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Analysis of water cooling process of steel strips on runout table Vakili, Soheyl 2011

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 ANALYSIS OF WATER COOLING PROCESS OF STEEL STRIPS ON RUNOUT TABLE  by Soheyl Vakili  B.Sc., Sharif University of Technology, 2003 M.Sc., Sharif University of Technology, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  September 2011  © Soheyl Vakili, 2011    ii    Abstract  This study engages in the thermal analysis of water jet cooling of a hot moving steel strip on a run-out table. General 3D FE programs are developed for the direct and inverse heat transfer analysis. Studies show that gradient-based inverse algorithms suffer from high sensitivity to measurement noise and instability in small time steps. These two shortcomings limit their application in modeling of the real problems. Artificial neural network (ANN), genetic algorithm (GA), and particle swarm optimization (PSO) methods are applied to the inverse heat conduction problem in order to overcome the challenges faced by the gradient-based methods. Among them, GA and PSO are found to be effective. CRPSO, a variation of PSO, shows the best computational performance. However, compared to the gradient-based methods, these algorithms are very slow. Thus, a set of modifications were performed in this research to accelerate their convergence rate. Sequential formulation using the future time steps, multi-objective optimization, and inexact pre-evaluation using surrogate models are some of these modifications. Inverse analysis of experimental data shows that heat transfer behavior on the plate is mainly a function of the surface temperature, and can be categorized into three zones: High, mid, and low temperature. The effects of jet line configuration, jet line spacing, and plate moving speed were studied. The most uniform distribution happens in the case of fully staggered configuration. In higher jet line distances, the interaction effects become less significant, and a more uniform distribution is observed. The plate speed affects the heat transfer rate under the impingement point for the higher surface temperatures. In the high entry temperatures, the impingement heat transfer rate is lower when the plate is moving at a higher velocity. The plate speed does not significantly change the heat transfer behavior in the parallel flow zone. Finally, the results of the heat transfer analysis were coupled with the microstructure and structure fields, to study the thermal stresses and deflection occurring in the strips during the cooling process. It was found that fully-staggered jet configuration, larger spacing between jet lines, and lower plate speeds result in a less deformed steel strip.    iii    Preface  Some of the contents of chapter 3 have been published in: • S. Vakili, and M. S. Gadala (2009) Effectiveness and Efficiency of Particle Swarm Optimization Technique in Inverse Heat Conduction Analysis, Numerical Heat Transfer, Part B: Fundamentals, vol. 56 (2) pp.119-141. • S. Vakili, and M. S. Gadala (2011) Low Cost Surrogate Model Based Evolutionary Optimization Solvers for Inverse Heat Conduction Problem, Submitted to International Journal of Heat and Mass Transfer. • S. Vakili, and M. S. Gadala (2011) A Modified Sequential Particle Swarm Optimization Algorithm with Future Time Data for Solving Transient Inverse Heat Conduction Problems, Numerical Heat Transfer, Part A: Applications, vol. 59 (12) pp.911-933. • S. Vakili, and M.S. Gadala (2010) Low Cost Particle Swarm Algorithm Using Surrogate Model Based Pre-Evaluation for Inverse Heat Conduction Analysis. ASME 2010 International Mechanical Engineering Congress & Exposition, Vancouver, BC, Canada, November 12-18. • S. Vakili, and M.S. Gadala (2008) Effectiveness of Particle Swarm Optimization Technique in Dealing with Noisy Data in Inverse Heat Conduction Analysis. International Simulation Multiconference, Istanbul, Turkey, July 13-16. •  S. Vakili, and M.S. Gadala (2008) Solving Inverse Heat Conduction Problems Using Particle Swarm Optimization Algorithm. The 16th Annual Conference of the CFD Society of Canada, Saskatoon, Saskatchewan, Canada, June 9 - 11.  Some of the contents of chapter 4 have been published in: • S. Vakili, and M.S. Gadala (2011) Boiling Heat Transfer of Multiple Impinging Jets on a Hot Moving Plate, Accepted for publication in Heat Transfer Engineering Journal.  Some of the contents of chapter 5 have been published in: • S. Vakili, and M.S. Gadala (2010), Finite Element Modeling of Thermal Stresses and Deformation in Steel Strips under Water Jet Impingement Cooling, the 23rd Canadian Congress of Applied Mechanics, Vancouver, BC, Canada, June 5-9.    iv    For all the above manuscripts, I have been responsible for performing the various parts of the research, analysis of the research data, and preparation of the manuscripts. Dr. M.S. Gadala has identified and designed the research program, provided feedback about the outcomes, and have helped in the review of the manuscript.    v    Table of Contents  Abstract ........................................................................................................................................................ ii Preface ......................................................................................................................................................... iii Table of Contents .......................................................................................................................................... v List of Tables............................................................................................................................................... ix List of Figures ............................................................................................................................................... x List of Symbols and Abbreviations ........................................................................................................... xiii Acknowledgements ................................................................................................................................... xvi Dedication ................................................................................................................................................ xvii 1. Introduction ............................................................................................................................................... 1 1.1 Motivations for Controlled Cooling of Steel ....................................................................................... 1 1.2 Research at UBC ................................................................................................................................. 3 1.3 The Need for Inverse Heat Conduction Solver ................................................................................... 6 1.4 Inverse Heat Conduction Analysis Methods ....................................................................................... 8 1.5 Jet Impingement Heat Transfer ......................................................................................................... 11 1.6 Thermal Stress and Deflection .......................................................................................................... 17 1.7 Objectives of This Study ................................................................................................................... 19 2. Three-dimensional Direct and Inverse Heat Conduction Analyses ........................................................ 21 2.1 Direct Three-Dimensional Heat Conduction Analysis ...................................................................... 21 2.1.1 The Need for Three-Dimensional Modeling .............................................................................. 21 2.1.2 Formulation ................................................................................................................................ 22 2.1.3 Numerical Modeling .................................................................................................................. 23 2.1.4 Special Considerations ............................................................................................................... 24 2.1.5 Implementation and Validation .................................................................................................. 24 2.2 Formulation of Inverse Analysis ....................................................................................................... 27 2.2.1 Objective Function ..................................................................................................................... 27 2.2.2 Gradient Based Calculation of Heat Flux................................................................................... 31 2.2.3 Convergence Criteria.................................................................................................................. 33 2.2.4 Flux Zoning Method (FZM) ....................................................................................................... 34 2.2.5 Flux Marching Method (FMM) .................................................................................................. 34 2.2.6 Different Techniques for Calculating Sensitivities .................................................................... 36 2.3 Results and Discussion ...................................................................................................................... 38    vi    2.3.1 Test Cases ................................................................................................................................... 38 2.3.2 Noisy Domains ........................................................................................................................... 44 2.3.3 Effects of the Regularization Parameter and Future Time Steps ............................................... 45 2.3.4 Effect of Time Step Size ............................................................................................................ 48 2.3.5 Comparison of 1D, 2D and 3D Algorithms ............................................................................... 48 2.4 Conclusion ......................................................................................................................................... 50 3. Gradient-Free Methods of Solving the Inverse Heat Conduction Problem ............................................ 51 3.1. Inverse Formulation ......................................................................................................................... 51 3.2. Artificial Neural Networks ............................................................................................................... 52 3.2.1. Feedforward Multilayer Perceptrons (FMLP) ........................................................................... 52 3.2.2. Radial Basis Function Networks (RBFN) ................................................................................. 53 3.2.3. Implementation in Inverse Heat Conduction Problem .............................................................. 54 3.4. Genetic Algorithm (GA) .................................................................................................................. 55 3.5. Particle Swarm Optimization ........................................................................................................... 57 3.5.1. Basic Concepts .......................................................................................................................... 57 3.5.2. Variations .................................................................................................................................. 62 3.5.3. Elite Particle and Elite Velocity ................................................................................................ 64 3.6. Application in Inverse Heat Conduction .......................................................................................... 64 3.7. Time Step Size ................................................................................................................................. 68 3.8. Efficiency ......................................................................................................................................... 69 3.9. Sequential vs. Whole Domain .......................................................................................................... 75 3.10. Noisy Domain Solution .................................................................................................................. 76 3.11. Future Time Step Regularization ................................................................................................... 83 3.12. Effect of Non-linearity ................................................................................................................... 90 3.13. Effect of Elite Members ................................................................................................................. 91 3.14. Multi-Criteria Formulation Based on Spatial Location .................................................................. 93 3.15. Surrogate Modeling ........................................................................................................................ 95 3.15.1. Motivation ............................................................................................................................... 95 3.15.2. Procedure ................................................................................................................................. 96 3.15.3. Surrogate Model Types ........................................................................................................... 96 3.15.4. Surrogate Modeling (SUMO) Toolbox ................................................................................... 98 3.15.5. Model Management ................................................................................................................ 98 3.15.6. Model Configuration ............................................................................................................. 100    vii    3.15.7. Results ................................................................................................................................... 100 3.15.8. Solution Time ........................................................................................................................ 101 3.15.9. Accuracy and Noise Resistance ............................................................................................ 102 3.15.10. Surrogate Model Management ............................................................................................ 104 3.16. Conclusion .................................................................................................................................... 107 4. Heat Transfer Behaviour under Multiple Circular Water Jets Impinging on a Moving Plate ............... 110 4.1. Experimental Setup ........................................................................................................................ 110 4.2. Uncertainty Analysis ...................................................................................................................... 113 4.3. Time Histories of Temperature and Heat Flux ............................................................................... 114 4.4. Heat Flux vs. Surface Temperature ................................................................................................ 120 4.5. Lateral Distribution of the Heat Flux ............................................................................................. 123 4.6. Lateral Distribution of the Surface Temperature ........................................................................... 125 4.7. Longitudinal Distribution of the Heat Flux .................................................................................... 127 4.8. Effect of the Plate Speed ................................................................................................................ 130 4.9. Conclusion ...................................................................................................................................... 132 5. Structural Effects of the Thermal Field during Water Jet Cooling ....................................................... 134 5.1. Phase Transformation Modeling .................................................................................................... 135 5.2. Thermal Stress Modeling ............................................................................................................... 137 5.3. Computational Modeling................................................................................................................ 139 5.4. Validation Cases ............................................................................................................................. 140 5.4.1. Validation Case I ..................................................................................................................... 140 5.4.2. Validation Case II ................................................................................................................... 141 5.5. Stationary Plate .............................................................................................................................. 143 5.6. Moving Plate .................................................................................................................................. 146 5.6.1. Single Passes ........................................................................................................................... 147 5.6.2. Multiple Passes ........................................................................................................................ 151 5.6.3. Effect of Plate Thickness ......................................................................................................... 155 5.7. Conclusion ...................................................................................................................................... 155 6. Conclusion and Future Work ................................................................................................................ 157 6.1. Conclusion ...................................................................................................................................... 157 6.1.1. 3D Finite Element Program for the Inverse Heat Conduction Analysis ................................. 157 6.1.2. Study of the Neural Network Algorithms for Inverse Heat Conduction Analysis .................. 157 6.1.3. Study of the GA and PSO Algorithms for Inverse Heat Conduction Analysis ....................... 158    viii    6.1.4. Improvements in the Efficiency and Effectiveness of Gradient-free Algorithms for IHC Analysis ............................................................................................................................................. 159 6.1.5. Surrogate Model Based Acceleration of GA and PSO Solvers ............................................... 159 6.1.6. Characterization of the Water Jet Impingement Cooling on a Moving Plate .......................... 160 6.1.7. Study of the Structural Effects of the Controlled Cooling of Steel Strips ............................... 161 6.2. Suggestions for Future Work ......................................................................................................... 162 6.2.1. Experimental Study of Higher Plate Speeds, Different Flow Rates, and Water Temperatures  ........................................................................................................................................................... 162 6.2.2. Correlations for Heat Transfer of Multiple Jets Impinging on a Hot Moving Plate ............... 162 6.2.3. Coupling of Hydrodynamics and Heat Transfer Models ........................................................ 162 6.2.4. Experimental Measurements and Correlations for Thermal Stresses and Deflections ........... 162 6.2.5. Developing Mechanistic Models for Moving Plate Jet Impingement Boiling ........................ 163 Bibliography .............................................................................................................................................. 164 Appendix A. Definition of Terms in the Finite Element Heat Conduction Formulation .......................... 175     ix    List of Tables Table  3.1. Comparison of the Typical Solution Time for Different Inverse Analysis Algorithms for the Whole Cooling Process on a Run-Out Table .............................................................................................. 70 Table  3.2. Performed Comparison Tests ..................................................................................................... 70 Table  3.3. Calculated t-Values for the Combination of 2 Solution Methods; tcritical = 1.73 ........................ 72 Table  3.4. Calculated U-Values for the Combination of 2 Solution Methods ............................................ 72 Table  3.5. Number of Computational Units of Different Gradient-Free Solution Methods, Their Speedup with Respect to GA, and Their Relative Cost with Respect to the Most Efficient Method ........................ 74 Table  3.6. Time Required to Perform One Computational Unit for each Test Case .................................. 75 Table  3.7. Comparison of Efficiency for Whole Domain and Sequential Implementations ....................... 76 Table  3.8. Effect of Self-Confidence Parameter: L2 Norm of the Error (W/m2) ......................................... 81 Table  3.9. Effect of the Self-Confidence Parameter on the L2 Norm of Error in the Solution ................... 82 Table  3.10. The L2 Norm of Error in the Solution in a Noisy Domain for Different Algorithms ............... 83 Table  3.11. The L2 Norm of Error in the Solution of Test Case I After Modifications in a Noisy Domain  ..................................................................................................................................................................... 89 Table  3.12. The L2 Norm of Error in the Solution of Test Cases IV and V After Modifications in a Noisy Domain ........................................................................................................................................................ 89 Table  3.13.The L2 Norm of Error in the Solution in an Exact Domain for Different Algorithms .............. 91 Table  3.14. The Average Speedup in Each Algorithm After Using Elite Particles and Velocities ............ 93 Table  3.15. Calculated U-Values to Study the Effect of Elite Particles on the Performance ...................... 93 Table  3.16. Required Time (in Seconds) to Construct the Model for the Linear Case ............................. 102 Table  3.17. Required Time (in Seconds) for 1000 New Predictions......................................................... 102 Table  3.18. R2 Values for Different Surrogate Models in the Exact and Noisy Domains for the Linear Case ........................................................................................................................................................... 103 Table  3.19. R2 Values for Different Surrogate Models in the Exact and Noisy Domains for the Nonlinear Case ........................................................................................................................................................... 104 Table  3.20. Effect of Using Surrogate Models on the Total Solution Time (s) for the Linear Case ......... 106 Table  3.21. Effect of Using Surrogate Models on the Total Solution Time (s) for the Nonlinear Case ... 107 Table  4.1. Test Matrix. .............................................................................................................................. 112 Table A. 1. Definition of the Terms in the FE General Heat Conduction Equation ................................. 175 Table A. 2. Definition of the Terms in the Heat Transfer Equation .......................................................... 176      x    List of Figures Figure  1.1. A Typical Production Line of Steel Rolls [2]. ............................................................................ 2 Figure  1.2. Three Different Types of Cooling on a Run-out Table [12]. ...................................................... 2 Figure  1.3. ROT Test Facility at UBC [24]. .................................................................................................. 3 Figure  1.4. Intrinsic thermocouple junction .................................................................................................. 4 Figure  1.5. Schematic Arrangement of Thermocouples [1]. ......................................................................... 5 Figure  1.6. Schematic View of Test Plate Setup [26]. .................................................................................. 5 Figure  1.7. Temperature Profiles During the Water Jet Impingement [25]. ................................................. 7 Figure  1.8. Different Boiling Regions Present During Top Jet Impingement [14]. .................................... 13 Figure  2.1. Contour Plot Showing Surface Heat Flux (W/m2) in a Staggered Nozzle Configuration [31] . 21 Figure  2.2. Geometry and mesh for the validation of the direct 3D code ................................................... 26 Figure  2.3. The results of validation for the 3D direct code ....................................................................... 27 Figure  2.4. FZM Model for Inverse Calculation of the Boundary Heat Flux ............................................. 34 Figure  2.5. Problem Geometry and Boundary Conditions, Test Case I ...................................................... 39 Figure  2.6. Test Case II; (a) Model, (b) Problem Description [1] ............................................................... 40 Figure  2.7. Temperature Distribution Inside the Slab ................................................................................. 41 Figure  2.8. (a) The Whole Block Consisting of Nine Thermocouples; (b) A Close View of the TC Hole from the Bottom .......................................................................................................................................... 42 Figure  2.9. (a) The Applied Heat Flux on the Top Surface; (b) The Thermocouple Readings Used for Inverse Analysis .......................................................................................................................................... 43 Figure  2.10. Test Case V; A Block Consisting of Eighteen Thermocouple Zones ..................................... 44 Figure  2.11. (a) The Applied Heat Flux on the Top Surface; (b) The Thermocouple Readings Used for Inverse Analysis; Second Line of Thermocouples, Test Case V ................................................................ 44 Figure  2.12. Effect of Regularization Parameter; 4 Future Steps; Error level = ±3°C; (a) α = 1e-12 (L2 Norm of Error = 4.63e5); (b) α = 1e-11 (L2 Norm of Error = 2.56e5); (c) α = 1e-10 (L2 Norm of Error = 1.18e5). ........................................................................................................................................................ 46 Figure  2.13. Effect of Future Time Steps; α = 1e-11; Error Level = ±3°C; (a) 4 Steps (L2 Norm of Error = 4.63e5); (b) 6 Steps (L2 Norm of Error = 1.72e5); (c) 8 Steps (L2 Norm of Error = 1.34e5) ..................... 47 Figure  2.14. Threshold of Divergence in the Solution of Test Case II Due to Small Time Step Size ........ 48 Figure  2.15. Results of the 1D, 2D, and 3D Inverse Algorithms in Predicting a Known Heat Flux Applied to a Direct 3D Problem ............................................................................................................................... 50 Figure  3.1. A Feedforward Network Topology ........................................................................................... 53 Figure  3.2. An RBF Network Topology ..................................................................................................... 54 Figure  3.3. Flowchart of a General Implementation of Genetic Algorithm (GA) ...................................... 56 Figure  3.4.  Velocity and Position Updates in the Basic Particle Swarm Optimization Algorithm ............ 61 Figure  3.5. Flowchart of the Basic Particle Swarm Optimization Procedure ............................................. 62 Figure  3.6. Test Case II; Time History of Heat Fluxes in a Typical Run-Out Table Application; Expected Results (Squares) vs. the RBF Network Results (Line) .............................................................................. 65 Figure  3.7. Individual Heat Flux Peaks vs. Time from a Typical Run-Out Table Application; Expected Results (Circles) vs. the RBF Network Results (Pluses) ............................................................................. 66 Figure  3.8. PSO Results, Test case I: (a) Without Regularization (L2 Norm of Error = 0.127), ................. 67 Figure  3.9. Test Case III; (a) The Top Plane (y = 1) Temperature Contours; Solid Lines: Expected Input; Dotted Lines: Output of the Inverse Analysis; (b) Temperature Distribution (Expected and Restored by PSO) on the Top Surface (y=1) Along the x=1.5 Line (L2 Norm of Error = 31.66) ................................... 67    xi    Figure  3.10. Heat Flux vs. Time: (a) Classical Approach, (b) PSO with the same computational expense, (c) Final results of PSO [112], (d) Final results of GA. .............................................................................. 69 Figure  3.11. Individual Heat Flux Peaks vs. Time from a Typical Run-Out Table Application; Expected Results (Circles) vs. the RBF Network Results (Pluses); Artificial Noise Added: c = ±0.1% ................... 77 Figure  3.12. Individual Heat Flux Peaks vs. Time from a Typical Run-Out Table Application; Expected Results (Circles) vs. the RBF Network Results (Pluses); Artificial Noise Added: c = ±0.3% ................... 78 Figure  3.13. Individual Heat Flux Peaks vs. Time from a Typical Run-Out Table Application; Expected Results (Circles) vs. the RBF Network Results (Pluses); Artificial Noise Added: c = ±0.5% ................... 78 Figure  3.14. Individual Heat Flux Peaks vs. Time from a Typical Run-Out Table Application; Expected Results (Circles) vs. the RBF Network Results (Pluses); Artificial Noise Added: c = ±1% ...................... 79 Figure  3.15. Test Case II; Effect of Regularization Parameter (1% added noise); a: α = 10-12; b: α = 10-11; c: α = 10-10 ................................................................................................................................................... 80 Figure  3.16. Test Case II; Effect of Self-Confidence Parameter (1% added noise); (a) c0=0.5; (b) c0=0.6; (c) c0=1.0; (d) c0=1.2 ................................................................................................................................... 81 Figure  3.17. PSO Behavior in Handling Noisy Data in Test Case I; (a) No Regularization; (b) Addition of Regularization Term (α = 10-10); (c) Tuning of Self-Confidence Parameter (c0 = 1.2); (d) Introduction of Future Time Step Data (10 time steps) ........................................................................................................ 86 Figure  3.18. PSO Behavior in Handling Noisy Data for Test Case IV; (a) No Regularization; (b) Addition of Regularization Term (α = 10-10); (c) Tuning of Self-Confidence Parameter (c0 = 1.2); (d) Introduction of Future Time Step Data (10 time steps) ................................................................................................... 87 Figure  3.19. PSO Behavior in Handling Noisy Data for the 2nd Set of Heat Fluxes in Test Case V; (a) No Regularization; (b) Addition of Regularization Term (α = 10-10); (c) Tuning of Self-Confidence Parameter (c0 = 1.2); (d) Introduction of Future Time Step Data (10 time steps) ........................................................ 88 Figure  3.20. Convergence of Scalar vs Vector Objective Function Formulations; (a) Test Case IV; (b) Test Case V ................................................................................................................................................. 95 Figure  3.21. Results of Surrogate Based PSO for (a) Test Case I; (b) Test Case IV ................................ 101 Figure  3.22. Effect of Surrogate Model Management on the Performance of Inverse Analyzer; (a) GA Solver; (b) PSO Solver .............................................................................................................................. 106 Figure  4.1. A Sample Plate, Showing the Dimensions and Thermocouple Locations .............................. 110 Figure  4.2. Nozzle Configuration with Respect to Thermocouples; (a) In-line; (b) Half-staggered; (c) Full-staggered [31]. ................................................................................................................................... 111 Figure  4.3. History of (a) Thermocouple Readings; (b) Surface Temperature; (c) Surface Heat Flux; High Entry Temperature .................................................................................................................................... 117 Figure  4.4. History of (a) Thermocouple Readings; (b) Surface Temperature; (c) Surface Heat Flux; Mid Entry Temperature .................................................................................................................................... 118 Figure  4.5. History of (a) Thermocouple Readings; (b) Surface Temperature; (c) Surface Heat Flux; Low Entry Temperature .................................................................................................................................... 119 Figure  4.6. Heat Flux and Heat Transfer Coefficient vs. Surface Temperature Across the Width of the Plate for Different Entry Temperature Ranges.......................................................................................... 122 Figure  4.7. Lateral Heat Flux Distribution at the Moment of Impingement of the First Jet Line; (a) No Stagger; (b) Half Stagger; (c) Full Stagger. .............................................................................................. 124 Figure  4.8. Lateral Temperature Distribution After the Impingement of the Second Jet Line; (a) No Stagger; (b) Half Stagger; (c) Full Stagger. .............................................................................................. 126    xii    Figure  4.9. Longitudinal Distribution of Heat Flux for the High Entry Temperature; (a) No Stagger, Jet Line Spacing: 25.4 cm; (b) Half Stagger, Jet Line Spacing: 25.4 cm; (c) Full Stagger, Jet Line Spacing: 25.4 cm; (d) Half Stagger, Jet Line Spacing: 50.8 cm. ............................................................................. 128 Figure  4.10. Longitudinal distribution of heat flux for the mid entry temperature; (a) No stagger, jet line spacing: 25.4 cm; (b) half stagger, jet line spacing: 25.4 cm; (c) full stagger, jet line spacing: 25.4 cm; (d) half stagger, jet line spacing: 50.8 cm ....................................................................................................... 129 Figure  4.11. Longitudinal Distribution of Heat flux for the Low Entry Temperature; (a) No Stagger, Jet Line Spacing: 25.4 cm; (b) Half Stagger, Jet Line Spacing: 25.4 cm; (c) Full Stagger, Jet Line Spacing: 25.4 cm; (d) Half Stagger, Jet Line Spacing: 50.8 cm .............................................................................. 130 Figure  4.12. (a) and (b) Impingement Heat Transfer Versus Entry Temperature; Jet Line Spacings of 25.4 and 50.8 cm, Respectively; (c) and (d) Jet Interaction Heat Transfer Versus Entry Temperature; Jet Line Spacings of 25.4 and 50.8 cm, Respectively; Solid Lines: No Stagger; Dashed Lines: Half Stagger; Dotted Lines: Full Staggered .................................................................................................................... 132 Figure  5.1. Different Kinds of Coupling between the Three Physical Fields in Cooling of Steel Strips .. 134 Figure  5.2. Schematic Representation of Cooling on the Run-out Table (Temperature vs. Time) ........... 136 Figure  5.3. Problem Dimensions and Temperature History in Validation Case I by Yoshida [86]. ......... 140 Figure  5.4. Validation Case I; Comparison between the Current Results and the Experimental and Numerical Results in the Literature [86]. .................................................................................................. 141 Figure  5.5. Geometry of Validation Case II [89] ...................................................................................... 141 Figure  5.6. Temperature Distribution in Validation Case II in (a) Transverse and (b) Longitudinal Directions [89] .......................................................................................................................................... 