Let

χ_{1} (x, t)

equal one in phase one and zero in phase two; its interface value is immaterial for area integrals. In the absence of phase change or interfacial mass transfer, the distributional phase mass conservation is

\partial_{t} (χ_{1} ρ) + \nabla \cdot (χ_{1} ρ u) = 0

. For a cross-section

S

of area

A

, define

α = \frac{1}{A} \int_{S} χ_{1} d A, ρ_{1} = \frac{1}{A α} \int_{S} χ_{1} ρ d A, u_{1} = \frac{\int _{S} χ _{1} ρ u _{x} d A}{A α ρ _{1}} .

(1)

Here the symbol

u_{x}

in the integral denotes the axial velocity component, not a derivative. Integrating through an arbitrary axial slice and using zero normal flux at the rigid pipe wall gives

\partial_{t} (ρ_{1} α) + \partial_{x} (ρ_{1} α u_{1}) = 0.

(2)

The velocity is mass weighted, so the averaged mass flux factors exactly; an unweighted phase mean would not in general have this property. This is the cross-section averaged phase mass balance.

For phase two use