Speeds for best range or endurance don't depend on altitude

This note is about Vbr (speed for best range) and Vbe (speed for best
endurance) - as well as specific range SR (nm per pound of fuel) and
specific endurance SE (hr per pound of fuel) themselves - and how these
quantities depend on (fixed pitch propeller) airplane gross weight W and
(density) altitude hRho. This was the subject of a recent post about
"throttle height." I added my two cents on specific range there, but I think
the subject is worth another, and better organized, nickel. I should have
included a page or two on this weight-and-altitude effect in Performance of
Light Aircraft, but I didn't. These things are important to understand in
case you ever find yourself low on fuel and far from home on a dark night.



Here are some numbers calculated for a Cessna 172. Units a [W]lbf,
[hRho]ft, [Vxx]KCAS, [SR]nm/lb fuel, [SE]hr/lb fuel.



1. W=2400 & hRho=0: Vbr=73, SR=2.40, Vbe=62, SE=0.0357

2. W=2400 & hRho=8000: Vbr=73, SR=2.40, Vbe=62, SE=0.0317

3. W=2000 & hRho=0: Vbr=67, SR=2.88, Vbe=57, SE=0.0470

4. W=2000 & hRho=8000: Vbr=67, SR=2.88, Vbe=57, SE=0.0416.



So your altitude doesn't matter for either Vbr or Vbe or SR. For best
endurance, however, lower altitude is better by a factor of square root of
sigma. Lower weight is best for both range and endurance and lower weight
results in lower values for both Vbr and Vbe.



The way this was calculated is much too complicated to discuss in ASCII, but
the chapter on Cruise and Partial-Throttle Performance in the book mentioned
above has all the formulas along with sample calculations. I use none of the
usual unrealistic approximations such as constant propeller efficiency eta,
constant lift coefficient CL, etc. The partial-throttle bootstrap extension
allows one to calculate how much engine torque is needed to fly level at any
given airspeed, altitude, and gross weight. Also allows one to find what
value of RPM is required to do that. I did assume a constant value of brake
specific fuel consumption rate (BSFC), but that is veridical for these low
power settings. (For this airplane BSFC takes a step up at around 122 HP,
76% power, from 0.45 to 0.51.)



Why is none of this in your POH? The POH cruise table is very poorly
designed. It uses the wrong independent variables for entering the tables.
(There is a problem with double valuedness of airspeed at low RPM values;
that's why the cruise tables don't go down there. In a sense this is the
partial throttle version of airplanes' having two speeds for full-throttle
level flight, the low one and the high one.)



The GAMA-format POH section 5 (Performance) does indeed show some advances
over most of its predecessors, but still has quite a ways to go to properly
embody brevity, simplicity, and safety. Well, it was a committee thing, and
as Robert A. Heinlein's character Lazarus Long said: "A committee is a life
form with six or more legs and no brain."



Hope this note helps.



John

--
John T Lowry, PhD
Flight Physics
5217 Old Spicewood Springs Rd, #312
Austin, Texas 78731
(512) 231-9391

What about wind? If Vbr is equal to the headwind component, range will be
zero. With one hour fuel. Vbr + 1 will give you a range of a mile. The
faster you go, the more range. At some speed though, increase in fuel
consumption offsets the gain.

Could you give us a simple and usable formula for the Skyhawk in your
example?

--
Roger Long

I forgot to mention, in that more recent post, that I was only considering
the case of no headwind, no tailwind. In the earlier post I did specify
that.

A sample result is shown for a 20-knot headwind (for a C172 at 6000 ft) on
pages 335-336 of Performance of Light Aircraft. It's essentially a
trial-and-error calculation. The 20-knot headwind raises Vbr=73 KCAS to
Vbrhw=78 KCAS and depressed the maximum specific range about 20%.

Specific range, in general, is (V-Vhw)/cP, where V is airspeed, Vhw is
headwind speed, c is brake specific fuel consumption rate, and P is engine
power. So a simple formula. But the optimal SOLUTION to that simple formula,
especially in the partial-throttle bootstrap approach, is far from simple.
That's because the formulas for engine torque and for RPM required for level
partial-throttle flight are somewhat complicated. Numerical solution, to
optimize, is easiest.

Everything Roger said about headwind effects is correct. Get enough headwind
and you'll fly backwards. In which case best range is given by landing
immediately!

John
--
John T Lowry
5217 Old Spicewood Springs Rd, #312
Austin, Texas 78731
(512) 231-9391

"Roger Long" om wrote in
message ...
What about wind? If Vbr is equal to the headwind component, range will be
zero. With one hour fuel. Vbr + 1 will give you a range of a mile. The
faster you go, the more range. At some speed though, increase in fuel
consumption offsets the gain.

Could you give us a simple and usable formula for the Skyhawk in your
example?

--
Roger Long

"John T Lowry" wrote in message


No comments except to say that it's good to have you back, John.


--
Jim Fisher

if your flying backwards how do you get to the airport to land ?


John T Lowry wrote:

Get enough headwind
and you'll fly backwards. In which case best range is given by landing
immediately!

Turn around? (grin)

Dave

Jeff wrote:
if your flying backwards how do you get to the airport to land ?


John T Lowry wrote:


Get enough headwind
and you'll fly backwards. In which case best range is given by landing
immediately!