142 Figure  5.7. Validation Case II; Comparison between the Current Results and the Numerical Results in the Literature [90] ........................................................................................................................................... 142 Figure  5.8: Schematic Arrangement of Thermocouples under Impinging Jet .......................................... 143 Figure  5.9: Top Surface Heat Flux History Obtained By Inverse Analysis of Temperature Readings at Thermocouple Locations. .......................................................................................................................... 144 Figure  5.10: Plate Geometry; Only the Shaded Area is Modeled. ............................................................ 144 Figure  5.11: Temperature Distribution in the Plate a Few Seconds after the Start of the Cooling Process  ................................................................................................................................................................... 145 Figure  5.12: Contours of (a) Z Displacement (Out of Plane) in Millimeters and (b) von Mises Stress in Pascals at the End of the Cooling Process ................................................................................................. 145 Figure  5.13. Maximum Out of Plane Displacement vs. Strip Thickness for the Stationary Plate ............ 145 Figure  5.14. The Deformed and the Edge of the Undeformed Strip in a Moving Case ............................ 146 Figure  5.15. Single Pass Results for High Temperature Range Cooling .................................................. 148 Figure  5.16. Single Pass Results for Mid Temperature Range Cooling. ................................................... 149 Figure  5.17. Single Pass Results for Low Temperature Range Cooling ................................................... 150 Figure  5.18. Results for a Plate Cooled Down to 500 °C .......................................................................... 152 Figure  5.19. Results for a Plate Cooled Down to 300 °C .......................................................................... 153 Figure  5.20. Results for a Plate Cooled Down to 100 °C .......................................................................... 154 Figure  5.21. Maximum Out of Plane Displacement vs. Strip Thickness for the Moving Plate ................ 155     xiii    List of Symbols and Abbreviations C Heat capacity matrix c0 Self-confidence parameter in particle swarm method; Atomic fraction c1, c2, c3, c4 Acceleration coefficients in particle swarm method cp Specific heat, J/kg.°C D Jet diameter deffγ Effective austenite grain size g Best global solution; Gravitational acceleration, m/s2 h Heat transfer coefficient, W/m.°C; Step size in Taylor series expansion H Vertical distance between nozzle exit to the plate; Latent heat of phase change i Time step index I Improvement in a predicted value in surrogate models J Number of spatial components of heat flux k Conductivity, W/m.°C K Heat conduction matrix K Constriction factor L Number of measurement points N Total number of time steps n Iteration step NE Number of function evaluations p Best solution of a particle P Pressure Q Flux load vector, J q Heat flux, W/m2 q Heat flux vector, W/m2 qb Heat generation per unit mass, W/kg r Normally distributed random vector; Number of future time steps R2 Deviation in the values, as defined by equation (3.26) S Total boundary domain T Temperature, °C T Temperature vector, °C    xiv    t Time, s; Student t-test value Ts Surface temperature, °C; Transformation start temperature U Mann-Whitney U value v Velocity of particles in particle swarm method; Center of receptive field unit in radial basis functions V Water velocity X Total sensitivity matrix x Position (value) of a particle in particle swarm method X Volumetric rate of phase transformation x, y, z Cartesian coordinates, m Y Measured temperature matrix  Greek Letters α Regularization parameter; Thermal expansion coefficient β Polynomial coefficients in surrogate modeling Δ Change in a quantity δ Norm of measurement errors ε Emissivity; Random error in measurements; Strain ρ Density, kg/m3 σ Stefan Boltzmann constant; Standard deviation; Stress ψ Radial basis function ω Weights in radial basis function surrogate model  Abbreviations ANN Artificial neural networks CRPSO Complete repulsive particle swarm optimization FE Finite Element FF Feedforward FMLP Feedforward multi-layer perceptron FMM Flux marching method FTS Future time step    xv    FZM  Flux zoning method GA  Genetic algorithm HTC Heat transfer coefficient IHCP Inverse heat conduction problem JMAK Johnson-Mehl-Avrami-Kolmogorov (phase transformation model) NN Neural networks PSO Particle swarm optimization RBF Radial basis function ROT Run-out table RPSO Repulsive particle swarm optimization TC Thermocouple TTT Temperature-time transformation 1D One dimensional 2D Two dimensional 3D Three dimensional  Superscripts calc calculated meas measured α Ferrite phase γ Austenite phase  Subscripts a Atmospheric j Index for space; Jet impingement n Nozzle exit s Stagnation     xvi    Acknowledgements  I offer my enduring gratitude to my supervisor, Dr. Mohamed S. Gadala, for his valuable insight and feedback, as well as his professional and personal support.  Many thanks to Dr. Fuchang Xu for his help in the first steps of this research, to Dr. Fateh Fazeli for his input on the microstructural modeling in chapter 5, to Hossein Mansour for his help in generating the computational grid for simulations, to Geoffrey Franco and Gary Lockhart for their roles in providing the experimental results, and to Dr. Matthias Militzer for coordinating the research in the UBC ROT research group.  And of course, special thanks to my dear family and friends. Without their support and guidance, I would be lost.     xvii    Dedication To my Mom, who dragged me back into the classroom after I skipped school in the first grade; and to my Dad, who coerced me into memorizing the multiplication table in the third grade.    1    1. Introduction 1.1 Motivations for Controlled Cooling of Steel Steel is one of the most widely used materials in today’s industries. The development of advanced high strength steels (AHSS) is very crucial in many different aspects of industry. Steel strips are used in various applications ranging from automobile bodies to drink cans. Roughly, 25 to 30% of this production includes the hot rolling process [1]. Hot rolled steel sheets can be used for piping and tubing, automotive parts, rail cars, and many more applications. The above factors stimulate the research and need for more detailed study on the procedure of steel production and how to control the resulting mechanical properties. The microstructure and mechanical properties of steel strips can be highly modified by the controlled cooling on the runout table. A finer grain can be achieved by using this process, which in turn results in an increase in strength, an improved notch toughness and resistance to brittle fracture, and contributes to a significant reduction in carbon content, while still maintaining the same strength level. The lower carbon content improves formability and contributes to superior notch toughness and weldability. In short, controlled cooling process will assure a balanced combination of mechanical properties. Controlled cooling can also eliminate the troublesome differences in strength among samples taken from the head, body and tail of a steel coil. Detailed study of the effect of cooling profile on the properties of steel began in the early sixties, mainly in the Swinden Laboratories in England [2]. Reference [3] is an early publication that describes the advantages of this kind of treatment. Recognizing the major influence of “time-temperature cycle during cooling” on the grain properties, a research program at J&L steel company started in 1961, in order to design an optimum cooling curve [2]. Since then there has been a great amount of effort to study and design the cooling process [1,4-11]. In a typical production line, the slabs are reheated in a furnace to a hot rolling temperature close to 850 °C, then rolled through roughing and finishing mills, and then they are cooled down to a coiling temperature of 550 – 650 °C while rolling on a so-called runout table (ROT). Figure  1.1 shows a schematic view of such a configuration in a steel mill.         lamin a com the s areas In su syste of ea anoth   still a predi proce sub p regar There are ar flow (cir parison of ame volume  of cooling, mmary, not m. Nowaday ch mechani er, and is m Although  lack of fu cting the tem ss. The com rocesses, t ding the l Figur  three main cular jet) an the perform  of water. H impact time  enough res s a combin sm and the ainly based Figure  1.2. T  accelerated ndamental u perature in plications i he lack of arge dimen e  1.1. A Typic  types of c d water curt ance of aspi owever, h , and surfac earch has b ation of the  amount of on their own hree Differen  cooling is nderstandin  the metal d n the proces real-life exp sions and al Production ooling syste ain, as show red sprays, e compared e temperatu een comple se methods  cooling in  trial and er t Types of Co already wid g of the pro uring coolin s arise from erimental temperature  Line of Steel ms used on n in Figure sprays, and the jets usi re, which pr ted yet on is used in s each of the ror experim oling on a Ru ely adopted cess, which g, and hind  the presen measuremen s of the i  Rolls [2].  the runou  1.2 [12]. Bl planar and c ng different ovided som the advanta teel industri m vary fro ents. n-out Table [  by steel ma  limits the ers the appr ce of a hug ts, as well ndustrial a  t table: wat azevic [5] p ircular jets  conditions e conflictin ges of each es. Relative m one stee  12]. nufacturers ability of a opriate desi e number of  as from th pplications. 2  er spray, erformed by using , such as g results.  cooling  position l mill to , there is ccurately gn of the  coupled e issues Several       resea 23]. H runou Follo temp minim 1.2 R a RO the A heade valve can b comp from adjus the p The a rchers have owever, th t table. Proper c wing a des eratures at izing the c esearch a By recogn T facility w dvanced M rs, one at t s, a furnace e set up ont arable with 50 to 90 m ted from 0.6 late is fixed. The maxi bove listed  studied the ere is still a ontrol of r ired time – the end, mi oolant consu t UBC izing the ne ith industria aterials Pr he top and , and device o the top he those used m. The vert  to 2 m. On  The alignm mum water  parameters  water jet c  need to stu oll and stri  temperatu nimizing te mption are Figure  1.3. R ed to furthe l scale has b ocessing La the other at s for heatin ader and can in steel indu ical distanc ly one noz ent angle of  flow capac  are very cl ooling proc dy many im p cooling re profile, a mperature g some of the OT Test Faci r investigate een constru b (AMPL).  the bottom g water and  be easily c stry. The di e between t zle is availa  the bottom ity is 138 l/ ose to those ess experim portant asp process req ttaining th radients in  main object lity at UBC [2  the water j cted at the U  It consists , a water p  measuring hanged if n stance betw he top nozz ble on the b nozzle to th min. The w  actually u entally and ects of contr uires achie e desired fi  the strip in ives. 4]. et impingem niversity of  of two w ump, pipe c  water temp ecessary. N een the top le exit and ottom head e plate may ater can be sed in steel /or numeric olled coolin ving sever nishing and  all directi  ent cooling  British Col ater banks ircuits, flow erature. The ozzle dimen nozzles is a the test plat er whose di be changed heated up t industry. T 3  ally [13- g on the al goals.  coiling ons, and  of steel, umbia in and two  control  nozzles sions are djustable e can be stance to . o 90 °C. he initial       temp samp of th meas of th conti K are of th locat the p is ab therm in Fi inerti show schem  eratures of les up to 12 Due to th e quick ev urements th ese method nuous meas  used in all e largest te ion, an inter late’s bottom out 1 mm ocouples, th gure  1.4. T a of junctio s a schema atic view o the test pla 00 °C), to m e considerab aporation o at highly re s normally urement tha experiments mperature, nal TC is in  surface. T below the e individua his type of n and henc tic arrangem f the test pla tes are roug ake it more le amount o f water on ly on a visu provide us t are require . These TC when used stalled in a b he measurin plate’s top l wires are junction, ca e decrease t ent of the te, includin Figure  1.4. In hly 700 – similar to an f steam gen  the hot su al inspection  with a sn d for this r s would nor at a temper lind hole w g junction i surface. In spot welded lled intrins he response  thermocou g the thermo trinsic therm 900 °C (th  actual stee erated durin rface of s  of the plat apshot of t esearch. Thu mally have ature highe ith a diamet s fixed onto order to re  directly to ic junction,  time of TC ples in the couple wire  ocouple junc e furnace is l mill. g these exp teel, metho e are not ap he thermal s, thermoco an error of a r than 277 er of 1.6 mm  the end sur duce the re the surface can greatly  close to z plate, and s and the m tion  capable of eriments, a ds such as plicable. A field, rathe uples (TCs pproximate °C. At eac  that is dril face of the sponse tim of the metal  reduce the ero [25]. F Figure  1.6 ounting dev 4   heating s a result infrared lso, most r than a ) of type ly 0.75% h sensor led from hole that e of the , as seen  thermal igure  1.5 shows a ice.         T resea UBC • • • he establish rch in this a  may be cate Studying “condition Studying jets under Studying condition Figure Figu ment of the rea for both gorized as: and model ing” of aus the thermal  cooling con and model s.  1.5. Schemati re  1.6. Schem industrial sc  stationery ing the ev tenite before  behaviour o ditions sim ing the aus c Arrangeme atic View of ale runout t and moving olution of  transforma f stationary ilar to those tenite decom nt of Thermo Test Plate Set able facility  plates [1,8 austenite m tion of optim  and movin found in ind position o  couples [1].  up [26].  in UBC has ,9,27-32]. T icrostructur um grain r g steel plat ustry. f steel und  facilitated he research e and the efinement. e, cooled by er complex 5  intensive  done at optimum  circular  cooling    6    • Establishing quantitative correlations between chemistry, microstructure, and property as a guide for steel grade (alloy) design and steel processing. • Developing new methods for inverse heat conduction problem (IHCP) to model the moving heat flux front on the cooled surface. • Investigation of the errors included in surface temperature measurements due to the presence of thermocouples and possible ways to remedy the problem. • Developing a special 2D program to model the runout table applications with direct and inverse heat transfer analysis capabilities. • Conducting experiments on the heat transfer of a single jet or multiple jets impinging on stationary and moving plates. • Studying the effect of the flow rate and inclination angle of a bottom jet. • Conducting experiments and numerical simulation of the jet impingement hydrodynamics. 1.3 The Need for Inverse Heat Conduction Solver Solving the original Navier-Stokes and energy equations directly, in order to have a proper estimate of the heat flux to the model is quite difficult, and requires a lot of simplifications. Instabilities in the vapour / liquid interface are very common due to the larger density of the liquid that overrides the vapour. Turbulence will be present and its modeling is extremely challenging. Heat generation inside the strip due to a phase change in steel is another factor that complicates the study of these problems. Water moving speed is usually approximated with very simplified models and correlations. The different physical fields, i.e. hydrodynamics, heat transfer, microstructural, etc. are highly coupled. Boiling heat transfer is also strongly affected by factors such as surface roughness that are very difficult to implement in the current numerical simulation models. While these mechanistic models [33,34] provide a valuable insight into the physics of the jet impingement boiling, there is still a need to study the jet impingement boiling process in larger scales and using the experimental data. Experimental measurements that may be used to adjust and verify the numerical models are also quite challenging. As shown in Figure  1.7, recent studies in the group [25] show that errors in the thermocouple measurement of the top plate surface may exceed 200 oC. As indicated in Reference [25], there are several parameters that contribute to these significant errors. One of these factors is the thermal mass of       the th due insul surfa betw distor meas of th on th   indus simu surfa ermocouple to the heat ating vapour ce. More im een their two tion in the uring the su ermocouples e surface. F An appro try-scale ex lations. This ce temperatu s and wires conduction  layer on th portantly, s  junctions, electric cur rface tempe  inside a pl igure  1.7. Tem ach that ma periments a  approach r res and/or h  on the plate  through th e surface an ince the th and water is rent and th ratures usin ate, and use perature Pro y overcom nd only util equires inve eat fluxes th  surface, wh e wires. A d create a co ermocouple  a conductiv us, some er g thermocou  numerical files During t e some of t ize inner su rse heat tran at would be ich causes lso, the pre ntact betwe s work thro e medium f ror in the ples, it is m simulations he Water Jet he above p b-surface te sfer conduc  used in con some chang sence of w en the liqui ugh creatin or electricit measuremen ore accurat to find the  Impingemen roblems is mperature tion (IHTC sequent sim e in the ther ires may b d water and g an electri y, there will ts. Thus, in e to use the boundary co t [25]. to use the r measuremen ) analysis to ulations. 7  mal field reak the the plate c current  be some stead of readings nditions  esults of ts in the  find the    8    1.4 Inverse Heat Conduction Analysis Methods In an inverse heat conduction problem, the boundary conditions, initial conditions, or thermophysical properties of material are not fully specified, and they are determined from measured internal temperature profiles. The main problem is that the effect of changes in boundary conditions are normally damped or lagged, i.e. the varying magnitude of the interior temperature profile lags behind the changes in boundary conditions and is generally of lesser magnitude. Therefore, such a problem would be a typically ill-posed problem and would normally be sensitive to the measurement errors. Thus, in general, the uniqueness and stability of the solution are not guaranteed [35-37]. Inverse problems can be formulated either as parameter estimation or function estimation problems. Inverse heat conduction problems, like most of the inverse problems encountered in science and engineering may be reformulated as an optimization problem. Therefore, many available techniques of solving the optimization problems are available as methods of solving the inverse problems. However, the corresponding objective function of the inverse problems can be highly nonlinear or non-monotonic, may have a very complex form, or in many practical applications, its analytical expression may be unknown. The objective function usually involves the squared difference between measured and estimated unknown variables. If Y and T are the vectors of the measured and estimated temperatures, then the objective function will be in the form of ܷ = ሾࢅ − ࢀሿ்ሾࢅ − ࢀሿ       ( 1.1) However, normally there is need for another term, called “regularization” in order to eliminate the oscillations in the results and make the solution more stable. The effect of this term and the strategy of choosing it will be discussed in details in the subsequent chapters. The above equation is only valid, if the measured temperatures and the associated errors have the following statistical characteristics [38]: ƒ The errors are additive, i.e.       Yi = Ti + εi     ( 1.2) where εi is the random error associated with the ith measurement. ƒ The temperature errors have zero mean. ƒ The errors have constant variance.    9    ƒ The errors associated with different measurements are uncorrelated. ƒ The measurement errors have a normal (Gaussian) distribution. ƒ The statistical parameters describing the errors, such as their variance, are known. ƒ Measured temperatures are the only variables that contain measurement errors. Measured time, positions, dimensions, and all other quantities are all accurately known. ƒ There is no more prior information regarding the quantities to be estimated. If such information is available, it should be utilized to improve the estimates. While classical methods, such as the least square regularization method [35,39], the sequential function specification method [36,39,40], the space marching method [41], conjugate gradient method [42,43], steepest descent method [44], online input estimation [45,46] and the model reduction model [47-49] are vastly studied in the literature, and applied to the problems in thermal engineering [50-56], including water cooling on run-out table [57], there are still some unsolved problems: • The solution often shows some kinds of overshoot and undershoot, which may result in non-physical answers. • Very high heat flux peak values such as those experienced in jet impingement cooling are normally damped and considerably underestimated. • Results are very sensitive to the quality of input. Measurement errors are intrinsic in laboratory experiments, so we need a more robust approach in solving the inverse problem. • The time step size that can be used with these methods is bounded from below, and cannot be less than a specific limit [35]. This causes temporal resolutions that are not sufficient for some real world applications, where changes happen very fast, such as a hot steel strip traveling at a high speed on a run-out table. State of the art and more recent optimization techniques may be used in the solution of the IHTC problem to aid in stability, solution time, and to help in achieving global minimum solutions. Some of these recent techniques are briefly reviewed in the following section: • Genetic Algorithm: This technique has been widely adopted to solve inverse problems [58-60]. Genetic algorithms (GAs) belong to the family of computational techniques originally inspired by the living nature. They perform random search optimization algorithms to find the global optimum to a given problem. The main advantage of GAs    10    may not necessarily be their computational efficiency, but their robustness, i.e. the search process may take much longer than the conventional gradient-based algorithms, but the resulting solution is usually the global optimum. Also, they can converge to the solution when other classical methods become unstable or diverge. However, this process can be time consuming since it needs to search through a large tree of possible solutions. Luckily, they are inherently parallel algorithms, and can be easily implemented on parallel structures. Another remedy is to use them in conjunction with other classical methods to overcome the disadvantages of both methods. There are mainly two approaches that may be used to overcome the difficulties of GA. One approach is to use the solution of classical approach with large time steps as an initial guess to GA. Since GAs do not require any information about the derivatives, they can be used with smaller time steps. This can also improve the quality of the results in terms of reducing the undershoots and overshoots, and reduce the dispersion and diffusion of the peak values. Another approach is to use the solution of classical approach as the initial guess, then using some previous knowledge of the final answer as a way to find the problematic parts of the curve, then using GAs to modify the values of those sections. • Neural Networks: Artificial neural networks can be successfully applied in the solution of inverse heat conduction problems [61-63]. They are capable of dealing with significant non-linearities and are known to be effective in damping the measurement errors. • Self-learning finite elements: This methodology combines neural network with a nonlinear finite element program in an algorithm which uses very basic conductivity measurements to produce a constitutive model of the material under study. Through manipulating a series of neural network embedded finite element analyses, an accurate constitutive model for a highly nonlinear material can be evolved [64,65]. It is also shown to exhibit a great stability when dealing with noisy data. • Maximum entropy method: This method seeks the solution that maximizes the entropy functional under given temperature measurements. It converts the inverse problem to a non-linear constrained optimization problem. The constraint is the statistical consistency between the measured and estimated temperatures. It can guarantee the uniqueness of the    11    solution. When there is no error in the measurements, maximum entropy method can find a solution with no deterministic error [66]. • Proper Orthogonal Decomposition: Here, the idea is to expand the direct problem solution into a sequence of orthonormal basis vectors, describing the most essential features of spatial and temporal variation of the temperature field. This can result in the filtration of the noise in the field under study [67]. • Particle Swarm Optimization (PSO): This is a population based stochastic optimization technique developed by Eberhart and Kennedy in 1995 [68], inspired by social behavior of bird flocking or fish schooling. Like GA, the system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike GA, PSO has no evolution operators such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. Compared to GA, the advantages of PSO are the ease of implementation and that there are few parameters to adjust. Some researchers showed that it requires less computational expense when compared to GA for the same level of accuracy in finding the global minimum [69]. PSO has been successfully applied in many areas, e.g. function optimization, artificial neural network training, fuzzy system control, and other areas where GA can be applied. It has also been successfully employed by many researchers in solving inverse heat transfer problem, mainly inverse radiation [70]. In this research, we will study the genetic algorithm, neural network, and particle swarm optimization techniques in more details. We will investigate their strengths and weaknesses, and try to modify them in order to increase their efficiency, i.e. to lower their computational cost, and to increase their effectiveness, i.e. to lower their sensitivity to the measurement errors, in solving inverse heat conduction problems. 1.5 Jet Impingement Heat Transfer Jet impingement cooling has been one of the most active areas of research in heat transfer for many years. However, due to the complex nature of the involved phenomena, and the ever increasing applications of this technique, it continues to attract a considerable amount of attention. The major industrial applications of jet impingement are cooling of electronic components and controlled cooling of metals after being subjected to hot processes. In each of    12    these applications, a large number of sub-processes are involved, and each of them is very complex in nature. Furthermore, in the case of controlled cooling of metals, the hot surface is normally moving, and there are multiple jets in various configurations impinging on its surface. Between different kinds of jets, round jets especially benefit from a higher specific cooling performance (ability of removing more heat per unit volume of the coolant), and can create more uniform cooling patterns on the plate surface, compared to the sprays and planar jets [71]. Hence, a more detailed study of the behavior of multiple circular jets impinging on a moving surface seems to be highly beneficial.  A good amount of research in this field has been focused on understanding the basic underlying mechanism of the jet impingement cooling. Thus, the studies are mainly focused on one single jet impinging on a stationary surface. While many researchers have tried to study the effects of multiple jets, or a moving surface, these two parameters are not adequately considered together in order to understand their coupled impacts. Also, previous studies usually categorize the wet area only into two zones: the impingement zone, and the parallel flow zone. While this may be an acceptable classification system for a stationary plate and a single nozzle, it is not very suitable for multiple jets impinging on a moving surface. The main problem with this classification is that there is no distinction between the parallel flow zone within a row of jets and the parallel flow zone between two consecutive jet rows. Moreover, many researchers have performed their experiments on a scaled down setup, while in this research we are using a large experimental industrial scale facility similar to the runout table facilities in material processing industries.         jet ca first z is ch some exten at a h boilin reduc pools trans main symm axisy Figu On its sim n be divide one is the i aracterized b  operating c d about 2 to With the igher level, g region. I ed heat tran , overriding fer in this zo The study  categories etrical with mmetrical a re  1.8. Differe plest form, d into five d mpingemen y single-ph onditions, s  4 times the movement o and the ons n the third sfer rate. T  the vapour ne occurs b  of the wate of stationar  respect to round the s nt Boiling Re  the area on ifferent hea t zone, and ase forced uch as plate  nozzle diam f the waterf et of boiling region, the hereafter, d  layer. Fina y radiation a r jet imping y and mov the water jet tagnation po gions Present  the top sur t transfer re is located ju convection  speed, flow eter [14,72- ront, the wa  is finally re  forced con ue to surfac lly, the last nd convecti ement cool ing surface  line in the int in the c  During Top face of a ho gions, as d st undernea and a very  rate, and j 74]. ter is heated ached, crea vection film e tension, t  region is w on to the su ing in the li s. For statio cases of wa ase of a sin Jet Impingem t plate that isplayed in th the impin effective co et cross sec , the surfac ting a relati  boiling o he water m here the pl rrounding ai terature can nary cases ter curtains gle circular  ent [14]. is cooled by Figure  1.8 [ ging jet. Th oling rate. B tion, this re e temperatur vely narrow ccurs, result ay agglome ate is still d r.  be divided , the cooled or a row of jet. Althoug 13   a water 14]. The is region ased on gion can e is kept  nucleate ing in a rate into ry, Heat into two  area is jets, and h, much    14    easier for experimentally investigation and numerical modeling, this case does not exist in industrial applications. What happens in a typical steel mill is water cooling of a moving hot surface. In the moving case, heat transfer caused by the impinging jets is no longer symmetrical. On one side, where the water flow and the work piece move in the same direction, the vapour layer thickness in the film boiling region decreases. On the other hand, the thickness of the vapour layer is increased in the side that the water flow and the hot surface move in the opposite direction [1]. Another factor that makes the analysis of the cooling process more complicated is the dependency of the thermal conductivity and the specific heat of the steel on temperature. Also, phase transformation in the steel affects the thermal field through the latent heat of transformation. Ochi et al. [15] experimentally investigated the transient boiling heat transfer to a circular water jet impinging on a hot plate. The temperature at the bottom was measured, and the temperature and flux at the top cooling surface were inversely calculated using the finite difference method. In the experiment, it was observed that a stable film boiling was maintained over the whole cooling surface of the heated plate, and the temperature fell linearly with time. Vader et al. [73,74] studied the convective nucleate boiling on a heated surface cooled by an impinging planar water jet under a steady-state condition. The temperatures were measured by TCs at the opposite dry face to the cooled top surface. The temperature and heat flux at the cooled surface were found by solving the steady state equation. In the calculations, the boundary conditions were assigned from the least squares cubic spline fit of the measured temperatures. Kumagi et al. [52] studied the transient cooling of a hot plate with an impinging water jet. Temperatures were measured at four points at different depths along the plate thickness and at nine locations in the width direction. An exponential equation with three coefficients was used to describe the temperature profile along the thickness, and then the coefficients were determined from the measured temperatures and the least squares method. The approximate temperature curve was extrapolated to the surface to obtain the surface temperature. The gradient at the surface produced the heat flux. Many researchers have studied the jet impingement under steady state conditions, i.e., the sample is constantly heated while being cooled by jet impingement. Some good examples are the works of Robidou et al. [10,75]. They use a 2D inverse heat conduction solver to find the local    15    heat fluxes and corresponding temperature. They found that in the forced convection regime, heat fluxes increase with an increase of subcooling, jet velocity, and decrease of the distance from the stagnation line, while in the fully developed nucleate boiling regime, no influence of jet velocity, subcooling, and nozzle to plate spacing on the heat flux is observed. Xu and Gadala [76] used their developed inverse algorithm [1] to study the heat transfer behaviour in the impingement zone under a circular water jet. They found out that the heat transfer behaviour in the stagnation zone is mainly affected by the water flow rate, with mild effect from the steel grades. Their developed code is 2D, so it cannot predict the effect of staggered jet configurations, in addition to not being able of modeling the heat transfer in the width of the strip. The study is also mainly focussed on the stagnation region in the stationary plate with only a single jet. One of the key research papers studying the interaction zone between multiple jets was published in 1977 by Ishigai et al. [77]. Unfortunately, this research is focused on the non- boiling heat transfer. They found that the radial location of interference is an important factor for both the hydrodynamics and thermal properties of the flow field. Different behaviors were attributed to different values of film Froude number and the Reynolds number based on the radius from the center of the jet. The reported thermal quantities are the surface temperature, and not the heat flux.  Sakhuja et al. [78] used experiments to study the effect of multiple jets on the boiling heat transfer on a steady plate. They found that at smaller jet spacing, the area that is covered by each jet is small, and crossflow causes a reduction in the heat transfer coefficient. On the other hand, a larger distance between the jets results in a smaller ratio of the impinging area to the coverage area, which again, reduces the overall heat transfer coefficient. So an optimum heat transfer is obtained when the jet spacing is in a bounded range. They found this range to be between 8 to 12 times of the one jet diameter.  Monde et al. [79] studied the effect of multiple jets on a stationary plate. Between two and four circular free surface jets were used. While they concluded that the degree of scatter in the results is typical for nucleate boiling, and thus, the number or location of jets has little or no effect on the heat transfer behavior, a more careful scrutiny of their data by Wolf et al. [13], suggested that they actually do have an effect, and additional investigation is required.    16     Slayzak et al. [80] studied the effects of interaction between adjacent free surface planar jets under non-boiling condition. They observed a strong upwelling of flow (called an interaction fountain) in the region between two planar jets. Interestingly, they found that the heat transfer under this fountain can be comparable to those related to the jet impingement region. They tried to explain this behavior based on the oscillatory nature of the interaction zone, which can encourage a strong mixing and intermittent disruption of the thermal boundary layers in the interaction region. They also studied the effect of interaction between adjoining rows of circular jets under non-boiling condition [81]. Again, an interaction fountain was observed. However, unlike the case of planar jets, where a well-defined interaction zone appears midway between the impinging jets, the interaction zone between circular jets was found to be irregular, and at a location different from the midway point. Also, there is no splattering or hydraulic jump occurring in their tests. Obviously, both the splattering and the hydraulic jump can increase the complexity of the flow and thermal fields.  