Isentropic Expansion and Compression

Compressors and Turbines

Unlike throttling valves (which just waste pressure), Compressors and Turbines (Expanders) are rotating machines designed to either inject mechanical work into a gas or extract useful mechanical work out of it.

• A Compressor uses a motor to inject shaft work into the gas, massively increasing its pressure and temperature.
• A Turbine (Expander) allows high-pressure gas to push against blades, extracting shaft work out of the gas, drastically dropping its pressure and temperature.

The Isentropic Ideal

To evaluate how perfectly a compressor or turbine is performing, engineers compare it against a theoretical thermodynamic benchmark known as an Isentropic Process.

An Isentropic process is one that is completely Reversible (no frictional losses, no turbulence) and Adiabatic (no heat escapes through the metal casing to the environment).

Because it is perfectly reversible and adiabatic, the Entropy ($S$) of the gas remains completely constant from the inlet to the outlet ($\Delta S = 0$).

Calculating Isentropic Work

For an ideal gas undergoing an isentropic compression or expansion, the relationship between pressure and temperature is governed by the specific heat ratio ($k = C_p / C_v$):

Because compressors inject so much energy so quickly, the gas exiting a compressor is extremely hot. The theoretical maximum temperature is the Isentropic Discharge Temperature ($T_2$).

The theoretical minimum work required to compress the gas (or maximum work extracted by a turbine) is proportional to the change in enthalpy, which is calculated as:

Real Machines and Isentropic Efficiency ($\eta$)

In the real world, nothing is perfectly reversible. Mechanical friction in the bearings, aerodynamic drag on the spinning impellers, and turbulent eddies in the gas all generate waste heat. This irreversibility means entropy always increases ($\Delta S > 0$).

Because of these losses, a real compressor requires more power than the theoretical isentropic calculation predicts. A real turbine generates less power than theoretical.

We quantify this imperfection using Isentropic Efficiency ($\eta_{is}$):

• For a Compressor: $\eta_{is} = \frac{W_{isentropic}}{W_{actual}}$ (Typically 75% to 85%)
• For a Turbine: $\eta_{is} = \frac{W_{actual}}{W_{isentropic}}$ (Typically 80% to 90%)

Because the actual work input for a compressor is higher, the actual discharge temperature of the gas will be significantly hotter than the ideal isentropic temperature.

Try the Calculator

Calculating actual discharge temperatures by hand for real gases is incredibly difficult due to the Z-factor. Use the Compressor Calculator on this page to let the simulator rigorously calculate the actual power required and the true discharge temperature!