Joined: 11 Dec 2005
|Posted: Mon Apr 09, 2012 3:42 am Post subject: Optimum Range, Altitude and Endurance for Jets
| Optimum Range:
Specific Air Range (SAR) is the maximum distance covered per unit of fuel.
SAR = TAS / fuel flow
Specific Fuel Consumption (SFC) is fuel consumption per unit of thrust. It is thus thrust-specific (TSFC), meaning that the fuel consumption is divided by the thrust.
SFC = Fuel flow / Thrust
Fuel Flow = SFC x Thrust
Combining the two formulas (by putting the value of Fuel Flow in the SAR formula) we get:
SAR = TAS / (SFC x Thrust)
In a level unaccelerating flight, Thrust = Drag.
SAR = TAS / (SFC x Drag)
According to the above formula we can observe two things:
1) SAR is inversely proportional to SFC (i.e. less SFC will give a better SAR)
2) SAR is dependant on TAS/Drag ratio (best TAS to Drag ratio occurs at the tangent to the drag curve, 1.32 VMD)
In reference to the best SFC, jet engines are designed to be most efficient at around 90% RPM.
Best range speed is at 1.32VMD.
At low level this speed will not be met with 90% thrust.
Thus SFC will not be the best at low level.
However at a higher levels, 1.32 VMD speed can be flown with 90% thrust, thus improving SFC.
The level where this can be done is the optimum altitude.
Flying at the optimum cruise altitude at 1.32VMD with engines at 90% rpm, the fuel burn will reduce weight.
Weight decreases - Lift requirement decreases - Lift induced drag decreases - VMD deceases.
Thus 90% thrust will accelerate the aircraft beyond 1.32 VMD (because drag reduces and thrust remains the same).
So to maintain 1.32 VMD (or for that matter any speed in a level flight) thrust needs to be reduced and the only way to do it without reducing engine RPM is to climb to a higher level.
Thus optimum altitude increases with reduction in mass.
Best option is to put the aircraft into a very Slow climb so that it is always at the optimum altitude where 90% thrust always gives 1.32VMD.
This steady continuous climb is called the cruise climb which is the optimum cruise profile for a jet.
However this may not be possible due to ATC restrictions.
Thus a compromise between a level cruise and a cruise climb is needed.
This compromise is called the Step Climb.
In a step climb you maintain the level and reduce the RPM to maintain the correct speed.
When weight reduces you climb to a higher level and repeat the process.
The SAR takes no account of wind.
However Gross Fuel Flow (GFF) caters for wind.
GFF = Fuel Flow / Groundspeed
Gross fuel flow (fuel mileage) can be given as kg/nm or nm/kg.
Endurance is the time that the aircraft can remain airborne.
This depends on fuel flow (SFC x Drag).
Thus optimum endurance is when SFC is lowest and Drag is minimum.
SFC will be lowest with thrust around 90% rpm.
Drag will be minimum at VMD.
Best endurance will be where these two parameters coincide i.e. 90% thrust maintains VMD.
At best range altitude 90% thrust maintains 1.32 VMD.
So for 90% thrust to maintain VMD (lower than 1.32 VMD), level will have to be raied.
Higher level will decrease the thrust (which will reduce speed) for the same (90%) rpm.
Thus best endurance altitude will be higher than the altitude for best range.
Best endurance speed in the AFM may be above VMD to get the advantage of speed stability which outweighs the reduction in endurance by flying faster.
Reduction in endurance is not much since the jet drag curves are very flat.