OPTIMUM ALTITUDE
Originally Posted by mutt
In the FMS equipped aircraft, try changing the Cost Index Value to 0, also input route winds from the flight plan, then review the optimum flight level.
Mutt
Mutt
Best regards,
Westhawk
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But does aoa remain constant during cruise at max range speed? Neglecting wind/temp changes etc.
Regarding the first (wind), the airspeed for max range will have to increase in a headwind and decrease in a tailwind. The limiting speeds (or AoA) will be the speed at which the drag curve begins to rise sharply for the headwind, and max endurance speed for the tailwind.
Regarding the second (calm air), AoA will not truly remain constant in any turbulence, because different amounts of lift (and therefore AoA) are required at different load factors at a given speed. Therefore, it is better said that the average or nominal AoA will remain essentially constant.
If you set an FMS to fly at ECON Cruise at Cost Index = 0 (Max Range), the speed will decrease with decreasing weight, and adjust for winds. I assume there is also a temperature factor, but I don't know what the adjustment will be.
Also remember that we fly with the "artificial" constraint of constant altitude (or step climbs to discrete altitudes). Ideally an FMS would also adjust altitude to the current EXACT altitude for the weight and temperature, but in reality it adjusts for the current altitude and gives its best estimate of the optimum altitude. When that optimum altitude equals one to which we can be cleared, we climb/descend to it.
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This started as wind trade - and I have been remiss recently for not keeping upto date on these things (here). Below is a link from a web page made up from a write up of mine (here in PPRUNE) of a few years back - it is probably still in the archive. It says most of what the original query wanted as an answer and is still as true today as it was a few years back when I wrote same. The laws of physics not having changed over much since then 
http://www.iasa-intl.com/wtt.html
MG
http://www.iasa-intl.com/wtt.html
MG
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Optimum Altitude
To go right back to the original question, a quick 'rundown' on range profiles used by jet aircraft is warranted. It is assumed that in referring to Optimum Altitude, reference is made to Fuel related optimum altitudes, the two most important of the fuel related profiles being Maximum Range Cruise (MRC) and Long Range Cruise (LRC). Maximum Range Cruise is an aeronautical 'Absolute', it is the profile which provides, as the name suggests, the absolute maximum range possible. Long Range Cruise, however, is a convenient speed (now falling out of favour) which gives much improved speeds over MRC, but at a fuel penalty of 1% as compared to MRC, i.e. LRC provides 99% of the range possible with MRC.
By definition, maximum range occurs when the lowest amount of fuel is consumed for the distance flown, which is directly related to Fuel Flow divided by True Air Speed if considering the still air case, or Fuel Flow divided by Ground Speed if considering the effect of wind.
Fuel Flow is directly related to Thrust, and Thrust must equal Drag in a stable cruise situation.
If we examine drag curves for an aircraft, it is possible to ascertain the relationship between Speed and Drag, and therefore Speed and Fuel Flow.
Below Mcrit, drag is directly related to Equivalent Air Speed (EAS), that is, for a given weight, the EAS for MRC will be constant right up to the Pressure Height where Mcrit is encountered. If the first diagram is examined,
Any line drawn from the Zero / Zero origin (0/0) to intersect the Drag curve in 2 places (e.g. the LRC example in the diagram) will have 2 intersections, with differing EAS and Drag / Fuel Flow for each intersection. The GRADIENT (Steepness) of the line drawn indicates the specific range of the aircraft. For example, a high Fuel Flow paired with a relatively low speed is indicative of poor specific range, a lower Fuel Flow at an increased speed is indicative of improving specific range. If we continue to reduce the gradient of the line from 0/0 to the point of tangency with the Drag Curve, we are at the lowest possible gradient, and it is the speed at the point of tangency that represents MRC. In the diagram (and it is a 'true' diagram, not a sketched one), for the particular weight for that drag curve, we can see that Maximum Range Speed is 248 knots EAS. For the example chosen, Mcrit for the aircraft is M0.73, so, up to 32,583 feet, where 248 KEAS = M0.73, MRC will be 248 KEAS at all altitudes for this weight. This will require a varying CAS schedule, i.e.
10,000 ft = 249.9 CAS : 20,000 ft = 252.9 CAS : 30,000 ft = 258.0 CAS : 32,583 ft = 259.8 CAS / M0.73
FLIGHT ABOVE Mcrit - With increasing Altitude (Pressure Height) beyond the point where MRC EAS passes Mcrit, additional wave drag is encountered, and drag divergence occurs, i.e. for the Pressure Height in question, there is a whole new drag curve, being the 'original' drag curve up to Mcrit, then followed by initially quite slow drag divergence with small Mcrit exceedance, but later diverging much more aggressively. If you refer to the second diagram, and it's expanded view centred on the area of interest, -
EXPANDED VIEW
Three 'new' drag curves will be seen for 35000, 40000, and 45000 feet. (The curves are, again, mathematically correct, not sketches). It will be seen that the point of tangency for the line drawn from the 0/0 origin tangential to the transonic drag curve results in MRC slightly above Mcrit at F/L 350, but increasing to M0.77 at 40,000 feet and M0.80 at 45,000 feet (the latter two being substantially above Mcrit). It's worth noting that, for the example given, drag at 40,000 and 45,000 feet is less than that for 35,000 feet, even though the Mach Number is high, wave drag is high, but the drag arising from the low speed polar is low. (The curve 'belongs' to a fairly high flying aircraft, most conventional airliners exhibit the same characteristics about 5,000 feet lower). Mutt, therein lies my assertion that in 99% of jet operations, we 'live' above Mcrit, weeelll, not always!