Filipovic, et al. [71] studied the heat transfer behavior of an array of round jets impinging on a moving surface. They have used some correlations for the boiling heat transfer in different regions, sometimes even from the stationary plates, and have made some assumptions regarding the extent of each zone. These correlations are applied, and the model is solved numerically. In their validation, they only consider the starting and coiling temperatures, and not the transient behavior of the thermal field. They found out that around 50% of the cooling is done by the film boiling mechanism, and the jet impingement regions are responsible for about 30% of the total heat extraction.  The uniformity of temperature distribution across the plate width when using multiple impinging jets was studied by Haraguchi and Hariki [82]. They used a radiation thermometer in their experimental setup on a stationary plate. They found the interaction zone to reduce the heat transfer coefficient around the impingement locations. To obtain a more uniform temperature distribution in the plate, they suggested decreasing the nozzle pitch. Unfortunately, this resulted in a reduction of heat transfer coefficient in those areas as well.  Liu [8] also studied the effect of jet interaction between adjacent water jets on the heat transfer behavior in a simulation of steel runout table. By visual inspection of the experiments, he observed that the dark areas that are associated with the high heat flux regions develop immediately at the stagnation points under the jets, and then gradually move toward the    17    interaction boundary, where they stabilize. He also observed a strong fountain at the interaction zone, accompanied by sizeable water splashes. However, the cooling intensity of water was weakened at the interaction region when boiling was the mechanism of heat transfer. He also studied the case where jets had different flow rates. In that case, the heat transfer rate under the weaker jet was reduced due to the existence of a cross flow created by the stronger jet.  Jondhale et al. [83] studied the heat transfer behavior on the top surface of a moving hot plate cooled by multiple impinging jets. They observed that there is a huge discrepancy between the heat transfer in the impingement zone and interaction zone, until the plate entry temperature reaches 200°C, when the water can wet the surface between the nozzles. They also observed that larger nozzle spacing in a jet row can actually increase the heat extraction rate under the jet, because it prevents having a submerged jet. The heat flux is however reduced in the interaction zone when having larger jet spacing.  Gradeck et al. [84] studied the effect of moving of the hot surface by impinging a planar water jet on a rotating cylinder with an initial temperature of 500 – 600 °C. They observed that the heat transfer is significantly changed when the surface is moving. The first effect was a homogenization of the boiling curves, and elimination of the “shoulder of flux”. They also observed a reduction in the value of the critical heat flux.  In this work, we are going to study the boiling heat transfer characteristics of two rows of three impinging jets on a moving hot plate. Experiments are conducted in an industrial scale setup, and thermocouples are used to measure the temperatures just under the plate surface. Then these temperatures are used as the input in an inverse heat conduction algorithm to find the boundary heat flux variations in space and time across the width and along the length of the moving flat plate. The effects of nozzle configuration, jet line spacing, and plate speed are going to be investigated. 1.6 Thermal Stress and Deflection As a result of an uneven temperature distribution in the strip, significant thermal stress occurs. This is even more magnified by the steel phase transformation. These stresses can result in local deformation and residual stresses. The flatness of the hot rolled products on the run-out table has not been sufficiently studied in the literature. Only a few researchers have investigated the thermal stresses of hot rolled strips [85].    18    Yoshida [86] was one of the first researchers to numerically study the non-flatness of hot rolled steel strips after cooling in a run-out table by applying finite-difference methods. Earlier, experiments had shown that the edge wave occurs after cooling in most cases, unless the strip is coiled at temperatures higher than 570 °C. He stated that the non-flatness occurs when the compressive residual thermal stress becomes larger than the critical buckling stress of a long thin plate. The thermal distribution was found using simple correlations for water and air cooling heat transfer coefficient, and the phase change was calculated based on a time-temperature- transformation (TTT) diagram. He found out that reducing the transverse variation of finishing temperature and cooling rate, coiling at higher temperatures and water cooling in the final sections of a run-out table can prevent the occurrence of edge waves. Among them, a uniform cooling rate across the strip was the most beneficial. Nakata and Yoshida [87] later extended the above study to further study the effect of the cooling uniformity in the strip flatness using both experimental measurements and numerical simulations. The thermal model was validated against the measured surface temperature distributions using an infrared camera. It was confirmed that the non-uniform cooling along the length of the strip can cause non-flatness in the strip. Han et al. [88], calculated the elastic strain, the volumetric strain caused by the thermal loads and phase transformation, the plastic strain, and the transformation induced superplastic strain, to estimate the total deformation of steel strips on a run-out table. The model was two- dimensional, and cooling conditions were set using simple correlations for two separate zones cooled by air or water. Zhou et al. [89] calculated the thermal expansion coefficient as a function of the cooling rate, phase composition and temperature. They used a plane stress finite element model based on the assumption that the through-thickness stress is negligible in thin strip. The validation was done only based on qualitative agreement of the predicted results by the results of a simple method in a confidential report. Thermal boundary conditions were applied in the form of simplified temperature distributions. They found that the temperature distribution in the length of the strip has little effect on the residual stress, but the thermal field across the width of the plate has a large contribution to the stress field. In a later study [90], they investigated the flatness of the plate under the controlled cooling process using three-dimensional shell elements. They    19    found that the severity of buckling in the strip increases with the transverse temperature gradient across the plate, i.e. it resulted in a lower wavelength and higher amplitude. As discussed above, most of the existing studies in the literature use simplified models for the thermal boundary conditions, usually based on some generalized correlations of heat transfer in two cooling zones (under impingement and parallel flow zones), or temperature measurements in few stages in the strip. They are also normally using simplified models for phase transformation. In the present study, we are going to use the results of our inverse heat conduction solver [76,91] to extract the boundary heat fluxes from the readings of thermocouples in an extensive set of experiments [1,31], and use a more refined and modern algorithm to calculate the phase transformation. We will study the cases of both the stationary and moving plates, and different jet configurations. 1.7 Objectives of This Study The following steps are taken in this research: 1) Developing a three-dimensional gradient-based inverse heat conduction program In order to be able to model different jet configurations accurately, and to obtain accurate heat flux values, a three-dimensional finite element heat conduction simulator is needed. The code should be capable of taking into account the heat generation caused by the steel phase transformation and the temperature and phase dependency of thermophysical properties. The model should be able to capture the interaction between several jets. Unlike the case of interacting planar jets, where a well-defined interaction zone exists in midway between the jets, interaction zones between arrays of circular jets are irregular and do not necessarily occur in the midway [81]. Thus, an iterative and sequential three-dimensional inverse heat conduction code is developed. The algorithm is based on the least-square technique, sequential function specification, and regularization. The code needs to be capable of handling unsteady cases, and moving boundary conditions. 2) Investigating various categories of stochastical inverse heat conduction solvers The above mentioned inverse solver, while being computationally efficient, suffers from some problems related to its gradient-based nature, such as sensitivity to the measurement errors, and instability in handling small time steps. Other forms of inverse solvers will be studied and modified to find a more effective algorithm. Modifications will be performed in order to improve not only the stability and effectiveness of the method, but also its computational expense.    20    3) Determination of surface heat flux and heat transfer behaviour in multiple jet impingement on a moving plate Existing numerical models are mainly focussed on the case of single water jet. While they are useful for understanding the physics and to validate the algorithm, they are far from the real situation in industry. Multiple jets with different configurations should be modelled to make the case closer to the industrial applications. The developed inverse algorithms will be used to determine the heat fluxes occurring during the water jet cooling process. We will be especially focused on cases where multiple jets (several jet lines, each including more than one nozzle) imping on a moving plate, which are closer to what really happens in an industrial run-out table. 4) Studying the effect of plate speed, jet line stagger, and jet line spacing on heat transfer Effect of these parameters on the maximum heat extraction, uniformity of cooling pattern and temperature distribution, and the overall efficiency of the run-out table will be studied. 5) Modeling the thermal stresses caused by the water jet cooling of stationary and moving plates As a result of non-uniform cooling on the plate, thermal stresses and deflection are expected. A model will be devised that will be capable of modeling the phase change process as well.        2. T progr Some solve 2.1 D main stable boun and d 2.1.1 fluxe as be surfa top su surfa differ Fi  hree-dime In this ch am. Then, a  modificatio d, and the e irect Thre The core ly used to e  as possibl dary conditi evelopment  The Need f In the cas s may be pr tween two s ce of a mov rface, there ces. This is ent than the gure  2.1. Con nsional D apter, we  classical gr ns and cons ffect of thes e-Dimensi of any inve valuate the e. It should ons, materia  of such a so or Three-D e of multip oduced. Thi uccessive j ing plate [3  is also a gr  due to the  top surface tour Plot Sho irect and discuss the adient-base iderations a e modificati onal Heat rse heat co value of obj also allow l properties lver is discu imensional le jets impi s can result et lines. Fig 1]. In additio adient acros  fact that th , mainly due wing Surface Inverse H developmen d inverse al re discussed ons are obse Conductio nduction co ective funct for the mod , and transie ssed.  Modeling nging on a in a large te ure  2.1 show n to this co s the thickn e cooling p  to the lack  Heat Flux (W eat Cond t of a 3D gorithm is d . Finally, so rved and di n Analysis de is a dire ions; thus, i eling of all nt behaviou moving sur mperature g s the distri nsiderable v ess of the p attern on th of pool boil /m2) in a Stag uction An finite elem eveloped ba me test case scussed.  ct heat con t needs to b  complicatio r. In this sec face, signifi radient acro bution of h ariation in late, betwee e bottom s ing regions gered Nozzle alyses ent heat co sed on the s are introd duction sol e as accura ns in the g tion, the for cantly diffe ss a jet line eat fluxes o the values a n the top an urface is es [28].   Configuratio 21  nduction 3D code. uced and ver. It is te and as eometry, mulation rent heat , as well n the top cross the d bottom sentially n [31]    22    2.1.2 Formulation The general governing equation for the 3D conduction heat transfer problems, shown in Figure 1, is written in the form: t Tcq z Tk zy Tk yx Tk x p b zyx ∂ ∂ =+ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ρ)()()(            ( 2.1) where T is the temperature, °C; qb is the heat generation per unit volume, W/m3; kx, ky, and kz are the conductivities in the x-, y-, and z-directions, respectively, W/m⋅°C; ρ  is the density, kg/m3; cp is the specific heat, J /kg⋅°C;  t is the time, s ; and x, y, and z are the Lagrangian coordinates of the point.   The boundary conditions may be one or a combination of the followings cases: Prescribed temperature: This is an example of a Dirichlet boundary condition (BC). The prescribed temperature Ts (°C) may be a function of time and boundary coordinate (spatial function): ),,,( tzyxTT s=       ( 2.2) Prescribed heat flow of flux: Specified heat flux (qs) may be a spatial function or a function of time: ),,,( tzyxq z Tk y Tk x Tk szyx =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ −  ( 2.3) where qs is the specified rate of heat flow per unit area (W/m2).  Prescribed heat flow is an example of Cauchy’s or Neumann boundary conditions.  If qs is zero, it will represent natural boundary conditions. Convection heat exchange: When there is a convective heat transfer on part of the body surface due to contact with a fluid medium, we have: )( fszyx TThz Tk y Tk x Tk −=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ −  ( 2.4) where h is the convection heat transfer or film coefficient (W/m2 ⋅ °C), which may be temperature dependent (nonlinear), Ts is the surface temperature (°C) and Tf is the fluid temperature (°C), which may be a spatial or time function. Radiation: Assuming grey body, this boundary condition is given by: [ ]44 rsrzyx TTzTkyTkxTk −=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ − εσ      ( 2.5)    23    where ε  is the emissivity of the surface of the body, σ  is the Stefan-Boltzmann constant (W/m2⋅°K4), Tsr is the absolute temperature of surface S5 (°K) and Tr is the known absolute temperature of the external radiative source (°K). The radiation boundary condition may be dealt with as a nonlinear convective boundary condition with an equivalent temperature dependent film coefficient,κ , where:     ( )( )rsrrsr TTTT ++= 22εσκ       ( 2.6) 2.1.3 Numerical Modeling Using a weighted residual Galerkin procedure, the final finite element equations may be written as: QKTTC =+&        ( 2.7) where C is the equivalent heat capacity matrix; K is the equivalent heat conduction matrix; T and T&  are vectors of the nodal temperature and its derivatives, respectively; and Q is the equivalent load vector. Detailed expressions of the matrices in equation (2.7) are given in Appendix-A. A general family of solution algorithms may be obtained by introducing a parameter α where      ( 0.10.0 ≤≤α ) such that )11 TT(T)T(T tαΔtttΔttαΔtt − Δ =− Δ = +++ tt α &           ( 2.8) TTT tΔttαΔtt )1( αα −+= ++         ( 2.9) If α = 0, an explicit Euler forward method is obtained; if α= ½, an implicit trapezoidal rule is obtained; and if α = 1, an implicit Euler backward method is obtained. Substituting equation (2.8) into equation (2.7) and applying Newton-Raphson iterations yields:  )1( ˆˆ )1(ˆˆ)( )1(1 −⎟⎠ ⎞⎜⎝ ⎛ +Δ+−−⎟⎠ ⎞⎜⎝ ⎛ +Δ++⎟⎠ ⎞⎜⎝ ⎛ +Δ+= − ⎥⎦ ⎤⎢⎣ ⎡ ⋅⎟⎠ ⎞⎜⎝ ⎛ Δ + Δ+ iccttirhttsbtti i t tt qQQQQQΔTCK ααα α α       ( 2.10) where 0≠α . The definition of all terms is given in Appendix-A and all quantities at time (t+αΔt) are calculated from a relation similar to equation (2.9). Depending on the value of α, the procedure may be either conditionally stable (α<0.5) or unconditionally stable 5.0≥α .    24    2.1.4 Special Considerations The thermal properties of steel normally change with a change in the temperature. This results in a nonlinear behaviour of the thermal field. The exact correlation depends on the chemical composition of each steel sample. For example, for DQSK steel, the relationship between the thermal conductivity and the temperature is: ࢑ = ૟૙. ૞ૠ૚ − ૙. ૙૜ૡ૝ૢ	 × ࢀ	(℃)	࢏࢔	ࢃ/࢓.℃     ( 2.11) and for SS316 steel, the correlation is:          	࢑ = ૚૚. ૚૝૚ + ૙. ૙૚૝ × ࢀ	(℃)	࢏࢔	ࢃ/࢓.℃     ( 2.12) These equations are valid from zero to 1000 °C. So the finite element code should be capable of handling these nonlinearities. It is also important to integrate the microstructure evolution models into the heat transfer analysis [1,92]. Since during the cooling process, the austenite will transform into other phases, a latent heat is released. While some researchers have ignored this effect [93], the accuracy of the simulations can be enhanced by incorporating this heat into the model [1,94]. In our implementation, following the procedure in Reference [1], the heat generation per volume is calculated by: FHQ &⋅= ( 2.13)  where H is the mole of material considered, and F& is the transformation rate of austenite to ferrite and pearlite per volume. This value is going to be calculated in a subroutine, and the values will be included in the source term Qb in equation (2.10). 2.1.5 Implementation and Validation In the developed program, the geometry can be discretized using isoparametric 3D elements. By default, the elements will have eight nodes and will resemble a brick. However, other 3D shapes can also be created. The program is capable of dealing with a mixture of element types. The program is also capable of dealing with all different types of boundary conditions, as mentioned above. The boundary conditions can be a function of space and/or time. The nonlinearities caused by the dependence of the thermophysical properties on the temperature can be handled in a step-wise staggered approach, i.e. the values of the parameters at the current    25    step are calculated based on the temperature at the previous step, and are assumed to be constant during the current step. The program is capable of handling both the steady and transient problems. Various verification cases were modeled and the results were compared with the results obtained by the commercial program ANSYS [95]. In all cases, the results were shown to be within less than 1% of each other. Some of the test cases are shown here. Geometry and Mesh Figure  2.2 shows a schematic of the geometry and the discretized domain. The width of the block (x direction) is 1.5 m, the height (y direction) is 0.5 m, and the length (z direction) is 5.0 m. The thermal conductivity is following equation (2.11), the density, ρ, is 7850 3mkg , and the heat capacity, Cp is 475 kgKJ . Test Case I The initial temperature of the block is 14 °C. The top surface of the block is subjected to a convection boundary condition with a film coefficient of 60 W/(m2.°C) and a fluid temperature of 40 °C. The comparison between the developed code and ANSYS is displayed in Figure  2.3(a). Test Case II Starting from an initial temperature of 8 °C, the top surface of the block is subjected to a constant heat flux of 100 W/m2. The comparison between the developed code and ANSYS is displayed in Figure  2.3(b). Test Case III Starting from an initial temperature of 1 °C, the top surface of the block is subjected to a radiation boundary condition with a surrounding temperature of 100 °C. The comparison between the developed code and ANSYS is displayed in Figure  2.3(c).          Test Starti speci result   Case IV ng from an fied temper s of the dev  initial tem ature bound eloped code Figure  2.2. G perature of ary conditio  and ANSY eometry and   1 °C, the n of Ttop = S for a point mesh for the v top surface 2x + 5z + 1  at y = 0.25 alidation of t  of the blo 0. The com  m is display he direct 3D c ck is subjec parison bet ed in Figur  ode 26  ted to a ween the e  2.3(d).        2.2 F 2.2.1 probl meas probl ormulatio  Objective F The boun em in whic ured temper em with so (a) (c) Figure n of Invers unction dary invers h we try t atures and me guessed   2.3. The resu e Analysis e heat cond o minimize the calculat  boundary lts of validatio  uction prob  the norm ed temperat conditions. n for the 3D lem can be of differen ures obtaine In order to (b) (d) direct code  formulated ce between d from a d  dampen th  as an opti  the experi irect solutio e oscillation 27  mization mentally n of the s in the    28    solution, and make the algorithm more stable in dealing with noisy measurement data, it is very common to include some kind of regularization terms that penalize the large magnitude error in the objective function. A common choice in inverse heat transfer problems is to use a scalar quantity based on the boundary heat fluxes, with a weighting parameter α, which is normally called the regularization parameter. The regularization term can be related to the values of heat flux, or their first or second derivatives with respect to time or space. Based on the previous experience [76], as well as our own trial of different regularization terms, we choose to use the heat flux values (zeroth-order regularization). The objective function is then ( )∑ ∑∑ = == ⎟⎠ ⎞⎜⎝ ⎛ +−= J j N i i j calci j N i measi j qTTf 1 1 22, 1 , α)()(q  ( 2.14)  where ,i meas jT and ,i calc jT are the values of expected (measured) and calculated temperatures at the ith time step and the jth spatial location, respectively; α is the regularization coefficient; and i jq  is the boundary heat flux vector at the ith time step and the jth spatial location, N is the total number of time steps, and J is the number of spatial components of temperature vectors. Due to the parabolic nature of the heat equation, there are basically two categories of approaches in solving the inverse transient heat conduction problems. One method is to try to find components of heat flux in each step of the solution. This is possible since the temperature distribution at the end of each time step is only a function of the temperature distribution at the previous time step and the boundary conditions during the present time step. These methods are normally called “sequential”. In these methods, in each step of the inverse analysis, the direct problem is not solved for the whole history, and is solved for only one (or a few) time steps. The optimization problem is then solved for the temperatures at that step, and then the problem is marched in time to the next steps. The other solution method is to treat the whole history of heat fluxes as an unknown in the optimization problem. In this formulation, which is usually called the “whole domain” approach, the direct problem will be solved over the whole time history.  Due to the diffusive nature of transient heat conduction and from a computational expense perspective, sequential estimation techniques are highly preferred over whole-domain inverse analysis techniques [35]. Unfortunately, there is an important drawback in using    29    sequential methods, and that is the high sensitivity of the solution to the experimental errors. That is why we will be using the future time steps concept. Due to the fact that inverse problems are generally ill-posed, the solution may not be unique and is generally sensitive to measurement errors. To decrease such sensitivity and improve the simulation in function specification methods, a number of future time steps (nFTS) are often utilized in the analysis of each time step [35]. This means that in addition to the measured temperature at the present time step T i, the measured temperatures at future time steps, FTSniii TTT +++ ,....,, 21 , are also used to approximate the heat flux qi. In this process, a temporary assumption is usually considered for the values FTSniii qqq +++ ,....,, 21 .  The simplest and most widely used one is to assume iki qq =+  for FTSnk ≤≤1 , which is also used in our work. Also in this work, we combine the regularization concept with the future time step data. This is analogous to the combined function specification-regularization method, initially introduced by Beck and Murio [96]. The sequential algorithm for time step M, when using r future time steps (the heat flux is known for t < tM), is presented below: 1. An assumption is made for the heat flux function at times tM, tM+1,…, tM+r (usually qM = qM+1=…=qM+r). 2. The direct problem is solved with measMT 1−  as the initial condition and the heat flux values from step 1 as its transient boundary conditions. The temperatures at these r time steps are calculated ( calcrM calc M calc M TTT ++ ,,, 1 L ). 3. The objective function is then evaluated using the heat fluxes from step 1, the calculated temperatures from step 2, and the known temperatures from the measurement ( measrM meas M meas M TTT ++ ,,, 1 L ). The new form of the objective function becomes: ( )∑ ∑∑ = + = + = ⎟⎠ ⎞⎜⎝ ⎛ +−= J j rM Mi i j calci j rM Mi measi j qTTf 1 22,, α)()(q  ( 2.15)    30    4. The algorithm is repeated until we reach a desired low value for the objective function, or after a certain number of iterations. The solution vector will be the spatial components of heat fluxes for the r time steps. 5. The values of qM, as obtained from step 4, are used, and the values for the rest of the heat fluxes (qM+1, …, qM+r) are discarded. 6. The above procedure is repeated (M = M+1) as you move to the next time step. In theory, it is possible to use multiple sensors instead of multiple future time steps to improve the stability of the method. However, this idea is not practical since adding thermocouples to the sample is not only more expensive, but also distorts the temperature field. For a detailed discussion of the effect of thermocouple hole on the thermal field, see References [25,97-101]. Using future time steps introduces a bias in the solution. This bias is normally observed in the form of rounding the spikes in oscillatory boundary condition histories, and an early prediction of these oscillations [35]. To obtain an appropriate level of regularization, the number of future time steps (or more accurately, the size of the look-ahead time window, i.e. the product of the number of future time steps and time step size) and the value of the regularization parameter must be chosen with respect to the errors involved in the temperature readings. The residual principle [36,102] can be used to determine these parameters based on the accuracy of thermocouples in the relative temperature range. In this method, the number of future time steps is increased until the RMS error of the estimation falls below the standard deviation of noise in the measured temperatures. Choosing proper values for the regularization parameter and the number of future time steps is a very challenging task, and although there are some methods to find the range of appropriate values, the fine tuning still remains more like an art rather than an exact science. Basically, there are five main approaches to choose a close to optimal value for these parameters [103]. They are the maximum likelihood (ML), the ordinary cross-validation (OCV), the generalized cross-validation (GCV), the L-curve, and the discrepancy principle (DP) methods. All these methods require some kind of knowledge about the solution characteristics. One important piece of information is the level of errors in the solution domain. Luckily, the thermocouples that are normally used in experiments are well documented, and their accuracy levels are well investigated and understood. In our experiments, we are using type K    31    thermocouples that have an error level of ±2.5 °C in temperatures below 333 °C, and an error level of ±0.75% of the measured temperatures between 333 °C and 1200 °C. Among these five methods, the discrepancy principle (DP) method is known to produce the best estimate of the regularization parameter for inverse heat conduction problems [103]. This method requires that the problem be solved so that the residual norm is the same as the norm of errors in the measurements, denoted by δ: ට∑ (ࢀ࢓ଙതതതത − ࢀࢉ࢏ )૛ࡸ࢏ୀ૚ = ࢾ     ( 2.16) When the values of the regularization parameter and the number of future time steps for one case are obtained, the same values can be normally used for similar problems. 2.2.2 Gradient Based Calculation of Heat Flux The heat flux and temperature vectors are TN ][ 21 qqqq K=      ( 2.17)                                      TiJ iii qqq ][ 21 K=q       ( 2.18)                             TiLm i m i m i m TTT ][ 21 K=T       ( 2.19) Ti Lc i c i c i c TTT ][ 21 K=T       ( 2.20) where L is the number of measurement points, J the number of heat flux components that can be determined for the flux space distribution on a surface. It should be noted that J must be less than or equal to L, the number of measurement points.  It should also be noted that the dimensions of the heat flux vector q i at each step are 1×J while the total heat flux vector q is J×N as it includes the data in N steps; and the temperature vector T i at each step is 1×L. Also, the temperature Tk is only determined or affected by the heat fluxes qm where m ≤ k.  Mathematically, we may express Tk as an implicit function of the heat flux:   ),,,( 21 kk c f qqqT K=                          ( 2.21) or in a successive form as:    32    ),( ),( ),( ),( 101 212 121 1 qTT qTT qTT qTT cc cc kk c k c kk c k c f f f f = = = = −−− − M                                                    ( 2.22) and the following equation is also valid   )()()( **222 2 *11 1 1 * kk k k ccck c k c qqq Tqq q Tqq q TTT − ∂ ∂ ++− ∂ ∂ +− ∂ ∂ += K      ( 2.23) The values with a ‘*’ superscript in the above equation may be considered as initial guess values ultimately lead to the temperature *kcT . Now we define the first derivative of temperature icT  with respect to heat flux q i as the sensitivity matrix: ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎣ ⎡ = ∂ ∂ = )()()( )()()( )()()( 21 22221 11211 iaiaia iaiaia iaiaia LJLL J J i i ci K MOMM K K q TX      ( 2.24) i s i cr rs q Tia ∂ ∂ =)( where i=1, 2…N, r =1, 2…L and s=1, 2…J. The sensitivity matrix Xi is an L×J matrix. The optimality of the objective function may be obtained by letting 0/ =∂∂ qf , and we get the following set of equations (note that q∂∂ /f should be done with respect to each component qi, with i=1, 2…N):  ( ) Nj T jN i i c i m T j i cjj N i j i c T j i c jjjjjj ,,2,1 )( * 1 ** 1 *** K= −−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ =− ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∑∑ = = = == qTT q qqI q T q T qqqqqq αα   ( 2.25) where *jq is the initial guess of heat fluxes, and Tci*  is the calculated temperature vector with the initial guess values.    33    Recalling equations (2.20) and (2.21), equation (2.22) may be rearranged and written in the following form: **))(( ** qTXqqIXX qqqq αα −Δ=−+== TT     ( 2.26) where X is labeled as the total sensitivity matrix for a multi-dimensional problem and has the following form: ⎥⎥ ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎢ ⎣ ⎡ = 12 12 1 0 00 000 XXX XX X X K OMM N       ( 2.27) and ( )TN*cNm*cm*cm TTTTTTT −−−=Δ K2211      ( 2.28) Note that the dimension of matrix X is (L×N)×(J×N) and ΔT has dimensions of (L×N). Also, note that the calculations in equation (2.23) may easily be done in the time domain and no function specification for iq is needed. If the total sensitivity is known, no iteration is required to get a final solution. 2.2.3 Convergence Criteria In each time step, the iterative procedure is used until the inversely predicted temperature Tc converges to the measured temperature Tm. Convergence criteria used to define the acceptance of the predicted temperature are based on an error norm defined by: Error-norm n =|| TΔ n||      ( 2.29) Two convergence criteria for ending the iteration process at each time step are used: Error-norm n ≤ δT      ( 2.30) or ε  norm-Error norm-Error-norm-Error n n1n ≤ +      ( 2.31) The values of δT and ε depend on the measurement error level. The rationale behind using absolute criteria is that while the norm at a given previous iteration is already very small, the relative norm criterion is still not satisfied in the last iteration.        2.2.4 in ve techn temp differ Meth the c subre movi 2.2.5 tries deter 1. Th the h 2. Th on th Whil FZM probl  Flux Zonin The boun ctor form. W ique is to erature mea  from one od (FZM) [ This appr ase where t gion. Howe ng boundary Fi  Flux Marc An altern to match the mines the he e heat flux eat flux on t e heat flux e prior node e in simple c , there are s ems are: g Method ( dary heat fl e need to a divide the surement. T to the next, 1]. oach is fairl he heat flux ver, some  heat flux, s gure  2.4. FZM hing Metho ative appro  movement at flux distr at each give he neighbou on the other  at the sever ases, and m everal serio FZM) ux, which is ssign this d  target sur he heat flux  as shown i y easy to u  is nearly c researchers uch as the c  Model for I d (FMM) ach to the F  of the heat ibution in ea n node at th ring upstrea  nodes at th al previous ostly in the us problems  the output iscretized v face into s es for each z n Figure  2. nderstand an onstant or have recog ase with the nverse Calcul ZM is the  flux on the ch region u e current ti m node at th e current tim time steps. oretical testi  that hinder of the deve ector form t everal zone one will be 4. This app d implemen does not dr nized it as  moving wa ation of the B Flux March  boundary. sing the foll me step is g e previous e step is ge ng, this met  its applicat loped inver o the target s, each co  constant fo roach is cal t. It is a ve amatically c inappropria terfront [1]. oundary Hea ing Method In its origin owing algor enerally ass time step. nerally assu hod may pe ion in real c se algorithm  surface. A rresponding r that zone, led the Flux ry good sol hange spati te in the c   t Flux  (FMM) [1 al form, thi ithm: umed to be med to be t rform better ases. Some 34  , will be common  to one and may  Zoning ution for ally in a ase of a ], which s method  equal to he value  than the  of these    35    a) The FMM heavily relies on the flow direction on the boundary. In three-dimensional cases, and especially in the cases with moving plates and multiple impinging jets in each row and multiple successive jet rows, it is increasingly hard to determine the upstream and downstream directions. The flow pattern is very complex, with stagnation zones in the area between two impinging jets. The shape and location of these areas are also irregular [81]. b) In order to obtain accurate results, the exact value of the water velocity must be known, since it is the speed at which the nodal heat flux values move from one node to the other. Again, there is currently no accurate way to predict these speeds, and while recent studies [32] have started looking in more details at the velocity of the water propagation, we are still far from having a robust and useful correlation for the water velocity, especially in cases with multiple jets and a moving surface. This is especially important when one notices that the “step mismatch”, i.e. moving of the heat flux from one node to the other at a speed higher (overstepping) or lower (understepping) than the real velocity, may produce lower quality results than the FZM [1]. c) One of the main characteristics of the heat transfer in the jet impingement cooling is the sharp heat flux peaks in the impingement zone. The FMM is, in nature, an interpolating and smoothing scheme, which tends to change the sudden behaviours into gradual changes. This has caused a dampening of the heat flux peak values, as well as a virtual negative heat flux before the real sharp increase [1]. Adjusting the waterfront speed may reduce the magnitude of this problem, but then it will be another parameter that is needed to be addressed and will increase the complexity of the program, and make it harder to be used as an off-the-shelf code. d) In the FMM, the following equation is used to find the time step at which the heat fluxes should move to the next node and the solution should be repeated [1]: ∆ݐ௜ = ௗೕ௩೔ೕ       ( 2.32) where Δti is the time step size in the ith step, dj is the jth spatial distance between thermocouples, and vij is the water moving speed over this distance in that time step. So in the cases of higher water moving speed, the denominator of the above equation becomes large, resulting in a very small time step size. This will result in either a large computational cost, or in many cases, to the instability of the gradient-based inverse algorithm, as will be discussed later in Section  2.3.4.    