Air Density and Density Altitude Calculations
updated: May 31, 2002
What is density altitude?
The density altitude is the altitude at which the density of the International Standard Atmosphere (ISA) is the same as the density of the air being evaluated. The Standard Atmosphere is simply a mathematical model of the atmosphere which is standardized so that predictable calculations can be made.
So, the basic idea of calculating density altitude is to calculate the actual density of the air, and then find the altitude at which that same air density occurs in the Standard Atmosphere.
In the following paragraphs, we'll go step by step through the process of calculating the actual density of the air, and then determining the corresponding density altitude.
Density and Density
Altitude:
Although the concept of density altitude is commonly used to help express the effects of aircraft performance, the really important quantity is actually the air density.
For example, the lift of an aircraft wing, the aerodynamic drag and the thrust of a propeller blade are all directly proportional to the air density. The downforce of a racecar spoiler is also directly proportional to the air density.. Similarly, the horsepower output of an internal combustion engine is related to the air density. The correct size of a carburetor jet is related to the air density, and the pulse width command to an electronic fuel injection nozzle is also related to the air density.
Density altitude has been a convenient yardstick
for pilots to compare the performance of aircraft at various altitudes, but it
is in fact the air density that is the fundamentally important quantity, and
density altitude is simply a way to express the air
density.
Units:
The 1976 International Standard Atmosphere is
mostly described in metric SI units, and I have chosen to use those same units
(in general). See
ref 8 and
ref 9 for conversion
factors to your favorite units.
Air Density
Calculations:
To begin to understand the calculation of air density, consider the ideal gas law:
(1) P*V = n*R*T
where: P = pressure
V = volume
n = number of moles
R = gas constant
T = temperature
Density is simply the number of molecules of the ideal gas in a certain volume, in this case a molar volume, which may be mathematically expressed as:
(2) D = n / V
where: D = density
n = number of molecules
V = volume
Then, by combining the previous two equations, the expression for the density becomes:
(3)
where: D = density, kg/m^{3}
P = pressure, Pascals ( multiply mb by 100 to get Pascals)
R = gas constant , J/(kg*degK) = 287.05 for dry air
T = temperature, degK = deg C + 273.15
As an example, using the ISA standard sea
level conditions of P = 101325 Pa and T = 15 deg C, the air density at sea
level, may be calculated as:
D = (101325) / (287.05 * (15 + 273.15)) = 1.2250 kg/m^{3}
This example has been derived for the dry air of the standard conditions. However, for real-world situations, it is necessary to understand how the density is affected by the moisture in the air.
The density of a mixture of dry air molecules and water vapor molecules may be expressed as:
(4)
where: D = density, kg/m^{3}
P_{d} = pressure of dry air, Pascals
P_{v}= pressure of water vapor, Pascals
Rd = gas constant for dry air, J/(kg*degK) = 287.05
Rv = gas constant for water vapor, J/(kg*degK) = 461.495
T = temperature, degK = deg C + 273.15
To determine the density of the air, it is necessary to know is the actual air pressure (also known as absolute pressure, or station pressure), the water vapor pressure, and the temperature.
It is possible to obtain a rough approximation of the absolute pressure by adjusting an altimeter to read zero altitude and reading the value in the Kollsman window as the actual air pressure, but this method only gives the correct reading if the ambient air temperature happens to be the same as standard temperature at your elevation. Near the end of this page I'll discuss how to use the altimeter reading to accurately determine the actual pressure.
Alternatively, there are many little electronic gadgets that can measure the actual air pressure directly, and quite accurately. The water vapor pressure can be determined from the dew point or from the relative humidity, and the ambient temperature can be measured in a well ventilated place out of the direct sunlight.
In the following section, we'll calculate the
portion of the total air pressure that is due to the water vapor in the air that
is being measuring.
Vapor Pressure:
A very accurate, albeit quite odd looking, formula for determining the saturation vapor pressure is a polynomial developed by Herman Wobus (see ref 2 ) :
(5) Es = e_{so} * p^{-8}
where: Es = saturation pressure of water vapor, mb
e_{so}=6.1078
p = (c_{0}+T*(c_{1}+T*(c_{2}+T*(c_{3}+T*(c_{4}+T*(c_{5}+T*(c_{6}+T*(c_{7}+T*(c_{8}+T*(c_{9}))))))))))
T = temperature, deg C
c_{0}=0.99999683
c_{1}=-0.90826951*10^{-2}
c_{2}=0.78736169*10^{-4}
c_{3}=-0.61117958*10^{-6}
c_{4}=0.43884187*10^{-8}
c_{5}=-0.29883885*10^{-10}
c_{6}=0.21874425*10^{-12}
c_{7}=-0.17892321*10^{-14}
c_{8}=0.11112018*10^{-16}
c_{9}=-0.30994571*10^{-19}
For situations where a slightly less
accurate formula is acceptable, the following equation offers good results,
especially at the higher ambient air temperatures where the saturation pressure
becomes significant for the density altitude calculations.
