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12 July, 04:52

Steel balls 10 mm in diameter are annealed by heating to 1150 K and then slowly cooling to 450 K in an air environment for which T[infinity] = 325 K and h = 25 W/m2 ·K. Assuming the properties of the steel to be k = 40 W/m·K, rho = 7800 kg/m3, and c = 600 J/kg·K, estimate the time required for the cooling process.

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  1. 12 July, 04:58
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    the time required for the cooling process = 0.1635 hours

    Explanation:

    We are given;

    Thermal conductivity; k = 40 W/m·K

    Density; ρ = 7800 kg/m³

    Specific heat capacity; c = 600 J/kg·K

    h = 25 W/m². k

    Diameter; D = 10mm = 0.01m

    T_ (∞) = 325K

    T_i = 1150K

    T = 450K

    The formula for time required for the cooling process is given by;

    t = (ρVc/hA) [In ((T_i - T_ (∞)) / (T - T_ (∞))) ]

    Where;

    V is volume = πD³/6

    A is Area = πD²

    Other terms remain are previously stated.

    Thus;

    t = (ρ (πD³/6) c / (h (πD²)) [In ((T_i - T_ (∞)) / (T - T_ (∞))) ]

    This is simplified into;

    t = (ρDc / (6h) [In ((T_i - T_ (∞)) / (T - T_ (∞))) ]

    Plugging in the relevant values to obtain;

    t = (7800*0.01*600 / (6*25) [In ((1150 - 325) / (450 - 325)) ]

    t = 312In (825/125)

    t = 588.766 seconds = 0.1635 hours
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