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18 February, 00:56

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?

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  1. 18 February, 02:55
    0
    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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