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12 February, 01:38

Consider a large plane wall of thickness L = 0.4 m, thermal conductivity k = 2.3 W/m·K, and surface area A = 30 m2. The left side of the wall is maintained at a constant temperature of T1 = 90°C, while the right side loses heat by convection to the surrounding air at T[infinity] = 25°C with a heat transfer coefficient of h = 24 W/m2·K. Assuming constant thermal conductivity and no heat generation in the wall, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the rate of heat transfer through the wall. Answer: (c) 9045 W

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  1. 12 February, 01:59
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    a. - k { (dT (L) / dx} = h{T (L) - T (s) }

    b. T (x) = 90 - 131.09 (x)

    c. 9045W

    Explanation:

    Given detail

    Thickness of wall L = 0.4 m

    Left side temperature of the wall T1 = 90°C,

    Thermal conductivity k = 2.3 W/m. k

    Right side temperature of the wall T (infinity) = T (s) = 25°C

    Heat transfer coefficient h = 24W/m2. K

    (a) Differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall,

    Qwall = - kA { (dT (0) / dx}

    {d²T / dx²} = 0

    T (0) = 90°C

    -k { (dT (L) / dx} = h{T (L) - T (s) }

    (b) relation for the variation of temperature in the wall by solving the differential equation.

    T (x) = xC1 + C2

    T (0) = 0*C1 + C2 = 90°C

    T (0) = C2 = 90°C

    -kC1 = hLC1 + hC2 - T (s) h

    -C1 (k+hL) = h (C2 - T (s))

    - C1 = {h (C2 - T (s)) / (k + hL) }

    C1 = - { 24 (90 - 25) / (2.3 + 24*0.4) }

    C1 = - 131.09°C

    T (x) = x * (-131.09) + 90

    T (x) = 90 - 131.09 (x)

    (c) rate of heat transfer through the wall

    Qwall = - 2.3 * 30 * (-131.09)

    Qwall = 9045W ans
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