Optimum Cruise Alt formula

[Trackdiamond]

Is there a reliable rule of thumb for reckoning the optimum cruise altitude in the manual flight planning case (computer flight plans packed) based on route distance, ground speed,and weight limits? Or the AFM perf graphs are the only tool?

This is just a curious question..no need for sarcastic answers please.Thanks for any formulae.

[Skyjob]

You asked for it (B737-800W):

The FMC generates optimum and maximum altitude data to assist the flight crew in choosing the best cruise altitude for a given trip. The calculated maximum altitude is used to ensure that the predicted path of the airplane remains within its performance capabilities. This maximum altitude is the lower of the thrust-limited altitude and the limits of the VNAV operational envelope.

The FMC calculates an economy cruise optimum altitude in one of two ways depending on the length of the planned trip. A "short trip" optimum altitude is calculated to minimize trip costs when the choice of cruise altitude is constrained by the climb and descent performance of the airplane. On longer trips, where cruise altitude choice is constrained only by the airplane's initial cruise altitude capability, a "long trip" optimum altitude is computed to minimize cost.

STOA
The short trip optimum altitude (STOA) is that altitude which yields the minimum trip cost when cruise altitude is constrained to be lower than the long trip optimum altitude by either the intersection of the climb and descent profiles, or minimum cruise time considerations. The STOA calculations are based on the standard trip profile as below. The STOA is displayed by the FMC as the optimum cruise altitude when the equivalent trip distance is less than 400 nautical miles.

SEA LEVEL TAKEOFF, CLIMBOUT TO 1500 FT
FLAP RETRACTION, ACCELERATION TO VREF40 +70 KCAS CLIMB AT 250 KCAS TO 10,000 FT
ACCELERATION TO ENROUTE CLIMB SPEED
CLIMB AT ENROUTE CLIMB CAS/MACH
CRUISE
DESCENT AT MACH/CAS SCHEDULE
DECELERATION TO 250 KCAS AT 10,000 FT
DESCENT AT 250 KCAS TO 1500 FT
DECELERATION AND FLAP EXTENSION AT 1500 FT APPROACH AND LANDING AT SEA LEVEL AIRPORT

The optimum altitude determined for each short trip has a corresponding optimum cruise time which is directly related to the computed equivalent trip distance. If the operator desires additional cruise time, the STOA may be decreased to make this accomodation. The minimum cruise time considered may be varied according to airline policy from one minute (FMC default) to twenty minutes.

Several parameters affect the short trip optimum altitude calculation. These include trip distance, cruise time, gross weight, temperature, wind, and airport elevations. Airplane speed effects are negligible. The altitude data for a baseline condition is used as a function of trip distance and cruise time. Approximate linear corrections are provided to account for the other variables. Wind effects are the most difficult to generalize. The FMC wind model is a linear variation in wind magnitude from zero on the ground to the input value at the input cruise altitude.

LTOA
The economy cruise long trip optimum altitude (LTOA) is used if the calculated equivalent trip distance is in excess of 400 nautical miles. The economy cruise LTOA is that altitude which the airplane should fly for lowest cost operation when flying in economy cruise mode at a given gross weight. The data stored in the FMC is computed by maximizing the inverse of the economy cruise cost function.


Function Maximized = VG / ((100 * CI) + WF)
where:
VG = ground speed in nautical miles per hour,
CI = cost index with units of dollars/hour, cents/pound
100 = cents to dollars conversion, WF = fuel flow in pounds per hour.

This function relates ground speed to the equivalent fuel flow "cost" of flying at that speed. The equivalent fuel flow consists of a pseudo fuel flow (100*CI) representing time costs, plus the actual fuel flow (WF) representing fuel costs. Therefore, the function which is maximized has the units of nautical air miles per pound (nam/lb). For the economy cruise mode the LTOA occurs where the result of this expression is maximized. This results in an optimum altitude yielding the minimum fuel burn at the specified speed.

The FMC stores the economy cruise LTOA as optimum gross weight/delta (W/δ) vs cost index/gross weight (CI/GW). The computation of optimum altitude includes an estimate of the climb fuel burn between the current altitude and optimum altitude, unless the airplane is above the optimum altitude. If the current altitude is above the optimum altitude, the computations are performed at the current gross weight. The economy cruise optimum altitude may be determined in the following manner:


Calculate CI/GW by dividing the cost index entered in the FMC by the current gross weight, in pounds.
Determine the optimum W for the CI computed in step 1
Determine the altitude corresponding to the optimum W/δ from step 2 and the current gross weight, in pounds.
Use the altitude determined in step 3 and the current altitude to estimate a climb fuel burn value using a linear climb fuel burn estimate.
The climb fuel burn can be estimated by the equation:
CLB FUEL = (A + B*WSOC/1000)*(HTOC - HSOC)
where:
A = -0.012243 (LB/FT),
B = 0.000657 (1000/FT),
WSOC = Gross weight at start of climb pressure Altitude (LB),
HTOC = Top of climb pressure altitude (FT),

HSOC = Start of climb pressure altitude (FT).
Subtract the value of CLB FUEL from the current gross weight to get the estimated TOC gross weight. Repeat steps 1 through 3 to get the final approximation of the LTOA.

