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20 February, 09:48

very long pipe of 0.05 m (r1) radius and 0.03 m thickness (r2 - r1) is buried at a depth of 2m (z) to transport liquid nitrogen. The top surface of the ground is flat and has a uniform temperature. The pipe wall has a very low thermal conductivity of 0.0035 Wm-1K-1 and receives a cooling power of 10 W/m to keep the liquid nitrogen at 77 K. If the thermal conductivity of the ground is 1 Wm-1K-1, what is the surface temperature of the ground

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  1. 20 February, 13:37
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    Therefore, the surface temperature of the ground is 352.2 K

    Explanation:

    Recall that

    r₁ = 0.05 m

    r₂ = 0.08 m

    T₁ = 77K

    Ki = 0.0035 Wm-1K-1

    Kg = Wm-1K-1

    z = 2m

    φ = 10 W/m

    Now,

    The heat transfer will occur in two resistance

    Ln (r₂/r₁) / 2π. kI = z/K * A

    so,

    φ = - (Ti - Tg) / ln (r₂/r₁) / 2πr. kI + z/kg * 2r₂ * I

    which is

    10 = (77-Tg / ln (0.08/0.05) / 2π * 0.0035 + 2/1*2*0.16

    10 = Δ T/2.137 + 6.25

    So,

    276.2 = (Tg - 77)

    Therefore Tg = 353.2K
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