Cross-section averaged phase mass balance (${\partial }_t(\alpha \bar{\rho })+{\partial }_x(\alpha \bar{\rho }\bar{u})=0$)

For a pipe of area $A$, let a phase indicator function be $\chi$. Define $\alpha =A^{-1}\int \chi dA$, $\bar{\rho }=(A\alpha {)}^{-1}\int \chi \rho dA$ and the mass-weighted axial velocity $\bar{u}=(A\alpha \bar{\rho }{)}^{-1}\int \chi \rho u_{x}dA$. Without interfacial mass transfer, integrate ${\partial }_t(\chi \rho )+\nabla \cdot (\chi \rho \mathbf{u})=0$ over a cross-section with impermeable lateral wall. The resulting averaged mass and flux are exactly $\alpha \bar{\rho }$ and $\alpha \bar{\rho }\bar{u}$, proving the displayed law. The complementary phase uses $1-\chi$ and $1-\alpha$. An unweighted phase velocity would not generally factor the mass flux correctly.