Note: Charts are required to determine LTOA as per above formula, but these are proprietary to the manufacturer, sorry these cannot be included here.

[phiggsbroadband]

Hi, This thread is similar to one in the Misc Section....


Do you find that your options are limited by the 'Coffin Corner' such that speeds must be kept between 370-500 Kts at FL350 and even less than that at higher FLs.
Flying in the Coffin Corner?Air France Flight 447 | Guest Blog, Scientific American Blog Network (http://blogs.scientificamerican.com/guest-blog/2011/05/12/flying-in-the-coffin-corner-air-france-flight-447/)


Apparently the U2 could have a speed range of just 5 Kts at maximum altitude.

[Trackdiamond]

Thanks for the informative replies guys.

Skyjob..many thanks but I was looking for Altitude reckoning at preflt planing stage..in cases of Computer flt plans not accessible and no previous experience on the route segment. I.e. you are going to a new destinantion.What Altitude to plan for given the distance.Also assuming the Quick Planning FL,Fuel,Time graphs not available..just a simple rule of thumb mental excercise.It is understood that FMS can compute all this.Just wanted to reert to manual methods.Assume FMS has failed that day too!

Sorry about the duplicated post.Will maintain it on this Techlog Forum for more visibility.

Thanks

[pattern_is_full]

The problem you face is that a "rule of thumb" pretty much has to involve a simple arithmetic calculation. Something you can do on the back of an envelope.

Optimum Altitude requires a recursive calculation, since, for example:

OptAlt will depend on ground speed, which will depend on winds aloft, which will depend on altitude

OptAlt will depend on fuel flow, which will depend on altitude.

OptAlt will depend on weight, which will depend on fuel flow, which will depend on altitude.......

With such a problem, the solution is to use "Monte Carlo methods" - simply do the calculation many times with different assumed cruise altitudes, and find the one that predicts the best speed or lowest fuel use (depending on which you are trying to optimize for).

This is what computers excel at (no pun intended) - doing, say, 20 grunt-work calculations in a fraction of a second, and picking the result with the optimax altitude. Even the tables (these days) are the result of computer calculations graphed out (with additional raw-data input from the company's test flights).

HOWEVER - I have seen a "rule of thumb" specific to the 747 variants that says OptAlt is 2000 feet (5-10%) below the ceiling for a given weight (presumably because it takes significant extra time and fuel to climb that last 2000 feet). Of course, you still need at least a simple table to figure that ceiling for any given weight. And you'd have to check winds (g/s) for that altitude and the neighboring ones to see if you get an improvement moving up or down a bit.

And that would appear to be for long flights with most of the distance travelled at cruise. Ignores climb costs except for that last 2000 feet.

I remember another (mentioned on this forum, I believe, for the B1900, but might have been some other TP) of 1000 feet per 10 miles of distance. I.E. 19,000 for a flight of 190 nm. There may have been a fudge factor I've forgotten, such as ((1000 feet per 10 miles) minus 2000 feet) = 17,000 for a 190nm trip.

[Ditched]

As a general rule of thumb on the 737NG take the trackmiles as a flightlevel. e.g. 300nm sector flight level 300 is about what you would fly.

Or take out FCOM 1 Performance Dispatch tables, no fancy computer needed!

[Trackdiamond]

Thank you patternisfull! Wonderful insights.

I like the BE1900 rule.The rule of thumb for jets perhaps has a distance demarcation to differentiate short from longhaul? Lke 400nm perhaps?

I have also come across something am sure deficiently simplistic..where the reasoning being for short haul spend 1/3 of the distance climbing, oe third cruise and the remaining third descending.working backward descent profile from 1/3 remaining distance reckons the cruise alt..If you can expand on this mathematically with required corrections for wind components and weight you might get a more intelligent solution.

Ditched ..hat sounds about right as the 1/3 formula leads to the same result...ofcourse wih weight and altitude capability limiting the final altitude on ultralonghauls.

[Skyjob]

The NG flies usually ~7-800 ft below Max FL as Optimum.
However the Maximum is weight dependent, as well as CI dependent (Weight & Speed function)
A rule of thumb can only be given for LTOA (400nm+) when weights involved are known. A rough estimate of rule burn to TOC can be achieved easily to allow the reduction from TOW and establish Cruise Weight.
STOA is not dependent on weight but on distance as the higher flight levels cannot be reached in time for descent.

[Trackdiamond]

Thanks Skyjob for your inputs.

When a rule of thumb for STOA can be established is it applicable on turboprops/corporate jets just as well as heavy jets?

The difference between Rec Max Alt and Optimum. Altitude..please can you explain ..as in Airbus FBW Case vs Boeing case...if different. Why are CIs different between Boring and Airbus?

[Skyjob]

STOA is aircraft dependent, as climb performance varies.
The formulae given were for 737NG as clearly stated in first post top line.
If you look at the STAO profile I gave, you will notice:

FLAP RETRACTION, ACCELERATION TO VREF40 +70 KCAS CLIMB AT 250 KCAS TO 10,000 FT

The Vref40 + 70 KCAS refers to the clean speed of the aircraft, based on Vref for flap 40 as a base reference for all (except flap 30) manoeuvring stages of flaps.
Of course when substituted for your type in question this could well work fine.