36    e) It is very crucial to notice that the problem with the FZM can be mitigated if smaller time steps are used. By refining the temporal resolution, the assumption that the heat flux is not moving in each smaller time step, will not significantly reduce the accuracy. Fortunately, the data provided by the data acquisition system is in very small intervals, so the limitation will not be caused by the experiments, but by the failure of the gradient-based inverse solvers to deal with smaller time steps. The remedy may be using inverse solvers that are not gradient-based. This will be studied in the next chapter. 2.2.6 Different Techniques for Calculating Sensitivities The form of the governing differential equation for both the temperature T (x, t) and the sensitivity coefficient X (x, t) is the same [35] and, therefore, the same finite element program may be used to calculate them. While this is efficient from the programming point of view, it may not be practical, especially when the temperature time history is somewhat long and the density of mesh is high, and also because of the fact that not all the coefficients at all points are needed to get the heat flux q. An alternative way is that the sensitivity matrices at each time step for the measurement points be calculated. Such a procedure would still require intensive calculation time and cost when the number of total time steps considered and/or the number of heat flux components are reasonably large. A perturbation algorithm [8] was used to obtain the sensitivity matrices. First, a given value is assumed for all components of the heat flux vector as input in the direct heat transfer calculation to get the temperature distribution for the given future steps at each thermocouple location. Then, one of the components of the heat flux vector is increased by a reasonable amount, such as 10%, to obtain new temperatures. While a perturbation of 10% looks larger than normal, we should notice that if the perturbation is small, the change in inside temperatures will be very small, due to the lagging and dampening of boundary changes in internal points, and the resulting sensitivity will be very small as well, and sometimes very close to zero. This can result in divergence or “division-by-zero” errors. The ratios of temperature difference at each thermocouple location to the applied increase in heat flux component are the sensitivity coefficients. Such perturbation is repeated for each component of the heat flux until all the sensitivity matrix components are obtained.    37    One important consideration in calculating the sensitivity is the nonlinearity. The whole sensitivity matrix is independent of the heat flux only if the thermal properties of the material are constant with temperature. For most materials, the thermophysical properties are temperature dependent.  In such cases, all properties should be updated at the beginning of each time step, which is time consuming, especially for large size models. Moreover, such changes in properties would not be very large and would not significantly change the magnitude of the sensitivity coefficients.  Also, updating the material properties at the beginning of each time step would be based on the temperatures Tk* obtained from the initially given values of the heat flux q*, which is essentially an approximation.  So, we may opt for updating the sensitivity matrix every M steps (in our numerical experiments, M=10). The results obtained under this assumption are very close to those obtained by updating the values at each step, so the assumption is justified. Another approach was also investigated in this research to replace the normal perturbation method in calculating the sensitivity coefficients. The suggested new approach is called the complex step method, and is basically based on a complex number, instead of a real number, perturbation. To introduce this method, we first write the Taylor series expansion of the real valued function f about both a real )( hx +  and a complex )( ihx + numbers. L+′′′+′′+′+=+ !3 )( !2 )()()()( 32 xfhxfhxfhxfhxf  ( 2.33) L+′′′−′′−′+=+ !3 )( !2 )()()()( 32 xfihxfhxfihxfihxf  ( 2.34) Based on these expansions, the following three approximations of the first derivative of f with respect to x is possible: L−′′′−′′−−+=′ !3 )( !2 )()()()( 2 xfhxfh h xfhxfxf  ( 2.35) L−′′′−−−+=′ !3 )( 2 )()()( 2 xfh h hxfhxfxf  ( 2.36) [ ] L+′′′++=′ !3 )()(Im)( 2 xfh h ihxfxf  ( 2.37) The first one is a normal forward difference scheme, which is similar to the perturbation scheme that was previously described, the second expression is a central difference scheme, and the third one that uses the complex number perturbation is the basis for our complex step method.    38    When compared with the forward difference scheme, the complex step method is of higher order (2 instead of 1), and it does not involve a difference operation, so we can choose small steps, with no loss of accuracy due to subtractive cancellation. In comparison with the central difference algorithm, the complex step again does not involve a difference operation, so we can choose small steps, with no loss of accuracy. Also, unlike the central difference scheme, the complex step method does not need to evaluate the function at a third point (x-h). In our inverse problem, the sensitivity matrix, X, is calculated using a perturbation method, which is similar to the forward difference algorithm: q qTqqT q TX Δ −Δ+ = ∂ ∂ = )()( **  ( 2.38) where q*  is the initial guess for the value of the heat flux, and Δq is the perturbation, which is typically about 10% of the value of heat flux. Using the complex step idea, the sensitivity will be calculated as [ ] q qiqT q TX Δ Δ+ = ∂ ∂ = )(Im *  ( 2.39) This approach was tested in our problem. We found that unless the problem is highly nonlinear (more than what heat conduction problems normally are), there is no tangible difference between the first order perturbation method, and the second order complex step algorithm. This approach also could not solve the problem of gradient based inverse heat conduction algorithms, such as time step size limitation, or sensitivity to the measurement errors. 2.3 Results and Discussion In this section, a few test cases are introduced and developed algorithms are tested in their capability of solving one-, two-, and three-dimensional problems. The effect of noise in the domain on the performance of the inverse algorithm is studied. Also, the effects of using regularization parameters, future time steps, and different time step sizes are also investigated. Finally, 1D, 2D, and 3D inverse algorithms are compared. 2.3.1 Test Cases Five test cases are introduced. They include one-, two-, and three-dimensional heat conduction problems, in transient or steady conditions, and both linear and nonlinear in nature.       Notic there Test input L = therm The h Figur The store meas Test to res exper e that these . Case I: 1D In this ca  heat flux at 1. The prob al conducti eat flux his e  2.5 shows direct probl d. The goa urements) a Case II: 2D In this tra emble the o iments, in o  test cases ar Transient P se, the triang  the left end lem was on vity varying ݇ tory is given (q the geometr Figure  2.5. P em is solve l is to m nd those tha  Transient nsient 2D p ne in water rder to dete e also going roblem ular heat flu  of the samp ce solved w  with the tem = 0.82 + ܶ  in the follo ⎪⎪⎩ ⎪⎪⎨ ⎧ +− − = 0 44.1 24.0 0 ) t t t y of the pro roblem Geom d and the t inimize the t can be obta Problem roblem, the  cooling of rmine the s  to be used x history, p le. For simp ith a cons perature at (ݔ = ܮ) wing form: ≥ <≤ <≤ < 44.1 184.0 024.0 24.0 t t t t blem, and it etry and Bou emperatures  difference ined by app heat flux at steel strips urface heat in the next opularized licity, the p tant k = 1  x = L as: 44. 84. s boundary c ndary Condit  at the righ  between lying an arb the top surf on the run-o flux, or the chapter, and by Beck et a arameters a (linear case onditions.  ions, Test Ca t end (x = these temp itrary input ace (see Fig ut table in  heat transf  will not be l. [35] is us re first take ), and then se I L) are obta eratures (s  heat flux. ure  2.6(a)) i a steel mill. er coefficien 39   repeated ed as the n as ρc = with the ( 2.40) ( 2.41) ined and imulated s chosen  In these t during       the w the p used therm diam junct the t cond adiab and t the st insid to ob Test 3, 1, subje ater cooling late and 1.0 to obtain th ocouples. A eter of 1.6 ion of the th op surface ition of the atic. The le he thermal c eel strips th e the object, tain the tran Case III: 3D The third and 10, in cted to a spe  process, th mm below t e boundary s shown in mm that is ermocouple of the plate top surface ft boundary onductivity at are used i  which is th sient heat flu (a) Figure  2.6  Steady Pr  test case is the x, y, an cified temp ermocouple he top surfa  conditions  Figure  2.6  drilled fro  is fixed to . The probl  is prescrib  is axisymm , k, is 44.5 W n our contro e assumed p x profile at . Test Case II oblem a 3D steady d z direction erature boun s are placed ce, as depic on the surfa (b), the ther m the botto the end surf em is solve ed heat flux etric. The d mK . Thes lled cooling osition of a  the top surf  ; (a) Model, (  heat condu s, respectiv dary condit  on the opp ted in Figur ce, based o mocouple is m surface ace of the h d as an ax . The other ensity, ρ, i e are very c  experimen thermocoup ace. b) Problem D ction proble ely. The to ion that vari osite side o e  2.6(b). Inv n the readi  located in of the plate ole, which i isymmetric  boundarie s 7850 mkg lose to the p t. Results ar le. Inverse a (b) escription [1] m in a slab p surface o es as ்ܶ௢௣ = f the plate, erse analys ngs of these a blind hol . The mea s about 1 m case. The b s are assum 3 , Cp is 47 hysical prop e obtained a nalysis is c  , with dimen f the slab (y 25ݔ + 10 40  or within is is then  internal e, with a surement m below oundary ed to be 5 kgKJ , erties of t a point onducted sions of  = 1) is 0ݖ + 10.       The o case. elem this p with Test steel thick radiu elem Figur resem of th out t therm therm direc the m limita ther bound  Figure  2.7 ent modeling Unlike th roblem, the respect to th Case IV: 3D A block c strip. The l ness are 12.7 s 0.5 mm a ents are us e  2.8(a) is a The boun ble the one e nine therm able with ocouple lo ocouple lo t finite elem esh was ref tions, we a aries are adi  displays th . Figu e previous te re are three e x and y co  Transient ontaining ni ength of the  mm and 6 nd height 5 ed to discr  close-up vi dary condit  in water co ocouple loc two rows o cations [91 cations insi ent code has ined until th re not prese abatic. The e temperatu re  2.7. Tempe st cases wh sensors loca ordinates (y  Problem – ne thermoco  block is 1 .65 mm, resp .65 mm is etize the d ew of one of ion of the t oling of stee ations. It is f staggered ]. Figure  2 de the plate  been valid e changes i nting them h material pr re distribut rature Distri ere there wa ted inside th  = 0.5, x = 1  Nine Therm uples is mo 14.3 mm (9 ectively. To taken out o omain. Figu  the TC hol op surface l strips. Figu  very simila  circular j .9(b) is the , obtained ated, and gri n the results ere. The ot operties are ion inside bution Inside s only one d e domain. T .5), and at t ocouples deled for ea  sections of  model the f the block. re  2.8(a) es. is prescribe re  2.9(a) sh r to an actu ets, imping  temperatu from direct d independe  were less t her boundar the same as the slab, ob the Slab ata point in hey are loc he z location ch pass of w  each 12.7 m thermocoup  Isoparamet shows the d heat flux ows the app al heat flux ing on the re history  finite elem nce studies han 5%. Ho ies are assu  in the prev tained by 3   the solution ated at the c s of 2, 5, an ater jet coo m). The w le hole, a cy ric eight-no whole dom  which is c lied heat flu happening o  third and at five of ent simulat  were perfor wever, due med to be a 41  ious test D finite  field, in enterline d 8. ling of a idth and linder of de brick ain, and hosen to x at five n a run-  seventh the nine ion. The med, i.e.  to space diabatic.       The d to be temp Thes exper of a top su Figu ensity, ρ, is  constant an erature (non e are the p iment. Resu thermocoup rface. re  2.8. (a) Th  7850 3mkg d equal to 4 linear probl k = 5.60 hysical prop lts are obtai le. Inverse a e Whole Bloc , Cp is 475 0 W/m.°C em) as − 0384.071 erties of th ned at the to nalysis is c (a) k Consisting o J/KgK, and (linear prob mWT× /9 e steel stri p of the cyl onducted to f Nine Therm the Bottom  the thermal lem), and la C°⋅ ps that are indrical hol  obtain the ocouples; (b)   conductivit ter changed used in ou e, which is transient he (b)  A Close View y, k, is first  to be depe r controlled the assumed at flux prof  of the TC H 42  assumed nding on ( 2.42)  cooling  position ile at the ole from       Figur Test show boun by im impin the to the b histor study analy which  e  2.9. (a) The Case V: 3D Another r n in Figure dary conditi pinging jet ging jets [9 p surface of oundary he ies for the n  the effect zer, especia  will be dis (a)  Applied Hea  Transient ow of therm   2.10, incre on is set to r s. However, 1], i.e. ther  each line is at fluxes on ew line of of the num lly to inve cussed in th t Flux on the Problem – E ocouples i asing the w esemble the  in this test e are two lin  set to resem  the top sur added therm ber of therm stigate the e next chapt  Top Surface; Analysis ighteen Th s added to idth of the  heat fluxes case, each th es of nine t ble the wat face of five ocouples. T ocouples o effect of m er. (b) The Therm ermocoup the previous  block to 2  occurring in ermocoupl hermocoup er jet coolin  of the ther he main rea n the perfo ulti-criteria (b) ocouple Rea les  test case i 5.4 mm. A  the water c e line is und les, and the g heat fluxe mocouples son for usin rmance of t objective fu dings Used fo n the y-dire gain the top ooling of st er a differen heat flux ap s. Figure  2. and the tem g this test c he designed nction form 43   r Inverse ction, as  surface eel strips t line of plied on 11 shows perature ase is to  inverse ulation,       Figur 2.3.2 calcu temp wher exact Figu e  2.11. (a) The  Noisy Dom To study lated bound eratures with e Tm is the v  temperatur re  2.10. Test  Applied Hea Anal ains the effects ary conditio  the follow irtual intern e, Texact; r i Case V; A Bl t Flux on the ysis; Second L  of measure ns, random ing equation TT exm = al temperatu s a normally ock Consistin Top Surface; ine of Therm  ment errors  errors are : ract ⋅+σ re that is u  distributed g of Eighteen  (b) The Ther ocouples, Tes  in interna imposed on sed in the in  random va Thermocoup mocouple Re t Case V l temperatu to the calc verse calcu riable with le Zones adings Used f res on the ulated exact lations inste  zero mean 44    or Inverse inversely  internal ( 2.43) ad of the and unit    45    standard deviation; and σ is the standard deviation. In this work, the maximum additive random error is ± 3 °C. There are several ways to make an inverse algorithm more stable when dealing with noisy data. For example, Gadala and Xu [57] have shown that increasing the number of “future time steps” in their sequential function specification algorithm results in greater stability. They have also demonstrated that increasing the regularization parameter, α, improves the ability of the algorithm to handle noisy data. However, these approaches were also shown to greatly increase the required number of iterations, and in many cases the solution may diverge. 2.3.3 Effects of the Regularization Parameter and Future Time Steps Both the regularization method, proposed by Tikhonov, and the future time steps concept in sequential function specification, proposed by Beck, have the same purpose of stabilizing the solution of the inverse heat conduction problem. In this research, these two methods have been combined, and their effects cannot be separated from each other. This approach is based on the idea of a regularized function specification method, and is shown to be more effective and more computationally efficient than using just one of these methods [104]. Notice that the future time steps concept is only applicable to the transient problems, so in test case III, which is a steady-state problem, only the Tikhonov regularization can be used. For the other test cases, while the exact value of the regularization parameter and the number of future time steps vary slightly, the same trends are always observed. We chose test case IV to show the effect of using a regularization parameter in Figure  2.12, and the effect of using more future time steps in Figure  2.13. As seen in Figure  2.12, increasing the value of the regularization coefficient results in a smoother profile for the predicted heat flux. Increasing the number of future steps has the same effect, as seen in Figure  2.13. These two effects can be combined. However, there are two main problems in using these techniques. First of all, the computational expense increases significantly when a higher number of future time steps are used. The other problem, which is more significant, is that there is a limit to the improvement that can be obtained by these two techniques. Increasing either of these parameters after a certain limit can cause divergence of the whole inverse solver.        Figu of re  2.12. Effect Error = 4.63e  of Regulariz 5); (b) α = 1e- ation Parame 11 (L2 Norm o (a) (b) (c) ter; 4 Future f Error = 2.5 Steps; Error l 6e5); (c) α = 1    evel = ±3°C; e-10 (L2 Norm (a) α = 1e-12 (  of Error = 1 46  L2 Norm .18e5).       Figure  2.13. Effe 4.63e5); ct of Future T (b) 6 Steps (L ime Steps; α 2 Norm of Err (a) (b) (c) = 1e-11; Erro or = 1.72e5); r Level = ±3°  (c) 8 Steps (L    C; (a) 4 Steps 2 Norm of Er  (L2 Norm of ror = 1.34e5) 47  Error =       2.3.4 unsta cond size, oscill size, may cooli the d to ha gradi to ov discu 2.3.5 temp  Effect of T It is well ble when th itioning of t huge non- ations that o the whole pr  Figure  2.14.  This time cause us to ng of steel p istance betw ve one data ent-based in ercome this ssed in mor  Compariso One of th erature grad ime Step Si  known tha e time step he sensitivit physical os ccurred in ogram dive Threshold of  step limita  miss some roducts on een two jet  point for verse solver  problem i e details in t n of 1D, 2D e reasons ients betwe ze t gradient b  size becom y matrix in cillations ar the solution rges. Divergence in tion can res of the main the run-out  lines is 10 each jet im s in problem s to use inv he next two  and 3D Al for using a en different ased algori es smaller smaller tim e produced  of Test Cas  the Solution ult in an un  features o table, if the cm, we need pingement m s with the erse solver chapters. gorithms  3D inverse thermocoup thms for so than a cert e steps. Bas  in the re e II). If we of Test Case I acceptably f the heat f sample is m  time steps oment. Th thermophys s that are n  solver ins le locations lving invers ain limit. Th ically, by re sults (see F  continue re  I Due to Sma low tempor lux profile. oving at a of less than is is quite ical properti ot gradient tead of a 2 . These gra e problems is is due t ducing the t igure  2.14 ducing the t ll Time Step S al resolutio For exampl speed of 10  0.01 s, just a low valu es of steel. -based. This D model [ dients can e 48   become o the ill- ime step  for the ime step ize n, which e, in the m/s, and  in order e for the One way  will be 1] is the xist in a    49    jet line between several impinging jets, or between two successive rows of jets. The simple 1D model assumes that the heat transfer is only happening in the normal direction to the plate. The 2D model also considers the heat flow between two neighbouring thermocouples. However, the 3D model is capable of modeling the heat flow in all directions, and is supposed to be a more physical simulation for the problem. Due to different modeling assumptions, especially different treatment of the boundary conditions, the values of heat fluxes that will be obtained from the same set of temperature readings will be different for each of these models. For example, because in the 1D and 2D models, one or two sides of the domain are assumed adiabatic, i.e. there is no heat exchange through them, the heat flux in the cooling process needs to cool down a smaller thermal mass, so the values that will be obtained in these cases for the peak heat flux will be lower. To investigate this, a set of known boundary heat fluxes are applied to the top surface of test case V, and the problem is directly simulated. The temperatures at the internal sensor locations are calculated and stored. Then the 1D, 2D, and 3D inverse algorithms are employed to use the stored temperatures and recover the missing boundary conditions. The time step size in the initial direct problem is selected to be smaller than the one used in inverse algorithms, in order to avoid the so-called “inverse crime” [105]. Figure  2.15 shows the result of the 1D, 2D, and 3D algorithms, in addition to the known applied heat flux. Note that due to the ill-posed nature of the inverse problem, even in the case of the 3D algorithm, some discrepancy between the expected heat flux and the results is observed. However, the 1D and 2D algorithms show a significant dampening in the predicted value of the heat flux. As mentioned previously, these algorithms study each thermocouple location in complete isolation (1D), or just consider the heat transfer in one lateral direction to the neighboring thermocouple(s) (2D). This means that a lower amount of thermal mass needs to be cooled. Thus, the predicted heat fluxes are smaller. Also, some of the details of the heat flux profiles, e.g. the shoulder before the first rise, and the small shoulder after it are not sensed in the 1D and 2D models. These effects are caused by the cooling of the neighboring thermocouple locations, and are not sensed in these algorithms.       Figur 2.4 C probl missi appli meth used. heat c  e  2.15. Result onclusion In this ch em were int ng boundar cations in t od is very se  In the next onduction p s of the 1D, 2  apter, equa roduced, an y condition he controlle nsitive to m  chapter, we roblem with D, and 3D Inv D tions for th d a gradient s based on d cooling o easurement  will try to out the sho  erse Algorith irect 3D Pro e direct and -based inve the reading f steel on  errors, and  find algorit rtcomings o ms in Predict blem  inverse sim rse solver w s of interna a run-out t becomes un hms that ar f the gradien ing a Known ulation of as extended l thermoco able. The re stable when e capable o t-based met Heat Flux Ap the heat co  to 3D to o uples, espec sults show  small time f solving th hods. 50   plied to a nduction btain the ially for  that the steps are e inverse    51    3. Gradient-Free Methods of Solving the Inverse Heat Conduction Problem As shown in Chapter-2, the gradient-based methods for solving the inverse heat conduction problem suffer from some shortcomings, such as instability in the presence of noise and in the smaller time step sizes. This can hinder their application in real-world problems, such as in the characterization of the jet impingement boiling heat transfer on the run-out table in a steel mill, where huge changes in the boundary conditions happen in short intervals, and the thermocouple readings are always erroneous. In this chapter, we try to investigate some gradient- free optimization methods and utilize them in the inverse heat conduction problem to mitigate these shortcomings, and to find a more suitable inverse solver for our applications. Three major techniques that are studied are: the artificial neural networks (ANN), the genetic algorithm (GA), and the particle swarm optimization (PSO). First, we introduce the basic algorithm and its formulation. Then, we test it in the solution of an inverse heat conduction problem, and try to optimize the performance parameters associated with that method to make it both more efficient, i.e. requiring lower computational expense, and more effective in handling noisy data. Finally, comparisons are made between different methods and their variations, and their advantages and disadvantages are studied. Also, in the last part of this chapter, a general modification is introduced that can be applied to the stochastical methods in inverse problems to accelerate their convergence. 3.1. Inverse Formulation The boundary inverse heat conduction problem can be formulated as an optimization problem in which we try to minimize the norm of difference between the experimentally measured temperatures and the calculated temperatures obtained from a direct solution of the problem with some guessed boundary conditions. In order to dampen the oscillations in the solution, and make the algorithm more stable in dealing with noisy measurement data, it is very common to include some kind of regularization terms that make the problem more stable, especially in the case of noisy domains. As mentioned in the previous chapter, based on previous experience [76], as well as our own trial of different regularization terms, we chose to use the heat flux values (zeroth-order regularization). The objective function will then be    52     ( 3.1) where and are the values of expected (measured) and calculated temperatures at the ith time step and the jth spatial location, respectively; α is the regularization coefficient; and  is the boundary heat flux vector at the ith time step and the jth spatial location, N is the total number of time steps, and J is the number of spatial components of temperature vectors. 3.2. Artificial Neural Networks Artificial Neural Networks (ANN) are motivated by the efficiency of brain in performing computations. These networks are made of a large number of processing units (neurons) that are interconnected through weighted connections, similar to synapses in the brain. In order for the network to perform the expected tasks, it should first go through a “learning” process. There are two main categories of learning: supervised, or unsupervised. In supervised learning, the network learning is achieved by practicing on pre-designed training sets, while in unsupervised learning, the network is presented with a set of patterns, and learns to group these patterns into certain categories. The supervised learning is useful in function fitting and prediction, while unsupervised learning is more applicable to pattern recognition and data clustering. Since the learning process in our application is a supervised one, we focus on this type of learning process. While there are several major classes of neural networks, in this research we studied only two of them, which are introduced in this section. 3.2.1. Feedforward Multilayer Perceptrons (FMLP) In a feedforward network, the nodes are arranged in layers, starting from the input layer, and ending with the output layer. In between these two layers, a set of layers called hidden layers, are present, with the nodes in each layer connected to the ones in the next layer through some unidirectional paths. See Figure  3.1 for a presentation of the topology. It is common to have a different number of elements in the input and output vectors. These vectors can occur either concurrently (order is not important), or sequentially (order is important). In inverse heat transfer applications, normally the order of elements is important, so sequential vectors are used. ( )∑ ∑∑ = == ⎟⎠ ⎞⎜⎝ ⎛ +−= J j N i i j calci j N i measi j qTTf 1 1 22, 1 , α)()(q ,i meas jT ,i calc jT i jq       3.2.2 See F given in wh with funct can b traini . Radial Ba The basic R igure  3.2 fo  by ich x is the σi as its unit ion, given a These net e designed ng set shoul F sis Function BFN inclu r a visual re input vector  width param s works norm  and trained d be availab igure  3.1. A F  Networks des only an presentation ( ) = ii rf x , and vi is th eter. The m ( ) = expif x ally require  much fast le in the beg eedforward N  (RBFN) input layer, . The form ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − i i σ vx  e vector den ost popular ⎠ ⎞ ⎜⎜⎝ ⎛ −− 2 2 2 i i σ vx  more neuro er. Howeve inning of th etwork Topo  a single hi of the radia oting the c  form of thi ⎟⎟  ns than the r, in order e process.  logy dden layer, l basis funct enter of the s function is  feedforwar to have go and an outp ion can be g receptive fie  the Gaussia d networks, od performa 53  ut layer. enerally ( 3.2) ld unit fi n kernel ( 3.3)  but they nce, the    54     Figure  3.2. An RBF Network Topology 3.2.3. Implementation in Inverse Heat Conduction Problem In order to use the artificial neural networks in the inverse heat conduction problem, we first started with a direct heat conduction finite element code, and applied several sets of heat fluxes in the boundary. The resulting temperatures in locations inside the domain, which correspond to the thermocouple locations in the experiments, were obtained. The neural network was then trained using the internal temperature history as an input, and the corresponding applied heat flux as the target. The assumption was that this way, the neural network should be able to act as an inverse analysis tool, and given a set of measured thermocouple readings, be able to reproduce the heat fluxes. The obtained results were, however, far from satisfactory. It seemed that the relationship between the actual values of temperatures and heat fluxes is a complicated one, which is very hard for the neural networks to understand and simulate, at least when using a reasonably small number of layers. Thus, we decided to reformulate the problem, and use the change in the temperature in each time step as the input. In this formulation, neural networks performed much better, and a good quality was achieved in the solution in a reasonable amount of time. Further investigations showed that if the time step size is varying, we can use a derivative of temperature with respect to the heat flux as the input, i.e. divide the temperature change by the time step size. The results were again satisfactory; however, more bookkeeping is needed, which complicates the implementation and makes the algorithm more prone to coding errors. This  f f f Output Nodes Hidden layer (RBF kernel) Input Nodes    55    practice is not normally recommended, unless it can result in a considerable reduction in the solution time. 3.4. Genetic Algorithm (GA) Genetic algorithm is probably the most popular stochastic optimization method. It is also widely used in many heat transfer applications, including inverse heat transfer analysis [106]. Figure  3.3 shows a flowchart of the basic GA. Similar to PSO, GA starts its search from a randomly generated population. This population evolves over successive generations (iterations) by applying three major operations. The first operation is “Selection”, which mimics the principle of “Survival of the Fittest” in nature. It finds the members of the population with the best performance, and assigns them to generate the new members for future generations. This is basically a sort procedure based on the obtained values of the objective function. The number of elite members that are chosen to be the parents of the next generation is also an important parameter. Usually, a small fraction of the less fit solutions are also included in the selection, to increase the global capability of the search, and prevent a premature convergence. The second operator is called “Reproduction” or “Crossover”, which imitates mating and reproduction in biological populations. It propagates the good features of the parent generation into the offspring population. In numerical applications, this can be done in several ways. One way is to have each part of the array come from one parent. This is normally used in binary encoded algorithms. Another method that is more popular in real encoded algorithms is to use a weighted average of the parents to produce the children. The latter approach is used in this research. The last operator is “Mutation”, which allows for global search of the best features, by applying random changes in random members of the generation. This operation is crucial in avoiding the local minima traps. More details about the genetic algorithm may be found in references [107,108].        roule Figu Among t tte selection re  3.3. Flowc he many va , intermedia hart of a Gen riations of te crossove eral Impleme GAs, in th r, and unifo ntation of Ge is study, w rm high-rat  netic Algorith e use a rea e mutation m (GA) l encoded [107]. The c 56  GA with rossover    57    probability is 0.2, and the probability of adjustment mutation is 0.9. These settings were found to be the most effective based on our experience with this problem. A mutation rate of 0.9 may seem higher than normal. This is because we start the process with a random initial guess, which needs a higher global search capability. However, if smarter initial guessing is utilized, a lower rate of mutation may be more effective. Genes in the present application of GA consist of arrays of real numbers, with each number representing the value of the heat flux at a certain time step, or a spatial location. 3.5. Particle Swarm Optimization We start this section by giving a description of the basic concepts of the algorithm. Then a brief description of the three variations of the PSO algorithm that are used in this study is given. Finally we investigate some modifications in the PSO algorithm to make it a more robust and efficient solver of the inverse heat conduction problem. 3.5.1. Basic Concepts Particle swarm optimization (PSO) is a high-performance stochastical search algorithm that can also be used to solve inverse problems. The method is based on the social behavior of species in nature, e.g., a swarm of birds or a school of fish. It was originally developed by Eberhart and Kennedy in 1995 [68]. In the basic PSO algorithm, if a member of the swarm finds a desirable position, it will influence the traveling path of the rest of the swarm members. Every member searches in its vicinity, and not only learns from its own experience (obtained in the previous iterations), but also benefits from the experiences of the other members of the swarm, especially from the experience of the best performer. The original PSO algorithm includes the following components [109]: • Particle Position Vector x: For each particle, this vector stores its current location in the search domain. These are the values for which the value of the objective function is calculated, and the optimization problem is solved. • Particle Velocity Vector v: For every particle, this vector determines the magnitude and direction of change in the position of that particle in the next iteration. This is the factor that causes the particles to move around the search space.    58    • Best Solution of a Particle p: For each particle, this is the position that has produced the lowest value of the objective function (the best solution with the lowest error in our case). So if f is the objective function that is supposed to be minimized, i is the index for each particle, and m is the iteration counter, then: ( )( )si ms m i xfp ≤≤ = 0 minarg  ( 3.4) • Best Global Solution g: This is the best single position found by all particles of the swarm, i.e., the single p point that produces the lowest value for the objective function, among all the swarm members. In other words, if n is the swarm size, then: ( )( )sk nkms m xfg ≤≤≤≤ = 1,0 minarg  ( 3.5) The number of particles in the swarm (n) needs to be specified at the beginning. Fewer particles in the swarm results in lower computational effort in each iteration, but possibly a higher number of iterations is required to find the global optimum. On the other hand, a larger population will have a higher computational expense in each iteration, but will likely require less iterations to reach the global optimum point. Earlier studies have shown that a smaller population is normally preferred [110,111]. This was also observed in our study; however, its effect on the convergence speed seems to be insignificant. The steps involved in the basic PSO algorithm are detailed below [109]: 1. Randomly initialize the positions and velocities for all the particles in the swarm. 2. Evaluate the fitness of each swarm member (objective function value at each position point). 