Thermal Heat Flux in an Isotropic Disk

Let us consider now the problem of heating of a circular plate with the sources of heat distributed over the surface. Let us prescribe the heat flux from both sides. Around the plate along the edges we will prescribe the constant temperature.

As an example we consider a thin plate with the thermal conductivity K=75 W/(m • oK) [Ductile Iron (SN)]. The radius of the plate is R=100 mm, thickness D=5 mm. The value of the density of the heat flux is F=60 W/m2. Temperature of the edges around the plate is T=293.15 oK. (see figure).

Input data and analytical solution can be viewed here.

Let us locate the sensors of the temperature as shown on the Figure and make a plot for them at r =0; 20; 40; 60 mm. In the given points we will compare the numerical solution obtained using AutoFEM Analysis with the analytical solution.

After carrying out calculation the following results are obtained:

Table 1. Parameters of finite element mesh

Table 2. Result "Temperature"

 

Conclusions:

The relative error of the numerical solution compared with the analytical solution not exceed 0.002% for linear elements.

The problem was solved almost exactly with minimum computational cost. It is important to note that at a point r=20 mm the solution with linear elements turned out to be more accurate. This was really possible because of the convergence properties of the solution in the finite element method (convergence in a sense of integral norm and smaller number of points/arguments plays a role here). However, in general it is impossible to predict appearance and location of these points.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

Read more about AutoFEM Thermal Analysis

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