Standard Atmosphere

 

The Standard Atmosphere in Aerodynamics

 

We said we were starting with the Ideal Gas Equation of State, P=RT. We will also make use of the Hydrostatic Equation, another relationship you have seen before in chemistry and physics:


 



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This tells us how pressure changes with height in a column of fluid. This tells us how pressure changes as we move up or down through the atmosphere.

These two equations, the Perfect Gas Equation of State and the Hydrostatic Equation, have three variables in them; pressure, density, and temperature. To solve for these properties at any point in the atmosphere requires us to have one more equation, one involving temperature. This is going to require our first assumption. We must have some relationship that can tell us how temperature should vary with altitude in the atmosphere.

Many years of measurement and observation have shown that, in general, the lower portion of the atmosphere, where most airplanes fly, can be modeled in two segments, the Troposphere and the Stratosphere. The temperature in the troposphere is found to drop fairly linearly as altitude increases. This linear decrease in temperature continues up

to about 36,000 feet (about 11,000 meters). Above this altitude the temperature is found to hold constant up to altitudes over 100,000 ft. This constant temperature region is the lower part of the Stratosphere. The troposphere and stratosphere are where airplanes operate, so we need to look at these in detail.

 

 

The Troposphere in Aerodynamics,

 

We model the linear temperature drop with altitude in the troposphere with a simple equation:

 

Talt = Tsealevel Lh

 

where “L” is called the “lapse rate”. From over a hundred years of measurements it has been found that a normal, average lapse rate is:

L = 3.56oR / 1000 ft = 6.5oK / 1000 meters .

This is often taught to pilots in a strange mixture of units as 1.98 degrees Centigrade per thousand feet!

 

Standard Atmosphere Hydrostatic

The other thing we need is a value for the sea level temperature. Our model, also based on averages from years of measurement, uses the following sea level values for pressure, density, and temperature.




 

TSL = 288 oK = 520 oR

So, to find temperature at any point in the troposphere we use:

T (oR) = 520 3.56(h),

 


where h is the altitude in thousands of feet, or

 

 

where h is the altitude in thousands of meters.

 

 

T (oK) = 288 6.5 (h)

 

We need to stress at this point that this temperature model for the Troposphere is merely a model, but it is the model that everyone in the aviation and aerospace community has agreed to accept and use. The chance of ever going to the seashore and measuring a temperature of 59oF is slim and even if we find that temperature it will surely change within a few minutes. Likewise, if we were to send a thermometer up in a balloon on any given day the chance of finding a “lapse rate” equal to the one defined as “standard” is slim to none, and, during the passage of a weather front, we may even find that temperature increases rather than drops as we move to higher altitudes. Nonetheless, we will work with this model and perhaps later learn to make corrections for non-standard days.

 

Now, if we are willing to accept the model above for temperature change in the Troposphere, all we have to do is find relationships to tell how the other properties, pressure and density, change with altitude in the Troposphere. We start with the differential form of the hydrostatic equation and combine it with the Perfect Gas equation to eliminate the density term.


 

which is rearranged to give

 

dP/P = (g/RT)dh.

 

Now we substitute in the lapse rate relationship for the temperature to get

 

dP/P = {g/[R(TSL-Lh)]}dh.

 

This is now a relationship with only one variable (P) on the left and only one (h) on the right. It can be integrated to give



 

In a similar manner we can get a relationship to find the density at any altitude in the troposphere



 

So now we have equations to find pressure, density, and temperature at any altitude in the troposphere. Care has to be taken with units when using these equations. All temperatures must be in absolute values (Kelvin or Rankine instead of Celsius or Fahrenheit). The exponents in the pressure and density ratio equations must be unitless. Exponents cannot have units!

We can use these equations up to the top of the Troposphere, that is, up to 11,000 meters or 36,100 feet in altitude. Above that altitude is the Stratosphere where temperature is modeled as being constant up to roughly 100,000 feet.

