# Frictional Pressure Drop Models for Two-Phase Flow

The pressure drop is very important for the design of two-phase devices, because it dictates the pump power that is required to drive the flow. As shown in

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

from Separated Flow Model for Two-Phase Flow, the pressure drop includes three parts: frictional, accelerational, and gravitational pressure drops. The accelerational and gravitational pressure drop terms the above two equations can be calculated based on physical properties and the void fraction. The frictional pressure drop for steady-state two-phase flow in a circular tube can be calculated by

$-\frac{d{{p}_{F}}}{dz}=\frac{4{{\tau }_{w}}}{D}\qquad\qquad(1)$

The frictional pressure drop for two-phase flow must be accounted for by using an empirical correlation, which can be based on either the homogeneous or separated flow model. While the former assumes that both liquid and gas move at the same velocity (slip ratio S is 1) and is also referred to as zero-slip model, the latter allows liquid and gas to move at different velocities.

## Correlations Based on the Homogeneous Model

Since the homogeneous model treats the two phases in the mixture as a pseudo-single-phase flow, the frictional pressure gradient can be calculated by (Beattie and Whalley, 1982)

$-\frac{d{{p}_{F}}}{dz}=\frac{4{{\tau }_{w}}}{D}=\frac{2{{f}_{{}}}{{{\dot{{m}''}}}^{2}}}{D\rho }\qquad\qquad(2)$

where f is the two-phase friction factor, which can be determined by empirical correlation for single phase flow. For many two-phase homogeneous flow considerations, particularly in the annular and bubble regimes, the following equation is proposed to determine the homogeneous friction factor, f:

$\frac{1}{\sqrt{f}}=3.48-4{{\log }_{10}}\left[ \frac{2\kappa }{D}+\frac{9.35}{\left( \operatorname{Re}\sqrt{f} \right)} \right]\qquad\qquad(3)$

where the term κ / D represents the surface roughness/diameter ratio. The Reynolds number can be calculated based on a homogeneous flow using

$\operatorname{Re}=\frac{\dot{{m}''}D}{\mu }\qquad\qquad(4)$

The homogeneous viscosity of the two-phase mixture is obtained by

$\frac{1}{\mu }=\frac{x}{{{\mu }_{v}}}+\frac{1-x}{{{\mu }_{\ell }}}\qquad\qquad(5)$

## References

Beattie, D.R.H., and Whalley, P.B., 1982, “A Simple Two-Phase Frictional Pressure Drop Calculation Method,” International Journal of Multiphase Flow, Vol. 8, pp. 83-87.

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