For an isentropic flow, entropy density can vary between different streamlines. If the entropy density is the same everywhere, then the flow is said to be homentropic.
Derivation of the isentropic relations
For a closed system, the total change in energy of a system is the sum of the work done and the heat added:
The reversible work done on a system by changing the volume is
where is the pressure, and is the volume. The change in enthalpy () is given by
Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), , and so All reversible adiabatic processes are isentropic. This leads to two important observations:
Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that
- , and
Using the general results derived above for and , then
So for an ideal gas, the heat capacity ratio can be written as
For a calorically perfect gas is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get
that is,
Using the equation of state for an ideal gas, ,
(Proof: But nR = constant itself, so .)
also, for constant (per mole),
- and
Thus for isentropic processes with an ideal gas,
- or