Tutorial supported by the REASON IST project of the EU Heat, like gravity, penetrates every substance of the universe; its rays occupy all parts of space. The theory of heat will hereafter form one of the most important branches of general physics. Joseph Fourier, 1824

Tutorial supported by the REASON IST project of the EU Heat equation

Tutorial supported by the REASON IST project of the EU Heat transfer in electronic circuits Heat equation ­initial conditions ­boundary conditions Initial conditions and boundary conditions should be cared about at measurements, too Many simulators care about conduction and convection only Heat exchange ­conduction ­convection ­radiation

Tutorial supported by the REASON IST project of the EU Heat exchange ­conduction q = -  T ­convection q = h (T – T  ) ­radiation q =   T 4

Tutorial supported by the REASON IST project of the EU Simulation techniques Analytical methods (e.g. based on Green’s functions) –Very quick solution –Limited geometries, simplified models Infinite half space + point source Multiple sources: superposition (linearity implicitly assumed) Semi-analytical methods –Still quick –More realistic models but still simplified Brick like structures with homogeneous material layers Fourier method (implemented by FFT) Fully numerical methods –Robust algorithms (FEM, FD) with sophisticated solvers –Difficult to build detailed models, long execution times –General, any geometry / boundary condition can be treated

Tutorial supported by the REASON IST project of the EU Example: IC chip thermal model Boundary conditions: Lateral surfaces: Bottom surface: Top surface: Heat equation: adiabatic surface heat flux heat exchange coefficient adiabatic surface adiabatic surface adiabatic surface heat exchange coefficient

Tutorial supported by the REASON IST project of the EU Multilayered model Non-ideal contacts: Ideal contacts: adiabatic surface heat flux heat exchange coefficient adiabatic surface adiabatic surface adiabatic surface heat exchange coefficient

Tutorial supported by the REASON IST project of the EU Green’s functions Methods for obtaining ­method of images ­Laplace transform ­separation of variables Possible interpretations ­response to instantaneous heat generation ­response to initial temperature distribution

Tutorial supported by the REASON IST project of the EU Initial distribution: Energy generation: Prescribed temperature: Prescribed heat flux: Convective condition: Green’s functions

Tutorial supported by the REASON IST project of the EU Temperature response dumping and lagging

Tutorial supported by the REASON IST project of the EU Green’s function solution methodology

Tutorial supported by the REASON IST project of the EU Solution comparison: analytical A C B