On the Convection-Dispersion Equation for a Finite Domain: Third-Type Boundaries as a Necessary Condition of the Conservation Law

Abstract

This paper resolves a longstanding discussion of a mathematical problem important in contaminant hydrogeology and chemical-reaction engineering, by discussing the foundations for a conceptual model of a dilute miscible solute undergoing longitudinal convection and dispersion with moderate rates of appearance and disappearance in a finite continuum. It is demonstrated that: (i) Hulburts conditions (a first-type entrance with a third-type exit) fail to satisfy overall mass conservation; (ii) the conditions of Wehner and Wilhelm which reduce to those of Danckwerts (a third-type entrance with a zero-gradient exit) satisfy overall mass conservation yet fail to satisfy internal consistency with the governing equation; (iii) only third-type boundaries simultaneously satisfy internal consistency and overall mass conservation which are, respectively, a necessary and sufficient condition for any solution to the governing equation. This result is extensible to quite general governing equations since the boundary conditions are shown to be independent of the fate mechanisms.