Analytical Solution of Equations for Chemical Transport with Adsorption, Longitudinal Diffusion, Zeroth-Order Production, and First-Order Decay
This Demonstration examines one-dimensional chemical transport in a porous medium as influenced by simultaneous adsorption, zeroth-order production, and first-order decay. The corresponding equation is :[more]
where is the effective dispersion coefficient, is the fluid phase concentration, is distance, is time, is the interstitial fluid velocity, is a retardation factor defined as , is a general decay constant defined as , and is the zeroth-order fluid phase source term. Here is the porous medium bulk density, is the ratio of adsorbed to fluid phase concentration, is the volumetric moisture content, is a first-order liquid phase decay constant, and is the first-order solid phase decay constant.
The transport equation is solved subject to the following initial and boundary conditions:
, where and are the constant initial fluid and surface boundary concentrations, taken here as .[less]
Reference  gives a complete derivation of the analytical solution using Laplace transforms. The solution is:
 M. T. Van Genuchten, "Analytical Solutions for Chemical Transport with Simultaneous Adsorption, Zero-Order Production, and First-Order Decay," Journal of Hydrology, 49(3–4), 1981 pp. 213–233. doi:10.1016/0022-1694(81)90214-6.