“Yes. A recent example from my experience. I obtained a solution terminating at 25S. The altitude after descent predicted by my model was 10,000ft, speed was corresponding to 310 KIAS. Nice numbers. All the turns had logical explanations. But I ran into a problem: insufficient fuel.”

I’d be interested in summary of the path logic even if it does not seem to work for fuel.

]]>Re: “I don’t have a need nor do I have the data to properly model fuel flow for a B777-200 for a condition of Cl<0.34. The fuel studies I am interested in are for speeds between Holding and LRC, and my efforts were to improve the accuracy of the fuel model in this range."

As I explained before, I am not really interested in improving accuracy to below 1% rms. That makes no sense because we do not know how many 'coffeepots' were operating. I am rather interested in the feasibility of paths at low altitudes and high KIAS, and that is why the range below Cl<0.34 is important to me.

]]>1. Since the table lists the required fuel load, the adjustment of 0.8%/10K SAT refers to fuel efficiency, not fuel flow, as you probably know.

2. Under the envisioned emergency, the plane would be flying level at 10,000 ft. I assume this is a geometric altitude, not a pressure altitude. As such, an increase is temperature from ISA would reduce the pressure, and for a given speed such as 250 KIAS, would increase the lift coefficient Cl and drag coefficient Cd.

I was looking again at Brian’s paper re: MH370 speed estimate.

It is very good paper, but maybe we should think of Arc3 to Arc5 (20:41 to 22:41) the “possible straight and steady speed” period. I am thinking maybe MH370 stayed around M.84 until near the boundary Indonesian air space and then cut back to holding or slower speed. Also maybe there was a final minor turn after 19:41.

]]>On the other hand, the FPPM for the 777-200ER/GE90-94B specifically says that fuel flows at cruise conditions increase by 3% for a 10K increase in TAT.

My guess is that the 3%/10K TAT correction is more relevant to MH370.

]]>@Oleksandr: I don’t have a need nor do I have the data to properly model fuel flow for a B777-200 for a condition of Cl<0.34. The fuel studies I am interested in are for speeds between Holding and LRC, and my efforts were to improve the accuracy of the fuel model in this range. If you are proposing a scenario in which you need to extend the range of your fuel model down to Cl=0.15, I encourage you to do so.

]]>Who are “they” ? ]]>

The 23 really should be less than your 20.96 L/D max, engines running, that is with engine inlet drag subtracted from gross thrust. The simulated glide L/D comparison would be even higher than the 23 without engine drag. Offsetting this some perhaps, Reynolds number in descent should decrease CD0, RN increasing by about 45% by sea level, same IAS.

I note Wikipedia lists the A380 L/D at 20 in the cruise, the 747, 17.7, so consistent with your 19.76 at M=.83.

Whatever, your model matches the fuel consumption expected within the operating range of interest.

I may be going over old ground here but as to what is expected, in the FCOM Performance Dispatch section I find mention of “Increase fuel required by 0.8% per 10˚C above ISA” but no mention of PDA. With other small drag variables like antenna configurations, trim, paint finish and cleanliness, ram air inlet and outflow valve openings it may be that in their estimates they add an increment for such customer variables also?

]]>Re: “Yes, it is possible to safely fly a B777-200 at conditions for which Cl=0.224 (but not the Cl=0.15 that you proposed).”

Cl=0.15 was a very rough approximate estimation derived from your Cl=0.34. And actually it turned out to be a very good estimate of the lower limit of Cl range in general, so I am puzzled by your resistance to accept it.

You wrote “To calculate Cl, I use a reference wing area of 4630 ft2. Observing all the conditions in Boeing’s LRC and Holding tables, I calculate a minimum Cl of 0.34 at the LRC speed of M0.636 at FL250 for 160 MT weight.” Then you asked “Exactly what speed, weight, and altitude do you believe would produce Cl=0.15?”. Bearing in mind you mentioned 160<174 tonnes, I assumed you were discussing a B777 in general, so I gave you an example. I can go further down assuming the altitude of 10,000 ft, weight of 140 tones, and overspeed 350 KIAS (if my understanding of FCOM that a pilot can go to 25KIAS above VMO is correct). This yields Cl = 0.165, which is a pretty close to the approximate figure I suggested earlier.

