# Separated Flow Model for Two-Phase Flow

Compared with the homogeneous model, the separated flow model has been used more widely, because it provides a better prediction of flow behavior with a manageable level of complexity. The separated flow model assumes that each phase displays different properties and flows at different velocities. It is a simpler version of the two-fluid model discussed in Chapter 4 because it is assumed that only velocities differ between the two phases, while the conservation equations are written only for the combined flow. In addition, the pressure across any given cross-section of a channel carrying a multiphase flow is assumed to be the same for both phases (Hewitt, 1998).

The mass flow rates of the liquid and vapor phases are obtained from (see Concepts and Notations for Two-Phase Flow):

${{\dot{m}}_{\ell }}={{\rho }_{\ell }}{{w}_{\ell }}{{A}_{\ell }}$
${{\dot{m}}_{v}}={{\rho }_{v}}{{w}_{v}}{{A}_{v}}$

The cross-sectional area of the channel occupied by liquid and vapor can be obtained by rearranging these two equations, i.e.,

${{A}_{\ell }}={{\dot{m}}_{\ell }}/({{\rho }_{\ell }}{{w}_{\ell }})\qquad\qquad(1)$
${{A}_{v}}={{\dot{m}}_{v}}/({{\rho }_{v}}{{w}_{v}})\qquad\qquad(2)$

The void fraction of the two-phase flow can be obtained by substituting eqs. (1) and (2) into the following equation

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

and the result is:

$\alpha =\frac{{{{\dot{m}}}_{v}}/({{\rho }_{v}}{{w}_{v}})}{{{{\dot{m}}}_{v}}/({{\rho }_{v}}{{w}_{v}})+{{{\dot{m}}}_{\ell }}/({{\rho }_{\ell }}{{w}_{\ell }})}\qquad\qquad(3)$

Considering the definition of quality

$x=\frac{{{{\dot{m}}}_{v}}}{{{{\dot{m}}}_{\ell }}+{{{\dot{m}}}_{v}}}$

the void fraction becomes

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

It can be seen that when the vapor velocity, wv, is greater than the liquid velocity, ${{w}_{\ell }}$ (which is often the case for vertical upward and horizontal cocurrent flow) the homogeneous model overpredicts the void fraction. On the other hand, when the vapor velocity, wv, is lower than the liquid velocity, ${{w}_{\ell }}$, as occurs in vertical downward flow, the homogeneous model underpredicts the void fraction.

The governing equations for steady-state two-phase flow in a channel for separated flow model have been given in Example 4.6. More generalized governing equations that are applicable to transient flow are presented here. The continuity equations for the vapor and liquid phases are, respectively

$\frac{\partial }{\partial t}({{\rho }_{v}}\alpha A)+\frac{\partial }{\partial z}({{\rho }_{v}}{{w}_{v}}\alpha A)={{{\dot{m}}'''}_{v}}\qquad\qquad(5)$
$\frac{\partial }{\partial t}[{{\rho }_{\ell }}(1-\alpha )A]+\frac{\partial }{\partial z}[{{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )A]={{{\dot{m}}'''}_{\ell }}\qquad\qquad(6)$

where ${{{\dot{m}}'''}_{v}}$ and ${{{\dot{m}}'''}_{\ell }}$ are the mass production rates of vapor and liquid due to phase change in the two-phase flow system. Conservation of mass requires that the summation of ${{{\dot{m}}'''}_{v}}$ and ${{{\dot{m}}'''}_{\ell }}$ equal zero, so the continuity equation for the two-phase system can be obtained by adding eqs. (5) and (6) together, i.e.,

$\frac{\partial }{\partial t}(\rho A)+\frac{\partial }{\partial z}\left[ {{\rho }_{v}}{{w}_{v}}\alpha A+{{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )A \right]=0\qquad\qquad(7)$

where ρ is the from density of the two-phase mixture

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

Considering the definition of mass flux:

