Compressor Polytropic Work

Specific work, discharge temperature, and gas power for a polytropic compression — ideal-gas basis, any n you specify.

Inputw = (n/(n−1)) · (R·T₁/M) · [(P₂/P₁)^((n−1)/n) − 1] T₂ = T₁ · (P₂/P₁)^((n−1)/n)

Inlet pressure (absolute) (kPa)or enter psia — converts to kPaDischarge pressure (absolute) (kPa)or enter psia — converts to kPaInlet temperature (K)or enter °F — converts to KPolytropic exponent nMolar mass (g/mol)Mass flow (optional) (kg/s) — optionalor enter lb/s — converts to kg/s

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The engineering

A real compressor stage is neither isothermal nor perfectly adiabatic — it follows something close to PVⁿ = constant, with n set by how much heat leaves through the casing and coolers. Give this card the suction state, pressure ratio, gas molar mass, and your polytropic exponent, and it returns specific work, head, discharge temperature, and (with a mass flow) the gas power the driver must supply before mechanical losses.

Sanity checks: for an uncooled stage, n runs slightly above the isentropic k when losses heat the gas (typical centrifugal air stage n ≈ 1.45–1.5 vs k = 1.4); intercooled or jacketed machines pull n below k, toward 1 for perfect isothermal compression. If your predicted discharge temperature exceeds roughly 150 °C on air service, most vendors will push you to intercool. The polytropic exponent ties to polytropic efficiency via ηₚ = [(k−1)/k] / [(n−1)/n].

This is an ideal-gas calculation. At high pressures or near the dew point, real-gas behavior matters — multiply by an average compressibility Z, or use the full Schultz method for a contract-grade number.

Where this math comes from

The PVⁿ generalization traces to W. J. M. Rankine's steam-engine analyses of the 1850s, but the word 'polytropic' belongs to Gustav Zeuner, the Dresden professor whose Technische Thermodynamik (1887) gave locomotive and compressor designers a single exponent to describe any curve between isothermal and adiabatic. It was an engineer's abstraction from the start — Zeuner cared about indicator diagrams, not statistical mechanics.

The modern compressor usage was cemented by John M. Schultz of Ingersoll-Rand, whose 1962 ASME paper 'The Polytropic Analysis of Centrifugal Compressors' worked out how to apply the polytropic path to real gases and became the backbone of the ASME PTC 10 test code. Perry's Chemical Engineers' Handbook carries the ideal-gas form this card computes as the standard first-pass sizing equation.

1859W. J. M. RankineManual of the Steam Engine treats expansion curves of the form PVⁿ = constant.
1887Gustav ZeunerCoins 'polytropic' in Technische Thermodynamik, unifying isothermal and adiabatic as special cases.
1962John M. SchultzASME paper extends polytropic compressor analysis to real gases — the industry-standard method.
1997ASMEPTC 10 performance test code pins polytropic head and efficiency as the acceptance metrics for centrifugal compressors.

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