3.2. Frictional pressure drop: Homogeneous model

The homogeneous model is the simplest model to calculate the frictional pressure drop in two-phase flows. We assume that the two phases are completely mixed and have the same velocity. The total mass flux of the mixture is given by

(3.7)\[ G = G_{G} + G_{L} = \alpha_{G} \rho_{G} v_{G} + \alpha_{L} \rho_{L} v_{L} = \rho_{G} j_{G} + \rho_{L} j_{L}\]

(it is a custom to use \(G\) for mass flux [Chi83].) The two-phase pressure drop for this flux is given by

(3.8)\[ - \left. \frac{dp}{dz} \right|_{TP} = \frac{\lambda_{H} G^{2}}{2 \rho_{H} D}\]

where \(D\) is the pipe diameter, and \(\lambda_{H}\) is the two-phase friction factor, which may be given in the Blasius form like

(3.9)\[ \lambda_{H} = C Re_{H}^{n}\]

The density of the homogeneous mixture is given by the harmonic mean:

(3.10)\[\begin{split} \frac{1}{\rho_{H}} = \frac{x}{\rho_{G}} + \frac{1 - x}{\rho_{L}} \end{split}\]

where \(x\) is the flow quality defined by

(3.11)\[ x = \frac{G_{G}}{G},~~~~1 - x = \frac{G_{L}}{G}\]

There are several options for the viscosity in the Reynolds number. If we take \(\mu_{L}\),

(3.12)\[ Re_{H} = \frac{G D}{\mu_{L}}\]

and

(3.13)\[ - \left. \frac{dp}{dz} \right|_{TP} = \frac{C \mu_{L} G^{2-n}}{2 \rho_{H} D^{1+n}}\]

Suppose a situation that the liquid phase flows with the total mass flux \(G\); we have

(3.14)\[ - \left. \frac{dp}{dz} \right|_{L0} = \frac{\lambda_{L} G^{2}}{2 \rho_{L} D} = \frac{C \mu_{L} G^{2-n}}{2 \rho_{L} D^{1+n}}\]

The square of the ratio of these pressure drops is termed the two-phase multiplier \(\phi_{L0}\):

(3.15)\[ \phi_{L0}^{2} = \frac{\left. dp /dz \right|_{TP}}{\left. dp / dz \right|_{L0}}\]

and for the homogeneous flow model with \(\mu_{H} = \mu_{L}\), we obtain

(3.16)\[ \phi_{L0}^{2} = \frac{\rho_{L}}{\rho_{H}} = 1 + \left\{ \frac{\rho_{L}}{\rho_{G}} - 1 \right\} x\]