In my opinion, it appears that Cl = 0.16 is physically possible to fly a B777 aircraft, but Cl=0.21 is a more realistic lower limit for the specific case of MH370. Agree? A curious thing is that Cl = 0.225 = (0.3+0.15)/2 – it seems we are converging.

]]>Yes, it is possible to safely fly a B777-200 at conditions for which Cl=0.224 (but not the Cl=0.15 that you proposed). I would not trust my drag model in this range. However, flying in this conditions would consume fuel at a much higher rate than we know is possible based on MH370’s fuel load and endurance.

The main purpose of my fuel study was to determine the fuel flow at speeds between the LRC and holding speeds. It is unlikely that MH370 flew with two engines for an extended time outside of this speed range.

]]>Just found your comment after I responded your earlier question.

What is a point to use the whole LRC and Holding tables for the calibration, if you want to restrict application of your FF model to the range of weights from 174 to 210 tonnes? You could possibly achieve a better RMS.

Also, in this case, the comparison with Gysbreght’s CI curve (at 240 tonnes) does not make sense.

Finally, would your model be applicable for Cl = 0.224 as long as you already admitted it is possible?

]]>“Exactly what speed, weight, and altitude do you believe would produce Cl=0.15?”

I have not estimated the exact minimum, but what about, for example:

Speed = 330 KIAS;

Altitude = 10,000 ft;

Weight = 160 tonnes?

If I am not mistaken Cl = 0.21 for these conditions.

]]>Exactly what speed, weight, and altitude do you believe would produce Cl=0.15?

]]>Re: “For the range of speeds, altitudes, and weights in the LRC and Holding tables, Cl is never less than 0.34. That’s why I think the extrapolated value to Cl=0 isn’t helpful.”

The ultimate goal of a fuel model is to provide fuel flow rates at various conditions. Many records in the ‘Holding’ table correspond to the speed of around 200 KIAS, which is approximately 1.6 times lower than VMO. Hence Cl~0.15 would also be of interest. If you know that your drag model is good at Cl = 0.3, but it also gives something unrealistic at Cl=0, can you still use it as a part of the FF model at Cl = 0.15?

———-

@Dennis,

Re: “Speaking of fuel consumption analysis. Does it matter? Can it help to refine the terminus?”

Yes. A recent example from my experience. I obtained a solution terminating at 25S. The altitude after descent predicted by my model was 10,000ft, speed was corresponding to 310 KIAS. Nice numbers. All the turns had logical explanations. But I ran into a problem: insufficient fuel.

]]>I am surprised the SSWG did not attempt accurate fuel modeling. Seems like an important constraint. Not sure at the end of the day how much it would contribute to terminal location area reduction.

]]>@DennisW: It is not trivial to model fuel flow to the level of accuracy required to discriminate between possible and impossible paths. The DSTG essentially chose to ignore constraints imposed by fuel load except to set limits on Mach number. The capability to model fuel flow is resident at Boeing, not the DSTG, and my guess is that capability is carefully protected for competitive reasons.

It is still unknown whether or not we can converge on one or more fuel models with sufficient accuracy that we can eliminate paths. @DrB started down this path, although with imperfect knowledge about PDA, CI, and temperature relationships. I think our group’s collective knowledge about these matters is increasing.

]]>Speaking of fuel consumption analysis. Does it matter? Can it help to refine the terminus?

The DSTG assumption 4 from page 60 of their book below:

begin cut-paste//

4. Infinite fuel: the fuel constraints on the aircraft can be applied to the pdf after-wards. In the simplest case, maximum reachable ranges could be used to censor impossible trajectories. However, analysis of candidate trajectories has indicated that the majority are feasible. Broad information about the fuel consumption rate of the aircraft has been used to inform the range of allowable Mach numbers.

end cut-paste//

I really don’t have a strong opinion. The DSTG does not use fuel consumption in their pdf algorithm. Certainly they had the benefit of ACARS data from 20 previous flights to model the fuel consumption of the aircraft very accurately. They chose not to do so. Why?