CI are set by the company management, they could well be the same within an operator for different types. Calculating the CI is something crew do not need to do, it will be provided for them as a default, or in case of advanced flight planning software packages, it can vary between flights.

The cost index denotes the relation of flight time related costs (e.g., crew, maintenance, or delay costs) and fuel cost. Moreover, cost index is a feature of the flight management system to minimize the combined fuel and time costs by varying the speed. Here, cost index is used as an input parameter to the ECON algorithm to determine the speed to fly to obtain the optimum trip cost.

Cost Index (Boeing units) = CostTime [$/h] / CostFuel [ct/lbs]

If the FMS is forced to fly a constant speed (a Mach number), all the FMS can do to reduce the trip fuel is to vary the optimum altitude and thus the recommended step climbs. However, entering a cost index and flying in ECON mode additionally allows for a speed variation.



Rec Max Alt or Boeing's Max Alt can be explained as:

Maintain a 0.3g buffet margin.
Fly in level flight at max cruise thrust.
Maintain a vertical speed of 300 FPM at climb thrust.


Optimum Altitude can be explained as:
(Airbus) The most fuel-efficient altitude for ECON CRZ speed based on gross weight, cost index, and temperature, using 500-foot increments and a minimum of 5 minutes cruise.
(Boeing) refer to my earlier expanded post

[phiggsbroadband]

Hi Skyjob... your quote...
Cost Index (Boeing units) = CostTime [$/h] / CostFuel [ct/lbs]

This seems to be an interesting concept, but can you explain what the CostTime and CostFuel units are? and are they really under the control of the pilots. Surely the Fuel cost is related to the country where the fuel is bought, and the Time cost is dependant upon whether the plane is urgently needed for its next flight.


For instance; It is not uncommon for inbound aircraft to LHR to be asked to slow to Minimum Clean many hundreds of miles out, in order to reduce the number of planes in the stacks, and to decrease the air-pollution over Southern England.

[VNAV PATH]

@ phiggsbroadsband, post just above :


Cost Index is indeed based on Cost time/ Cost Fuel ratio , with values not necessarly updated by company . Some companies have not updated their CI for years. But they adjust basic CI to their fuel /speed policy .


Pilots just adjust the CI of FMS, in flight to adapt needed time of arrival . FMS speed will ajust accordingly.


CI = 0 gives the maxi range speed.

[single chime]

CI is airline+aircraft type+city pair- specific. As a rule of thumb, pricy fuel=low CI; pricy crew=high CI. Nowadays, fuel is expensive everywhere, so CI are usually low.

[Trackdiamond]

Many thanks Skyjob and all who have contributed to this thread. Enligtening!

[Chris Scott]

In case it's of any interest, we used to "thumb" through a rather well-presented cruise performance booklet in my VC10 days. Oddly enough, I don't remember it including a graph for "Optimum Altitude", as such - unlike all the jets I flew subsequently. (Yes, the last one also presented it in the FMS.) Here's a link to our VC10 page for M0.82 (the other pages covered up to M0.86).

http://www.pprune.org/aviation-history-nostalgia/484448-duxford-scrap-super-vc10-2.html#post8275311

I guess one definition of Opt Alt, for a given Mach, is the altitude at which the fuel burn is least for each AIR mile. The graphs show that as KG/N.A.M. (nautical air mile). For each of the various weights, the curves of altitude versus KG/N.A.M. bottom-out at a buffet margin of around 1.4g. As you can see, the "I.O.A.T." (TAT) apparently makes no difference, because the higher fuel-flow in hotter air is offset by the higher TAS at the same Mach.

Needless to say, the kgs per nautical GROUND mile at the proposed new F/L are another matter. You never really know what the wind is going to be up there - unless some preceding crew can tell you. And that's particularly true when you are near the trop (er, tropopause).

I should point out, however, that the VC10's Conway engines had a very low bypass-ratio (hence the excruciating fuel-flows) :} , and were powerful enough to get you well up into coffin-corner if you were tired of living - hence the pecked lines for the 1.35g, 1.25g and 1.15g buffet-margins. We tended to wait until we could climb on the 1.35 buffet, unless we were going to lose an important opportunity from ATC...

PS
At risk of stating the obvious, perhaps I should add that the above considerations don't relate to the optimum altitude for the short sector or diversion case. Even if there are sufficient track miles to climb to optimum altitude and descend to the destination, the benefits of doing so are less pronounced for a high-bypass turbofan than with a low-bypass engine or (particularly) a turbo-jet.

[ZCMN25]

I have also a question about Flight Levels that is not or must not be allocated as a Cruising Level. It is a FL195, can anyone tell me why FL195 must not allocated as a Cruising Level?

[safelife]

Rule of thumb for A330: 565-GW

[misd-agin]

737-800 is typically 1700' between OPT and MAX.

[misd-agin]

For longer flights -

320 family w/sharklets - 510 minus GW

170,000 lbs? 510 - 170 = 340