By these means, it may be seen that for a given weight, MRC speed can be established for any useable altitude. That satisfies the requirements of aerodynamic efficiency, what must now be examined is engine and Fuel Flow efficiency.
Jet engines operate at their optimum Thrust Specific Fuel Consumption (TSFC), i.e. the least amount of fuel consumed to produce each unit of thrust, at quite high engine speed. This is typically in the region 90-93% N1, any lower or higher engine speed will result in poorer TSFC, with the worst 'off-optimum' penalty being found at higher than optimum engine speeds. Although the aircraft may be flown at the appropriate MRC at a low altitude, e.g. 10,000 or 20,000 feet, the amount of thrust available will be well in excess of that required, and will have to be 'throttled' to a lower than optimum TSFC engine speed. In short, although at the aerodynamic optimum for the Weight and Altitude, the engine will be well 'off optimum' TSFC. As altitude is increased, engine speed must be increased to maintain the required thrust in the reducing density, with consequential increase in engine TSFC efficiency. When the aircraft reaches a level where optimum TSFC engine speed is required to maintain the MRC speed, the aircraft is at OPTIMUM LEVEL.
As weight burns off, the AoA required will be less due to the lower weight, and MRC speed reduces (usually, but not always). The drag curve moves down, and to the left (by a lesser amount). To maintain MRC speed, engine thrust must therefore be reduced, to below optimum TSFC engine speeds. Although range will be increasing if the level is maintained due to the lower fuel flow, we are not doing as well as we could, we are 'off optimum'. The solution, in an ideal world of no ATC constraints, would be to maintain the engines at optimum TSFC N1, and allow the aircraft to cruise climb at MRC CAS or Mach Number. In short, Optimum Level will steadily increase in still air, due to the 'need' to maintain the engines at optimum speed if we are to obtain the maximum possible range.
Similar arguments apply to other fuel related Optimum Levels, for example, LRC.
EFFECT OF WIND - Wind effect may also be seen on the same Drag Curve. The Horizontal Axis of the Graph is speed (EAS in this case), and the effect of wind component is simply to shift the 0/0 origin. The 'tricky' bit is that the graph is for EAS whereas Wind Component is a TAS effect, and a wind component must be converted to the EAS equivalent. A 100 knot wind component is equivalent to 56 knots EAS at 35,000 feet at ISA. For a Headwind, if the line of tangency is projected from 56 Kt EAS to the Drag Curve, the new MRC for Ground Miles per unit of fuel may be found (at a HIGHER Mach Number). Conversely, for a Tailwind, if the line of tangency is projected from -56 Kt EAS (Left of the origin) to the Drag Curve, the new MRC for Ground Miles per unit of fuel may be found (at a LOWER Mach Number). Modern generation FMCs do calculate this, but a risky mix of 'real world' ground data remains mixed
with still air data in most FMCs, beware!
A piece of cake!
Regards,
Old Smokey
By definition, maximum range occurs when the lowest amount of fuel is consumed for the distance flown, which is directly related to Fuel Flow divided by True Air Speed if considering the still air case, or Fuel Flow divided by Ground Speed if considering the effect of wind.
Fuel Flow is directly related to Thrust, and Thrust must equal Drag in a stable cruise situation.
If we examine drag curves for an aircraft, it is possible to ascertain the relationship between Speed and Drag, and therefore Speed and Fuel Flow.
Below Mcrit, drag is directly related to Equivalent Air Speed (EAS), that is, for a given weight, the EAS for MRC will be constant right up to the Pressure Height where Mcrit is encountered. If the first diagram is examined,
Any line drawn from the Zero / Zero origin (0/0) to intersect the Drag curve in 2 places (e.g. the LRC example in the diagram) will have 2 intersections, with differing EAS and Drag / Fuel Flow for each intersection. The GRADIENT (Steepness) of the line drawn indicates the specific range of the aircraft. For example, a high Fuel Flow paired with a relatively low speed is indicative of poor specific range, a lower Fuel Flow at an increased speed is indicative of improving specific range. If we continue to reduce the gradient of the line from 0/0 to the point of tangency with the Drag Curve, we are at the lowest possible gradient, and it is the speed at the point of tangency that represents MRC. In the diagram (and it is a 'true' diagram, not a sketched one), for the particular weight for that drag curve, we can see that Maximum Range Speed is 248 knots EAS. For the example chosen, Mcrit for the aircraft is M0.73, so, up to 32,583 feet, where 248 KEAS = M0.73, MRC will be 248 KEAS at all altitudes for this weight. This will require a varying CAS schedule, i.e.
