1. Multi-layer Sphere Temperature Analysis Adam Hickman Brennan Crellin

2. Introduction Rio Tinto seeks a method for determining the slag/matte level of a molten furnace bath. The decided solution is to house electronic sensors in a container capable of withstanding the conditions of the molten fluid for between 20-120 min. The decided container shape is a sphere. The sphere will have four layers; each layer providing different desired attributes. Two metallic layers will increase average density, a ceramic (or vacuum) layer will insulate, and a wax layer will absorb energy to impede heat transfer to the electronics.

3. Problem Establish a method for temperature analysis of the four-layer sphere. This will allow optimization of a sphere that keeps the electronics below 250 o C. The method should be robust enough to allow for material properties and layer thickness to be changed.

4. Method (Finite Difference) Initial efforts involved adapting the heat equation to the problem in the form presented in Equation 2.27 of the text. Because δΦ and δθ are constant, the equation becomes: Then the method of 5.10.1, Discretization of the Heat Equation, was used. This method was abandoned because it only solves for interior nodes and assumes a solid sphere.

5. Method (Finite Difference) Final analysis used the general form of the heat equation, identified as Equation 5.81 in the text: Ė storage = Ė in + Ė generated Using this equation we analyzed the energy balance for multiple cases.

6. Solution The equation was discretized to calculate node temperatures at time steps. The equation was derived for four node types, contributing to the the total solution: material change nodes, interior nodes, the center node (n max ), and the n max-1 node. Below is the material change node equation: *see the other equations in Appendix A

7. Solution A Screenshot of the working excel solution:

8. Conclusion The working excel solution allows for rapid numerical analysis of the sphere for various material properties and thicknesses. This analytical tool assisted in solving the overall problem to optimize the design of the sphere for the Rio Tinto application. We found that a sphere of 20cm diameter will last for between 50-60 min. given the chosen materials. Future work will include further optimization of materials and sphere size.

9. Appendix A Interior node: Center node (n max) : Node (n max-1 ):