Aspherical 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

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