3. At iteration m, the velocity of the particle i, is updated as: ( ) ( )1 0 1 1 2 2m m m m m mi i i i iv c v c r p x c r g x+ = + − + −  ( 3.6) where mix and miv are the position and velocity of particle i at the m-th iteration, respectively; mip and mg are the best positions found up to now by this particle (local memory) and by the whole swarm (global memory) so far in the iterations, respectively; c0 is called the inertia coefficient or    59    the self-confidence parameter and is usually between zero and one; c1 and c2 are the acceleration coefficients that pull the particles toward the local and global best positions; and r1 and r2 are random vectors in the range of (0,1). The ratio between these three parameters controls the effect of the previous velocities and the trade-off between the global and local exploration capabilities. 4. Update the position of each particle using the updated velocity and assuming unit time: 1 1m m m i i ix x v + + = +  ( 3.7) 5. Repeat (2) – (4) until a convergence criterion (an acceptable fitness value or a certain maximum number of iterations) is satisfied. The first step of the above algorithm may be improved, if instead of using random numbers, we initialize the algorithm with some meaningful initial guess, i.e. a set of values close to the ones that we expect the final solution to be. We chose not to do so in this research, because we wanted to compare the efficiency of different algorithms, and for that reason, decided to let them perform with minimum supervision. On the other hand, efficiency of the gradient based algorithms is less affected by the values of the initial guess. However, in those methods one may fall into the local minima of the function. The iterations in this research were carried out until the best value of the objective function normalized with respect to the best value of the objective function in the first iteration was less than 10-6 or the average value of the best objective function became less than the variance of added artificial noise. There are some considerations that must be taken into account when updating the velocity of the particles (step 3 of the above algorithm). First, we need a value for the maximum velocity. A rule of thumb requires that, for a given dimension, the maximum velocity, max,iv , should be equal to one-half the range of possible values for the search space. For example, if the search space for a specific dimension is the interval [0, 100], we will take a maximum velocity of 50 for this dimension. If the velocity obtained from Equation ( 3.6) is higher than max,iv , then we will substitute the maximum velocity with 1+miv . The reason for having this maximum allowable velocity is to prevent the swarm from “explosion” (divergence). Another popular way of preventing divergence is a technique called “constriction”, which dynamically scales the velocity update [109]. The first method was used in the previous research by the authors [112].    60    However, further investigation showed that a better performance is obtained when combining the constriction technique while limiting the maximum velocity. In this research, the velocity updates are done using constriction and can be written as: ( ) ( )( )1 1 1 2 2m m m m m mi i i i iv K v c r p x c r g x+ = + − + −  ( 3.8) where K is the constriction factor, and is calculated as [109]: 2 2 2 4 K ϕ ϕ ϕ = − − −  ( 3.9) where φ = c1 + c2. Here, following the recommendations by Clerc [109], the initial values for c1 and c2 are set to 2.8 and 1.3, respectively. These values will be modified in subsequent iterations, as discussed below. As mentioned above, the relation between the self-confidence parameter, c0, and the acceleration coefficients, c1 and c2, determines the trade-off between the local and global search capabilities. When using the constriction concept, the constriction factor is responsible for this balance. As we progress in time through iterations, we get closer to the best value. Thus, a reduction in the value of the self-confidence parameter will limit the global exploration, and a more localized search will be performed. In this research, if the value of the best objective function is not changed by a certain number of iterations (10 iterations in our case), the value of K is multiplied by a number less than one (0.95 for our problems) to reduce it (i.e. oldnew KK 95.0= ). These numbers are mainly based on the authors’ experience, and the performance is not very sensitive to their exact values. Some other researchers have used a linearly decreasing function to make the search more localized after the first few iterations [110]. These techniques are called “dynamic adaptation”, and are very popular in recent implementations of PSO [113]. Also, in updating the positions, one can impose a lower and upper limit for the values, usually based on the physics of the problem. For example, in a heat transfer application, if we know that cooling is happening at the boundary, then the heat fluxes are going to be negative, and we can assign an upper limit of zero for these values. If the position values fall outside this range, several treatments are possible. In this research, we set the value to the limit that has been passed by the particle. Other ideas include substituting that particle with a randomly chosen    61    particle in the swarm, or penalizing this solution by increasing the value of the objective function. The above steps are illustrated in Figure  3.4. Figure  3.5 shows a flowchart of the whole process of basic PSO algorithm for each time step.  Figure  3.4.  Velocity and Position Updates in the Basic Particle Swarm Optimization Algorithm  current velocity m ix ip mg 1+m ix m iv 1+m iv new velocity particle memory effect best performer effect    62     Figure  3.5. Flowchart of the Basic Particle Swarm Optimization Procedure 3.5.2. Variations Unfortunately, the basic PSO algorithm may get trapped in a local minimum, which can result in a slow convergence rate, or even premature convergence, especially for complex problems with many local optima. Therefore, several variants of PSO have been developed to improve the performance of the basic algorithm [114]. Some variants try to add a chaotic acceleration factor to the position update equation, in order to prevent the algorithm from being Iter = 1; Iterations i = 1; Particles Randomly initialize positions (heat fluxes)x1 and velocities v1 Evaluate )( Iterixf  (Vector of objective function values) )()( ,, Iter ji Iter ji pfxf < Iter ji Iter ji xp ,, = )()( , Iter j Iter ji gfxf < Iter ji Iter j xg ,= i = n i=i+1 Iter = Iter + 1 Yes Yes No No Set )( 1pf and )( 1gf to very large numbers Iter=Itermax or f(gIter)<tol Yes No No Solution gIter Update Position 1, +Iter jix Update Velocity 1, +Iter jiv Yes j=1; spatial components j = J j=j+1 Yes No    63    trapped in local minima [115]. Others try to modify the velocity update equation to achieve this goal. One of these variants is called the Repulsive Particle Swarm Optimization (RPSO), and is based on the idea that repulsion between the particles can be effective in improving the global search capabilities and finding the global minimum [70,116]. The velocity update equation for RPSO is ( ) ( )1 0 1 1 2 2 3 3m m m m m mi i i i j i rv c v c r p x c r p x c r v+ = + − + − +  ( 3.10) where mjp is the best position of a randomly chosen particle among the swarm, c3 is an acceleration coefficient, r3 is a random vector in the range (0,1), and rv is a random velocity component. Here c2 is -1.43, and c3 is 0.5. These values are based on recommendations by Clerc [109]. Based on our own experience, minor changes in these values do not significantly impact the performance. The newly introduced third term on the right-hand side of Eq. ( 3.10), with a negative coefficient ( 2c ), causes repulsion between the particle and the best position of another randomly chosen particle. The third term’s role is to prevent the population from being trapped in a local minimum. The fourth term generates noise in the particle’s velocity in order to take the exploration to new areas in the search space. Once again, we are gradually decreasing the weight of the self-confidence parameter. Note that the third term on the right-hand side of Eq. ( 3.6), i.e., the tendency toward the global best position, is not included in a repulsive particle swarm algorithm in most of the literature. The repulsive particle swarm optimization technique does not benefit from the global best position found. A modification to RPSO that also uses the tendency towards the best global point is called the “Complete Repulsive Particle Swarm Optimization” or CRPSO [112]. The velocity update equation for CPRSO will be: ( ) ( ) ( )1 0 1 1 2 2 3 3 4 4m m m m m m m mi i i i i j i rv c v c r p x c r g x c r p x c r v+ = + − + − + − +  ( 3.11) In CRPSO, by having both an attraction toward the particle’s best performance, and a repulsion from the best performance of a random particle, we are trying to create a balance between the local and global search operations.    64    3.5.3. Elite Particle and Elite Velocity This modification has been proposed by Fourie and Groenwold [117] in structural optimization applications, and is borrowed from the genetic algorithm, where the gene that has the best characteristics never leaves the population. Here, we replace the particle with the worst value of objective function with the best global solution, g. Also, if a particle’s velocity caused an improvement in the best global solution during the mth iteration, it is allowed to continue with the same velocity, without being influenced by other particles. In other words, if mkv caused an improvement on g, then: 1 0 m k m m k k g x x g c v+ = = + ( 3.12) 3.6. Application in Inverse Heat Conduction As the first step in our study of these techniques, we investigate their capability in solving a general inverse heat conduction problem similar to the problem on the run-out table, such as the test cases introduced in the previous chapter. In order to save space and since the results of PSO and GA are very similar, we only present the graphs of some of the test cases. However, other test cases will be used later in this manuscript to study the different modifications in the original algorithm. We start by applying the artificial neural networks to the inverse heat conduction problem. This is different from GA and PSO, since those methods perform a stochastical search and are similar in many aspects, while the artificial neural networks are more like a correlation between the inputs and outputs. Figure  3.6 shows the result of the application of the radial basis function neural networks for the whole history of the heat fluxes on the runout table. Temperatures start at 700 ºC and go down to 176 ºC. The heat flux vs. time profile is plotted in Figure  3.6. As can be seen from this figure, neural networks are generally capable of dealing with the whole range of the cooling history.    65     Figure  3.6. Test Case II; Time History of Heat Fluxes in a Typical Run-Out Table Application; Expected Results (Squares) vs. the RBF Network Results (Line)  However, this method has limitations, as observed in Figure  3.6, and in more details in Figure  3.7. The latter figure shows a close-up view of the peaks of heat flux that happen during the cooling process on the run-out table, i.e. the peaks in Figure  3.6. The circles are the expected heat flux, and the plusses are the result of NNs. The top left sub-figure is the first peak heat flux in time, and then it moves to the right, and then to the next row. Note that each even sub-figure (2nd, 4th, and so on) is a very smaller peak which is associated with the second row of jets. These peaks are not very obvious in Figure  3.6, due to the scaling. Going through the subfigures of heat fluxes, it is apparent that the success or failure of NNs is not that much related to the temperature range, or the magnitude of heat fluxes, but on the actual shape of the heat flux profile. If the heat flux has a clear thin peak and two tails before and after the peak, the NN is doing a good job. However, the existence of other details in the heat flux profile reduces the quality of the NN predictions. Also, considering the ill-posed nature of the problem, and all the complications that are involved, we can generally say that in most cases (about 75% of the cases) it does a decent job. Overall we can say that NNs are more useful in getting a general picture of the solution, rather than producing a very accurate and detailed answer to the IHCP. Also, it should be noted that if parameters such as the number of jets, plate speed, or the distance between jet rows are changed, we need to train a new neural network. -1.E+07 -5.E+06 0.E+00 5.E+06 1.E+07 0 1 2 3 4 5 6 7 Su rf ac e H ea t F lu x (W ) Time (s) Expected NN Results       Fi detai regul to tes the n gure  3.7. Indi On the o ls of the m arization for t cases I and ext subsectio vidual Heat F Resu ther hand, G issing boun  these meth  III. As me ns, while st lux Peaks vs. lts (Circles) v A and PSO dary condit ods to work ntioned befo udying the e Time from a s. the RBF Ne  algorithm ions. Notice  properly. S re, the solu ffect of mo Typical Run- twork Result s show reas  that we st ee Figure  3. tions to the difications i Out Table Ap s (Pluses) onably goo ill need to 8 and Figur other test ca n the algorit plication; Ex d prediction have some e  3.9 for the ses will be hms. 66   pected s of the form of  solution shown in        Fig Dotte Figure  3.8 ure  3.9. Test C d Lines: Outp o (a) . PSO Results (b) (a) ase III; (a) T ut of the Inve n the Top Sur , Test case I: ( With Regular he Top Plane rse Analysis; face (y=1) Alo  a) Without R ization (L2 No   (y = 1) Temp (b) Temperat ng the x=1.5 egularization rm of Error erature Cont ure Distribut Line (L2 Norm (b)  (L2 Norm of = 0.014) (b) ours; Solid Li ion (Expected  of Error = 3 Error = 0.127 nes: Expected  and Restored 1.66) 67   ),   Input;  by PSO)    68    3.7. Time Step Size One of the main problems of the classical approaches, such as the sequential function specification method, is their instability when small time steps are used. Unlike direct problems where the stability requirement gives the upper limit of the time step size, in inverse problems the time step is bounded from below. Figure  3.10(a) [112] shows the oscillation in the results obtained by the function specification method and a time step size of 0.01 (s), which corresponds to the onset of instability. For time steps smaller than this, the whole process diverges. Luckily, PSO, GA, and NNs can successfully produce the results for the same time step size, as shown in Figure  3.10(b) for PSO. Note that the oscillations here are not due to the instability caused by the time step size, and can be improved by performing more iterations, as is shown in Figure  3.10(c). Figure  3.10(d) shows the final results of the GA method. It is, however, important to mention that the time requirements for these techniques are much higher than those of the classical function specification approaches.        Figu 3.8. E funct radia comp comp therm gradi re  3.10. Heat fficiency In this s ion specific l basis funct are the time ares the so al history p ent-based fu (a) (c) Flux vs. Time Fin ection, we ation metho ion neural n  that is requ lution time roblem of c nction spec : (a) Classica al results of P first compa d with GA, etworks. W ired to get a  for differe ooling steel ification m   l Approach, ( SO [112], (d) re the solu  the three v e assume th  certain acc nt inverse  on a run-ou ethod. The b) PSO with t  Final results tion time r ariations of at there is uracy in the analysis alg t table. The stochastical (b) (d) he same comp  of GA. equired for  PSO, and no noise in  heat flux p orithms for  fastest solu  methods su utational exp  the gradie the feedforw the solution redictions. T  solving th tion techniq ch as GA 69    ense, (c) nt-based ard and , and we able  3.1 e whole ue is the and PSO    70    variants suffer from high computational cost. RBF neural networks perform much faster than GA and PSO, but they are still slower than the gradient-based methods, such as function specification. Table  3.1. Comparison of the Typical Solution Time for Different Inverse Analysis Algorithms for the Whole Cooling Process on a Run-Out Table  Function Specification Method GA PSO RPSO CRPSO FMLP RBFN Solution Time (s) 1406 8430 6189 5907 6136 7321 2316 However, due to the limitations mentioned in the previous chapter for the gradient-based methods, and in the previous sections of this chapter for the neural networks, in many cases one has to choose between GA and PSO variations. Then, among the slower algorithms, the efficiency of the algorithm becomes even more important. Thus, in the remainder of this section, we will compare the performance of GA and PSO variations in more details. The six different comparison tests performed are presented in Table  3.2. Because of the stochastic nature of these methods, they may perform differently for various runs. Therefore, a statistical approach is needed to compare their performances. Table  3.2. Performed Comparison Tests Test # 1 2 3 4 5 6 Method1 GA GA GA PSO PSO RPSO Method2 PSO RPSO CRPSO RPSO CRPSO CRPSO The statistical t-test [118] is used to compare the number of function evaluation calls (number of times that the direct solver is executed) of 10 runs of each algorithm, which means a sample size of 10 is used. The purpose of performing this test is to determine whether or not the difference between our two averages is large in comparison with the standard deviation. All of the runs used in the performance evaluation test are tested for the accuracy of the results, and have proven to produce acceptable results. If we represent the number of function evaluations for methods 1 and 2 with NE1 and NE2, respectively, then the average number of function evaluations for method 1 is    71    1 1 1 1 1 n N N n i E E i∑ = =  ( 3.13) where 1n is the number of executions of method 1 (in these examples, n1 = n2 = 10). The same equation holds for method 2. The standard deviation of each sample can be calculated as ( ) ( )1 21 1 1 1 1 1 i n E E i E N N s N n = − = − ∑  ( 3.14) The t-value can then be calculated as ( ) 21 21 11 nnNs NNt E EE + − =  ( 3.15) where s is the standard deviation of both samples and is calculated as ( ) ( ) ( ) ( ) ( ) 2 11 21 2 2 21 2 1 −+ −+− = nn NsnNsn Ns EEE  ( 3.16) The objective of the test is to check whether generally 12 EE NN < . The null hypothesis in this test will be: 12 EE NN ≥ . The test is a one-sided test of significance of comparison of two means [118], with a significance level of 5%, and a 10+10-2=18 degrees of freedom, which gives a critical t-value of 1.73. Table  3.3 shows the results of the comparison tests. The shaded values show the cases where the null hypothesis is accepted, i.e., the second method does not perform better than the first one. In the other cases, which are the majority of the tests, using the second method improves the performance, i.e., reduces the number of function evaluations, at least in 95% of the occurrences.     72    Table  3.3. Calculated t-Values for the Combination of 2 Solution Methods; tcritical = 1.73  Test Case I Test Case II Test Case III Test Case IV Test Case V Test 1 4.88 10.08 8.26 10.20 13.43 Test 2 6.60 12.52 9.41 15.05 15.46 Test 3 10.78 17.76 13.41 19.55 12.44 Test 4 1.49 3.32 1.27 4.39 8.21 Test 5 4.63 7.59 7.92 9.71 10.90 Test 6 3.04 3.45 7.75 10.01 9.22  Since an important requirement for the validity of the t-test results is the normality of the distributions for both categories that are compared, and it normally requires a larger number of simulations, the t-test statistics may not be a perfectly sound comparison tool. Thus, we have also used the Mann-Whitney non-parametric statistical test [118] to investigate our results. Table  3.4 shows the U-values for the Mann-Whitney U-test. Again the lowest U values occur at the same cases as the t-test, and only these two aren’t acceptable with a significance level of 5%, for a one-sided Mann-Whitney U-test.  Table  3.4. Calculated U-Values for the Combination of 2 Solution Methods  Test Case I Test Case II Test Case III Test Case IV Test Case V Test 1 97 100 100 100 100 Test 2 100 100 100 100 100 Test 3 100 100 100 100 100 Test 4 72 87.5 62 88 94 Test 5 94 100 100 94 100 Test 6 84 88 99 94 94    73    The following observations can be made regarding the results: 1. PSO variations are generally less computationally expensive than GA in solving inverse heat conduction problems. 2. The advantage of PSO over GA is more pronounced in more complex cases (more unknowns and fewer data points). 3. Between the three variations of PSO, the difference is less significant, but generally, CRPSO performs better than the other two variations. Table  3.5 shows the required number of computational units (i.e., solutions of the direct problem, and obtaining the norm of difference between that solution and the expected one), the relative speedup of each variation of PSO with respect to GA, and the relative computational cost of each method. From the table, speedups of up to 36% are possible when using some PSO variations instead of GA in an inverse heat conduction problem.     74    Table  3.5. Number of Computational Units of Different Gradient-Free Solution Methods, Their Speedup with Respect to GA, and Their Relative Cost with Respect to the Most Efficient Method  GA PSO RPSO CRPSO Test Case I No. of Direct Calculations 5075 4582 4427 4155 Speedup 1.00 1.11 1.15 1.22 Relative Cost 1.22 1.10 1.06 1.00 Test Case II No. of Direct Calculations 13499 11382 10659 9939 Speedup 1.00 1.19 1.27 1.36 Relative Cost 1.36 1.14 1.07 1.00 Test Case III No. of Direct Calculations 23903 21119 20872 19611 Speedup 1.00 1.13 1.15 1.22 Relative Cost 1.22 1.08 1.06 1.00 Test Case IV No. of Direct Calculations 31050 24187 23451 23218 Speedup 1.00 1.28 1.32 1.34 Relative Cost 1.34 1.04 1.01 1.00 Test Case V No. of Direct Calculations 57438 47497 44321 42475 Speedup 1.00 1.21 1.30 1.35 Relative Cost 1.35 1.12 1.04 1.00  Table  3.6 shows the required time to perform one computational unit for each test case. Note that these values have no impact on the performance comparison of the tested optimization algorithms, and are similar for all the optimization methods.     75    Table  3.6. Time Required to Perform One Computational Unit for each Test Case  Test Case I Test Case II Test Case III Test Case IV Test Case V Time for one computational unit (s) 1.65 10.82 5.41 5.06 8.11 The difference between the behaviors of PSO and GA can be attributed to the different underlying mechanisms in these two evolutionary algorithms. For example, mutation in GA occurs in all directions, while the mutation-like behavior in PSO is directional. 3.9. Sequential vs. Whole Domain Due to the parabolic nature of the heat equation, there are basically two categories of approaches in solving the inverse transient heat conduction problems. One method is to try to find components of heat flux in each step of the solution. This is possible since the temperature distribution at the end of each time step is only a function of the temperature distribution at the previous time step and the boundary conditions during the present time step. These methods are normally called “sequential”. In these methods, in each step of the inverse analysis, the direct problem is not solved for the whole history, and is solved for only one (or a few) time steps. The optimization problem is then solved for the temperatures at that step, and then the problem is marched in time to the next steps. The other solution method is to have the whole history of heat fluxes as the unknown in the optimization problem. In this formulation, which is usually called the “whole domain” approach, the direct problem will be solved over the whole time history. Due to the diffusive nature of transient heat conduction, from the computational expense perspective, sequential estimation techniques are highly preferred over the whole-domain inverse analysis techniques [35]. Unfortunately, there is an important drawback in using sequential methods, and that is the high sensitivity of the solution to experimental errors. That is why we will be using the concept of future time steps. Table  3.7 shows the average solution time for ten simulations of the test cases using different solution methods and whole domain and sequential formulations. Test case III is a steady-state problem, and thus is not included here. As seen in the table, sequential formulation always accelerates the solution process. Here, the speedup is defined as the whole domain    76    solution time over the sequential solution time, and is usually somewhere between 30 and 60 percent. Table  3.7. Comparison of Efficiency for Whole Domain and Sequential Implementations   Whole Domain Solution Time (s) Sequential Solution Time (s)  Speedup Test Case I PSO 8316 5904 1.41 RPSO 8107 5107 1.59 CRPSO 8020 5293 1.52 Test Case II PSO 15052 11759 1.28 RPSO 14501 11069 1.31 CRPSO 14115 10942 1.29 Test Case IV PSO 79978 58384 1.37 RPSO 76784 56820 1.35 CRPSO 74266 55208 1.34 Test Case V PSO 121870 81348 1.50 RPSO 109539 82910 1.32 CRPSO 116449 81824 1.42 3.10. Noisy Domain Solution To investigate the behavior of different inverse algorithm variations in dealing with noise in the data, a known boundary condition is first applied to the direct problem. The temperature at some internal point(s) will be calculated and stored. Then random errors are imposed onto the calculated exact internal temperatures with the following equation: rTT exactm ⋅+= σ  ( 3.17) where Tm is the virtual internal temperature that is used in the inverse calculations instead of the exact temperature, Texact; r is a normally distributed random variable with zero mean and unit standard deviation; and σ is the standard deviation. Virtual errors of 0.1%, 0.3%, 0.5%, and 1% of the temperature magnitude are investigated in this research. Since the temperature is changing from 700 to 176 ºC, the magnitude of the noise can be as high as 7 ºC for the initial peaks.       Gene may not l other (plus histor ±1%, Fig  We start b rally, the st be possible t ook promisi  methods, s es) versus t y of the pla  respectively ure  3.11. Ind Results (C y studying ability of th o tune the p ng in terms uch as PSO he expected te. The am . ividual Heat F ircles) vs. th the effective e neural net arameters to  of noise re . Figure  3.  results (cir ount of add lux Peaks vs e RBF Netwo ness of the works is on  make them sistance, si 11 - Figure cles) for ind ed noise in . Time from a rk Results (Pl neural netw the same or  a little bit m nce such m  3.14 show ividual hea these figure  Typical Run- uses); Artifici orks in han der as othe ore stable, odifications the results t flux peak s is ±0.1%, Out Table Ap al Noise Adde dling noisy r inverse me  but general  exist for a of the RBF s during the  ±0.3%, ±0 plication; Ex d: c = ±0.1% 77  domains. thods. It ly it does lmost all network  cooling .5%, and  pected        Fig Fig ure  3.12. Ind Results (C ure  3.13. Ind Results (C ividual Heat F ircles) vs. th ividual Heat F ircles) vs. th lux Peaks vs e RBF Netwo lux Peaks vs e RBF Netwo . Time from a rk Results (Pl . Time from a rk Results (Pl  Typical Run- uses); Artifici  Typical Run- uses); Artifici Out Table Ap al Noise Adde Out Table Ap al Noise Adde plication; Ex d: c = ±0.3% plication; Ex d: c = ±0.5% 78   pected   pected         Fig  noisy time have the a the re first e to the recon accur close speci ure  3.14. Ind Results ( There are  data. For e steps” in the also demon lgorithm to quired num xamine the  PSO metho Figure  3. structed he ate results a  to those re fication app ividual Heat F Circles) vs. th  several wa xample, Ga ir sequentia strated that handle noisy ber of iterat  effect of th d, to improv 15 shows t at flux, usi re obtained ported in [5 roach and P lux Peaks vs e RBF Netwo ys to make dala and Xu l function s increasing  data. How ions, and in e regulariza e the effect he effect o ng the bas  for a range 7], i.e., the SO. . Time from a rk Results (P  an inverse  [57] have s pecification the regulariz ever, the la  many cases tion parame iveness of th f varying ic particle  of values  proper valu  Typical Run- luses); Artific  algorithm hown that i algorithm r ation param tter approac  the solutio ter, and then e inverse a the regulari swarm opti of α = 10-12 es of α are Out Table Ap ial Noise Add more stable ncreasing th esulted in g eter, α, im h was show n may diver  investigate lgorithm in d zation para mization te  to 10-10. Th  very simil plication; Ex ed: c = ±1%  when deal e number o reater stabil proves the a n to greatly ge. In this w  an approac ealing with meter value chnique. St ese results ar for the s 79   pected ing with f “future ity. They bility of  increase ork, we h unique  noise.  on the able and are very equential       Figur noisy and t again 10 (Fi value are sh e  3.15. Test C Another f  data is the he accelera , with the sa gure  3.15(c  of the self- own below (a) ase II; Effect actor that ca value of the tion coeffici me level of )). The acce confidence .  of Regulariza n affect the  self-confid ents. To sh  noise as in leration coe parameter, c  (c) tion Paramet c: α = 10-10  performanc ence param ow the effe Figure  3.15 fficients are 0, is change er (1% added e of a PSO eter, c0, or ct of this p , and a regu  set to the d d from the (b)   noise); a: α = inverse appr the ratio bet arameter, te larization pa efault valu default valu  10-12; b: α = oach in dea ween this p st case II i rameter equ e of 1.42. T e of 0.7. Th 80   10-11; ling with arameter s studied al to 10- he initial e results       Figu  re  3.16. Test Ta (a) (c) Case II; Effec ble  3.8. Effec c0 α = 10-12 5 α = 10-10 4 t of Self-Conf t of Self-Conf 0.5 .88e+5 5 .85e+5 4   idence Param c0=1.0; (d) c0= idence Param 0.6 .24e+5 4 .30e+5 3 eter (1% add 1.2 eter: L2 Norm 0.7 1 .47e+5 3.8 .19e+5 2.3 (b) (d) ed noise); (a)  of the Error .0 1. 1e+5 2.98 6e+5 2.10  c0=0.5; (b) c0 (W/m2) 2 e+5 e+5 81    =0.6; (c)    82    As can be seen in Figure  3.16 (for α = 10-10), and more quantitatively in Table  3.8 (for α = 10-10 and 10-12), increasing the value of the self-confidence parameter results in better handling of the noisy data. This trend was observed for values up to approximately 1.3, after which the results become worse, and diverge. The same trend was also observed in the other test cases. One possible explanation is that increasing the ratio of the self-confidence parameter with respect to the acceleration coefficients results in a more global search in the domain, and therefore increases the capability of the method to escape from the local minima caused by the noise, and find values closer to the global minimum. This effect was observed to be weaker in highly noisy domains. However, in the presence of a moderate amount of noise (which is the case in most experimental setups), increasing the self-confidence ratio results in more effectiveness in dealing with noisy data. Table  3.9 shows the same trend when using different variations of the PSO algorithm on test cases I, IV, and V. As can be seen in Table  3.9, the best effectiveness is normally obtained by RPSO, closely followed by CRPSO. Considering the higher efficiency of CRPSO, it is still recommended for the inverse heat conduction analysis.  Table  3.9. Effect of the Self-Confidence Parameter on the L2 Norm of Error in the Solution C0 0.7 0.8 0.95 1.1 1.2 Test Case I PSO 0.07466 0.06722 0.06113 0.05651 0.05397 RPSO 0.05546 0.05010 0.04362 0.04096 0.03728 CRPSO 0.05798 0.05163 0.04415 0.04262 0.04043 Test Case II PSO 8.105e+4 7.532e+4 7.079e+4 6.823e+4 6.257e+4 RPSO 7.577e+4 7.064e+4 6.685e+4 6.346e+4 5.816e+4 CRPSO 7.611e+4 6.739e+4 6.117e+4 5.999e+4 5.822e+4 Test Case III PSO 1.641e+5 1.477e+5 1.322e+5 1.282e+5 1.222e+5 RPSO 1.328e+5 1.272e+5 1.203e+5 1.184e+5 1.158e+5 CRPSO 1.361e+5 1.288e+5 1.221e+5 1.184e+5 1.143e+5   Table  3.10 shows the value of L2 norm of error in the solution, for ±1% added noise, and for different algorithms. It can be seen that the RBF neural networks perform better than the    83    function specification method, and somewhere between the genetic algorithm and PSO variants. The most noise resistant algorithms are PSO variants, and the least stable algorithm is the gradient-based function specification method.  Table  3.10. The L2 Norm of Error in the Solution in a Noisy Domain for Different Algorithms  Function Specification Method GA PSO RPSO CRPSO FMLP RBFN L2 Norm of Error 9.14e4 6.61e4 5.24e4 4.82e4 5.02e4 8.91e4 5.91e4 3.11. Future Time Step Regularization There are also other ways to make an inverse algorithm more stable when dealing with noisy data. One of them is using the concept of future time steps that was also discussed in the function specification method in the previous chapter. Due to the fact that inverse problems are generally ill-posed, the solution may not be unique and would, in general, be sensitive to measurement errors. To decrease such sensitivity and to improve the simulation in function specification methods, a number of future time steps (nFTS) are often utilized in the analysis of each time step [35]. This means that in addition to the measured temperature at the present time step T i, the measured temperatures at future time steps, , are also used to approximate the heat flux qi. In this process, a temporary assumption would usually be considered for the values of .  The simplest and the most widely used one is to assume  for , which is also used in our work. Also in this work, we combine the regularization concept with the future time step data. This is analogous to the combined function specification-regularization method, initially introduced by Beck and Murio [96]. The sequential algorithm for time step M when using r future time steps (the heat flux is known for t < tM) is presented below: 1. An assumption is made for the heat flux function at times tM, tM+1,…, tM+r (usually qM = qM+1=…=qM+r). FTSniii TTT +++ ,....,, 21 FTSniii qqq +++ ,....,, 21 iki qq =+ FTSnk ≤≤1    84    2. Direct problem is solved with  as the initial condition and the heat flux values from step 1 as its transient boundary conditions. The temperatures at these r time steps are calculated ( ). 3. The objective function is then evaluated using the heat fluxes from step 1, the calculated temperatures from step 2, and the known temperatures from the measurement ( ). The new form of the objective function will be:  ( 3.18) 4. Steps 1 to 3 are repeated for each swarm member, and the GA or PSO algorithm is repeated until we reach a desired low value for the objective function, or after a certain number of iterations. The solution vector will be the spatial components of heat fluxes for the r time steps. 5. Values of qM are used as obtained from step 4 and the values for the rest (qM+1, …, qM+r) are discarded. 6. The above procedure (M = M+1) is repeated as you move to the next time step. In theory, it is possible to use multiple sensors instead of multiple future time steps to improve the stability of the method. However, this idea is not practical since adding thermocouples to the sample is not only more expensive, but also distorts the temperature field. Using future time steps introduces a bias in the solution. This bias is normally observed in the form of rounding the spikes in oscillatory boundary condition histories, and an early prediction of these oscillations [35]. To obtain an appropriate level of regularization, the number of future time steps (or more accurately, the size of the look-ahead time window, i.e. the product of the number of future time steps and time step size) and the value of the regularization parameter must be chosen with respect to the errors involved in the temperature readings. The residual principle [36,102] can be used to determine these parameters based on the accuracy of thermocouples in the relative temperature range. In this method, the number of future time steps meas MT 1− calc rM calc M calc M TTT ++ ,,, 1 L meas rM meas M meas M TTT ++ ,,, 1 L ( )∑ ∑∑ = + = + = ⎟⎠ ⎞⎜⎝ ⎛ +−= J j rM Mi i j calci j rM Mi measi j qTTf 1 22,, α)()(q    85    is increased until the RMS error of the estimation falls below the standard deviation of noise in the measured temperatures. Figure  3.17 shows the results of applying the basic PSO-based inverse analyzer to the first test case, when a moderate amount of noise is added to the temperature data. As can be seen in part (a) of the figure, the method with default settings and no regularization term shows a very unruly behavior. While addition of the zeroth-order Tikhonov regularization term and the tuning of the self-confidence parameter can enhance the behavior, as seen in parts (b) and (c) of Figure  3.17, there is a limit to these modifications, i.e. after a certain limit the method will become very slow or will diverge. However, using the future time step concept, can increase the stability of the algorithm, as seen in Figure  3.17(d). Table  3.11 also shows the L2 norm of error in the solution after each stage of algorithm modifications. Figure  3.18, Figure  3.19 and Table  3.12 show the same trend for the first and second set of heat fluxes applied to the three- dimensional problem. As expected, the norm of error in the solution in each case decreases after each set of modifications is applied. Also, investigating the numbers in Table  3.11 and Table  3.12 shows that among the three variations of PSO, RPSO is the most effective one in handling noisy data, closely followed by CRPSO. This is in agreement with the findings in Reference [112], shown earlier in this chapter. As can be seen from the figures, in the cases without the future time step method, the predicted heat fluxes overestimate the peak heat flux values, but when future time steps are used, due to the anticipation of the temperature in the following time steps, the predicted peak better matches the expected results.        