4

100

Mathematical Modeling

of Fluid Systems

and Thermal Systems

4–1 Introduction

This chapter treats mathematical modeling of fluid systems and thermal systems. As the

most versatile medium for transmitting signals and power, fluids—liquids and gases—

have wide usage in industry. Liquids and gases can be distinguished basically by their rel-

ative incompressibilities and the fact that a liquid may have a free surface, whereas a gas

expands to fill its vessel. In the engineering field the term pneumaticdescribes fluid

systems that use air or gases and hydraulicapplies to those using oil.

We first discuss liquid-level systems that are frequently used in process control. Here

we introduce the concepts of resistance and capacitance to describe the dynamics of such

systems. Then we treat pneumatic systems. Such systems are extensively used in the au-

tomation of production machinery and in the field of automatic controllers. For instance,

pneumatic circuits that convert the energy of compressed air into mechanical energy enjoy

wide usage. Also, various types of pneumatic controllers are widely used in industry. Next,

we present hydraulic servo systems. These are widely used in machine tool systems, aircraft

control systems, etc. We discuss basic aspects of hydraulic servo systems and hydraulic

controllers. Both pneumatic systems and hydraulic systems can be modeled easily by using

the concepts of resistance and capacitance. Finally, we treat simple thermal systems. Such

systems involve heat transfer from one substance to another. Mathematical models of

such systems can be obtained by using thermal resistance and thermal capacitance.

Outline of the Chapter. Section 4–1 has presented introductory material for the

chapter. Section 4–2 discusses liquid-level systems. Section 4–3 treats pneumatic

systems—in particular, the basic principles of pneumatic controllers. Section 4–4 first

discusses hydraulic servo systems and then presents hydraulic controllers. Finally,

Section 4–5 analyzes thermal systems and obtains mathematical models of such systems.

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