Speed for maximum range

[karimi]

A little help from those crazy frenchies...

Maxi Specific Range= V/FF (V=Velocity=TAS)
Specific Comsuption (Csp)=FF/T (T=Thrust)

-->Maxi Range= V/T x Csp

For a given Csp (wich depends on Alt (lowest one at OPT Alt) and on RPM (lowest around 80%N1)) we can figure out the incidence which will give us the Maxi Range:
We all know the basic forces acting on an aircraft in straight and level flight:
W= 1/2xdxV2xSxCL d=air density
-> V= SQRT (2W/d S CL)
We also have for straight and level flight:
T=W/F F is the "Finesse" of the aircraft (in french) or the Lift/Drag Ratio= CL/CD

----> Maxi Range= SQRT (2W/d S CL) x F/W xCsp
=...............................x CL/ WxCD xCsp
= SQRT (2/W d S) x SQRT CL/CD xCsp

So we can see that for a given Weight, air density,wing area,
the incidence of Maxi Range is the one wich gives SQRT CL /CD
Maxi.
In practical as previously mentioned, it is approximately
1,3 Incidence of Finesse Max (Max Lift/Drag).

Sorry for all those computation but all the Graphs come from there...

[barit1]

LRC tables?

It's been a while, but my recollection is that Long Range Cruise tables are published on the theoretical basis of 98% of max range. The rationale is that at true max range, the speed stability is really poor and thus large thrust lever excursions are needed to try to hold speed, and the benefit of the slightly lower speed is thus lost.

Can anyone confirm this is still true?

[Old Smokey]

LRC is based upon 99% of Maximum Range Cruise.

Speed stability has nothing to do with MRC/LRC, it is a problem at Vmd, i.e. the optimum speed for Holding and Best Climb Angle, which is why the published speeds for these is a 'touch' on the high side, with minimal loss of optimum Holding / Climb Angle performance.

Regards,

Old Smokey

[Dick Whittingham]

The figure of 1.32 we have been talking about is derived exactly from the mathematics of the drag curves, as has been shown above. It is actually the fourth root of 3, rounded up a bit. HOWEVER, this is true for aircraft only if the drag co-efficients used in the formulae are constant over the range of speed we are talking about.

So long as the co-efficients are constant then for all cases of Cdp and Cdi the 1.32 relationship will apply. That is for all values of Cdp and Cdi from zero to infinity and any combination of these.

There are two well known cases where the co-efficients are not constant, and 1.32 no longer applies. These are at the rise in Cdp when wave drag comes in and in the "laminar bucket", an area around best L/D for some airfoils where wing laminar flow gives a local improvement in Cl/Cd

When the derived curves are drawn a tall and narrow total drag curve shows a poor L/D ratio and a fat low curve a good L/D ratio. In both cases, so long as the co-eficients are constants the tangent from the origin will meet the total drag curve at 1.316 times Vmd

Dick W

[Keith.Williams.]

It is certainly the case that if DP is proportional to EAS squared and DI is proportional to 1/EAS squared, throughout the entire speed range, and we ignore all other possibilities, then the best range will be at 1.316 VMD. Unfortunately this is rarely the case.

For DP to be strictly proportional to EAS squared, it is necessary for CDP to be constant. And for DI to be strictly proportional to 1/EAS squared it is necessary for CDI to be proportional to 1/EAS to the fourth power.

But above MCDR we get an increase in drag that is not described
by these equations. If this increase is very large then the right side of the drag curve will be very steep. It is entirely possible that this will cause best range to occur at a speed lower than 1.316 VMD.

If however the aircraft is designed for high speed flight, with for example area rule, supercritical wings, or swing wings, then the increase in drag above MCDR is unlikely to be so severe and MCDR will itself be greater. This will make the right side of the drag curve less steep, and more like the theoretical shape.

The use of things such as winglets will also affect both the DP and DI, thereby changing the shape of the drag curve. The exact effects in this case will depend upon the speed for which the winglets have been optimised.

Because of all of these possible factors, it is far too simplistic to simply state "the tangent to the drag curve is always at 1.32vmd, it's calculus".

As I said in my original contribution to this string "the 1.32 VMD figure is just conventional wisdom". It is reasonable starting point when teaching the subject or making initial guestimates. But to get really accurate results we must take all of the relevant factors into account.

[Dick Whittingham]

But no one said that the tangent was ALWAYS at 1.32 Vmd.

But if, as Alex and everyone else has pointed out, the curves are made up of an x squared and a one over x squared summed up then it is precise mathematics and not "conventional wisdom" that holds. I hope everyone is now quite clear about this.

Equally, I think everyone on this thread now understands how the mathematical result varies when Cdp and Cdi are not constants.

Dick W

[autoflight]

Let us not forget headwind / tailwind. If headwind is equal to LRC TAS, G/S will be zero and you will go nowhere. In this case, a higher speed would be an advantage, because at least you will be proceeding. In a high tailwind, you can afford to reduce speed to be in the tailwind for a longer period of time. So, the wind component might not be quite as much as TAS, but the principle still applies. Fly faster in a headwind & slower in a tailwind for max range.