The Optimum Cruise Altitude

[Bill Von Sennet]

al_morro


The optimum cruise altitude
« on: Aug 20th, 2008, 8:32am »

Hi Tom !

I would like to know what are the factors that determine the optimum cruise altitude of an aircraft, and what is the maximum pitch angle ( and AOA) accepted at cruise altitude (knowing that the dynamic pressure decreases with altitude leading to a decrease in lift, at constant speed).

Thanks alot.


Tom Goodrick

Re: The optimum cruise altitude
« Reply #1 on: Aug 20th, 2008, 11:49am »

First, the dynamic pressure only decreases with altitude if you fly the airplane the wrong way. You mention "constant speed" and it is clear you mean constant true airspeed. Nobody flies at constant true airspeed. They fly at constant indicated airspeed which guarantees that the dynamic pressure is constant. I should add that they try to fly at constant indicated airspeed though the engine cannot always produce enough power to allow this.

The pitot tube that measures air pressure avtually measures dynamic pressure which, according to Bernoulli's theroies is the difference betweeh total pressure and static pressure along a stream line. So it is measureing 0.5 * Density * speed squared. From that analog relations are used to calculate the airspeed. But, since no simultaneous measurement of density is possible, the calculation - or calibration - assumes sea level density for a "Standard Day". Therefore the indicated airspeed is directly related to the dynamic pressure but differs from the true airspeed by the square root of the ratio of density to standard sea level density.

This is why you must always fly using Indicated Airspeed, "KIAS," as the guide. Pilots are only interested in Indicated airspeed. The navigator is the only guy interested in True Airspeed, "KTAS," for estimating time of arrival.

Stall speed is always set in KIAS, never KTAS. The plane will stall at the same KIAS regardless of altitude.

Second, power or thrust always decrease with altitude but in ways that differ from the variation in dynamic pressure so we cannot generally hold the same indicated airspeed.

When we fly a Bonanza, we see 168 KIAS and KTAS at sea level on 75% power and about 173 KTAS at 5,000 ft where 75% is the max power attainable. At 5,000 ft we will see about 162 KIAS. If you climb higher, the power at full throttle will decrease and true airspeed will decrease even when you try to keep indicated airspeed the same.

In a Learjet, you lift off at about 150 KIAS and you fly at 250 KIAS to 10,000 ft (mandated by law) and then go to a higher speed such as 300 KIAS (if there is no forecast of turbulance) and 4000 fpm until about 20,000 ft where you must set 3000 fpm to keep the same KIAS, and 30,000 ft where you must set about 2,000 fpm to keep a decent KIAS (about 250). At 30,000 ft you start using Mach as the guide and try to maintain Mach 0.7 to 0.75 during the remaining climb. Normal cruise altitude starts at about 41,000 ft and increases as fuel is burned off. The true airspeed will be about 450 KTAS while the indicated airspeed is about 230 KIAS.

Because all aerodynmic forces and moments depend on dynamic pressure, the pilot strives to maintain the same dynamic pressure in the same stages of flight. One value is used for climb and descent and another value is used for cruise with many aircraft. The climb dynamics dictate some variations for jets. where you have large variations both in thrust and in weight between takeoff and cruise.

"Optimum Cruise Altitude" is determined as the best balance between engine performance and drag. The thrust of a jet engine varies with the pressure ratio (ambient to sea level). At 40,000 ft the max thrust is about 19-20% which is the same as the pressure ratio.

Because T=D and L=W in level flight, T/W= 1/ (L/D). This means that the required thrust per unit weight varies with the inverse of the lift/drag ratio. The lift/drag ratio depends on angle of attack and is the primary indicator of aerodynamic efficiency. CL and CD depends on angle of attack which depends on dynamic pressure - balancing those moments. Thus there is an indicated airspeed that is truly optimum. Unfortunately, this varies with weight which is continually changing in a jet. You see it gets complicated. Cruise is generally managed at a less than optimum lift/drag ratio that gives "good" speed with "decent" economy.


Angle of attack is generally the lowest at cruise speed. This means that induced drag is the least important at cruise speed and, therefore, fancy winglets have little to do with cruise. They do help during climb. On my panels I have a digital angle of attack indicator that works fine in cruise.

