Temperature and Mach number

[dashtrashtoo!]

I know there are multiple threads on similar topics- but before you all shoot me down and tell me to do a search- I have!!! But I have a specific question which has either not been answered or I am not smart enough to work out all the crazy mathematical derivations and calculations that have been posted!

My question is this: I have read somewhere (I think, I'm starting to think I was dreaming, because of course I can't find it again) that temperature changes do not affect Mach number, as temperature affects both TAS and the local speed of sound in the same manner. I think this is not true as TAS is not affected by 39 times the square root of the absolute temperature.

So, what is the effect of changing temperatures on Mach number?

And remember, I am about as far from an aeronautical engineer or mathematician as one can get, so kindy speak please! :confused:

[Milt]

Keeping it simple - the speed of sound in air increases with increasing temperatue. Rough numbers are 760 mph at sea level for a temp of 15 C and 660 mph at the tropopause for a temp
of -56 C

Not sure whether the same goes for the speed of sound in water?
Any Sonar experts out there?

[dashtrashtoo!]

I know that the local speed of sound varies at a rate of 39 times the square root of the abosolute temperature i.e temp goes up, local speed of sound goes up and vice versa.

But what is the relationship of temperature and TAS (other than temp goes up, TAS goes up), and the combined effects of temperature and TAS and temperature and local speed of sound on Mach number?

Thanks

[Old Smokey]

A change in temperature has absolutely no effect upon Mach Number, Calibrated Airspeed, or Equivalent Airspeed.

A change in temperature will have a direct effect upon the Local Speed of Sound, and thus, for a given Mach Number (or CAS and EAS), TAS will change.

For a given Mach Number, TAS in Knots may be found from the formula -

TAS = 38.975 X Mach Number X Square Root of Absolute Temperature.

(Where Absolute Temperature °K = SAT °C + 273.15)

There is a side issue which may be misleading you. If an aircraft is in stable flight with a constant thrust and Mach Number, encountering warmer air will cause the Mach Number to decrease. That is entirely due to the effect of the warmer air upon Net Engine Thrust, and nothing to do with Mach number. If thrust is restored to it's original value (A new N1, EPR etc.), the original Mach Number, Mach, CAS, and EAS will all be restored, whilst TAS is higher. Colder air will have the opposite effect.

Regards,

Old Smokey

[dashtrashtoo!]

They're exactly the words I was looking for- thanks Old Smokey, you're a Tech Log legend! :cool:

[L Peacock]

Old smokey

Just to round this off, do you have the equation that links CAS & TAS?

Cheers

[Old Smokey]

L Peacock,

I'd be happy to just as soon as I get home from the present trip.

Beware! It's not as simple a process as you might think.

Aw shucks dashtrashtoo!, you've made me a legend in my own mind.:eek:

Danny, love the new layout - Blue was always my favourite colour:ok:

Regards,

Old Smokey

[XPMorten]

Here you go.

http://www.xplanefreeware.net/morten/DOCS/CAS.jpg

http://www.xplanefreeware.net/morten/DOCS/Mach2.jpg

Cheers,

M

[Mad (Flt) Scientist]

A change in temperature has absolutely no effect upon Mach Number, Calibrated Airspeed, or Equivalent Airspeed.

A change in temperature will have a direct effect upon the Local Speed of Sound, and thus, for a given Mach Number (or CAS and EAS), TAS will change.

There is a side issue which may be misleading you. If an aircraft is in stable flight with a constant thrust and Mach Number, encountering warmer air will cause the Mach Number to decrease. That is entirely due to the effect of the warmer air upon Net Engine Thrust, and nothing to do with Mach number. If thrust is restored to it's original value (A new N1, EPR etc.), the original Mach Number, Mach, CAS, and EAS will all be restored, whilst TAS is higher. Colder air will have the opposite effect.

Just to slightly finesse that sequence of events.

If a plane were flying along and encountered a temperature change (increase) the inertial speed will not, initially, change (that is the "real" speed of the lump of metal hurtling through the air). So if there are no wind changes, neither will the TAS change; instead, everything else will change to adjust for the new temperature and the (temporarily) constant TAS (with due allowance for sensor lags, in the real world).

Then, because the aircraft is now no longer at its original EAS, it will have an imbalance between thrust and drag (and if thrust also changes as noted above, there's further imbalance to account for). Therefore the aircraft will now accelerate or decelerate until the appropriate thrust-drag balance is attained. If the thrust were restored to the original steady state value, you'll get the same Mach, CAS, EAS as before, as OS describes.

