Solidification / MeltingMoving Boundary Problems:A finite difference method
Final ProjectMANE 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
CL
CL
2
2
Tw < Freezing
Tw < Freezing
Symmetry B.C. Tinitial > Freezing0
y
H
x
H
y
u
x
u
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.
y
u
x
uk
yxt
H tyxu ,,Find: Such that:
rp uuCH
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
0001.1
561.1
0001.1
9999.0
0.1__
init
solfusion
melt
freeze
liqpsolpsolliqsol
T
H
T
T
CCkk
22
22
max 4
1
yx
yx
k
Ct p
Determining solid/liquid interface
Tem
p
Y-coord @ x=0
Solid Liquid Interface Temp < Freezing Temp
Results T(x,y,t) #grid pts. = 11x11, time = 0.0001
Results table: 11x11 mesh
x distance
time 0 0.1 0.2 0.3 0.4 0.5 0.60.05 0.7536 0.7536 0.7536 0.7536 0.7536 0.7535 0.75340.10 0.6590 0.6590 0.6590 0.6589 0.6586 0.6577 0.65010.15 0.5963 0.5963 0.5963 0.5949 0.5911 0.5827 0.50040.20 0.5001 0.5001 0.5001 0.5001 0.5001 0.4999 0.39990.25 0.4649 0.4649 0.4649 0.4600 0.4001 0.3999 0.30 0.4000 0.4000 0.4000 0.3999 0.3669 0.2358 0.35 0.3525 0.3525 0.3525 0.3001 0.2801 0.40 0.2999 0.2999 0.2999 0.2611 0.1203 0.45 0.2000 0.2000 0.2000 0.1776 0.50 0.1558 0.1558 0.1558
x distance
time 0 0.1 0.2 0.3 0.4 0.5 0.60.05 0.8125 0.8106 0.8048 0.7940 0.7764 0.7476 0.69040.10 0.6979 0.6965 0.6921 0.6836 0.6683 0.6392 0.56060.15 0.6157 0.6141 0.6095 0.6000 0.5810 0.5201 0.20 0.5473 0.5453 0.5394 0.5268 0.4789 0.25 0.4865 0.4838 0.4755 0.4567 0.3894 0.30 0.4302 0.4263 0.4146 0.3654 0.35 0.3766 0.3708 0.3534 0.2859 0.40 0.3337 0.3158 0.2623 0.45 0.2816 0.2585 0.1893 0.50 0.2376 0.2056 0.1097
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
time 0 0.1 0.2 0.3 0.4 0.5 0.60.05 -7% -7% -6% -5% -3% 1% 9%0.10 -6% -5% -5% -4% -1% 3% 16%0.15 -3% -3% -2% -1% 2% 12% 0.20 -9% -8% -7% -5% 4% 0.25 -4% -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
Results – Non real solidPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 11x11, time = 0.0001
Results Table showing solid-liquid interface #grid pts. = 41x41, dt = 0.00005
x-coord.
time (sec) 0 0.1 0.2 0.3 0.4 0.5 0.6
0.0500 0.7664 0.7664 0.7664 0.7664 0.7664 0.7662 0.7653
0.1000 0.6719 0.6719 0.6718 0.6713 0.6699 0.6662 0.6500
0.1500 0.5991 0.5989 0.5980 0.5955 0.5905 0.5750 0.4819
0.2000 0.5251 0.5251 0.5251 0.5250 0.5132 0.4625 0.0000
0.2500 0.4749 0.4749 0.4749 0.4500 0.4250 0.1744 0.0000
0.3000 0.4244 0.4227 0.4141 0.3929 0.3176 0.0000 0.0000
0.3500 0.3694 0.3675 0.3500 0.3140 0.0000 0.0000 0.0000
0.4000 0.3157 0.3126 0.2902 0.2121 0.0000 0.0000 0.0000
0.4500 0.2500 0.2500 0.2179 0.0000 0.0000 0.0000 0.0000
0.5000 0.2000 0.1921 0.1219 0.0000 0.0000 0.0000 0.0000
0.5500 0.1250 0.1177 0.0000 0.0000 0.0000 0.0000 0.0000
0.6000 0.0417 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
ResultsPlot of Temperature: #grid pts. = 41x41, dt = 0.00005
ResultsPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 41x41, dt = 0.00005
Results – Non real solidPlot of Enthalpy: Red Solid – Blue Liquid#grid pts. = 61x61, dt = 0.000025
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