In a fluid flow, if the temperature change associated with the freestream is more than 5%, then it is imperative to study the heat transfer aspects. For the lowspeed flows \(\mathrm {\left( M<0.5\right) }\), the thermodynamic considerations are not required due to large heat capacity of the fluid compared to its kinetic energy. The temperature will remain constant even if the whole kinetic energy is converted into heat and thus, the static and the stagnation temperatures of the fluid are equal. But, when the freestream Mach number is greater than 0.5, the change in energy of the flow will be substantially large. Hence, the kinetic energy of the flow should also be taken into account along with internal energy. In addition, for \(\mathrm {M>0.5}\), the difference between static and stagnation temperatures is large and therefore, the thermodynamic concepts should also be considered in the analysis of compressible fluids.
A system is defined as a quantity of matter or a region in space, chosen for study; and the mass or the region outside the system is known as its surroundings. A real or an imaginary surface separating the system and surroundings is called the boundary, which may be fixed or movable. The system and its surrounding are collectively called the universe.
The systems are classified into two: a closed system and an open system. A closed system, also called control mass, has fixed amount of mass and does not allow the transport of mass across the boundary. The energy, however, in the form of heat or work can cross the boundary. Furthermore, if energy is also not allowed to cross the boundary, the system is called an isolated system. The boundary of a closed system may be fixed or movable.
The first law of thermodynamics is essentially the law of conservation of energy, which states that the total energy of an isolated system remains constant. The energy can be neither created nor destroyed; however, it can change its form. Mathematically, the first law of thermodynamics is expressed as
$$\begin{aligned} \mathrm {dU}&= \mathrm {\delta Q\delta W} \end{aligned}$$
It states that “the heat added to the system minus work done by the system is equal to the change in internal energy of the system.” Moreover, for a given
\(\mathrm {dU}\), an infinite number of processes (paths) are possible to cause a change of state. However, for compressible fluids, we are interested in the following three processes only.

Reversible Process—It is the process that can be reversed without leaving any trace on the surroundings. That is, both the system and the surroundings are returned to their initial states at the end of the reverse process.

Adiabatic Process—It is the process in which no heat transfer is occurring across the boundary of the system.

Isentropic Process—It is the process that is both reversible and adiabatic.
The Fourier’s law of heat conduction states that “the heat flux per unit area in a given direction is proportional to the temperature gradient in the same direction.” That is,
$$\begin{aligned} \mathrm {q_{x}}&= \mathrm {\kappa \frac{dT}{dx}} \end{aligned}$$
The most useful form of energy equation for compressible flows is
$$\begin{aligned} \mathrm {h_{0}}&= \mathrm {h+\frac{v^{2}}{2}} \end{aligned}$$
This limitation on the direction of the process is imposed by the second law of thermodynamics. The two most important forms of second law are given below. The first statement regards to a heat engine and the second one regards to a heat pump. These statements, although, have no mathematical proofs, but, so far their violation is not recorded.
Kelvin–Planck Statement—It is impossible to construct a cyclically operating device, which produces no other effect than the extraction of heat from a single thermal reservoir and delivers an equivalent amount of work.
Clausius Statement—It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lowtemperature body to a hightemperature body.
Suppose
\(\mathrm {Q_{Source}}\) and
\(\mathrm {T_{Source}}\) are the heat output from and absolute temperature of a hightemperature heat source, respectively; and
\(\mathrm {Q_{Sink}}\) and
\(\mathrm {T_{Sink}}\) are the heat input to and absolute temperature of a lowtemperature heat sink, respectively. Assuming the working substance as an ideal gas and all the processes to be reversible, the thermodynamic efficiency can be written as
$$\begin{aligned} \mathrm {\eta }&= \mathrm {1\frac{Q_{Sink}}{Q_{Source}}} \end{aligned}$$
It can be shown that the maximum efficiency is achieved if the process works on Carnot cycle.
$$\begin{aligned} \mathrm {\eta }&= \mathrm {1\frac{T_{Sink}}{T_{Source}}} \end{aligned}$$
Entropy can be regarded as a measure of disorder or randomness in a system. The change in entropy is given by
$$\begin{aligned} \mathrm {dS}&= \mathrm {\frac{\delta Q}{T}+dS_{irrev}} \end{aligned}$$
Since
\(\mathrm {dS_{irrev}>0}\), the above equation is also written as
This is known as the Clausius inequality.
A relation between thermal properties such as pressure, temperature, and density is known as the thermal equation of state. For a perfect gas, the thermal equation of state is the ideal gas law, given as \(\mathrm {p=\rho RT}\)
For any gas, the commonly referred calorical properties are internal energy, enthalpy, and entropy. Any relation between the calorical properties and the thermal properties is called the calorical equation of state: \(\mathrm {c_{\forall }=\left( \frac{{{\partial }}u}{{{\partial }}T}\right) _{\forall }}\) and \(\mathrm {c_{p}=\left( \frac{{{\partial }}h}{{{\partial }}T}\right) _{p}}\).
A perfect gas is the gas which has intermolecular spacing so large that the intermolecular forces are neglected. For a perfect gas, both \(\mathrm {c_{p}}\) and \(\mathrm {c_{\forall }}\) are constant and independent of temperature. Such a gas is termed as calorically perfect gas. A perfect gas is always thermally as well as calorically perfect. That is, it must satisfy both the thermal equation of state and the calorical equations of state. Moreover, a calorically perfect gas will always be thermally perfect but vice versa is not true. Hence, thermal perfectness is a precondition for caloric perfectness.
From kinetic theory of gases, the specific heat ratio
\(\mathrm {\left( \gamma \right) }\) can be represented in terms of degrees of freedom
\(\mathrm {\left( n\right) }\) of gas molecules as
$$\begin{aligned} \mathrm {{{\gamma }}}&= \mathrm {\frac{n+2}{n}} \end{aligned}$$