Thermal Analysis Contents - 1 TABLE OF CONTENTS 1 THERMAL ANALYSIS 1.1 Introduction... 1-1 1.2 Mathematical Model Description... 1-3 1.2.1 Conventions and Definitions... 1-3 1.2.2 Conduction... 1-4 1.2.2.1 Energy-Balance Equation... 1-4 1.2.2.2 Transport Law... 1-5 1.2.2.3 Mechanical Coupling: Thermal Strains... 1-5 1.2.2.4 Fluid Coupling: Thermally Induced Pore Pressures... 1-6 1.2.2.5 Initial and Boundary Conditions... 1-6 1.2.3 Advection... 1-7 1.2.3.1 Energy Balance for Convective-Diffusive Heat Transport... 1-7 1.2.3.2 Fluid Mass Balance (Slightly Compressible Fluid)... 1-7 1.2.3.3 Transport Laws... 1-8 1.2.3.4 Thermal-Mechanical-Pore Pressure Coupling... 1-8 1.2.3.5 Initial and Boundary Conditions... 1-9 1.3 Numerical Formulation... 1-10 1.3.1 Conduction... 1-10 1.3.1.1 Finite-Difference Approximation to Space Derivatives... 1-10 1.3.1.2 Nodal Formulation of the Energy-Balance Equation... 1-10 1.3.1.3 Explicit Finite-Difference Formulation... 1-12 1.3.1.4 Implicit Finite-Difference Formulation... 1-14 1.3.1.5 Thermal-Stress Coupling... 1-17 1.3.1.6 Thermal-Pore Pressure Coupling... 1-17 1.3.2 Advection... 1-18 1.3.2.1 Stability and Accuracy... 1-19 1.4 Solving Thermal-Only and Coupled-Thermal Problems... 1-20 1.4.1 Thermal Conduction-Only Analysis... 1-20 1.4.2 Thermal Advection-Conduction Analysis... 1-21 1.4.2.1 Steady-State Conduction Solution... 1-22 1.4.2.2 Forced Advection and Free Advection Simulations... 1-22 1.4.2.3 Synchronization of Fluid and Thermal Times... 1-22 1.4.3 Thermal-Mechanical Analysis... 1-23 1.4.4 Thermal-Mechanical Pore-Pressure Coupling... 1-25 1.5 Input Instructions for Thermal Analysis... 1-26 1.5.1 FLAC Commands... 1-26 1.5.2 FISH Variables... 1-36 1.6 Systems of Units for Thermal Analysis... 1-37
Contents - 2 Thermal Analysis 1.7 Verification Problems... 1-39 1.7.1 Thermal Conduction... 1-39 1.7.1.1 Conduction through a Composite Wall... 1-39 1.7.1.2 Steady-State Temperature Distribution along a Rectangular Fin 1-45 1.7.1.3 Thermal Response of a Heat-Generating Slab... 1-51 1.7.1.4 Transient Temperature Distribution in an Orthotropic Bar... 1-58 1.7.1.5 Spherical Cavity with Applied Heat Flux... 1-66 1.7.2 Thermal Conduction Mechanical... 1-72 1.7.2.1 Semi-Infinite Slab with Applied Heat Flux... 1-72 1.7.2.2 Infinite Line Heat Source in an Infinite Medium... 1-80 1.7.3 Thermal Conduction Poro Mechanical... 1-92 1.7.3.1 Thermal-Pore Pressure Coupled Response... 1-92 1.7.3.2 Heating of a Half-Space... 1-95 1.7.4 Thermal Conduction-Advection... 1.7.4.1 One-Dimensional Solution of Thermal Transport by Forced Convection 1-110 and Conduction... 1.7.4.2 Steady-State Convection in a Saturated Porous Medium Heated 1-110 from above... 1.7.4.3 Steady-State Convection in a Saturated Porous Medium Heated 1-123 from below... 1-126 1.8 References... 1-159
Thermal Analysis Contents - 3 TABLES Table 1.1 System of SI units for thermal problems... 1-37 Table 1.2 System of Imperial units for thermal problems... 1-38 Table 1.3 Problem specifications... 1-39 Table 1.4 Comparison of FLAC results and the analytical solution... 1-45
