Hot Air Balloons - Calculate Lifting Weights

Calculate hot air ballons lifting forces.

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 (m³, ft³)

ρ_c = density cold surrounding air (kg/m³, slugs/ft³)

ρ_h = density hot balloon air (kg/m³, slugs/ft³)

a_g = acceleration of gravity (9.81 m/s², 32.174 ft/s²)

Example - Lifting Force created by a Hot Air Balloon

A hot air balloon with volume 10 m³ (353 ft³) is heated to 100 ^oC (212 ^oF). The temperature of the surrounding air is 20 ^oC (68 ^oF). The air density at temperature 100 ^oC is 0.946 kg/m³ (0.00184 slugs/ft³) and the air density at temperature 20 ^oC is 1.205 kg/m³ (0.00234 slugs/ft³).

The lifting force can be calculated as

F_l = (10 m³) [(1.205 kg/m³) - (0.946 kg/m³)] (9.81 m/s²)

= 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/s²)

= 2.6 kg

The calculation of lifting force can be done in Imperial units as

F_l = (353 ft³) [(0.00234 slugs/ft³) - (0.00184 slugs/ft³)] (32.174 ft/s²)

= 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.