This tutorial presents the mathematical model implemented in the Mass Transfer Sorption with Solute Decay applet. The model describes one-dimensional advective-dispersive transport in a semi-infinite porous medium with two-site kinetic non-equilibrium sorption and optional first-order decay in all phases (Cameron & Klute 1977; Toride et al. 1993).
The solution method is Gauss-Chebyshev numerical quadrature as derived by Toride et al. (1993), which is incorporated in the CXTFIT-1 code. No closed-form analytical expression exists in general; the applet evaluates the integral numerically at each output point.
The physical system is a one-dimensional column of porous medium, semi-infinite in the positive x-direction. A finite-duration pulse of solute at concentration $C_f$ enters at $x = 0$ over the time interval $0 < t \le t_f$; after $t_f$ the inlet concentration drops to zero. The solute undergoes advection, hydrodynamic dispersion, two-site sorption, and optional first-order decay.
Mass conservation for a solute in a porous medium with two sorbed phases (equilibrium $s_e$ and kinetic $s_k$) and first-order decay gives:
where $c$ is the dissolved concentration [M/L³], $s_e$ and $s_k$ are the equilibrium and kinetic sorbed concentrations [M/M], $D$ is the hydrodynamic dispersion coefficient [L²/T], $v$ is the pore-water velocity [L/T], $\theta$ is porosity [—], $\rho$ is bulk density [M/L³], and $\mu_l$, $\mu_{se}$, $\mu_{sk}$ are the first-order decay rates [1/T] in each phase.
The two-site model (Cameron & Klute 1977) partitions the total sorbed concentration $s = s_e + s_k$ between two types of sites:
Equilibrium Sites (Type 1)
A fraction $f$ of the total sorption capacity is assumed to be in instantaneous equilibrium with the dissolved phase via a linear isotherm:
where $K_d$ [L³/M] is the linear distribution coefficient and $f$ (0 ≤ $f$ ≤ 1) is the equilibrium site fraction.
Kinetic Sites (Type 2)
The remaining fraction (1 − $f$) exchanges solute with the dissolved phase via a first-order rate process:
where $\alpha$ [1/T] is the first-order mass transfer (rate) coefficient. The kinetic sites drive non-equilibrium behavior: when $\alpha \to \infty$ the kinetic sites reach equilibrium instantaneously (equivalent to $f = 1$); when $\alpha$ is small the kinetic sites respond slowly, causing tailing on breakthrough curves.
The column is initially solute-free:
At the far end, concentration vanishes:
At the inlet, a third-type (flux) boundary condition is applied:
This condition conserves solute mass at the inlet by equating the advective-dispersive flux to the applied flux, rather than simply fixing the concentration (Danckwerts condition).
Substituting Eqs. (2) and (3) into Eq. (1) and defining dimensionless parameters yields a coupled PDE–ODE system. Toride et al. (1993) derived the solution in terms of convolution integrals. The applet evaluates these integrals using Gauss-Chebyshev quadrature with 75 quadrature points (ported from the original CXTFIT-1 FORTRAN code).
Dimensionless Parameters
The solver works with the following dimensionless groups:
| Symbol | Definition | Physical meaning |
|---|---|---|
| $P$ | $vL/D$ | Péclet number (L = column length) |
| $R$ | $1 + \rho K_d/\theta$ | Retardation factor |
| $\beta$ | $(\theta + f\rho K_d)\,/\,(\theta + \rho K_d)$ | Partition coefficient between mobile and total capacity |
| $\omega$ | $\alpha(1-\beta)R\,L/v$ | Dimensionless mass transfer number |
| $\mu_1^*$ | $(\theta\mu_l + f\rho K_d\mu_{se})\,L\,/\,(\theta v)$ | Dimensionless decay for equilibrium phase |
| $\mu_2^*$ | $(1-f)\rho K_d\mu_{sk}\,L\,/\,(\theta v)$ | Dimensionless decay for kinetic phase |
Finite Pulse by Superposition
The finite-pulse solution is obtained by superposition. The concentration at any point and time is:
where $c_\infty(x,\tau)$ is the semi-infinite step-input solution evaluated at lag time $\tau$, and $\mathbf{1}_{t>t_f}$ is the unit step function. The applet applies this superposition automatically.
| Symbol | Name | Units |
|---|---|---|
| $c$ | Dissolved (liquid-phase) concentration | M/L³ |
| $s_e$ | Equilibrium sorbed concentration | M/M |
| $s_k$ | Kinetic sorbed concentration | M/M |
| $s$ | Total sorbed concentration: $s = s_e + s_k$ | M/M |
| $v$ | Pore-water velocity (average linear velocity) | L/T |
| $D$ | Hydrodynamic dispersion coefficient | L²/T |
| $\theta$ | Porosity (volumetric water content) | — |
| $\rho$ | Dry bulk density | M/L³ |
| $K_d$ | Linear distribution (partition) coefficient | L³/M |
| $f$ | Fraction of equilibrium sorption sites (0 ≤ $f$ ≤ 1) | — |
| $\alpha$ | First-order mass transfer rate (kinetic sites) | 1/T |
| $\mu_l$ | First-order decay rate in the liquid phase | 1/T |
| $\mu_{se}$ | First-order decay rate on equilibrium sorption sites | 1/T |
| $\mu_{sk}$ | First-order decay rate on kinetic sorption sites | 1/T |
| $C_f$ | Inlet source concentration | M/L³ |
| $t_f$ | Duration of the finite concentration pulse | T |
| $R$ | Retardation factor: $R = 1 + \rho K_d/\theta$ | — |
| $x$ | Spatial coordinate (positive downstream) | L |
| $t$ | Time | T |
- Cameron, D. R. & Klute, A. (1977). Convective-dispersive solute transport with a combined equilibrium and kinetic adsorption model. Water Resources Research, 13(1), 183–188.
- Toride, N., Leij, F. J., & van Genuchten, M. Th. (1993). A comprehensive set of analytical solutions for nonequilibrium solute transport with first-order decay and zero-order production. Water Resources Research, 29(7), 2167–2182.
- van Genuchten, M. Th. & Alves, W. J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. USDA Technical Bulletin No. 1661. U.S. Department of Agriculture, Washington, D.C.
- Parker, J. C. & van Genuchten, M. Th. (1984). Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport. Water Resources Research, 20(7), 866–872.