Consider the flat plate of Problem 3.101, but with the heat sinks at different temperatures, T(0) = To and T(L) = T L , and with the bottom surface no longer insulated. Convection heat transfer is now allowed to occur between this surface and a fluid at Toc, with a convection coefficient h.

(a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate.

(b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks.

(c) For q0 = 20,000 W/m2, To = 100°C, TL = 35°C, T∞ = 25°C, k = 25 W/m ∙ K, h = 50 W/m2 ∙ K, L = 100 mm, t = 5 mm, and a plate width of W = 30 mm, plot the temperature distribution and determine the sink heat rates, qx(0) and qx(L). On the same graph, plot three additional temperature distributions corresponding to changes in the following parameters, with the remaining parameters unchanged: (i) q = 30,000 W/m2, (ii) h = 200 W/m2 ∙ K, and (iii) the value of q for which q,(0) = 0 when h = 200 W/m2 ∙ K.