Fig Re ure  3.17. PSO gularization T  Behavior in erm (α = 10- Handling Noi 10); (c) Tuning Future Tim sy Data in Te  of Self-Confi e Step Data st Case I; (a) N dence Param (10 time steps o Regulariza eter (c0 = 1.2) ) tion; (b) Add ; (d) Introduc 86   ition of tion of       Figu Re  re  3.18. PSO B gularization T ehavior in H erm (α = 10- andling Noisy 10); (c) Tuning Future Tim  Data for Tes  of Self-Confi e Step Data t Case IV; (a) dence Param (10 time steps  No Regulari eter (c0 = 1.2) ) zation; (b) Ad ; (d) Introduc 87   dition of tion of       Fig Regu       ure  3.19. PSO larization; (b  Behavior in ) Addition of R = 1.2); (d) Handling Noi egularizatio  Introduction sy Data for th n Term (α = 1 of Future Tim  e 2nd Set of H 0-10); (c) Tuni e Step Data eat Fluxes in ng of Self-Co (10 time steps  Test Case V nfidence Para ) 88   ; (a) No meter (c0    89     Table  3.11. The L2 Norm of Error in the Solution of Test Case I After Modifications in a Noisy Domain Method No Regularization Tikhonov Regularization Tikhonov Regularization + Tuning of the c0 parameter Tikhonov Regularization + Tuning of the c0 parameter + Future time steps PSO 0.10845 0.07466 0.05397 0.02108 RPSO 0.10070 0.05546 0.03728 0.01855 CRPSO 0.10148 0.05798 0.04043 0.01876   Table  3.12. The L2 Norm of Error in the Solution of Test Cases IV and V After Modifications in a Noisy Domain Method No Regularization Tikhonov Regularization Tikhonov Regularization + Tuning of the c0 parameter Tikhonov Regularization + Tuning of the c0 parameter + Future time steps 1st Set of Heat Fluxes PSO 1.504E+05 8.105E+04 6.257E+04 3.179E+04 RPSO 1.393E+05 7.577E+04 5.816E+04 2.813E+04 CRPSO 1.418E+05 7.611E+04 5.822E+04 2.837E+04 2nd Set of Heat Fluxes PSO 2.712E+05 1.641E+05 1.222E+05 6.305E+04 RPSO 2.448E+05 1.328E+05 1.158E+05 5.400E+04 CRPSO 2.495E+05 1.361E+05 1.143E+05 5.440E+04    90    While using Tikhonov regularization, tuning the value of the self-confidence parameter and using future time steps can improve the quality of the solution when dealing with noisy data, but it also increases the number of parameters that should be controlled. Luckily, once these parameters are set for one simulation of a specific test case, they can be used again, and there is no need to determine the most effective values in each run. Also, these modifications can increase the required solution time, so there will always be a compromise between the level of accuracy and the computational expense, which should be considered according to the requirements of each application. 3.12. Effect of Non-linearity While methods such as GA and PSO are based on a stochastical search of the whole domain, and therefore are not significantly affected by nonlinearity, the neural networks are more sensitive to nonlinearity. In other words, since the same drop in temperature values can be caused by different values of heat flux, a neural network that is trained with the relationship between the temperature change values and heat flux magnitudes may not be correctly capable of recognizing this nonlinear pattern, and as a result the performance will suffer. To investigate this effect, two kinds of expressions are used for thermal conductivity in this study. In one, we assume a constant thermal conductivity of 40 W/m.°C, while in the other a temperature- dependent expression is used: W/m.°C ( 3.19) As expected, the nonlinearity will weaken the performance of both feedforward and radial basis function neural networks. The effect is seen as the training of the network stalls after a number of epochs. In order to deal with this, increasing the number of hidden layers, increasing the number of neurons in each layer, and choosing different types of transfer functions were investigated. However, none of these methods showed a significant improvement in the behavior of the network. The other methods of solving the inverse problem that are introduced in this research, are much less sensitive to the effect of nonlinearity. Table  3.13 compares the error in the solution for both the linear and nonlinear cases, if the same numbers of iterations, generations, and epochs are used for different methods of solving the inverse heat conduction. As can be seen, the neural networks perform very poorly in the nonlinear cases, while the other methods, either gradient based or stochastical, are immune to the problems caused by Tk ×−= 03849.0571.60    91    nonlinearity. Basically, neural networks, at least in the form that is used in this research, see nonlinearity as a kind of noise. It should be noted that neural networks can be useful in making rough estimates of the answer, or combined with some other techniques employed as an inverse solver for nonlinear cases [64].   However, on their own, they are not a suitable choice for an accurate prediction of the boundary conditions in a nonlinear inverse heat conduction problem. Table  3.13.The L2 Norm of Error in the Solution in an Exact Domain for Different Algorithms  Function Specification Method GA PSO RPSO CRPSO FMLP RBFN Linear 1.81e2 7.62e2 3.85e2 3.42e2 3.17e2 9.90e2 5.35e2 Nonlinear 2.14e2 7.71e2 4.46e2 5.12e2 4.26e2 3.57e4 2.76e4 3.13. Effect of Elite Members In this section, the effect of including the elite particle’s position and velocity in the performance of the sequential PSO-based inverse solver is studied. We first take a look at the average speedup as the result of introducing the elite members into the algorithm. Here, the speedup is defined as the ratio of the required solution time in the case without the elite members to the solution time after the elite members are utilized. Obviously, a higher speedup means that the concept of using the elite particles was more effective. Table  3.14 shows the average speedup of 10 runs for each algorithm. As can be seen from the table, in some cases, an improvement of 14% can be achieved as a result of using elite particles and velocities. While the introduction of this concept usually improved the performance, this improvement did not take place in all the simulations. One must especially remember that the performance of PSO, like any other random search algorithm, can vary from one simulation to another. In order to take care of this randomness in the simulations, a Mann-Whitney non- parametric statistical test [118] is used to compare the solution time for the 10 runs of each algorithm for each test case and assess the improvement caused by the inclusion of the elite particle concept. All of the runs used in the performance evaluation test are tested for result accuracy, and proved to produce acceptable solutions.    92    The objective of the test is to check whether, generally, the algorithm with the elite member is faster than the same variation without the elite member. The null hypothesis in this test is that the introduction of the elite particle concept does not improve the solution time, i.e.: . The test is a one-sided test of significance of the comparison of two means [118]. If for each simulation with elite members, we count the number of simulations without elite members that are slower than it, and then sum these counts for all the simulation set, we obtain the U-value. A significance level of 5%, and an equal number of samples for both sides ( ) gives a critical U-value of 73. Table  3.15 shows the U-values for the Mann-Whitney U-test. The only case that does not pass the test is the basic PSO algorithm applied to the last test case. As seen in the table, RPSO always performs better when the elite particles are introduced. Based on the results of this table, including the elite particle in the PSO algorithm to solve the inverse heat conduction problem is justified. memberselitewithmemberselitewithout meSolutionTimeSolutionTi −−−− ≤ 1021 == nn    93    Table  3.14. The Average Speedup in Each Algorithm After Using Elite Particles and Velocities  Whole Domain Sequential Test Case I PSO 1.126 1.129 RPSO 1.173 1.133 CRPSO 1.138 1.131 Test Case II PSO 1.020 1.174 RPSO 1.152 1.147 CRPSO 1.011 1.279 Test Case III PSO 1.132 1.237 RPSO 1.205 1.065 CRPSO 1.129 1.086 Test Case IV PSO 1.052 1.105 RPSO 1.112 1.116 CRPSO 1.093 1.104 Test Case V PSO 1.043 1.050 RPSO 1.052 1.096 CRPSO 1.082 1.101  Table  3.15. Calculated U-Values to Study the Effect of Elite Particles on the Performance  Test Case I Test Case II Test Case III Test Case IV Test Case V PSO 100 88 88 84 72 RPSO 100 100 100 100 100 CRPSO 100 88 84 97 88 3.14. Multi-Criteria Formulation Based on Spatial Location It is common in inverse heat conduction applications to have more than one sensor inside the sample. In the case of inverse boundary problems, the number of these sensors is often equal to the number of spatial components of the missing boundary heat flux. The formulation of Eq. ( 3.1) uses a norm of the difference between the calculated and measured temperatures that take    94    into account all of the thermocouples. In other words, the objective is to minimize some sort of overall difference between these two sets of temperatures. Thus, when searching for the best solution, the method does not distinguish between different spatial components. One way to devise a more intelligent search algorithm is to define a multi-criteria formulation, in which each of the spatial components of the missing boundary condition is treated as an individual search space. This way, if one of the spatial components of the heat flux vector gets close to the expected value, it will not be discarded due to the poor quality of the other components. In this formulation, the objective function will be a vector, and each of its components will represent one of the sensor locations. We can write this objective function as:  ( 3.20) It should be noted that this is not equal to decoupling of the thermocouples, because the direct problem would still be solved as a whole. The performance of multi-criteria optimization (vector objective function) is compared with the case where the objective function is a single scalar quantity. Test cases IV and V are solved, and the history of the best value of the objective function in the swarm is normalized with respect to the best value of the objective function in the first iteration, and is plotted versus the iteration number. As can be seen in Figure  3.20, this modification greatly improves the convergence rate of the PSO-based inverse analyzer. Comparing the two parts of Figure  3.20, we observe that while the increased number of thermocouples causes a lower rate of convergence, both when the vector or scalar formulation is used, the relative speedup of the vector formulation stays in the same range. ( ) ( ) ( ) ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎝ ⎛ +− +− +− = ∑∑ ∑∑ ∑∑ + = + = + = + = + = + = rM Mi i J calci J rM Mi measi J rM Mi icalci rM Mi measi rM Mi icalci rM Mi measi qTT qTT qTT qf 22,, 2 2 2, 2 , 2 2 1 2, 1 , 1 α)( α)( α)( )( M       Figu 3.15. 3.15. simp simu volum proce nume geom more or pa but o norm spend value ident solut re  3.20. Conv  Surrogate 1. Motivati In many ly want to f lations. Wh e, finite el ss is norma rical simul etrical (two  unreasonab rticle swarm nly the sort ally discard  a large am s. Instead a ify the prom ions. (a) ergence of Sc  Modeling on engineering ind the opti ile convent ement, etc.) lly very com ations usin -dimensiona le in the cas . In these te ing of the m ed, especial ount of co n inexact p ising memb alar vs Vecto   application mum design ional metho  are capable putational g simplifie l instead of e of rank-b chniques, th embers ma ly in the in mputationa re-evaluatio ers, and lat r Objective F Case V s, when we  of a system ds of num  of produc ly expensive d physical 3D) models ased evoluti e actual val tters. The e itial steps o l resources n approach er perform t unction Form  are faced w , it is nece erical simu ing very acc . While it  (inviscid , they are st onary algor ue of the obj xact magnit f the algori on exactly using surro he full eval (b) ulations; (a) T ith an inv ssary to per lation (fini urate result is very com flow inste ill time-dem ithms, such ective func ude of the thm. So, it  evaluating gate models uations only est Case IV; erse problem form multip te differenc s in most c mon to sim ad of visc anding. Thi as genetic a tion is not im objective fu seems logic these cost  may be ad  for these c 95   (b) Test , or we le direct e, finite ases, the plify the ous), or s is even lgorithm portant, nction is al not to function opted to andidate    96    3.15.2. Procedure The procedure for constructing the surrogate models has five major stages: 1. Design of experiments. In this stage, we choose the samples in the design space that are going to be used for constructing the surrogate model. Since the number of chosen samples is much smaller than all the possible cases, it should be a good representative of the domain. Some of the methods that can be used for the design of experiments are random, error-based, density- based, gradient-based, and some hybrids of these. A random sample selection scheme is used in this study. 2. Full numerical simulation of the selected samples. The high-fidelity and expensive numerical simulation technique is used to evaluate the samples that were chosen in the previous stage. 3. Model selection. In this step, we choose between several types of available surrogate models, such as polynomial, radial basis function, neural networks, etc. Choosing the most suitable surrogate model type is a problem-related decision in most cases. There are a number of criteria that can be used to rank different surrogate modeling techniques. Some of the most common measures are the accuracy, efficiency, and robustness of the models. Other criteria may include transparency, i.e. capability of explicitly showing the relationship between the input and output, and the ease of implementation [119]. In this research, we are going to compare the implemented methods in term of robustness and efficiency, which are more crucial in our inverse heat conduction applications. 4. Model identification. Here we tune the parameters of the selected model type to maximize the capability of the algorithm. Another evolutionary optimization algorithm may be applied to find the best set of parameters. The available options are different for each model type. 5. Model validation. In this stage, we make sure that the model is capable of producing acceptable results for some other samples that were not used in the previous steps. Some available algorithms are validation set, cross-validation, leave-one-out, and model difference. Cross-validation is used in this research. 3.15.3. Surrogate Model Types  Four types of surrogate models were tested in this research: Polynomial models, Radial Basis Function models, Kriging models, and Feedforward Neural Networks. Some researchers have previously compared the performance of several surrogate model types in solving some    97    benchmark optimization problems [119]. In this study, we directly focus on their performance in inverse heat conduction problems. Polynomial Models A second-order polynomial model is the most popular polynomial model, and is also used in this research. It can be written in the following form: 0 1 1 1 ˆ i i n i j i j i n i j n y x x xβ β β − + + ≤ ≤ ≤ ≤ ≤ = + +∑ ∑  ( 3.21) where ŷ is the estimated response, x is the vector of inputs, β values are the coefficients that need to be estimated, and i and j are the indices of the input parameters. This is normally done using the least square method [120]. Radial Basis Function Using the same definitions for x and y as above, the radial basis function model can be expressed as: ( ) 1 ˆ i i i i n y x cωψ ≤ ≤ = −∑  ( 3.22) where ωi is the selected weight for each parameter, ci denotes the ith basis function center, and ψ is the basis function. The main strength of this model arises from the fact that it is linear in terms of the basis function weights; however it is capable of predicting highly nonlinear responses [120]. Kriging Models One of the most popular types of surrogate models is the Kriging method. In this variation, the basis function is of the form: ߰(௜) = expቌ−෍ߠ௝ቚݔ௝(௜) − ݔ௝ቚ ௣ೕ ௞ ௝ୀଵ ቍ ( 3.23) The vector ࣂ = 	 ሼߠଵ, ߠଶ, … , ߠ௞ሽ represents the width of the basis function, and it can change from variable to variable. Also, the exponent values, p, can also vary from one variable to another. Basically, a higher value for the exponent increases the smoothness of the function around that    98    variable [120]. In estimating the output of the Kriging model, it is necessary to perform matrix inversions [121]. This can significantly increase the computational costs of this method when the dimensions are high. Feedforward Neural Networks The MATLAB implementation of neural networks is also used as a surrogate model. A feedforward multilayer perceptron uses three or more layers of neurons (nodes) with nonlinear activation functions. The learning process in this type of neural network takes place by altering the connection weights after each segment of data is processed, based on the amount of error in the output compared to the expected result. 3.15.4. Surrogate Modeling (SUMO) Toolbox In this research, the SUrrogate MOdeling (SUMO) MATLAB toolbox [122] is used to construct the surrogate models. This open source toolbox is developed by the IBCN research group of the Department of Information Technology (INTEC) at Ghent University. It has excellent extensibility, making it possible for the user to add, customize and replace any element of the modeling procedure. It also has a wide range of built-in test functions and test cases, as well as support for many different model types. In SUMO, first, an initial design is generated and evaluated. Then, a set of models is constructed, and the accuracy of these models is approximated using a set of measures, such as cross-validation or an external validation test set. Each type of model has a number of parameters which can be customized, e.g. degrees of freedom for rational models, smoothness for RBF models, number and size of hidden layers in neural networks, etc. These parameters are adjusted using an optimization method, and additional models are produced until no additional improvement can be made by changing the model parameters. If the desired accuracy has not so far been reached, an adaptive sampling algorithm is employed to generate a new set of sample locations, and the algorithm starts from the beginning. 3.15.5. Model Management One drawback of using surrogate models in the context of evolutionary algorithms is the convergence to a “false optimum”, i.e. a solution that is an optimum to the approximate surrogate model, and not to the original objective function [123]. In order to overcome this    99    problem, it is crucial to use the surrogate model with the original objective function. The approach that is used to select some (or all) of the individuals to be used with the original fitness function is called model management [124]. It can be divided into two major forms: • Fixed Evolution Control. A fixed percentage of the individuals are chosen to be evaluated by the original model. The individual members can be chosen randomly, or based on the value of the approximate objective function, i.e. choosing the ones that seem to be more promising based on the pre-evaluation using the surrogate model. If a lower portion of the individuals are chosen, it results in a faster iteration, but it is possible to miss the global optimum, and a larger number of iterations may be required. • Adaptive Evolution Control. In this strategy, the number of members that are chosen to be evaluated by the original model can be different in each iteration. The selection criteria can be a function of the values generated by the surrogate model, the iteration number, the degree of improvement in the results, or other factors based on the nature of the problem and the chosen optimization algorithm. In this research, we investigate the usage of two different strategies of adaptive evolution control. The first one is based on the concept of “index of improvement” [125]. If ࣹ௠௜௡ is the current minimum of the objective function, and መࣹ(ݔ) is the predicted value of the objective function by a new point x, then the index of improvement can be defined as ܫ(ݔ) = ቊ 0 ݂݅ መࣹ(ݔ) > ࣹ௠௜௡	ࣹ௠௜௡ − መࣹ(ݔ) ݋ݐℎ݁ݎݓ݅ݏ݁  ( 3.24) If a point has a large value of this index of improvement, then it is more likely to improve the value of the objective function and should be evaluated using the full simulation method. In our research we started testing all the samples with ܫ(ݔ) ≥ 0.1	ࣹ௠௜௡ in the initial iterations, and then gradually relaxed this condition in later iterations, where the values are closer to each other, and improvements are smaller. The second type of adaptive evolution control, employed in this research, is based on the iteration number, i.e. a smaller ratio of the solutions were fully tested in the first iterations, and then this number was gradually increased to include nearly all of the solutions in the final iterations. This strategy is based on the observation that the candidate solutions in the later iterations generally have better qualities, and also, that the search in these later stages is more localized. Since the values of the objective function are going to be closer to each other, surrogate modeling may not be able to correctly sort them, and a high fidelity solution is more desirable.    100    3.15.6. Model Configuration In the inverse problems with multiple sensors, such as in test cases IV and V, there are two different strategies in designing the surrogate model. One is to construct a simple model for a single thermocouple location and its respective boundary heat flux. In this case, we will need to run this model for the number of sensors in the problem. The other strategy is to create a larger model that takes the whole boundary heat flux matrix as its input (with rows for different time steps and columns for different sensor locations), and its output is also a matrix including the temperature responses in all the thermocouple locations. In this strategy, only one surrogate model is needed, but it will be larger and more complicated. This latter form is a more physical representation of the real problem. In this research, for test cases IV and V, we use both strategies. The first one will be called “3D – 1 sensor”, and the second method will be called “3D – 9 sensors” and “3D – 18 sensors” for test cases IV and V, respectively. We will also study the application of surrogate modeling in a simple case, by investigating test case I. In this test case, since there is only one sensor in the domain, only one kind of surrogate model is used; “1D – 1 sensor”. Their performances will be compared in terms of the time needed to construct the model, the prediction time, the total solution time, and their accuracy in predicting the results and in dealing with noisy domains. 3.15.7. Results The iterations were continued until the normalized value of the best objective function with respect to the best value of the objective function in the first iteration became smaller than 10-7. Hence, the qualities of the results in all cases were very similar. However, the time and computational expense required for the method to reach such a solution were very different. Figure  3.21 shows the typical results of the applied algorithm.       3.15. techn mode Effic How to the to the comp simu runni for co    Figur 8. Solution First, we iques (poly ling strateg iency is sim ever, in surr  time it tak  time requir are both k lations are d ng Microso nstructing a (a) e  3.21. Resul Time  are going nomial, rad ies (1 senso ply related t ogate model es to constru ed to perfor inds of eff one on a PC ft Windows nd impleme ts of Surrogat to compar ial basis fun r, 9 sensors o the amou ing there ar ct a model m new pred iciency of  with an In XP Professi nting the m  e Based PSO e the four ction, Krig , and 18 sen nt of time re e two distinc . The second ictions using the four su tel Pentium onal. Table odels, respe for (a) Test C previously ing, and fee sors) in ter quired for a t kinds of e  kind of co  the constru rrogate mo D 2.80 GH  3.16 and Ta ctively. (b) ase I; (b) Tes mentioned dforward n ms of comp  method to fficiency. T mputational cted model deling tech z CPU, with ble  3.17 sh t Case IV surrogate m eural netwo utational ef reach a cert he first one  efficiency . In this rese niques. Al  2 GB of R ow the requ 101   odeling rks) and ficiency. ain goal. is related is related arch, we l of the AM, and ired time    102    Table  3.16. Required Time (in Seconds) to Construct the Model for the Linear Case  Polynomial RBF Kriging FF Neural Network 1D – 1 sensor < 1 4.42 43.70 9.63 3D – 1 sensor 12.14 15.46 306.73 43.81 3D – 9 sensors 96.68 125.38 2183.97 316.53 3D – 18 sensors 146.47 187.44 2915.53 550.76  Table  3.17. Required Time (in Seconds) for 1000 New Predictions  Polynomial RBF Kriging FF Neural Network 1D – 1 sensor < 1 9.13 6.65 9.55 3D – 1 sensor < 1 11.19 7.43 14.47 3D – 9 sensors 5.14 90.75 68.44 119.45 3D – 18 sensors 9.05 147.92 125.24 205.45  As can be seen from Table  3.16 and Table  3.17, the polynomial regression model is the fastest model to develop and implement. Radial basis function and feedforward neural networks have comparable implementation times (both much more than the polynomial method), but in terms of model construction, the radial basis function model is faster. The construction time for the Kriging model is high, especially for the cases with more sensors. As expected, using a more complex model (more sensors) will increase the time required for both constructing the model and making predictions. 3.15.9. Accuracy and Noise Resistance Another important measure for evaluating the surrogate modeling methods is their accuracy in predicting the exact values, and also their effectiveness in dealing with noisy domains. To examine this latter property, we add artificial noise to the results of the direct solution for the points that are used to construct the model. Then the model is evaluated for    103    another set of points to see how it is affected by the noise. Random errors are imposed onto the calculated exact internal temperatures with the following equation: rTT exactm ⋅+= σ  ( 3.25) where Tm is the modified temperature that is used instead of the exact temperature, Texact; r is a normally distributed random variable with zero mean and unit standard deviation; and σ is the standard deviation. To compare the methods, the R2 value is calculated for 20 samples: ( ) ( )∑∑ == −−−= n i i n i ii yyyyR 1 2 1 22 ˆ1  ( 3.26) where iŷ is the predicted value, corresponding to the expected value iy , and y is the mean of the expected values. Obviously, a higher value of R2 (closer to 1) is more desirable. Table  3.18 shows the value of R2 for different methods, for both the exact and noisy domains, for the linear case, i.e. when the thermal conductivity is a constant value. Table  3.19 shows the same results, but for the nonlinear case, i.e. the temperature dependent thermal conductivity.  Table  3.18. R2 Values for Different Surrogate Models in the Exact and Noisy Domains for the Linear Case  Polynomial RBF Kriging FF Neural Network 1D – 1 sensor exact 0.99 0.99 0.99 0.99 noisy 0.90 0.95 0.84 0.96 3D – 1 sensor exact 0.99 0.98 0.99 0.99 noisy 0.88 0.93 0.81 0.96 3D – 9 sensors exact 0.99 0.99 0.99 0.99 noisy 0.90 0.94 0.85 0.97 3D – 18 sensors exact 0.99 0.99 0.99 0.99 noisy 0.91 0.94 0.85 0.97     104    Table  3.19. R2 Values for Different Surrogate Models in the Exact and Noisy Domains for the Nonlinear Case  Polynomial RBF Kriging FF Neural Network 1D – 1 sensor exact 0.69 0.87 0.92 0.78 noisy 0.64 0.80 0.70 0.75 3D – 1 sensor exact 0.58 0.86 0.90 0.75 noisy 0.56 0.82 0.71 0.73 3D – 9 sensors exact 0.60 0.86 0.91 0.77 noisy 0.57 0.83 0.71 0.74 3D – 18 sensors exact 0.60 0.87 0.92 0.77 noisy 0.57 0.83 0.72 0.75  As seen from Table  3.18, when there is no noise in the domain, and the problem is linear, the three methods produce results with very similar levels of accuracy. However, in the case of noisy domains, the feedforward neural networks are the safest choice, followed closely by RBF. Polynomials and Kriging show the worst performance. Comparing Table  3.19 with Table  3.18 shows that the nonlinearity in the problem considerably lowers the accuracy of all surrogate techniques. This is especially pronounced in the case of polynomials. The Kriging model performs better than the rest in the case where the data is exact. However, addition of noise significantly reduces its accuracy. In the nonlinear problem with noisy data, the RBF model is the most accurate one. Since most real-world inverse heat conduction problems are nonlinear and noisy, the RBF method seems most suitable. 3.15.10. Surrogate Model Management Since the most significant part of the computational expense is created by the full exact solution of the direct problem, the effect of model management (number of the most promising particles that have been solved exactly using finite element techniques) is studied in Figure  3.22. The RBF model is used in this section, since in the previous section, it was found to be the most robust surrogate model for inverse heat conduction problems. The “Adaptive I” strategy is based on the concept of the index of improvement, as mentioned in Equation ( 3.24). The “Adaptive II”    105    strategy is based on changing the percentage of exactly evaluated members based on the iteration number. The figure shows the value of the objective function attained by the best performing particle in each iteration, normalized to the best value in the first generation, versus the total number of full direct simulations of the direct problem using the 3D finite element code for test case IV. As is seen in Figure  3.22, the computational expense is reduced considerably if only a smaller portion of promising members are evaluated (e.g., 20% in this problem). However, choosing a very small percentage (10% in this example) is inefficient, since it requires more iterations, and the total number of direct solutions will be more than the 20% case. Also notice that the strategy of evaluating 100% of the particles is equivalent to not using surrogate modeling at all. In this sense, we can say that using surrogate modeling can speed up the PSO- based inverse algorithm up to around 3 times and the GA-based inverse algorithm up to around 4.5 times. Table  3.20 shows the effect of using the surrogate models on CPU time. A considerable amount of speedup is observed when using a suitable ratio for the number of exactly evaluated swarm members. Also, it seems that using simpler model configurations (1 sensor instead of 9 or 18) is preferable. This is reasonable considering that the nature of surrogate modeling is based on approximation, and not a full detailed solution. As can be seen from Table  3.21, the speedup of the nonlinear case is similar to the one in the linear case, and RBF seems to perform a very good job there as well. This can also be explained as the main role of the surrogate models is ranking of candidates, and even though in the nonlinear case, the accuracy of the prediction is lower, the RBF model can still choose which candidates will perform better.        Figur   e  3.22. Effect Table  3.20. E % of fully e 1D – 3D – 3D – 3D – 1 (a) of Surrogate ffect of Usin valuated m  1 sensor  1 sensor 9 sensors 8 sensors Model Manag g Surrogate M embers 10 GA 58 PSO 33 GA 15 PSO 89 GA 15 PSO 89 GA 15 PSO 89 ement on the (b) PSO Solv odels on the 0% 40% 31 3341 94 1986 301 8869 51 5006 301 11404 51 6257 301 11852 51 7010   Performance er Total Solution 20% 10% 2317 338 1268 181 5460 734 2685 349 5253 697 2891 3374 5323 612 3090 318 (b)  of Inverse A  Time (s) for Adaptive 6 2241 1 1152 5 4060 8 2194 7 3820 1889 0 3569 7 1812  nalyzer; (a) G  the Linear C  I Adaptiv 3712 2105 9895 4715 8324 3822 6809 3683 106  A Solver; ase e II            107     Table  3.21. Effect of Using Surrogate Models on the Total Solution Time (s) for the Nonlinear Case % of fully evaluated members 100% 40% 20% 10% Adaptive I Adaptive II 1D – 1 sensor GA 6036 3584 2354 3397 2422 4070 PSO 3483 1995 1268 1864 1162 2256 3D – 1 sensor GA 16071 9394 5511 8139 4189 10103 PSO 9174 5282 2991 3537 2473 5325 3D – 9 sensors GA 16071 11821 5625 7117 4281 9360 PSO 9174 6409 3178 3424 2102 4306 3D – 18 sensors GA 16071 12818 5634 6713 3629 7254 PSO 9174 7054 3251 3567 1915 3710 3.16. Conclusion In order to overcome the shortcomings associated with the gradient-based inverse heat conduction solvers in our steel cooling applications, such as the instabilities due to the thermocouple reading noise, and low temporal resolution, some gradient-free algorithms were investigated in this chapter. These methods are artificial neural networks, genetic algorithms, and particle swarm optimization. The artificial neural networks are capable of capturing the whole thermal history on the run-out table, but are not very effective in restoring the detailed behavior of the boundary conditions. Also, they behave poorly in nonlinear cases and where the boundary condition profile is different. Thus, their application in the run-out table heat transfer analysis will be limited to a general prediction of the whole history. GA and PSO are more effective in finding a detailed representation of the time-varying boundary conditions, as well as in nonlinear cases. However, their convergence takes longer. Comparison was made between GA and PSO variations in terms of both the computational efficiency and effectiveness in dealing with noisy data. PSO was found to improve efficiency    108    (i.e., reduce the required computational effort), especially in more complex test cases. A variation of the basic PSO, called CRPSO in this research, showed the best performance among the three versions. The effectiveness of PSO was also studied in the presence of noise. PSO proved to be effective in handling noisy data, especially when its performance parameters were tuned. The proper choice of the regularization parameter helped PSO deal with noisy data, similar to the way it helps the classical function specification approaches. An increase in the self- confidence parameter was also found to be effective, as it increased the global search capabilities of the algorithm. RPSO was the most effective variation in dealing with noise, closely followed by CRPSO. The latter variation is recommended for inverse heat conduction problems, as it combines the efficiency, i.e. a lower computational expense, and effectiveness, i.e. resistance to the measurement errors, required by these problems. Sequential formulation, i.e. solving the inverse problem for each separate time step was found more efficient than a whole domain implementation. Also, in the case of multi- dimensional problems with multiple sensors, the proposed multi-objective optimization was very successful in accelerating the solution. The relative speedup seemed to be independent of the number of spatial components. Introducing the concept of elite members to the algorithm also resulted in a faster convergence; however, the improvement was of less magnitude. It also looked more effective in simpler cases, i.e. smaller problems with fewer internal measurement points. In order to stabilize the algorithm, especially making it more effective in dealing with noisy measurement data, the concept of future time steps was borrowed from the function specification methods, and was applied to the sequential algorithm. It was very helpful in making the method more stable, especially when used in combination with a Tikhonov regularization term and tuning of the self-confidence parameter in PSO. The developed modified formulation for the inverse heat conduction problem and PSO implementation seem to be promising in terms of stability and performance for the cases similar to those studied in this research. While evolutionary optimization methods are very effective in dealing with inverse heat conduction problems in terms of stability and handling of noisy data, their very high computational expenses hinder the wide usage of these algorithm. Inexact pre-evaluation using surrogate models can be used to enhance the speed of evolutionary algorithms in solving the inverse heat conduction problems. In this strategy, the entire candidate solutions are first roughly evaluated using a much simpler surrogate model, and only the top performing members are    109    exactly evaluated using the full direct solvers, e.g. finite element method. Polynomial methods are the cheapest selection, both during the model construction phase, and the implementation. Radial basis function and feedforward neural networks have similar construction time, but in making predictions, RBF models perform faster. Kriging model is the slowest, especially when the number of unknowns increases. All the investigated surrogate models perform very well in predicting linear cases with no added noise. Nonlinear domains are much harder to predict. The most suitable model for noisy nonlinear cases is the RBF model. Since real-world cases of inverse heat conduction problem are normally nonlinear and noisy, this method is recommended for these applications. The best performing model management strategy is an adaptive selection scheme, based on the expected improvement of the objective function. Also, the most appropriate way to construct the surrogate model is to make a separate model for each sensor. If done properly, surrogate modeling can speed up the stochastical methods of solving the inverse heat conduction problem up to around 5 times. Finally, based on the performance that was measured on the test cases in this research, a sequential implementation of the CRPSO algorithm, using a multi-criteria formulation of the objective function, and using future time steps, Tikhonov regularization, and a tuned self- confidence parameter, in conjunction with a single-sensor RBF surrogate model, using the adaptive selection based on the expected improvement of the objective function, is the most appropriate solution method for solving inverse heat conduction problems similar to the ones studied in this research, with characteristics like the ones encountered on a steel run-out table.     