For most purposes the properties of the air we fly in are described by the International Standard Atmosphere (ISA). This makes some assumptions about what is a normal day, then computes the air properties at different altitudes. In this blog I am going to give you the fastest explanation of how the ISA works out the properties of the air above us. Then in the next blog I will derive a formula to show how things change when the day is not standard.
ISA assumes that the atmosphere is in layers, and each layer has a predefined change of temperature with height. Below 11,000 metres (the Troposphere) the temperature reduces with height, then above that level (the Stratosphere) the temperature is constant. The standard goes on to higher layers used by rockets and balloons but for aviation we can stop there. In fact, for this blog we’re not going above 11,000 metres to keep things simple.
In the ISA Troposphere, sea level temperature is +15C and the temperature reduces by 6.5C per 1,000 metres (about 2C per 1,000 ft). The pressure at sea level is 1,013 mBar and the acceleration due to gravity is 9.81 m/sec2.
Calculating the Atmosphere
Little Pieces of Air Piled High
This bit I think is brilliant. I have borrowed a diagram from Professor Mustafa Cavcar’s paper which shows how we can take a small piece of air with unit area and height dh. (As this is only a small piece of air, we use the mathematical term dh to represent a small change in height). The mass of air in this cylinder is determined by the density of the air, ρ, and its volume, 1xdh (remember, an area of 1 with a depth dh). The force on this mass of air is determined by the earth’s gravity, so is given by the term ρ x g x dh
Now, the change in pressure from the bottom of the piece of air to the top of the piece of air has to be enough to balance the weight of this piece of air.
Each time we go up a little bit in height, the pressure reduces a little bit because the stack of air above us which we have to keep up is a little bit smaller.
Gas Equation of State
When I was at university, my old thermodynamics lecturer was a Scotsman who had a catch phrase that sticks in the mind. “I say to my wife, ‘What does pV equal?’ and she says ‘nRT’”. I can hear his rolled R to this day. This gives us the gas equation of state:
Here, a gas at pressure p, and volume V has a number of moles n, temperature T (degrees Kelvin) and R is the ideal gas constant. This equation is useful for ideal gasses; however, for real systems some people prefer this form of the equation:
In this form P is the density and R Specific is the specific gas constant, which from now on we will simply refer to as R. This constant is a fixed number for a given gas, and for air is 287. The units of R are horrible, so we ignore this and pretend it’s just a number.
Using this equation with the equation for dp for a little piece of air we obtain the following:
But in the Troposphere the temperature varies with height, with an assumed constant lapse rate, L:
As all the terms except h and p are constants, this is how we relate altitude to pressure. The pressure altimeter in an aircraft just performs this equation by measuring the static pressure outside and displaying a height reading. As well as relating height to pressure, the same basic formulae can be used to work out air density, temperature and speed of sound. Different functions apply above 11km, but the principles are similar, just that the maths is a bit easier!
For the know-it-all’s out there, yes, I did make some simplifications. I ignored the difference between geopotential and geometric heights. That is, the fact that as you get away from the centre of the earth, the acceleration due to gravity reduces. This is mostly because the ISA standard is actually geopotential heights and not geometric and partly because the effect is relatively small. At 7,000 ft the geometric altitude is 7,002.3 ft. Also I ignored the effect of water vapour (ISA assumes dry air) and some of the constants I listed did not have all their decimal places. The variation in temperature with altitude was assumed constant whereas inversions and other effects can undermine this assumption.