Determine (a) the temperature at the 16 equally spaced points shown in the accompanying sketch to an accuracy of three significant figures and (b) the rate of heat flow per meter thickness. Assume two-dimensional heat flow and k = 1 W/(mK).

Question

GIVEN

A two-dimensional object with specified surface temperatures

FIND

(a) The temperature at the 16 specified locations

(b) The heat flow per meter thickness

ASSUMPTIONS

Steady state

SKETCH

Answer 1

Because of symmetry, it is only necessary to consider 1/8 th of the figure as shown below



(a) Temperature Distribution

There are three nodes remaining for which we must determine the temperature. For these nodes, we

need energy balance equations for the control volumes. The control volumes are shown as dashed lines

surrounding each node.

For the node at i = 2, j = 1

The x axis is a line of symmetry so no heat flows into the control volume across it



Since we have chosen ?x = ?y, this equation simplifies to



For the node at i = 2, j = 2, we use



And for the node at i = 2, j = 3, we have for energy balance



Substituting the known boundary temperatures, these equations simplify to



These three equations can be solved by elimination. Substitute Equations (1) and (3) into Equation (2)

to give



Substitute this result into Equation (3) to get



and then from Equation (1) we find



(b) Heat flow

The total heat flow for the object can be calculated from



which simplifies to