maximum range speed (jet)

[adri737]

Hello everybody!

I've two questions concerning the maximum range speed of a jet aircraft.
When we take a look at the graph drag vs TAS , the lowest point on the drag curve is the Vmd, and the tangent from the origin of the curve is the speed for max range

1°) Why, mathematically speaking, the tangent from the origin will give the point of the "(TAS/DRAG)max"?

2°)Can someone explain me why the speed for max range is equal to 1,32 Vmd for a jet?

I tried to find some infos on the internet but the only one i found was the relationship Vmp/Vmd.

Thanks a lot :)

[CI54]

I'll try and answer your second question...

The jet engine (or turbofan engine in recent years) is a lot more efficient at higher speed as compared to the wings. After all, these are not gliders, therefore the most fuel efficient speed is a lot higher compared to the minimum drag speed because at those speed, the engine is sipping as little fuel as possible while covering the most distance forward...

That is the easiest explanation I can give. But if the question is why is it that the max range speed is exactly 1.32 Vmd, I'm afraid my knowledge doesn't go that deep...

[oxenos]

Q1. The tangent to the curve gives the lowest ratio of drag to TAS. Try taking figures off a graph and working out the ratio. Further up the curve will give a higher TAS, but at the expense of a disportionately higher drag. Further down the curve will give lower drag but with a greater loss of TAS.
Q2. There is, I believe, no mathematical reason why it should be 1.32. Since modern jets are pretty much clones of each other I suspect that 1.32 is a working figure which will apply to most aircraft

[de facto]

For those who already finished and forgot their ATPL::8

Max range Cruise B737: cost index 0 ECON.
Best L/D descent B737 :cost index 0 ECON.

[Oktas8]

Q1: In a jet, fuel flow is proportional to thrust. In S&L flight, thrust = drag. So min drag speed is equal to minimum consumption rate of fuel, which is a good speed for endurance.

Best range speed combines minimising fuel flow but also covering the ground at a reasonable speed, with minimum kg/hr fuel and maximum nm/hr. Since on the drag curve there is one value of thrust for each speed in S&L, best range speed occurs at only one value of minimum (kg/hr divided by nm/hr). If in school you were ever taught that the gradient of a line = "rise over run", you will see why the minimum gradient line represents minimum FF/TAS, just touching the drag curve at one point.

Long answer to a short question. Hope it helps.

[rudderrudderrat]

Hi adri737,

Can someone explain me why the speed for max range is equal to 1,32 Vmd for a jet?
A copy of Carson's mathematical reasoning is here:
http://cafefoundation.org/v2/pdf_tech/MPG.engines/AIAA.1980.1847.B.H.Carson.pdf

The relationship between Vmax range and Vmin drag is proportional to the fourth root of 3.

[adri737]

Thank you very much for your quick replies guys :ok: !!

[JY9024]

Great answers, I got asked this today while in the cruise, the best I could come up with is somewhere between Green Dot and VMO!!
JY

[bookworm]

2°)Can someone explain me why the speed for max range is equal to 1,32 Vmd for a jet?

Here comes the calculus... (^ is to-the-power-of)

If the drag curve has the shape D = V^2 + 1/V^2 (parasite drag + induced drag), then its minimum can be found by differentiating and finding the zero. So the minimum is at 2*V - 2/V^3 = 0, in other words V = 1. So for that curve, minimum drag occurs at V = 1. Assuming constant thrust-specific fuel consumption, that gives you best endurance.

The curve D/V represents the drag per unit distance travelled, and if you want to maximise range you minimise drag per unit distance travelled. D/V = V + 1/V^3 so the minimum is where 1 - 3/V^4 = 0, or V = 3^(1/4) = 1.32. Assuming constant thrust-specific fuel consumption, that gives you best range.

You can stick prefactors in front of the terms in the drag curve, but they cancel out and don't change the ratio between the minima of the two curves.

So the 1.32 ratio applies when:

* the drag curve has the shape V^2 + 1/V^2 (usually a pretty good approximation)
and
* thrust specific fuel consumption is constant (a truly lousy approximation for modern engines, but loved by text books).

[rudderrudderrat]

So the minimum is at 2*V - 2/V^2 = 0,

Should it not be 2*V - 2/V^3 = 0 ?

[bookworm]

Should it not be 2*V - 2/V^3 = 0 ?

Yes, sorry, well spotted. I'll correct it above to avoid confusion.

[adri737]

Thanks a lot!! now I understand from where the 1.32 comes from:ok:

[adri737]

I did the calculation, if someone wish the PDF file just ask me :)

[adri737]

I did the calculation, so if someone wants the PDF file just ask me:)

[Old Smokey]

An excellent and praise-worthy post from bookworm, whose post should put to rest queries fom those who wondered at the origin of the 1.32 'rule' for the theoetical jet transpot.:D

Bookworm has qualified his/her response well, adding 2 very significant 'it all depends' caveats with the last two notes.

I'd like to add a further consideration for the jet transport, and that is consideration for wave drag (the High Speed Polars) for flight above Mcrit. Bookworm has specified (correctly) that "D = V^2 + 1/V^2 (parasite drag + induced drag)", fine for flight below Mcrit (typically at altitudes below typical jet cruising levels), but at levels where 1.32 Vmd is above Mcrit, a third component for Total Drag must be considered, where -

Total Drag = Parasite drag + Induced drag + Wave drag.

An approximation for Wave Drag for a generic and theoretical jet is (V - Vmcrit)^3, thus the Total Drag (theoretical) is best summarised as -

D = V^2 + 1/V^2 + (V - Vmcrit)^3.......... (Which accounts for relatively small wave drag increments for a short time above Mcrit, but increasing substantially thereafter).

After applying calculus to find the Maxima and minima after this third drag component is considered, it inevitably leads to Speed for Maximum Range being BELOW 1.32 Vmd, and significantly below at higher altitudes.

99% of Jet operations are above the level where Mcrit comes into play, with Max Range Speed inevitably just above Mcrit. Thus, whilst the 1.32 Vmd 'rule' is a good approximation at lower levels, it becomes redundant at typical cruise levels for jet aircraft.

I am definately NOT criticising bookworm's excellent (and well qualified) post, I am ADDING to it.:ok:

Regards,

Old Smokey