Advanced9 min read6 / 9

Turboprops & turboshafts

Push the bypass logic to its limit — put all the useful energy on the shaft and almost none in the jet — and the engine stops being a thrust machine and becomes a power machine. Geared to a propeller it is a turboprop; spinning a helicopter rotor, a pump or a generator it is a turboshaft. Same core, same maths, different customer.

The free power turbine

Both engines add a second turbine downstream of the gas generator’s own. The first turbine still balances the compressor exactly as in the turbojet analysis ( $\tau_t$ from the work balance). The free power turbine then extracts essentially everything that remains above ambient, leaving just enough pressure for a soft residual jet. Mechanically it shares no shaft with the core — which is why a helicopter’s rotor can hold constant speed while the gas generator spools up and down beneath it.

Per kilogram of airflow, the ideal shaft power is the energy the turbojet would have put into its jet, minus the little kept for the exhaust:

\frac{P_{\text{shaft}}}{\dot m} \;=\; c_p\,T_{t5}\left(1 - \frac{1}{\tau_r\tau_c\tau_t}\right)\eta_{\text{extract}} \;\approx\; \frac{V_{9,\text{turbojet}}^2}{2}

(1)

and the two figures of merit change accordingly — power replaces thrust:

\text{PSFC} = \frac{\dot m_f}{P_{\text{shaft}}} \quad \left[\tfrac{\text{g}}{\text{kW·h}}\right], \qquad F_{\text{prop}} = \frac{\eta_{\text{prop}}\,P_{\text{shaft}}}{V_0}

(2)

$\text{PSFC}$: power-specific fuel consumption; good turboshafts reach ~270–330 [g/(kW·h)]
$\eta_{\text{prop}}$: propeller efficiency, ~0.8–0.88 in its design regime [–]
$F_{\text{prop}}$: propeller thrust — note the division by V₀ [N]

Why propellers give up near Mach 0.6–0.7

A propeller blade’s tip moves through the air at the vector sum of rotation and flight speed:

V_{\text{tip}} = \sqrt{(\Omega R)^2 + V_0^2}

(3)

Rotation alone already puts tips near Mach 0.7–0.8. Add flight speed and the tips go supersonic while the aircraft is still deeply subsonic — shocks form, drag on the blade soars, efficiency collapses, and the noise becomes intolerable (ask anyone who has heard supersonic-tip props at an airshow). Compressibility — the same physics as the aerodynamics course’s critical-Mach article — sets the propeller’s speed limit, not the engine behind it.

Turboshaft: the same engine with no V₀ at all

A helicopter’s “flight speed” through its own rotor is nearly zero, so the residual-jet question disappears entirely: useful output is pure shaft power, PSFC is the only economy number, and power-to-weight is king — a modern turboshaft delivers roughly five times the power per kilogram of a piston engine, which is why piston helicopters are now only trainers. The demo’s turboshaft page therefore swaps the thrust readouts for shaft power and PSFC, and its cutaway shows the concentric-shaft gearbox doing the real work.