Course SD 2225 Heat transfer by conduction in a 2D metallic plate Pau Mallol, Georgios Spanopoulos, Alan Vargas KTH, April 2008
Course SD 2225 Heat transfer by conduction in a 2D metallic plate Pau Mallol, Georgios Spanopoulos, Alan Vargas KTH, April 2008
Physical Background
Heat transfer: thermal energy in transit due to a spatial temperature difference within/between media.
Modes of heat transfer:
Differential Equation
Heat sources Convection
with air Radiation Assumptions:
- no heat sources in plate
- no convection
- no radiation
- constant conduction
thermal conductivity k Poisson’s Equation The equation that governs the process is:
Boundary Conditions
One side is thermally insulated, whereas the rest kept at a certain constant temperature
Meshing
a) 19 x 13 b) 49 x 31 c) 124 x 76 3 meshes with both COMSOL and MATLAB
COMSOL: Resolution & Results
COMSOL: Resolution & Results
COMSOL: Resolution & Results
67.346667 67.346702 67.347493 Benchmark
Temperature [oC]
Point (0.2,0.6)m 0.939 0.157 0.094 Computing
Time [s] 9424 1519 247 Number of elements FINE
124 X 76 MEDIUM
49 X 31 COARSE
19 X 13
MATLAB: Discretization
DD: 2nd order Finite Difference Method DG: using same stepsize h in both directions
MATLAB: Discretization
DB: 1st and 2nd order Finite Difference Method 1st order 2nd order DD (cont.): discretized DE
MATLAB: Linear Sytems of Eq.
Analytical 2D problem results to be 1D problem after discretization.
MATLAB: Linear Sytems of Eq.
There are MxN UNKNOWNS, the discretized temperatures in all points of the grid. STIFFNESS & STABILITY ? System is of very SPARSE nature -> treat it this way to save computational effort. -7.9964 -7.9861 -7.9215 λMIN -0.0016 -0.0064 -0.0381 λMAX FINE
124 X 76 MEDIUM
49 X 31 COARSE
19 X 13 A Elliptic DE has been reduced to a linear system of MxN EQUATIONS to be solved.
MATLAB: Resolution & Results
MATLAB: Resolution & Results
COMSOL & MATLAB: comparison
COMSOL insensitive to mesh fineness. MATLAB depends strongly upon mesh fineness -> ACCURACY
COMSOL & MATLAB: comparison
COMSOL is more efficient with big systems. 0.939 0.157 0.094 COMSOL
Time [s]
1.826 0.082 0.041 MATLAB
Time [s] 9424 1519 247 Number of elements FINE
124 X 76 MEDIUM
49 X 31 COARSE
19 X 13
Conclusions
Max/Min temperatures not consistent in COMSOL (depend on mesh); MATLAB is OK. COMSOL: easier, faster, more accurate and efficient than MATLAB. But COMSOL is particular use and MATLAB offers infinite possibilities (general). STABILITY: numerical systems to these PDE’s are always stable, no matter what h. ACCURACY: in COMSOL does not depend on h, in MATLAB strongly depends on h -> limitation: backward slash operator A\b size of A limited to about 10000.
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