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Zuber-Findlay notation

In this section we introduce some general concepts used in connection with one-dimensional two-phase flows. We will focus our attention on the notation for cross-sectional averages introduced by Zuber and Findlay [15].

For the moment we state that quantities without the averaging symbols $\mbox{$<\!{\;\;}\!>$}$ are local time averaged quantities.

The local time averaged volumetric fluxes j_g and $j_\ell$ measured in units of [m/s] are defined in terms of the local time averaged void fraction $\alpha$ and phase velocities u_g and $u_\ell$ as

$$j_\ell \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;(1-\alpha) u_\ell$$ (6.1)

$$j_g \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\alpha u_g$$ (6.2)

The indices $\ell$ and g indicate the liquid and vapor phases respectively. The total local volumetric flux j in units of [m/s] is defined as

$$j \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;j_\ell + j_g$$ (6.3)

The volumetric flow rate of liquid and vapor, $Q_\ell$ and Q_g [ ${\mbox{m}}^3/{\mbox{s}}$], may be defined in terms of the volumetric fluxes as follows

$$Q_\ell \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\int\limits_{A_c} j_\ell dA$$ (6.4)

$$Q_g \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\int\limits_{A_c} j_g dA$$ (6.5)

where A_c is the cross-sectional area of the channel in [${\mbox{m}}^2$]. The total volumetric flow rate, Q, is defined by

$$Q \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;Q_\ell + Q_g$$ (6.6)

At this point we introduce the cross-sectional average notation of Zuber-Findlay by stating the following definitions

$$\mbox{$<\!{\zeta}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex ... ...iptstyle\triangle$}}\;\frac{1}{A_c} \int\limits_{A_c} \zeta dA$$ (6.7)

$$\mbox{$<\!{\zeta_\ell}\!>$}_\ell \;\hbox{$=$\kern-0.68em\ra... ...ox{$<\!{(1-\alpha)\zeta_\ell}\!>$}}{1-\mbox{$<\!{\alpha}\!>$}}$$ (6.8)

$$\mbox{$<\!{\zeta_g}\!>$}_g \;\hbox{$=$\kern-0.68em\raise1.1... ...\frac{\mbox{$<\!{\alpha\zeta_g}\!>$}}{\mbox{$<\!{\alpha}\!>$}}$$ (6.9)

These definitions enable us to redefine the volumetric flow rates as

$$Q_\ell = A_c \mbox{$<\!{j_\ell}\!>$} = A_c \mbox{$<\!{(1-\a... ... A_c \mbox{$<\!{u_\ell}\!>$}_\ell (1-\mbox{$<\!{\alpha}\!>$})$$ (6.10)

$$Q_g = A_c \mbox{$<\!{j_g}\!>$} = A_c \mbox{$<\!{\alpha u_g}\!>$} = A_c \mbox{$<\!{u_g}\!>$}_g \mbox{$<\!{\alpha}\!>$}$$ (6.11)

$$Q = A_c ( \mbox{$<\!{j_\ell}\!>$} + \mbox{$<\!{j_g}\!>$} ) ... ...pha}\!>$}) + \mbox{$<\!{u_g}\!>$}_g \mbox{$<\!{\alpha}\!>$} )$$ (6.12)

Assuming uniform densities of liquid and vapor across the flow area, ie

$$\rho_\ell = \mbox{$<\!{\rho_\ell}\!>$}_\ell$$ (6.13)

and

$$\rho_g = \mbox{$<\!{\rho_g}\!>$}_g$$ (6.14)

we can express the mass flow rates of liquid, vapor and mixture, $\mbox{$\dot{m}$}_\ell$, $\mbox{$\dot{m}$}_g$ and $\mbox{$\dot{m}$}$ in [ ${\mbox{kg}}/{\mbox{s}}$], as

$$\mbox{$\dot{m}$}_\ell \;\hbox{$=$\kern-0.68em\raise1.1ex \... ..._\ell \mbox{$<\!{u_\ell}\!>$}_\ell (1-\mbox{$<\!{\alpha}\!>$})$$ (6.15)

$$\mbox{$\dot{m}$}_g \;\hbox{$=$\kern-0.68em\raise1.1ex \hbo... ...} = A_c \rho_g \mbox{$<\!{u_g}\!>$}_g \mbox{$<\!{\alpha}\!>$}$$ (6.16)

$$\mbox{$\dot{m}$}\;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$... ...>$}) + \rho_g \mbox{$<\!{u_g}\!>$}_g \mbox{$<\!{\alpha}\!>$} )$$ (6.17)

The average mass fluxes, $\mbox{$<\!{G_\ell}\!>$}$, $\mbox{$<\!{G_g}\!>$}$ and $\mbox{$<\!{G}\!>$}$ in [ ${\mbox{kg}}/({\mbox{m}}^2\cdot{\mbox{s}})$], are defined by

$$\mbox{$<\!{G_\ell}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex ... ...\ell \mbox{$<\!{u_\ell}\!>$}_\ell (1-\mbox{$<\!{\alpha}\!>$})$$ (6.18)

$$\mbox{$<\!{G_g}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex \h... ...\!>$} = \rho_g \mbox{$<\!{u_g}\!>$}_g \mbox{$<\!{\alpha}\!>$}$$ (6.19)

$$\mbox{$<\!{G}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbo... ...le\triangle$}}\;\mbox{$<\!{G_\ell}\!>$} + \mbox{$<\!{G_g}\!>$}$$ (6.20)

The average flow quality, $\mbox{$<\!{x}\!>$}$ in [--], is defined by

$$\mbox{$<\!{x}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbo... ...!>$}} = \frac{\rho_g\mbox{$<\!{j_g}\!>$}}{\mbox{$<\!{G}\!>$}}$$ (6.21)

We are now going to derive the fundamental void fraction-quality relation of two-phase flow by noting that

$$\frac{\mbox{$<\!{j_g}\!>$}}{\mbox{$<\!{j_\ell}\!>$}} = \fra... ...ll} \frac{\mbox{$<\!{\alpha}\!>$}}{1-\mbox{$<\!{\alpha}\!>$}}$$ (6.22)

and

$$\frac{\mbox{$<\!{G_g}\!>$}}{\mbox{$<\!{G_\ell}\!>$}} = \fra... ...}{1-\mbox{$<\!{x}\!>$}} = \frac{\rho_g Q_g}{\rho_\ell Q_\ell}$$ (6.23)

Combining these two and isolating $\mbox{$<\!{\alpha}\!>$}$ yields

$$\mbox{$<\!{\alpha}\!>$} = \frac{\mbox{$<\!{x}\!>$}}{ \mbox{... ...left[ \frac{\rho_g}{\rho_\ell}\right] (1-\mbox{$<\!{x}\!>$}) }$$ (6.24)

where the slip ratio^6.2, S, is defined by

$$S \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscripts... ...}\;\frac{\mbox{$<\!{u_g}\!>$}_g}{\mbox{$<\!{u_\ell}\!>$}_\ell}$$ (6.25)

Having treated the basic notation in regard to one-dimensional two-phase flow we move to a derivation of the general conservation equations of one-dimensional separated flow.

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