TAS, MACH & OAT

[simonijs]

Only recently I "happened" to notice that in PSX the commanded MACH-nr seems to change with a change in temperature, rather than the TAS. I would expect this to be other way round. As outside air temperature changes both TAS and Speed of Sound [a] are affected. However, the ratio between the two remains the same. A theoretical example:

ERF: 328900 kg cruising at M0,844 at 34000 ft:

t = -50,4C                ρ = 0,3910 kg/m3      TAS = 490,6 kts     a = 581,6 kts       M = 0,844
t = -51,4C                ρ = 0,3927 kg/m3      TAS = 489,5 kts     a = 580,3 kts       M = 0,844
t = -52,4C (ISA)       ρ = 0,3945 kg/m3       TAS = 488,4 kts     a = 579,0 kts       M = 0,844
t = -53,4C               ρ = 0,3963 kg/m3       TAS = 487,3 kts     a = 577,7 kts       M = 0,844
t = -54,4C               ρ = 0,3981 kg/m3       TAS = 486,2 kts     a = 576,3 kts       M = 0,844

In PSX and for the above conditions, I see the following (data taken from the PSX Embry Riddle Flight Data Recorder):

t = -41,5   TAS = 498,56         M = 0,8405
t = -40,5   TAS = 498,46         M = 0,8385
t = -39,5   TAS = 498,00         M = 0,8360

With increasing temperature, in PSX TAS also seems to slightly decrease (instead of increasing).

I think that with temperature changes the MACH-nr should remain the same, and that only TAS should change – and hence groundspeed. They then both decrease as weight decreases. 

Regards, Simon

[Hardy Heinlin]

I think that with temperature changes the MACH-nr should remain the same

No, the ECON Mach number is calculated by the FMC and for the ECON calculation the FMC takes the OAT into account: When the OAT varies, the command Mach will vary too (if the other ECON factors like tailwind, cost index etc. remain constant).

And in SEL SPD mode the command Mach remains constant even when the OAT changes.

In your second table the Mach decreases as well, so you can't say the TAS decreases while the Mach remains constant.


Regards,

|-|ardy

[simonijs]

When OAT changes, TAS will change by SQRT(T) (in Kelvin), Speed of Sound (a) will change by the same SQRT(T), Thrust and Drag are influenced by T, they all work together as if it were mother nature. If, as in PSX, TAS is kept constant while only (a) is influenced by changes in OAT, then yes: Mach-nr will change. But I don't think this is how it works. Mach-nr is the ratio of TAS/a, both take the effect of the same temperature change, hence the ratio does not change.

I wasn't talking about another mode than ECON, both the theoretical example as the data from PSX were using a CI.

Flying a CI means flying with a (more or less) constant value of CL (or: constant ratio of CL over CD). Then using an alternative form of the Lift formula
 
                   

shows that flying at one Flight Level (or following along one line of Static Pressure) at a constant CL the Mach-nr only changes as W changes. Changes in OAT affect both TAS and (a) in the same way, leaving M as it was.

My second table...: I can't follow your reasoning. My whole point was, that in PSX the Mach-nr changes while TAS is kept constant (observation 1 & 2). That is what that table shows. It also shows that TAS does not increase by approximately 1 kt for every degree of temperature rise (observation 2 & 3). In that table, there is a five minute interval between the recorded data, aircraft mass was reduced by approximately 1000 kg between each interval. While OAT increases, in PSX TAS is decreasing...

Flight Planning software like SimBrief deal with temperature changes as shown in my first table. On relatively short distances where weight does not change that much, Mach is a constant number, TAS changes by 1 kt for every degree of change in OAT. Both then change as weight is reduced by fuel burn.

Regards, Simon

[Hardy Heinlin]

I'm sorry, I don't understand the goal of your thoughts.

My sim has been using this formula here for 30 years:



I guess this formula is well-known; it's not my invention. I won't change it :-)


|-|ardy

[Hardy Heinlin]

t = -41,5  TAS = 498,56        M = 0,8405
t = -39,5  TAS = 498,00        M = 0,8360
With increasing temperature, in PSX TAS also seems to slightly decrease (instead of increasing).

When the air gets warmer and your thrust control is manually fixed, the actual thrust will decrease. Thus, the aircraft will slow down, i.e. all kinds of forward velocities will decrease – be it Mach or TAS or EAS or whatever. When the thrust is zero, all velocities are zero too. So TAS has to decrease. You can't expect the TAS to increase while the thrust is changing from high to low – which is a trend towards zero.