Here is a table of angle of attack (AOA) versus cruise speed for the Beech Bonanza V35B. All data was taken at 5,000 ft on a standard day.

KIAS KTAS ETrim AOA Power
162_173__-0.8_1.6__75.04%
154_164__-0.1_1.8__64.88%
144_154___0.8_2.2__55.04%
131_141___2.4_2.7__42.88%
127_136___3.1_2.9__38.56%
85__91___15.6_7.1__38.28%
68__75___24.0_14.0_28.12%

In real life, most aircraft are designed so that they will fly level (zero pitch) at cruise. They set the wing at an incidence angle to the fuselage axis to achieve this. In FS we try to do this by adjusting the lift-AoA table. In FS9 we cannot simply adjust the wing incidence because the brainless folks governing the development of FS removed the incidence angle as an adjustment.

[Tom Goodrick]

The above statement was written a little quickly one morning in response to Al Morro's question. It is essentially correct but it is complicated and portions of it may require more explanation. please feel free to ask questions and I'll try to prepare good answers.

One other aspect of this problem relates to time-to-climb and flight length. on many short trips, the "optimum altitude" as determined by airlines using compleicated mathematical routines will be substantially lower than the altitude on a long trip for best economy. Some discussion of this will be found under "CRUSING" under FS PILOTING TECHNIQUES.

[Tom Goodrick]

I feel I should say something about MAXIMUM CRUISE ALTITUDE. There is a chance al morro may have intended to ask about maximum rather than optimum altitude. For maximum altitude the aircraft is near its maximum angle of attack at cruise because a maximum lift coefficient is required to holde the weight of the aircraft when the density is very low.

The maximum altitude is not a practical consideration in flight. Many factors join to make the maximum altitude unattainable to most pilots. The reason for this is that you would probably run out of either fuel or life support before you reached it. If you try to climb any aircraft as high as you can, you'll see that the rate of climb decays considerably. For that reason, you'll not find "Maximum Altitude" listed as a specification for any aircraft in modern aviation literature. Back in the early days of flying, the aviation press was full of superlatiuves. They used the terms Maximum Altitude and Maximum Speed. But as the general experience with aviation increased, these terms were dropped. Maximum speed is also of no practical consideration. (Max cruise speed is, however, still discussed, especially by people trying to sell airplanes. We'll discuss that eventually.)

Incidentally, by "life support" I mean't mainly oxygen though it does get awfully cold up there too. If you takeoff in your Bonanza with a new engine and maybe some custom anti-stall devices to test your max altitude, you'll probably have two or three other people in addition to full tanks. The people require continuous oxygen above 12,000 ft. (The service ceiling of a V35B Bonanza is 17,858 ft). Oxygen tanks for that many people may have a limit of 30 minutes.

Aircraft specifications do include the term "Service Ceiling" which is essentially a more practical term than maximum altitude. It is defined as the altitude at which the fully loaded aircraft can climb at 200 feet per minute. Few pilots ever get close to that altitude in aircraft. That is not practical either.

What is causing the limitation in altitude is the "tug of war" or attempted balance of forces between the wing and the engine. To sustain itself in level flight, the aircraft engine must develop enough thrust to balance the drag produced by the aircraft and, mainly, by its wing. As the altitude increases, any air-breathing engine runs out of thrust. The lift force is the product of the lift coefficient, the wing area and the dynamic pressure (which depends both on density and true airspeed or depends solely on indicated airspeed). You can see that if we fly at a minimum indicated airspeed, adequate to make enough lift to support the weight of the aircraft, we can still have a fairly high true airspeed if we are high enough. However, that same minimum indicated airspeed is what we call "Stall Speed" so that should tell you something. Hang on that minimum indicated airspeed and you are on the verge of stall. Do anything to require more lift, like a turn, and you will stall.

Now consider whether you can expect the engine to provide adequate power to overcome the drag. At that maximum lift coefficient you have both the profile or zero-lift drag coefficient and the maximum induced drag drag coefficient which varies with the square of the lift coefficient. This does not mean the drag force is a maximum as the dynamic pressure is low. But it does show what the composition of forces involves. The drag coefficient would be about six times higher than at your normal cruise condition.

