Solute transport equations

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

The steady-state solute transport equation can be written as follows ... 

A differential solute transport equation derived for Levy motions would facilitate solute transport studies in the same way that the ADE facilitated applications of the Brownian motion model. Recently, Zaslavsky (1994) suggested a procedure to derive such an equation using fractional derivatives that in effect account for the memory of solute particles. Zaichev and Zaslavsky (1997), Benson (1998), and Chaves (1998) modified Zaslavsky s procedure to account properly for mathematical properties of fractional derivatives in the one-dimensional case. The simplest form of the one-dimensional equation assumes symmetrical dispersion ... 

To this point, several assumptions have been built into the theoretical development. The soil medium is being characterized as a homogeneous volume allowing the one-dimensional solute transport equation... 

Mass balance equations of pesticide fate and transport are developed for the surface and subsurface zones in PRZM. In the surface zone, avenues of loss include soluble loss in runoff, percolation to the next zone, sortoed loss in erosion, and decay in both phases. In the subsurface zones, losses include plant uptake and percolation in the soluble phase, and decay in both phases. A backward difference, Implicit numerical scheme is used to solve the partial differential solute transport equations, with a time step of one day and a spatial increment specified by the user. 

The accompanying solute transport equation is the finite difference equivalent of Equation 2. The upper boundary condition is represented by a flux equation,... 

Many one-dimensional solute-transport models have been developed and used to analyze column data. For a recent review, see Grove and Stollenwerk (13). Four different models were used in the study discussed in this article to simulate the shape of the column-breakthrough curves. All four models contain a one-dimensional solute-transport equation and use the Freundlich equation to describe sorption. They differ in the rate mechanism that is assumed to control transport of Mo(VI) from flowing phase to solid surface. The essential features of each model are summarized in Table III. 

The code solves four main governing equations the equilibrium equation, the flow equation, the heat transport equation, and the solute transport equation. These four equations for deformable porous media are... 

From the large number of mathematical models for the transport of transformation products with kinetic reactions that can be considered in the Rockflow system we have chosen a first-order chemical nonequilibrium model to simulate the sorption reaction. It can be described by the governing solute transport equation with rate-limited sorption and first-order decay in aqueous and sorbed phases. This model includes the processes of advection, dispersion, sorption, biological degradation or radioactive decay of the contaminant in the aqueous and/or sorbed phases. Figure 6.1 illustrates the conceptual model for sequential decay of a reactive species. 

Because the gases in the membrane are in low concentration and do not interact significantly with each other, a dilute-solution transport equation can be used... 

Yotsukura, N. (1977). Derivation of solute-transport equations for a turbulent natural-channel flow. J. Res. U.S. Geol. Surv. 5, 277-284. 

Adsorption may also be modeled as a nonequilibrium process using nonequilibrium kinetic equations. In a kinetic model, the solute transport equation is linked to an appropriate equation to describe the rate that the solute is sorbed onto the solid surface and desorbed from the surface (Fetter, 1999). Depending on the nonequilibrium condition, the rate of sorption may he modeled using an irreversible first-order kinetic sorption model, a reversible linear kinetic sorption model, a reversible nonlinear kinetic sorption model, or a bilinear adsorption model (Fetter, 1999). 

This section summarizes modeling studies reported since the reviews of Sharland and Turnbull. First, models are discussed which considered the pit surface to be actively dissolving, and did not take into account repassivation of surfaces in the pit. Then, the results of two modeling studies are described which included both dissolution and passivation. We choose to highlight repassivation in this way, since it underlies the important issue of pit stability. All models were based on the dilute solution transport equations in the previous section, except as indicated. 

Hebert s model for tunnel growth predicted the tuimel shape on the basis of the repassivation model just described." Starting from the initial condition of a cubic etch pit, the model calculated the evolution of the pit shape resulting from dissolution and sidewall passivation. The dissolution rate was taken directly from experimental measurements." Since the passivation kinetics were potential-dependent, it was necessary to accmately predict the potential at the tunnel tip. This required the use of concentrated solution transport equations, for the first time in a pitting model. All transport and kinetic parameters used in the model were taken from independent sources. The calculations showed that pits growing at the bulk solution repassivation potential spontaneously transformed into tunnels by sidewall passivation (Fig. 6). The tuimels then grew with parallel walls until the concentration at the tip approached saturation, at... 

Figure 7. Comparison of chloride concentration at the dissolving surface of a pit, calculated using models based on (a) dilute solution and (b) concentrated solution transport equations. The dashed line is calculated from the dilute solution model assuming a binary electrolyte in the pit. (Reproduced with permission from Ref. [50]. Copyright 1999, The Eletrochemical Society.)...

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