quick LSS calculations?

[gilbertmchris]

Hi

I was asked today what a quick way of working out the local speed of sound is from 'C to knots...

643+(1.2C)

Any quicker ways in your head?

[PBL]

....quick way of working out the local speed of sound is from 'C to knots...

643+(1.2C)


It is always good to specify your units. You mean speed of sound is
643 + 1.2T kts, where T is temperature measured in °C.

This is one of the best linear approximations to speed of sound in dry air. Between -40°C and 17°C it yields the speed of sound to the nearest kt. From 17°C to 37°C your formula is one kt too high, and at 40°C two kts too high. Similarly, at -50°C it is one kt too high, and at -60°C two kts too high.

I note, though, that this formula mixes up the units.

For those who wish to work it out for themselves, the exact formula is SpdSound = Sqrt(gammaDryAir x gasConstantDryAir x Temp°K), where gammaDryAir is the ratio between specific heat at constant pressure and specific heat at constant volume of dry air, about 1.4; the gasConstantDryAir is about 287 in SI units (J per kt per °K), and °K is about °C + 273. (cf. e.g., John D. Anderson Jr., Introduction to Flight, 6th edition, McGraw-Hill 2008, eqn 4.54 in Section 4.9. The value of gammaDryAir is in Section 4.6, and that of gasConstantDryAir is in Section 2.3).

What you get from this formula is speed of sound in meters per second (SI units). You can just plug the numbers in and press "Sqrt" on your calculator. If you want a series approximation, first note that the formula is approximately 20 x Sqrt(273 + x°C), or 331 x Sqrt(1 + x/273). Then you can expand this last term binomially and ignore quadratic and higher terms. Or leave the quadratic term in if you wish for better approximation at temperatures above 15°C and below -40°C. If I do this, I get SpdSnd = 331 + 0.575.x - 0.000556.x^2 + epsilon, where "epsilon" are the cubic and higher terms. The quadratic term makes a difference outside +/-20°C.

If you want to get SpdSnd in kts, you need to calculate in feet per second using gasConstantDryAir or 1716 ft-lbs per slug per °Rankine (where °Rankine is approximately (°F + 460). Then you need to convert ft/sec into kts (you won't go far wrong if you take 1 ft/sec = 0.6 kt). I am not exactly "bilingual" in SI/English units (to use Anderson's quaint way of putting it), so I prefer to calculate in SI and convert m/s to kts directly.

PBL

[Superpilot]

I like that!

[fantom]

That deserves a prize.

[Denti]

Exhaustive reply indeed.

By the way PBL, you know that Bielefeld doesn't exist, do you? Or are you one of THEM? ;)

[PBL]

That deserves a prize.
For showing that I can hack freshman aero? Or for turning something simple into something complicated with no apparent effort? Or simply because Awards are like piles. Sooner or later, every bum gets one?

By the way PBL, you know that Bielefeld doesn't exist, do you?

Well, of course I do! My colleague Fabio Magnifico produced the film which proved it (after the book). Which, I understand, grossed the most of them all at our local CineMaXX in the last few months.

PBL

[Pugilistic Animus]

one other way to see it c =3RT/M; from equipartition principle of kinetic molecular theory ... M= molar mass

:)

[Old Smokey]

Excellent response from PBL, one that deserves a "cut and paste" to the data files.

It is a good response for those seeking the background to the derivation of the simpler, yet highly accurate formulae used daily. Gamma, for example at 1.4 does indeed vary slightly, but is used as a constant by people such as Boeing and NASA. I am a product of the Douglas (the Santa Monica Douglas people, remember them?) process, and they too used 1.4 as a constant.

What it all boils down to, if you accept constants with minor variation as a constant, is a fairly simple formula which uses Static Air Temperature (SAT) in Degrees Celsius and Mach Number (M) -

TAS = 38.975 * SQR (SAT + 273.15) *M

This can be easily accomplished with a hand-held calculator (and is also used by most FMC/FMS units). It provides for TAS by applying the Mach Number, but for LSS simply use 1 for Mach Number.

Simple, but very accurate:ok:

PBL, Denti, Bielefeld gibt es doch!

Regards,

Old Smokey

[Pugilistic Animus]

just to clarify 'c' generally is used for any gas and 'a' for air; that formula is a general formula from fluid mechanics..although totally useless for flight :p

Old Smokey great to see you so active here again--- you're too damn busy:)