(6)
where: Es = saturation pressure of water vapor, mb
Tc = temperature, deg C
c_{0} = 6.1078
c_{1} = 7.5
c_{2} = 237.3
See ref 2 and ref 11 for additional vapor pressure formulas.
Here's a calculator that evaluates the
saturation vapor pressure using equations 5 and 6 as given above:
Saturated Vapor Press Calculator |
The Smithsonian reference tables (see ref 1) give the following values of saturated vapor pressure values at specified temperatures. Entering these known temperatures into the calculator will allow you to evaluate the accuracy of the calculated results.
Deg C | Es, mb |
30 | 42.430 |
20 | 23.373 |
10 | 12.272 |
0 | 6.1078 |
-10 | 2.8627 |
-30 | 0.5088 |
Armed with the vapor pressure equations, the next step is to determine the actual value of vapor pressure.
When calculating the vapor pressure, it is often more accurate to use the dew point temperature that the relative humidity. Although relative humidity can be used to determine the vapor pressure, the value of relative humidity is strongly affected by the ambient temperature, and is therefore constantly changing during the day as the air is heated and cooled.
In contrast, the value of the dew point is much more stable and is often nearly constant for a given air mass. Therefore, using the dew point as the measure of humidity allows for more stable and therefore potentially more accurate results.
Actual Vapor Pressure from the Dew
Point:
To determine the actual vapor pressure, simply use the dew point as the value of T in equation 5 or 6. That is, at the dew point, Es = Pv.
(7) Es = Pv at the dew point
Actual Vapor Pressure from Relative Humidity:
Relative humidity is defined as the ratio (expressed as a percentage) of the actual vapor pressure to the saturation vapor pressure at a given temperature.
To find the actual vapor pressure, simply multiply the saturation vapor pressure by the percentage and the result is the actual vapor pressure. For example, if the relative humidity is 40% and the temperature is 30 deg C, then the saturation vapor pressure is 42.43 mb and the actual vapor pressure is 40% of 42.43 mb, which is 16.97 mb.
Density Calculations:
Now that the actual vapor pressure is known, we can calculate the density of the combination of dry air and water vapor as described in equation 4.
The total measured atmospheric pressure is the sum of the pressure of the dry air and the vapor pressure:
(8) P = Pd + Pv
where: P = total pressure
Pd = pressure due to dry air
Pv = pressure due to water vapor
So, rearranging that equation, we see that Pd = P-Pv. Now we have all of the information that is required to calculate the air density.
Calculate the air density:
Now armed with those equations and the actual air pressure, the vapor pressure and the temperature, the density of the air can be calculated..
Here's a calculator that determines the air density from the actual pressure, dew point and air temperature using equations 4, 6, 7 and 8 as defined above:
Air Density Calculator |
Some examples of calculations using air density:
Example 1) The lift of an aircraft wing
may be described mathematically (see ref 8)
as:
L = c_{1} * d * v^{2}/2 * a
where: L = lift
c_{1} = lift coefficient
d = air density
v = velocity
a = wing area
From the lift equation, we see that the lift of a wing is directly proportional to the air density. So if a certain wing can lift, for example, 3000 pounds at sea level standard conditions where the density is 1.2250 kg/m3, then how much can the wing lift on a warm summer day in Denver when the air temperature is 95 deg (35 deg C), the actual pressure is 24.45 in-Hg (828 mb) and the dew point is 67 deg F (19.4 deg C)? The answer is about 2268 pounds.
Example 2) The engine manufacturer Rotax advises that their carburetor main jet should be adjusted according to the air density (see ref 6). Specifically, if the engine is jetted properly at air density d1, then for operation at air density d2 the new jet diameter j2 is given mathematically as (Note: in some equations, where the exponent may not be obvious, the symbol ** is used to denote exponentiation.):
j_{2} = j_{1} * (d_{2}/d_{1})^{**(1/4)}
where: j_{2} = diameter of new
jet
j_{1 } = diameter of jet that was proper at density d1
d_{1} = density at which the original jet j1 was
correct
d_{2} = the new air density
That is, Rotax says that the correct jet diameter should be sized according to the fourth root of the ratio of the air densities (i.e. take the square root twice).