It rather dpends on whether you're considering the transient response or not: in other words, whether you are "performance" or "S&C" oriented person.

[Dan Winterland]

38.94 times the square root of degrees Kelvin is a bit more accurate for finding the local speed of sound, rather than 39.

I must get out more.:hmm:

[Milt]

Dan

How accurate do you want to be? I see a difference of 0.15% between 38.94 and 39.

A bit like the journalist or the pure scientist who notices an aircraft flying at about 1000 ft and converts it to be 304.800 metres. Amazing how often we see a serious approximation converted to an absolute value to the nth degree which automatically causes most of us to lose credibility.

[XPMorten]

Those formulas are official Airbus material.... :ok:

Cheers,

M

[chornedsnorkack]

Shouldn´t the IAS and Mach number also depend on the composition of air, for equal temperature and static pressure?

Yes, the ratio of nitrogen to oxygen in air is roughly constant. As are the quantities of minor ingredients like argon and carbon dioxide.

But what about water?

The molecular mass of water is 18, while that of nitrogen is 28 and for oxygen, 32. This means that for equal total static pressure and temperature, moist air should hold less oxygen and nitrogen, and have lower density, than dry air. Less density means less lift and IAS for equal TAS.

The effects on sound speed should be yet more complicated...

For one thing, water vapour has lower molecular mass, which should mean faster speed of sound.

For another, there are the issues with water having an additional rotational degree of freedom, and more allowed vibrations, which IIRC ought to mean slower speed of sound...

And then I cannot quite figure out what happens to sound in wet adiabatic conditions...

Anyway, the speed of sound in water varies in a weird manner - goes through maximum at a certain temperature... But since it is so huge, I do not think planes should worry about Mach numbers in water - coming near water at such speeds, they have worries in air already, and no chance on water contact.

Froude numbers is quite another matter!

[selfin]

Magical little guide to this topic here (http://www.aviation.org.uk/pdf/Aircraft_Performance_Flight_Testing.pdf)

Airspeed starts on page 49(30).

[Old Smokey]

SPEED CALCULATIONS

Sorry for the late reply L Peacock, but better late than never. I warned you that it was complex, but here it is in a slightly simplified fashion.

All calculations are for Pressure Height, Valid from -1000 feet to 82021 feet of Pressure Height, with the Tropopause at 36089.24 feet.

To convert from Metres to Feet –
Feet = Metres / 0.3048

All Temperatures are in °C, to convert from Fahrenheit to Celcius –
Temperature °C = (°F-32) X 5/9

All working is in °Absolute (Kelvin), to convert from °C to °K (Kelvin) –
Absolute Temperature °K = °C + 273.15

All Pressures are in hPa, to convert from Inches of Mercury (“Hg) to hPa –
Pressure (hPa) = Pressure (Inches of Mercury) X 33.869

It will be necessary to know the ISA Standard Temperature (Ts) for the Pressure Height, and the Static Pressure (Ps) for the Pressure Height.

For Sea Level –
Standard Temperature (To) °K = 288.15°K
Standard Pressure (Po) = 1013.25 hPa

For the Pressure Height –
Standard Temperature (Ts) °K = 288.15 – PH X .0019812 to 36089.24 feet, and 216.65°K thereafter (to 82021 feet).

It is necessary to calculate the Mean Temperature (Tm) of the column of air to the Pressure Height, Note that this is NOT the arithmetic mean –

Tm = ((SQR 288.15 +SQR Ts)/2)^2 to 36089.24 feet.
(It is not necessary to make this calculation above 36089.24 feet).

Up to 36089.24 ft : Ps = 1013.25 / 10 ^ (PH / 220.82682 / Tm)
Above 36089.24 ft : Ps =1282.03 / 10 ^ (PH / 47912.5808)

Some Examples from these formulae –
10000 ft : Ts = 268.338°K : Tm = 278.156°K : Ps = 696.49 hPa
35000 ft : Ts = 218.808°K : Tm = 252.289°K : Ps = 238.50 hPa
45000 ft : Ts = 216.650°K : Tm = 216.650°K : Ps = 147.46 hPa (Tm above Tropopause)

The purists will note minor errors here, the formulae are extremely complex, and simplified for practical purposes. The Ps errors amount to accuracy within the equivalent of 10 feet of Pressure Height to 35000 feet, and within 15 feet from 35000 to 60000 feet, resulting in not more than 0.1 Kt error in speed.