For any aircraft, the Service Ceiling is a spec that must be determined by testing. When the aircraft is undergoing flight testing, it goes through a series of tests to determine numbers that can be predicted by design equation but are not known until actually demonstrated in testing. If you wish, you can do this in FS9. Just fly high until you can sustain a climb speed of only 200 fpm. I would assume you would do this with takeoff at full gross weight. Of course, it would be impossible in real life to have the aircraft at full gross weight when you reach such a high altitude as you would have burned off a lot of fuel getting there. But I think all specs are based on full gross weight.

For those who may want to know more, we can discuss power available and power required for planes with props and thrust available and thrust required for planes with jet engines. That might be added below sometime soon. Ignore it if you don't want the added confusion. The discussion above is all most people care to know.

[Tom Goodrick]

Here are some actual values of the Service Ceiling specified for certain aircraft that you might fly in FS9:

Cessna 152__________14,700 ft
Cessna 172 (160hp)___14,200 ft
Cessna 182__________16,500 ft
Piper Ch 6 300________16,250 ft
Cessna Stationair 6____14,800 ft
Cessna 182 RG________14,300 ft
Baron B55____________19,300 ft
Baron 58_____________18,600 ft
Rockwell Shrike Cmdr___19,400 ft
Cessna 340___________29,800 ft
Cessna 411A__________31,350 ft

Mooney Bravo_________25,000 ft

Turboprops Certified Ceilings or "Max Operating Alt"
Rockwell 690B___________32,800 ft
MU-2__________________28,000 ft
King Air B200____________35,000 ft
Merlin 300______________31,000 ft
King Air 350_____________35,000 ft


JETS Certified Ceilings:
Beech 400______________45,000 ft
Learjets 31, 45, 60_______51,000 ft
Learjet 35______________41,000 ft

MD-90_________________37,000 ft
Boeing Airlines__________Not Specified

Notes: Most turbocharged piston aircraft like the Mooney are given a certified ceiling that is the highest altitude they are allowed to fly. Since all flights above 18,000 ft are IFR flights with altitudes contiunously read by ATC, you don't ever fly above a certified ceiling intentionally. Your flyling career would be adversely affected. The Cessna 340 and Cessna 414 are pressurized as well as turbocharged. You would normallay not exceed the altitude at which the cabin is at 8,000 ft which is about 21,000 ft.

Turboprops are jets that are capable of operating at very high altitude. But they are limited by controllability issues. This also applies to many fanjets. A case in point is the Learjet 35. When it was first put on the market, it had a certified altitude of 51,000 ft. Then a few of them disappeared from radar screens and were found in small pieces. The ceiling was reduced to 41,000 ft which is common for many small jets. The fact that other Learjets are rated to 51,000 ft means very little. You can only climb that high with a light cabiin load after most fuel has burned off. This gives you a slight economy boost at the end of the trip before a steep glide to the destination.

Jets are limited by controllability and stability which is significantly diminished at high altitude. These are also related to Mach number. A slight lack of care in performing a maneuver can lead to exceeding the Mach limit which will cause loss of control effectiveness and can lead to catastrophic events such as separation of elevators and ailerons.

[Smithy]

Very informative info there Tom and as I havent read every part of it yet I may be asking a question already answered, but~

How does one determine the cruise speed

not only for your planes specified, but in general?

cheers mate



*ok maybe it was already answered, but~

[Tom Goodrick]

There are some fancy equations that can be plotted using Excel. We try to avoid those nasty equations on the Forum.

I always work from published specs for the aircraft when I develop FD files. I have many books and magazines covering most aircraft in the world for several years back. I subscribe to FLYING and AOPA PILOT magazines for data on new aircraft.

But the best way for the user to do it is to pick a set of altitudes. For a normally-aspirated engine I'd pick 3,000 through 10,000 at 1000 ft intervals. Climb to the lowest altitude and set 75% power. Today we have gauges on the panel that show power directly. (In the real world they are called fuel or engine management computers, rather cheap.) Let the autopilot control the plane and see what airspeed you get (indicated and true) for 75% power. Lean the mixture properly. Repeat at each altitude. Jot down the feul flow as well as the airspeeds. At some point the engine will not produce 75% so you use max throttle. When I set up the FD files, I use a published cruise spec (altitude, true airspeed and fuel flow) and make the model match each spec at the one specified altitude. Then it is usually very close to real values at all other altiudes if you have other conditions to check. This check must be made in zero wind and standard temperature profile - "Clear Weather".

