Isentropic process

In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process is useful in engineering as a model of and basis of comparison for real processes. This is idealized as reversible processes do not occur in reality; thinking of a process as both adiabatic and reversible would show that the initial and final entropies are the same, thus, the reason it is called isentropic (entropy does not change). Thermodynamic processes are named based on the effect they would have on the system (ex. isovolumetric: constant volume, isenthalpic: constant enthalpy). Even though in reality it is not necessarily possible to carry out an isentropic process, some may be approximated as such.
The word "isentropic" can be interpreted in another way, since its meaning is deducible from its etymology. It means a process in which the entropy of the system remains unchanged, as mentioned this requires the process to be both adiabatic and reversible. However, this could also occur in a system where the work done on the system includes friction internal to the system, and heat is withdrawn from the system in just the right amount to compensate for the internal friction, so as to leave the entropy unchanged. However, in relation to the universe, the entropy of the universe would increase as a result.
Background.
The second law of thermodynamics states that
where formula_2 is the amount of energy the system gains by heating, formula_3is the temperature of the surroundings, and formula_4 is the change in entropy. The equal sign refers to a reversible process, which is an imagined idealized theoretical limit, never actually occurring in physical reality, with essentially equal temperatures of system and surroundings. For an isentropic process, which by definition is reversible, there is no transfer of energy as heat because the process is adiabatic, "δQ" = 0. In an irreversible process of transfer of energy as work, entropy is produced within the system; consequently, in order to maintain constant entropy within the system, energy must be removed from the system as heat during the process.
For reversible processes, an isentropic transformation is carried out by thermally "insulating" the system from its surroundings. Temperature is the thermodynamic conjugate variable to entropy, thus the conjugate process would be an isothermal process, in which the system is thermally "connected" to a constant-temperature heat bath.
Isentropic processes in thermodynamic systems.
The entropy of a given mass does not change during a process that is internally reversible and adiabatic. A process during which the entropy remains constant is called an isentropic process, written formula_5 or formula_6. Some examples of theoretically isentropic thermodynamic devices are pumps, gas compressors, turbines, nozzles, and diffusers.
Isentropic efficiencies of steady-flow devices in thermodynamic systems.
Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process. The parameter that describes how efficiently a device approximates a corresponding isentropic device is called isentropic or adiabatic efficiency.
Isentropic efficiency of turbines:
Isentropic efficiency of compressors:
Isentropic efficiency of nozzles:
For all the above equations:
Isentropic devices in thermodynamic cycles.
Note: The isentropic assumptions are only applicable with ideal cycles. Real cycles have inherent losses due to compressor and turbine inefficiencies and the second law of thermodynamics. Real systems are not truly isentropic, but isentropic behavior is an adequate approximation for many calculation purposes.
Isentropic flow.
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 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 formula_15 is the pressure, and formula_16 is the volume. The change in enthalpy (formula_17) is given by
Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), formula_19, and so formula_20 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
Using the general results derived above for formula_25 and formula_26, then
So for an ideal gas, the heat capacity ratio can be written as
For a calorically perfect gas formula_30 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, formula_33,
also, for constant formula_38 (per mole),
Thus for isentropic processes with an ideal gas,
Table of isentropic relations for an ideal gas.
Derived from
where: