Hot Air Balloons  Calculate the Lifting Force
Calculate hot air ballon lifting force.
Hot Air Lifting Force
The lifting force from a hot air balloon depends on the density difference between balloon air and surrounding air, and the balloon volume. The lifting force can be calculated as
F _{ l } = V (ρ _{ c }  ρ _{ h } ) a _{ g } (1)
where
F _{ l } = lifting force (N, lb_{f} )
V = balloon volume (m^{3}, ft^{3} )
ρ _{ c } = density cold surrounding air (kg/m^{3}, slugs /ft^{3} )
ρ _{ h } = density hot balloon air (kg/m^{3}, slugs /ft^{3} )
a _{ g } = acceleration of gravity (9.81 m/s^{2}, 32.174 ft/s^{2})
Example  Lifting Force created by a Hot Air Balloon
A hot air balloon with volume 10 m^{3} (353 ft^{3} ) is heated to 100 ^{o}C (212 ^{o}F) . The temperature of the surrounding air is 20 ^{o}C (68 ^{o}F). The air density at temperature 100 ^{o}C is 0.946 kg/m^{3} (0.00184 slugs/ft^{3} ) and the air density at temperature 20 ^{o}C is 1.205 kg/m^{3} (0.00234 slugs/ft^{3} ) .
The lifting force can be calculated as
F _{ l } = (10 m^{3} ) [(1.205 kg/m^{3} )  (0.946 kg/m^{3} )] (9.81 m/s^{2})
= 25.4 N
Weight  or gravity force  can be calculated as
F _{ g } = m a _{ g } (2)
where
F _{ g } = weight  gravity force (N, lb_{f} )
m = mass (kg, slugs )
Since lifting force of a flying air balloon equals weight (F _{ l } = F _{ g } )  the lifted mass can be expressed by combining (1) and (2) to
m = F _{ l } / a _{ g }
= (25.4 N) / (9.81 m/s^{2})
= 2.6 kg
The calculation of lifting force can be done in Imperial units as
F _{ l } = (353 ft^{3} ) [(0.00234 slugs/ft^{3} )  (0.00184 slugs/ft^{3} )] (32.174 ft/s^{2})
= 5.7 lb_{f}
Hot Air Balloon  Specific Lifting Force
Specific lifting force (force per unit air volume) created by an hot air balloon  balloon temperature vs. surrounded air temperature  are indicated in the charts below.
SI  units
Imperial Units
Example  Specific Lifting Force from a Hot Air Balloon
If the balloon temperature is 60 ^{o}C and the surrounded temperature is 20 ^{o}C  the chart indicates a specific lifting force
3.3 N/m^{3}
Hot Air Balloon Lifting Force Calculator
This calculator can be used to calculate the lifting force of a volume with lower density than surrounding air.
 density helium : 0.1664 kg/m^{3}
 density hydrogen : 0.0899 kg/m^{3}
 more gases
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