For example, if the correct jet at sea level
standard conditions is a number 160 and the jet number is a measure of the jet
diameter, then what jet should be used for operations on the warm summer day in
Denver described above? The ideal answer is a jet number 149, and in practice
the closest available jet size is then selected.
Example 3) In the same service bulletin mentioned above, Rotax says that their engine horsepower will decrease in proportion to the air density.
hp_{2} = hp_{1} * (d_{2}/d_{1})
where: hp_{2} = the new horsepower at density d_{2}
hp_{1} = the old horsepower at density d_{1}
If a Rotax engine was rated at 38 horsepower at sea level standard conditions, what is the available horsepower according to that formula when the engine is operated at a temperature of 30 deg C, a pressure of 925 mb and a dew point of 25 deg C? The answer is approximately 32 horsepower. (Click this link for details on the SAE method of correcting horsepower.)
Back on the trail of Density Altitude...
The definition of density altitude is the altitude at which the density of the 1976 International Standard Atmosphere is the same as the density of the air being evaluated. So, now that we know how to determine the air density, we can solve for the altitude in the International Standard Atmosphere that has the same value of density.
The International Standard Atmosphere is a
mathematical description of a theoretical column of air (see ref 5). To get the proper results,
it is necessary to use the following constants that are specified in the 1976
International Standard Atmosphere document:
P_{o} = 101325 sea level standard pressure, Pa
T_{o} = 288.15 sea level standard temperature, deg K
g = 9.80665 gravitational constant, m/sec^{2}
L = 6.5 temperature lapse rate, deg K/km
R = 8.31432 gas constant, J/ mol*deg K
M = 28.9644 molecular weight of dry air, gm/mol
In the ISA, the lowest region is the
troposphere which extends from sea level up to 11 km (about 36,000 ft). The
model that will be developed here is only valid in the troposphere. The
equations that define the air in the troposphere are:
(9)
(10)
(11)
where: T = ISA temperature in deg K
P = ISA pressure in Pa
D = ISA density in kg/m^{3} H = ISA geopotential altitude in km
One way to determine the altitude at which a certain density occurs is to rewrite the equations and solve for the variable H, which is the geopotential altitude.
So, it is now necessary to rewrite equations 9,
10, and 11 in a manner that expresses altitude H as a function of density D.
After a bit of gnashing of teeth and general turmoil, the exact solution for H
as a function of D, may be written as:
(12)
Using the numerical values of the ISA constants, that expression may be
evaluated as:
where H = geopotential altitude, km
D = air density, kg/m^{3 }
Now that H is known as a function of D, it is easy to solve for the Density Altitude of any specified air density.
It is interesting to note that equations 9, 10 and 11 could also be evaluated to find H as a function of P as follows:
where H = geopotential altitude, km^{} P = actual air pressure, Pascals
Now that we can determine the altitude for a
given density, it may be useful to consider some of the definitions of
altitude.
Different Flavors of Altitude:
There are three commonly used varieties of altitude (see ref 4). They are: Geometric altitude, Geopotential altitude and Pressure altitude.
Geometric altitude is what you would measure with a tape measure, while the Geopotential altitude is a mathematical description based on the potential energy of an object in the earth's gravity. Pressure altitude is what an altimeter displays when set to 29.92.
The ISA equations use geopotential altitude, because that makes the equations much simpler and more manageable. To convert the result from the geopotential altitude H to the geometric altitude Z, the following formula may be used:
(13)
where E = 6356 km, the radius of the earth (for 1976 ISA)
H = geopotential altitude, km
Z = geometric altitude, km
Density Altitude Calculator:
The following calculator uses equation 12 to convert an input value of air density to the corresponding altitude in the 1976 International Standard Atmosphere. Then, the results are displayed as both geopotential altitude and geometric altitude, which are very nearly identical at lower altitudes.
Note that since these equations are designed to model the troposphere, this calculator will give an error message if the calculated value of altitude is beyond the bounds of the troposphere, which extends from sea level up to a geopotential altitude of 11 km.
Density Altitude Calculator 1 |
Here's a calculator that uses the actual
pressure, air temperature and dew point to calculate the air density as well as
the corresponding density altitude:
Density Altitude
Calculator 2 |
Density Altitude calculations using Virtual Temperature:
As an alternative to the use of equations which describe an atmosphere made up up the combination of air and water vapor, it is possible to define a virtual temperature and then consider the atmosphere to be only dry air.
