 # Course SD Heat transfer by conduction in a 2D metallic plate

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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
The equation that governs the process is: Heat sources Radiation Convection with air Assumptions: - no heat sources in plate - no convection - no radiation - constant conduction thermal conductivity k Poisson’s Equation

Boundary Conditions One side is thermally insulated, whereas the rest kept at a certain constant temperature

Meshing 3 meshes with both COMSOL and MATLAB a) 19 x 13 b) 49 x 31
c) 124 x 76

COMSOL: Resolution & Results

COMSOL: Resolution & Results

COMSOL: Resolution & Results
COARSE 19 X 13 MEDIUM 49 X 31 FINE 124 X 76 Number of elements 247 1519 9424 Computing Time [s] 0.094 0.157 0.939 Benchmark Temperature [oC] Point (0.2,0.6)m

MATLAB: Discretization
DG: using same stepsize h in both directions DD: 2nd order Finite Difference Method

MATLAB: Discretization
DD (cont.): discretized DE DB: 1st and 2nd order Finite Difference Method 1st order 2nd order

MATLAB: Linear Sytems of Eq.
Analytical 2D problem results to be 1D problem after discretization.

MATLAB: Linear Sytems of Eq.
Elliptic DE has been reduced to a linear system of MxN EQUATIONS to be solved. There are MxN UNKNOWNS, the discretized temperatures in all points of the grid. STIFFNESS & STABILITY ? A COARSE 19 X 13 MEDIUM 49 X 31 FINE 124 X 76 λMAX λMIN System is of very SPARSE nature -> treat it this way to save computational effort.

MATLAB: Resolution & Results

MATLAB: Resolution & Results

COMSOL & MATLAB: comparison
COMSOL insensitive to mesh fineness. MATLAB depends strongly upon mesh fineness -> ACCURACY

COMSOL & MATLAB: comparison
COARSE 19 X 13 MEDIUM 49 X 31 FINE 124 X 76 Number of elements 247 1519 9424 COMSOL Time [s] 0.094 0.157 0.939 MATLAB 0.041 0.082 1.826 COMSOL is more efficient with big systems.

Conclusions 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 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).

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