[simonijs]

I guess this formula is well-known; it's not my invention. I won't change it :-)
|-|ardy

I know - and use - that formula too (it is included in the next image). It shows that temperature affects the Speed of Sound. But temperature affects TAS as well, through density. They both change in the same way, by the SQRT(T). The ratio between the two does not change.

When the air gets warmer and your thrust control is manually fixed, the actual thrust will decrease. Thus, the aircraft will slow down, i.e. all kinds of forward velocities will decrease – be it Mach or TAS or EAS or whatever. When the thrust is zero, all velocities are zero too. So TAS has to decrease. You can't expect the TAS to increase while the thrust is changing from high to low – which is a trend towards zero.

Temperature also affects D(rag). If thrust decreases because of a rise in temperature, then D will decrease because of that same rise in temperature (D ~ 1/T). They would still be balanced.

The next image is of the actual OFP of yesterday's flight KLM868 (PH-BHL, B789, RJBB-EHAM). Unfortunately it does not show TAS. But it does show quite a temperature change.



At T_O_S the OAT = -50, Mach = 0,839 and GS = 469 kts. The calculated TAS (by me) = 488,4 kts. With info on TT and Wind/Velocity data, this matches the GS of 469 kts.
At CUTEL, Mach is still 0,839, OAT = -57 and GS = 473. Calculated TAS (by me) = 480,7 kts.

So, a temperature change of 7 degrees results in a change of TAS of 7,7 kts (~1,1 kt per degree as was already shown in my first table).

My bet is, that in the real aircraft the FMS adjusts Thrust to maintain the commanded ratio of TAS over Speed of Sound (--> M). In the screenshot that is M0,839. In PSX, the FMS adjusts the Mach-nr. in order to maintain the same TAS. And I don't think PSX is correct here.

Regards,
Simon

[Hardy Heinlin]

I thought you were talking about the flight model. Now I see you're just talking about the FMC, i.e. about the FMC's ECON SPD algorithm. I take note of your suggestion. Thank you.


Regards,

|-|ardy

[simonijs]

Well, thank you for looking into it!

And just to be sure...: in the Mach-formula - that you provided above - it seems as if a change in OAT is only applied to the Speed of Sound part of that formula.

I don't know if that is to be pinned under flight model, or FMC's ECON SPD algorithm.

Regards,
Simon

[Hardy Heinlin]

Hi Simon,

I've checked this stuff and ...

I'm sorry, I can't make any modifications there.

Too risky. Too hypothetical.

Thanks for your feedback anyway ...


Regards,

|-|ardy

[United__744]

Temperature changes; Mach number changes. Aircraft changes its speed accordingly.

You are correctly associating TAS and Mach as having the same relationship, but what you neglected was CAS which changes as well, relative to TAS/Mach.

As the temperature rises, the density decreases, and so the Mach number decreases. The aircraft will accelerate to hold the desired Mach number, and TAS will increase. CAS falls with increasing temperature, and will show a lower value than before the temperature change for the same Mach number.

Reverse the trends for cooling air.

[Hardy Heinlin]

It's not about the physical formula. It's about the cost index based FMC ECON command speed algorithm. The algorithm varies the command Mach also depending on the tailwind component, for example, even though the wind has nothing to do with Mach, physically. (For example.)

[United__744]

It's not about the physical formula. It's about the cost index based FMC ECON command speed algorithm. The algorithm varies the command Mach also depending on the tailwind component, for example, even though the wind has nothing to do with Mach, physically. (For example.)

Ahh! I missed the CI part!

[simonijs]

Hi Hardy, and anyone else interested in this subject

Apologies for a delayed reply: I have been away from home and my computers since the early weeks of May until last week. It gave me 'some' time to think this over.
Repeating my initial point first: when temperature changes (up or down), TAS and Speed of Sound [hereafter: the letter 'a'] change as well, the Mach-number remains unchanged.

This can be easily verified, using a simple Excel-document (https://www.dropbox.com/scl/fi/tvki5r4hbp04i3cc4roeb/TAS-MACH-OAT.xlsx?rlkey=vbqu3v25a0d2do4ne660xt7iv&dl=1). It starts off with the numbers from the situation: 34000 ft (FL340, which was maintained during the situation), aircraft mass: 328915 kg, OAT: -41,5 C and CL = 0,48275. I have protected the workbook, except for these four Cells (C1-C4). All formulas, used for the calculations, are shown on the righthand side of the worksheet. They are not hypothetical.

In my first post, temperatures were increasing from ± -41,5 to ± -39,5 C. By just changing that in the temperature field, one will see that TAS & a change; however, the Mach-number does NOT change. The other editable fields all influence both TAS and Mach-number. Important ones for this subject: Aircraft mass and Lift Coefficient CL. Just reduce the aircraft mass by 1000 kg or just increase the Lift Coefficient CL by 0,001 and see what happens to TAS and Mach.