110    4. Heat Transfer Behaviour under Multiple Circular Water Jets Impinging on a Moving Plate In this chapter, we study the boiling heat transfer characteristics of two rows of three impinging jets on a moving hot plate. Experiments are conducted in an industrial scale setup, and thermocouples are used to measure the temperatures just under the plate surface. Then these temperatures are used as the input in our inverse heat conduction algorithm to find the boundary heat flux variations in space and time across the width and along the length of the moving flat plate. The effects of nozzle configuration, jet line spacing, and plate speed are investigated. The results in this chapter are obtained using our developed CRPSO algorithm, as introduced in the previous chapter. The sequential implementation is always used, and multi-criteria formulation of the objective function, future time steps, Tikhonov regularization, and a tuned self-confidence parameter are adopted to increase the stability of the method. Also, a single-sensor RBF surrogate model, using the adaptive selection based on the expected improvement of the objective function is utilized to reduce the computational costs. 4.1. Experimental Setup The general characteristics of the experimental run-out table facility were described in chapter 1. In the experiments that are analyzed in this chapter, there are eighteen thermocouples attached to each plate. The ones that are located along the lateral direction are 12.6 mm apart from each other. The distance between the two sets of thermocouples is either 25.2 cm, or 50.4 cm, based on the distance between jet lines for that experiment. Figure  4.1 shows a top view of the TC configuration on the plate.   Figure  4.1. A Sample Plate, Showing the Dimensions and Thermocouple Locations  43 cm 120 cm 25.2 cm 22.7 cm TC9 TC8 TC7 TC6 TC5 TC4 TC3 TC2 TC1       data The s In ou succe possi confi flow temp  Fig  be ca wher accel our e wher 7.6 m Thermoco acquisition b ame softwa r experimen ssive jet lin ble jet confi gurations ar rate is con erature is 25 ure  4.2. Nozzl The jet ve lculated by e Vj is the j eration, m/s xperiments t The diam e Dn is the n m. uple signal oards, and re also contr ts, the nozz es is either gurations: i e shown in F stant at 15  °C. e Configurati locity (impa et velocity, 2; and H is t he jet veloc eter of the w ozzle diame s, as well as are sent to a ols the wate les are plac 25.4 cm or 5 n-line (no st igure  4.2. T  l/min, wh on with Respe ct velocity) m/s; Vn is he vertical d ity will be 5 ater jet Dj c ter. In our c V j = jD =  the flow rat  PC that use r flow rate t ed 1.5 m ab 0.8 cm. Th agger), half he plate spe ich gives a ct to Thermo staggered [3  is the veloc the water v istance from .49 m/s. an be calcul ase, the wat gHVn 2 2 ± jnn VVD ⋅ es from the s DASYLa hat exits the ove the plat e nozzle dia  staggered, ed is set to n outlet ve couples; (a) I 1]. ity at which  elocity at no  the nozzle ated by  er jet diamet  flow meter, b® 7.0 data  nozzles. e. The spac meter is 19 and fully st either 0.35 o locity of 0 n-line; (b) Ha  the water h zzle exit, g  exit to the er at the sta  are gathere acquisition ing between mm. There aggered. Th r 1.0 m/s. T .88 m/s. Th lf-staggered; its the plate  is the grav plate surfac gnation poin 111  d by two software.  the two are three ese three he water e water  (c) Full- , and can ( 4.1) itational e, m. For ( 4.2) t will be    112    The pressure at the stagnation point is given by  ( 4.3) where Pa is the atmospheric pressure and ρ is the density of water. In this case, the pressure at the stagnation point will be about 116,389 Pa. The saturation temperature of water at this condition can be found from a saturation table of water, and is around 103.6 °C. The plate passes through the system several times, so the input plate temperature varies from around 700 °C, down to just slightly more than 100 °C. This gives us the opportunity of studying different boiling heat fluxes, due to the various levels of temperature gradient. The experimental database used in this research is developed by Franco [31]. Twelve different experimental setups are analyzed, each with a different combination of jet line spacing, nozzle stagger, and plate speed. Table  4.1 shows these setups. In the following sections, we are going to study the effect of these parameters on the heat transfer behavior on the plate surface.  Table  4.1. Test Matrix. Set-up Jet Line Spacing (cm) Nozzle Stagger (mm) Plate Speed (m/s) S1 25.4 0 0.35 S2 25.4 0 1.0 S3 25.4 25.4 0.35 S4 25.4 25.4 1.0 S5 25.4 50.8 0.35 S6 25.4 50.8 1.0 S7 50.8 0 0.35 S8 50.8 0 1.0 S9 50.8 25.4 0.35 S10 50.8 25.4 1.0 S11 50.8 50.8 0.35 S12 50.8 50.8 1.0  2 2 1 jas VPP ρ+=    113    In order to determine the heat fluxes caused by impinging water jets on the surface of hot moving plate, we need to find the surface temperatures. Unfortunately, measuring the surface temperatures directly is not accurate. Our initial experiments with placing thermocouples on the plate surface showed an error of around 250 ºC, when the surface temperature was around 700 ºC [25]. Other methods such as infrared thermography may not be influenced by the water interaction on the surface, but are very hard to get in our application, since the water vapor covers the plate surface. Also these methods provide a snapshot of the surface temperature at each step rather than a continuous recording of temperature history during time. As a result, researchers use thermocouple readings from inside the plate to obtain the boundary heat flux values. The inverse algorithms that were developed in the previous chapters are used to obtain the surface temperature and heat fluxes during the water cooling process. 4.2. Uncertainty Analysis As it was mentioned before, type K thermocouples that are used in this study, have an error of approximately 2 °C. There is also some uncertainty in the values of thermophysical properties, such as heat conduction coefficient, and heat capacity. Thermocouple position and depth are the other sources of uncertainty. Since these values are the inputs of the inverse heat conduction analysis, their uncertainties will propagate through the algorithm, and will result in an uncertainty band around the output values, such as the surface heat flux and temperature. While it is possible to use analytical equations to calculate the uncertainty in boundary heat fluxes for simple linear inverse heat conduction problems, in case of nonlinear problems with multiple sensors, a numerical procedure is normally required [126]. This approach is called the “computerized uncertainty analysis” [127]. While there are several methods available for this approach [128], in this research, we have used a finite difference method, based on the sequentially perturbing the inputs, as detailed in [127]. Our calculations indicated that the uncertainty in the values of surface heat flux and temperature is between ±8% for the parallel flow zones, away from the stagnation point, and ±16% under the impinging jet. Also, it should be noted that the several sets of repeatability measurements were performed, and the data showed excellent repeatability [31].    114    4.3. Time Histories of Temperature and Heat Flux Based on our observations, the most significant factor affecting the heat transfer behavior is the range of plate surface temperature when entering the jet line. Thus, we have presented many of our results in three different entry temperature ranges: High (500 – 750 °C), mid (250 – 500 °C), and low (100 – 250 °C). Figure  4.3 - Figure  4.5 show the time history of thermocouple readings (inside temperature) from the experiments, and surface temperature just above the thermocouple and the heat flux at the location of thermocouple, both obtained by our inverse analysis, at five cross-section locations, for three different entry temperature ranges: high temperature (around 700 °C, Figure  4.3), mid temperature (around 400 °C, Figure  4.4), and low temperature (around 120 °C, Figure  4.5). The data shown here is from the S6 setup, i.e. a jet line spacing of 25.4 cm, a nozzle stagger of 50.8 cm (full stagger), and a plate speed of 1.0 m/s are used. That means that for the first pass, the 3rd thermocouple is under the impingement, and for the 2nd pass, the 7th one is in the impingement zone. As it can be seen in Figure  4.3(a), the inside temperature is initially around 700 °C (A). All temperatures fall after the line reaches the first jet line (B). The magnitude of the temperature drop is much higher for the TC location under the jet (E). For the rest of the TCs, the magnitude of the drop is more or less of the same order, but very gradually decreases when we move away from the impingement point (TCs 1 and 7). When the plate moves away from the jet line, the temperature partially recovers, due to the thermal mass of the plate (C). However, the existence of the second jet line stops the recovery process, and reduces the temperature once again (D). Note that the second jet line is impinging on a plate with a quite non-uniform temperature distribution, which is also lower than the initial entry temperature. Due to this non-uniformity, the history of the second jet line shows a larger diversity in behavior. Once again, the most significant temperature drop occurs at the impingement location (TC 7 this time). After moving away from the 2nd jet line, temperature again recovers, and due to the lack of a new jet line, the temperature distribution starts to become uniform again (G). Figure  4.3(b) shows the surface temperatures at the location of each TC. These values are obtained by using the inverse analysis. As expected these values are smaller than the inner temperatures, and if examined carefully, exhibit an earlier response to the jet line. This is of course due to the time lag that exists for the boundary conditions to be sensed inside the plate. It is also observed that the amount of the temperature drop is larger for the surface compared to the    115    temperature drop inside the plate. This is also related to the damping of changes in boundary condition inside the plate. These two effects (time lag and damping) are always present in inverse heat transfer analysis and contribute to the complexity of such problems. Figure  4.3(c) displays the surface heat flux at the location of each TC for the same experiment, again obtained by the inverse analysis. The heat fluxes are initially very small, and close to zero (A). After the TCs reach the first jet line, there is a rise in the amount of heat fluxes (B), with the impingement zone, experiencing the largest value of heat flux (E). The same thing happens for the second jet line, however since now the temperature distribution is not uniform anymore; there is a larger diversity in the heat flux behaviors. Figure  4.4(a), (b), and (c) show similar results as Figure  4.3(a), (b) and (c), respectively, but for an entry temperature of around 400 °C. The trends are qualitatively similar to the higher entry temperature case, but values are different. This is especially obvious when comparing the heat flux curves. As is seen in Figure  4.4(c), the peak heat flux value reaches a value more than twice of the peak heat flux of Figure  4.3(c). This shows that a more effective boiling mechanism is present in the mid range of surface temperatures. This can be related to a thinner vapor layer on the plate surface in this temperature range. This layer is thicker in the high temperature range, which means a film boiling heat transfer mechanism. In the mid temperature range, the momentum of the vertical impinging jet can break the thinner layer apart, and liquid can touch a wider area, so a partially nucleate boiling mechanism is responsible for higher heat fluxes. Finally, Figure  4.5(a) represents the last series of TC readings for the low entry temperature range. With an initial inner temperature of 120 °C (A), i.e. just about 20 °C higher than the boiling point, the plate is subjected to the first row of jets. As shown in Figure  4.5(b), this causes the surface temperatures to drop below the boiling point. Interestingly, in this case, there is not a major difference between the cross sectional locations, i.e. the impingement and parallel flow zones exhibit a more uniform temperature drop. This is more clear in Figure  4.5(c). As is seen in this figure, the peak values of heat flux (B) are not as different as previous cases. This is in accordance with the findings of Jondhale [83] that the significance of boiling heat transfer in the impingement zone compared to the boiling in the parallel flow zone is reduced when the surface temperature falls below 200 ºC. This is even more pronounced when the (even colder) plate passes the second jet line (D). Interestingly, in both cases of peak heat flux (B and D), the maximum heat flux is not at the location of impingement, but at somewhere between the    116    two adjacent impinging circular jets. This is in agreement with the findings of Slayzack et al. [81] about the water fountain in the interaction zone between two jets. Therefore, the high entry temperature range causes a relatively thick vapor layer that acts as insulation, and the heat transfer mechanism is film boiling, unless when an impinging jet with high momentum breaks down that layer. In the mid entry temperature range, the thinner vapor layer is a weaker insulation, and the dominant mode of heat transfer is mostly transition boiling. In the low entry temperature range, the local surface temperature falls quickly below the boiling point, and the heat transfer mode is non-boiling forced convection.        Figure  4.3. History of (a) Thermocouple Rea E (a) (b) (c) dings; (b) Su ntry Temper rface Temper ature ature; (c) Surface Heat Flu 117  x; High       Figure  4.4. History of (a) Thermocouple Re E (a) (b) (c) adings; (b) Su ntry Temper rface Temper ature ature; (c) Surface Heat Flu 118  x; Mid       Figure  4.5. History of (a) Thermocouple Rea E (a) (b) (c) dings; (b) Su ntry Temper rface Temper ature ature; (c) Surface Heat Flu 119  x; Low    120    4.4. Heat Flux vs. Surface Temperature Figure  4.6 show the variation of surface heat flux and heat transfer coefficient (h) versus the surface temperature, during a two jet line experiment for high, mid, and low entry temperatures, and for five of the thermocouples across the width of the plate. Remember that both the surface heat flux and surface temperature are obtained by running an inverse analysis on the readings of internal thermocouples. The surface heat transfer coefficient is simply obtained by dividing the values of heat flux by the temperature difference between the surface and the cooling water. Since the cooling water in all these experiments has a temperature of 25 °C, the heat transfer coefficient will be ( ) ( )25−=−= surface surface watersurface surface T q TT q h  ( 4.4) Figure  4.6(a) and (b), respectively show the heat flux and heat transfer coefficient vs. surface temperature for the high entry temperature range (around 700°C). The letters on the graph correspond to the same letters in Figure  4.3. As we move from (A) to (B), the temperature will fall very fast, as the impingement point will produce the highest heat flux (E) for the thermocouple that is directly under the jet. After passing the first jet line, the surface temperature partially recovers (C). The same thing happens during the impingement of the second jet line, however, with a lower initial surface temperature and a more non-uniform distribution. The plate finally exits both jet lines with a surface temperature of around 670 °C (G). Figure  4.6(c) and (d) show the same thing but for the mid entry temperature range (around 400 °C). Here the behavior is very much like the previous case, however, the heat transfer rate is much higher. This is especially significant in the under impingement area (E and F). Also during the temperature recovery after impingement (moving toward points C and G), there is a dent in the graph, i.e. the temperature starts to recover at a high rate, then stops, a little bit of cooling occurs, and then temperature recovers at a slower rate. This should be related to the lower thermal energy that is stored in the plate. Finally, Figure  4.6(e) and (f) show the same quantities, but for a low entry temperature experiment (around 130 °C). Here the behavior is very different from the previous two cases, especially because after the first jet line, the surface temperature falls below the boiling point, so for the second jet line, the heat transfer mechanism is not boiling, but a forced convection by the water flowing over the plate. Again, for both jet lines, maximum heat transfer is not happening    121    in the impingement point, but somewhere between the two jets that the interaction fountain is formed.        Figure  4.6. Heat F (a) (c) (e) lux and Heat    Transfer Coe for Differen fficient vs. Su t Entry Temp rface Temper erature Rang (b) (d) (f) ature Across es the Width of 122  the Plate    123    4.5. Lateral Distribution of the Heat Flux Since we only have nine thermocouples that are clustered in the central part of the plate (see Figure  4.1), we are going to use the three nozzle configurations (see Figure  4.2) to study the variation of the heat flux under and between the nozzles. The lateral location between the jets is normalized with respect to the distance between the first and last thermocouples (101.6 mm) with TC1 and TC9 at the normalized locations of zero and one, respectively. The lateral distribution of heat flux is displayed in Figure  4.7, for the moment of impingement of the first jet line. In the three cases, the jet impinges on thermocouples 5 (Figure  4.7 (a)), 4 (Figure  4.7 (b)), and 3 (Figure  4.7 (c)). For each case, three heat flux distributions are displayed, one for each of the three temperature ranges (high, mid, and low). As is observed in these figures, the maximum heat flux values happen at the mid entry temperature range. Here, we see that the heat flux distribution is more non-uniform in the mid entry temperature range as well, with the peak heat flux occurring at the point of impingement. Also, we see that at low temperature range, the maximum heat flux is not occurring at the impingement point. The heat flux distribution is nearly symmetrical around the impingement point for the inline nozzle configuration. However, half-staggered and full-staggered nozzle configurations result in more complicated flow fields that result in an unsymmetrical heat transfer behavior around the impingement zone. This is especially more pronounced when the entry temperature is in the high or low temperature ranges. These profiles are going to be especially useful in studying the thermal stresses in plates during a cooling process on the runout table in a material processing mill, which may sometime result in a deformation in the cooled plate.       Figure  4.7. Lateral Heat Flux Distribution Stagger; (b) (a) (b) (c)  at the Mome Half Stagger; nt of Impinge  (c) Full Stagg    ment of the F er. irst Jet Line; 124  (a) No    125    4.6. Lateral Distribution of the Surface Temperature Figure  4.8 shows the surface temperature distribution across the plate after the second jet line has passed versus the normalized lateral location between TC1 and TC9. This data is also going to be highly useful in studying the thermal stresses in the plate, as well as making sure that different parts of the plate follow a similar temperature history profile, and thus, the mechanical properties are going to be uniform. Again for each nozzle configuration, three lateral temperature distributions are displayed, one for each of the three temperature ranges (high, mid, and low). As it is seen in Figure  4.8(c), the most uniform distribution is happening in the case of fully staggered nozzle configuration. This is especially true at low entry temperature range. The most severe case of non-uniform temperature distribution is usually the case of mid entry temperature. Since this figure shows the temperature profile after both jet lines have hit the plate, one can also judge the time that it takes for the plate to reach a more homogeneous temperature distribution. As is seen in parts (b) and (c) of Figure  4.8, where the lateral location of the second line of nozzles is different than the first, the effect of the first jet line is dissipated more slowly in mid surface temperature range. This can be due to the fact that at mid surface temperature range, there is a very high heat flux occurring at the impingement point, resulting in a very effective temperature drop.       Figure  4.8. Lateral Temperature Distribution (b) Half (a) (b) (c)  After the Imp  Stagger; (c) F ingement of ull Stagger.    the Second Jet Line; (a) No 126   Stagger;    127    4.7. Longitudinal Distribution of the Heat Flux As discussed earlier, although many researchers do not distinguish the two different parallel flow zones, the flow and heat transfer fields between two successive jet rows is different from the ones between two nozzles in one jet line. One reason is the jet rows have more distance from each other compared to the nozzles in one jet line. Also, the plate velocity vector is now oriented in a different direction with respect to the spreading water. Nozzle stagger configuration and jet line spacing are the two parameters that affect the longitudinal distribution of heat flux. Figure  4.9 shows the heat flux distribution between two successive jet lines for some of the thermocouples (across the plate width) for the high temperature range. Figure  4.9(a), (b), and (c) show the heat flux distribution for jet line spacing of 25.4 cm, and for no stagger, half- staggered, and full-staggered configurations. Figure  4.9(d) shows the distribution for the half- staggered configuration, but for a jet line spacing of 50.8 cm. As it can be seen in this figure, the heat flux is more or less uniform between two successive jet lines in this temperature range. Only in the half-staggered case, with a jet line spacing of 25.4 cm (Figure  4.9(b)), there is an increase in the heat flux somewhere between the two jet lines, probably due to some kind of a stagnation line. Comparing parts (b) and (d) of this figure shows that increasing the jet line spacing may result in the dissipation of this heat flux spike. Also, notice that in all cases the heat flux between two jet lines is smaller than the one between two impinging jets in one jet line. Figure  4.10 displays the same graphs but for the mid temperature range. As shown in these figures (in this temperature range and for a low nozzle spacing, cases-a, b, and c) the heat flux in the interaction zone between two jet lines rises to a value that is comparable to the value of heat flux in the interaction zone between two nozzles of a same jet line. This effect is, however, dissipated when the jet line spacing is increased, probably due to a weaker interaction fountain between two successive jet rows. Also notice that the heat flux values are higher to the right of the peak values in the impingement point. It means that the heat transfer in the downstream of the impingement line (where the plate motion and the water spreading directions are the same) is higher. Figure  4.11 shows the same graphs for the low entry temperature range. Here, there is a considerable amount of heat flux happening in the region between two jets. The reason is that in this region, the water remains longer on the surface before evaporation and the pool of water is still cooling the plate when the thermocouple is between two impinging jets. Also the vapor layer       on th neces some but in of en point Figur Spaci e surface is sary in orde  of the therm  the paralle try tempera .  e  4.9. Longitu ng: 25.4 cm; (  not presen r to break it ocouples, l flow zone ture, the pe (a) (c) dinal Distrib b) Half Stagg t or is very  down in the the peak hea after the jet ak heat flux ution of Heat er, Jet Line S Half Stagge thin, so the  low entry t t flux is no line. This is  across the  Flux for the H pacing: 25.4 c r, Jet Line Sp   vertical mo emperature t happening  similar to th  plate width igh Entry Te m; (c) Full St acing: 50.8 cm mentum of s. Notice tha  on the wat e observati  is not und (b) (d) mperature; ( agger, Jet Lin .  the water j t in this cas er impingem on that for t er the impi a) No Stagger e Spacing: 25 128  et is not e and for ent line, his range ngement   , Jet Line .4 cm; (d)       Fig spaci ure  4.10. Lon ng: 25.4 cm; ( (a) (c) gitudinal dist b) half stagge ribution of he r, jet line spa stagger    at flux for the cing: 25.4 cm , jet line spaci   mid entry te ; (c) full stagg ng: 50.8 cm (b) (d) mperature; (a er, jet line spa ) No stagger, cing: 25.4 cm  129   jet line ; (d) half       Figur Spaci 4.8. E betw which whol e  4.11. Longit ng: 25.4 cm; ( ffect of th The ratio een two imp  in turn ch e heat transf (a) (c) udinal Distri b) Half Stagg e Plate Sp  between th inging jet li anges the s er process. bution of Hea er, Jet Line S Half Stagge eed e plate spe nes to hit th urface temp Also, plate t flux for the pacing: 25.4 c r, Jet Line Sp  ed and jet e plate, and erature for speed has an Low Entry Te m; (c) Full St acing: 50.8 c line spacin  hence the a the second  effect on t (b) (d) mperature; ( agger, Jet Lin m g determin mount of te  jet line, an he values of a) No Stagger e Spacing: 25 es the time mperature r d hence, af  the heat flu 130   , Jet Line .4 cm; (d)  it takes ecovery, fects the x versus    131    the surface temperature. Figure  4.12(a) and (b) display the value of heat flux under the impingement zone, each for 6 sets of experiments, with jet line spacings of 25.4 and 50.8 cm, respectively. The symbols on the graph show the four different combinations of plate speed and jet line spacing, and the connecting line pattern is representing the nozzle stagger configuration. Solid lines represent the cases with no stagger, while dashed and dotted lines stand for the half- staggered and full staggered configurations, respectively. As it can be observed in these figures, there is not a clear difference between different setups in the temperatures below 400 °C. However, in the temperatures higher than this there is a clear difference between six setups with a plate speed of 0.35 m/s (higher branch) and the ones with a plate velocity of 1 m/s (lower branch). In other words, in the high entry temperatures, the heat transfer rate is much lower when the plate is moving at a higher velocity. This can be related to the fact that at high plate speeds, the vapor layer that is formed on the surface, and works like an insulator, is more strongly attached to the plate, so the impinging jet has a bigger challenge in breaking down the vapor layer and getting in touch with plate. As a result, a more stable film boiling heat transfer will be present, which causes lower heat transfer rates. This temperature range in which the plate speed affects the heat transfer rate is in accordance with the experiments done by Gradeck et al. [84]. Obviously, in order to better understand this behavior, more experiments are needed with a wider range and variety of plate speeds. Notice that this effect is only limited to the impingement zone. Figure  4.12 (c) and (d) display the variation of the surface heat flux between two nozzles versus the surface temperature. Here, however, there is not any clear correlation between the plate speed and the heat transfer rate.       Figu and Spa 4.9. an in spaci condu re  4.12. (a) an  50.8 cm, Res cings of 25.4 Conclusion Boiling h dustrial scal ng are varie ction proc (a) (c) d (b) Imping pectively; (c) and 50.8 cm,  eat transfer e experimen d in the exp edure. Surfa ement Heat T and (d) Jet In Respectively; Li on a hot pla tal setup is eriments. Th ce tempera ransfer Versu teraction Hea Solid Lines: N nes: Full Stag te cooled by  studied. No ermocouple ture is the s Entry Tem t Transfer Ve o Stagger; D gered  multiple w zzle config  readings ar most signi (b) (d) perature; Jet rsus Entry T ashed Lines: ater jets and uration, plat e analyzed u ficant facto  Line Spacing emperature; J Half Stagger;  multiple je e speed, an sing an inv r affecting 132  s of 25.4 et Line  Dotted t lines in d jet line erse heat the heat    133    transfer behaviour on the plate. At high temperatures, the plate surface is covered with a condensed vapour layer that acts as an insulation blanket, and requires an impinging jet with high momentum to break it down. This translates into a stable film boiling regime, with reduced values of heat fluxes. Heat transfer rate increases at medium temperatures where the vapour thickness is smaller. For low entry temperatures in a multi jet line experiment, the heat transfer is lowered again. In the low temperature region, the maximum heat flux does not occur at the impingement point, but at a location between the jets, due to formation of an interaction water fountain. The most important effect of nozzle stagger is the uniformity of heat transfer across the width of the plate. The most uniform distribution is happening in the case of fully staggered configuration. The heat transfer between two successive rows of jet lines is affected by the jet line distance. In higher jet line distances, the interaction effects become less significant, and a more uniform distribution is observed. However, when the jet lines are closer, the effect of interaction is more pronounced. In medium and high temperatures, there is a spike in the heat flux values in the interaction zone. In the low temperature range, because the water does not evaporate quickly, a pool of water forms on the plate surface, resulting in a flattened heat flux versus time profile. The plate speed affects the heat transfer rate under the impingement point for the higher surface temperatures. In the high entry temperatures, the impingement heat transfer rate is lower when the plate is moving at a higher velocity. The plate speed however does not significantly change the heat transfer behavior in the other parts of the plate surface.        cooli that u previ simu used valid probl mutu [129] chapt mech this c heat signi Final and i weak  F 5. 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A e controlle lates the ste ing the stres , for both hosen based re are three ructural, an  5.1 shows, the other tw gly affects  in the volu sition also l capacity. e temperatu  on the ther model the st etween the Th ield durin hermal fiel  finite elem d cooling of el phase com s and defle the stationa  on the test  different ty d structural the levels o physics as the mechan me. This al changes the However, t re depende mal and mi rong and m ree Physical g Water d during the ent based a  steel strips position in ction values ry and mov  case in the pes of phy fields, and of these cou  described i ical field th so is describ  thermal pr his last eff ncy of the crostructura edium coup Fields in Coo Jet Coolin  water impi pproach is p  (as discuss  each time . The DQSK ing plates. literature. sics involve all three of plings are n details lat rough chan ed in detail operties, suc ect is relati  thermal pr l fields is ve lings, but ne  ling of Steel S 134  g ngement resented ed in the step, and  steel is  For the d in this them are different er in this ging the s later in h as the vely less operties. ry weak, glect the trips    135    5.1. Phase Transformation Modeling There are three main approaches to find the state of phase transformation during the cooling: 1. Using the laws of thermodynamics, kinetics, and diffusion. 2. Using the continuous cooling transformation (CCT) diagram: ⎪⎪⎭ ⎪⎪⎬ ⎫ ⎪⎪⎩ ⎪⎪⎨ ⎧ ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ − − −= n fTsT TsTbXX exp10   ( 5.1) where X0 is described as the volumetric rate of intended phase at transformation termination, Ts and Tf are the transformation start and finish temperatures, respectively, and n and b are constants. Because the cooling rate at each point is different, we will have different values for Tf, Ts, n and b. This approach was developed by Boyadjiev, et al. [130]. 3. Using the isothermal temperature-time transformation (TTT) diagrams: The initial time should be defined as the time that the transformation begins under continuous cooling conditions. The first category of the models are the most accurate ones, but have not been used extensively in the literature due to the lack of information about the exact characteristics of thermal field history and the model calibration coefficients. Previous research (in UBC group) has provided the required information to use in the above models [131,132]. The algorithm that is used here is based on the work of Liu, et al. [132], and falls in the first family of the models. In this algorithm, the static recrystallization kinetics is simulated by the Johnson-Mehl-Avrami- Kolmogorov (JMAK) theory. The transformation start temperature was also found following the method suggested in the work of Militzer, et al. [131].         Figur Starti Δt. T wher (TDc and auste and T TS, w wher fα. Th fracti Figure  5.2. S The first s e  5.2, TN  is ng from this hen, the foll e ) ⎜⎝ ⎛ = 453000 c0 is the at nite and fer i is the tem hen the valu e ݀௘௙௙ఊ  is the The next ese calcula on as: chematic Rep tep in this a the tempera  point, we d owing summ =aR + + 9.8339 1 T c omic fractio rite, respect perature at t e of Ra beco  effective au step would b tions start a resentation o lgorithm is ture where t ivide the co ation value (∑ = n i ic TD 1 2 ( )⎟⎠ ⎞− ex 273 1 00 c n, ܿ௘௤ఊ  and ively, which he ith time mes larger γ eqc R* 0= stenite grai e to calcula t T = TS. In f Cooling on finding the t he nucleatio oling path to  is calculate ) ( )( )− − eieq ieq cTc Tc αγ γ (⎢⎣ ⎡ −− 17767p  ܿ௘௤ఈ  are th  are known step in degr than a critic ( ) γ eff i d cT c 0 025. − n size, and i te the fracti  each time the Run-out T ransformati n starts, and  small segm d along the ( )Δiq tT c0 )⎜⎝ ⎛26436 0c e equilibriu  as a functio ees Celsius al value, R*,  s again know on of austen step, we def  able (Tempe on start temp  is known a ents, each w cooling path − + 0 273 1 T m concentr n of tempe . Transform  where: n for each ite that is tr ine the equ rature vs. Tim erature, TS. nd equal to ith the leng , starting fro ⎥⎦ ⎤⎟⎠ ⎞00022. ations of c rature for ea ation starts, steel compo ansformed t ilibrium tran 136  e)  In 797 °C. th of m TN : ( 5.2) ( 5.3) arbon in ch steel,  i.e. Ti = ( 5.4) sition. o ferrite, sformed    137    ( ) ( ) ( )ieqieq ieqeq TcTc cTc f αγ γ α − − = 0  ( 5.5) and the real transformed fraction can be obtained as: eqfFf ααα ⋅=  ( 5.6) where at T = Ts the value of ܨఈ is 0.001, and at each next time step, it is updated through:  t dt dFFF oldnew Δ+= ααα  ( 5.7) where ( )( )ααα FTbdt dF i −= 1  ( 5.8) and ( ) ( ) ⎭⎬ ⎫ ⎩⎨ ⎧ − −−−= α α f cTTb ii 1 24.11860001.04.4exp 0  ( 5.9) For a more detailed discussion of this model, as well as the physical significance of various terms, refer to [131,132]. 