In some cases you take a cue from the published info on the aircraft. For example the published specs for the Mooney Bravo we all have show two conditions, 25,000 ft and 10,000 ft at the prescirbed max continuous cruise power of 89% (obviously established by a Marketing Director). The data show:
25,000 ft 214 KTAS 20.5 gph 89% power
10,000 ft 188 KTAS 20.4 gph 89% power

They also show
25,000 ft 187 KTAS 13.3 gph at 60% power
10,000 ft 168 KTAS 12.8 gph at 60% power

Maybe the Marketing Director would be dumb enough to fly at 25,000 ft with a cheap oxygen mask but few others would. The aircraft is mostly flown at 10,000 ft or below and at 75% power instead of 89% or 60%. I think that gives you about 180 KTAS at 10,000 ft. Coming down from 25,000 ft would take close to an hour.

[Tom Goodrick]

I decided to play with the "fancy" equation for cruise speed. There is a text I picked up just 20 years ago (one of my most recoent acquisitions) that presents the equation in a concise form. I had derieved it independently but in a form that was a little more messy. This time I looked at the equation and saw a better way of stating it than either the author or I had done previously. It shows indicated airspeed as a function of design constants and the parameter T/W or thrust weight ratio. That's nice and would seem to make it independent of altitude. I simply backed up a step in the derivation by the author, Francis J Hale, to where he had the equation solved for the dynamic pressure at cruise and then I incerted the indicated airspeed and solved for it. That removed the dependency on the density.

But when we look at the thrust/weight ratio, that is highly dependent on the altitude because it decays with the pressure ratio at altitude. So while we can get indicated airspeed at a value of the thrust/weight ratio, we must look at altitude to see how much thrust is available from the engine.

The equation as stated is mainly useful for jets although it is possible to go back into the equation's derivation and introduce the power for a piston engine. But then the solution for a velocity becomes much more complicated. I'll do it later when totally lacking for excitement.

I put the equation into a spreadsheet and then added a section where the thrust is adjusted for the pressure ratio at certain altitudes and also put in the correction to true airspeed that can be made for the selected altitudes. Hale had noted that the equation gives results that tend to go astray of Mach constraints. I put in an upper limit in the table for Mach 0.8 which is just below the limit for most bizjets and airliners.

The result is a rather interesting set of curves. I ran curves at sea level, 20,000 ft, 30,000 ft, 40,000 ft and 50,000 ft - the extent of normal bizjet altitudes. There is a minimum acceptable thrust/weight ratio at each altitude to keep the eqaution from getting into imaginary numbers. Also, of course there are two solutions - one slow and one fast - as commonly found in aircraft performance relations. We need only worry about the high speed solution as those are most interesting.

I also had to invent a new term while making these plots. It is MSLET or Mean Sea Level Equivalent Thrust. It is on the absissa of the plot. Instead of plotiing with the actual thrust, this shows the sea level thrust that would be equivalent or would occur at the same throttle setting as the actual thrust at altitude. It simplifies the plots.

The performance curves show that the best speed can be obtained at sea level. This seems a little strange as most jets try hard to fly high. here are some peak values for the Learjet 45 using the design values I have in my FD file for it:
MSL: 528 KTAS (Mach 0.8)
20,000 ft: 491 KTAS (Mach 0.8)
30,000 ft: 468 KTAS (at T/W=0.4 which was the max I allowed)
40,000 ft: 384 KTAS (at T/W=0.4)
50,000 ft: no solution.

I am going to run some flight tests with the existing FD files to see if FS shows any of these values.

There are some practical considerations that get a little complicated. First the weight changes significantly during flight. It is far from a constant. The limit of 0.4 for T/W is with half the allowable fuel. It could be higher with less fuel. indeed the LJ45 "book" says it can only climb to 51,000 ft when low on fuel near the end of its cruise. "Why bother?" is a good question.

