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The Calculation of Air Density in Various Units * VOL. 30, No. 9, NOVEMBER, 1949 319 The Calculation of Air Density in Various Units * NELSON R. WILLIAMS U. S. Naval Ordnance Test Station, Inyo kern, California IR density enters into many important Law, the density p can be computed from: problems in meteorology, related sciences, p = (p - 0.37Se)/RdT A and engineering. For example, at the Naval Ordnance Test Station, knowledge of air = C(p - 0.37Se)/T « C(p - Vse)/T, (1) density is necessary for determining the drag co- where p is total pressure of the air, e is partial efficient of a missile and correction curves for the pressure of water vapor, Rd is the gas constant for refraction of light in the atmosphere. The me- dry air = 2.870 X 106 ergs/gm °K, T is the abso- teorologist, in his increasing contact with other lute temperature (°K) of the air, and the constant scientists and industrial engineers, often is called C is a combination of 1 /Rd and the factor neces- upon to calculate air density in units other than sary for conversion to the particular unit desired. those used in meteorology. He is then confronted The value given for Rd agrees very well with the by the necessity of laboriously working out con- experiment. The accuracy of humidity measure- version factors to compute the desired density, ments does not justify carrying beyond the third since meteorologists commonly do not use density decimal place the factor by which e is multiplied. directly as a parameter, and therefore the formu- From the definition of the mixing ratio w, the las in the meteorology textbooks usually have to vapor pressure can be expressed by be solved explicitly for the density, and the con- stants evaluated according to the units of meas- pw urement. Therefore, the preparation of a table 6 = 0.622 +w' with conversion factors for computing density in various units was thought to be required. Substituting this value for e in (1) : For the purpose of density calculations dry air from sea level to at least 50 km (31 miles) may be p -cp(l- 0.378 considered a uniform mixture of gases having a constant percentage by mass of the main con- Since w rarely exceeds 0.03 and the humidity stituents, Nitrogen, Oxygen, Argon, and Carbon correction is always small, w in the denominator Dioxide f [1-4]. In fact, the National Advisory can be neglected in comparison with 0.622. The Committee for Aeronautics, in preparing tentative resulting equation, which is sufficiently exact for tables for the upper atmosphere, assumes that density calculations, is: variations in chemical composition are too small to have an appreciable effect upon the computa- p = Cp( 1 - 0.61 w)/T. tion of density and pressure below altitudes of If w is expressed in grams of water vapor per 80 km (50 miles) in the daytime and 105 km (65 kilogram of dry air, miles) at night. Above these levels the dissocia- tion of oxygen molecules by solar radiation must p = Cp( 1 - 0.00061W)/r be considered. The percentages of other gases, » cp( 1 - 6 X 10-*w)/T. (2) such as Neon, Helium, Krypton, and Hydrogen are so small that they may be disregarded in TABLE I gives the values of the C factor for density computations. The amount of Carbon computing densities in different units. The tem- Dioxide varies slightly, but this does not alter the perature T must always be in °K. By taking the density significantly. appropriate ratio of the C factors, the table can also be used to convert a given density from one FORMULAS "USED unit to another. The C factor for the slugs/ft3 Reference to standard textbooks [2, 5] shows unit is based upon the standard sea-level accelera- that by use of the Perfect Gas Law and Dalton's tion of gravity, 32.174 ft/sec2, at latitude 45°. * The opinions and assertions contained herein are the private ones of the writer and are not to be construed as EXAMPLES official, or reflecting the view of the Navy Department or the Naval Service at large. Example 1.