arXiv subjects

W. J. Golz

Publications and source records attributed to W. J. Golz.

3 recordsLinked to original sources

arXiv2004

Solute Transport in a Porous Medium: A Mass-Conserving Solution for the Convection-Dispersion Equation in a Finite Domain

This dissertation resolves a longstanding discussion of a mathematical problem important in contaminant hydrogeology and chemical-reaction engineering, the proper mathematical description for a miscible solute undergoing longitudinal convective-dispersive transport with production, decay, and sorption in a porous medium. Initial and input concentrations may be any continuously differentiable functions and the problem is stated for a finite domain. This domain yields a mass balance which requires Robin (i.e., third-type) boundaries, which describe a continuous flux but a discontinuous resident-concentration. The discontinuity in the resident concentration at the outflow boundary yields an underdetermined system when the exit concentration is not experimentally measured. This is resolved by defining the unknown effluent concentration from a semi-infinite problem which satisfies a Dirichlet (i.e., first-type) condition at the origin. The solution is represented in a uniformly convergent series of real variables. The large volume of antecedent literature on finite solutions for convective-dispersive transport equations grew out of the historical precedents set by Danckwerts (1953) and Wehner and Wilhelm (1956) whom made simplifying assumptions of continuous boundary concentrations. This dissertation includes the demonstration that continuous-concentration hypotheses, whether rendered as Dirichlet or homogeneous Neumann (i.e., second-type) conditions, satisfy external mass conservation yet fail to provide solutions that are internally consistent with the governing equation.

math.AP↗

arXiv2004

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

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.

math.AP↗

arXiv2004

The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest. Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a well-posed problem by forcing a homogeneous Neumann exit, widely known as the Danckwerts [1] condition. We provide a solution to the Neumann problem and use it to produce an estimate which demonstrates that the Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.

math.AP↗