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Solidification / Melting Moving Boundary Problems: A finite difference method

Solidification / Melting Moving Boundary Problems: A finite difference method. Final Project MANE 6640 – Fall 2009 Wilson Braz. Background. Solidification has obvious application to engineering problems such as: Casting, metallurgy, soil mechanics, freezing of food, etc.

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Solidification / Melting Moving Boundary Problems: A finite difference method

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  1. Solidification / MeltingMoving Boundary Problems:A finite difference method Final Project MANE 6640 – Fall 2009 Wilson Braz

  2. Background • Solidification has obvious application to engineering problems such as: • Casting, metallurgy, soil mechanics, freezing of food, etc. • Solidification may be modeled using a moving boundary. • Several techniques for solving the moving boundary problem. • Isotherm Migration Method (IMM), method of lines, finite element, finite difference, enthalpy method, and others.

  3. C C L L The problem: • Use 2-D Finite Difference method to analyze the solidification of square plate Symmetry B.C. Tinitial > Freezing Tw < Freezing 2 2 Tw < Freezing

  4. The Problem - Continued.Comparison of Results • Results from 2 different sources are in disagreement • Method was coded in MATLAB • Results were compared with those given in sources

  5. Approach - General • Enthalpy method • Use an explicit, finite difference routine to numerically solve • Develop numeric approximation equations, discretize domain, set initial conditions, set boundary conditions, march through domain, step through time. Find: Such that:

  6. Approach Technique • Material properties vary depending on state (liquid or solid) • Conditional statements test for material state using temperature. • Apply appropriate values for material properties depending on state. • Calculate ‘H(x,y)’ using finite differencing • Find ‘u(x,y)’ given using new ‘H(x,y)’

  7. Non-Real material properties, initial and boundary conditions: • To simplify calculation, and to compare directly with published results, the following material properties were used: • Mesh size varied • Time increment set to satisfy CFL condition

  8. Determining solid/liquid interface Solid Liquid Interface Temp < Freezing Temp Temp Y-coord @ x=0

  9. Results T(x,y,t)#grid pts. = 11x11, Dtime = 0.0001

  10. Results table: 11x11 mesh Values of the y-coordinate on the solid-liquid interface for fixed values of x at various times Values using method coded in MATLAB Values taken from John Crank NOTE: Values of x-coordinate shown in left table were found by liniearly interpolating location where T = 1.0000. Method used on right table is unknown Comparison

  11. Results – Non real solidPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 11x11, Dtime = 0.0001

  12. Results Table showing solid-liquid interface #grid pts. = 41x41, dt = 0.00005

  13. ResultsPlot of Temperature: #grid pts. = 41x41, dt = 0.00005

  14. ResultsPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 41x41, dt = 0.00005

  15. Results – Non real solidPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 61x61, dt = 0.000025

  16. Difficulties & Limitations with this approach • Trouble matching results presented by John Crank, and Ernesto Gutierrez-Miravete • Suspect an issue with initial calculation of H(x,y,0), or u(x,y,0) • 1st time step shows temperature jump up to ~2

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