Concepts from Thermodynamics

1.1 Introduction

The basis of any physical theory is a set of experimental results. From these special primary observations, general principles are abstracted, which can be formulated in words or in mathematical equations. These principles are then applied to correlate and explain a group of physical phenomena and to predict new ones.

The experimental basis of thermodynamics is formalized in the so-called principal laws. The law of conservation of energy, which thermodynamics shares with mechanics, electrodynamics, etc., is one of these principal laws. It introduces the concept of internal energy of a system. The other principal laws of thermodynamics introduce and define the properties of entropy and temperature, the two concepts which are particular and fundamental for thermodynamics.

The principles laid down in these fundamental laws apply to the relations between equilibrium states of matter in bulk. For instance, thermodynamics yields the relation between the specific heats at constant pressure and at constant volume; it relates the temperature dependence of the vapor pressure to the latent heat of evaporization; it gives upper bounds for the efficiency of cyclic processes, etc.

Fluid mechanics of perfect fluids, i.e., fluids without viscosity and heat conductivity, is an extension of equilibrium thermodynamics to moving fluids. The kinetic energy of the fluid has now to be considered in addition to the internal energy which the fluid possesses when at rest. The ratio of this kinetic energy per unit mass to the internal energy per unit mass is a characteristic dimensionless quantity of the flow problem and in the simplest cases is directly proportional to the square of the Mach number. Thermodynamic results are taken over to perfect fluid flow almost directly.

Fluid mechanics of real fluids goes beyond classical thermodynamics. The transport processes of momentum and heat are of primary interest here, and a system through which momentum, heat, matter, etc., are being transported is not in a state of thermodynamic equilibrium, except in some rather trivial cases, such as uniform flow of matter through a fixed system.

But, even though thermodynamics is not fully and directly applicable to all phases of real fluid flow, it is often extremely helpful in relating the initial and final conditions. This complex of problems is best illustrated with a simple example. Assume a closed, heat-insulating container divided into two compartments by a diaphragm. The compartments contain the same gas but at different pressures p1 and p2, and different temperatures T1 and T2. If the diaphragm is removed suddenly, a complicated system of shock and expansion waves occurs, and finally subsides due to viscous damping. Thermodynamics predicts the pressure and temperature in this final state easily. Fluid mechanics of a real fluid should tackle the far more difficult task of computing the pressure, temperature, etc., as a function of time and location within the container. For large times, pressure and temperature will approach the thermodynamically given values. Sometimes we need only these final, equilibrium values and hence can make very good use of thermodynamic reasoning even for problems that involve real fluid flow.

In fluid mechanics of low-speed flow, thermodynamic considerations are not needed: the heat content of the fluid is then so large compared to the kinetic energy of the flow that the temperature remains nearly constant even if the whole kinetic energy is transformed into heat.

In modern high-speed flow problems, the opposite can be true. The kinetic energy can be large compared to the heat content of the moving gas, and the variations in temperature can become very large indeed. Consequently the importance of thermodynamic concepts has become steadily greater. The chapter therefore includes material that is more advanced and not needed for the bulk of the later chapters. Articles that are starred can be omitted at first reading without loss of continuity.

1.2 Thermodynamic Systems

A thermodynamic system is a quantity of matter separated from the "surroundings" or the "environment" by an enclosure. The system is studied with the help of measurements carried out and recorded in the surroundings. Thus a thermometer inserted into a system forms part of the surroundings. Work done by moving a piston is measured by, say, the extension of a spring or the movement of a weight in the surroundings. Heat transferred to the system is measured also by changes in the surroundings, e.g., heat may be transferred by an electrical heating coil. The electric power is measured in the surroundings.

The enclosure does not necessarily consist of a solid boundary like the walls of a vessel. It is only necessary that the enclosure forms a closed surface and that its properties are defined everywhere. An enclosure may transmit heat or be a heat insulator. It may be deformable and thus capable of transmitting work to the system. It may also be capable of transmitting mass. Every real wall has any one of these properties to a certain degree. There do not exist perfectly rigid walls, for example, and similarly there is no perfect heat insulator. However, it is convenient to use an idealized enclosure, consisting of parts which have well-defined properties such as complete heat insulation, etc.

For our purposes it is sufficient to deal with fluids only. The systems that we shall consider here are:

(a) A simple, homogeneous system composed of a single gas or liquid.

(b) A homogeneous mixture of gases.

(c) A heterogeneous system composed of the liquid and gaseous phase of a single substance.

1.3 Variables of State

If a system is left alone for a sufficiently long time, that is, if no heat and no mass is transferred to it and no work is done on it during this time, it will reach a state of equilibrium. All microscopically measurable quantities will become independent of time. For example, the pressure p, the volume V, and the temperature θ, can be measured, and in equilibrium do not depend upon time.

Variables that depend only upon the state of the system are called variables of state, p and V are evidently such variables, and these two are already familiar from mechanics. For a complete thermodynamic description of a system, we need new variables of state, foreign to mechanics. Thus it is a result of experience that the pressure of a system is not only a function of...