However, temperatures do not change in a split second. In this case, it took 12 minutes for the temperature to go from -41,5 to -39,3. The aircraft mass had reduced to 326667 kg. So now:

1.    the temperature rise causes TAS & a to increase; the Mach-number remains unchanged
2.    aircraft weight is reduced, which causes TAS & Mach-number to reduce

Looking at the formulas, formula 1 is repeated as formula 7 and rewritten. From this recomposed Lift formula, the product CL*M^2 [hereafter: 'the product'] will decrease as aircraft weight decreases. This may be verified in Cell E33 in my Excel document. This product is an important parameter, for instance to calculate OPT ALT (formula 8 ) and 1,3G Buffet Limit Pressure Altitude. The temperature effect is eliminated in this way (see 1 above).

I  think - and this is hypothetical - that both CL and M^2 need to reduce simultaneously as weight reduces.

This (https://www.dropbox.com/scl/fi/nzhuq9jn20bscxna4bdlg/erau1-2026-APR-06-14-30-13.csv.xlsx?rlkey=x81jjjmc33d90vlywrw9nqhpo&dl=1) is to another Excel-document in which data of the situation, described above, is taken from the FDR in PSX (ERAU). I have edited the file to only show relevant data every 20 seconds.
It shows that CL, calculated by me in column O, initially is increasing (orange-coloured cells), thereby reducing both TAS & Mach-number. Whenever CL is decreasing, TAS and Mach-nr are increasing. In the recorded data CL and M never decrease both at the same time. To me it looks like CL is manipulated here to control TAS. If so, I don't think this is correct. The product is still decreasing but an increase in – for instance – CL requires an 'exaggerated' reduction of TAS (to comply with the overall requirement for the product to reduce as weight reduces). This causes TAS to sit at ± 498 kts, while a is increasing because of temperature: hence, the Mach-number decreases. There was little wind influence, if at all.

From column T onwards and for the weight change between specific time stamps, I calculated the change in the product. Using a steady and calculated decrease in CL, new values for Mach (using formula 7) and TAS were calculated. Now Mach-nr is also steadily decreasing (column W, in very tiny bits) as weight decreases, and TAS (column X) is steadily increasing as temperature increases. As a result, in twelve minutes time TAS increased from 498,5 kts to 500,4 kts, while the Mach-number reduced from M0,84045 (rounded 0,840) to 0,8396 (rounded: 0,840). The respective PSX numbers (in columns H and I) at time stamp 00:12:00 were 497,5 kts and M0,8346 (rounded to 0,835).

The screenshot in post #5, taken from KLM's OFP, shows a rounded M0,839 while the temperature dropped from -50 to -57. The (calculated) TAS must have reduced from ± 488 kts to ± 481 kts. It behaves how I calculated things.

Formula 6 in my first Excel-document is rewritten as well, by replacing v^2 with the result of v^2 from formula 1. In this way the temperature-density effect is eliminated for both D and TR.
While using my proposed method of steadily decreasing CL, the induced drag coefficient CD,I is also steadily decreasing, keeping the ratio CD/CL nearly constant.

I have absolutely no knowledge of how Boeing programmed the FMC to deal with these temperature changes, nor how this is done in PSX. But for now I'd like to maintain my point of view that PSX may not be correct here. Perhaps our real world pilots on the forum could make some notes of TAS and M-nr. when OAT changes (on short distances, to avoid the influence of large weight reductions).

Kind regards,
Simon

[simonijs]

Temperature changes; Mach number changes. Aircraft changes its speed accordingly.
You are correctly associating TAS and Mach as having the same relationship, but what you neglected was CAS which changes as well, relative to TAS/Mach.

The discussion, I started, was about TAS and Speed of Sound. They are related, providing a Mach-nr. Both TAS and Speed of Sound are affected by temperature changes, in the same way.

I wasn't discussing - nor neglecting - CAS: a change in temperature has no effect on CAS. You may want to look at the first Excel workbook, mentioned in my previous post, in which a third worksheet is added especially for you. This worksheet contains the formula that converts TAS into CAS, the (result of the) calculation is repeated on the first Worksheet, in cell F16.

Below you find a SimBrief generated United Airlines OFP. Just enter the numbers from the OFP in my Excel-document: 34000 ft (in Cell C1), 292700 kg (Cell C2, the average of 293,0 & 292,4 MT), -49 (Cell C3), and 0,439424 (Cell C4). Then change the temperature to -51. See what happens to TAS, Mach-nr. and CAS and verify with Excel and OFP.



You then may want to reconsider your opinion.

Kind regards,
Simon