# Lesson 2 - Problem Definition: Mass in a Well-Mixed Tank

We will introduce the details of the GoldSim Contaminant Transport Module by building a simple example: a model of a single well-mixed compartment (e.g., a tank).

This example is very simple indeed.  It consists of a large tank completely filled with water. Suspended in the water is a fine-grained sand (it remains suspended due to constant mixing).  A small quantity of two chemicals (named X and Y) is introduced into the tank.  X and Y both partition between water and the sand (Y to a greater extent than X). X is a conservative species (it does not decay).  Y, however, decays over time (but no decay products are accounted for).

The tank is assumed to be continuously and instantaneously mixed. This means that when we add the two chemicals they are instantaneously mixed across the entire volume of water in the tank. That is, there are never any spatial concentration gradients in the tank (i.e., at any given time, the concentration of any particular chemical in the water is always the same everywhere in the tank).

In addition, we are assuming that the partitioning of the two chemicals between water and the sand can be treated as equilibrium (instantaneous) reversible linear partitioning.  That is, it is assumed that when the chemicals enter the tank, they are immediately partitioned between the water and the sand everywhere in the tank according to an equilibrium partition coefficient that describes the ratio of the concentration on the sand to the concentration in the water at equilibrium (as discussed in Unit 3, Lesson 5).

Finally, we will assume that Y decays according to a first-order reaction (in which the rate is linearly proportional to the mass in the tank).  In describing such a reaction, the decay rate is often represented by the variable λ and can be described in terms of a half-life (the time for the mass to decrease by half).  The half-life can be expressed as ln(2)/ λ.

Our goal is to compute the total mass of X and Y in the tank, and the concentration of X and Y in water and on the sand.  The concentration (and mass) of X will not change with time (since mass is not being transported out of the tank, and X does not decay). The concentration (and mass) of Y, however, will change with time due to decay.

The various input parameters describing this system are summarized below:

Variable Value
Volume of Water in Tank 5 m3
Mass of Sand in Tank 0.1 kg
Initial Mass of X and Y 100 g
Sand Partition Coefficient for X 2 m3/kg
Sand Partition Coefficient for Y 5 m3/kg
Half-Life of Y 50 day
Density of Sand 1600 kg/m3

Starting in the next Lesson, we will build this model together.