Review of the Simplest Example

Let us go over the leading points that this example has brought out. 

We first choose variables sufficient to describe the situation. This choice is tentative, for we may need to omit some or recruit others at a later stage (e.g., if V is constant, it can be dismissed as a variable). In general, variables fall into two groups independent (in our example, time) and dependent (volume and concentration) variables. The term lumped is applied to variables that are uniform throughout the system, as all are in our simple example because we have assumed perfect mixing. If we had wished to model imperfect mixing, we would have had either to introduce a number of different zones (each of which would then be described by lumped variables) or to introduce spatial coordinates, in which case the variables are said to be distributed.2 Lumped variables lead to ordinary equations distributed variables lead to partial differential equations. 

When we make a balance to obtain a differential equation, we are invoking a natural law, the conservation of matter in our case. If the net flux of any conserved quantity into a lumped system over its boundaries is F, the rate of generation within the system is G, and the amount contained in it is H, then the balance gives 

We shall derive the equations for distributed systems from this equation later. 

In expressing F, G, and H in terms of the dependent variables, we use certain properties of the materials involved (e.g., that the reaction is first order, so G = Vkc). These are sometimes called constitutive relations because they invoke the constitutions of the various components. They are not principles applying to everything, as are natural laws, but apply only to the materials in question. 

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