Density Alt and ISA Question

[SpreadEagle]

This is not a question how to use the formulas. More where do you get the numbers.

Lots of people use 4 feet per degree 'rule of thumb' but where do you get the 4 from?

If this is from the other rule of thumb using 120ft/degree and then dividing it by 30 millibars to get 4, whats the formula to give you the 120 figure please?

I need a few less rules of thumb and a better understanding right now. Any help would be much appreciated.

:ok: - A Rule of thumb!

[paco]

Just add or subtract 120 feet for every degree above or below ISA, respectively (techinally it should be 118.8). You can find out what the ISA temp should be by lining up the 10s on you whizzwheel and looking in the little windows

Phil

[FlyingStone]

First some data you need to have in order to calculate the DA accurately:

Temperature at sea level (ISA): T0ISA = 273,15 K + 15 K = 288,15 K
Temperature (let's say ISA+1): T = T0ISA + 1K = 289,15 K
Static pressure at sea level (ISA): p0ISA = 1013,25 hPa = 101325 Pa
Static pressure (let's say it's the same as ISA since you are only interested in change of DA in light of temperature): p = p0ISA = 101325 Pa
Temperature gradient: a = 0.0065 K/km (notice: kilometers!!)
Gas constant: R = 287 J/(kg * K)
Gravity acceleration: g = 9,81 m/s^2
Air density at sea level (ISA): Rho0 = 1.225 kg/m^3

First, we are interested in effect of temperature increase on air density:

Rho = p / (R * T) = 1.2207 kg/m^3

Now we can calculate the density altitude with the following formula:

DA = (T0ISA / a) * (1 - (Rho / Rho0))^((R * a) / (g - R * a)) = 36.258 m

All we have to do now is to convert DA from meters to feet:

DAf = DA * 3.28 ft/m = 118.928 ft

Very close to the 118.8 ft/K that is mentioned throughout the ATPL books. Nice to see that at least few people care to really understand what is behind rules of thumb and ATPL theory, not just learn the answers to the questions :ok:

On Wikipedia (http://en.wikipedia.org/wiki/Density_altitude) you can find another formula, which gave me the result of 117.794 ft/K and probably suffers from rounding errors.

[SpreadEagle]

Flying stone, thanks. Awesome answer. Just what I wanted. I had scoured the internet and just kept seeing people bunging 120 into the formula with no explanation as to why.

That is the 120 mystery solved. :D

I am assuming the 4ft mystery number comes from the pressure decreasing by 4% per 1000ft, giving you 4ft/1000ft per degree.

[FlyingStone]

Glad to hear :) I too didn't really understand density altitude (and TAS/Mach for that matter) until I've been introduced to these formulas.