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

the continuity equation can be rewritten as

$\frac{\partial }{\partial t}(\rho A)+\frac{\partial }{\partial z}(\dot{{m}''}A)=0\qquad\qquad(8)$

The momentum equations for the vapor and liquid phases are

\begin{align} & \frac{\partial }{\partial t}\left( {{\rho }_{v}}{{w}_{v}}\alpha A \right)+\frac{\partial }{\partial z}\left( {{\rho }_{v}}w_{v}^{2}\alpha A \right) \\ & =-\alpha A\frac{\partial p}{\partial z}-g{{\rho }_{v}}\alpha A\cos \theta -{{\tau }_{w,v}}{{P}_{w,v}}+{{F}_{\ell ,v}} \\ \end{align}\qquad\qquad(9)
\begin{align} & \frac{\partial }{\partial t}\left[ {{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )A \right]+\frac{\partial }{\partial z}\left[ {{\rho }_{\ell }}w_{\ell }^{2}(1-\alpha )A \right] \\ & =-(1-\alpha )A\frac{\partial p}{\partial z}-g{{\rho }_{\ell }}(1-\alpha )A\cos \theta -{{\tau }_{w,\ell }}{{P}_{w,\ell }}+{{F}_{v,\ell }} \\ \end{align}\qquad\qquad(10)

where τw,v and ${{\tau }_{w,\ell }}$ are shear stresses at the wall for vapor and liquid, respectively. Pw,v and ${{P}_{w,\ell }}$ are the portion of the perimeter that is in contact with vapor and liquid, respectively. ${{F}_{\ell ,v}}$ and ${{F}_{v,\ell }}$ are the interactive forces between the liquid and vapor phases, which satisfy ${{F}_{\ell ,v}}=-{{F}_{v,\ell }}$ as required by Newton’s third law. Assuming constant wall shear stress τw around the periphery of the channel, the momentum equation of the two-phase system can be obtained by adding eqs. (9) and (10), i.e.,

$A\frac{\partial \dot{{m}''}}{\partial t}+\frac{\partial }{\partial z}\left[ {{\rho }_{v}}w_{v}^{2}\alpha A+{{\rho }_{\ell }}w_{\ell }^{2}(1-\alpha )A \right]=-A\frac{\partial p}{\partial z}-g\rho A\cos \theta -{{\tau }_{w}}P\qquad\qquad(11)$

Substituting superficial mass fluxes:

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

into eq. (11), the momentum equation becomes

$A\frac{\partial \dot{{m}''}}{\partial t}+\frac{\partial }{\partial z}\left\{ A\left[ \frac{G_{v}^{2}}{{{\rho }_{v}}\alpha }+\frac{G_{\ell }^{2}}{{{\rho }_{\ell }}(1-\alpha )} \right] \right\}=-A\frac{\partial p}{\partial z}-g\rho A\cos \theta -{{\tau }_{w}}P\qquad\qquad(12)$

Considering that the mass flux of the vapor phase can be expressed as

${{G}_{v}}=x\dot{{m}''}$ [eq. $x=\frac{{{G}_{v}}}{{\dot{{m}''}}}$

and the [Concepts and Notations for Two-Phase Flow|mass flux of the liquid]] is

${{G}_{\ell }}=\dot{{m}''}-{{G}_{v}}=(1-x)\dot{{m}''}$

the momentum equation in the separated flow model becomes

$\frac{\partial \dot{{m}''}}{\partial t}+\frac{1}{A}\frac{\partial }{\partial z}\left\{ \dot{{m}''}A\left[ \frac{{{x}^{2}}}{{{\rho }_{v}}\alpha }+\frac{{{(1-x)}^{2}}}{{{\rho }_{\ell }}(1-\alpha )} \right] \right\}=-\frac{\partial p}{\partial z}-g\rho \cos \theta -\frac{{{\tau }_{w}}P}{A}\qquad\qquad(13)$