DENSITY RATIO

Density Ratio is the ratio of the Density at the Pressure Height to that at Sea Level. Density is Pressure divided by temperature. For Simple Speed calculations, the Inverse Density Ratio (IDR) is more useful, i.e. the ratio of Sea Level Density to the Density at Altitude.

IDR = Po / To X Ts / Ps

For the 10,000 ft example –
IDR = 1013.25 / 288.15 X 278.156 / 696.49 = 1.404337638

NOTE – The data used here was for ISA temperature. In ‘off’ ISA conditions, actual Temperature SAT (as °K of course) should be used in lieu of the standard temperature (Ts) at altitude, e.g. in ISA+10°C conditions in the example, use 288.156 (SAT) instead of the standard 278.156 for the Ts input.

DENSITY AIRSPEED, AN APPROXIMATE

For Low Altitude and Low speed operations, e.g. up to 200 KIAS and 10,000 feet, correction for Density alone suffices for a reasonably accurate TAS calculation. This calculation assumes that CAS = EAS, i.e. dynamic pressure alone (EAS) is close to Impact Pressure (CAS). Although used as TAS, calculation of TAS using CAS and Density alone is, strictly speaking, Density Airspeed, DAS.

DAS = CAS X SQR (IDR)

For the example of 150 CAS at 10000 feet,
DAS = 150 X SQR 1.404337638 = 177.8 Kt.

This compares to an actual TAS of 174.1, an error of 3.7 Kt, acceptable to most people. In an ideal world, where aircraft had EAS indicators, this method could be used at all altitudes with accuracy. At higher speeds and altitudes, increasing Mach Number causes a much greater difference between Impact and Dynamic Pressure, and such a method becomes unacceptable. For example, a CAS of 300 Kt at 35000 feet yields a DAS of 538.8 Kt, Vs a TAS of 503.6 Kt, an unacceptable 35.2 Kt error. Thus, at higher speeds and altitudes, compressibility MUST be considered (the ‘f’ factor), and compressibility relates directly to Mach Number. This leaves 2 options, either calculate the ‘f’ factor (a useless number for further operations), or the Mach Number, which has considerable use in other areas. This latter option is chosen for further calculations.

ACCURATE SPEED CALCULATIONS

Excepting low and slow speed calculations, compressibility MUST be considered. An example will be given here for an aircraft at CAS = 300 Kt at 35000 feet, and SAT of -45°C (ISA + 9.342°C).

The steps are as follows –

As previously described, find the Static Pressure (Ps) at the Pressure Height. It is vital that the Static Pressure be calculated using the STANDARD Temperature for 35000 feet, NOT the actual Temperature (SAT).

For the example, as previously found, Ps = 238.50 hPa at 35,000 feet.

CONVERT CAS TO MACH NUMBER

Mach No = SQR (5 X (Po / Ps X (CAS^2 / K + 1) ^ 3.5 – Po / Ps + 1) ^ (2/7) – 5)
Where : Po = 1013.25 (Sea Level Standard Pressure),
Ps = Standard Pressure at the Pressure Height, and
K= 2188648.141 (a constant))

Note that Temperature has no influence upon the conversion from CAS to Mach Number, only CAS and Pressure.
For CAS = 300 at 35000 ft, Mach No. = 0.8733

CONVERT MACH NUMBER TO TAS

NOW, use the actual Absolute Temperature (SAT), -45°C = 228.15°K

TAS = Mach No. X 38.975 X SQR (Absolute Temperature)

For the example, Mach 0.8733 at 228.15°K = 514.1 Kt TAS.

CALCULATING EAS (If you’re interested)

Essentially, this is the reverse of the Density Airspeed (DAS) calculation, but using the ACTUAL Density Ratio, that is, using the Actual Temperature instead of the Standard Temperature.

EAS = TAS X SQR (Ps / SAT°K X To / Po)

For the example, EAS = 514.1 X SQR (238.50 / 228.15 X 288.15 / 1013.25) = 280.3 Kt.
Thus, although the Airspeed Indicator displays 300 Knots, the actual ‘Aerodynamic Value’ of the airspeed (Dynamic Pressure) is only 280.3 Kt.

That’s the complete speed picture. TAT, RAT etc., haven’t been covered. To summarise the procedure –

Calculate the Standard Temperature at the Pressure Height –
Ts (°C) = 15 – PH X .0019812 to 36089.24 ft, then -56.5°C above.

