96_Heat Flux Example

In this post, I aim to solve an example that involves determining the temperature at each point along an insulated rod with a total lengh $L$.

 

 

The left end is fixed at ${100^ \circ }{\text{C}}$, and a heat flux of $q = 5000{\text{W/}}{{\text{m}}^2}$ acts on the right end. Here, ${K_{xx}}$ is ${\text{6W/(m}}{{\text{ - }}^{\text{o}}}{\text{C)}}$, and the cross-section is a circle with an area of ${\text{A = 0}}{\text{.1}}{{\text{m}}^2}$. Calculate the temperature at $L{\text{/}}4,\,\,L{\text{/}}2,\,\,3L{\text{/}}4$ and $L$.

 

 

Given the assumptions of no heat transfer at the surface due to insulation around the perimeter, a constant temperature at the left end, and a constant heat flux at the right end, the following equation can be derived.

 

 

and

 

$Q=0$(no heat source), $q^{*}=0$(no heat flux).

 

 

By assembling the global stiffness matrix through the element stiffness matrix and constructing the global force matrix, it can be represented as follows.

 

 

Since $T_1$ is ${100^ \circ }{\text{C}}$, the values for $T_2$, $T_3$ and $T_4$ can also be determined by solving the equations.

 

 

By substituting the determined temperatures, the nodal heat source for the first equation can be calculated as follows.

 

 

According to the above equation, the nodal heat source is negative, indicating that heat is exiting from the left end. Additionally, this value is equal to the heat source entering from the right end.