Mechanics of Fluids

488 Compressible flow of gases
The conditions under which a gas may be assumed perfect are discussed
in Section 1.4 and attention is there drawn to the precise definition of the
constant R and to its dimensional formula.
From Section 3.5.1 we recall the First Law of Thermodynamics, from
which was derived the Steady-Flow Energy Equation (3.13). Since thermal
energy and mechanical energy are interchangeable, amounts of either may
be expressed in terms of the same SI units. The inclusion of the mechanical
equivalent of heat, J, in equations expressed in SI units is therefore unnecessary.
It is in any case simply a conversion factor equal to unity and is
required only when data about heat and mechanical energy are expressed
with different units.
When the physical properties of a gas (e.g. its pressure, density and temperature)
are changed, it is said to undergo a process. The process is said
to be reversible if the gas and its surroundings could subsequently be completely
restored to their initial conditions by transferring back to (or from)
the gas exactly the amounts of heat and work transferred from (or to) it during
the process. A reversible process is an ideal never achieved in practice.
Viscous effects and friction dissipate mechanical energy as heat which cannot
be converted back to mechanical energy without further changes occurring.
Also, heat passes by conduction from hotter to cooler parts of the system
considered, and heat flow in the reverse direction is not possible. In practice,
then, all processes are, in various degrees, irreversible. A process may be considered
reversible, however, if velocity gradients and temperature gradients
are small, so that the effects of viscosity and heat conduction are negligible.
If, in the course of a process, no heat passes from the gas to its surroundings
or from the surroundings to the gas, then that process is said to be adiabatic.
During an adiabatic process the internal energy of the gas may change: the
gas may be in a thermally insulated container, and yet some of the kinetic
energy of particles may be converted to internal energy by viscous action.
An adiabatic process therefore is not necessarily reversible.
Work done on a fluid by its surroundings may compress it or increase its
kinetic energy (including the energy of eddies in turbulence) and potential
energy. During an infinitesimal reversible (i.e. frictionless) process the work
done on the fluid is therefore
p(−δV) + δ(KE) + δ(PE)
where p represents its absolute pressure, V its volume, and KE and PE
represent the kinetic energy and potential energy respectively. (The minus
sign appears because δ means ‘a very small increase of’.)
The First Law of Thermodynamics may be stated as:
Heat transferred to fluid + Work done on fluid = Increase in internal energy
+ Increase in Kinetic and Potential Energies
(In this statement all the terms have the dimensions [ML 2 T −2 ].)
Therefore
∴ δQrev + p(−δV) = δ(internal energy)
It is usual practice to express equations of thermodynamics which are
derived from or relate to the First Law of Thermodynamics in the form