1. Solidification / Melting Moving 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. The problem:  Use 2-D Finite Difference method to analyze the solidification of square plate C L C L 2 2 T w < Freezing Symmetry B.C. T initial > 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 Temp Y-coord @ x=0 Solid Liquid Interface Temp < Freezing Temp

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

10. Results table: 11x11 mesh x distance time00.10.20.30.40.50.6 0.050.7536 0.75350.7534 0.100.6590 0.65890.65860.65770.6501 0.150.5963 0.59490.59110.58270.5004 0.200.5001 0.49990.3999 0.250.4649 0.46000.40010.3999 0.300.4000 0.39990.36690.2358 0.350.3525 0.30010.2801 0.400.2999 0.26110.1203 0.450.2000 0.1776 0.500.1558 x distance time00.10.20.30.40.50.6 0.050.81250.81060.80480.79400.77640.74760.6904 0.100.69790.69650.69210.68360.66830.63920.5606 0.150.61570.61410.60950.60000.58100.5201 0.200.54730.54530.53940.52680.4789 0.250.48650.48380.47550.45670.3894 0.300.43020.42630.41460.3654 0.350.37660.37080.35340.2859 0.400.33370.31580.2623 0.450.28160.25850.1893 0.500.23760.20560.1097 Values of the y-coordinate on the solid-liquid interface for fixed values of x at various times Values using method coded in MATLABValues taken from John Crank time00.10.20.30.40.50.6 0.05-7% -6%-5%-3%1%9% 0.10-6%-5% -4%-1%3%16% 0.15-3% -2%-1%2%12% 0.20-9%-8%-7%-5%4% 0.25-4% -2%1%3% 0.30-7%-6%-4%9% 0.35-6%-5%0%5% 0.40-10%-5%14% 0.45-29%-23%6% 0.50-34%-24%42% Comparison 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

11. Results – Non real solid Plot of Enthalpy: Red Solid – Blue Liquid #grid pts. = 11x11, time = 0.0001

12. Results Table showing solid-liquid interface #grid pts. = 41x41, dt = 0.00005 x-coord. time (sec)00.10.20.30.40.50.6 0.05000.7664 0.76620.7653 0.10000.6719 0.67180.67130.66990.66620.6500 0.15000.59910.59890.59800.59550.59050.57500.4819 0.20000.5251 0.52500.51320.46250.0000 0.25000.4749 0.45000.42500.17440.0000 0.30000.42440.42270.41410.39290.31760.0000 0.35000.36940.36750.35000.31400.0000 0.40000.31570.31260.29020.21210.0000 0.45000.2500 0.21790.0000 0.50000.20000.19210.12190.0000 0.55000.12500.11770.0000 0.60000.04170.0000

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

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

15. Results – Non real solid Plot 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) 1 st time step shows temperature jump up to ~2