An Introduction to Air Density
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Saturation 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 value of the saturation vapor pressure, 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.
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, Pv = Es.
(7a) Pv = Es at the dew point
where Pv= pressure of water vapor (partial pressure)
Es = saturation vapor pressure ( multiply mb by 100 to get Pascals)
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.
(7b) Pv = RH * Es
where
Pv=
pressure of water vapor (partial pressure)
RH = relative humidity (expressed as a decimal value)
Es = saturation vapor pressure ( multiply mb by 100 to get Pascals)
Dry Air Pressure:
Now that the water vapor pressure is known, we are nearly ready to calculate the density of the combination of dry air and water vapor as described in equation 4a, but first, we need to know the pressure of the dry air.
The total measured atmospheric pressure (also called actual pressure, absolute pressure, or station pressure) is the sum of the pressure of the dry air and the vapor pressure:
(8a) P = Pd + Pv
where: P = total pressure
Pd = pressure due to dry air
Pv = pressure due to water vapor
So, rearranging that equation:
(8b) Pd = P - Pv
where: P = total pressure
Pd = pressure due to dry air
Pv = pressure due to water vapor
Now that we have the pressure due to water vapor and also the pressure due to the dry air, we have all of the information that is required to calculate the air density using equation 4a.
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:
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Air Density Calculator |
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.
Some examples of calculations using air density:
Example 1) The lift of an aircraft
wing may be described mathematically (see
ref 8) as:
L = c1 * d * v2/2 * a
where: L = lift
c1 = 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 (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
j1 = diameter of jet that was proper at density d1
d1 = density at which the original jet j1 was correct
d2 = 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. (Note: according to Poiseuille's Law, the volumetric flow rate through a circular cross section is proportional to the fourth power of the diameter.)
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.
hp2 = hp1 * (d2/d1)
where: hp2 = the new horsepower at density d2
hp1 = the old horsepower at density d1
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.)
Importance of Air Density:
So far, we've been discussing real physical attributes which can be precisely measured, with air density being the weight per unit volume of an air mass. The air density, as shown in the previous examples, affects the lift of a wing, the fuel required by an engine, and the power produced by an engine. When precision is required, air density is a much better measure than density altitude.
Air density is a physical quality which can be accurately measured and verified. On the other hand, density altitude is a rather conceptual quantity which depends upon a hypothetical "standard atmosphere" which may or may not accurately correspond to the actual physical conditions at any given location. Nonetheless, density altitude has a long heritage and remains a common (although rather hypothetical) representation of air density.
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 1976 International Standard Atmosphere (ISA) is a mathematical description of a theoretical atmospheric column of air which uses the following constants (see ref 16):
Po = 101325 sea level standard pressure, Pa
To = 288.15 sea level standard temperature, deg K
g = 9.80665 gravitational constant, m/sec2
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), and the model which will be developed here
is only valid in the troposphere.
The following equations describe temperature, pressure and density of the air in the ISA troposphere:
(9)
(see ISA pg 10, Eqn 23)
(10)
(see ISA pg 12, Eqn 33a)
(11)
(see ISA pg 15, Eqn 42)
where: T = ISA temperature in deg K
P = ISA pressure in Pa
D = ISA density in kg/m3
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 which expresses altitude H as a function of density
D. After a bit of gnashing of teeth and general turmoil using algebraic
substitutions of those three equations, 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/m3
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.
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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:
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Density Altitude
Calculator 2 |
Density Altitude calculations using Virtual Temperature:
As an alternative to the use of equations which describe the atmosphere as being made up of a combination of dry air and water vapor, it is possible to define a virtual temperature for an atmosphere of 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 4a:
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:
(14)
where Tv = virtual temperature, deg K
T = ambient temperature, deg K
c1 = ( 1 - (Mv / Md ) ) = 0.37800
E = vapor pressure, mb
P = actual (station) pressure, mb
where Md is molecular weight of dry air = 28.9644
Mv is molecular weight of water = 18.016
(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:
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Virtual Temperature
Calculator |
The virtual temperature Tv 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
Tv = 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. (For more information about ambient air pressure measurements see the pressure measurement page.)
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:
(16)
By numerically evaluating the constants and converting to customary units of altitude and pressure, the equation may be written as:
Pa = [ASk1 - ( k2 * H ) ]1/k1
where Pa = 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:
Pa = [AS0.1903 - (1.313 x 10-5) x H]5.255
where Pa = actual (station) pressure, inches Hg
AS = altimeter setting, inches Hg
H = station elevation, feet
(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.
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Altimeter Values to Actual Pressure |
Using National Weather Service Barometric Pressure:
Now you're probably wondering about converting sea-level corrected barometric pressure, as reported in a weather forecast, 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.
Simpler Methods of Calculation...
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 simple approximations which have been developed over the years.
