Isentropic Flow Calculator

Background.

When a gas expands without exchanging heat and without friction, its temperature falls by an amount fixed entirely by how far the pressure drops. Halve the pressure of air this way and its absolute temperature falls by about 18%: air at 300 K comes out near 246 K, well below freezing. Run the process the other way, compressing instead of expanding, and the same law explains why a compressor outlet runs hot and why a bicycle pump warms in your hand.

Such a process is called isentropic, because the entropy of the gas stays constant. It is an idealisation, but a central one. Nozzles, diffusers, turbines and compressors are all designed against it, and the gap between a real machine and the isentropic version of the same pressure change is what engineers call isentropic efficiency. Friction, turbulence and shock waves all add entropy, so a real compression ends hotter than the ideal one and a real expansion ends less cold.

The relation used here, T₂/T₁ = (p₂/p₁)^((γ−1)/γ), holds for an ideal gas whose heat capacities do not change with temperature. γ is the ratio of the gas's heat capacity at constant pressure to that at constant volume, and it sets how strongly temperature follows pressure. For air near room temperature it is about 1.4, giving an exponent of 2/7. For a monatomic gas such as helium or argon it is 5/3, giving 2/5, so helium cools more for the same pressure drop.

The answer is a ratio, not a temperature. Multiply it by the starting temperature in kelvin, never in degrees Celsius. The pressures must be absolute, since a gauge reading leaves out the atmosphere's 101,325 Pa and changes the ratio. The same physics, applied to a rising parcel of dry air, gives the atmosphere's cooling of about 9.8 °C per kilometre of ascent.

What is isentropic flow calculator?

The isentropic temperature ratio is the factor by which the absolute temperature of an ideal gas changes when its pressure changes reversibly and adiabatically, meaning with no heat exchanged and no losses. It has no units. A value below 1 means the gas cools as it expands; a value above 1 means it heats as it is compressed. It depends only on the pressure ratio p₂/p₁ and on the heat-capacity ratio γ = c_p/c_v, not on the size of the equipment or the speed of the flow.

How to use this calculator.

1. 01Enter the starting and final pressures as absolute pressures in pascals. Add 101,325 Pa to gauge readings, and multiply values in bar by 100,000.
2. 02Enter γ for the gas: about 1.4 for air at ordinary temperatures, or 5/3 (about 1.667) for monatomic gases such as helium and argon. It must be greater than 1.
3. 03Read the temperature ratio and multiply it by the starting temperature in kelvin to get the ideal final temperature. Subtract 273.15 to convert back to °C.
4. 04For a real compressor, turbine or nozzle, treat the answer as the loss-free limit: the actual outlet will be warmer than this, whether the gas was compressed or expanded.

The formula.

T₂/T₁ = (p₂/p₁)^((γ − 1)/γ). Subscripts 1 and 2 mark the start and end of the process, p is absolute pressure and T is absolute temperature. The exponent comes from combining the ideal-gas law with the isentropic relation p·v^γ = constant, which follows from the first law of thermodynamics for a reversible process with no heat transfer and constant heat capacities. γ is the ratio of specific heats, c_p/c_v. Only the ratio of the two pressures matters, so any unit works as long as both pressures share it; the page asks for pascals. The exponent is small, 0.2857 for air, so temperature responds far less than proportionally: compressing air tenfold raises its absolute temperature by a factor of about 1.93, not 10. The page requires positive pressures and γ above 1, and it leaves out shocks, friction, heat transfer and real-gas behaviour.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NASA Glenn Research Center, Isentropic Flow Equations.
2. [2]NACA Report 1135, Equations, Tables, and Charts for Compressible Flow.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.