A spherical hot-air balloon is initially filled with air at 120 kPa and 20°C with an initial diameter of 5 m. Air enters this balloon at 120 kPa and 20°C with a velocity of 3 m/s through a 1-m-diameter opening. How many minutes will it take to inflate this balloon to a 17-m diameter when the pressure and temperature of the air in the balloon remain the same as the air entering the balloon?

t = 1064 s

Explanation:

This exercise will work with the equations of fluid mechanics, let's start by calculating the filling flow rate

Q = A v

Area of the circle is

r = d / 2 = 0.50 m

A = π R²

A = π 0.50²

A = 0.785 m²

Q = 0.785 3

Q = 2,356 m · / s

Let's calculate the initial and final volume

Ro = 5/2 m = 2.5 m

Rf = 17/2 m = 8.5 m

V = 4/3 π r³

Vo = 4/3 π 2.5³

Vo = 65.45 m³

Vf = 4/3 π 8.5³

Vf = 2572.44 m³

The difference is the volume to be filled

ΔV = Vf - Vo

ΔV = 2572.44 - 65.45

ΔV = 2506.99 m3

As we have the flow, let's calculate the time

Q = V / t

t = V / Q

t = 2506.99 / 2,356

t = 1064 s