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Multi-layer Sphere Temperature Analysis

Multi-layer Sphere Temperature Analysis. Adam Hickman Brennan Crellin. Introduction. Rio Tinto seeks a method for determining the slag/matte level of a molten furnace bath.

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Multi-layer Sphere Temperature Analysis

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  1. Multi-layer SphereTemperature 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 250oC. • 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: Ėstorage = Ėin + Ėgenerated • 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 (nmax), and the nmax-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 (nmax): • Node (nmax-1):

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