A bit of a physics question

jr1104 · 7 Jan 2010 at 13:28

Probably not the best place to put this, but I figured someone might know.

I'm trying to calculate the change in density in different parts of the International Standard Atmosphere. The formula I'm using works fine for the troposphere, where the lapse rate is negative, but when it gets to the stratosphere, it all goes weird, due to the positive lapse rate. I'm wondering if anyone knew what I'm supposed to do here, or if there is a different equation to use?

Thanks

Smithy · 7 Jan 2010 at 13:38

What's the formula you're using?

jr1104 · 7 Jan 2010 at 13:54

d/d0=(t/t0)^((g/LR)-1) where d is density, t is temp and L is the lapse rate.

I'm trying it a slightly different way now, by working out temperature and pressure, then using d = P/Rt, but I still have the same problem with the positive lapse rate not working in my equations.

Smithy · 7 Jan 2010 at 15:16

So you're attempting to derive a formula to calculate the change in density with respect to base height? I gather this is what you're trying to achieve since the ISA is simply a model for atmospheric conditions at given altitudes.

jr1104 · 7 Jan 2010 at 15:47

Well, I have the formulas, and I'm wanting to calculate the density at different heights, in order to use it to calculate aerodynamic drag. But because the object is passing through different layers of the atmosphere, I need to use different base temperature and lapse rate values. The problem is that when calculating the pressure, using formula 1 below, the positive lapse rate means that the pressure increases, which obviously it shouldn't do if I am gaining altitude.

formula 1: P/P(0) = T/T(0) ^ (g/LR)

I've got around this for now by entering the lapse rate as negative all the time, and this gives me the graph I expected, but it seems like that isn't the correct thing to do.

Smithy · 8 Jan 2010 at 14:33

It wouldn't be the correct thing to do. The lapse rate defines the temperature variable at a given altitude, manipulating the sign is going to give an unrealistic value for temperature and will produce an incorrect solution.

In your case, it would be best to use the formula calculated from the ideal gas law for density; rho = p/R.T

You'll need to re-write that expression as a function of altitude. Introducing molar mass;

rho = p.M/R.T

Where;
p = absolute atmospheric pressure
R = universal gas constant R = 8.31447 J/(mol·K)
M = molar mass of dry air M = 0.0289644 kg/mol
T = temperature (K)

The initial calculation of pressure is where the variable of altitude also has an influence. You'll often see an expression for temperature given as;
T = T0 - L.h

Where;
T = temperature (K)
T0 = base temperature (K)
L = lapse rate
h = height (Km)

However, this is only applicable to the troposphere. There are other models for calculating temperature, the following is taken from NASA's website.

For h < 11km (troposphere)
T = 15.04 - .00649 * h

For 11km < h < 25km (lower stratosphere)
T = -56.46

For h > 25km (upper stratosphere)
T = -131.21 + .00299 * h

Where;
T = temperature (Celsius)
h = height (metres)

Convert the resulting temperature to Kelvin to apply it to the density formula above.

Problems with accuracy easily occur here, for instance water vapour has an effect on the density of air. Further calculations and application of formula are required to compensate. Also remember that the above formulae are only models, but are usually accepted for their accuracy.

Which course are you studying?