 

 

The Stratosphere in Aerodynamics,

 

We can use the temperature lapse rate equation result at 11,000 meters altitude to find the temperature in this part of the Stratosphere.

Tstratosphere = 216.5oK = 389.99oR = constant

The equations for determining the pressure and density in the constant temperature part of the stratosphere are different from those in the troposphere since temperature is constant. And, since temperature is constant both pressure and density vary in the same manner.

The term on the right in the equation is “e” or 2.718, evaluated to the power shown, where h1 is the 11,000 meters or 36,100 ft (depending on the unit system used) and h2 is the altitude where the pressure or density is to be calculated. T is the temperature in the stratosphere.

Using the above equations we can find the pressure, temperature, or density anywhere an airplane might fly. It is common to tabulate this information into a standard atmosphere table. Most such tables also include the speed of sound and the air viscosity, both of which are functions of temperature. Tables in both SI and English units are given below.

 

 

A look at these tables will show a couple of terms that we have not discussed. These are the speed of sound “a”, and viscosity “μ”. The speed of sound is a function of temperature and decreases as temperature decreases in the Troposphere. Viscosity is also a function of temperature.

The speed of sound is a measure of the “compressibility” of a fluid. Water is fairly incompressible but air can be compressed as it might be in a piston/cylinder system. The speed of sound is essentially a measure of how fast a sound or compression wave can move through a fluid. We often talk about the speeds of high speed aircraft in terms of Mach number where Mach number is the relationship between the speed of flight and the speed of sound. As we get closer to the speed of sound (Mach One) the air becomes more compressible and it becomes more meaningful to write many equations that describe the flow in terms of Mach number rather than in terms of speed.

Viscosity is a measure of the degree to which molecules of the fluid bump into each other and transfer forces on a microscopic level. This becomes a measure of “friction” within a fluid and is an important term when looking at friction drag, the drag due to shear forces that occur when a fluid (air in our case) moves over the surface of a wing or body in the flow.

Two things should be noted in these tables about viscosity. First, the units look sort of strange. Second, the viscosity column is headed with μ X 10x. The units are the proper ones for viscosity in the SI and English systems respectively; however, if you talk to a chemist or physicist about viscosity they will probably quote numbers with units of “poise”. The

10xnumber in the column heading means that the number shown in the column has been multiplied by 10xto give it the

value shown. This is, to most of us, not intuitive. What this means is that in the English unit version of the Standard Atmosphere table, the viscosity at sea level has a value of 3.719 times ten to the minus 7.

So now we can find the properties of air at any altitude in our model or “standard” atmosphere. However, this is just a model, and it would be rare indeed to find a day when the atmosphere actually matches our model. Just how useful is this?

In reality this model is pretty good when it comes to pressure variation in the atmosphere because it is based on the hydrostatic equation which is physically correct. On the other hand, pressure at sea level does vary from day to day with weather changes, as the area of concern comes under the various high or low pressure systems often noted on weather maps. Temperature represents the greatest opportunity for variation between the model and the real atmosphere, after all, how many days a year is the temperature at the beach 59oF (520oR)? Density, of course, is a function of pressure and temperature, so its “correctness” is dependent on that of P and T.

 

On the face of things, it appears that the Standard Atmosphere is somewhat of a fantasy. On the other hand, it does give us a pretty good idea of how these properties of air should normally change with altitude. And, we can possibly make corrections to answers found when using this model by correcting for actual sea level pressure and temperature if

 

needed. Further, we could define other “standard” atmospheres if we are looking at flight conditions where conditions are exceptionally different from this model. This is done to give “Arctic Minimum” and “Tropical Maximum” atmosphere models.

In the end, we do all aircraft performance and aerodynamic calculations based on the normal standard atmosphere and all flight testing is done at standard atmosphere pressure conditions to define altitudes. The standard atmosphere is our model and it turns out that this model serves us well.

One way we use this model is to determine our altitude in flight.