General conservation equations of separated flow

In this section we will derive a simple one-dimensional two-fluid model of fully separated flow in a fashion traditionally used in literature--see, for instance, [18] and [19].

The conservation equations can be derived phenomenologically using Figure 6.2. Considering the control volumes which hold the liquid phase and vapor phase separately and the following quantities at $z+\Delta z$

$\begin{eqnarray*} {\mbox{$\dot{m}$}_\ell + \frac{\partial}{\partial z} ( \mbox{... ...)) \Delta z &, & p + \frac{\partial}{\partial z} (p) \Delta z \end{eqnarray*}$

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

where

are the mass flow rates of liquid and vapor respectively [ ${\mbox{kg}}/{\mbox{s}}$ ].
>: are the vapor weighted mean phase velocities of liquid and vapor respectively [ ${\mbox{m}}/{\mbox{s}}$ ].
>: is the cross-sectional area of the channel [ ${\mbox{m}}^2$ ].
>: is the cross-sectional averaged void fraction [--].

and making several simplifying assumptions of which the most obvious are

.: We consider a vertical channel with the z-axis pointing upwards.
.: Uniform pressure at any cross section, ie $\mbox{$<\!{p_\ell}\!>$}_\ell = \mbox{$<\!{p_g}\!>$}_g = p_i = p$ .
.: Essentially uniform (or flat) profiles of among others u, $\rho$ and h.
.: Only the liquid phase is in contact with the channel wall.

the following conservation equations are obtained (see [18, chap. 5] and [19, chap. 2])

$\begin{figure} % latex2html id marker 24261\rule{\textwidth}{0.2mm} \rule{0cm}... ...ure}\hspace{1em}Schematic illustration of idealized separated flow.}\end{figure}$

Mass:

$\begin{displaymath} \frac{\partial}{\partial t}\left[A_c(1-\mbox{$<\!{\alpha}\!... ...\ell}\!>$}_\ell \rho_\ell\right] = \mbox{$\dot{m}$}_{g \ell}' \end{displaymath}$

(6.26)

and

$\begin{displaymath} \frac{\partial}{\partial t}\left[A_c\mbox{$<\!{\alpha}\!>$}... ...ox{$<\!{u_g}\!>$}_g\rho_g\right] = \mbox{$\dot{m}$}_{\ell g}' \end{displaymath}$

(6.27)

Momentum:

$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[ \mbox{$<\!{u_\ell}... ...x{$<\!{u_\ell}\!>$}_\ell + (1-\eta_m)\mbox{$<\!{u_g}\!>$}_g) \end{eqnarray*}$

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

and

$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[ \mbox{$<\!{u_g}\!>... ...x{$<\!{u_\ell}\!>$}_\ell + (1-\eta_m)\mbox{$<\!{u_g}\!>$}_g) \end{eqnarray*}$

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

Energy:

$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[(\mbox{$<\!{e_\ell}... ...ell}\!>$}_\ell ^2 ) + \mbox{$<\!{u_\ell}\!>$}_\ell \tau_i P_i \end{eqnarray*}$

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

and

$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[(\mbox{$<\!{e_g}\!>... ...\mbox{$<\!{u_g}\!>$}_g^2 ) - \mbox{$<\!{u_g}\!>$}_g\tau_i P_i \end{eqnarray*}$

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

In the above mentioned equations (6.27)-(6.32) the symbols have the following interpretation

time [s].
>: axial coordinate with the axis pointing upwards (opposite of the gravity vector, $\hspace{0.2ex}\underline{g}{}\hspace{0.15ex}$ ) [m].
>: cross-sectional area of the channel [ ${\mbox{m}}^2$ ].
>: cross-sectional averaged void fraction [--].
>: densities of the liquid and vapor respectively [ ${\mbox{kg}}/{\mbox{m}}^3$ ].
>: Void fraction weighted cross-sectional averages of the phase velocities of the liquid and vapor respectively [ ${\mbox{m}}/{\mbox{s}}$ ].
>: mass transfer rate per unit length from liquid to vapor and from vapor to liquid respectively at the liquid-vapor interface [ ${\mbox{kg}}/({\mbox{m}}\cdot{\mbox{s}})$ ].
>: pressure [Pa].
>: wall shear stress [ ${\mbox{N}}/{\mbox{m}}^2$ ].
>: friction perimeter, ie the perimeter on which the wall shear force acts [m].
>: shear stress at the liquid-vapor interface [ ${\mbox{N}}/{\mbox{m}}^2$ ].
>: interfacial perimeter, ie the perimeter on which the interfacial shear force acts [m].
>: so-called evaporative momentum transfer term [--]. According to [16, p.73-80] the choice $\eta_m = \frac{1}{2}$ ensures the desirable feature of isentropic flow.
>: wall heat flux [ ${\mbox{W}}/{\mbox{m}}^2$ ].
>: heated perimeter, ie the perimeter through which the wall heat flux enters the flow [m].
>: heat flux at the liquid-vapor interface [ ${\mbox{W}}/{\mbox{m}}^2$ ].
>: internal heat generations in the liquid and vapor phases respectively [ ${\mbox{W}}/{\mbox{m}}^3$ ].
>: Void fraction weighted cross-sectional averages of the internal energy of the liquid and vapor phases respectively [ ${\mbox{J}}/{\mbox{kg}}$ ].

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