# Hot Air Balloons - Calculate the 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*. The temperature of the surrounding air is

^{o}C (212^{o}F)*20*The

^{o}C (68^{o}F).*air density*at temperature

*100*is

^{o}C*0.946 kg/m*and the air density at temperature

^{3}(0.00184 slugs/ft^{3})*20*is

^{o}C*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*- the chart indicates a specific lifting force

^{o}C* 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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