Propulsive efficiency

In aircraft and rocket design, overall propulsive efficiency $eta$ 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:

$eta\; =\; eta\_c\; eta\_p$, [ [http://www.hq.nasa.gov/pao/History/SP-468/ch10-3.htm ch10-3 ] ]

where $eta\_c$ is the cycle efficiency and $eta\_p$ 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.
*Other engine internal inefficiencies such as friction losses, power taken off to drive accessories like compressors and generators etc. etc. These typically give waste heat that represent 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

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, [K.Honicke, R.Lindner, P.Anders, M.Krahl, H.Hadrich, K.Rohricht. Beschreibung der Konstruktion der Triebwerksanlagen. Interflug, Berlin, 1968] is

:$eta\_p\; =\; frac\{2\}\{1\; +\; frac\{c\}\{v$

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.

Rocket engines

Rocket engine's $eta\_c$ 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 being lost as heat energy in the exhaust.

Rocket engines have a slightly different propulsive efficiency $eta\_p$ than airbreathing jet engines since rockets are able to exceed their exhaust velocity, and the lack of intake air changes the form of the equation somewhat. See diagram.

:$eta\_p=\; frac\; \{2\; frac\; \{u\}\; \{c\; \{1\; +\; (\; frac\; \{u\}\; \{c\}\; )^2\; \}$Rocket Propulsion elements- seventh edition, pg 37-38]

Propeller engines

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:

$550\; P\_e\; =\; frac\{eta\_c\; H\; h\; J\}\{3600\}$,

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:

$eta\_c\; =\; \{14\}/\{C\_p\}$,

expressed as a percentage.

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

$eta\; =\; \{12\}/\{C\_p\}$.

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 $eta$ of the R-3350 occurs at a lower speed because of the aforementioned loss of propulsive efficiency of the propeller as Mach approaches 1.

ee also

*Specific fuel consumption
*Jet engine
*Propeller engine

References

*cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468|url=http://www.hq.nasa.gov/pao/History/SP-468/cover.htm|accessdate=2006-04-22
*cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468 Appendix E|url=http://www.hq.nasa.gov/pao/History/SP-468/app-e.htm|accessdate=2006-04-22

Notes