Mass conservation

Mass conservation requires that a material region retain its mass when no mass-producing process is included. For mass density $\rho$ and velocity field $\mathbf{v}$, the local continuity equation isor equivalently $D\rho /Dt+\rho \nabla \cdot \mathbf{v}=0$. The second form says that a moving fluid element loses mass density as its volume expands, rather than losing mass.

A pipe narrowing in a steady flow gives a simple test: the mass flow through every cross-section is the same, so $\rho AU$ is constant for a uniform section profile. Constant mass density then makes $AU$ constant. In Lagrangian coordinates, the equivalent relation is $\rho J={\rho }_0$, where $J=\det F$ is the local volume ratio. A fixed observation region can gain mass through its boundary even though the material moving through it obeys mass conservation.

Conservation of total material and isotope mass. For feed, product and enrichment tails on one mass basis, $F=P+W$ and $F{x}_f=P{x}_p+W{x}_w$.