One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m ∙ K. For these conditions, the temperature distribution has the form, T(x) = a + bx + cx². The surface at x = 0 has a temperature of T (0) ≡ T_o = 120°C and experiences convection with a fluid for which T_∞; = 20°C and h = 500 W/m² ∙ K. The surface at x = L is well insulated.

(a) Applying an overall energy balance to the wall, calculate the internal energy generation rate, q.

(b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution.

(c) Consider conditions for which the convection coefficient is halved, but the internal energy generation rate remains unchanged. Determine the new values of a, b, and c, and use the results to plot the temperature distribution. Hint: recognize that T (0) is no longer 120°C.

(d) Under conditions for which the internal energy generation rate is doubled, and the convection coefficient remains unchanged (h = 500 W/m² ∙ K), determine the new values of a, b. and c and plot the corresponding temperature distribution. Referring to the results of parts (b), (c), and (d) as Cases I, 2, and 3, respectively, compare the temperature distributions for the three cases and discuss the effects of h and q on the distributions.

Transcribed Image Text:

T = 120°C T()- T= 20°C h= 500 W/m?-K 4. k = 5 Wim-K 111 Fluid L= 50 mm