|Thermodynamics and Propulsion|
where is a constant that depends on the substance. We can determine the constant for any substance if we know how much heat is transferred. Since heat is path dependent, however, we must specify the process, i.e., the path, to find .
Two useful processes are constant pressure and constant volume, so we will consider these each in turn. We will call the specific heat at constant pressure , and that at constant volume , or and per unit mass.
In the derivation of , we considered only a constant volume process, hence the name, ``specific heat at constant volume.'' It is more useful, however, to think of in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than as a quantity related to heat transfer in a special process. In fact, the derivatives above are defined at any point in any quasi-static process whether that process is constant volume, constant pressure, or neither. The names ``specific heat at constant volume'' and ``specific heat at constant pressure'' are therefore unfortunate misnomers; and are thermodynamic properties of a substance, and by definition depend only the state. They are extremely important values, and have been experimentally determined as a function of the thermodynamic state for an enormous number of simple compressible substances2.1.
where is the number of moles of gas in the volume . Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a substance as an ideal gas constitutes a model of the actual physical situation, and the limits of model validity must always be kept in mind.
One of the other important features of an ideal gas is that its internal energy depends only upon its temperature. (For now, this can be regarded as another aspect of the model of actual systems that the ideal gas represents, but it can be shown that this is a consequence of the form of the equation of state.) Since depends only on ,
In the above equation we have indicated that can depend on . Like the internal energy, the enthalpy is also only dependent on for an ideal gas. (If is a function of , then, using the ideal gas equation of state, is also.) Therefore,
If we are interested in finite changes of internal energy or enthalpy, we integrate,
Over small temperature changes ( ), it is often assumed that and are constant. Furthermore, there are wide ranges over which specific heats do not vary greatly with respect to temperature, as shown in SB&VW Figure 5.11. It is thus often useful to treat them as constant. If so
These equations are useful in calculating internal energy or enthalpy differences, but it should be remembered that they hold only if the specific heats are constant.
We can relate the specific heats of an ideal gas to its gas constant as follows. We write the first law in terms of internal energy,
and assume a quasi-static process so that we can also write it in terms of enthalpy, as in Section 2.3.4,
Equating the two first law expressions above, and assuming an ideal gas, we obtain
An expression that will appear often is the ratio of specific heats, which we will define as
Below we summarize the important results for all ideal gases, and give some values for specific types of ideal gases.
In general, for substances other than ideal gases, and depend on pressure as well as on temperature, and the above relations will not all apply. In this respect, the ideal gas is a very special model.
In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume. The simple relations between changes in energy (or enthalpy) and temperature are a consequence of the behavior of an ideal gas, specifically the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances.2.4
From the first law, with , , and ,
Also, using the definition of enthalpy,
The underlined terms are zero for an adiabatic process. Rewriting (2.7) and (2.8),
Combining the above two equations we obtain
Equation (2.9) can be integrated between states 1 and 2 to give
For an ideal gas undergoing a reversible, adiabatic process, the relation between pressure and volume is thus:
We can substitute for or in the above result using the ideal gas law, or carry out the derivation slightly differently, to also show that
We will use the above equations to relate pressure and temperature to one another for quasi-static adiabatic processes (for instance, this type of process is our idealization of what happens in compressors and turbines).