How to Calculate the Weight of Air and Model Hot Air Balloon Lift - overflite
Hot air balloons fly because balloon and heated air inside weigh less than surrounding ambient air displaced by balloon. Difference in air weights is gross lift. Difference minus weight of balloon is net lift.
Absolute Fahrenheit is used to calculate air weights at different temperatures. It is the number of degrees above Absolute Zero. Which is around minus 460 degrees Fahrenheit.
As prime example, 48 Fahrenheit is the same as 508 Absolute Fahrenheit, and 175 degrees Fahrenheit is the same as 635 Absolute Fahrenheit.
The standard weight of normal average sea level air is .0765 pounds per cubic foot at 59 F (519 Absolute Fahrenheit). Or 1.224 ounces. When air temperature goes up, air weight goes down. By inverse proportion. Do the math. Multiply 519 by 1.224 to get 635. This means that the heated air will weigh exactly one ounce per cubic foot at 635 Absolute Fahrenheit. Or 175 F.
At 48 F (508 A) a cubic foot of normal pressure sea level air weighs almost exactly 1 1/4 ounces. Heated by 127 degrees, to 175 F (635 A), it weighs almost exactly an ounce, or 4/5 of its original weight, for a gross lift of a 1/4 ounce per cubic foot.
When air temperature goes up, air weight goes down, by an inverse proportion. From the ambient of 48 F (508 A) to 175 F (635 A), the increase is 635/508, or 5/4, since both numbers are divisible by 127. So the heated air will weigh 4/5 of its original weight. Or a reduction of 1/5. From 1 1/4 ounces to an ounce. For a lift of a 1/4 ounce per cubic foot.
Calculating the Weight of Air, using the "635 Factor"
Mathematically, with two variables, if one goes down at the same rate the other one goes up, the variables can be multiplied times each other, and their product will be constant. So, at standard normal pressure sea level, ie. 29.92 inches of mercury (1013.25 Millibars), the following equations are true:
Absolute Air Temperature * Cubic Foot Air Weight = "635" (or other specific number, adjusted for air pressure)
Cubic Foot Air Weight = "635" / Absolute Air Temperature
Absolute Air Temperature = "635" / Cubic Foot Air Weight
Formulas for Model Hot Air Balloon Lift:
Gross Lift = ( "635" / Ambient Temperature ) - ( "635" / Heated Air Temperature )
Gross Lift = ( "635" / Ambient Temperature ) - ( "635" / (Ambient Temperature + Heat Rise ) )
Gross lift can also be expressed as a fraction, as follows:
Gross Lift = "635" * ( Heated Air Temp. - Ambient Temp. ) / ( Heated Air Temp. * Ambient Temp. )
Gross Lift = ( "635" * Heat Rise) / ( Ambient Temperature * ( Ambient Temperature + Heat Rise) )
Alternately, the heated air temperature and the heat rise can be expressed in terms of gross lift, as follows:
Heated Air Temperature = ( "635" * Ambient ) / ( "635" - ( Gross Lift * Ambient ) )
Heat Rise = ( Gross Lift * Ambient ^ 2)
/ ( "635" - ( Gross Lift * Ambient ) )
Effects of Different Ambient Temperatures on Gross Lift
As example, if the ambient temperature goes up 20 degrees, from 48 F (508 A) to 68 F (528 A), the increase in temperature and the decrease in air weight is about 4%, and sea level air weighs around 1.20 ounces per cubic foot. Heated by 127 degrees, to 207 F (655 A), the temperature increases by 655/528, or about 24%, and the air weight decreases proportionately to about .97 ounces per cubic foot. So, the gross lift is about .23 ounces per cubic foot, or about 8% less than at an ambient of 48 F.
Alternately, if the ambient goes down 20 degrees to 28 F (488 A), the air weighs around 4% more, or about 1.30 ounces per cubic foot. Heated 127 degrees to 155 F (615 A) the temperature increases by 615/488, or about 26% and the air weight decreases proportionately to about 1.03 ounces per cubic foot. The result is a gross lift of about .27 ounces per cubic foot, or about 8% more than at an ambient of 48 F, and about 16% more than at an ambient of 68 F ambient.
In reality though, the changes in lift are believed to be slightly less than predicted. In cold ambients there is "more air to heat," and at higher ambients there is "less air to heat." This causes slightly lower heat rises in cold weather, and slightly higher heat rises in warm weather, of up to several degrees, for a given amount of engine power.
Adjusting the "635 Factor" to Account for Weather and Elevation
Elevation reduces air pressure, air weight, and the temperature where air weighs exactly an ounce per cubic foot by around 3% per thousand feet, slightly more for lower elevations, and slightly less for higher elevations. This reduces the "635 factor" by around 20 degrees per thousand feet. Adjustments for altitudes of up to 25,000 feet, that are over 99% accurate, can be calculated using the following formula:
Air Pressure Adjustment = 1 - ( ( Elevation -- in thousands ) / ( 27 + ( Elevation -- in thousands / 2 ) ) )
Note: For altitudes of up to 35,000 feet,
adjustments that are close to 99% accurate can be
calculated by increasing the addition factor to
28; for up to 40,000 feet by increasing the factor
to 29, and for up to 45,000 feet by increasing the
factor to 30, etc. See
Standard Atmospheric Pressure Tables - (2).
Accordingly, for a given heat rise, the resulting lift will be approximately the same as the percentage of sea level air. In reality though, higher elevation air is believed to get somewhat more lift than predicted, since higher elevation air has "less air to heat," resulting in a greater heat rise from a given amount of engine power than at sea level.
Guidelines for Balloon Weights at Sea Level:
For unpowered paper balloons the maximum weight shouldn't be much more than around 1/8th ounces per cubic foot, since they will tend to cool down fairly quickly, and you want to have reasonable flight time.
For manned hot air balloons, you can figure that the gross lift is probably around 1/5th of an ounce per cubic foot, assuming the air is heated by around 100 degrees in mild weather, and slightly more in warm weather. In this example, 80 cubic feet of volume is required for each pound of gross lift. So, to lift a thousand pounds, 80,000 cubic feet of volume would be required.