Travelling waves during the transport of reactive solute in porous media : combination of Langmuir and Freundlich isotherms

Recently, it has been shown that in the case of nonlinear solute adsorption the displacement may be in the form of a travelling wave. In this paper, we investigate whether a travelling wave type of behaviour can be expected when two different types of sorption sites can be distinguished with different isotherms and kinetics. Illustrations are given for cases where the overall isotherm comprises two contributions that follow the Langmuir and the Freundlich equations, respectively. Boundary conditions are chosen that ensure a decrease in concentration in the direction of flow. Depending on the value of the Freundlich power (p) the travelling wave may exist. For p = 1, the travelling wave always exists, whereas for 1 <p = 2 it depends on the values of the other adsorption parameters and whether a lower bound of the upstream concentration (at x = -8) is exceeded. For p = 2, the existence of the travelling wave requires that the upstream concentration does not exceed an (specified) upper bound. Besides illustrating some waves we show that two different rate functions that have the Freundlich isotherm as their limit for an infinite rate parameter result in qualitatively different travelling waves.