The energy equations for the vapor and liquid phases are

$\frac{\partial }{\partial t}\left[ {{\rho }_{v}}\alpha A\left( {{h}_{v}}+\frac{w_{v}^{2}}{2}+gz\cos \theta \right) \right]+\frac{\partial }{\partial z}\left[ {{\rho }_{v}}{{w}_{v}}\alpha A\left( {{h}_{v}}+\frac{w_{v}^{2}}{2}+gz\cos \theta \right) \right]$
$={{P}_{v}}{{{q}''}_{w}}+{q}'''\alpha A+\alpha A\frac{\partial p}{\partial t}+{{{q}'''}_{\ell ,v}}\qquad\qquad(14)$
$\frac{\partial }{\partial t}\left[ {{\rho }_{\ell }}(1-\alpha )A\left( {{h}_{\ell }}+\frac{w_{\ell }^{2}}{2}+gz\cos \theta \right) \right]+\frac{\partial }{\partial z}\left[ {{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )A\left( {{h}_{\ell }}+\frac{w_{\ell }^{2}}{2}+gz\cos \theta \right) \right]$
$={{P}_{\ell }}{{{q}''}_{w}}+{q}'''(1-\alpha )A+(1-\alpha )A\frac{\partial p}{\partial t}+{{{q}'''}_{v,\ell }}\qquad\qquad(15)$

where the heat flux at the surface of the channel, q''w, is assumed to be the same for the perimeter of the channel, whether it is in contact with liquid or vapor. ${{{q}'''}_{\ell ,v}}$ and ${{{q}'''}_{v,\ell }}$ are the interphase heat transfers that satisfy ${{{q}'''}_{\ell ,v}}=-{{{q}'''}_{v,\ell }}$. The energy equation for the two-phase mixture is then obtained by adding eqs. (14) and (15), i.e.,

\begin{align} & A\frac{\partial }{\partial t}\left[ {{\rho }_{v}}\alpha \left( {{h}_{v}}+\frac{w_{v}^{2}}{2}+gz\cos \theta \right)+{{\rho }_{\ell }}(1-\alpha )\left( {{h}_{\ell }}+\frac{w_{\ell }^{2}}{2}+gz\cos \theta \right) \right] \\ & +\frac{\partial }{\partial z}\left\{ A\left[ {{\rho }_{v}}{{w}_{v}}\alpha \left( {{h}_{v}}+\frac{w_{v}^{2}}{2}+gz\cos \theta \right)+{{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )\left( {{h}_{\ell }}+\frac{w_{\ell }^{2}}{2}+gz\cos \theta \right) \right] \right\} \\ & =P{{{{q}''}}_{w}}+{q}'''A+A\frac{\partial p}{\partial t} \\ \end{align}\qquad\qquad(16)

Substituting the following mass fluxes

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

into eq. (16), the energy equation becomes

\begin{align} & A\frac{\partial }{\partial t}\left[ {{\rho }_{v}}\alpha \left( {{h}_{v}}+\frac{G_{v}^{2}}{2\rho _{v}^{2}{{\alpha }^{2}}}+gz\cos \theta \right)+{{\rho }_{\ell }}(1-\alpha )\left( {{h}_{\ell }}+\frac{G_{\ell }^{2}}{2\rho _{\ell }^{2}{{(1-\alpha )}^{2}}}+gz\cos \theta \right) \right] \\ & +\frac{\partial }{\partial z}\left\{ A\left[ {{G}_{v}}\left( {{h}_{v}}+\frac{G_{v}^{2}}{2\rho _{v}^{2}{{\alpha }^{2}}}+gz\cos \theta \right)+{{G}_{\ell }}\left( {{h}_{\ell }}+\frac{G_{\ell }^{2}}{2\rho _{\ell }^{2}{{(1-\alpha )}^{2}}}+gz\cos \theta \right) \right] \right\} \\ & =P{{{{q}''}}_{w}}+{q}'''A+A\frac{\partial p}{\partial t} \\ \end{align}\qquad\qquad(17)

