Propulsive efficiency

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In aircraft and rocket design, overall propulsive efficiency η is the efficiency, in percent, with which the energy contained in fuel is converted into propulsive energy. Mathematically, it is represented as:

η = η_cη_p,^[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.
• 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.

[edit] Estimating propulsive efficiency

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

[edit] Jet engines

Heat added per unit time, Q, is calculated as:

,

where H is the heating value of fuel in British thermal units/pound, h is the fuel flow rate in pounds per hour, and J is the Joule constant of 778 foot-pounds force per BTU.

For SI units, the formula is:

where H is in joules per kilogram, h in is kilograms per second, or with h in kg/h, divide by 3600 as above. For these units, J = 1.

For a jet engine, either turbojet or turbofan, the propulsive power P is calculated as:

where T is thrust in pounds-force and V is airstream velocity in feet per second and P is power in ft·lbf/s, or T is in newtons and V is in meters per second which results in P in watts. (NB: This velocity is always below the speed of sound. Supersonic aircraft use special air intake configurations to reduce the incoming air to subsonic speeds.)

Thus, propulsive efficiency can be estimated as follows:

.

Given that specific fuel consumption is C_T = h / T and estimating H to be 18,500 BTU/lb, the equation, expressed as percentage, is simplified to:

or ,

where M is the velocity as a Mach number (again, always below Mach 1.0).

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

[edit] Rocket engines

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 being lost as heat energy in the exhaust.

Rocket engines have a slightly different propulsive efficiency η_pthan 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.

^[2]

[edit] 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:

,

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:

η_c = 14 / C_p,

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:

η = 12 / C_p.

[edit] 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.

[edit] References

• Loftin, LK, Jr.. Quest for performance: The evolution of modern aircraft. NASA SP-468. Retrieved on 2006-04-22.
• Loftin, LK, Jr.. Quest for performance: The evolution of modern aircraft. NASA SP-468 Appendix E. Retrieved on 2006-04-22.

[edit] Notes

1. ^ ch10-3
2. ^ Rocket Propulsion elements- seventh edition, pg 37-38