Lesson 10 - Modeling Mass Transport Between Environmental Compartments

With the exception of the Example models we looked at in Unit 4 to provide an overview of the Contaminant Transport Module, all of the other Examples and Exercises we have looked at so far have involved well-mixed tanks. This was not accidental.  A well-mixed tank provides an excellent example of the kind of physical system that can be readily represented by a Cell pathway (in fact, the icon for the Cell pathway is a well-mixed tank).

You may indeed have a need to simulate an actual well-mixed tank using GoldSim (e.g., in a water treatment facility or some other kind of plant).  More often, of course, you will need to simulate a natural environmental compartment such as a pond, reservoir, or lake, or perhaps an atmospheric or soil compartment. Although these are not tanks, it is important to understand that it is often completely appropriate to treat such environmental compartments as well-mixed and model them using Cell pathways. In some cases, the entire compartment can be treated using a single Cell pathway, while in others, it may not be appropriate to treat an entire environmental compartment as well-mixed, and you will need to discretize the compartment using multiple Cells.

To understand this, keep in mind that the key assumption of a Cell pathway is that it is well-mixed.  But what does that actually mean?  A compartment can be considered to be well-mixed if the time frame of the mixing is rapid relative to the time frame of other processes you are simulating.  For example, imagine that a pond has two distinct layers (e.g., due to density differences). Whether or not you need to represent those layers using two different Cells is a function of how quickly the layers mix relative to how quickly the entire pond flushes. If the layers mix rapidly relative to the rate at which the entire pond flushes, you probably can represent the pond using a single Cell.  However, if the layers mix very slowly, you may need to represent each layer as a separate Cell in order to model the system accurately.  In this Lesson, we will use a simple Example to illustrate this concept.

The system we will consider looks like this:

Two ponds are located next to each other.  The ponds maintain constant volumes and water flows through the system at a constant rate.  Contaminated water flows from a stream into the first pond, and the first pond discharges via another stream into the second pond. The second pond then discharges to another stream (which then leaves the system of interest).  The first pond is rather deep, and as a result it has two distinct layers (a top layer of warmer water and a larger bottom layer of colder water). Each layer is well-mixed, and the mixing between the layers (which is assumed to occur at a constant specified rate) determines how well-mixed the entire pond is. The second pond is shallower, and hence is assumed to be very well-mixed. Mass of a contaminant is introduced for a fixed amount of time into the top layer of the first pond and then this moves through the two ponds.  We are interested in the simulating the peak concentration in the second pond that results from this mass input.

The various input parameters describing this system are summarized below:

Let’s open the model now so we can examine it. Go to the “Examples” subfolder of the “Contaminant Transport Course” folder you should have downloaded and unzipped to your Desktop, and open a model file named ExampleCT8_Two_Ponds.gsm.

The Inputs Container contains all of the inputs described in the table above.  The only input that is worth examining here is the inflow concentration (which is used to specify a Defined Concentration in a Source Cell that flows into Pond1.  This is represented using a Selector.  A Time History Result element showing that variable can be found in the Inputs Container.  Run the model and double-click on it now:

This results in  a 5-day  slug of mass (in which mass is added at a rate of 100 m3/day * 0.1 mg/l).  As we shall see, this mass is added to the upper layer of Pond1.

Close this result and return to Edit Mode.  Then navigate to the Ponds Container.  Inside that Container you will see three sub-Containers.  Each one of these models the system in a different way.  Click into the top Container (Low_Mixing_Two_Cells):

In this Container Pond1 is represented by two Cells.  The Cells are connected by equal and opposite advective mass flux links to represent the mixing. Pond1_Upper_Layer flows into Pond1_Lower_Layer, and Pond1_Lower_Layer flows into Pond1_Upper_Layer at the same rate. Water also flows from Pond1_Upper_Layer to Pond2, and then on to a Sink Cell. The contaminant mass (discussed above) is added to Pond1_Upper_Layer.

Return one level up to the Ponds Container.  If you look into the High_Mixing_Two_Cells Container you will see that it is identical to the Low_Mixing_Two_Cells Container, with the only exception being that a different mixing rate is used between the two layers. 

Finally, if you look into the One_Cell Container you will see that Pond1 is represented by a single Cell:

This, of course, is equivalent to assuming that Pond1 is completely well-mixed (there are no separate layers).

Hence, these three separate Containers provide three different ways to represent this system.  The Time History Result element (in the Ponds Container) plots the concentration in Pond2 for all three systems, allowing us to compare the behavior.  Run the model and double-click on this Result element now:

To better understand these results, based on the input parameters (Pond1 volume and the various flow rates) we can compute the time it takes a particular volume to “turnover” or mix (volume/rate) for the two processes being modeled (flushing by flow through the system and mixing of the two layers).

• The time required for Pond1 to be flushed by the flow rate through the system is about 6 days.
• The time required for the two layers to be well-mixed assuming a low mixing rate is about 12 days.
• The time required for the two layers to be well-mixed assuming a high mixing rate is about 1 day.

Hence, for the high mixing assumption, the time required for the two layers to mix is significantly less than the time required for the pond to flush.  This would imply that for that mixing rate assumption, the mixing is rapid relative to the other processes being modeled. In fact, if we compare the high mixing rate result to the result obtained using just one Cell (and hence complete mixing), we see that the results are quite close (for practical purposes, identical). This is because in this case, the concentrations in the upper and lower layers are quite close:

From this we could conclude that if the high mixing rate assumption is correct, Pond1 could be accurately modeled with a single Cell.

Note, however, that for the low mixing assumption, the time required for the two layers to mix twice as long as the time required for the pond to flush.  Hence, this would imply that for that mixing rate assumption, the mixing between layers must be accounted for. In fact, if we compare the low mixing rate result to the result obtained using just one Cell, we see the results are quite different. This is because in this case, the concentrations in the upper and lower layers are quite different:

From this we could conclude that if the low mixing rate assumption is correct, Pond1 could NOT be accurately modeled with a single Cell, and two Cells would be required. 

Although this Example is quite simple, it provides an indication of how you would go about determining how to use Cells to appropriately represent different kinds of environmental compartments. We will examine more complex examples in later Units.

Before leaving this Lesson, let’s briefly revisit some of the biggest simplifying assumptions in this model:

• The flow rate through the system was constant.  In more realistic model, this flow rate would vary time.  In addition, there would be other inflows (e.g., rainfall and runoff) and outflows (e.g., evaporation) from the ponds.
• The volumes of the ponds are constant.  In a more realistic model, these would vary with time.
• The travel time between the two ponds is instantaneous (and mass is not dispersed as it travels between the ponds).  Of course, in some systems this may be appropriate, but in others there may be a significant delay (and mass may be dispersed or otherwise impacted) as it moves between the ponds.
• The mixing in Pond1 was treated in a very simple way.  First, the rate was treated as a constant.  In a more realistic model, the mixing rate would likely change with the seasons. For example, it is likely that for part of the year (e.g., winter) it might be quite high, resulting in the system being well-mixed, and then may gradually decrease as the temperature in the upper layer increased through the summer.  The change back from a two-layer (poorly-mixed) system to a one-layer (well-mixed)  system could then happen quite suddenly in the fall (e.g., due to a single storm event).

As we will see in later Units, all of these things can readily be represented in GoldSim.