**Objectives**

This lab exercise aims to explore the solution a 2-D Finite Element problem assembled via the Galerkin Weak Statement. The specific problem involves a dielectric heat source within a larger heat sink. The non-uniform meshing is generated via the MATLAB Delauney triangulation routine.

**Problem Statement **

The given problem considers a two-material medium. The larger medium acts as a high-conductivity heat sink. It is an ellipse

with major axis, a=1, and minor axis, b=0.75. The smaller medium acts as a dielectric heat source with low-conductivity. It is a circle with radius, r=0.2, placed within the ellipse with center coordinates (x,y)=(0.422,-0.222). A provided MATLAB file generates a Delauny triangulation mesh. The circular heat source region has a finer mesh since higher temperature gradients are expected due to the material's higher thermal conductivity. Figure 1 below shows the generated MATLAB meshing for the problem.

Figure 1: Delauny triangulation mesh for a circular heat source within an elliptical heat sink.

Considering the above problem constraints, the problem becomes one of 2-D heat conduction with two non-overlaping domains. The appropriate mathematical formulation is as follows:

the boundary condition is thus of Dirichlet type with a fixed temperature around the edges of the elliptical domain.

A MATLAB code is provided by the instructor to generate a solution for this problem. In this lab, the conductivities of the two domains will be varied to explore the temperature profile solutions. Table 1 below outlines the six cases that are explored. The basic method is to vary the material thermal conductivity of the circular region with respect to the elliptical region.

Table 1: Problem property variation matrix for the FE problem analysis.

===================================================================== Thermal Conductivities | Sources | Boundary Temp k_{a}k_{b}| s_{a}s_{b}| T_{b}_________________________|___________________|_______________________ 1.0 0.01 0 200 100 1.0 0.10 0 200 100 1.0 0.75 0 200 100 1.0 1.00 0 200 100 1.0 10.00 0 200 100 1.0 0.10 0 1000 100 =====================================================================

It should be noted that the last case explores only how a variation in the source term changes the problem solution.

**Discussion of Results **

Figures 1a and 1b show the temperature distribution across the 2-D domain. In this case, the elliptical heat-sink region has a thermal conductivity of 1.0 while the circular dielectric heat source has a thermal conductivity of 0.1. As expected, the heat source experiences very high temperature gradients due to the relatively poor conductive properties of the material. The heat sink, however, conducts the generated heat evenly across the domain with very little resulting temperature gradient. As seen, the Dirichlet boundary conditions ensure that the temperature at the edge of the heat sink is 100.

Figure 1a: Planform view of temperature solution (k_{source}=0.01 and k_{sink}=1.0).

Figure 1b: Perspective view of temperature solution (k_{source}=0.01 and k_{sink}=1.0).

In Figures 2a and 2b, the thermal conductivity of the circular dielectric heat source has been increased to 0.10. Now there is a smaller relative difference between the thermal conductivities of the two materials. The circular dielectric heat source region now conducts more heat to the elliptical region. This is seen by the lower temperatures of the heat source (from ~300 to ~124) and higher temperatures of the heat sink (from ~100 to ~103) in comparison to the previous case.

Figure 2a: Planform view of temperature solution (k_{source}=0.1 and k_{sink}=1.0).

Figure 2b: Perspective view of temperature solution (k_{source}=0.1 and k_{sink}=1.0).

In Figures 3a and 3b, the thermal conductivity of the circular dielectric heat source has been increased to 0.75. The heat source conducts even more heat to the sink than previously experienced. The peak temperatures of the source region are thus even lower (from ~124 to ~107) while the temperatures of the sink region are higher (from ~103 to ~105).

Figure 3a: Planform view of temperature solution (k_{source}=0.75 and k_{sink}=1.0).

Figure 3b: Perspective view of temperature solution (k_{source}=0.75 and k_{sink}=1.0).

In Figures 4a and 4b, both the circular heat source and elliptical heat sink regions have a thermal conductivity of 1. As expected this case shows a smooth temperature profile from the source to the boundaries of the sink. Since the heat is evenly conducted throughout the entire problem domain, the peak temperature is the least of all the previous cases (~106.4).

Figure 4a: Planform view of temperature solution (k_{source}=1.0 and k_{sink}=1.0).

Figure 4b: Perspective view of temperature solution (k_{source}=1.0 and k_{sink}=1.0).

In Figures 5a and 5b, the situation is reversed. Now the circular heat source region has a high thermal conductivity of 1.0 and the elliptical region has a low thermal conductivity of 0.1. Now the heat source region experiences a fairly uniform temperature profile due to the poor thermal conductivity while the elliptical region experiences the highest temperature profiles yet seen. In comparison to the second case with opposite conductivities, the maximum temperature is much less (~145 compared to ~300). Even though the elliptical domain has a poor conductivity, it still conducts more heat away from the source due to its larger size.

Figure 5a: Planform view of temperature solution (k_{source}=1.0 and k_{sink}=0.1).

Figure 5b: Perspective view of temperature solution (k_{source}=1.0 and k_{sink}=0.1).

Finally, Figures 6a and 6b reconsider the second case (heat sink k=1.0, heat source k=0.1) with an increase in the heat source (s=1000 instead of s=200). As seen when compared with Figures 2a and 2b, the higher heat source produces a relatively same temperature profile. Now, however, the temperatures have almost doubled.

Figure 6a: Planform view of temperature solution with a source term increase from 200 to 1000 (k_{source}=0.01 and k_{sink}=1.0).

Figure 6b: Perspective view of temperature solution with a source term increase from 200 to 1000 (k_{source}=0.01 and k_{sink}=1.0).

**Conclusion **

In conclusion, a conduction heat transfer study has been completed for a dielectric heat source/sink system of materials. A variation of the material thermal conductivities has allowed an exloration of the FE temperature distribution solution. Through this, fun has been had by all involved.

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