Since ${{G}_{v}}=x\dot{{m}''}$ and ${{G}_{\ell }}=\dot{{m}''}-{{G}_{v}}=(1-x)\dot{{m}''}$, eq. (17) can be modified as

\begin{align} & A\frac{\partial }{\partial t}\left[ {{\rho }_{v}}{{h}_{v}}\alpha +{{\rho }_{\ell }}{{h}_{\ell }}(1-\alpha ) \right]+\frac{\partial }{\partial z}\left\{ \dot{{m}''}A\left[ x{{h}_{v}}+(1-x){{h}_{\ell }} \right] \right\} \\ & =P{{{{q}''}}_{w}}+{q}'''A-\frac{\partial }{\partial z}\left\{ \frac{{{{\dot{{m}''}}}^{3}}A}{2}\left[ \frac{{{x}^{3}}}{\rho _{v}^{2}{{\alpha }^{2}}}+\frac{{{(1-x)}^{3}}}{\rho _{\ell }^{2}{{(1-\alpha )}^{2}}} \right] \right\}-\dot{{m}''}Ag\cos \theta \\ & -A\frac{\partial }{\partial t}\left\{ \frac{{{{\dot{{m}''}}}^{2}}}{2}\left[ \frac{{{x}^{2}}}{2{{\rho }_{v}}\alpha }+\frac{{{(1-x)}^{2}}}{2{{\rho }_{\ell }}(1-\alpha )} \right] \right\}+A\frac{\partial p}{\partial t} \\ \end{align}\qquad\qquad(18)

For steady-state two-phase flow in a circular tube with constant cross-sectional area (A = constant), the continuity equation (8) results in $d\dot{{m}''}/dz=0$. The momentum equation (7) reduces to

$-\frac{dp}{dz}=\frac{4{{\tau }_{w}}}{D}+\dot{{m}''}\frac{d}{dz}\left[ \frac{{{x}^{2}}}{{{\rho }_{v}}\alpha }+\frac{{{(1-x)}^{2}}}{{{\rho }_{\ell }}(1-\alpha )} \right]+g\rho \cos \theta \qquad\qquad(19)$

The three terms on the right-hand side of eq. (19) represent pressure drops due to friction, dpF / dz, acceleration, dpa / dz, and gravity, dpg / dz. To predict the pressure drop of two-phase flow using the separated flow model, empirical correlations for friction and void fraction are needed, as is evident from eq. (19).

The energy equation in the separated flow model for steady-state two-phase flow in a circular tube with constant cross-sectional area can be obtained by simplifying eq. (18), i.e.,

$\frac{d}{dz}\left[ x{{h}_{v}}+(1-x){{h}_{\ell }} \right]=\frac{4{{{{q}''}}_{w}}}{\dot{{m}''}D}+\frac{{{q}'''}}{{\dot{{m}''}}}-\frac{{{{\dot{{m}''}}}^{2}}}{2}\frac{d}{dz}\left[ \frac{{{x}^{3}}}{\rho _{v}^{2}{{\alpha }^{2}}}+\frac{{{(1-x)}^{3}}}{\rho _{\ell }^{2}{{(1-\alpha )}^{2}}} \right]-g\cos \theta \qquad\qquad(20)$

For a two-phase flow system with condensation or evaporation, where kinetic and potential energy, as well as internal heat generation, can be neglected, eq. (20) also reduces to

$\frac{dx}{dz}=\frac{4{{{{q}''}}_{w}}}{{\dot{m}}''D{{h}_{\ell v}}}$

from Homogeneous Flow Model for Two-Phase Flow. Therefore, the energy equations for the homogeneous model and the separated flow model are the same if kinetic and potential energy, as well as internal heat generation, can be neglected.

## References

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

Hewitt, G.F., 1998, “Multiphase Fluid Flow and Pressure Drop,” Heat Exchanger Design Handbook, Vol. 2, Begell House, New York, NY.