# Lesson 12 - Unit 8 Summary

In previous Units we have been discussing representing parts of our system using well-mixed compartments.  Mathematically these kinds of systems can be represented using a set of coupled ordinary differential equations. In these systems, there were two key assumptions:

• Each compartment could be assumed to be well-mixed (i.e., concentrations were the same throughout the compartment); and
• The mass transport from one location to the other was a function only of the mass (or concentration) in the “upstream” compartment.

However, in many cases, it will not be appropriate to represent parts of the system in this way.  In some parts of the system, both of these assumptions will be difficult to defend.  In particular,

• Concentrations will vary spatially in a continuous manner so that assuming a relatively large well-mixed region would be inappropriate; and
• The mass transport rate at any point cannot be described in terms of a single concentration or mass, but must be described in terms of a concentration gradient (a concentration difference over some distance).

These parts of a system (with an explicit spatial component) require a different approach to representing and solving the mass transport equations. Mathematically, we need to represent these systems not as ordinary differential equations, but as partial differential equations, in that concentration is a function of both time and space. This was the first of two Units that describes how to model such systems in GoldSim. Specifically, we discussed the process of diffusion, by which a substance tends to spread steadily from regions of high concentration to regions of lower concentration due to the random thermal motions of suspended atoms, molecules and particles.

We started the Unit by first revisiting a very simple advective system (rather than a diffusive system) in order to illustrate the problem of numerical diffusion.

Next we discussed the diffusion equation (Fick’s Second Law) and described how this equation is numerically represented in GoldSim. By doing so, we could see the ways that diffusive transport is different than advective transport.  In particular, we noted that whereas the mass transfer rate from one Cell to another for advection depends only on the concentration of the upstream Cell, the mass transfer rate from one Cell to another for diffusion depends on the concentrations in both Cells (their difference).

After describing how diffusive mass flux links are created in GoldSim, we looked at an Example of diffusion through a tube (or column), and revisited the issue of numerical diffusion.

We then discussed how diffusive engineered barriers (in the form of caps, covers or surrounding material) are often used to control the release of contaminants, and explored how we could represent other diffusive geometries for such barriers (other than linear).

Toward the end of the Unit, we discussed how the kinetics of many processes are often controlled by diffusion. In particular, we discussed the importance of diffusive boundary layers.

Finally, we closed the Unit by discussing several dimensionless factors that can be used to modify the diffusive conductance under certain circumstances.

The various examples in this Unit should have made it very clear that diffusion, while conceptually simple, can be a very complex process indeed!

In the next Unit, we will discuss another type of transport process in which concentration is a function of both time and space (and hence, like diffusion, needs to be represented not by ordinary differential equations, but by partial differential equations): advective-dispersive transport.