# Concepts and Notations for Two-Phase Flow

Since liquid-vapor two-phase flow with phase change will be the focus in this section, one phase is designated liquid and the other as vapor. The vapor or holdup void fraction represents the time-averaged volumetric fraction of vapor in a two-phase mixture, i.e.,

$\alpha =\frac{\int_{{{V}_{v}}}{dV}}{\int_{V}{dV}}=\frac{{{V}_{v}}}{{{V}_{v}}+{{V}_{\ell }}}\qquad\qquad(1)$

It should be noted that $\alpha ={{\varepsilon }_{v}}$.

For liquid-vapor two-phase flow in a pipe, the volume of each phase consists of the cross-sectional area of the flow tube covered by that phase times a differential length element. Since the differential length is common in both phases, the void fraction can be considered the time-averaged area fraction:

$\alpha =\frac{\Delta z\int_{{{A}_{v}}}{dA}}{\Delta z\int_{A}{dA}}=\frac{{{A}_{v}}}{{{A}_{v}}+{{A}_{\ell }}}\qquad\qquad(2)$

The density of the two-phase mixture is defined as the average mass per unit volume:

$\rho =\frac{\int_{{{V}_{\ell }}}{{{\rho }_{\ell }}dV}+\int_{{{V}_{v}}}{{{\rho }_{v}}dV}}{{{V}_{\ell }}+{{V}_{v}}}\qquad\qquad(3)$

If the density of each phase is constant, the following relationship can be obtained by employing eq. (1):

$\rho =(1-\alpha ){{\rho }_{\ell }}+\alpha {{\rho }_{v}}\qquad\qquad(4)$

The phase velocity is the mean velocity of each phase and is defined as the volumetric flow rate of that phase through its cross-sectional area. This area-averaged velocity should also be considered as time-averaged velocity to eliminate random fluctuations, i.e.,

${{\left\langle {{w}_{\ell }} \right\rangle }^{\ell }}=\frac{{{Q}_{\ell }}}{{{A}_{\ell }}}\qquad\qquad(5)$
${{\left\langle {{w}_{v}} \right\rangle }^{v}}=\frac{{{Q}_{v}}}{{{A}_{v}}}\qquad\qquad(6)$

where ${{Q}_{\ell }}$ and Qv are the volumetric flow rates of the liquid and vapor phases. The phase velocities defined in eqs. (5) and (6) are intrinsic-averaged velocity and will be represented by ${{w}_{\ell }}\text{ and }{{w}_{v}}$ in future references for ease of notations. The superficial velocity, or volumetric flux, of each phase is defined as the volumetric flow rate of that phase divided by the total cross-sectional flow area in question, i.e.,

${{j}_{\ell }}=\frac{{{Q}_{\ell }}}{{{A}_{\ell }}+{{A}_{v}}}\qquad\qquad(7)$
${{j}_{v}}=\frac{{{Q}_{v}}}{{{A}_{\ell }}+{{A}_{v}}}\qquad\qquad(8)$

The superficial velocity defined in eqs. (7) and (8) is equal to the velocity that each phase would have if it were to flow alone in the channel at its specified mass flow rate. The relationship between the superficial velocity and the phase velocity of the liquid phase can be obtained by combining eqs. (5) and (7) and using the definition of the void fraction in eq. (2), i.e.,

${{j}_{\ell }}={{w}_{\ell }}(1-\alpha )\qquad\qquad(9)$

The relationship between the superficial velocity and the phase velocity of the vapor phase can be obtained in a similar manner:

${{j}_{v}}={{w}_{v}}\alpha \qquad\qquad(10)$

Since the void fraction in two-phase flow always ranges between zero and one, it can be seen from eqs. (9) and (10) that the phase velocities for each phase are greater than the corresponding volumetric flux. The total volumetric flux of the two-phase mixture, j, can be expressed as

$j={{j}_{\ell }}+{{j}_{v}}\qquad\qquad(11)$

The volumetric flow fraction, β, is defined as the volumetric flow rate of the vapor divided by the total volumetric flow rate:

$\beta =\frac{{{j}_{v}}}{{{j}_{\ell }}+{{j}_{v}}}=\frac{{{Q}_{v}}}{{{Q}_{\ell }}+{{Q}_{v}}}\qquad\qquad(12)$

which is a particularly convenient quantity for the experimentalist since the volumetric flow rates can be readily calculated or measured.

It has been observed experimentally that the one-dimensional phase velocity of the vapor is normally greater than the one-dimensional phase velocity of the liquid in flowing two-phase systems. A slip ratio is defined as the ratio of the phase velocity of the vapor to that of the liquid, i.e.,

$S=\frac{{{w}_{v}}}{{{w}_{\ell }}}\qquad\qquad(13)$

For homogeneous flow, the velocities for the liquid and vapor phases are identical and therefore the slip ratio S = 1, i.e.,

${{w}_{v}}={{w}_{\ell }}\qquad\qquad(14)$

Substituting eqs. (9) and (10) into eq. (14) yields

$\frac{{{j}_{v}}}{\alpha }=\frac{{{j}_{\ell }}}{1-\alpha }\qquad\qquad(15)$

The void fraction for homogeneous flow is then

$\alpha =\frac{{{j}_{v}}}{{{j}_{\ell }}+{{j}_{v}}}\qquad\qquad(16)$

which indicates that α = Β for homogeneous flow.

The quality is defined as the vapor (gas) content of the two-phase flow. It is a parameter that identifies the dryness or the wetness of the two-phase system.

$x=\frac{{{{\dot{m}}}_{v}}}{{{{\dot{m}}}_{\ell }}+{{{\dot{m}}}_{v}}}\qquad\qquad(17)$

where $\dot{m}$ is the mass flow rate of the liquid or vapor (gas) phase, obtained by

${{\dot{m}}_{\ell }}={{\rho }_{\ell }}{{w}_{\ell }}{{A}_{\ell }}\qquad\qquad(18)$
${{\dot{m}}_{v}}={{\rho }_{v}}{{w}_{v}}{{A}_{v}}\qquad\qquad(19)$

The superficial mass flux or mass velocity of liquid and vapor is defined as

${{G}_{\ell }}=\frac{{{{\dot{m}}}_{\ell }}}{A}={{\rho }_{\ell }}{{j}_{\ell }}={{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )\qquad\qquad(20)$
${{G}_{v}}=\frac{{{{\dot{m}}}_{v}}}{A}={{\rho }_{v}}{{j}_{v}}={{\rho }_{v}}{{w}_{v}}\alpha \qquad\qquad(21)$

The total mass flux in two-phase flow is then

$\dot{{m}''}={{G}_{\ell }}+{{G}_{v}}={{\rho }_{v}}{{j}_{v}}+{{\rho }_{\ell }}{{j}_{\ell }}={{\rho }_{v}}{{w}_{v}}\alpha +{{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )\qquad\qquad(22)$

The quality, x, can also be defined in terms of mass flux as

$x=\frac{{{G}_{v}}}{{\dot{{m}''}}}\qquad\qquad(23)$

Substituting eq. (23) into eqs. (20) and (23), one can relate the superficial velocity to the quality as

${{j}_{\ell }}=\frac{\dot{{m}''}(1-x)}{{{\rho }_{\ell }}}\qquad\qquad(24)$
${{j}_{v}}=\frac{\dot{{m}''}x}{{{\rho }_{v}}}\qquad\qquad(25)$

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.