Analytical traveling wave solutions for transport with nonlinear and nonequilibrium adsorption

Abstract

Transport was modeled for a soil with dual porosity, or with chemical nonequilibrium, assuming first‐order kinetics. The equilibrium sorption equation in the immobile region is nonlinear. Two equilibrium equations for sorption were considered, that is, the Langmuir and the Van Bemmelen‐Freundlich equations. The sorption equation in the mobile region is assumed to be linear. Analytical solutions were obtained that describe the traveling wave displacement found for initial resident concentrations that are smaller than the feed concentration and for infinite displacement times, neglecting the coupled effects of dispersion and nonequilibrium conditions. These waves travel with a fixed shape and a fixed velocity through the homogeneous flow domain. Besides expressions for the front shape, expressions for the front thickness and the front position were also presented. Differences with respect to the linear sorption case are the smaller front thickness and the non‐Fickian type of displacement. The non‐Fickian behavior is intrinsic to the traveling wave assumption as the front does not spread with the square root of time. The analytical solutions obtained for the equilibrium and for the nonequilibrium situations are mathematically equivalent. Only the effective diffusion/dispersion coefficient needs to be adapted to account for nonequilibrium effects, as for linear dual‐porosity models. Apart from early time behavior, the traveling wave solutions agree well with numerical approximations. The front steepness depends sensitively on the degree of nonlinearity. The sensitivity on the dispersion coefficient and first‐order rate coefficient may be large but depends on which mechanism controls front spreading.