Thermal Analysis Contents - 1

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Geoffrey Wilcox
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1 Thermal Analysis Contents - 1 TABLE OF CONTENTS 1 THERMAL ANALYSIS 1.1 Introduction Mathematical Model Description Conventions and Definitions Conduction Energy-Balance Equation Transport Law Mechanical Coupling: Thermal Strains Fluid Coupling: Thermally Induced Pore Pressures Initial and Boundary Conditions Advection Energy Balance for Convective-Diffusive Heat Transport Fluid Mass Balance (Slightly Compressible Fluid) Transport Laws Thermal-Mechanical-Pore Pressure Coupling Initial and Boundary Conditions Numerical Formulation Conduction Finite-Difference Approximation to Space Derivatives Nodal Formulation of the Energy-Balance Equation Explicit Finite-Difference Formulation Implicit Finite-Difference Formulation Thermal-Stress Coupling Thermal-Pore Pressure Coupling Advection Stability and Accuracy Solving Thermal-Only and Coupled-Thermal Problems Thermal Conduction-Only Analysis Thermal Advection-Conduction Analysis Steady-State Conduction Solution Forced Advection and Free Advection Simulations Synchronization of Fluid and Thermal Times Thermal-Mechanical Analysis Thermal-Mechanical Pore-Pressure Coupling Input Instructions for Thermal Analysis FLAC Commands FISH Variables Systems of Units for Thermal Analysis

2 Contents - 2 Thermal Analysis 1.7 Verification Problems Thermal Conduction Conduction through a Composite Wall Steady-State Temperature Distribution along a Rectangular Fin Thermal Response of a Heat-Generating Slab Transient Temperature Distribution in an Orthotropic Bar Spherical Cavity with Applied Heat Flux Thermal Conduction Mechanical Semi-Infinite Slab with Applied Heat Flux Infinite Line Heat Source in an Infinite Medium Thermal Conduction Poro Mechanical Thermal-Pore Pressure Coupled Response Heating of a Half-Space Thermal Conduction-Advection One-Dimensional Solution of Thermal Transport by Forced Convection and Conduction Steady-State Convection in a Saturated Porous Medium Heated from above Steady-State Convection in a Saturated Porous Medium Heated from below References

3 Thermal Analysis Contents - 3 TABLES Table 1.1 System of SI units for thermal problems Table 1.2 System of Imperial units for thermal problems Table 1.3 Problem specifications Table 1.4 Comparison of FLAC results and the analytical solution

4 Contents - 4 Thermal Analysis FIGURES Figure 1.1 Composite wall Figure 1.2 Idealization of the wall for the FLAC model Figure 1.3 Zone distribution Figure 1.4 Steady-state temperature distribution Figure 1.5 Temperature vs distance comparison between FLAC (Table 1) and analytical solution (Table 2) Figure 1.6 Temperature distribution of a rectangular fin Figure 1.7 FLAC model showing history locations Figure 1.8 Temperature evolution Figure 1.9 Temperature distribution at steady state Figure 1.10 FLAC (Table 1) and analytical (Table 2) temperature distributions at steady state Figure 1.11 Heat-generating slab showing initial and boundary conditions Figure 1.12 FLAC model of slab Figure 1.13 FLAC zone distribution and boundary conditions Figure 1.14 Temperature distributions for different times Figure 1.15 Temperature evolution in the center of the slab Figure 1.16 Temperature distribution at steady state Figure 1.17 Problem geometry Figure 1.18 Model for FLAC analysis Figure 1.19 FLAC zone distribution Figure 1.20 FLAC and analytical temperature distribution through the bar Figure 1.21 Temperature distribution after 500 hours Figure 1.22 Temperature distribution after 1000 hours Figure 1.23 FLAC grid and applied flux Figure 1.24 Temperature distribution at 2500 seconds Figure 1.25 FLAC and analytical temperature histories at three locations Figure 1.26 Semi-infinite slab with applied heat flux Figure 1.27 FLAC conceptual model Figure 1.28 FLAC zone distribution and boundary conditions Figure 1.29 Solution process by alternately turning thermal and mechanical logic on and off Figure 1.30 Temperature distribution after 1 second Figure 1.31 FLAC and analytical temperature distribution at 0.2 second and 1 second Figure 1.32 Vertical stress distribution after 1 second Figure 1.33 FLAC and analytical vertical stress distribution at 0.2 second and 1 second Figure 1.34 FLAC s conceptual axisymmetric model Figure 1.35 FLAC grid for infinite line heat source (note window distortion) Figure 1.36 Close-up view of FLAC grid near source