]]>If I read your plot correctly, TSFC is roughly 20% higher than it is supposed to be at cruise altitude and speed.

———-

@Victor,

It can be expected that the interpolation in the range 0.3<=Cl<=0.50 is reasonable, but I am not so sure about extrapolation to above 0.50 or below 0.30. Also, now I am not sure whether these curves really correspond to Cl = 0.3, 0.4 and 0.5 or 0.2, 0.3 and 0.4 respectively.

The derived TSFC is almost constant over most of the thrust range (total 2-engine thrust). Perhaps Obert’s curves have been derived from FCOM data **assuming** constant TSFC?

The data for holding speeds are even stranger.

]]>Here is the plot of drag coefficient versus Mach number for various Cl. The Cd curves were derived using a quadratic fit of the curves for Cl=0.3, 0.4, and 0.5. I believe the curves are accurate for 0.3<=Cl<=0.55 and for M<=0.87.

]]>Thanks for the additional plot. That is what I expected.

Do you find it realistic that Cd for Cl=0 is higher than for Cl=0.3,0.4, and 0.5 at, say, M=0.85? Also that Cd for Cl=0 is comparable to Cd for Cl = 0.30 at M=0.65?

I posted the first version of my xls-calculator together with some references on April 4, 2017 at 7:52 am; one of them “compressibledrag.pdf”. I think Cl=0 should correspond to the minimum drag to make model physically meaningful.

]]>Re “I am not certain of exactly how the enroute fuel range is calculated, and so I have not and do not plan to make a comparison. I see a lot more value in matching the fuel flow data, where a comparison can be made at a particular altitude, weight, and speed.”

What is a difference how it is exactly calculated? You have the average speeds and weights over 30-minute intervals or so, at respective altitudes. As a matter of fact, having a model, you can make even more accurate estimates of consumed fuel to compare it with FCOM. Anyway, it is up to you what set of data you want to use for calibration and what set for validation.

]]>Please note that this is the flow per engine. The maximum flow per engine in the LRC table is 4746 kg/hr, corresponding to 300 MT at FL330 at M0.838.

]]>If a model predicts the correct flow rate and a reasonable (L/D) ratio, then the predicted TSFC is also reasonable. For my model, the RMS error in fuel flow is about 1.13%, and even better for typical values of airplane weight. The predicted (L/D) is around 20 for typical cruise conditions. That implies the calculated TSFC is in the acceptable range. My model is semi-empirical with underlying physical models, which is why I suspect it does well over a broad range of conditions.

“A quadratic dependency is the best only if we don’t have any other information or a theory.”

I don’t know of a better relationship or theory that is based on actual data from a B777-200.

“Could you also include comparison with “Enroute Fuel Range” table in your last plot?”

I am not certain of exactly how the enroute fuel range is calculated, and so I have not and do not plan to make a comparison. I see a lot more value in matching the fuel flow data, where a comparison can be made at a particular altitude, weight, and speed.

]]>Just wondering, how your Cd(Cl,M) curve looks like for Cl=0?

It seems your curves Cl = 0.3, 0.4, 0.5 quite closely match those from “compressibledrag.pdf” for Cl = 0.2, 0.3, 0.4 respectively.

A quadratic dependency is the best only if we don’t have any other information or a theory.

Could you also include comparison with “Enroute Fuel Range” table in your last plot? I see that your FF range is limited to ~4,800 kg/hr, while this table includes significantly higher fuel flow rates.

]]>In fact, I have used this drag model with a simplified turbine engine model to produce the flow rate data with great success. I can now re-produce all the LRC and Holding data available from Boeing with an RMS error of only 1.13%. That includes the data from LRC speed at 43,000 ft all the way down to Holding speed at 1500 ft.

I’ll document and share the model in the near future.

]]>“I have digitized the Cd(M) data from Obert (available as three curves for Cl=0.3,0.4,0.5) and now have a very accurate method of determining Cd as a function of M and Cl, assuming a quadratic dependency of Cd on Cl at a given Mach number.”