—Radiosonde measurements at lati- tNo allowance is made for local contamination, such as from industrial fumes. tude 45° and altitude 5,000 feet above mean sea Unauthenticated | Downloaded 09/27/21 12:35 AM UTC 320 BULLETIN AMERICAN METEOROLOGICAL SOCIETY TABLE I. CALCULATION OF AIR DENSITY IN VARIOUS UNITS P = C(p- 0.37Se)/T « C(p - He)/Tf or p = Q>(1 - 0.00061w)/r « Cp( 1 - 6 X 10~%)/T, (w in gm/kg). Temperature in °K C factor to get density in: lbs/ft3 Pressure in: gm/cm3 gm/liter or kg/m3 gm/m3 slugs/ft3 ("weight density") Dynes/cm2 (baryes) 3.484 X 10~7 3.484 X IO"4 3.484 X 10"1 2.175 X 10~5 6.760 X IO"7 Millibars 3.484 X IO"4 3.484 X 10"1 3.484 X 102 2.175 X IO"2 6.760 X IO"4 (1 mb = 1,000 dynes/cm2) Mm Mercury (0°C) 4.645 X IO"4 4.645 X 10"1 4.645 X 102 2.900 X IO"2 9.013 X IO"4 Inches Mercury (32°F) 1.180 X IO"2 1.180 X 10 1.180 X 104 7.367 X 10"1 2.290 X 10~2 Lbs/inch2 2.402 X IO"2 2.402 X 10 2.402 X 104 1.500 4.661 X IO"2 Atmospheres 3.530 X 10"1 3.530 X 102 3.530 X 105 1.204 X 10 6.850 X 10"1 level show: pressure 850 mb, temperature 20°C is then: and mixing ratio 10 gm/kg. In order to calculate p = 4.645 X 10-4(760 - 0.378 X 11.5)/288.18 a drag coefficient, the air density, in slugs/ft3, 3 3 must be known. Solution: = 1.218 X 10~ gm/cm = 1.218 gm/liter or kg/m3. p = (6.760 X IO"4) X 850(1 - 0.00061 X 10)/293 The density of dry air at this temperature and = (1.9611 X IO"3) X 0.9939 pressure is 1.225 X 10-3 gm/cm3. Thus, even = 1.9491 X 10"3 slugs/ft3. with a relative humidity of 90%, the effect of The quantity 1.9611 X 10~3 is the density of dry water vapor on the density is very small at this air, and this must be multiplied by the factor temperature. 0.9939 to obtain the density of moist air. Even Example 3.—The density of dry air at 1 at- with the relatively high moisture content of 10 mosphere pressure and 0°C is 1.293 X 10~3 gm/ gm/kg, the effect of the water vapor on the density cm3. What is the density (strictly speaking, the changes the third digit only slightly. Therefore "weight density") in pounds per cubic foot. So- in many practical problems the moisture term can lution :— be neglected. Actually, the accuracies of most observations do not justify carrying the density 2 204 V 10 3.530 X 10-'•a^XlO-*) = 0.0807 lbs/ft*. calculations beyond the third figure. In this par- ticular example the results were carried out to five places for comparison purposes only. The ACKNOWLEDGMENT factor 6.760 X 10~4 was computed using the stand- ard sea-level acceleration of gravity at 45°, namely The assistance of Miss Louise C. Lener and Mr. 32.174 ft/sec2. If the value for gravity at 5,000 Frederick Schuett in checking the calculations is feet were used, the calculated density would be gratefully acknowledged. slightly greater, but the correction is insignificant compared to observational errors. However, cer- REFERENCES tain types of problems, such as the computation [1] Paneth, F. A., Quart. Jr. of Royal Met. Soc., v. 65, of a standard atmosphere to high altitudes, require 1939, p. 304. corrections for the variation of gravity with lati- [2] Berry, F. A., Bollay, E., Beers, N. R., Handbook of tude and altitude. Meteorology. McGraw-Hill, New York, 1945. [3] Warfield, C. N., Tentative Tables for the Proper- Example 2.—Compute the density of air at the ties of the Upper Atmosphere. NACA Technical standard sea-level pressure of 760 mm mercury Note No. 1200. National Advisory Committee for and standard temperature of 15°C, but with a Aeronautics, Washington, 1947. [4] Grimminger, G., Analysis of Temperature, Pressure, relative humidity of 90%. Solution:—From TA- and Density of the Atmosphere Extending to Ex- BLE 85 of the Smithsonian Meteorological Tables treme Altitudes. Rand Corp., Santa Monica, Calif., [6], the vapor pressure is 11.5 mm. The vapor [5] Holmboe, J., Forsythe, G. E., and Gustin, W., Dy- pressure can also be obtained from psychrometric namic Meteorology. J. Wiley, N. Y., 1945. [6] Smithsonian Meteorological Tables. Smithsonian slide rules, graphs, or other tables. The density Institution, Washington, 5th Revised Edition, 1939. 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