Blackbird
Topic Author
Posts: 3384
Joined: Wed Oct 06, 1999 10:48 am

### Mach Number / Temperature Rise

Isn't there a formula for determining basically how the temperature will rise as the mach number increases?

If I recall, there was a 100 in the formula...

Anyone know the formula?

Andrea V. Kamarov

timz
Posts: 6233
Joined: Fri Sep 17, 1999 7:43 am

### RE: Mach Number / Temperature Rise

Temperature of what?

LeanOfPeak
Posts: 496
Joined: Thu Oct 21, 2004 2:18 am

### RE: Mach Number / Temperature Rise

Do you mean stagnation temperature?

T0 = T [1 + (g+1)/2 M2]

The temperature that would result if the flow were slowed to zero velocity adiabatically?

Newark777
Posts: 8284
Joined: Thu Dec 02, 2004 6:23 am

### RE: Mach Number / Temperature Rise

ad·i·a·bat·ic Audio pronunciation of "adiabatically" ( P ) Pronunciation Key (d--btk, d--)

Of, relating to, or being a reversible thermodynamic process that occurs without gain or loss of heat and without a change in entropy.

Harry

[Edited 2005-07-27 04:46:46]
Why grab a Heine when you can grab a Busch?

HT1000
Posts: 35
Joined: Sun Jun 26, 2005 9:09 am

### RE: Mach Number / Temperature Rise

Might be this one :

T = (TAS/100)² - (1/10).(X) with X = (TAS/100)²

T is not TAT but the difference between TAT and SAT. (In °C)
Works for TAS < 240 KT

Few Were Born With It. Even Fewer Know What To Do With It.

vikkyvik
Posts: 12043
Joined: Thu Jul 31, 2003 1:58 pm

### RE: Mach Number / Temperature Rise

I'm not really sure what you mean, but I'll give you the generic formula for speed of sound (which depends on the temp):

a = sqrt(gamma*R*T)

where:
a = speed of sound (in meters per second)
T = temp in Kelvin (zero degrees Celsius = 273.15 degrees Kelvin)

and specific to air:
gamma = adiabatic constant = 1.4
R = specific gas constant = 287

so for a temperature of 20 deg. C in air (68 deg F):

a = sqrt(1.4*287*293.15) = 343.2 m/s (about 773 mph)

~Vik
I'm watching Jeopardy. The category is worst Madonna songs. "This one from 1987 is terrible".

LeanOfPeak
Posts: 496
Joined: Thu Oct 21, 2004 2:18 am

### RE: Mach Number / Temperature Rise

Aaaagh...

g is a ratio of specific heats, so it is indeed unit-less.

PLEASE put J/kgK after R = 287.

Blackbird
Topic Author
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Joined: Wed Oct 06, 1999 10:48 am

### RE: Mach Number / Temperature Rise

I was basically getting at stagnation temps...

AVKam

vikkyvik
Posts: 12043
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### RE: Mach Number / Temperature Rise

LeanOfPeak,

My fault - didn't mean to offend your sensibilities

I knew it wasn't unitless, but just didn't notice the omission (it's been awhile since I went over this stuff).

~Vik
I'm watching Jeopardy. The category is worst Madonna songs. "This one from 1987 is terrible".

LeanOfPeak
Posts: 496
Joined: Thu Oct 21, 2004 2:18 am

### RE: Mach Number / Temperature Rise

It's not so much my sensibilities. With the commonality of Imperial units in the aerospace sector, the omission of units introduces the possibility for substantial error, as it leaves the impression of a dimensionless constant applicable regardless of the set of units being used (i.e. Mach number, Reynolds number, Froude number, Fibonnacci number [That one's a joke, so no one take it seriously as a dimensionless constant, though it is a real term], Poisson's ratio, and lift, drag, and frictional coefficients).

You can ask the folks who set up the Mars Climate Orbiter regarding the consequences of that one.

Airmansv
Posts: 11
Joined: Sun Apr 03, 2005 9:49 pm

### RE: Mach Number / Temperature Rise

LeanOffPeak has the correct equation.

Total (stagnation) to Static Temperature equals
1+ (1/2 *(GAMMA-1))* Mn**2, where gamma equals Cp / Cv, and is 1.4 under standard conditions.

You cannot measure static temperature by the way when you are moving. It is the total temperature that you measure. Hence one calculates Mn from total to static pressure ratio equation, and use it in the above equation having measured total temperature to calculate static temperature.

lehpron
Posts: 6846
Joined: Tue Jul 10, 2001 3:42 am

### RE: Mach Number / Temperature Rise

This has been buggin me since I learned of this equation:

 Quoting LeanOfPeak (Reply 2):Do you mean stagnation temperature? T0 = T [1 + (g+1)/2 M2] The temperature that would result if the flow were slowed to zero velocity adiabatically?

Regarding that, say a plane were moving at M5 at 80km (somehow) where the temperature is 165 Kelvin. Accordingly, the stagnation temperature should be 990 Kelvin. But the local density of air is a million times smaller than sea level. How does it get hot with almost no air out there?

Even the space just above the Earth has been measured to have an average of 4K, to the space shuttle, the stagnation average would be 104K @ M25 even when its almost a vacume out there!
The meaning of life is curiosity; we were put on this planet to explore opportunities.

vikkyvik
Posts: 12043
Joined: Thu Jul 31, 2003 1:58 pm

### RE: Mach Number / Temperature Rise

 Quoting Lehpron (Reply 11):Regarding that, say a plane were moving at M5 at 80km (somehow) where the temperature is 165 Kelvin. Accordingly, the stagnation temperature should be 990 Kelvin. But the local density of air is a million times smaller than sea level. How does it get hot with almost no air out there? Even the space just above the Earth has been measured to have an average of 4K, to the space shuttle, the stagnation average would be 104K @ M25 even when its almost a vacume out there!

I'm not totally sure, but I'll take an educated guess. Remember that temperature is the average kinetic energy of the molecules, while heat is the total energy. So a very sparse atmosphere can have a high temperature, but very little stored heat energy as compared to a dense atmosphere. Also, with a sparse atmosphere, one needs to add very little heat to increase the temperature. So the overall kinetic energy that is lost when the flow is stagnated should be more than enough to raise the average kinetic energy (temperature) of a sparse atmosphere.

~Vik
I'm watching Jeopardy. The category is worst Madonna songs. "This one from 1987 is terrible".

LeanOfPeak
Posts: 496
Joined: Thu Oct 21, 2004 2:18 am

### RE: Mach Number / Temperature Rise

For starters, neither a shock wave nor friction is an adiabatic process.

phollingsworth
Posts: 635
Joined: Sat Mar 06, 2004 6:05 am

### RE: Mach Number / Temperature Rise

 Quoting Newark777 (Reply 3):ad·i·a·bat·ic Audio pronunciation of "adiabatically" ( P ) Pronunciation Key (d--btk, d--) adj. Of, relating to, or being a reversible thermodynamic process that occurs without gain or loss of heat and without a change in entropy.

Except that isn't the correct definition. That would be an isentropic process (entropy remains the same). Adiabatic is only the first part. All isentropic processes are adiabatic, but not all adiabatic processes are isentropic, think standard treatment of shock-waves (i.e. frozen flow). Shock-waves are inherently highly viscous phenomena and are often not truly adiabatic; however, in may cases treating them adiabatically results in estimations that are well within the acceptable range of error.

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