Thanks for the curves CD(CL,M). I was thinking of the method you proposed, however I am not sure if it is a correct approach. If you had a chance to take a look at “compressibledrag.pdf”, then the dependency of Cd on Cl does not seem to be quadratic at M>0.75.

——-

@Gysbreght, @Victor:

SFC = 0.557 lb/hr/lb at FL350, M=0.83, Thrust = 13000 lb, RR Trent 892.

Just one data point useful to verify or calibrate a fuel flow model.

The curves show a strong dependency of drag on Mach number, as expected from the underlying curves presented by Obert. This dependency increases at higher values of lift, as shown by the spreading of the curves. At low Mach numbers, the maximum (L/D) is 20.96, as shown by the tangent dashed line. At M=0.83, the maximum (L/D) is reduced to 19.76, as shown by the tangent dotted line.

]]>On further reflection (while starting along that road), I don’t think you can get a TSFC map that way.

All you can get is two lines running across the Mach lines, one line for Holding speed and another line for LRC.

]]>Professor Obert’s B777 data could be used with the FCOM performance data to obtain engine TSFC. A map of TSFC/Theta^X = f(Thrust/Delta, M) would look somewhat like

this.

Delta = atmospheric pressure ratio

Theta = atmospheric temperature ratio

X = theta exponent

Professor Obert’s B777 data could be used with the FCOM performance data to obtain engine TSFC. A map of TSFC*Theta^X = f(Thrust/Delta, M) would look somewhat like

this.

Delta = atmospheric pressure ratio

Theta = atmospheric temperature ratio

X = theta exponent

*“As I understand, the 777-200ER has the same airframe and wing as the 777-200. Difference being increase in MTOW (and therefore [heavier] engines capable of higher thrust for take off and climb. If this is the case should we not expect the same form drag and lift for any particular pressure alt/TAS/gross weight – and thus thrust required in cruise? And if that is correct, then any difference between the two variants is down to the engine efficiency (FF) to generate the required thrust?”*

The engines on the -200 and -200ER are essentially the same, except that the -892 engine used on the -200ER is rated to operate at a higher take-off/climb thrust to satisfy the increased thrust requirements of that aircraft’s higher maximum take-off weight. Given the drag characteristics of the two aircraft variants are the same, the cruise FF should be the same for a given weight & altitude.

]]>Enter the following into SkyVector

MUTMI 3393S MUTMI YPLM

Its shows 1726 nM from MUTMI to 3393S (a little ways inside Arc6) and 1637 nM from MUTMI to Learmonth. So it looks to me that Learmonth is borderline reachable from MUTMI, pending a more accurate fuel model assessment.

It also assumes MH370 had a FMT at about 1840 to get to MUTMI relatively promptly without much loiter.

]]>Thank you for your curves of CD = f(CL,M). Using those curves I found D/Wmin=.0496 at M.78; CL=0.542.

For the graph I extrapolated the curves a bit for CL>0.50.

]]>As for the “Enroute Fuel Range from a Check Point” Tables in the FCOM, the values show better efficiency because they no longer need to include the large amounts used during the initial climb.

]]>Paper linked below has some comparisons relative to the BADA and ICAO data. Does not appear to be all that accurate.

http://www.mit.edu/~hamsa/pubs/ChatiBalakrishnanICAS2016.pdf

]]>Upon reviewing my fuel model from two years ago, I see a mistake I made in fitting the drag curve. Recognizing the mistake and re-fitting the curves might allow me to improve the accuracy of the results even further.

]]>Asking that question is answering it. The article doesn’t identify aircraft model, engine and flight condition. I’ve always used words like “probably” and “possibly” when referring to it as a B777 plot. In the context of the article’s discussion of Cost Index, only the upper range of speeds (above MRC) is relevant.

Below MRC the curve could be accurate, but does not need to be. It would be interesting to find the speed for maximum L/D at FL350; 240 MT, based on your curves of CD=f(CL, M).

]]>