6. Dynamical Models and System Behavior

Causal physical models are answers to the question “why:” Why is a system in a certain state? Why do processes run a certain way? A complete model of systems and processes is simply a combination of all relations—laws of balance and constitutive laws we have collected so far—necessary for a particular example. The purpose of a model is to determine quantities describing a situation at a moment, or to predict the outcome of processes.

Dynamical models

Dynamical models combine laws of balance with the appropriate constitutive laws. They are created by a combination of steps described above in Systems analysis I: Laws of balance, and Systems analysis II: Pressure and pressure differences, with the particular laws for special processes found in Constitutive laws.

If the model describes a dynamical situation, it may be expressed with the help of a system dynamics tool. A system dynamics diagram represents the necessary laws of balance and constitutive laws (Figure 1). Laws of balance are “drawn” graphically by combinations of stocks and flows.

Analytical solutions

Systems made up of containers and pipes show relatively simple behavior. Complex behavior is commonly the result of the interaction of several simple elements. For the simplest systems (those having constant values of capacitance and resistance) analytic solutions of the model equations can be obtained. In the case of draining straight-walled tanks through horizontal pipes with laminar flow we get

If an empty tank is charged, the solution of the model is

Time constants

The behavior (fluid level as a function of time) for the simple cases of draining and filling of a tanks is shown in the accompanying graphs (Figure 1). The solutions of the model are exponential functions. A measure of how fast (or slow) the process is, is the time it would take for the tank to drain or to fill were the level to continue to change at the initial rate. This time is called the capacitive time constant tau_C of the system. In one time constant, the level of fluid in the system shown on the left in Figure 1 drops to 1/e = 0.37 times the initial level: