Propulsive efficiency

In aircraft and rocket design, overall propulsive efficiency η is the efficiency, in percent, with which the energy contained in a vehicle's propellant is converted into useful energy, to replace losses due to air drag, or gravity, or to accelerate the vehicle.

Mathematically, it is represented as:

^[1]

where η_c is the cycle efficiency and η_p is the propulsive efficiency.

• The cycle efficiency, in percent, is the proportion of heat energy in the fuel that is converted to mechanical energy by the engine. It is always less than the Carnot efficiency because of heat that is necessarily lost in the engine exhaust and internal inefficiencies such as friction losses, power to drive accessories like compressors and generators, etc. These typically generate waste heat that represents loss of power.
• The propulsive efficiency, in percent, is the proportion of that mechanical energy that is actually used to propel the aircraft. It is always less than 100% because of kinetic energy loss to the exhaust, and less-than-ideal efficiency of the propulsive mechanism, whether a propeller, a jet exhaust, or a fan. In addition, propulsive efficiency is greatly dependent on air density and airspeed. For example, propulsive efficiency of propellers falls dramatically as the Mach number approaches 1.0 because of compressibility effects on the propeller blades; and there is always some energy lost due to the change in airspeed of the air itself from the propulsive process.

Estimating propulsive efficiency

Jet engines

Dependence of the energy efficiency (η) from the exhaust speed/airplane speed ratio (c/v) for airbreathing jets

For all jet engines the propulsive efficiency (essentially energy efficiency) is highest when the engine emits an exhaust jet at a speed that is the same as, or nearly the same as, the vehicle velocity. The exact formula for air-breathing engines as given in the literature,^[2] is

where c is the exhaust speed, and v is the speed of the aircraft.

A corollary of this is that, particularly in air breathing engines, it is more energy efficient to accelerate a large amount of air by a little bit than a small amount by a large amount, even though the thrust is the same. Note however, that duct jet engines give no thrust when the inlet and exhaust speeds are exactly the same.

Dependence of the propulsive efficiency (η_p) upon the vehicle speed/exhaust speed ratio (v/c) for rocket engines

Rocket engines

A rocket engine's η_c is usually high due to the high combustion temperatures and pressures, and long nozzle employed. The value varies slightly with altitude due to atmospheric pressure on the outside of the nozzle/engine, but can be up to 70%. Most of the remainder is lost as heat energy in the exhaust.

Rocket engines have a slightly different propulsive efficiency (η_p) than airbreathing jet engines as the lack of intake air changes the form of the equation. This also means that rockets are able to exceed their exhaust velocity. See diagram.

^[3]

As with ducted jet engines, matching the exhaust speed and the vehicle speed gives optimum efficiency in principle. Although in practice rocket exhausts are high and typically fixed, this can be a useful theoretical consideration. Unlike ducted engines, rockets give thrust even when the two speeds are equal.

Propeller engines

The calculation is somewhat different for reciprocating and turboprop engines which rely on a propeller for propulsion since their output is typically expressed in terms of power rather than thrust. The equation for heat added per unit time, Q, can be adopted as follows:

where P_e is engine output in horsepower, converted to foot-pounds/second by multiplication by 550. Given that specific fuel consumption is C_p = h/P_e and using the aforementioned substitutions for H and J, the equation is simplified to:

expressed as a percentage.

Assuming a typical propulsive efficiency η_p of 86% (for the optimal airspeed and air density conditions for the given propeller design), maximum overall propulsive efficiency is estimated as:

Examples

One of the most efficient aircraft piston engines ever built was the Wright R-3350 Turbocompound radial. Thanks to recapturing some of the exhaust energy through three turbochargers coupled to the driveshaft, it was able to achieve an overall propulsive efficiency of about 32% at Mach 0.5. This is about the same as a modern civilian turbofan engine at Mach 0.8. The peak η of the R-3350 occurs at a lower speed because of the aforementioned loss of propulsive efficiency of the propeller as Mach approaches 1.

See also

• Specific fuel consumption (thrust)
• Jet engine
• Propeller engine

References

• Loftin, LK, Jr.. "Quest for performance: The evolution of modern aircraft. NASA SP-468". http://www.hq.nasa.gov/pao/History/SP-468/cover.htm. Retrieved 2006-04-22. 
• Loftin, LK, Jr.. "Quest for performance: The evolution of modern aircraft. NASA SP-468 Appendix E". http://www.hq.nasa.gov/pao/History/SP-468/app-e.htm. Retrieved 2006-04-22. 

Notes

1. ^ ch10-3
2. ^ K.Honicke, R.Lindner, P.Anders, M.Krahl, H.Hadrich, K.Rohricht. Beschreibung der Konstruktion der Triebwerksanlagen. Interflug, Berlin, 1968
3. ^ George P. Sutton & Oscar Biblarz, Rocket Propulsion Elements, pg 37-38 (seventh edition)