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 if the temperature distribution on the top surface of the bar varies sinusoidally from 40°C at the left edge to a maximum of 250°C in the center and back to 40°C at the right edge.

Question

GIVEN

Square bar with one surface at fixed temperature, one surface with a specified temperature

distribution, and two surfaces with convective boundary conditions

FIND

(a) Temperature at 9 shown nodes

SKETCH

Answer 1

Define the following symbols



Since the temperature varies sinusoidally across the top surface we have



Now, define the temperature at the four nodes on the top edge as



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



Finally, for the control volume at node 9



This set of equations can be solved iteratively. Here we used a spreadsheet to employ the Gauss-Seidel

iteration method. 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.