Finite Element Thermography

Preface ix
Introduction xv

Part 1. Thermal Diffusion 1

Chapter 1. Steady State 3
1.1. Partial differential equations of heat transfer in a solid 3
1.2. Variational method 6
1.3. Rayleigh-Ritz method 10
1.4. Finite element model 12
1.5. Finite element assembly 18
1.6. Convergence of the finite element method 20
1.7. Solution to the system of linear equations 24
1.8. Thermal bridge 24

Chapter 2. Boundary Conditions. 27
2.1. Dirichlet boundary conditions 27
2.2. Neumann boundary conditions 32
2.3. Convection 34

Chapter 3. Transient Response 49
3.1. General formulation 49
3.2. Temporal integration 51
3.3. The capacity matrix 53
3.4. Temperature evolution in a solid with adiabatic boundaries 54
3.5. Solid immersed in a fluid 56

Chapter 4. Isoparametric Elements 59
4.1. The concept of Coons patch 59
4.2. Trilinear hexahedron 61
4.3. Two-parameter surfaces or Coons patches 63
4.4. Three-dimensional isoparametric element 67
4.5. Using fins to cool a solid 71

Part 2. Thermal Radiation. 73

Chapter 5. Black Bodies 75
5.1. Planck's law 75
5.2. Stefan-Boltzmann law 78
5.3. Short waves and long waves 80
5.4. Kirchhoff's law 82
5.5. Solid angle 82
5.6. The solar constant 86
5.7. View factor 86
5.8. Calculation of the view factor using the ray tracing method 88

Chapter 6. The Sun and Short Waves 95
6.1. The ellipse and its anomalies 96
6.2. Keplerian orbits 100
6.3. The position of the Sun 103
6.4. Sidereal time and hour angle 106
6.5. Position of the Sun based on latitude and time 108
6.6. Attenuation of solar radiation by the atmosphere 110
6.7. Calculation of solar irradiance 112
6.8. Radiosity equations 120
6.9. Extended view factors 124

Chapter 7. The Earth and Long Waves 127
7.1. Longwave heat transfer 127
7.2. Three conventional surfaces 129
7.3. Cube section with a radiative cavity 131
7.4. Cube with a radiative cavity 133

Chapter 8. Mean Radiant Temperature 137
8.1. A cube and two spherical projections 137
8.2. Mollweide projection 138
8.3. Position of a point on the sphere in the Mollweide projection 140
8.4. Visualization of shortwave radiosities in the cube 142
8.5. Longwave radiation in a cube 143
8.6. Mollweide projections of the cube and cells of equal area 145
8.7. Use of the ray tracing technique 147
8.8. Cube with five cold sides and one hot side 151

Conclusion 153
References 155
Index 161