The virtual temperature is the temperature that dry air would have if its pressure and specific volume were equal to those of a given sample of moist air. It's often easier to use virtual temperature in place of the actual temperature to account for the effect of water vapor while continuing to use the gas constant for dry air.
The results should be exactly the same as in the previous method, this is just an alternative method.
There are two steps in this scheme: first calculate the virtual temperature and then use that temperature in the corresponding altitude equation.
The equation for virtual temperature may be derived by manipulation of the density equation that was presented earlier as equation 4:
Recalling that P = Pd + Pv, which means that Pd = P - Pv, the equation may be rewritten as
Finally, a new temperature T_{v}, the virtual temperature, is defined such that
By evaluating the numerical values of the constants, setting P_{v} = E, noting that R_{d} = R*1000/M_{d} and that R_{v}=R*1000/M_{v}, then the virtual temperature may be expressed as:
(14)
where T_{v }= virtual temperature, deg K
T = ambient temperature, deg K
c_{1} = ( 1 - (M_{v} / M_{d} ) ) = 0.37800
E = vapor pressure, kg/m^{3 } P = actual (station) pressure, mb
where M_{d} is molecular weight of dry air =
28.9644
M_{v} is molecular weight of water = 18.016
(Note that for convenience, the units in Equation 14 are not purely SI units, but rather are customary units for the vapor pressure and station pressure.)
The following calculator uses equation 6 to find the vapor pressure, then calculates the virtual temperature using equation 14:
Virtual
Temperature
Calculator |
The virtual temperature T_{v} may
used in the following formula to calculate the density altitude. This formula is
simply a rearrangement of equations 9, 10 and 11:
(15)
Using the numerical values of the ISA constants, equation 15 may be
rewritten using the virtual temperature as:
where H = geopotential density altitude, km
T_{v} = virtual temperature, deg K
P = actual (station) pressure, Pascals
Using the Altimeter Setting:
When the actual pressure is not known, the altimeter reading may be used to determine the actual pressure.
The altimeter setting is the value in the Kollsman window of an altimeter when the altimeter is adjusted to read the correct altitude. The altimeter setting is generally included in National Weather Service reports, and can be used to determine the actual pressure using the following equations:
According to NWS ASOS documentation, the actual pressure P_{a} is
related to the altimeter setting AS by the following equation:
(16)
By numerically evaluating the constants and converting to customary units of altitude and pressure, the equation may be written as:
P_{a} = [AS^{k1} - ( k2 * H ) ]^{1/k1 }
where P_{a} = actual (station) pressure, mb
AS = altimeter setting, mb
H = geopotential station elevation, m
k1 = 0.190263
k2 = 8.417286*10^{-5}
When converted to English units, this is the relationship between station
pressure and altimeter setting that is used by the National Weather Service ASOS
weather stations (see ref 10 ) as:
P_{a} = [AS^{0.1903} - (1.313 x 10^{-5}) x H]^{5.255 }
where P_{a} = actual (station) pressure, inches Hg
AS = altimeter setting, inches Hg
H = station elevation, feet
Using these equations, the altimeter setting may be readily converted to actual pressure, then by using the actual pressure along with the temperature and dew point, the local air density may be calculated, and finally the density may be used to determine the corresponding density altitude.
Given the values of the altimeter setting (the value in the Kollsman window) and the altimeter reading (the geometric altitude), the following calculator will convert the altitude to geopotential altitude, and solve equation 16 for the actual pressure at that altitude.
Altimeter Setting to Actual Pressure |
Using National Weather Service Barometric Pressure:
Now you're probably wondering about converting sea-level corrected barometric pressure, as commonly reported by the National Weather Service, to actual air pressure for use in calculating density altitude. Well the good news is that yes, sea level barometric pressure can be converted to actual air pressure. The bad news is that the result may not be very accurate.
If you want accurate density or density altitude calculations, you really need to know the actual air pressure.
In order to compare surface pressures from various parts of the country, the National Weather Service converts the actual air pressure reading into a sea level corrected barometric pressure. In that way, the common reference to sea level pressure readings allows surface features such as pressure changes to be more easily understood.
But, unfortunately, there really is no fool-proof way to convert the actual air pressure to a sea level corrected value. There are a number of such algorithms currently in use, but they all suffer from various problems that can occasionally cause inaccurate results (see ref 7).
It has been estimated that the errors in the sea level pressure reading (in mb) may be on the order of 1.5 times the temperature error for a station like Denver at 1640 meters. So, if the temperature error was 10 deg C, then the sea level pressure conversion might occasionally be in error by 15 mb. At the very highest airports such as Leadville, Colorado at an elevation of 3026 meters (9927 ft), perhaps the error might be on the order of 30 mb.
