Compute its maximum payload at sea level

Question

A modern-day zeppelin holds 9,860 m³ of helium. Compute its maximum payload at sea level. (Assume the helium and air to be at 0°C and 1 atm.)

Answer 1

Given the question of the maximum payload at sea level, we wish to solve for the weight that the zepplin can hold. This translates to the net bouyancy force that is allowable by the aircraft and the temprature and altitude.

We will make the assumption that the weight of the zepplin itself if negligible at sea level and only consider the force of the helium against air.

If we draw a free body diagram, the boyancy force is directly perpedicular to the ground in the up direction, the force of the helium is opposite of the boyancy force and the force of the payload is also opposite of the boyancy force.

We then have a static force equation as follows,

0 = F_b - F_He - F_Payload

⇒ F_Payload = F_b -F_He

Forces are defined as:

External Force = mass * gravity, and mass can be rewritten in terms of density and volume so,

External Force = density * Volume * gravity

Specifically for the problem,

F_b = ρ_air * V* g,

F_He = ρ_He * V* g

From data online, the denisty of Helium and Air at sea level (1 atm) and (0ºC) is

ρ_air = 1.29 kg/m³

ρ_{He =} 0.1786 kg/m³

g = 9.81 m/s

If we then solve for the force of the payload then divide by gravity to find the payload mass we get,

m_payload= 10,987.8 kg

This weight is reasonable as it is almost the same mass of a city bus or an African elephant!