In a long, 30-cm square bar shown in the accompanying sketch, the left face is maintained at 40°C and the top face is maintained at 250°C. The right face is in contact with a fluid at 40°C through a heat transfer coefficient of 60 W/(m2 K) and the bottom face is in contact with a fluid at 250°C through a heat transfer coefficient of 100 W/(m2 K). If the thermal conductivity of the bar is 20 W/(mK), calculate the temperature at the 9 nodes shown in the sketch.

Question

GIVEN

• Square bar with two surfaces at fixed temperature and two surfaces with convective boundary conditions

FIND (a) Temperature at 9 shown nodes

SKETCH

Answer 1

Define the following symbols



the temperature at nodes 1, 2, 4, and 5 is just the average of the temperature at

the neighbor nodes



The remaining control volumes have convective boundary conditions and we need to develop

individual energy balance equations for each.

For the control volume surrounding node 3



which can be solved for T3 as follows



For the control volume at node 6



For the control volume at node 7



For the control volume at node 8



This set of equations can be solved iteratively. The table below shows the results of the first 25

iterations after which the calculation appears to converge. Values for the 9 nodal temperatures at the

zero iteration are the first guess.