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20 December, 09:54

g The electric power needs of a community are to be met by windmills with 40-m-diameter rotors. The windmills are to be located where the wind is blowing steadily at an average velocity of 6 m/s. Determine the minimum number of windmills that need to be installed if the required power output is 2100 kW. The gas constant of air is 0.287 kPa·m3/kg·K.

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  1. 20 December, 12:31
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    Given Data

    The diameter of the wind mills is d = 40m

    Velocity of the air is V = 6 m / s

    Required power output is: P ₀ = 2100 k W

    Expression to calculate the exergy of the air is

    E = V ² / 2

    Substitute the value in above expression

    E = (6 m / s) ² / 2

    E = 18 m ² / s ² x (1kJ/kg / 1000m²/s²)

    E = 0.018 k J / k g

    Expression to calculate the density of the air is

    P v = m R T

    m / v = P / RT ⋯ ⋯ (I)

    Here

    m is the mass of the air,

    v is the volume of the air,

    P is the atmospheric pressure,

    T is the standard temperature at the atmospheric pressure and

    R is the gas constant

    As the density is

    ρ = m / V

    Substitute the value in expression (I)

    ρ = 101 kP a / (0.287 k J / k g ⋅ K) (298 K)

    ρ = 1.180 k g / m ²

    Expression to calculate the mass flow rate is

    m = ρ A V ⋯ ⋯ (I I)

    Here A is the area of the windmill

    Expression to calculate the A is

    A = π / 4 d ²

    Substitute the value in above expression

    A = π / 4 (40 m ²)

    A = 1256.63 m ²

    Substitute the value in expression (II)

    m = (1.180 k g / m ³) (1256.63 m ²) (6 m / s)

    m = 8896.94 k g / s

    Expression to calculate the maximum power available to the windmill is

    P w = m (V ² / 2)

    Substitute the value in above expression

    P w = 8896.94 k g / s ((6m/s) ²/2)

    P w = 160144.92 W * (1 W / 1000 k W)

    P w = 160.144 k W

    Expression to calculate the number of windmills required is

    n = P o / P w

    Substitute the value in above expression

    n=2100kw/160.144kw

    n=13.11
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