5.2. Thermal Stress Modeling If the temperature in a homogenous body changes by ΔT, the total strain in the body is consisted of two parts, the free thermal expansion, plus the strains related to the stress state in the body, and can be expressed by the following equations: ( ) ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ ⎨ ⎧ = = = ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎠ ⎞⎜⎝ ⎛ +−+= ⎥⎦ ⎤⎢⎣ ⎡ +−+= ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎠ ⎞⎜⎝ ⎛ +−+= Gzxσzxε Gyzσyzε Gxyσxyε EyσxσνzσαΔTzε EzσxσνyσαΔTyε EzσyσνxσαΔTxε  ( 5.10)    138    where ε is the strain, σ is the stress, ν is the Poisson’s ratio, E is the Young’s modulus, and G is the shear modulus. Solving the above equations for stress yields: ( ) ( ) ( ) ( ) ( ) ( ) ⎪⎪⎩ ⎪⎪⎨ ⎧ === Δ−+++= Δ−+++= Δ−+++= xzxzyzyzxyxy yxzx zxyx zyxx T T T μεσμεσμεσ βεελεμλσ βεελεμλσ βεελεμλσ ;; 2 2 2  ( 5.11) where λ, μ, and β are defined as: ( )( ) ( ) ( )⎪⎩ ⎪⎨ ⎧ −= += −+= ναβ νμ νννλ 21 12 211 E E E  ( 5.12) The equations of equilibrium for temperature change in the body are: ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ =+ ∂ ∂ + ∂ ∂ + ∂ ∂ =+ ∂ ∂ + ∂ ∂ + ∂ ∂ =+ ∂ ∂ + ∂ ∂ + ∂ ∂ 0 0 0 Z zyx Y zyx X zyx zyzxz yzyxy xzxyx σσσ σσσ σσσ  ( 5.13) where X, Y, and Z are the body forces per unit volume. The boundary conditions are given by: ⎪⎩ ⎪⎨ ⎧ =++ =++ =++ Szyzxz Syzyxy Sxzxyx Znml Ynml Xnml σσσ σσσ σσσ  ( 5.14) where l, m, and n are the direction cosines of the outward drawn normal and Xs, Ys, and Zs are the surface forces per unit area. In cases where the boundary conditions, temperature distribution, geometry, or material properties are more complicated, using analytical methods to solve for the thermo-mechanical properties is impossible. Using numerical methods, such as finite elements is very common to study these problems. While in the above equations, we assumed that the change in the temperature affects the strain values only through the thermal expansion coefficients, however in these controlled cooling applications, there are normally two main sources for the strain in the steel. One is the thermal strain that is related to the temperature gradient and the thermal expansion coefficient of the mixture of phases (obtained using the rule of mixtures). The other one is the volumetric    139    strain associated with the phase transformation, i.e. the change in the volume of the same amount of steel during austenite to ferrite transformation. In this research, we combine the effects of these sources by creating a total coefficient of expansion: changephasethermaltotal −+= ααα  ( 5.15) where αtotal is the total expansion coefficient, αthermal is the weighted average of the thermal expansion coefficients of the phases in each location in the steel sample, and αphase-change is the phase change expansion coefficient. This latter component can be calculated in each time step i as: T V VV i ii changephase Δ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − = − − − 3 1 1α   ( 5.16) where Vi and Vi-1 are the mixture specific volumes in the current and the last steps, respectively. In each step, the mixture specific volume can be obtained using the rule of mixtures: ( )iii XVXVV −⋅+⋅= 1γα   ( 5.17) and αV  and γV are the specific volumes of ferrite and austenite, respectively. These values are related to the lattice parameters of ferrite and austenite (aα and aγ, respectively), as: ( ) 321 αα aV =  and ( ) 341 γγ aV =  ( 5.18) The lattice parameters vary with temperature, and in the case of austenite also with the carbon fraction value [133]. 5.3. Computational Modeling A commercial finite element software is used to model the coupled thermal and structural field. Shell elements are used to discretize the domain. On the top surface of the plate, the heat fluxes are applied as the thermal boundary condition. After the thermal and microstructure simulations are performed, results are written to a file, which is subsequently read by the structural simulation. Since the effect of structural field on the thermal and microstructural fields is weak (see Figure  5.1), there is no need to feed the structural results back into the thermal/microstructural models. In all the following validation cases, as well as the new simulations, grid and time step independency has been investigated, i.e the mesh is refined until changes in the results were less than 5%.       5.4. V simil avail 5.4.1 uses movi mode fields boun at dif stage nume agree alidation In order ar cases fro able in the li . Validation This test a convection ng plate, a l, and a tw  in the pla dary conditi ferent locat  cooling. Figure  5.3. P  Figure  5. rical simul ment is obs  Cases to validate m literature terature.  Case I case is base  correlation simplified ti o-dimension te. Figure ons. Residu ions and un roblem Dime 4 shows the ation result erved betwe the modelin  are simula d on the wo  to model th me-tempera al finite di  5.3 shows al stresses ar der differen nsions and Te  results of s), and the en the result g of therma ted here, a rk of Yoshi e thermal b ture-transfo fference me a schematic e then plott t cooling st mperature H Yoshida’s s results obt s. l stresses o nd the resu da [86]. As oundary co rmation (TT thod to sim  of the pr ed versus th rategies, e.g istory in Vali tudy (strain ained by th n the run-o lts are com was mentio ndition on t T) for the p ulate the th oblem, and e distance fr . uniform, e  dation Case I  gauge rea e present s ut table, tw pared with ned in Chap he top surfa hase transf ermal and s the simple om the edge arly stage, by Yoshida [8 dings as we tudy. A ve 140  o of the the ones ter 1, he ce of the ormation tructural  thermal  of plate and later 6]. ll as the ry good       Figur 5.4.2 a sim show cond proce the r differ  e  5.4. Validati . Validation The work plified temp s the geome itions. Agai ss. ABAQU esidual stre ent location on Case I; Co  Case II  done by Zh erature-tim try, and Fig n, a TTT d S finite ele sses and di s. F -140 -90 -40 10 60 0 R es id ua l s tr es s ( M Pa ) Str mparison bet Result ou, et al. [9 e curve as th ure  5.6 dis iagram is ment code i splacements igure  5.5. Geo 0.1 0 D ain gauge ween the Cur s in the Liter 0] is the bas e cooling b plays the tra used for fin s used to sim  are report metry of Vali .2 0.3 istance from Yoshid rent Results a ature [86]. is of the sec oundary con nsverse and ding the p ulate the t ed for diffe dation Case I 0.4 0.5  the edge (m a Cu nd the Exper ond validat dition in the  longitudin hase transf hermal and rent coolin  I [89] 0.6 ) rrent study  imental and N ion case. Th ir model. F al thermal b ormation du structural fi g strategies 0.7 141  umerical ey apply igure  5.5 oundary ring the elds, and , and at       Figur maxi Figu e  5.6. Temper Figure  5. mum differe re  5.7. Valida (a) ature Distrib 7 shows Zh nce is less t tion Case II; -10 0 10 20 30 40 50 60 70 0 R es id ua l s tr es s ( M Pa ) ution in Valid ou’s result han 7%, wh Comparison 0.1 Di Zhou, ation Case II [89]  s, as well ich validates between the C Literature [ 0.2 0.3 stance from  et al.  in (a) Transv as the resul  our solutio urrent Resul 90] 0.4 the edge (m) Current stu (b) erse and (b) L ts obtained n method. ts and the Nu 0.5 dy ongitudinal D  by this stu  merical Resul 0.6 142  irections dy. The ts in the       5.5. struct our k are im impin The p the ti heate are fe fluxe readi  Stationary In this sec ural field in nowledge, h Figure  5.8 plemented ging on its late surface p of the ther d up to 900 d into the C s for each th ngs of the fi Fi  Plate tion, we hav  a hot statio as not been  shows the inside of it a center, we a  is 240 by 2 mocouple is °C, and then RPSO inver ermocouple rst to the eig gure  5.8: Sche e used the h nary steel pl studied in th steel plate u t different r re assuming 40 mm, and  1 mm unde  is cooled d se solver th  location, i.e hth thermoc matic Arrang eat fluxes o ate cooled b e literature. nder the imp adii. Since t  that the tem  is 6 mm thi r the top sur own by the at was devel . different r ouple, in or ement of The btained by X y a circular  inging jet, a he plate is s perature dis ck. The ther face of the p impinging w oped in Cha adial locatio der from lef rmocouples u u and Gad impinging w nd the eigh tationary an tribution wi mocouples a late. The p ater jet. Th pter 3, whic ns. Figure  5 t to right.  nder Imping ala [76] to s ater jet. Th t thermocou d the water j ll be symme re 16 mm a late is origin ermocouple h produces .9 shows th ing Jet 143  tudy the is, to ples that et is tric. part, and ally  readings the heat e       Fi  struct jet is doma secon out o over conto defle the d gure  5.9: Top Based on ural analysi  impinging o in, shell ele Figure  5. ds after the f the plane the plate re urs of von ction vs. the ifference in  Surface Hea  the symme s. Figure  5. n the cente ments are us 11 shows th  start of the displacemen aches a ne Mises stres  strip thickn temperature Figure  5.1 t Flux History The try assump 10 shows th r and the w ed. e temperat  cooling pro t (in mm) a arly uniform s at the sam ess at the e  across the p 0: Plate Geom  Obtained By rmocouple Lo tion, only o e whole ge ater moves r ure distribu cess. Figure t the end o  value of e time. Fig nd of the co late increas etry; Only th  Inverse Ana cations. ne quarter ometry and adially ove tion on the   5.12(a) sh f the coolin around 130 ure  5.13 sh oling proce es, which re e Shaded Ar  lysis of Temp of the plat the modele r plate. In o top surface ows the con g process w  °C. Figure ows the ma ss. As expec sults in a hig  ea is Modeled erature Readi e is modele d area (shad rder to discr  of the pla tours of the hen the tem   5.12(b) sh ximum out ted, in thick her deform . 144  ngs at d in the ed). The etize the te a few  vertical, perature ows the of plane er strips ation.       Fig  Figur  ure  5.11: Tem e  5.12: Conto Figure  5.13 perature Dis (a) urs of (a) Z D . Maximum O tribution in t isplacement ( at the E ut of Plane D he Plate a Few  Out of Plane) nd of the Coo isplacement v  Seconds afte  in Millimeter ling Process s. Strip Thick  r the Start of (b) s and (b) von  ness for the S  the Cooling P  Mises Stress tationary Pla 145  rocess  in Pascals te       5.6. M mode previ confi each intere realit of the coilin both one temp fluxe simu the u  oving Pl In this sec l the structu ous chapter guration (in There are pass in isol sting from y. The other  furnace, an g temperatu of these str of the three erature). Th s until it re lations, inste Figure  5. ndeformed o Figure ate tion, we us ral field du will also be line, half-sta  several way ation, startin an academic  approach w d then appl re, or any ategies are u  temperatu en, we start aches temp ad of a mov 14 shows an ne.   5.14. The De e the results ring the co  studied her ggered, and s that one c g with a fla  point of v ould be to y the bound lower tempe sed. First, w re zones di  from the h eratures of ing plate, th  oblique vie formed and th  of Chapter oling proce e. These pa  full-stagger an study the t plate with iew, it is no start with a h ary conditio rature that e apply th scussed in ot plate com around 500 e heat fluxe w of the de e Edge of the 4 (heat flux ss. The para rameters are ed), and jet  history of t  no residua t a good re ot flat plate ns in time, u we might b ree isolated the previou ing out of , 300, and s move over formed mov  Undeformed es vs. time f meters that  the plate m line spacing he structura l stress in it presentation  with no re ntil the tem e interested  passes of th s chapter ( the furnace 100 °C. N  the plate w ing strip, a  Strip in a M or moving p  were studi oving spee . l field. We c . While this  of what ha sidual stress perature rea  in. In this e heat flux high, mid, , and apply ote that in ith respect t s well as the oving Case 146  lates) to ed in the d, the jet an study  may be ppens in , just out ches the research, , each in and low  the heat all these o time.  edge of     147    5.6.1. Single Passes Single pass simulations were performed for three temperature zones (high, mid, and low temperatures). Figure  5.15 - Figure  5.17 show the results of these simulations for different setups. In each case, we are assuming that the plate is entering the cooling station with uniform temperature distribution and no initial stress or deflection.     148     Figure  5.15. Single Pass Results for High Temperature Range Cooling 395 445 495 545 595 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 24 26 28 30 32 34 36 38 40 0 25.4 50.8 M ax  D isp la ce m en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 27 28 29 30 31 32 33 34 35 36 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    149     Figure  5.16. Single Pass Results for Mid Temperature Range Cooling. 155 165 175 185 195 205 215 225 235 245 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 4 6 8 10 12 14 16 18 20 0 25.4 50.8 M ax  D isp la ce m en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 9 10 11 12 13 14 15 16 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    150     Figure  5.17. Single Pass Results for Low Temperature Range Cooling   9 14 19 24 29 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 0 1 2 3 4 5 6 0 25.4 50.8 M ax  D is pl ac em en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 0 0.5 1 1.5 2 2.5 3 3.5 4 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    151    As seen in the above figures, higher maximum temperature difference across the plate width results in a higher maximum von Mises stress, and maximum displacement. In each of these figures, values are typically larger for the no-stagger (inline) cases (S1, S2, S7, and S8), followed by the half-staggered ones (S3, S4, S9, and S10). The fully-staggered setups (S5, S6, S11, and S12) have the lowest values of the lateral temperature variations, as well as the lowest values of the von Mises stress and displacement. The most significant factor in controlling the stress and flatness in the strips is the jet configuration, and the most desirable configuration is the one that promotes a more uniform distribution of temperature across the plate width. Among the cases with the same nozzle configuration, plate speed is the next important factor in determining the values of stress and deformation. Plates traveling at slower speeds are cooled down more uniformly, and will have less stress and deflection in them. Finally, longer jet spacing gives the opportunity of a more uniform cooling and causes less deflection in the plate. However, the effect of jet line spacing and plate speed is much less significant than the effect of jet stagger. As expected, the cooling in the low temperature zone has the lowest effect on the plate deformation. It should be noted that in this temperature range, there is practically no phase transformation and already all of the steel is transformed to ferrite. 5.6.2. Multiple Passes For the results in the previous section, we were assuming an initial uniform temperature distribution, and no initial stress or deflection in the plate. In Figure  5.18 - Figure  5.20, we will study the effect of the initial non-uniform temperature distribution and stress and deflection in the plate. In order to do this, we start with a uniform hot plate with no stress and deflection, coming out of furnace at about 750 °C, and then through multiple passes through the cooling station, we lower the temperature to 500, 300, and 100 °C, and display the results in Figure  5.18, Figure  5.19, and Figure  5.20, respectively.    152     Figure  5.18. Results for a Plate Cooled Down to 500 °C 215 235 255 275 295 315 335 355 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 19 21 23 25 27 29 31 33 35 37 39 0 25.4 50.8 M ax  D is pl ac em en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 19 21 23 25 27 29 31 3 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    153     Figure  5.19. Results for a Plate Cooled Down to 300 °C 235 245 255 265 275 285 295 305 315 325 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 2 2.5 3 3.5 4 4.5 5 5.5 6 0 25.4 50.8 M ax  D is pl ac em en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 0 0.5 1 1.5 2 2.5 3 3.5 4 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    154     Figure  5.20. Results for a Plate Cooled Down to 100 °C   75 85 95 105 115 125 135 0 25.4 50.8 M ax  v on  M is es  S tr es s ( M Pa ) Jet Stagger (mm) 0.5 1 1.5 2 2.5 3 0 25.4 50.8 M ax  D is pl ac em en t ( m m ) Jet Stagger (mm) 27 32 37 0 25.4 50.8 ac ro ss  th e Pl at e (° C ) Jet Stagger (cm) Plate Speed =1 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 1 m/s, Jet Line Spacing = 50.8 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 25.4 cm Plate Speed = 0.35 m/s, Jet Line Spacing = 50.8 cm 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 25.4 50.8 M ax  Δ T ac ro ss  th e Pl at e (° C ) Jet Stagger (mm)    155    The trend for the effect of jet stagger, jet line spacing, and plate speed are similar to those observed in section  5.6.1. It is also observed that lowering the final coiling temperature will result in lowering the stress and deflection. This is due to the fact that thermal contraction of ferrite will cancel out some of the volumetric expansion due to the austenite to ferrite transformation. 5.6.3. Effect of Plate Thickness Figure  5.21 shows the variation of the maximum out of plane displacement for the moving plate versus the strip thickness, for case S1, when continuously cooled down to around 300 °C. The trends for all of the moving plate cases are similar, and resemble the trend for the stationary plate, as depicted in Figure  5.13.   Figure  5.21. Maximum Out of Plane Displacement vs. Strip Thickness for the Moving Plate  5.7. Conclusion In this chapter, the structural behavior of a hot steel strip when subjected to cooling by circular water jets was studied, by combining the heat transfer, phase transformation, and structural models. The heat transfer model uses the heat fluxes obtained by an inverse analysis of the thermocouple readings, and the phase change model is a detailed simulation using the 0 1 2 3 4 5 6 0 2 4 6 8 10 12 M ax im um  O ut  o f P la ne  D isp la ce m en t ( m m ) Strip Thickness (mm)    156    Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory. The model is validated using sample cases in the literature. The cooling of both the stationary and the moving plates were studied. It was found that the maximum temperature difference across the plate width is directly affecting the maximum von Mises stress and displacement. Also, these values are typically larger for the no-stagger (inline) cases, followed by the half-staggered ones. The fully-staggered setups have the lowest values of the lateral temperature variations, as well as the lowest values of the von Mises stress and displacement. The most significant factor in controlling the stress and flatness in the strips is the jet configuration, and the most desirable configuration is the one that promotes a more uniform distribution of temperature across the plate width. Plate speed is the next important factor in determining the values of stress and deformation. Plates traveling at slower speeds are cooled down more uniformly, and will have less stress and deflection in them. Finally, longer jet spacing gives the opportunity of a more uniform cooling and causes less deflection in the plate. It was also observed that increasing the plate thickness, increases the maximum deflection, due to a larger temperature gradient through the thickness.     157    6. Conclusion and Future Work 6.1. Conclusion The controlled cooling of steel strips on run-out table is a crucial procedure that can result in obtaining desired mechanical properties. Despite being used in steel mills for decades, the process is still mainly based on trial-and-error. This is due to the complicated nature of the events that happen, and the highly coupled nature of the thermal, fluid flow, microstructure, and mechanical fields. Thus, there is still a need to study this process in more details. The present study is a part of a comprehensive research project undertaken in UBC [1,8,9,27-32]. Here, we first established a reliable inverse analysis tool that can be used to obtain the boundary heat fluxes from the temperature history of thermocouples implemented inside the plate. Then, these heat fluxes were studied to understand the heat transfer behavior of boiling water impinged on a moving hot plate through multiple circular jets. Finally, the heat fluxes were coupled with the microstructure and mechanical fields to study the thermal stresses and deflection or warping in the steel plates. Recommendations were made to the industry about the ways to improve the cooling performance, and reduce the adverse structural effects. The following subsections summarize the findings and contributions of this research. 6.1.1. 3D Finite Element Program for the Inverse Heat Conduction Analysis A gradient-based algorithm for the inverse heat conduction analysis is developed. In this method, the least-square method, future time step technique, zeroth order regularization, and sequential function specification are used. The program can account for the heat generation during the phase transformation as a source term in the heat transfer equation. The results showed that the method is highly sensitive to the measurement errors, and becomes unstable in small time steps, thus limiting its application in real run-out table studies. As a remedy for the shortcomings related to the gradient-based inverse heat conduction solvers in our steel cooling applications, such as the instabilities due to the thermocouple reading noise and low temporal resolution, three gradient-free algorithms were studied. These methods are the artificial neural networks, genetic algorithms, and particle swarm optimization. 6.1.2. Study of the Neural Network Algorithms for Inverse Heat Conduction Analysis Several types of artificial neural networks were studied in this research. Our investigations showed that a suitable formulation for the inverse heat conduction problems in    158    neural networks should relate the gradient (and not the values) of the inside temperature to the boundary heat fluxes. Also, it was found that the radial basis function is the most suitable type of neural networks for inverse heat conduction problems. The artificial neural networks were shown to be capable of capturing the whole thermal history on the run-out table, but not very effective in restoring the detailed behavior of the boundary conditions. Also, they behave poorly in nonlinear cases and where the boundary condition profile is different. Thus, their application in the run-out table heat transfer analysis will be limited to a general prediction of the whole history. 6.1.3. Study of the GA and PSO Algorithms for Inverse Heat Conduction Analysis While GA has been previously used for inverse heat conduction analysis, the application of PSO to these problems was first studied in this research. GA and PSO are more effective in finding a detailed representation of the time-varying boundary conditions, as well as in nonlinear cases. However, their convergence takes longer. For the first time, comparison was made between GA and PSO variations in terms of both the computational efficiency and effectiveness in dealing with the noisy data in inverse heat conduction problems. PSO was found to improve efficiency (i.e., reduce the required computational effort), especially in more complex test cases. In addition to the basic PSO algorithm, two other variations were studied in this research. One of them, the complete repulsive PSO (CRPSO), was introduced for the first time in this work. CRPSO showed the best performance among three other versions. The effectiveness of PSO was also studied in the presence of noise in inverse heat conduction applications. PSO proved to be effective in handling noisy data, especially when its performance parameters were tuned. The proper choice of the regularization parameter helped PSO deal with noisy data, similar to the way it helps the classical function specification approaches. This research showed that an increase in the self-confidence parameter is also effective, as it increases the global search capabilities of the algorithm. RPSO was found to be the most effective variation in dealing with noise, followed closely by CRPSO. The latter variation was recommended for inverse heat conduction problems, as it combines the efficiency and effectiveness required by these problems.    159    6.1.4. Improvements in the Efficiency and Effectiveness of Gradient-free Algorithms for IHC Analysis This research showed that while both GA and PSO are capable of overcoming many of the shortcomings faced by the classical inverse analysis algorithms, they suffer from a slow convergence rate. Several modifications were proposed for the first time in this research that resulted in a faster algorithm for inverse analysis of heat transfer. Sequential formulation, i.e. solving the inverse problem for each separate time step was found more efficient than a whole domain implementation. Also, in the case of multi-dimensional problems with multiple sensors, we proposed a multi-objective optimization formulation that was very successful in accelerating the solution. The relative speedup seemed to be independent of the number of spatial components. Introducing the concept of elite members to the PSO algorithm also resulted in a faster convergence; however, the improvement was of less magnitude. It also looked to be more effective in simpler cases, i.e. smaller problems with fewer internal measurement points. Also, the concept of future time steps was borrowed from the function specification methods, and for the first time was applied to the sequential stochastic inverse algorithms. It proved to be very helpful in making the method more stable, especially when used in combination with a Tikhonov regularization term and tuning of the self-confidence parameter in PSO. The developed modified formulation for the inverse heat conduction problem and PSO implementation seem to be promising in terms of stability and performance, and make an effective and efficient algorithm for inverse heat conduction problem. 6.1.5. Surrogate Model Based Acceleration of GA and PSO Solvers In order to further improve the convergence characteristics of the proposed stochastic inverse heat conduction algorithms, the concept of inexact pre-evaluation using surrogate models was introduced and used to enhance the speed of evolutionary algorithms in solving the inverse heat conduction problems. In this strategy, the entire candidate solutions are first roughly evaluated using a much simpler surrogate model, and only the top performing members are exactly evaluated using the full direct solvers, such as finite element method. It was found that polynomial methods are the cheapest selection, both during the model construction phase, and the implementation. Radial basis function (RBF) and feedforward neural networks have similar construction time, but in making prediction, RBF models perform faster. Kriging model is the slowest, especially when the number of unknowns increases. All the investigated surrogate    160    models perform very well in predicting linear cases with no added noise. Nonlinear domains are much harder to predict. The most suitable model for noisy nonlinear cases is the RBF model. Since real-world cases of inverse heat conduction problem are normally nonlinear and noisy, this method was recommended for these applications in this study. An adaptive selection scheme based on the expected improvement of the objective function was found to be the best performing model management strategy. Also, making a separate model for each sensor was found to be the most appropriate way to construct the surrogate model. If done properly, surrogate modeling can speed up the stochastical methods of solving the inverse heat conduction problem up to around 5 times. Creating the structure for a stable, accurate, and fast inverse heat conduction solver is one of the major contributions of this research. 6.1.6. Characterization of the Water Jet Impingement Cooling on a Moving Plate Boiling heat transfer on a hot plate cooled by multiple water jets and multiple jet lines in an industrial scale experimental setup was studied in this research. Nozzle configuration, plate speed, and jet line spacing were varied in the experiments. Thermocouple readings were analyzed using a 3D inverse heat conduction procedure. It was found that the surface temperature is the most significant factor affecting the heat transfer behaviour on the plate. At high temperatures, the plate surface is covered with a condensed vapour layer that acts like an insulation blanket, and requires an impinging jet with high momentum to break it down. This translates into a stable film boiling regime, with reduced values of heat fluxes. Heat transfer rate was shown to increase at medium temperatures where the vapour thickness is smaller. For low entry temperatures in a multi jet line experiment, the heat transfer is lowered again. One of the very interesting findings of this study was the fact that in the low temperature region, the maximum heat flux does not occur at the impingement point, but at a location between the jets, due to the formation of an interaction water fountain. The most important effect of nozzle stagger is the uniformity of heat transfer across the width of the plate. The most uniform distribution is happening in the case of fully staggered configuration. The heat transfer between two successive rows of jet lines is affected by the jet line distance. In higher jet line distances, the interaction effects become less significant, and a more uniform distribution is observed. However, it was shown that when the jet lines are closer, the effect of interaction is more pronounced. In this study, it was observed that in medium and high temperatures, there is a spike in the heat flux values in the interaction zone. In the low    161    temperature range, because the water does not evaporate quickly, a pool of water forms on the plate surface, resulting in a flattened heat flux versus time profile. The plate speed affects the heat transfer rate under the impingement point for the higher surface temperatures. In the high entry temperatures, the impingement heat transfer rate is lower when the plate is moving at a higher velocity. The plate speed however does not significantly change the heat transfer behavior in the other parts of the plate surface. 6.1.7. Study of the Structural Effects of the Controlled Cooling of Steel Strips It is desirable that the strips maintain their flatness throughout the controlled cooling process. Thus, it was necessary to study the effect of cooling patterns on the structure of the strips. A literature survey shows that the previous researches in this field are very preliminary, and include major simplifications. One of the major simplifications in the previous studies is the oversimplified phase transformation model. For the first time in this work, an advanced phase transformation model, developed in UBC, is used in association with the thermal and structural models. In this algorithm, the static recrystallization kinetics is simulated by the Johnson-Mehl- Avrami-Kolmogorov (JMAK) theory. This model is based on the real physics of the problem, and enhances the accuracy of our simulations. Another improvement in the structural modeling in this research is the use of real heat flux data as the boundary conditions in the thermal field. The other researches assume a very simplified distribution of temperature or heat flux across the plate, and assume a linear decrease in temperature during the cooling on the run-out table. This research uses the results of a 3D inverse analysis of the thermal field, and applies an accurate heat flux boundary condition on the surface of the plate. Also, it was shown that the most significant factor in controlling the stress and flatness in the strips is the jet configuration, and the most desirable configuration is the one that promotes a more uniform distribution of temperature across the plate width. Plate speed was found to be the next important factor in determining the values of stress and deformation. Plates traveling at slower speeds are cooled down more uniformly, and will have less stress and deflection in them. Finally, longer jet spacing gives the opportunity of a more uniform cooling and causes less deflection in the plate. It was also observed that increasing the plate thickness, increases the maximum deflection, due to a larger temperature gradient through the thickness.    162    6.2. Suggestions for Future Work 6.2.1. Experimental Study of Higher Plate Speeds, Different Flow Rates, and Water Temperatures Using a higher plate speed will make the simulation closer to the real case happening in a steel mill. Also, water flow rate and temperature are the other parameters that can be used in a steel mill, in order to control the characteristics of the heat transfer. Thus, an experimental study of the effect of these parameters will be beneficial. 6.2.2. Correlations for Heat Transfer of Multiple Jets Impinging on a Hot Moving Plate More data points are needed to create correlations for the important heat transfer characteristics, such as the maximum heat flux and total extracted heat, as functions of the plate speed, number and configuration of the jets, and the water temperature and flow rate. 6.2.3. Coupling of Hydrodynamics and Heat Transfer Models A proper heat transfer model requires information about the water velocity and pressure distribution on the plate surface. The experimental and numerical study of the hydrodynamics of the water jet impingement on a moving plate has been started in UBC [32]. It should be coupled with the heat transfer studies, in order to provide a more comprehensive model for the phenomena in the jet impingement cooling of the hot moving plates. 6.2.4. Experimental Measurements and Correlations for Thermal Stresses and Deflections The thermal stresses and deflection on a water cooled hot plate should be investigated experimentally. The results of the existing numerical model should be compared with the experimental values. Also, similar to the heat transfer case, correlations should be developed in order to find the effect of cooling parameters on the structural field. The effect of steel composition and physical properties on the structural results should also be investigated to make future predictions possible, regardless of the type of steel that is used.    163    6.2.5. Developing Mechanistic Models for Moving Plate Jet Impingement Boiling The knowledge that is gained by studying the heat transfer and hydrodynamics of the jet impingement on a moving plate can be used to develop and validate a mechanistic model for the different zones of the jet impingement boiling heat transfer.     164    Bibliography [1]     F. Xu, Finite Element Simulation of Water Cooling Process of Steel Strips on Runout Table, Ph.D. Thesis, University of British Columbia, 2005. [2]     M. Korchynsky, Development of Controlled Cooling Practice, in: Accelerated Cooling of Steel : Proceedings of a Symposium Sponsored by the Ferrous Metallurgy Committee of the Metallurgical Society of AIME and Held in Pittsburgh, Pennsylvania, August 19-21, Anonymous PA: Warrendale, pp. 3-14, 1985. [3]     E. R. Morgan, T. E. Dancy, and M. Korchynsky, Improved High Strength Low Alloy Steels through Hot Strip Mill Controlled Cooling, Blast Furnace and Steel Plant, vol. 53 (10) pp.921-929, 1965. 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[ ] [ ] [ ]∫= S sTsr dSNNK κ  Thermal conductivity matrix due to radiation B.C. { } [ ]∫= V TBb dVNqQ  Heat flux vector due to internal heat generation { } [ ]∫= S Tsss dSNqQ  Heat flux vector due to input surface flux { } [ ]∫= S Tsfh dSNhTQ  Heat flux vector due to convection B.C. { } [ ]∫= S Tsrr dSNTQ κ  Heat flux vector due to radiation B.C. { }T , { }T&  Vector of global nodal temperatures and temperature gradients, respectively. ))(( 22 srrsrr TTTT ++= εσκ  Equivalent heat transfer coefficient due to radiation [ ]{ }eii TBTNT =∂ ∂ =⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ )( xx  Partial derivative of temperature [ ] ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ = z y x ksym k k K 0 00  Element conductivity matrix [ ] [ ]nNNNN L21=  Approximation or shape function , n is the number of nodes per element          176    Table A. 2. Definition of the Terms in the Heat Transfer Equation Term Name Expression )1(ˆ −Δ+ ihtt Q Nodal heat flux contribution due to convection B.C, nonlinear and transient effects ( ) dSh h Ti S itt f ttSSitthtt ⋅−= ∫ −Δ+Δ+−Δ+Δ+ − )1()1()1(ˆ TTNNQ Note: )1()1( −Δ+−Δ+ =⋅∫ i h T htt S SSitt dSh KNN )1(ˆ −Δ+ irtt Q Nodal heat flux contribution due to radiation B.C., nonlinear and transient effects ( ) dS h Ti S itt r ttSSitthtt ⋅−= ∫ −Δ+Δ+−Δ+Δ+ − )1()1()1(ˆ TTNNQ κ Note: )1()1( −Δ+−Δ+ =⋅∫ i h T rtt S SSitt dS KNNκ )1(ˆ −Δ+ ictt Q Nodal heat flux contribution due to conductivity, nonlinear and transient effects dV V ittittTctt i ⋅= ∫ −Δ+−Δ+Δ+ − )1()1()1(ˆ TBΚBQ Note: ∫ −Δ+−Δ+ =V CttittT iKdV )1()1( BΚB )1(ˆ −Δ+ ictt q Nodal heat flux contribution due to thermal capacity, nonlinear and transient effects ( )[ ] [ ] t dVtc tittitt tittT V ittctt i Δ−≡ ⋅Δ−⋅= −Δ+−Δ+ −Δ+−Δ+Δ+ ∫− / /)(ˆ )1()1( )1()1()1( TTC TTNNq ρ   

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