So, let's see if the LJ45 really can go faster at sea level. I suspect it can. I think the fly high rules were developed to save fuel, not to go fast.

~~~~~~~~~~~~~~~~~~~~~~~~

Well, tests backed up these results, but they turned up another problem. I'd forgotten jets also have a dynamic pressure limit in addition to the Mach limit. For most jets this limit is about 350 KIAS. So we have to eliminate all conditions where the KIAS is beyond 350. This will be the limit for most conditions below 30,000 ft. Also, when running fast on the deck (offshore east of Miami), almost all my engine parameters were in the red when I got to 350 KIAS. So, the equation works but ....

[Tom Goodrick]

Late last night - a particularly dumb time to start a gauge project - I made a gauge that shows Thrust/Weight in percent as a twin-engine jet flies. When the bugs were out of it, I took the LJ45 to 20,000 ft and levelled off. At takeoff (from 640 msl) I saw a max of 35.39%. Level at 20,000 ft in steady cruise I saw 10.00% with 275 KIAS and 361 KTAS. The Excel Sheet shows 363 KTAS at T/W=10.52%.

One reason I made the gauge was to check the value of T/W at the same throttle setting at different altitudes. That will come today. I am encouraged that it was so close.

When I finish up a few little things with the Sheet, I'll send it to anyone who requests one by email.

[Tom Goodrick]

OK. There has been more confirmation of the validity of the equation and of my spreadsheet and assumptions. The most surprising thing predicted by the equation is that, at each altitude, there is a minimum sustainable level speed that is well above stall speed. This shows flight possible in descrete bands as you climb in altitude. Consider the following test results with the LJ4:

At MSL:
Sheet: 192 to 317 KTAS.
Test: 178 to 320 KTAS (limit VMO)

At 20,000 ft:
Sheet: 243 to 436 KTAS
Test: 245 to 440 KTAS

At 30,000 ft:
Sheet 288 to 468 KTAS
Test: 293 to 470 KTAS

The minimum speed seems to be a valid characteristic of jets. Remember this is caused by a difference under a radical going negative and causing imaginary numbers!

One other thing is that in the test it became clear that the jet has poor speed stability. Here we mean that, on autopilot with heading and altitude hold with thrust set manually, This is something I have found with many jets doing cross-country flights. I usually would set just HDG and ALT mode, put the power at a desired level and ignore the flight for a while. But I have found it works better to set SPD mode. With auto throttle enabled, it will hold the speed well and will thus perform in a predictable fashion. Otherwise the speed can wander and the aircraft can get into trouble.

[Tom Goodrick]

It's too bad this will get stuck at the end of this long article where few people will read it. So anyway, I recently started flying FS2002 on my laptop. Unfortunately none of my XML gauges work on the laptop even though I know I developed many of them when flying FS2002. I needed to do some speed checks to find the best altitude for cruise. First I determined the proper settings for each aircraft to match specified speed, altitude and fuel flow for each aircraft. Then I made flights in standard conditions (clear weather) setting the fuel flow for 75% power at each of these altitudes. Speed at full throttle was used if I could not get the desired fuel flow for 75%.

Altitude_Cessna 182____Cessna 182RG__Mooney Bravo _ Beech Baron 58
________15.0 gph______15.0 gph______17.2 gph______2x16.2 gph
2000_____135_________150___________162________196
4000_____138_________153___________164________200
6000_____140_________156___________167________203
8000_____142_________156___________171________200
10000____138_________151___________174________196

The Bravo is the only one that is turbocharged. It could fly higher and fatser but that is impractical because you and everyone on board would need to wear oxygen masks and because descents must be made at 500 fpm to protect the ears of all on board. Pressurized planes can fly high because they can descend fast without hurting ears and no one has to wear an oxygen mask.

Some specs for the Bravo show cruise at higher speeds using 85% to 89% power. That is not conducive to good engine life. This is not done by people who want good engine life.

These results are very typical for non-turbocharged engines where the power maxes out at about 6000 - 7000 ft. But I found in my experiemnts with these aircraft in FS2002 that using a high value of the power scalar will mess up this altitude/power relationship. There is a way of calculating the engine parameters needed to make a given max power value. This should be done. (I need to review it.) Without that, use the fuel flow for 75% power as a guide to setting power. Adjust drag to match cruise speeds with only a little use of the power and thrust scalars.

Some people want you to adjust the mixture for "best economy." It is best to adjust for best power (lean to maximize fuel flow) and then readjust the throttle to get the fuel flow for 75% power. Using 60% power instead of 75% power gives good economy. This is what is generally mean't by "Economy Cruise Power." Given no other guidance just reduce the 75% fuel flow by 20% and use that as your economy power setting.