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 its volume. A new variable of state, θ, the temperature, has to be introduced. For a simple system,

p = p(V, θ) (1·1)

Following R. H. Fowler, one states the "zeroth law of thermodynamics":

There exists a variable of state, the temperature θ. Two systems that are in thermal contact, i.e., separated by an enclosure that transmits heat, are in equilibrium only if θ is the same in both.

Consequently, with the help of Eq. 1.1, we can use the pressure and the volume of an arbitrary system as a thermometer.

When we discuss the exchange of work or heat between a system and its surroundings, we find the need for a variable of state E, the internal energy, which measures the energy stored in the system. The first law of thermodynamics introduces E, as will be seen later.

Furthermore we shall find it necessary to introduce a variable of state S, the entropy, which, for example, is needed to decide whether a state is in stable equilibrium. The second law of thermodynamics introduces S and defines its properties.

For a simple system E and S are functions of p, V, θ. But, since p can be expressed by V and θ, using Eq. 1.1, it is sufficient to write:

E = E(V, θ) (1·2)

S = S(V = θ) (1·3)

Relations like Eqs. 1.1, 1.2, and 1.3 are called equations of state. Specifically Eq. 1.1 is called the "thermal equation of state"; Eq. 1.2, the "caloric equation of state." A specific substance is characterized by its equations of state. The forms of these equations cannot be obtained from thermodynamics but are obtained from measurements or else, for a particular molecular model, from statistical mechanics or kinetic theory.

Any variable of state is uniquely defined for any equilibrium state of the system. For example, if a system changes from one state of equilibrium, say A, to another state B, then EB – EA is independent of the process by which the change occurred. The important consequences of this property of the variables of state will become evident later.

One distinguishes between intensive and extensive variables of state. A variable is called extensive if its value depends on the mass of the system. The mass If of a system is thus an extensive quantity, and so are E, V, and S. For example, the internal energy E of a certain mass of a gas is doubled if the mass is doubled; the energy of a system that consists of several parts is equal to the sum of the energies of the parts.

Variables of state that do not depend upon the total mass of the system are called intensive variables, p and θ are typical intensive variables. For every extensive variable like E we can introduce an intensive variable e, the energy per unit mass or specific energy. Similarly we can define a specific volume v, specific entropy s, etc. Specific quantities will be denoted by lower-case letters.

1.4 The First Principal Law

Consider a fluid contained in a heat-insulating enclosure, which also contains a paddle wheel that can be set into motion by a falling weight. The pressure of the system is kept constant. The temperature θ and the volume V are measured initially (state A). The weight is allowed to drop a known distance, and θ and V are measured again after the motions in the system have died down and a new state of equilibrium B has been established.

In this way a certain amount of work W, equal to the decrease in potential energy of the weight, has been done on the system. Conservation of energy requires that this work is stored within the system. Hence there exists a function E(V, θ) such that

EB – EA = W (1·4)

It is also possible to use work to produce an electric current and to supply this work to the system in the form of heat given off by a heating coil. Both of these experiments were performed by Joule in his classical studies on the mechanical equivalent of heat. A given amount of work done on the system yields the same difference in internal energy regardless of the rate at which the work is done and regardless of how it is transmitted.

One can furthermore relax the condition of complete heat insulation and allow also the passage of a certain amount of heat Q through the enclosure. Q can be defined calorically by the change in temperature of a given mass of water, or one can use Joule's experiments to define Q entirely in mechanical terms. It is important, however, to define Q and W in terms of changes measured in the surroundings.

We can thus formulate the first law:

There exists a variable of state E, the internal energy. If a system is transformed from a state of equilibrium A to another one, B, by a process in which a certain amount of work W is done by the surroundings and a certain quantity of heat Q leaves the surroundings, the difference in the internal energy of the system is equal to the sum of Q and W,

EB – EA = Q + W (1·5)

It is often convenient to discuss a simple idealized enclosure, the cylinder-piston arrangement of Fig. 1.1. The cylinder walls are assumed rigid. We can assume them to be heat-insulating or. capable of heat transmission, depending on the process that we wish to study. Work can be done by the surroundings only by the displacement of the piston. W is defined as in mechanics in terms of a force vector F and a displacement dr,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1·6)

The force acting on the piston is parallel to the displacement; thus, introducing the pressure p and the piston surface area A, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1·7)

with the convention that dV is positive if the volume of the system increases. It is not difficult to show that Eq. 1.6 leads to Eq. 1.7 even in the case of pressures acting on a deformable enclosure of any shape. (Shear forces can be introduced also; this is done later in discussing the equations of motion of a real fluid in Chapter 13.) For a small change of state, we can write Eq. 1.5 in differential form,

dE = dQ + dW (1·8)

or, using Eq. 1.7,

dE = dQ - pdV (1·8a)

Equation 1.8a can also be written for unit mass:

De = dq – p dv (1·8b)

Now E is a variable of state, whereas Q and W depend on the process followed in changing the state. This is sometimes indicated by writing δW and δQ instead of dW and dQ. We shall not follow this custom here.

1.5 Irreversible and Reversible Processes

A change of state of a system is possible only by a process for which

ΔE = Q + W

The first law does not restrict the possible processes any further.

Now in the paddle wheel experiment of Joule it is evidently impossible to reverse the direction of the process. One cannot induce the wheel to extract the energy ΔE from the system and to lift the weight. The process is irreversible. It is very easy to find other similar situations, and indeed all natural or "spontaneous" processes are irreversible. If one scrutinizes these irreversible processes, it becomes evident that the deviation of the system from equilibrium during the process is of primary importance. A motion like the stirring of a fluid, sudden heating, etc., induces currents in the system. The term current refers to the flux of a quantity like heat, mass, momentum, etc. A current of heat flows, if there exists a finite temperature difference; a current of mass flows, if there exist differences in concentration of one component; a current of momentum flows, if there exist differences in velocity.

A system is in a state of equilibrium if it is free of currents. A process leading from one state to another is reversible if the system remains during the whole process in equilibrium; i.e., if the work W and the heat Q are added in such a way that no currents are produced. Such an ideal reversible process can actually be closely approximated in an experiment. For example, instead of using the paddle wheel, W could be transferred to an insulated system by a slow displacement of a piston, so that the pressure and temperature remain uniform within the system during the whole process. (Exercise 1.9 gives a simple and instructive example of an irreversible process.)

The changes of state discussed here lead from one static condition of the system to another. It is often much more convenient to consider processes that proceed at a steady rate. This is true for many measurements in thermodynamics and is essential for fluid mechanics. Thus, instead of dealing with a paddle wheel in a closed "calorimeter," as in Joule's experiment, we may consider a heat-insulated duct in which a fluid flows at a steady rate through a turbine wheel or fan. The system consists now of a certain mass of fluid which passes through the fan. Instead of dealing with a system before and after the motion of the paddle wheel, we now deal with the fluid upstream and downstream of the fan. Our definitions of thermodynamic equilibrium can be extended to this case easily. For direct comparison with thermodynamic processes like Joule's experiment, we have to require the fluid to flow very slowly so that its kinetic energy is negligible. In the next chapter we shall drop this restriction and extend the same considerations to high-speed fluid flow.