Part $2$: Nodal Analysis

A bar with the known thermal conductivity of $k=140\frac{W}{m}K$ has a trapezoidal cross section.

The left and right faces are at temperatures $T_h=100C$ and $T_0=0C,$ respectively.

Problems

A$)$ Solve for the temperature at each node and label on a schematic of the

bar.

B$)$ Using the shown temperature color gradient, color the bar to the best of

your ability to represent the temperatures you labeled on each node.

C$)$ Determine the heat transfer rate per unit bar length using a finite difference

approach with $Dx=Dy=10mm.$

D$)$ Compare the heat rate to that of a bar of a $20mm30mm$ rectangular

cross section where the height of the domain is $20$ mm $.$

Comments:

$(1)$ The trapezoidal and rectangular bars are of the same mass. Hence, the trapezoidal bar has

a greater heat transfer rate per unit mass.

$(2)$ The heat rate per unit length can be calculated in various ways. For example, it could be

determined by calculating the heat rate in the horizontal direction per unit length between nodes

$11$ and $12,7$ and $8,$ plus that between nodes $2$ and $3.$