Course SD Heat transfer by conduction in a 2D metallic plate
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1 Course SD 2225 Heat transfer by conduction in a 2D metallic plate Pau Mallol, Georgios Spanopoulos, Alan Vargas KTH, April 2008
2 Physical BackgroundHeat transfer: thermal energy in transit due to a spatial temperature difference within/between media.Modes of heat transfer:
3 Differential Equation The equation that governs the process is:Heat sourcesRadiationConvectionwith airAssumptions:- no heat sources in plate- no convection- no radiation- constant conductionthermal conductivity kPoisson’s Equation
4 Boundary ConditionsOne side is thermally insulated, whereas the rest kept at a certain constant temperature
5 Meshing 3 meshes with both COMSOL and MATLAB a) 19 x 13 b) 49 x 31 c) 124 x 76
8 COMSOL: Resolution & Results COARSE19 X 13MEDIUM49 X 31FINE124 X 76Number of elements24715199424ComputingTime [s]0.0940.1570.939BenchmarkTemperature [oC]Point (0.2,0.6)m
9 MATLAB: Discretization DG: using same stepsize h in both directionsDD: 2nd order Finite Difference Method
10 MATLAB: Discretization DD (cont.): discretized DEDB: 1st and 2nd order Finite Difference Method1st order2nd order
11 MATLAB: Linear Sytems of Eq. Analytical 2D problem results to be 1D problem after discretization.
12 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 ?ACOARSE19 X 13MEDIUM49 X 31FINE124 X 76λMAXλMINSystem is of very SPARSE nature -> treat it this way to save computational effort.
15 COMSOL & MATLAB: comparison COMSOL insensitive to mesh fineness.MATLAB depends strongly upon mesh fineness -> ACCURACY
16 COMSOL & MATLAB: comparison COARSE19 X 13MEDIUM49 X 31FINE124 X 76Number of elements24715199424COMSOLTime [s]0.0940.1570.939MATLAB0.0410.0821.826COMSOL is more efficient with big systems.
17 ConclusionsSTABILITY: 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 aboutMax/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).