Convert Ts (°C) to Absolute Temperature –
Ts (°K) = Ts (°C) + 273.15

Calculate the Mean Temperature of the air column (not above 36089.24 ft) –
Tm = ((SQR 288.15 + SQR Ts °K) / 2 ) ^ 2

Calculate the Standard Pressure (Ps) at the Pressure Height –
Up to 36089.24 ft : Ps = 1013.25 / 10 ^ (PH / 220.82682 / Tm)
Above 36089.24 ft : Ps =1282.03 / 10 ^ (PH / 47912.5808)

Convert CAS to Mach Number –
Mach No = SQR (5 X (Po / Ps X (CAS^2 / K + 1) ^ 3.5 – Po / Ps + 1) ^ (2/7) – 5)
Where K= 2188648.141 (a constant))

Convert Mach Number to TAS –
TAS = Mach No. X 38.975 X SQR (SAT °K)

Calculate EAS (if you want) –
EAS = TAS X SQR (Ps / SAT°K X To / Po)

A lot of these formulae can be merged to provide for more convenient 'one-liners' if you choose. I WARNED YOU!

Regards,

Old Smokey

[L Peacock]

Smokey, It's taken me six months to discover this post. Thanks for the effort and sorry for the belated gratitude. Just what I was after (then and now!).:D

[Rivet gun]

There's various ways of expressing the speed equations, but my favorites are those which express speeds as a function of the standard sea level speed of sound (ao). These have the advantage that they work in any units, i.e if you want the answer in knots use ao = 661.48 kts, or if you want m/s use ao = 340.35 m/s etc.

For pressures, we are interested only in pressure ratios, which are dimensionless, so pressure units and absolute values are irrelevant. For temperature you can use Kelvin or Rankine so long as you are consistant. For altitude (at least in the troposphere) you can use any units so long as you use the ISA lapse rate in the same units.

So we have:

Speed of sound (a)
a= ao x sqrt(Ts/To)

where
Ts is static air temperature (K)
To is 288.15 K

From which we get TAS in terms of Mach number (M).

TAS = M x ao x sqrt(Ts/To)

Now, by analogy we can express EAS in terms of Mach number.

EAS = M x ao x sqrt(Ps/Po)

If you wanted to know EAS in flight it would be a simple matter to make a spredsheet of ao x sqrt(Ps/Po) against flight levels, then multiply this by indicated Mach number.

To get the presssure ratio Ps/Po, for the troposphere:

Ps/Po = (1- h x L/To) ^ 5.25588

where
h is flight level
L is ISA lapse rate (0.1981224 K/100ft)

To get Mach number from CAS you need the equation given by Old Smokey which requires the inverse of the pressure ratio and a constant K.

M = sqrt(5 X (Po / Ps X (CAS^2 / K + 1) ^ (7/2) – Po / Ps + 1) ^ (2/7) – 5)

Note that K is also a function of ao.

K = 5 x ao ^ 2 (I calculate K = 2187771, I think Old Smokey's value is a little out)


If you want to calibrate an airspeed indicator from a manometer or accurate pressure gauge you need CAS as a function of impact pressure (qc). Here you do need the absolute value of Po in whatever units you are measuring qc so as to get the qc/Po ratio.

CAS = ao x sqrt(5 x ((qc/Po +1) ^ (2/7)-1))

So if you remember that ao = 661.48 knots it might come in handy one day!

[gearpins]

M.N=TAS/local speed of sound.
TAS=Pitot pr-static pr (hence subject to density )
local speed of sound is also a funtion of density
hence in the above eq the effect of density is nullified (numerator/denominator). any change in temp will affect density at both ends

[rhovsquared]

wow I missed this one too and the 238 page, PDF , i'm gonna print that And i'm going to print this entire thread - wow Old Smokey !!!!
P.S I'm glad I took One Year of Calc based physics, and math through multivariable calculus, Physical chemistry I and II [ P- chem I,comprises 1. Gas laws :\ 2.Chemical thermodynamics and Kinetics:\ ] And P- chem II :\ :\ Quantum mechanics :\ Spectroscopy :\ and Statisical thermodynamics :\ :\ :mad: All ending with a nice digression on Kinetic molecular theory. :ugh: :\ :ugh: and read Aerodynamics for Naval Aviator's
Otherwise it would be all Greek to me :}
rhovsquared :D