Conservation of a Scalar Extensive in Differential Form

Abstract

A directional, off-centered, cell balance equation for a scalar substance is developed in a Eulerian volume overlaid with a Lagrangean advection channel. A small but non-zero $\Delta x$, $\Delta y$, $\Delta z$ transport cell is considered with one-dimensional advection in x direction with $v_x = v$ velocity component that may change with x. Diffusion and convection is assumed simultaneously in the entire cell of ${\Delta{V}} = {\Delta{V}}_a + {\Delta{V}}_s$ in x, y, and z directions, where ${\Delta{V}}_a$ and ${\Delta{V}}_s$ are the advection and stagnant volumes, respectively. The stagnant volume is distributed evenly along $\Delta x$, characterized by volume fraction of $S = {\Delta{V}}_s/{\Delta{V}} = {\Delta{A}}_s/{\Delta{A}}$.