The Standard Atmosphere in
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
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!
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
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.