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Air Density and Density Altitude Calculations updated: Jul 2, 2007
Density Altitude On-Line Calculators:
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. And finally, at the very end of this article, we'll compare the accurate density altitude calculations with the results of a greatly simplified equation that ignores the effects due to water vapor in the air. Some different meanings of the word "altitude": An aircraft altimeter measures only air pressure... nothing else. If the air pressure changes, due to temperature or humidity, then an aircraft altimeter will of course change to indicate the actual air pressure. Nonetheless, the aircraft altimeter is simply measuring air pressure. As odd as it may seem, an aircraft altimeter does not actually measure altitude, it only measures pressure. Hence, the name "pressure altitude" is properly applied to any aircraft altimeter reading. For pilots, it is very important to understand that an aircraft altimeter only measures air pressure (not altitude). This point is especially important to understand with the advent and use of GPS. An aircraft flying at a specific pressure altitude (as indicated by an altimeter) may note some other altitude displayed on the GPS (which measures actual distance above mean sea level). In some cases this difference is small... but in some cases it could be enough to cause a mid-air collision if a pilot was flying on a GPS altitude rather than pressure altitude. (To solve that problem, some GPS units do include an air pressure sensor so that they can indicate pressure altitude.) Therefore, it is crucial to always verify what is meant by "altitude", and differentiate a pressure-based measurement of "pressure altitude" from a distance-based measurement of actual altitude. Density altitude is a concept based on solely on air density, and is neither "pressure altitude" nor "mean sea-level altitude", but is strictly "density altitude". Now... on to Density Altitude.....
Although the concept of density altitude is commonly used to help express the effects of aircraft performance, the really underlying property of interest 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 one way to express the air density. (Note: If you're just hunting for a simple, but not very accurate, approximation for density altitude, be sure to study the "Simpler Methods of Calculation" section near the end of this article.)
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. To begin to understand the calculation of air density, consider the ideal gas law:
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:
Then, by combining the previous two equations, the expression for the density becomes:
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:
To determine the density of the air, it is necessary to know is the actual air pressure (also known as absolute pressure, total air 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 (also called actual air pressure,
absolute pressure, or station 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 ) :
where: Es = saturation pressure of water vapor, mb
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:
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.
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 rather than 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 regardless of the normal daily temperature changes. Therefore, using the dew point as the measure of humidity allows for more stable and therefore potentially more accurate results.
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.
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.
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:
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.
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:
Moist Air is Less Dense... As you may have noticed, moist air is less dense than dry air. It may seem reasonable to try to argue against that simple fact based on the observation that water is denser than dry air... which is certainly true, but irrelevant. Solids, liquids and gasses each have their own unique laws, so it is not possible to equate the behavior of liquid water with the behavior of water vapor.
The
ideal gas law says that a certain volume of air at a certain
pressure has a certain number of molecules. That's just the way this
world works, and that simple fact is expressed as the ideal gas law,
which was shown above in equation 1.
Note
that this is the gas law... not a liquid law, nor a solid law, but a
gas law. Hence comparisons to a liquid are of little help in
understanding what is going on in the air, and may simply result in
more confusion.
According to the ideal gas law, a cubic meter of air around you,
wherever you are right now, has a certain number of molecules in it,
and each of those molecules has a certain weight.
Most
of the air is made up of nitrogen molecules N2 with a somewhat
lesser amount of oxygen O2 molecules, and then other molecules such
as water vapor.
Since
density is weight divided by volume, we need to consider the weight
of each of the molecules in the air. Nitrogen has an atomic weight
of 14, so an N2 molecule has a weight of 28. For oxygen, the atomic
weight is 16, so an O2 molecule has a weight of 32.
Now
along comes a water molecule, H2O. Hydrogen has an atomic weight of
1. So the molecule H20 has a weight of 18. Notice that a water
molecule is lighter weight than either a nitrogen molecule or an
oxygen molecule.
Therefore, when a given volume of air, which contains only a certain
number of molecules, has some water molecules in it (which are very
light weight), it will weight less than the same volume of air
without any water molecules.
