Compartment modelling is a means of constructing a differential equation for a complicated process by considering just the inputs and outputs of the process, during a small time interval. The basic ideas are developed in the context of a model describing the mixing of a dye and water. Compartment models are then formulated for the pollution in a lake and the temperature of a domestic hot water system. The latter model uses ideas about the flow of heat from Chapter 12. The differential equations obtained are mainly of the first-order linear constant-coefficient type.

A mixing problem

One of the aims of modelling is to isolate the most important factors in a problem and ignore those which may not be important. Even very complicated processes can initially be analysed using very simple mathematical models which may later be extended to more complex and realistic models by incorporating more features. In problems involving the mixing of two or more substances, simple models may be formulated by considering the input and output to a compartment containing the quantity of interest.

The following problem will be used to illustrate these ideas. The problem is illustrated in Figure 13.1.1.

Statement of problem

In a dye factory a large vat is used to mix dye and water. The water flows in at a rate of 6 litres/minute and the dye flows in at a rate of 2 litres/minute.