternatively, one may define the point of void departure by heat flux correlations [33], which divide the total wall heat flux into parts including vaporization and condensation.

We can model the flow quality, $\mbox{$<\!{x}\!>$}$, during subcooled boiling in two ways. The first way--the so-called mechanistic model--is based upon heat transfer correlations which divide the wall heat flux into parts one part which vaporizes and one part which rises the mean liquid bulk enthalpy. The second way the so-called profile fit model assumes a functional relationship between the mixture enthalpy, $\mbox{$<\!{h}\!>$}$, and the flow quality, $\mbox{$<\!{x}\!>$}$.

The profile fit model is the simplest to use and adequate for steady-state conditions [18]. One of the simplifying features of the profile fit model when compared to the mechanistic model is that the calculation procedure for the flow quality is uncoupled from the heat transfer calculations.

We will use the Levy profile fit model recommended by [18] in our core flow model. The Levy profile fit model states that

$$\mbox{$<\!{x}\!>$}= \mbox{$<\!{x_e}\!>$} - \mbox{$<\!{x_e}\... ...$<\!{x_e}\!>$}_d} - 1 \right] \qquad{\mbox{ for }}\qquad z>z_d$$ (6.84)

where the mean equilibrium quality, $\mbox{$<\!{x_e}\!>$}$, is defined by

$$\mbox{$<\!{x_e}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex \h... ...scriptstyle\triangle$}}\;\frac{\mbox{$<\!{h}\!>$}-h_f}{h_{fg}}$$ (6.85)

and $\mbox{$<\!{x_e}\!>$}_d$ is the equilibrium quality at the void departure point z_d which is defined as

$$\mbox{$<\!{x_e}\!>$}_d = - \frac{h_f-h_{\ell,d}}{h_{fg}}$$ (6.86)

where the point of void departure is estimated from, for instance, the Saha-Zuber correlation (6.93).

Introducing the expressions (6.95) and (6.96) into the profile fit expression (6.94) give us the true flow quality $\mbox{$<\!{x}\!>$}$ in term of the mixture enthalpy, $\mbox{$<\!{h}\!>$}$, ie

$$\mbox{$<\!{x}\!>$}= \frac{\mbox{$<\!{h}\!>$}- h_f}{h_{fg}} ... ...t[ - \frac{\mbox{$<\!{h}\!>$}-h_f}{h_f-h_{\ell,d}} -1 \right]$$ (6.87)

The last term in (6.97) derives from the non-equilibrium. We note that the last term goes towards zero for large $\mbox{$<\!{h}\!>$}$, $\mbox{$<\!{h}\!>$}\gg h_f$, ie when we reach an equilibrium flow.

It is important to understand that the mean or bulk liquid enthalpy, $\mbox{$<\!{h_\ell}\!>$}_\ell$, and the subcooled flow quality, $\mbox{$<\!{x}\!>$}$ are interrelated. When the mixture enthalpy has been obtained from the mixture energy equation (6.58) and the subcooled flow quality calculated from the profile fit (6.97) the mean liquid enthalpy, $\mbox{$<\!{h_\ell}\!>$}_\ell$, is given by (see (6.51))

$$\mbox{$<\!{h_\ell}\!>$}_\ell = \frac{\mbox{$<\!{h}\!>$}- \mbox{$<\!{x}\!>$}\mbox{$<\!{h_g}\!>$}_g}{1-\mbox{$<\!{x}\!>$}}$$ (6.88)

Strictly speaking the Levy profile fit has only been confirmed for a uniform axial heat flux, ie not for a non-uniform heat flux occurring in a BWR. With this restriction in mind Hsu [34, p. 6-61] recommends to derive a profile fit which corresponds to the non-uniform axial heat flux in question following a procedure given by Ahmad [36].

It is customary to use the saturated flow correlations for the void fraction and the wall shear stress also for subcooled boiling.

We will use the Dix correlation (6.75) for evaluating C₀ and the correlation proposed by Lahey and Moody (6.78) for $\overline{u}{}_{g j}$. Alternatively one may choose the values originally used by Levy [19, p. 232]

$$\begin{eqnarray*} C_0 &=& 1.13\\ \overline{u}{}_{gj} &=& 1.18 \left[ \frac{ (\rho_\ell-\rho_g)\sigma g}{\rho_\ell^2}\right]^{1/4} \end{eqnarray*}$$

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

If we were to use these correlations for C₀ and $\overline{u}{}_{gj}$ together with the correlations given in section 6.7.2.1 we would have to face a discontinuity in both the distribution parameter and the mean drift flux (which result in a discontinuity in the void fraction)--a very non-physical phenomenon and unpleasant to deal with in regard to the numerical solution.

In regard to the wall shear stress we are in practice forced to use saturated boiling correlations since the theoretical and especially experimental investigations are very limited [19, p. 233].

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