How it works — the temperature method
A compressor's efficiency cannot be read from pressures alone: the same pressure ratio costs different work depending on how lossy the compression is, and the fingerprint of those losses is the discharge temperature. Measure \(P_1, T_1, P_2, T_2\) reliably and the polytropic exponent — hence the polytropic efficiency — falls out of two logarithms.
1 · Polytropic exponent n from the measured duty
A reversible adiabatic path with heat transfer folded into a constant exponent follows \(P\,v^{n}=\text{const}\), which combined with the gas law gives the temperature–pressure relation:
$$\frac{T_2}{T_1}=\left(\frac{P_2}{P_1}\right)^{\frac{n-1}{n}} \;\;\Longrightarrow\;\; n=\frac{1}{1-\dfrac{\ln(T_2/T_1)}{\ln(P_2/P_1)}}$$Two sanity gates before trusting \(n\): \(T_2>T_1\) (a compressor heats the gas; otherwise the logarithm is meaningless) and \(P_2>P_1\) (it is a compressor). This tool refuses to report an efficiency if either fails.
2 · Polytropic efficiency — and why it exceeds the isentropic one
The polytropic efficiency compares the reversible work of an infinitesimal-step ideal path with the actual work. Equating \(\frac{n-1}{n}=\frac{k-1}{k\,\eta_p}\):
$$\eta_p=\frac{k-1}{k}\cdot\frac{n}{n-1} \qquad\text{and inversely}\qquad \frac{n-1}{n}=\frac{k-1}{k\,\eta_p}$$The isentropic efficiency compares with a single finite pressure jump instead:
$$\eta_s=\frac{T_{2s}-T_1}{T_2-T_1} =\frac{r_p^{\frac{k-1}{k}}-1}{r_p^{\frac{n-1}{n}}-1},\qquad T_{2s}=T_1\,r_p^{\frac{k-1}{k}}$$Why \(\eta_p>\eta_s\) for the same machine: an ideal infinitesimal path reuses the "already-paid" work of earlier steps, while the isentropic reference must climb the whole pressure jump in one ideal step. The gap narrows as \(r_p\to1\) (the two definitions agree point-by-point) and widens with pressure ratio — which is why vendor curves quote polytropic efficiency across the whole map while test codes report both.
3 · Polytropic head and gas power
\(Z\) is the compressibility factor (1.0 for the ideal-gas screen; use a real-gas \(Z\) at high pressure). The gas power is the actual work rate delivered to the gas — shaft power is higher by the mechanical and seal losses.
Verdict thresholds (🟢🟡🔴)
- 🟢 ηp = 70–80% — the typical centrifugal-compressor band for a well-behaved stage.
- 🟡 ηp = 60–70% — below typical: check for fouling, off-BEP operation, or side loads.
- 🔴 ηp < 60% — poor performance or a measurement problem.
- 🟡 ηp = 80–85% — above the typical band: plausible at low pressure ratio (ηp drifts up as \(r_p\to1\)), but verify.
- 🔴 ηp > 85% — data suspect: ηp is approaching ηs, which usually means \(T_2\) was read badly (radiation/conduction on the probe, stratification, or a transient).
- 🔴 \(T_2\le T_1\) or \(P_2\le P_1\) — not a compression duty; no efficiency exists.
Temperature-method caveat: the entire answer lives in \(\ln(T_2/T_1)\), so \(T_2\) must be a stable, representative, calibrated measurement — a 1 K error at 420 K moves ηp by roughly half a point. ASME PTC 10 style practice: multi-point/averaged probes, verified location outside boundary layers, steady-state hold before reading.
Worked example (anchor case)
Air (\(M\) = 28.97, k = 1.4), \(P_1\) = 100 kPa, \(T_1\) = 300 K, \(P_2\) = 300 kPa, \(T_2\) = 420 K, \(Z\) = 1:
| Step | Value |
|---|---|
| \(r_p=300/100\) | 3.000 |
| \(n=\dfrac{1}{1-\ln(420/300)/\ln 3}\) | 1.4414 |
| \(\eta_p=\frac{0.4}{1.4}\times\frac{1.4414}{0.4414}\) | 93.3% (small-rp duty — flagged, see verdict logic) |
| \(T_{2s}=300\times3^{0.2857}\) | 410.6 K → \(\eta_s\) = 92.2% |
| \(W_p=1\times3.2656\times286.99\times300\times(3^{0.3063}-1)\) | 112.5 kJ/kg |
| Actual work \(=W_p/\eta_p=1.005\times(420-300)\) | 120.6 kJ/kg ✓ (cross-check via \(c_p\Delta T\)) |
| Gas power at \(\dot m=1\) kg/s | 120.6 kW |
The \(c_p\Delta T\) cross-check closing to within rounding is the whole point of the temperature method: the numbers are internally consistent no matter how good or bad the machine is.
Symbol table
| Symbol | Meaning | Units |
|---|---|---|
| \(P_1,\,P_2\) | inlet / discharge pressure | kPa / psi |
| \(T_1,\,T_2\) | inlet / discharge temperature (absolute) | K / °R |
| \(T_{2s}\) | isentropic discharge temperature | K / °R |
| \(r_p\) | pressure ratio \(P_2/P_1\) | — |
| \(n,\,k\) | polytropic / isentropic exponent | — |
| \(\eta_p,\,\eta_s\) | polytropic / isentropic efficiency | % |
| \(M\) | molecular weight | kg/kmol |
| \(R\) | specific gas constant \(=8314.5/M\) | J/(kg·K) |
| \(Z\) | compressibility factor | — |
| \(W_p\) | polytropic head | kJ/kg / ft·lbf/lb |
| \(\dot m\) | mass flow | kg/s / lb/min |
| \(P_{gas}\) | gas power (delivered to the gas) | kW / hp |
Engineering criteria applied
- PTC 10 style conventions — absolute temperatures only; \(T_2\) is the discharge temperature measured at a representative, steady location; states are the flange-to-flange inlet and discharge of the compressing section.
- Ideal-gas screen — \(Z=1\) and constant \(k\); for high-pressure or heavy gases substitute a real-gas \(Z\) and mixture \(k\) — the structure of the method is unchanged.
- Sanity gates first — \(T_2>T_1\) and \(P_2>P_1\) are checked before any efficiency is reported.
- Bands — 70–80% typical (green); 60–70% below typical (amber); <60% poor (red); 80–85% above typical (amber); >85% data suspect (red).