Some Mathematics Transport Equations

Before turning to the transport equation for Po t), let us add some remarks about the mathematical properties of the basic operators of the theory. 

The transport equations for g and are used in two-fluid models for multiphase flows. A fc-fluid model can be developed by treating all particles with the same internal coordinates as a fluid. Thus, for example, if all particles are identical except that some have mass Ml and the others have mass M2 (which implies that they have different solid densities), then we can treat the particle phase as two fluids. Mathematically, this follows directly from the form of the NDF for this case. 

Developing concentration profiles in a soil column for cyclical boundary loading functions is important for several reasons. One reason is that the solution increases the repertoire of mathematical models that are available for which someone may find a use. A second reason is that the solution can be used by those who set up experiments to estimate parameters such as the dispersion coefficient. Another reason is that analytical solutions to tracer transport equations are desired because they can be used easily lot some simple flow cases to quickly estimate what... 

All of the models used in these studies are based upon the convection-dispersion equation for solute transport through porous media and thus are constrained by the inherent limitations of this mathematical representation of actual processes. These limitations, analyzed in some detail in a number of recent papers (9.10.11.12.13). are real for many field conditions. On the other hand, alternative approaches (e.g. stochastic transfer models) are still in an early state of development for solute transport applications. Consequently, we have initiated our modeling efforts with the traditional transport equations. Hopefully, improved approaches will be developed in the near future. 

The general phenomenon of polymer adsorption/retention is discussed in some detail in Chapter 5. In that chapter, the various mechanisms of polymer retention in porous media were reviewed, including surface adsorption, retention/trapping mechanisms and hydrodynamic retention. This section is more concerned with the inclusion of the appropriate mathematical terms in the transport equation and their effects on dynamic displacement effluent profiles, rather than the details of the basic adsorption/retention mechanisms. However, important considerations such as whether the retention is reversible or irreversible, whether the adsorption isotherm is linear or non-linear and whether the process is taken to be at equilibrium or not are of more concern here. These considerations dictate how the transport equations are solved (either analytically or numerically) and how they should be applied to given experimental effluent profile data. 

---