5 Thermal Analysis Contents - 5 Figure 1.37 Temperature distribution at 5 years Figure 1.38 FLAC and analytical temperature distribution at 1 and 5 years Figure 1.39 Radial displacement contours at 5 years Figure 1.40 FLAC and analytical radial displacements at 1 and 5 years Figure 1.41 Radial stress contours at 5 years Figure 1.42 FLAC and analytical radial stresses at 1 and 5 years Figure 1.43 Tangential stress contours at 5 years Figure 1.44 FLAC and analytical tangential stresses at 1 and 5 years Figure 1.45 FLAC grid for heating of a half-space Figure 1.46 Close-up view of FLAC grid near source Figure 1.47 FLAC and analytical temperature profiles Figure 1.48 FLAC and analytical pore pressure profiles Figure 1.49 FLAC and analytical out-of-plane stress profiles Figure 1.50 FLAC and analytical temperature histories Figure 1.51 FLAC and analytical pore pressure histories Figure 1.52 FLAC and analytical out-of-plane stress histories 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) 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) 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 versus distance profiles at steady state for convection and conduction acting in opposite directions 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 Figure 1.58 FLAC grid and location of monitoring points Figure 1.59 Initial temperature contours conduction solution Figure 1.60 Initial pore pressure contours conduction solution Figure 1.61 Evolution of temperature with time at 5 monitoring points Ra= Figure 1.62 Temperature contours and flow vectors after 17,000 supersteps Ra = Figure 1.63 Temperature contours after 27,000 supersteps Figure 1.64 Evolution of temperature at 5 control points after 27,000 supersteps Figure 1.65 Evolution of temperature at 5 control points after 77,000 supersteps Figure 1.66 Temperature contours on a plane parallel to the x-axis after 77,000 supersteps Figure 1.67 Pore pressure contours and flow vectors after 77,000 supersteps

6 Contents - 6 Thermal Analysis Figure 1.68 Temperature contours, analytical steady-state solution, Rayleigh = 4π Figure 1.69 Steady-state temperature contours and flow vectors for 8 1 box, Ra = Figure 1.70 Close-up view of flow vectors for 8 1 box, Ra = Figure 1.71 Steady-state temperature contours and flow vectors for a 1 1 box, Ra = Figure 1.72 Steady-state temperature contours and flow vectors coarse grid Figure 1.73 Steady-state temperature contours and flow vectors medium grid Figure 1.74 Steady-state temperature contours and flow vectors fine grid

7 Thermal Analysis Contents - 7 EXAMPLES Example 1.1 Conduction through a composite wall Example 1.2 Steady-state temperature distribution along a rectangular fin Example 1.3 Thermal response of a heat-generating slab Example 1.4 Transient temperature distribution in an orthotropic bar Example 1.5 Spherical cavity with applied heat flux Example 1.6 Semi-infinite slab with applied heat flux Example 1.7 Infinite line heat source in an infinite medium Example 1.8 Thermal-pore pressure coupled response Example 1.9 Heating of a half-space Example 1.10 Forward and backward forced convection Example 1.11 Natural advection Example 1.12 Convection in a porous square medium heated from below Example 1.13 Steady-state convection in a long porous box heated from below Example 1.14 Steady-state convection in a porous medium (Ra = 508) Example 1.15 Grid sensitivity analysis

8 Contents - 8 Thermal Analysis

4 Undrained Cylindrical Cavity Expansion in a Cam-Clay Medium

Undrained Cylindrical Cavity Expansion in a Cam-Clay Medium 4-1 4 Undrained Cylindrical Cavity Expansion in a Cam-Clay Medium 4.1 Problem Statement The stress and pore pressure changes due to the expansion