And further complicating matters, without knowing the details of the algorithm that was used to calculate the sea level pressure, it is likely that there will be some additional error introduced in the process of converting the sea level pressure back to the desired actual station pressure.
These error estimates are probably on the extreme side, but it seems reasonable to say that the density altitude calculations made using the National Weather Service sea level pressure calculations may have an uncertainty of ±10% or more.
When using pressure data from the National
Weather Service, be certain to find out if the pressure is the altimeter setting
or the sea-level corrected pressure. They may be quite different in some
situations.
Density Altitude Algorithm...
Here is a list of the steps performed by my Density Altitude Calculator :
1. convert ambient temperature to deg C,
2. convert geometric (survey) altitude to geopotential altitude in meters,
3. convert dew point to deg C,
4. convert altimeter setting to mb.
5. use temp to calculate vapor pressure in mb,
6. use geopotential altitude and altimeter setting to calculate the absolute pressure in mb,
7. use absolute pressure, vapor pressure and temp to calculate air density in kg/m3,
8. use the density to find the ISA altitude in meters which has that same density,
9. convert the ISA geopotential altitude to geometric altitude in meters,
10. convert the geometric altitude into the desired units and display the results.
On-Line Calculators:
Click here for Density Altitude Calculator with English units only.
Click here for Density Altitude Calculator with Metric units only.
Click here for Density Altitude Calculator using relative humidity rather than dew point.
Click here for Density Altitude Calculator with both English and Metric units.
Click here for new Engine Tuner's Calculator that includes relative horsepower, air density, density altitude, virtual temperature, absolute pressure, vapor pressure, relative humidity and dyno correction factor!
Simpler Methods of Calculation...
If you want to know the actual density altitude, it will need to be calculated in a manner similar to what I have described above.
There are many forms of simpler approximations and generalizations that have been developed over the years, but they are not really density altitude, they are just numbers that are kinda like density altitude. When the air is dry, the approximations and simplifications can be fairly accurate but in real life situations with moisture in the air they can be quite inaccurate.
enjoy....
References:
1. List, R.J. (editor), 1958, Smithsonian Meteorological Tables, Smithsonian Institute, Washington, D.C.
2. Thermodynamic subroutines by Schlatter and Baker .... lots of Fortran algorithms and excellent references
3. El Paso National Weather Service ... equations in perl cgi scripts by Tim Brice
4. http://mtp.jpl.nasa.gov/notes/altitude/altitude.html ... different flavors of altitude explained
5. http://www.pdas.com/hydro.htm ... the basic equations for the 1976 Standard Atmosphere
6. http://www.aviasport.com/Documentacion/ROTAX/Boletines/8UL87.PDF ... Rotax service bulletin
7. http://www.crh.noaa.gov/unr/edusafe/mslp/ ... discussion of sea-level conversion problems
8. http://www.digitaldutch.com/unitconverter/index.htm ... conversion factors
9. http://physics.nist.gov/Pubs/SP811/appenB8.html ... SI conversion factors from NIST
10. http://meted.ucar.edu/export/asos/Pressure.HTML ... ASOS algorithms
11. http://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htm
... NASA vapor pressure
Some related web links:
http://atmos.nmsu.edu/education_and_outreach/encyclopedia/humidity.htm ... humidity equations
http://www.digitaldutch.com/atmoscalc/ ... ISA calculator on-line
http://www.grc.nasa.gov/WWW/K-12/airplane/short.html ... index of education materials from NASA
ISA Standard Conditions ... listing of sea level conditions and a table of variation with altitude
El Paso NWS - calculators ... atmospheric calculators using Tim Brice's cgi scripts
http://www.grc.nasa.gov/WWW/K-12/airplane/short.html ... NASA airfoil simulator... this is fantastic
http://www.usatoday.com/weather/wdenalt.htm ... lots of pages of weather related info and formulas
http://www.srv.net/~airpig/dnsalt.html
... airman Mike's page, a very basic look at density altitude
http://www.reefnet.on.ca/gearbag/wwwatm.html
.... pressure equations
http://www.natmus.dk/cons/tp/atmcalc/atmoclc1.htm
... moisture calculations
http://efml.stanford.edu/FAMbook/Chap2.pdf...
equations in pdf format
http://hurri.kean.edu/~yoh/calculations/satvap/satvap.html
... saturation equations plus calculator
http://hurri.kean.edu/~yoh/calculations/moisture/Equations/moist.html
... moisture calculations
http://www.weathergraphics.com/da/
... amazing low-cost software for weather analysis