Isentropic Relations for an Ideal Gas - Temperature and density
In thermodynamics, an isentropic process is an idealized thermodynamic process that is adiabatic and in which the work transfers of the system are frictionless; there is no transfer of heat or of matter and the process is reversible. Such an idealized process is useful in engineering as a model of and basis of comparison for real processes.
The word 'isentropic’ is occasionally, though not customarily, interpreted in another way, reading it as if its meaning were deducible from its etymology. This is contrary to its original and customarily used definition. In this occasional reading, it means a process in which the entropy of the system remains unchanged, for example because work done on the system includes friction internal to the system, and heat is withdrawn from the system, in just the right amount to compensate for the internal friction, so as to leave the entropy unchanged.
In fluid dynamics, an isentropic flow is a fluid flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.
Note that energy can be exchanged with the flow in an isentropic transformation, as long as it doesn’t happen as heat exchange. An example of such an exchange would be an isentropic expansion or compression that entails work done on or by the flow.
For an isentropic flow, entropy density can vary between different streamlines. If the entropy density is the same everywhere, then flow is said to be homentropic.
|T2||temperature at state 2 (K)|
|T1||temperature at state 1 (K)|
|ρ2||density at state 2 (kg/m3)|
|ρ1||density at state 1 (kg/m3)|
|γ||ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air) (dimensionless)|