Optimal Control of Induction Heating Processes Edgar Rapoport Vulia Pleshiutseua (löfh Taylor & Francis ^^ J Taylor & Francis Group Boca Raton London New York CRC is an imprint of the Taylor & Francis Group, an informa business
Table of Contents Chapter 1 Introduction to Theory and Industrial Application of Induction Heating Processes 1 1.1 Short Description of Operating Principles of Induction Heaters on the Level of Basic Physical Laws 1 1.1.1 B asic Electromagnetic Phenomena in Induction Heating 1 1.1.2 Basic Thermal Phenomena in Induction Heating 5 1.2 Mathematical Modeling of Induction Heating Processes 7 1.2.1 Mathematical Modeling of Electromagnetic and Temperature Fields 8 1.2.2 Basic Model of the Induction Heating Process 12 1.3 Typical Industrial Applications and Fundamental Principles of Induction Mass Heating 19 1.4 Design Approaches of Induction Mass Heating 25 1.5 Technological Complex "Heater-Equipment for Metal Hot Working" 29 1.6 Technological and Economic Advantages of Induction Heating 31 References 33 Chapter 2 Optimization Problems for Induction Heating Processes 35 2.1 Overview of Induction Heating Prior to Metal Hot Working Operations as a Process under Control 35 2.2 Cost Criteria 38 2.3 Mathematical Models of a Heating Process 41 2.4 Control Inputs 45 2.5 Constraints 49 2.5.1 Constraints on Control Inputs '. 50 2.5.2 Technological Constraints on Temperature Distribution during the Heating Process 51 2.5.3 Constraints Related to Specifics of Subsequent Metal Working Operations 53 2.6 Disturbances 54 2.7 Requirements of Final Temperature Distribution within Heated Workpieces 57 2.8 General Problem of Time-Optimal Control 59 2.9 Model Problems of Optimal Control Respective to Typical Cost Functions 66
2.9.1 Problem of Achieving Maximum Hearing Accuracy 66 2.92 Problem of Minimum Power Consumption 69 References 72 Chapter 3 Method for Computation of Optimal Processes for Induction Heating of Metals 73 3.1 Universal Properties of Temperature Distribution within Workpieces at End of Time-Optimal Induction Heating Processes 73 3.2 Extended Discussion on Properties of Final Temperature Distribution for Time-Optimal Induction Heating Processes 79 3.3 Typical Profiles of Final Temperature Distribution and Set of Equations for Computation of Optimal Control Parameters 82 3.4 Computational Technique for Time-Optimal Control Processes 90 3.5 Application of the Suggested Method to Model Problems Based on Typical Cost Functions 97 3.6 Examples 101 3.6.1 Solution of Time-Optimal Control Problem 101 3.6.2 Solution of Minimum Power Consumption Problem 110 3.7 General Problem of Parametrical Optimization of Induction Heating Processes 111 References 116 Chapter 4 Optimal Control of Static Induction Heating Processes 119 4.1 Time-Optimal Control for Linear One-Dimensional Models of Static IHP with Consideration of Technological Restraints 119 4.1.1 General Overview of Optimal Heating Power Control 120 4.1.2 Power Control during the Holding Stage 127 4.1.3 Computational Technique for Optimal Heating Modes, Taking into Consideration Technological Constraints 131 4.1.4 Examples 140 4.2 Time-Optimal Problem, Taking into Consideration the Billet Transportation to Metal Forming Operation 142 4.2.1 Problem Statement 142 4.2.2 Computational Technique for the "Transportation" Problem of Time-Optimal Heating 149 4.2.3 Technological Constraints in "Transportation" Problem 156 4.2.4 Examples 163 4.3 Time-Optimal Heating under Incomplete Information with Respect to Controlled Systems 165 4.3.1 Problem Statement 166 4.3.2 Technique for Time-Optimal Problem Solution under Interval Uncertainties 168
4.4 Heating Process with Minimum Product Cost 174 4.4.1 Problem of Metal Scale Minimization 176 4.4.1.1 Overview of Optimal Heating Modes 176 4.4.1.2 Two-Parameter Power Control Algorithm of Scale Minimization 178 4.4.2 Minimization of Product Cost 187 4.5 Optimal Control of Multidimensional Linear Models of Induction Heating Processes 192 4.5.1 Linear Two-Dimensional Model of the Induction Heating Process 193 4.5.2 Two-Dimensional Time-Optimal Control Problem 198 4.5.3 Time-Optimal Control of Induction Heating for Cylindrical Billets 200 4.5.4 Time-Optimal Control of Induction Heating of Rectangular-Shaped Workpieces 215 4.5.4.1 Surface Heat-Generating Sources 215 4.5.4.2 Optimization of Internal Source Heating 231 4.5.4.3 Exploration of Three-Dimensional Optimization Problems for Induction Heating 236 4.6 Optimal Control for Complicated Models of the Induction Heating Process 240 4.6.1 Overview 240 4.6.2 Approximate Method for Computation of the Optimal Induction Heating Process for Ferromagnetic Billets 242 4.6.3 Optimal Control for Numerical Models of Induction Heating Processes 245 References 254 Chapter 5 Optimal Control of Progressive and Continuous Induction Heating Processes 257 5.1 Optimization of Continuous Heaters at Steady-State Operating Conditions 258 5.1.1 Overview of Typical Optimization Problems and Methods for Their Solution 258 5.1.2 Design of Minimum Length Inductor 263 5.1.3 Optimization of the Continuous Heating of Ferromagnetic Materials 273 5.1.4 Optimization of the Continuous Heating Process Controlled by a Power Supply Voltage 281 5.2 Optimization of Progressive Heaters at Steady-State Operating Conditions 288 5.2.1 Key Features of Optimization Problems for Progressive Heaters 288
5.2.2 Optimization of Induction Heater Design and Operating Modes 290 5.2.3 Optimal Control of a Single-Section Heater 298 5.2.4 Two-Position Control of Slab Induction Heating 306 References 308 Chapter 6 Combined Optimization of Production Complex for Induction Billet Heating and Subsequent Metal Hot Forming Operations 309 6.1 Mathematical Models of Controlled Processes 310 6.2 General Problem of Optimization of a Technological Complex 315 6.3 Maximum Productivity Problem for an Industrial Complex "Induction Heater-Extrusion Press" 317 6.4 Multiparameter Statement of the Optimization Problem for Technological Complex "Heating-Hot Forming" 322 6.5 Combined Optimization of Heating and Pressing Modes for Aluminum Alloy Billets 325 6.5.1 Time-Optimal Heating Modes 325 6.5.2 Time-Optimal Pressing Modes 325 6.5.2.1 Temperature Distribution within Pressurized Metal 327 6.5.2.2 Optimal Program of Extrusion Speed Variation 329 6.5.3 Computational Results 330 6.5.3.1 Optimization of Billet Gradient Heating 332 6.6 About Optimal IHI Design in Technological Complex "Heating-Hot Forming" 334 References 339 Conclusion 341 Index 343