13.6.1: Multi–Phase Averaged Variables Definitions

The total mass flow rate through the tube is the sum of the mass flow rates of the two phases

\[ \dot{m} = \dot{m}_G + \dot{m}_L
\label{phase:eq:massFrac}
\]

It is common to define the mass velocity instead of the regular velocity because the "regular'' velocity changes along the length of the pipe. The gas mass velocity is

\[ G_G = \dfrac{\dot{m}_G}{A}
\label{phase:eq:GmasV}
\] Where \(A\) is the entire area of the tube. It has to be noted that this mass velocity does not exist in reality. The liquid mass velocity is

\[ G_L = \dfrac{\dot{m}_L}{A}
\label{phase:eq:LmasV}
\] The mass flow of the tube is then

\[ G = \dfrac{\dot{m}}{A}
\label{phase:eq:TmasV}
\] It has to be emphasized that this mass velocity is the actual velocity. The volumetric flow rate is not constant (since the density is not constant) along the flow rate and it is defined as

\[ Q_G = \dfrac{G_G}{\rho_G} = U_{sG}
\label{phase:eq:GqRate}
\] and for the liquid

\[ Q_L = \dfrac{G_L}{\rho_L}
\label{phase:eq:LqRate}
\] For liquid with very high bulk modulus (almost constant density), the volumetric flow rate can be considered as constant. The total volumetric volume vary along the tube length and is

\[ Q = Q_L + Q_G
\label{phase:eq:TqRate}
\] Ratio of the gas flow rate to the total flow rate is called the 'quality' or the "dryness fraction'' and is given by

\[ X = \dfrac{\dot{m}_G}{\dot{m}} = \dfrac{G_G}{G}
\label{phase:eq:X}
\] In a similar fashion, the value of \((1 - X)\) is referred to as the "wetness fraction.'' The last two factions remain constant along the tube length as long the gas and liquid masses remain constant. The ratio of the gas flow cross sectional area to the total cross sectional area is referred as the void fraction and defined as

\[ \alpha = \dfrac{A_G}{A}
\label{phase:eq:alpha}
\] This fraction is vary along tube length since the gas density is not constant along the tube length. The liquid fraction or liquid holdup is

\[ L_H = 1 - \alpha = \dfrac{A_L}{A}
\label{phase:eq:1ninusAlpha}
\] It must be noted that Liquid holdup, \(L_H\) is not constant for the same reasons the void fraction is not constant. The actual velocities depend on the other phase since the actual cross section the phase flows is dependent on the other phase. Thus, a superficial velocity is commonly defined in which if only one phase is using the entire tube. The gas superficial velocity is therefore defined as

\[ U_{sG} = \dfrac{G_G}{\rho_G} =
\dfrac{X\,\dot{m}\,}{\rho_G\,A} = Q_G
\label{phase:eq:Usg}
\] The liquid superficial velocity is

\[ U_{sL} = \dfrac{G_L}{\rho_L} =
\dfrac{\left(1-X\right)\,\dot{m}\,}{\rho_L\,A} = Q_L
\label{phase:eq:Usl}
\] Since \(U_{sL} = Q_L\) and similarly for the gas then

\[ U_m = U_{sG} + U_{sL}
\label{phase:eq:QlQg}
\] Where \(U_m\) is the averaged velocity. It can be noticed that \(U_m\) is not constant along the tube. The average superficial velocity of the gas and liquid are different. Thus, the ratio of these velocities is referred to as the slip velocity and is defined as the following

\[ SLP = \dfrac{U_G}{U_L}
\label{phase:eq:skip}
\] Slip ratio is usually greater than unity. Also, it can be noted that the slip velocity is not constant along the tube. For the same velocity of phases (\(SLP=1\)), the mixture density is defined as

\[ \rho_m = \alpha\, \rho_G + (1-\alpha)\, \rho_L
\label{phase:eq:RhoMix}
\] This density represents the density taken at the "frozen'' cross section (assume the volume is the cross section times infinitesimal thickness of \(dx\)). The average density of the material flowing in the tube can be evaluated by looking at the definition of density. The density of any material is defined as \(\rho = m/V\) and thus, for the flowing material it is

\[ \rho = \dfrac{\dot{m} } {Q}
\label{phase:eq:rhoAvreI}
\] Where \(Q\) is the volumetric flow rate. Substituting equations (1) and (7) into equation (16) results in

\[ \rho_{average} = \dfrac {\overbrace{X\,\dot{m}}^{\dot{m}_G} + \overbrace{(1-X)\,\dot{m} }^{\dot{m}_L}}
{Q_G + Q_L} \, = \,
\dfrac{X\,\dot{m} + (1-X)\,\dot{m} }
{\underbrace{\dfrac{X\,\ddot{m}}{\rho_G }} _{Q_G} + \underbrace{\dfrac{(1-X)\,\dot{m}} {\rho_L}} _{Q_L} }
\label{phase:eq:rhoAvreM}
\]
Equation (17) can be simplified by canceling the \(\dot{m}\) and noticing the \((1−X) +X = 1\) to become

Averaged Density

[
\label {phase:eq:rhoAvreF}
\rho_{\text{average}} = \dfrac{1}{\dfrac{X}{\rho_G} + \dfrac{(1-X)}{\rho_L} }
\]

The average specific volume of the flow is then

\[ v_{\text{average}} = \dfrac{1}{\rho_{\text{average}}} =
\dfrac{X}{\rho_G} + \dfrac{(1-X)}{\rho_L} =
X\,v_G + (1-X)\,v_L
\label{phase:eq:spesificVolume}
\]
The relationship between \(X\) and \(\alpha\) is

\[ X = \dfrac{\dot{m}_G}{\dot{m}_G + \dot{m}_L } =
\dfrac{\rho_G\, U_G\, \overbrace{A\, \alpha}^{A_G} }
{\rho_L U_L \underbrace{A (1-\alpha) }_{A_L} +
\rho_G\, U_G\, A\, \alpha } =
\dfrac{\rho_G\, U_G\, \alpha}
{\rho_L\, U_L\, (1-\alpha) + \rho_G\, U_G\, \alpha }
\label{phase:eq:X-alpha}

\[ If the slip is one \(SLP=1\), thus equation (20) becomes

\[ X = \dfrac{\rho_G\, \, \alpha} {\rho_L\,(1-\alpha) + \rho_G\, \alpha }
\label{phase:eq:X-alphaUG=UL}
\]

Contributors and Attributions

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.