Heat Transfer in the Solid

In our next model we considered heat transfer in the solid. First, we posed a cross-sectional geometry similar to that of our reactor.

We then introduced the following boundary conditions:

• The boundaries at A are insulated:

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• The boundary at B is isothermal at T = 1.0.
• The boundary at C involves convective cooling

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We had two methods we were able to use to find solutions for this model. The first method was to use the finite difference method. It can be shown that at an interior node in this model the heat equation yields the following difference equation:

We constructed a mesh of nodes across our geometry, included appropriate equations on the boundaries and solved the problem iteratively in Excel.

The second method we had available was Matlab PDE Toolbox, which uses the finite element method. Once we constructed the model and boundary conditions within Matlab, we achieved a similar solution to the first method. It should be noted that for this problem the PDE Toolbox was more robust than Excel. As the thermal conductivity decreased the spreadsheet began to have difficulties converging on a solution, while Matlab showed no such problems. Click below for our results using Excel and Matlab PDE Toolbox.

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