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Course SD 2225 Heat transfer by conduction in a 2D metallic plate

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:.

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Course SD 2225 Heat transfer by conduction in a 2D metallic plate

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  1. Course SD 2225Heat transfer by conductionin a 2D metallic plate Pau Mallol, Georgios Spanopoulos, Alan Vargas KTH, April 2008

  2. Physical Background • Heat 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 sources Radiation Convection with air • Assumptions: • - no heat sources in plate • - no convection • - no radiation • - constant conduction • thermal conductivity k Poisson’s Equation

  4. Boundary Conditions • One 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

  6. COMSOL: Resolution& Results

  7. COMSOL: Resolution & Results

  8. COMSOL: Resolution & Results

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

  10. MATLAB: Discretization • DD (cont.): discretized DE • DB: 1st and 2nd order Finite Difference Method 1st order 2nd 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 ? • System is of very SPARSE nature -> treat it this way to save computational effort.

  13. MATLAB: Resolution & Results

  14. MATLAB: Resolution & Results

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

  16. COMSOL & MATLAB: comparison • COMSOL is more efficient with big systems.

  17. 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 10000. • 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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