Contents - 4 Thermal Analysis FIGURES Figure 1.1 Composite wall... 1-39 Figure 1.2 Idealization of the wall for the FLAC model... 1-42 Figure 1.3 Zone distribution... 1-43 Figure 1.4 Steady-state temperature distribution... 1-44 Figure 1.5 Temperature vs distance comparison between FLAC (Table 1) and analytical solution (Table 2)... 1-44 Figure 1.6 Temperature distribution of a rectangular fin... 1-45 Figure 1.7 FLAC model showing history locations... 1-47 Figure 1.8 Temperature evolution... 1-49 Figure 1.9 Temperature distribution at steady state... 1-50 Figure 1.10 FLAC (Table 1) and analytical (Table 2) temperature distributions at steady state... 1-50 Figure 1.11 Heat-generating slab showing initial and boundary conditions... 1-51 Figure 1.12 FLAC model of slab... 1-53 Figure 1.13 FLAC zone distribution and boundary conditions... 1-53 Figure 1.14 Temperature distributions for different times... 1-56 Figure 1.15 Temperature evolution in the center of the slab... 1-57 Figure 1.16 Temperature distribution at steady state... 1-57 Figure 1.17 Problem geometry... 1-58 Figure 1.18 Model for FLAC analysis... 1-60 Figure 1.19 FLAC zone distribution... 1-64 Figure 1.20 FLAC and analytical temperature distribution through the bar... 1-65 Figure 1.21 Temperature distribution after 500 hours... 1-65 Figure 1.22 Temperature distribution after 1000 hours... 1-66 Figure 1.23 FLAC grid and applied flux... 1-68 Figure 1.24 Temperature distribution at 2500 seconds... 1-71 Figure 1.25 FLAC and analytical temperature histories at three locations... 1-71 Figure 1.26 Semi-infinite slab with applied heat flux... 1-72 Figure 1.27 FLAC conceptual model... 1-73 Figure 1.28 FLAC zone distribution and boundary conditions... 1-74 Figure 1.29 Solution process by alternately turning thermal and mechanical logic on and off... 1-77 Figure 1.30 Temperature distribution after 1 second... 1-78 Figure 1.31 FLAC and analytical temperature distribution at 0.2 second and 1 second.. 1-78 Figure 1.32 Vertical stress distribution after 1 second... 1-79 Figure 1.33 FLAC and analytical vertical stress distribution at 0.2 second and 1 second. 1-79 Figure 1.34 FLAC s conceptual axisymmetric model... 1-82 Figure 1.35 FLAC grid for infinite line heat source (note window distortion)... 1-86 Figure 1.36 Close-up view of FLAC grid near source... 1-86
Thermal Analysis Contents - 5 Figure 1.37 Temperature distribution at 5 years... 1-88 Figure 1.38 FLAC and analytical temperature distribution at 1 and 5 years... 1-88 Figure 1.39 Radial displacement contours at 5 years... 1-89 Figure 1.40 FLAC and analytical radial displacements at 1 and 5 years... 1-89 Figure 1.41 Radial stress contours at 5 years... 1-90 Figure 1.42 FLAC and analytical radial stresses at 1 and 5 years... 1-90 Figure 1.43 Tangential stress contours at 5 years... 1-91 Figure 1.44 FLAC and analytical tangential stresses at 1 and 5 years... 1-91 Figure 1.45 FLAC grid for heating of a half-space... 1-97 Figure 1.46 Close-up view of FLAC grid near source... 1-97 Figure 1.47 FLAC and analytical temperature profiles... 1-107 Figure 1.48 FLAC and analytical pore pressure profiles... 1-107 Figure 1.49 FLAC and analytical out-of-plane stress profiles... 1-108 Figure 1.50 FLAC and analytical temperature histories... 1-108 Figure 1.51 FLAC and analytical pore pressure histories... 1-109 Figure 1.52 FLAC and analytical out-of-plane stress histories... 1-109 Figure 1.53 Comparison of temperature versus distance at three different times for convection and conduction acting in the same direction (solid lines) and conduction alone (symbols)... 