Example 1) The lift of an aircraft
wing may be described mathematically (see
ref 8) as:
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 (see ref 6 ) advises that their carburetor main jet diameter should be adjusted according to the air density . 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: j2 = j1 * (d2/d1) (1/4)
where: j2 = diameter of new jet 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 in example 1 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.
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. (See also details on the SAE method of correcting horsepower.)
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. To get the proper
results, it is necessary to use the following constants that are specified
in the 1976 International Standard Atmosphere document:
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:
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:
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:
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 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 Tv, the virtual temperature, is defined such that
By evaluating the numerical values of the constants, setting Pv = E, noting that Rd = R*1000/Md and that Rv=R*1000/Mv, then the virtual temperature may be expressed as:
where Md is molecular weight of dry air = 28.9644 (Note that for convenience, the units in Equation 14 are not purely SI units, but rather are US 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:
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 Pa is
related to the altimeter setting AS by the following equation:
By numerically evaluating the constants and converting to customary units of altitude and pressure, the equation may be written as:
(Note: several other equations for converting actual pressure to altimeter setting are given in ref 12.) 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.
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 :
My On-Line Density Altitude 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!
If you really want to know the actual density altitude, it will need to be calculated in the general manner that has been described above. However, there are many forms of simpler approximations and generalizations that have been developed over the years, but please note that they are not really the density altitude, they are just numbers that approximate the density altitude. In some situations, the density altitude approximations can be fairly accurate, but in some real life situations with high moisture content in the air, the approximations can sometimes be quite inaccurate. The simpler form of the approximations is obtained by simply ignoring the actual moisture content in the air. Nonetheless, for those who really want a simpler equation, here is an equation used by the National Weather Service (see ref 13) to calculate the approximate density altitude without any need to know the humidity, dew point or vapor pressure:
This simplified equation is, basically, just equation (12) above rewritten in US customary units with no pressure contribution due to water vapor pressure. The following calculator can be used to compare the results of
the accurate calculations (in geometric altitude, as described earlier on this web page) with
the results from the preceding simplified equation:
The results for dry air (very low dew point) are nearly identical, while the greatest errors in the simplified equation are when there is a lot of water vapor in the air, i.e. high temperature accompanied by a high dew point. For example, on a hot, rainy summer afternoon here in Colorado, 95 deg F with a dew point of 95 deg, at an altitude of 5050 feet and an altimeter setting of of 29.45 , the actual pressure is 24.445 in-Hg and the actual Density Altitude is 9753 feet, while the simplified equation gives a result of 8933 feet.... an error of 820 feet. So, if you don't mind errors approaching 10% when the air is saturated with a lot of water vapor (that is, on a hot day with the dew point approaching the ambient temperature), then the simplified equation, which is much easier to calculate, may suit your needs. But if really want the utmost accuracy, then you'll have to deal with the gory details of vapor pressure.
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 6. http://wahiduddin.net/calc/refs/8UL87.pdf ... Rotax Service Bulletin 7. http://www.crh.noaa.gov/unr/?n=mslp ... clearing confusion over sea level pressure analysis 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://www.nwstc.noaa.gov/DATAACQ/d.ALGOR/d.PRES/PRESalgoProcess.html ... see step 8... (or see The Automated Surface Observing System (ASOS) Algorithm Tutorial ). 11. http://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htm ... NASA vapor pressure 12. There are some additional altimeter setting algorithms at http://www.wmo.ch/pages/prog/www/IMOP/publications/IOM-19-Synoptic-AWS.pdf and http://www.srh.noaa.gov/elp/wxclc/formulas/altimeterSetting.html Also see http://www.softwx.com/weather/uwxutils.html for weather equations, including additional methods for converting station pressure to altimeter setting. 13. Precision Digital Barometer Spec .. PDF file... National Weather Service document that includes equations for altimeter setting, and a simple approximation for density altitude.
Some related web links: http://www.luizmonteiro.com ... a large collection of aviation related calculators 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 El Paso NWS - calculators ... atmospheric calculators using Tim Brice's cgi scripts http://www.grc.nasa.gov/WWW/K-12/airplane/foil2.html ... NASA airfoil simulator... this is fantastic
http://www.usatoday.com/weather/wdenalt.htm ... lots of pages of weather
related info and formulas
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