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3 October, 09:16

A 1 m long aluminum wire of diameter of 1 mm is submerged in an oil bath of temperature 28 °C. The wire has an electrical resistance of 0.02 Ω/m, which generates heat when current is applied to the wire. The convection coefficient between the wire and oil bath is 470 W/m2∙K. Assume no radiation effects. List and justify any assumptions you make.

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  1. 3 October, 10:35
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    T = 163.45 C

    t = 25.1 s

    Explanation:

    Given:

    The complete question is given below:

    A 1 mm long wire of diameter D=1 mm is submerged in an oil bath of temperature T∞=28∘C. The wire has an electrical resistance per unit length of R′e=0.02Ω/m. If a current of I=100 A flows through the wire and the convection coefficient is h=470W/m2⋅K,

    The properties of the wire are ρ=8000kg/m3, c=500J/kg⋅K, and k=20W/m⋅K.

    Find:

    what is the steady-state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature that is within 1∘C of the steady-state value?

    Solution:

    - First we will check whether we can use Lumped capacitance method can be applied by calculating Biot Number Bi, as follows:

    Bi = h*D / 4*k

    Bi = 470*2.5*10^-4 / 4*20

    Bi = 0.00146875 ... < 0.1

    - From the condition validated above using Biot number we can apply Lumped capacitance method:

    - Using Energy Balance we have:

    Q_convec = Q_gen

    A_wire*h * (T - T∞) = I^2*R′e

    - Re - arrange for T:

    T = T∞ + I^2*R′e / A_wire

    - Where, A_wire is the cross sectional area of the wire as follows:

    A_wire = pi*D

    Hence,

    T = T∞ + I^2*R′e / h*pi*D

    Plug in values:

    T = 28 + 100^2*0.02 / pi * (10^-3) * 470

    T = 163.45 C

    - Next using the derived results from Lumped capacitance for a rod we have:

    dT/dt = 4*I^2*R′e / p*c_p*pi*D^2 - (4*h/p*c_p*D) * (T - T∞)

    - The solution of the above differential equation is derived as:

    Ln ((T - T∞ - (I^2*R′e / h*pi*D)) / (T_i - T∞ - (I^2*R′e / h*pi*D))) = - 4*h*t/p*c_p*D

    - Plug in values and solve for t.

    Ln ((163.45-28-135.4510154) / (28-28-135.4510154))) = - 4*470/8000*500*0.001

    Ln (7.49643679*10^6) = - 0.47*t

    t = 11.8 / 0.47

    t = 25.1 s
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