Chapter 4, Problem 4.52P

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 600K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of $50{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ 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.

1. Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures $T_2,T_4,$ and $T_7\left(\text{K}\right).$
2. Calculate the heat loss per unit length from the channel.