1. Polytropic Head & Thermodynamic Work
For real hydrocarbon gases with variable compressibility \(Z(P,T)\), compressor thermodynamic head is evaluated using Schultz's polytropic method per ASME PTC 10:
\[ H_p = Z_{avg} \left( \frac{R}{MW} \right) T_1 \cdot \left[ \frac{r_p^{(n-1)/n} - 1}{(n - 1) / n} \right] \]
where the polytropic temperature exponent \(m = \frac{n-1}{n}\) is related to the gas isentropic exponent \(k = C_p/C_v\) and polytropic efficiency \(\eta_p\) by:
\[ m = \frac{n - 1}{n} = \frac{k - 1}{k \cdot \eta_p} \]
The discharge temperature \(T_2\) and gas power consumption \(P_{shaft}\) are:
\[ T_2 = T_1 \cdot (r_p)^m \qquad P_{gas} = \frac{\dot{m} \cdot H_p \cdot g}{\eta_p} \qquad P_{shaft} = \frac{P_{gas}}{\eta_{mech}} + P_{loss} \]
2. Dynamic Surge Margin (\(SM\%\))
Surge occurs when gas throughput drops below the aerodynamic stalling limit of the impeller blades, causing pressure breakdown and cyclical flow reversal. The Surge Margin at any operating speed is defined as:
\[ SM\% = \left( \frac{Q_{actual} - Q_{surge}(N)}{Q_{actual}} \right) \times 100\% \]
Per API 617, continuous steady-state operation requires \(SM\% \ge 10\%\) to \(15\%\). When \(SM\%\) drops below the Surge Control Line, antisurge valves open to recycle cooled discharge gas back to the suction scrubber.
3. Aerodynamic Fan Laws (Affinity Scaling)
Compressor performance at off-design speeds is scaled from design conditions via aerodynamic similarity:
\[ \frac{Q_2}{Q_1} = \frac{N_2}{N_1} \qquad \frac{H_2}{H_1} = \left( \frac{N_2}{N_1} \right)^2 \qquad \frac{P_2}{P_1} = \left( \frac{N_2}{N_1} \right)^3 \]