Consider the square channel shown in the sketch operating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of 50 W/m2 ∙ K. From a symmetrical element of the channel, a two-dimensional grid has been constructed and the nodes labeled. The temperatures for nodes 1, 3, 6, 8, and 9 are identified.

(a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, and T7 (K).

(b) Calculate the heat loss per unit length from the channel.

T= 300 K h = 50 Wim?. K Ax = Ay = 0.01 m T = 600 K -k = 1 W/m-K 7, = 430 K T3 = 394 K Tg = Tg = 600 K TE = 492 K