10,000 ft = 249.9 CAS : 20,000 ft = 252.9 CAS : 30,000 ft = 258.0 CAS : 32,583 ft = 259.8 CAS / M0.73
FLIGHT ABOVE Mcrit - With increasing Altitude (Pressure Height) beyond the point where MRC EAS passes Mcrit, additional wave drag is encountered, and drag divergence occurs, i.e. for the Pressure Height in question, there is a whole new drag curve, being the 'original' drag curve up to Mcrit, then followed by initially quite slow drag divergence with small Mcrit exceedance, but later diverging much more aggressively. If you refer to the second diagram, and it's expanded view centred on the area of interest, -
EXPANDED VIEW
Three 'new' drag curves will be seen for 35000, 40000, and 45000 feet. (The curves are, again, mathematically correct, not sketches). It will be seen that the point of tangency for the line drawn from the 0/0 origin tangential to the transonic drag curve results in MRC slightly above Mcrit at F/L 350, but increasing to M0.77 at 40,000 feet and M0.80 at 45,000 feet (the latter two being substantially above Mcrit). It's worth noting that, for the example given, drag at 40,000 and 45,000 feet is less than that for 35,000 feet, even though the Mach Number is high, wave drag is high, but the drag arising from the low speed polar is low. (The curve 'belongs' to a fairly high flying aircraft, most conventional airliners exhibit the same characteristics about 5,000 feet lower). Mutt, therein lies my assertion that in 99% of jet operations, we 'live' above Mcrit, weeelll, not always!
By these means, it may be seen that for a given weight, MRC speed can be established for any useable altitude. That satisfies the requirements of aerodynamic efficiency, what must now be examined is engine and Fuel Flow efficiency.
Jet engines operate at their optimum Thrust Specific Fuel Consumption (TSFC), i.e. the least amount of fuel consumed to produce each unit of thrust, at quite high engine speed. This is typically in the region 90-93% N1, any lower or higher engine speed will result in poorer TSFC, with the worst 'off-optimum' penalty being found at higher than optimum engine speeds. Although the aircraft may be flown at the appropriate MRC at a low altitude, e.g. 10,000 or 20,000 feet, the amount of thrust available will be well in excess of that required, and will have to be 'throttled' to a lower than optimum TSFC engine speed. In short, although at the aerodynamic optimum for the Weight and Altitude, the engine will be well 'off optimum' TSFC. As altitude is increased, engine speed must be increased to maintain the required thrust in the reducing density, with consequential increase in engine TSFC efficiency. When the aircraft reaches a level where optimum TSFC engine speed is required to maintain the MRC speed, the aircraft is at OPTIMUM LEVEL.
As weight burns off, the AoA required will be less due to the lower weight, and MRC speed reduces (usually, but not always). The drag curve moves down, and to the left (by a lesser amount). To maintain MRC speed, engine thrust must therefore be reduced, to below optimum TSFC engine speeds. Although range will be increasing if the level is maintained due to the lower fuel flow, we are not doing as well as we could, we are 'off optimum'. The solution, in an ideal world of no ATC constraints, would be to maintain the engines at optimum TSFC N1, and allow the aircraft to cruise climb at MRC CAS or Mach Number. In short, Optimum Level will steadily increase in still air, due to the 'need' to maintain the engines at optimum speed if we are to obtain the maximum possible range.
Similar arguments apply to other fuel related Optimum Levels, for example, LRC.
EFFECT OF WIND - Wind effect may also be seen on the same Drag Curve. The Horizontal Axis of the Graph is speed (EAS in this case), and the effect of wind component is simply to shift the 0/0 origin. The 'tricky' bit is that the graph is for EAS whereas Wind Component is a TAS effect, and a wind component must be converted to the EAS equivalent. A 100 knot wind component is equivalent to 56 knots EAS at 35,000 feet at ISA. For a Headwind, if the line of tangency is projected from 56 Kt EAS to the Drag Curve, the new MRC for Ground Miles per unit of fuel may be found (at a HIGHER Mach Number). Conversely, for a Tailwind, if the line of tangency is projected from -56 Kt EAS (Left of the origin) to the Drag Curve, the new MRC for Ground Miles per unit of fuel may be found (at a LOWER Mach Number). Modern generation FMCs do calculate this, but a risky mix of 'real world' ground data remains mixed
with still air data in most FMCs, beware!
A piece of cake!
Regards,
Old Smokey
Last edited by Old Smokey; 23rd May 2006 at 06:34.
