A plane wall (p = 4000 kg/m³, c p = 500 J/kg ∙ K, k = 10 W/m ∙ K) of thickness L = 20 mm initially has a linear, steady-state temperature distribution with boundaries maintained at T₁ = 0°C and T₂ = 100°C. Suddenly, an electric current is passed through the wall, causing uniform energy generation at a rate q = 2 X 10⁷ W/m³. The boundary conditions T₁ and T₂ remain fixed.

(a) On T-x coordinates, sketch temperature distributions for the following cases: (i) initial condition

(t < 0); (ii) steady-state conditions (t →∝), assuming that the maximum temperature in the wall exceeds T 2 ; and (iii) for two intermediate times. Label all important features of the distributions.

(b) For the system of three nodal points shown schematically (1, m, 2), define an appropriate control volume for node m and, identifying all relevant processes derive the corresponding finite-difference equation using either the explicit or implicit method.

(c) With a time increment of ∆t = 5s, use the finite-difference method to obtain values of Till for the first 45 s of elapsed time. Determine the corresponding heat fluxes at the boundaries that is, q"_s: (0, 45 s) and q"_x (20 mm, 45 s).

(d) To determine the effect of mesh size, repeat your analysis using grids of 5 and 11 nodal points (it = 5.0 and 2.0 mm, respectively).

Transcribed Image Text:

-T2 = 100°C T = 0°C -12 0, = 2 x 10 Wim L = 20 mm Lax