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Void fraction

In this section we derive the void fraction model originally introduced by Zuber and Findlay. We will consequently call the model the Zuber-Findlay void fraction model. This model is chosen on the following grounds

• The model has found a wide acceptance in the nuclear industry.
• The model includes effects of non-uniform cross-sectional void distribution.

We introduce the local drift velocities, u_gj and $u_{\ell j}$ [ ${\mbox{m}}/{\mbox{s}}$], by the following expressions

$$u_{gj} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;u_g - j$$ (6.49)

and

$$u_{\ell j} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;u_\ell - j$$ (6.50)

These velocities express the phase velocities relatively to the center of volume velocity j (the total volumetric flux, (6.3)).

The local drift fluxes, j_gj and $j_{\ell j}$ [ ${\mbox{m}}/{\mbox{s}}$], are defined by

$$j_{gj} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptsc... ...e\triangle$}}\;\alpha u_{gj} = \alpha (u_g-j) = j_g - \alpha j$$ (6.51)

$$j_{\ell j} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scri... ...lpha)u_{\ell j} = (1-\alpha)(u_\ell -j) = j_\ell -(1-\alpha)j$$ (6.52)

and express the volumetric fluxes of vapor and liquid, respectively, through a cross-sectional plane moving with the velocity equal to the center of volume velocity j.

Now, using these newly introduced quantities we can derive the Zuber-Findlay void fraction model. Using the identity

$$j_g \equiv \alpha j + \alpha(u_g-j)$$ (6.53)

and averaging over the cross-sectional area (see (6.7)) we obtain

$$\mbox{$<\!{j_g}\!>$} = \mbox{$<\!{\alpha j}\!>$} + \mbox{$<\!{\alpha(u_g-j)}\!>$}$$ (6.54)

The fundamental assumption is that the local quantities $\alpha$ and j may vary across the cross section such that, in general,

$$\mbox{$<\!{\alpha j}\!>$} \ne \mbox{$<\!{\alpha}\!>$}\mbox{$<\!{j}\!>$}$$ (6.55)

Since we are not interested in the local variation we define the correlation coefficient

$$C_0 \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscrip... ...<\!{\alpha j}\!>$}}{\mbox{$<\!{\alpha}\!>$}\mbox{$<\!{j}\!>$}}$$ (6.56)

which is called the distribution parameter^6.8 [--] and a new velocity

$$\overline{u}{}_{gj} \;\hbox{$=$\kern-0.68em\raise1.1ex \hb... ...\frac{\mbox{$<\!{\alpha(u_g-j)}\!>$}}{\mbox{$<\!{\alpha}\!>$}}$$ (6.57)

which is called the drift velocity [ ${\mbox{m}}/{\mbox{s}}$]. Physically the distribution parameter accounts for the non-uniform distribution of void and flow whereas the drift velocity accounts for the local slip between the two phases and is closely related to the terminal rise velocity of the vapor phase through a stagnant liquid.

Introducing the correlation coefficients (6.66) and (6.67) in the averaged equation (6.64) yields

$$\mbox{$<\!{j_g}\!>$} = C_0 \mbox{$<\!{\alpha}\!>$}\mbox{$<\!{j}\!>$} + \overline{u}{}_{gj} \mbox{$<\!{\alpha}\!>$}$$ (6.58)

or

$$\mbox{$<\!{\alpha}\!>$} = \frac{\mbox{$<\!{j_g}\!>$}}{C_0\mbox{$<\!{j}\!>$}+\overline{u}{}_{gj}}$$ (6.59)

For practical calculations it is useful to express $\mbox{$<\!{\alpha}\!>$}$ in terms of the averaged flow quality $\mbox{$<\!{x}\!>$}$. According to the last expression in (6.21) we have

$$\mbox{$<\!{j_g}\!>$} = \frac{1}{\rho_g} \mbox{$<\!{G}\!>$}\mbox{$<\!{x}\!>$}$$ (6.60)

and

$$\mbox{$<\!{j_\ell}\!>$} = \frac{1}{\rho_\ell} \mbox{$<\!{G}\!>$}(1-\mbox{$<\!{x}\!>$})$$ (6.61)

Inserting these quantities into (6.69) we obtain

$$\mbox{$<\!{\alpha}\!>$} = \frac{ \frac{1}{\rho_g} \mbox{$<... ...\mbox{$<\!{x}\!>$}}{\rho_\ell} \right] + \overline{u}{}_{gj}}$$ (6.62)

and dividing by $\mbox{$<\!{G}\!>$}/\rho_g$ we obtain the Zuber-Findlay void fraction model

$$\mbox{$<\!{\alpha}\!>$} = \frac{\mbox{$<\!{x}\!>$}}{ C_0\le... ...t] + \frac{ \rho_g \overline{u}{}_{gj}}{\mbox{$<\!{G}\!>$}} }$$ (6.63)

Upon comparing (6.73) with the general void fraction expression (6.24) we note that the slip implicitly given in the Zuber-Findlay void fraction model, $S_{\mbox{\protect\scriptsize ZF}}$, is

$$S_{\mbox{\protect\scriptsize ZF}} = C_0 + \frac{\mbox{$<\!{... ...verline{u}{}_{gj}}{ (1-\mbox{$<\!{x}\!>$})\mbox{$<\!{G}\!>$}}$$ (6.64)

In order to use the Zuber-Findlay void fraction model we have to specify the quantities C₀ and $\overline{u}{}_{gj}$ through external constitutive relations.

We find a myriad of different correlations for C₀ and $\overline{u}{}_{gj}$ in the literature. One of the proven set of correlations for normal BWR-operation, ie co-current upwards flow, consists of the one by Dix [18, p. 207] for C₀ which states

$$C_0 = \mbox{$<\!{\beta}\!>$}\left( 1 + \left[ \frac{1}{\mbox{$<\!{\beta}\!>$}} - 1 \right]^b \right)$$ (6.65)

where the exponent b is defined by

$$b \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\left[ \frac{\rho_g}{\rho_\ell} \right]$$ (6.66)

and $\mbox{$<\!{\beta}\!>$}$ is defined by

$$\mbox{$<\!{\beta}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex ... ...<\!{x}\!>$}+ \frac{\rho_g}{\rho_\ell} (1-\mbox{$<\!{x}\!>$})}$$ (6.67)

The correlation for $\overline{u}{}_{gj}$ given by Lahey and Moody [18] states

$$\overline{u}{}_{gj} = 2.9 \left[ \frac{ (\rho_\ell-\rho_g)\sigma g}{\rho_\ell^2}\right]^{1/4}$$ (6.68)

External constitutive relations exist which are applicable to every flow, ie for low-flow or counter-current flow as well as normal BWR-operation conditions [30]. However, in general use of these relations require iterative solutions and consequently are more impractical to use [31, p. 6-1].

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