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.  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.

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

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

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:

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) ’

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

Determining solid/liquid interface Temp x=0 Solid Liquid Interface Temp < Freezing Temp

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

Results table: 11x11 mesh x distance time x distance time 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 time % -6%-5%-3%1%9% %-5% -4%-1%3%16% % -2%-1%2%12% %-8%-7%-5%4% % -2%1%3% %-6%-4%9% %-5%0%5% %-5%14% %-23%6% %-24%42% Comparison NOTE: Values of x-coordinate shown in left table were found by liniearly interpolating location where T = Method used on right table is unknown

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

Results Table showing solid-liquid interface #grid pts. = 41x41, dt = x-coord. time (sec)

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

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

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

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