Rate limited desorption in solute transport with decay

It is known that in some contaminated aquifers the rate of contaminant desorption from the solid phase to the aqueous phase limits the remediation rate (Travis and Doty, 1990). In such circumstances, the solute transport equation subject to rate limited desorption can be used in assessing the effectiveness of any remedial technologies. We propose to model desorption as a one-site process with a linear first order equation and decay (in the aqueous phase) as a first order process and analyze such a problem. We will impose conditions that relate to a contaminated aquifer where initially the solutes in the aqueous and solid phases are in linear equilibrium such that there is no flux of the contaminant at the inlet and no dispersive flux of the contaminant at the outlet. In non-dimensional form, the mathematical model (in one dimension) can be written as,

with boundary conditions

and initial conditions

Equation (1) is the advective-dispersive equation describing solute transport and equation (2) is the rate equation for sorbed concentration. The constants , ,  and D are all positive and they are related to desorption rate, the ratio- bulk density/volumetric water content-, decay rate and dispersion coefficient respectively. Since the model presented above is linear, one should be able to analyze it exactly with regard to any change in desorption rate or decay rate. Our objective regarding this problem is to carry out such an analysis.