1-121 Figure 1.54 Comparison of temperature versus distance at three different times for convection and conduction acting in opposite directions (solid lines) and conduction alone (symbols)... 1-121 Figure 1.55 Figure 1.56 Comparison of numerical (symbols) and analytical (solid lines) temperature versus distance profiles at steady state for convection and conduction acting in the same direction... Comparison of numerical (symbols) and analytical (solid lines) temperature 1-122 versus distance profiles at steady state for convection and conduction acting in opposite directions... 1-122 Figure 1.57 Comparison of numerical (symbols) and analytical (solid lines) pore pressure profiles at steady state for a porous saturated layer heated from above... 1-125 Figure 1.58 FLAC grid and location of monitoring points... 1-127 Figure 1.59 Initial temperature contours conduction solution... 1-127 Figure 1.60 Initial pore pressure contours conduction solution... 1-128 Figure 1.61 Evolution of temperature with time at 5 monitoring points Ra=42... 1-129 Figure 1.62 Temperature contours and flow vectors after 17,000 supersteps Ra = 42.. 1-129 Figure 1.63 Temperature contours after 27,000 supersteps... 1-130 Figure 1.64 Evolution of temperature at 5 control points after 27,000 supersteps... 1-130 Figure 1.65 Evolution of temperature at 5 control points after 77,000 supersteps... 1-131 Figure 1.66 Temperature contours on a plane parallel to the x-axis after 77,000 supersteps 1-132 Figure 1.67 Pore pressure contours and flow vectors after 77,000 supersteps... 1-132
Contents - 6 Thermal Analysis Figure 1.68 Temperature contours, analytical steady-state solution, Rayleigh = 4π 2... 1-133 Figure 1.69 Steady-state temperature contours and flow vectors for 8 1 box, Ra = 42. 1-139 Figure 1.70 Close-up view of flow vectors for 8 1 box, Ra = 42... 1-140 Figure 1.71 Steady-state temperature contours and flow vectors for a 1 1 box, Ra = 508 1-144 Figure 1.72 Steady-state temperature contours and flow vectors coarse grid... 1-148 Figure 1.73 Steady-state temperature contours and flow vectors medium grid... 1-149 Figure 1.74 Steady-state temperature contours and flow vectors fine grid... 1-149
Thermal Analysis Contents - 7 EXAMPLES Example 1.1 Conduction through a composite wall... 1-40 Example 1.2 Steady-state temperature distribution along a rectangular fin... 1-47 Example 1.3 Thermal response of a heat-generating slab... 1-54 Example 1.4 Transient temperature distribution in an orthotropic bar... 1-60 Example 1.5 Spherical cavity with applied heat flux... 1-68 Example 1.6 Semi-infinite slab with applied heat flux... 1-74 Example 1.7 Infinite line heat source in an infinite medium... 1-82 Example 1.8 Thermal-pore pressure coupled response... 1-93 Example 1.9 Heating of a half-space... 1-98 Example 1.10 Forward and backward forced convection... 1-111 Example 1.11 Natural advection... 1-123 Example 1.12 Convection in a porous square medium heated from below... 1-134 Example 1.13 Steady-state convection in a long porous box heated from below... 1-140 Example 1.14 Steady-state convection in a porous medium (Ra = 508)... 1-145 Example 1.15 Grid sensitivity analysis... 1-150
Contents - 8 Thermal Analysis