# Hot Air Balloon Lifting Force

## Calculate potential lifting force in hot air volume

### 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},

*/ft*

*slugs*^{3})

* ρ_{c}* = 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*is heated to

^{3})*100*. The temperature of the surrounding air is

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

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

*air density**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

* Fg = m a _{g} (2)*

*where *

*Fg = 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

*balloon volume (m ^{3})*

* balloon volume density (kg/m ^{3})*

* surrounding air density (kg/m ^{3})*

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