For example, a particularly convenient form of density altitude approximation is obtained by simply ignoring the actual moisture content in the air. Here is such an equation which has been 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:
17)
where: DA = density altitude, feet
Pa = actual pressure (station pressure), inches Hg
Tr = temperature, deg R (deg F + 459.67)
This simplified equation (17) is, basically, just equation (12) 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:
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Comparison of |
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.
To explore the effects of water vapor, consider, for example, a hypothetical ambient temperature of 95 deg F, with a dew point of 95 deg, at an altitude of 5050 feet and an altimeter setting of 29.45 , the actual air pressure would be 24.445 in-Hg and the actual Density Altitude would be 9753 feet, while the simplified equation gives a result of 8933 feet.... an error of 820 feet. The actual air density in this case would be reduced by about 3%, compared to dry air.
Or, for a hypothetical 95 deg F foggy day at sea level, with a dew point of 95 deg F and an altimeter setting of 29.92, the actual density altitude is 2988 ft, while the simplified equation gives a result of 2294 ft... an error of 694 ft. Similar to the previous example, the actual air density in this would be reduced by about 3%, compared to dry air.
Those examples are quite extreme, but in actual practice it is quite common to see errors on the order of 200 to 400 ft along the sea coast and in the sweltering mid-west, which may be inconsequential, or may be significant, depending upon your specific situation.
So, if you don't mind some error when the air has a lot of water vapor, then the simplified equation, which is much easier to calculate, may suit your needs.
But if you really want the utmost accuracy in determining the density altitude, then you'll have to deal with the gory details of vapor pressure and compute the "real" density altitude.
Based on the reported observations from a variety of US airports, it appears that the ASOS and AWOS-3 automated weather observation systems (which report weather conditions including density altitude at many airports in the US) use a simplified equation, and gives essentially the same results as equation 17 above. That is, it appears that the current ASOS/AWOS density altitude does not account for effects of moisture in the air.
You can compare the actual Density Altitude with the ASOS/AWOS-3 reported values using the calculator at: Density Altitude Calculator - with selectable units.
However, before you get too distressed by such seemingly "sloppy" ASOS/AWOS calculations, keep in mind that the International Standard Atmosphere is merely a conceptual model which may or may not accurately represent the conditions at any given location on any given day. That is, "density altitude" and "standard atmosphere" are theoretical concepts which are based upon a number of assumptions about the atmosphere, and may or may not accurately depict the actual physical conditions at any actual location, no matter how accurate the calculations may be.
Actually, it would be far more meaningful, useful and precise if ASOS/AWOS reported the actual air density in kg/m3, and if pilot's handbooks were also expressed as actual air density in kg/m3.
Hopefully, someday all of the aircraft performance tables/charts and weather reporting systems will simply use the actual air density and thereby avoid this arcane concept of density altitude... but, for now, we're stuck with "density altitude".
If we really want to be precise and consistent, we must focus our attention on the actual air density, not density altitude.
Density Altitude Calculation Algorithm...
For those who want to do their own density altitude calculations, here's a list of the steps performed by my on-line 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. calculate the saturation vapor pressure, given the ambient temperature
6. calculate the actual vapor pressure given the dew point temperature
7. use geopotential altitude and altimeter setting to calculate the absolute pressure in mb,
8. use absolute pressure, vapor pressure and temp to calculate air density in kg/m3,
9. use the density to find the ISA altitude in meters which has that same density,
10. convert the ISA geopotential altitude to geometric altitude in meters,
11. convert the geometric altitude into the desired units and display the results.
My On-Line Density Altitude and Engine Tuner's Calculators:
Click here for Density Altitude Calculator using dew point.
Click here for Density Altitude Calculator using relative humidity
Click here for Engine Tuner's Calculator which includes air density, density altitude, relative horsepower, virtual temperature, absolute pressure, vapor pressure, relative humidity and dyno correction factor.
enjoy....
Richard Shelquist
Longmont, Colorado
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 ... weather related formulas
4. http://mtp.mjmahoney.net/www/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/images/epz/wxcalc/altimeterSetting.pdf 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.
14. For more details about the effects of non-ideal compressibility and vapor pressure not measured over liquid water, see Techniques and Topics in Flow Measurement, Frank E. Jones, p37 and also Comité International des Poids et Mesures CIPM-2007 (or CIPM-81/89). PDF file of Revised Formula (CIPM-2007) To convert the CIPM-2007 density to the forms given in my equation 4a and 4b, note that Xv = RH * f * Psv/P, with RH = Pv/Psv. Let f = 1, which then gives Xv = Pv/P. Then let Z=1, and simply rearrange the equation to yield the forms given in my 4a and 4b.
15. Evaporation into the Atmosphere, Wilfried Brutsaert, p37. (PDF excerpt)
16.
1976 International Standard Atmosphere (PDF file)
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/foil3.html ... NASA airfoil simulator... fun tool
http://www.usatoday.com/weather/wdenalt.htm ... lots of pages of weather
related info and formulas
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/ ... low-cost software for
personal weather analysis
Copyright 1998-2012, All Rights Reserved, Richard Shelquist, Shelquist Engineering.